Modeling and Simulation of Infectious Diseases: Microscale Transmission, Decontamination and Macroscale Propagation 3031180526, 9783031180521

Table of contents :
Preface
Contents
About the Author
List of Figures
1 Preliminaries: Basic Mathematics, Optimization and Machine-Learning
1.1 Elementary Notation and Mathematical Operations
1.1.1 Vectors, Products and Norms
1.1.2 Basic Linear Algebra
1.1.3 Integral Transformations
1.2 Temporal Discretization Methods
1.2.1 Isolating a Single Particle
1.3 Basic Machine-Learning and Optimization
1.3.1 Gradient-Based Methods
1.3.2 Difficulties
1.4 Genetic-Based Machine-Learning Algorithm
1.4.1 Algorithmic Specifics
2 Part 1: Macroscale Disease Propagation
2.1 Introduction
2.1.1 Classical Basic Models
2.1.2 SIR Sub-population Models
2.1.3 Generalization of the SIR Family of Models
2.1.4 Agent-Based Models and Objectives
2.2 Direct Agent-Based Interaction Models
2.2.1 Agent-to-Agent Interaction and Rules of Engagement
2.2.2 Algorithm
2.2.3 Computational Acceleration
2.3 A Model Problem
2.4 Extensions
2.4.1 An Example of a Swarm Formulation
2.5 An Algorithm for Movement in a Region
2.5.1 Preliminary Numerical Example
3 Part 2: Microscale Disease Transmission and Ventilation System Design
3.1 Introduction
3.2 Analytical Characterization: Simplified Stokesian Model
3.2.1 Analysis of Particle Velocities
3.2.2 Analysis of Particle Positions
3.2.3 Settling (Airborne) Time
3.3 Computational Approaches for More Complex Models
3.3.1 More Detailed Characterization of the Drag
3.3.2 Simulation Parameters
3.4 Ventilation System Design
3.4.1 Assumptions
3.4.2 Incorporation of Masks
3.4.3 Incorporation of Vents
3.4.4 Overall Model
3.5 Genetic-Based Machine-Learning Ventilation Optimization
3.5.1 Model Problem
3.6 Summary and Extensions
4 Part 3: Ultraviolet Viral Decontamination
4.1 Introduction
4.1.1 Objectives
4.2 Electromagnetic Energy Propagation
4.2.1 Beam-Ray Decomposition
4.2.2 Reflection and Absorption of Rays
4.3 Electromagnetic Wave Propagation and Rays
4.3.1 Plane Harmonic Wave Fronts
4.3.2 Natural (Random) Electromagnetic Energy Propagation
4.3.3 Reflection and Absorption of Energy-Fresnel Relations
4.3.4 Reflectivity
4.4 Model Problem and Response Trends
4.4.1 Tracking of Beam-Decomposed Rays
4.4.2 Test Surface
4.5 Numerical/Quantitative Examples
4.6 Summary and Discussion
5 Part 4: Vaccine Design and Immune-System Response
5.1 Introduction
5.1.1 Brief History
5.1.2 Types of Vaccines
5.1.3 Vaccine Efficacy
5.1.4 Objectives of This Work
5.2 A Flexible Immune-Response Digital-Twin
5.3 Rapid Voxel Based Computation
5.4 Numerical Simulation of the Coupled System
5.4.1 Discretization of the c- and s-Fields
5.4.2 Iterative (implicit) Solution Method
5.5 Operation Counts in a Voxel-Based Method
5.6 Numerical Examples
5.7 Genetic-Based Machine-Learning Framework
5.7.1 Algorithmic Settings
5.7.2 Parameter Search Ranges and Results
5.8 Discussion and Summary
Epilogue
Appendix A Artificial Neural Networks
References

Citation preview

Tarek I. Zohdi

Modeling and Simulation of Infectious Diseases Microscale Transmission, Decontamination and Macroscale Propagation

Modeling and Simulation of Infectious Diseases

Tarek I. Zohdi

Modeling and Simulation of Infectious Diseases Microscale Transmission, Decontamination and Macroscale Propagation

Tarek I. Zohdi University of California Berkeley, CA, USA

ISBN 978-3-031-18052-1 ISBN 978-3-031-18053-8 (eBook) https://doi.org/10.1007/978-3-031-18053-8 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

In Remembrance: Dedicated to my father, Magd E. Zohdi (1933–2020) and the millions of victims of the COVID-19 pandemic.

Preface

The COVID-19 pandemic that started in 2019–2020 has been responsible for millions of deaths and has led to a huge interest in modeling and simulation of infectious diseases. This monograph is an outgrowth of this era. In order to make this monograph as concise and accessible as possible to a wide audience, it is comprised of a short introductory chapter on basic mathematics, optimization and machine learning and then four subsequent parts that focus on aspects of modern modeling and simulation tools for the analysis of infectious diseases. Specifically, the monograph is organized as follows: • Preliminaries-Basic Mathematics, Optimization and Machine learning: The systems encountered in this field of study invariably lead to models with differential equations that have many parameters. This requires tools for numerical simulation and parameter identification, which are formulated as inverse/optimization problems for cost/error function minimization. The cost functions associated with parameter identification for complex physical systems are usually nonconvex and nonsmooth, making application of gradient-based methods extremely difficult or impossible. This motivates nonderivative, machine learning-based, algorithms. • Part 1-Macroscale Disease Propagation: The increase in readily available computational power raises the possibility that direct agent-based modeling can play a key role in the analysis of epidemiological population dynamics. Specifically, Part 1’s objective is to first introduce the reader to classical qualitative approaches for disease spread and then to present more modern methods, based on agent-based methods to investigate the emergent structure of SIR-type (Susceptible–Infected–Removed/Recovered) populations, on a global planetary scale. As an illustrative model problem, a planet-wide system is developed, based on interaction between discrete entities (agents), where each agent on the surface of the planet is initially uninfected. Infections are then seeded on the planet. Contracting an infection depends on the characteristics of each agent, i.e., their susceptibility and contact with the seeded, infected, agents. Agent mobility on the planet is dictated by social policies, for example “shelter in place,” and “complete lockdown”. The global population is then allowed to evolve according to infected

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states of agents, over many generations, leading to an SIR population. This chapter follows work found in Zohdi [173]. • Part 2-Microscale Disease Transmission and Ventilation System Design: One of the central issues in the propagation of infectious diseases is the potential infection zone produced by a respiratory emission, such as a virus-laden cough. In Part 2, mathematical models are developed to simulate the progressive time evolution of the distribution of particles produced by respiratory emissions. The models ascertain the range, distribution and settling time of the particles under the influence of gravity and drag from the surrounding air, which are needed for constructing social distancing policies and workplace protocols. This leads to another key issue—the design of ventilation systems to mitigate airborne transmission of infectious diseases, produced by respiratory emissions. Accordingly, the model is then extended to couple: – the progressive time evolution of the spatiotemporal distribution of particles produced by respiratory emissions, – the use of masks impede respiratory particle emission propagation and – the use of genetic-based machine learning to optimally design ventilation systems to capture particles not stopped by masks. This chapter follows work found in Zohdi [172, 176]. • Part 3-Ultraviolet Viral Decontamination: Part 3 focuses on viral decontamination by ultraviolet (UV) irradiation technologies. Specifically, Part 3 develops an efficient and rapid computational method to simulate a UV pulse, in order to ascertain the decontamination efficacy of UV irradiation for a surface. It is based on decomposition of a pulse into groups of rays, which are then tracked as they progress toward the target contact surface. This allows one to quickly quantify the decontamination efficacy across the topology of the structure. This chapter follows work found in Zohdi [174]. • Part 4-Vaccine Design and Immune Response: In Part 4, a computational framework is developed that researchers in the field can easily implement and subsequently use as an efficient tool to study the immune response to a vaccine injection. There are three main components: – Digital-twin construction: The work develops an approach that efficiently simulates the time-transient proliferation of cells/antibodies (proteins) and regulator/antigens (deactivated toxin) to an injected vaccine within tissue possessing complex heterogeneous microstructure. – Efficient computation: A technique based on voxel (3D “volume pixels”) representation of tissue microstructure and corresponding digital solution methods is developed for the calculations, allowing extremely fast methods to be used to construct derivatives and to iteratively solve the system with minimal memory requirements.

Preface

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– Machine learning: The rapid and efficient computation allows for many vaccines to be tested quickly and uses a genetic-based machine learning algorithm to optimize the system. This is particularly useful for rapid design of next-generation vaccines and boosters for disease strain mutations. This chapter follows work found in Zohdi [179]. A key concept that arises throughout the monograph is that of a “digital-twin” of physical reality, i.e., a digital replica of a complex system that can be inexpensively and safely manipulated and optimized in a virtual setting and then deployed in the physical world afterward, thus reducing the potential costs of experiments and accelerating the development of new technologies. Berkeley, USA September 2022

Tarek I. Zohdi

Acknowledgements This work has been partially supported by the University of California Berkeley, College of Engineering, the Will C. Family Hall Chair in Engineering, the USDA AI Institute for Next Generation Food Systems (AIFS), USDA award number 2020-67021-32855 and Lam Research. Many thanks go to Aidan Gould, Maya Horii and Zachary Yun for their efforts in proofreading drafts of this work.

Contents

1 Preliminaries: Basic Mathematics, Optimization and Machine-Learning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Elementary Notation and Mathematical Operations . . . . . . . . . . . . . . 1.1.1 Vectors, Products and Norms . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.2 Basic Linear Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.3 Integral Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Temporal Discretization Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.1 Isolating a Single Particle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Basic Machine-Learning and Optimization . . . . . . . . . . . . . . . . . . . . . 1.3.1 Gradient-Based Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.2 Difficulties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Genetic-Based Machine-Learning Algorithm . . . . . . . . . . . . . . . . . . . 1.4.1 Algorithmic Specifics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 1 2 3 5 5 7 7 8 9 9

2 Part 1: Macroscale Disease Propagation . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1 Classical Basic Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.2 SIR Sub-population Models . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.3 Generalization of the SIR Family of Models . . . . . . . . . . . . . 2.1.4 Agent-Based Models and Objectives . . . . . . . . . . . . . . . . . . . . 2.2 Direct Agent-Based Interaction Models . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Agent-to-Agent Interaction and Rules of Engagement . . . . . 2.2.2 Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.3 Computational Acceleration . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 A Model Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.1 An Example of a Swarm Formulation . . . . . . . . . . . . . . . . . . . 2.5 An Algorithm for Movement in a Region . . . . . . . . . . . . . . . . . . . . . . 2.5.1 Preliminary Numerical Example . . . . . . . . . . . . . . . . . . . . . . .

13 13 14 15 16 18 19 19 21 21 22 26 27 29 30

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3 Part 2: Microscale Disease Transmission and Ventilation System Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Analytical Characterization: Simplified Stokesian Model . . . . . . . . . 3.2.1 Analysis of Particle Velocities . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2 Analysis of Particle Positions . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.3 Settling (Airborne) Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Computational Approaches for More Complex Models . . . . . . . . . . 3.3.1 More Detailed Characterization of the Drag . . . . . . . . . . . . . . 3.3.2 Simulation Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Ventilation System Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.1 Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.2 Incorporation of Masks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.3 Incorporation of Vents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.4 Overall Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Genetic-Based Machine-Learning Ventilation Optimization . . . . . . . 3.5.1 Model Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6 Summary and Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

33 33 35 35 37 37 38 38 39 43 46 47 47 47 48 48 50

4 Part 3: Ultraviolet Viral Decontamination . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.1 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Electromagnetic Energy Propagation . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1 Beam-Ray Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.2 Reflection and Absorption of Rays . . . . . . . . . . . . . . . . . . . . . 4.3 Electromagnetic Wave Propagation and Rays . . . . . . . . . . . . . . . . . . . 4.3.1 Plane Harmonic Wave Fronts . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.2 Natural (Random) Electromagnetic Energy Propagation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.3 Reflection and Absorption of Energy-Fresnel Relations . . . . 4.3.4 Reflectivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Model Problem and Response Trends . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.1 Tracking of Beam-Decomposed Rays . . . . . . . . . . . . . . . . . . . 4.4.2 Test Surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Numerical/Quantitative Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6 Summary and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

55 55 56 57 57 58 59 60 61 61 63 64 65 65 66 67

5 Part 4: Vaccine Design and Immune-System Response . . . . . . . . . . . . . 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.1 Brief History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.2 Types of Vaccines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.3 Vaccine Efficacy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.4 Objectives of This Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 A Flexible Immune-Response Digital-Twin . . . . . . . . . . . . . . . . . . . . 5.3 Rapid Voxel Based Computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Numerical Simulation of the Coupled System . . . . . . . . . . . . . . . . . .

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5.4.1 Discretization of the c- and s-Fields . . . . . . . . . . . . . . . . . . . . . 5.4.2 Iterative (implicit) Solution Method . . . . . . . . . . . . . . . . . . . . . Operation Counts in a Voxel-Based Method . . . . . . . . . . . . . . . . . . . . Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Genetic-Based Machine-Learning Framework . . . . . . . . . . . . . . . . . . 5.7.1 Algorithmic Settings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.7.2 Parameter Search Ranges and Results . . . . . . . . . . . . . . . . . . . Discussion and Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

79 82 86 87 89 91 92 94

Epilogue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Appendix A: Artificial Neural Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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5.5 5.6 5.7

5.8

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

About the Author

Tarek I. Zohdi received his Ph.D. in 1997 in Computational and Applied Mathematics from the University of Texas at Austin. He was a postdoctoral fellow at the Technical University of Darmstadt in Germany from 1997 to 1998 and then a lecturer (C2-Oberingenieur) at the Gottfried Leibniz University of Hannover in Germany from 1998 to 2001, where he received his Habilitation in General Mechanics (Allgemeine Mechanik). He joined the faculty of the Department of Mechanical Engineering at the University of California, Berkeley, in 2001. He is a Chancellor’s Professor of Mechanical Engineering and holder of the W. C. Hall Family Endowed Chair in Engineering. He also holds a Staff Scientist position at Lawrence Berkeley National Labs and an Adjunct Scientist position at the Children’s Hospital Oakland Research Institute. His main research interests are in modeling, simulation and optimization of nonconvex multiscale–multiphysics problems for industrial applications. He has published over 200 archival refereed journal papers and eight books: (1) Introduction to computational micromechanics (T. Zohdi and P. Wriggers, SpringerVerlag), (2) An introduction to modeling and simulation of particulate flows (T. Zohdi, SIAM), (3) Electromagnetic properties of multiphase dielectrics: a primer on modeling, theory and computation (T. Zohdi, Springer-Verlag), (4) Dynamics of charged particulate systems: modeling, theory and computation (T. Zohdi, SpringerVerlag) (5 and 6) A finite element primer for beginners-the basics (T. Zohdi, SpringerVerlag, two distinct editions), (7) Modeling and simulation of functionalized materials for additive manufacturing and 3D printing: continuous and discrete media (T. Zohdi, Springer-Verlag) and (8) Modeling and simulation of infectious diseases: microscale transmission, decontamination and macroscale propagation (T. Zohdi, Springer-Verlag), as well as eight handbook/book chapters and five encyclopedia chapters. In 2000, he received the Zienkiewicz Prize and Medal, which are awarded once every two years, to one postgraduate researcher under the age of 35, by The Royal Institution of Civil Engineers in London, to commemorate the work of Prof. O. C. Zienkiewicz, for research which contributes most to the field of numerical methods in engineering. In 2002, he received the Best Paper of the Year 2001 Award in London, at the Lord’s Cricket Grounds, for a paper published in Engineering Computations, pertaining to modeling and simulation of the propagation of failure in particulate aggregates of material. In 2003, he received the Junior Achievement xv

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Award of the American Academy of Mechanics. The award is given once a year, to one postgraduate researcher, to recognize outstanding research during the first decade of a professional career. In 2008, he was elected Fellow of the International Association for Computational Mechanics (IACM), and in 2009, he was elected Fellow of the United States Association for Computational Mechanics (USACM). The USACM is the primary computational mechanics organization in the United States and the International Association for Computational Mechanics is the primary international organization in this field. In 2011, he was selected as “Alumnus of the Year” by the Department of Mechanical Engineering at Louisiana State University (LSU), where he did his undergraduate studies. In 2017, he was awarded the University of California, Berkeley Distinguished Teaching Award, in recognition for faculty that have established a sustained and varied record of teaching excellence. This is the highest award for teaching in the University. In 2019, he was elected as Fellow of the American Academy of Mechanics (AAM)-only one new Fellow is inducted into the nation and the Americas into the AAM each year. In 2020, he received the prestigious Humboldt-Forschungspreis (Humboldt Research Prize). The prize, given by the Alexander von Humboldt Foundation of the German Government, recognizes renowned researchers outside of Germany whose “fundamental discoveries, new theories or insights have had a significant impact on their own discipline and who are expected to continue producing cutting-edge achievements in the future.” He received it in the area of Mechanics in recognition of lifetime achievements. Overall, he has given more than 200 other plenary, keynote and contributed lectures at conferences, universities and other research institutions worldwide. He is an Editor-in-Chief of the leading journal in his field, Computer Methods in Applied Mechanics and Engineering (CMAME) serves on 11 editorial boards of international journals. He is also the co-founder and co-editor-in-chief of the journal Computational Particle Mechanics (CPM), as well as an editor of the Computational Mechanics book series (Wiley). He has organized or co-organized over 30 international conferences and workshops and been appointed/invited to the Scientific Advisory Boards of over 40 international conferences. He was elected President of the USACM in 2012 and served from 2012 to 2014. Since 2009, he has served as a representative of the USACM on the General Council of the IACM, which is the governing committee of the primary international organization in his field of research and was elected to the Executive Council of IACM in 2020 (seven were elected worldwide in 2020). In 2014, he was appointed by the United States National Academy of Science (NAS) and the National Research Council (NRC) as a member of the US National Committee for Theoretical and Applied Mechanics (USNC/TAM) representing the USACM (4/15/2014–10/31/2018). USNC/TAM is the primary national governing body for Mechanics in the United States. This committee operates under the auspices of the US Board on International Scientific Organizations (BISO) and the Policy and Global Affairs Division of the NRC. In 2018, he was elected to Member-at-Large status of the USNC/TAM by the National Academy of Sciences. For more information, see http://www.me.berkeley.edu/people/faculty/tarek-i-zohdi/.

List of Figures

Fig. 1.1 Fig. 1.2 Fig. 1.3 Fig. 2.1

Fig. 2.2

Fig. 2.3 Fig. 2.4 Fig. 2.5

Fig. 2.6

Fig. 2.7

Fig. 2.8 Fig. 2.9

Machine-learning and digital-twin models . . . . . . . . . . . . . . . . . . The overall process for the MLA-genetic algorithm . . . . . . . . . . . The basic action of a genetic algorithm . . . . . . . . . . . . . . . . . . . . . A model problem of a “planet” with a population, which experiences sudden localized infections. Left: a model schematic and Right: a computational model (blue representing currently uninfected and green representing infected (see Zohdi [173])) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The classical process of developing a continuum model from an inherently discrete system, which is then re-discretized into nodes. Information is lost in this “homogenization” process, thus motivating agent-based models (Zohdi [162]) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A schematic of the growth of subpopulations (see Zohdi [173]) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The infection zone and flow chart (see Zohdi [173]) . . . . . . . . . . Spherical coordinates are used for agent placement. In this example, agent high-density concentration centers are at the z-poles (see Zohdi [173]) . . . . . . . . . . . . . . . . . . . . . . . . . . . Starting from left to right and top to bottom, the progressive growth of an SIR population. Shown are after t = 0, t = 0.2T, t = 0.4T, t = 0.6T, t = 0.8T and t = T time, with mobility parameters: Aφ = 0.004 and Aθ = 0.004 (see Zohdi [173]) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Evolution of infected and dead/recovered with mobility parameters: Aφ , Aθ = 0.000, 0.001, 0.002, 0.003, 0.004 and 0.005 (see Zohdi [173]) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Creating a “forbidden” zone (see Zohdi [173]) . . . . . . . . . . . . . . . Agent-obstacle-target model problem (see Zohdi [173]) . . . . . . .

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Fig. 2.10

Fig. 3.1 Fig. 3.2

Fig. 3.3

Fig. 3.4

Fig. 3.5

Fig. 3.6

Fig. 3.7 Fig. 3.8

Fig. 4.1 Fig. 4.2

List of Figures

Starting from left to right and top to bottom, the progressive movement of a group of agents (blue) avoiding obstacles (green) to get to the target (red, see Zohdi [173]) . . . . . . . . . . . . . Model problem for release of cough particles (see Zohdi [172, 176]) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cough simulation (from a starting height of 2 m, for v f = (0, 0, 0)): successive frames indicating the spread of particles. a Large particles travel far and settle quickly and b Small particles do not travel far and settle slowly (see Zohdi [172, 176]) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Zoom on cough simulation (from a starting height of 2 m, for v f = (0, 0, 0)): successive frames indicating the spread of particles. a Large particles travel far and settle quickly and b Small particles do not travel far and settle slowly (see Zohdi [172, 176]) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Left: Room configuration model problem considered in this work. Middle: a zoom on mask microscale fibers. Right: generic portable air-filtration systems (see Zohdi [172, 176]) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Left: The model problem studied in this work, including masks and ventilation systems. Shown are the particles captured by the mask and the particles that get through, color coded by size. Also shown are the location of the vents for this specific scenario (which will be optimized in this work). Right: Zoom on the release of cough particles, color coded by size (Zohdi [172,176]) . . . . . . Flow patterns with the model. Left: flow velocity magnitude. Right: flow velocity magnitude with streamlines (see Zohdi [172, 176]) . . . . . . . . . . . . . . . . . . . . The genetic-based MLA flowchart (Zohdi [147, 168, 170, 171, 175–180, 183]) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Optimization for successively longer time limits of T = 1.75, 2.0, 2.25 and 2.5 s, showing the reduction of the cost function for the 16 parameter set. Shown is the best performing gene (design parameter set, in red) as a function of successive generations, as well as the average performance of the entire population of genes (design parameter set, in green). Successively allowing longer simulation times allows the vents to trap more particles (see Zohdi [172, 176]) . . . . . . . . . . . . . . . . . . . . . . An electromagnetic pulse applied to a surface (see Zohdi [174]) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Top view for a surface with amplitude A = 0.3. Colors indicate the absorption, normalized by the incoming. radiation level (see Zohdi [174]) . . . . . . . . . . . . . . . . . . . . . . . . . .

31 35

41

42

43

46

48 49

52 57

67

List of Figures

Fig. 4.3

Fig. 5.1

Fig. 5.2 Fig. 5.3

Fig. 5.4

Fig. 5.5

Fig. 5.6

Fig. 5.7

Sequence of frames for a surface with amplitude A = 0.3 (the amplitude was enhanced by a factor of 20 in graphics to more easily see the effects of the topology on absorption). Colors indicate the absorption, normalized by the incoming radiation level (see Zohdi [174]) . . . . . . . . . . . . Model Problem: an injection into a representative volume element of heterogeneous “marbled” tissue.The injection site is given high concentration of antibodies/cells and antigens/regulator (see Zohdi [179]) . . . . . . . . . . . . . . . . . . . . Left, a tissue microstructure and, right, a voxel representation of the tissue microstructure (see Zohdi [179]) . . . Sequentially finer voxel representations of slightly overlapping particles in a matrix: for 41 × 41 × 41 (206763 D O F) voxel-mesh, for 61 × 61 × 61 (680943 D O F) voxel-mesh and for 81 × 81 × 81 (1594323 D O F) voxel-mesh. Top row: Microstructure/marbled tissue (particles and matrix). Second row: Just tissue (matrix). Third row: Just marbling (particles). Bottom row: Just interfaces (between marbling and matrix, see Zohdi [179]) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The overall iterative (left) solution and the matrix-free approach using a moving front through the voxels(right) During the iterative solution process, the most current value of a voxel is used in any calculation, for example a construction of the Laplacian, or any other term in the governing differential equations (see Zohdi [179]) . . . . . . Left, the morphology of the microstructure and, right, morphology of the microstructure and mesh (see Zohdi [179]) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The injection viewed from the exterior and the morphology of the microstructural-marbling. From left to right and top to bottom: Cell concentration (c) and growth from an injection at the surface (see Zohdi [179]) . . . . . . . . . . . . The evolution of the time-step size over time (see Zohdi [179]) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xix

68

76 77

78

85

89

90 91

xx

Fig. 5.8

Fig. 5.9

Fig. A.1

List of Figures

Shown are the cost function for the best performing gene (red) as a function of successive generations, as well as the average cost function of the entire population of genes (green). We allowed the genetic-based MLA to readapt every 10 generations, leading to the (slight) nonmonotone reduction of the cost function. Often, this action is more efficient than allowing the algorithm not to readapt, since it probes around the current optimum for better local alternatives. In this case, the algorithm makes slow progress until generation 10, when a readaptation/recentering occurred, and then slowed reduced the cost function over 200 generations from approximately  ≈ 2 to  = 0.4097. The algorithm produces a massive reduction of error from  ≈ 50 to  ≈ 0.4-a factor of 125 (see Zohdi [179]) . . . . . . . . . . . . . . . . ZOOM: Shown are the cost function for the best performing gene (red) as a function of successive generations, as well as the average cost function of the entire population of genes (green). We allowed the genetic-based MLA to readapt every 10 generations, leading to the (slight) nonmonotone reduction of the cost function. Often, this action is more efficient than allowing the algorithm not to readapt, since it probes around the current optimum for better local alternatives. In this case, the algorithm makes slow progress until generation 10, when a readaptation/recentering occurred, and then slowed reduced the cost function over 200 generations from approximately  ≈ 2 to  = 0.4097. The algorithm produces a massive reduction of error from  ≈ 50 to  ≈ 0.4-a factor of 125 (see Zohdi [179]) . . . . . . . . . . . . . . . . Top: An ANN comprised of (1) Five layers (one input layer and four hidden layers) (2) 35 activation neurons (3+5+7+9+11) and (3) 223 weighted synapses. The color-coding represents the value of the weights. Bottom: Various neuron activation functions: (1) Linear (2) Sigmoid and (3) Double Sigmoid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

93

94

100

Chapter 1

Preliminaries: Basic Mathematics, Optimization and Machine-Learning

This chapter provides the essential mathematical concepts needed for the subsequent applications encountered in this monograph.

1.1 Elementary Notation and Mathematical Operations 1.1.1 Vectors, Products and Norms In this work, boldface symbols imply vectors or tensors. A fixed Cartesian coordinate system will be used throughout this monograph. The unit vectors for such a system are given by the (fixed) mutually orthogonal triad (e1 , e2 , e3 ). For the inner product of two vectors u and v we have in three dimensions u·v =

3 

vi u i = u 1 v1 + u 2 v2 + u 3 v3 = ||u|||v||cosθ,

(1.1)

i=1

where ||u|| =



u 21 + u 22 + u 23

(1.2)

represents the Euclidean norm in IR 3 and θ is the angle between them. We recall that a norm has three main characteristics for any two bounded vectors u and v (||u|| < ∞ and ||v|| < ∞): • ||u|| ≥ 0, ||u|| = 0 if and only if u = 0, • ||u + v|| ≤ ||u|| + ||v|| and • ||γu|| = |γ|||u||, where γ is a scalar. Two vectors are said to be orthogonal if u · v = 0. The cross (vector) product of two vectors is © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 T. I. Zohdi, Modeling and Simulation of Infectious Diseases, https://doi.org/10.1007/978-3-031-18053-8_1

1

2

1 Preliminaries: Basic Mathematics, Optimization and Machine-Learning

   e1 e2 e3    u × v = −v × u =  u 1 u 2 u 3  = ||u||||v||sinθ n,  v1 v2 v3 

(1.3)

where n is the unit normal to the plane formed by the vectors u and v. The temporal differentiation of a vector-valued function is given by d du 1 (t) du 2 (t) du 3 (t) u(t) = e1 + e2 + e3 = u˙ 1 e1 + u˙ 2 e2 + u˙ 3 e3 . dt dt dt dt

(1.4)

The spatial gradient of a scalar-valued function (a dilation to a vector) is given by  ∇x φ =

 ∂φ ∂φ ∂φ e1 + e2 + e3 . ∂x1 ∂x2 ∂x3

(1.5)

The gradient of a vector-valued function is a direct extension of the preceding def∂u i inition. For example, ∇x u has components of ∂x . The divergence of a vector (a j contraction to a scalar) is defined by 

∂ ∂ ∂ ∇ x · u = e1 + e2 + e3 ∂x1 ∂x2 ∂x3



 · (u 1 e1 + u 2 e2 + u 3 e3 ) =

∂u 2 ∂u 3 ∂u 1 + + ∂x1 ∂x2 ∂x3

 .

(1.6)

The curl of a vector is defined as    e1 e2 e3   ∂ ∂ ∂  ∇x × u =  ∂x1 ∂x2 ∂x3  .  u1 u2 u3 

(1.7)

The triple product of three vectors is ⎞ ⎛  w1 w 2 w 3    w · (u × v) = ⎝ u 1 u 2 u 3 ⎠ = (w × u) · v.  v1 v2 v3 

(1.8)

This represents the volume of a parallelpiped formed by the three vectors.

1.1.2 Basic Linear Algebra If we consider the second order tensor A with its matrix representation ⎡

⎤ A11 A12 A13 [ A] = ⎣ A21 A22 A23 ⎦ . A31 A32 A33 def

(1.9)

1.1 Elementary Notation and Mathematical Operations

3

The matrix [ A] is said to be symmetric if [ A] = [ A]T and skew-symmetric if [ A] = −[ A]T . A first order contraction (inner product) of two matrices is defined by

A · B = [ A][B] which has components o f

N 

Ai j B jk = Cik ,

(1.10)

j=1

where it is clear that the range of the inner index j must be the same for [ A] and [B]. The second order inner product of two matrices is A : B = Ai j Bi j = tr ([ A]T [B]).

(1.11)

1.1.3 Integral Transformations The divergence of a vector-valued function (a contraction to a scalar-valued function) is defined by N  u i,i , (1.12) ∇x · u = i=1

whereas for a second order tensor (a contraction to a vector): ∇x · A has components of

N 

Ai j, j .

(1.13)

j=1

The gradient of a vector (a dilation to a second order tensor) is: ∇x u has components of u i, j ,

(1.14)

whereas for a second order tensor (a dilation to a third order tensor): ∇x A has components of Ai j,k .

(1.15)

The gradient of a scalar (a dilation to a vector): ∇x φ has components of φ,i .

(1.16)

The scalar product of two second order tensors, for example the gradients of first order vectors, is defined as

4

1 Preliminaries: Basic Mathematics, Optimization and Machine-Learning

∂vi ∂u i ∂x j ∂x j   

∇x v : ∇x u =

def

= vi, j u i, j

i, j = 1, 2, 3,

(1.17)

in Cartesian bases

where ∂u i /∂x j , ∂vi /∂x j are partial derivatives of u i and vi , and where u i , vi are the Cartesian components of u and v and ∇x u · n has components of

,

u i, j n j   

i, j = 1, 2, 3.

(1.18)

in Cartesian bases

where n is vector, such as a normal vector. For a scalar, we have   ∇x φ d = φn d A 

φ,i d =

u i, j d =



ui ni d A

(1.21)

Bi j n j d A,

(1.22)

∂



 Bi j, j d =



∂

(1.20)

 u i,i d =



and analogously for a tensor B   ∇x · B d = B · ndA

u i n j d A. ∂



∂

(1.19)





∂

φn i d A ∂



The divergence theorem for vectors is   ∇x · u d = u · n d A 





∂

and for a vector   ∇x u d = u ⊗ n d A 



∂

where n is the outward normal to the bounding surface. These standard operations arise throughout the analysis. A generalization of these last results is 

 ∇x ∗ B d = 

n ∗ B d A,

(1.23)

∂

where, when ∗ = ·, we have the divergence theorem and when ∗ = × we have the “cross-product” theorem.1 For proofs, see Chandrasekharaiah and Debnath [27] or Malvern [90]. 1

Also, we have the point-wise product rule: d da da (a ∗ b) = ∗b+a∗ . dt dt dt

(1.24)

1.2 Temporal Discretization Methods

5

1.2 Temporal Discretization Methods An important tool for simulation of the upcoming systems that are built on differential equations. For example, we now specifically address the second-order particle-based systems of interest later.

1.2.1 Isolating a Single Particle Each particle’s equation of motion is given by m v˙ = ,

(1.25)

where  is the force provided from interactions with other particles the external environment. Expanding the velocity in a Taylor series about t + φt we obtain (0 ≤ φ ≤ 1) v(t + t) = v(t + φt) +

dv 1 d2v | (1 − φ)2 (t)2 + O(t)3 |t+φt (1 − φ)t + dt 2 dt 2 t+φt

(1.26)

and v(t) = v(t + φt) −

1 d 2v dv |t+φt φt + |t+φt φ2 (t)2 + O(t)3 . dt 2 dt 2

(1.27)

Subtracting the two expressions yields dv v(t + t) − v(t) ˆ |t+φt = + O(t), dt t

(1.28)

ˆ where O(t) = O(t)2 when φ = 21 . Inserting this into the equation of motion yields t 2 ˆ (t + φt) + O(t) . (1.29) v(t + t) = v(t) + m Note that adding a weighted sum of Eqs. 1.26 and 1.27 yields v(t + φt) = φv(t + t) + (1 − φ)v(t) + O(t)2 ,

(1.30)

which will be useful shortly. Now expanding the position of the center of mass in a Taylor series about t + φt we obtain

6

1 Preliminaries: Basic Mathematics, Optimization and Machine-Learning

r(t + t) = r(t + φt) +

dr 1 d2 r |t+φt (1 − φ)t + | (1 − φ)2 (t)2 + O(t)3 dt 2 dt 2 t+φt

(1.31)

and r(t) = r(t + φt) −

1 d2 r dr |t+φt φt + |t+φt φ2 (t)2 + O(t)3 . dt 2 dt 2

(1.32)

Subtracting the two expressions yields r(t + t) − r(t) ˆ = v(t + φt) + O(t). t

(1.33)

Inserting Eq. 1.30 yields 2 ˆ r(t + t) = r(t) + (φv(t + t) + (1 − φ)v(t)) t + O(t)

(1.34)

and thus using Eq. 1.29 yields r(t + t) = r(t) + v(t)t +

φ(t)2 2 ˆ (t + φt) + O(t) . m

(1.35)

The term (t + φt) can be approximated by (t + φt) ≈ φ(r(t + t)) + (1 − φ)(r(t)),

(1.36)

yielding r(t + t) = r(t) + v(t)t +

φ(t)2 2. ˆ (φ(r(t + t)) + (1 − φ)(r(t))) + O(t) m

(1.37)

We note that • When φ = 1, then this is the (implicit) Backward Euler scheme, which is very stable (very dissipative) and O(t)2 locally in time, • When φ = 0, then this is the (explicit) Forward Euler scheme, which is conditionally stable and O(t)2 locally in time, • When φ = 0.5, then this is the (implicit) “Midpoint” scheme, which is stable and 2 ˆ = O(t)3 locally in time. O(t) Typically, if the systems are relatively simple, and if small times steps can be used, Explicit methods are preferred. We will use such methods in this monograph. However, in passing we remark that efficient methods for complex systems have been developed by use of adaptive iterative schemes in Zohdi [146–183].

1.3 Basic Machine-Learning and Optimization

7

1.3 Basic Machine-Learning and Optimization Throughout this monograph, we will use optimization and machine-learning methods. Our primary objective is to construct digital-twins of physical systems and then to optimize their performance using machine-learning algorithms. We start by denoting the system performance error, known as the cost or error function, by the symbol 0 ≤ () ≤ ∞ 0 ≤ () = ||(desir edr esult) − (achievedr esult)|| = cost or err or, (1.38) where “desired result” is an observation to match or a system response that is sought after, and the “achieved result” is the response produced by the mathematical model for a given set of system “design” variables. The ideal objective is to drive the cost/error to zero (or a minimum), by manipulating N design variables, denoted by the design vector: ⎫ ⎧ 1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎨ 2 ⎪ def (1.39)  = 3 ⎪ ⎪ ⎪ ⎪ ... ⎪ ⎪ ⎪ ⎪ ⎭ ⎩ N Here we follow Zohdi [147, 168, 170, 171, 175–180, 183] in order to minimize cost functions. Cost functions are often nonconvex in design parameter space and often nonsmooth. Their minimization is usually extremely difficult or impossible with direct application of gradient methods. This motivates nonderivative search methods, for example those found in machine-learning algorithms (MLA). One of the most basic subset of MLA’s are so-called Genetic Algorithms (GA, see Holland [62], as well as Goldberg [52], Davis [32], Onwubiko [107] and Goldberg and Deb [53]). Typically, one will use a GA first in order to isolate multiple local minima, and then either (1) use a gradient-based algorithm afterwards or (2) reset the GA to concentrate its search around these locally convex regions. GA are one of the most robust MLA methods, and are widely used in industry. We first start by reviewing classical gradient-based methods.

1.3.1 Gradient-Based Methods Gradient-based optimization proceeds by forcing the gradient of ∇ () = 0. Expanding (linearizing) around a first guess i yields   ∇ (i+1 ) ≈ ∇ (i ) + ∇ ∇ (i ) · (i+1 − i ) + higher or der ter ms ≈ 0 (1.40)

8

1 Preliminaries: Basic Mathematics, Optimization and Machine-Learning

or in more streamlined matrix notation, defining the Hessian, [IH] = ∇ (∇ ()) and {g} = ∇ (), thus [IH]{} + {g} = 0. (1.41) Following a standard gradient (Newton-type) multivariate search, a new design increment is computed, (1.42)  = (1 , 2 , ... N ), for a design vector, , by solving the following system, [IH]{} = −{g}, where [IH] is the Hessian matrix (N × N ), with components Hi j =

∂ 2 () , ∂i ∂ j

(1.43)

{g} is the gradient (N × 1), with components gi =

∂() ∂i

(1.44)

and where {} is the design increment (N × 1), with components i . After the design increment has been solved for, one then forms an updated design vector, new = old + , and the process is repeated until ||∇ || ≤ T O L. Explicitly, the incremental system is ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

∂ 2 () ∂1 ∂1 ∂ 2 () ∂2 ∂1 ∂ 2 () ∂3 ∂1 ∂ 2 () ∂4 ∂1

...... ......

∂ 2 () ∂1 ∂2 ∂ 2 () ∂2 ∂2 ∂ 2 () ∂3 ∂2 ∂ 2 () ∂4 ∂2

...... ......

∂ 2 () ∂1 ∂3 ∂ 2 () ∂2 ∂3 ∂ 2 () ∂3 ∂3 ∂ 2 () ∂4 ∂3

...... ......

∂ 2 () ∂1 ∂4 ∂ 2 () ∂2 ∂4 ∂ 2 () ∂3 ∂4 ∂ 2 () ∂4 ∂4

...... ......

∂ 2 () ∂ 2 () ∂ 2 () ∂ 2 () ∂ N ∂1 ∂ N ∂2 ∂ N ∂3 ∂ N ∂4

.....

⎤

⎧ ⎥ ⎪ 1 ⎪ ...... ⎥ ⎥⎪ ⎪ 2 ⎥⎪ ⎪ ⎨ 3 ...... ⎥ ⎪ ⎥ 4 ⎥ ...... ⎥ ⎪ ⎥⎪ ⎪ ..... ...... ⎥ ⎪ ⎪ ..... ⎥⎪ ⎪ ...... ⎦ ⎩  N .....

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ . =− ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ..... ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ..... ⎪ ⎪ ⎭ ⎪ ⎪ ⎪ ⎪ ⎭ ⎩ ∂() ⎪ ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨

∂() ∂1 ∂() ∂2 ∂() ∂3 ∂() ∂4

(1.45)

∂ N

1.3.2 Difficulties Numerical construction of derivatives Because the cost/error functions are usually not a closed-form expression, the derivatives must be constructed numerically by repeatedly running simulations with different input states. For example: • For the first derivative of  at (1 , 2 , 3 ): ∂ (1 + 1 , 2 , 3 ) − (1 − 1 , 2 , 3 ) ≈ ∂1 21

(1.46)

1.4 Genetic-Based Machine-Learning Algorithm

9

• For the second derivative at (1 , 2 , 3 ): ∂ ∂1



∂ ∂1



 ≈

∂ ∂1



|

1+

1 2

,2 ,3

−



∂ ∂1



|

1−

1 2

,2 ,3

1   (1 + 1 ), 2 , 3 − (1 , 2 , 3 ) 1 = 1 1   (1 , 2 , 3 ) − (1 − 1 , 2 , 3 ) − . 1

(1.47)

• For the cross-derivative of (1 , 2 ): ∂ ∂2



∂ ∂1

 ≈

∂ ∂2



(1 + 1 , 2 , 3 ) − (1 − 1 , 2 , 3 ) 21



(1.48)

1 ((1 + 1 , 2 + 2 , 3 ) − (1 − 1 , 2 + 2 , 3 )) 41 2 1 − ((1 + 1 , 2 − 2 , 3 ) − (1 − 1 , 2 − 2 , 3 )) . 41 2 ≈

An exhaustive review of these methods can be found in the texts of Luenberger [89] and Gill, Murray and Wright [51]. Unfortunately, for most complex systems, cost functions such as  are nonconvex and nonsmooth in design parameter space. Their minimization is usually impossible with direct application of gradient-based methods. This motivates nonderivative search methods, such as those found in machinelearning algorithms (MLA). Nonconvexity and nondifferentiability As mentioned earlier, one of the most popular and basic subsets of MLA are so-called Genetic Algorithms (GA). Machine-learning algorithms are well-suited for nonconvex, nonsmooth, multicomponent and multistage systems (Figs. 1.1, 1.2 and 1.3). For a review of GA, see the pioneering work of Holland [62, 63], as well as Goldberg [52], Davis [32], Onwubiko [107] and Goldberg and Deb [53]. A description of the algorithm will be provided next (following Zohdi [147, 168, 170, 171, 175–180, 183]).

1.4 Genetic-Based Machine-Learning Algorithm 1.4.1 Algorithmic Specifics Following Zohdi [147, 168, 170, 171, 175–180, 183], the algorithm is as follows: • STEP 1: Randomly generate a population of S starting genetic strings, i , (i = 1, 2, 3, . . . , S) :

10

1 Preliminaries: Basic Mathematics, Optimization and Machine-Learning

UPDATE SYSTEM SETTINGS

MACHINE−LEARNING TO TRAIN AND

DIGITAL REPLICA OF A PHYSICAL SYSTEM:

OPTIMIZE THE SYSTEM

"DIGITAL TWIN"

EXTRACT OPTIMAL SYSTEM SETTINGS/ SYSTEM PARAMETERS

Fig. 1.1 Machine-learning and digital-twin models

GENERATE GENETIC POPULATION

COMPUTE GENETIC PERFORMANCE IN DIGITAL TWIN SIMULATOR

RANK POPULATION

BEST WORST

BREED STRONG GENES

ELIMINATE WEAK GENES

REPEAT FOR NEXT GENERATION

CHILDREN

ELIMINATE

PARENTS INTRODUCE NEW GENES

CHILDREN NEW GENES

Fig. 1.2 The overall process for the MLA-genetic algorithm

⎧ i ⎫ 1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎨ i2 ⎪ def i i  = 3 ⎪ ⎪ ⎪ ... ⎪ ⎪ ⎪ ⎪ ⎭ ⎩ i ⎪ N

(1.49)

≤ The system parameter search is conducted within the constrained ranges of (−) 1 (+) (−) (+) (−) (+) 1 ≤ 1 , 2 ≤ 2 ≤ 2 and 3 ≤ 3 ≤ 3 , etc.

1.4 Genetic-Based Machine-Learning Algorithm

11 TWO RANKED "PARENTS"

Λ1 Λ2 Λ3 Λ4 Λ5 Λ6 Λ7 Λ8

ETC ...

SUCCESSIVE CHILDREN

Π

RANDOM CONVEX COMBINATIONS

TWO GENERATED "CHILDREN" GENE POOL OF PARAMETER SETS

Λ

Fig. 1.3 The basic action of a genetic algorithm

• STEP 2: Compute fitness of each string: (i ), (i=1, …, S) • STEP 3: Rank genetic strings: i , (i=1, …, S) • STEP 4: Mate nearest pairs and produce two offspring, (i=1, …, S):

def

λi =  ◦ i + (1 − ) ◦ i+1

⎧ φ1 i1 ⎪ ⎪ ⎪ ⎪ ⎨ φ2 i2 def = φ3 i3 ⎪ ⎪ ... ⎪ ⎪ ⎩ φ N iN

⎫ ⎪ ⎪ ⎪ ⎪ ⎬

⎫ ⎧ (1 − φ1 )i+1 ⎪ ⎪ 1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎨ (1 − φ2 )i+1 2 + (1 − φ3 )i+1 3 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ... ⎪ ⎪ ⎪ ⎭ ⎩ i+1 ⎭ (1 − φ N ) N

(1.50)

and

def

λi+1 =  ◦ i + (1 − ) ◦ i+1

⎧ ψ1 i1 ⎪ ⎪ ⎪ ⎪ ⎨ ψ2 i2 def = ψ3 i3 ⎪ ⎪ ... ⎪ ⎪ ⎩ ψ N iN

⎫ ⎪ ⎪ ⎪ ⎪ ⎬

⎫ ⎧ (1 − ψ1 )i+1 ⎪ ⎪ 1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎨ (1 − ψ2 )i+1 2 + (1 − ψ3 )i+1 3 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ... ⎪ ⎪ ⎪ ⎭ ⎩ i+1 ⎭ (1 − ψ N ) N

(1.51)

where for this operation, the φi and ψi are random numbers, such that 0 ≤ φi ≤ 1, 0 ≤ ψi ≤ 1, which are different for each component of each genetic string • STEP 5: Eliminate the bottom M strings and keep top K parents and their K offspring (K offspring+K parents+M=S) • STEP 6: Repeat STEPS 1–5 with top gene pool (K offspring and K parents), plus M new, randomly generated, strings We remark that this algorithm is easily parallelizable. The following observations are important: • Typically, one will use a GA first in order to isolate multiple local minima, and then use either (1) a gradient-based algorithm around these locally convex regions or (2) reset the GA to concentrate its search around these locations (the best performing parameter sets) every few generations. • If one selects the mating parameters φ s and ψ s to be greater than one and/or less than zero, one can induce “mutations”, i.e. characteristics that neither parent

12

1 Preliminaries: Basic Mathematics, Optimization and Machine-Learning

possesses. However, this is somewhat redundant with introduction of new random members of the population. • If one does not retain the parents in the algorithm above, it is possible that inferior performing offspring may replace superior parents. Thus, top parents should be kept for the next generation. This guarantees a monotone reduction of the cost function. Furthermore, retained parents do not need to be re-evaluated-making the algorithm less computationally less expensive in the next generation. Numerous studies of the author have shown that advantages parent retention outweighs inbreeding, for sufficiently large population sizes. • There are other machine-learning type paradigms that complement genetic-based approaches, such as Artificial Neural Networks (ANN). ANN have received huge attention in the scientific community over the last decade and are based on layered input-output type frameworks that are essentially adaptive nonlinear regressions of the form O = B(I, w), where O is a desired output and B is the ANN comprised of (1) synapses, which multiply inputs (Ii , i = 1, 2, . . . , M) by weights (wi , i = 1, 2, . . . , N ) that represent the input relevance to the desired output, (2) neurons, which aggregate outputs from all incoming synapses and apply activation functions to process the data and (3) training, which calibrates the weights to match a desired overall output. In the Appendix, more details on ANN are provided.

Chapter 2

Part 1: Macroscale Disease Propagation

Summary The increase in readily available computational power raises the possibility that direct agent-based modeling can play a key role in the analysis of epidemiological population dynamics. The objective of this chapter is to develop a robust agent-based computational framework to investigate the emergent structure of SIRtype (Susceptible-Infected-Removed/Recovered) populations and variants thereof, on a global planetary scale. To accomplish this objective, we develop a planet-wide model based on interaction between discrete entities (agents), where each agent on the surface of the planet is initially uninfected. Infections are then seeded on the planet in localized regions. Contracting an infection depends on the characteristics of each agent-i.e. their susceptibility and contact with the seeded, infected agents. Agent mobility on the planet is dictated by social policies, such as “shelter in place”, “complete lockdown”, etc. The global population is then allowed to evolve according to the infected states of agents over many time periods, leading to an SIR population. The work illustrates the construction of the computational framework and the relatively straightforward application with direct, non-phenomenological, input data. Numerical examples are provided to illustrate the model construction and the results of such an approach. This chapter follows work found in Zohdi [173].

2.1 Introduction There are numerous issues associated with the COVID-19 pandemic. Macroscale (planetary) disease propagation, in addition to the related issues of logistical and political responses, is a central theme. Accordingly, the objective of this work is to develop a computationally-amenable agent-based model to investigate the behavior of an infected population by directly working at the individual-to-individual level of interaction. The wide-spread availability of computational power now raises the possibility that robust agent-based modeling can play a significant role in the analysis of infectious disease propagation. The key feature of agent-based modeling is that discrete entities (agents) are used to directly represent a population © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 T. I. Zohdi, Modeling and Simulation of Infectious Diseases, https://doi.org/10.1007/978-3-031-18053-8_2

13

14

2 Part 1: Macroscale Disease Propagation UNINFECTED

SURFACE PATCH

INFECTED

GLOBE

Fig. 2.1 A model problem of a “planet” with a population, which experiences sudden localized infections. Left: a model schematic and Right: a computational model (blue representing currently uninfected and green representing infected (see Zohdi [173]))

(Fig. 2.1). This enables the detailed analysis of epidemiological population dynamics and the ability to investigate the emergent structure of SIR-type (SusceptibleInfected-Removed/Recovered) populations. This method can also be used to investigate more complex scenarios, for example due to initially localized infections within a population on a global planetary scale, including the effects of social responses.

2.1.1 Classical Basic Models Before proceeding with the construction of an agent-based approach, it is useful to review basic concepts in the analysis of population dynamics, which dates back over two centuries to the work of Thomas Malthus. In 1798, he postulated that a population, denoted p, at a future time (t + t), is related to the current population (at time t) by p(t + t) = λ p(t), (2.1) where λ = 1 + (b − d)t, where b is a birth rate parameter and where d is a death rate parameter. One may write p(t + t) − p(t) = (λ − 1) p(t),

(2.2)

leading to, in the limit as t → 0, dp p = (b − d)t ⇒ = (b − d)dt. p p Integrating and applying the initial condition p(t = 0) = p(0) yields

(2.3)

2.1 Introduction

15

ln( p(t)) = c + (b − d)t ⇒ p(t) = ec+(b−d)t ⇒ p(t) = p(0)e(b−d)t .

(2.4)

Variants/extensions of this simple model include interacting subpopulations. One family of models are of particular interest in this work, so-called SIR-type (Susceptible-Infected-Removed/Recovered), described next. Subsets of the population are assigned either S, I, or R status. The genesis of such models is the work from 1927 of Kermack and McKendrick [76].

2.1.2 SIR Sub-population Models SIR models identify three subpopulation classes of individuals, with an assumption that the overall population is constant, since the pandemic/epidemic time scales are faster than birth time scales. The following are key: • S=“Susceptible”, which can contract diseases, • I=“Infected”, which can transmit the disease and who are infected and • R=“Recovered/Removed” (dead or immune), where the typical assumptions include: (a) the gain in the infected class is at a rate proportional to the number of infected and susceptible, that is k1 S I , k1 > 0 (b) the rate of removal of the infected is proportional to the number of infected, k2 I , k2 > 0 and (c) the incubation period is short enough to be negligible. In addition, if it is assumed that the various classes are uniformly mixed (no spatial dependence), one has for the susceptible population: dS = −k1 S I, dt

(2.5)

while for the infected subpopulation dI = k1 S I − k2 I, dt

(2.6)

dR = k2 I. dt

(2.7)

and for the removed population

Adding all of the populations together yields dS dI dR + + = 0, dt dt dt

(2.8)

S + I + R = p,

(2.9)

where

16

2 Part 1: Macroscale Disease Propagation

where p is the total population, S(0) = So > 0, I (0) = Io > 0, R(0) = 0, k1 > 0 is the infection rate and k2 > 0 is the death rate. A crucial question is, under what conditions does an infection grow, i.e. an “epidemic” occur? From Eq. 2.6, if k1 S I − k2 I > 0 ⇒ k1 S I > k2 I ⇒ and thus

k1 S > 1, k2

dI > 0. dt

(2.10)

(2.11)

Equation 2.10 provides the threshold for the susceptible population k1 k2 S>1⇒S> k2 k1

(2.12)

to allow growth to occur. The parameter k2 /k1 is sometimes called the infectious contact rate, while its reciprocal is called the recovery/removal rate. This simple model is one of the most basic to describe epidemics. For reviews, we refer the reader to Murray [100].

2.1.3 Generalization of the SIR Family of Models The basic SIR model can be extended in the following ways: • The SIS model: Susceptible-Infected-Susceptible (again), typified by the common cold. • The SIRD model: Susceptible-Infected-Recovered-Deceased, which distinguishes between recovered and dead. • The MSIR model: Maternal-Susceptible-Infected-Recovered, where the “M class” stands for immunity derived from the mother. • The SEIR model: Susceptible-Exposed-Infected-Recovered, which distinguishes between “infected” and “exposed”. • The SEIS model: Susceptible-Exposed-Infected-Susceptible (again), typified by the common cold, yet distinguishes between “infected” and “exposed”. • The MSEIR model: Maternal-Susceptible-Exposed-Infected-Recovered, which incorporates features of the models above. • The MSEIRS model: Maternal-Susceptible-Exposed-Infected-RecoveredSusceptible, which incorporates all of the features of the models above. For overviews and details on these various models, we refer the reader to Kermack and McKendrick [76], Murray [100], Hethcote [60, 61], Harko et al. [57], Baily [6], Altizer and Nunn [1], Miller [96], Miller [97], Osemwinyen and Diakhaby [108], Brauer and Castillo [20] Anderson [3], Barlett [8], May and Anderson [94], Capasso

2.1 Introduction Fig. 2.2 The classical process of developing a continuum model from an inherently discrete system, which is then re-discretized into nodes. Information is lost in this “homogenization” process, thus motivating agent-based models (Zohdi [162])

17 FUNDAMENTALLY DISCRETE

SURFACE PATCH

HOMOGENIZE (PDE GENERATION) NUMERICALLY DISCRETIZE GRID POINTS

[24] and Vynnycky and White [133]. Various additional features can be included, such as • • • • • •

Variable contact rates, Adult vaccinations, Child vaccinations, Newborn vaccinations, Effects of age and Vector transmission (for example from mosquitos).

Virtually all subsequent, more complex spatio-temporal extensions construct homogenized continuum (PDE-based) models, incorporating the above models. These approaches require extensive, complex discretization techniques and are of limited value for studies on population dynamics with underlying complex interaction between subpopulations (Fig. 2.2). Such models have marginal predictive capability and are computationally expensive, due to the extremely fine discretization needed to achieve tolerable numerical accuracy. Independent of the numerical difficulties, such modeling approaches attempt to develop continuum type field equations, by passing to the spatio-temporal limit (as t → 0, x → 0) and make somewhat unrealistic assumptions in order to obtain tractable partial differential equations which yield, at best, qualitative estimates of the true population behavior. One must question the process of first homogenizing an inherently discrete population’s characteristics in order to develop PDE-based continuum models and then re-discretizing them into nodal values. This process is not bijective, in other words, one does not recover the original discrete system (Fig. 2.2). Information is lost in this process. Also, because of the simplifying assumptions on interaction, births, age structuring, etc., that are typically made, the resulting PDE-discretized equations are not as physically meaningful as

18

2 Part 1: Macroscale Disease Propagation INITIAL CONDITION

POSSIBLE OUTCOME

DYNAMIC SUBPOPULATIONS

Fig. 2.3 A schematic of the growth of subpopulations (see Zohdi [173])

the true discrete population interactions that they were originally based upon. Essentially, in dealing with small subpopulations, or populations with complex mixing and high heterogeneity, the assumptions behind regularization techniques leading to continuum models may be difficult to justify. This motivates developing agent-based methods, which are based on directly computing the interactions between individuals or population subgroups (Fig. 2.3).

2.1.4 Agent-Based Models and Objectives Agent-based models attempt to characterize the interaction between individuals, directly incorporating stochastic methods, and have been used across many different disciplines, such as: biology, business, economics, social sciences, robotics and technology. The objective is to obtain deeper insight into the connection between micro-interaction and emergent macroscale behavior. The method first appeared in the 1940s, but was computationally infeasible until the 1990s. Although there are many forms of agent-based models, computationally, they are quite similar to particle interaction models in mathematical physics. The use of the term agent is often attributed to Holland and Miller’s 1991 paper “Artificial Adaptive Agents in Economic Theory” [63]. The 1990s and early 2000s led to many approaches, often connected to the concepts of swarms of agents and aggregate movement. For reviews of the literature, see Bonabeau [18]. The attractive features of the approaches are the ability to model individual interactions nonphenomenologically, and to allow the system to evolve autonomously. The utility of such approaches is that one can trivially modify the “rules of engagement”, population sizes, reproduction rates, etc, and provide quantitative spatial and temporal information. Clearly, such a computational technique is easy to implement, and it is no extra effort to increase the number of characteristic population parameters. We refer the reader to Zohdi [173] for reviews. Our objective in the current work is to apply similar concepts to pandemic model-

2.2 Direct Agent-Based Interaction Models

19

ing, where the interaction is via infection transmission. Specifically, we develop a robust agent-based computational framework to investigate the emergent structure of SIR-type (Susceptible-Infected-Removed/Recovered) populations and more complex extensions on a global planetary scale, where each agent on the surface of the planet is initially uninfected. Infections are then seeded on the planet in localized regions. Contracting an infection depends on the characteristics of each agent-i.e. their susceptibility and contact with the seeded, infected agents. Agent mobility on the planet is dictated by social policies, such as “shelter in place”, “complete lockdown”, etc. The global population is then allowed to evolve according to infected states of agents over many time periods, leading to an SIR population. The work illustrates the construction of the computational framework and provides numerical examples. Remarks: In the field of astro-biology, there are related concerns about Earthbased microbes potentially infecting other worlds, by being carried on spacecraft missions. The recent discovery of water on Mars (as well as the moons of Saturn and Jupiter) has heightened such concerns. It has been reported that, on Mars, liquid water can possibly form seasonally in locations where snow is present on soils with saline, producing brine (Martinez and Renno [93]). Since terrestrial bacteria can grow in brine, infection of the Marian biosphere could be possible. The reverse is also true, since returning spacecraft from Mars could bring back non-terrestrial organisms to Earth. NASA and other space agencies have consistently reported that Bacillus spores, subjected to years vacuum in space, cosmic radiation and extreme temperatures, can survive, if they are shielded by the exterior of a spacecraft. We refer readers to Beaty [11], Fischer et al [41], Martinez and Renno [93], Summons et al [122], Michalski [95] and Debus [33] for details.

2.2 Direct Agent-Based Interaction Models Following Zohdi [162], we now construct a model problem based on discrete ruledriven interaction between agents of subpopulations. One can consider an agent as an individual or a small group of individuals (a “meta-individual”).

2.2.1 Agent-to-Agent Interaction and Rules of Engagement Consider the following construction, for the “rules of engagement” for intermeshed infected and uninfected subpopulations, which are in close proximity to one another (Fig. 2.4): • If two agents of the subpopulations, denoted (I = in f ected) and (U = unin f ected), come within a certain distance,

20

2 Part 1: Macroscale Disease Propagation INITIALIZE AGENT CHARACTERISTICS (a) SUSCEPTIBILITY (b) INFECTED (c) AGE (d) DEAD/RECOVERED (e) ETC.

INFECTION ZONE

INITIALIZE AGENT POSITIONS

COMPUTE RELATIVE PROXIMITIES

COMPUTE INFECTION TRANSMISSION

UPDATE INCUBATION PERIODS

Fig. 2.4 The infection zone and flow chart (see Zohdi [173]) Fig. 2.5 Spherical coordinates are used for agent placement. In this example, agent high-density concentration centers are at the z-poles (see Zohdi [173])

z AGENT PLACEMENT

Θ

INCREASING AGENT DENSITY

Φ

r

DECREASING AGENT DENSITY

y

x POLES ARE CONCENTRATION CENTERS

) ||r i(I ) − r (U j || ≤ d I U ,

(2.13)

then the two are said to engage in “contact”. • If the uninfected agent is susceptible, then the agent becomes infected. • The “susceptibility” of the uninfected subpopulations is heterogeneous, with agents preset to “susceptible” or “not susceptible” at the beginning of the simulation. • The incubation period for an infected agent to either recover or die is TI . • Once an agent in the population perishes or becomes immune, it cannot affect the rest of the population (Fig. 2.5).

2.2 Direct Agent-Based Interaction Models

21

2.2.2 Algorithm The algorithm is as follows: • • • • • • • •

STEP 1: Select: (a) The number of agents in the population: Na . (b) The threshold infection distance: d I U (c) The susceptibility of the agents. (d) The total simulation time, T . T . (e) The cycle time, t (thus, the number of time-periods becomes t STEP 2: Generate the initial population locations on the globe. STEP 3: For each population, loop over each agent in the infection radius, according to the “rules of engagement” in the previous section, and compute the interaction of the pair. • STEP 4: Compute the survivors and deaths of the existing agents for the time period. • STEP 5: Repeat STEPS 2-4 for the next time period until the overall simulation time is complete. The relative ease at which one can generate such a population, and step it through several time periods, is readily apparent.

2.2.3 Computational Acceleration There are a variety of techniques to accelerate the computations. The primary computational expense is neighbor-to-neighbor contact checks (an O(Na2 ) operation). To mitigate this, one can construct so-called “interaction” or “Verlet” lists of neighboring agents that an agent may be in contact with at any given time. One can retain this Verlet list for a preset number of time-steps and then update it when appropriate. The approach is relatively straightforward to implement and can speed up the computations dramatically (see Pöschel and Schwager [109] and Zohdi [157, 158]). Alternative computational acceleration approaches can be achieved via sorting and binning methods, which proceed by partitioning the whole domain into bins. The agents are sorted by the bins in which they reside. The agent interaction proceeds, bin by bin, where the agents within a bin potentially only interact with agents within their bin or neighboring bins. Parallel processing is a further acceleration that can be employed whereby groups of agents or bins are sent to each processor and updated periodically, sharing the agent information between processors every few time-steps. In this work, we only implemented the Verlet list algorithm.

22

2 Part 1: Macroscale Disease Propagation

2.3 A Model Problem As an example, consider a population with 20,000 agents (Fig. 2.6). We consider six cases of increasingly mobile subpopulations that are initially uniformly mixed across the globe, with agents being infected or uninfected and being susceptible or not susceptible. We employed the following parameters:

Fig. 2.6 Starting from left to right and top to bottom, the progressive growth of an SIR population. Shown are after t = 0, t = 0.2T, t = 0.4T, t = 0.6T, t = 0.8T and t = T time, with mobility parameters: Aφ = 0.004 and Aθ = 0.004 (see Zohdi [173])

2.3 A Model Problem

• • • • • • • •

23

The globe/planet radius, R = 1. The total simulation time: T = 30. The infection distance: d I U = 0.01R. The population initially infected= 1%. The population that is susceptible= 25%. The incubation period for an agent to either recover or die is TI = 5. Update cycle time=t = 0.1. The location of each agent was achieved using a random spherical coordinate scheme (see Fig. 2.9), whereby θ (inclination angle) is a random number between 0 ≤ θ ≤ π and φ (azimuth angle) is a random number between 0 ≤ φ ≤ 2π, yielding the following in Cartesian coordinates: r x = Rsin(θ(t = 0))cos(φ(t = 0))

(2.14)

r y = Rsin(θ(t = 0))sin(φ(t = 0))

(2.15)

r z = Rcos(θ(t = 0)).

(2.16)

and

and

• The mobility is given by (the location at an instant of time later) r x = Rsin(θ(t + t))cos(φ(t + t))

(2.17)

r y = Rsin(θ(t + t))sin(φ(t + t))

(2.18)

r z = Rcos(θ(t + t)).

(2.19)

and

and

This produces dense population centers at the z-poles. The updated values for the angles are given by Inclination mobility :θ(t + t) = θ(t) + θ,

(2.20)

where θ = Aθ × β, β being a random number between −1 ≤ β ≤ 1 and Aθ is a mobility amplitude parameter and Azimuthal mobility :φ(t + t) = φ(t) + φ,

(2.21)

where φ = Aφ × γ, γ being a random number between −1 ≤ γ ≤ 1 and Aφ is a mobility amplitude parameter.

24

2 Part 1: Macroscale Disease Propagation

Fig. 2.7 Evolution of infected and dead/recovered with mobility parameters: Aφ , Aθ = 0.000, 0.001, 0.002, 0.003, 0.004 and 0.005 (see Zohdi [173])

Figure 2.6 illustrates the progressive growth of SIR subpopulations. Shown are results at t = 0, t = 0.2T, t = 0.4T, t = 0.6T, t = 0.8T and t = T times, with mobility parameters: Aφ = 0.004 and Aθ = 0.004. Figure 2.7 shows the evolution of infected and dead/recovered with mobility parameters: 0.0 ≤ Aφ ≤ 0.005 and 0 ≤

2.3 A Model Problem

25

Aθ ≤ 0.005. With increases in mobility, there is an immediate and direct impact on the spread of the infection, emanating primarily from the dense population centers at the z-poles. The utility of the presented computational approach is that one can trivially modify the “rules of engagement”, population sizes, etc., and provide quantitative spatial and temporal information. Clearly, such a computational technique is easy to implement. Numerous additional features can be added easily. For example, one could add population growth in a variety of ways, such as, algorithmically: • If an agent of a population survives beyond a certain number of time periods, it then produces offspring, and then perishes. • The offspring are placed within an “offspring radius”, centered at the spatial location of the parent. The number of children possible that an individual can have, at maturity, is given by o f f spring = integer(φ × M), (2.22) where 0 ≤ φ ≤ 1 is a random number and where M is the maximum number of children possible. The function “integer” extracts the nearest integer from (φ × M). • After giving birth to the offspring once, the agent cannot have offspring again. We note that, if desired, incorporation of “forbidden regions” i.e. “uninhabitable zones” within the domain is relatively easily to enforce by checking at each timestep whether an individual has entered such an area (Fig. 2.8). If entered, then the individual is moved back outside, and a new position is recalculated with a different trajectory. Another extension to the overall modeling is to provide more details on the movement of the agents. For example, as described at the outset of this work, we could have allowed for agents to move according to more physical rules. It is here where one can draw on swarm movement models discussed earlier in the work. This is discussed further next.

Fig. 2.8 Creating a “forbidden” zone (see Zohdi [173])

26

2 Part 1: Macroscale Disease Propagation

Fig. 2.9 Agent-obstacletarget model problem (see Zohdi [173])

z OBSTACLES AGENTS

TARGET

y x ZOOM GLOBE

2.4 Extensions One direction in which to extend this research is in the swarm-like movement of agents. The origins of swarm modeling are in the description of biological groups’ (flocks of birds, schools of fish, crowds of human beings, etc.) responses to predators or prey (Breder (1952 [21]). Early approaches that rely on decentralized organization can be found in Beni [13], Brooks [23], Dudek et al. [36], Cao et al. [25], Liu and Passino [87] and Turpin et al. [126]. Usual models incorporate a tradeoff between long-range interaction and short-range repulsion between individuals, dependent on the relative distance between individuals (see Gazi and Passino [47, 47, 49], Bender and Fenton [12] or Kennedy and Eberhart [75]). The most basic model is to treat each individuals as a point mass (Zohdi [147]), which we adopt here, and to allow the system to evolve, based on Newtonian mechanics, using a combination of shortrange and long-range interaction forces (Gazi and Passino [47, 47, 49], Bender and Fenton [12], Kennedy and Eberhart [75] and Zohdi [147, 152, 162, 166, 170, 173]).1 For some creatures, the “visual field” of individuals may play a significant role, while if the agents are robots or Unmanned Aerial Vehicles (UAVs), the communication can be electronic. We refer the reader to Zohdi [147, 152, 162, 166, 170, 173] for reviews and provide an example of such a formulation next. Remark: In some systems, agents interact with a specific set of other agents, regardless of whether they are far away (Feder [38]). This appears to be the case for Starlings (Sturnus vulgaris). In Ballerini et al [7], the authors concluded, that such birds communicate with a certain number of birds surrounding it and that those 1

There are other modeling paradigms, for example mimicking ant colonies (Bonabeau et al. [16]), which exhibit foraging-type behavior and trail-laying-trail-following mechanisms for finding food sources (see Kennedy and Eberhart [75] and Bonabeau et al. [16], Dorigo et al. [35], Bonabeau et al. [16], Bonabeau and Meyer [17] and Fiorelli et al. [40]).

2.4 Extensions

27

interactions are governed by topological distance and not metric distance. Interested readers are referred to Ballerini et al [7].

2.4.1 An Example of a Swarm Formulation In order to illustrate how swarm movement is modeled, following Zohdi [147, 152, 162, 166, 170], we treat the agents as point masses, i.e. we ignore their dimensions. For each agent (Ns in total) the equations of motion are m i v˙ i = m i r¨ i =  itot = F(N iat , N iao , N iaa ),

(2.23)

where the position of a point (agent) in space is given by the vector r i , the velocity is given by v i = r˙ i , the acceleration is given by ai = v˙ i = r¨ i , and where  itot represents the total forces acting on an agent i, N iat represents the interaction between agent i and desired targets, N iao represents the interaction between agent i and obstacles and N iaa represents the interaction between agent i and other agents. In the context of a pandemic, examples of targets could be hospitals, food distribution centers, etc., while obstacles could be physical barriers, such as buildings. Agent-target interaction Consider agent-target interaction 1/2 def at  = di j , ||r i − T j || = (ri1 − T j1 )2 + (ri2 − T j2 )2 + (ri3 − T j3 )2

(2.24)

where T j is the position vector to target j and the direction to each target is ni→ j =

T j − ri . ||r i − T j ||

(2.25)

For each agent (i), we compute a weighted direction to each target nˆ i→ j = (wt1 e−a1 di j − wt2 e−a2 di j )ni→ j , at

at

(2.26)

where the wti are weights reflecting the importance of the target, ai are decay parameters. The directions are summed (and normalized later in the analysis) to give an overall direction to move towards N iat =

Nt  j=1

Agent-obstacle interaction Now consider agent-obstacle interaction

nˆ i→ j .

(2.27)

28

2 Part 1: Macroscale Disease Propagation

1/2 def ao  ||r i − O j || = (ri1 − O j1 )2 + (ri2 − O j2 )2 + (ri2 − O j2 )2 = di j ,

(2.28)

where O j is the position vector to obstacle j and the direction to each obstacle is O j − ri . ||r i − O j ||

ni→ j =

(2.29)

For each agent (i), we compute a weighted direction to each obstacle nˆ i→ j = (wo1 e−b1 di j − wo2 e−b2 di j )ni→ j , ao

ao

(2.30)

where the woi are weights reflecting the importance of the obstacle, bi are decay parameters. The directions are summed (and normalized later in the analysis) to give an overall direction to move towards N iao =

No 

nˆ i→ j .

(2.31)

j=1

Agent-agent interaction Now consider agent(i)-agent( j) interaction 1/2 def aa  ||r i − r j || = (ri1 − r j1 )2 + (ri2 − r j2 )2 + (ri3 − r j3 )2 = di j ,

(2.32)

and the direction to each agent r j − ri . ||r i − r j ||

ni→ j =

(2.33)

For each agent (i), we compute a weighted direction to each agent nˆ i→ j = (wa1 e−c1 di j − wa2 e−c2 di j )ni→ j , aa

aa

(2.34)

where the wai are weights reflecting the importance of the agents, ci are decay parameters. The directions are summed (and normalized later in the analysis) to give an overall direction to move towards N iaa =

Na 

nˆ i→ j .

(2.35)

j=1

Summation of interactions We now aggregate the contributions by weighting their overall importance with weights for agent/target interaction, Wat , agent/obstacle interaction, Wao and

2.5 An Algorithm for Movement in a Region

29

agent/agent interaction, Waa :2 N itot = Wat N iat + Wao N iao + Waa N iaa . We normalize the result ni∗ =

N itot . ||N itot ||

(2.36)

(2.37)

The forces are then constructed by multiplying the thrust force available by the propulsion system (foot, bicycle, car, etc.), Fi , by the overall normal direction  itot = Fi ni∗ .

(2.38)

We then integrate the equations of motion: mi v˙ i =  itot , yielding v i (t + t) = v i (t) +

t tot  (t) mi i

(2.39)

(2.40)

and r i (t + t) = r i (t) + tv i (t).

(2.41)

||v i (t + t)|| > vmax ,

(2.42)

Note that if then we define v iold (t + t) = v i (t + t) and the velocity is rescaled v inew (t + t) = vmax

v iold (t + t) , ||v iold (t + t)||

(2.43)

with v i (t + t) = v inew (t + t).

2.5 An Algorithm for Movement in a Region Consider the following algorithm: 1. Initialize the locations of the targets: T i = (Tx , Ty , Tz )i , i=1, 2,…N T =targets. 2. Initialize the locations of the obstacles: O i = (Ox , O y , Oz )i , i=1, 2,…N O =obstacles. 3. Initialize the locations of the agents: r i = (r x , r y , r z )i , i=1, 2,…Na =agents. 2

The parameters in the model will be optimized shortly.

30

2 Part 1: Macroscale Disease Propagation

4. For each agent (i), determine the distance and directed normal to each target, obstacle and other agents. 5. For each agent (i), determine interaction functions N iat , N iao , N iaa and ni∗ . 6. For each agent (i), determine force acting upon it,  itot = Fi ni∗ . 7. For each agent (i), integrate the equations of motion (checking constraints) to produce v i (t + t) and r i (t + t). 8. Determine if any targets have been reached by checking the distance between agents and targets (2.44) ||r i − T j || ≤ T olerance. For any T j , if any agent has satisfied this criteria, then immobilize i (fix ri ). 9. The entire process is then repeated for the next time-step.

2.5.1 Preliminary Numerical Example As a preliminary example, consider the following parameters: • • • • • • • • • • •

Mass = 75 kg, 100 agents, 1 target, 10 obstacles, T = 2000 s, t = 0.001 s, Initial agent velocity, v i (t = 0) = 0 m/s, Initial agent domain (10,10,0) m, Thrust force available by the system, Fi = 107 N, Domain of (500,500,0) m, Maximum velocity agent vmax = 2 m/s.

The vector of system parameter inputs def

i = {1 , 2 ... N } = {Wma , Wao , Waa , wt1 , wt2 , wo1 , wo2 , wa1 , wa2 , a1 , a2 , b1 , b2 , c1 , c2 },

(2.45) given by a test parameter vector def

i = {10.00, 5.00, 7.50, 0.25, 0.50, 0.60, 0.80, 0.30, 0.85, 0.15, 0.50, 1.00, 0.75, 0.90, 0.60},

(2.46) selected within the following intervals: • • • • •

Overall weights: 0 ≤ Wat , Wao , Waa ≤ 10, Target weights: 0 ≤ wt1 , wt2 ≤ 1, Obstacle weights: 0 ≤ wo1 , wo2 ≤ 1, Member weights: 0 ≤ wa1 , wa2 ≤ 1 and Decay coefficients: 0 ≤ a1 , a2 ≤ 1, 0 ≤ b1 , b2 ≤ 1, 0 ≤ c1 , c2 ≤ 1.

2.5 An Algorithm for Movement in a Region

31

Fig. 2.10 Starting from left to right and top to bottom, the progressive movement of a group of agents (blue) avoiding obstacles (green) to get to the target (red, see Zohdi [173])

Figure 2.10 illustrates the results of this parameter choice. Remark: We emphasize that the rapidly computable models in this chapter facilitate the concept of a digital-twin of physical reality, i.e., a digital replica of a population that can be safely manipulated and optimized in a virtual setting and then deployed afterwards in the physical world, with the goal being to reduce costs of experiments and to accelerate the development of new technologies.

Chapter 3

Part 2: Microscale Disease Transmission and Ventilation System Design

Summary In this chapter, mathematical models are developed to simulate the progressive time-evolution of the distribution of particles produced by respiratory emissions. The models ascertain the range, distribution and settling time of the particles under the influence of gravity and drag from the surrounding air, which are needed for constructing social distancing policies and workplace protocols. This leads to another key issue—the design of ventilation systems to mitigate airborne transmission of infectious diseases, produced by respiratory emissions. Accordingly, the model is then extended to couple: • the progressive time-evolution of the spatio-temporal distribution of particles produced by respiratory emissions, • the use of masks to impede respiratory particle emission propagation and • the use of genetic-based machine-learning to optimally design ventilation systems to capture particles not stopped by masks. This chapter follows work found in Zohdi [172, 176].

3.1 Introduction It is well-established that COVID-19 primarily spreads from person-to-person by respiratory droplets produced when an infected person coughs or sneezes. Subsequently, the droplets come into contact with the eyes, nose or mouth of a nearby person or a person touches an infected surface, then makes contact with their eyes, nose or mouth. Since the virus is small, 0.06–0.14 microns in diameter, it can be contained in or attached to such emitted droplets. Droplets as small as one micron can carry enough viral load to cause an infection. A particular concern is the interaction of droplets with ventilation systems, which potentially could enhance the propagation of pathogens. This has implications on situation-specific safe distancing and the design of building filtration systems, air distribution, heating, air-conditioning and © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 T. I. Zohdi, Modeling and Simulation of Infectious Diseases, https://doi.org/10.1007/978-3-031-18053-8_3

33

34

3 Part 2: Microscale Disease Transmission and Ventilation System Design

decontamination systems, for example using UV-c and related technologies. In order to facilitate such system redesigns, fundamental analysis tools are needed that are easy to use. Accordingly, this work develops a simulator for the analysis of cough particle tracking. In its most basic form, a cough can be considered as a high-velocity release of a random distribution of particles of various sizes, into an ambient atmosphere. We refer the reader to Wei and Li [136], Duguid [37], Papineni and Rosenthal [110], Wei and Li [137], Zhu et al. [142], Chao et al. [30], Morawska et al. [99], VanSciver et al. [130], Kwon et al. [78], Tang et al. [123], Xie et al. [128], Gupta et al. [55], Wan et al. [134], Villafruela et al. [131], Nielson [102] Zhang and Li [141] and Lindsley et al. [86] for extensive reviews of coughs and other respiratory emissions. Following formulations for physically similar problems associated with particulate dynamics from the fields of blasts, explosions and fire embers (Zohdi [161, 164, 167, 171]), we make the following assumptions: • We assume the same initial velocity magnitude for all particles under consideration, with a random distribution of outward directions away from the source of the cough. This implies that a particle non-interaction approximation is appropriate. Thus, the inter-particle collisions are negligible. This has been repeatedly verified by “brute-force” collision calculations using formulations found in Zohdi [151, 153, 155, 159]. • We assume that the particles are spherical with a random distribution of radii Ri , i = 1, 2, 3 . . . N = par ticles. The masses are given by m i = ρi 43 π Ri3 , where ρi is the density of the particles. • We assume that the cough particles are quite small and that the amount of rotation, if any, contributes negligibly to the overall trajectory of the particles. The equation of motion for the ith particle in the system is grav

m i v˙ i =  i

drag

+ i

,

(3.1)

with initial velocity v i (0) and initial position r i (0). The gravitational force is grav = m i g, where g = (gx , g y , gz ) = (0, 0, −9.81) m/s2 . i • For the drag, we will employ a general phenomenological model drag

i

=

1 ρa C D ||v f − v i ||(v f − v i )Ai , 2

(3.2)

where C D is the drag coefficient, Ai is the reference area, which for a sphere is Ai = π Ri2 , ρa is the density of the ambient fluid environment and v f is the velocity of the surrounding medium which, in the case of interest, is air. We will assume that the velocity of the surrounding fluid medium (v f ) is given, implicitly assuming that the dynamics of the surrounding medium are unaffected by the particles.1 In order to gain insight, initially, we will discuss the closely related, analytically tractable, Stokesian model next. 1

We will discuss these assumptions further, later in the work.

3.2 Analytical Characterization: Simplified Stokesian Model

35

Fig. 3.1 Model problem for release of cough particles (see Zohdi [172, 176])

Remarks: As mentioned, there are a large number of physically similar phenomena to a cough, such as the particulate dynamics associated with blasts, explosions and fire embers. We refer the interested reader to the wide array of literature on this topic; see Plimpton [112], Brock [22], Russell [116], Shimanzu [118], Werrett [138], Kazuma [73, 74], Wingerden et al. [140] and Fernandez-Pello [39], Pleasance and Hart [111], Stokes [121] and Rowntree and Stokes [115], Hadden et al. [56], Urban et al. [127] and Zohdi [171] (Fig. 3.1).

3.2 Analytical Characterization: Simplified Stokesian Model 3.2.1 Analysis of Particle Velocities For a (low Reynolds number) Stokesian model, the differential equation for each particle is (Fig. 3.4) mi

dv i = m i g + ci (v f − v i ) dt

(3.3)

36

3 Part 2: Microscale Disease Transmission and Ventilation System Design

where ci = μ f 6π Ri , where μ f is the viscosity of the surrounding fluid (air) and the local Reynolds number for a particle is Re = 2Ri ρa ||v μf viscosity. This can be written in normalized form as def

f

−v i ||

ci dv i ci f + vi = g + v . dt mi m    i  ai

and μ f is the fluid

(3.4)

bi

This can be solved analytically to yield, for example in the z direction (with constant fluid velocity)   c bi z bi z − i t vi z (t) = vi zo − e mi + , (3.5) a ai z    i z  Bi z

Ai z

where • ai z =

ci mi

=

• Bi z =

f

9μ f vz 2ρi Ri2 2ρi Ri2 f vi zo − (gz 9μ f + vz ), 2ρ R 2 f (gz 9μi f i + vz ),

• bi z = gz + • Ai z =

9μ f , 2ρi Ri2 f

ci vz mi

= gz +

,

where the same holds for the y and x directions. The trends are • As t → ∞ vi z (t = ∞) →

2gz ρi Ri2 + vzf , 9μ f

(3.6)

• As Ri → 0 vi z (Ri →= 0, t →= ∞) → vzf . • The decay rate is controlled by mcii = ambient velocities extremely quickly.

9μ f , 2ρi Ri2

(3.7)

indicating that small particles attain

Some special cases: • With no gravity: vi z (t) = (vi zo − vzf )e

c

− mi t i

+ vzf .

(3.8)

3.2 Analytical Characterization: Simplified Stokesian Model

37

• With no damping: dvi z = gz ⇒ vi z (t) = vi zo + gz t. dt

(3.9)

Again, we note that the equations are virtually the same for the x and y directions, with the direction of gravity and fluid flow being the main differentiators.

3.2.2 Analysis of Particle Positions From the fundamental equation, relating the position r i to the velocity d ri = vi , dt

(3.10)

c dri z − i t = vi z = Ai z e mi + Bi z dt

(3.11)

we can write for the z direction

with the same being written for the x and y directions. Integrating and applying the initial conditions yields ri z (t) = ri zo +

c mi − i t Ai z (1 − e mi ) + Bi z t. ci

(3.12)

f

If gz = 0 and vz = 0, then 9μ

ri z (t) = ri zo + vi zo 2ρi

− f2 t Ri2 (1 − e 2ρi Ri ). 9μ f

(3.13)

As t → ∞ ri z (∞) = ri zo + vi zo 2ρi

Ri2 . 9μ f

(3.14)

As Ri → 0, the travel distance is dramatically shorter. The converse is true, larger particles travel farther.

3.2.3 Settling (Airborne) Time The settling, steady-state velocity can be obtained directly from

38

3 Part 2: Microscale Disease Transmission and Ventilation System Design

dv i + ai v i = bi , dt by setting

dv i dt

(3.15)

= 0, one can immediately solve for the steady-state velocity v i (∞) =

2ρi Ri2 bi = g +vf. ai 9μ f

(3.16)

The trends are • As Ri → 0, then v i (∞) → v f , 2ρ R 2 • As v f → 0, then v i (∞) → 9μi f i g. In summary • Large particles travel far and settle quickly and • Small particles do not travel far and settle slowly. Remark: The ratio of the Stokesian drag force to gravity is 9μ f ||v f − v i || || drag,Stokesian || = , || grav || 2ρi Ri2 g

(3.17)

which indicates that for very small particles, drag will dominate the settling process and for larger particles, gravity will dominate.

3.3 Computational Approaches for More Complex Models 3.3.1 More Detailed Characterization of the Drag In order to more accurately model the effects of drag, one can take into account that the empirical drag coefficient varies with Reynolds number. For example, consider the following piecewise relation (Chow [30]): • • • • •

24 , For 0 < Re ≤ 1, C D = Re For 1 < Re ≤ 400, C D = Re24 0.646 , 5 For 400 < Re ≤ 3 × 10 , C D = 0.5, For 3 × 105 < Re ≤ 2 × 106 , C D = 0.000366Re0.4275 and For 2 × 106 < Re < ∞, C D = 0.18, def

where, as in the previous section, the local Reynolds number for a particle is Re = 2Ri ρa ||v f −v i || and μ f is the fluid viscosity.2 We note that in the zero Reynolds number μf 2

The viscosity coefficient for air is μ f = 0.000018 Pa-s.

3.3 Computational Approaches for More Complex Models

39

limit, the drag is Stokesian. In order to solve the governing equation, grav

m i v˙ i =  i

drag

+ i

1 = m i g + ρa C D ||v f − v i ||(v f − v i )Ai , 2

(3.18)

we integrate the velocity numerically 1 v i (t + t) = v i (t) + mi ≈ v i (t) +

t+t 

grav

( i

drag

+ i

) dt

t

t grav drag  i (t) +  i (t) . mi

(3.19)

The position is the obtained by integrating again: t+t 

r i (t + t) = r i (t) +

v i (t) dt ≈ r i (t) + tv i (t).

(3.20)

t

This approach has been used repeatedly for a variety of physically similar drift-type problems in Zohdi [161, 164, 167, 171]. Remark: The piecewise drag law of Chow [30] is a mathematical description for the Reynolds number over a wide range and is a curve-fit of extensive data from Schlichting [117].

3.3.2 Simulation Parameters In order to illustrate the model, the following simulation parameters were chosen: • • • •

Starting height, 2 m, Total simulation duration, 4 s, Time-step size, t = 10−6 seconds, Cough velocity, V c (t = 0) = 30 m/s (taken from the literature which indicates 10 m/s ≤ V c ≤ 50 m/s), • Density of particles, ρi = 1000, kg/m3 , 3 • Density of air, ρa = 1.225,  Pn kg/m and T otal = i=1 m i = 0.0005 kg. • Total mass, M

Particle generation A mean particle radius was chosen to be R¯ = 0.0001 m with variations according to Ri = R¯ × (1 + A × ζi ),

(3.21)

40

3 Part 2: Microscale Disease Transmission and Ventilation System Design

where A = 0.9975 and a random variable −1 ≤ ζi ≤ 1. The algorithm used for particle generation was: • M=0 • Start loop: i = 1, Pn • Ri = R¯ × (1 + A × ζi ) • M = M + m i = M + ρi 43 π Ri3 • If M ≥ M T otal then stop (determines Pn =particles) • End loop Initial trajectories We determined the initial trajectories from the following algorithm: • Specify relative direction “cone” parameters: N c = (N xc , N yc , Nzc ), • For each particle, i = 1, 2, 3, . . . , Pn , construct a (perturbed) trajectory vector: N i = (N xc + Acx × ηi x , N yc + Acy × ηi y , Nzc + Acz + ×ηi z ) = (Ni x , Ni y , Ni z ), (3.22) where −1 ≤ ηi x ≤ 1, 0 ≤ ηi y ≤ 1 and −1 ≤ ηi z ≤ 1. • For each particle, normalize the trajectory vector: ni =

1 (Ni x , Ni y , Ni z ). ||N i ||

(3.23)

• For each particle, the velocity vector is constructed by a projection onto the normal vector: v i = V c ni .

(3.24)

Numerical results An extremely small (relative to the total simulation time) time-step size of t = 10−6 s was used. Further reductions of the time-step size produced no noticeable changes in the results, thus the solutions generated can be considered to have negligible numerical error. The simulations took under 10 s on a standard laptop. The algorithm generated 59941 particles ranging from 2.5 × 10−7 m ≤ Ri ≤ 2 × 10−4 m (i.e. 0.25 microns ≤ Ri ≤ 200 microns). We used a trajectory cone of N c = (0, 1, 0) and Ac = (1, 0.5, 1) in the example given. Figures 3.2 and 3.3 illustrate the results for f the parameters above (for v y = 0). If particles contacted the floor, they were immobilized. The maximum distance travelled from the source located at (0, 0, 2) was 2.72 m (achieved by large particles). Table 3.1 shows variation in the headwind. For strong tailwind, the larger particles land further away from the cough source. As the analytical theory asserts, successive frames indicate that: (a) Large particles travel far and settle quickly and (b) Small particles do not travel far and settle slowly (when there are no ambient velocities). As observed in the simulations, the settling

3.3 Computational Approaches for More Complex Models

41

Fig. 3.2 Cough simulation (from a starting height of 2 m, for v f = (0, 0, 0)): successive frames indicating the spread of particles. a Large particles travel far and settle quickly and b Small particles do not travel far and settle slowly (see Zohdi [172, 176])

42

3 Part 2: Microscale Disease Transmission and Ventilation System Design

Fig. 3.3 Zoom on cough simulation (from a starting height of 2 m, for v f = (0, 0, 0)): successive frames indicating the spread of particles. a Large particles travel far and settle quickly and b Small particles do not travel far and settle slowly (see Zohdi [172, 176])

3.4 Ventilation System Design

43

Table 3.1 Maximum distance from the source at the end of T = 4 s f

v y (m/s)

Max-distance (m)

Comments

−2.0 −1.0 0.0 1.0 2.0

8.002 4.210 2.721 4.937 8.736

Due Due Due Due Due

COUGH

to to to to to

small small large large large

particles moving particles moving particles moving particles moving particles moving

backwards backwards forwards forwards forwards

VENTS

z

y FIBER

x

Fig. 3.4 Left: Room configuration model problem considered in this work. Middle: a zoom on mask microscale fibers. Right: generic portable air-filtration systems (see Zohdi [172, 176])

of the small particles is still not achieved by the end of the simulation time (here 4 s). Accordingly, the simulations were also run for extremely long periods to ascertain that the “mist” of small particles remained airborne for several minutes (as predicted by the theory). For strong opposing headwind, small particles move backwards, and still remain airborne for extended periods of time. This is by far the most dangerous case, since this will encounter other persons at the torso level. We also note that ratio of the general drag to gravity indicates: 3C D ρa ||v f − v i ||2 || drag,general || = , grav || || ρi Ri g

(3.25)

which indicates that at high velocities, the dynamics are dominated by drag.

3.4 Ventilation System Design As we have discussed, it is well-established that COVID-19 primarily spreads from person-to-person by respiratory droplets that are produced when an infected person coughs or sneezes. Since the virus is small, 0.06–0.14 microns in diameter, it can be contained in or attached to such emitted droplets. Droplets as small as one micron can carry enough viral load to cause an infection. A particular concern is the inter-

44

3 Part 2: Microscale Disease Transmission and Ventilation System Design

action of droplets with ventilation systems, which can capture emitted pathogens, but can also potentially could enhance their propagation. This has implications on situation-specific safe distancing and the design of building filtration systems, air distribution, heating, air-conditioning and decontamination systems, for example using UV-c and related technologies. In order to facilitate such system redesigns, fundamental analysis tools are needed that are easy to use. It is now widely accepted that masks are an integral part of controlling the spread of airborne infectious diseases, such as COVID-19. There are three general categories of face masks: • Cloth Masks (for the general public)-provides some protection to the wearer, but mainly serves to stop the spread of viruses, • Surgical Masks (for healthcare workers and the general public)-provides partial protection and • N95 Respirators (primarily for healthcare workers)-provides high level of protection. The most effective is the N95 mask, which is excellent at trapping very small particles (below 0.1 micron), since the particle motion is random (Brownian) due to collisions with other particles, air molecules, etc., which will force them to collide with an electrically charged “sticky” fiber. It is also excellent in trapping large particles (above 0.3 microns), since their motion is effectively linear across the mask (because of their inertia) and will collide with a fiber. It has more difficulty trapping medium-sized particles (0.1 microns–0.3 microns), which travel with airflow patterns. However, N95 masks use charged fibers (electrets) to attract particles of all sizes. N95 means that it traps at least 95% of the particles. Thus, three main features (1) Van der Waals forces at the very small scale (2) many layers of fibers and (3) charged fibers make such masks effective. However, there are problems, for example due to illfitting masks, repeated re-use, due to shortages, use of alcohols and solvents for mask cleaning, which may damage the mask or neutralize the fiber charge, etc. The proper donning and doffing (removal from face) of a mask is nontrivial, due to strap elongation, facial hair, etc. Thus, one must assume that some particles get through, regardless of the mask type. Thus, in workplaces, some type of filtration is needed, generally of HEPA (High Efficiency Particulate Absorbing) grade, which remove 99.95% of the particles equal to 0.3 microns in size, with efficiency increasing for particles larger or smaller than this size. Such filters were created by the German military in WWII for gas masks, whereby increases in mask efficiency were noted by simply adding a piece of paper (cellulose) to the mask. Such systems began to be available commercially in the 1950s for decontamination control in hospitals, nuclear facilities, homes vehicles, etc. They are comprised of randomly arranged fiberglass fibers between 0.5 and 2.0 microns. As with N95 masks, they are designed to capture a range of particles by the same mechanisms as N95 masks. Another common term is “MERV” Minimum Efficiency Reporting Value (1987 ASHRAE), which is designed to report the worst case performance of a filter, ranging 0.3–10 microns, with a scale rating of from 1 to 16 (highest at 95%). In summary:

3.4 Ventilation System Design

45

• General masks: • Surgical Masks (for healthcare workers and the general public)-provides partial protection and • Cloth Masks (for the general public)-provides some protection to the wearer, serves to stop the spread of viruses. • N95 masks: • N95 Respirators (primarily for healthcare workers)-provides high level of protection, • N95 masks use charged fibers (electrets) to attract particles of all sizes. N95 = traps at least 95% of particles, • N95 are excellent for trapping very small particles (below 0.1 micron), since their motion is random (Brownian) due to collisions with other particles, air molecules, etc., forces them to collide with electrically charged “sticky” fibers, • N95 are excellent in trapping large particles (above 0.3 microns), since their motion is effectively linear across the mask (because of their inertia) and will collide with a fiber and • N95’s main features (1)Van der Waals forces at small scale and (2) many layers of fibers and (3) charged fibers. • Problems: • N95: poor in trapping medium-sized particles (0.1 microns–0.3 microns), which travel with airflow patterns, • Ill-fitting masks, re-use due to shortages, use of solvents for mask cleaning which neutralize the fiber charge and • Improper donning and doffing of a mask is nontrivial, due to strap elongation, facial hair, etc. It is essential that ventilation systems are now given much more attention than ever before, in particular HEPA (High Efficiency Particulate Absorbing) grade, which remove 99.95% of 0.3 micron particles. Accordingly, this work seeks to develop a framework to optimize ventilation systems. One can consider this as a guide to either redesign vents, design for workplace layout or a guide for the placement of mobile vents. In this work, mathematical models are developed to simulate the progressive time-evolution of the distribution of particles produced by a cough. Two main enhancements to the previous models, are explored: • The use of masks to impede respiratory particle release and • The use of machine-learning to optimize ventilation (flow) systems to capture particles not stopped by masks Specifically, the previous models are combined with genetic machine-learning methods to construct optimal ventilation systems. The genetic representation allows for optimization of ventilation flowrates and placements, in order to sequester particles released from respiratory emissions such as coughs, sneezes, etc (Fig. 3.5).

46

3 Part 2: Microscale Disease Transmission and Ventilation System Design

Fig. 3.5 Left: The model problem studied in this work, including masks and ventilation systems. Shown are the particles captured by the mask and the particles that get through, color coded by size. Also shown are the location of the vents for this specific scenario (which will be optimized in this work). Right: Zoom on the release of cough particles, color coded by size (Zohdi [172, 176])

3.4.1 Assumptions Similar to before, we make the following assumptions (Zohdi [172, 176]): • We assume the same initial velocity magnitude for all particles under consideration, with a random distribution of outward directions away from the source of the cough. This implies that a particle non-interaction approximation is appropriate. Thus, the inter-particle collisions are negligible. This has been repeatedly verified by “brute-force” collision calculations using formulations found in Zohdi [151, 153, 155, 159]. • We assume that the particles are spherical with a random distribution of radii Ri , i = 1, 2, 3 . . . N = par ticles. The masses are given by m i = ρi 43 π Ri3 , where ρi is the density of the particles. • We assume that the cough particles are quite small and that the amount of rotation, if any, contributes negligibly to the overall trajectory of the particles. The equation of motion for the ith particle in the system is grav

m i v˙ i =  i

drag

+ i

+  imask ,

(3.26)

with initial velocity v i (0) and initial position r i (0). The gravitational force is grav = m i g, where g = (gx , g y , gz ) = (0, 0, −9.81) m/s 2 . i • For the drag, we will employ a general phenomenological model drag

i

=

1 ρa C D ||v f − v i ||(v f − v i )Ai , 2

(3.27)

3.4 Ventilation System Design

47

where C D is the drag coefficient, Ai is the reference area, which for a sphere is Ai = π Ri2 , ρa is the density of the ambient fluid environment and v f is the velocity of the surrounding medium which, in the case of interest, is air. We will assume that the velocity of the surrounding fluid medium (v f ) is given, implicitly assuming that the dynamics of the surrounding medium are unaffected by the particles.3

3.4.2 Incorporation of Masks The mask model will include an extra term in the drag resistance for particles in the domain of the mask ahead of the face:  imask = −Rv i Ais ,

(3.28)

where As = 4π R 2 .

3.4.3 Incorporation of Vents To account for multiple interacting vents assume a radial distance decay model for flow field from each vent, j = 1, 2, . . . N of the form: v

v j = v oj e−d||r f −r i || f

(r iv − r i ) , ||r iv − r i ||

(3.29)

where r vj is the position of the jth vent and d is a decay factor. Figure 3.6 illustrates the flow patterns with this model for 4 randomly placed vents. These will be optimized later in the work.

3.4.4 Overall Model In summary, the overall model is: grav

m i v˙ i =  i

drag

+ i

1 +  imask = m i g + ρa C D ||v f − v i ||(v f − v i )Ai − Rv i Ais . 2

(3.30)

3

We will discuss these assumptions further, later in the work.

48

3 Part 2: Microscale Disease Transmission and Ventilation System Design

Fig. 3.6 Flow patterns with the model. Left: flow velocity magnitude. Right: flow velocity magnitude with streamlines (see Zohdi [172, 176])

3.5 Genetic-Based Machine-Learning Ventilation Optimization The rapid rate at which these simulations can be completed enables the ability to explore inverse problems which seek to determine what parameter combinations can deliver a desired result (Fig. 3.7). Following the genetic-based machine-learning algorithm introduced in Chap. 1 (Zohdi [147, 168, 170, 171, 175–180, 183]), we formulate the objective as a cost function minimization problem that seeks system parameters that match a desired response (1 , ...16 ) =

P A RT I C L E S R E M AI N I N G . T O T AL R E L E AS E D

(3.31)

We systematically minimize Eq. 3.31, min  , by varying the design paramedef def ters: i = {i1 , i2 , i3 , ..., iN } = { panel si ze, spacing, angles...}. The system (+) parameter search is conducted within the constrained ranges of (−) 1 ≤ 1 ≤ 1 , (+) (−) (+) (−) 2 ≤ 2 ≤ 2 and 3 ≤ 3 ≤ 3 , etc. These upper and lower limits would, in general, be dictated by what is physically feasible.

3.5.1 Model Problem The system was optimized, Eq. 3.31, varying the following (16) parameters: • Ventilation locations: 4 vents, each with 3 (x1 , x2 , x3 ) locations = 12 variables and • Ventilation flow rates : 4 vents, each with 1 flow rate = 4 variables.

3.5 Genetic-Based Machine-Learning Ventilation Optimization

49

SET THE FLOW PARAMETER GENE =Λ

RUN VENTILATION SIMULATION

STORE PERFORMANCE Π POPULATION

GENERATIONS

REPEAT FOR NEXT GENE

RANK GENES IF COST FUNCTION Π < TOL

STOP

Fig. 3.7 The genetic-based MLA flowchart (Zohdi [147, 168, 170, 171, 175–180, 183])

The design parameters  = {1 , 2 ... N } were optimized over the search intervals (16 variables): i− ≤ i ≤ i+ , i = 1, 2, ...16. Specifically, the system minimized the cost function by varying 16 parameters (4 vents each with 3 (x1 , x2 , x3 ) locations and one flow rate) within the following ranges: Flow rate of vent 1:v1− = 0 ≤ v1o ≤ v1+ = 1.0, Flow rate of vent 2: v2− = 0.5 ≤ v2o ≤ v2+ = 1.5, Flow rate of vent 3:v3− = 0.0 ≤ v3o ≤ v3+ = 1.0, Flow rate of vent 4:v4− = 0.5 ≤ v4o ≤ v4+ = 1.5, Position of vent 1: (x1 , x2 , x3 )− = (−0.5, 1.0, 2.0) ≤ (x1 , x2 , x3 ) ≤ (x1 , x2 , x3 )+ = (1.5, 3.0, 4.0), • Position of vent 2: (x1 , x2 , x3 )− = (−1.5, 2.0, 2.0) ≤ (x1 , x2 , x3 ) ≤ (x1 , x2 , x3 )+ = (0.5, 4.0, 4.0), • Position of vent 3: (x1 , x2 , x3 )− = (−1.5, 1.0, −1.0) ≤ (x1 , x2 , x3 ) ≤ (x1 , x2 , x3 )+

• • • • •

= (1.0, 3.0, 1.0) and

• Position of vent 4: (x1 , x2 , x3 )− = (−0.5, 2.0, −1.0) ≤ (x1 , x2 , x3 ) ≤ (x1 , x2 , x3 )+ = (1.5, 4.0, 1.0).

Figure 3.8 shows the reduction of the cost function for the 16 parameter set. This cost function  represents the fraction of uncaptured particles. Shown is the best performing gene (design parameter set, in red) as a function of successive generations, as well as the average performance of the entire population of the genes (designs, in green). We used the following genetic-based MLA settings: • • • •

Number of design variables: 16, Population size per generation: 24, Number of parents to keep in each generation: 6, Number of children created in each generation: 6,

50

3 Part 2: Microscale Disease Transmission and Ventilation System Design

• Number of completely new genes created in each generation: 12 • Number of generations for re-adaptation around a new search interval: 10 and • Number of generations: 50. Table 5.1 and Fig. 3.8 illustrate the results. The algorithm was automatically reset around the best gene every 10 generations. The entire 50 generation simulation, with 24 genes per evaluation (1200 total designs) took a few minutes on a laptop, making it ideal as a design tool. In Fig. 3.8, the optimization results are for successively longer time limits of T = 1.75, 2.0, 2.25 and 2.5 s for the 16 parameter set. Successively allowing longer simulations times allows the vents to trap more particles. We note that, for a given set of parameters, each complete simulation takes on the order of 0.25 seconds, several thousand parameters sets can be evaluated in well under an hour, without even exploiting the inherent parallelism of the genetic-based MLA. The operational speed of the model makes it a simulation tool that is easily adoptable as a digital-twin that can be inexpensively and safely tested, manipulated and optimized in a virtual setting and then deployed in the physical world afterwards, thus reducing experimental testing and accelerating new ventiliation system development (Table 3.2).

3.6 Summary and Extensions The focus of this work was centered around developing a framework for optimizing ventilation systems, based on the concept of digital-twins of physical reality, utilizing an analysis first found in Zohdi [172, 176], which developed models for the range, distribution and settling time of the cough particles under the influence of gravity and drag from the surrounding air. Numerical examples were provided to illustrate the framework. For general cough conditions, there can be cases where there are large changes in the surrounding fluid flow, due to the motion of the particles and cough. The result is a system of coupled equations between the particles and the fluid, requiring spatio-temporal discretization, such as high-fidelity Finite Element or Finite Difference methods, of the classical equations governing the surrounding fluid mechanics (Navier Stokes)

Balance o f mass :

∂ρ = −∇x ρ · v − ρ∇x · v, ∂t

Balance o f momentum : ρ(

∂v + (∇x v) · v) = ∇x · σ + f , ∂t

Constitutive Law : σ = −P1 + λtr D1 + 2μ D = −P1 + 3κ tr3D 1 + 2μ D , (3.32) where ρ(x) is the density field of the fluid, v(x) is the fluid velocity field, σ (x) is the fluid stress field, D(x) is the fluid velocity gradient field, f (x) is the body

1

0.514

0.728

0.514

0.491

T lim

0.175

2.000

2.225

2.500

2.219

2.343

2.184

2.194

2

2.800

2.931

2.712

2.661

3

0.832

0.765

0.746

0.781

4

6 3.419 3.426 3.307 3.296

5

−0.286

−0.199

−0.263

−0.313

Table 3.2 The top system parameter performers (1 − 16 )

2.488

2.429

2.383

2.411

7

1.345

1.205

1.275

1.369

8

10

−0.718 2.343

−0.752 2.466

−0.772 2.434

−0.740 2.368

9

0.233

0.228

0.228

0.245

11

0.722

0.736

0.731

0.671

12

14

−2.824 2.039

−3.903 2.057

−1.047 2.018

−3.543 2.057

13

0.656

0.669

0.657

0.689

15

1.482

1.494

1.485

1.485

16

0.000

0.025

0.125

0.365



3.6 Summary and Extensions 51

52

3 Part 2: Microscale Disease Transmission and Ventilation System Design

BEST POPULATION

BEST POPULATION

BEST POPULATION

BEST POPULATION

Fig. 3.8 Optimization for successively longer time limits of T = 1.75, 2.0, 2.25 and 2.5 s, showing the reduction of the cost function for the 16 parameter set. Shown is the best performing gene (design parameter set, in red) as a function of successive generations, as well as the average performance of the entire population of genes (design parameter set, in green). Successively allowing longer simulation times allows the vents to trap more particles (see Zohdi [172, 176])

force field, P(x) is the fluid pressure field, λ(x) and μ(x) are fluid material property fields.4 It is important to emphasize that physically compatible boundary data must be applied, and this is not a trivial matter for compressible flow. Additionally, the first law of thermodynamics should be included (along with equations for various chemical reactions), which reads as ρ w˙ − σ : ∇x v + ∇x · q − ρz = 0,

(3.33)

where w(x) is the stored energy in the fluid, q(x) is the heat flux field, z is the heat source field per unit mass. Generally such models are ineffective for rapid It is customary to specify v and P on the boundary, and to determine ρ on the boundary through the Equation of State. P is given by an Equation of State. 4

3.6 Summary and Extensions

53

real-time use, but are quite useful for detailed offline background analyses, where a rapid response is a nonissue. The continuum discretization is usually combined with a Discrete Element Method for the particle dynamics. There are a variety of such approaches, for example, see Avci and Wriggers [5], Onate et al. [104, 105], Leonardi et al. [80], Onate et al. [106], Bolintineanu et al. [14] and Zohdi [151, 159]. Such models are significantly more complex than the models used in the current work. More detailed analyses of fluid-particle interaction can be achieved in direct, brute-force, numerical schemes, treating the particles as part of the fluid continuum (as another fluid or solid phase), and thus meshing them in a detailed manner. In such an approach (for example see Avci and Wriggers [5]) • A fluid-only problem is solved, with (instantaneous) boundary conditions of v f (x) = v i (x) at each point on the fluid-particle boundaries, where the velocity of the points on the boundary are given by v i (x) = v icm + ωi × Rcm→sur f. (x),

(3.34)

where v icm is the center of mass and ωi is the angular velocity for each of the individual particles and Rcm→sur f. is a vector from the mass center to the surface. • For each particle, one would solve: drag

m i v˙ i =  i

+ other f or ces

(3.35)

+ other moments

(3.36)

and drag

Ii ω˙ i = M i

where the forces and moments would have a contribution from the fluid drag (with particle occupying domain i and outward surface normal n) which is defined as  drag i = σ · n d A, (3.37) ∂ i

and  drag Mi

=

Rcm→sur f. × σ · n d A,

(3.38)

∂ i

• At a time-step, the process is iteratively driven by solving the fluid-only problem first, then the particles-only problem, and repeated until convergence in an appropriate norm. Along these lines, in Zohdi [151, 159], more detailed, computationally intensive models were developed to characterize the motion of small-scale particles embedded in a flowing fluid where the dynamics of the particles affects the dynamics

54

3 Part 2: Microscale Disease Transmission and Ventilation System Design

of the fluid and vice-versa. In such a framework, a fully implicit Finite-Difference discretization of the Navier-Stokes equations was used for the fluid and a direct particle-dynamics discretization is performed for the particles. Because of the large computational difficulty and expense of a conforming spatial discretization needed for large numbers of embedded particles, simplifying assumptions are made for the coupling, based on semi-analytical computation of drag-coefficients, which allows for the use of coarser meshes. Even after these simplifications, the particle-fluid system was strongly-coupled. The approach taken in that work was to construct a sub-model for each primary physical process. In order to resolve the coupling, a recursive staggering scheme was constructed, which was built on works found in Zohdi [151, 153, 155, 159]. The procedure was as follows (at a given time increment): (1) each submodel equation (fluid or particle-system) is solved individually, “freezing” the other (coupled) fields in the system, allowing only the primary field to be active, (2) after the solution of each submodel, the associated field variable was updated, and the next submodel was solved and (3) the process is then repeated until convergence. The time-steps were adjusted to control the rates of convergence, which is dictated by changes in the overall physics. Specifically, the approach was a staggered implicit time-stepping scheme, with an internal recursion that automatically adapted the time-step sizes to control the rates of convergence within a time-step. If the process did not converge (below an error tolerance) within a preset number of iterations, the time-step was adapted (reduced) by utilizing an estimate of the spectral radius of the coupled system. The developed approach can be incorporated within any standard computational fluid mechanics code based on finite difference, finite element, finite volume or discrete/particle element discretization (see Labra and Onate [79], Onate et al. [104, 105], Avci and Wriggers [5]). However, while useful in many industrial applications where high precision is required, the use of such a model for the coarser applications of interest in this work is probably unwarranted. Remark: In the next chapter, we study a closely related theme, ultraviolet decontamination of pathogens.

Chapter 4

Part 3: Ultraviolet Viral Decontamination

Summary This chapter focuses on viral decontamination by ultraviolet (UV) irradiation technologies and develops an efficient and rapid computational method to simulate a UV pulse in order to ascertain the decontamination efficacy of UV irradiation for a surface. It is based on decomposition of a pulse into a groups of rays, which are then tracked as they progress towards the target contact surface. The algorithm computes the absorption at the point of contact and color codes it relative to the incoming irradiation. This allows one to quickly quantify the decontamination efficacy across the topology of a structure. This chapter follows work found in Zohdi [174].

4.1 Introduction Viral decontamination based on UV technology has become ubiquitous, with many variants now being proposed, in response to the outbreak of COVID-19. UV light varies in wavelength from 10 nm to 400 nm, thus making it shorter than visible wavelengths and larger than x-rays. Short wave UV light (UV-c) can damage viral DNA and sterilize surfaces making it useful in the medical industry. This was first noted in 1878 (Downes and Blunt [34]) when the effect of short-wavelength light killing bacteria was discovered. By 1903 it was known that the most effective wavelengths were around 250 nm (UV-c), for which Niels Finsen won a Nobel Prize (for skin-based tuberculosis eradication using UV light). By approximately 1960, the effect that ultraviolet radiation can destroy DNA in living microorganisms was established (see Bolton and Colton [15] for reviews). While many types of decontamination technologies are of interest (see references Anderson et al. [3], Battelle [9], Boyce et al. [19], Card et al. [26], Heimbuch and Harish [58], Heimbuch et al. [59], Ito and Ito [70], Lin et al. [83], Kanemitsu [72], Lindsley et al. [84], Lore et al. [88], Marra et al. [91], Mills et al. [98], Nerandzic et al. [101], Viscusi et al. [132] and Tseng and Li [124]), this work will focus on UV-c technologies. The literature asserts that UV-c irradiation doses of above 1 J/cm2 at 254 nm peak wavelength © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 T. I. Zohdi, Modeling and Simulation of Infectious Diseases, https://doi.org/10.1007/978-3-031-18053-8_4

55

56

4 Part 3: Ultraviolet Viral Decontamination

inactivates SARS-CoV-2 and achieves above a 99% biocidal efficacy on Bacillus subtilis spores.1 However, the literature also presents evidence that it is difficult to ensure that all surfaces are completely decontaminated, due to shadowing effects. Accordingly, this work focuses on fast simulation tools for viral decontamination by ultraviolet (UV) irradiation technologies. Specifically, an efficient and rapid computational method is developed to simulate a UV pulse, in order to ascertain the decontamination efficacy of UV irradiation for a surface. Remark: For maximum efficacy, purely UV-c protocols should be one component of a multistage decontamination process involving a combination of (1) hydrogen peroxide vapor and gas plasma, (2) heat and humidity and (3) UV-c irradiation.

4.1.1 Objectives This work develops an efficient and rapid computational method to simulate a UV pulse in order to ascertain the decontamination efficacy of UV irradiation for a surface. It is based on decomposition of a pulse into a groups of rays, which are then tracked as they progress towards the target contact surface. The algorithm computes the absorption at the point of contact and color codes it relative to the incoming irradiation. This allows one to rapidly quantify the decontamination uncertainty by identifying regions where the absorption is inadequate and serves as a guide for practitioners to ascertain where problems may occur, before proceeding to experiments. Additionally, the reflections are calculated, and can be used to ascertain bystander safety. The interest here is on the absorption of an initially coherent pulse (Fig. 4.1), represented by multiple collimated (parallel) rays (initially forming a planar wave front), where each ray is a vector in the direction of the flow of energy (the rays are parallel to the initial wave’s propagation vector). We make the following observations: • It is assumed that the features of the surface to be irradiated are at least an order of magnitude larger than the wavelength of the incident radiation (essentially specular surfaces), therefore “geometrical” ray tracing theory is applicable, and is wellsuited for the systems of interest. It is important to emphasize the regimes of validity of such a model are where the surface features are larger than the UV wavelengths. For example, if we were to use UV-rays (10−8 m ≤ λ ≤ 4 × 10−7 m), the features in this analysis would be assumed to possess scales larger than approximately 4 × 10−6 m. For systems containing features smaller than this, one can simply use the model as a qualitative guide. • Ray-tracing is a method that is employed to produce rapid approximate solutions to wave-equations for high-frequency/small-wavelength applications, where the primary interest is in the overall propagation of energy.2 While 1 J/cm2 is effective for decontamination, it is harmful to humans. Resolving diffraction (which ray theory is incapable of describing) is unimportant for the applications of interest.

1 2

4.2 Electromagnetic Energy Propagation

57

BEAM DECOMPOSITION

INCOMING RAYS

Ii incident SURFACE TO BE DECONTAMINATED

Θ i

Θr Ir reflected

Θa I a absorbed Fig. 4.1 An electromagnetic pulse applied to a surface (see Zohdi [174])

• Ray-tracing methods proceed by initially representing wave fronts by an array of discrete rays. Thereafter, the problem becomes one of a primarily geometric character, where one tracks the changing trajectories and magnitudes of individual rays which are dictated by the reflectivity and the Fresnel conditions (if a ray encounters a material interface). Other high frequency irradiation regimes can also be simulated in the same manner, such as X-rays and gamma rays, provided that the scattering target has the appropriate (larger) length-scale. Even in the case where this clear separation of length scales is not present, this model still provides valuable information on the propagation of the beam and the reflected response of the dispersed system. • Ray-tracing methods are well-suited for computation of scattering in complex systems that are difficult to mesh/discretize (such as the Finite Difference Time Domain Method or the Finite Element Method).

4.2 Electromagnetic Energy Propagation 4.2.1 Beam-Ray Decomposition In order to connect the concept of a ray with a pulse/beam, since I¯ is the energy per unit area per unit time (power), we obtain the power associated with an entire pulse/beam by multiplying the irradiance by the cross-sectional area of an initially coherent beam, I¯ Ab , where Ab is the cross-sectional area of the beam (comprising

58

4 Part 3: Ultraviolet Viral Decontamination

all of the rays). The energy per unit time (power) for a ray in the pulse/beam is then given by I = I¯ Ar = I¯ Ab /Nr ,

(4.1)

where Nr is the number of rays in the beam (Fig. 4.1) and Ar can be considered the area associated with a ray. Essentially, rays are a mathematical construction/discretization of a pulse/beam We refer the reader to Gross [54] and Zohdi [149, 169, 174] for details.

4.2.2 Reflection and Absorption of Rays Following a framework found in Zohdi [149, 169, 174], we consider a ray of light incident upon a material interface which produces a reflected ray and a transmitted/absorbed (refracted) ray (Fig. 4.1). The amount of incident electromagnetic power def is given by Ii and the reflected is Ir , is given by the total reflectance (ratio), IR = IIri , where 0 ≤ IR ≤ 1. We have the following observations: • The angle between the point of contact of a ray (Fig. 4.1) and the outward normal to the surface at that point is the angle of incidence, θi . The classical reflection law states that the angle at which a ray is reflected is the same as the angle of incidence and that the incoming (incident, θi ) and outgoing (reflected, θr ) ray lays in the same plane, and θi = θr . • The classical refraction law states that, if the ray passes from one medium into a second one (with a different index of refraction), and, if the index of refraction of the second medium is less than that of the first, the angle the ray makes with thenormal def to the interface is always less than the angle of incidence, where nˆ = vvai = ai μμai = sinθi , θa sinθa

being the angle of the absorbed ray (Fig. 4.1), va is the propagation speed in the absorbing medium, vi is the propagation speed in the incident medium, a is the electric permittivity of the absorbing medium, μa magnetic permeability of the absorbing medium, i is the electric permittivity in the incident medium and μi magnetic permeability in the incident medium. • Recall, all electromagnetic radiation travels, in a vacuum, at the speed c ≈ 2.99792458 × 108 ± 1.1 m/s. In any another medium v = √1μ for electromagnetic waves.3 • We define nˆ as the ratio of the refractive indices of the ambient (incident) medium (ni ) and absorbing medium (na ), nˆ = na /ni , where μˆ is the ratio of the magnetic permeabilities of the surrounding incident medium (μi ) and scattering/absorbing medium (μa ), μˆ = μa /μi . Thus, we have 3

The free space electric permittivity is o =

magnetic permeability is μo = 4π

× 10−7

1 c2 μo

= 8.8542 × 10−12 CN−1 m−1 and the free space

WbA−1 m−1

= 1.2566 × 10−6 WbA−1 m−1 .

4.3 Electromagnetic Wave Propagation and Rays

nˆ =

na = ni



59

a μa ⇒ a μa = (ˆn)2 i μi . i μi

(4.2)

• For a pulse of light, the reflectivity IR can be shown to be (see [54] for example) IR =

Ir Ii

=

1 2

 nˆ 2 μˆ nˆ 2 μˆ

1

cosθi −(ˆn2 −sin2 θi ) 2

2

 +

1

cosθi +(ˆn2 −sin2 θi ) 2

1

cosθi − μ1ˆ (ˆn2 −sin2 θi ) 2

2  ,

1

cosθi + μ1ˆ (ˆn2 −sin2 θi ) 2

(4.3)

where Ii is the incoming irradiance, Ir the reflected irradiance, nˆ is the ratio of the refractive indices of the of absorbing (na ) and incident media (ni ), where the refractive index is defined as the ratio of the speed of light in a vacuum (c) to that of the medium (v), where the speed of electromagnetic waves is c = √1o μo , where  is the electric permittivity and μ is the magnetic permeability. • We consider applications with non-magnetic media and frequencies where the magnetic permeability is virtually the same for both the incident medium (usually the atmosphere) and the scattering/absorbing medium. Thus, for the remainder of the work, we shall take μˆ = 1 (μo = μi = μa ), thus nˆ =

na = ni



a μa ⇒ a μa = (ˆn)2 i μi ⇒ a = (ˆn)2 i . i μi

(4.4)

This yields  IR =

Ir Ii

=

1 2

1

nˆ 2 cosθi −(ˆn2 −sin2 θi ) 2

1

nˆ 2 cosθi +(ˆn2 −sin2 θi ) 2

2

 +

1

cosθi −(ˆn2 −sin2 θi ) 2

1

cosθi +(ˆn2 −sin2 θi ) 2

2  .

(4.5)

• Notice that as nˆ → 1 we have complete absorption, while as nˆ → ∞ we have complete reflection. The total amount of absorbed power by the material is (1 − IR)Ii . The next section supplies the theory underpinning electromagnetic wave propagation and rays.

4.3 Electromagnetic Wave Propagation and Rays Following a framework found in Zohdi [149, 169, 174], the propagation of electromagnetic waves in free space can be described by a simplified form of Maxwell’s equations (see Jackson [71], Zohdi [156]) ∇ × E = −μo ∂H ∂t ,

and

∇ × H = o ∂E ∂t ,

(4.6)

60

4 Part 3: Ultraviolet Viral Decontamination

where ∇ · H = 0, ∇ · E = 0, E is the electric field, H is the magnetic field, o is the free space permittivity and μo is the free space permeability. Using standard vector identities, one can show that ∇ × (∇ × E) = −μo o ∂∂tE2 ,

and

∇ × (∇ × H) = −μo o ∂∂tH 2 ,

1 ∂2 E , c2 ∂t 2

and

∇2H =

2

2

(4.7)

and that ∇2E =

where the speed of electromagnetic waves is c =

1 ∂2 H , c2 ∂t 2

(4.8)

√ 1 . All electromagnetic radiation o μo 8

travels, in a vacuum, at the speed c ≈ 2.99792458 × 10 ± 1.1 m/s. In any another medium, for electromagnetic waves, the propagation speed is v = √1μ , where  and μ are the electric permittivity and magnetic permeability of that medium, respectively.4

4.3.1 Plane Harmonic Wave Fronts Now consider the special case of plane harmonic waves, for example of the form E = Eo cos(k · x − ωt)

and

H = H o cos(k · x − ωt),

(4.9)

where x is an initial position vector to the wave front, where k is the direction of as the angular propagation. We refer to the phase as φ = k · x − ωt, and ω = 2π τ frequency, where τ is the period. For plane waves, the wave front is a plane on which φ is constant, which is orthogonal to the direction of propagation, characterized by k. In the case of harmonic waves, we have k × E = μo ωH

and

k × H = −o ωE,

(4.10)

and k · E = 0 and k · H = 0. The three vectors, k, E and H constitute a mutually E×H . Electromagorthogonal triad.5 The direction of wave propagation is given by ||E×H|| netic waves traveling through space carry electromagnetic energy which flows in the direction of wave propagation. The energy per unit area per unit time flowing perpendicularly into a surface in free space is given by the Poynting vector S = E × H.

4

The free space electric permittivity is o =

1 c2 μo

= 8.8542 × 10−12 CN−1 m−1 and the free space

magnetic permeability is μo = 4π = 1.2566 × 10−6 WbA−1 m−1 . By combining the relations in Equation 4.10 one obtains ||k|| = ωc . × 10−7

5

WbA−1 m−1

4.3 Electromagnetic Wave Propagation and Rays

61

4.3.2 Natural (Random) Electromagnetic Energy Propagation Since at high-frequencies E, H and S oscillate rapidly, it is impractical to measure instantaneous values of S directly. Consider the harmonic representations in Eq. 4.9 which leads to S = Eo × H o cos2 (k · x − ωt), and consequently the average value over a longer time interval (T ) than the time scale of rapid random oscillation, S T = Eo × H o cos2 (k · x − ωt) T = 21 Eo × H o ,

(4.11)

leading to the definition of the irradiance def

I = ||S|| T = 21 ||Eo × H o || =

1 2



o ||Eo ||2 . μo

(4.12)

Thus, the rate of flow of energy is proportional to the square of the amplitude of the electric field.

4.3.3 Reflection and Absorption of Energy-Fresnel Relations We consider a plane harmonic wave incident upon a plane boundary separating two different materials, specifically vacuum and surface, which produces a reflected wave and an absorbed (refracted) wave (Fig. 4.1). Two cases for the electric field vector are considered: • (1) electric field vectors that are parallel (||) to the plane of incidence and • (2) electric field vectors that are perpendicular (⊥) to the plane of incidence. In either case, the tangential components of the electric and magnetic fields are required to be continuous across the interface. Consider case (1). We have the following general vectorial representations E|| = E|| cos(k · x − ωt) e1

and

H || = H|| cos(k · x − ωt) e2 ,

(4.13)

where e1 and e2 are orthogonal to the propagation direction k. By employing the law of refraction (ni sinθi = na sinθa ) we obtain the following conditions relating the incident, reflected and absorbed components of the electric field quantities E||i cosθi − E||r cosθr = E||a cosθa

and

H⊥i + H⊥r = H⊥a .

(4.14)

Since, for plane harmonic waves, the magnetic and electric field amplitudes are E related by H = vμ , we have

62

4 Part 3: Ultraviolet Viral Decontamination

E||i + E||r =

μi vi μi na ˆ def n E||a = E||a = E||a , μa va μa ni μˆ

(4.15)

where μˆ = μμai , nˆ = nnai and where vi , vr and va are the values of the velocity in the incident, reflected and absorbed directions.6 By again employing the law of refraction, we obtain the Fresnel reflection and transmission/absorption coefficients, generalized for the case of unequal magnetic permeabilities def

r|| =

def

E||r E||i

=

nˆ μˆ cosθi −cosθa nˆ μˆ cosθi +cosθa

and

E||a E||i

a|| =

=

2cosθi . cosθa + μnˆˆ cosθi

(4.16)

Following the same procedure for case (2), where the components of E are perpendicular to the plane of incidence, we have r⊥ =

E⊥r E⊥i

=

cosθi − μnˆˆ cosθa

and

cosθi + μnˆˆ cosθa

a⊥ =

E⊥a E⊥i

=

2cosθi . cosθi + μnˆˆ cosθa

(4.17)

Our primary interest is in the reflections. We define the reflectances as def

IR|| = r||2

and

def

IR⊥ = r⊥2 .

(4.18)

Particularly convenient forms for the reflections are r|| =

nˆ 2 μˆ nˆ 2 μˆ

1

1

cosθi −(ˆn2 −sin2 θi ) 2

and

1 cosθi +(ˆn2 −sin2 θi ) 2

r⊥ =

cosθi − μ1ˆ (ˆn2 −sin2 θi ) 2

1

cosθi + μ1ˆ (ˆn2 −sin2 θi ) 2

.

(4.19)

Thus, the total energy reflected can be characterized by def

IR =



Er Ei

2 =

2 2 E⊥r + E||r

Ei2

=

I||r + I⊥r . Ii

(4.20)

If the resultant plane of oscillation of the (polarized) wave makes an angle of γi with the plane of incidence, then E||i = Ei cosγi

and

E⊥i = Ei sinγi ,

(4.21)

and it follows from the previous definition of I that I||i = Ii cos2 γi

and

I⊥i = Ii sin2 γi .

(4.22)

Substituting these expression back into the expressions for the reflectances yields

6

Throughout the analysis we assume that nˆ ≥ 1.

4.3 Electromagnetic Wave Propagation and Rays

IR =

63

I||r I⊥r 2 cos2 γi + sin γi = IR|| cos2 γi + IR⊥ sin2 γi . Ii Ii

(4.23)

For natural or unpolarized electromagnetic radiation, the angle γi varies rapidly in a random manner, as does the field amplitude. Thus, since cos2 γi (t) T =

1 2

and

sin2 γi (t) T = 21 ,

(4.24)

and therefore for natural electromagnetic radiation I||i =

Ii 2

and

I⊥i = I2i .

I||r I||i

and

r⊥2 =

(4.25)

and therefore r||2 =

 E 2 2 ||r 2 E||i

=



2 E⊥r 2 E⊥i

2

=

I⊥r . I⊥i

(4.26)

Thus, the total reflectance becomes IR =

1 1 (IR|| + IR⊥ ) = (r||2 + r⊥2 ), 2 2

where 0 ≤ IR ≤ 1. For the cases where sinθa = relations as r|| = where, j =

nˆ 2 μˆ nˆ 2 μˆ

> 1, one may rewrite reflection

sinθi nˆ

1

cosθi −j(sin2 θi −ˆn2 ) 2

1 cosθi +j(sin2 θi −ˆn2 ) 2

(4.27)

1

and

r⊥ =

cosθi − μ1ˆ j(sin2 θi −ˆn2 ) 2

1

cosθi + μ1ˆ j(sin2 θi −ˆn2 ) 2

,

(4.28)

√ −1, and in this complex case7 def

IR|| = r|| r¯|| = 1,

and

def

IR⊥ = r⊥ r¯⊥ = 1,

(4.29)

where r¯|| and r¯⊥ are complex conjugates. Thus, for angles above the critical angle θi∗ , all of the energy is reflected. Notice that as nˆ → 1 we have complete absorption, while as nˆ → ∞ we have complete reflection. The amount of absorbed irradiance by the surface is Ia = (1 − IR)Ii .

4.3.4 Reflectivity To observe the dependency of IR on nˆ and θi we can explicitly write

7

The limiting case

sinθi∗ nˆ

= 1, is the critical angle (θi∗ ) case.

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4 Part 3: Ultraviolet Viral Decontamination

IR =

1 2

 nˆ 2 μˆ nˆ 2 μˆ

1

cosθi −(ˆn2 −sin2 θi ) 2

1

cosθi +(ˆn2 −sin2 θi ) 2

2

 +

1

cosθi − μ1ˆ (ˆn2 −sin2 θi ) 2

2  .

1

cosθi + μ1ˆ (ˆn2 −sin2 θi ) 2

(4.30)

We observe: • As nˆ → ∞, IR → 1, no matter what the angle of incidence’s value. We note that as nˆ → 1, provided that μˆ = 1, IR → 0, i.e. all incident energy is absorbed (it is transparent). • With increasing nˆ , the angle for minimum reflectance grows larger. As mentioned previously, for the remainder of the work, we shall take μˆ = 1 (μo = μi = μa ), thus  na a μa nˆ = = ⇒ a μa = (ˆn)2 i μi ⇒ a = (ˆn)2 i . (4.31) ni i μi • The previous assumption yields  IR =

Ir Ii

=

1 2

1

nˆ 2 cosθi −(ˆn2 −sin2 θi ) 2

1

nˆ 2 cosθi +(ˆn2 −sin2 θi ) 2

2

 +

1

cosθi −(ˆn2 −sin2 θi ) 2

2 

1

cosθi +(ˆn2 −sin2 θi ) 2

.

(4.32)

Remark: We now recall Eq. 4.1 connects the concept of a ray with a pulse/beam and the following: • Since I¯ is the energy per unit area per unit time (power), we obtain the power associated with an entire pulse/beam by multiplying the irradiance by the crosssectional area of an initially coherent beam, I¯ Ab , where Ab is the cross-sectional area of the beam (comprising all of the rays). • The energy per unit time (power) for a ray in the pulse/beam is then given by I = I¯ Ar = I¯ Ab /Nr , where Nr is the number of rays in the beam (Fig. 4.1) and Ar can be considered the area associated with a ray. • The reflection relation, Eq. 4.30, can then be used to compute changes in the magnitude of the reflected rays (and the amount absorbed), with directional changes given by the laws of reflection. We refer the reader to Gross [54] and Zohdi [149, 169, 174] for details.

4.4 Model Problem and Response Trends From this point forth, we assume that the ambient medium behaves as a vacuum. Accordingly, there are no energetic losses as the rays move through the surrounding medium.

4.4 Model Problem and Response Trends

65

4.4.1 Tracking of Beam-Decomposed Rays Starting at t = 0 and ending at t = T , the simple overall algorithm to track rays is as follows, at each time increment: 1. Check for intersections of rays with surfaces (hence a reflection), and compute the ray magnitudes and orientation if there are reflections (for all rays that are ref experiencing a reflection, Ij , j = 1, 2, ...Rays), 2. Increment all ray positions (rj (t + t) = rj (t) + tvj (t), j = 1, 2, ..., Rays), 3. Increment time forward (t = t + t) and repeat the process for the next time interval. In order to capture all of the ray reflections that occur: • The time-step size t is dictated by the offset height of the source. A somewhat ad-hoc approach is to scale the time-step size by the speed of ray propagation H , where H is the height of the source and 0.0001 ≤ ξ ≤ according to t = ξ ||v|| 0.01. Typically, the results are insensitive to ξ that are smaller than this range. • Although outside the scope of this work, one can also use this algorithm to compute the thermal response by combining it with heat transfer equations via staggering schemes (Zohdi [149, 156]).

4.4.2 Test Surface The discrete-ray approach is flexible enough to simulate a wide variety of systems. As a test surface, we consider a topology to be irradiated described by F(x1 , x2 , x3 ) = 0. The outward surface normals, n, needed during the scattering calculations, are easy to characterize by writing n=

∇F . ||∇F||

(4.33)

The components of the gradient are ∇F =

∂F ∂F ∂F e1 + e2 + e3 . ∂x1 ∂x2 ∂x3

(4.34)

It is advantageous to write the surface in parametric form: F(x1 , x2 , x3 ) = G(x1 , x2 ) − x3 = 0.

(4.35)

x3 = G(x1 , x2 ).

(4.36)

which leads to

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4 Part 3: Ultraviolet Viral Decontamination

The gradient becomes ∇F =

∂G ∂G e1 + e2 − e3 . ∂x1 ∂x2

(4.37)

In order to determine whether a ray has made contact with a surface domain, one checks if the x3 component of a ray (rj ) is less that x3 of the surface.

4.5 Numerical/Quantitative Examples We have the following set up for a series of tests: • The initial velocity vector for all initially collimated (parallel) rays comprising the beam was v = (c, 0, 0), where c = 3 × 108 m/s is the speed of light in a vacuum. • We used a parametrized test surface given by     2ω2 πx2 2ω1 πx1 sin , x3 = 2 + A sin L1 L2

(4.38)

with L1 = L2 = 1, ω1 = 1.5 and ω2 = 0.75, where we vary amplitude, A. We also added a flat cut-off so that the surface had a half-sine wave character (Fig. 4.3). • The number of rays in the beam was steadily increased from Nr = 100, 200, etc, until the results were insensitive to further refinements. This approach indicated that between approximately 9500 ≤ Nr ≤ 10000 parallel rays in rectangular crosssectional plane of the beam. The rays were randomly placed within the beam (Figs. 4.1 and 4.3), to correspond to unpolarized incoming energy, and yielded stable results across the parameter study range. • Figure 4.3 shows a sequence of frames of the detailed response of a surface to 10000 rays. Figure 4.2 shows a top view. Table 4.1 shows the steady loss of absorption efficacy with contact surface amplitude oscillation (waviness). The algorithm computes the absorption at the point of contact and color codes it relative to the incoming irradiation. This allows one to quickly quantify the decontamination across the topology of the structure. • This approach also allows an analyst to explore nonuniform beam profiles, for example exponential central irradiance decay: I (d ) = I (d = 0)e−ad , where d is the distance from the center of the initial beam, where in the case of a = 0, one recaptures a flat beam, I (d ) = I (d = 0).8

r inc Note that algorithmically, we can the set total initial irradiance via N i=1 Ii (t = 0)Ar = P Watts.To achieve this distribution, one would first place rays randomly in the plane, and then scale the individual I inc by e−ad and the normalized the average so that the total was P watts.

8

4.6 Summary and Discussion

67

Fig. 4.2 Top view for a surface with amplitude A = 0.3. Colors indicate the absorption, normalized by the incoming. radiation level (see Zohdi [174])

4.6 Summary and Discussion Because UV-c based decontamination methods are becoming widely used in industry, with many variants being proposed, fast computational analysis and design tools are needed to ascertain their effectiveness. Accordingly, this chapter developed a discrete-ray model to allow for propagation of energy encountering a surface, based on the decomposition of irradiation into a groups of rays, which are then tracked as they progress towards the target. This facilitates: • Quick quantification of the decontamination efficacy across the topology of the structure (color coding the efficacy relative to the incoming irradiation). • Parametric studies to the changes in absorption as a function of changes in surface geometry. The simulations take on the order of one minute on a laptop. This type of approach makes it quite suitable for use in conjunction with mobile decontamination systems and allows provides a simpler alternative to a direct, computationally intensive, discretization of a continuum description using Maxwell’s equations with a Finite Element or Finite Difference method. We emphasize that this model and rapid simulation technique facilitates the concept of a digital-twin of physical reality that can be inexpensively and safely manipulated and optimized in a virtual setting and then

68

4 Part 3: Ultraviolet Viral Decontamination

Fig. 4.3 Sequence of frames for a surface with amplitude A = 0.3 (the amplitude was enhanced by a factor of 20 in graphics to more easily see the effects of the topology on absorption). Colors indicate the absorption, normalized by the incoming radiation level (see Zohdi [174])

4.6 Summary and Discussion

69

Table 4.1 The loss of absorption efficacy with contact surface amplitude oscillation (waviness) Surface amplitude Average surface absorption 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

0.8888 0.8879 0.8808 0.8675 0.8507 0.8327 0.8126 0.7878 0.7679 0.7485 0.7315

deployed in the physical world afterwards, thus reducing the costs of experiments and accelerating new decontamination technology development. Remark: In the next chapter, we conclude the monograph with a study of modeling and simulation of vaccine design and immune response.

Chapter 5

Part 4: Vaccine Design and Immune-System Response

Summary In this chapter, a computational framework is developed that researchers in the field can easily implement and subsequently use as an efficient tool to study the immuneresponse to a vaccine injection. There are three main components: 1. Digital-twin construction: The work develops an approach that efficiently simulates the time-transient proliferation of cells/antibodies (proteins) and regulator/antigens (deactivated toxin) to an injected vaccine within tissue possessing complex heterogeneous microstructure. Here, we use the terms “cells” and “antibodies”, as well as “regulator” and “antigen” interchangeably. The approach utilizes two strongly-coupled conservation laws: • Conservation law 1: comprises (a) rate of change of cells/antibodies, (b) cellular/antibody migration, (c) cellular/antibody proliferation controlled by a cell/antibody mitosis regulating chemical (antigen), (d) cell/antibody apoptosis and • Conservation law 2: comprises (a) rate of change of the cell/antibody mitosis chemical regulator/antigen, (b) regulator/antigen diffusion, (c) regulator production by cells/antibody and (d) regulator/antigen decay. 2. Efficient computation: A technique based on voxel (3D “volume pixels”) representation of tissue microstructure and corresponding digital solution methods is developed for the calculations, which avoids computationally expensive steps involved in usual Finite Element procedures such as topologically conforming meshing, mapping, volume integration, stiffness matrix generation and matrixbased solution methods. The process proceeds by converting the tissue microstructure into voxels. The problem then becomes “digital” on a regular “voxel-grid”, directly manipulating voxel values, allowing extremely fast methods to be used to construct derivatives and to iteratively solve the system with minimal memory requirements. 3. Machine-learning: The rapid and efficient computation allows for many vaccines to be tested quickly and uses a genetic-based machine-learning algorithm to opti© The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 T. I. Zohdi, Modeling and Simulation of Infectious Diseases, https://doi.org/10.1007/978-3-031-18053-8_5

71

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5 Part 4: Vaccine Design and Immune-System Response

mize the system. This is particularly useful for rapid design of next-generation vaccines and boosters for disease strain mutations. This chapter follows work found in Zohdi [179].

5.1 Introduction A vaccine is a treatment that supplies acquired immunity to a particular disease. Vaccines greatly reduce the risk of infection by working with the body’s natural defense to safely develop immunity to the disease. They typically contain a substance that resembles the actual disease micro-organism, that is not harmful, for example, weakened forms of the infectious disease, such as disabled forms of the microbes and weakened forms of the associated toxin or surface proteins. The substance (“vaccine”) stimulates and trains the immune system to attack and destroy the infectious disease. The World Health Organization (WHO) indicates that there are 25 preventable infectious diseases that have licensed vaccines, such as polio, measles, tetanus, chickenpox, etc.

5.1.1 Brief History In 1798, Edward Jenner used cowpox (“smallpox of cows”, variolae vaccinae) to make humans immune to smallpox by opening a small wound and placing cowpox into the wound), making the person slightly ill, but eventually immune to smallpox. In the mid-1700s, John Williams also used weaken smallpox (weakened by subjecting samples to smoke, camphor and underground burial) to similarly inoculate people. It is notable to remark that these types of approaches had been used in the eastern population centers (such as Constantinople) for centuries before. In 1881, Pasteur suggested that these types of processes be called “vaccinations”, in honor of Jenner. As an example of vaccine efficacy, consider the measles in the United States. In 1958, before the measles vaccine, there were 768,094 cases of the measles (552 deaths). The next year (1959) after mass inoculation, there were 150 cases. It is estimated that 1,000,000 lives are saved from measles per year worldwide by vaccination [139].

5.1.2 Types of Vaccines There are a variety of vaccine types: • Attenuated-type: live attenuated (cultivated) micro-organisms with disabled virulence,

5.1 Introduction

73

• Inactivated-type: previously virulent micro-organisms that have been previously irradiated or chemically destroyed -they are an empty bacterial cell envelope, • Toxoid-type: inactivated toxins that come from the microorganisms, • Subunit-type: fragments of micro-organisms, for example only the surface proteins, • Conjugate-type: combines a weak antigen with a strong antigen as a carrier, so that the immune system has a stronger response to the weak antigen, • Outer membrane vesicle-type: OMV’s are naturally immunogenic and can be manipulated to produce vaccines, • Heterotypic-type: “Jennerian Vaccines” use pathogens of other animals that do not cause the full disease in humans, but give immunity, • Viral vector-type: use a safe virus to insert pathogen genes into the body to produce antigens to stimulate an immune response and • mRNA-type: composed of the nucleic acid pf RNA packaged within a vector such as lipid nanoparticles-the synthetic RNA stimulates the immune system. Vaccine licensing follows multiple phases in order to demonstrate safety as a given dose, effectiveness in preventing infections for target populations and an enduring preventive effects. Prominent regulatory bodies include the WHO, FDA (Food and Drug Association) and EMA (European Medicine Association). For Covid-19, vaccines include: • Pfizer-Biotech: mRNA-type, • Moderna: mRNA-type, • AstraZeneca: heterotypic-type genetically-modified chimpanzee adenovirus (such as colds), • Sputnik: vector-type genetically-modified common cold viruses, • Sinopharm: inactivated-type Covid-19 viruses and • Johnson and Johnson: vector-type genetically modified human adenovirus.

5.1.3 Vaccine Efficacy The key to any vaccine efficacy (such as the above) is that the immune system, when properly trained on a weakened version of the infectious disease (or surrogate), “remembers” its strategy and is able to fight off the real version by recognizing that protein coat on the virus and preparing a response by (1) neutralizing the target “agent” before it enters the cells and (2) recognizing and destroying infected cells before the disease can multiply in vast numbers. Two key terms are: • Antigen: a toxin or foreign substance that induces an immune response in the body-for example the production of “antibodies”. • Antibody: a blood protein produced in response to and countering an “antigen”. Antibodies chemically combine with entities that the body identifies as alienbacteria viruses and other foreign bodies in the blood, etc.

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5 Part 4: Vaccine Design and Immune-System Response

The following are common efficacy problems: • vaccine attenuation over time (thus warranting “boosters”), • host immune deficiency due to age, ethnicity and individual genetic variations, etc., • lack of “B-cells”, which generate antibodies to react and bind with pathogen antigens, • slow development of immunity and • antibodies which cannot completely disable the pathogen (although the vaccine still has a positive effect since it reduces the mortality rate). Further issues controlling the effectiveness are (1) the type of disease, (2) strain of the disease and the (3) vaccination schedule. There are of course potential adverse effects from a vaccine such as fever, pain and muscle aches.

5.1.4 Objectives of This Work Advances in data science and the associated areas of machine-learning and digitaltwins are transforming society and engineering. An objective of this work is to harness these advances for vaccine design. Specifically, there have been dramatic advances in technologies associated with autonomous operations across many industries. These have the potential to drastically improve industrial efficiency, quality and safety. Some approaches are methodical and systematic, while others are ad-hoc and haphazard. In the world of systems engineering, increasingly sophisticated and integrated approaches for digital systems are appearing at a rapid rate. This involves the application of machine-learning to digital-twins, whereby they learn from their mistakes/errors and constantly evolve to improve in a virtual environment. Digitaltwins can also run in tandem with real systems and thereby serve as controllers. These concepts, which leverage the data-centric world, and the corresponding modeling paradigms are increasingly used in fields outside of traditional Computational Mechanics. Accordingly, in this work, a computational approach is developed that researchers in the field can easily implement and subsequently use as an efficient tool to study the immune-response to a vaccine injection. There are three main components to this work: 1. Digital-twin construction: The work develops a computational framework that efficiently simulates the time-transient proliferation of cells/antibodies (proteins) and regulator/antigens (deactivated toxin) to an injected vaccine within tissue possessing complex heterogeneous microstructure. We use the terms “cells” and “antibodies” interchangeably and “regulator” and “antigen” interchangeably. The approach utilizes two strongly-coupled conservation laws: (1) one for the antibodies and (2) one for the antigens. 2. Efficient computation: A technique based on voxel (3D “volume pixels”) representation of tissue microstructures and corresponding digital solution methods

5.2 A Flexible Immune-Response Digital-Twin

75

is developed for the calculations, which avoids computationally expensive steps involved in usual Finite Element procedures such as topologically conforming meshing, mapping, volume integration, stiffness matrix generation and matrixbased solution methods. The process proceeds by converting the tissue microstructure into voxels. The problem then becomes “digital” on a regular “voxel-grid”, directly manipulating voxel values, allowing extremely fast methods to be used to construct derivatives and to iteratively solve the system with minimal memory requirements. 3. Machine-learning: The rapid and efficient computation allows for many vaccines to be tested quickly and uses a genetic-based machine-learning algorithm to optimize the system. This is particularly useful for rapid design of next-generation vaccines and boosters for disease strain mutations. Numerical examples are provided to illustrate the results, with the overall goal being to provide a computational framework to rapidly design and deploy a vaccine for a targeted response.

5.2 A Flexible Immune-Response Digital-Twin We start by developing a flexible model for the time-transient proliferation of cells/antibodies (proteins) and regulator/antigens (deactivated toxin) to an injected vaccine within tissue possessing heterogeneous “marbled” microstructure (Fig. 5.1). We develop two strongly-coupled conservation laws: • Conservation Law 1: comprises (a) rate of change of cells/antibodies, (b) cellular/antibody migration, (c) cellular/antibody proliferation controlled by a cell/antibody mitosis regulating chemical (antigen), (d) cell/antibody apoptosis and • Conservation Law 2: comprises (a) rate of change of the cell/antibody mitosis chemical regulator/antigen, (b) regulator/antigen diffusion, (c) regulator production by cells/antibody and (d) regulator/antigen decay. Throughout the construction of the model, we consider infinitesimal deformations, ˙ = ∂() |. In other words, the domain does not change its shape or geometry with () ∂t changes in concentration. The “cell/antibody” balance (c) per unit volume and a cell/antigen’ mitosis regulating chemical (s) denoted by the normalized concentration of c (cells/antibodies), in an arbitrary subvolume of material contained within , denoted ω, consists of a concentration (storage) term c, an inward normal migration flux term, −m · n, a proliferation term, r (s), and a cell/antibody apoptosis term, τ (c) < 0, leading to   ∂ c dω = − m(c) · n da ∂t ω ∂ω       antibody storage

 + 

ω

 r (s) dω

+







τ (c) dω . ω





antibody migration antibody pr oli f eration antibody apoptosis

(5.1)

76

5 Part 4: Vaccine Design and Immune-System Response ANTIGENS/REGULATOR

INJECTION

ANTIBODIES/CELLS

HETEROGENEOUS TISSUE SAMPLE

Fig. 5.1 Model Problem: an injection into a representative volume element of heterogeneous “marbled” tissue.The injection site is given high concentration of antibodies/cells and antigens/regulator (see Zohdi [179])

and simultaneously the balance of a cell/antibody mitosis regulating chemical/antigen (s)     ∂ s dω = − f (s) · n da + p(c) dω + γ(s) dω , (5.2) ∂t ω ω ω ∂ω             antigen storage

antigen di f f usion antigen pr oduction

antigen loss

where s is the cell/antibody mitosis regulator/antigen concentration, − f · n is an inward normal migration flux term, p(c) is a production term and γ(s) < 0 is a regulator loss term. After using the divergence theorem on the flux terms, since the volume ω is arbitrary, one obtains a diffusion-reaction model in strong form (assuming a Fickian-type law, m = −ID · ∇c and f = −IK · ∇s) ∂c = ∇ · ID · ∇c + r (s) + τ (c) ∂t

(5.3)

and simultaneously the balance of a mitosis regulating chemical (s) ∂s = ∇ · IK · ∇s + p(c) + γ(s). ∂t

(5.4)

There is extensive literature on the construction of the functions r (s), τ (c), p(c) and γ(s) for specific types of problems, such as wound healing and infection response. See Murray [100] for an extensive review, with early experimental studies dating back at least to Lindquist [84], Van den Brenk [129], Crosson et al. [31], Zieske et al.

5.3 Rapid Voxel Based Computation

77

Fig. 5.2 Left, a tissue microstructure and, right, a voxel representation of the tissue microstructure (see Zohdi [179])

[145], Franz et al. [43] and Sherratt and Murray [120]. Such a coupled system can represent a variety of biological systems, such as growth in biological scaffolding, proliferation of damaged cellular tissue, etc. The modeling of this process has a close similarity to multicomponent diffusion-reaction industrial processes, and we refer the reader to Zohdi [146–154] (Fig. 5.2).

5.3 Rapid Voxel Based Computation In many methods that analyze micro-heterogeneous materials, the computation of the response of multiple representative volume elements (RVE) of heterogeneous materials are required. The RVE domain is usually taken to be cubical, but contains complex microstructure, for example randomly distributed particulates representing functionalizing dopants in a binding matrix material. Two frequent applications where this occurs are: • Material design optimization, where the microstructures are constantly being changed during the search process. Usually, the Finite Element Method (FEM) is the default method used for such an analysis. • Multiscale methods, where the FEM is used on the macroscale and then it is also used on microscale RVE-like problems which are solved at select locations throughout the domain. This is sometimes referred to as F E 2 . The fundamental problem is that the FEM is computationally expensive for microscale problems with a cubical RVE domain, due to the “ingredients” involved in the usual Finite Element process: meshing, mapping, volume integration, stiffness matrix generation and matrix-based solution methods. This computational expense is even more unwarranted when there is an interest in time-dependent regimes. An alternative family of techniques that is naturally suited for these types of problems are socalled “Digital/Voxel-Image” methods, which convert the material microstructure

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5 Part 4: Vaccine Design and Immune-System Response

Fig. 5.3 Sequentially finer voxel representations of slightly overlapping particles in a matrix: for 41 × 41 × 41 (206763 D O F) voxel-mesh, for 61 × 61 × 61 (680943 D O F) voxel-mesh and for 81 × 81 × 81 (1594323 D O F) voxel-mesh. Top row: Microstructure/marbled tissue (particles and matrix). Second row: Just tissue (matrix). Third row: Just marbling (particles). Bottom row: Just interfaces (between marbling and matrix, see Zohdi [179])

into voxels (3D “volume pixels”; see Fig. 5.3). The problem then becomes “digital” on a regular “voxel-grid”, directly manipulating voxel values. Extremely fast methods can then be used to construct derivatives and the solve the system.

5.4 Numerical Simulation of the Coupled System

79

The use of volume pixels, so-called “voxels” (Foley et al. [42]), is widespread in the visualization and analysis of medical and scientific data (Chmielewski et al. [29]) and in the video-gaming industry. The well-known video-game “Outcast”, and others in the 1990s employed this graphics technique for effects such as reflection and bumpmapping and usually for terrain rendering, although other techniques have overtaken it as the method of choice. However, the most widely used application of a voxel is to represent material properties. For example, in CT scans, so-called Hounsfield units are used which measure the opacity of material to X-rays (Novelline [103]). There are approximately 30 different types of values acquired from MRI or ultrasound. Thus, in many cases, the voxels are already supplied, and it makes little sense to employ the usual Finite Element machinery: topologically conforming meshing, mapping, volume integration, stiffness matrix generation, etc. In this work, we illustrate the process of voxelization, derivative construction and solution methods. Additionally, we also supply an analysis of the operation counts. Numerical examples are provided to illustrate the approach.

5.4 Numerical Simulation of the Coupled System The present section develops a flexible and robust solution strategy to resolve the coupled system. There are two main components to the computational approach: • Spatio-temporal discretization of the diffusive continuum model, • Iterative staggering to solve the coupled system, whereby the time-steps are adaptively adjusted to control the error associated with the incomplete resolution of the concentration fields.

5.4.1 Discretization of the c- and s-Fields The concentration field will require spatial discretization with some type of mesh, for example using a finite difference, finite volume or finite element method. Temporal approximation For the c-concentration field, we write ∂c def = ∇ · ID · ∇c + r (s) + τ (c) = L . ∂t

(5.5)

We discretize for time=t + φt, and using a trapezoidal “φ − scheme” (0 ≤ φ ≤ 1) c(t + t) ≈ c(t) + t (φL(t + t) + (1 − φ)L(t)) . Similarly for s-field,

(5.6)

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5 Part 4: Vaccine Design and Immune-System Response

∂s def = ∇ · IK · ∇s + p(c) + γ(s) = M. ∂t

(5.7)

s(t + t) ≈ s(t) + t (φM(t + t) + (1 − φ)M(t)) .

(5.8)

and

Spatial discretization of the fields Numerically, the components of the gradient of c are approximated by central finite difference stencils of the basic form, for example for c, for example ∂x∂ 1 ∂c | ∂x1 (x1 ,x2 ,x3 )

≈

c(x1 +x1 ,x2 ,x3 )−c(x1 −x1 ,x2 ,x3 ) 2x1

(5.9)

for each of the (x1 , x2 , x3 )-directions, in order to form the terms needed. This is a second-order accurate stencil. For a generic second order scheme spatial derivative for an arbitrary flux component in the x1 direction (components of ∇ · F) F1 (x1 + ∂F1 |(x1 ,x2 ,x3 ) ≈ ∂x1

x1 , x2 , x3 ) 2

− F1 (x1 − x1

x1 , x2 , x3 ) 2

,

(5.10)

where generically, for example with an arbitrary material coefficient for example ID(x) = ID(x)1 ⎛

⎞

⎜ ∂c ⎟ x1 c(x1 + x1 , x2 , x3 ) − c(x1 , x2 , x3 ) ⎜ ⎟ , x2 , x3 ) ≈ ID(x1 + ⎜ID ⎟| x ⎝ ∂x1 ⎠ (x1 + 2 1 ,x2 ,x3 ) 2 x1       F1

∂c ∂x1 |(x + x1 ,x ,x ) 2 3 1 2

(5.11) and ⎛

⎞

⎜ ∂c ⎟ x1 c(x1 , x2 , x3 ) − c(x1 − x1 , x2 , x3 ) ⎜ ⎟ , x2 , x3 ) ⎜ID ⎟ |(x − x1 ,x ,x ) ≈ ID(x1 − ⎝ ∂x1 ⎠ 1 2 2 3 2 x1       F1

∂c ∂x1 |(x − x1 ,x ,x ) 2 3 1 2

(5.12) where ID(x1 + and

x1 1 , x2 , x3 ) ≈ (ID(x1 + x1 , x2 , x3 ) + ID(x1 , x2 , x3 )), 2 2

(5.13)

5.4 Numerical Simulation of the Coupled System

ID(x1 −

81

x1 1 , x2 , x3 ) ≈ (ID(x1 , x2 , x3 ) + ID(x1 − x1 , x2 , x3 )). 2 2

(5.14)

∂c These approximations are made for all components and combinations in ID ∂x j appearing in the field equations. The mathematical representation of the derivatives can be summarized in the following manner ( j = 1, 2, 3), for example for j = 1: )−c(x1 −x1 ,x2 ,x3 ) ∂c ≈ ID(x1 , x2 , x3 ) c(x1 +x1 ,x2 ,x32x 1. Voxel-Gradient: ID ∂x 1 1 2. Voxel-Laplacian:

∂ ∂x1

∂c ID ∂x1

 ≈

  ∂c | ID ∂x 1

x

(x1 + 2 1 ,x2 ,x3 )

  ∂c | − ID ∂x x 1 (x1 − 1 ,x2 ,x3 ) 2

x1  

1 x1 c(x1 + x1 , x2 , x3 ) − c(x1 , x2 , x3 ) ID(x1 + , x2 , x3 ) = x1 2 x1  

x1 c(x1 , x2 , x3 ) − c(x1 − x1 , x2 , x3 ) 1 ID(x1 − , x2 , x3 ) − x1 2 x1

3. Voxel-Interface: ID(x1 ±

x1 1 2 , x 2 , x 3 ) ≈ 2 (ID(x 1 ± x 1 , x 2 , x 3 ) + ID(x 1 , x 2 , x 3 ))

Remark: To illustrate second-order accuracy, consider a Taylor series expansion for an arbitrary function c c(x1 + x1 , x2 , x3 ) = c(x1 , x2 , x3 ) + +

∂c 1 ∂2c |(x1 ,x2 ,x3 ) x1 + |(x ,x ,x ) (x1 )2 x1 ∂x1 2 ∂x12 1 2 3

1 ∂3c |(x ,x ,x ) (x1 )3 + O((x1 )4 ) 6 ∂x13 1 2 3

(5.15)

and c(x1 − x1 , x2 , x3 ) = c(x1 , x2 , x3 ) − −

∂c 1 ∂2c |(x1 ,x2 ,x3 ) x1 + |(x ,x ,x ) (x1 )2 x1 ∂x1 2 ∂x12 1 2 3

1 ∂3c |(x ,x ,x ) (x1 )3 + O((x1 )4 ). 6 ∂x13 1 2 3

(5.16)

Subtracting the two expressions yields ∂c c(x1 + x1 , x2 , x3 ) − c(x1 − x1 , x2 , x3 ) |(x1 ,x2 ,x3 ) = + O((x1 )2 ). (5.17) ∂x1 2x1 All other derivatives follow from this basic process, which is relatively standard in the all stencil-based discretizations. For the s-field, the discretization is the same. Remark: At the length-scales of interest, it is questionable whether the ideas of a sharp material interface are justified. Accordingly, later, we simulated the system with and without Laplacian smoothing, whereby one smooths the material data by post-

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processing the original material data, voxel by voxel, to produce a smoother material ˆ (using the previous voxel approximations and representation, for example, for ID nodal subscript notation):  1  IDi+1, j,k − 2IDi, j,k + IDi−1, j,k (xi )2   1 IDi, j+1,k − 2IDi, j,k + IDi, j−1,k + (x j )2   1 IDi, j,k+1 − 2IDi, j,k + IDi, j,k−1 = 0 + (xk )2

∇ 2 ID =

(5.18)

ˆ i, j,k , given by which yields a smoother value of IDi, j,k , denoted ID ˆ i, j,k = ∇ 2 ID = 0 ⇒ ID

 1 IDi+1, j,k + IDi−1, j,k + IDi, j+1,k + IDi, j−1,k + IDi, j,k+1 + IDi, j,k−1 . 6

(5.19) The same process was applied to the other parameters, generically denoted, A(x), by enforcing ∇x2 A = 0, as well as for any other material data. The simulations were run with and without data smoothing, with the results being negligibly different for sufficiently fine voxel-meshes.

5.4.2 Iterative (implicit) Solution Method Implicit time-stepping methods, with time-step size adaptivity, built on approaches found in Zohdi [146–159], will be used throughout the upcoming analysis. In order to introduce basic concepts, we consider a first order vector-valued differential equation ˙ = F(W ), W

(5.20)

which, after being discretized using a trapezoidal “φ-method” (0 ≤ φ ≤ 1)   W L+1 = W L + t φF(W L+1 ) + (1 − φ)F(W L ) ,

(5.21)

yields the following abstract form A(W L+1 ) = B. Explicitly, W is the vector of all voxel values in the system

(5.22)

5.4 Numerical Simulation of the Coupled System ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨

c1 c2 c3 ... cN def W = s1 ⎪ ⎪ ⎪ ⎪ s2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ s3 ⎪ ⎪ ⎪ ⎩ ... sN

83 ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬

(5.23)

⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭

It is convenient to write A(W L+1 ) − B = G (W L+1 ) − W L+1 + R = 0,

(5.24)

where R is a remainder term that does not depend on the solution, i.e. R = R(W L+1 ). A straightforward iterative scheme can be written as W L+1,K = G (W L+1,K −1 ) + R,

(5.25)

where K = 1, 2, 3, ... is the index of iteration within time-step L + 1. The convergence of such a scheme is dependent on the behavior of G. Namely, a sufficient condition for convergence is that G is a contraction mapping for all W L+1,K , K = 1, 2, 3... In order to investigate this further, we define the iteration error as def

 L+1,K = W L+1,K − W L+1 .

(5.26)

A necessary restriction for convergence is iterative self consistency, i.e. the “exact” (discretized) solution must be represented by the scheme G(W L+1 ) + R = W L+1 .

(5.27)

Enforcing this restriction, a sufficient condition for convergence is the existence of a contraction mapping  L+1,K = ||W L+1,K − W L+1 || = ||G (W L+1,K −1 ) − G (W L+1 )|| ≤ η L+1,K ||W L+1,K −1 − W L+1 ||,

(5.28)

where, if 0 ≤ η L+1,K < 1 for each iteration K , then  L+1,K → 0 for any arbitrary starting value W L+1,K =0 , as K → ∞. This type of contraction condition is sufficient, ˙ = F(W ) but not necessary, for convergence. Inserting these approximations into W leads to   W L+1,K ≈ t φF(W L+1,K −1 ) + t (1 − φ)F(W L ) + W L ,       G(W L+1,K −1 )

(5.29)

R

whose contraction constant is scaled by η ∝ φt. Therefore, if convergence is slow within a time-step, the time-step size, which is adjustable, can be reduced by an

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5 Part 4: Vaccine Design and Immune-System Response

appropriate amount to increase the rate of convergence. Decreasing the time-step size improves the convergence, however, we want to simultaneously maximize the timestep sizes to decrease overall computing time, while still meeting an error tolerance on the numerical solution’s accuracy. In order to achieve this goal, we follow an approach found in Zohdi [146] originally developed for continuum thermo-chemical multifield problems in which one first approximates η L+1,K ≈ S(t) p

(5.30)

(S is a constant) and secondly one assumes the error within an iteration to behave according to (S(t) p ) K  L+1,0 =  L+1,K ,

(5.31)

K = 1, 2, ..., where  L+1,0 is the initial norm of the iterative error and S is intrinsic to the system.1 Our goal is to meet an error tolerance in exactly a preset number of iterations. To this end, one writes (S(ttol ) p ) K d  L+1,0 = Ctol ,

(5.32)

where Ctol is a (coupling) tolerance and where K d is the number of desired iterations.2 If the error tolerance is not met in the desired number of iterations, the contraction constant η L+1,K is too large. Accordingly, one can solve for a new smaller step size, under the assumption that S is constant,  1  tol ( CL+1,0 ) pKd (5.33) . ttol = t 1 L+1,K (  L+1,0 ) pK The assumption that S is constant is not critical, since the time-steps are to be recursively refined and unrefined throughout the simulation. Clearly, the expression in Eq. 5.33 can also be used for time-step enlargement, if convergence is met in less than K d iterations.3 Specifically, the solution steps are, within a time-step (Fig. 5.4): • (1): Start a global fixed iteration (set i = 1, ..., Nn (voxel counter) and K = 0 (iteration counter)) • (2): If i > Nn then go to (4) • (3): If i ≤ Nn then: (a)Compute the concentration ciL+1,K (b)Go to (2) for the next voxel (i = i + 1) • (4): Repeat steps 1-3 for the voxels, i = 1, ..., Nn . • (5): Measure error (normalized) quantities (where wc is a weight on the cell contribution and ws is a weight on the regulator contribution) For the class of problems under consideration, due to the quadratic dependency on t, p ≈ 1. Typically, K d is chosen to be between five to ten iterations, although this is problem and analyst dependent. 3 At the implementation level, since the exact solution is unknown, the following relative error term 1 2

def

is used,  L+1,K = W L+1,K − W L+1,K −1 .

5.4 Numerical Simulation of the Coupled System

85 VALUES TO BE COMPUTED(UNKNOWN)

VOXELS

ITERATE

UPDATE

VALUES ALREADY COMPUTED (KNOWN)

Fig. 5.4 The overall iterative (left) solution and the matrix-free approach using a moving front through the voxels(right) During the iterative solution process, the most current value of a voxel is used in any calculation, for example a construction of the Laplacian, or any other term in the governing differential equations (see Zohdi [179])

def

(a)  L+1,K = wc

 Nn

L+1,K − ciL+1,K −1 || i=1 ||ci  Nn L+1,K || i=1 ||ci

N p + ws

 L+1,K where T O L is an error tolerance. ⎛T O L 1 ⎞ T O L pK d def ⎝ (  L+1,0 ) ⎠. (c)  K = 1 L+1,K ( ) pK  L+1,0

L+1,K − siL+1,K −1 || i=1 ||si N p L+1,K || i=1 ||si

def

(b) E K =

• (6): If the tolerance is met: E K ≤ 1 and K < K d then (a) Increment time: t = t + t (b) Construct the next time-step: (t)new =  K (t)old , (c) Select the minimum size: t = M I N ((t)lim , (t)new ) and go to (1) • (7): If the tolerance is not met: E K > 1 and K < K d then (a) Update the iteration counter: K = K + 1 (b) Reset the voxel counter: i = 1 (c) Go to (2) • (8): If the tolerance is not met (E K > 1) and K = K d then (a) Construct a new time-step: (t)new =  K (t)old (b) Restart at time t and go to (1)

Time-step size adaptivity is critical, since the system’s dynamics can dramatically change over the course of time, possibly requiring quite different time-step sizes to control the iterative error. However, to maintain the accuracy of the time-stepping scheme, one must respect an upper bound dictated by the discretization error, i.e., t ≤ t lim . Note that in step (5),  K may enlarge the time-step if the error is lower than the preset tolerance. At a given time, once the process is complete, the time is incremented forward and the process is repeated. The overall goal is to deliver solutions where the iterative error is controlled and the temporal discretization accuracy

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5 Part 4: Vaccine Design and Immune-System Response

dictates the upper limit on the time-step size (t lim ). Clearly, there are various combinations of solution methods that one can choose from. For example, for the overall field coupling, one may choose implicit or explicit staggering and within the staggering process, either implicit (0 < φ ≤ 1) or explicit time-stepping (φ = 0), and, in the case of implicit time-stepping, iterative or direct solvers. Furthermore, one could employ internal iterations for each field equation, then update, more sophisticated metrics for certain components of the error, etc, etc. For example, we utilized an error measure that used the concentrations at the voxels of the Finite Difference grid, but other metrics are certainly possible. For details see Zohdi [146–160, 162–180]. Remark: Because the internal system solvers within the staggering scheme are also iterative and use the previously converged solution as their starting value to solve the system of equations, a field that is relatively insensitive at given stage of the simulation will converge in very few internal iterations (perhaps even one). Staggering schemes are widely used in the computational mechanics literature, dating back, at least, to Zienkiewicz [143] and Zienkiewicz et al. [144]. For in depth overviews, see the works of Lewis and Schrefler (Lewis et al. [81] and Lewis and Schrefler [82]) and a series of works by Schrefler and collaborators: Schrefler [119], Turska and Schrefler [125], Bianco et al. [10] and Wang and Schrefler [135]. Remark: During the iterative solution process, the most current value of a voxel is used in any calculation, for example a construction of the Laplacian, or any other term in the governing differential equations. At the length-scales of interest, it is questionable whether the ideas of a sharp material interface are justified. Accordingly, we simulated the system with and without Laplacian smoothing, whereby one smooths the material data by post-processing the material data, voxel by voxel, to produce a smoother material representation. The simulations were run with and without data smoothing, with the results being negligibly different for sufficiently fine meshes.

5.5 Operation Counts in a Voxel-Based Method The cost of constructing an array for a temporal update using a voxel calculation is: • P × V , where V is the number of voxels and S = O(10) is, associated with summing up the terms needed to construct L and M. Specifically, there are six Laplacian terms needed in Eqs. 5.3 and 5.4. To construct each Laplacian term one must perform four operations. Thus, S=24 operations in total. • Thus, K × S × V ≈ 24K V , is the total count per time-step, where K is the number of iterations in a time-step. The cost of constructing an array for temporal update using an FEM calculation is associated with (1) meshing the microstructure (mappings, etc) (2) numerically integrating the weak form (3) generating a stiffness matrix and (4) solving a system of equations. Specifically (Zohdi [182]):

5.6 Numerical Examples

87

• Construction of the stiffness matrix: P × E, where E is the number of elements and P = O(700) stems from mapping and integrating the terms needed to construct the stiffness matrix associated with the weak form O(10), of which there are 35 entries [8 × 8 = 64] in a (symmetric) linear diffusion element stiffness matrix (linear hexahedra), which must be computed for both equations, yielding P = 70 × 10 = 700. • Iterative solution: I × Q, where I is the number of iterations associated with, for example, a Conjugate-Gradient solver and Q is the cost of a matrix-vector multiplication. Q is on the order of N q , where N q is the number of voxels in the system and 1 < q ≤ 2, and depends on the sparsity of the stiffness matrix. For example, for linearized elasticity, using an element-by-element multiplier (not counting preconditioning), for linearized elasticity and using linear hexahedra, Q = 70N . Thus, I × Q ≈ 70I N . • Thus, a comparison of the total operation counts between the Voxel method and FEM is roughly 24K V V oxel ≈ . FEM 700E + 70I N

(5.34)

Equation 5.34 clearly shows the favorable ratio of operation counts of the voxel approach relative to FEM. In more complex (nonlinear) problems, where the stiffness matrix would have to be reformed after each iteration, the term 700E would need to be multiplied by I .

5.6 Numerical Examples As an example, we consider a cubical test domain with an initial interior concentration of zero cells and zero regulator. We inject both cells and regulator at a given location at the top (Fig. 5.1). The injection site has a controlled concentration of both cells and regulator over time. The boundary conditions for the cells and regulator were held to be zero, other than at the injection site. Key qualitative ratios can be identified by considering the special case of steady state and spatially uniform fields: ∂c = ∇ · ID · ∇c + r (s) + τ (c) = 0 ⇒ r (s) + τ (c) = 0 ∂t

(5.35)

and ∂s = ∇ · IK · ∇s + p(c) + γ(s) = 0 ⇒ p(c) + γ(s) = 0. ∂t

(5.36)

Assuming linear relationships yields (τ (c) = −τ c < 0) r (s) + τ (c) = r s − τ c = 0 ⇒ c =

rs τ

(5.37)

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5 Part 4: Vaccine Design and Immune-System Response

and (γ(s) = −γs < 0) p(c) + γ(s) = pc − γs = 0 ⇒ c =

γs . p

(5.38)

Thus, while it is unrealistic that the field would ever be uniform, these ratios are key to understanding what controls the growth of c. We considered a heterogeneous “marbled” domain where the medium has a microstructure comprised of randomly distributed spheres (occupying approximately 25% volume fraction) in a homogeneous matrix-an idealization of “marbled” tissue. The following parameters were used (with standard metric units used throughout): • • • • • • • • • •

size of the domain was 0.01 × 0.01 × 0.01 m, injection site, a controlled cell concentration c(t) = co eat , co = 1, a = 0.01, injection site, a controlled regulator concentration s(t) = so eat , so = 1, a = 0.01, injection site was 0.005 × 0.00125 m (elliptical cross-section) and 0.0025 m deep, total simulation time was T = 20 s, cell proliferation term, r (s) = +ˆr s, with a different rˆ for each material phase, cell apoptosis term, τ (c) = −τˆ c, with a different τˆ for each material phase, regulator production term, p(c) = + pc, ˆ with a different pˆ for each material phase, regulator loss term, γ(s) = −γs, ˆ with a different γˆ for each material phase, homogeneous base, rˆo = 20, rˆ1R = rrˆˆ1 = 10, o

• homogeneous base, τˆo = 0.1, τˆ1R =

τˆ1 τˆo

• homogeneous base, pˆ o = 0.001, pˆ 1R = • • • •

= 10, pˆ 1 pˆ o

= 100,

γˆ o = 0.1, γˆ 1R = γγˆˆ 1 = o ID = D1, D = 10−6 ,

100, homogeneous base, homogeneous base, homogeneous base, IK = K 1, K = 10−7 , heterogeneous/marbled case, rˆo = 20, rˆ1R = rrˆˆ1 = 10, rˆ2R =

• heterogeneous/marbled case, • heterogeneous/marbled case, • heterogeneous/marbled case, • heterogeneous/marbled case, • heterogeneous/marbled case,

rˆ2 = 1, rˆo τˆ1 τˆ2 τˆo = 0.1, τˆ1R = τˆ = 10, τˆ2R = τˆ = 1, o o pˆ o = 0.001, pˆ 1R = ppˆˆ1 = 100, pˆ 2R = ppˆˆ2 = 1, o o γˆ o = 0.1, γˆ 1R = γγˆˆ 1 = 100, γˆ 2R = γγˆˆ 2 = 1, o o ID1 = D1 1, D1 = 10−6 and ID2 = D2 1, D2 = IK 1 = K 1 1, K 1 = 10−7 and IK 2 = K 2 1, K 2 = o

10−7 , 10−8 .

The time-steps were initially started to be quite small in order to allow the system to evolve the time-step size during the beginning of the simulation. A trapezoidal time-stepping parameter of φ = 0.5 was chosen. In this case, we started the timestep size at 0.001 s and allowed it to be enlarged up to 20 times that size, if the algorithm and error estimates warranted it (which was the case in the examples given). During the computations with the heterogeneous “marbled” media, the spatial discretization meshes were repeatedly refined until the solutions did not exhibit any more sensitivity to further refinement of the grid-spacing. We started with meshes such as a 21 × 21 × 21 mesh, arising from having a cubical mesh with 10 voxels from the centerline plane of symmetry and one voxel in the middle, and then repeatedly refined in the following sequential manner: 1. Mesh # 1: a 21 × 21 × 21 mesh, which has 9261 degrees of freedom per field, for a total of 18,522 degrees of freedom,

5.7 Genetic-Based Machine-Learning Framework

89

Fig. 5.5 Left, the morphology of the microstructure and, right, morphology of the microstructure and mesh (see Zohdi [179])

2. Mesh # 2: a 41 × 41 × 41 mesh, which has 68,921 degrees of freedom per field, for a total of 137,842 degrees of freedom, 3. Mesh # 3: a 61 × 61 × 61 mesh which has 226,981 degrees of freedom per field, for a total of 453,962 degrees of freedom, etc. Approximately between a 41-level and a 61-level mesh, the results stabilized, indicating that the results are essentially free of any appreciable numerical error. Figure 5.5 illustrates the morphology of the microstructure as resolved by the grid. Figure 5.6 shows cross-sections of the concentration of cells domain over time. Figure 5.7 depicts the evolution of the time-step size over time. The computations are designed so that they take a few minutes on a standard laptop. The selected parameter choices were provided to illustrate the overall working on the model, and a wide variety of parameter choices are possible, depending on the application. This is discussed further next.

5.7 Genetic-Based Machine-Learning Framework The rapid rate at which these simulations can be completed enables the ability to explore inverse problems seeking to determine what parameter combinations can deliver a desired result. Following Zohdi [147, 168, 170, 171, 175–180, 183], we formulate the objective as a cost function minimization problem that seeks system parameters that match a desired response by minimizing a cost/error function (). Specifically, we use

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5 Part 4: Vaccine Design and Immune-System Response

Fig. 5.6 The injection viewed from the exterior and the morphology of the microstructuralmarbling. From left to right and top to bottom: Cell concentration (c) and growth from an injection at the surface (see Zohdi [179])

5.7 Genetic-Based Machine-Learning Framework

91

Fig. 5.7 The evolution of the time-step size over time (see Zohdi [179])

 = w1

||AN T I B O D I E S − T A RG E T1 || ||AN T I G E N S − T A RG E T2 || + w2 . ||T A RG E T1 || ||T A RG E T2 ||

(5.39)

The system parameter search is conducted within the constrained ranges of (−) 1 ≤ (−) (+) (−) (+) 1 ≤ (+) ,  ≤  ≤  and  ≤  ≤  , etc. These upper and lower 2 3 1 2 2 3 3 limits would, in general, be dictated by what is physically feasible. The system parameters to vary and optimize are the 20 parameters associated with injection concentration of antibodies, injection concentration of antigens, antibody reaction rate constants, antigen reaction rate constants, antibody diffusion constants, antigen diffusion constants, antibody regulator constants and antigen regulator constants.

5.7.1 Algorithmic Settings Following the genetic-based machine-learning algorithm introduced in Chap. 1 (following Zohdi [147, 168, 170, 171, 175–180, 183]), in the upcoming example, the design parameters  = {1 , 2 ... N } are optimized over the search intervals (20 variables): i− ≤ i ≤ i+ , i = 1, 2, ...20. Specifically, we varied the injection concentration of antibodies, injection concentration of antigens, antibody reaction rate constants, antigen reaction rate constants, antibody diffusion constants, antigen diffusion constants, antibody regulator constants and antigen regulator constants. Figures 5.8 and 5.9 show the reduction of the cost function for the 20 parameter set. Shown are the best performing gene (design parameter set, in red) as a function of successive generations, as well as the average performance of the entire population of the genes (designs, in green). We used the following genetic-based MLA settings:

92

• • • • • • •

5 Part 4: Vaccine Design and Immune-System Response

Number of design variables: 20, Population size per generation: 24, Number of parents to keep in each generation: 6, Number of children created in each generation: 6, Number of completely new genes created in each generation: 12, Number of generations for re-adaptation around a new search interval: 10, Number of generations: 200.

5.7.2 Parameter Search Ranges and Results We considered a 20 parameter vaccine design. The following search parameter ranges were used (with w1 = 1 and w2 = 1): • • • • • • • • • • • • • • • • • • • •

+ 1 = Injection concentration of Antibodies:− 1 = 0.01 ≤ 1 ≤ 1 = 10, − + 2 = Injection concentration of Antigens:2 = 0.1 ≤ 2 ≤ 2 = 10, + 3 = Antibody reaction rate constant of media 1:− 3 = 0.1 ≤ 3 ≤ 3 = 10, − 4 = Antibody reaction rate constant of media 2:4 = 0.1 ≤ 4 ≤ + 4 = 10, + = 0.1 ≤  ≤  5 = Antibody reaction rate constant of media 3 :− 5 5 5 = 10, − + 6 = Antigen reaction rate constant of media 1:6 = 0.1 ≤ 6 ≤ 6 = 10, + 7 = Antigen reaction rate constant of media 2:− 7 = 0.1 ≤ 7 ≤ 7 = 10, − 8 = Antigen reaction rate constant of media 3 :8 = 0.1 ≤ 8 ≤ + 8 = 10, + = 0.1 ≤  ≤  = 10, 9 = Antibody diffusion constant of media 1:− 9 9 9 + = 0.1 ≤  ≤  = 10, 10 = Antibody diffusion constant of media 2:− 10 10 10 + = 0.1 ≤  ≤  = 10, 11 = Antibody diffusion constant of media 3 :− 11 11 11 + = 0.1 ≤  ≤  = 10, 12 = Antigen diffusion constant of media 1:− 12 12 12 + 13 = Antigen diffusion constant of media 2:− 13 = 0.1 ≤ 13 ≤ 13 = 10, + 14 = Antigen diffusion constant of media 3 :− 14 = 0.1 ≤ 14 ≤ 14 = 10, − 15 = Antibody regulator constant of media 1:15 = 0.1 ≤ 15 ≤ + 15 = 10, + = 0.001 ≤  ≤  16 = Antibody regulator constant of media 2:− 16 16 16 = 0.1, − 17 = Antibody regulator constant of media 3 :17 = 0.001 ≤ 17 ≤ + 17 = 0.1, + = 0.1 ≤  ≤  = 10, 18 = Antigen regulator constant of media 1:− 18 18 18 + = 0.001 ≤  ≤  = 0.1, 19 = Antigen regulator constant of media 2:− 19 19 19 + = 0.001 ≤  ≤  = 0.1, 20 = Antigen regulator constant of media 3 :− 20 20 20

Figures 5.8 and 5.9 illustrate the results for the cost function for the best performing gene (red) as a function of successive generations, as well as the average performance cost function of the entire population of genes (designs, in green), using design weights of w1 = 1 and w2 = 0.1. We allowed the genetic-based MLA to readapt every 10 generations, leading to the (slightly) nonmonotone reduction of the cost function. Often, this action is more efficient than allowing the algorithm not to readapt, since it probes around the current optimum for better local alternatives. Table 5.1 shows the final design parameters. The entire 200 generation simulation, with 24 genes per evaluation (4800 total vaccine designs) took a few minutes on a

5.7 Genetic-Based Machine-Learning Framework

93

BEST POPULATION

Fig. 5.8 Shown are the cost function for the best performing gene (red) as a function of successive generations, as well as the average cost function of the entire population of genes (green). We allowed the genetic-based MLA to readapt every 10 generations, leading to the (slight) nonmonotone reduction of the cost function. Often, this action is more efficient than allowing the algorithm not to readapt, since it probes around the current optimum for better local alternatives. In this case, the algorithm makes slow progress until generation 10, when a readaptation/recentering occurred, and then slowed reduced the cost function over 200 generations from approximately  ≈ 2 to  = 0.4097. The algorithm produces a massive reduction of error from  ≈ 50 to  ≈ 0.4-a factor of 125 (see Zohdi [179])

laptop, making it ideal as a design tool. We note that, for a given set of parameters, a complete simulation takes a fraction of a second, thus thousands of parameter sets can be evaluated in an hour, without even exploiting the inherent parallelism of the genetic-based MLA. The algorithm produces a massive reduction of error from  ≈ 50 to  ≈ 0.4-a factor of 125. Remark-Design of a booster: For diseases that are constantly evolving, such as the current strains of COVID19, the utility of the presented method becomes even clearer, since one can start the process from the current vaccine design to develop a modified booster. The speed at which the calculations can be achieved makes the model and simulation tool easily utilized as a digital-twin which can be safely and inexpensively optimized in a virtual setting and deployed in the physical world afterwards, thus accelerating the development of new vaccines.

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5 Part 4: Vaccine Design and Immune-System Response

Fig. 5.9 ZOOM: Shown are the cost function for the best performing gene (red) as a function of successive generations, as well as the average cost function of the entire population of genes (green). We allowed the genetic-based MLA to readapt every 10 generations, leading to the (slight) nonmonotone reduction of the cost function. Often, this action is more efficient than allowing the algorithm not to readapt, since it probes around the current optimum for better local alternatives. In this case, the algorithm makes slow progress until generation 10, when a readaptation/recentering occurred, and then slowed reduced the cost function over 200 generations from approximately  ≈ 2 to  = 0.4097. The algorithm produces a massive reduction of error from  ≈ 50 to  ≈ 0.4-a factor of 125 (see Zohdi [179]) Table 5.1 The system parameters (1 − 20 ) for the best performing design (gene) with w1 = w2 = 1 1 2 3 4 5 6 7 8 9 10 0.119 11

0.168 12

7.185 13

9.043 14

8.114 15

2.562 16

6.988 17

1.873 18

0.543 19

8.641 20

6.042

4.673

9.197

9.408

3.730

0.078

0.072

9.720

0.0016 0.061

 0.4097

5.8 Discussion and Summary In summary, the purpose of this work was to present a flexible computational modeling framework that researchers in the field can easily implement and subsequently use as an efficient tool to study the immune-response to a vaccine injection. The framework is flexible enough to allow researchers to input virtually any type vaccine and immune-system interaction. However, in the present formulation, notably absent are the effects of deformation and stress in the system. At a minimum, this would require a third field equation governing the balance of linear momentum, ∇x · σ + f = ρ˙v , where σ is the Cauchy stress, f are the body forces, ρ is the density and v is the velocity, in addition to constitutive laws for soft tissue (see the extensive works of Fung [44–46] Holzapfel [64, 65] or Humphrey [67, 68]). At finite deformations, the previous conservation laws can be generatived in the following manner:

5.8 Discussion and Summary

d dt



   dJ dc dc J +c dωo = J + c J ∇x · v dωo dt dt dt ωo ωo ωo     ∂c ∂c = (5.40) + v · ∇x c + c∇x · v dω = + ∇x · (cv) dω ∂t ∂t

c dω = ω

95

d dt

ω





c J dωo =

ω

thus ∂c + ∇x · (cv) = ∇ · ID · ∇c + r (s) − τ (c). ∂t

(5.41)

∂s + ∇x · (sv) = ∇ · IK · ∇s + p(c) − γ(s). ∂t

(5.42)

and

Clearly, specific material data is needed for tissue. In this regard, we again refer the reader to Murray [100] for an extensive review, with early experimental studies dating back at least to Lindquist [84] Van den Brenk [129], Crosson et al. [31], Zieske et al. [145], Franz et al. [43] and Sherratt and Murray [120]. Generally, because the distribution of water, biological fluids and chemical species within such tissue are dependent on the deformation of the solid, coupled multifield computations are necessary to realistically simulate such systems. For example, in many models of muscle tissue, it is usually assumed that the response depends on the concentration of a mobile chemical species present, for example, intracellular calcium Ca 2+ , and U is the stretch along the muscle fiber, relative to a reference sarcomere length. A basic form suggested is σ = σ(Ca 2+ , U ), where σ is the total Cauchy stress (active and passive), which combines the mechanical (passive) contribution and the actively generated muscle tension. We refer the reader to Rachev and Hayashi [113], Humphrey [67, 68], Klepach et al. [77] and Ambrosi et al. [2] for reviews. Incorporation of such effects is under investigation by the author.

Epilogue

The COVID19 pandemic that started in 2019-2020 has led to a gigantic increase in modeling and simulation of infectious diseases. There are numerous topics associated with this epoch-changing event, such as (a) disease propagation, (b) transmission, (c) decontamination, (d) vaccines, etc. This is an evolving field. The targeted objective of this monograph was to expose researchers to key topics in this area, in a very concise manner. It is not meant to be a comprehensive overview of the field of modeling and simulation of infectious diseases. The topics selected for discussion have evolved with the progression of the pandemic. Beyond the introductory chapter on basic mathematics, optimization and machine-learning, the monograph covered four themes in modeling and simulation of infectious diseases, specifically: • • • •

Part 1: Macroscale disease propagation, Part 2: Microscale disease transmission and ventilation system design, Part 3: Ultraviolet viral decontamination and Part 4: Vaccine design and immune response.

It is important to emphasize that the rapid speed at which the simulations operate makes the presented computational tools easily deployable as digital-twins, i.e., digital replicas of complex systems that can be inexpensively and safely optimized in a virtual setting and then used in the physical world afterwards, thus reducing the costs of experiments and also accelerating development of new technologies. This monograph represents a small contribution to the vigorous and growing research in this area. I am certain that, despite painstaking efforts, there remain errors of one type or another in this monograph. Therefore, I would be grateful if readers who find such flaws could contact me at [email protected].

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 T. I. Zohdi, Modeling and Simulation of Infectious Diseases, https://doi.org/10.1007/978-3-031-18053-8

97

Appendix A Artificial Neural Networks

Artificial Neural Nets (ANN) are based on layered input-output type frameworks that are essentially adaptive nonlinear regressions of the form O = B(I1 , I2 , ..., I M , w1 , w2 , ..., w N ),

(A.1)

where O is a desired output and B is the ANN comprised of: • Synapses, which multiply inputs (Ii , i = 1, 2, ..., M) by weights (wi , i = 1, 2, ..., N ) that represent the input relevance to the desired output, • Neurons, which aggregate outputs from all incoming synapses and apply activation functions to process the data and • Training, which calibrates the weights to match a desired overall output. For example, Figure A.1 illustrates a detailed ANN comprised of (1) Five layers (one input layer and four hidden layers) (2) 35 activation neurons (3+5+7+9+11) and (3) 223 weighted synapses. The primary issue with ANNs is the calibration or “training” of the synapse weights. The key components of an ANN can be summarized as follows (which is centered around training): • STEP 1: Guess a set of trial weights, given by the vector wi=1 , for the synapses and insert into the ANN (detailed construction shown shortly) B(I, wi ) = Oi ,

(A.2)

which produces an overall trial output. • STEP 2: Compute the error: def

E i = ||Odesir ed − Oi ||,

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 T. I. Zohdi, Modeling and Simulation of Infectious Diseases, https://doi.org/10.1007/978-3-031-18053-8

(A.3)

99

100

Appendix A: Artificial Neural Networks

A(x)

A(x)

x

LINEAR

A(x)

x

SIGMOID/LOGISTICAL

x

DOUBLE−SIGMOID

Fig. A.1 Top: An ANN comprised of (1) Five layers (one input layer and four hidden layers) (2) 35 activation neurons (3+5+7+9+11) and (3) 223 weighted synapses. The color-coding represents the value of the weights. Bottom: Various neuron activation functions: (1) Linear (2) Sigmoid and (3) Double Sigmoid

where Odesir ed is the desired output, which could come from experimental/field data or results from a complex computational model of a system, where a reduced complexity ANN may be useful to represent the system. • STEP 3: The minimization of the error by adjusting the weights: wi+1 = wi + wi+1

(A.4)

• STEP 4: Repeat Steps 1-3 until the best set of weights are found to minimize the error. The determination of the synapse weights can be cast as a nonconvex optimization problem, whereby the cost/error function represents the normed difference between observed data and the output of the ANN for a selected set of weights. The objective is to select a set of weight which minimizes the cost/error. One family of methods that are extremely well-suited for this process are genetic-based machine-learning

Appendix A: Artificial Neural Networks

101

algorithms. The goal of this short communication is to illustrate this process. The specifics of ANN construction are as follows: • Step 1: Assign initial starting “guessed” weights to the synapses; this is the number of unknowns to be updated/optimized: wi • Step 2: Collect (sum) the input (Ii , i = 1, ...synapses, Figure A.1) from each synapse input to each neuron: • Neuron 1: I1 w1→N1 + I2 w2→N1 + ... = S N1 , Neuron 2: I1 w1→N2 + I2 w2→N2 + ... = S N2 , etc. • Step 3: For each neuron, apply an activation function (Figure A.1) to process input: Neuron 1: A N1 (S N1 ),Neuron 2: A N2 (S N2 ), etc. • Linear activation: A(x) = x which provides proportional feedback • Sigmoid/logistical activation: A(x) = 1+e1 −x , reinforces input for x → ∞ and deletes input for x → −∞ −x , reinforces input for x → ∞ and negates • Double-sigmoid activation: A(x) = 1−e 1+e−x input for x → −∞ • Step 4: The output function sums contributions from the last layer: def

Nlast w Ni →O A Ni (S Ni )) = Oi . O(i=1

(A.5)

• Step 5: Compute the error E i = ||O Desir ed − Oi ||, in the appropriate problem specific norm, where O Desir ed is observed data to be matched. • Step 6: Repeat Steps 1-5 with an updated (improved) set of weights until def def a cost function (i ) = E i is minimized by varying a design vector i = {i1 , i2 , ..., iN } = {w1i , w2i , ..., wiN }. This is the key step, referred to as training or calibration. There are a variety of approaches to minimize the error, for example by utilizing the genetic-based machine-learning algorithm (MLA) introduced earlier, which is well-suited for nonconvex optimization. This proceeds by minimizing , by varydef ing the design parameters, i = {i1 , i2 , i3 , ..., iN }, where the search is con(+) (−) (+) ducted within the constrained ranges of (−) 1 ≤ 1 ≤ 1 , 2 ≤ 2 ≤ 2 , (−) (+) 3 ≤ 3 ≤ 3 , etc. These upper and lower limits are dictated by what is physically feasible. The hybrid use of digital-twins, genetic-based machine-learning and Artificial Neural Networks is currently under investigation by the author.

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