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Frontiers in Mathematical Modelling Research
 1685074308, 9781685074302

Table of contents :
Mathematics Research Developments
Frontiers in MathematicalModelling Research
Contents
Preface
Some Key Features of the Book
Chapter 1Introduction to MathematicalModelling inApplications
Abstract
Introduction
Types of Mathematical Modelling Approaches
Applications of Modelling in Science and Engineering
Analysis of MathematicalModel
QualitativeAnalysis of a Deterministic Model
Lypunov Global Stability Theory
LaSalle’s Invariance Principle
QualitativeAnalysis of a StochasticModel
Existence and Uniqueness Theorem
Stability of Equilibrium Solutions for SDE
Stochastic Stability Theorem
Layout of this Book
Chapter 2: Fault Diagnosis of Rotating Machines Based on the Math-ematicalModel of a Rotor Bearing-Mass System
Chapter 3: Experimental and Mathematical Modelling to Investigatethe Kinetic Behavior of Plasmid DNA Production by Escherichia ColiDH5
Chapter 4: MathematicalModelling of Robotic Digitalised Production
Chapter 5: MathematicalModelling and Simulation of a RobotManipulator
Chapter 6: Mathematical Modeling Applied to Control the EmergingDeadly Nipah Fever in Bangladesh
Chapter 7: Mathematical Modeling of the Closed-Loop Performanceof a Continuous Bioreactor under a Feedback Polynomial-TypeController
Chapter 8: Mathematical Study of Human Movement andTemperature in the Transmission Dynamics ofDengue Disease Between Two Patches
Chapter 9: A Numerical Model of Malaria Fever Transmission withOrganized Vector Populace and Irregularity
Chapter 10: MathematicalModeling in Food and Agricultural Areas
Chapter 11: Mathematical Modelling of Complex Systems usingStochastic Partial Differential Equations: Review and Developmentof Mathematical Concepts
Conclusion
References
Chapter 2Fault Diagnosis of Rotating Machines Basedon the MathematicalModel of a RotorBearing-Mass System
Abstract
Introduction
MathematicalModel of the RBMS
Transfer Functions
Observability for Unbalance Effects
Observability under Misalignment
Observability Due to Unbalance and Misalignment
RBMS Design
Validation Results
Conclusion
Acknowledgment
References
Chapter 3Experimental and MathematicalModelling to Investigate the KineticBehavior of Plasmid DNA Production byEscherichia coli DH5
Abstract
Introduction
Methods
Bacterial Strain and Plasmid
Medium and Inoculum
Cultivation
DCW and Glycerol Determination
Plasmid DNA Quantification
NPT II and Organic Acids Quantification
Model Development
Thermodynamic Analysis
Results
pDNA Production Experiments
Parametric Identification and Simulations
Thermodynamics and the Effect of Temperature on theParameters
Conclusion
References
Chapter 4Mathematical Modelling inRobotic Digital Production
Abstract
Introduction
Analytical Framework
Related Work
Research Purpose and Objectives
Method
Section 1. Dynamic and Static Modeling of Processes ofCreation of High-Tech Digital Production Structures
Setting the Task of Increasing the Enterprise Potential by CreatingHigh-Tech Structures
Development of a Generalized Economic Mathematical Model
Prime Cost of Production
Analysis and Peculiarities of the Implementation of the EconomicMathematical Model
Modification of the Economic Mathematical Model for a Given ProductInnovation Release Program
Modification of the Economic Mathematical Model at a Given Level ofAutomation
Static Modeling of Processes of Creation of High-Tech Structures
Section 2. Discrete Programming Method in the Strategy ofModeling the Output and Capacity Utilization of High-TechDigital Production Structures
Optimization of the Production Program of Flexible DigitalManufacturing Organizational and Production Structures
Management of Processes of Mastering Production Capacities ofthe Developed Highly Automated Organizational and ProductionStructures
Section 3. Modeling of Processes for Selection of EconomicallyExpedient Limits for Robotics of High-Tech Structures inDigital Production
Development of an Economic Mathematical Model for DeterminingEconomically Feasible Boundaries for Robotics of Mass Production
Development of the Economic Mathematical Model for DeterminingEconomically Feasible Boundaries of Robotics of DiversifiedProduction
Section 4. Mathematical Modeling of Solving theCombinatorial Tasks of Minimizing Costs in the Processof Creating High-Tech Digital Production Structures
Setting the Task of Modeling the Optimal Composition of HighlyAutomated Production Units
Algorithm of Creation of Highly Automated Production Units ofthe High-Tech Organizational and Production Structure
Computational Mechanism of Implementation of the Algorithm ofCreation of Highly Automated Production Units of the High-TechOrganizational and Production Structure
Results and Discussion
Conclusion
References
Chapter 5Mathematical Modelling and Simulation ofa Robot Manipulator
Abstract
Introduction
MathematicalModelling
Forward Kinematics
Inverse Kinematics
Differential Kinematics
Singularities
Trajectories
Numerical Simulation
Acknowledgment
Conclusion
References
Chapter 6MathematicalModeling Appliedto Control the Emerging DeadlyNipah Fever in Bangladesh
Abstract
Introduction
MathematicalModel Formulation
Analysis of the Model
Boundedness
Basic Reproduction Number
Equilibrium Analysis
Global Stability of the Endemic Equilibrium Point E
Incorporating Optimal Control to the Model
Characterization of Optimal Controls
Numerical Simulations
Conclusion
Acknowledgments
References
Chapter 7Mathematical Modelling ofthe Closed-Loop Performance ofa Continuous Bioreactor undera Feedback Polynomial-Type Controller
Abstract
Introduction
Methods
ABE Fermentation Model
Bifurcation Analysis
Stability of the Process
Cubic Control Design
Results
Bifurcation Analysis and Open-Loop Stability
Closed Loop Analysis
System Stability Analysis
Conclusion
References
Chapter 8Mathematical Studyof Human Movement and Temperaturein the Transmission Dynamics ofDengue Disease between Two Patches
Abstract
Introduction
Model Formulation and Analysis
Existence and Stability of Disease Free Equilibrium Point
Basic Reproduction Number
Numerical Results and Discussion
Conclusion
References
Chapter 9A Numerical Model of Malaria FeverTransmission with Organized VectorPopulace and Irregularity
Abstract
1. Introduction
2. Mathematical Model
3. Dynamical Behaviour without Noise
4. Dynamical Behaviour with Noise
5. Numerical Simulations
Conclusion
Acknowledgments
Conflict of Interest
References
Chapter 10MATHEMATICAL MODELLINGIN FOOD AND AGRICULTURAL AREAS
ABSTRACT
INTRODUCTION
MATHEMATICAL MODELING IN THE FOOD AREA
Heat and Mass Transfer Models
Heat Transfer
Mass Transfer
Diffusive Mass Transfer
Fick’s Law
Maxwell-Stefan Theory
Effective Diffusivity
Convective Mass Transfer
Microbial and Enzymatic Inactivation
Temperature Profiles and Thermal Conductivity Coefficient
Food Drying
Other Food-Related Areas
Mathematical Modeling in the Agricultural Area
Biochemical Reactions
Growth Performance
Plants Processes Dynamic
Plants Demand Components
Other Agricultural-Related Areas
Plant Diseases
Pest Control
Animal Care
Conclusion and Future Outlooks
References
Chapter 11Mathematical Modelling of ComplexSystems Using Stochastic PartialDifferential Equations: Review andDevelopment of Mathematical Concepts
ABSTRACT
Introduction
Models, Mathematics and Modelling
Motivation
SPDEs and Modelling
Development of Polynomial Chaos Expansion
One Dimensional PCE
Multi-Dimensional PCE
Calculation of PCE
The Intrusive Projection Method (The GalerkinProjection Method)
The Non-Intrusive Projection Method
Implementation of Polynomial Chaos Expansion
Discretization Scheme in Time and Space
Example 1: First Order Stochastic Process
Construction of PCE Model of Example 1
Example 2: Stochastic Differential Equation
Simulating Sample Realizations of a Brownian Motion (BM)
The Euler-Maruyama (EM) Method
Example 3: The Stochastic Advection Diffusion Equation (SADE)
The Wick Product
The Wick Product in Physics
The Wick Product in Stochastic Analysis
Conclusion and Discussion
Appendix A
References
About the Editors
M. Haider Ali Biswas, PhD
M. Humayun Kabir, PhD
Index
Blank Page

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Mathematics Research Developments

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Mathematics Research Developments Partial Differential Equations: Theory, Numerical Methods and Ill-Posed Problems Michael V. Klibanov and Jingzhi Li (Authors) 2022. ISBN: 978-1-68507-592-7 (Hardcover) 2022. ISBN: 978-1-68507-727-3 (eBook) Advanced Engineering Mathematics and Analysis. Volume 2 Rami A. Maher (Author) 2022. ISBN: 978-1-68507-605-4 (Hardcover) 2022. ISBN: 978-1-68507-691-7 (eBook) Outliers: Detection and Analysis Apra Lipi, Kishan Kumar, and Soubhik Chakraborty (Authors) 2022. ISBN: 978-1-68507-554-5 (Softcover) 2022. ISBN: 978-1-68507-587-3 (eBook) Generalized Inverses: Algorithms and Applications Ivan Kyrchei (Editor) 2021. ISBN: 978-1-68507-356-5 (Hardcover) 2021. ISBN: N/A (eBook) Advanced Engineering Mathematics and Analysis. Volume 1 Rami A. Maher (Author) 2021. ISBN: 978-1-53619-869-0 (Hardcover) 2021. ISBN: 978-1-68507-435-7 (eBook) Navier-Stokes Equations and their Applications Peter J. Johnson (Editor) 2021. ISBN: 978-1-53619-967-3 (Softcover) 2021. ISBN: 978-1-68507-162-2 (eBook)

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M. Haider Ali Biswas and M. Humayun Kabir

Frontiers in Mathematical Modelling Research

Copyright © 2022 by Nova Science Publishers, Inc. https://doi.org/10.52305/VLST3329 All rights reserved. No part of this book may be reproduced, stored in a retrieval system or transmitted in any form or by any means: electronic, electrostatic, magnetic, tape, mechanical photocopying, recording or otherwise without the written permission of the Publisher. We have partnered with Copyright Clearance Center to make it easy for you to obtain permissions to reuse content from this publication. Simply navigate to this publication’s page on Nova’s website and locate the “Get Permission” button below the title description. This button is linked directly to the title’s permission page on copyright.com. Alternatively, you can visit copyright.com and search by title, ISBN, or ISSN. For further questions about using the service on copyright.com, please contact: Copyright Clearance Center Phone: +1-(978) 750-8400 Fax: +1-(978) 750-4470 E-mail: [email protected].

NOTICE TO THE READER The Publisher has taken reasonable care in the preparation of this book, but makes no expressed or implied warranty of any kind and assumes no responsibility for any errors or omissions. No liability is assumed for incidental or consequential damages in connection with or arising out of information contained in this book. The Publisher shall not be liable for any special, consequential, or exemplary damages resulting, in whole or in part, from the readers’ use of, or reliance upon, this material. Any parts of this book based on government reports are so indicated and copyright is claimed for those parts to the extent applicable to compilations of such works. Independent verification should be sought for any data, advice or recommendations contained in this book. In addition, no responsibility is assumed by the Publisher for any injury and/or damage to persons or property arising from any methods, products, instructions, ideas or otherwise contained in this publication. This publication is designed to provide accurate and authoritative information with regard to the subject matter covered herein. It is sold with the clear understanding that the Publisher is not engaged in rendering legal or any other professional services. If legal or any other expert assistance is required, the services of a competent person should be sought. FROM A DECLARATION OF PARTICIPANTS JOINTLY ADOPTED BY A COMMITTEE OF THE AMERICAN BAR ASSOCIATION AND A COMMITTEE OF PUBLISHERS. Additional color graphics may be available in the e-book version of this book.

Library of Congress Cataloging-in-Publication Data ISBN:  H%RRN

Published by Nova Science Publishers, Inc. † New York

Contents

Preface

…………………………………………………………. vii

Chapter 1

Introduction to Mathematical Modelling in Applications………………………………………….…..1 M. Haider Ali Biswas and M. Humayun Kabir

Chapter 2

Fault Diagnosis of Rotating Machines Based on the Mathematical Model of a Rotor Bearing-Mass System ………………………………….……………… 23 Juan L. Mata-Machuca, Ricardo Aguilar-López, Julio Tapia-Reyes and Jorge Fonseca-Campos

Chapter 3

Experimental and Mathematical Modelling to Investigate the Kinetic Behavior of Plasmid DNA Production by Escherichia coli DH5 ……………….. 47 Fernando Grijalva-Hernández, María del Carmen Montes-Horcasitas, Jaime Ortega-López, Edgar N. Tec-Caamal and Ricardo Aguilar-López

Chapter 4

Mathematical Modelling in Robotic Digital Production………………………………………………75 Marina Batova, Vyacheslav Baranov and Irina Baranova

Chapter 5

Mathematical Modelling and Simulation of a Robot Manipulator………………………………………….. 149 Juan L. Mata-Machuca, Ricardo Aguilar-López and Jorge Fonseca-Campos

Chapter 6

Mathematical Modeling Applied to Control the Emerging Deadly Nipah Fever in Bangladesh …….. 175 M. Haider Ali Biswas, Mst. Shanta Khatun, M. Nazmul Hasan and M. Humayun Kabir

vi

Contents

Chapter 7

Mathematical Modelling of the Closed-Loop Performance of a Continuous Bioreactor under a Feedback Polynomial-Type Controller ……………. 205 Ricardo Aguilar-López, Edgar N. Tec-Caamal, Juan C. Figueroa-Estrada, Alma R. DomínguezBocanegra and María Isabel Neria-González

Chapter 8

Mathematical Study of Human Movement and Temperature in the Transmission Dynamics of Dengue Disease between Two Patches ……………... 223 Ganga Ram Phaijoo and Dil Bahadur Gurung

Chapter 9

A Numerical Model of Malaria Fever Transmission with Organized Vector Populace and Irregularity …247 Kalyan Das and M. N. Srinivas

Chapter 10

Mathematical Modelling in Food and Agricultural Areas …………………………………... 273 Maicon Sérgio N. dos Santos, Carolina E. Demaman Oro, Rogério M. Dallago, Giovani L. Zabot and Marcus V. Tres

Chapter 11

Mathematical Modelling of Complex Systems Using Stochastic Partial Differential Equations: Review and Development of Mathematical Concepts……… 333 Parul Tiwari, Don Kulasiri and Sandhya Samarasinghe

About the Editors…………………………………………………..…373 Index

…..………………………………………………………377

Preface Mathematical modeling is the process of trying to precisely define a nonmathematical situation, real-life phenomena of changing world and the relationships between the situations in the language of mathematics, and finding out mathematical formulations or patterns within these situations and phenomena. Mathematical modeling in terms of nonlinear dynamic equations is described as a conversion activity of real problems in a mathematical form. The interactions between the mathematical and biological sciences have been increasing rapidly in recent years. Both traditional topics, such as population and disease modeling, and new ones, such as those in genomics arising from the accumulation of DNA sequence data, have made mathematical modeling in biomathematics an exciting field. The best predictions of numerous individuals and scientific communities have suggested that this growing area will continue to be one of the most dominating and fascinating driving factors to capture the global change phenomena and design a sustainable management for a better world. Frontiers in Mathematical Modelling Research provides the most recent and up-to-date developments in the mathematical analysis of real world problems arising in engineering, biology, economics, geography, planning, sociology, psychology, medicine and epidemiology of infectious diseases. Mathematical modeling and analysis are important, not only to understand disease progression, but also to provide predictions about the evolution of the disease and insights about the dynamics of the transmission rate and the effectiveness of control measures. One of the main focuses of the book is the transmission dynamics of emerging and re-emerging infectious diseases and the implementation of intervention strategies. It also discusses optimal control strategies like pharmaceutical and non-pharmaceutical interventions and their

M. Haider Ali Biswas and M. Humayun Kabir

viii

potential effectiveness on the control of infections with the help of compartmental mathematical models in epidemiology. This book also covers a wide variety of topics like dynamic models in robotics, chemical process, biodynamic hypothesis and its application for the mathematical modeling of biological growth and the analysis of diagnosis rate effects and prediction of zoonotic viruses, data-driven dynamic simulation and scenario analysis of the spread of diseases. Frontiers in Mathematical Modelling Research will play a pivotal role as helpful resource for mathematical biologists and ecologists, epidemiologists, epidemic modelers, virologists, researchers, mathematical modelers, robotic scientists and control engineers and others engaged in the analysis of the transmission, prevention, and control of infectious diseases and their impact on human health. It is expected that this self-contained edited book can also serve undergraduate and graduate students, young scholars and early career researchers as the basis for meaningful directives of current trends of research in mathematical biology.

Some Key Features of the Book     

Offers analytical and numerical techniques for dynamic models of nonlinear differential equations covering diverse area of applications Discusses mathematical modeling and their applications in treating infectious diseases or analyzing their spreading rates Covers the application of differential equations for analyzing robotic problems in systems and control engineering Examines fault diagnosis of rotating machines by mathematical model of a rotor bearing-mass system Focuses on the bioreactor mathematical model to simulate the nonlinear behavior of the reactor in chemical process.

In: Frontiers in Mathematical Modelling ... ISBN: 978-1-68507-430-2 c 2022 Nova Science Publishers, Inc. Editors: M. Biswas and M. Kabir

Chapter 1

Introduction to Mathematical Modelling in Applications M. Haider Ali Biswas1,∗ and M. Humayun Kabir2 1 Mathematics Discipline, Science Engineering and Technology School, Khulna University, Khulna, Bangladesh 2 Department of Mathematics, Jahangirnagar University, Dhaka, Bangladesh

Abstract This chapter presents a basic foundation and description of mathematical modelling and its applications in different aspects of real life problems. The key features of this chapter are to address the real life situations in terms of mathematical reasoning under appropriate nonlinear dynamical equations and emphasize validating the well-posedness of the solutions for the underlying models both in deterministic and stochastic contexts. Finally, the main contributions of different chapters of this book have been summarized in a very precise fashions.

Keywords: mathematical modelling, deterministic and stochastic modelling, qualitative features, dynamical behaviors, stability analysis ∗

Corresponding Author Email: [email protected].

2

M. Haider Ali Biswas and M. Humayun Kabir

Introduction Models are used to describe our beliefs about how the world functions. In mathematical modelling, those beliefs can be translated into the language of mathematics. There are numerous advantages: 1. Mathematics is a very precise language. This helps us to formulate ideas and identify underlying assumptions. 2. Mathematics is a concise language, with well-defined rules for manipulations. 3. All the results that mathematicians have proved over hundreds of years are at our disposal. 4. Computers can be used to perform numerical simulations. There is a large element of compromise in mathematical modelling. The majority of interacting systems in the real world are far too complicated to model in their entirety. Hence the first level of compromise is to identify the most important parts of the system. These will be included in the model, the rest will be excluded. The second level of compromise concerns the amount of mathematical manipulation which is worthwhile. Although mathematics has the potential to prove general results, these results depend critically on the form of equations used. Small changes in the structure of equations may require enormous changes in the mathematical methods. Using computers to handle the model equations may never lead to elegant results, but it is much more robust against alterations. Mathematical modelling can be used for a numerous reasons. How well any particular objective is achieved depends on both the state of knowledge about a system and how well the modelling is done. Examples of the range of objectives are: 1. Developing scientific understanding - through quantitative expression of current knowledge of a system (as well as displaying what we know, this may also show up what we do not know); 2. Test the effect of changes in a system; 3. Aid decision making, including (i) tactical decisions by managers; (ii) strategic decisions by planners.

Introduction to Mathematical Modelling in Applications

3

Types of Mathematical Modelling Approaches There are several categories of modelling approach and the category of model has an important role to develop its structure. One division between models is based on the type of outcome they predict. Deterministic models ignore random variation, and so always predict the same outcome from a given starting point. On the other hand, the model may be more statistical in nature and so may predict the distribution of possible outcomes. Such models are said to be stochastic [1, 2, 3]. A second method of distinguishing between types of models is to consider the level of understanding on which the model is based. The simplest explanation is to consider the hierarchy of organizational structures within the system being modeled. A model which uses a large amount of theoretical information generally describes what happens at one level in the hierarchy by considering processes at lower levels these are called mechanistic models, because they take account of the mechanisms through which changes occur. In empirical models, no account is taken of the mechanism by which changes to the system occur. Instead, it is merely noted that they do occur, and the model tries to account quantitatively for changes associated with different conditions. The two divisions above, namely deterministic/stochastic and mechanistic/empirical, represent extremes of a range of model types. In between lie a whole spectrum of model types. Also, the two methods of classification are complementary. For example, a deterministic model may be either mechanistic or empirical (but not stochastic). Examples of the four broad categories of models implied by the above method of classification are: Table 1. Categories of models [1] Deterministic

Stochastic

Empirical Predicting cattle growth from a regression relationship with feed intake Analysis of variance of variety yields over sites and years

Mechanistic Planetary motion, based on Newtonian mechanics Genetics of small populations based on Mendelian inheritance

4

M. Haider Ali Biswas and M. Humayun Kabir

One further type of model, the system model, is worthy of mention. This is built from a series of sub-models, each of which describes the essence of some interacting components. The above method of classification then refers more properly to the sub-models: different types of submodels may be used in any one system model. Much of the modelling literature refers to ‘simulation models’. Why are they not included in the classification? The reason for this apparent omission is that ‘simulation’ refers to the way the model calculations are done - i.e., by computer simulation. The actual model of the system is not changed by the way in which the necessary mathematics is performed, although our interpretation of the model may depend on the numerical accuracy of any approximations.

Applications of Modelling in Science and Engineering Mathematical modelling is aimed to reflect the diversified aspects of the real phenomena in terms of their interaction and dynamics with the aid of mathematics. It creates a bridge between theoretical analysis and experimental study in making them enrich as one of the most vital pillars of science and engineering. In recent years, mathematical modeling plays a pivotal role in the filed of climate change, environmental management, marine resource managements, sustainable adaptation, and human resource management of unemployment problems for sustainable economic development, etc. We refer readers to [6, 7, 8, 9, 10, 11, 12] for more recent developments in those growing areas. The significant contribution of mathematical modelling is becoming more and more apparent. Biology and bio-medicine are another crucial area of applications where mathematical modelling has long history in investigating the cell-virus interactions inside the human body and the basic transmission of emerging and re-emerging infectious diseases.Mathematical modelling has been applied to epidemic analysis for effective control and better prediction to reduce the ongoing recent outbreak of COVID-19 [13, 14, 15] and several other contagious zoonotic diseases, like nipah virus infection (NiV) [16] as well as chronic diseases, e.g., HIV [17, 18], Cancer and Chronic Liver Cirrhosis [19, 20]. Emerging development of scientific computations is one of the main reasons for this rising accomplishment while it permits the transformation of a mathematical model. A mathematical model can be solved explicitly occasionally and it is translated into an algorithm so that it can be dealt with

Introduction to Mathematical Modelling in Applications

5

more powerful computers depending on the nature of the problem. A biological system consists of a set of components: diversified physical objects that interact each other and they are classified in terms of their shapes and functions. The physical segregation either can be real (for instance, a membrane) or imaginary, that is accessible to matter, and energy (that is, an open system). The isolation of biological systems from a thermodynamical equilibrium causes a functional segregation. The exchange between matter and energy with the respective environment is an important requirement to undergo the chemical-physical processes which take place far from equilibrium. Thus a living system concerns a reference to the respective environment with which it interconnects is defined. It may be noted here that the system disappears when it is focused on the elements only neglecting the interactions between them and with the environment due to the aggregation of elements caused by lack of interaction. For this purpose, a mathematical model is a conceptual or mathematical representation of a biological system that facilitate to understand and quantify it. The difference between conceptual and mathematical resides only on the way the representation is formulated. A model is consistently a simplified portrayal of the reference system, which is wished to understand and quantify by the scientists. It ultimately undergoes as a systematizing the available knowledge and understanding of a given phenomenon and the respective facts. Mathematical and computational modelling approaches are the obvious tool to understand the biological phenomena theoretically. However, things are now beginning of an exciting new era of research in which mathematicians and computer scientists will be challenged by the innovative problems posed by biologists.The mathematicians and computer scientists will apply their tools and techniques to apprise the experimentalists, who are responsible to verify the respective outcomes and predictions of the model. This is how, there is an opportunity to deal with many of the severe problems in the biosciences through a critical interdisciplinary interactions. Moreover, mathematical modelling is one of the most effective tools extensively applied to engineering problems. It help s to conceive the significant qualitative features of these phenomena, to process and predict data, and to propose complex engineering systems. A novel model can be able to detect possible upcoming difficult issues in an engineering activity and predict the complications of endorsed engineering solutions in

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M. Haider Ali Biswas and M. Humayun Kabir

the endlong. Several mathematical models have successfully been applied to well-known of engineering problems, for instance, heat flows, resource consumption, construction design, robotics, motor design, and industrial management, etc. They have portrayed a significant predictions for which such engineering problems have been solved effectively. Exploring correlations between mathematical models in diversified applied areas, that is, science and engineering is invariably beneficial for the potential progress of mathematical techniques and for all domain of modelling.

Analysis of Mathematical Model Qualitative Analysis of a Deterministic Model In order to perform a qualitative analysis, we consider the following mathematical model based on ordinary differential equations dx = f (x), x = (x1 , x2 , . . . , xn) ∈ Rn , f = (f1 , f2 , . . . , fn ) , dt

(1)

under the initial condition x(0) = x0 .

(2)

The steady states x∗ ∈ Rn of the model (1) can be obtained by solving f (x∗ ) = 0. To investigate the local stability of the steady states x∗ by linearizing the system (1) in the following manner. We choose a small perturbation of the steady states x∗ under the transformation x(t) = x∗ + ε(t), where ε(t) is a small perturbation [2]. As a result, the system (1) takes the following form d (x∗ + ε) = f (x∗ + ε). dt

(3)

The Taylor expansion of the function f (x∗ + ε) gives f (x∗ + ε) = f (x∗ ) + εf 0 (x∗ ) +

ε2 00 ∗ ε3 f (x ) + f 000 (x∗ ) + . . . . (4) 2 6

Neglecting the higher order terms, f (x∗ + ε) can be approximated as follows: f (x∗ + ε) ≈ εf 0 (x∗ ). (5)

Introduction to Mathematical Modelling in Applications

7

So, the equation (3) can be rewritten as dε = f 0 (x∗ )ε. dt

(6)

The general solution of (6) is given by ε(t) = ε0 ef

0 (x∗ )t

,

(7)

where ε0 = ε(0) = x(0) − x∗ . This indicates that ε(t) can be regulated by the value of f 0 (x∗ ). It is evident that for f 0 (x∗ ) < 0, ε(t) → 0 as t → ∞ which confirms that the steady state x∗ is locally asymptotically stable. On the other hand, ε(t) → ∞ as t → ∞ for f 0 (x∗ ) > 0 and that confirms that the steady state x∗ is locally unstable. More specifically, the sign of f 0 (x∗ ) can be determined by the eigenvalues of the following Jacobian matrix   ∂f ∂f1 ∂f1 1 ∂x1 ∂x2 . . . ∂xn  ∂f2 ∂f2 ∂f2   ∂x1 ∂x2 . . . ∂x  n  J= . . (8) . . . .. .. ..   ..  ∂fn ∂fn ∂fn ∂x1 ∂x2 . . . ∂xn

The eigenvalues of J can be determined by solving its characteristic equation |J − λI| = 0, for instance, λ1 , λ2, . . . , λn. According to the nature and sign of the eigenvalues, the stability of the steady states x∗ can be summarized in the following way: If the real part of eigenvalues are negative, Re(λi) < 0, ∀i (i = 1, 2, . . .n) then the steady state x∗ is locally asymptotically stable. If Re(λi) > 0, for some i (i = 1, 2, . . .n), then the steady state x∗ is locally unstable. Moreover, the global stability of the steady states x∗ can be investigated by using different approaches, e.g., Lyapunov function, LaSalle’s Invariance Principle, etc. Lypunov Global Stability Theory Lyapunov global stability theory is used to make conclusions about trajectories of a system x˙ = f (x) without finding its trajectories. Suppose that V : Rn → R is a Lypunov function satisfying the following criteria 1. V is positive definite, and

8

M. Haider Ali Biswas and M. Humayun Kabir 2. V˙ (z) < 0 for all z 6= 0, V˙ (0) = 0,

then every trajectory of x˙ = f (x) converges to zero as t → ∞, that is, the system is globally asymptotically stable. Suppose that V : Rn → R is a Lypunov function satisfying the following criteria 1. V is positive definite, and 2. V˙ (z) ≤ −αV (z) for all z and a constant α, then there exists an M such that every trajectory of x˙ = f (x) satisfies αt x(t) ≤ M e− 2 converges to zero as t → ∞ that is known as globally exponential stability. LaSalle’s Invariance Principle Suppose there is a neighborhood D of x∗ and a continuously differentiable (time-independent) positive definite function V : D → R whose orbital derivative V˙ is negative semidefinite. Let I be the union of all complete orbits contained in {x ∈ D : V˙ (x) = 0}. (9) Then there is a neighborhood U of x∗ such that for every x0 ∈ U, w(x0) ⊂ I.

Qualitative Analysis of a Stochastic Model Consider the following stochastic population model based on a stochastic differential equation (SDE) dX(t) = f (X(t), t)dt + g(X(t), t)dW (t), t ∈ [t0 , T ], T > 0,

(10)

with the initial condition X(0) = X0 ,

(11)

where X  0 is taken  to be a random variable independent of W (t) − W (t0 ) 2 and E |X0 | < +∞. Here, X is a random variable which is a realvalued function defined on the sample space Ω, that is, X : Ω → R and f

Introduction to Mathematical Modelling in Applications

9

& g are termed the drift and diffusion coefficient functions, respectively. The exact solution to the SDE (10) in integral form is Z t Z t X(t) = X0 + f (X(s), s)ds+ g(X(s), s)dW (s), t ∈ [t0 , T ]. (12) t0

t0

Moreover, a one-dimensional stochastic process {X(t)}t∈[t0,T ] is termed a solution of (10) if the following characteristics hold i. process {X(t)} is continuous and At -adapted; ii. process {f (X(t), t)} ∈ L1 ([t0 , T ]) and the process {g(X(t), t)} ∈ L2 ([t0 , T ]); and iii. Equation (12) holds for every t ∈ [t0 , T ] almost surely. In order to assure the existence and uniqueness of solution [3] of the SDE (10), one may use the following existence and uniqueness theorem. Existence and Uniqueness Theorem If the coefficient functions f, g of SDE (10) satisfy the conditions |f (x, t) − f (y, t)| + |g(x, t) − g(y, t)| ≤ k |x − y|

(13)

for some constant k and all t ∈ [t0 , T ], T > 0; and   |f (x, t)|2 + |g(x, t)|2 ≤ k2 1 + |x|2

(14)

uous M2 ([t0 , T ]) bounded solutions of (10), then ! ˜ t = 0 = 1, P sup Xt − X

(15)

for some constant k2 and all t ∈ [t0 , T ], T > 0, then there exists a continuously adapted  solution Xt of (10) such that X(t) = X0 and 2 ˜ t are both continsupt0 ≤t≤T E |Xt| < +∞. Moreover, if Xt and X

0≤t≤T

and thus the solution Xt is unique.

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M. Haider Ali Biswas and M. Humayun Kabir

In addition, the solution Xt, with X(t0 ) = X), is stochastically bounded if for each ε > 0 there exists a γε = γε (t0 , X0 ) such that inf P (|Xt | ≤ γε ) > 1 − ε.

t∈[t0,T ]

(16)

If in (16) γε depends only on X0 , then Xt is said to be uniformly stochastically bounded. Stability of Equilibrium Solutions for SDE With the aid of the Lyapunov approach in determining the stability of SDE (10), one may consider the following general stochastic stability theorem. The stochastic stability theorem was developed under the following assumptions: a. the assumptions of the existence and uniqueness theorem are satisfied; b. the initial value X(t0 ) = X0 ∈ R is taken to be, with probability 1, a constant; and c. for any X0 independent of Wt , t ≥ t0 , Equation (10) has a unique global solution Xt(t0 , X0 ) with continuous sample paths and finite moments; and d. f (0, t) = 0 and g(0, t) = 0 for t ≥ t0 so that the unique solution Xt ≡ 0 corresponds to the initial value X(t0 ) = X0 = 0. Assume that f (0, t) = 0 for all t ≥ t0 . Then (10) is said to have a trivial or equilibrium solution Xt ≡ 0 corresponding to the initial value X0 = 0. Stochastic Stability Theorem i. Suppose assumptions (a)-(d) are in effect and that there exists a positive definite stochastic Lyapunov function v(Xt, t) defined on Nh × [t0 , +∞) that is everywhere continuously twice differentiable in Xt and once in t. In addition, Lv (X − t, t) ≤ 0, t ≥ t0 , 0 ≤ X − t ∈ Nh , where L is given by the following form: Lv(Xt, t) =

∂v 1 ∂ 2v ∂v . + f (Xt , t) + g(Xt, t)2 ∂t ∂Xt 2 ∂Xt2

(17)

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Then the trivial or equilibrium solution of (10) is stochastically stable. ii. If, in addition, V (Xt, t) is decrescent and Lv(Xt, t) is negative definite, then the equilibrium solution is stochastically asymptotically stable. iii. If the assumptions of (ii) hold for a radially unbounded function v(Xt, t) defined everywhere on [t0 , +∞)×R, then the equilibrium solution is globally stochastically asymptotically stable [4, 5].

Layout of this Book The book consists of eleven chapters including introduction. A brief summary of all the chapters are presented below. Chapter 2: Fault Diagnosis of Rotating Machines Based on the Mathematical Model of a Rotor Bearing-Mass System Mathematical model of a rotor bearing- mass system (RBMS) is one of the important bases to deal with fault diagnosis techniques in rotating machines under unbalance and misalignment. The fault diagnosis is approached as an observation problem where faults are reconstructed by means of the Luenberger observer. In order to validate the performance of the given methodology a laboratory setup prototype has been implemented employing a mechatronic framework. The fault diagnosis system consists of: two bearings, a shaft, a disk, two inertial measurement units, a computer, a single-phase induction motor and a start-stop module. The disk is placed along the shaft, in the middle of the two bearings in order to cause a rotor fault due to unbalance, on the other hand, misalignment is caused by means of a misaligned bearing. Chapter 3: Experimental and Mathematical Modelling to Investigate the Kinetic Behavior of Plasmid DNA Production by Escherichia Coli DH5α The kinetic behavior of plasmid DNA production by Escherichia coli DH5? can be investigated by developing a mathematical model, based on classical mass balance principles and experimental determinations of biomass, glycerol, plasmid, neomycin phosphotransferase II, acetate, lactate, formate, and succinate concentrations at different temperatures in batch cul-

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M. Haider Ali Biswas and M. Humayun Kabir

ture. This model includes 34 parameters and its effect on behavior of model was done by a parametric sensitivity analysis and 11 parameters (maximum rates) were selected to calculate their values at each temperature. Relationships between operating temperature and the maximum rates of growth, plasmid production, neomycin phosphotransferase II production and degradation, acetate, succinate, lactate, formate production and consumption were used to estimate the activation energy (Ea), enthalpy (∆H#), entropy (∆S#) and free Gibbs energy (∆G#). The reliability quantification of model by global linear correlation between proposed model and experimental data was 0.944 at 37C. The maximum rates included in model show an increase by increasing temperature, except for maximum rate of acetate consumption, which decreased. According to Ea analysis of the metabolic pathways involved in the pDN A production by Escherichia coli DH5? for aerobic growth on glycerol minimal medium, it was determined that the production is thermodynamically favored under this order, NPT II (metabolic burden)¿organic acids (overflow metabolic) > pDN A. This study can be applied an industrial scale as a reliable tool, since the thermodynamic parameters estimated are independent of the reactor scale. Chapter 4: Mathematical Modelling of Robotic Digitalised Production In the digital economy, flexible robotic structures and information systems are given priority which aims to create the mathematical modelling and automation of production. The set of economic and mathematical models for high-tech structures based on digital production were developed. The research was particularly focused on the economics of enterprises using robotic digital production. The challenges in choosing the optimal enterprise-modernisation strategy, that reflects innovative specifics of the digital transformation, were considered. The model’s optimality criterion is a dynamic performance indicator Net Present Value (NPV). The heuristic algorithm based on the Johnson method was proposed and the mathematical model for forming an optimal production program based on the analysis of the order portfolio was developed. Also mathematical models for mastery of the production capacity by high-tech structures were proposed. The static mathematical model for evaluating efficiency was developed. The mathematical modelling for optimal composition selection and loading of

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robotic structures was performed. The target function of optimality is cost minimization of creating and using robotic structures. The developed models have a high level of detail. They contain several groups and classes of variables that characterize technological innovations, production equipment, robotics, automation level, production areas, production time, costs, intellectual resources, etc. The simulation results can be used to solve production challenges, such as selection of the optimal ratio between flexibility and performance of a robotic structure and economic challenges, such as choosing for robot type and the defined robotic-production program. Chapter 5: Mathematical Modelling and Simulation of a Robot Manipulator Robotics has had an exponential development in terms of its implementation and development in industrial and everyday applications, mainly with the aim of replacing the human being in activities that involve the performance of repetitive tasks, greatly reducing the time of almost any process, the costs and the physical effort applied. In this way, its study in universities and research institutions have opened the gap to promote innovation and manipulator optimization, or to demonstrate the disadvantages or limitations that some type of robotic solution may have in certain applications. In this work, the main problem is the mathematical modelling and simulation of a 4 degree of freedom (DOF) robotic manipulator with trajectorytracking control. First of all, the modelling is addressed, it is based on the forward kinematics, the inverse kinematics and the differential kinematics. Then, as a proposed solution to the control problem, it has been suggested to carry out the control by means of independent joints, which basically consists of taking the variables that place the robot in the configuration that reaches the desired position by injecting them to the controller along with feedback from the position sensors such as controller error. Finally, the results obtained with the proposed independent joint controllers show acceptable operation characteristics during simulation. Chapter 6: Mathematical Modeling Applied to Control the Emerging Deadly Nipah Fever in Bangladesh Nipah virus (NiV), a Paramyxoviridae family member is a zoonotic virus, that is, it can transmit from animals (e.g., bats, sheep, and pigs, etc.) to

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people. Among other countries, its outbreak attracts considerable attention in Bangladesh. It is well known that NiV is a newly detected highly emerging pathogen and no proper drugs and/or vaccines are available yet for its treatments. As a result, it is essential for mathematicians and biologists to understand the disease dynamics in the human body in order to develop effective methods for prevention and control. Mathematical models have become one of the important tools in analyzing the spread and control of infectious diseases like NiV [16]. In order to understand such a transmission dynamics of NiV, we propose a mathematical model considering four compartments: susceptible, exposed, infected, and recovered classes. We confirm the boundedness of solutions of the model. To classify the epidemic and endemic cases, we determine the basic reproduction number of the model. The local stabilities at the disease-free equilibria and endemic equilibria are investigated that identify the qualitative behavior of solutions. Since the transmission interactions of NiV in a population are very complex, so it is difficult to comprehend the large-scale dynamics of disease spread. In that case, an optimal control technique is applied to obtain a better control strategy. We introduce two control measures, e.g., campaign for social awareness against NiV infection and social distancing from NiV infected individuals in the dynamic model. The existence of optimal solutions is verified using Pontryagin’s maximum principle. Numerical simulation is performed to confirm the analytical results which exhibits that the infected individuals are reduced and susceptible individuals are increased significantly when massive awareness campaign against NiV is performed. Furthermore, numerical simulation affirms that the infection of NiV minimizes remarkably when these two control measures are incorporated. Finally, it is asserted that our proposed control strategy provides significant reduction of the transmission of NiV infection in absence of effective vaccines and drugs. Chapter 7: Mathematical Modeling of the Closed-Loop Performance of a Continuous Bioreactor under a Feedback Polynomial-Type Controller This study proposes a feedback control law with a third order polynomial term in its structure, which reduces the control error in a small neighborhood around the desired reference path. The performance of the controller

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was evaluated in a continuous acetone, butanol and ethanol (ABE) fermentation process with an experimentally corroborated kinetic model for tracking purposes. The proposed controller was able to increase the butanol concentration in the reactor using the dilution rate (D) as control input. A step trajectory for butanol production at 8 g L-1 and 9.4 g L-1 was followed with success under the control law action, showing a realizable control effort. A stability analysis employing the zero dynamics methodology under the Lyapunov framework, allowed to determine the stability of the non-controlled variables under closed loop operation. A comparison with a standard Proportional-Integral (PI) controller and numerical experiments allowed to assess both satisfactory performance and the possible real time implementation of the proposed controller. Chapter 8: Mathematical Study of Human Movement and Temperature in the Transmission Dynamics of Dengue Disease Between Two Patches Dengue is an infectious disease that is spreading all over the world. Human movement leads to the spread of the disease in new places and temperature affects the life-cycle and biting behavior of the mosquitoes which transmit dengue disease. In the current work, a compartmental model is proposed to study the effect of temperature and human movement in the spread of dengue disease in the two patch environment. The human population is classified into susceptible, exposed, infectious and recovered classes, and the vector population into susceptible, exposed and infectious classes. It is observed that the Disease Free Equilibrium Point of the model is locally asymptotically stable when the basic reproduction number R0 < 1 and this point is unstable when the number R0 > 1. Model parameters are considered temperature- dependent. Simulated results show that human movement and temperature have a significant impact in the evolution and spread of the disease. Both human movement and temperature have a significant impact on the transmission of the disease. Restricting human movement from low-risk patch to the high-risk patch contributes in decreasing the disease prevalence. But restriction of human movement from the high-risk patch to the low-risk patch leads to an increase in the prevalence of the disease. So, effective management of human movement between the patches can be helpful in reducing the prevalence of the disease.

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M. Haider Ali Biswas and M. Humayun Kabir

Chapter 9: A Numerical Model of Malaria Fever Transmission with Organized Vector Populace and Irregularity Malaria fever is an infectious disease caused by the plasmodium parasite, which is transmitted to humans by female mosquito bites. According to estimates from the World Health Organization, three billion and two hundred million people were at danger of sickness, with twenty-four lakhs of new cases and four lakhs and thirty-eight thousand deaths. Any way suburban of Saharan Africa remains the weakest district with high pace of passing because of intestinal sickness. To lessen the effect of intestinal sickness on the planet, numerous logical endeavors were finished including numerical modeling and analytical findings. It has been discovered that reducing the number of mosquitos has minimal effect on the research of disease transmission of intestinal sickness in high-transmission areas. It should also be noted that natural and climatic factors have a substantial impact on vector-borne disease transmission and spread. Climate elements, such as temperature, precipitation, moistness, wind, and length of sunshine, have a significant impact on mosquito population irregularity and circadian cadence, as well as other biological and social features. Furthermore, because eggs, hatchlings, and pupae are not related with the transmission cycle, the mosquito life cycle is typically ignored in most numerical models. This is a useful decomposition of the frame work, however the results of these models do not predict the force of jungle fever in most endemic areas. In this way, the presence pattern of mosquitoes and the influence of irregularity, which are crucial portions of the elements of jungle fever transmission, must be considered. In this paper, we develop a numerical model of nonself-sufficient customary differential conditions that depicts the features of malaria fever transmission as well as the vector population’s age structure. Mosquitoes’ gnawing speed is seen as a positive intermittent capability that is dependent on climatic circumstances. We obtain the model’s fundamental multiplication proportion and show that it is the edge boundary between the illness’s termination and industriousness. Hence, by applying the hypothesis of correlation and the hypothesis of uniform constancy, we demonstrate that on the off chance that the essential propagation proportion is under 1, at that point the infection free balance is universally asymptotically steady and in the event that it is more noteworthy than 1, at that point there exists at any rate one sure intermittent arrangement. We

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also looked at the effect of noise on the model system’s dynamics. The system’s stochastic analysis reveals that environmental noise has an impact on the dynamical behaviour of solutions. MaTLab is being used to carry out numerical simulations. Chapter 10: Mathematical Modeling in Food and Agricultural Areas Mathematical modeling examines the simulation of real systems to predict their behavior. The principal importance of the use of mathematical modeling is in the development and improvement of equipment and processes. Currently, it is possible to observe the use of mathematical modeling in several industrial processes because it provides a standard or mathematical formula to explain the process under analysis. In this chapter, mathematical modeling applied in food and agricultural related areas is discussed, as well as some recent research that achieved promising results are highlighted. Mathematical models based on basic characteristics of food, such as enthalpy, thermal conductivity, specific heat, and thermal diffusivity, among other properties and characteristics, are discussed. Furthermore, the chapter includes heat and mass transfer models in industrial processes of food development and the enzymatic and microbiological inactivation that occurs in food during processing. The mathematical models most used in the food drying process and the mathematical models used in other food-related areas are included and discussed. Also, the mathematical modeling approach has also been widely used in agricultural systems, particularly in dynamic processes that involve the development and establishment of plants in the ambient. For many years, the statistical approach has been implemented in agricultural researches. However, the advancement of computational and simulation technology has allowed the rise of mathematical applications in the agricultural and animal sciences. Thus, the exploration of empirical and mechanistic models has added value in the advances of studies that investigate the effects of fundamental elements for plant growth, such as climatic factors, nutrients, and water availability. The chapter includes the topics referring to biochemical reactions in plants, growth performance, processes dynamic, and demand components. Also, different overly complex fields of study that require optimization and visionary and innovative strategies, such as plant diseases, pest control, and animal care, are discussed. Finally, future outlooks are presented to con-

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tribute to the countless opportunities and applicability of highly promising techniques and methodologies in food and agricultural areas. Chapter 11: Mathematical Modelling of Complex Systems using Stochastic Partial Differential Equations: Review and Development of Mathematical Concepts Modelling real life stochastic phenomena is difficult due to heterogeneity in associated parameters or insufficient information, or both; hence it is important to develop stochastic models to understand the behaviors of complex Systems. Stochastic Partial Differential Equations (SPDEs) [4, 5], are used to model physical, engineering, and biological systems in which, small scale effects and related uncertainties are modelled as stochastic processes. In this chapter, we discuss the modelling of complex systems using SPDEs, giving reasons for such developments and focusing on numerical solutions. We primarily discuss Polynomial Chaos Expansion (PCE) approach. In recent years, several extensions of PCE, such as Generalized PCE using the Askey scheme [4], have been developed to overcome the slow computational speed of the method for non-Gaussian random variables. When there are viable restrictions on model predictability due to inter-dependencies between the physical system and the associated parameters, PCE provides a robust mathematical structure and acceptable probabilistic solutions. We also discuss the use of Wick products [5] in stochastic analysis. Using the Wick product instead of the usual point-wise multiplication makes it possible to study anticipating processes. The Wick product is innate in stochastic analysis (with respect to Wiener chaos space), as it is implicit in the Ito integral. For anticipating processes, the Skorohod integral is similar to the Ito integral. Thus, Wick calculus works equally well for adapted and non-adapted processes.

Conclusion Mathematical modelling is an undeniable issue to deal with real-life problems. Two different modelling techniques, e.g., deterministic and stochastic approaches have been presented to exhibit their diversified applications in science and engineering. The qualitative features of solutions of a deterministic model have been discussed that are applied to validate the model.

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Moreover, a qualitative analysis of a stochastic model has been portrayed under the stochastic stability theory. Finally, a layout of all the chapters of this book has been presented in brief.

References [1] Glenn Marion, An Introduction to Mathematical Modelling, Lecture Note, Bioinformatics and Statistics, Scotland (2008). [2] Burghes D.N., Borrie M.S., Modelling with Differential Equations, First Edition, Ellis Horwood Limited, England, UK (1981). [3] Strogatz Steven H., Nonlinear Dynamics and Chaos: with applications to physics, biology, chemistry, and engineering, Perseus Books (1994). [4] Panik Michael J., Stochastic Differential Equations: An Introduction with Applications in Population Dynamics Modeling. Wiley & Sons, Inc. (2017). [5] Has’minskiy. Stochastic Stability of Differential Equations. Alphen: Sijtjoff and Noordhoff (1980). [6] Mandal S., Islam M.S., Biswas M.H.A., Akter S., A Mathematical Model Applied to Investigate the Potential Impact of Global Warming on Marine Ecosystems, Applied Mathematical Modelling, 101: 1937, (2022). [7] Mandal S., Islam M.S., Biswas M.H.A., Modelling with Strategies to Control the Adverse Effects of Global Warming on Marine Ecosystems, Model. Earth Syst. Environ., (2021). [8] Mandal S., Islam M.S., Biswas M.H.A., Akter S., Modeling the Optimal Mitigation of Potential Impact of Climate Change on Coastal Ecosystems, HELIYON, 7(7), (2021). [9] Mandal S., Islam M.S., Biswas M.H.A., Modeling the Potential Impact of Climate Change on Living Beings near Coastal Area, Modeling Earth Systems and Environment, 7:1783?1796, (2021).

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[10] Roy B., Roy S.K., Biswas M.H.A., Effects on prey-predator with different functional responses. Int. J. Biomath., 10(8): 1750113-22, (2017). [11] Khatun M.R., Biswas M.H.A., Mathematical modeling applied to renewable fishery management, Mathematical Modelling of Engineering Problems, 6(1): 121-128 (2019). [12] Mallick U.K., Biswas M.H.A., Mathematical Approach with Optimal Control: Reduction of Unemployment Problem in Bangladesh, Journal of Applied Nonlinear Dynamics, 9(2): 231-246, (2020). [13] Biswas M.H.A., Islam M.A., Akter S., Mondal S., Khatun M.S., Samad S.A., Paul A.K., Khatun M.R., Modelling the Effect of SelfImmunity and the Impacts of Asymptomatic and Symptomatic Individuals on COVID-19 Outbreak, CMES-Computer Modeling in Engineering & Sciences, 125(3): 1033-1060, (2020). [14] Biswas M.H.A., Paul A.K., Khatun M.S., Mondal S., Akter S., Islam M.A., Khatun M.R., Samad S.A., Modeling the Spread of COVID-19 Among Doctors from the Asymptomatic Individuals, Mathematical Engineering, pp. 3960,(2021). [15] Kabir M.H., Gani M.O., Mandal S., Biswas M.H.A., Modeling the dispersal effect to reduce the infection of COVID-19 in Bangladesh, Sensors International, 1: 100043, (2020). [16] Mondal M.K., Hanif M., Biswas M.H.A., A Mathematical Analysis for Controlling the Spread of Nipah Virus Infection. International Journal of Modelling and Simulation, 37(3): 185-197, (2017). [17] Biswas M.H.A., Haque M.M., Mallick U.K., Optimal Control Strategy for the Immunotherapeutic Treatment of HIV Infection with State Constraint, Optimal Control, Applications and Methods, 40(4): 807818, (2019). [18] Sahani S.K., Islam A., Biswas M.H.A., Mathematical Modeling Applied to Understand the Host-Pathogen Interaction of HIV Infection in Bangladesh, Surveys in Mathematics and its Applications, 12: 165 178, (2017).

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[19] Khatun M.S., Biswas M.H.A., Optimal Control Strategies for Preventing Hepatitis B Infection and Reducing Chronic Liver Cirrhosis Incidence, Infectious Disease Modeling, 5: 91-110, (2020). [20] Khatun M.S., Biswas M.H.A., Modeling the Effect of Adoptive T cell Therapy for the Treatment of Leukemia, Comp and Math Methods, 2(2): e1069, (2020).

In: Frontiers in Mathematical Modelling ... ISBN: 978-1-68507-430-2 c 2022 Nova Science Publishers, Inc. Editors: M. Biswas and M. Kabir

Chapter 2

Fault Diagnosis of Rotating Machines Based on the Mathematical Model of a Rotor Bearing-Mass System Juan L. Mata-Machuca1,∗, Ricardo Aguilar-L´opez2,†, Julio Tapia-Reyes3 and Jorge Fonseca-Campos4 1 Department of Advanced Technologies, UPIITA, Instituto Politecnico Nacional, Mexico City, Mexico 2 Department of Biotechnology and Bioengineering, CINVESTAV-IPN, Mexico City, Mexico 3 Mechatronic Engineering, UPIITA, Instituto Politecnico Nacional, Mexico City, Mexico 4 Department of Basic Sciences, UPIITA, Instituto Politecnico Nacional, Mexico City, Mexico

Abstract This work deals with fault diagnosis techniques in rotating machines under unbalance and misalignment based on the mathematical model of a rotor bearing-mass system (RBMS). The fault diagnosis is approached as an observation problem where faults are reconstructed by means of the Luenberger observer. In order to validate the performance of the given methodology a laboratory setup ∗ †

Corresponding Author’s Email: [email protected]. Corresponding Author’s Email: [email protected].

24

J. L. Mata-Machuca, R. Aguilar-L´opez, J. Tapia-Reyes et al. prototype has been implemented employing a mechatronic framework. The fault diagnosis system consists of: two bearings, a shaft, a disk, two inertial measurement units, a computer, a single-phase induction motor and a start-stop module. The disk is placed along the shaft, in the middle of the two bearings in order to cause a rotor fault due to unbalance, on the other hand, misalignment is caused by means of a misaligned bearing.

Keywords: rotor bearing-mass system, fault diagnosis, rotating machines, observers, mechatronics approach, unbalance, misalignment

Introduction Misalignment and unbalance forces are present in all rotating machines and these may produce unallowed deviations of a process’s normal operation condition according to its usual or accepted behavior [1]-[4]. These fault conditions are frequent causes of mechanical vibration which may produce critical damages in operating machinery, then it is important the fault diagnosis in order to formulate a maintenance or repair strategy that involves safety, time, regulations and even economic resources [5]-[8]. For example, in machining processes, the lifetime of cutting tools used in manufacturing processes can be extended, indeed, it is possible to prevent wear of gear couplings and other elements in the machine [9]-[11]. Fault diagnosis represents an important role on modern industrial processes since a lot of applications require to operate machines continuously [12]-[15]; some of these machines contain rotating elements that perform several tasks and operate at different speeds, such as motors, generators and transmission shafts [16]. An important aspect in the observer approach is the evaluation of the variables involved in the behavior of the system [17][20]. The implementation of observer-based systems in the study of mechanical vibrations in rotors is applicable to predictive maintenance since it allows to carry out corrective actions in a way that the times of maintenance are minimized and that can be guaranteed that the machine element will not suffer damages under certain conditions [21]. The development of fault diagnosis systems oriented to mechanical vibrations in rotating elements due to unbalance and misalignment is very useful in machining processes as is shown in [22] in which employed a process based on the model for the control of vibrations in a micro-milling

Fault Diagnosis of Rotating Machines ...

25

machine using magnetic supports. The regulation of the magnetic flux is performed through a proportional-integral-derivative (PID) controller and the estimation of the cutting force on the micro-milling machine is made using the signals of the magnetic actuators. The problem of force estimation is expressed as a problem of estimating inputs; where the cutting forces are unknown and solution is given through a Wiener adaptive filter. Due to the Luenberger observer it was possible to carry out the task of detecting and locating faults in a rigorous manner [23], where a fault detection and diagnosis system is developed through neural networks. In [24] a band breaking is detected through a proportional observer of reduced order designed with algebraic differential techniques. The development of [24] explains how faults can be diagnosed in electromechanical systems from the identification of parameters such as resistance, inductance and magnetic flux in a dc motor. Reference [25] deals with detection and isolation of faults using the Jeffcott rotor as a mechanical model, here the mathematical model was obtained with the representation of bond graphs; one of the fault conditions in [25] is the fault by unbalance when adding masses in the shaft and also when there is a fault in the system of measurement of revolutions of the rotor. Reference [26] proposes the use of an active disk implemented in a Jeffcott rotor for the control of vibrations. To be able to identify the balance mass, two non-linear controllers were applied. The algebraic method of parameter estimation was used because of the need to estimate the controller that manipulates the speed of rotation in the rotor. In this book chapter the fault diagnosis is approached as an observation problem [27, 28], where faults are reconstructed by means of the Luenberger observer, employing signals from inertial measuring units. The proposed technique is based on mathematical model of a RBMS under the conditions of misalignment and unbalance. Then, the faults in the RBMS can be diagnosed considering the behavior of a rotatory machine element by changing operating conditions, in two ways: (1) if a weight is added to the shaft generating unbalance, and (2) if there is a lateral misalignment. Moreover, in order to validate the performance of the given methodology a laboratory setup prototype has been implemented, where the output signals are obtained by means of two inertial measurement units MPU6050 and for the data acquisition the development board STM32F407VG is used. Furthermore, the state observer is developed in Labview and the human machine interface is executed in Matlab.

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Mathematical Model of the RBMS Rotating machines are essentially composed of three main components which are: the rotor, the supports and the support structure [29]. A rotor is the essential component in any type of rotating machine, the analysis of any machine must be directed to find the dynamic properties of this type of elements. Furthermore, a rotor is a rotating machine element that includes the shaft, disk, bearings, gears, couplings, and other elements that are mounted on the shaft. For the analysis, it is necessary to define a stationary coordinate system made up of the axes Ox, Oy and Oz , see Figure 1. We can note that Oz is the rotation axis. The rotor’s center of mass is capable of shifting along the axes Ox , Oy and Oz onto u, v and w, respectively.

Figure 1. Reference frames. Considering the displacements u, v and w due to the rotation around the z axis the resulting vectors of angular displacement can be found as shown in Figure 2, where, θ is the angular position with respect to u axis, ψ is the angular position with respect to v axis and φ is the angular position with respect to w axis.

Fault Diagnosis of Rotating Machines ...

27

Figure 2. Description of the rotor variables. The mechanical system displayed in Figure 3 shows a shaft simply supported with a disk at its half length. Where P is the location of the mass that causes unbalance, Ω is the angular velocity around the z axis, C is geometrical center located along the rotation axis, and O is the origin of the inertial frame of reference.

Figure 3. Unbalance representation. The system seen from the x − y plane is shown in Figure 4, where ε is known as eccentricity. The coordinates of P with respect to the origin O are obtained as follows,     xp(t) xc (t) + ε cos(Ωt) P − O = rp(t) = = (1) yp (t) yc (t) + ε sin(Ωt) The time derivative of (1) is given by,   x˙ c (t) − εΩ sin(Ωt) r˙p(t) = y˙c (t) + εΩ cos(Ωt)

(2)

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J. L. Mata-Machuca, R. Aguilar-L´opez, J. Tapia-Reyes et al.

Figure 4. Description of x − y plane. Then, the motion equations of the rotor are, ( m¨ xc + kxc = mεΩ2 cos(Ωt) + Fx m¨ yc + kyc = mεΩ2 sin(Ωt) + Fy

(3)

System (3) assumes the unbalancing action due to a mass added to the disk as an external force with components Fx and Fy . Due to the symmetry of the rotor, the response in the coordinate x is the same as in y. The residual unbalance is given by, ( m¨ x + kx = mεΩ2 cos(Ωt) (4) m¨ y + ky = mεΩ2 sin(Ωt) The second equation of (4) has the following solution, y(t) = Y sin(Ωt − φ)

(5)

where, Y is the amplitude, Ω is the angular velocity around the z axis and φ is the angle of phase shift that exists between the geometric center and the center of mass. Substituting the value of y in Equation (4), with φ = 0, Y Ω2 = 2 ε w n − Ω2

(6)

Fault Diagnosis of Rotating Machines ... k where, wn2 = . m The mathematical model including gyroscopic effects is,  m¨ u + kT u + kC ψ = 0    m¨ v+k v−k θ = 0 T

C

I θ¨ + Ip Ωψ˙ − kC v + kR θ = 0    d¨ Id ψ − Ip Ωθ˙ + kC u + kR ψ = 0

29

(7)

where, Ip is the polar moment of inertia, Id is the diametrical moment of inertia, kT , kC , and kR are stiffness coefficients. These coefficients consider translation, rotation and coupling for a half-length load on a simply supported beam. 3EI(a3 + b3 ) kT = (8) a3 b3 kC =

3EI(a2 − b2 ) a2 b2

(9)

3EI(a2 + b2 ) (10) ab Parameters a and b correspond to the application length of the load, E es the modulus of elasticity of the rotor’s material and I is the circular section’s moment of inertia. If gyroscopic effects do not have any contribution (IP Ω = 0) and kC = 0, we have that,  m¨ u + kT u = 0    m¨ v + kT v = 0 (11) I θ¨ + kR θ = 0    d¨ Id ψ + k R ψ = 0 kR =

The solution of system (11) is expressed by 8 roots as follows,

30

J. L. Mata-Machuca, R. Aguilar-L´opez, J. Tapia-Reyes et al. r  kT   s1,2 = j     rm    kR    s3,4 = j I rd  kT    s5,6 = −j    rm      s7,8 = −j kR Id

(12)

These roots represent the natural response without external excitation. In the same way, when the rotational kinetic energy is IP Ω 6= 0, the solution of system (7) is obtained considering the following equations,  θ(t) = θ0 e−st (13) ψ(t) = ψ0 e−st where, θ(0) = θ0 and ψ(0) = ψ0 . Substituting (13) into (7) yields to,  (Id s2 + kR )θ0 − Ip Ωsψ0 = 0 Ip Ωsθ0 + (Id s2 + kR )ψ0 = 0

(14)

Simplifying (14) we have, (Id s2 + kR )2 + (IpΩs)2 = 0 Then, the natural response considering gyroscopic effects is, r √ s1,2 = ±j

s3,4 = ±j

r

2 Id kR +Ip 2 Ω2 +Ip Ω 2 Id 2

Ip 2 Ω2 +4 Id kR

2 Id kR +Ip 2 Ω2 +Ip Ω 2 Id 2

Ip 2 Ω2 +4 Id kR



(15)

(16)

The response of the system has been found without any excitation force that disturbs the system, it is necessary to know the response when the system is under the action of unbalance and misalignment, so that the influence of these two types of disturbance can be combined with the mathematical model. The unbalance force by the rotor mass is,

Fault Diagnosis of Rotating Machines ...

31

F = mεΩ2

(17)

For the misalignment condition, we consider the center line of the rotor is deflected as shown in Figure 5.

Figure 5. Misalignment representation. The misalignment on plane x − z is given by,  m¨ u + kT u = mεΩ2 cos(Ωt)    m¨ v + kT v = mεΩ2 sin(Ωt) I θ¨ + Ip Ωψ˙ + kR θ = 0    d¨ Id ψ − Ip Ωθ˙ + kR ψ = (Id − Ip)βΩ2 cos(Ωt)

(18)

The model under misalignment on planes x−z and y−z and unbalance is described by,    

m¨ u + kT u m¨ v + kT v ¨ I θ + Ip Ωψ˙ + kR θ    d¨ Id ψ − Ip Ωθ˙ + kR ψ

= mεΩ2 cos(Ωt) = mεΩ2 sin(Ωt) = (Id − Ip )βΩ2 sin(Ωt) = (Id − Ip )βΩ2 cos(Ωt)

= Fx = Fy = Fθ = Fψ

(19)

Transfer Functions

We apply the Laplace transform to misalignment equations of (18), with zero initial conditions, Id s2 θ(s) + Ip Ωψ(s) + kR θ(s) = 0 Id s2 ψ(s) − Ip Ωsθ(s) + kR ψ(s) = Fψ (s)

(20)

32

J. L. Mata-Machuca, R. Aguilar-L´opez, J. Tapia-Reyes et al.

By equation (20) we obtain θ(s), θ(s) =

−Ip Ωsψ(s) Id s2 + kR

(21)

The transfer function for the input fψ (t) is, ψ(s) Id s2 + kR = 2 4 2 Fψ (s) Id s + (2kRId + Ip2 Ω)s2 + kR

(22)

and the transfer funtion for the unbalace force, 1 X(s) = 2 Fx (s) s + ωn2

(23)

Observability for Unbalance Effects We have the mathematical model under unbalance, ( m¨ u + ku = mεΩ2 cos(Ωt) = Fx m¨ v + kv = mεΩ2 sin(Ωt) = Fy Let us define the state variables,  z1    z2 z    3 z4

= = = =

u u˙ v v˙

(25)

Then, we have the state space representation of the form,  z˙ = Az + B u y= Cz with z = [z1 z2 z3 z4 ]0 , u = [Fx Fy ]0 and  0 1 0  −wn2 0 0 A=  0 0 0 0 0 −wn2

(24)

 0 0   1  0

(26)

Fault Diagnosis of Rotating Machines ... 

   B=   C=



0 1 m 0 0

0

33



 0    0  1 

m

1 0 0 0 0 0 1 0



We take as outputs the positions of the center of mass. The observability is computed via the Kalman criterion, that is to say, system (26) is observable if the observability matrix,   C  CA    2   O =  CA  (27)   ..   . n−1 CA

has full rank. Hence, the observability matrix of system (26) is, 

     O=     

1 0 0 0 0 0 1 0 0 1 0 0 0 0 0 1 −wn2 0 0 0 0 0 −wn2 0 2 0 −wn 0 0 0 0 0 −wn2

           

(28)

By (28), we conclude that system (26) is observable with respect to the positions of the center of mass since, rank(O) = 4

(29)

34

J. L. Mata-Machuca, R. Aguilar-L´opez, J. Tapia-Reyes et al.

Observability under Misalignment The mathematical model for misalignment is defined by system (19),  Id θ¨ + Ip Ωψ˙ + kR θ = (Id − Ip )βΩ2 sin(Ωt) = Fθ (30) Id ψ¨ − Ip Ωθ˙ + kR ψ = (Id − Ip )βΩ2 cos(Ωt) = Fψ We apply the state variables,  z1    z2 z    3 z4

The state space representation is,   0 1 0    k     − R 0 0    Id    z˙ =  0 0  0  IpΩ kR   0 −   Id  Id      0 1 0 0   y= z 0 0 0 1

= = = =

θ θ˙ ψ ψ˙

0 Ip Ω − Id 1 0

(31)





      z +    

0 1 Id 0

0



 0    u 0  1 

0

(32)

Id

with z = [z1 z2 z3 z4 ]0 , u = [Fθ Fψ ]0 and the outputs are angular velocities. The observability matrix is, 2

6 6 6 6 6 6 6 6 6 O=6 6 6 6 6 6 6 6 4

0 0

1 0 0

0 0 0

0

Ip Ω Id

0

a

− kIR d Ip Ω kR Id 2

− kIR d

I Ωk − pI 2 R d kR 2 Id 2

+

0

Ip 2 Ω2 kR Id 3

0

0

b



0 kR Id 2

+

Ip 2 Ω2 kR Id 3

Ip Ω Id

0 0 Ip 2 Ω2 2 „ Id « I 2 Ω2 k Ip Ω IR + p 2

− kIR d

0

2

3

0 1

Ip Ω kR Id 2

+



Id

d

Id

0

7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 (33)

with, a=−

kR Ip 2 Ω2 − Id Id 2

(34)

Fault Diagnosis of Rotating Machines ...

b=−

Ip Ω k R − Id 2

Ip Ω



kR Id

+

Ip 2 Ω2 Id 2

Id

35



(35)

and, rank(O) = 4

(36)

Then, system (38) is observable with respect to the angular velocities.

Observability Due to Unbalance and Misalignment Consider system (19). Let us define the change of variables, 8 z1 = u > > < z2 = u˙ > z˙ = u ¨ > : 2 z˙1 = z2

z3 = v z4 = v˙ z˙4 = v¨ z˙3 = z4

z5 = θ z6 = θ˙ z˙6 = θ¨ z˙5 = z6

z7 = ψ z8 = ψ˙ z˙8 = ψ¨ z˙7 = z8

(37)

The state space system under unbalance and misalignment is, 8 > > > > > > > > > > > > > > > > > > > > > z˙ = > > > > > > > > > > > > > > > > > > > > > > > > > > < > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > y= > > > :

2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4

0 kT − m 0

2

6 6 6 6 6 6 6 6 +6 6 6 6 6 6 6 6 4 2

1 6 0 6 4 0 0

1

0

0

0

0

0

0

0

0

0

0

0

0

0

0

1

0

0

0

0

0

0

0

0

1

0

0

0

0

0

0

0

0 kT − m 0

0

0

0

0

0

0

0

0

0 kR − Id 0

0

0

0

0

0

0 1 m 0

0

0

0

0

0

0

0

0

0

0

0

0 1 m 0

0

0

0

0

0

0 1 Id 0

0

0

0

0

0 0 0 0

0 1 0 0

0 0 0 0

0

0 0 0 0

0 1 Id 0 0 1 0

0 Ip Ω Id



0 kR Id

7 7 7 7 0 7 7 7 0 7 7z 7 0 7 Ip Ω 7 7 − Id 7 7 7 1 5 0

3

(38)

7 7 7 7 7 7 7 7 7 u 7 7 7 7 7 7 7 5 0 0 0 0

3

3 0 0 7 7 z 0 5 1

with z = [z1 z2 z3 z4 z5 z6 z7 z8 ]0 , u = [Fu Fv Fθ Fψ ]0 .

36

J. L. Mata-Machuca, R. Aguilar-L´opez, J. Tapia-Reyes et al.

Since equations of unbalance and misalignment are decoupled the observability condition is satisfied.

RBMS Design RBMS construction has been proposed as follows. The disk was made of steel SAE 1040 and has a total mass of 1.0049 kg, a diameter of 0.1 m and a thickness of 0.02 m. The disk has 12 perforations every 30◦ around its periphery to cause unbalance forces, see Figure 6 for more details.

Figure 6. Disk dimensions (mm). Figure 7 shows a representation of the proposed RBMS where the numbers correspond to, 1. Single phase induction motor, 127 V, 6 poles, 60 Hz. 2. Shaft of length L=0.22 m, SAE 1040. 3. Bearings SFK. 4. MPU6050 inertial measurement units. 5. Disk, m=1.0049 kg and diameter of 0.1 m, SAE 1040.

Fault Diagnosis of Rotating Machines ... 6. Start button. 7. Stop button. 8. Coupling between the shaft and the motor, CLX-16-8-F.

Figure 7. RBMS. (a) Computer aided design, (b) laboratory setup. For the RBMS the following assumptions are considered, 1. The rotor is axially symmetric. 2. The rotor has a constant angular velocity. 3. The deformation depends only on its stiffness coefficient.

37

38

J. L. Mata-Machuca, R. Aguilar-L´opez, J. Tapia-Reyes et al. 4. Damping or aerodynamic effects are not included in the supports; it is also assumed that the supports are isotropic.

We split the fault diagnosis system in four main blocks as is shown in Figure 8. The fault diagnosis algorithm is applied on the RBMS under two conditions, misalignment and unbalance. The fault diagnosis system is equipped with two MPU6050 transducers in order to measure the signals of interest. These signals will be processed in a STM32FG07VG development board that stores the obtained data. In addition, the development board will send the data to a computer to determine the vibration’s severity and ensure the fault diagnosis system receives the information necessary for its operation, such as position, speed and acceleration at the geometrical center of the RBMS.

Figure 8. Functions of the fault diagnosis system.

Fault Diagnosis of Rotating Machines ...

39

The components of the fault diagnosis system are shown in Figure 9. The objective of the signal acquisition is to obtain the amplitude of vibration as well as the frequency. After obtaining the signals from the MPU 6050 sensors using a microcontroller, angular velocity and linear acceleration data is stored in the SD card. In this work, 1500 samples were used, each one was taken in intervals of 1 ms.

Figure 9. Fault diagnosis system. The acquisition system is composed by: data communication card RS232, card for data storage, processing interface in Labview and processing interface in Matlab. The development card chosen to perform the task of data acquisition

40

J. L. Mata-Machuca, R. Aguilar-L´opez, J. Tapia-Reyes et al.

is the STM32F407VG, its main specifications are: memory up to 1 M B, LCD interface, oscillator up to 25 M Hz and maximum internal frequency of 168 M Hz, 32 bit timers, USB 2.0 interface, RS232 interface, SD protocol Interface, CAN protocol interface, SPI and I2C connection, and DSP unit. Some of the MPU6050 inertial measurement unit characteristics are: ranges ±2g, ±4g, ±8g, ±16g, ranges ±250 dps (degrees per second) to 20000 dps, 6 Axis, and power consumption of V cc = 3 − 3.6 V , I = 500µA. Data sheet states that to calibrate a solid-state accelerometer, it is necessary to know parameters obtained for that specific accelerometer. Assuming that hardware design is complete ant the microprocessor is programmed to receive data in digital format through SPI or I2C protocol, the next step is to convert that data in meaningful information. For a given accelerometer there is a 16 bit unsigned integer digital output. For a gyroscope you have a sensitivity of ±250 dps of 0.00875 dps/LSB obtained from the data sheet. For example, when the gyroscope is still, the x − y − z axis provide the outputs: X axis F F 96LSB = −106LSB = −106 × 0.00875 = −0.93dps Y axis 0045LSB = 69LSB = 69 × 0.00875 = −0.6dps Z axis F F CCLSB = −52LSB = −0.46dps Therefore, the sensor output (when used as a gyroscope) can be expressed as: Rt = SC(Rm − R0 )(5.63) (39)

where, Rt es the real value expressed in dps. Rm es the gyroscope measurement expressed as an integer. R0 is the zero change level when there is no angular variation applied SC is the scale factor (sensitivity). Once the previous values are known it is recommended to average a group of values from 50 to 100. Applying the following equation the true value of the measure is:     0  Rx − Rx0 Rx SCx 0 0  Ry  =  0 SCy 0  R0y − Ry0  (40) 0 Rz 0 0 SCz Rz − Rz0

Fault Diagnosis of Rotating Machines ...

41

where, Rx, Ry , Rz are the values measured by the gyroscope. Rx0, Ry 0, Rz 0 are values at zero level of variation SCx , SCy , Scz Scale factor (sensitivity) per axis R0x, R0y , R0z . Angular velocity final value.

Validation Results The Luenberger observer was implemented on variable w, results of the observer’s implementation are shown when an unbalance mass of 40g was added to the disk. In the observer tests it can be seen that the system has a sinusoidal behavior due to the input provided by the motor as is shown in Figure 10. Figure 10(a) describes the displacements in w due to the unbalance mass of 40g. In Figure 10(b) can be seen the performance of the fault diagnosis scheme since the deviation is detected and identified by the Luenberger observer.

Figure 10. Fault diagnosis. (a) Displacement in w; (b) Observer results. These situations, at an industrial level, have an impact on the production time and economic losses due to damages and maintenance. Based on the analysis shown in this book chapter, control strategies aimed at reduc-

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J. L. Mata-Machuca, R. Aguilar-L´opez, J. Tapia-Reyes et al.

ing or minimizing vibrations caused by misalignment and unbalance can be applied.

Conclusion In this chapter the problem of fault diagnosis in rotating machines was tackled. The proposed method is based on the mathematical model of a rotor bearing-mass system in the presence of faults produced by misalignment and unbalance. The observability condition for unbalance effects is satisfied when positions of the center of mass are measured. In the same way, the observability condition due to misalignment is fulfilled if the measured outputs are angular velocities. Since equations of unbalance and misalignment are decoupled the observability condition is satisfied for the state space model of the RBMS, then the faults due to misalignment and unbalance can be estimated. To validate the results, the fault diagnosis was made experimentally by means of the Luenberger observer. Moreover, the design and implementation of the RBMS have been developed in detail.

Acknowledgment This work has been supported by the Secretar´ıa de Investigaci´on y Posgrado of the Instituto Polit´ecnico Nacional (SIP-IPN) under the research grant 20211509.

References [1] Simm, A., Wang, Q., Huang, S., & Zhao, W. (2016). Laser based measurement for the monitoring of shaft misalignment. Measurement, 87, 104-116. [2] Lal, M., & Tiwari, R. (2018). Experimental identification of shaft misalignment in a turbo-generator system. Sdhan, 43(5), 80. [3] Tonks, O., & Wang, Q. (2017). The detection of wind turbine shaft misalignment using temperature monitoring. CIRP Journal of Manufacturing Science and Technology, 17, 71-79.

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[4] Gama, A. L., de Lima, W. B., & de Veneza, J. P. S. (2017). Detection of shaft misalignment using piezoelectric strain sensors. Experimental Techniques, 41(1), 87-93. [5] Kumar, A., & Kumar, R. (2020). Development of LDA based indicator for the detection of unbalance and misalignment at different shaft speeds. Experimental Techniques, 44(2), 217-229. [6] Da Silva, R. R., Costa, E. D. S., De Oliveira, R. C., & Mesquita, A. L. (2017). Fault diagnosis in rotating machine using full spectrum of vibration and fuzzy logic. Journal of Engineering Science and Technology, 12(11), 2952-2964. [7] Mart´ınez-Morales, J. D., Palacios-Hern´andez, E. R., & CamposDelgado, D. U. (2018). Multiple-fault diagnosis in induction motors through support vector machine classification at variable operating conditions. Electrical Engineering, 100(1), 59-73. [8] Guan, Z., Chen, P., Zhang, X., Zhou, X., & Li, K. (2017). Vibration Analysis of Shaft Misalignment and Diagnosis Method of Structure Faults for Rotating Machinery. International Journal of Performability Engineering, 13(4). [9] Liang, Y. C., Wang, S., Li, W. D., & Lu, X. (2019). Data-driven anomaly diagnosis for machining processes. Engineering, 5(4), 646652. [10] Madhusudana, C. K., Budati, S., Gangadhar, N., Kumar, H., & Narendranath, S. (2016). Fault diagnosis studies of face milling cutter using machine learning approach. Journal of Low Frequency Noise, Vibration and Active Control, 35(2), 128-138. [11] Aralikatti, S. S., Ravikumar, K. N., Kumar, H., Nayaka, H. S., & Sugumaran, V. (2020). Comparative study on tool fault diagnosis methods using vibration signals and cutting force signals by machine learning technique. Structural Durability & Health Monitoring, 14(2), 127. [12] Jalan, A. K., Patil, S., & Mittal, G. (2020). A Review on Fault Diagnosis of Misaligned Rotor Systems. International Journal of Performability Engineering, 16(4).

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[13] Yu, W., & Zhao, C. (2019). Online fault diagnosis for industrial processes with Bayesian network-based probabilistic ensemble learning strategy. IEEE Transactions on Automation Science and Engineering, 16(4), 1922-1932. [14] Yu, W., & Zhao, C. (2019). Broad convolutional neural network based industrial process fault diagnosis with incremental learning capability. IEEE Transactions on Industrial Electronics, 67(6), 5081-5091. [15] Cordoneanu, D., & Nitu, C. (2018). A Review of Fault Diagnosis in Mechatronics Systems. In International Conference of Mechatronics and Cyber-Mixmechatronics (pp. 173-184). Springer, Cham. [16] Pezzani C. M., Bossio J. M., Castellino A. M., Bossio G. R., De Angelo C. H., Bearing Fault Detection in Wind Turbines with Permanent Magnet Synchronous Machines, IEEE Latin America Transactions, vol. 12, no. 7, pp. 1199-1205, 2014. [17] Piltan, F., & Kim, J. M. (2018). Bearing fault diagnosis by a robust higher-order super-twisting sliding mode observer. Sensors, 18(4), 1128. [18] Zemzemi, A., Kamel, M., Toumi, A., & Farza, M. (2018). A new robust observer design for nonlinear systems with application to fault diagnosis. Transactions of the Institute of Measurement and Control, 40(13), 3696-3708. [19] Yi, L., Liu, Y., Yu, W., & Zhao, J. (2020). A novel nonlinear observer for fault diagnosis of induction motor. Journal of Algorithms & Computational Technology, 14, 1748302620922723. [20] El Merraoui, K., Ferdjouni, A., & Bounekhla, M. H. (2020). Observer-based IM stator fault diagnosis: Experimental validation. Periodicals of Engineering and Natural Sciences, 8(2), 870-883. [21] Wang, H., Ye, X., & Yin, M. (2016). Study on predictive maintenance strategy. International Journal of Science and Technology, 9(4), 295300. [22] Roiger S., Model-based Process Monitoring and Control of Micromilling using Active Magnetic Bearings, PhD Thesis, 2011.

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[23] Shiao M., Bond graph model based fault detection and isolation of loaded Jeffcott rotor system, Thesis, School of Electrical and Electronic Engineering, SIMTech, 2009. [24] Genta G., Dynamics of Rotating systems, Springer, 2005. [25] Wang Z., Schittenhelm R.S., Rinderknecht S., Observer design for unbalance excited rotor systems with gyroscopic effect. Proc. IEEE International Conference on Mechatronics and Automation (ICMA), pp. 712, 2012. [26] Mart´ınez-Guerra R., Mata-Machuca J.L., Fault detection and diagnosis in nonlinear systems: a differential and algebraic viewpoint, Serie: Understanding Complex Systems, Springer, 2014. [27] Mart´ınez-Guerra R., Mata-Machuca J.L., Rinc´on-Pasaye J., Fault diagnosis viewed as a left invertibility problem. ISA Transactions, vol. 52, no. 5, pp. 652-661, 2013. [28] Mata-Machuca J.L., Martnez-Guerra R., Rinc´on-Pasaye J., Fault diagnosis in nonlinear dynamical systems based on left invertibility condition: a real-time application to three-tank system. Proc. American Control Conference (ACC2011), pp. 810-815, 2011. [29] Friswell L. M., Dynamics of rotating machines, 2a ed. Londres: Cambridge University press, 2010. [30] Mart´ınez-Guerra R., Garrido R., Palacios R., Mendoza J., Fault detection in a belt-drive system using a proportional reduced order observer. Proc. American Control Conference, pp. 3106-3110, 2004.

In: Frontiers in Mathematical Modelling … ISBN: 978-1-68507-430-2 Editors: M. Biswas and M. Kabir © 2022 Nova Science Publishers, Inc.

Chapter 3

Experimental and Mathematical Modelling to Investigate the Kinetic Behavior of Plasmid DNA Production by Escherichia coli DH5 Fernando Grijalva-Hernández, María del Carmen Montes-Horcasitas, Jaime Ortega-López, Edgar N. Tec-Caamal and Ricardo Aguilar-López Biotechnology and Bioengineering Department, Center for Research and Advanced Studies of the National Polytechnic Institute, Mexico City, Mexico

Abstract The kinetic behavior of plasmid DNA production by Escherichia coli DH5α was investigated by developing a mathematical model, based on classical mass balance principles and experimental determinations of biomass, glycerol, plasmid, neomycin phosphotransferase II, acetate, lactate, formate, and succinate concentrations at different temperatures in batch culture. This model includes 34 parameters and its effect on behavior of model was done by a parametric sensitivity analysis and 11 parameters

48

F. Grijalva-Hernández, M. del Carmen Montes-Horcasitas et al. (maximum rates) were selected to calculate their values at each temperature. Relationships between operating temperature and the maximum rates of growth, plasmid production, neomycin phosphotransferase II production and degradation, acetate, succinate, lactate, formate production and consumption were used to estimate the activation energy (Ea), enthalpy (ΔH#), entropy (ΔS#) and free Gibbs energy (ΔG#). The reliability quantification of model by global linear correlation between proposed model and experimental data was 0.944 at 37°C. The maximum rates included in model show an increase by increasing temperature, except for maximum rate of acetate consumption, which decreased. According to Ea analysis of the metabolic pathways involved in the pDNA production by Escherichia coli DH5α for aerobic growth on glycerol minimal medium, it was determined that the production is thermodynamically favored under this order, NPT II (metabolic burden) ˃ organic acids (overflow metabolic) ˃ pDNA. This study can be applied an industrial scale as a reliable tool, since the thermodynamic parameters estimated are independent of the reactor scale.

Keywords: activation energy, heat stress, kinetic production, sensitivity analysis, thermodynamic parameters

Introduction DNA plasmid (pDNA) is a technology that has proven favorable results for use in gene therapy, vaccines against infectious, genetic diseases and cancer [1]. Many candidates of pDNA vaccines for different diseases are in clinical trials. However, doses needed to achieve adequate immune response are in the order of milligrams and requiring the development of highly productive processes [2]. To effectively and efficiently operate the fermentation process, the kinetic characteristics of entire plasmid production process is required. During pDNA production process by Escherichia coli, the activities of the microorganisms respond to changes in the environmental conditions, which are accompanied by variations in the metabolic behavior of the microorganisms. Although the current trend is to use antibiotic-free plasmid in pDNA vaccines production due to the risk of serious hypersensitivity reactions in patients, during recent years, pDNA containing kanamycin resistance gene (nptII) have been still used [3-5], because kanamycin is not commonly used to treat human infections [6-8] and NPTII does not possess any of the characteristics associated with

Experimental and Mathematical Modelling …

49

allergenic proteins and toxicity in humans [9]. One of the most important variables in pDNA production process is temperature. The influence of temperature on the kinetics parameters of pDNA production have been reported in many works [10-12]. To gain insight into the time-variant process dynamics, some kinetic models associated with key conditions process and composition of growth media for pDNA production have been proposed [3, 13]. García-Rendón et al. [3], proposed a model to describe cell growth, plasmid production and substrate consumption in an exponential-fed perfusion fermentation of E. coli DH5α, only used to investigate the key culture strategies and scale-up projections; likewise, Lopes et al. [13], built an interpretative kinetic model for batch cultures using a minimal medium defined with glucose, glycerol and with a mixture of these, however, despite the model showed a good performance, the adjustment made for its calibration may not have been adequate, due to the carbon contribution by yeast extract and bactotryptone (30 g/L) used in the experiments is not contemplated; this is a disadvantage of semi-defined culture medium to propose a kinetic model, since the value of the 17 parameters contained in the mathematical structure can be underestimated or overestimated. However, a numerical study of the influence of temperature on the pDNA production and products related to the overflow metabolism (acetate, lactate, formate and succinate) and the metabolic burden (mainly synthesis of NPT II) has not been reported; likewise, the models proposed to describe the production of pDNA do not involve all the important inputs (carbon source and organic acids consumption) and outputs (pDNA, NPT II and organic acids production) that have already been reported [14], therefore, in this work a phenomenological mathematical model is proposed to quantify and describe the effect of the temperature on the kinetic parameters of the plasmid production, including all those inputs and outputs, allowing the calculation of several thermodynamic parameters of the system. The results accordingly provide a better understanding of the effects of temperature on the activities of the cell for further development of the process.

50

F. Grijalva-Hernández, M. del Carmen Montes-Horcasitas et al.

Methods Bacterial Strain and Plasmid The host-plasmid combination was E. coli DH5α transformed with the plasmid pVAX-NH36 (Invitrogen by Life Technologies, USA), which contains DNA fragment encoded NH36 antigen of Leishmania donovani [15]. pVAX-NH36 contains kan+ and a pUC origin (pMB1-derived) for high-copy number replication [16]. E. coli DH5α-pVAX1-NH36 was propagated on LB agar plates with soy peptone and kanamycin (50 µg/mL). A single colony from this plate was used to inoculate a 1000 mL baffled flask with aeration system containing 200 mL of chemically defined mineral medium (CDMM, described below) plus glycerol 12.5 g/L. The shake flask was incubated in an orbital shaker (New Brunswick, USA) carried out at 30°C, 250 rpm and 0.5 vvm. As cell cryo-protection, 20 mL of glycerol 80% (w/v) was added when the culture reached an optical density (OD600nm) of 8 and aliquots of 10 mL were frozen immediately on dry ice and store at -70°C (CDMM seed bank).

Medium and Inoculum The CDMM for growth E. coli DH5α previously modified [5] had the following composition (g/L): K2HPO4, 5.8; KH2PO4. 7.5; antifoam 204, 0.125; (NH4)2SO4, 5.92; MgSO4.7H2O, 2.3; NaCl, 2; FeCl.6H2O, 2.85 mg; and 14.7 mL of trace mineral solution (TMS). The composition of TMS was the following, in g/L: 2 g ZnCl2.4H2O, 2 CoCl.6H2O, 2 g Na2MoO4.2H2O, 1.9 g CuCl2.2H2O, 1.6 g H3BO3, 1.6 g MnSO4.H2O, 0.6 g citric acid, and 1 g CaCl2. The fermentations were performed with CDMM plus glycerol and thiamine 25 g/L and 40 µg/mL, respectively.

Experimental and Mathematical Modelling …

51

10 mL of cryo-conserved CDMM seed bank was inoculated into a 750 mL jar bioreactor containing 200 mL of CDMM plus 12.5 g/L glycerol, 50 µg/mL kanamycin and 40 µg/mL thiamine and carried out at 30°C, 1200 rpm, 1 vvm. The culture was used as inoculum (20% v/v) when an increase of %DO was monitored, reaching a complete consumption of glycerol and OD600nm of 24 (DCW of 9 g/L).

Cultivation Duplicated fermentations were carried out in a 2 L jar bioreactor containing 1 L of CDMM plus 200 mL of inoculum and carried out at 37°C, 1200 rpm, 1 vvm. The dissolved oxygen concentration was maintained above 30% air saturation by enriching air with pure oxygen. The experiments were carried out in isothermal conditions at 30, 33, 35, 40 and 42°C and monitored by removing 5 mL samples every 1 h for DCW, glycerol, plasmid, NPT II and organic acids (acetate, lactate, formate and succinate). Data obtained at 37°C were used to perform a nonlinear parameter adjustment by the Levenberg-Marquardt algorithm.

DCW and Glycerol Determination A calibration curve built previously (gDCW/L=0.38×OD600nm provided that OD600 0 (70)

or 𝛾𝑗𝑞 −1 𝛾𝑗𝑞

∙ 𝑍𝑗𝑡𝑒𝑐ℎ ∙

𝑝𝑛 𝑡𝑗 𝑝𝑛 𝑝𝑜 𝑡𝑗 +𝑡𝑗

+ [𝑇𝑅𝑗 ∙ 𝑘𝑗𝑠ℎ −

1 𝛾𝑗𝑞

𝑟𝑜𝑏 ∙ 𝑍𝑗𝑞 ]>0

(71)

Hence, 𝑝𝑛

𝑡𝑗 𝑝𝑛

𝑍𝑟𝑜𝑏

𝑗𝑞 − 𝑝𝑜 > ( 𝛾 −1

𝑡𝑗 +𝑡𝑗

𝑇𝑅𝑗 ∙𝑘𝑗𝑠ℎ ∙𝛾𝑗𝑞 𝛾𝑗𝑞 −1

𝑗𝑞

)∙

1 𝑍𝑗𝑡𝑒𝑐ℎ

(72)

In this expression, the value of 𝑝𝑛

𝑡𝑗 𝑝𝑛

𝑝𝑜

𝑡𝑗 +𝑡𝑗

(73)

reflects the proportion of auxiliary time that is not overlapped by the main processing time in the total norm of auxiliary time. The first member in the right part of the inequality characterizes the ratio of costs associated with the operation of the industrial robot and costs associated with the operation of technological equipment. The second member in the right part of the inequality represents the ratio of the worker's salary (taking into account the change of the operation’s duration in the robotic variant) to the costs associated with the operation of technological equipment (Baranov et al., 2015b). Thus, it is possible to draw a conclusion about expediency of robotizing of the operation in the created high-tech organizational and production structure. The expediency of the operation is achieved if the

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Marina Batova, Vyacheslav Baranov and Irina Baranova

relative saving of auxiliary time, not overlapped by the main processing time, exceeds the difference between the relative saving of labor of the worker and the robot used for servicing the operation. Thus, the condition for efficiency of robotics of the arbitrarily chosen operation performed in the high-tech organizational and production structure is obtained. This result can be used to solve economic tasks of robotics related not solely to machine tools, but also transport and storage tools as well as other types of work performed in the high-tech organizational and production structures of the company (Pelevin and Tsudikov. 2017; Martin, 2015).

Development of the Economic Mathematical Model for Determining Economically Feasible Boundaries of Robotics of Diversified Production In this case, an annual product innovation plan is formed for the production system to be robotized. During the development of economic mathematical models to determine the economically viable boundaries of robotizing of diversified production, only the operational time ‘stocks’ are taken into account, and the batch launch of manufacturing of product innovations is considered a fixed value. Based on these assumptions, for an enterprise high-tech organizational and production structure, the question of expediency of robotics of a single unit of technological equipment is considered. In this case, it is assumed that the operating time ‘stocks’ of technological equipment released as a result of robotics can be used to produce product innovations that are not initially assigned to this production structure. By analogy with the previous case, we choose the maximum Net Present Value as an indicator of the efficiency of robotics of technological equipment. This will correspond to the minimum annual costs for the creation and operation of the robotic unit. The difference between the results and the costs resulting from the year of operation of the robotic unit will be the savings on current operating costs. These savings characterize the reduction of the prime cost of manufactured product innovations. It is assumed that the source of such savings is the robotics coefficient in the

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121

technological process. At the same time, we assume that all investments related to the robotics of the production system are made at the beginning of implementation of the technologically oriented project. Then, by reducing the share of manual labor and releasing workers from the production process, the cost of product innovations will be reduced by the following amount: 𝑜𝑝

𝑜𝑝

∆𝑃𝐶 𝑟𝑤 = ∑ 𝑇𝑅𝑗 ∙ 𝑡𝑗 ∙ 𝑘𝑗 𝑁𝑗𝑎𝑛𝑛

(74)

where tjop  the operative time of the j-th operation in the variant to be robotized. An additional reduction in the cost of product innovations due to the release of technological equipment from the production structure to be robotized will be: 1

𝑝𝑛 𝛾𝑗𝑞 −1 ∙ 𝛾𝑗𝑞

∆𝑃𝐶 𝑟𝑡𝑒 =  ∑ 𝑁𝑗𝑎𝑛𝑛 ∙ 𝑡𝑗 𝐶 𝑠𝑝𝑎𝑐𝑒 ∙ 𝑆 𝑡𝑒𝑐ℎ )

(𝑏 𝑡𝑒𝑐ℎ ∙ 𝐼𝐶 𝑡𝑒𝑐ℎ + 𝑊 𝑒𝑙𝑒𝑐 + 𝑊 𝑡𝑜𝑜𝑙 + (75)

where   the annual actual time ‘stocks’ of technological equipment and the q-th industrial robot; The prime cost increase at inclusion in the production system of the q-th industrial robot servicing the technological equipment will be: 1

𝑜𝑝

∆𝑃𝐶𝑞𝑟𝑜𝑏 =  ∑ 𝑁𝑗𝑎𝑛𝑛 ∙ 𝑡𝑗 ∙ (𝑏𝑞𝑟𝑜𝑏 ∙ 𝐼𝐶𝑞𝑟𝑜𝑏 + 𝑊𝑟𝑜𝑏𝑞𝑒𝑙𝑒𝑐 + 𝑊𝑟𝑜𝑏𝑞𝑡𝑜𝑜𝑙 + 𝐶 𝑠𝑝𝑎𝑐𝑒 ∙ 𝑆𝑞𝑟𝑜𝑏 )

(76)

The formation of the mathematical model is based on the following idea. In the production system J technological operations are performed. Each of these operations is marked by the index j. The norm of operational time set by the technologist, for the j-th operation in the basic version is tjop,bas.

122

Marina Batova, Vyacheslav Baranov and Irina Baranova 𝑜𝑝

𝑡𝑗

𝑝𝑛

= 𝑡𝑗𝑎 + 𝑡𝑗

(77)

where tjа – automatic execution time of the j-th operation. The task of choosing operations, robotizing of which is economically expedient, is solved. It is assumed that each j-th operation selected for robotics will be served by the q-th industrial robot. Since the robot works faster than a human being, the robot reduces the auxiliary uninterrupted time tjpn by the value of jq. In the robotized version, the auxiliary uninterrupted time becomes equal to: 𝑝𝑛,𝑟𝑜𝑏

𝑡𝑗

𝑝𝑛

=

𝑡𝑗

(78)

𝛾𝑗𝑞

If the condition to reduce the auxiliary uninterrupted time is not met, robotics becomes economically unsound. Robotics can then be justified only through analysis and economic evaluation of social factors. The value tjа of the basic and robotic variants does not change, because the replacement of the basic (technological) equipment is not assumed. 𝑜𝑝,𝑟𝑜𝑏

Then, the norm of operative time in the robotized variant (𝑡𝑗

) will

be: 𝑜𝑝,𝑟𝑜𝑏

𝑡𝑗

𝑜𝑝

= 𝑡𝑗 −

𝑝𝑛

𝑡𝑗

𝛾𝑗𝑞

(79)

or 𝑜𝑝,𝑟𝑜𝑏

𝑡𝑗

𝑝𝑛

= 𝑡𝑗𝑎 + 𝑡𝑗



𝛾𝑗𝑞 −1 𝛾𝑗𝑞

(80)

Where jq - a coefficient that takes into account the change in duration of the j-th operation during its robotizing by the q-th industrial robot. Consequently, 1

𝑜𝑝

1

𝑝𝑛

∆𝑃𝐶𝑞𝑟𝑜𝑏 =  ∑ 𝑁𝑗𝑎𝑛𝑛 ∙ (𝑡𝑗 − 𝛾 ∙ 𝑡𝑗 ) ∙ (𝑏𝑞𝑟𝑜𝑏 ∙ 𝐼𝐶𝑞𝑟𝑜𝑏 + 𝑗𝑞

𝑊𝑟𝑜𝑏𝑞𝑒𝑙𝑒𝑐 + 𝑊𝑟𝑜𝑏𝑞𝑡𝑜𝑜𝑙 + 𝐶 𝑠𝑝𝑎𝑐𝑒 ∙ 𝑆𝑞𝑟𝑜𝑏 )

(81)

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Mathematical Modelling in Robotic Digital Production

Dynamics of investments in the robotics of diversified production project are caused by decrease in capital investments into the production system at the release of technological equipment and production spaces (Kqtech). As a result of acquisition of robotics by the enterprise, there is an increase in capital investments (Kqrob). The amount of change of the listed elements of investment expenses of the enterprise will be: 1

𝑝𝑛

∆𝐾𝑞𝑡𝑒𝑐ℎ =  ∙ (𝐼𝐶𝑗𝑡𝑒𝑐ℎ + 𝐶 𝑎𝑑𝑑 ∙ 𝑆𝑗𝑡𝑒𝑐ℎ ) ∙ ∑ 𝑁𝑗𝑎𝑛𝑛 ∙ 𝑡𝑗 ∆𝐾𝑞𝑟𝑜𝑏 =

1



𝑜𝑝



𝛾𝑗𝑞 −1

∙ (𝐼𝐶𝑞𝑟𝑜𝑏 + 𝐶 𝑎𝑑𝑑 ∙ 𝑆𝑞𝑟𝑜𝑏 ) ∙ ∑ 𝑁𝑗𝑎𝑛𝑛 ∙ (𝑡𝑗 −

(82)

𝛾𝑗𝑞

1 𝛾𝑗𝑞

𝑝𝑛

∙ 𝑡𝑗 ) (83)

Thus, the company’s annual cost savings due to robotizing of the production system can be structured as the following economic mathematical model: 𝑜𝑝

𝑜𝑝

𝑊 𝑠𝑎𝑣 = ∑ {𝑇𝑅𝑗 ∙ 𝑡𝑗 ∙ 𝑘𝑗 ∙ 𝑁𝑗𝑎𝑛𝑛 +

1



𝑝𝑛

∙ 𝑁𝑗𝑎𝑛𝑛 ∙ 𝑡𝑗



𝛾𝑗𝑞 − 1 𝛾𝑗𝑞

∙ (𝑏 𝑡𝑒𝑐ℎ ∙ 𝐼𝐶 𝑡𝑒𝑐ℎ + 𝑊 𝑒𝑙𝑒𝑐 + 𝑊 𝑡𝑜𝑜𝑙 + 𝐶 𝑠𝑝𝑎𝑐𝑒 ∙ 𝑆 𝑡𝑒𝑐ℎ ) −

1



1 𝑜𝑝 𝑝𝑛 ∙ 𝑁𝑗𝑎𝑛𝑛 ∙ (𝑡𝑗 − ∙𝑡 ) 𝛾𝑗𝑞 𝑗 ∙ (𝑏𝑞𝑟𝑜𝑏 ∙ 𝐼𝐶𝑞𝑟𝑜𝑏 + 𝑊𝑟𝑜𝑏𝑞𝑒𝑙𝑒𝑐 + 𝑊𝑟𝑜𝑏𝑞𝑡𝑜𝑜𝑙 + 𝐶 𝑠𝑝𝑎𝑐𝑒 ∙ 𝑆𝑞𝑟𝑜𝑏 ) + [(𝑘𝑝𝑡𝑒𝑐ℎ + 𝐸) ∙ 𝐼𝐶 𝑡𝑒𝑐ℎ + (𝑘𝑝 𝑠𝑝𝑎𝑐𝑒 + 𝐸) ∙ 𝐶 𝑎𝑑𝑑 1 𝑝𝑛 𝛾𝑗𝑞 − 1 ∙ 𝑆 𝑡𝑒𝑐ℎ ] ∙ ∙ 𝑁𝑗𝑎𝑛𝑛 ∙ 𝑡𝑗 ∙  𝛾𝑗𝑞 − [(𝑘𝑝𝑞𝑟𝑜𝑏 + 𝐸) ∙ 𝐼𝐶𝑞𝑟𝑜𝑏 + (𝑘𝑝 𝑠𝑝𝑎𝑐𝑒 + 𝐸) ∙ 𝐶 𝑎𝑑𝑑 ∙ 𝑆𝑞𝑟𝑜𝑏 ] 1 1 𝑜𝑝 𝑝𝑛 ∙ ∙ 𝑁𝑗𝑎𝑛𝑛 ∙ (𝑡𝑗 − ∙ 𝑡 )}  𝛾𝑗𝑞 𝑗 (84)

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Marina Batova, Vyacheslav Baranov and Irina Baranova

The cost savings of an enterprise can be calculated per unit of time, for example, per hour. Let us denote Ztech as hourly costs for process equipment and Zqrob as hourly costs for the q-th industrial robot. The expression of cost savings in the implementation of a company's robotics project for diversified production will assume the following form: 𝑜𝑝

𝑜𝑝

𝑝𝑛

𝑊 𝑠𝑎𝑣 = ∑ 𝑇𝑅𝑗 ∙ 𝑡𝑗 ∙ 𝑘𝑗 + 𝑍 𝑡𝑒𝑐ℎ ∙ ∑ 𝑁𝑗𝑎𝑛𝑛 ∙ 𝑡𝑗 ∑ 𝑁𝑗𝑎𝑛𝑛 ∙ (𝑡𝑗𝑜𝑝 −

1 𝛾𝑗𝑞



𝛾𝑗𝑞 −1 𝛾𝑗𝑞

− 𝑍𝑞𝑟𝑜𝑏 ∙

𝑝𝑛

∙ 𝑡𝑗 )

(85)

For the project of robotics of diversified production to be economically feasible, it is necessary to ensure that the following condition is met: ∑ 𝑇𝑅𝑗 ∙ 𝑡𝑗𝑜𝑝 ∙ 𝑘𝑗𝑜𝑝 + 𝑍 𝑡𝑒𝑐ℎ ∙ ∑ 𝑁𝑗𝑎𝑛𝑛 ∙ 𝑡𝑗𝑝𝑛 ∙ 1

𝑜𝑝

𝛾𝑗𝑞 −1 𝛾𝑗𝑞

− 𝑍𝑞𝑟𝑜𝑏 ∙ ∑ 𝑁𝑗𝑎𝑛𝑛 ∙

𝑝𝑛

𝑡𝑗 + 𝑍𝑞𝑟𝑜𝑏 ∙ ∑ 𝑁𝑗𝑎𝑛𝑛 ∙ 𝛾 ∙ 𝑡𝑗 ≥ 0 𝑗𝑞

(86)

Hence, ∑ 𝑁𝑗𝑎𝑛𝑛 ∙ 𝑡𝑗𝑜𝑝 ∙ (𝑍𝑞𝑟𝑜𝑏 − 𝑇𝑅𝑗 ∙ 𝑘𝑗𝑜𝑝 ) ≤ ∑ 𝑁𝑗𝑎𝑛𝑛 ∙ 𝑡𝑗𝑝𝑛 ∙ (𝑍 𝑡𝑒𝑐ℎ − 𝑍𝑡𝑒𝑐ℎ 𝛾𝑗𝑞

+

𝑍𝑞𝑟𝑜𝑏 𝛾𝑗𝑞

)

(87)

The resulting economic mathematical model can be used to solve a number of problems associated with the implementation of technologically oriented innovation projects in the field of robotics of diversified production. For example, in a given production program of product innovations processed in a high-tech organizational and production structure of diversified production, to determine the efficiency of the industrial robot of the j-th model. The challenges of selecting the robot type for high-tech organizational and production structure of diversified production and defining the

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production program of the robotic complex can also be solved. The created program of using the industrial robot of the predetermined model should provide maximum value of savings of annual expenses for creation and operation of a robotized complex of diversified production. In this case, a linear programming task arises and the target function is defined on the basis of the following expression: 𝑜𝑝

𝑜𝑝

𝑝𝑛

𝑚𝑎𝑥 ∑ 𝑁𝑗𝑎𝑛𝑛 ∙ {𝑡𝑗 ∙ (𝑇𝑅𝑗 ∙ 𝑘𝑗 − 𝑍𝑞𝑟𝑜𝑏 ) + 𝑡𝑗 𝑍𝑞𝑟𝑜𝑏 𝛾𝑗𝑞

𝑍 𝑡𝑒𝑐ℎ (𝛾𝑗𝑞 −1)

∙(

𝛾𝑗𝑞

)}

− (88)

provided that the following restrictions imposed on the target function are fulfilled: 𝑁𝑗𝑎𝑛𝑛 ≥ 0 𝑎𝑛𝑑 𝑁𝑗𝑎𝑛𝑛 ≤ 𝐴𝑗𝑎𝑛𝑛

(89)

where Njann  are the required variables; Ajann  annual program of production of the i-th part by the production department (site, shop, etc.). The algorithm aimed at solving such a task is based on the principles of directed search (Greshilov, 2014; Struchenkov, 2016).

Section 4. Mathematical Modeling of Solving the Combinatorial Tasks of Minimizing Costs in the Process of Creating High-Tech Digital Production Structures Setting the Task of Modeling the Optimal Composition of Highly Automated Production Units In modern conditions, one of the most significant functional strategies of an enterprise becomes an innovative component of the competitive strategy (Radiyevsky, 2013). In the process of innovation activity, enterprises create production systems focused on flexible automation and the use of innovative technologies (process innovations) and produce

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Marina Batova, Vyacheslav Baranov and Irina Baranova

competitive products on the global market. This strategy leads to the emergence of an innovation subsystem that dominates the management system. This subsystem is based on technological innovations that generate organizational and managerial, resource, financial, economic and other types of innovations (Baranov et al., 2016). The process innovations created by an enterprise, which take the form of technological capital, give an impulse for further development of innovative processes, in particular, the processes of formation of high-tech robotic organizational and production structures (Nikolaev et al., 2010). This leads to the appearance of not only modern highly automated fixed assets, but also organizational and managerial innovations, existing as a set of mathematical models, methods and information tools for building highly automated structures (Baranov et al., 2016). Thus, technological capital generates the ability of an enterprise to use rational ways of organizing production processes based on the optimal combination of various components of capital (financial, intellectual, etc.) (Fatkhutdinov, 2015; Shishmarev, 2015). In its turn, technological capital, acting in the form of various intellectual property objects (patented technologies, unique know-hows, etc.), activates organizational and economic relations, both at the level of the enterprise as a whole and at the level of its individual structural components (Baranov et al., 2016). Given the importance of the innovation subsystem and its technological capital for the competitiveness of the enterprise, the task of modeling the optimal composition of production units of high-tech organizational and production structures arises (Struchenkov, 2016). The formation of such units presupposes the introduction of highly automated equipment and industrial robots into their composition, which performs the functions of transport and transfer devices. As a result of the design of high-tech organizational and production structures and their production units, the creation of a high production potential of the enterprise is ensured. This potential, covering the totality of fixed assets, intangible assets, organizational and managerial innovations and a number of other elements, is characterized by the size of

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127

production capacity, i.e., maximum ability of highly automated units to fulfill a given production program. In the highly automated systems that are being created, the problem of servicing several parallel units of technological equipment by a robot arises. Moreover, for each of these units, the execution of one operation is assigned for a certain time interval. For the convenience of mathematical modeling, we will consider equipment performing machining operations. For the operation performed in a highly automated production unit by the j-th unit of technological equipment, the corresponding time parameters are set. These parameters are:   

auxiliary time (tjaux), i.e., time of robot servicing the j-th unit of the equipment; time of automatic operation of the j-th unit of the equipment (tjа); operative time of execution of the j-th operation (tjop), the value of which will be 𝑜𝑝

𝑡𝑗

= 𝑡𝑗𝑎 + 𝑡𝑗𝑎𝑢𝑥

(90)

We will formulate the problem of modeling the optimal composition of a high-tech organizational and production structure as follows. Within the framework of the structure, it is required to create a set of highly automated production units that satisfy a number of conditions. These conditions consist in the fact that it is necessary to select for each unit the number of units of technological equipment in such a way that during the automatic operation of any unit of equipment the robot can serve all other units included in the formed set of equipment. The result of solving the problem formulated by us means such an organization of the production process, in which, when servicing the technological equipment by the robot, the minimum downtime of the equipment and the maximum load coefficient of the robot are ensured. Therefore, for any technological operation performed in a highly automated production unit (robotic set), the following ratio must be met:

128

Marina Batova, Vyacheslav Baranov and Irina Baranova 𝑡𝑗𝑎 ≥ 𝑇 𝑡𝑜𝑡 − 𝑡𝑗𝑎𝑢𝑥

(91)

Where Ttot  total robot load time in the equipment maintenance cycle. This time is determined based on the following ratio of time parameters: 𝑎𝑢𝑥 𝑇 𝑡𝑜𝑡 = ∑𝜔 𝑗=1 𝑡𝑗

where   the amount of equipment included in the highly automated production unit, which is serviced by one robot. When determining the total time the robot is busy in the service cycle, the summation is performed over the entire amount of equipment included in a highly automated production unit, which is serviced by one robot. High robot load coefficient servicing the process equipment is a necessary condition to ensure uninterrupted operation of the equipment included in the highly automated production unit. However, such a condition is not sufficient. Formulation of sufficient condition of ensuring uninterrupted operation of highly automated production unit is connected with setting limits on values of operational time of each operation performed by equipment of the unit (tjop). Let's consider that in the basic production system of the enterprise, on the basis of the elements of which highly automated production units are formed, proportional operations are performed, i.e., operations for which the ratio of 𝑜𝑝

𝑡𝑗

= 𝐵𝑗 ∙ 𝑑𝑚𝑖𝑛

(92)

is true, where Bj are integers; dmin  the minimal operational time out of the whole set of operations performed by the equipment included in the highly automated production unit formed within the high-tech organizational and production structure. In case of implementation in the highly automated production unit of proportional operations, it is possible to determine analytically the value

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of that minimum time interval, during which the service of all units of technological equipment, included in the set, is repeated in the same order. This time interval will represent the cycle of service performed by the robot, working with the technical equipment of the highly automated production unit. With proportional operations, the duration of the service cycle of the equipment carried out by the robot in the formed highly automated production unit is calculated in the following manner: 𝑇𝑐𝑦𝑐𝑙𝑒 =  ∙ 𝑑𝑚𝑖𝑛

(93)

where   is the least common multiple of the Bj coefficients. In the course of the process equipment maintenance cycle carried out by the robot the number of implementations of the j-th operation will be the value determined by the ratio 

(94)

𝐵𝑗

In doing so, the robot spends the time tjaux. to perform the j-th operation. Therefore, the robot load coefficient will be: 𝑙𝑜𝑎𝑑 𝑘𝑟𝑜𝑏 =

 𝑇𝑐𝑦𝑐𝑙𝑒

∙ ∑𝜔 𝑗=1

𝑡𝑗𝑎𝑢𝑥 𝐵𝑗

=

1 𝑑𝑚𝑖𝑛

∙ ∑𝜔 𝑗=1

𝑡𝑗𝑎𝑢𝑥 𝐵𝑗

(95)

Using the results of the production process organization, which consists of proportional operations, we can build a number of approximate algorithms for creating highly automated production structures (robotic sets), and then, combining them into a single system, form a high-tech robotic organizational and production structure.

Algorithm of Creation of Highly Automated Production Units of the High-Tech Organizational and Production Structure The task of organizing uninterrupted maintenance of technological equipment by an industrial robot in the course of designing robotic organizational and production structures and in the process of their

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operation is formulated in different ways. At the design stage, it is required to find the optimal option for dividing the entire set of technological equipment of the basic production system of an enterprise to be robotized into highly automated production units (equipment sets), so that each of these sets is serviced by one robot. In the process of operation of high-tech robotic organizational and production structures, the solution to the problem consists in the optimal assignment of operations to existing units for a planned period of time. In both cases it is necessary to consider units of technological equipment interchangeable in relation to performed operations. If this condition is not observed for all units of production system, the task of optimization of composition of highly automated production units should be solved in relation to each group of interchangeable equipment separately (Baranov, 2016b). While setting the task of optimizing the composition of highly automated production units, it should be taken into account that some types of industrial robots are stationary (Baranov, 2016b). These robots have no freedom of movement. Therefore, for stationary robots, a limit should be imposed on the number of process equipment serviced by them. The task of optimizing the composition of highly automated production units is therefore more complex. This is due to the fact that the planned production volume for some operations may not provide the full load of equipment. Therefore, for each technological operation, besides the values tjop and tjaux, it is necessary to set the planned load coefficient of the equipment performing this operation (Kjload) (Baranov, 2016b). Then the task of optimizing the composition of highly automated production units by dividing the interchangeable equipment of the basic production system of the enterprise to be robotized into robotic production units at the design stage can be formulated as follows. Firstly, it is necessary to form an array of raw data. Most importantly, within the framework of creation of such an array we should be aware of the number of technological equipment units (J). Each of these units is assigned to execution of the j-th operation with known values of tjop, tjaux, Kjload. Secondly, the maximum allowed quantity of equipment (r), which can be

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located in the service area of each robot. For a robot, servicing an unlimited working zone, we can assume that the maximum allowable amount of equipment that can be in such a zone of each robot is equal to the number of operations. To solve such a problem it is necessary to find a variant of dividing the given set of equipment into highly automated production units (kits) providing an optimum of the selected target function. This optimum is determined on condition that the actual load coefficient of the j-th unit equipment (kjload) working as part of the highly automated production units (robotic set) satisfies the inequality of the following kind: 𝐾𝑗𝑙𝑜𝑎𝑑 ≤ 𝑘𝑗𝑙𝑜𝑎𝑑 ≤ 𝐾𝑗𝑙𝑜𝑎𝑑 + 

(96)

where  is the set value of deviation of the actual load coefficient relative to the planned value. As a target efficiency function, it is reasonable to choose one that provides the maximum of Net Present Value (NPV) or internal rate of return (IRR), achieved in the life cycle interval of the project to create hightech robotic organizational and production structures. Besides, as a target function of efficiency it is possible to use the functionality that minimizes the complex of integral costs (investment and current costs) within the life cycle interval of the technologically oriented investment project implemented by the enterprise. Such costs are associated with the acquisition and use of elements of the designed highly automated production units (equipment, robotics, distributed information systems, transport and storage devices, etc.). In the process of operation of the formed highly automated production units, the formulation of the task acquires another form. In this case, the number of formed units () and the number of operations (J) to be performed in the planned period should be known in advance. These operations are characterized by the same parameters as in the previous case. Moreover, the number of operations (J) should not exceed the value of the total number of technological equipment of a high-tech robotic

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organizational and production structure, i.e., equipment included in all highly automated production units (sets formed at the design stage). To solve such a problem, it is necessary to fix operations to highly automated production units in such a way that would ensure the minimum current costs for maintenance and operation of formed sets. At the same time, the requirements to the value of the actual load coefficient of the j-th unit of equipment (kjload) remain the same as in the previous case. These requirements formalize the conditions of the enterprise’s mandatory implementation of the planned production program. The formulated variants of tasks that have to be solved within the framework of optimization of composition of highly automated production units are combinatorial tasks (Greshilov, 2014; Kolemaev, 2014; Korte and Vygen, 2018). There are two ways to precisely solve the problems of such nature. The first method involves a complete review of possible options for including equipment in the highly automated production units (robotic kits). The second variant is connected with designing possible variants of uniting the operations to the process equipment of highly automated production units. With a large number of technological equipment included in the production system, such tasks belong to the class of combinatorial tasks, which require considerable time to be solved. That is why for the enterprise management it is of practical interest to develop approximate methods to solve them (Baranov et al., 2018). We propose an approximate method for solving such problems. The algorithm is based on the fact that the total number of operations performed (J) is given, each of which is characterized by parameters. The permissible excess of the actual load coefficient of technological equipment in relation to the planned value does not exceed a predetermined value (). It is necessary to fix the set number of operations to highly automated production units in such a way that in each unit uninterrupted service of the equipment is carried out by the robot (Baranov, 2016b). At the same time, the distribution of operations between the robotic units should be performed in such a way that the actual load coefficients of each piece of equipment are, firstly, not less than the planned (calculated) load

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coefficients (Baranov et al., 2018) and, secondly, differ from them not more than by the value of predetermined deviation value ().

Source: author’s research.

Figure 1. Block diagram of the algorithm.

The algorithm proposed by us  the block diagram which is shown in Figure 1  includes the following sequence of steps:

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Step 1: For each j-th operation, the recurrence period (τj) is calculated, i.e., the time interval at which the load coefficient of the equipment performing this operation (with the accuracy of the rounding of the division result) is equal to Kjload. Such a calculation assumes the determination of an integer part of the values of tjop. Step 2: the minimum value (dmin) is selected from the set of calculated repetition period values. Step 3: the actual load coefficient of the equipment performing the jth operation is calculated. The following model is used for the calculation: 𝑜𝑝

𝑘𝑗𝑙𝑜𝑎𝑑 = 𝐵

𝑡𝑗

𝑗 ∙𝑑𝑚𝑖𝑛

(97)

In this model, the Bj value is chosen as an integral part of the ratio 𝑑

𝜏𝑗 𝑚𝑖𝑛

.

At the same time, it should be taken into account that the technological operation can be performed in the formed set provided that the actual load coefficient of the j-th unit of equipment (kjload) is within the limits of permissible deviation from the value of the planned load coefficient (Kjload). Step 4: the final check of acceptability of the selection of equipment set for the highly automated production unit is performed. The primary aim of the check is to control the basic condition of organization of uninterrupted service of the equipment by the robot, i.e., in observance of mathematical equality of the following kind: 𝑇 𝑙𝑜𝑎𝑑 = 𝑑𝑚𝑖𝑛 . Step 5: correction of the number of equipment units, included into the highly automated production unit (robotic set). The correction supposes the reduction of the number of equipment units, included into the highly automated production unit, and is carried out in two cases (Baranov et al., 2018). Firstly, if at the previous step of the algorithm it turns out that 𝑇 𝑙𝑜𝑎𝑑 > 𝑑𝑚𝑖𝑛 , and secondly, if the number of technological equipment in the set exceeds the permissible number of equipment units, which can be serviced by the robot.

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Computational Mechanism of Implementation of the Algorithm of Creation of Highly Automated Production Units of the High-Tech Organizational and Production Structure To implement the described algorithm, one can build a corresponding computational mechanism. In the course of creating such a mechanism, the task of dividing the given set of technological equipment into highly automated production units (robotic sets) is replaced by the task of sequential separation of units from the basic production system to be robotized. This mechanism is implemented in the following sequence: Step 1: For each j-th operation performed in the production system to be robotized, a repetition period is calculated as j Step 2: the whole set of technological operations performed in the production system to be robotized is ordered in ascending order j. Step 3: the implementation of the process of forming the highly automated production units (robotized sets). In this case, the value of τj for the first operation in the sequence should be used as the parameter dmin. Further, other operations are added to the selected one. When forming a set of operations, a number of parameters are taken into account, including:   

fulfillment of the operation proportionality condition; acceptability of the value of actual load coefficient of the equipment performing the j-th operation (kjload); restrictions on the number of technological equipment, serviced by the robot within the zone of its activity.

It is obvious that the value of deviation of the actual equipment loading coefficient of the j-th unit (kjload) from the value of the planned loading coefficient (Kjload) affects the formation of a set of operations performed using the equipment included into the highly automated production units (robotic sets). By varying this parameter within some limits, admissible from the point of view of the given design conditions for these units, it is

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possible to obtain various options for organizing uninterrupted maintenance of equipment by the robot. Each of these options will correspond to a certain value of the target efficiency function selected when setting the task (Baranov, 2016b). The choice of the option that provides a value close to the optimum of the objective function allows us to speak about the quasi-optimal nature of the approximate method proposed by us for creating highly automated production units of a high-tech organizational and production structure. The algorithm for solving the problem of assigning operations to technological equipment during the operation of highly automated production units (robotic sets) is similar to the considered algorithm. The actual number of pieces of technological equipment included in various highly automated production units (robotic kits) acts as a limit on the amount of this equipment located in the robot's zone of activity (Baranov et al., 2018). Quasi-optimization of the solution is achieved on the set of admissible solutions generated by means of variation of the value of deviation of the actual load coefficient of the j-th unit of equipment (kjload) from the value of the planned load coefficient (Kjload) within the limits from 0 to the maximum admissible value. The essential difference of this algorithm from the previously considered one is that due to the predetermined number of highly automated production units (robotic sets), the set of feasible solutions may turn out to be empty. Such a situation is possible even if the number of operations performed within a separate highly automated production unit (robotic set) is less than the number of technological equipment of this unit. In this case, obtaining at least one admissible solution of the task will be connected with the change of initial data. This change may concern a number of parameters, including changes in composition and structure of technological operations, production program, planned equipment load coefficient, etc. (Baranov et al., 2018). Such changes dictate the necessity to improve the mechanism of control over the processes of mastering the production capacities, both the ones created as a result of designing the highly automated units, and the high-tech organizational and production structures of the enterprise as a whole.

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Results and Discussion The set of dynamic models proposed in the chapter provides for the economically reasonable selection of the best alternative to create high-tech digital structures at the enterprise level. The models include a compact description of interaction between the innovative modernization factors affecting the adoption of the optimal decision. The resulting set of models is combined, as it deploys integer variables along with continuous parameters and the model limits are of the nonlinear form. The developed modifications of the economic mathematical model enable the enterprise to reduce costs on the design of high-tech structures at the set values of the production program or the automation level of the technological equipment. The discrete programming task related to the development of the optimal program of the high-tech production structures has been solved within the modeling process of the production capacity management strategy. The analysis of various heuristic algorithms allowed substantiating the selection of the algorithm built on the Johnson method. The developed model was used to solve the selection problem of the optimal ratio between the flexibility and productivity of high-tech production structures. For highly automated organizational and production structures of hightech enterprises the task was solved for the management of their production capacity mastering processes. Mastering of the production capacities was described as a transient process by an inhomogeneous difference equation. The integral solution was presented as a sum of the global solution of the homogeneous equation and the particular solution of the inhomogeneous equation. Layout implementation of the problem solution model and algorithm in a high-level programming language demonstrated high robustness of this model, allowing the reduction of requirements applied to the quality of source data. The problems related to selection of economically expedient robotics limits of high-tech structures with regard to various types of production processes were solved via the mathematical modeling. The modeling results were used to solve a number of problems emerging in the

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production robotics projects. In particular, the application efficiency of the specific robot model was evaluated at the given production program of the high-tech organizational and production structure. The problem was solved for the selection of the robot type for the high-tech organizational and production structure of the diversified production, as well as the linear programming task related to the determination of the production program of a robotized complex. The theoretical framework and methodology for the optimization of the robotized organizational and production structures were proposed. The mathematical modeling was performed and the solution algorithm was developed for the combinatorial cost minimization tasks in case of hightech structures created at the enterprise. The solution of this class of problems allowed minimizing the excessive resources in created robotized organizational and production structures and ensuring the maximum possible loading of their production capacities. Information systems supporting the adoption of a wide range of investment solutions are to be established for the efficient implementation of the developed economic mathematical models. When solving such tasks, information technologies become the key efficiency factors of the innovative modernization strategy. The use of intellectual information systems and technologies, as well as Business Intelligence systems combining technologies and tools for the business analysis of data becomes topical in the longer term. In order to solve the issues related to the design and integration of the decision-making support systems into the software package structure of digital enterprises, including the software and hardware packages for the processing and analysis of Big Data, technically, economically and financially substantiated programs are to be developed to introduce the information systems into the corporate information structure. Such programs will guarantee efficient business performance management through the implementation of CRM systems. At that, a focus on the process approach to management will actualize the solution of the business process regulation and modeling tasks.

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Management of complex projects gains importance and requires further development in terms of the life cycle of complex systems created at the enterprise level. Modern informational support tools for the management of business processes implemented on advanced digital platforms are to be involved to solve such tasks. The solution of the said task will definitely improve the integration efficiency of the economic mathematical models proposed by us into the project management field. In this regard, the information management tooling developments, including CALS technologies and CASE tools become highly promising, while the study of CAD/CAM/CAE systems integration issues represents a key efficiency tool for the robotics processes. To achieve this, the use of CASE technologies with the application of Erwin tools, development of detailed information models for business processes in the field of robotized production organization and operation are to be studied in detail.

Conclusion The developed models have a high level of detail. They contain several groups and classes of variables that characterize technological innovations, production equipment, robotics, automation level, production areas, production time, costs (including investment in modernization and subsequent production), intellectual resources, etc. Using the high-level programming language Java, we developed an enterprise software package based on the developed models. The implementation of the algorithm in Java ensures high robustness of the created models, thereby reducing the requirements for the quality of the source data. The simulation results can be used to solve production challenges, such as selection of the optimal ratio between flexibility and performance of a robotic structure. Application of the linear programming built into the models also could be used to solve economic challenges, such as choosing for robot type and the defined robotic-production program.

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In: Frontiers in Mathematical Modelling ... ISBN: 978-1-68507-430-2 c 2022 Nova Science Publishers, Inc. Editors: M. Biswas and M. Kabir

Chapter 5

Mathematical Modelling and Simulation of a Robot Manipulator Juan L. Mata-Machuca1,∗, Ricardo Aguilar-L´opez2,† and Jorge Fonseca-Campos3 1

Department of Advanced Technologies, UPIITA, Instituto Politecnico Nacional, Mexico City, Mexico 2 Department of Biotechnology and Bioengineering, CINVESTAV-IPN, Mexico City, Mexico 3 Department of Basic Sciences, UPIITA, Instituto Politecnico Nacional, Mexico City, Mexico

Abstract Robotics has had an exponential development in terms of its implementation and development in industrial and everyday applications, mainly with the aim of replacing the human being in activities that involve the performance of repetitive tasks, greatly reducing the time of almost any process, the costs and the physical effort applied. In this way, its study in universities and research institutions have opened the gap to promote innovation and manipulator optimization, or to demonstrate the disadvantages or limitations that some type of robotic solution may have in certain applications. In this ∗ †

Corresponding Author’s Email: [email protected]. Corresponding Author’s Email: [email protected].

150 J. L. Mata-Machuca, R. Aguilar-L´opez and J. Fonseca-Campos work, the main problem is the mathematical modelling and simulation of a 4 degree of freedom (DOF) robotic manipulator with trajectory-tracking control. First of all, the modelling is addressed, it is based on the forward kinematics, the inverse kinematics and the differential kinematics. Then, as a proposed solution to the control problem, it has been suggested to carry out the control by means of independent joints, which basically consists of taking the variables that place the robot in the configuration that reaches the desired position by injecting them to the controller along with feedback from the position sensors such as controller error. Finally, the results obtained with the proposed independent joint controllers show acceptable operation characteristics during simulation.

Keywords: 4 DOF robotic manipulator, mathematical modelling, simulation, trajectory-tracking control

Introduction In recent years it has become evident that robotic technology has captivated the world with its multiple applications. Frequently, an adjective is added to the term robot, which allows to establish its characteristics or field of application, for example, robotic manipulator, humanoid robots, domestic robots, aerial robots, land mobile robots, etc. [1, 2, 3]. Robotics has had an exponential development in terms of its implementation and development in industrial and everyday applications, mainly with the aim of replacing the human being in activities that involve the performance of repetitive tasks, greatly reducing the time of almost any process, the costs and the physical effort applied [4, 5, 6, 7]. In this way, its study has opened the gap to promote innovation and optimization, or to demonstrate the disadvantages or limitations that some type of robotic solution may have in certain applications [8, 9, 10, 11]. Currently, there is a wide variety of techniques for controlling the position of robotic manipulators, which consider the operator as a crucial part of the control of the entire system, making its functions and modes of operation available to them [12, 13, 14]. Particularly, the robotic manipulators have a great versatility of use due to the fact that almost in any industrial or technological development it is necessary to carry out the task of changing an item’s position [15, 16, 17, 18].

Mathematical Modelling and Simulation of a Robot Manipulator 151 According to the definition of robotic manipulator given by the International Organization for Standardization, a robotic manipulator is a machine capable of moving or dragging an object, and is also composed of coupled segments which maintain a relative motion between them. Analyzing the word manipulator in more detail, it refers to the actions of manipulating, moving, handling or translating, to give some examples. When the term robotic is attributed to the word previously described, the term as a whole is seen as a controlled, programmed and constructed machine to move an element to a determined position or area [19]. The fundamental structure of any manipulator is the open kinematic chain [19]. A kinematic chain is defined as open when there is only one sequence of links connecting the two ends of the chain. The most common geometries for an open chain robot manipulator are: (1) cartesian, realized by three prismatic joints which often conform an orthogonal coordinate system and (2)SCARA, this geometry consists in two revolute joints that allow the robot to move over one plane. As summary, we focus on solving the following issues, 1) Robot morphology. It is necessary to delimit properties such as: workspace, dimensions, joint types, end effector, etc. The task to be performed in this case will be a drawing in the x − y plane; this will help us delimit characteristics such as: velocity, desired precision, load capacity, etc. 2) Kinematic model. To understand the movement, it is necessary the kinematic analysis, both direct and inverse. 3) Path Generation. This implies the need of a user entering information to the robot. 4) Position control. Using the kinematic model, the trajectory-tracking applying a PID controller is achieved. Based on the above, this work proposes the mathematical modelling and simulation of a 4 DOF robotic manipulator for trajectory-tracking. The essence of the control of the proposed robotic system is based on the Denavit-Hartenberg convention for the kinematic study of the manipulator, in addition to the application of a control by independent joints that

152 J. L. Mata-Machuca, R. Aguilar-L´opez and J. Fonseca-Campos acquires the respective input variables with modes of operation that consider the correct operation of the tracking process.

Mathematical Modelling As is shown in Figure 1, the 4 DOF robotic manipulator consists of three types of articulations: shoulder (2 DOF); elbow (1 DOF); and wrist (1 DOF). Figure 1(a) describes the computer-aided design (CAD) using SolidWorksr and Figure 1(b) is the equivalent representation in Matlabr . The main dimensions are indicated on Figure 2, which are taken as: l1 = 35 cm, l2 = 27.5 cm, l3 = 22.5 cm, l4 = 7 cm, lp = 7 cm and lo = 4 cm. Now, we establish the kinematic analysis.

Forward Kinematics An absolute reference frame is placed at the base of the robot, and each of the links’ locations is described with respect to the reference system in order to solve the problem of direct kinematics, see Figure 3. The DenavitHartenberg method is applied since the task will be performed only in the x − y plane. It is possible to establish a fixed reference system located at the base of the robot and describe the location of each of the links with respect to said reference system in order to solve the problem of direct kinematics. Then, we proceed to find the homogeneous transformation matrix T that relates the position and orientation of the end effector of the manipulator with respect to the fixed reference system located at its base. This T matrix will be a function of the joint coordinates. To describe the location of each link, a frame of reference is assigned to each one, that is, the frame i is attached to the link i. It is worth mentioning that each joint has an axis with respect to which its movement is described and by convention, this is the z axis of the reference frame. Between consecutive links there are matrices commonly denoted by A that represent the position and orientation of link i with respect to link i − 1. We apply the Denavit-Hartenberg algorithm for each coordinate system as is shown in Table 1.

Mathematical Modelling and Simulation of a Robot Manipulator 153

(a)

(b)

Figure 1. 4 DOF robotic manipulator. (a) CAD, (b) representation in Matlabr. As mentioned above, the Denavit-Hartenberg convention is based on homogeneous transformation matrices (Ai ),

154 J. L. Mata-Machuca, R. Aguilar-L´opez and J. Fonseca-Campos

(a)

(b)

Figure 2. Robot dimensions. (a) Lateral view, (b) front view. Table 1. Denavit-Hartenberg parameters Link 1 2 3 4

di l1 0 0 l4

ai 0 l2 l3 0

αi 90◦ 0 0 0

qi q1 q2 q3 q4

Ai = Rotz,qi T ransz,di T ransx,ai Rotx,αi

(1)

Mathematical Modelling and Simulation of a Robot Manipulator 155

Figure 3. Coordinate systems’ assignation. where, each matrix of (1) is defined as,  cos(qi ) − sin(qi ) 0  sin(qi ) cos(qi ) 0 Rotz,qi =   0 0 1 0 0 0   1 0 0 0  0 1 0 0   T ransz,di =   0 0 1 di  0 0 0 1   1 0 0 ai  0 1 0 0   T ransx,ai =   0 0 1 0  0 0 0 1

 0 0   0  1

(2)

(3)

(4)

156 J. L. Mata-Machuca, R. Aguilar-L´opez and J. Fonseca-Campos

Rotx,αi



1 0 0  0 cos(αi) − sin(αi ) =  0 sin(αi ) cos(αi ) 0 0 0

 0 0   0  1

(5)

Then, using equation (1) and the parameters of Table 1 we obtain the matrices,   C1 0 S1 0  S1 0 −C1 0   A1 =  (6)  0 1 0 l1  0 0 0 1   C2 −S2 0 C2 l2  S2 C2 0 S2 l2   A2 =  (7)  0 0 1 0  0 0 0 1   C3 −S3 0 C3 l3  S3 C3 0 S3 l3   A3 =  (8)  0 0 1 0  0 0 0 1   C4 −S4 0 0  S4 C4 0 0   (9) A4 =   0 −1 0 l4  0 0 0 1 We use the notation: cos(qi ) = Ci , sin(qi ) = Si . Now, the transformation matrix T40 is calculated as,   r11 r12 r13 Px r21 r22 r23 Py   T40 = A1 A2 A3 A4 =  r31 r32 r33 Pz  0 0 0 1

(10)

Mathematical Modelling and Simulation of a Robot Manipulator 157 where,   r11     r 21    r  31     r12     r22    r32 r  13     r23      r33    Px     Py    P z

= −C4 (C1 S2 S3 − C1 C2 C3 ) − S4 (C1 C2 S3 + C1 C3 S2 ) = −C4 (S1 S2 S3 − C2 C3 S1 ) − S4 (C2 S1 S3 + C3 S1 S2 ) = C4 (C2 S3 + C3 S2 ) + S4 (C2 C3 − S2 S3 ) = S4 (C1 S2 S3 − C1 C2 C3 ) − C4 (C1 C2 S3 + C1 C3 S2 ) = S4 (S1 S2 S3 − C2 C3 S1 ) − C4 (C2 S1 S3 + C3 S1 S2 ) = C4 (C2 C3 − S2 S3 ) − S4 (C2 S3 + C3 S2 ) = S1 = −C1 =0 = l2 C1 C2 + l2 S1 − l3 C1 S2 S3 + l3 C1 C2 C3 = l2 C2 S1 − l2 C1 − l3 S1 S2 S3 + l3 C2 C3 S1 = l1 + l2 S2 + l3 C2 S3 + l3 C3 S2

(11)

All results of Equations (1)-(11) can be verified using the Matlab code of Table 2. From (10), the upper left sub-matrix (3 × 3),   r11 r12 r13  r21 r22 r23  r31 r32 r33 indicates the orientation of the end effector with respect to the base frame, and the upper right sub-matrix (3 × 1),   Px  Py  Pz provides its position also with respect to the base’s coordinate system.

Inverse Kinematics Once the position and orientation of the end effector are known, it is necessary to know the joint angles needed to reach that position, for this end

158 J. L. Mata-Machuca, R. Aguilar-L´opez and J. Fonseca-Campos Table 2. Matlab code for the forward kinematics Table 2. Matlab code for the forward kinematics.

Mathematical Modelling and Simulation of a Robot Manipulator 159 we use the scheme of Figure 4, where, q p = Px2 + Py2 q r = p2 − l42   l4 θc = arctan r   Py α = arctan Px

Figure 4. Top view of the robotic manipulator. Then, q1 can be found as, 

Py q1 = α + θc = arctan Px



  l4 + arctan r

(12)

Now, we consider the following relations, r 2 = Px2 + Py2

(13)

160 J. L. Mata-Machuca, R. Aguilar-L´opez and J. Fonseca-Campos R2 = r 2 + Pz2 = l22 + l32 + 2l2 l3 cos(q3 )

(14)

Equation (14) is deduced by Figure 5.

Figure 5. “Elbow up” and “Elbow down” configurations. Substituting (13) into (14), cos(q3 ) =

Px2 + Py2 + Pz2 − l22 − l32 2l2 l3

(15)

and using (16)

sin(q3 ) , then, by equations (15) and (16), we obtain cos(q3 )   p   1 − cos2 (q3 )  q3 = arctan  ± (17)  P 2 + P 2 + P 2 − l2 − l2  x y z 2 3 2l2 l3 ! p 2l2 l3 1 − cos2 (q3 ) q3 = arctan ± 2 (18) Px + Py2 + Pz2 − l22 − l32

Since tan(q3 ) = q3 ,

p sin(q3 ) = ± 1 − cos2 (q3 )

We note that q3 has two solutions due to the configurations of the robotic manipulator (elbow up and elbow down) as is shown in Figure 5. Indeed, we define q2 in terms of α and β, q2 = β − α

(19)

Mathematical Modelling and Simulation of a Robot Manipulator 161 where, 

Pz β = arctan r and





Pz



 = arctan  q ± Px2 + Py2 

l3 sin(q3 ) α = arctan l2 + l3 cos(q3 )



Replacing (20) and (21) into (19)     P l sin(q ) z 3 2  − arctan q2 = arctan  q l2 + l3 cos(q2 ) ± Px2 + Py2

(20)

(21)

(22)

Finally, we can see from Figure 6 that q4 represents the orientation of the end effector and it is defined by q4 = qe − q3 − q2

where qe is measured from the horizontal plane to the end effector.

Figure 6. Representation of q4 .

(23)

162 J. L. Mata-Machuca, R. Aguilar-L´opez and J. Fonseca-Campos

Differential Kinematics Let us consider Px , Py and Pz obtained by the matrix T40 , and qe given by (23), x = l2 C1 C2 + l2 S1 − l3 C1 S2 S3 + l3 C1 C2 C3 = fx y = l2 C2 S1 − l2 C1 − l3 S1 S2 S3 + l3 S1 C2 C3 = fy z = l1 + l2 S2 + l3 C2 S3 + l3 C3 S2 = fz qe = q2 + q3 + q4 = fqe The differential kinematics model can be represented as,     x˙ q˙1  y˙   q˙2       z˙  = J  q˙3  q˙e q˙4

(24)

where, J is the Jacobian matrix described by,  ∂f x  ∂q1   ∂fy   ∂q 1  J =  ∂fz   ∂q1   ∂f qe ∂q1

∂fx ∂q2 ∂fy ∂q2 ∂fz ∂q2 ∂fqe ∂q2

∂fx ∂q3 ∂fy ∂q3 ∂fz ∂q3 ∂fqe ∂q3

∂fx ∂q4 ∂fy ∂q4 ∂fz ∂q4 ∂fqe ∂q4

            

(25)

The partial derivatives of fx , fy , fz and fqe with respect to q1 , q2 , q3

Mathematical Modelling and Simulation of a Robot Manipulator 163 and q4 , are ∂fx ∂q1 ∂fx ∂q2 ∂fx ∂q3 ∂fx ∂q4 ∂fy ∂q1 ∂fy ∂q2 ∂fy ∂q3 ∂fy ∂q4 ∂fz ∂q1 ∂fz ∂q2 ∂fz ∂q3

=

l2 C1 − l2 C2 S1 − l3 C2 C3 S1 + l3 S1 S2 S3

=

−C1 (l3 S23 + l2 S2 )

=

−l3 S23 C1

=

0

=

l2 S1 + l2 C1 C2 + l3 C1 C2 C3 − l3 C1 S2 S3

=

−S1 (l3 S23 + l2 S2 )

=

−l3 S23 S1

=

0

=

0

=

l3 C23 + l2 C2

=

l3 C23 ∂fz ∂q4 ∂fφ ∂q1 ∂fφ ∂q2 ∂fφ ∂q3 ∂fφ ∂q4

=

0

=

0

=

1

=

1

=

1

where, Cxy = cos(x + y) and Sxy = sin(x + y). Hence, the Jacobian matrix is, 2

l2 C1 − l2 C2 S1 − l3 C2 C3 S1 + l3 S1 S2 S3 6 l2 S1 + l2 C1 C2 + l3 C1 C2 C3 − l3 C1 S2 S3 J =6 4 0 0

−C1 (l3 S23 + l2 S2 ) −S1 (l3 S23 + l2 S2 ) l3 C23 + l2 C2 1

−l3 S23 C1 −l3 S23 S1 l3 C23 1

3 0 0 7 7 0 5 1

164 J. L. Mata-Machuca, R. Aguilar-L´opez and J. Fonseca-Campos

Singularities When the robot is located at a singularity, in order for the end of the manipulator to move at constant speed, it is required to request joint movements at unapproachable speeds by the joint actuators. The robot singularities are obtained by means of the determinant of the Jacobian matrix, det(J) = −l2 l3 (l2C2 S3 − l3 S2 + l3 C32 S2 + l3 C2 C3 S3 ) where, C32 = cos2 (q3 ). There is a singular configuration of the manipulator when the determinant of the Jacobian is equal to 0. X There is a singularity when s3 = 0, this is fulfilled for q3 = 0 and q3 = π. Figure 7 shows the robot positions for both singularities. The determinant is plotted to observe the behavior of the singularities. Figure 8 shows that, det(J) = 0 when q3 = 0, q3 = π and q3 = 2π. In general, there are singularities when q3 = nπ for n ∈ Z. To avoid the robot from reaching singularities or having collisions between the links, the range of movement of each motor coupled to the different joints is limited. For more details see Figure 9. Therefore, the work area is defined by X Link 1: [-81◦ to 180◦ ] Rotation on z axis. X Link 2: [-4◦ to 94◦ ] Rotation on xy plane. X Link 3: [-184◦ to -90◦ ] Rotation on xy plane. X Link 4: [-170◦ to 150◦] Rotation on xy plane.

Mathematical Modelling and Simulation of a Robot Manipulator 165

(a)

(b)

Figure 7. Singularities. (a) q3 = 0, (b) q3 ≈ π. Trajectories To perform a certain task, the robot must move to an initial point to and end point. This movement can be performed following various spacial trajectories. From all possible paths, commercial robots use only the easiest ones to perform, as is shown in Figure 10. The reference trajectory of Figure 10 is proposed as a time dependent

166 J. L. Mata-Machuca, R. Aguilar-L´opez and J. Fonseca-Campos 107

2.5 2 1.5 1

det(J)

0.5 0 -0.5 -1 -1.5 -2 -2.5 0

1

2

3

4

5

q3

Figure 8. Determinant of the Jacobian matrix.

Figure 9. Work area.

6

2

7

Mathematical Modelling and Simulation of a Robot Manipulator 167 expression as follows, x = r cos(t) + h y = r sin(t) + k

(26)

where, r is the radius and (h, k) are the coordinates of the center of the circumference.

Figure 10. Parameterization of a circumference as a function of time. Dimensions in mm.

Numerical Simulation The control problem for robot manipulators is to determine the required inputs to get the end effector to reach the desired position to perform a specific task. The control inputs are commonly the torques that must be applied to the robot’s joints so that they move to the desired position, these inputs depend on the actuator that will provide such torques, which in many cases are electric actuators. In this work, permanent magnet direct current motors are used, which can be modeled as,

168 J. L. Mata-Machuca, R. Aguilar-L´opez and J. Fonseca-Campos d R Kv 1 i(t) = − i(t) − w(t) + v(t) dt L L L d Kt B 1 w(t) = i(t) − w(t) − TL (t) dt J J ZJ d q = w(t) dt

(27) (28) (29)

where, i(t) is the armature current (A), R is the total motor resistance (Ω), L is the total motor inductance (H), Kv is the voltage constant (V/rad/s), w(t) is the angular velocity (rad/s), v(t) is the input voltage (V), Kt is the torque constant (N·m/A), J is the inertia moment (kg·m2 ) and TL is the load torque (N·m). Independent joint control is proposed as a solution of trajectorytracking, where the control scheme of Figure 11 is applied. The user will be able to modify the positions and orientations of the joints as well as choosing between various reference trajectories.

(a)

(b)

Figure 11. Control scheme. (a) Blocks diagram, (b) Matlab-Simulink diagram. The PID structure is, ui (t) = Kp ei (t) + KI

Z

ei (τ )dτ + Kd e˙i (t)

(30)

Mathematical Modelling and Simulation of a Robot Manipulator 169

(a)

(b)

Figure 12. (a) Desired q1 vs attained q1 , (b) Desired q2 vs attained q2 . where, ui (t) is the control input and ei (t) = qi,ref − qi , and Kp , KI and Kd are the control gains. The controller develops an output that flows to the dynamic model of the motor’s joint to be controlled, which contains the parameters that describe the motor and relate the electrical variables to the mechanical variables. This control technique consists in taking from the robot’s kinematic model the variables that place the robot in the desired configuration (Figure 12) and use them as an input of the controller along with the measured position. The controller output is then passed on to the dynamic model of the motor that drives such joint. Figure 12 depicts the desired q1 and q2 for the robot along with the attained q1 and q2 , respectively. By separating each joint, a maximum disturbance is calculated, which

170 J. L. Mata-Machuca, R. Aguilar-L´opez and J. Fonseca-Campos

(a)

(b)

Figure 13. Controller output. (a) u1 , (b) u2 . was achieved by placing the robot in the configuration that generates the greatest load for each actuator. For the first actuator, which moves the base of the robot, it has no load due to the weight of the links since gravity is parallel to the axis of the motor, which means that the only disturbance of the motor for the most part is generated by the inertia of robot which is maximum when the links are aligned and parallel to the x-y plane and this configuration jointly yielded a maximum acceleration a torque of 6.1 N·m. For link 2, the maximum torque due to gravity is 1.37 N·m. The simulation is carried out in a time of 10 seconds, so it was decided to add a disturbance with the maximum value of inertia of the arm multiplied by a sinusoidal function. The controller output is also reviewed, which shows us the required

Mathematical Modelling and Simulation of a Robot Manipulator 171

Figure 14. Proposed trajectory (blue line) vs trajectory obtained by the controller (dashed red line). torque of the actuator which must be within the motor’s capabilities. Figure 13 describes the controller outputs, we can observe a maximum torque for control input u1 approximately equal to 0.6 N·m, and for control input u2 a maximum torque near to 2 N·m. To validate the obtained models and controllers, a Matlab simulation was performed. Through the simulation, it is possible to observe that using the proposed controller, the robotic manipulator follows the circular trajectory inside the work area, see Figure 14.

Acknowledgment This work has been supported by the Secretar´ıa de Investigaci´on y Posgrado of the Instituto Polit´ecnico Nacional (SIP-IPN) under the research grant 20211509.

Conclusion In this chapter the mathematical modeling and simulation of a robot manipulator of 4 DOF with an application for tracking point-to-point trajectories were addressed. Regarding kinematic modeling, the robotics calcula-

172 J. L. Mata-Machuca, R. Aguilar-L´opez and J. Fonseca-Campos tions obtained allowed us to know the behavior of the robotic manipulator. Tracking of trajectories was achieved through a control scheme by independent joints. Although the results obtained with the proposed independent joint controllers showed good operation characteristics during simulation, as future work, we will verify the results via experimental implementations.

References [1] Lewis, F. W., Jagannathan, S., & Yesildirak, A. (2020). Neural network control of robot manipulators and non-linear systems. CRC press. [2] Baek, J., Jin, M., & Han, S. (2016). A new adaptive sliding-mode control scheme for application to robot manipulators. IEEE Transactions on Industrial Electronics, 63(6), 3628-3637. [3] He, W., Ge, W., Li, Y., Liu, Y. J., Yang, C., & Sun, C. (2016). Model identification and control design for a humanoid robot. IEEE Transactions on Systems, Man, and Cybernetics: Systems, 47(1), 45-57. [4] Pham, A. D., & Ahn, H. J. (2018). High precision reducers for industrial robots driving 4th industrial revolution: state of arts, analysis, design, performance evaluation and perspective. International Journal of Precision Engineering and Manufacturing-Green Technology, 5(4), 519-533. [5] Pellegrinelli, S., Orlandini, A., Pedrocchi, N., Umbrico, A., & Tolio, T. (2017). Motion planning and scheduling for human and industrialrobot collaboration. CIRP Annals, 66(1), 1-4. [6] Oyekan, J. O., Hutabarat, W., Tiwari, A., Grech, R., Aung, M. H., Mariani, M. P., & Dupuis, C. (2019). The effectiveness of virtual environments in developing collaborative strategies between industrial robots and humans. Robotics and Computer-Integrated Manufacturing, 55, 41-54. [7] Wu, J., Zhang, D., Liu, J., Jia, X., & Han, X. (2020). A computational framework of kinematic accuracy reliability analysis for industrial robots. Applied Mathematical Modelling, 82, 189-216.

Mathematical Modelling and Simulation of a Robot Manipulator 173 [8] Mousavi, S., Gagnol, V., Bouzgarrou, B. C., & Ray, P. (2018). Stability optimization in robotic milling through the control of functional redundancies. Robotics and Computer-Integrated Manufacturing, 50, 181-192. [9] Yang, C., Ma, H., & Fu, M. (2016). Advanced technologies in modern robotic applications. Springer Singapore. [10] Chen, D., Zhang, Y., & Li, S. (2017). Tracking control of robot manipulators with unknown models: A jacobian-matrix-adaption method. IEEE Transactions on Industrial Informatics, 14(7), 30443053. [11] Jin, L., Li, S., Xiao, L., Lu, R., & Liao, B. (2017). Cooperative motion generation in a distributed network of redundant robot manipulators with noises. IEEE Transactions on Systems, Man, and Cybernetics: Systems, 48(10), 1715-1724. [12] Sariyildiz, E., Sekiguchi, H., Nozaki, T., Ugurlu, B., & Ohnishi, K. (2018). A stability analysis for the acceleration-based robust position control of robot manipulators via disturbance observer. IEEE/ASME Transactions on Mechatronics, 23(5), 2369-2378. [13] Oliveira, J., Oliveira, P. M., Boaventura-Cunha, J., & Pinho, T. (2017). Chaos-based grey wolf optimizer for higher order sliding mode position control of a robotic manipulator. Nonlinear Dynamics, 90(2), 1353-1362. [14] Baigzadehnoe, B., Rahmani, Z., Khosravi, A., & Rezaie, B. (2017). On position/force tracking control problem of cooperative robot manipulators using adaptive fuzzy backstepping approach. ISA transactions, 70, 432-446. [15] Homayounzade, M., & Khademhosseini, A. (2019). Disturbance observer-based trajectory following control of robot manipulators. International Journal of Control, Automation and Systems, 17(1), 203211. [16] Andreev, A., & Peregudova, O. (2019). Trajectory tracking control for robot manipulators using only position measurements. International Journal of Control, 92(7), 1490-1496.

174 J. L. Mata-Machuca, R. Aguilar-L´opez and J. Fonseca-Campos [17] Bouakrif, F., & Zasadzinski, M. (2016). Trajectory tracking control for perturbed robot manipulators using iterative learning method. The International Journal of Advanced Manufacturing Technology, 87(58), 2013-2022. [18] Jin, L., Li, S., Yu, J., & He, J. (2018). Robot manipulator control using neural networks: A survey. Neurocomputing, 285, 23-34. [19] Jin, L., Li, S., Yu, J., & He, J. (2018). Robot manipulator control using neural networks: A survey. Neurocomputing, 285, 23-34. [20] Siciliano, B., & Khatib, O. (Eds.). (2016). Springer handbook of robotics. Springer.

In: Frontiers in Mathematical Modelling ... ISBN: 978-1-68507-430-2 c 2022 Nova Science Publishers, Inc. Editors: M. Biswas and M. Kabir

Chapter 6

Mathematical Modeling Applied to Control the Emerging Deadly Nipah Fever in Bangladesh M. Haider Ali Biswas1,∗, Mst. Shanta Khatun1, M. Nazmul Hasan2 and M. Humayun Kabir2 1 Mathematics Discipline, Science Engineering and Technology School, Khulna University, Khulna, Bangladesh 2 Department of Mathematics, Jahangirnagar University, Dhaka, Bangladesh

Abstract Nipah virus (NiV), a Paramyxoviridae family member is a zoonotic virus, that is, it can transmit from animals (e.g., bats, sheep, and pigs, etc.) to people. Among other countries, its outbreak attracts considerable attention in Bangladesh. It is well known that NiV is a newly detected highly emerging pathogen and no proper drugs and/or vaccines are available yet for its treatments. As a result, it is essential for mathematicians and biologists to understand the disease dynamics in the human body in order to develop effective methods for prevention and control. Mathematical models have become one of the important tools in analyzing the spread and control ∗

Corresponding Author’s Email: [email protected].

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M. Haider Ali Biswas, Mst. Shanta Khatun et al. of infectious diseases like NiV. In order to understand such a transmission dynamics of NiV, we propose a mathematical model considering four compartments: susceptible, exposed, infected, and recovered classes. We confirm the boundedness of solutions of the model. To classify the epidemic and endemic cases, we determine the basic reproduction number of the model. The local stabilities at the disease-free equilibria and endemic equilibria are investigated that identify the qualitative behavior of solutions. Since the transmission interactions of NiV in a population are very complex, so it is difficult to comprehend the large-scale dynamics of disease spread. In that case, an optimal control technique is applied to obtain a better control strategy. We introduce two control measures, e.g., campaign for social awareness against NiV infection and social distancing from NiV infected individuals in the dynamic model. The existence of optimal solutions is verified using Pontryagin’s maximum principle. Numerical simulation is performed to confirm the analytical results which exhibits that the infected individuals are reduced and susceptible individuals are increased significantly when massive awareness campaign against NiV is performed. Furthermore, numerical simulation affirms that the infection of NiV minimizes remarkably when these two control measures are incorporated. Finally, it is asserted that our proposed control strategy provides significant reduction of the transmission of NiV infection in absence of effective vaccines and drugs.

Keywords: Nipah virus, SEIR model, stability analysis, optimal control, numerical simulations

Introduction Zoonotic diseases which can jump from other animals to humans or vice versa are particularly more troublesome and deadly. Zoonotic diseases are unique as they are mainly caused by pathogens such as fungi, bacteria, parasites, and viruses, etc. These pathogens typically survive in a reservoir host, which has immunity to the pathogen. The list of possible reservoir hosts capable of transmitting the disease to humans is large; however the most common carriers are apes, insects, rodents mosquitoes, and bats, etc. The diseases are then transferred to humans who come in contact with an infected animal through bites or scratches, an infected animals’ environ-

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ment, or animal secretions such as saliva, feces, or mucus. These diseases often have a higher virulence because of the lack of any immunity within the humans and the ease of transmission. Recently, different types of zoonotic diseases are emerging because of an increase interaction between human and wildlife. The increase in farming and deforestation has resulted in humans and wildlife into the same habitat. A prime example is the emergence of the Nipah virus, which is a member of the Henipavirus genus in the Paramyxoviridae family having subfamily Paramyxovirinae and it has become a growing concern of South-East Asia. NiV infection in human was first recognized in a large outbreak of 276 cases were reported in Peninsular Malaysia and Singapore during September 1998 to June 1999 [1-3]. At that time, most of the patients infected by NiV who had contact with pigs [4] and the death rate was 39% for the patients identified primarily [1, 5]. Clinical studies noted that diffuse vasculitis most prominently involving the central nervous system with intense immunostaining of endothelial cells with anti-NiV hyperimmune serum [2]. The virus was first detected in a patient of Sungai Nipah village in Malaysia and the name “Nipah” was first introduced according to the name of that village [1-3]. The human outbreak of Nipah infection ceased after widespread deployment of personal protective equipment to people having contact with sick pigs, restriction on livestock movements, and culling over 900,000 pigs [6]. Large fruit bats of the genus Pteropus appear to be the natural reservoir of NiV. In Malaysia, the seroprevalence of neutralizing antibodies to NiV in colonies of Pteropus vampyrus and Pteropus hypomelanus ranged from 7 to 58% [7]. Antibodies against henipaviruses have been identified in Pteropus bats wherever they have been tested including Cambodia, Thailand, India, Bangladesh, and Madagascar [8-12]. NiV was idetified from urine specimens collected underneath a P. hypomelanus roost and from partially eaten fruit dropped during feeding activity in Malaysia [13], from urine collected underneath a Pteropus lylei roost in Cambodia [12], and from saliva and urine of Pteropus lylei in Thailand. Experimental infection of Pteropus bats with NiV does not cause illness in the bats [14]. Surveys of rodents and other animals have not been identified on the other wildlife reservoirs for NiV [9]. Over 50 species of Pteropus bats live in South and South-East Asia. Pteropus giganteus, the only Pteropus species found in Bangladesh, is widely roamed across the country and frequently has antibodies to NiV [9, 15]. In South-East Asia, NiV infection

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Figure 1. Schematic diagram of the transmission of Nipah Virus (NiV). has become one of the major epidemic diseases. Such an epidemic arises in the winter season and appears worse because of people’s lack of knowledge and awareness about NiV. That is why, NiV infection may become one of the greatest public health concerns. A precise characterization of the molecular epidemiology, phylogenetic relationship, and evolution of NiV, a Bangladesh perspective is discussed in [16]. A schematic representation of the transmission of Nipah Virus (NiV) is depicted in Figure 1 to have a precise reflection of NiV infection and its transmission as well. Several researchers paid attention to reveal the dynamics of the transmission of NIV infections from epidemiological and public health viewpoints [17-20]. Different approaches to control the transmission of NiV were presented in a rigorous way in terms of the characterization of NiV and the level of its outbreak [20-23]. A dataset [24] for the infection of NiV in Bangladesh is presented in Table 1 which reflects the scenario of the transmission of NiV in Bangladesh. Table 1 represents that the death rate for

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NiV infected individuals is very high compare to other epidemic diseases. NiV caused an outbreak of severe febrile encephalitis in humans with a high mortality rate, whereas, in pigs, encephalitis and respiratory diseases with a relatively low mortality rate [1]. This virus also caused systemic infections in humans, pigs, and other mammals. Table 1. Outbreaks of NiV infections in Bangladesh [24, 25] Year 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 2014

No. of infected people 13 0 12 67 12 0 20 10 4 17 24 6 24 27

No. of deaths 9 0 8 50 11 0 13 9 1 15 24 6 21 14

Mortality rate 69% 0 67% 75% 92% 0 65% 90% 25% 88% 100% 100% 87.5% 52%

In the light of the foregoing, there needs an urgent and serious coordinated response from all viewpoints in order to combat the NiV infection effectively. As a part of this coordinated approach, mathematical modeling can play a pivotal role to control the spread of NiV infection because modeling approach can successfully describe the mechanism of the spreading dynamics of NiV like other contagious disease. Mathematical investigations in the NiV infections were performed in [26]. A Minimization of the total number of infectious individuals and the cost which is related to creating awareness and treatment [27]. Only the control strategies to reduce the infection of NiV was discussed considering bats and humans [28]. Biswas [29,30] first developed a mathematical model to describe and study the transmission dynamics of NiV and discuss the control strategy of the spread of infection with the help of optimal control theory. However,

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exposed population for the infection of NiV was not taken in to account in [30]. But exposed compartment play a pivotal role in an epidemic model as an SEIR model. In this respect, we propose a four compartmental model introducing exposed population that consists of a four nonlinear ordinary differential equations. We discuss the boundedness of solutions of the model to assure the feasibility of the solutions. A basic reproduction number for the model is determined so that severity of NiV can be classified [31, 32]. We analyze local stability of the equilibria of the model while the global stability of the endemic equilibria is also described that can reflect the endemic case of NiV. In addition, we impose optimal control strategy [33] in the model to minimize the number of susceptible & infected populations and maximize the recovered population under the control of massive awareness campaign against NiV through print & electronic medias and social media as well. Moreover, the strict maintenance of social distancing from the infected individuals by NiV is incorporated as another control strategy in the proposed model. Under these two control parameters, a corresponding standard optimal control problem [33] of the proposed model is described in a precise manner. A numerically simulation is performed to demonstrate the analytical findings and the influence of the control strategy on the infection of NiV and recovery from NiV as well. Numerical results reveal that the infected population can be reduced remarkably under the considered control strategies. Finally we assert some concluding remarks based on the analytical results and numerical investigations under the control strategies.

Mathematical Model Formulation In this section, a mathematical model is formulated to describe the transmission dynamics of NiV. The progression of NiV within the total population can be simplified into four different compartments.These four compartments represent four different groups of people under the following assumptions: (A1) The susceptible (S) are those people who are vulnerable to come into contact with NiV. (A2) The exposed (E) are those people who have come into contact with

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the NiV but are not yet infected. (A3) The infected (I) are those people who have become infected with NiV and are able to transmit the virus to others. (A4) The recovered (R) are those people who have recovered from NiV disease or having permanent immunity. We assume that the total number of population is constant and represented by N = S + E + I + R.

Figure 2. Schematic representation of the assumptions (A1)-(A4). The classical model for micro parasite dynamics like NiV is the flow of virus among susceptible, exposed (but not infectious), infectious, and recovered compartments. The schematic diagram of the transmission of NiV and the inter-compartmental relations is presented in Figure 2. The description and values of all parameters used in our proposed model are presented in Table 2. Under the assumptions (A1)-(A4) and the parameters presented in Table 2 taking into account, the mathematical formulation of the transmission dynamics of NiV in terms of a SEIR model that can be represented in the form of following nonlinear systems of ordinary differential equations:  dS   = Λ − βSI − µS,   dt    dE    = (1 − p)βSI − (α + µ + ε1 )E, dt (1) dI    = pβSI + αE − (δ + µ + ε )I, 2   dt      dR = ε E + ε I − µR. 1 2 dt

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Here we assume that susceptible are born into the population at rate Λ. Susceptible are infected at rate SI and move either into the exposed (or latent) class E or directly into infected class I with fast progression rate p. Exposed individuals progress to active disease when they are infectious at a constant α. In the infectious state, individuals suffer an increased death rate due to disease. Table 2. Description of parameters involved in the model (1) Parameters Λ β α δ µ ε1 ε2 p

Physical Interpretation Recruitment rate Effective contract rate Progression rate of infection Death rate due to disease Natural death rate Recovery rate of exposed individuals due to awareness Recovery rate of infectious individuals due to treatment The rate of fast progression (p < 1)

Analysis of the Model In order to analyze the model (1), we consider the following initial value problem (IVP) incorporating initial conditions  dS   = Λ − βSI − µS,   dt    dE    = (1 − p)βSI − (α + µ + ε1 )E, dt (2) dI    = pβSI + αE − (δ + µ + ε )I, 2   dt      dR = ε E + ε I − µR, 1 2 dt

with the initial conditions S(0) = S0 ≥ 0,

E(0) = E0 ≥ 0,

I(0) = I0 ≥ 0,

R(0) = R0 ≥ 0. (3)

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Boundedness Theorem 1. All the solutions of the IVP (2)-(3) in R4+ with non-negative initial conditions are always bounded, for all t ≥ 0 Proof. Let us consider total population of the individuals is given by N = S + E + I + R so that we find dN ds dE dI dR = + + + = Λ−µ(S +E +I +R)−δI = Λ−µN −δI, dt dt dt dt dt that can be rewritten as dN dN + µN = Λ − δI, ⇒ + µN ≤ Λ. dt dt

(4)

Therefore, it yields N (t) ≤

Λ + ce−µt , µ

(5)

where c is an arbitrary constant.   Λ It follows that N (t) ≤ max N (0), µ . Since S(0) ≥ 0, E(0) ≥ 0, I(0) ≥ 0, R(0) ≥ 0, such that N (0) = S(0) + E(0) + I(0)+ R(0) ≥ 0 and letting t → ∞, we have limt→∞ supN (t) ≤ Λµ , which is independent of the initial conditions. We observe that all the solutions of the model initiating in R4+ eventually lie in the region defined by X = {(S, E, I, R) ∈ R4+ : N = Λ µ + κ} for some κ > 0. This complete the proof of the boundedness of the solutions and hence the system under consideration is dissipative.

Basic Reproduction Number The basic reproduction number R0 is the potential measurement for spreading disease in a population. Mathematically, R0 is a threshold parameter for the stability of a disease-free equilibrium and is related to the peak and final density of an epidemic. It is defined as the expected number of secondary cases of infection which would occur due to a primary case in a completely susceptible population [26]. If R0 < 1, then a few infected individuals introduced into a completely susceptible population will, on average, fail to replace themselves, and the disease will not spread. On the other hand, when R0 > 1, then the number of infected individuals will

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increase with each generation and the disease will spread. Moreover, it is noted that the basic reproduction number is a threshold parameter for invasion of a disease organism into a completely susceptible population; if the disease begins to spread, conditions favoring spread will change and R0 may no longer be a good measure of disease transmission. However, in many disease transmission models, the peak prevalence of infected hosts and the final size of the epidemic are increasing functions of R0 , making it a useful measure of spread. Now, according to the biological interpretation, we can define the basic reproduction number We can define the basic reproduction number R0 by the method of next generation matrix method [27] in the following form R0 = β ×

1 α × , α + ε1 + µ δ + ε2 + µ

(6)

where α+εα1 +µ is the probability that an individual has become infectious (moved from E to I), and δ+ε12 +µ is the average time that a person is infectious (spends in I class). The equation (6) can be rewritten as the following form αβ R0 = , (7) φ1 φ2 where φ1 = α + ε1 + µ and φ2 = δ + ε2 + µ.

Equilibrium Analysis In this section, we investigate the existence of non-negative equilibria of the system (1) by solving the following nonlinear algebraic system  Λ − βIS − µS = 0,    (1 − p)βIS − φ E = 0, 1 (8)  pβIS + αE − φ I = 0, 2    ε1 E + ε2 I − µR = 0.

By solving the system (8), we find the following two non-negative equilibria of the model (1) only: one is disease-free equilibrium E1 and other one is endemic equilibrium E∗ , where φ1 = α+µ+ε1 and φ2 = δ+µ+ε2 ,   Λ E1 , 0, 0, 0 , and E∗ (S ∗ , E ∗, I ∗, R∗ ) , µ

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where φ1 φ2 (1 − p) (Λαβ(1 − p) + pβΛφ1 − µφ1 φ2 ) , E∗ = , αβ(1 − p) + pβφ1 αβφ1 (1 − p) + pβφ21 (Λαβ(1 − p) + pβΛφ1 − µφ1 φ2 ) I∗ = , βφ1 φ2 (αε2 (1 − p) + ε1 φ2 (1 − p) + pε2 φ1 ) (Λαβ(1 − p) + pβΛφ1 − µφ1 φ2 ) R∗ = . αβµφ1 (1 − p) + pµβφ21 φ2 S∗ =

The semitrivial equilibria E1 always exist while the nontrivial equilibrium point E∗ exists underlying the condition Λαβ(1 − p) + pβΛφ1 > µφ1 φ2 . A local stability analysis of the system (2) is performed around the equilibria presented in the previous section. To do so, we linearize the ¯ E, ¯ I, ¯R ¯ so that the following system (1) around the equilibrium point S, Jacobian matrix can be obtained   −(β I¯ + µ) 0 −β S¯ 0 ¯ ¯    ¯ E, ¯ I, ¯R ¯ =  (1 − p)β I −φ1 (1 − p)β S 0  . J S, (9) ¯ ¯  pβ I α pβ S − φ2 0  0 ε1 ε2 −µ   Theorem 1. Around the equilibrium point E1 Λµ , 0, 0, 0 the system (1) is locally asymptotically stable if the inequalities φ1 φ2 >

pβΛφ1 µ

+ (1 −

p) αβΛ µ

pβΛ µ

< φ1 + φ2 and

hold simultaneously.

  Proof. Around the disease-free equilibrium point E1 Λµ , 0, 0, 0 , we investigate the eigenvalues of the following Jacobian matrix   ¯ −µ 0 −β S 0    0 −φ1 (1 − p)β S¯ 0  Λ . , 0, 0, 0 =  (10) J  0 α pβ S¯ − φ2 0  µ 0 ε1 ε2 −µ Around the disease-free equilibrium point E1 , the Jacobian matrix (10) takes the following form,   −µ 0 −β Λµ 0  0 −φ (1 − p)β Λ 0    1 µ J(E1 ) =  (11) . Λ  0 α pβ µ − φ2 0  0 ε1 ε2 −µ

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The characteristic equation for (11) gives the eigenvalues λ1 = −µ, λ2 = −µ while λ3 and λ4 are the roots of the following quadratic equation   pβΛ pβΛφ1 αβΛ 2 − φ1 − φ2 λ + φ1 φ2 − − (1 − p) = 0. (12) λ − µ µ µ

and φ1 φ2 >

pβΛφ1 µ

pβΛ µ

< φ1 + φ2 αβΛ + (1 − p) µ . It follows that the equilibrium point E1 pβΛ pβΛφ1 + (1 − p) αβΛ µ < φ1 + φ2 and φ1 φ2 > µ µ and is

The roots of the equation (12) have negative real part if is locally stable if locally unstable otherwise.

Theorem 2. The endemic equilibrium point E∗ (S ∗ , E ∗, I ∗ , R∗ ) of the system (1) is locally asymptotically stable if the following inequalities hold (i) I ∗ > pS ∗ , S ∗ < (ii) S ∗ >

1 2pβ ,

2pβ 3+β 2 I ∗ +p . p2 β(1+2β 2)

Proof. At the interior equilibrium point E∗ , the Jacobian matrix is given by   −(βI ∗ + µ) 0 −βS ∗ 0  (1 − p)βI ∗ −φ1 (1 − p)β 0  . J(E∗ ) =  (13) ∗  pβI ∗ α pβS − φ2 0  0 ε1 ε2 −µ One of the eigenvalues of J(E∗ ) is λ1 = −µ and other three eigenvalues are given by the roots of the equation λ3 + σ1 λ2 + σ2 λ + σ3 = 0,

(14)

where σ1 = −(c11 + c22 + c33 ), σ2 = c11 c22 + c22 c33 + c11 c33 + c12 c21 + c13 c31 , σ3 = c13 c21 c32 − c11 c22 c33 − c12 c21 c33 − c13 c31 c22 , and   c11 c12 c13 J(E∗ ) = c21 c22 c23  . c31 c32 c33 According to the Routh-Hurwitch criteria [34,35], all three roots of the characteristic equation (14) have negative real negative real part under the

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inequalities I ∗ > pS ∗ , S ∗
, 2pβ p2 β(1 + 2β 2 )

which satisfy the following conditions σ1 > 0, σ3 > 0 and σ1 σ2 − σ3 > 0.

Global Stability of the Endemic Equilibrium Point E∗ Theorem 3. The interior equilibrium point E∗ is globally asymptotically stable if α2 < 4φ1 φ2 . Proof. To investigate the global stability of the endemic equilibrium point E∗ , we consider the following Lyapunov function: L(S, E, I, R) = L1 (S) + L2 (E) + L3 (I) + L4 (R),

(15)

where, 

 S L1 (S) = S − S − S ln , S∗   E L2 (E) = E − E ∗ − E ∗ ln , E∗   I ∗ ∗ L3 (I) = I − I − I ln , I∗   R L4 (R) = R − R∗ − R∗ ln . R∗ ∗



(16)

We now compute the derivatives of L1 (S), L2(E), L3(I), and L4 (R) along the solutions E∗   S ∗ ˙∗ L˙ 1 (S) = 1 − S = (S − S ∗ )[Λ − βSI − µS] S   ∗ ˙L2 (E) = 1 − E E˙ ∗ = (E − E ∗ )[(1 − p)βSI − φ1 E] E   (17) I ∗ ˙∗ L˙ 3 (I) = 1 − I = (I − I ∗ )[pβSI + αE − φ2 I] I   R∗ ˙ .L4 (R) = 1 − R˙∗ = (R − R∗ )[ε1 E + ε2 I − µR]. R

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So, the equation (15) can be rewritten in the following form using (17) ˙ L(S, E, I, R) = (S − S ∗ )[−β(SI − S ∗ I ∗ ) − µ(S − S ∗ )] + (E − E ∗ )[(1 − p)β(SI − S ∗ I ∗ ) + φ1 (E − E ∗ )] + (I − I ∗ )[pβ(SI − S ∗ I ∗ ) + α(E − E ∗ ) − φ2 (I − I ∗ )] + (R − R∗ )[ε1 (E − E ∗ ) + ε2 (I − I ∗ ) − µ(R − R∗ )]. This gives, ˙ L(S, E, I, R) = −β(SI − S ∗ I ∗ )[(S − S ∗ ) − (1 − p)(E − E ∗ ) − p(I − I ∗ )] − µ(S − S ∗ )2 − φ1 (E − E ∗ )2 − φ2 (I − I ∗ )2 − µ(R − R∗ )2 + α(E − E ∗ )(I − I ∗) + (R − R∗ )[ε1(E − E ∗ ) + ε2 (I − I ∗)].

Since

S S∗

>

E E∗

>

I I∗

and R0 < 1 then we have

˙ L(S, E, I, R) ≤ −[µ(S−S ∗ )2 +φ1 (E−E ∗ )2 +φ2 (I −I ∗ )2 +µ(R−R∗ )2 −α(E−E ∗ )(I −I ∗ ).

˙ This implies that L(S, E, I, R) ≤ 0 if α2 < 4φ1 φ2 . Hence the endemic equilibrium point E∗ is globally asymptotically stable.

Incorporating Optimal Control to the Model We now turn to our model of the system (1) when the control measures are taken into account. We recall here that in Bangladesh the NiV infections are transmitted by drinking of raw date palm sap contaminated with NiV and by the close physical contact with Nipah infected patients. In this regards, we introduce two additional variables representing control functions denoted by u = (u1 , u2 ) in the dynamics (1). As we discussed before, no proper treatment (neither by vaccination nor appropriate drugs) is available for NiV infections till now. The only ways to control the disease and/or prevent people from being infected by nipah virus are (i) before outbreak: huge mass awareness and educational campaigns among the people of the risky areas so that they can be motivated from not drinking raw date sap and (ii) after outbreak: the family members and relatives of the infected patients as well as the health-care givers (e.g., doctors and nurses) should follow ’social distances’ [29] so that no more human to human (H2H) infections occur. So our controls u1 (t) measures the effort needed to increase

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mass and educational campaigns, reducing the effective transmission rate β and u2 (t) measures the effort required social distancing while administering antiviral drug treatment and/or giving health cares to novel the infected individuals. We assume that our control functions are bounded and Lebesgue measurable on the interval [0, T ], where T denotes a pre-selected length of time during which these controls are applied. Furthermore, wherever a full effort is being placed on mass campaign or social distancing at time t, we would have that u1 (t) and u2 (t) must be equal to one. Moreover, when no effort is being placed in these controls at time t, then u1 (t) and u2 (t) are equal to zero. Under the above settings, We define the controls taking values in measurable control set U = {(u1 (t), u2(t)) : 0 ≤ ui (t) ≤ 1, i = 1, 2, a.e. t ∈ [0, T ]}. Our objective is to chosen as the cost functional Z T 1 (A1 S(t) + A2 I(t) + (B1 u21 (t) + B2 u22 (t)))dt (18) Minimize 2 (u1 (t),u2(t))∈U 0 where A1 , A2 , B1 and B2 are weight parameters balancing the costs. The aim is to minimize the total number of infected people over a certain fixed time and at the same time to minimize the systematic costs for the mass campaigns and/or educational programs as well as the social distances. These controls that are introduced in the model system (1) are in the form of reducing susceptible and infected population [11]. So, based on that we can formulate the model with the control variables as follows:  dS   = Λ − βSI − µS − u1 S,   dt    dE    = (1 − p)βSI − (α + µ + ε1 )E, dt (19) dI    = pβSI + αE − (δ + µ + ε )I − u I, 2 2   dt      dR = ε E + ε I − µR + u S + u I, 1 2 1 2 dt where the control parameters, u1 represents the awareness campaign for NiV and u2 represents the social distancing from NiV infected individual. The objective of the optimal control is to maximize the number of recovered population and also minimize the infected individuals along with the

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cost of the controls. Under the Pontryagin’s maximum principle [33], the objective function is given by

Minimize (u1,u2 )∈U

J=

Z

T 0



A1 S(t) + A2 I(t) +

subject to N˙ (t) = f(t, N, U ) ∀ t ∈ [0, T ],

  1 B1 u21 (t) + B2 u2 (t)2 ) dt 2

u(t) ∈ U a.e., N (0) = N0 , (20)

where, ` ´ ` N (t) = S(t), E(t), I(t), R(t) , N (0) = S0 , E0 , I0 , R0 ), and U = {u(t) : 0 ≤ u ≤ 1}, u(t) = (u1 (t), u2 (t)),

and A1 , A2 , B1 , B2 are positive weight. The term Bu2 is the cost of control efforts on reducing or minimizing the infections.

Characterization of Optimal Controls According to the Pontryagin’s maximum principle, let us convert the system into a maximizing pointwise Hamiltonian H with respect to (u1 , u2 ) ∈ U,  1 H(t, N, u, λ) = A1 S(t) + A2 I(t) + B1 u21 + B2 u22 2 + λ1 {Λ − βSI − µS − u1 S} (21) + λ {(1 − p)βSI − (α + µ + ε )E} 2

1

+ λ3 {pβSI + αE − (δ + µ + ε2 )I − u2 I} + λ4 {ε1 E + ε2 I + u1 S + u2 I − µR}, where λ1 , λ2 , λ3, and λ4 be the adjoint variables or costate variables. Now applying Pontryagin’s maximum principle [36] and the existence results for the optimal control [33], we obtain the following proposition. Proposition 1. Consider (u∗1 , u∗2 ) be the optimal control pair of (20) which minimizes J(u1 , u2 ) over U , there exists adjoint variables λ1 , λ2, λ3 and

Mathematical Modeling Applied to Control ... λ4 satisfying, where ∂H ∂S ∂H ∂E ∂H ∂I ∂H ∂R

dλ1 dt

∂H dλ2 ∂H dλ3 ∂H = − ∂N , dt = − ∂N , dt = − ∂N , 1 2 3

191 dλ4 dt

∂H = − ∂N 4

= A1 + λ1 (−βI − µ − u1 ) + λ2 (1 − p)βI + λ3 pβI + λ4 u1 , = −λ2 (α + µ + ε1 ) + λ3 α + λ4 ε1 , = A2 − λ1 βS + λ2 (1 − p)βS + λ3 (pβS − (δ + µ + ε2 )) + λ4 (ε2 + u2 ), = −λ4 µ,

and with the transversality condition, λ1 (T ) = 0, λ2(T ) = 0, λ3(T ) = 0 and λ4 (T ) = 0. ∗ Using optimality condition we have, ∂H ∂u = 0 at u and subsequently we find  ∂H   = B1 u1 − λ1 S + λ4 S = 0 at u∗1 , and  ∂u1 ∂H    = B2 u2 − λ3 I + λ4 I = 0 at u∗2 . ∂u2 Hence it follows that

u∗1 =

(λ3 − λ4 )I (λ1 − λ4 )S and u∗2 = . B1 B2

(22)

On the interior of the control set U , the following characterization of the optimal control holds,   n (λ1 − λ2 )S o ∗ , u1 = min 1, max 0, B1   n (λ3 − λ4 )I o u∗2 = min 1, max 0, . B2 We note that the initial time condition and final condition have in the state system and costate system respectively. This can be rewritten in the following form (u∗1 , u∗2 ) =

„ „ « „ « « n n (λ1 − λ2 )S o (λ3 − λ4 )I o , min 1, max 0, . min 1, max 0, B1 B2

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Numerical Simulations In this section, we perform numerical simulations of the system (1) to understand the transmission dynamics of NiV. For this purpose, we use Runge-Kutta fourth order method to obtain numerical solutions of the IVP (2)-(3). Furthermore, we aim to investigate and interpret the effect of control parameters incorporated to the system (1), namely, the number of quarantined individuals, η and the enhanced personal hygiene due to public enlightenment, γ. Figure 3 illustrates time series of solutions of the IVP (2)-(3) without any control and that demonstrates the solution of the  IVP, (S,E, I, R)(t) converges to the disease-free equilibrium point E1 Λµ , 0, 0, 0 after large time when Λαβ(1 − p) + pβΛφ1 < µφ1 φ2 , that is, β < βc =

µφ1 φ2 αΛ(1−p)+pΛφ1 .

This confirms the local stability   of the disease-free equilibrium point E1 Λµ , 0, 0, 0 . In order to reveal the dynamics  of (S, E, I, R)(t) around the disease-free equilibrium point Λ E1 µ , 0, 0, 0 of the IVP (2)-(3), phase portraits of the IVP in different cases are shown in Figure 4. Figure 5 represents time series of so250

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Figure 3. Time Series of solution (S, E, I, R)(t) of the IVP (2)-(3) where the parameters, Λ = 2.0, α = 0.012, β = 0.003 (< βc ), ε1 = 0.01, ε2 = 0.41, p = 0.98, µ = 0.01, δ = 0.83.

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lutions of the IVP (2)-(3) without any control and that demonstrates the solution of the IVP, (S, E, I, R)(t) converges to the endemic equilibrium point E ∗ (S ∗ , E ∗, I ∗ , R∗ ) after large time underlying the inequality Λαβ(1 − p) + pβΛφ1 > µφ1 φ2 , that is, the parameter β satisfies β > µφ1 φ2 βc = αΛ(1−p)+pΛφ , where rest of the parameters are suitably fixed. This 1 confirms the global stability of the equilibrium point E ∗ (S ∗ , E ∗, I ∗, R∗ ). In order to reveal the dynamics of (S, E, I, R)(t) of the IVP (2)-(3) around the endemic equilibrium point E ∗ (S ∗ , E ∗, I ∗ , R∗), phase portraits of the IVP in different situations are depicted in Figure 6.

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Figure 4. Phase portrait of the IVP (2)-(3). We now concentrate to inspect the influence of the control parameters on the solution of the system with control (19). For this purpose, we solve the corresponding IVP (23)-(24) using Runge-Kutta fourth order method to understand the dynamical behavior of its solutions with respect to the control parameters u1 and u2 .

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Figure 5. Time Series of solution (S, E, I, R)(t) of the IVP (2)-(3) where the parameters, Λ = 2.0, α = 0.012, β = 0.0199 (> βc ), ε1 = 0.01, ε2 = 0.41, p = 0.98, µ = 0.01, δ = 0.83.

 dS    dt = Λ − βSI − µS − u1 S,     dE    = (1 − p)βSI − (α + µ + ε1 )E, dt dI    = pβSI + αE − (δ + µ + ε2 )I − u2 I,   dt      dR = ε E + ε I + u S + u I − µR, 1 2 1 2 dt with the initial conditions S(0) = S0 ≥ 0,

E(0) = E0 ≥ 0,

I(0) = I0 ≥ 0,

(23)

R(0) = R0 ≥ 0. (24)

Figure 3 and 5 represent the dynamical behavior of the solutions of susceptible, exposed, infected, and recovered without any control. However, we

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Figure 6. Phase portrait of the IVP (2)-(3). aim to reduce the number of infection of NiV so that the number of susceptible individuals an also be decreased. Moreover, the number of recovered individuals is also expected to increase because permanent immunity is considered once they are infected by NiV as we assumed in the model formulation. In order to understand the influence of the control parameters u1 and u2 on the solution of the IVP (23)-(24), we first consider the case: u1 6= 0 and u2 = 0 and subsequently solve the IVP using RK4 method. We find that the susceptible individuals are reduced than that for without control due to the control parameter u1 , as presented in Figure 7. Beside this, infected people are also reduced while the recovered people are increased once the control parameter u1 is imposed. In other words, we can claim that the susceptible individuals and infected individuals by NiV can be reduced significantly if an effective awareness campaign against NiV

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infection is performed throughout the country. In that case, recovered population can be increased under a reasonable awareness campaign against NiV. 9 1.25

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Figure 7. Dynamical behaviour of solution (S, E, I, R)(t) of the IVP (23)(24) under the control parameters u1 6= 0 and u2 = 0. On the other hand, we focus on the case: u1 = 0 and u2 6= 0 and subsequently solve the IVP (23)-(24). We inspect that the infected population is significantly reduced than that for without control under the control parameter u2 , as presented in Figure 8. In addition, recovered population is increased while the susceptible population population is increased once the control parameter u2 is imposed. To be specific, it is confirmed that the infected population by NiV can be reduced notably when social distancing is properly maintained from NiV infected individuals throughout the country. In that case, recovered people

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Figure 8. Dynamical behaviour of solution (S, E, I, R)(t) of the IVP (23)(24) under the control parameters u1 = 0 and u2 6= 0. can be increased underlying a proper social distance from NiV infected patient. In addition, susceptible people increases under that situation because there is no influence of awareness campaign against NiV, social distancing can only be maintained once an individual is infected by NiV. We now emphasize on the collective influence of the control parameters u1 and u2 on the dynamics of susceptible, exposed, infected, and recovered population. For this purpose, we concentrate the case: u1 6= 0 and u2 6= 0 and investigate the qualitative behavior of solutions of the IVP (23)-(24) by solving it numerically. It is perceived that both the susceptible individuals and infected individuals decrease significantly under the control parameters u1 and u2 , as depicted Figure 9. In a similar manner, exposed peopole is also reduced

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Figure 9. Dynamical behaviour of solution (S, E, I, R)(t) of the IVP (23)(24) under the control parameters u1 6= 0 and u2 6= 0. remarkably for this case. Furthermore, Figure 9 demonstrates that the recovered individuals increase when the control parameters u1 and u2 are taken in to account. More specifically, it is affirmed that the infection of NiV can be minimized remarkably when a suitable awareness campaign against NiV infection is well-performed and social distancing is strictly maintained once individuals are infected by NiV. Furthermore, susceptible individuals can also be reduced under the imposed controls. Hence, it follows that the awareness campaign against NiV and social distancing from NiV infected individuals play a pivotal role to control the severity of Nipah Virus (NiV). Finally, our modeling approach under optimal control strategy has a reasonable agreement with the epidemiological observations as a non-pharmaceutical interventions and that will have an effective role to

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prevent and control the NiV infection in Bangladesh.

Conclusion Nipah virus (NiV) is one of the most public health concerns across the globe as there are no available preventive vaccine and treatments to prevent and protect it. However, non-pharmaceutical interferences, e.g., awareness campaigns, social distancing, etc. have an effective role to control the transmission of NiV. In order to address this issue, we have proposed a four-compartmental SEIR epidemic model based on ordinary differential equations. We have established boundedness of solutions of the model to conceive the behavior of its solutions. For understanding the dynamics among susceptible, exposed, infected, and recovered compartments, it has been investigated the occurrence of non-negative equilibria E1 (disease fee equilibrium) and E∗ (endemic equilibrium). To conceive the qualitative behavior of solutions of the model, a local stability analysis of the equilibria has also been performed. In addition, we have discussed the global stability of the endemic equilibrium E∗ which results a specific condition for endemic situation in terms of the parameters. To establish the effective control measures for reducing the transmission of NiV, we have introduced optimal control strategy in the proposed model. The two control strategies: the social awareness campaign against NiV infection (u1 ) and the social distancing from NiV infected individuals (u2 ) have been incorporated. The existence of optimal solutions has also been verified under Pontryagin’s maximum principle. Furthermore, a numerical simulation has been performed to demonstrate the analytical results. It has been shown that the solution of the corresponding IVP of the model (1) converges to E1 and E∗ for β < βc and β > βc , respectively. Our modeling approach reflects that the transmission of NiV can be remarkably reduced when a social awareness campaign against NiV infection is performed effectively, that is, infection of NiV significantly decreases. Once individuals are infected by Nipah virus, its spreading can be controlled if the social distancing from infected individuals is strictly maintained, that is, the infection of NiV is remarkably minimized while the recovered individuals are increased. It has also been claimed that the transmission of NiV can be minimized significantly when both the non-pharmaceutical controls are preformed, that is, due to the collective influence of both the control strategies imposed to

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the model. Eventually, there is a quite reasonable agreement between the results obtained by the proposed model and the epidemiological insights of NiV.

Acknowledgments The authors greatly acknowledge the partial financial support provided by TWAS, Italy with the grant ref.no.: 17-392RG/MATHS/AS-IFR3240297753.

References [1] Chua K. B., Nipah virus outbreak in Malaysia. J Clin Virol. 26: 265– 275 (2003). [2] Chua K. B. et al. Nipah virus: a recently emergent deadly Paramyxovirus. Science. 288: 1432–1435 (2000). [3] Paton N. I., Leo Y. S., Zaki S. R., Outbreak of Nipah-virus infection among abattoir workers in Singapore. The Lancet. 354: 1253–1256 (1999). [4] Parashar U. D., Sunn L. M., Ong F., Case-control study of risk factors for human infection with a new zoonotic paramyxovirus, Nipah virus, during a 1998-1999 outbreak of severe encephalitis in Malaysia. J Infect Dis. 181: 1755–1759 (2000). [5] Goh K. J., et al, Clinical features of Nipah virus encephalitis among pig farmers in Malaysia. N. Engl J. Med. 342: 1229-1235 (2000). [6] Uppal P. K., Emergence of Nipah virus in Malaysia. Ann N Y Acad Sci. 916: 354-357 (2000). [7] Daszak P., Plowright R., Epstein J. H., The emergence of Nipah and Hendra virus: pathogen dynamics across a wildlife-livestock human continuum. In: Collinge S., Ray C., editors. Disease ecology. Oxford: Oxford University Press; p. 186-201 (2006).

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[20] Lo M. K., Lowe L., Hummel K. B., et al. Characterization of Nipah virus from outbreaks in Bangladesh, 2008–2010. Emerg Infect Dis. 18: 248-255 (2012). [21] Neilan R. M., Lenhart S., An introduction to optimal control with an application in disease modeling. DIMACS Ser Discrete Math. 75: 67-81 (2010). [22] Nipah, outbreak in Lalmonirhat district. ICDDR, B. Health Sci Bull. 9: 13-18 (2011). [23] Stula M., Doko A., Maras J., Country’s Internet Spreading Rate Modelling with Fuzzy Cognitive Map. Int J. Model Simul. 32: 287-295 (2012). [24] National guideline for management, prevention and control of Nipah virus infection including encephalitis. Directorate General of Health Services, Ministry of Health and Family Welfare, Government of the People’s Republic of Bangladesh; 2015. [25] Biswas M. H. A., Haque M. M. and Duvvuru G., A mathematical model for understanding the spread of Nipah fever epidemic in Bangladesh. Proceedings of the International Conference on Industrial Engineering and Operations Management (IEOM), p. 1-8. DOI: 10.1109/IEOM.2015.7093861 (2015). [26] Sultana J., Chandra N. Podder, Mathematical analysis of Nipah virus infections using optimal control theory, Journal of Applied Mathematics and Physics, 4: 1099-1111 (2016). [27] Mondal M. K., Hanif M., Biswas M. H.A., A mathematical analysis for controlling the spread of Nipah virus infection , IJMS, 37(3): 185197 (2017). [28] Mondal M. K., Hanif M., Biswas M. H. A., A Mathematical Analysis of the Transmission Dynamics of Ebola Virus Diseases, Journal of Nonlinear Analysis and Optimization: Theory & Applications, 7(2): 57-66 (2016).

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[29] Biswas M. H. A.m Model and control strategy of the deadly Nipah virus (NiV) infections in Bangladesh. Res. Rev. Biosci. 6:370–377 (2012). [30] Biswas M. H. A.m Optimal control of Nipah virus (NiV) infections: a Bangladesh scenario. J. Pure. Appl. Math. 12:77-104 (2014). [31] Diekman O., Heesterbeek J. A. P, Metz V., On the definition and computation of the basic reproduction ratio R0 in the model of infectious disease in heterogonous populations. J Math Biol. 28: 365-382 (1990). [32] Diekmann, O., Heesterback, J. A. P., Roberts, M.G., The construction of next generation matrices for compartmental epidemic models, J. Royal Soc. Interface, 7(47): 873-885 (2010). [33] Biswas M. H. A., Necessary Conditions for Optimal Control Problems with State Constraints: Theory and Applications, PhD Thesis, Faculty of Engineering, University of Porto, Portugal, (2013). [34] Routh E. J., A treatise on the stability of a given state of motion: particularly steady motion. Macmillan and Company, London (1877). [35] Hurwitz A., et al. On the conditions under which an equation has only roots with negative real parts. Selected papers on Mathematical trends in Control Theory 65:273 - 284 (1964). [36] Pontryagin L. S., Boltyanskii V. G., Gamkrelidze R. V., Mischenko E. F., The Mathematical Theory of Optimal Processes, John Wiley, New York, (1962).

In: Frontiers in Mathematical Modelling … ISBN: 978-1-68507-430-2 Editors: M. Biswas and M. Kabir © 2022 Nova Science Publishers, Inc.

Chapter 7

Mathematical Modelling of the Closed-Loop Performance of a Continuous Bioreactor under a Feedback Polynomial-Type Controller Ricardo Aguilar-López*, Edgar N. Tec-Caamal1, Juan C. Figueroa-Estrada1, Alma R. Domínguez-Bocanegra1 and María Isabel Neria-González2 1

Biotechnology and Bioengineering Department, CINVESTAV-IPN, Mexico City, Mexico 2 Chemical and Biochemical Engineering Division, Tecnológico de Estudios Superiores de Ecatepec, Ecatepec de Morelos, México

*

Corresponding Author’s Email: [email protected] (Corresponding author).

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Abstract This study proposes a feedback control law with a third order polynomial term in its structure, which reduces the control error in a small neighborhood around the desired reference path. The performance of the controller was evaluated in a continuous acetone, butanol and ethanol (ABE) fermentation process with an experimentally corroborated kinetic model for tracking purposes. The proposed controller was able to increase the butanol concentration in the reactor using the dilution rate (D) as control input. A step trajectory for butanol production at 8 g L-1 and 9.4 g L-1 was followed with success under the control law action, showing a realizable control effort. A stability analysis employing the zero dynamics methodology under the Lyapunov framework, allowed to determine the stability of the noncontrolled variables under closed loop operation. A comparison with a standard Proportional-Integral (PI) controller and numerical experiments allowed to assess both satisfactory performance and the possible real time implementation of the proposed controller.

Keywords: butanol, ABE fermentation, cubic controller, continuous bioreactor, product inhibition

Introduction Biorefinery studies are generally focused on biofuels production [1, 2]. Among these, butanol is a biofuel that is gaining more attention due to its potential to replace ethanol as gasoline additive and its significance as industrial solvent. Butanol (C4H9OH) is a four-carbon primary alcohol, which can be obtained from lignocellulosic raw materials by means of microbial fermentation [3-5], where Clostridium strains are widely employed for the production of butanol on an industrial scale. In a typical ABE (acetone-butanol-ethanol) fermentation using glucose as carbon source, it is commonly observed that pH drops due to the production of organic acids including butyric, acetic and lactic ones (acidogenic stage). This undergoes a metabolic shift, which after a threshold of pH of the medium induces the production of butanol, acetone and ethanol (solventogenic stage) [6-8]. It is well known that the ABE fermentation is severely limited by product inhibition at butanol concentrations over one percent v/v that can significantly impact the cell growth, and consequently

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the process productivity [6, 9]. Therefore, optimizing the ABE fermentation process remains a challenge to overcome in the industry. For instance, in pilot scale processes, cell recycling, cell immobilization or in situ extractive fermentations to minimize product inhibition are commonly used strategies for increasing cell density and productivity. Notwithstanding, to date, with the aforementioned process schemes the best productivities obtained for the ABE fermentations are less than two percent (4.46 g L-1 h-1) and the solventglucose yields are less than 25%. Thus, showing the lack of efficiency in the implementation of these strategies [5]. Generally, most of the regulation tasks in the industry is realized via classical proportional-integral-differential (PID) feedback strategies [1012]. Therefore, the standard approach for designing a controller based on the state space multivariable control theory is to construct an empirical linear model by employing input/output response of the considered reactor, or alternatively, linearize the reactor model with a first order Taylor series approximation, so that linear design strategies can be applied [13]. Under the aforementioned approach, standard PID controllers should perform well when the non-linear process operates at a point with a neighborhood of linearity large enough to encompass excursions due to disturbances. Despite this, these control methodologies give little consideration to the characteristics of the non-linear behavior of the process, and therefore are highly dependent on adjustment procedures and extensive testing. As a consequence, in several cases the designer is forced to save efficiency to assure adequate levels of safety and operation, or equivalently, to provide a large enough neighborhood where the linear control strategies perform appropriately. The industrial reactor design is commonly conditioned by economic objectives, which usually take the process towards the operation at a highly sensitive state. As a result, the non-linear nature of the reactor becomes more relevant and poses a more difficult control problem. In some cases, the reactor must operate at an unstable stationary point or in an induced periodic transient [14-19]. The above observations suggest that more powerful non-linear control techniques, based on more systematic procedures, could find application in the design and operation of CSTRs. The stabilization of fermentation processes, specifically under continuous

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regime, has been analyzed for several years, and have been tackled by process engineers via the application of linear PID, adaptive, predictive, I/O linearizing, fuzzy, neural and optimal controllers with success [20-22]. The application of nonlinear controllers to bioprocesses can be explained by the highly nonlinear dynamic behavior, which are characterized to be a high order systems, also with high parametric sensitibity, these lead to steady state multiplicity, unstabilities, input multiplicity, between others nonlinear effects, from the above, the linear control approaches, generally have poor performances, because they are designed in a narrow operation conditions, where the bioprocess can show linear dynamic behavior [23-25]. The objective of this work is to develop a class of non-linear feedback controller with a third order polynomial (cubic) structure of the control error, in order to provide an asymptotic stabilization of the considered process to improve the butanol concentration in a continuous ABE fermentation. A numerical framework is presented to demonstrate the satisfactory performance of the proposed controller.

Methods ABE Fermentation Model Biological systems can be described by experimentally validated mathematical models and analyzed by simulations techniques through the evaluation of its dynamic behavior under different operating conditions. In modelling tasks of biosystems, unstructured kinetic models are commonly employed due to its easy construction and implementation. In our work, a mathematical model reported by Kalafatakis et al. [9] that describes the conversion of glycerol into butanol using Clostridium pasteurianum was used. The model considers the type of glycerol and the inhibition effect of butanol on cell growth, which was represented by an inhibition term (μ2) in the Eq. (5) [24]. Figure 1 shows a schematic representation of the system set-up and the implementation of a control law. The kinetic model was used as a benchmark production plant, which was considered to simulate and analyze continuous operation. For this, an isothermal, homogeneous

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and perfectly mixed reactor was assumed. The reactor model (Eqs. 1-3) is based on classical mass balance approach to describe the time evolution of the main state variables, i.e., biomass (x1), glycerol (x2) and butanol (x3), concentrations as follows: 𝑥̇ 1 =-D[x1 ]+μ1 (x2 )μ2 (x3 )2 x1 -kd (x1 ) ẋ 2 =D[x2_in -x2 ]-

ẋ 3 =-D[x3 ]-

μ1 (x2 )μ2 (x3 )2 x

1

Y1

μ1 (x2 )μ2 (x3 )2x Y2 1

Y1

(1)

(2)

(3)

with μ1 (x2 )= μmax

x2 ks +x2

k

μ2 (x3 )= k +xi i

3

(4)

(5)

where, μmax = 0.6 h-1; ks = 1.13 g L-1; ki = 7.65 g L-1; kd = 0.007 h-1; Y1 = 0.57; Y2 = 3.56 are the kinetic parameters. This model has been experimentally validated in batch and continuous operation [8].

Figure 1. Schematic representation of the control action under continuous operation of the ABE fermentation.

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Bifurcation Analysis Bifurcation analysis provides an overview of the steady-state behavior of a dynamical process by varying input parameters [27,28]. In particular, to characterize the butanol production in a continuous process at steady state, this analysis was performed. In general, a bifurcation of a dynamical system of the form x˙ = f(x, ω), x ϵ Rn , ω ∈ Rk takes place when one or more of the parameters (ω) change. In this work, the dilution rate (D = F/V, h-1) and the glycerol inlet concentration (x2in, g L-1) were used as bifurcation parameters to construct the bifurcation diagrams. As can be noticed, both bifurcation parameters are usually operational variables, which can be manipulated and selected in order to lead the fermentation process for a specific steady-state operation target, as usual for this kind of processes [29, 30]. This analysis was carried out in the MATCONT 5.3 software package that works within the framework of the MATLAB® (R2016a) platform.

Stability of the Process One of the most important features that have to be analyzed of a reacting system subject to a control law is the stability of the inverse of the process model, also known as zero dynamics. For biochemical process, control affine schemes are commonly used, which can be expressed in vector form as follows: ẋ =f(x)+G(x)u where, x∈Rn is the vector of states, u∈Rq , 𝑓(𝑥): Rn → Rn is a non-linear and smooth vector field, and G(x): Rn×q is a matrix that relates the control and manageable variables. Being the control input vector, the vector field and the trajectories bounded. To analyze the zero dynamics of the system,

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it is necessary to determine its inverse, however, for non-linear systems this step is often impossible. Furthermore, for partially controlled affine systems, one can determine stability features of the zero dynamics when the system is controlled by a subset of q states Xc, following the behavior of the (n-q) uncontrolled states XU, according to Eq. (6). x ̇ =f (x)+GC (x)u ẋ =f(x)+G(x)u→ { C C xU =fD (x)+GD (x)u

(6)

Once manageable inputs are selected, it is assumed that the regulated variables will remain steady at the required set point. From this, if the evolution of the dynamic behavior of the uncontrolled variables is not stable, it can be inferred that the zero dynamics are not stable as well. All uncontrolled variables should trend to an attractor to ensure complete stability [31]. Thus, the time derivative of the uncontrolled states must present a negative sign or zero value (decreasing trend) at the operational adjustment point to ensure the stability of the system.

Cubic Control Design In this section is developed a class of non-linear feedback controller with a third-order polynomial structure to control an ABE fermentation process at desired reference points and ensure the stability of the closed-loop system. The proposed controller regulates the inflow (dilution rate, D = F/V) to the bioreactor in order to control the mass balance of butanol. The controller is given by the following equation: D = k1 en -k2

(7)

where, for the control error, e=x3 -x3sp, x3sp is the corresponding set point. The exponent of the control error is n = 3, k1 and k2 are the named controller gains, which can be adjusted according to the controller response. More details are given below.

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Now, considering a regulation case, where the set point is constant, the dynamics of the control error is defined by ė =ẋ 3. From this, by substituting equation (6) in equation (3), the closed-loop form of the butanol mass balance equation is defined as: ė =

μ1 (x2 )μ2 (x3 )2 x Y2 1

Y1

+(x3in -x3 )(k1 en -k2 )

(8)

Let us to consider the following assumptions: by mass balance conservation principles, the mass concentrations remain bounded in a compact set defined as ℵ = {xi / xi ∈[0,xi,max ]} for i = 1,2,3. Therefore the term ‖x3in -x3 ‖≤℘ and the non-linear kinetic term is bounded as μ1 (x2 )μ2 (x3 )2 x



1Y2

Y1

‖ ≤ F. From these assumptions and by applying the

Cauchy-Schwartz inequality to maximize equation (8), the following is obtained: ‖ė ‖ ≤ F+℘(|k1 | ‖en ‖-|k2 |)

(9)

Now, by selecting k2 ≅℘-1 F, the following inequality is generated: ‖ė ‖ ≤ ℘|k1 | ‖en ‖

(10)

By solving inequality: 1

‖e‖ ≤ (|(1-n)|℘|k1 |t)|(1-n)|

(11)

Note that for n > 1 and large enough, the norm of the control error can be decreased leading to the butanol trajectory to stay within the vicinity of the required set point. As observed from inequality (Eq. 9), the control gain k2 helps to compensate the nonlinearity of the kinetic reaction term, thus obtaining the closed-loop structure of inequality from Eq. (9), and the final structure of the inequality from Eq. (10), which can be defined as practical

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stability of the biosystem, where the trajectories are theoretically brought to a reference point, as closed as desired.

Results Bifurcation Analysis and Open-Loop Stability To find suitable operational regions of the ABE process, a bifurcation analysis was performed. Figures 2(a-c) show the bifurcation diagrams (at steady state) for biomass, glycerol and butanol, employing the dilution rate and the glycerol inlet concentration as bifurcation parameters. The figures show that when a glycerol inlet concentration close to 60 g L-1 and a dilution rate value of 0.062 h-1, it is possible to obtain a glycerol consumption of 97.5%. It can be noted that under these conditions, biomass concentrations close to 30 g L-1 and a production of butanol of 9.4 g L-1 can be obtained, this is a natural response of the fermentation, such that with an increase of the corresponding substrate the biomass growth is also increased, improving the butanol generation, under the restrictiction of the inhibitory substrate concentration value. From this analysis, it was observed that the butanol concentration can be increased up to 62%. Similar operating conditions near to this operational region can be selected as suitable operating set points for controlling purposes. For our case study, 8 g L-1 and 9.4 g L-1 of butanol were selected as reference points. Can be noticed that with the bifurcation analysis the rank of the dilituin rate can also be determinated, is observed that higher values of the dilution rate, the residence time of the reactives in the bioreactor is diminished together with the glycerol conversion. Moreover, it is widely known that an equilibrium point of the dynamic system is stable if all the eigenvalues (λ) have negative real parts of their linear representation by the Jacobian matrix of the system. An equilibrium point is unstable if at least one of the eigenvalues has a positive real part [31]. The eigenvalues corresponding to the equilibrium points of biomass, glycerol and butanol presented in Figures 1(a-c), resulted for all cases with negative real values (data not shown), which indicates that the equilibrium

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points presented from this analysis are stable, under the selected operation conditions. However, under small changes of process parameters the corresponding equilibrium points can be altered, leading the bioreactor production to undesirable process conditions, such as washout regimen, low conversion regimen, input multiplicity and so on.

Figure 2. Bifurcation diagrams for biomass, glycerol and butanol employing the dilution rate and the inlet concentration of glycerol as bifurcation parameters.

Closed Loop Analysis Numerical experiments were carried out to show the closed-loop behavior of the bioreactor. The initial conditions were x1 = 0.05 g L-1, x2 = 40 g L-1, x3 = 0.0 g L-1 and x2in = 60 g L-1. The proposed controller is based on a single input-single output (SISO) structure, where the butanol concentration is considered as the controlled variable, while the input flow defined by the dilution rate (D), is considered as the control input. The controller parameters were estimated as k1 = 300 g3 L-3 and k2 = 0.016 h-1 in accordance with the previous theoretical results. The parameter n obeys the theoretical property given by the inequality of Eq. (11), but for practical application n = 3 is proposed, resulting in a cubic controller. The n value was selected considering the best output control response and the corresponding control effort. Furthermore, numerical experiments were

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programmed so that after 60 h the control law is activated with a first set point of x3sp = 8 gL-1. Subsequently, after 100 h, a second set point, x3sp = 9.4 gL-1 is activated. It is important to note that the selected set points are in accordance with the results from the bifurcation analysis, in order to select operating points with a high butanol production and avoid the inhibition effect. As mentioned, the control task is the increase of the butanol concentration via the dilution rate manipulation via the proposed control law. However, as can be observed in the mathematical model structure of the bioreactor, the dilution rate affect to all the state equations and then their corresponding dynamic response, for this is necessary to analyse the dynamic response of the properly uncontrolled variables under variation of the dilution rate to prevent undesirable effects or unstabilities of this uncontrolled mass concentrations. The above mentioned undesirable effects can include inhibitory glycerol concentrations and biomass washout in the bioreactor.

Figure 3. Closed loop dynamics of the biomass (a), glycerol (b) and biomass under the action of the cubic control law and a linear PI controller, employing the dilution rate (d) as control input.

The dynamics of the uncontrolled variables (biomass and glycerol) are shown in Figure 3a and b. The control action on the butanol variable can be observed in Figure 3c. Note that both reference points were reached in a finite time interval, which indicates the satisfactory performance of the

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controller. Figure 3d shows the control effort applied to the dilution rate to reach the set points and increase the butanol concentration up to 45.6%. For comparison purposes and to show some advantages of the proposed controller, the performance of a linear PI controller (Eq. 11) is presented. te

D=Kp (e+ ∫0 τ dt )

(12)

The PI controller is implemented under the same conditions as describe above. The tuning of the Kp and τ gains of the PI control was calculated based on the Internal Model Controller (IMC) guidelines [32]. The calculated dimensionless values were Kp = 0.2 and τ = 5. Figures 3a and (b) show that for both uncontrolled variables (biomass and glycerol), the concentrations take longer to stabilize using the classical PI control law. Figure 3c shows the dynamics of the butanol concentration under the action of both controllers. As can be observed, the proposed controller is capable to lead the butanol trajectory to the required set points without overshoots and long settling time, while the PI controller generates a large overshoot when is activated. The cubic controller showed better performance due to the shorter time to reach stability of the butanol concentration in both reference points. Furthermore, it is needed to analyze the corresponding control effort of the dilution rate for both controllers. Figure 3d depict the dynamic behavior of the control inputs. As can be seen, the proposed controller is able to lead the butanol concentration to the required trajectory with a realizable effort, reaching numerical values of D = 0.11 h-1 and D = 0.06 h-1, which is adequate to maintain high butanol productivity in the continuous operation. Can be observed that the proposed controller tends to increase the residence time of the corresponding chemical compounds in the bioreactor by diminishing the dilution rate values. The increase of the residence time increase via chemical reaction the butanol concentrations. Note that the PI controller requires to diminish the input flow at values near to zero, which significantly affects the productivity of the reactor, leading to an undesirable operational condition. For a more detailed analysis, the

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integral time squared error (ITSE) was also calculated for both controllers. As expected, the cubic controller performs better due to its lower ITSE value, as seen in Figure 4.

Figure 4. Determination of the Integral time Squared Error (ITSE) for both controllers.

System Stability Analysis The proposed stability analysis of the uncontrolled variables is very intuitive, if the closed-loop values of the time derivatives of biomass and glycerol concentrations under the proposed controller are positive, this means an increase of the corresponding concentrations, leading to an unstable behaviors and to the upper bound values of these concentrations, on the other hand, if the time derivatives are negatives the mass concentrations decreases lower generation or consumption of biomass and glycerol concentrations respectively. Finally, at time derivatives with zero values indicate that the concentrations reach an stable steady state. The zero dynamics of the system was evaluated through numerical simulations to determine the stability of the uncontrolled variables of the process, that is biomass and glycerol (see Figure 5). During the first 20 hours, the glycerol (x2 ) shows a stable dynamic, while the biomass (x1 ) presented an unstable behavior under this open loop operation time. After 40 hours of operation, the time derivative of the uncontrolled variables presents a decreasing trend, ensuring in this way the stability of the system at open loop. After 60 hours the proposed controller is activated with the

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first set point (8 g/L), which leads to a momentary instability of the two variables (x1 and x2), taking approximately 3 hours to stabilize the system. Finally, the second set point is activated after 100 hours. Similarly, the variables stabilize again in approximately 2 hours. The time derivatives of the uncontrolled states before and after the control action tend to zero, i.e., lim ẋ 1 , ẋ 2 →0. The butanol trajectory is controlled under the proposed t→∞

control law, while the biomass and glycerol concentrations tend to stabilize. This indicate that the bioreactor operation is closed-loop stable.

Figure 5. Zero dynamic performance of uncontrolled variables, biomass (x1 ) and glycerol (x2 ). The controller is turn on after 60 h for the first set point and the switch for the second set point is at 100 h.

Conclusion In this work is proposed a class of polynomial controller with a simple structure and two parameters, which can be tuned under the guidelines provided in the text, to generate a asymptotic-type convergence of the control error as is analytically proved. The proposed single imput-simgle output controller is applied to a mathematical model of a continuous stirred tank reactor for butanol production, considered as a benchmark. The constructed bifurcation diagrams by selecting the dilution rate and the glycerol input concentration as bifurcation parameters, allowed to determine feasible steady-state operating regions of the process and stablish the required set points to obtain high butanol concentrations. This

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analysis also showed that the system does not present multiplicity of stationary states. The structure of the proposed controller allowed to compensate the non-linear behavior in the reactor by selecting the control gain k2 similar to the upper bound of the kinetic model. Furthermore, the polynomial term induced stable behavior, where the tracking error can be made small enough by proper selection of parameter n. The proposed controller is able to reach satisfactorily the required trajectory for the butanol concentration with an adequate control effort, showing a better overall performance than a standard PI controller, increasing the butanol concentration in a 62%. Finally, the zero dynamics methodology allowed to determine that the uncontrolled variables, i.e., biomass and glycerol were stable under closed loop operation, concluding the the bioreactors is stabilizable.

References [1]

[2]

[3]

[4] [5]

Ranjan, A., Moholkar, V. S. 2012. Biobutanol: science, engineering, and economics, International Journal of Energy Research, 36:277323. Zabed, H., Sahu, J. N., Suely, A., Boyce, A. N., Faruq, G. 2017. Bioethanol production from renewable sources: Current perspectives and technological progress, Renewable and Sustainable Energy Reviews, 71:475-501. Sanchez, A., Valdez-Vazquez, I., Soto, A., Sánchez, S., Tavarez, D. 2017. Lignocellulosic n-butanol co-production in an advanced biorefinery using mixed cultures, Biomass and Bioenergy, 102:1-12. Green, E. M. 2011. Fermentative production of butanol-the industrial perspective, Current Opinion in Biotechnology, 22:337-343. Velázquez-Sánchez, H. I., Aguilar-López, R. 2018. Novel kinetic model for the simulation analysis of the butanol productivity of Clostridium acetobutylicum ATCC 824 under different reactor configurations, Chinese Journal of Chemical Engineering, 26:812821.

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Ricardo Aguilar-López, Edgar N. Tec-Caamal et al. Maddipati, P., Atiyeh, H. K., Bellmer, D. D., Huhnke, R. L. 2011. Ethanol production from syngas by Clostridium strain P11 using corn steep liquor as a nutrient replacement to yeast extract, Bioresource Technology, 102:6494-6501. Velázquez-Sánchez, H. I., Dominguez-Bocanegra, A. R., AguilarLópez, R. 2019. Modelling of the pH dynamic and its effect over the Isopropanol-Butanol-Ethanol fermentation by Clostridium acetobutylicum pIPA3-Cm2, Fuel. 235:558-566. Ibham Veza, Mohd Farid Muhamad Said, Zulkarnain Abdul Latiff. 2021. Recent advances in butanol production by acetone-butanolethanol (ABE) fermentation. Biomass and Bioenergy. 144, 105919. Kalafatakis, S., Skiadas, I. V., Gavala, H. N. 2019. Determining butanol inhibition kinetics on the growth of Clostridium pasteurianum based on continuous operation and pulse substrate additions, Journal of Chemical Technology and Biotechnology, 94:1559-1566. Skupin, P., Metzger, M. 2018. PI control for a continuous fermentation process with a delayed product inhibition, Journal of Process Control. 72:30-38. Pachauri, N., Singh, V., Rani, A. 2017. Two degree of freedom PID based inferential control of continuous bioreactor for ethanol production, ISA Transactions, 68:235–250. Rani, A., Singh, V., Gupta, J. R. P. 2013. Development of soft sensor for neural network-based control of distillation column, ISA Transactions, 52:438–449. Ray, W. H. 1981. Advanced Process Control, McGraw-Hill, New York. Sinha, A., Mishra, R. K. 2018. Control of a nonlinear continuous stirred tank reactor via event triggered sliding modes, Chemical Engineering Science, 187:52–59. Padmanabhan, L., Lapidus, L. 1977. Control of chemical reactors, in Chemical Reactor Theory. A Review. Lapidus, L., Amundson N. R. (Eds.). Prentice-Hall, Englewood Cliffs, NJ.

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[16] Bailey, J. E. 1977. Periodic phenomena, in Chemical Reactor Theory. A Review. Lapidus, L., Amundson N. R. (Eds.). PrenticeHall, Englewood Cliffs, NJ. pp. 758-813. [17] Spitz, J., Laurence, R., Chappalear, D. 1977. An experimental study of a polymerization reactor in periodic operation, AIChE Symposium Series, 72, 86-100. [18] Hermann, R., Krener, A. J. 1977. Nonlinear controllability and observability, IEEE Transactions on Automatic Control, 44, 728740. [19] Karstens K, Trippel S, Götz P. 2021. Process Engineering of the Acetone-Ethanol-Butanol (ABE) Fermentation in a Linear and Feedback Loop Cascade of Continuous Stirred Tank Reactors: Experiments, Modeling and Optimization. Fuels. 2, 2:108-129. [20] Ali Reza, Z., Mehdi, R. 2017. Fuzzy optimization approach for the synthesis of polyesters and their nanocomposites in in-situ polycondensation reactors, Industrial and Engineering Chemistry Research, 56:11245-11256. [21] Flores-Hernández, A. A., Reyes-Reyes, J., Astorga-Zaragoza, C. M., Osorio-Gordillo, G. L., García-Beltrán, C. D. 2018. Temperature control of an alcoholic fermentation process through the Takagi– Sugeno modeling, Chemical Engineering Research and Design, 140:320-330. [22] Kravaris, C., Savoglidis, G. 2012. Tracking the singular arc of a continuous bioreactor using sliding mode control, Journal of the Franklin Institute, 349:1583-1601. [23] Musmade, B., Patre, B. 2015. Sliding mode control design for robust regulation of time-delay processes, Transactions of the Institute of Measurement and Control, 37:699-707. [24] Zhao, D., Zhu, Q., Dubbeldam, J. 2015. Terminal sliding mode control for continuous stirred tank reactor, Chemical Engineering Research and Design, 94:266–274. [25] Devrim B. Kaymak. 2019. Design and Control of an Alternative Process for Biobutanol Purification from ABE Fermentation. Ind. Eng. Chem. Res. 58, 5, 1957–1965.

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[26] Gordeev, L. S., Koznov, A. V., Skichko, A. S., Gordeeva, Y. L. 2017. Unstructured mathematical models of lactic acid biosynthesis kinetics: A review, Theoretical Foundations of Chemical Engineering, 51:175-190. [27] Sun, Z., Kota, A., Sarsani, S., West, D. H., Balakotaiah, V. 2018. Bifurcation analysis of methane oxidative coupling without catalyst, Chemical Engineering Journal, 343:770–788. [28] Gomez-Acata, R. V., Lopez-Perez, P. A., Maya-Yescas, R., AguilarLopez, R. 2012. Bifurcation analysis of continuous aerobic nonisothermal bioreactor for wastewater treatment, IFAC Proceedings Volumes, 45:24–29. [29] Zhao, T., Tashiro, Y. & Sonomoto, K. 2019. Smart fermentation engineering for butanol production: designed biomass and consolidated bioprocessing systems. Appl. Microbiol. Biotechnol. 103, 9359–9371. [30] Kwabena Darkwah, Barbara L. Knutson, and Jeffrey R. Seay. 2020. A Perspective on Challenges and Prospects for Applying Process Systems Engineering Tools to Fermentation-Based Biorefineries. ACS Sustainable Chem. Eng. 6, 3, 2829–2844. [31] Figueroa-Estrada, J. C., Neria-González, M. I., Rodríguez-Vázquez, R., Tec-Caamal, E. N., Aguilar-López, R. 2020. Controlling a continuous stirred tank reactor for zinc leaching, Minerals Engineering, 157:106549. [32] Ogunnaike, B. A., Ray, W. H., 1994. Process dynamics modeling and control, Oxford University Press, N.Y.

In: Frontiers in Mathematical Modelling ... ISBN: 978-1-68507-430-2 c 2022 Nova Science Publishers, Inc. Editors: M. Biswas and M. Kabir

Chapter 8

Mathematical Study of Human Movement and Temperature in the Transmission Dynamics of Dengue Disease between Two Patches Ganga Ram Phaijoo∗ and Dil Bahadur Gurung Department of Mathematics, School of Science, Kathmandu University, Nepal

Abstract Dengue is an infectious disease that is spreading all over the world. Human movement leads to the spread of the disease in new places and temperature affects the life-cycle and biting behavior of the mosquitoes which transmit dengue disease. In the current work, a compartmental model is proposed to study the effect of temperature and human movement in the spread of dengue disease in the two patch environment. The human population is classified into susceptible, exposed, infectious and recovered classes, and the vector population into susceptible, exposed and infectious classes. It is observed that the Disease Free Equilibrium Point of the model is locally asymptotically stable when the basic reproduction number R0 < 1 and this point is unstable when the number R0 > 1. Model ∗

Corresponding Author’s Email: [email protected]

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Ganga Ram Phaijoo and Dil Bahadur Gurung parameters are considered temperature-dependent. Simulated results show that human movement and temperature have a significant impact in the evolution and spread of the disease. Both human movement and temperature have a significant impact on the transmission of the disease. Restricting human movement from low-risk patch to the high-risk patch contributes in decreasing the disease prevalence. But restriction of human movement from the high-risk patch to the low-risk patch leads to an increase in the prevalence of the disease. So, effective management of human movement between the patches can be helpful in reducing the prevalence of the disease.

Keywords: basic reproduction number, dengue, temperature, two patch model, stability

Introduction Infectious diseases are posing a constant threat to humans worldwide. Dengue disease is rapidly spreading mosquito-borne viral disease affecting tropical regions of the world. It has imposed a significant social burden on human populations. It is affecting more than 100 countries. About 50 100 millions dengue cases are reported yearly [1]. The disease is communicated to humans through the bites of Aedes mosquitoes. Four serologically distinct but closely related viruses called DEN 1 − DEN 4 cause dengue. When a human gets infected from one of the four serotypes of the dengue viruses, he/she acquires immunity against the virus of the same serotype but no long term immunity against other serotypes of dengue virus [2, 3]. Modeling is an influential tool to study infectious diseases mathematically. The models are helpful to predict the future outbreak of the disease and it can help to propose control strategies against the diseases. [4] developed compartmental models to study transmission dynamics of communicable diseases. In epidemiology, the model has been one of the best mathematical tools to study communicable diseases. So, their work was republished in 1991 [5]. Nowadays the model has been used to study diseases like malaria, dengue, chikungunya, Zika, ebola etc. Esteva and Vargas analyzed dengue disease transmission dynamics using SIR model taking constant [6] and variable [7] human populations. They considered Susceptible, Infected and Recovered classes for humans and, only two classes Susceptibles and Infected for mosquito population. Recovery from the disease in

Mathematical Study of Human Movement and Temperature ... 225 the mosquito population was not considered due to the short life span. Numerous mathematical study of dengue disease has been made to observe the dynamics of the disease from different views and angles. The succession of two dengue cases caused by two sero-types of viruses is studied by [8]. DHF (Dengue Hemorrhagic Fever) compartment is taken into account to study dengue disease assuming two viruses namely strain 1 and strain 2 [9]. [10] used real data to study dengue cases of different years. [11] used SIR compartmental model to study dengue disease transmission dynamics. [12] discussed about memory in the dengue disease model. Many researchers have included incubation periods in their study while studying diseases like dengue, malaria etc. Both intrinsic and extrinsic incubation periods influence the evolution and the transmission of dengue disease. [13] proposed SEIR compartmental model of dengue disease taking exposed classes of both vector and host population. [14] analyzed the role of extrinsic incubation period. Also, [15] investigated the dengue disease with extrinsic and intrinsic incubation periods in mosquito and humans respectively. [16] used SEIR - SEI compartmental model to analyze the control measures considering exposed classes in both host and vector populations. Human movement has become the main cause of the spread of epidemic diseases. When people move they may carry pathogens in or on their body and may also transport vectors of the diseases. So, it is important to understand how the human movement affects the epidemiology of the diseases and how the movement can be managed to control the diseases. Various research works are done to observe the role of human mobility in the spread of infectious diseases [17], [18], [19], [20]. [21] established the threshold criteria for local and global stability of equilibrium point on disease model when the population dispersed. [22] investigated the threshold criteria in a patchy system. They studied the effect of reproduction number on the persistence of the disease. [23] described the disease propagation when people travel in patches. [24] studied the transmission of malaria disease in a patchy environment generalizing the Ross-Macdonald model to n-patches. [25] analyzed the dengue disease transmission and spread with human movement to observe the role of residence time. Many infectious diseases are temperature-dependent. The temperature has an influential impact on the transmission dynamics of the diseases. Mosquitoes’ life cycle and their vectorial capacities are affected by tem-

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perature levels. Dengue incidence can be forecasted by using temperature and rainfall. [26] developed a model of dengue disease based on weather to forecast dengue cases. [27] studied the survival of Aedes mosquitoes. Study of vectorial capacity and effects of temperature for dengue epidemic potential is made by [28]. They focused on understanding how the change in climatic factors influences the spread of dengue. [29] claimed that the evolution of dengue disease cases relies on the seasonal change of the climate; very high and very low temperature scales are not favourable for the disease. The incubation period, temperature level and human movement affect the prevalence and spread of dengue disease significantly. So, we propose SEIR-SEI compartmental model to study dengue disease transmission dynamics in two patches incorporating human movement and temperature.

Model Formulation and Analysis We propose two patch SEIR-SEI compartmental model. Present model is based on [30] model. We have used this model to address vector host interaction in dengue disease dynamics. To formulate the deterministic epidemic compartmental model, we consider that humans move from one patch to the other. Due to the short-range flight capacity of the mosquitoes, we neglect vector movement. The total population, Nih of humans is further divided into Sih (Susceptible host), Eih (Exposed host), Iih (Infectious host) and Rhi (Recovered host) classes. Total mosquito population, Niv is further divided into Siv (Susceptible vector), Eiv (Exposed vector) and Iiv (Infectious vector) classes (i = 1, 2). We assume that humans move at the rates mSij (Susceptible host’s I movement rate), mE ij (Exposed host’s movement rate), mij (Infectious host’s movement rate) and mR ij (Recovered host’s movement rate) from j th to the ith patch (i, j = 1, 2 and i 6= j). Figure 1 shows dengue disease transmission dynamics in two patches. Model parameters are discussed in Table 1. The temperature-dependent parameters are presented in (1). Here, we consider that the model parameters bi , βih , βiv , dvi , νiv (i = 1, 2) are temperature-dependent [28, 29]. The temperature dependency relations are:

Mathematical Study of Human Movement and Temperature ... 227

Figure 1. Flow chart of the two patch model. Table 1. Description of the model parameters Symbols Ahi dhi νih γih βih Avi dvi νiv βiv bi

Description host’s recruitment rate host’s death rate progression rate of hosts from exposed to infectious class host’s recovery rate vector to host transmission probability vector’s recruitment rate vector’s death rate progression rate of exposed vector to the infectious class host to vector transmission probability biting rate

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bi (T )

=

βiv (T )

=

βih(T )

=

0.0043T + 0.0943 (21◦ C ≤ T ≤ 32◦ C) ( 0.0729T − 0.9037 (12.4◦ C ≤ T ≤ 26.1◦ C) 1

(26.1◦ C < T < 32.5◦ C)

√ 0.001044T (T − 12.286) 32.461 − T

(1)

(12.286◦ C < T < 32.461◦ C) νiv (T )

=

4 + e5.15−0.123T

dvi (T )

=

0.8692 − 0.159T + 0.01116T 2 − 3.408 × 10−4 T 3 + 3.809 × 10−6 T 4

(12◦ C < T < 36◦ C)

(10.54◦ C ≤ T ≤ 33.41◦ C) and i = 1, 2.

Present two patch dengue disease transmission model is described by the following system: For Patch 1: dS1h dt

= Ah1 −

dE1h dt

=

dI1h dt dRh1 dt dS1v dt dE1v dt dI1v dt

b1 β1h h v S1 I1 + mS12 S2h − mS21 S1h − dh1 S1h h N1

b1 β1h h v h E h h h h S1 I1 + mE 12 E2 − m21 E1 − (ν1 + d1 )E1 h N1

= ν1h E1h + mI12 I2h − mI21 I1h − (γ1h + dh1 )I1h h R h h h = γ1h I1h + mR 12 R2 − m21 R1 − d1 R1

= Av1 − =

b1 β1v v h S1 I1 N1h

− dv1 S1v

b1 β1v v h S I − (ν1v + dv1 )E1v N1h 1 1

= ν1v E1v − dv1 Iiv

(2)

Mathematical Study of Human Movement and Temperature ... 229 For Patch 2: dS2h dt

= Ah2 −

dE2h dt

=

dI2h dt dRh2 dt dS2v dt dE2v dt dI2v dt

b2 β2h h v S I + mS21 S1h − mS12 S2h − dh2 S2h N2h 2 2

b2 β2h h v h E h h h h S2 I2 + mE 21 E1 − m12 E2 − (ν2 + d2 )E2 N2h

= ν2h E2h + mI21 I1h − mI12 I2h − (γ2h + dh2 )I2h h R h h h = γ2h I2h + mR 21 R1 − m12 R2 − d2 R2

= Av2 − =

(3)

b2 β2v v h S2 I2 − dv2 S2v N2h

b2 β2v v h S I − (ν2v + dv2 )E2v N2h 2 2

= ν2v E2v − dv2 I2v

Where, Sih (t) + Eih + Iih (t) + Rhi (t) = Nih (t), Niv (t) where i = 1, 2.

Siv (t) + Eiv + Iiv (t) =

Existence and Stability of Disease Free Equilibrium Point An equilibrium when infected hosts and infected mosquitoes are absent, is called disease free equilibrium (DFE). When E1h = 0, E1v = 0, I1h = 0, I1v = 0, Rh1 = 0 and E2h = 0, E2v = 0, I2h = 0, I2v = 0, Rh2 = 0, we obtain the following DFE Df (S1h∗ , S2h∗, 0, 0, 0, 0, 0, 0, S1v∗, S2v∗ , 0, 0, 0, 0) of the system of equations (2) - (3), given by the following system of equations ASh = B CSv with A = S2h∗ ]T ,



dh1 + ms21 −mS12 −mS21 dS2 + mS12

(4)

= D (5)   v  d1 0 , C = , Sh = [S1h∗ , 0 dv2

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Sv = [S1v∗ , S2v∗ ]T , B = [Ah1 , Ah2 ]T , D = [Av1 , Av2 ]T Here, S1h∗ =

Ah1 dh2 + Ah1 mS12 + Ah2 mS12 , dh1 dh2 + dh1 mS12 + dh2 mS21

S1v∗ =

Av1 dv1

S2h∗ =

Ah2 dh1 + Ah2 mS21 + Ah2 mS21 , dh2 dh2 + dh2 mS21 + dh1 mS12

S2v∗ =

Av2 dv2

Basic Reproduction Number Basic reproduction number (also called basic reproductive ratio) is defined as the average number of secondary infections caused by an infectious person in a susceptible community [31–33]. It is considered as a metric to decide if the disease will spread or will die out from society. The basic reproduction number is the threshold for many epidemiological models. From the definition, if the basic reproduction number R0 is less than unity, the disease vanishes and the disease will take hold when the number is greater than unity. Next Generation Matrix is used to compute the number R0 of a compartmental model. This method was first introduced by [31] and refined for epidemiological models by [33]. Let p, q respectively denote disease- and non-disease compartments, and u and v respectively be population sizes in the respective compartments p and q. Suppose that secondary cases of the disease increases in the kth compartment at the rate Fk and Vk is the rate at which the the compartment is reduced. Then duk dt dv` dt

= Fk (u, v) − Vk (u, v), k = 1, 2, 3, · · · , p

(6)

= g` (u, v), ` = 1, 2, 3, · · · , q

(7)

After linearization of the ODE model about disease free equilibrium point 0 , the two matrices F and V are obtained.     ∂Fk ∂Vk F = (0 ) , V = (0 ) ∂u` ∂u`

Mathematical Study of Human Movement and Temperature ... 231 The matrix F V −1 is called the next generation matrix [33] and the number R0 is computed as the spectral radius of the matrix F V −1 . R0 = ρ(F V −1 ). Here, ρ(A) denotes the spectral radius of the matrix A. For the system of equations (2) - (3), 

          F =          2 6 6 6 6 6 6 6 6 6 6 4

0 0 0 0

0

0

b1 β1h S1h ∗ N1h

0

0 0 0 0

0

0

0

0 0 0 0

b1 β1v S1v ∗ N1h

b2 β2h S2h ∗ N2h

0

0

0

0 0 0 0

0

0

0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

E h dh 1 + m21 + ν1 −mE 21 0 0 −ν1h 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

−mE 12 E h dh 2 + m12 + ν2 0 0 0 −ν2h 0 0

b2 β2v S2v ∗ N2h 0 0 0 0

V= 0 0 v dv 1 + ν1 0 0 0 −ν1v 0

0 0 0 dv + ν2v 2 0 0 0 −ν2v

0 0 0 0 I + γh + m dh 1 21 1 −mI 21 0 0

                    

0 0 0 0 −mI 12 I h dh 2 + m12 + γ2 0 0

0 0 0 0 0 0 dv 1 0

And, R0 =

r

´ 1 1` m1 R201 + m2 R202 + 2 2

q (m1 R201 + m2 R202 )2 − 4m3 R201 R202

(8)

0 0 0 0 0 0 0 dv 2

3 7 7 7 7 7 7 7 7 7 7 5

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where, =

s

R02

=

s

m1

=

h h g1 n1 (mI12 mE 21 ν2 + ν1 g2 n2 ) h I I E ν1 (−m12 m21 + g1 g2 )(−m12 mE 21 + n1 n2 )

m2

=

I h h g2 n2 (mE 12 m21 ν1 + g1 n1 ν2 ) E ν2h(−mI12 mI21 + g1 g2 )(−mE 12 m21 + n1 n2 )

m3

=

g1 n1

= =

R01

b21 S1h∗ S1v∗ β1h β1v ν1h ν1v 2 dv1 N1h (dh1

h v v + mI21 + γ1h )(dh1 + mE 21 + ν1 )(d1 + ν1 )

b22 S2h∗ S2v∗ β2h β2v ν2h ν2v 2

h v v dv2 N2h (dh2 + mI12 + γ2h )(dh2 + mE 12 + ν2 )(d2 + ν2 )

(mI12 γ1h

+

g3 dh2

dh1 + mI21 + γ1h , h dh1 + mE 21 + ν1 ,

+

g3 γ2h

g 1 n1 g 2 n2 h h h h + g2 dh1 )(mE 12 ν1 + n3 d2 + n3 ν2 + n2 d1 )

g2 = dh2 + mI12 + γ2h , h n2 = dh2 + mE 21 + ν2 ,

g3 = mI21 + γ1h h n3 = mE 21 + ν1

Here, R01 and R02 are respective local basic reproduction numbers of the two patches. Theorem 1 (Stability). The disease free equilibrium of the model (2) - (3) is locally asymptotically stable for R0 < 1 and it is unstable for R0 > 1. Proof. About the disease free equilibrium point Df , Jacobian matrix for the model equations (2) - (3) is   P Q J= 0 F −V Here,  dh1 + mS12 −mS12 0 0  −mS21  dh2 + mS12 0 0  −P =  h R R  0 0 d1 + m21 −m12  0 0 −mR dh2 + mR 21 12 

Mathematical Study of Human Movement and Temperature ... 233 The matrix −P has each column sum positive and each non-diagonal element is non-positive. So, −P is a non-singular M - matrix and spectral abscissa s(P ) < 0 [34]. Also, the matrix F is non-negative matrix and V is a non-singular M -matrix. So, s(F − V ) < 0 when ρ{F V −1 } < 1 [33], i.e., R0 < 1. Thus, eigenvalues of the system of equations (2) - (3) will have negative real part when R0 < 1 and so, the point Df is locally asymptotically stable for R0 < 1. Also, R0 > 1 implies s(F − V ) > 0 [33]. It assures that at least one eigenvalue lies in right half plane. So, Df is unstable for R0 > 1.

Numerical Results and Discussion Human movement and temperature affect the spread and persistence of dengue disease significantly. We carried out the numerical results to observe the impact of human movement and temperature in the evolution and spread of dengue disease in two patches. We used the following parameter values for the simulation purpose: N1h = 50000, dh1 = dh2 = 0.00004029, ν1h = ν1h = 0.1667, γ1h = = 0.0714, N2h = 20000. Parameters bi , βih , βiv , dvi , νiv are discussed in (1). Human movement from one patch to the other causes an increase in the prevalence of the disease. Figure (2) and Figure (3) depict that prevalence of the disease increases when hosts move from one patch to the other. The infective host population size increases at the beginning. It is because of the interaction of the susceptible host population with the infective vector population. Afterwards, the population size starts decreasing because of the recovery of the infected hosts from the disease and some hosts may die due to the natural cause. Figure (4) and (5) are simulated to study the effect of temperature in the transmission of dengue disease. We have assumed patch 1 to have temperature in the range of 20o C − 30o C and patch 2 to have the temperature in the range of 15o C − 25o C. Infectious host population size is observed to be increasing in both patches with the higher temperature level. Disease incidence is seen higher in patch 1 when the temperature is 30oC and in patch 2 when the temperature in the patch is 25o C. Thus, these simulaγ2h

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Figure 2. Infectious humans of patch 1. tions show that temperature contributes to increasing the prevalence of the disease. Basic reproduction number is a metric that helps to determine if the disease spreads in a population. A greater value of the number means higher the prevalence and transmission rate of the disease. Figure (6) and Figure (7) are simulated to see the impact of human movement on basic reproduction number R0 when temperature in patch 1 is 30o C and that in patch 2 is 20oC. For the temperature levels used in the simulation, R01 > R02. That means, there is a higher risk of the disease transmission in patch 1 than in patch 2. So, Figure (6) shows that R0 increases with the increase in human movement rate from low-risk patch 2 to the high-risk patch 1. At the same time, the number decreases with the increase in human movement rate from high-risk patch 1 to the low-risk patch 2. Figure (8) and Figure (9) show the impact of temperature on local basic reproduction numbers R01 and R02 with different human movement rates.

Mathematical Study of Human Movement and Temperature ... 235

Figure 3. Infectious hosts of patch 2. The value of the basic reproduction number increases with the increase in temperature level. It is observed that the value of R01 increases with the increase in movement rate and with the increase in temperature level. The basic reproduction number R01 attains its maximum value when the temperature of the patch 1 is 29.3oC [28]. In case of second patch, value of R02 increases with movement rate and temperature. R02 attains its maximum value at the temperature 25o C. Figure (10) - Figure (15) are simulated to observe the impact of human movement from one patch to the other in one direction only. The disease prevalence can be controlled by altering the human movement rate from one patch to the other. Figure (10) shows that human movement from high-risk patch 1 to low-risk patch 2 restricting the human movement from patch 2 to patch 1 causes the reduction in infective human population size in patch 1. The host movement makes the low-risk patch to become highrisk region (Figure (11)).

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Figure 4. Infectious hosts of patch 1 with different temperature levels.

Figure 5. Infectious hosts of patch 2 with different temperature levels.

Mathematical Study of Human Movement and Temperature ... 237

Figure 6. Basic reproduction number and movement rates of infected hosts.

Figure 7. Basic reproduction number and movement rates of susceptible hosts.

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Figure 8. Basic reproduction number of patch 1 with infectious hosts’ movement from patch 1 to patch 2.

Figure 9. Basic reproduction number of patch 2 with infectious hosts’ movement from patch 2 to patch 1.

Mathematical Study of Human Movement and Temperature ... 239

Figure 10. Infectious hosts of patch 1 without host movement from patch 2 to patch 1.

Figure 11. Infectious hosts of patch 2 without host movement from patch 2 to patch 1.

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Figure 12. Infectious hosts of patch 1 without host movement from patch 1 to patch 2.

Figure 13. Infectious hosts of patch 2 without host movement from patch 1 to patch 2.

Mathematical Study of Human Movement and Temperature ... 241

Figure 14. Basic reproduction numbers of patch 1 and patch 2 against host I movement rate, m = mS12 = mE 12 = m12 without host movement from patch 1 to patch 2.

Figure 15. Basic reproduction numbers of patch 1 and patch 2 against host I movement rate, m = mS21 = mE 21 = m21 without host movement from patch 2 to patch 1.

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Similarly, restriction of human movement from high-risk patch 1 to low-risk patch 2 helps to reduce the disease prevalence in low-risk patch 2 Figure (13) and causes the high disease prevalent patch to be even more disease prevalent Figure (12). Thus, effective management of human movement is helpful to control the cases of dengue disease. Figure (14) and Figure (15) justify that when there is human movement in one direction only from high to low-risk patch, there is a reduction in the value of R01 and an increase in the value of R02 . Meanwhile, when humans move from low to high-risk patch, R01 increases and R02 decreases.

Conclusion Infectious diseases like dengue have always been a major part of humans and the whole human society. The diseases are increasing their geographic range due to human mobility for different purposes. The disease prevalence is increasing due to the change in temperature levels as the vectors are getting favourable temperature level to survive and transmit the dengue disease. In the present work, a two patch model of dengue disease is proposed to observe the impact of human movement and temperature. We observed that human movement can increase and decrease disease prevalence. Temperature increases the prevalence of the disease, but extreme high temperature is not favourable for dengue disease. The simulated results show that the maximum disease prevalence occurs at about 29.3oC temperature and the disease prevalence decreases afterwards with higher temperature levels. The study shows that basic reproduction R0 depends on temperature levels and human movement rates. Local basic reproduction numbers R01 and R02 can be reduced by managing the human movement between high and low-risk patches. Further, the mathematical study shows that the disease free equilibrium of the present model is locally and asymptotically stable if the reproduction number is less than unity and it is unstable when the number is greater than unity.

Mathematical Study of Human Movement and Temperature ... 243

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[32] Hethcote, H. W. (2000). The mathematics of infectious diseases, SIAM Review, Vol. 42, pp. 599 – 653. [33] van den Driessche, P. and Watmough, J. (2002). Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosci., Vol. 180, pp. 29 – 48. [34] Berman, A. and Plemmons, R. J. (1979). Non-negative matrices in mathematical sciences, Academic press. [35] Arino, J., Durcot, A. and Zongo, P. (2012). A meta population model for malaria with transmission blocking partial immunity in hosts, J. Math. Biol., Vol. 64, pp. 423 – 448. [36] Aron, J. L. (1988). Mathematical modeling of immunity to malaria, Math. BioSci., Vol. 90, pp. 385 – 396.

In: Frontiers in Mathematical Modelling … ISBN: 978-1-68507-430-2 Editors: M. Biswas and M. Kabir © 2022 Nova Science Publishers, Inc.

Chapter 9

A Numerical Model of Malaria Fever Transmission with Organized Vector Populace and Irregularity Kalyan Das1, and M. N. Srinivas2,† 1

Department of Basic and Applied Sciences, National Institute of Food Technology Entrepreneurship and Management, HSIIDC Industrial Estate, Kundli, Haryana, India 2 Department of Mathematics, School of Advanced Sciences, Vellore Institute of Technology, Vellore, Tamil Nadu, India

Abstract Malaria fever is an infectious disease caused by the plasmodium parasite, which is transmitted to humans by female mosquito bites. According to estimates from the World Health Organization, three billion and two hundred million people were at danger of sickness, with twenty-four lakhs of new cases and four lakhs and thirty-eight thousand deaths. Any way suburban of Saharan Africa remains the weakest district with high pace of passing because of intestinal sickness. To lessen the effect of intestinal sickness on the planet, numerous logical endeavors were finished including numerical modeling and  †

Corresponding Author’s Email: [email protected] (Corresponding author). Corresponding Author’s Email: [email protected] (Corresponding author).

248

Kalyan Das and M. N. Srinivas analytical findings. It has been discovered that reducing the number of mosquitos has minimal effect on the research of disease transmission of intestinal sickness in high-transmission areas. It should also be noted that natural and climatic factors have a substantial impact on vector-borne disease transmission and spread. Climate elements, such as temperature, precipitation, moistness, wind, and length of sunshine, have a significant impact on mosquito population irregularity and circadian cadence, as well as other biological and social features. Furthermore, because eggs, hatchlings, and pupae are not related with the transmission cycle, the mosquito life cycle is typically ignored in most numerical models. This is a useful decomposition of the framework, however the results of these models do not predict the force of jungle fever in most endemic areas. In this way, the presence pattern of mosquitoes and the influence of irregularity, which are crucial portions of the elements of jungle fever transmission, must be considered. In this paper, we develop a numerical model of non-self-sufficient customary differential conditions that depicts the features of malaria fever transmission as well as the vector population’s age structure. Mosquitoes’ gnawing speed is seen as a positive intermittent capability that is dependent on climatic circumstances. We obtain the model’s fundamental multiplication proportion and show that it is the edge boundary between the illness’s termination and industriousness. Hence, by applying the hypothesis of correlation and the hypothesis of uniform constancy, we demonstrate that on the off chance that the essential propagation proportion is under 1, at that point the infection free balance is universally asymptotically steady and in the event that it is more noteworthy than 1, at that point there exists at any rate one sure intermittent arrangement. We also looked at the effect of noise on the model system’s dynamics. The system’s stochastic analysis reveals that environmental noise has an impact on the dynamical behaviour of solutions. MATLab is being used to carry out numerical simulations.

Keywords: mathematical model, malaria, stability, steady state, stochasticity

1. Introduction Malaria is a long-standing disease that poses a number of health risks. The tropical regions of Africa, Asia, and America are ideal for the disease’s rapid dissemination. The plasmodium parasite is the cause of this disease. This infection is transmitted to humans via the bite of a female mosquito. Within a few days of an infected mosquito bite, medical symptoms such as a rise in body temperature, exhaustion, discomfort, shivering, and sweating may appear. There is currently no effective vaccine in

A Numerical Model of Malaria Fever Transmission …

249

development, and some existing antimalarial drugs are losing their efficacy due to parasite drug resistance. The main model of intestinal sickness transmission was created by Ross [1]. As per Ross, in the event that the mosquito populace can be decreased to under a specific edge, at that point intestinal sickness can be destroyed. Following that, Macdonald tweaked the model and added super illness. He claimed that reducing mosquito numbers had minimal effect on the research of disease transmission of intestinal ailment in high-transmission areas [2]. These days, a few numerical models have been created to lessen the intestinal sickness passing rate on the planet [3, 4]. Disregarding the endeavors made, it is as yet hard to foresee future intestinal sickness force, especially taking into account environmental change. It is clear that natural and climatic factors have a significant impact on vector-borne infection transmission and dispersion. Irregularity and circadian cadence of mosquito populace, just as other environmental and social highlights, are unequivocally impacted by climatic factors, for example, temperature, precipitation, moistness, wind, and span of sunshine [5]. Furthermore, because eggs, hatchlings, and pupae are not related with the transmission cycle, the mosquito life cycle is frequently disregarded in most numerical models. That is a valuable improvement of the framework however tragically the consequences of these models don’t foresee jungle fever force in most endemic areas. In this manner the presence pattern of mosquitoes and the influence of irregularity, which are major portions of the factors of intestinal disease transmission, must be considered. Moulay et al. [6] recently devised a numerical model depicting mosquito population aspects that takes into account auto guideline wonders of eggs and hatchlings stages. They identified a limit and demonstrated that it controls the growth of the mosquito population. We intend to develop a numerical model that depicts the features of jungle fever transmission while taking into account climatic factors and the mosquito’s life cycle. The impact of the model displaying mosquito population elements on the model of intestinal disease transmission is examined. Using the correlation hypothesis and the notion of uniform determination, we analyse the global solidity of the nontrivial illness free harmony [7–10] and the presence of positive occasional

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arrangements. The following is the order in which this work will be completed. In Section 2, we develop a numerical model for our problem. Segment 3 explains the model’s components without clamor. Segment 4 furnishes the elements of the model with commotion. Dissemination examination of the numerical model is talked about in segment 5. Computational recreations are acted in Section 6. In the last segment, Section 7, we close and give a few comments and future works.

2. Mathematical Model Based on the earlier research and innovations done in [16-25], we inspired and proposed an organized jungle fever model including irregularity to represent the vital sickness. Fever caused by mosquito populace and later advanced stages in humans are further are further categorized into different classes. Based on state and characteristics, the populace of human are further modeled as: “the susceptible class - Sh , uncovered or exposed class - Eh , irresistible or infectious class -

I h , and recuperated or recovered class - Rh .”

There are primarily two key stages in mosquito populace based on generations: the development stage and the oceanic stage. Following that, the further categorization of mosquito populace as (i) juvenile, (ii) adult. Juvenile populace again categorized as: “eggs class – 𝐸, hatchlings and pupae class – 𝐿.” Adult populace again categorized as: “susceptible class -

Sm , uncovered or exposed class - Em , and irresistible or infectious class I m .” For all time t , “The total populace of human and adult mosquitoes are N h ,

At respectively.”  E  E (t )  b 1   A   s  d  E  11 (t ) K1  

(2.1)

 L  L(t )  s 1   E   sL  d L  L  2 2 (t ) K2  

(2.2)

A Numerical Model of Malaria Fever Transmission …

251

Figure 1. Schematic representation of malaria model system.

Sh (t )     Rh   d h  kh  Sh  33 (t )

(2.3)

Eh (t )  kh Sh   d h    Eh   4 4 (t )

(2.4)

I h (t )   Eh   d h  d p  rh  I h  55 (t )

(2.5)

Rh (t )  rh I h   d h    Rh  66 (t )

(2.6)

Sm (t )  sL L   d m  km  Sm  77 (t )

(2.7)

Em (t )  km Sm   vm  dm  Em  88 (t )

(2.8)

I m (t )  vm Em  d m I m  99 (t )

(2.9)

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Kalyan Das and M. N. Srinivas

where N h (t )  E (t )  L(t )  Sh (t )  Eh (t )  I h (t )  Rh (t ) ; A(t )  Sm (t )  Em (t )  I m (t )

Here “  is the constant recruitment rate for humans, d h is the human death rate,

 is the transmission rate of humans from Eh to I h , d p is the

percentage of people who die as a result of a disease,

rh is the human

recuperation rate,  is the per capita rate of loss of immunity for humans,

S L is the rate of transition from L to adult, dm is the adult vector death rate,

vm is the transmission rate of mosquitoes from

Em to I m , K1 represents the

available breeding places have been taken up by eggs., K2 represents larvae occupying available breeder sites, s is the transfer rate from 𝐸 to 𝐿, b is eggs laying rate, d is the rate of egg death, d L is the larvae death rate,

 (t )  1 (t ), 2 (t ), 3 (t ), 4 (t ), 5 (t), 6 (t), 7 (t), 8 (t), 9 (t)  a pleasant nine-dim additive Gaussian white noise process E  i  t   0; i  1, 2, 3, 4, 5, 6, 7,8,9 E  i  t   j  t     ij  t  t  ; i, j  1, 2, 3, 4, 5, 6, 7,8,9 where

symbol;  is the delta-Dirac function.

 ij is the Kronecker

i represents the amplitudes of the

noise on the system where i  1, 2, 3, 4, 5, 6, 7,8, 9 and   t  is a Gaussian white noise process at time t .”

3. Dynamical Behaviour without Noise By the motivation of [25] and [18-23], we are studying the effect of noise on the malaria disease system (2.1-2.9). In this process, we are providing some observations done by [25] in this section without additive white noise. The authors [25] define a numerical model of non-autonomous normal differential circumstances that depicts the features of Malaria fever transmission as well as the vector population’s age structure. Mosquito

A Numerical Model of Malaria Fever Transmission …

253

biting rate is seen as a positive occasional capacity that is influenced by meteorological factors. The model’s important propagation proportion is determined, and it is shown to be the limit boundary between the illness’s termination and cleverness. Furthermore, authors [25] shown that “if the fundamental reproduction ratio is less than 1, the disease-free equilibrium is globally asymptotically stable, whereas if it is greater than 1, at least one positive periodic solution emerges.” At last, mathematical simulations are completed to delineate their insightful outcomes.

4. Dynamical Behaviour with Noise Ecological systems are characterized by a number of forces that are not constant in time but change, such as climate and natural disturbances. Due to the unpredictability entailed in weather systems, climate disruptions, a significant portion of climatic variables is random (with perhaps the exception of mechanisms heavily influenced by relativistic oscillations). The occurrence of random drivers in bio-geophysical systems promotes the investigation as to how a randomized setting might alter and influence the behaviour of biological ecosystems. We now investigate the influence of random environmental variations on stability using the stochastic model (2.1)-(2.9). The model’s parameters swing around their average values due to random variations. With additive white noises, we consider the randomness of the model (2.1)-(2.9). Whatever component      t  in the framework will affect the outcome of the Gaussian distribution disruption, where “  is the amplitude of the noise and   t  is a Gaussian white noise process at time t .” The parametric and nonparametric models, on the other hand, have the identical equilibrium states, which would then oscillate near actual average states. Here “ N h (t )  Sh (t )  Eh (t )  I h (t )  Rh (t ) ;

A(t )  Sm (t )  Em (t )  I m (t ) ”

(4.1)

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Kalyan Das and M. N. Srinivas

The behavior of the system (2.1)-(2.9) all over the internal equilibrium point E  E  , L , Sh* , Eh* , I h* , Rh* , Sm* , Em* , I m*  is the subject of this study in accordance with the method proposed by Carletti [26] and Nisbet and Gurney [27]. Let * * E (t )  u1 (t )  E * ; L(t )  u2 (t )  L ; Sh (t )  u3 (t )  Sh ;

Eh (t )  u4 (t )  Eh* ; I h (t )  u5 (t )  I h* ; Rh (t )  u6 (t )  Rh* ;

Sm (t )  u7 (t )  Sm* ; Em (t )  u8 (t )  Em* ; I m (t )  u9 (t )  I m* (4.2)

and by focusing solely on the effects of linear stochastic perturbations and as a result, the model (2.1)-(2.9) is reduced to the linear system shown below.  bE *   bE *   bE *  u1 (t )    u   u   7   8   u9  11 (t )  K1   K1   K1 

(4.3)

 sL*  u2 (t )     u1  2 2 (t )  K2 

(4.4)

u3 (t )  33 (t )

(4.5)

u4 (t )  4 4 (t )

(4.6)

u5 (t )  55 (t )

(4.7)

u6 (t )  66 (t )

(4.8)

u7 (t )  77 (t )

(4.9)

A Numerical Model of Malaria Fever Transmission …

255

u8 (t )  88 (t )

(4.10)

u9 (t )  99 (t )

(4.11)

Taking the Fourier transform of (4.3) - (4.11) we get,  bE *   bE *   bE *   u7 ( )    u8 ( )    u9 ( )  K1   K1   K1 

11 ( )  (i )u1 ( )  

 SL*   u1 ( )  (i )u2 ( )  K2 

22 ( )  

(4.12)

(4.13)

33 ( )  (i )u3 ( )

(4.14)

44 ( )  (i)u4 ()

(4.15)

55 ( )  (i )u5 ( )

(4.16)

66 ( )  (i )u6 ( )

(4.17)

77 ( )  (i)u7 ()

(4.18)

88 ( )  (i )u8 ( )

(4.19)

99 ( )  (i )u9 ( )

(4.20)

The matrix form of (4.12) and (4.20) is M   u      

(4.21)

256

Kalyan Das and M. N. Srinivas  A1   B1  C1   D1 where, M     E1   F1 G  1  H1 I  1

A2 B2 C2 D2 E2 F2 G2 H2 I2

A3 B3 C3 D3 E3 F3 G3 H3 I3

A4 B4 C4 D4 E4 F4 G4 H4 I4

A5 B5 C5 D5 E5 F5 G5 H5 I5

A6 B6 C6 D6 E6 F6 G6 H6 I6

A7 B7 C7 D7 E7 F7 G7 H7 I7

A8 B8 C8 D8 E8 F8 G8 H8 I8

A9   B9  C9   D9  ; E9   F9  G9   H9  I 9 

u    [u1 ( ), u2 (), u3 (), u4 (), u5 (), u6 (), u7 (), u8 (), u9 ()]T     [ 11 ( ),  2 2 ( ), 33 ( ),  4 4 ( ),  5 5 ( ),  6 6 ( ),  7 7 ( ),  8 8 ( ),  9 9 ( )]T

A1  i; A2  0; A3  0; A4  0; A5  0; A6  0; A7  A8  A9  (bE * / K1 ) ; B1  ( sL* / K 2 ); B2  i; B3 ( )  B4 ( )  B5 ( )  B6 ( )  B7 ( )  B8 ( )  B9 ( )  0 ;

C1  0; C2  0; C4  0; C5  0; C6  0; C7  0; C8  0; C9  0; C3  i ;

D1  0; D2  0; D3  0; D5  0; D6  0; D7  0; D8  0; D9  0; D4  i; E1  0; E2  0; E3  0; E4  0; E6  0; E7  0; E8  0; E9  0; E5  i;

F1  0; F2  0; F3  0; F4  0; F5  0; F7  0; F8  0; F9  0; F6  i; G1  0; G2  0; G3  0; G4  0; G5  0; G6  0; G8  0; G9  0; G7  i; H1  0; H 2  0; H 3  0; H 4  0; H 5  0; H 6  0; H 7  0; H 9  0; H8  i; I1  0; I 2  0; I 3  0; I 4  0; I 5  0; I 6  0; I 7  0; I8  0; I 9  i;

As a result, the solution to (4.21) is u    K ( )   , where K ( )  inverse of  M    (4.22)

The solution components of (4.22) are given by 9

ui     K ij    j   ; i  1, 2,3, 4,5, 6, 7,8,9 j 1

(4.23)

A Numerical Model of Malaria Fever Transmission …

257

The spectrum of ui , i  1,2,3,4,5,6,7,8,9 are given by 9

Sui      j Kij   ; i  1, 2,3, 4,5, 6, 7,8,9 2

j 1

As

a

result,

the magnitudes are given by ui , i  1, 2,3, 4,5,6,7,8,9 u 2  i

9

1 2



 j 1 

of

the

variable’s

variations

Kij ( ) d; i  1, 2, 3, 4, 5, 6, 7,8, 9 . 2

j

That is to say, the variations of ui , i  1, 2,3, 4,5,6,7,8,9 are obtained as u 2  1







1 2

     2 2 2 2   1 B11 ( ) d    2 B12 ( ) d   3 B13 ( ) d    4 B14 ( ) d     2



2





1 2



B35 ( ) d  2

5

4









B26 ( ) d  2

6





9

2  B19 ( )  







B27 ( ) d  2

7





B28 ( ) d  2

8



 2  B29 ( ) d   





9









B36 ( ) d  2

6





B37 ( ) d  2

7





B38 ( ) d  2

8



 2  B39 ( ) d   





9



1  2 2 2 2   1 B41 ( ) d    2 B42 ( ) d   3 B43 ( ) d    4 B44 ( ) d 2     B45 ( ) d 





B46 ( ) d  2

6







B47 ( ) d  2

7









B48 ( ) d  2

8







9



 2 B49 ( ) d  

    1  2 2 2 2   1 B51 ( ) d    2 B52 ( ) d   3 B53 ( ) d    4 B54 ( ) d 2     

B55 ( ) d  2

5







5

2





2

5

u 2 















u 2 



8 B18 ( ) 

     2 2 2 2   1 B31 ( ) d    2 B32 ( ) d   3 B33 ( ) d    4 B34 ( ) d    









3



2

 2 2 2 2   1 B21 ( ) d    2 B22 ( ) d   3 B23 ( ) d    4 B24 ( ) d     

B25 ( ) d 

1 2





7 B17 ( ) d 



2

5

u 2 



2











6 B16 ( ) d 



u 2  



5 B15 ( ) d 







B56 ( ) d  2

6







B57 ( ) d  2

7







B58 ( ) d  2

8







9

 2 B59 ( ) d  

258

Kalyan Das and M. N. Srinivas u 2  6







1 2

     2 2 2 2   1 B61 ( ) d    2 B62 ( ) d   3 B63 ( ) d    4 B64 ( ) d    

B65 ( ) d  2

5



 u7 2 







1 2



B66 ( ) d  2

6





B67 ( ) d  2

7





B68 ( ) d  2

8









9

 2 B69 ( ) d  

     2 2 2 2  B (  ) d    B (  ) d    B (  ) d   4 B74 ( ) d   1 71 2 72 3 73       

 2 2 2 2 2   5 B75 ( ) d   6 B76 ( ) d   7 B77 ( ) d   8 B78 ( ) d   9 B79 ( ) d        



u 2  8







    1  2 2 2 2   1 B81 ( ) d    2 B82 ( ) d   3 B83 ( ) d    4 B84 ( ) d 2    

     2 2 2 2 2  B (  ) d    B (  ) d    B (  ) d    B (  ) d   5 85 6 86 7 87 8 88      9 B89 ( ) d  



u 2  9









    1  2 2 2 2   1 B91 ( ) d    2 B92 ( ) d   3 B93 ( ) d    4 B94 ( ) d 2    

B95 ( ) d  2

5





B96 ( ) d  2

6







B97 ( ) d  2

7







B98 ( ) d  2

8







9



 2  B99 ( ) d   

(4.24) where Bmn  X mn  iYmn ; m, n  1, 2, 3, 4, 5, 6, 7,8, 9 R( )  iI ( )

X11   8 ; Y11  0; X 12  0; Y12  0; X 13  0; Y13  0;

X14  Y14  X15  Y15  X16  Y16  X17  0; Y17  (bE * 7 / K1 ) X18  0; Y18  (bE * 7 / K1 ) ; X19  0; Y19  (bE * 7 / K1 )

X 21  0; Y21  sL* 7 / K2 ; X 22   8 ; Y22  0; X 23  0; Y23  0;

X 24  0; Y24  0; X 25  0; Y25  0; X 26  0; Y26  0;

X 27  0; Y27  (bsL* E* 6 / K1K2 ) ; X 28  (bsL* E * 6 / K1K 2 ); Y28  0;

A Numerical Model of Malaria Fever Transmission …

259

X 29  (bsL* E * 6 / K1K 2 ); Y29  0

X 31  0; Y31  0; X 32  0; Y32  0;

X 33   8 ; Y33  0; X 34  0; Y34  0;

X 35  0; Y35  0; X 36  0; Y36  0; X 37  0; Y37  0; X 38  0; Y38  0; X 39  0; Y39  0;

X 41  0; Y41  0; X 42  0; Y42  0; X 43  0; Y43  0; X 44   8 ; Y44  0; X 45  0; Y45  0; X 46  0; Y46  0; X 47  0; Y47  0; X 48  0; Y48  0; X 49  0; Y49  0;

X 51  0; Y51  0; X 52  0; Y52  0; X 53  0; Y53  0; X 54  0; Y54  0; X 55   8 ;

Y55  0; X 56  0; Y56  0; X 57  0; Y57  0; X 58  0; Y58  0; X 59  0; Y59  0; X 61  0; Y61  0; X 62  0; Y62  0; X 63  0; Y63  0; X 64  0; Y64  0; X 65  0; Y65  0;

X 66   8 ; Y66  0; X 67  0; Y67  0; X 68  0; Y68  0; X 69  0; Y69  0;

X 71  0; Y71  0; X 72  0; Y72  0; X 73  0; Y73  0; X 74  0; Y74  0;

X 75  0; Y75  0; X 76  0; Y76  0; X 77   8 ; Y77  0;

X 78  0; Y78  0; X 79  0; Y79  0; X 81  0; Y81  0; X 82  0; Y82  0; X 83  0; Y83  0; X 84  0; Y84  0; X 85  0; Y85  0;

X 86  0; Y86  0; X 87  0; Y87  0; X 88   8 ; Y88  0; X 89  Y89  0; X 91  0; Y91  0; X 92  0; Y92  0; X 93  0; Y93  0; X 94  0; Y94  0; X 95  0; Y95  0;

X 96  0; Y96  0; X 97  0; Y97  0; X 98  0; Y98  0;

X 99   8 ; Y99  0; Thus (4.24) becomes,  u1 2 

1 2



2





1

 R ( )  I

2



 

 

 

 

 

 

 

 1 X 112  Y112  2 X 12 2  Y12 2  3 X 132  Y132  4 X14 2  Y14 2  5 X15 2  Y15 2  ( )   6 X 16 2  Y16 2  7 X 17 2  Y17 2  8 X 18 2  Y18 2  9 X 19 2  Y19 2 

 d  

260

Kalyan Das and M. N. Srinivas  u2 2 

1 2

 u3 2 

1 2

 u4 2 

1 2

 u5 2 

1 2

 u6 2 

1 2

 u7 2 

1 2

 u8 2 

1 2

 u9 2 

1 2





1

 R ( )  I 2

2

 



1

 R ( )  I 2

2

 

2

 1 X 312  Y312  ( )  

2



    X   X   X   X 2

      X   X   X   X

 1 X 412  Y412  ( )  



 Y32 2

2

32

6



1

 R ( )  I



      X   X   X   X

    Y   X Y   X Y     X Y   X

 d   Y   d  Y    Y   d  Y  

 1 X 212  Y212  2 X 22 2  Y22 2  3 X 232  Y232  4 X 24 2  Y24 2  5 X 25 2  Y25 2  ( )   6 X 26 2  Y26 2  7 X 27 2  Y27 2  8 X 28 2  Y28 2  9 X 29 2  Y29 2 

 Y36 2

2

36

2

6

2 42 2

46

    X   X

 Y42 2  Y46 2



33

7

37

3

7

 Y37 2

2

2 43 2

47

      X   X

 Y332

2

3

2

4

34

8

38

2

4

44

 Y47 2

8

48

2

5

35

9

39

2

38

 Y432

      X   X

2

34

2

2

2

39

2

44

2

2

5

45

9

49

2

48

      X   X

2

35

2

45

2

2

49

 1 X 512  Y512  2 X 52 2  Y52 2  3 X 532  Y532  4 X 54 2  Y54 2  5 X 55 2  Y55 2 1  R2 ( )  I 2 ( )   6 X 56 2  Y56 2  7 X 57 2  Y57 2  8 X 58 2  Y58 2  9 X 59 2  Y59 2   





1

 R ( )  I 2

2

 

1

 R ( )  I 2

2





2

2





1

 R ( )  I 2



2

6

 

2

 Y62 2

2

 Y66

62 66

2

3

7

 

2

 Y632

2

 Y67

63 67

 

2

4

8

 

2

 Y64 2

2

 Y68

64 68

 

2

5

9

 

2

 Y65 2

2

 Y69 2

65 69





 





 

 

 

 

 

 

 

  d 





 

 

 

 

 

 

 

  d 

 1 X 712  Y712   2 X 72 2  Y72 2  3 X 732  Y732  4 X 74 2  Y74 2  5 X 75 2  Y75 2  ( )   6 X 76 2  Y76 2  7 X 77 2  Y77 2  8 X 78 2  Y78 2  9 X 79 2  Y79 2 

1

 R ( )  I

 1 X 612  Y612  ( )  

2

 d     d 

  d 

 1 X 812  Y812  2 X 82 2  Y82 2  3 X 832  Y832  4 X 84 2  Y84 2  5 X 85 2  Y85 2  ( )   6 X 86 2  Y86 2  7 X 87 2  Y87 2  8 X 88 2  Y88 2  9 X 89 2  Y89 2 

 1 X 912  Y912  2 X 92 2  Y92 2  3 X 932  Y932  4 X 94 2  Y94 2  5 X 95 2  Y95 2  ( )   6 X 96 2  Y96 2  7 X 97 2  Y97 2  8 X 98 2  Y98 2  9 X 99 2  Y99 2 

where M ( )  R( )  iI ( ) ; R    0 ; I    

9

If we analyze the influence of noise on any of the populations, and we want to know how the system (2.1)-(2.9) behaves with either, 1  0 or

2  0 or 3  0 or 4  0 or 5  0 or 6  0 or 7  0 or 8  0 or 9  0 then the population variances are: If 1  2  3  4  5  6  7  8  0 then, 

 X  1  Y 2 2  u1  9  1918 d ;  u2  9  2918 d ;  u9 2  9  2 d ; 2   2   2   2

2

 ui 2  0; i  3,4,5,6,7,8. If 1  2  3  4  5  6  7  9  0 then,



A Numerical Model of Malaria Fever Transmission … 

 u1 2 

8  Y182 ; 2 8  X 282 ; 2 8 1 d d  u2   18 d  u8  2   2 2  18 2  

 u 2  0; i  3, 4,5,6,7,9. i

If 1  2  3  4  5  6  8  9  0 , then

 u1 2 

  7  Y17 2 ; 2 7 Y27 2 ; 2 7 1   d   d  d  u7 u2 2   2 2  18 2  18

 u 2  0; i  3,4,5,6,8,9. i

If 1  2  3  4  5  7  8  9  0 then,  u6 2 

6  1 ; 2 d  u  0; i  1, 2,3, 4,5,7,8,9. 2   2 i

If 1  2  3  4  6  7  8  9  0 then,  u4 2 

4  1 ;  2  0; i  1, 2,3,5,6,7,8,9. d ui 2   2

If 1  2  4  5  6  7  8  9  0 then,

 u3 2 

3  1 2 d ;  ui  0; i  1, 2, 4,5,6,7,8,9. 2  2  

If 1  3  4  5  6  7  8  9  0 then,

 u2 2 

2  1 2 d ;  ui  0; i  1,3, 4,5,6,7,8,9. 2   2

If 2  3  4  5  6  7  8  9  0 then,

261

262

Kalyan Das and M. N. Srinivas

 u1 2 

1  1 2 1  Y212 2 ; d    d ;  ui  0; i  3,4,5,6,7,8,9. u 2  18  2   2   2

For modest levels of mean square fluctuations, population variances imply population stability, whereas larger values of population variances suggest population instability.

5. Numerical Simulations Albeit some intriguing insightful consequences of model (2.1-2.9) have been acquired in the earlier section, it turns out to be considerably harder to give the investigation of the model (2.1-2.9) in detail in light of its intricacy. However, the mathematical reenactment can assist us with giving a profundity investigation of the model (2.1-2.9), specifically, the effect of white noise on the framework (2.1-2.9).

Figure 2. Time series evaluation of human and mosquito population with initial values of populations 2400; 1200; 1500; 50; 200; 50; 3000; 100; 500.

A Numerical Model of Malaria Fever Transmission …

263

Figure 3. Time series evaluation of egg and larva population with initial conditions 2400; 1200.

Figure 4. Time series evaluation of human population with initial values 1500; 50; 200; 50.

264

Kalyan Das and M. N. Srinivas

Figure 5. Time series evaluation of mosquito population with initial values 3000; 100; 500.

Figure 6. Time series evaluation of egg, larva and mosquito population with initial values 2400; 1200; 3000; 100; 500.

A Numerical Model of Malaria Fever Transmission …

265

Figure 7. Time series evaluation of egg, larva, Susceptible(H), Exposed(H), Infected(H), Recovered(H), Susceptible(M), Exposed(M), Infected(M) populations with initial values 2400; 1200; 3000; 100; 500.

Figure 8. Time series evaluation of populations with intensities 15, 15, 15, 15, 15, 10, 10, 10, 10 and with initial values 2400; 1200; 1500; 50; 200; 50; 3000; 100; 500.

266

Kalyan Das and M. N. Srinivas

Figure 9. Time series evaluation of populations with intensities 200, 200, 200, 200, 200, 100, 100, 100, 100 and with initial conditions 2400; 1200; 1500; 50; 200; 50; 3000; 100; 500.

Figure 10. Time series evaluation of populations with intensities 2000, 2000, 2000, 2000, 2000, 1000, 1000, 1000, 1000 and with initial values 400; 1200; 1500; 50; 200; 50; 3000; 100; 500.

A Numerical Model of Malaria Fever Transmission …

267

Figure 11. Time series evaluation of populations with intensities 5000, 5000, 5000, 5000, 5000, 3000, 3000, 3000, 3000 and with initial values 2400; 1200; 1500; 50; 200; 50; 3000; 100; 500.

Conclusion We proposed a seasonal stochastic model of malaria transmission in this article, with limitations such as the larva and pupa classes not being separated and the effect of climatic change on mosquito life cycles not being taken into account. In order to quantitatively identify the impact of external hazards on the mathematical function (2.1)-(2.9), stochastic perturbations are applied on the system (2.1)-(2.9) with respect to white noise around its positive equilibrium. From Figures (8-11) it is remarked that with the various intensities of noise up-to certain tolerance limit, stochastic effect has a significant impact on the size of epidemic and the trajectories ultimately converge to stability instead of initial oscillation to the system. The demographic variations demonstrate the populace’s consistency under slight mean - squared deviations, according to the stochastic analysis, but the population variances for bigger values of mean square fluctuations imply population instability. In brief, there is no fundamental difference in dynamics rather that the oscillation for the

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randomly fluctuating environment on the system by additive white noises and it can be able to check the fluctuation of the disease by controlling the strength of the noises. Also from Figure 2, it is provided that the time series evaluation of populations of all compartments. Time series evaluations of populations in compartment wise are provided in Figures 3, 4, 5, 6, 7. Until around the past century, the most widely used computational equations for malaria infection included predictable state and action systems that would include mean condition in an uncertain population density, such as Macdonald’s formulations and subsequent versions incorporating resistance. Such systems do not account for stochastic events, which are especially important when transmitting levels are reduced. With relatively little transmitting, a deterministic system could attain a stable level. As a consequence, we created a computational equations of malaria infection and behavior in this setting that included and excluded large variations. In particular, the stochastic system shows a stable state with moderate noise intensity rates. Weather and climatic disruptions, for example, affect disease systems in ways that have not been constant throughout time rather oscillate. Climate variations and disruptions have a significant bearing in unpredictable behavior, as illustrated in the present study’s simulation results. Due to the extreme unpredictability intrinsic in climate changes, a major portion of climatic variables is randomized, with perhaps the exception of systems influenced by predictable oscillations.

Acknowledgments The support from Department of Basic and Applied Science, NIFTEM and NIFTEM Knowledge Centre is greatly acknowledged.

Conflict of Interest The authors have no conflict so that the publication of the manuscript can be interrupted.

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Ross, R. The prevention of malaria, London, John Murray, 1911. Macdonald, G. The epidemiology and control of malaria, Oxford University press, London, 1957. [3] Chiyaka, C., J. M. Tchuenche, W. Garira, and S. Dube, A mathematical analysis of the effects of control strategies on the transmission dynamics of malaria, Applied Mathematics and Computation, vol. 195, no. 2, pp. 641–662, 2008. [4] Ngwa G. A. and W. S. Shu, A mathematical model for endemic malaria with variable human and mosquito populations, Mathematical and Computer Modelling, vol. 32, no. 7-8, pp. 747– 763, 2000. [5] Beck-Johnson, L. M., W. A. Nelson, K. P. Paaijmans, A. F. Read, M. B. Thomas, and O. N. Bjørnstad, The effect of temperature on Anopheles mosquito population dynamics and the potential for malaria transmission, PLoS ONE, vol. 8, no. 11, Article ID e79276, 2013. [6] Ouedraogo, W., B. Sangar´e, and S. Traor´e, Some mathematical problems arising in biological models: a predator-prey model fishplankton, Journal of Applied Mathematics and Bioinformatics, vol. 5, no. 4, pp. 1–27, 2015. [7] Chukwu, E. N. On the boundedness and stability properties of solutions of some differential equations of the fifth order, Annali di Matematica Pura ed Applicata. Serie Quarta, vol. 106, pp. 245–258, 1975. [8] Sinha, S. Stability result of a sixth order non-linear system, vol. 7, pp. 641–643, 1971. [9] Tunc, C. On the stability and boundedness of solutions in a class of nonlinear differential equations of fourth order with constant delay, Vietnam Journal of Mathematics, vol. 38, no. 4, pp. 453–466, 2010. [10] Tunc, C. New results on the stability and boundedness of nonlinear differential equations of fifth order with multiple deviating

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Kalyan Das and M. N. Srinivas arguments, Bulletin of the Malaysian Mathematical Sciences Society, vol. 36, no. 3, pp. 671–682, 2013. Chitnis, N., J. M. Hyman, and J. M. Cushing, Determining important parameters in the spread of malaria through the sensitivity analysis of a mathematical model, Bulletin of Mathematical Biology, vol. 70, no. 5, pp. 1272–1296, 2008. Wang W. and X. Zhao, Threshold dynamics for compartmental epidemic models in periodic environments, Journal of Dynamics and Differential Equations, vol. 20, no. 3, pp. 699–717, 2008. Wang, J., S. Gao, Y. Luo, and D. Xie, Threshold dynamics of a huanglongbing model with logistic growth in periodic environments, Abstract and Applied Analysis, vol. 2014,Article ID 841367, 10 pages, 2014. Xiao-Qiang, Z., CMS Books in mathematics/Ouvrages de math´ematiques de la SMC, Springer Verlag, New York, NY, USA, 16th edition, 2003. Magal P. and X. Q. Zhao, Global attractors and steady states for uniformly persistent dynamical systems, SIAM Journal on Mathematical Analysis, vol. 37, no. 1, pp. 251–275, 2005. Moulay, D., M. A. Aziz-Alaoui, and M. Cadivel, The chikungunya disease: modeling, vector and transmission global dynamics, Mathematical Biosciences, vol. 229, no. 1, pp. 50–63, 2011. Lou Y. and X. Q. Zhao, A climate-based malaria transmission model with structured vector population, SIAM Journal on Applied Mathematics, vol. 70, no. 6, pp. 2023– 2044, 2010. Das, K., Reddy, K. S., Srinivas, M. N., & Gazi, N. H. (2014). Chaotic dynamics of a three species prey-predator competition model with noise in ecology. Applied Mathematics and Computation, 231, 117-133. Das, K., Srinivas, M. N., & Huda Gazi, N. (2019). Diffusion Dynamics and Impact of Noise on a Discrete-Time Ratio-Dependent Model: An Analytical and Numerical Approach. Mathematical and Computational Applications, 24(4), 103.

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[20] Das, K., Srinivas, M. N., Madhusudanan, V., & Pinelas, S. (2019). Mathematical analysis of a prey-predator system: An adaptive backstepping control and stochastic approach. Mathematical and Computational Applications, 24(1), 22. [21] Das, K., Srinivas, M. N., Srinivas, M. A. S., & Gazi, N. H. (2012). Chaotic dynamics of a three species prey–predator competition model with bionomic harvesting due to delayed environmental noise as external driving force. Comptes Rendus Biologies, 335(8), 503-513. [22] Reddy, K. S., Das, K., Srinivas, M. N., & Kumar, C. P. (2016). Stochastic Nonlinear Aspect of Noise in a Two Predators–One Prey Mathematical Model Induced Cyclic Oscillations. Nonlinear Phenomena in Complex Systems, 19(4), 315-329. [23] Das, K., Srinivas, M. N., Kabir, M. H., & Gani, M. O. (2020). Noiseinduced control of environmental fluctuations in a three-species predator–prey model. Modeling Earth Systems and Environment, 1-21. [24] Srinivas, M. N., Das, K., Srinivas, M. A. S., & Gazi, N. H. (2011). Prey predator fishery model with stage structure in two patchy marine aquatic environment. Applied Mathematics, 2(11), 1405. [25] Traoré, B., Sangaré, B., & Traoré, S. (2017). A mathematical model of malaria transmission with structured vector population and seasonality. Journal of Applied Mathematics, Article ID 6754097, 2017. [26] Carletti M. Numerical simulation of a Campbell-like stochastic delay model for bacteriophage infection. An IMA J Mat Med Biol 2006; 23: 297–310. [27] Nisbet R. M., Gurney W. S. C. Modelling fluctuating populations. New York: John Wiley; 1982.

In: Frontiers in Mathematical Modelling … ISBN: 978-1-68507-430-2 Editors: M. Biswas and M. Kabir © 2022 Nova Science Publishers, Inc.

Chapter 10

MATHEMATICAL MODELLING IN FOOD AND AGRICULTURAL AREAS Maicon Sérgio N. dos Santos1, Carolina E. Demaman Oro2, Rogério M. Dallago2, Giovani L. Zabot1 and Marcus V. Tres1,* 1

Laboratory of Agroindustrial Processes Engineering (LAPE), Federal University of Santa Maria (UFSM), Center DC, Cachoeira do Sul - RS, Brazil 2 Department of Food Engineering, URI Erechim, Fátima, Erechim - RS, Brazil

ABSTRACT Mathematical modeling examines the simulation of real systems to predict their behavior. The principal importance of the use of mathematical modeling is in the development and improvement of equipment and processes. Currently, it is possible to observe the use of mathematical modeling in several industrial processes because it provides a standard or *

Corresponding Author’s Email: [email protected].

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M. Sérgio N. dos Santos, C. E. Demaman Oro et al. mathematical formula to explain the process under analysis. In this chapter, mathematical modeling applied in food and agricultural related areas is discussed, as well as some recent research that achieved promising results are highlighted. Mathematical models based on basic characteristics of food, such as enthalpy, thermal conductivity, specific heat, and thermal diffusivity, among other properties and characteristics, are discussed. Furthermore, the chapter includes heat and mass transfer models in industrial processes of food development and the enzymatic and microbiological inactivation that occurs in food during processing. The mathematical models most used in the food drying process and the mathematical models used in other food-related areas are included and discussed. Also, the mathematical modeling approach has been widely used in agricultural systems, particularly in dynamic processes that involve the development and establishment of plants in the ambient. For many years, the statistical approach has been implemented in agricultural researches. However, the advancement of computational and simulation technology has allowed the rise of mathematical applications in the agricultural and animal sciences. Thus, the exploration of empirical and mechanistic models has added value in the advances of studies that investigate the effects of fundamental elements for plant growth, such as climatic factors, nutrients, and water availability. The chapter includes the topics referring to biochemical reactions in plants, growth performance, processes dynamic, and demand components. Also, different overly complex fields of study that require optimization and visionary and innovative strategies, such as plant diseases, pest control, and animal care, are discussed. Finally, future outlooks are presented to contribute to the countless opportunities and applicability of highly promising techniques and methodologies in food and agricultural areas.

Keywords: crop modeling, food processing, modeling methodology, optimization strategies, simulation background

INTRODUCTION With the intensification of cultivation and strategies that enhance the productivity of crops to sustain the growing global demand, it is extremely necessary to adopt tools that combine the various events and processes that occur in an agricultural system. The technological development of agricultural and food engineering requires the application of methods that consider the maximum components and variables to be adjusted to obtain adequate planning of agricultural production (Djekic et al., 2019; Kocira, 2018). The development of strategic tools, such as the introduction of a

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systematic mathematical approach, is interesting and allows the comprehension of a set of observations that characterize a system and all the variables that act on it (Payvandi et al., 2014). Conceptually, a model is the mathematical representation of the dynamic physical, chemical, and biological events and phenomena in a quantitative approach (Möbius and Laan, 2015). Knowledge of the factors that affect the transfer of heat and mass in food is extremely important. A strategy that has long been used to predict the behavior of the different systems that can be achieved is to describe them using mathematical models. This allows better control of the process and further optimization. However, this is not a trivial task as these systems are mixtures of several components. The study of heat transfer in liquid and non-solid foods depends, in the first instance, on the study of food rheology. In this sense, it is common to use food-simulating fluids to know their physical properties through mathematical models. Foods that are typically heated by conduction are canned meats, vegetables, and highly compressed fruits in their packaging, as well as starchy products in a gelatinized state. Foods heated simply by convection are fruit juices, soups, and dairy products, among others. Those heated by conduction and convection are solid products immersed in brine or syrup and soups containing vegetables or meat. The challenge of mathematical modeling of food products is about products that when heated initially behave as convective, but during heating, they start to have a typically conductive behavior, due to the change in their structure, such as an increase in viscosity. An example of these foods is those that contain starch, which gelatinizes during heating (Gumerato, Schmidt, and Goldoni, 2007). Therefore, the process to be modeled can help to determine the methodology to be used in industrial terms through the effects of the different components and the consequent predictions. This chapter presents the main formulas used for mathematical modeling in the food industry since each case is specific to a given product. The focus is on heat and mass transfer models in industrial food development processes, food drying, and the enzymatic and microbiological inactivation that occurs in food during processing.

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The use of mathematical models introduced to agricultural systems comprises a wide and complex approach. These models promote a quantitative context from intracellular processes that occur in small leaf units, to the study of pedological dynamism and resource flows to the atmosphere (Baghalian et al., 2014; Vereecken et al., 2016; Hornig, 2018; Payvandi et al., 2014). Models that explore agricultural systems and the processes that surround them have been studied since the 1960s. However, the scientific academy search exploring less traditional and reductionist approaches and seek to study a higher number of system components and diverse edaphoclimatic characteristics (Jones et al., 2017). Recent advances in mathematical modeling in agriculture have been directed at the manipulation of mathematical models that consider the implications of climate change in the fragmentation of ecosystems and the impacts of this phenomenon on the success of the system (Zhao et al., 2013). A scenario of wide changes has led to a wide imbalance in areas of agricultural production, mainly due to the occurrence of pests and diseases on a large scale, mortality and reduction in animal reproductive potential, and restricted plant growth and grain quality (Rojas-Downing et al., 2017). Thus, some studies report the importance of the mathematical approach in technological systems to optimize and add value to the harvesting and crop processing procedures (Bulgakov et al., 2018; Singh et al., 2018). Understanding mathematical modeling as an optimization process for agricultural complexes allows important decisions to be made regarding crop management and productivity, such as irrigation scheduling, chemical applications, and integrated management planning, minimizing unnecessary errors and consumption (Mardani Najafabadi et al., 2019; Basir et al., 2019). Thus, the integration of strategic factors that focus on the progress of the investigative process of numerous components that comprise agricultural systems is recommended, mainly from a sustainable and economic point of view (Turner et al., 2016). This chapter provides relevant insight into the main dynamic phenomena that occur in plants and recent studies that address the use of systematic mathematical modeling in agricultural systems. Initially,

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studies that explored the quantitative approach in metabolic and biochemical reactions of plants, such as the photosynthetic process and elements translocation, are presented. Considering that several factors act directly on the plant potential growth, scientific studies that aim to adjust the traditional plant growth models are portrayed. Furthermore, significant mathematical models are pointed out for the application of the determination of evapotranspiration and plant respiration processes and their influence on the environment and equations directed to the transport performance and allocation of essential nutrients to plants and their development. Finally, an overview of trending subjects in the scientific field, such as animal welfare, the potential for pest dissemination, and the incidence of severe diseases is revealed, as a way of providing information that characterizes the different processes under a mathematical interpretation.

MATHEMATICAL MODELING IN THE FOOD AREA Heat and Mass Transfer Models In food processing, heat transfer is always accompanied by mass transfer, which can have important implications for weight loss and product quality (Pham, 2006). Generally, the transfer is directed towards moisture, as in the case of drying and freezing/thawing. The main modes of heat transfer are conduction, convection, and radiation. Mass transfer, on the other hand, is diffusive or convective. Heat conduction/transfer is generally described using Fourier's law and mass transfer can be modeled using Fick's law (Khan et al., 2020). However, the realistic mathematical modeling of mass transfer in a multicomponent mixture is notoriously complex due to the presence of diffusive phenomena that are not described by Fick's law. To overcome this complexity, Maxwell-Stefan equations and generalized Fick's law can represent all possible interactions in multicomponent mixtures and the effects of cross-diffusion are considered (Dal’Toé et al., 2015).

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Heat Transfer Heat transfer can be accomplished through conduction, convection, and radiation. Conduction (Equations 1 and 2) can be defined by Fourier’s equation as the process by which energy is transferred from a hightemperature region to a lower temperature region within a medium (solid, liquid, or gas) or between different media in direct contact. Convection (Equation 3) can be defined as the process by which energy is transferred from hot portions to cold portions of a fluid, or between a surface and a fluid, through the combined action of heat conduction, energy storage, and movement. Newton's equation for convection (Equation 3) is widely used in convection processes, mainly natural and forced, and in some cases in processes related to boiling and condensation. Radiation (Equation 4) can be defined as the process by which heat is transferred from a surface at a high temperature to a surface at a lower temperature when such surfaces are separated in space, even though there is a vacuum between them. The energy thus transferred is called thermal radiation and is made in the form of electromagnetic waves (Incropera et al., 2013). 𝑞̇ = −𝑘𝐴 𝑞̇ =

𝑑𝑇 𝑑𝑥

𝑘.𝐴 ∆𝑇 𝐿

(1) (2)

where 𝑞̇ is the conduction heat flow, 𝑘 is the material thermal conductivity, 𝐴 is the area of the section through which heat flows through conduction, 𝑑𝑇⁄𝑑𝑥 is the temperature gradient in the section, that is, the rate of change of temperature 𝑇 with distance, in the x-direction of the heat flow, and 𝐿 is the thickness of the material. 𝑞̇ = ℎ𝐴∆𝑇

(3)

where ∆𝑇 is the temperature difference between the surface and the fluid, and ℎ is the convection heat transfer coefficient or film coefficient.

Mathematical Modelling in Food and Agricultural Areas 𝑞̇ = 𝜎𝐴1 𝐹12 (𝑇14 − 𝑇24 )

279 (4)

where 𝜎 is the Stefan-Boltzmann constant (4,88 x 10-8 kcal h-1 m² K4), and 𝐹12 is the Form Factor, which depends on the relative geometry of the materials and their emissivity. Based on these equations, the necessary substitutions can be made according to the proposed problem, which may be in simple walls, walls combined in parallel or series, and cylindrical and spherical configurations. Also, heat transfer can be calculated considering conduction, convection, and radiation in a combined system. There is a wide variety of empirical and theoretical equations for estimating the coefficient of heat exchange by convection in which a good scenario analysis is of great importance. Ggross errors in the use of equations to predict the values of heat exchange coefficients can be avoided. As the fluids have low thermal conductivity, a smaller boundary layer represents less resistance to heat transport and the result is a higher coefficient of thermal exchange by convection. In other words, the higher the turbulence of the system, the smaller the boundary layer and, consequently, the higher the heat exchange coefficient. The thermal exchange coefficient is a thermophysical property that has a specific characteristic for each material and depends on the purity, temperature, and physical state in which the material is found. The Nusselt number (𝑵𝒖) is a dimensionless number, which represents the improvement of heat transfer through a fluid layer as a result of convection. It is the ratio between heat exchange by convection and heat exchange by conduction in the fluid's boundary layer. A 𝑵𝒖 number of 𝑵𝒖 = 𝟏 for a fluid layer represents the transfer of heat through the layer by pure conduction as if the fluid was completely at rest. The higher the 𝑵𝒖 number, the more effective the convection is. Physically, the 𝑵𝒖 number represents the ratio between the heat transfer of a fluid by convection (the transfer of the fluid in motion) and conduction (which can be considered an extreme case of convection, that is, the convection of a fluid at rest). Considering a fluid layer of thickness 𝑳, the coefficient of heat exchange by convection (𝒉), and the thermal

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conductivity (𝒌) of the fluid surrounding the material, Equation 5 is presented. Also, the 𝑵𝒖 number is closely related to the Péclet (𝑷𝒆) number and the two numbers are used to describe the proportion of the thermal energy supplied to the fluid and the thermal energy conducted within the fluid (Incropera et al., 2013). 𝑁𝑢 =

𝐡𝐞𝐚𝐭 𝐭𝐫𝐚𝐧𝐬𝐟𝐞𝐫 𝐛𝐲 𝐜𝐨𝐧𝐯𝐞𝐜𝐭𝐢𝐨𝐧 𝐡𝐞𝐚𝐭 𝐭𝐫𝐚𝐧𝐬𝐟𝐞𝐫 𝐛𝐲 𝐜𝐨𝐧𝐝𝐮𝐜𝐭𝐢𝐨𝐧

=

ℎ𝐿 𝑘𝑓𝑙𝑢𝑖𝑑

(5)

The Prandtl (𝑷𝒓) number is a dimensionless number that approximates the ratio of moment diffusivity (kinematic viscosity) and thermal diffusivity of a fluid, expressing the relationship between the diffusion of the amount of movement and the diffusion of the amount of heat within the fluid itself, being a measure of the efficiency of these transfers in the hydrodynamic and thermal limit layers. For liquid metals, the 𝑷𝒓 number is very small, usually in the range of 0.01 to 0.001. This means that thermal diffusivity, which is related to the rate of heat transfer by conduction, dominates unambiguously. This very high thermal diffusivity results from the very high thermal conductivity of metals, which is approximatley100 times higher than that of water. The analog for the mass transfer of the 𝑷𝒓 number is the Schmidt (𝑺𝒄) number. The Rayleigh (𝑹𝒂) number is defined as the product of the Grashof (𝑮𝒓) number, which describes the relationship between buoyancy and viscosity within a fluid, and the Prandtl number, which describes the relationship between dynamic diffusivity and diffusivity thermal. Thus, these numbers (𝑮𝒓, 𝑷𝒓, 𝑹𝒂) are the variables that influence the 𝑵𝒖 number for the calculation of the heat transfer coefficient by natural convection.

Mass Transfer The term mass transfer refers to the process in which the migration of matter from one point to another occurs in continuous space-time. In the case of diffusion mass transfer in the absence of other gradients (such as temperature, pressure, electrical potential, etc.), the molecules of a given

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species, within the same phase, will move, due to the existence of a gradient of concentration. This gradient causes a flow (molar or mass) of the solute in the mixture. In fluid media, another mass transfer mechanism occurs, in which macroscopic movement of part of the fluid occurs, which is a mechanism called convection.

Diffusive Mass Transfer Fick’s Law In a binary system, diffusion is described by Fick's first law (Equation 6), which relates the diffusive flow, ⃗𝐽⃗1 , to the gradient of the molar fraction of

component 𝒊. ⃗𝐽⃗1 = −𝑐𝑡 𝐷12 ∇𝑥1

(6)

where 𝑐𝑡 is the total molar concentration, 𝐷12 is Fick's diffusion coefficient, and 𝛻𝑥1 is the molar composition gradient of component 1. Similarly, the diffusive flow for species 2 is given by Equation 7. ⃗⃗⃗ 𝐽2 = −𝑐𝑡 𝐷21 ∇𝑥2

(7)

Knowing that for a binary case the sum of the flows is equal to zero, ⃗𝐽⃗1 + ⃗⃗⃗ 𝐽2 = 0, and the sum of the molar fractions is equal to 1, 𝑥1 + 𝑥2 = 1, it becomes evident that the binary diffusion coefficients must be the same, 𝐷12 = 𝐷21 = 𝐷. Thus, there is only one coefficient describing the molecular diffusion process, a driving force, and an independent diffusive flow. However, the equations for the diffusive flow of a ternary system depend not only on its gradient of composition but also on the gradient of a second component. In this way, the diffusive flows become non-linear, and the generalized diffusive flow equation for systems with 𝒏 components, the flow of the nth component can be expressed by Equation 8. ⃗⃗⃗ ⃗ 𝐽𝑛 = − ∑𝑛−1 𝑖=1 𝐽𝑖

(8)

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In binary mixtures, the diffusive flow for species i is obtained through Fick's law, which is summarized as the simple product between a diffusion coefficient and the concentration gradient of species 𝒊. However, in multicomponent mixtures the flow of species 𝒊 depends, in addition to its concentration gradient, on the concentration gradient of the other species present in the mixture, resulting in non-linear flow equations represented in the matrix format, where Fick's law is generalized. Maxwell-Stefan Theory The characteristic diffusive effects of multicomponent mixtures, known as cross-diffusion, are responsible for the existence of non-diagonal diffusion coefficients. These effects are called: osmotic diffusion (existence of diffusive flow even in the absence of concentration gradients), diffusion barrier (the diffusive flow is null even if there is a concentration gradient), and reverse diffusion (the diffusive flow is contrary to its gradient concentration). The estimation of multicomponent diffusion coefficients is carried out employing the Maxwell-Stefan theory, which allows the diffusion coefficients that reflect the interaction of pairs of species in a mixture to be obtained. The calculation of these coefficients is complex and involves the determination of a series of unknown parameters, this being done through the use of correlations, which need to be properly chosen based on the nature of the chemical species and the physical conditions of the system. Thus, the diffusive flow in a multicomponent mixture is obtained using the Maxwell-Stefan theory together with the generalized Fick's law (Dal’Toé et al., 2015). Maxwell-Stefan's theory is based on the concepts of the kinetic theory of gases and takes into account the contributions of the conservation of movement, resulting from the collision between molecules, and the forces acting on the molecules. For an ideal gas system, these contributions can be represented as described in Equation 9.

∇𝑝1 = −𝑓12 𝑥1 𝑥2 (𝑢 ⃗⃗⃗⃗1 − 𝑢 ⃗⃗⃗⃗2 )

(9)

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where 𝜵𝒑𝟏 is the partial pressure of the system when the molecules of species 1 try to move between the molecules of species 2 at a relative speed of 𝑢 ⃗⃗⃗⃗1 − 𝑢 ⃗⃗⃗⃗2 , 𝑥1 is the molar fraction of species 1, and 𝑓12 can be defined as an inverse drag coefficient, given by 𝒟12 = 𝑃⁄𝑓12 . By allocating this term in Equation 10, the Maxwell-Stefan equation is obtained for diffusion in an ideal gas binary system, at constant temperature and pressure. ⃗⃗⃗⃗⃗ −𝑢 ⃗⃗⃗⃗⃗ ) 1 𝑥 𝑥 (𝑢 ⃗⃗⃗⃗ 𝑑1 = (𝑃) ∇𝑝1 = − 1 2𝒟 1 2 12

(10)

where ⃗⃗⃗⃗ 𝑑1 is the driving force for diffusion and 𝒟12 is called the MaxwellStefan diffusion coefficient for 1-2. Considering the constant pressure along the diffusion path and the ideal gas phase, the driving force of the equation can be written in terms of the composition of the system. Thus, there is a simplified form of the Maxwell-Stefan equation (Equation 11). ∇𝑥𝑖 = −

⃗⃗⃗⃗𝑖 −𝑢 ⃗⃗⃗⃗𝑗 ) 𝑥𝑖 𝑥𝑗 (𝑢 𝒟𝑖𝑗

(11)

Through this equation, it is noted that 𝐷12 = 𝐷21, due to the restriction 𝛻𝑥1 + 𝛻𝑥2 = 0. Maxwell-Stefan's theory, when applied to binary systems, results in quite simple equations. However, its complexity increases as the number of components under analysis increases. Generalizing the previous equation for a system with 𝒏 ≥ 𝟑 components and writing it as a function of the diffusive flow, we have in the same way as in the generalized Fick's law, only 𝒏 − 𝟏 independent flows, and with this restriction, it is possible to write the following Equation 12, in which 𝐵𝑖𝑗 are coefficients that depend on the composition of the system and the diffusion coefficients of Maxwell-Stefan, 𝒟𝑖𝑗 . ⃗⃗ 𝑐𝑡 ⃗⃗⃗ 𝑑𝑖 = −𝐵𝑖𝑖 ⃗𝐽𝑖 − ∑𝑛−1 𝑗=1 𝐵𝑖𝑗 𝐽𝑗 𝑗≠𝑖

(12)

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The introduction of non-ideality modifies the term 𝑑𝑖 which, instead of being caused by the pressure acting on the molecules, is now written in the function of the chemical potential of the species, the true driving force for mass diffusion. Since, in practical terms, the chemical potential is not an easily manipulated quantity, it is convenient to convert it to a measurable basis. In this case, the composition of the system.

Effective Diffusivity The complexity in the use of the Maxwell-Stefan theory and the generalized Fick's law led to the investigation of more simplified ways to calculate flows in a multicomponent mixture, such as the use of effective diffusivities. The flow, in this case, similar to Fick's first law, depends only on the concentration gradient of the species itself, being written for a generic species, as described in Equation 13. ⃗𝐽𝑖 = −𝑐𝑡 𝒟𝑖𝑒𝑓 ∇𝑥𝑖 𝑖 = 1,2, … , 𝑛

(13) 𝑒𝑓

where 𝑐𝑡 is the molar concentration of the mixture and 𝒟𝑖 the effective diffusion coefficient of species i that represents the diffusion of i in the mixture. The effective diffusion coefficient brings together in a single coefficient the interactions between species 𝒊 and all other components of the mixture. Depending on the desired application, the formula can be rewritten and adjusted. Khan et al. (2020) modeled heat and mass transfer using Fourier's law (14) and Fick's law (15), respectively, considering the spatial distribution of air velocity during intermittent microwave convective drying with a computational Fluid Dynamics (CFD) model. 𝜌𝐶𝑝

𝜕𝑇 𝜕𝑡

𝜕𝑐 𝜕𝑡

= ∇(𝑘𝑒𝑓𝑓 ∇𝑇) + 𝑄𝑚

+ ∇(−𝐷𝑒𝑓𝑓 ∇𝑐) = 0

(14) (15)

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where 𝑇 is the temperature at time 𝒕, 𝝆 is the density, 𝐶𝑝 is the specific heat, 𝑘𝑒𝑓𝑓 is the thermal conductivity, 𝑄𝑚 is the volumetric heat source, 𝑐 is the instantaneous moisture concentration, 𝑡 is time, and 𝐷𝑒𝑓𝑓 is the effective moisture diffusivity (Khan et al., 2020).

Convective Mass Transfer Natural convection generally occurs in concentrated solutions, when the flow of matter generated by the difference in concentration causes movement in the fluid that increases the transport speed of the solute. In cases where the effect of the medium speed on the distribution of solute concentration is caused by some external agent, forced convection occurs. In this way, Fick's first law can be extended to the case where the convective contribution is present. In the case of a binary mixture, Equation 16 is used (Incropera et al., 2013). 𝑛𝐴 = −𝐷𝐴𝐵 ∇𝜌𝐴 + 𝑤𝐴 (𝑛𝐴 + 𝑛𝐵 )

(16)

where 𝐷𝐴𝐵 is the diffusion coefficient of component A in medium B, 𝜌𝐴 is the mass concentration of component A in the mixture, 𝑛𝐴 is the total mass flow of component A, 𝑛𝐵 is the total mass flow of component B, and 𝑤𝐴 is the mass fraction of component A in the mixture. The second mathematical model that can be used involves the convective mass transfer coefficient 𝑘𝑚 . It is used mainly for fluids in movement close to a surface or when two relatively immiscible fluids come into contact with each other (Equation 17). 𝑛𝐴 = 𝑘𝑚 (𝜌𝐴𝑆 − 𝜌𝐴 )

(17)

where 𝜌𝐴𝑆 is the equilibrium concentration of component 𝑨 in the medium at a given temperature and pressure, and 𝜌𝐴 represents the mass concentration of the solute at some point in the fluid phase.

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Microbial and Enzymatic Inactivation Several mathematical models are available in the literature to represent the complex phenomenon of microbial and enzymatic inactivation in relation to time and isothermal conditions, which include dissociation, aggregation, denaturation, chemical decomposition, and coagulation (BustamanteVargas et al., 2019). The mathematical models predict the changes that can occur in food during processing and storage to prevent losses in these stages. The main enzymes that are the target of inactivation in the food industry are polyphenol oxidases, peroxidases, pectinases, lipases, lipoxygenases, catalases, and dehydrogenases. Polyphenol oxidases (PPOs) have a metallic active site with one (or more) coordinated copper, which binds and cleaves the molecular oxygen of the substrate. This indicates enzymatic activity and results in oxidation (Raymundo-Pereira et al., 2020). The enzymatic oxidation results in dark pigments in fruits and vegetables, which are most often not desirable (Villamil-Galindo, Van de Velde, and Piagentini, 2020). However, PPOs have a high thermal resistance (Ke et al., 2021) and their deactivation kinetics in different foods should be studied. Peroxidase is an oxidase that also causes the browning of fruits and vegetables (Li and Tang, 2020). Pectinases are a group of enzymes that degrade pectin based on preference for the substrate, such as pectate or pectic acid (Gaio et al., 2017). These enzymes are generally used to clarify juices and wines, improving filtration, and decreasing viscosity (Gaio et al., 2017; Bustamante-Vargas et al., 2019). However, commercial pectinases can negatively affect product quality because of impurities and secondary enzyme activities (Merín and de Ambrosini, 2015). Lipases and lipoxygenases are endogenous enzymes that are involved in the degradation of nutrients in food products (Xu et al., 2016), leading mainly to rancidity. Lipases catalyze several reactions, such as hydrolysis, aminolysis, alcoholysis, esterification, and interesterification (Ficanha et al., 2021). Lipoxygenases (LOXs) contain non-heme iron in their structure and LOXs action results in the oxidation of essential fatty acids, and consequently, reactive free radical molecules (Janve et al., 2014).

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Catalases (CATs) are metalloenzymes type I (the heme-containing) or type II (the non-heme), also called “pseudocatalases” (Grigoras, 2017). Catalases can catalyze peroxidatic reactions, like peroxidases. Also, dehydrogenases are a type of oxidoreductase enzyme that catalyze oxidation-reduction reactions (Zhang et al., 2019). Therefore, the inactivation of these enzymes is considered a critical processing step for food preservation. Among unit operations, bleaching and pasteurization are the most common for enzyme inactivation. However, depending on the type of product and the time required to perform inactivation, complete inactivation can lead to overheating and an unwanted uncharacteristic flavor. Equations 18-23 show the kinetic models generally applied to predict residual enzymatic activity when some type of denaturation occurs, in which 𝑨𝒐 is the initial enzymatic activity, 𝑨 is the enzymatic activity in time or cycle 𝒏, 𝒌 is the inactivation rate constant at a given pressure and temperature conditions, and 𝒕 is the time taken for the experiment. The models are first-order (Equation 18), distinct isozymes (Equation 19), twofraction (Equation 20), multicomponent first-order (Equation 21), fractional conversion (Equation 22), and Weibull distribution (Equation 23) (Gomes, Sarkis, and Marczak, 2018). The kinetic model of Equation 2 takes into account the existence of several isoenzymes, one labile (𝑳) and the other stable (𝑺) or resistant (𝑹), which are grouped into two fractions (Benito-Román et al., 2020). The fraction of the activity of the thermolabile isoenzyme group in relation to the total activity of the enzyme is represented by the coefficient 𝒂. The Weibull distribution pattern (Equation 23) is based on the assumption that the momentary rate of thermal sensitivity to heat is only a factor in the intensity of transient heating and residual activity, but not the rate at which the residual activity was achieved under the conditions examined, where 𝒃 is the scale and 𝒏 is the shape factors (Shalini, Shivhare, and Basu, 2008). As some models are essentially the same, it is necessary to consider statistical and physical criteria to choose the model that best represents the data obtained. 𝐴⁄𝐴0 = 𝑒𝑥𝑝−𝑘𝑡

(18)

288

M. Sérgio N. dos Santos, C. E. Demaman Oro et al. 𝐴⁄𝐴0 = 𝐴𝐿 . 𝑒𝑥𝑝−𝑘𝐿 𝑡 + 𝐴𝑅 . 𝑒𝑥𝑝−𝑘𝑅𝑡

(19)

𝐴⁄𝐴0 = 𝑎. 𝑒𝑥𝑝−𝑘𝐿 𝑡 + (1 − 𝑎)𝑒𝑥𝑝−𝑘𝑅𝑡

(20)

𝐴⁄𝐴0 = [𝑒𝑥𝑝−𝑘1 𝑡 + 𝑟. 𝑒𝑥𝑝−𝑘2 𝑡 ]/(1 + 𝑟)

(21)

𝐴⁄𝐴0 = 𝐴𝑅 + (𝐴0 − 𝐴𝑅 )𝑒𝑥𝑝−𝑘𝑡

(22)

𝑠 = 𝑠0 . 𝑒𝑥𝑝−𝑏.𝑡

𝑛

(23)

Shalini et al., (2008) demonstrated that the Weibull distribution model was the most efficient to describe the parameters of peroxidase deactivation in mint leaves and is appropriate for predicting the behavior of the enzyme at the temperatures evaluated (70-100 °C). Gomes et al., (2018) also evaluated the Weibull distribution model and reported that the times required to inactivate 90% of Tetsukabuto pumpkin peroxidase were 1.5 and 2.6 min for ohmic and conventional bleaching, respectively. However, the same model does not describe the behavior of different enzymes, as reported by Siguemoto et al. (2018). The authors demonstrated that good results were found for the enzymes polyphenol oxidase and peroxidase, while the pectin methylesterase data was underestimated by the model for the treatment of cloudy apple juice processed in a continuous flow microwave-assisted unit. The main microorganisms that are the target of inactivation in the food industry are Escherichia coli, Listeria, Salmonella, Staphylococcus, Bacillus, Clostridium, Yersinia, Aeromonas, Cladosporium, Saccharomyces cerevisiae, Zygosaccharomyces bailii, Neosartorya fischeri, Lactobacillus, among others (Souza and Koutchma, 2021). For microorganisms, the results are usually expressed as log reduction (Equation 24), where 𝑵 is the number of viable cells after treatment and 𝑵𝟎 is the number of viable cells before treatment (Fauzi, Farid, and Silva, 2017). 𝑁

𝐿𝑜𝑔𝑟𝑒𝑑𝑢𝑐𝑡𝑖𝑜𝑛 = 𝐿𝑜𝑔10 (𝑁 ) 0

(24)

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First-order kinetics is the model used to describe both enzymatic and microbial inactivation, where chemical changes are proportional to the concentration and described by an inactivation rate (𝒌) (Equation 25), which results in the Napierian logarithm (Equation 26). When converted to decimal logarithms, they can report the number of decimal reductions (Equation 27) (Serment-Moreno et al., 2014). −

𝑑𝐶 𝑑𝑡

= 𝑘𝐶

(25)

𝑙𝑛 𝐶 = −𝑘𝑡

(26)

𝐶

0

𝑁

𝑘

𝐿𝑜𝑔10 𝑁 = − 2.303 𝑡 0

(27)

Based on these equations, several models of microbial inactivation have been proposed in the literature (Weibull Biphasic, Log-Logistic, Bigelow Model, and Eyring–Arrhenius Models, among others) and their application depends on the conditions of pressure, temperature, equipment, sample, and industrial processing conditions.

Temperature Profiles and Thermal Conductivity Coefficient Thermal conductivity is a physical property of materials of high importance when evaluating heat transfer by conduction. Thermal conductivity quantifies the ability of materials to conduct heat. Materials with high thermal conductivity conduct heat more quickly than materials with low thermal conductivity. Fourier's law is the mathematical basis in the description of thermal heat conduction, as it highlights an important property of materials: thermal conductivity (𝒌). The k factor is a property of the material and indicates the amount of heat that will flow through a unitary area if the temperature gradient is unitary. The thermal conductivity in the 𝒚 and 𝒛 directions has similar definitions (𝒌𝒚 and 𝒌𝒛 ). If the material is isotropic, the thermal conductivity does not depend on the direction of the heat transfer and it can be written: 𝒌𝒙 = 𝒌𝒚 = 𝒌𝒛 = 𝒌.

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In addition to thermal conductivity, other physical properties have a high influence on heat transfer by conduction: thermal diffusivity (𝛼) and specific heat (𝐶𝑝 ), and heat capacity (𝐶). According to Incropera (2013), thermal diffusivity measures the ability of a certain material to conduct thermal energy with its ability to store it. Specific heat, by the contrary, defines the thermal variation of a given material when receiving a certain amount of heat. In mathematical terms, the heat capacity is defined as the product of the mass by the specific heat (Incropera et al., 2013). The thermal conductivity allows the calculation of the temperature profile in the layers of the material of interest. The study of the temperature profile is important to adjust industrial parameters, for example, to describe the cooking process of certain food and the time needed to reach a given temperature in the center of it. Foods are complex matrices because several components and transformations are present, and knowledge of the temperature profile in each layer becomes essential.

Food Drying Drying is a thermodynamic process, through which the moisture content is reduced, from the inside of the material to the surface through the supply of heat. Food drying has been gaining popularity due to sensory acceptance, and for presenting ready-to-eat foods with significant nutritional value (Khan et al., 2020). The mathematical model to be built for the drying process depends on the level of detail desired. Therefore, the drying process can be approached according to its simplicity or complexity. We can take into consideration (i) only the transfer of heat and mass between a surface and a fluid without detailing the temperature and mass profile in a fluid, (ii) a detailed profile of the temperature and mass in a fluid from complex calculations, or (iii) a detailed mass and temperature profile for the fluid and its surroundings. Each type of dryer, such as an industrial tray, roller or tunnel dryers, atomizers, freeze dryers, and dryers that employ high-vacuum, ultratemperature, extrusion, fluidized bed, microwave or radiofrequency methods, among others, are used to achieve different process needs, which

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are generally inextricably linked to the product. The drying process has many variables, such as initial product moisture, final product moisture, mass diffusion coefficient, the relative exchange area, specific heat, volume variation (shrinkage), air temperature, airflow, equilibrium humidity, material thickness, the vapor pressure of drying air, temperature and speed of water diffusion in the product, among others, that need to be considered. In response to the drying process, the drying curve shows the variation of the initial and final content (equilibrium) of the material's moisture as a function of time, which depends on the relationship of water to the solid structure and the transport of water inside the material to its surface. This factor cannot be generalized for all foods and its response depends on many variables. Knowledge of the relationships between temperature, relative moisture, and the water content of biological products is essential for the correct performance of the product's drying, storage, and conservation process. The equilibrium water content, also called hygroscopic balance, is the limited water content that a product reaches when exposed to a controlled environment (temperature and relative humidity condition) for a sufficiently long time. There are several semiteoric and empirical models that describe the hygroscopic balance of foods and some of the most used ones are shown in Table 1. Mathematical models can also be used in sublimation drying techniques, such as freeze-drying. Nakagawa and Ochiai (2015) developed a model to predict the sublimation progresses multi-dimensionally in freeze-dried peeled apple cubes. The authors observed that the heat transfer coefficients remained constant under the different heating and condenser temperatures, as well as in evacuation rates. However, the results suggested that the mass transfer coefficient values were mostly influenced by the evacuation rate (Nakagawa and Ochiai, 2015).

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M. Sérgio N. dos Santos, C. E. Demaman Oro et al. Table 1. Semiteoric and empirical models used to describe the hygroscopic balance of food matrices

Model

Equation

𝑃𝑣 2𝜎𝑣𝑐𝑜𝑠𝛼 𝑙𝑛 ( ) = − 𝑃𝑣𝑠 𝑟𝑅𝑇𝑎𝑏𝑠

Kelvin

𝑋𝑒 𝐶𝐿 𝑎𝑤 = 𝑋𝑚 1 + 𝐶𝐿 𝑎𝑤

Langmuir

𝑎𝑤 (1 − 𝑎𝑤 )𝑋 1 𝑎𝑤 (𝐶𝐵𝐸𝑇 − 1) = + 𝑋𝑚 𝐶𝐵𝐸𝑇 𝑋𝑚 𝐶𝐵𝐸𝑇

Brunauer, Emmett e Teller (BET) Guggenheim, Anderson e de Bôer (GAB)

𝑋𝑒 =

𝐺𝑘𝑋𝑚 𝑎𝑤 (1 − 𝑘𝑎𝑤 )(1 − 𝑘𝑎𝑤 + 𝐺𝑘𝑎𝑤 )

Henderson

1 − 𝑈𝑅 = exp[−𝑎𝑇𝑎𝑏𝑠 𝑋𝑒𝑐 ]

HendersonThompson

1 − 𝑈𝑅 = exp[−𝑎(𝑇 + 𝑏)𝑋𝑒𝑐 ]

Chung e Pfost

𝑋𝑒 = 𝑎 − 𝑏 ln[−(𝑇 + 𝑐) ln(𝑈𝑅)]

Cavalcanti Mata Henderson modified by Cavalcanti Mata Sabbah

𝑋𝑒 =

ln(1 − 𝑈𝑅) − 𝑐 𝑎 𝑇𝑏

1 − 𝑈𝑅 = exp[−𝑎(𝑇 𝑏 )𝑋𝑒𝑐 ]

𝑋𝑒 = 𝑎

𝑏 𝑎𝑤 𝑇𝑐

Variables 𝑃𝑣 : product water vapor pressure 𝑃𝑣𝑠 : vapor pressure at saturation temperature 𝜎: surface tension of water 𝑣: volume 𝑟: radius of the cylindrical capillary 𝑅: Universal gas constant 𝑇𝑎𝑏𝑠 : absolute temperature 𝑋𝑒 : equilibrium water content of the product, decimal dry basis 𝑋𝑚 : water content in the monolayer, % dry basis 𝐶𝐿 : Langmuir constant 𝑎𝑤 : water activity, dimensionless 𝐶𝐵𝐸𝑇 : BET constant 𝐺: Guggenhein constant 𝑘: constant related to the energy of interaction between the multiple layers of water 𝑎 𝑒 𝑐: parameters that depend on the product 𝑈𝑅: relative moisture 𝑏: parameter that depends on the product

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Other Food-Related Areas Mathematical modeling in the food area can be used in the entire production chain, from the raw material to the storage of the final product. Some variables can be assumed based on fluids and mixtures that simulate the viscosity and other properties of the food under study. In addition to the approaches presented in the present study, some authors report interesting results in the literature using mathematical modeling in foodrelated areas. For example, Garre et al. (2019) proposed a mathematical model to describe the distribution of food products within a particular trade network, as a response to food incidents, to evaluate the risks in different geographical locations (Garre et al., 2019). Another mathematical modeling work that deserves to be highlighted is in the modeling of fermentation products. For example, the process of ethanol production during kefir fermentation from goat’s milk (Wawrzyniak et al., 2019). Understanding the process and kinetic parameters involved in fermentation allows reproduction on an industrial scale while maintaining quality control. Other mathematical models described in the literature refer to sodium chloride penetration profile during cheese brining (Van Wey et al., 2014), 3D model of chicken meat roasting (Rabeler and Feyissa, 2019), the release of food active compounds from viscoelastic matrices (Oroná et al., 2021), and cooking with alcoholic beverages (Snitkjær and Risbo, 2019), among others. Through the information exposed in the current chapter, infinite possibilities of study within the area of foods are remarkable. That is why it is important to measure all parameters to achieve realistic mathematical modeling of the process under study.

Mathematical Modeling in the Agricultural Area Biochemical Reactions Plant metabolism encompasses diverse biochemical reactions involving a portion of substrates and products, resulting in a metabolic network of

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cellular production (Küken and Nikoloski, 2019). The synthesis of plant natural products (PNPs) from enzymatic metabolism and production of secondary metabolites has been a major advance in the scientific field and involves a complex chain of high interest for obtaining desired compounds (Cravens et al., 2019). The direction of studies that involve a high diversity of components that are essential for the effectiveness of biochemical and cellular level events is relevant to agricultural systems (Wang et al., 2019). Thus, a detailed perception of reactions that occur in plants and are essential for plant growth and reaching high productivity is necessary. The performance of highly efficient mathematical models in the simulation of complex biochemical processes, such as photosynthesis and translocation of photoassimilates and essential elements, provides the basis for understanding the cellular changes of plants and culminates in the foundation of the energy and photosynthetic structure of plants (Nägele and Weckwerth, 2012). Numerous processes encompass the cellular nature of plants and act directly for the activation of fundamental biochemical reactions, synthesis of compounds, and transfer of essential elements to plant functional units (Figure 1). Recently, a study simulated the transport of sucrose and the growth of rice grains in a complex transport network of phloem in a source-drain relation, through the use of mechanistic mathematical modeling (Seki et al., 2015). Considering that grain filling is directly influenced by the transport and translocation capacity of nutrients and the source-drain relation, the application of models of dynamics of sucrose transport and grain growth (Equation 28) and grain yield (Equation 29) provided an accurate analysis regarding the characterization of grains and panicles under different conditions under intracellular and extracellular variables: 𝑑𝑠0 (𝑡) 𝑑𝑡

1

𝑡

= 𝑣 {𝑎(𝛾0 − ℎ0 ) [1 − 𝑏] + ∑𝑖𝜖𝑫(𝑠𝑖 (𝑡)[−J𝑖 (𝑡)]) − 0

𝑠0 (𝑡) ∑𝑖𝜖𝑫[J𝑖 (𝑡)]}

(28)

Figure 1. Characterization of the main biochemical and metabolic reactions and activities that occur in a plant organism.

296 where

M. Sérgio N. dos Santos, C. E. Demaman Oro et al. 𝑑𝑠0 (𝑡) 𝑑𝑡

is the timewise increment of sucrose solute due to sucrose

loading in the collection phloem; 𝒂 is the proportion of sucrose loaded into the collection phloem; 𝛾0 is the initial value for the mean rate of sucrose assimilation; ℎ0 is the initial value for the respiration rate; 𝑫 is the neighboring nodes of node 0; 𝐽𝑖 (𝑡) is the solution flow between the leaf and one of its neighboring nodes; and 𝑣0 is the volume of node 0. 𝑌(𝑡𝐻 ) = ∑𝑘𝜖 (1,…,𝑀)𝑠.𝑡.𝑤𝑘 (𝑡𝐻 )>𝑊∗

𝑤𝑘 (𝑡𝐻) (0 𝑊

≤ Y(𝑡𝐻 ) ≤ M) (29)

where 𝑴 is the number of grains; 𝒕𝑯 is the harvesting time; and 𝒘∗ is the minimum grain size above which market value is guaranteed. Also, a study was conducted to evaluate the photosynthetic potential of Saccharum officinarum at different levels of silicon (Si), since this element is essential for biochemical and physiological reactions and acts directly on the mechanism to combat water stress in plant species (Verma et al., 2020). A model (Equation 30) of cumulative photosynthetic responses was proposed and it was observed that Si dynamics in species is an efficient approach to understand the mechanisms of tolerance of Saccharum officinarum to water deficit: 𝑡

𝑃𝑡 = ∫0 𝑒

𝜂 𝑡

[𝜇 ln 𝑡− ]+𝜉𝑑𝑡

(30)

where 𝑃𝑡 is the cumulative photosynthetic response from time 𝑡 = 0 to time 𝒕 = 𝒕; 𝒕 is the time; and 𝜇, 𝜂, 𝜉 are individual order constants.

Growth Performance The use of mathematical models that provide a quantitative comprehension of plant growth and development is one of the main techniques for predicting and validating observations made in field experiments. The application of accurate models reflects the influence of essential components in the system, as well as the level of relationship between

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these elements, facilitating the prediction of morphophysiological events involving plants and ways of controlling these phenomena (Jones et al., 2017). Several factors directly affect the plant growth performance, which corroborates the necessity for a dynamic analysis from the point of view of the agroecosystems (Figure 2).

Figure 2. Representation of components that act on plant growth dynamics and their interactions with the environment.

Considering the use of models, the current literature uses an approach of optimization of classic descriptive equations, above all that result in defined sigmoidal curves, to provide information about plant behavior over a certain period (Liu et al., 2018). This improvement process is conducted through the constant addition of variables that can affect the evolution of plant growth. In general, the behavior of the plant growth trajectory is expressed by a sigmoidal curve, expressed by the direct relationship between plant cell activity and time (Figure 3). However, the approach of symmetry of the curve performance results in a lack of accuracy in plant growth behavior.

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Figure 3. Representation of the standard growth curve and the main phases related to plant growth.

Thus, studies whose objective is to adjust the growth curve for different conditions, through the application of asymmetric growth models, have been reported in the literature. A study was conducted for the development of new sigmoidal growth (NSG) model assuming an asymmetric growth curve and for comparison with different classic sigmoidal models (logistic (Equation 31), Gompertz (Equation 32), Richards (Equation 33), and ontogenetic growth equations (Equation 34) (Cao et al., 2019). The results showed that the model (Equation 35) was efficient to evaluate the growth evolution of plants and animals and presented an R2 of 0.98 when compared to classic models: 𝑤

𝑚𝑎𝑥 𝑤 = 1+𝑒 −𝑘(𝑡−𝑡 𝑚)

(31)

where 𝒘 is the biomass at time 𝒕; 𝒘𝒎𝒂𝒙 is the biomass asymptotic maximum value; 𝒌 is the rate of growth; and 𝒕𝒎 is the inflection point when the growth rate reaches the peak, and the biomass reaches half of its asymptotic maximum value. 𝑤=𝑤

𝑚𝑎𝑥 𝑒

−𝑒−𝑘(𝑡−𝑡𝑚 )

(32)

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where 𝒕𝒎 is the time when the growth rate reaches the peak and the biomass reaches 𝒘𝒎𝒂𝒙⁄𝒆. 𝑤=

𝑤𝑚𝑎𝑥

(33)

1⁄ 1+𝑣𝑒 −𝑘(𝑡−𝑡𝑚) 𝑣

where 𝒗 is the model constant; 𝒕𝒎 is the time when the growth rate reaches the peak and the biomass reaches; and 𝒘𝒎𝒂𝒙 is the biomass asymptotic maximum value. 1 4 𝑤 = 𝑤𝑚𝑎𝑥 {[(1 − 𝑤0 /𝑤𝑚𝑎𝑥 )1/4 ]. 𝑒𝑥𝑝 [(−𝑎𝑡/4𝑤𝑚𝑎𝑥 )]}

4

(34)

where 𝒂 is a constant; and 𝒘𝟎 is the initial biomass at 𝒕 = 𝟎.

𝑤 = 𝑤𝑚𝑎𝑥 − 1 𝑎𝑏(𝑛−𝑡)1+ ⁄𝑏 {𝑡(𝑡−𝑚)+𝑏{[𝑡(2𝑛+3𝑡)−𝑚(𝑛+4𝑡)]+𝑏2 [−3𝑚(𝑛+𝑡)+2(𝑛2 +𝑛𝑡+𝑡 2 ]}}

(1+𝑏)(1+2𝑏)(1+3𝑏)

(35) where 𝑎, 𝑏, 𝑚, 𝑛, and 𝑤𝑚𝑎𝑥 are model parameters, in which 𝒕 ≤ 𝒎, 𝒕 = 𝒎; 𝒕 ≥ 𝒏, 𝒕 = 𝒏. The representation of a new sigmoidal (NSG) model that accurately considers the trajectory of the plant growth cycle, estimating the maximum biomass values in the harvesting process to enhance the reliability of yield data was developed (Liu et al., 2018). Furthermore, a comparison was affected with the beta sigmoidal growth (BSG) model, as a way to contrast the growth trajectory of different agricultural species (sweet sorghum, mung beans, sunflower, peanut, cotton, garden pea, black soybean, and Adzuki bean). The results showed a high R2 (up to 0.98) for both models. However, NSG presented more accurate estimates when compared to BSG, considering specific criteria adopted in the study. In this way, the presented models that estimate the maximum biomass (Eq. 9), BSG model (Equation 36), and NSG model (Equation 37) appear as important tools for

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monitoring the phenological cycle of agricultural species and allows the adoption of strategic crop management decision making.

𝑑𝑤 𝑑𝑡

=

𝑡𝑚 −𝑡𝑏 𝛿 𝑡𝑒 −𝑡 𝑡−𝑡𝑏 𝑡𝑒−𝑡𝑏 𝑐𝑚 [(𝑡 −𝑡 ) (𝑡 −𝑡 ) ] 𝑒 𝑚 𝑚 𝑏

(36)

where 𝒘 is biomass at time 𝒕; 𝜹 is a constant; 𝒄𝒎 is the maximum growth rate, reached at time 𝒕𝒎 ; 𝒕𝒃 is the time of starting growth, and 𝒕𝒆 is the time of ending growth. 𝑡𝑚

𝑡𝑒 −𝑡 𝑡 𝑡 −𝑡 ) (𝑡 ) 𝑒 𝑚 𝑒 −𝑡𝑚 𝑚

𝑤 = 𝑤𝑚𝑎𝑥 (1 + 𝑡

(0 ≤ 𝑡𝑚 < 𝑡𝑒 )

(37)

where 𝒕𝒆 is the time when biomass reaches the value of 𝒘𝒎𝒂𝒙 . 1

𝑤 = 3 𝑔(𝑡) +

𝑐 2 (𝜑(𝑡)+𝜃(𝑡) 𝜂(𝑡)−16 𝜂(𝑡)) 4𝑘 3

(38)

where 𝒄, 𝒌 are independent constants.

Plants Processes Dynamic The dynamic processes involving the functioning of the metabolic system of agricultural species, such as respiration, and transpiration, and evapotranspiration (leaf transpiration + soil evaporation) introduce the comprehension of mass and energy exchanges between plants and the environment (Silva and Lambers, 2020; Zhang et al., 2020). However, the literature reports an important gap regarding the processes that involve breathing and evapotranspiration and the complexity of its components in an ecosystem (Dusza et al., 2020; Hashem et al., 2020). Regarding the evapotranspiration process, this component is characterized as a primordial dynamic and vital mechanism for plants, referring to the association of two distinct procedural components: plant transpiration, pertinent to the suppression of water from the plant towards

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the atmosphere; and soil evaporation, related to the removal of the same resource, from the soil surface to the atmospheric layer (Allen et al., 2011). The availability of tools that provide precision and speed in the processes of obtaining data is extremely important, especially due to the reliability required in the information obtained. Currently, the development of models referring to ETo calculations is seen as a major factor, mainly due to the accuracy and applicability of the data in irrigation programming and management strategies. However, due to a range of existing methods for developing ETo estimates, the choice of the most appropriate resource depends, above all, on the number of necessary meteorological parameters and the availability of these parameters (Alencar et al., 2000). Accuracy and precision in evapotranspiration estimates are fundamental components for adequate water management in agricultural systems and subsequent assessment of the soil water balance (Salama et al., 2015). However, the difficulty of an accurate determination of evapotranspiration becomes an important factor to be considered. Therefore, over the years, many approaches have been developed to estimate reference evapotranspiration, such as Penman-Monteith (Allen et al., 1998) (Equation 39), Makkink (Makkink, 1957) (Equation 40), Turc (Turc, 1961) (Equation 41), Penman (Penman, 1948) (Equation 42), Priestley and Taylor (Priestley and Taylor, 1972) (Equation 43), JensenHaise (Jensen, 1963) (Equation 44), Hargreaves-Samani (Hargreavis and Samani, 1985) (Equation 45), Tanner and Pelton (Tanner and Pelton, 1960) (Equation 46), among others. Each approach differs in the meteorological variables necessary to calculate the evapotranspiration estimate: ∗

𝐸𝑇𝑜 = 0,408𝛥(𝑄 − 𝐺) +

[𝑦 900 𝑢2.

𝑒𝑠−𝑒 ] 𝑇+273

𝛥+𝑦(1+0,34 𝑢2)

(39)

where 𝑬𝑻𝒐 is the reference evapotranspiration (mm); ∆ is the tangent of the saturation curve at daily temperature (kPa ºC-1); 𝜸 is the psychrometric constant (0,066 kPa ºC-1); 𝜸∗ is the corrected psychrometric constant for resistances (kPa ºC-1); 𝑸∗ is the radiation balance (MJ m-2 day-1); 𝑮 is the

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soil heat flow (MJ m-2 day-1); 𝒖𝟐 is the wind speed in 2 m height (m s-1); and (𝒆𝒔 − 𝒆) = 𝒅 being the saturation deficit (kPa). 𝛥

𝐸𝑇𝑃 = 0,61 (𝛥+𝛾) (𝐾 ↓/2,45) − 0,12

(40)

where 𝑲 ↓ is the global solar radiation (MJ m-2 day-1). 𝐸𝑇𝑜 = 0.013 × [𝑇𝑚𝑎𝑥/(𝑇𝑚𝑎𝑥 + 15) × (50 + 23.88 𝐾 ↓)

(41)

where 𝑇𝑚𝑎𝑥 is the maximum daily air temperature (ºC).

𝐸𝑇𝑜 =

𝑠 γ

[( )(

Q∗ )+Ea] 2,45 s +1 γ

(42)

where 𝑬𝒂 is the aerodynamic factor of evapotranspiration (mm). Δ )(𝑄 ∗ Δ+γ

𝐸𝑇𝑃 = 1,26 × (

− 𝐺 ⁄2.45)

(43)

where 𝟐. 𝟒𝟓 = 𝑳 being latent heat of vaporization. 𝐸𝑇𝑃 = (𝐾 ↓/2.45)(0.0252T + 0.078)

(44)

where 𝑻 is the average daily air temperature (ºC). 𝐸𝑇𝑃 = 0.0023 × 𝐾𝑜 ↓/[2,45 (𝑇𝑚𝑎𝑥 − 𝑇𝑚𝑖𝑛 )0.5 × (T + 17.8)] (45) 𝐸𝑇𝑃 = 1.12 𝑄 ∗ /(2.45 − 0.11)

(46)

The combined use of mathematical models for the determination and comparison of ETo in different edaphoclimatic conditions is wide and extremely addressed in the scientific literature. These parameters introduce studies conducted in important agricultural regions, such as Brazil (Gurski et al., 2018; Ferraz et al., 2020), Hungary (Trajkovic et al., 2020), United

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States (Gao et al., 2017), India (Pandey et al., 2016; Poddar et al., 2018), China (Song et al., 2019), and Senegal (Djaman et al., 2015). Thus, a comprehension of the respiratory complex of an ecosystem is extremely important, since this environment directly influences CO2 emissions and the environment temperatures (Huntingford et al., 2017). The introduction of mathematical models, mainly on a large scale, considering different local conditions, provides a consistent and reliable quantitative observation of the effect of several variables on vital processes of plants (Wang et al., 2017). A study was conducted to verify the influence of the temperature increase in more than 200 plant species in seven biomes (tundra, boreal forest, temperate, temperate woodland, temperate rainforest, and two types of tropical rainforests) through the mathematical approach (Heskel et al., 2016). The application of mathematical models (Equation 47-48) that relate the process of plant respiration and the climatic conditions of the different environments was efficient to verify the direct effect of the earth geo-climatic components on plants: ln 𝑅 = 𝑎 + 0.1012𝑇 − 0.0005𝑇 2

(47)

where 𝑹 is the rate at a given leaf 𝑻; 𝒕 is the leaf response, and 𝒂 is an independent coefficient. 𝑅𝑡 = 𝑅𝑇𝑟𝑒𝑓 × 𝑒

2 [0.1012(𝑇−𝑇𝑟𝑒𝑓 )−0.0005(𝑇 2 −𝑇𝑟𝑒𝑓 )]

(48)

where 𝑅𝑇𝑟𝑒𝑓 is exp (𝒂 + 𝟎. 𝟏𝟎𝟏𝟐 𝑻𝒓𝒆𝒇 − 𝟎. 𝟎𝟎𝟎𝟓 𝑻𝟐𝒓𝒆𝒇 ). Another study provided an equational approach to plant respiration using growth algorithms, maintenance components, and the photosynthetic complex (Equation 49-52) (Gifford, 2003). The use of these variables was efficient for the behavior of the C-cycle in a scenario of climate change and adverse conditions: 𝑅 = 𝑅𝑠 (𝑇) × 𝑊

(49)

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where 𝑹 is the respiration rate per plant or unit area of soil (g d-1); 𝑻 is the temperature; 𝑾 is the dry weight per plant or unit of land area (g), and 𝑹𝒔 is the specific respiratory rate per unit plant weight (g g-1). 𝑅 = 𝑅𝑚 + 𝑅𝑝 = 𝑟𝑚 𝑊 + 𝑟𝑝 𝑃

(50)

where 𝑹 is the respiration rate per unit ground area (g C m–2 d–1); 𝑹𝒎 is the maintenance-dependent component of respiration (g C m–2 d–1); 𝑹𝒑 is a photosynthesis-dependent component of respiration (g g-1), and 𝑷 is the photosynthesis rate per unit ground area (g C m–2 d–1). 𝑅 = 𝑟𝑚 𝑊 + 𝑟𝑔 𝐺

(51)

where 𝑮 = 𝑷 − 𝑹 is the plant growth rate, and 𝒓𝒈 is a grow coefficient. 𝑅𝑠=𝑟𝑚 + (𝑟𝑔 × 𝑅𝐺𝑅)

(52)

where 𝑹𝒔 = 𝑹⁄𝑾 being the relating specific respiration rate, and 𝑹𝑮𝑹 = 𝑮⁄𝑾 is the relative growth rate.

Plants Demand Components Mineral nutrients, such as nitrogen, phosphorus, and potassium are key components for growth and development (Chun et al., 2017). In general, the absorption of these nutrients is completed in the form of inorganic ions, from the root system and is translocated to different parts of the plant (Raddatz et al., 2020). These elements have several functions and are fundamental to numerous biochemical and physiological phenomena that occur in plants. The use of mineral nutrients to enhance the productivity of plant species is necessary, mainly because these elements have a direct impact on plant metabolism and growth performance (Xu et al., 2020). Thus, studies that explore the mathematical approach in the transfer and

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control mechanisms of nutrients in plant systems are extremely relevant in the agricultural scenario. A study was developed with the main objective of validating a mathematical model that quantitatively addresses the water and nutrient absorption activities of the root system and photosynthesis of plant parts of Petunia hybrida (Feller et al., 2015). The results were extremely pertinent to the comprehension of the relations between the parts of the plant and their interactions through the translocation of sugars and phosphate (Pi), in which the Equations 53-57, represent, respectively, solute transport rate from shoot to root compartment, sugar transport rate from shoot to root compartment, Pi uptake rate from the soil, Pi transport rate from root to shoot compartment, and Pi transport rate from shoot to root compartment: 𝑠ℎ→𝑟 (𝑡) 𝑇𝐻2𝑂 =

𝑠ℎ (𝑡)−𝐶 𝑟 (𝑡),0) 𝑚𝑎𝑥 (𝐶𝑠𝑢 𝑠𝑢 𝑃 𝑑𝐻2𝑂 𝑅𝑡𝑢𝑏𝑒 (𝑉 𝑝𝑙 (𝑡))/𝑅𝑇

𝑠ℎ→𝑟 (𝑡) 𝑠ℎ→𝑟 (𝑡) 𝑇𝑠𝑢 = 𝑇𝐻2𝑂

𝑛(𝑉 𝑝𝑙 (𝑡))

𝑠ℎ (𝑡) 𝐶𝑠𝑢 𝑃 𝑑𝐻2𝑂

𝑠𝑜𝑖𝑙 (𝑡)) 𝑈𝑝ℎ (𝑡) = 𝑈𝑚𝑎𝑥 𝑀𝑈 (𝐶𝑝ℎ 𝑀𝑎𝑠 (𝑉 𝑟 (𝑡))

(53)

(54)

(55)

𝑟→𝑠ℎ 𝑠ℎ→𝑟 (𝑡) (𝑡) = (𝐸(𝑡)𝑆 𝑠ℎ (𝑡) + 𝑇𝐻2𝑂 𝑇𝑝ℎ + 𝑃 𝑟 𝑑𝐻2𝑂 𝑉 𝑠ℎ (𝑡))𝑝𝑚𝑎𝑥

𝑟 (𝑡)−𝐶 𝑟 𝑚𝑎𝑥 (𝐶𝑝ℎ 𝑝ℎ.𝑔 ,0)

(56)

𝑃 𝑑𝐻2𝑂

𝑠ℎ→𝑟 (𝑡) 𝑠ℎ→𝑟 (𝑡)𝑝 𝑠ℎ 𝑇𝑝ℎ = 𝑇𝐻2𝑂 𝑚𝑎𝑥

𝑠ℎ (𝑡)−𝐶 𝑠ℎ 𝑚𝑎𝑥 (𝐶𝑝ℎ 𝑝ℎ.𝑔 ,0) 𝑃 𝑑𝐻2𝑂

(57)

where 𝑉 𝑠ℎ (𝑡) is the shot volume at time 𝒕; 𝑉 𝑟 (𝑡) is the root volume at time 𝑠ℎ→𝑟 (𝑡) 𝒕; 𝑃(𝑡) is the rate of photosynthesis; 𝑇𝑠𝑢 is the sugar transport rate 𝑟→𝑠ℎ (𝑡): from shoot to root; 𝑇𝑝ℎ Pi transport rate from root to shoot; 𝑠ℎ→𝑟 (𝑡) 𝑃 𝑇𝑝ℎ is the Pi transport rate from shoot to root; 𝑑𝐻2𝑂 is the quantity of

solute per cm3ofplant; 𝑻 is the temperature; 𝑹 is a gas constant; 𝒏 is the

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𝑟→𝑠ℎ (𝑡) number of phloem tubes; 𝑇𝑝ℎ is the volume of water transpired by 𝑠ℎ→𝑟 (𝑡) the leaves; 𝑇𝑝ℎ is the volume of solute transported downwards in the 𝑠ℎ phloem; 𝐶𝑝ℎ.𝑔 is the concentration of phosphate required per unit of plant 𝑟 volume; and 𝐶𝑝ℎ.𝑔 is the concentration of phosphate required per root

volume. Furthermore, a mathematical approach was applied to the morphophysiological investigation of changes in the root structure of Arabidopsis thaliana in response to nitrogen availability (Araya et al., 2016). For the calibration and validation of the model, a series of data related to the characteristics of the root system of the species submitted to different concentrations of nitrate or ammonium was used. The models verified the temporal relation between the effect on plants and the N concentrations, at doses of 30, 300, and 3000 mmol L-1 NO3- (Equation 58-63): 𝑒 𝛽3

𝑃𝑅 (𝑡) = 1+𝑒 −(𝛽1 +𝛽2 𝑡)

(58)

𝐿𝑅𝑁 (𝑡) = 𝑒 δ1 𝑃𝑅(𝑡)δ2

(59)

𝐸𝐿𝑅𝑁 (𝑡) = 𝐿𝑅𝑁 (𝑡)

𝑝3 1+𝑒 −(𝑝1+𝑝2𝑡)

𝐿𝑅 (𝑡) = 𝑒 α1 𝐸𝐿𝑅𝑁(𝑡 − 𝑑)α2 𝐿𝑅𝑁 (𝑡) 𝑃𝑅 (𝑡) 𝐸𝐿𝑅𝑁 (𝑡) 𝑝 = 1+𝑒 −(𝑝31+𝑝2𝑡) 𝐿𝑅𝑁 (𝑡)

𝐿𝑅𝐷 (𝑡) = 𝐹𝐿𝑅𝐸 (𝑡) =

(60) (61) (62) (63)

where 𝑳𝑹(𝒕) is the frequency of lateral root; 𝑭𝑳𝑹𝑬 (𝒕) is the frequency of lateral root emergence; 𝑬𝑳𝑹𝑵 (𝒕) is the emerged lateral root number; 𝑳𝑹𝑵 (𝒕) is the total lateral root number; 𝑷𝑹(𝒕) is the primary root length. The following independent parameters were estimated by regression analysis: 𝜷𝟏 , 𝜷𝟐 , and 𝜷𝟑 for primary root length, 𝜹𝟏 and 𝜹𝟐 for total lateral

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root number, 𝒑𝟏 , 𝒑𝟐 , and 𝒑𝟑 for emerged lateral root number, and 𝒂𝟏 , 𝒂𝟐 , and 𝒅 for total lateral root length. The use of reagents with a high degree of purity is fundamental for the development of experiments aimed at nutritional deficiency in plant species (Silva and Lambers, 2020). One study aimed to develop mathematical models to minimize the costs of reagents with a high degree of purity in experiments of nutritional deficiency in plants (Garcia et al., 2019). Two linear programming models (A, using 16 reagents in the preparation of the culture solution; and B, with 27 reagents in the culture solution) were developed, considering the restrictions for each of the nutrients in the culture solution and the costs of the reagents. The results obtained showed that it is possible to reduce costs by up to approximately 26%, contributing to important decision-making. Equations 64 and 65 constitute the objective function, a model based on restrictions on the concentration of nutrients, and a condition of non-negativity: 𝐹𝑂𝑚𝑖𝑛 (𝑅$) ∑𝑛𝑖=1 𝑃𝑖 𝑥𝑖

(64)

∑𝑛𝑘 𝑖=1 𝑅𝑘𝑖 = 𝐶𝑘 , ∀𝑘 {1, 2, 3, … , 13}

(65)

Where: 𝒏 is the number of reagents adopted; 𝑲 are the nutrients tested, such as N, P, K, Ca, Mg, S, Fe, B, Mn, Zn, Cu, and Mo; 𝑷 is the price of reagent (R$ g-1); 𝒙𝒊 is the decision variable representing the quantity of reagent 𝒊 (g); 𝑹𝒌𝒊 is the quantity of nutrient 𝒌 available and present in reagent 𝒊, expressed as the mass ratio of the atomic weight of atoms of the nutrient present in the reagent to the molecular mass of reagent; and 𝑪𝒌 is the desired concentration of nutrient 𝒌 expressed as grams per liter of culture solution (g L-1).

Other Agricultural-Related Areas The dynamics of processes involving nutritional demand, phenological growth and development, and the biochemical reactions in plants are extremely important to understand the complexity that involves these

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species in the soil-plant-atmosphere system. Recent models have been applied to predict the behavior of plants regarding external conditions and the extent to which these factors directly influence their performance. However, mathematical studies have been conducted to evaluate the effectiveness of prediction models under the influence of pathogens and unwanted pests in the environment. Equations based on plant-vectorpathogen models have been adopted for yield prediction, integration of control strategies, and management costs. This section aims to develop a quick insight into the use of mathematical modeling applied in a dynamic occurrence of pests, diseases, and animal care aimed at agricultural and livestock systems.

Plant Diseases A detailed epidemiological study on plant diseases is essential to understand the population dynamics of pathogens and to predict the magnitude of the damage caused by these microorganisms in plants. The development and application of mathematical models that aim to represent the dynamics of disease-causing pathogens in plants in a period has been verified in the literature over the years. A study involving the occurrence of Fusarium verticillioides in maize was developed through mathematical modeling of the quantification of the phytotoxin fumonisins, produced by the fungus (Bernd et al., 2008). The results showed that the models were efficient in predicting the occurrence of fumonisins, benefiting the corn harvest and post-harvest processes, and reducing damage and loss of grain quality. Besides, another study was conducted to develop vector-plantvirus models in agricultural species with continuous replanting, in addition to establishing the best technique or integration of techniques for better plant performance (Bokil et al., 2019). Considering possible scenarios of infection and susceptibility to plant infection, the model indicated that control measures should be taken at the beginning of the infection. The source of fungal inoculum, the dispersion capacity, the latent period, and the host resistance performance are very important parameters for the development of models for prediction and estimation of the

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occurrence and severity of fungal diseases in plants (Van Maanen and Xu, 2003). A scientific study applied to bacteria (Candidatus Liberibacter) in citrus and virus (transmitted by the Bemisia tabaci biotype B vector) in tomatoes showed that the detection of the primary inoculum combined with the host plants resistance is an excellent alternative for the adoption of preventive management strategies in polycyclic diseases (Filho et al., 2016). For this, the consideration of variables such as rates of primary and secondary infection, amount of inoculum, and the number of attacked and healthy plants are fundamental to the hypotheses presented by mathematical models. Mathematical validation has also been applied to soil organisms that affect the root system of plants. Nematode diffusion and migration prediction models have been developed, considering parameters such as population density, initial concentration, special fraction, and distribution of these microorganisms (Feltham et al., 2002). Based on these observations, the pressure under the plants can be evaluated. Furthermore, the calibration and validation of models referring to nematode dynamics in the soil-plant system hve been verified in the scientific literature. The development and application of the SIMBA-NEM model, on a field scale, showed that this strategy is largely functional for simulating the dynamics of Radopholus similis and Pratylenchus coffeae in banana (Tixier et al., 2006). The mathematical equation involves population growth considering a range of factors, such as competition between nematode species, cultivation system, temporal characterization, soil cover, and physical and chemical characteristics of the soil, providing information on the best management strategies to be adopted. The dynamics of the relationships of biotic and abiotic factors in the occurrence and consequent control of unwanted pathogens perpetuate benefits in the adoption of strategies and decision-making. Many models of environmental impacts and their influence on the population of pathogens were developed through regression or empirical equations, considering factors such as disease incidence and severity, meteorological variables, and population characterization of vectors (Bokil et al., 2019). A study showed that the phenomenon of El Niño Southern oscillation (ENOS), responsible for the abnormal heating of the tropical portion of the

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Pacific Ocean, amplifies the occurrence of avocado wilt (Verticillium sp.) (Ramírez-Gil et al., 2020). Based on the characteristics of the plant and the dynamics of the pathogen in the system, different techniques to mitigate its effects on plants were adopted, reducing the plant susceptibility to adverse conditions. Also, the incorporation of variables such as temperature, precipitation, and wind in deterministic models shows these factors affect the severity of the pathogen population in the agricultural system and alter the balance of the environment (Murwayi et al., 2017). However, considering the heterogeneity of agricultural species and their susceptibility to certain pathogens, some parameters are fundamental for the modeling of disease prediction and control equations, such as the temporal and spatial dynamics and anthropomorphic influence on the environment (Cunniffe et al., 2015). Climatic conditions and future projections have been addressed due to the susceptibility to the occurrence of diseases and the pressure that pathogens exert on host plant organisms (Juroszek and Von Tiedemann, 2015). Some studies have shown that the severity of the impacts caused by the presence of pathogens in an agricultural system shows a significant increase in the coming years, such as Plasmopara viticola in grapes (Salinari et al., 2006), Sclerotinia minor in spring crops, such as soybeans and corn (Manici et al., 2014), and Blumeria graminis and Puccinia sp. in wheat (Juroszek and von Tiedemann, 2013).

Pest Control The occurrence of pests in an agricultural system severely compromises the productive potential of crops, considering the high capacity for species dispersion and, often, difficult control. Several factors benefit the emergence and dissemination of these organisms, such as the moisture content in the soil-plant-atmosphere system, solar radiation, and temperature (Delcour et al., 2015). The development of modeling linked to the behavior of pest species in the environment has as main objective the domain of strategies or combination of strategies that aim to keep the pest populations at levels below those that cause economic damage

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(Chávez et al., 2017). Theoretically, this concept corresponds to integrated pest management (IPM), which aims to adopt a set of control methods (e.g., biological control, cultural treatments, and chemical or conventional control through the use of specific insecticides), aiming at a balanced system with low economic and environmental risks. Chemical and biological control strategies have been reported through a mathematical approach that considers factors such as the initial populations of pests and natural enemies, propagation and growth rate, and the relationship between these animals and the environment (Chávez et al., 2018). A study involving the dynamics of the fall armyworm (Spodoptera frugiperda), one of the main corn pests, based on the daily thermal necessity of the animal for its development was conducted in an area with Bt and non-Bt corn cultivation (Garcia et al., 2019). The results obtained from the computational models made it possible to define the dissemination capacity of the pest and establish previous management schedules, considering variables such as insect resistance to transgenic crops, cultivation area, and movement in the larval phase. Another study involving S. frugiperda provided the comprehension of the potential distribution of the species on the Asian continent through a model based on the Receiver Operating Characteristic (ROC) that relates the species ability to spread with space-time variables, such as temperature, precipitation, and isothermality (Baloch et al., 2020). Thus, several other species with high potential for damage to crops have been studied based on the same approach, such as Helicoverpa armigera (Hübner) (Milonas et al., 2016), Phakopsora pachyrhizie (Del Ponte et al., 2006), Bemisia tabaci (Suekane et al., 2018), Anastrepha fraterculus (Frighetto et al., 2019), Diabrotica speciosa (Garcia et al., 2020), Chrysodeixis includens (Souza et al., 2019), and Spodoptera litura (Fand et al., 2015). The improvement of impulsive differential equation (IDE) models, which cover a high number of variables and fluctuation levels of the parameters used, is a reliable tool and has expressed accurate estimates of the model and its components with stochastic effects (Akman et al., 2015). The use of this strategy was also applied in an SI (susceptible-infected) model whose objective was the impulsive release of infectious pests and

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chemical pulverization in a dynamic predator-prey process (Jiao et al., 2009). The results showed that the methodological and model approach, considering variables such as the transmission and incidence rates, was an efficient strategy for pest control management. The quantification of the impacts caused by pests on agricultural species through the application of a mathematical approach is extremely significant, especially for interpreting the dissemination capacity of these organisms and defining a control plan. Recent advances seek to conduct scientific research to link the spread of pests with multiple biotic and abiotic factors, pest-natural enemy relations, and the influence of climate change on the spread of pests (Tonnang et al., 2017). Furthermore, various factors have been explored such as the adoption of technological complexes for the prediction of population density of pests, the process of synchronization in a wide plant-pest-enemy-natural relationship, influence of edaphoclimatic conditions in the pest population in scenarios situations future climatic conditions, the association of different variables in remote sensing tools, and economic risks through the inclusion of geographic and climatic factors (Donatelli et al., 2017; Tonnang et al., 2017).

Animal Care Considering the population increase and the food demand an efficient animal production chain that concentrates on the complexity of the different components of the system is necessary. The recent progress in the use of mathematical language in the field of animal nutrition and care shows the importance of using this strategy to provide the necessary food composition for animals, mitigating unnecessary costs and nutritional excess. Empirical models are insufficient for accurate modeling of the animal production system. As a result, scientific literature has explored the application of highly detailed mechanistic models, ranging from animal metabolic reactions to growth performance and nutrient elimination (Hanigan and Daley, 2020). The use of mathematical equations has been directed towards meeting the nutritional and energy needs of animals based on a diet rich in compounds such as proteins, carbohydrates, and fat

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(Dumas et al., 2008). Thus, the expansion of mathematical models for determining nutritional needs in different species has been considered. The application of a mechanistic model (CNCPS-S - Cornell Net Carbohydrate and Protein System) in the prediction of nutritional needs in olives and goats showed high precision and determination coefficient (R2) up to 0.99, considering internal and external factors and specific needs of animals, such as lactation, pregnancy, and body reserves (Tedeschi et al., 2010). The use of this language in the zootechnical field is related to the evaluation and forecast of the growth of animals submitted to a controlled diet. A study involving Nellore bulls submitted to tropical grazing with different supplements (protein and mineral) was developed using mathematical models to describe animal growth (Fernandes et al., 2012). The results showed that parameters such as the initial weight, supplementation period, age, and growth rate were fundamental for the adequacy and precision of a model called multiphase to the local conditions in which the animals are submitted. Mathematical studies directed at the analysis of pig growth showed that the phenotype of animals is an important parameter to define the best strategies for crossing and obtaining larger animals (Stass, 2020). Also, quantitative analyzes have been applied to verify the growth limitations in pigs, considering the nutritional conditions of different locations. A study conducted considering different breeds of pigs and territories showed that the use of the InraPorc model (comprising variables such as age, body weight, and protein consumption and deposition) was efficient in providing innovative insights into the factors that affect the growth of animals in different conditions (Brossard et al., 2019). The use of quantitative models has also been applied in areas of toxicology and veterinary pharmacology. The application of models such as the physiologically based pharmacokinetic model (PBPK) showed satisfactory results regarding the behavior of the animal circulatory system (Lautz et al., 2019). PBPK is a simulator and prediction model that allows a characterization of the processes of absorption, distribution, metabolism, and elimination (ADME) of certain products in different biological systems

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considering physiological, biochemical, and physicochemical factors, such as body weight, flow of blood, and enzyme activity (Lin et al., 2016). The dynamics of the transmission and contagion of diseases and pathogens among animals play an important role to reduce health and economic impacts. A stochastic mathematical model of the type “SVEIMQR” performed the simulation of the transmission dynamics of contagious bovine pleuropneumonia (CBPP) in a herd of cattle (Ssematimba et al., 2015). The application of this tool resulted in the development of probabilities of up to 97.5% of elimination of the disease, considering susceptibility, exposure, and infection, creating a schedule of tests and vaccination, and the diagnosis of new control strategies. Similar efficacy was achieved through the use of a dynamic model of CBPP transmission in animals separated by classes, according to susceptibility to disease, vaccination, level of infection and recovery, and exposure to the pathogen (Aligaz and Munganga, 2019). Considering the intervention for the application of treatment with vaccines and antibiotics, the model showed that up to 85% of infected animals must receive treatment against the disease within 49.5 days for effective control of CBPP. However, some gaps related to the models still need an additional approach, mainly related to a more mechanistic representation of feed consumption, in addition to the nutrient cycle and the consequences caused by the microbial occurrence in the production system (Black, 2014). Furthermore, future projections indicate that the development of mathematical models must be directly related to technological advances over the years and the innovation of these tools must focus on obtaining highly reliable data and creating new resources, instead of investing time in modifications of existing models (Tedeschi, 2019).

Conclusion and Future Outlooks Mathematical modeling in the food industry has been reported in the literature to better understand the behavior of foods in a given experimental condition. In this chapter, the main mathematical models have been described regarding heat and mass transfer, food drying, and microbial and

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enzymatic inactivation. These models are important for predicting chemical and biochemical reactions that can happen during food processing. Thus, the prediction of mathematical parameters and their interactions with the food matrix becomes an indispensable tool for the food industry. The scientific importance directed to the use of the application of models and mathematical simulations in the agricultural field has been widely reported in the literature recently. Over the years, the constant increase in variables that influence plant phenomena and interact with each other in the soil-plant-atmosphere system can be visualized. One of the principles of the development of computational language in the context of agriculture is to obtain highly accurate arguments in the face of tests and hypotheses performed at a field level. The prediction of plant performance and its interaction with the environment in adverse situations becomes an excellent alternative for directing resources and establishing strategies to enhance the productive system. In this chapter, recent mathematical models were investigated that served as an academic-scientific foundation to support innumerable hypotheses that surround the interactions of plants and the environment in which they are inserted. Edaphoclimatic conditions, nutrient translocation demand, and gradient, and the impacts of external organisms on plant balance are some of the factors that play a key role in the physical, chemical, and biological status of plants. However, difficulties and unexpected results are often achieved in field experiments, especially under unusual conditions or quite different from a typical situation. Recent advances in the computational language procedure have considered, for example, different scenarios of climate change and their influence on the dynamics of agricultural systems. The development of mechanistic models that adapt different performances of biotic and abiotic factors in plant growth events in the face of situations of geo-climatic changes provides an accurate diagnostic of the processes that involve agroecosystems. Scientific evidence shows that unexpected conditions favor the occurrence of pests and diseases in plants, causing high impacts on the productive potential of the species and economic losses. The improvement of models that consider the behavior of these organisms in situations of

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change in the environment is a major challenge for the coming years. With the necessity to expand production and maintain cultivation areas, plant exploitation tends to be more intense and the inevitability of sustaining the growing food demand requires attention to the care of the crops. However, new pathogens and pests with high damage and dissemination potentials have been increasingly reported, resulting in economic losses and growth in different regions. In this context, modeling and simulation in agriculture are characterized as a viable alternative to estimate the impacts of these and several other elements on plants and to provide decision-making for correct strategies. The development of a new generation of tools that provide direct interaction between the field and technology is an important legacy to be proposed to the academic-scientific community and its contribution to ecological dynamics is invaluable.

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In: Frontiers in Mathematical Modelling … ISBN: 978-1-68507-430-2 Editors: M. Biswas and M. Kabir © 2022 Nova Science Publishers, Inc.

Chapter 11

Mathematical Modelling of Complex Systems Using Stochastic Partial Differential Equations: Review and Development of Mathematical Concepts Parul Tiwari1, Don Kulasiri1, and Sandhya Samarasinghe1,2 1

Centre for Advanced Computational Solutions, Lincoln University, New Zealand 2 Department of Environmental Management, Lincoln University, New Zealand

ABSTRACT Modelling real life stochastic phenomena is difficult due to heterogeneity in associated parameters or insufficient information, or both; hence it is important to develop stochastic models to understand the behaviours of complex Systems. Stochastic Partial Differential Equations (SPDEs) (Holden et al. 2010; Nicholas 2017), are used to model physical, engineering, and biological systems in which small scale effects and related 

Corresponding Author’s Email: [email protected].

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Parul Tiwari, Don Kulasiri and Sandhya Samarasinghe uncertainties are modelled as stochastic processes. In this chapter, we discuss the modelling of complex systems using SPDEs, giving reasons for such developments and focusing on numerical solutions. We primarily discuss Polynomial Chaos Expansion (PCE) approach (Wan et al. 2004). In recent years, several extensions of PCE, such as Generalised PCE using the Askey scheme (Xiu and Em Karniadakis 2003), have been developed to overcome the slow computational speed of the method for non-Gaussian random variables. When there are viable restrictions on model predictability due to interdependencies between the physical system and the associated parameters, PCE provides a robust mathematical structure and acceptable probabilistic solutions. We also discuss the use of Wick products (Venturi et al. 2013) in stochastic analysis. Using the Wick product instead of the usual pointwise multiplication makes it possible to study anticipating processes. The Wick product is innate in stochastic analysis (with respect to Wiener chaos space), as it is implicit in the Ito integral. For anticipating processes, the Skorohod integral is similar to the Ito integral. Thus, Wick calculus works equally well for adapted and non-adapted processes.

Keywords: stochastic partial differential equations, polynomial chaos expansion, Brownian motion, orthogonal polynomials, wick product

Introduction Models, Mathematics and Modelling A model is a systematic and clear description of a real system. It facilitates understanding, enables predictions and control of the system. Models serve various purposes. A few of the objectives are explained in Figure 1.

Figure 1. Objectives of a model.

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Mathematics is as old as humanity. Models demonstrate our reliance on mathematics, computation, and scientific laws to translate those our conceptual understanding into mathematical formulations. Modelling the behaviour of complex systems is a never-ending goal in both science and engineering. Fast and efficient computational algorithms have greatly increased the efficacy of such modelling. The role of modelling is to minimize the differences between the conceptual model and observations made of the real system thus modelled. Models are used as a robust tool and to explain the working phenomena of complex systems (Figure 2) but, we must consider the limitations of these models as well. In modelling, a researcher attempts to abstract the key features of a process. This process may be economic, natural, or otherwise. Due to their adaptive nature, complex systems can evolve overtime as a reaction of change in other interacting systems. Predicting results or future states for these systems is truly demanding because of the complicated interactions and adaptations that occur within these systems.

Figure 2. A schematic of different phases of credible mathematical modelling.

Assumptions play a critical role in mathematical modelling in simplifying the complexities associated with a real system. The purpose of model building should be clear and the use of various tools such as flow diagram sometimes can be important. Formulated equations or differential

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equations represent the core of model, and physical characteristics are modelled as the states or variables. Outputs are the most often approximate solutions, and a computer programming code is considered as a generic model.

Motivation It is not always possible to obtain exact solutions for all mathematical models, especially when stochastic processes are concerned. Stochastic models help the modeller to characterize the randomness in several ways. Suitable mathematical modelling and efficient numerical simulation, both play a crucial role in predicting the behaviour of the model. Stochastic models account for uncertainties due to the varying behavioural characteristics. In recent years, numerical solutions for SPDEs have received much attention. SPDEs with a multiscale structure can accurately simulate high-dimensional multiscale dynamical systems that are sensitive to noise. While researchers typically use Monte Carlo based methods (Barth, Schwab, and Šukys 2016) to solve SPDEs, these methods suffer from a slow rate of convergence. Random/stochastic PDE models contain random terms which have uncertainties related to space or time, or both. It is quite possible that some of the parameters are unknown or there is insufficient information. As a result of these uncertainties, it is hard to obtain solutions to the stochastic processes, particularly when using a single realization. The researcher thus must perform a large number of iterations to obtain a good approximation. This is the primary reason why the computational cost of stochastic models is high. Because of progress in the field of simulation, computing and its uses in science and engineering, researchers are now able to solve a large number of complex problems involving stochastic processes. However, uncertainties associated with the physical parameters make it difficult to provide objective confidence measures of the numerical predictions. Though the computer simulations provide feasible and reliable information, yet stability, convergence and the efficiency of numerical algorithms should be studied rigorously.

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SPDEs and Modelling SPDEs have been applied in diverse areas: for instance, the propagation of an electric field potential in a neuron (Ma et al. 2019), to examine the structure of an interest rate model in economics and finance (Verschuren 2020), in non-linear filtering (Lototsky, Mikulevicius, and Rozovskii 1997), in advection dispersion reaction equations (Z. Zhang et al. 2012), transport in porous media (R. Ghanem 1998), turbulence model (Zeng, Wang, and Maccall 2020), wave propagation (Kalpinelli, Frangos, and Yannacopoulos 2013), heat equations (Chiba 2012). Numerous other examples can be found in biology (Bellmann 1979), structural mechanics (Matthies et al. 1997), modern integrated circuits (Kaintura, Dhaene, and Spina 2018), epidemiological sciences (Britton et al. 2015; MacKenzie and Bishop 2001) and chemical master equations (McQuarrie 1967; Smadbeck and Kaznessis 2014). Multiple numerical methods have been developed to simulate SPDEs. One has to rely on suitable numerical methods which can define the model with an acceptable level of error. Probabilistic methods are used extensively to deal with complexity of these model. These methods either use a statistical or non-statistical approach. The statistical methods are based on sampling techniques, including stratified sampling, or Latin hypercube sampling. Monte Carlo (MC) based methods (Dereich and Li 2016) are commonly used to solve SPDEs for probabilistic uncertainty analysis, with diverse applications. These methods are most frequently and widely used for forward propagation of the input randomness to see the system response. In spite of being flexible, MC methods suffer with slow convergence rate, proportional to √1⁄𝑁 (where 𝑁 is the number of samples used in Monte Carlo simulations). Furthermore, these methods sometimes fail to utilize the possible regularity that the solution might have, particularly with respect to the input parameters. A few other multilevel algorithms such as Quasi Monte Carlo (Croci, Giles, and Farrell 2019), Markov-chain Monte Carlo (MCMC) (Herbei, Paul, and Berliner 2017; Jasra et al. 2017) and importance of sampling (Ebener et al. 2019; Salins and Spiliopoulos 2016)

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are suggested so that accuracy and convergence of numerical methods can be improved. One option which considers uncertain parameters and field variables is spectral representation (Kim, Debusschere, and Najm 2007). Spectral representation presents an efficient alternate to MC-based techniques, such as the polynomial chaos expansion (PCE) methods (Wiener 1938; Cameron and Martin 1947), stochastic collocation (SC) methods (Kamrani 2016), and stochastic reduced order models (SROMs) (le Matre et al. 2002). Among all the spectral expansion techniques, PCE methods are prevalent in the literature for three main reasons: they are fast and efficient, they can handle different types of systems involving random variables of different distributions and can deal with estimation of second and higher order moments with ease of integration using non-intrusive approaches. PCE methods have a strong mathematical basis and functional relationship for statistical quantities; mean, variance and density functions can be obtained from these representations. Numerous research articles are being published based on PCE methods for the last four decades and existing literature shows the significance of these methods. These publications note the accuracy, efficiency, modelling power, and flexibility of PCE methods. Wiener (R. G. Ghanem and Spanos 1991) defined the polynomial chaos (PC) theory and developed functional basis expansion based on Hermite polynomials and Gaussian random variables to prove the convergence in L2-sense. R. G. Ghanem and Spanos (R. G. Ghanem and Spanos 1991) describe the implementation of polynomial chaos in a finite element context. (Le Matre et al. 2002) extended these techniques to study thermodynamic systems. (Xiu and Shen 2008a) used generalized polynomial chaos (gPC) for uncertainty quantification to study the interactions of fluid structures. Uncertainty quantification of electrochemical microfluid systems is carried by (Debusschere et al. 2003) using this approach. Generalized polynomial chaos adopts orthogonal polynomials to represent general random processes; nonlinear behaviour of the system dynamics can be captured by using higher order polynomial chaos expansions due to significant parametric perturbations. In addition, higher order PCE is capable of

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inspecting nonlinear sensitivity effects because multiple parameters can be considered all together.

Development of Polynomial Chaos Expansion To understand the concept and use, it is first necessary to describe the systematic approach of polynomial chaos expansion.

One Dimensional PCE Let us consider the heat flow and associated variation of temperature with position and time in a rod of length L by following the ideal assumptions: The rod is perfectly insulated; heat flows from a higher temperature to a lower temperature and there are no internal heat sources or sinks. Classical one-dimensional heat flow is given by the partial differential equation, 𝜕𝑢 𝜕𝑡

𝜕2 𝑢

= 𝑘 𝜕𝑥 2

(1)

where 𝑢(𝑥, 𝑡) is the temperature at a distance 𝑥 at any given time 𝑡 and 𝑘 is the thermal diffusivity given as follows: 𝑘=

𝑘0 𝑠𝜌

(2)

subject to boundary and initial conditions: 𝑢(0, 𝑡) = 𝑢(𝐿, 𝑡) = 𝑢0 ; 𝑢(𝑥, 0) = 𝑓(𝑥)

(3)

The solution of equation (1) is a classic example of theory of partial differential equations. But here, we are focusing on the question, “what if 𝑘 is random”? Diffusivity is a material property and we shall consider the material to be heterogeneous.

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To understand PCE’s conceptual framework, we shall consider parameter uncertainty as a stochastic quantity, with a known probability distribution function. For instance, in the case of one-dimensional heat flow by considering the heterogeneous material, say thermal diffusivity, k is a random quantity. Empirically, k is written as 𝑘 = 𝑘̅ + 𝑘 ∗

(4)

where 𝑘̅ is the mean diffusivity and 𝑘 ∗ denotes the uncertainty in the value. If we assume that 𝑘 is sampled from a Gaussian PDF with mean 𝜇 and standard deviation 𝜎, then 𝑘 is defined as 𝑘 = 𝜇 + 𝜎 𝜉,

(5)

here 𝜉 is a normalized Gaussian random variable with zero mean and of unit variance. Using PCE, the spectral representation of 𝑘 is given as: 𝑝 𝑘 = ∑𝑖=0 𝑘𝑖 𝜑𝑖 (𝜉)

(6)

Here 𝑘𝑖 ′𝑠 are deterministic coefficients, p is the order of the expansion, and 𝜑𝑖 ′𝑠 are the orthogonal basis functions of a random variable 𝜉. As we assume the PDF of 𝑘 to be Gaussian, the basis functions can be expressed as Hermite polynomials. Hermite polynomials of order 𝑖, can be determined using Equation (7) 𝜑𝑖 (𝜉) = 𝑒 𝜉

2 /2

(−1)𝑖

2 𝜕𝑖 (𝑒 −𝜉 /2 ) 𝜕𝜉 𝑖

(7)

Hermite polynomials 𝜑𝑛 (𝑥) satisfy the following recurrence relation 𝜑𝑛+1 (𝑥) = 𝑥𝜑𝑛 (𝑥) − 𝑛𝜑𝑛−1 (𝑥), 𝑛 > 0 and

(8)

Mathematical Modelling of Complex Systems … ∞

∫−∞ 𝜑𝑛 (𝑥)𝜑𝑚 (𝑥)𝑤(𝑥)𝑑𝑥 = √2𝜋 𝑛! 𝛿𝑚𝑛 where 𝑤(𝑥) = 𝑒 −𝜉 delta defined as:

2 /2

341 (9)

is the weight function and 𝛿𝑚𝑛 is the Kronecker

0, 𝑚 ≠ 𝑛 𝛿𝑚𝑛 = { 1, 𝑚 = 𝑛

(10)

Orthogonal properties of hermite polynomials and the related theorems are given in Appendix A. The first six Hermite polynomials are given by the following equations:𝜑0 (𝑥) = 1, 𝜑1 (𝑥) = 𝑥, 2 𝜑2 (𝑥) = 𝑥 − 1,

𝜑3 (𝑥) = 𝑥 3 − 3𝑥, 𝜑4 (𝑥) = 𝑥 4 − 6𝑥 2 + 3, and 𝜑5 (𝑥) = 𝑥 5 − 10𝑥 3 + 15𝑥. Projecting 𝜑𝑖 (𝜉) on both sides of Equation (6) and taking expectation, the deterministic coefficients 𝑘𝑖 are given by:

, 𝑖 = 0,1, … … … , 𝑛

2 ∞ 1 ∫ 𝑘𝜑𝑖 (𝜉) 𝑒 −𝜉 /2 𝑑𝜉 −∞

𝑘𝑖 = =

(11)

Further, using PCE, the solution of Equation (1) can be represented as 𝑢(𝑥, 𝑡) = ∑𝑛𝑖=0 𝑢𝑖 (𝑥, 𝑡)𝜑𝑖 (𝜉)

(12)

where the 𝑢𝑖 ′s are the unknown coefficients analogous to the known coefficients of k. Again, with the orthogonality of 𝜑𝑖 ′s, the 𝑢𝑖 ′s are determined as:

342

Parul Tiwari, Don Kulasiri and Sandhya Samarasinghe 𝑢𝑖 =

,𝑖

= 0,1, … … … , 𝑛

(13)

< 𝑢, 𝜑𝑖 > is calculated using formula explained in equation (11) and < 𝜑𝑖 2 > is the norm of 𝜑𝑖 . Table 1. The Wiener-Askey scheme of orthogonal polynomials Type of variable

Continuous

Discrete

Random variable (𝝃) Gaussian uniform gamma beta binomial Poisson negative binomial hypergeometric

Polynomials used for 𝝋(𝝃) Hermite Legendre Laguerre Jacobi Krawtchouk-chaos Charlier-chaos Meixner-chaos Hahn-chaos

Range of random variable (−∞, ∞) [𝑎, 𝑏] [0, ∞) [𝑎, 𝑏] {0,1, ..., N} {0, 1, 2, ...} {0, 1, 2, ...} {0, 1, ..., N}

Note: A few important questions arise here. a) Why do we need the orthogonality of the polynomials? b) Should we always use Hermite polynomials as an orthogonal basis? Details for the orthogonal properties of the functional basis are explained in Appendix A. The orthogonality of the polynomials enables us to evaluate the spectral expansion of a random variable by projecting onto the basis function. This orthogonal projection minimizes the meansquare error (MSE) in the random parameter (in this case, 𝑘) on the space spanned by (𝜑𝑖 )𝑛𝑖=0. Furthermore, the choice of the orthogonal polynomials depends on the nature of the Probability Distribution Function (PDF) of the random parameter being considered. Several other functional representations are also used and explained in Table (1) for continuous and discrete PDFs. The convergence rate strongly depends on the choice of basis functions (see Xiu and Em Karniadakis 2003b).

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Dependency between the order of polynomial and the type of PDF for a higher convergence rate As an example, we may consider a one-dimensional ordinary differential equation 𝑑𝑢(𝑡) 𝑑𝑡

= −𝛼𝑢(𝑡) , 𝑢(0) = 1 and 𝛼 is an uncertain parameter

following a standard Gaussian PDF i.e., 𝛼~𝒩(0,1). Figure (3) shows the convergence rate based on the choice of orthogonal polynomials. It is clear from the graph that the convergence rate is faster in case of using Hermite polynomials for a Gaussian PDF of a random variable.

Figure 3. Convergence of a different orthogonal polynomial based Gaussian PDF.

Multi-Dimensional PCE Let us again consider Equation (1), in which (𝑥, 𝑡) ∈ 𝐷 × (0. 𝑇], 𝐷 ∈ ℝ𝑑 , 𝜉 ∈ 𝐿2 (Ω) whose elements have finite variance of the second order i.e., 𝐸(𝜉 2 ) < ∞, 𝐸 denotes the expectation operator and is measurable w. r. to 𝜎 − algebra, also the set 𝐿2 (Ω) is equipped with the inner product
= 𝐸(𝜉1 𝜉2 ); the function ℊ: 𝑥 × 𝑡 × 𝜉 → ℝ is a forcing term and 𝑢: 𝑥 × 𝑡 → ℝ is a solution of the Equation (1) which defines a random field over the spatial and temporal variables due to random variable vector 𝜉. Define a set of multi-indices 𝛼 = (𝛼1 , 𝛼2 , … … … … , 𝛼𝑖 , … … ) ∈ ℕ𝑁 𝑐 = : ℑ is a set of non-negative integers with a finite number of non-zero entries. Thus, ℑ = (𝛼𝑖 , 𝑖 ≥ 1), 𝛼𝑖 ∈ (0,1,2, … ), 𝛼 = ∑∞ 𝑖=1 𝛼𝑖 < ∞ The stochastic solution of Equation (1), using polynomial chaos expansion, is expressed as: 𝑢 ≈ 𝑢(𝑥, 𝑡) = ∑𝛼∈ℑ 𝑢𝛼 (𝑥, 𝑡)𝜑𝛼 (𝜉)

(14)

where 𝜑𝛼 (𝜉) is a complete orthonormal system in 𝐿2 (Ω, ℱ, Ρ) generally known as trial basis, and 𝑢𝛼 (𝑥, 𝑡) are the spatial and temporal deterministic coefficients, known as polynomial chaos coefficients, satisfying a system of deterministic equations, and can be obtained via different numerical methods. The choice of the trial basis 𝜑𝛼 (𝜉) is crucial in dealing with SPDEs and is dependent on the type of distribution of the random variables. It is not influenced by the type of system. If random variables are independent, their joint PDF corresponds to the product of the distributions of each random variable. The choice of the trial basis distinguishes homogeneous chaos from generalized polynomial chaos (gPC). For homogeneous chaos, Hermite polynomials are used to form basis functions in terms of Gaussian random variables. In contrast, as shown in Table (1), in gPC several other random variables are used. These are based on the Wiener –Askey scheme (Xiu et al. 2002). There is more flexibility in choosing polynomial in gPC rather than using only Hermite polynomials. For example: in equation (14), a combination of Jacobi polynomials with beta random variables (Table (1) summarizes the combination required to satisfy the property of a complete orthonormal system and follows Cameron-Martin theorem. For multi-indices 𝛼 = (𝛼1 , 𝛼2 , … … ) ∈ ℑ, a multi-variable orthogonal polynomial for a Gaussian random variable 𝜉 = (𝜉1 , 𝜉2 , … … … ) is defined as: 𝜑𝛼 (𝜉) = ∏∞ (15) 𝑖=1 𝐻𝛼𝑖 (𝜉𝑖 )

Mathematical Modelling of Complex Systems …

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where 𝐻𝛼𝑖 (𝜉𝑖 ) are the normalized Hermite polynomials of order 𝑛. It is worth mentioning here that, 𝐸(𝐻𝛼 (𝜉)) = 0, 𝛼 ≠ 0 and 𝐸(𝐻0 (𝜉)) = 𝐻0 (𝜉) = 1. The 𝜑𝛼 (𝜉)’s are n-dimensional polynomials that are mutually orthogonal with respect to the probability density function 𝜌(𝜉) and ∞

< 𝜑𝑖 (𝜉), 𝜑𝑗 (𝜉) >= ∫−∞ 𝜑𝑖 (𝜉)𝜑𝑗 (𝜉)𝜌(𝜉)𝑑𝜉 =∥ 𝜑𝑖 ∥2 𝛿𝑖𝑗 (16) where 𝛿𝑖𝑗 is given by equation (10). Since    , R.H.S of Equation (15) has finite terms, we need to choose the optimal truncation to solve equation (4) numerically based on   n and the order of , O( )  p , where n and p are the highest order of

Hermite polynomials and the maximum number of Gaussian random variables respectively. Thus, the truncated solution of Equation (14) is given as 𝑢(𝑥, 𝑡) = ∑𝛼∈ℑ𝑛,𝑝 𝑢𝛼 (𝑥, 𝑡)𝜑𝛼 (𝜉)

(17)

The total number of terms in the above expansion is 𝑝+𝑛 𝑝+𝑖−1 ∑𝑛𝑖=0 ( )=( 𝑝 ) 𝑖

(18)

For example, if 𝑛 = 4 and 𝑝 = 6, total elements of a truncated basis 𝜑𝛼 (𝜉) is

10! i.e., 210 elements. Some are calculated in Table 2. 6!4!

The number of terms will increase very fast with both n and p . The trial base 𝜑𝛼 (𝜉) based on 𝜉𝑖 with a higher subscript 𝑖 is less important than based on 𝜉𝑖 with a lower subscript 𝑖. Thus, it is much better to use lower order Hermite polynomials 𝜉𝑖 with higher subscripts instead of using the same order for all 𝜉𝑖 , 𝑖 ≤ 𝑝. Luo (2006) adopts a sparse index truncation scheme.

346

Parul Tiwari, Don Kulasiri and Sandhya Samarasinghe Table 2. Orthonormal set of Hermite polynomials for a multi-index 

 0 1 1 1 2 2 2 2 3 3 3 3 3 4 4 4

 (0,0,0,0,0,0) (1,0,0,0,0,0) (0,1,0,0,0,0) (0,0,1,0,0,0) and so on (2,0,0,0,0,0) (0,2,0,0,0,0) (1,1,0,0,0,0) (0,1,1,0,0,0) and so on (3,0,0,0,0,0) (0,3,0,0,0,0) (1,1,1,0,0,0) (2,1,0,0,0,0) (0,0,2,1,0,0) and so on (4,0,0,0,0,0) (2,2,0,0,0,0) (3,1,0,0,0,0) and so on

𝝋𝜶 (𝝃) 𝜑0 (𝜉) = 1 𝜑1 (𝜉) = 𝐻1 (𝜉1 ) = 𝜉1 𝜑1 (𝜉) = 𝐻1 (𝜉2 ) = 𝜉2 𝜑1 (𝜉) = 𝐻1 (𝜉3 ) = 𝜉3 𝜑2 (𝜉) = 𝐻2 (𝜉1 ) = 𝜉12 − 1 𝜑2 (𝜉) = 𝐻2 (𝜉2 ) = 𝜉22 − 1 𝜑2 (𝜉) = 𝐻1 (𝜉1 )𝐻1 (𝜉2 ) = 𝜉1 𝜉2 𝜑2 (𝜉) = 𝐻1 (𝜉2 )𝐻1 (𝜉3 ) = 𝜉2 𝜉3 𝜑3 (𝜉) = 𝐻3 (𝜉1 ) = 𝜉13 − 3𝜉1 𝜑3 (𝜉) = 𝐻3 (𝜉2 ) = 𝜉23 − 3𝜉2 𝜑3 (𝜉) = 𝐻1 (𝜉1 )𝐻1 (𝜉2 )𝐻1 (𝜉3 ) = 𝜉1 𝜉2 𝜉3 𝜑3 (𝜉) = 𝐻2 (𝜉1 )𝐻1 (𝜉2 ) = (𝜉12 − 1) 𝜉2 𝜑3 (𝜉) = 𝐻2 (𝜉3 )𝐻1 (𝜉4 ) = (𝜉32 − 1) 𝜉4 𝜑4 (𝜉) = 𝐻4 (𝜉1 ) = 𝜉14 − 6𝜉12 + 3 𝜑4 (𝜉) = 𝐻2 (𝜉1 )𝐻2 (𝜉2 ) = (𝜉12 − 1)(𝜉22 − 1) 𝜑4 (𝜉) = 𝐻3 (𝜉1 )𝐻1 (𝜉2 ) = (𝜉13 − 3𝜉1 )𝜉2

Calculation of PCE Stochastic governing equations are transformed into a set of deterministic equations by applying intrusive and non-intrusive projection methods and then discretized via standard numerical techniques. Here PC/gPC serve as a complete basis to represent stochastic processes.

The Intrusive Projection Method (The Galerkin Projection Method) In intrusive approaches, random variables are generally approximated using a predefined PDF of another random variable. This provides a set of coupled equations (Ghanem 1998, Todor and Schwab 2007). For a stochastic diffusion equation, this occurs when the diffusivity (permeability or conductivity) is modelled as a random field. The

Mathematical Modelling of Complex Systems …

347

stochastic Galerkin Projection method applies the model equations to obtain the output. The main objective of this method is to minimize the errors measured by the residual of equations. It presumes knowledge of model equations and PDF of PC expansion. Applying the Galerkin projection method to Equation (1), Step 1: Express the random variables 𝑘 and 𝑢 as polynomial chaos expansion as 𝑘 = ∑𝑛𝑖=0 𝑘𝑖 𝜑𝑖 (𝜉), 𝑢(𝑥, 𝑡) = ∑𝑛𝑗=0 𝑢𝑗 (𝑥, 𝑡)𝜑𝑖 (𝜉)

(19)

Step 2: Substituting the above expansions in Equation (1), we get ∑𝑛𝑗=0

𝜕 𝑢 (𝑥, 𝑡)𝜑𝑗 (𝜉) 𝜕𝑡 𝑗

= ∑𝑛𝑖=0 𝑘𝑖 𝜑𝑖 (𝜉)

𝜕2 ∑𝑛 𝑢 (𝑥, 𝑡)𝜑𝑗 (𝜉) 𝜕𝑥 2 𝑗=0 𝑗

(20)

Step 3: Projecting this equation onto each polynomial basis 𝜑𝑙 (𝜉), 𝑙 = 0, … , 𝑛; 𝜕 𝑢 (𝑥, 𝑡)𝜑𝑗 (𝜉)𝜑𝑙 (𝜉) = 𝜕𝑡 𝑗 𝜕2 ∑𝑛𝑖=0 ∑𝑛𝑗=0 𝑘𝑖 2 𝑢𝑗 (𝑥, 𝑡)𝜑𝑖 (𝜉)𝜑𝑗 (𝜉)𝜑𝑙 (𝜉) 𝜕𝑥

∑𝑛𝑗=0

(21)

Step 4: Taking expectation and utilizing the orthogonality of basis functions, we obtain < 𝜑𝑗 2 >

𝜕𝑢𝑙 𝜕𝑡

= ∑𝑛𝑖=0 ∑𝑛𝑗=0 𝑘𝑖

𝜕 2 𝑢𝑙 𝜕𝑥 2

< 𝜑𝑖 𝜑𝑗 𝜑𝑙 >

(22)

defining, 𝑎𝑗𝑙 = ∑𝑛𝑖=0 𝑘𝑖 𝑒𝑖𝑗𝑙 , where 𝑒𝑖𝑗𝑙 =< 𝜑𝑖 𝜑𝑗 𝜑𝑙 > Re-writing the above equation, < 𝜑𝑗 2 >

𝜕𝑢𝑗 𝜕𝑡

= ∑𝑛𝑗=0

𝜕 2 𝑢𝑗 𝜕𝑥 2

𝑒𝑖𝑗𝑙 , ∀ 𝑗 ∈ [0, 𝑛]

(23)

348

Parul Tiwari, Don Kulasiri and Sandhya Samarasinghe The coefficients 𝑒𝑖𝑗𝑙 , along with < 𝜑𝑗 2 >, can be determined using the

definition of 𝜑𝑗 . Equation (23) is a set of (n+1) coupled deterministic differential equations and can be solved using any traditional methods, including the Gauss-Elimination or Gauss-Seidel methods. However, this approach may be challenging in the case of non-linearity in the system. We also need to solve the set of (n+1) ordinary differential equations per state variable. In addition, the approximation of time derivatives involves the errors which could lead to long time integration problems. Thus, there may be issues related to specific stability conditions and stabilization schemes for the numerical codes.

The Non-Intrusive Projection Method Another approach for dealing with PCE is the non-intrusive approach. Here simulations are used as black boxes. The non-intrusive spectral projection method (NISP) relies on computing the inner products. In contrast to Galerkin projection, non-Intrusive methods focus mainly on the approximation of desired statistics such as the mean, variance and PDF. A researcher does not need full knowledge of the model equations, nor all of the model unknowns. Non-intrusive methods reuse code and rely on deterministic computations. Broadly, these methods include random sampling-based projection methods such as Monte Carlo, Quasi-Monte Carlo and collocation methods (Le Maıtre et al. 2002, Rajabi et al. 2015, Ghiocel and Ghanem 2002). PCE coefficients are calculated by generating a large number of random variables according to the model strategy. The computational procedure for non-intrusive methods is the same as of intrusive projection methods except when it comes to calculating the inner product. The NISP approach uses inner products to project the response against each basis function. Orthogonality properties of polynomials are used to obtain each coefficient. The inner product can be evaluated numerically either using random sampling approach for the integral or by applying quadrature formula for deterministic sampling. A classic method used in deterministic

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sampling is the Gauss-Hermite quadrature method. It uses a different number of quadrature nodes. For understanding, to determine the coefficients 𝑢𝑖 , 𝑢𝑖 =



1

= ∫𝐷 𝑢(𝜉)𝜑𝑖 (𝜉)𝑃(𝜉)𝑑𝜉 𝑖

(24)

where 𝑃(𝜉) is the PDF of the chosen random variable. The formula for evaluating the above integral isa) Random sampling: 1

∫𝐷 𝑢(𝜉)𝜑𝑖 (𝜉)𝑃𝜉 (𝜉)𝑑𝜉 = 𝑠 ∑𝑛𝑠=1 𝑢(𝜉𝑠 )𝜑𝑖 (𝜉𝑠 )

(25)

b) Deterministic sampling/The quadrature rule: 𝑞

∫𝐷 𝑢(𝜉)𝜑𝑖 (𝜉)𝑃𝜉 (𝜉)𝑑𝜉 = ∑𝑠=1 𝑞𝑠 𝑢(𝜉𝑠 )𝜑𝑖 (𝜉𝑠 )

(26)

There is another class of non-intrusive methods. The regression-based method relies on a selected set of points instead of evaluating the integrals (Webster and Sokolov, 2000). More advanced strategies are employed to make these methods computationally effective. To reduce the complexity, smarter tensorization strategies are applied using Sparse grid methods. Total degree truncation of the PC basis is used in place of partial degree truncation. The Smolyak formula is used for quadrature interpolation.

Implementation of Polynomial Chaos Expansion Discretization Scheme in Time and Space To solve the governing equation, we need to discretize it in space and time. For a given space discretization, grid steps are taken as 0 = 𝑥0 < 𝑥1
0, 𝑥 ∈ ℝ

(50)

This is the simplified model of turbulence which has been studied in the literature by several authors (Kerner 1999; Herman and Gardels 1963; Chowdhury, Santen, and Schadschneider 2000) who take additive spacetime white noise 𝑊̇ (𝑥, 𝑡) as the driving force. The mathematical framework that is being used for Equation (50) cannot be used to solve the equation with multiplicative white noise (Istvan Gyongy and Nualart 1999). It is given as 𝜕𝑢 𝜕𝑡

2

𝜕𝑢 𝜕 𝑢 𝜕𝑢 + 𝑢 𝜕𝑥 = 𝜕𝑥 2 + 𝜕𝑥 𝑊̇ (𝑥, 𝑡) , 𝑡 > 0, 𝑥 ∈ ℝ

(51)

The reason is that the solution, if it exists, is not regular enough to explain pointwise multiplications of 𝑢

𝜕𝑢 𝜕𝑥

and

𝜕𝑢 𝜕𝑥

𝑊̇ (𝑥, 𝑡).

But, using the concept of Wick product, the study of Equation (38) is possible in the framework of white noise theory. The Wick product preserves the martingale property and provides a new approach to the calculation of Ito.

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Parul Tiwari, Don Kulasiri and Sandhya Samarasinghe

The Wick Product in Physics The following representation of the Wick product in physics is based on Gjessing’s explanation (Gjessing et al. n.d.). If (Ω, ℱ, Ρ) is a probability space, 𝑋 is a random variable. Let us define a formal derivative of 𝑋 by considering a formal power series in 𝑋 as 𝜕 𝑛 (∑∞ 𝑛=0 𝑎𝑛 𝑋 ) 𝜕𝑋

𝑛 = ∑∞ 𝑛=0(𝑛 + 1)𝑎𝑛+1 𝑋

(52)

The recursive formula to find the Wick product of 𝑋 𝑚 is denoted as : 𝑋 𝑚 : is defined as 𝜕 (: 𝑋 𝑚 : ) 𝜕𝑋

= 𝑚(: 𝑋 𝑚−1 : ),

: 𝑋0: = 1

and

𝐸[: 𝑋 𝑚 : ] = 0, 𝑚 =

1,2, … … … Hence, : 𝑋 ∶= 𝑋 − 𝐸(𝑋) 2

: 𝑋 2 ≔ 𝑋 2 − 2𝑋𝐸(𝑋) − 𝐸(𝑋 2 ) + 2(𝐸(𝑋)) : 𝑋 3 ≔ 𝑋 3 − 3𝑋 2 𝐸(𝑋) − 3𝑋𝐸(𝑋 2 ) + 6𝑋𝐸(𝑋)2 − 𝐸(𝑋)3 + 6𝐸(𝑋)𝐸(𝑋 2 ) − 6𝐸(𝑋 3 ) {

The Wick Product in Stochastic Analysis If (𝐒)∗ is a Hida distribution space and F(ω) and G(ω) are two elements of (𝐒)∗ defined as 𝐹(ω) = ∑𝛼 𝑎𝛼 𝐻𝛼

(53)

G(ω) = ∑𝛽 𝑏𝛽 𝐻𝛽

(54)

and

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363

where 𝛼 and 𝛽 are the multi-indices as defined in section 2.2, then the Wick product of F and 𝐺 is defined as 𝐹(ω) ◊ 𝐺(ω) = ∑𝛼,𝛽 𝑎𝛼 𝑏𝛽 𝐻𝛼+𝛽

(55)

and 𝐹(ω) ◊ 𝐺(ω) ∈ (𝐒)∗. A detailed explanation and the application of the Wick product can be found in Øksendal (1996), Hu and Yan (2009), Venturi et al. (2013).

Conclusion and Discussion Due to the unavailability of closed form solutions, it is challenging to obtain an efficient and accurate solutions for SPDEs. Where a closed form solution exists, it may not be able to study the characteristics of the solution efficiently. Mild solutions study the effect of perturbations in equations that can be formulated as abstract evolution equations. As the system is complex, solutions of the governing equations is a random field rather than a simple function. Random field also exhibits the implicit irregularity of the system. In this way, SPDEs are more capable of apprehending the behaviour of interesting phenomena under study. Further successful applications of PCE methods are in uncertainty optimization, stability control and in reliability engineering (L. Zhang et al. 2020; S. and S. 2006). These methods are helpful in finding an optimal and efficient way to develop the response surface of a model under uncertainty conditions. Nevertheless, intrusive approaches are more flexible even though the governing equations are mostly coupled (Xiu and Shen 2008b). Since the origin of PCE methods, several advancements are being done in terms of representation of random variables. One of the progresses is to decompose the stochastic space using local polynomial chaos expansions rather than global (Tryoen et al. 2010; le Maître and Knio n.d.). In this chapter, we discussed polynomial chaos expansion methods which has been widely studied and applied in different branches of science and engineering. Nevertheless, there is a need of exhaustive and detailed

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overviews of up to date and advanced methods in this field. In addition, due to high computational costs, there is also need of filling the gap between the academia and industry applications. With PCE, the random input function can be represented as a stochastic metamodel. The original PCE methods follow an intrusive approach and requires huge number of changes in available deterministic codes in order to analyse the model. Furthermore, full control is limited to model equations. Researchers have developed nonintrusive methods which do not modify the actual model. It is always not possible to obtain knowledge about the distribution of an uncertain parameter, thus we must transform that distribution into a standard known distribution and then apply PCE approaches. As an advancement of these methods, data driven polynomial chaos expansion is now being used to deal with the problems with large data bases.

Appendix A (i) Inner Product Let Ϝ ∈ ℝ and 𝑋 be an Ϝ vector space. A map  ,   : 𝑋 × 𝑋 → Ϝ is called an inner product of 𝑋 if a) < 𝑥 , 𝑥 > ≥ 0 for all 𝑥 ∈ 𝑋 and < 𝑥 , 𝑥 ≥ 0 iff 𝑥 = 0 [Positive definite] b) < 𝑥 , 𝑦 > = < 𝑦 , 𝑥 > for Ϝ = ℝ for all 𝑥, 𝑦 ∈ 𝑋 [Symmetric] c) < 𝑥 , 𝑦1 + 𝑦2 > =< 𝑥, 𝑦1 > +< 𝑥, 𝑦2 > for all 𝑥, 𝑦1 , 𝑦2 ∈ 𝑋 and < 𝑥 , 𝜆𝑦 > = 𝜆 < 𝑥 , 𝑦 > for all 𝑥, 𝑦 ∈ 𝑋 and 𝜆 ∈ Ϝ [Linear in 2nd argument] (ii) Norm If  ,   is an inner product, then norm of 𝑋 is defined as X

,

  x, x 

Mathematical Modelling of Complex Systems …

365

(iii) Hilbert Space If ( X , 

,

) is a Banach space, then ( X ,  ,  ) is called a Hilbert

space. (iv) Hermite polynomials and related properties Hermite polynomials 𝐻𝑛 (𝑥) satisty the following second order ordinary differential equation 𝑑2 𝑦 𝑑𝑥 2

𝑑𝑦

− 𝑥 𝑑𝑥 + 𝑛𝑦 = 0

and can be obtained using the three-term recurrence relation 𝐻𝑛+1 (𝑥) = 𝑥𝐻𝑛 (𝑥) − 𝑛𝐻𝑛−1 (𝑥) A few of the Hermite polynomials are plotted in Figure (7). Important properties of these polynomials area) Generating Function Hermite polynomials 𝐻𝑛 (𝑥) are the coefficients of of 𝑒

1 𝑥𝑡− 𝑡 2 2

𝑡𝑛 𝑛!

in the expansion

, that means 1 2

𝑒 𝑥𝑡−2𝑡 = ∑∞ 0 𝐻𝑛 (𝑥)

𝑡𝑛 𝑛!

b) Orthogonality Let ( X ,  ,  ) be an inner product space. Any 𝑥, 𝑦 ∈ 𝑋 are said to be orthogonal if < 𝑥 ,𝑦 > = 0

366

Parul Tiwari, Don Kulasiri and Sandhya Samarasinghe Hermite polynomials are orthogonal with respect to the kernel function

or weight function 𝑤(𝑥) = 𝑒 −𝑥

2 /2

. Orthogonality is given by



∫−∞ 𝐻𝑚 (𝑥)𝐻𝑛 (𝑥)𝑤(𝑥)𝑑𝑥 = √2𝜋 𝑛! 𝛿𝑚𝑛 0, 𝑚 ≠ 𝑛 and 𝛿𝑚𝑛 = { and is known as a Kronecker delta. 1, 𝑚 = 𝑛 c) Completeness For any given Hilbert space of functions 𝑓(𝑥) satisfying 



2

f ( x) w( x)dx   , Hermite polynomials form an orthogonal basis in



which the inner product is given as ∞ ̅̅̅̅̅̅ 𝑤(𝑥) 𝑑𝑥 and an orthogonal basis < 𝑓(𝑥), 𝑔(𝑥) > = ∫−∞ 𝑓(𝑥)𝑔(𝑥)

for 𝐿2 (𝑅, 𝑤(𝑥)𝑑𝑥) forms a complete orthogonal system. (v) Trigonometrical Basis functions 𝐵𝑖 (𝑠) are the orthonormal basis in 𝐿2 (Ω). The trigonometric sine and cosine functions can be used as basis functions. Cosine basis functions are given as: 2 ℎ

(𝑘−1)𝜋𝑠

√ cos ( 𝐵𝑖 (𝑠) =



) , 𝑖 ≥ 2, 𝑘 ≥ 2

1

{

√ , 𝑖 = 1, 𝑘 = 1 ℎ

(vi) Cameron Martin Theorem This theorem allows us to build a strong connection between a Brownian motion drift and a standard Brownian motion. If we have a Brownian motion drift and a given probability measure or the physical measure, then we can find another probability measure under which these original random motion with drift behaves as a standard Brownian motion.

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We can link a Brownian motion with drift and a standard Brownian motion just by changing the probability measure on the measuring space. The Cameron Martin Theorem guarantees that the polynomial chaos representation converges to a functional, in the 𝐿2 −sense, or in other words, to a limit in the mean sense, to the distribution that we have as long as the target distribution has a finite second moment.

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About the Editors M. Haider Ali Biswas, PhD Professor Biswas is currently affiliated with Khulna University, Bangladesh as a Professor of Mathematics under Science Engineering and Technology School and he served as the Head of Mathematics Discipline from 2015 to 2018. Prof. Biswas obtained his BSc (Honors) in Mathematics and MSc in Applied Mathematics in the year 1993 and 1994 respectively from the University of Chittagong, Bangladesh, MPhil in Mathematics in the year 2008 from the University of Rajshahi, Bangladesh and PhD in Electrical and Computer Engineering from the University of Porto, Portugal in 2013. He has more than 22 years of teaching and research experience in the graduate and postgraduate levels at different public universities in Bangladesh. He published three books, seven book chapters and more than 200 research papers in peer reviewed journals and international conferences. Biswas supervised (is supervising) more than 80 undergraduate students (Undergraduate Project Thesis), 30 MSc Students (MSc Thesis and Project Thesis), 3 MPhil Students and 5 PhD Students at Different Public Universities including Khulna University in Bangladesh. Prof. Biswas has worked on several R&D projects in home and abroad as PI and/or Researcher. In particular, he conducted several research projects funded by Khulna University Research Cell, the Ministry of Science and Technology, Bangladesh, University Grants Commission of Bangladesh and The World Academy of Science (TWAS), Trieste, Italy. His present research interests include Optimal Control with Constraints, Nonsmooth Analysis, ODEs and Dynamical Systems, Mathematical Modeling, Mathematical Ecology, Environmental modeling and Climate change, Mathematical Biology and Biomedicine, and

374

About the Editors

Epidemiology of Infectious Diseases. Since the last ten years, Prof. Biswas has been working on the applications of mathematical models for designing and implementing those to real life problems, especially for the sustainable/optimal management under the changing environment due to global warming. He is the life/general members of many Societies around the world including Institute of Mathematics and its Applications (IMA), UK and European Mathematical Society (EMS). Dr. Biswas is the founder member of Mathematical Forum Khulna and served as the General Secretary of the Forum in 2013-2015. Professor Biswas has delivered numerous Keynote speeches, served as invited, plenary and panel speakers at different international conferences/seminars/workshops in home and abroad. Professor Biswas was nominated as the Member of the Council of Asian Science Editors (CASE) for 2017-2020 and the Associate Member of the Organization for Women in Science for the Developing World (OWSD) in 2017. Professor Biswas has been serving as a founder President of Bangladesh Society for Mathematical Biology (BSMB) since 2020. Corresponding Editor’s Email: [email protected]

M. Humayun Kabir, PhD Dr. Kabir completed BSc (Hons.) and MSc from the Department of Mathematics, Jahangirnagar University, Bangladesh. He also received MPhil in Applied Mathematics from the same Department. Dr. Kabir obtained MSc in Industrial & Applied Mathematics from the Department of Mathematics, Università Degli Studi di Milano, Italy under the Erasmus Mundus Europe Asia (EMEA) program. He achieved PhD in Mathematical Sciences from the Meiji Institute for Advanced Study of Mathematical Sciences (MIMS), Meiji University, Japan as a JSPS-GCOE research Fellow under the supervision of Professor Emeritus Masayasu Mimura. Dr. Kabir has been working as a faculty member in the Department of Mathematics, Jahangirnagar University, Bangladesh since 2009. Dr. Kabir received an Excellent Research Award at the 8th Taiwan-Japan Joint Workshop for Young Scholars in Applied Mathematics 2017 that was hosted by Hiroshima University, Japan. He has been a Visiting Research Fellow (MIMS Research Promoter) of the Center for Mathematical Modeling & Applications (CMMA), Meiji University, Japan since April 2018. His research interest includes a mathematical model-aided understanding of biological phenomena and numerical simulation. He

About the Editors

375

delivered numerous invited and contributory talks in many international conferences and workshops. He published a good number of articles in reputed journals and conference proceedings. Furthermore, he is actively involved in the editorial boards of many journals. He is also serving as the founder secretary of Bangladesh Society for Mathematical Biology.

Index # 4 DOF robotic manipulator, 150, 151, 152, 153

A ABE fermentation, 206, 207, 209, 211 accurate models, 297 activation energy, 11, 48, 64, 65 activation entropy, 68 active compound, 293, 326 algorithm, 4, 51, 61, 76, 104, 106, 107, 108, 112, 113, 126, 133, 134, 135, 136, 137, 138, 140 annual cost savings, 117, 118, 123 automation of production, 12, 75, 76, 83 automation of production systems, 83

B basic reproduction number, 180, 183, 184, 223, 230, 234, 239 biological systems, 5, 314, 334 biomass, 11, 47, 53, 54, 56, 57, 58, 60, 61, 63, 64, 72, 209, 213, 214, 215, 216, 217, 218, 219, 222, 299, 300 Brownian motion, 334, 353, 354, 356, 366, 367 butanol, 14, 206, 208, 209, 210, 211, 212, 213, 214, 215, 216, 218, 219, 220, 222

C capacity utilization, 110 capital investments, 83, 85, 89, 90, 91, 94, 96, 99, 100, 117, 123 chemical reactor, 220 climate change, 4, 268, 276, 304, 316, 319, 324, 328 climatic factors, 17, 248, 249, 274, 312 condition for efficiency of robotics, 120

conduction, 275, 277, 278, 279, 280, 289, 290 conductivity, 17, 274, 278, 279, 280, 285, 289, 290, 347 continuous bioreactor, 220, 221 control measures, vii, 14, 176, 199, 225, 245, 309 crop management, 276, 300 crop modeling, 274 current company costs, 92

D Dengue fever, 15, 223, 224, 226, 243, 244, 245 differential equations, viii, 54, 269, 339, 348 diffusion, 277, 280, 281, 282, 283, 284, 285, 291, 309, 319, 346, 353, 356 diffusivity, 17, 274, 280, 285, 290, 339, 340, 346 digital production, 12, 75, 76, 78, 82, 83 digital transformation, 12, 75, 77, 78, 81, 82, 105, 142, 143, 148 direct material costs, 86 discrete programming, 76, 105, 106, 137 dynamic modeling, 81

E economic and mathematical models, 12 economic efficiency, 76, 77, 79 economic mathematical models, 79, 138 Escherichia coli DH5, 12, 48, 53, 69, 70, 71, 72 evapotranspiration, 277, 301, 302, 319, 321, 322, 323, 326, 327, 328, 329, 330, 332

F fault diagnosis, viii, 11, 23, 24, 25, 37, 38, 41, 43, 44

378

Index

flexible robotics, 76 food processing, 274, 277, 315 food-simulating fluids, 275

G Gaussian random variables, 18, 344 Gibbs energy, 12, 48

H heat stress, 48 heat transfer, 275, 277, 278, 279, 280, 289, 290, 291, 319 heuristic algorithm, 76, 137 highly automated systems, 127

I innovation, 13, 76, 81, 84, 86, 88, 91, 93, 96, 97, 99, 100, 105, 106, 107, 118, 120, 125, 126, 127, 141, 150, 314 innovative modernization projects, 76, 77 investment costs, 85

J Johnson method algorithm, 106

K kinetic behavior, 47 kinetic model, 14, 49, 53, 206, 208, 219, 287, 324, 328 kinetic parameters, 49, 63, 69, 209 kinetic production, 48

L life cycle, 16, 77, 132, 139, 143, 226, 248, 249, 267 linear programming, 125, 140, 307 Lyapunov function, 10

M malaria, 224, 225, 242, 243, 248, 251, 253, 267, 268, 269, 270, 271

management strategy, 79 mathematical model(s), vii, viii, x, 1, 2, 4, 5, 6, 11, 12, 13, 14, 17, 25, 29, 32, 33, 41, 47, 49, 69, 75, 76, 77, 78, 79, 80, 82, 84, 86, 87, 88, 90, 94, 95, 96, 97, 98, 100, 102, 105, 113, 118, 122, 123, 125, 126, 137, 138, 139, 140, 141, 150, 151, 171, 176, 179, 180, 208, 215, 218, 222, 248, 269, 270, 271, 274, 275, 276, 277, 285, 286, 290, 293, 294, 296, 297, 303, 305, 307, 308, 309, 313, 314, 315, 317, 318, 319, 320, 322, 326, 328, 330, 331, 335, 336, 374 mathematical modelling, 1, 2, 5, 12, 13, 76, 150, 151, 322, 335, 336 modeling methodology, 274

N Nipah virus, 13, 175, 177, 198, 200 numerical simulations, 16, 191, 248

O operative-calendar planning, 105 optimal control, vii, 14, 176, 179, 180, 189, 191, 199, 201, 202, 208 optimal production program, 76, 105 optimal ratio between flexibility and productivity, 108 optimality criterion, 12, 91, 93, 95, 96, 97, 99 optimization strategies, 274 orthogonal polynomials, 334, 338, 342, 343 orthogonality, 341, 342, 347, 360

P pathogens, 176, 225, 308, 310, 314, 316, 325 photosynthesis, 294, 304, 305, 306, 331 plant growth, 274, 276, 277, 294, 297, 298, 299, 304, 320, 332 plant growth and development, 297 plasmid DNA production, 47, 69, 71 polynomial chaos expansion, 338, 339, 357, 363, 364 polynomial controller, 218

Index process innovations, 84, 88, 109, 113, 126 processing funding capacity, 88 product inhibition, 56, 206, 207, 220 production plan, 82, 104, 106, 107, 108, 208

Q quantitative approach, 275, 277

R robot load coefficient, 128, 130 robotics, viii, 6, 12, 76, 77, 79, 80, 82, 85, 113, 114, 115, 117, 119, 120, 121, 122, 123, 124, 125, 132, 138, 139, 140, 171, 174

S SEIR model, 181, 201, 245 sensitivity analysis, 11, 48, 270 simulation, viii, xi, 4, 13, 17, 76, 140, 150, 151, 170, 171, 176, 199, 219, 234, 268, 271, 273, 274, 294, 314, 316, 318, 320, 336, 374 simulation background, 274 stability, 6, 7, 8, 10, 14, 15, 18, 64, 103, 180, 185, 192, 199, 203, 206, 210, 211, 213, 216, 217, 248, 253, 262, 267, 269, 336, 348, 363 stability analysis, 14, 185, 199, 206, 217 steady state, 6, 7, 208, 210, 213, 217, 248, 270 stochastic partial differential equations, 334 stochasticity, 248

technological innovation, 12, 76, 81, 82, 85, 86, 89, 90, 92, 93, 94, 99, 115, 126, 139 temperature, 11, 12, 15, 16, 42, 48, 49, 57, 58, 59, 63, 64, 65, 66, 67, 68, 69, 71, 72, 73, 223, 224, 226, 233, 234, 239, 241, 242, 244, 248, 249, 269, 278, 279, 280, 283, 285, 287, 289, 290, 291, 292, 302, 303, 304, 306, 310, 311, 320, 323, 328, 330, 331, 339 thermodynamic parameters, 12, 48, 49, 57, 64, 67, 68 total capital investments, 92 trajectory-tracking control, 150

V variables, 13, 15, 33, 34, 49, 66, 76, 82, 84, 86, 94, 95, 125, 137, 139, 150, 152, 168, 190, 206, 209, 210, 211, 215, 216, 217, 218, 219, 253, 268, 274, 275, 280, 291, 293, 296, 298, 302, 303, 304, 309, 310, 311, 312, 313, 315, 334, 336, 338, 344, 345, 346, 347, 348, 351, 356, 363 variations, 48, 84, 253, 257, 267, 268, 354 vector, 15, 16, 43, 70, 71, 105, 210, 223, 225, 226, 227, 233, 243, 248, 249, 252, 253, 270, 271, 308, 309, 326, 344, 364 virus infection, 200, 201, 202 viruses, viii, 224, 225

W wick product, 334

X xylem, 326

T target efficiency function, 101, 131

379

Z zinc, 222