Mathematical Models for Neglected Tropical Diseases Essential Tools for Control and Elimination, Part B [1st Edition] 9780128099728, 9780128099711

Table of contents :
Content:
Advances in ParasitologyPage i
Series EditorPage ii
Front MatterPage iii
CopyrightPage iv
ContributorsPages ix-x
PrefacePages xi-xviMaria-Gloria Basáñez, Roy M. Anderson
Chapter One - Mathematical Modelling of Trachoma Transmission, Control and EliminationOriginal Research ArticlePages 1-48A. Pinsent, I.M. Blake, M.G. Basáñez, M. Gambhir
Chapter Two - Progress in the Mathematical Modelling of Visceral LeishmaniasisOriginal Research ArticlePages 49-131K.S. Rock, R.J. Quinnell, G.F. Medley, O. Courtenay
Chapter Three - Soil-Transmitted Helminths: Mathematical Models of Transmission, the Impact of Mass Drug Administration and Transmission Elimination CriteriaOriginal Research ArticlePages 133-198J.E. Truscott, H.C. Turner, S.H. Farrell, R.M. Anderson
Chapter Four - Studies of the Transmission Dynamics, Mathematical Model Development and the Control of Schistosome Parasites by Mass Drug Administration in Human CommunitiesOriginal Research ArticlePages 199-246R.M. Anderson, H.C. Turner, S.H. Farrell, J.E. Truscott
Chapter Five - River Blindness: Mathematical Models for Control and EliminationOriginal Research ArticlePages 247-341M.G. Basáñez, M. Walker, H.C. Turner, L.E. Coffeng, S.J. de Vlas, W.A. Stolk
Chapter Six - The Role of More Sensitive Helminth Diagnostics in Mass Drug Administration Campaigns: Elimination and Health ImpactsOriginal Research ArticlePages 343-392G.F. Medley, H.C. Turner, R.F. Baggaley, C. Holland, T.D. Hollingsworth
Chapter Seven - Lessons Learned From Developing an Eradication Investment Case for Lymphatic FilariasisOriginal Research ArticlePages 393-417R.J. Kastner, C.M. Stone, P. Steinmann, M. Tanner, F. Tediosi
IndexPages 419-429
Contents of Volumes in This SeriesPages 431-448

Citation preview

VOLUME NINETY FOUR

ADVANCES IN PARASITOLOGY Mathematical Models for Neglected Tropical Diseases: Essential Tools for Control and Elimination, Part B

SERIES EDITOR D. ROLLINSON Life Sciences Department The Natural History Museum, London, UK [email protected]

J. R. STOTHARD Department of Parasitology Liverpool School of Tropical Medicine Liverpool, UK [email protected]

EDITORIAL BOARD T. J. C. ANDERSON Department of Genetics, Texas Biomedical Research Institute, San Antonio, TX, USA
NEZ ~ M. G. BASA Professor of Neglected Tropical Diseases, Department of Infectious Disease Epidemiology, Faculty of Medicine (St Mary’s Campus), Imperial College London, London, UK S. BROOKER Wellcome Trust Research Fellow and Professor, London School of Hygiene and Tropical Medicine, Faculty of Infectious and Tropical, Diseases, London, UK R. B. GASSER Faculty of Veterinary and Agricultural Sciences, The University of Melbourne, Parkville, Victoria, Australia N. HALL School of Biological Sciences, Biosciences Building, University of Liverpool, Liverpool, UK J. KEISER Head, Helminth Drug Development Unit, Department of Medical Parasitology and Infection Biology, Swiss Tropical and Public Health Institute, Basel, Switzerland

R. C. OLIVEIRA Centro de Pesquisas Rene Rachou/ CPqRR - A FIOCRUZ em Minas Gerais, Rene Rachou Research Center/CPqRR - The Oswaldo Cruz Foundation in the State of Minas Gerais-Brazil, Brazil R. E. SINDEN Immunology and Infection Section, Department of Biological Sciences, Sir Alexander Fleming Building, Imperial College of Science, Technology and Medicine, London, UK D. L. SMITH Johns Hopkins Malaria Research Institute & Department of Epidemiology, Johns Hopkins Bloomberg School of Public Health, Baltimore, MD, USA R. C. A. THOMPSON Head, WHO Collaborating Centre for the Molecular Epidemiology of Parasitic Infections, Principal Investigator, Environmental Biotechnology CRC (EBCRC), School of Veterinary and Biomedical Sciences, Murdoch University, Murdoch, WA, Australia X.-N. ZHOU Professor, Director, National Institute of Parasitic Diseases, Chinese Center for Disease Control and Prevention, Shanghai, People’s Republic of China

VOLUME NINETY FOUR

ADVANCES IN PARASITOLOGY Mathematical Models for Neglected Tropical Diseases: Essential Tools for Control and Elimination, Part B Edited by


NEZ ~ MARIA GLORIA BASA Department of Infectious Disease Epidemiology, Faculty of Medicine, Imperial College London, London, UK

ROY M. ANDERSON Department of Infectious Disease Epidemiology, Faculty of Medicine, Imperial College London, London, UK

AMSTERDAM • BOSTON • HEIDELBERG • LONDON NEW YORK • OXFORD • PARIS • SAN DIEGO SAN FRANCISCO • SINGAPORE • SYDNEY • TOKYO Academic Press is an imprint of Elsevier

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Notices

Knowledge and best practice in this field are constantly changing. As new research and experience broaden our understanding, changes in research methods, professional practices, or medical treatment may become necessary. Practitioners and researchers must always rely on their own experience and knowledge in evaluating and using any information, methods, compounds, or experiments described herein. In using such information or methods they should be mindful of their own safety and the safety of others, including parties for whom they have a professional responsibility. To the fullest extent of the law, neither the Publisher nor the authors, contributors, or editors, assume any liability for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions, or ideas contained in the material herein. ISBN: 978-0-12-809971-1 ISSN: 0065-308X For information on all Academic Press publications visit our website at https://www.elsevier.com

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CONTRIBUTORS R.M. Anderson London Centre for Neglected Tropical Disease Research, London, United Kingdom; School of Public Health, Imperial College London, London, United Kingdom R.F. Baggaley London School of Hygiene and Tropical Medicine, London, United Kingdom M.G. Bas
an ~ ez Imperial College London, London, United Kingdom I.M. Blake Imperial College London, London, United Kingdom L.E. Coffeng University Medical Center, Rotterdam, Rotterdam, The Netherlands O. Courtenay University of Warwick, Coventry, United Kingdom S.H. Farrell London Centre for Neglected Tropical Disease Research, London, United Kingdom; School of Public Health, Imperial College London, London, United Kingdom M. Gambhir Monash University, Melbourne, VIC, Australia C. Holland Trinity College Dublin, Dublin, Ireland T.D. Hollingsworth University of Warwick, Coventry, United Kingdom R.J. Kastner Swiss Tropical and Public Health Institute, Basel, Switzerland; University of Basel, Basel, Switzerland G.F. Medley London School of Hygiene and Tropical Medicine, London, United Kingdom A. Pinsent Monash University, Melbourne, VIC, Australia R.J. Quinnell University of Leeds, Leeds, United Kingdom K.S. Rock University of Warwick, Coventry, United Kingdom

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P. Steinmann Swiss Tropical and Public Health Institute, Basel, Switzerland; University of Basel, Basel, Switzerland W.A. Stolk University Medical Center, Rotterdam, Rotterdam, The Netherlands C.M. Stone Swiss Tropical and Public Health Institute, Basel, Switzerland; University of Basel, Basel, Switzerland M. Tanner Swiss Tropical and Public Health Institute, Basel, Switzerland; University of Basel, Basel, Switzerland F. Tediosi Swiss Tropical and Public Health Institute, Basel, Switzerland; University of Basel, Basel, Switzerland J.E. Truscott London Centre for Neglected Tropical Disease Research, London, United Kingdom; School of Public Health, Imperial College London, London, United Kingdom H.C. Turner London Centre for Neglected Tropical Disease Research, London, United Kingdom; School of Public Health, Imperial College London, London, United Kingdom S.J. de Vlas University Medical Center, Rotterdam, Rotterdam, The Netherlands M. Walker Imperial College London, London, United Kingdom

PREFACE This is the second (Part B) of two volumes for Advances in Parasitology on the topic of ‘Mathematical Models for Neglected Tropical Diseases: Essential Tools for Control and Elimination’. Since the publication of Part A in 2015, considerable progress has been made, in this research field, and under the fillip of both the NTD Modelling Consortium (http://www. ntdmodelling.org/) and the London Centre for Neglected Tropical Disease Research (LCNTDR) (http://www.londonntd.org/), to bring together diverse modelling groups and approaches: (1) to scrutinize the effectiveness of current interventions for achieving the goals set in 2012 by the World Health Organization (WHO) (http://apps.who.int/iris/bitstream/10665/ 70809/1/WHO_HTM_NTD_2012.1_eng.pdf) for the control or elimination of 17 NTDs by 2020, and (2) to investigate how best to deploy alternative/complementary strategies should the former be deemed insufficient in all endemic settings. In January 2013, the WHO published its second report on NTDs (http://www.who.int/neglected_diseases/2012report/ en/), aiming to sustain the drive fuelled by the original roadmap and in May that year, the 66th World Health Assembly adopted resolution WHA66.12, calling on member states to intensify and integrate measures, as well as to plan investments to improve the health and social well-being of populations affected by NTDs (http://www.who.int/neglected_ diseases/WHA_66_seventh_day_resolution_adopted/en/). To this end, mathematical models of NTD transmission and control have a crucial role to play in guiding programmes and helping policy makers to plan those strategies that have the highest impact with available intervention tools and to identify those that are the most cost effective. Models also help in designing monitoring and evaluation protocols to assess progress towards the 2020 goals. In October 2015, the NTD Modelling Consortium published a thematic collection of papers on quantitative analyses of the current strategies to achieve the 2020 goals for NTDs (https://www.biomedcentral.com/ collections/ntdmodels2015), and in January 2016 the LCNTDR launched a collection of papers to highlight recent advances in scientific research for NTD control (http://www.parasitesandvectors.com/series/LCNTDR). This volume complements both these thematic collections by offering state-of-the-art reviews on modelling bacterial (trachoma), protozoan

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(leishmaniasis), and helminth (intestinal nematode, schistosome and filarial) infections with the aim of identifying approaches to support the control and elimination goals proposed by the WHO and endorsed by the London Declaration on NTDs (http://unitingtocombatntds.org/resource/londondeclaration). Part B starts with a chapter by Amy Pinsent and colleagues reviewing mathematical models of trachoma (caused by the bacterium Chlamydia trachomatis) in the context of the four-pronged approach adopted by the Global Elimination of blinding Trachoma (GET 2020) alliance. This approach relies on the SAFE strategy (S for trichiasis surgery; A for antibiotic treatment; F for facial cleanliness and E for environmental improvement) for eliminating trachoma (the foremost cause of infectious blindness worldwide). This chapter highlights that although a large body of work has focussed on modelling the effectiveness of antibiotic administration (A) in reducing or interrupting trachoma transmission, much less work has been conducted to model the effect of nonpharmaceutical interventions (the F and E components of SAFE). Two further gaps were identified, the former relating to a scarcity of models linking infection and disease to enable rigorous evaluation of progress towards GET 2020 (elimination of blinding trachoma) and the latter on the little consideration that has been given to modelling the cost effectiveness of the SAFE interventions. The chapter by Kat Rock and coworkers reviews progress in the mathematical modelling of the life-threatening form of leishmaniasis, visceral leishmaniasis (VL). VL (also known as kala-azar and caused by members of the protozoan parasite species complex Leishmania donovani) has been targeted for elimination as a public health problem on the Indian subcontinent (where it is deemed to be mainly an anthroponosis). In parallel to issues discussed in the chapter on trachoma, quantifying progress towards elimination of an NTD as a public health problem is not a trivial pursuit. It requires a fundamental understanding of the relationship between infection and disease as well as of the progression across disease stages. In VL, disease progression through different stages and the influence of asymptomatic infection and of post-kala-azar dermal leishmaniasis (PKDL) in diagnosis and infectivity to vectors are not yet clearly understood. The authors highlight the success of recent control efforts, which have prompted the original goal of elimination as a public health problem to be brought forward from 2020 to 2017. However, they also recommend caution, as modelling and transmission studies have indicated that asymptomatically infected individuals might

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hinder elimination efforts and/or obscure the true number of leishmaniasis cases. Following on with the theme of NTDs whose amelioration as a public health problem (control rather than elimination) constitutes the 2020 goal, James Truscott and colleagues review the underlying biology and epidemiology of the three main intestinal nematode (ascariasis, trichuriasis and hookworm) infections included under the umbrella term of soil-transmitted helminthiases (STHs). These authors review efforts to model the transmission cycle of the (directly transmitted) helminth species (Ascaris lumbricoides, Trichuris trichiura and Necator americanus/Ancylostoma duodenale) responsible for STHs and the effects of preventative chemotherapy on their control and potential elimination. Although considered together under the STH term, recent modelling work conducted by the authors has shown that the different epidemiological characteristics of these nematode species means that a one-size-fits-all mass drug administration (MDA) approach is not equally suitable for the attainment of control (or elimination) goals. When linked to intervention cost-effectiveness analyses, these models can helpfully inform public heath policy. Research gaps identified include a poor understanding of the diagnostic performance of the tools currently used to detect and quantify infection, which consequently hinders efforts to test and improve the models. Schistosomiasis – a term encompassing intestinal and urinary schistosomiasis caused by a number of (snail-transmitted) schistosome species – is earmarked for elimination (of infection) in selected African countries according to the WHO 2012 roadmap on NTDs. Roy Anderson and coauthors review modelling frameworks for schistosomiasis transmission and control, with particular reference to the role of exposure and/or acquired immunity on age profiles of infection and reinfection by Schistosoma mansoni and Schistosoma haematobium. These authors also discuss the influence of predisposition to infection and to treatment uptake (adherence/compliance) on model outcomes of intervention impact and present deterministic and stochastic approaches. In a similar fashion to the chapter on trachoma, the authors highlight that key epidemiological measures, such as the basic reproduction number and the force of infection (in helminths the rate at which new worms are acquired), are seldom derived from field data and presented in published modelling studies. Given the ambitions of the WHO to achieve elimination of schistosomiasis transmission in defined regions, the authors advise caution, echoing the conclusions of the chapter on VL. Stochastic model outputs indicate potentially long durations of parasite persistence in

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some communities even after achieving levels of therapeutic coverage above those required to break transmission. In common with the paper on STH modelling, this chapter also discusses issues regarding the performance of available diagnostics, a theme running through the volume and that is the subject of the chapter by Medley and coauthors. A review of control programmes and associated models for onchocerciasis (river blindness) is presented by Maria-Gloria Bas
an ~ez and colleagues. Human onchocerciasis (a Simulium-transmitted, anthroponotic filarial infection) has been targeted for elimination in selected African countries by 2020 and in 80% of endemic countries by 2025. These authors focus on the comparison between the EPIONCHO (deterministic) and ONCHOSIM (stochastic microsimulation) models and discuss how these two distinct frameworks are addressing the key questions of where can onchocerciasis be eliminated with current intervention strategies by 2020/2025, and what alternative strategies could help to accelerate elimination. Currently, onchocerciasis programmes deliver annual or 6-monthly MDA with ivermectin, and the importance of achieving high therapeutic coverage and minimizing systematic nonadherence is highlighted. However, ivermectin is deemed to have little efficacy in reducing the survival of the long-lived (Onchocerca volvulus) adult worms (although it affects the worm’s ability to produce live progeny). Therefore, the authors discuss a number of alternative and complementary strategies (including vector control) and their potential modalities of delivery to infected individuals and endemic populations. Graham Medley and colleagues devote the following chapter to introduce the properties of current diagnostic assays for STHs and highlight the programmatic consequences of their dwindling sensitivity as control progresses and infection prevalence falls. Using A. lumbricoides as a case study – due to the availability of high-quality data – the authors investigate how more sensitive diagnostics would affect important features of MDA programmes such as: (1) their implementation and duration; (2) the number of treatment rounds that would be deployed (including the question of distributing unnecessary treatments); (3) the health impact of the programme, and (4) the probability of achieving elimination. Although focussed on the STHs in general, and ascariasis in particular, the issues addressed in the chapter are relevant to all the NTDs discussed in this volume. Finally, Fabrizio Tediosi and coauthors use lymphatic filariasis (LF) as an example from which to learn lessons when developing an eradication

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investment case (EIC). LF is a mosquito-transmitted infection caused by a number of filarial nematodes, namely Wuchereria bancrofti, Brugia malayi and Brugia timori (with the former being responsible for >90% of LF cases). The Global Programme for the Elimination of Lymphatic Filariasis (GPELF) was established from the outset to aim at elimination of the infection (the transmission of W. bancrofti involves only humans and mosquitoes) under the broader Global Alliance for the Elimination of Lymphatic Filariasis (GAELF). In this chapter, the authors highlight the value and implications of undertaking a comprehensive analysis of the overall situation prior to embarking on an elimination or eradication programme and developing an EIC. Considerations for LF elimination/eradication and its associated modelling include an assessment of the level and commitment of the necessary investments; the availability and efficacy of treatments (including their efficacy in killing the long-lived adult worms); the feasibility of scaling up the coverage of MDA to all endemic communities, the quantification of the health impact of the programme, its cost effectiveness and the consideration of broader economic benefits. With the fifth anniversary of the WHO NTD roadmap and the London Declaration on NTDs approaching in January 2017, the publication of this volume is timely. The WHO and Uniting to Combat NTDs (http:// unitingtocombatntds.org/) will host an NTD summit in Geneva, Switzerland, in April 19–22, 2017. The purpose of this joint summit will be both to celebrate the progress made thus far and to rally the NTD community towards the 2020 milestones and beyond, recognizing that increased efforts are required to push towards 2020 across all NTDs. Thanks to a number of programmes of unprecedented scale and exemplary public–private partnership, several NTDs are receding, whereas others are well positioned to achieve the 2020 targets. However, perceived success can be a double-edged sword, giving the impression that the job is done or leading to donor fatigue. In fact, some programmes are in the process of winding down or have closed altogether (e.g., the African Programme for Onchocerciasis Control, APOC), placing the NTD community at a crucial crossroad. Substantial implementation, funding and R&D gaps remain and the recently launched Expanded Special Project for Elimination of NTDs (ESPEN) that will succeed APOC requires considerable additional investment. As the coordination of NTD programmes is devolved to endemic countries, the diseases will have to compete with a myriad of other national health priorities at a pivotal time in the road to elimination. This time, however, is also

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accompanied by a growing interest in developing and understanding the features of novel diagnostics, drugs and vector control strategies for neglected diseases. This second volume on the mathematical modelling of NTDs has, therefore, been designed to support the NTD community and policy makers, by describing a set of quantitative tools to enhance scientific understanding of how best to suppress transmission and reduce morbidity at the community level. Quantitative analyses, based on mathematical frameworks that capture the known biology and epidemiology of the aetiological agents responsible for NTDs, help to identify clearly which are the optimal and cost-effective strategies for their control. This volume focuses on the strengths and weaknesses of current models, on the estimation of key parameters from field epidemiological studies and on future research needs. We thank the support of the editorial team of Advances in Parasitology and hope that our readers will enjoy this volume as much as we have enjoyed preparing it. Maria-Gloria Bas
an ~ ez and Roy M. Anderson September 2016

CHAPTER ONE

Mathematical Modelling of Trachoma Transmission, Control and Elimination ~ ezx, M. Gambhir* A. Pinsent*, 1, I.M. Blakex, M.G. Bas
an *Monash University, Melbourne, VIC, Australia x Imperial College London, London, United Kingdom 1 Corresponding author: E-mail: [email protected]

Contents 1. Introduction 1.1 Clinical and epidemiological features 1.2 Trachoma control and elimination 1.3 The role of mechanistic and statistical models

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1.3.1 Deterministic and stochastic modelling 1.3.2 Modelling infection and modelling disease

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1.4 Aims and objectives of this review 2. Methods 3. Results 3.1 Characteristics of identified studies 3.2 Deterministic and stochastic models 3.3 Data sets used and model fitting to data 3.4 Transmission intensity and the basic reproduction number 3.4.1 Estimating the force of infection 3.4.2 Estimating the basic reproduction ratio

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3.5 Acquired immunity: recovery rate and infectivity 3.6 Infection and active disease, modelling of disease sequelae 3.7 Simulation of interventions, forecasting infection and disease, and analysis of their cost-effectiveness 4. Discussion 4.1 Using modelling to determine the feasibility of the GET 2020 goals 4.2 Conclusions Acknowledgements References

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Abstract The World Health Organization has targeted the elimination of blinding trachoma by the year 2020. To this end, the Global Elimination of Blinding Trachoma (GET, 2020) Advances in Parasitology, Volume 94 ISSN 0065-308X http://dx.doi.org/10.1016/bs.apar.2016.06.002

© 2016 Elsevier Ltd. All rights reserved.

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alliance relies on a four-pronged approach, known as the SAFE strategy (S for trichiasis surgery; A for antibiotic treatment; F for facial cleanliness and E for environmental improvement). Well-constructed and parameterized mathematical models provide useful tools that can be used in policy making and forecasting in order to help to control trachoma and understand the feasibility of this large-scale elimination effort. As we approach this goal, the need to understand the transmission dynamics of infection within areas of different endemicities, to optimize available resources and to identify which strategies are the most cost-effective becomes more pressing. In this study, we conducted a review of the modelling literature for trachoma and identified 23 articles that included a mechanistic or statistical model of the transmission, dynamics and/or control of (ocular) Chlamydia trachomatis. Insights into the dynamics of trachoma transmission have been generated through both deterministic and stochastic models. A large body of the modelling work conducted to date has shown that, to varying degrees of effectiveness, antibiotic administration can reduce or interrupt trachoma transmission. However, very little analysis has been conducted to consider the effect of nonpharmaceutical interventions (and particularly the F and E components of the SAFE strategy) in helping to reduce transmission. Furthermore, very few of the models identified in the literature review included a structure that permitted tracking of the prevalence of active disease (in the absence of active infection) and the subsequent progression to disease sequelae (the morbidity associated with trachoma and ultimately the target of GET 2020 goals). This represents a critical gap in the current trachoma modelling literature, which makes it difficult to reliably link infection and disease. In addition, it hinders the application of modelling to assist the public health community in understanding whether trachoma programmes are on track to reach the GET goals by 2020. Another gap identified in this review was that of the 23 articles examined, only one considered the cost-effectiveness of the interventions implemented. We conclude that although good progress has been made towards the development of modelling frameworks for trachoma transmission, key components of disease sequelae representation and economic evaluation of interventions are currently missing from the available literature. We recommend that rapid advances in these areas should be urgently made to ensure that mathematical models for trachoma transmission can robustly guide elimination efforts and quantify progress towards GET 2020.

1. INTRODUCTION 1.1 Clinical and epidemiological features Trachoma is one of 17 neglected tropical diseases (NTDs) prioritized by the World Health Organization (WHO) for control and elimination through preventive chemotherapy or intensified disease management strategies (WHO, 2015a). NTDs are mostly responsible for chronic infections/ conditions that can cause severe morbidity in affected individuals, leading

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to long-term disability but are deemed to be associated with relatively low mortality in comparison to acute, epidemic infectious diseases (WHO, 2015a; Hotez et al., 2014). Transmission of NTDs is facilitated by living conditions that are associated with poverty, such as poor housing and sanitation, and limited access to clean water and basic health care (Hotez et al., 2009). Trachoma is the leading global cause of infectious blindness and is currently estimated to affect 84 million people across 51 endemic countries (WHO, 2012). An estimated 1.8 million people are visually impaired as a result of the disease, of which 0.5 million people are irreversibly blind (WHO, 2012; WHO, 2015b). Active inflammatory diseasedtrachomatous follicular and trachomatous inflammatory [TF, TI, according to the WHO simplified grading scheme (Taylor et al., 2014)]dis caused by infection with the bacterium Chlamydia trachomatis. Repeated infection with these bacteria leads to an immunopathological response characterized by scarring of the inner part of the eyelid, and an eventual curling-in of the eyelashes, which abrades the corneal surface. This can lead to trachomatous trichiasis (TT), corneal opacity (CO) and blindness. Excess mortality is also reported to be associated with blinding trachoma (Hotez et al., 2014). Estimates of the Disability-Adjusted Life Years (DALYs) due to trachoma have been variable. The Global Burden of Disease (GBD) 1990 Study estimated the burden of trachoma (all ages) to be 144,000 (95% uncertainty interval [95% UI],104,000e189,000) DALYs, whereas the GBD 2010 Study reported a value of 334,000 (95% UI 243,000e438,000) (Murray et al., 2012). Other authors have set this figure at least at one million DALYs (Evans and Ranson, 1995) or as high as 3.6 million, with the highest proportion (72%) contributed by sub-Saharan Africa (Frick et al., 2003). Among the major causes of blindness in 2010, trachoma represented 5.2% in subSaharan Africa (Naidoo et al., 2014). An accurate quantitative estimate of the burden of trachoma remains, however, challenging due to several factors, including scarce data availabilitydwhich limits the ability to estimate accurately the number of people infecteddand an unresolved issue as to whether trichiasis should be considered as a disabling disease sequela (Burton and Mabey, 2009). The economic impact of trachoma in terms of lost productivity is estimated to range between US $2.9 and $5.3 billion annually, rising to US $8 billion if trichiasis is included in the estimate (WHO, 2015b). Countries reported to have the highest prevalence of infection are located in East Africa and the Sahel belt; however, trachoma is also prevalent in Southeast Asia, the Middle East, the Indian subcontinent and Latin

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America (Burton and Mabey, 2009), although the distribution and the prevalence of infection are far more heterogeneous in these regions in comparison to sub-Saharan Africa. While trachoma was previously prevalent in Europe and North America only 100 years ago, improvements in sanitary and living conditions resulted in the gradual disappearance of infection (Burton and Mabey, 2009). Infection with C. trachomatis is spread through two primary routes. The first is direct personal contact which could be direct hand contact with an infected individual or through contact with clothing which has contacted infectious discharge (Burton and Mabey, 2009). The second route involves eye-seeking flies (e.g., Musca sorbens) which have contacted the discharge from an infected person’s eyes or nose (Emerson et al., 2004). For transmission of infection to be sustained, it must be consistently transmitted from person to person. The severity of disease experienced by an infected individual varies with age and hence their duration of exposure to infection (Bailey et al., 1999; Grassly et al., 2008). Infection with C. trachomatis becomes shorter in duration and reduced in frequency as individuals age; therefore, the highest burden of C. trachomatis is observed in young children (Bailey et al., 1999; Grassly et al., 2008). Repeated infection with age (continuous exposure) leads to conjunctival scarring, ultimately leading to TT, CO and blindness as mentioned previously (Burton and Mabey, 2009; West et al., 1991). Several epidemiological surveys have suggested that severe sequelae in the form of TT and CO disproportionally affect women in comparison to men, as a result of women having a higher exposure to the reservoir source of infection, which is reported to be young children (Courtright and West, 2004; West et al., 1991). A number of risk factors for trachoma transmission have been identified, including (1) secretions from the eye which other individuals may come into contact with, and which may also attract flies which help to facilitate transmission (Emerson et al., 2004; Ngondi et al., 2008); (2) overcrowding within the household, which increases the frequency of contact between individuals potentially leading to more frequent infection events (Abdou et al., 2007; Ngondi et al., 2008) and (3) limited supplies of clean water resulting in infrequent face washing, general poor hygiene practice and lack of easy access to latrines, which can lead to a buildup of faecal matter in the environment which attracts eye-seeking flies (Emerson et al., 2004). Transmission intensity of trachoma within a community is classified according to the prevalence of active disease in 1- to 9-year olds. Communities are considered hyperendemic if the prevalence of active disease in this age group

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is >20%, mesoendemic if prevalence is >10% but mean), where most hosts harbour none or few worms and a few harbour many. The pattern is well described by the negative binomial probability model with aggregation parameter k which varies inversely with the degree of aggregation or clumping. Values of k above 5 indicate 20 Frequency of observaon

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6-10 11-20 21- 40 Number of worm pairs

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Figure 6 The frequency distribution of the number of worm pairs of Schistosoma mansoni found at autopsy (Cheever, 1968).

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Figure 7 Relationship between the prevalence of infection (fraction infected) and the mean worm burden for various values of the aggregation parameter k as defined for the negative binomial distribution (Eq. (1)). When k > 5 the distribution is Poisson in form (¼ random distribution).

that the worms are randomly distributed in the host population (variance z mean). For the data recorded in Fig. 6, k is estimated as 0.2, reflecting a high degree of aggregation (Cheever, 1968). This distribution predicts the following relationship between the proportion infected, P, and the mean worm burden, M, for a given k value,
   M k P ¼1 1þ (1) k As displayed in Fig. 7, this relationship helps to explain the patterns recorded in Figs 2 and 4. The relationship suggests that small differences in prevalence (particularly where this is high and k is small reflecting a high degree of worm aggregation) may be associated with considerable differences in intensity of infection and, therefore, possibly of the prevalence and severity of clinical disease (Cheever, 1968). As such, the monitoring and evaluation of MDA programmes must be based on intensity measures, not just on prevalence, since the latter will not reflect the true magnitude of the impact on intensity. A relationship between prevalence and mean egg output can be constructed and used to indirectly estimate k and other parameters associated with the worm’s natural history within the host. For different host populations or subpopulations, values for prevalence and mean number of eggs per gram (epg) can be calculated. Mean egg output can be related to

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mean worm burden, which can, in turn, be related to measured prevalence through Eq. (1). This method has been used to calculate k and worm fecundity parameters, leading to results comparable to direct estimation from autopsy studies (Chan et al., 1995).

2.4 Reinfection posttreatment After a period of continued exposure to infection post-chemotherapeutic treatment with the most widely used drug, praziquantel, which has on average a high efficacy (Table 2), people tend to reacquire infection. Thus, despite abundant evidence of immunological responses to parasite antigens, continual exposure leads to reinfection, especially in children. The absence of strong acquired immunity, despite high exposure, is poorly understood, but of obvious significance to the study of transmission dynamics and the impact of MDA. There is limited evidence of a slow buildup of some acquired immunity in adults (Wilkins et al., 1987; Chandiwana et al., 1991), although the evidence for S. haematobium infections is possibly stronger (Butterworth, 1998), but this is insufficient to prevent infection even in elderly individuals with a long history of exposure. Schistosome parasites have large genomes and release many secretions that are thought to modulate the effectiveness of immunological attack by the human host.

2.5 Predisposition For most human helminth infections including the schistosomes, postchemotherapy, those heavily infected at the point of treatment tend to reacquire high burdens of parasites after a period of reinfection. Predisposition for heavy (or light) infection may be due to a number of influences including Table 2 Reported drug efficacy figures for praziquantel against the main human schistosome infections Parasite Drug efficacy References

Schistosoma mansoni Schistosoma haematobium

92.5% 85.3e100%

S. mansoni S. haematobium

79e90% 63e85%

Schistosoma japonicum Schistosoma intercalatum

80e90% 89%

Kihara et al. (2007) Ojurongbe et al. (2014), Tchuente et al. (2004) Utzinger et al. (2000) Utzinger et al. (2000), Tchuente et al. (2004) Utzinger et al. (2000) Utzinger et al. (2000)

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Table 3 Human helminths where evidence for predisposition to heavy infection has been recorded Evidence for Parasite Country predisposition References

Ascaris lumbricoides A. lumbricoides Enterobius vermicularis Necator americanus

India Burma India India

Yes Yes Yes Yes

Ancylostoma duodenale Trichuris trichiura

India St Lucia

Yes Yes

Schistosoma mansoni

Kenya

Yes

Haswell-Elkins et al. (1987) Hliang (1989) Haswell-Elkins et al. (1987) Schad and Anderson (1985), Haswell-Elkins et al. (1987) Schad and Anderson (1985) Haswell-Elkins et al. (1987), Bundy et al. (1985) Wilkins et al. (1987)

host genetic background and social, behavioural or environmental factors. The precise causes for the observed patterns for human helminths are not well understood, but behavioural factors are undoubtedly important for schistosome infections (Bensted-Smith et al., 1987). Table 3 records studies that have observed predisposition for human helminth infections.

3. POPULATION PROCESSES Before turning to the development of mathematical models of transmission and control, a brief review of some of the key population processes provides guidance for model construction and parameterization.

3.1 Parasite life spans As noted earlier, much simplification can be made in the study of transmission dynamics by noting the large discrepancies in the average duration of stay in different compartments in the two-host life cycle of the schistosome species as recorded in Table 1. The key features of the dynamics can be captured by simply constructing equations of the population turnover of the adult worms in the human host, taking due note of human demography and the influence of the other stages in the snail intermediate host and the free-living larval stages on the net rate of transmission pertaining to a defined human community. The dynamics of the worm population in the human host in response to perturbations created, for example, by MDA will be largely driven by the timescale of the adult worm life expectancy. For example, return

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Table 4 Estimates of the life span of adult worms in the human host Parasite Life span in years References

Schistosoma mansoni S. mansoni

2.7e4.5, mean 3.3 5.7e10.5

Anderson and May (1985a) Fulford et al. (1995)

times to precontrol steady states of the mean worm burden are largely set by this parameter (Anderson and May, 1985a, 1991). Unfortunately, parasite life expectancy is difficult to measure in practice. Published attempts either depend on the duration of egg output in an untreated person who moves to live in an infection-free environment, or by the use of various statistical approaches to the analysis of observed epidemiological patterns or mathematical model fitting procedures, particularly reinfection posttreatment (Wallerstein, 1949; Berberian et al., 1953; Fulford et al., 1995). Table 4 summarizes some of these published estimates. The average value lies in the range 3e5 years. As we shall see later, the precise value has a strong influence on the predicted dynamics of transmission.

3.2 Human demography Since all infected persons contribute to the transmission intensity prevailing in a defined habitat via water contact by all age groups and passage of faecal material or urine into snail habitats, human demography with respect to the age distribution of the affected population will have a significant influence on net transmission and the impact of MDA targeted at specific age groups such as SAC. An illustration of this is provided in Fig. 8, where the age distribution of Uganda in 2011 is plotted. Only 31.2% of the human population is in the SAC group (5e14 years). Simple calculations, based on age intensity profiles (based on epg of faeces) for S. mansoni in two different countries, Uganda and Brazil, plus the prevailing demography of the human populations, permit estimates to be made of the percentage of the worms (or more precisely e egg output) in the SAC group. The figures are 39.7% for the Ugandan study and 27.4% for the Brazilian study. Such calculations, simple as they are, are of great importance to the design of MDA programmes. If only targeted at SAC, a large fraction of the worm population is not exposed to chemotherapy. This facilitates the persistence of infection within such communities. However, it can be argued that this reservoir of infection, to some extent, guards against the evolution of dug resistance by continually diluting the gene pool of the parasite with offspring not exposed to the selective pressure applied by treatment.

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Proportion of the population (%)

25

20

15

10

5

0 0–4

10–14 20–24 30–34 40–44 50–54 60–64 70–74 80–84 90–94 100+

Age group in years

Figure 8 Age distribution of Uganda in 2011. Note that the proportion of the population in the school-aged children (SAC e 5e14 years of age) age classes is 31.2% (Anderson et al., 2013).

3.3 Age-related exposure to infection Given that water contact is required for the human host to be exposed to infection, due to the aquatic lifestyle of the snail intermediate host, agerelated changes in such contact can play an important role in determining the shape of observed age intensity profiles, independent of any buildup of acquired immunity with age. Many accounts of age-related water contact have been published (Chandiwana, 1987; Wilkins et al., 1987; Chan et al., 2000). One example is presented in Fig. 9, which records observation data on human behaviour in St Lucia (Jordan, 1972). Note how the severity of changes with age well mimics the age intensity patterns displayed in Figs 2 and 3, giving strength to the argument that ecology (human behaviour) is more important than immunology.

3.4 Density dependence Density dependence in birth and death rates is a well-established principle in population ecology, where per capita birth rates fall and death rates rise as population density increases for reasons related to the competition for available resources. For parasites, these effects may arise from acquired immunity where the severity of immunological attack rises as parasite burden increases and/or from resource limitation. In the case of schistosomes, the only welldocumented density-dependent process is that of worm fecundity in the human host and, indeed, in experimental mammalian hosts, such as mice

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3500

Total duraon of water contact over 8 days (minutes)

3000 2500 2000 1500 1000 500

0 0-4

5-9

10-14

15-19

20-29

30-39

40-49

50 +

Age group in years

Figure 9 Total duration of water contact over an 8-day observation period in St Lucia (Jordan, 1972). 1

0.8 0.7 (Thousands)

Eggs/female/gram of faeces

0.9

0.6 0.5 0.4 0.3 0.2 0.1 0 0

200

400

600

Female Worm Burden

Figure 10 Density-dependent fecundity in Schistosoma mansoni based on autopsy data with the best fit exponential decay curve (Cheever, 1968).

and baboons (Medley and Anderson, 1985). Fig. 10 records such a pattern for S. mansoni in humans from the Cheever (1968) autopsy studies. The exponential model fits such data adequately where the per female worm egg output f(M) as a function of worm burden M is given by, f ðM Þ ¼ aegM

(2)

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where a and g are parameters estimated from fitting the model to data (Medley and Anderson, 1985). The term exp(g) is a useful summary quantity of the severity of density dependence.

3.5 Mating probabilities

Probability of being mated ɸ

Schistosome parasites are dioecious with separate sexes and as noted first by Macdonald (1965), the need to mate to produce viable offspring creates a threshold in parasite population density below which mating frequency is too low to maintain transmission. This threshold is often referred to as the ‘breakpoint’ in transmission. The concept is made more complex by the parasitic mode of life since male and female worms must be present in the same host for successful sexual reproduction. Therefore, the probability distribution of parasites per host greatly influences mating success. Macdonald assumed a Poisson probability distribution, but as shown in and earlier section, the observed distributions are highly clumped and the negative binomial model provides a good description of observed pattern. Aggregation enhances the likelihood of encountering a member of the opposite sex within the human host and therefore reduces the level of the breakpoint. Whether or not the parasites are monogamous or polygamous will also influence the level of the breakpoint as first noted by May (1977), and this concept is illustrated in Fig. 11, which plots the breakpoint based on the

k=0.01 k=0.1 k=0.3 k=1.0 Random

Mean worm burden

Figure 11 Mating probabilities and the breakpoint in transmission. The plot records the value of the probability of being mated for different assumptions for the degree of worm aggregation within the host population (the magnitude of the negative binomial k which varies inversely with the degree of aggregation) under the assumption of polygamy. Note: Mathematical models predict that stable equilibria of either a persistent parasite infection or extinction are separated by an unstable equilibrium. The level of the unstable point is determined by the value of k and the reproductive biology (monogamous or polygamous) of the parasite (May, 1977; Anderson and May, 1985a). Values close to zero are predicted for moderate-to-high worm burdens and low values of k.

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assumption of monogamy for various degrees of worm aggregation within the host population (the magnitude of k). Schistosomes that live in pairs in permanent copulation have been assumed to be monogamous and mate for life. Robust evidence for this is limited, but in the model development described in later sections this assumption is made, although its relaxation to a state of polygamy is also examined.

3.6 Basic reproduction number A central concept in the study of the epidemiology of infectious diseases is that of the basic reproduction number R0. For macroparasites, R0 is defined as the average number of female offspring produced by a female worm (or worm pair) that both infected the definitive human host and survived to reproductive maturity, in a totally susceptible population. For the parasite to persist, R0 > 1. In simple terms, R0 can be defined as the product of reproductive output multiplied by the probability of transmission success, divided by the mortality terms throughout the life cycle. More will be said of this in the following mathematical model section, but the measurement of R0 and the impact of MDA on the effective reproduction number, R, in a defined population is central to an understanding of the transmission dynamics and control of schistosome infections. High values of R0 reflect high transmission intensity in a given setting, and concomitantly, reflect the need for frequent drug treatment to break transmission.

4. MATHEMATICAL MODELS OF THE BASIC DYNAMICS OF TRANSMISSION The structure of this section is set to move from simple deterministic models through increasing degrees of complexity to numerical studies of individual-based stochastic models. The real world is replete with many complexities, but simple models shed much light on the key processes influencing transmission and the impact of control measures for helminth parasites. In addition, given the relatively simple dynamics of this class of infectious diseases, they produce predictions that are fairly reliable in a qualitative sense. In other words, they are well able to accurately predict the overall epidemiological patterns which are of importance in setting overall public health policy for the control of transmission. More complex models are required to produce quantitative predictions on the precise impact of defined control measures.

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4.1 Simple deterministic model with no age structure The short life expectancies of the free-living and snail host stages of the life cycle, relative to the many-year life expectancy of the adult worm in its human host (Table 1) imply that a sensible discussion of the dynamics can be based on a single equation for the adult worms. The simplest deterministic model for the dynamics over time of the mean worm burden in the human, M(t), is given by a simple differential equation (Anderson and May, 1985a; Anderson, 1980), " # 1T T f dM ðtÞ 2 1 2 ¼ m1 M ðtÞ 1 1 (3) dt 2T2 fM ðtÞ þ 1 Here, ½ arises from an assumed 1:1 sex ratio, 1/m1 is adult worm life expectancy, and in the terminology of Macdonald (1965), the parameters T1 and T2 denote transmission from snail to human and human to snail, respectively, where, T1 ¼ b1 l2 N2 =½m1 ðb1 N1 þ m5 Þ

(4)

T2 ¼ f b2 l1 N1 =½m4 ðb2 N2 þ m2 Þ

(5)

and Here l2 is the rate of cercarial shedding per infected snail, b1 is the transmission coefficient for a cercaria to infect a human, m5 is the death rate of the cercaria and N1 is human population size. As such the term b1/(b1N1 þ m5) is the probability that a cercaria infects a human. N2 is the population size of the snail host, l1 is the rate of egg production per mated female (not assumed to be density dependent as yet), b2/(b2N2 þ m2) is the probability that an egg produces a miracidium which infects a susceptible snail, f is the proportion of latent-infected snails that survive to release cercaria, m2 is the death rate of miracidia and m4 is the death rate of shedding snails. The function f in Eq. (3) denotes the mating probability. As derived by May (1977), this function for a dioecious species with a 1:1 sex ratio which is monogamous and distributed in a negative binomial manner with clumping parameter k within the human population is given by, h i Z 2p ð1  cos qÞ fðM ; kÞ ¼ 1  ð1  aÞð1þkÞ =2p dq (6) 0 ð1 þ a cos qÞð1þkÞ where a ¼ M/(k þ M). Note that for k small, as it usually is, f / 1 for moderate-to-high mean worm burdens.

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For a polygamous parasite the equivalent expression derived by May (1977) is somewhat simpler (as plotted in Fig. 11), fðM; kÞ ¼ 1  ½1 þ M =2kð1þkÞ

(7)

For the model defined in Eq. (3), the basic reproduction number R0 is defined as, 1 R0 ¼ T1 T2 f 2

(8)

This expression for R0 clearly illustrates how all the various life cycle population parameters (birth, death and infection rates) influence its magnitude. Hairston attempted to measure all these individual parameters with limited success. As illustrated in a later section, a more practical approach is to estimate R0 directly, from epidemiological data on age intensity of infection profiles and from reinfection data posttreatment, by fitting the mathematical models to the observed trends. As a preliminary approach, if we assume that f z 1, this model permits analytical exploration, to show that two equilibria exist, one of parasite extinction M* ¼ 0 and another for stable endemic infection M* > 0, where,    1 1  M ¼ T1 T2 f  1 (9) T2 f 2 2 Put more simply, in terms of R0 this is, M  ¼ T1 ðR0  1Þ=R0

(10)

This equation confirms that R0 > 1 for parasite persistence. When the mating probability, f, is less than unity in value, Eq. (3) cannot be solved analytically, given the complexity of the expression defined in Eq. (6). However, it can be shown that at equilibrium the system has three possible solutions, two of which are stable and represent parasite extinction (M* ¼ 0) and stable endemic infection M* > 0. The two are separated by an unstable state M*u. This is the transmission breakpoint first identified by Macdonald. As noted earlier, he assumed a Poisson distribution of parasites which meant that the unstable point was well above zero in value. For the observed negative binomial patterns, this unstable mean worm load is very low and close to zero for k < 0.1. A graphical illustration of this is given in Fig. 11, which plots the probability mating term, f, as a function of the value of k and the mean equilibrium worm burden M*, to show that the

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value is close to unity for high degrees of aggregation (k ¼ 0.01), even when the mean worm burden is very low. One useful result that can be derived from this model with a mating probability is how the percentage of unfertilized eggs may increase as the mean worm burden falls due to females not encountering a male, under, for example, intense MDA (May, 1977; Anderson and May, 1991). In the case of polygamy (the most pessimistic assumption, since only one male in the host could fertilize many females, while for monogamy each female must find its own male), if x is the fraction of people only passing unfertilized eggs (where y ¼ 1x is the fraction passing fertilized eggs), the relationship between x, y and the degree of parasite aggregation k given a 1:1 sex ratio in births is, n h ik o. 1 x ¼ 1  y  2ð1  yÞk  1 y (11) As MDA programmes expand in terms of coverage, plotting this relationship will give an idea of at what level of the prevalence of unfertilized eggs is the system on the point of transmission cessation. As noted elsewhere, the predicted relationship well matches observed trends for intestinal helminths in South Korea (Anderson and May, 1991). The observation of high f values at low k values and low mean worm burdens raises the question of whether it is worth including the mating probability term in mathematical models, given the complexity of the functions, to gain insights into the dynamics of transmission. For general insights, the answer is probably no. However, for numerical studies of more complex models that include age structure, age-dependent transmission and exposure to infection and/or acquired immunity, the answer is yes. The reason for the latter is the importance of the unstable equilibrium when assessing the impact of MDA programmes. Once crossed, by sufficient drug coverage, the parasite extinction state is the attractor in the system and hence the boundary of drug coverage to place the dynamical system into the region of extinguishing parasite transmission. As shown later, it is possible to plot the value of the unstable state as a function of the proportion of different age groups of the human community effectively treated per annum (or other time interval). In what follows in this subsection we examine the general analytical insights to be gained when it is assumed that f ¼ 1 for all values of M* such that no unstable boundary exists. The transmission boundary between the only two equilibria, M* ¼ 0 and M* > 1, is R0 < 1.

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Under this assumption the analytical solution for M* > 0 for the hybrid deterministic model, given that the distribution of parasite numbers per person is given a probability distribution assumption (negative binomial) but this distribution is treated in a deterministic manner since k is assumed to be independent of M(t), is given by, h 1 i. M  ¼ k R0kþ1  1 ð1  zÞ (12) Here z is the density-dependent adult worm fecundity term given by z ¼ exp(g) (Eq. (2)). This shows that the average worm load at equilibrium is simply determined by the magnitude of R0, the degree of worm aggregation, k, and the severity of density dependence on adult worm fecundity, z. Given the assumption that k is independent of M*, then the equilibrium prevalence of infection P* is given by Eq. (1). Eq. (12) can be used to obtain rough estimates of R0, given observations on the overall mean worm burden, the degree of parasite aggregation and the severity of density dependence on fecundity (both the latter from the autopsy study data collected by Cheever, 1968). However, this approach is not advised given the convex shapes of observed age intensity profiles for schistosomes which reflect age-dependent exposure and/or acquired immunity. The other general insights that can be derived from this simple model concerns how the system might behave under perturbation from the equilibrium by, say, MDA. For those who design and implement MDA programmes, the key questions are what fraction must be treated per unit of time and what might be an optimal interval between rounds of treatment? If we assume the system is in endemic equilibrium, then the time, tf, taken to reach a fraction f of the equilibrium abundance in growth to the equilibrium is approximately given by, lnð1  f Þ tf ¼  ¼ lnð1  f ÞL m1

(13)

where 1/m1 ¼ L is the adult parasite life expectancy in the human host. This approximation shows clearly that ‘bounce back’ time is tightly controlled by the adult worm life expectancy. For the major helminths of humans, which include onchocerciasis, lymphatic filariasis, schistosomes and soil-transmitted helminths such as Ascaris lumbricoides, in order, adult worm life expectancies go from long (10 years or more) to short (around 1 year), with the schistosomes in the middle at around 3.5e5 years (Table 4). Bounce back post a

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round of treatment will therefore be fast for Ascaris, medium for schistosomes and slow for the filarial worms. The actual time required to return to the pretreatment equilibrium will depend on the fraction of the total population treated, but Eq. (13) suggests that for schistosomes, post a round of treating a high fraction of the population, it will take a number of years to recover to pretreatment levels. As such, effective control where a high fraction are treated could be induced by biennial or longer-interval treatments. In other words, the control by MDA of helminth infection would be easier for filarial worms (if efficacious macrofilaricidal drugs existed) and schistosomes than for soil-transmitted helminths. A further insight concerns the faction to be treated to reduce the effective reproductive number, R, to below unity in value to cross a transmission threshold. The fraction, gc, that must be treated per unit of time with a drug of efficacy h (fraction of worms killed) to achieve this is given by, 
 expð1  R0 Þ gc ¼ 1  h (14) L This equation makes clear that the task of breaking transmission (R < 1) is simply determined by the magnitude of R0 reflecting the intensity of transmission in a defined habitat, adult parasite life expectancy, L (defined in the same time units as gc), and drug efficacy, h (see Table 2). A plot of the impact of the treated fraction against prevalence and average intensity is shown in Fig. 12. As the frequency of treatment rises, mean intensity declines in an approximate slow exponential manner, but prevalence shows little change until the fraction treated approaches the critical value (Eq. (14)).

Figure 12 The predicted impact of the fraction treated with chemotherapy, gc, on the prevalence and mean intensity of infection (Eq. (14)). [R0 ¼ 2, k ¼ 0.024, h ¼ 0.9, 1/m1 ¼ 7 years, z ¼ 0.99].

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Simple deterministic models can also be employed to examine different ways to deliver treatment in a population where infection is endemic. So far, Eq. (14) is based on the assumption that the people treated at each round of chemotherapy are chosen at random. They could be chosen according to worm burden (or egg output), in what is termed selective chemotherapy (Warren, 1982). Anderson and May (1982) have examined this problem and derived an expression for the average proportion of the mean worm burden M* killed, Km, by a single round of selective treatment, where, (

ðkþ1Þ ) 1 Km ¼ ga h 1  zð1  aÞ 1 þ ð1  zÞM k ,( (15)
k ) 1 1  ð1  aÞ 1 þ ð1  zÞM  k Here ga is the average proportion treated, under the assumption of a negative binomial distribution of worms per person, and its definition depends on the function chosen to define how selective the treatment programme is. Anderson and May chose a continuous function, g(i), to define the probability that a person with i worms gets treated, where,
  i gðiÞ ¼ w 1  ð1  aÞexp  (16) I Here w and a are constants (both 0) or remain constant (as ¼ 0) with age. The value of a0 was estimated to be equal to zero for Cameroon (Filipe et al., 2005). 2.1.1.2 Mating probability of female adult worms

Following May (1977), the probability that a female worm is mated depends on: (1) the sex ratio (assumed to be 1:1 in Onchocerca volvulus, Schulz-Key, 1990); (2) the sexual system (assumed to be polygamous, Schulz-Key and Karam, 1986) and (3) the distribution of adult worms among the human host population (assuming that males and females are distributed together following a negative binomial with overdispersion parameter kW, Duerr et al., 2004). The mating probability, f(W,kW), is an increasing (positive density-dependent) function of mean female (nonfertile, N plus fertile, F) worm burden, W ¼ N þ F, whose shape and rate of increase (from 0 to 1) is inversely influenced by the value of kW (the smaller the value, the stronger the degree of parasite overdispersion and the faster the mating probability increases with mean worm burden) (Anderson and May, 1985),   W ðkW þ1Þ f½W ; kW  ¼ 1  1 þ : (4) kW 2.1.1.3 Excess human mortality

In addition to excess mortality due to onchocerciasis-associated blindness (Kirkwood et al., 1983), there is a density-dependent relationship between microfilarial load and relative risk of mortality (Little et al., 2004b; Walker et al., 2012b). Some versions of EPIONCHO have incorporated the relationship between microfilarial load and blindness incidence (Little et al., 2004a), and between microfilarial load and human excess mortality (Little et al., 2004b; Walker et al., 2012b) in order to link the core infection model with a disease model for the estimation of disease burden and costeffectiveness of control interventions (Turner et al., 2014a, 2014b). Both relationships are parameterized using: (1) the microfilarial load lagged by 2 years (reflecting that blindness and excess mortality are associated with

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past rather than current infection) and (2) the mean number of microfilariae per skin snip (as opposed to per mg of skin), which is designated M 0 (as opposed to M ) and is derived from the modelled microfilarial load (per mg of skin) assuming an arithmetic mean skin snip sample weight of 1.7 mg estimated using data from Collins et al. (1992). The relative risk of blindness associated with infection is given by a simple log-linear function of M 0 (lagged by 2 years, i.e., M 0 ðt 2Þ, but with the time dependence omitted for brevity). By contrast, relative risk of mortality associated with (past) infection is nonlinearly related to M 0 and host age, a, by the expression exp½ f ðM 0 Þaw  where, f ðM 0 Þ ¼

b1 M 0b3 ; ð1 þ b2 M 0b3 Þ

(5)

and parameters b ¼ {b1,b2,b3} and w were estimated by fitting to the longitudinal OCP dataset collected from 1974 through 2001 (Walker et al., 2012b). Results indicate that for a given microfilarial load the relative risk of mortality is statistically significantly greater in those aged less than 20 years (Fig. 3).

Figure 3 Observed and fitted relative mortality risk with Onchocerca volvulus microfilarial skin load according to age group. Individuals 0. In foci with

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seasonal transmission, the endemic equilibrium conditions given in Section 2.1.3.1 correspond to the annual biting rate (ABR), the instantaneous biting rate averaged over the year, Z1 ABR ¼

BRðtÞdt

(17)

0

2.1.4 Posttreatment parasite dynamics 2.1.4.1 Ivermectin

In EPIONCHO, the dynamics of skin microfilarial load and of the proportion of adult female worms producing live microfilariae following ivermectin treatment with the standard dose of 150 mg/kg are modelled according to the results of the systematic review, metaanalysis and modelling presented by Bas
an ~ez et al. (2008). Briefly, microfilaridermia is reduced by half after 24 h, by 85% after 72 h, by 94% after 1 week and by 98%e99% after 1e2 months (microfilaricidal effect), the latter also corresponding to the time when the fraction of females harbouring live microfilariae is at its lowest (embryostatic effect), reduced by around 70% from its original value. After the first 2 months following treatment, microfilariae gradually reappear in the skin following the resumption of microfilarial production by the female worms. The microfilaricidal effect (excess mortality of microfilariae due to treatment), sM1 ðsÞ, is modelled as, sM1 ðsÞ ¼ ðs þ vÞu ;

(18)

where s is time after treatment, v is a constant added to time after treatment to allow for a very large, yet finite, microfilaricidal effect at the point of treatment and u is a shape parameter for the per capita death rate of microfilariae following treatment. The embryostatic effect (the treatmentinduced per capita rate at which fertile females become nonfertile), l1(s), is modelled as, l1 ðsÞ ¼ lMAX expð4sÞ; 1

(19)

where lMAX is the maximum rate of treatment-induced sterility, and 4 is the 1 rate of decay of this effect with time after treatment. At present, EPIONCHO does not assume that standard dose ivermectin treatment has a direct macrofilaricidal (killing of the adult worms) effect, but an antimacrofilarial effect (a cumulative reduction in the rate of microfilarial production by adult females of 7% per dose) has been assumed in more

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recent versions (Turner et al., 2014c) to account for results presented by Gardon et al. (2002) and Cupp and Cupp (2005). The modelling of this cumulative, antimacrofilarial effect is described in Section 2.1.5. 2.1.4.2 Moxidectin

The temporal dynamics of skin microfilarial loads from the ivermectin treatment arm in the phase II moxidectin study (Awadzi et al., 2014) were within the range observed by Bas
an ~ez et al. (2008) (Fig. 4A). Moxidectin treatment was assumed to exert the same types of effects on the parasite as ivermectin. Therefore, moxidectin’s effects were parameterized by fitting the functions used by Bas
an ~ez et al. (2008) (Eqs (18) and (19) above) to the percentage reduction in skin microfilarial densities from pretreatment, measured 8 days, 1, 2, 3, 6, 12 and 18 months after a single dose of 8 mg moxidectin (91e186 mg/kg) (Fig. 4B).

Figure 4 The dynamic effect of a single dose of ivermectin (A) and moxidectin (B) on skin microfilarial load. The data points are derived from skin microfilarial loads (the mean of four microfilarial counts) collected from (A) 45 individuals treated with ivermectin and (B) 38 individuals treated with moxidectin, as part of the Phase II clinical safety trial of moxidectin for the treatment of onchocerciasis (Awadzi et al., 2014). The effect of a single dose of ivermectin previously fitted to microfilarial load data ~ez et al. (2008) is shown as a solid line in (A). The microfilarial dynamics induced by Bas
an by ivermectin are not re-estimated here and hence provide a validation of the previous parameterization. The dynamical effect of moxidectin was fitted to the trial data on ~ez microfilarial loads from treated participants using the same approach as in Bas
an et al. (2008) and is shown as a solid red line in (B). Error bars are the 95% confidence intervals which in some circumstances were narrower than the plotted data point and so are not discernible (Turner et al., 2015b).

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2.1.4.3 Coverage and adherence

Whilst (therapeutic) coverage describes the proportion of the population treated at a particular treatment round, adherence (compliance) describes the degree to which individuals adhere correctly to the treatment schedule. In EPIONCHO, the human host population (and subsequently the parasite population) is partitioned into different treatment groups according to how regularly they receive ivermectin treatment: (1) a fully compliant group who takes treatment every round; (2) two semicompliant groups who take treatment every other round alternately, and (3) a systematically noncompliant, fourth, group who never takes treatment. 2.1.4.4 Infection dynamics

The following set of partial differential equations (omitting time and age dependencies on the left hand side for simplicity) describes the dynamics of infection intensity in human and vector hosts under treatment with ivermectin (or moxidectin), with subscript s denoting host sex and d denoting treatment adherence category, vNs;d vNs;d 1 þ ¼ m b Us ða pÞdH ½Lðt pÞLðt pÞexpð mH pÞ 2 vt va þ ½l0 þ l1 ðsÞFs;d ðt; aÞ  ð6þ sW ÞNs;d ðt; aÞ

(20)

vFs;d vFs;d þ ¼ 6 Ns;d ðt; aÞ  ½l0 þ l1 ðsÞ þ sW Fs;d ðt; aÞ vt va

(21)

  vMs;d vMs;d þ ¼ f Ws;d ðt; aÞ; kW εjd ðtÞFs;d ðt; aÞ vt va  ½sM0 þ sM1 ðsÞMs;d ðt; aÞ

(22)

    vLs;d vLs;d þ ¼ b Us ðaÞdV Ms;d ðt; aÞ Ms;d ðt; aÞ  sL Ms;d ðt; aÞ Ls;d ðt; aÞ vt va (23) Z XX qs hd rðaÞUs ðaÞLs;d ðt; aÞda (24) LðtÞ ¼ s

d

a

Function l1(s) denotes the excess per capita rate at which fertile females become nonfertile following treatment (embryostatic effect), with s being the time since last treatment (Eq. (19) above); f[Ws,d(t,a),kW] the mating probability as described in Eq. (4) and Section 2.1.4.5; sM1 ðsÞ is the excess per capita death rate of microfilariae following ivermectin or moxidectin

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treatment (microfilaricidal effect, Bas
an ~ez et al., 2008, Eq. (18)), and jd(t) the average value of the factor modifying (decreasing) female worm fertility in adherence group d when a cumulative effect of treatment is considered (Section 2.1.5). 2.1.4.5 Mating probability

It is assumed that the distribution of adult worms among hosts of the same adherence group is adequately described by a negative binomial distribution (NBD) with mean (female) worm load, Ws,d(t,a), and overdispersion parameter, kW. Assuming that: (1) male and female worms are distributed together, (2) they are polygamous, i.e., a single male has the potential to fertilize all females within a host (Schulz-Key and Karam, 1986; Hildebrandt et al., 2012) and (3) they have a balanced (1:1) worm sex ratio, the probability that a female worm is mated (May (1977) is,     Ws;d ðt; aÞ ðkW þ1Þ f Ws;d ðt; aÞ; kW ¼ 1  1 þ : (25) kW Note that the degree of overdispersion of the adult worm population (inversely measured by the value of kW) is assumed to be unaffected by treatment. 2.1.4.6 Cumulative effect of treatment on female worm fertility

At any time after the start of a simulated treatment programme, the worm population in adherence group d comprises worms previously exposed to different numbers of ivermectin (or moxidectin) treatments. This is because (1) worms continually infect hosts throughout the treatment programme and (2) hosts in different adherence groups receive different numbers of treatments at different times. If ivermectin or moxidectin are assumed to suppress cumulatively the fertility of female O. volvulus, then the average reduction in fertility of the worm population will critically depend on the fraction of worms exposed to different numbers of treatments. To this end, n was defined as the maximum number of previous exposures to ivermectin, and n þ 1 submodels were formulated to track worm populations acquired during discrete time intervals throughout the course of a simulated treatment programme. Note that n varies among adherence groups (for example, for systematic noncompliers n ¼ 0), and exposure group (number of treatments to which worms have been exposed, j), as some worms, acquired after the final treatment, will be unexposed to treatment ( j ¼ 0). (The possibility of unexposed worms gives rise to the n þ 1 (as opposed to n) submodels.)

276

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an

Consider a treatment programme starting at time s0 (that is, the first dose of ivermectin or moxidectin is administered at time t ¼ s0 ). Worms exposed to all n treatments ( j ¼ n) are those that were acquired at time t < s0 . By redefining the rate of establishment of female adult worms as (first term of the right hand side of Eq. (20)), 1 Ls ðt; aÞ ¼ m b Us ða pÞdH ½Lðt pÞLðt pÞexpð mH pÞ; 2

(26)

the rate of establishment of adult female worms that will be exposed to all n treatments (i.e., those acquired before the commencement of treatment) in adherence group d can be expressed as,  Ls ðt; aÞ for 0 < t < s0 Ls;d;j¼n ðt; aÞ ¼ (27) 0 otherwise: By contrast, unexposed female worms ( j ¼ 0) are those acquired after the last treatment which, if the n treatments were administered at frequency f (where f ¼ 1 represents annual treatment and f ¼ 2 represents biannual treatment), indicates that infection occurred at t > s0 þ (n  1)/f. (In this chapter we explore an annual (ivermectin, moxidectin) or a 6-monthly (ivermectin) treatment frequency.) That is,  Ls ðt; aÞ for s0 þ ðn  1Þ=f < t < N Ls;d;j¼0 ðt; aÞ ¼ : (28) 0 otherwise: It follows that the rate of establishment of adult worms exposed to the intervening numbers of treatments j ¼ 1, 2,., n  1 is given by,  Ls ðt; aÞ for s0 þ ðn  1  jÞ=f < t < s0 þ ðn  jÞ=f Ls;d;j¼0 ðt; aÞ ¼ 0 otherwise: (29) These conditions are used to define partial differential equations for the mean number of female adult worms, Ws,d,j(t, a), in each exposure group j ¼ 0, 1,., n, vWs;d;j ðt; aÞ vWs;d;j ðt; aÞ þ ¼ Ls;d;j ðt; aÞ  sW Ws;d;j ðt; aÞ: vt va

(30)

Note that for the purposes of tracking adult worms exposed to different numbers of treatments, the fertility status (fertile/nonfertile) of female worms is not distinguished. Taking the expectation of Ws,d,j(t,a) with respect to host age a and sex s yields,

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X Z Ws;d ðtÞ ¼ q rðaÞWs;d;j ðt; aÞda; s s

(31)

a

where r(a), the probability density function of host age, a, is rðaÞ ¼

mH expðmH aÞ : 1  expðmH am Þ

(32)

Summing over treatment exposure groups gives the mean number of worms per host in adherence group d, Wd ðtÞ ¼

j¼n X

Wd;j ðtÞ:

(33)

j¼0

The fraction of the total female worm population in treatment exposure group j, denoted ud,j(t), is now given by, ud;j ðtÞ ¼

Wd;j ðtÞ : Wd ðtÞ

(34)

Each subsequent exposure to ivermectin or moxidectin (after the first exposure) was assumed to cause a 7% reduction in female worm fertility (varied in the sensitivity analysis between 1% e weak cumulative effect e and 30% e strong cumulative effect), such that the fertility of female worms exposed to j treatments, Jj, is given by,  1 for j ¼ 0 (35) Jj ¼ ð1  zÞj1 for j > 0 with parameter z ¼ 0.01 (minimum value), 0.07 (nominal value) or 0.3 (maximum value) (Table 2). The maximum value was motivated by the findings of the modelling study conducted by Plaisier et al. (1995), fitting to data from Alley et al. (1994) on the first community trial of annual ivermectin treatment in the then highly hyperendemic focus of Asubende in Ghana using ONCHOSIM. Note that for j ¼ 0 (and for j ¼ 1), Jj ¼ 1 indicates that EPIONCHO assumes that worms previously unexposed to treatment ( j ¼ 0), or exposed to a single, first dose ( j ¼ 1) have, respectively, full fertility, or the potential to regain full fertility. Subsequent treatments may cause a cumulative reduction of female worm fertility. The average value of the factor modifying the per capita microfilarial production rate of (fertile) female worms in adherence group d, jd(t), is

n

f s l1(s)

lMAX 1 4 sM1 ðsÞ n

Time since last ivermectin (or moxidectin) treatment Excess per capita rate at which fertile females become non-fertile following treatment (embryostatic effect) Maximum rate of treatment-induced sterility Rate of decay of treatment-induced sterility with time after treatment Excess per capita death rate of microfilariae following ivermectin treatment (microfilaricidal effect) Constant added to time after treatment to allow for a very large, yet finite, microfilaricidal effect at the point of treatment Shape parameter for the per capita death rate of microfilariae following treatment

0 for those hosts never taking treatment to 15 (annual) or 30 (biannual) for those taking all treatments Ivermectin: Annual or biannual Moxidectin: Annual years

Turner et al. (2013a)

e

lMAX expð4 sÞ year1 1

Bas
an ~ez et al. (2008)

Ivermectin: 32.4 year1 Moxidectin: 462 year1 Ivermectin: 19.6 year1 Moxidectin: 4.83 year1 (s þ n)u year1

Turner et al. (2015b)

Ivermectin: 0.0096 years Moxidectin: 0.04 years

Turner et al. (2015b)

Ivermectin: 1.25 Moxidectin: 1.82

Turner et al. (2015b)

Turner et al. (2013a)

Turner et al. (2015b) Bas
an ~ez et al. (2008)

~ez et al. M.G. Bas
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u

Maximum number of previous exposures to ivermectin by worms in a given adherence group Frequency of treatment

278

Table 2 Definition and values of parameters and variables for ivermectin or moxidectin treatment effects in EPIONCHO Symbol Definition of variables and parameters Expression, average value and units Sources

Ls,d,j(t,a)

Ws,d,j(t,a)

ud,j(t) Jj z

jd(t)

Time at which treatment programme starts The rate of establishment of female adult worms at time t in hosts of age a, sex s, treatment adherence group d and exposure group (number of treatments to which worms have been exposed) j Mean number of female adult worms at time (t) in hosts of age (a); sex s, treatment adherence group d and treatment exposure group j The fraction of the total worm population in treatment exposure group j The fertility of adult worms in treatment exposure group j The per dose reduction in fertility caused by treatment when a cumulative effect is assumed The average value of the factor modifying female worm fertility in adherence group d

e

e

Eqs (27)e(29)

Turner et al. (2013a)

Eqs (30) and (31)

Turner et al. (2013a)

Eq. (34)

Turner et al. (2013a)

Eq. (35)

Turner et al. (2013a)

0.01, 0.07, 0.30

Turner et al. (2015b)

Eq. (36)

Turner et al. (2013a)

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s0

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calculated using the fraction of the total worm population in each treatment exposure group ud,j(t) (Eq. (32)) and Jj (Eq. (33)), jd ðtÞ ¼

j¼n X

Jj ud;j ðtÞ:

(36)

j¼0

Definitions and values of parameters for treatment effects in EPIONCHO are given in Table 2. 2.1.5 Model outputs 2.1.5.1 Intensity of infection

In this chapter infection intensity refers to microfilarial load in those aged 20 years (to facilitate comparison with ONCHOSIM results, which use the community microfilarial load [CMFL]). However, CMFL (Remme et al., 1986) refers to a geometric mean microfilarial load per skin snip in those aged 20 years, rather than to an arithmetic mean microfilarial load per mg of skin, as used in EPIONCHO. Microfilarial load in those aged 20 years is calculated from Eq. (22) above by integrating over age (from a ¼ 20 to a ¼ am, the maximum human age of 80 years recorded in the Cameroonian datasets) and summing over sex s and adherence group d, M ðtÞ20 ¼

XX s

d

a¼a Z m

r0ðaÞMs;d ðt; aÞda;

qs hd

(37)

a¼20

r0 (a)

is the probability density function of host age between 20 and where am ¼ 80 years, r0 ðaÞ ¼

mH expð mH aÞ ; ½expð mH 20Þ  expðmH am Þ

(38)

and mH is the per capita death rate of humans. 2.1.5.2 Prevalence of infection

According to the current version of EPIONCHO, overall microfilarial prevalence in adherence group d (pd(t)) is derived by using a relationship between prevalence and microfilarial load at the community level in Cameroon described in Bas
an ~ez and Boussinesq (1999) and reparameterized by Turner et al. (2014a). This relationship assumes that skin microfilarial load per person is distributed according to an NBD with mean Md(t) and overdispersion parameter kM. The best fit to the microfilarial prevalence

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vs. intensity relationship was obtained when kM was allowed to be a function of the mean. Assuming that the degree of microfilarial overdispersion does not depend on adherence group, pd(t) is given by,  kM ½Md ðtÞ Md ðtÞ (39) pd ðtÞ ¼ 1  1 þ kM ½Md ðtÞ where Md(t) is given by, Md ðtÞ ¼

X s

Z rðaÞMs;d ðt; aÞda;

qs

(40)

a

and kM is given by, kM ½Md ðtÞ ¼

k0 Md ðtÞ : 1 þ k1 Md ðtÞ

(41)

The overall population prevalence at time t was obtained by summing pd(t) across adherence groups, X hd pd ðtÞ: (42) pðtÞ ¼ d

2.2 ONCHOSIM ONCHOSIM is a stochastic, individual-based model for the transmission and control of onchocerciasis. This model describes a dynamic human population, consisting of a discrete number of individuals. The computer program tracks change in the composition of the human population and in the infection status of each individual in the population over time (t, in 1month time steps) and age (a). The transmission of infection between individuals is captured by a deterministic submodel, accounting for the Simulium fly population dynamics and the fate of the parasite in the fly. The model was developed in close collaboration with the OCP (Plaisier et al., 1990). A formal description of the model, presented previously by Habbema et al. (1996a) and Coffeng et al. (2014a), is included here to facilitate access to the readers and to allow a direct comparison with EPIONCHO. Table 3 lists ONCHOSIM’s parameters, notation, values and sources. 2.2.1 Human population demography The human population dynamics is governed by birth and death processes. We define F(a) as the probability to survive to age a (apart from excess

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282

mortality due to onchocerciasis-associated blindness). The values used are as follows:

age (a) F(a)

0 1.000

5 0.804

10 0.772

15 0.760

20 0.740

30 0.686

50 0.509

90 0.000

Survival at intermediate ages is obtained by linear interpolation. The expected number of births (per year) at a given moment t is given by, Rb ðtÞ ¼

na X

Nf ðk; tÞ$rb ðkÞ;

(43)

k¼1

where Nf (k,t) is the no. of women in age group k at time t, rb(k) is the annual birth rate in age-group k: 0.109 babies per year for women between 15 and 20 years; 0.300 between 20 and 30 years; 0.119 between 30 and 50 years; 0.0 for all other ages and na is the number of age groups considered. Each month, Rb(t) is adapted according to the number of women and their age-distribution. Depending on the size of the initial population and birth and death rates, the human population may increase. The program allows the specification of a ‘maximum population size’, in order to keep the size population representative for the type of community being simulated and limit computation time. As soon as the simulated population exceeds this maximum, a random fraction of the population is sampled and removed from the population. This can be thought of as emigration. In most published model applications, the model is used to simulate a village population with a maximum size of 440. This value is well within the ranges (23e828, with a median of 171), recorded in the OCP database (O’Hanlon et al., 2016). The population distribution resulting from the aforementioned parameters closely follows the age distribution in Sub-Saharan Africa as shown elsewhere (Coffeng et al., 2014a). 2.2.2 Parasite population regulation in humans 2.2.2.1 Exposure to fly bites

The number of bites mbri(t) a person i gets in month t (in the absence of vector control) is given by, mbri ðtÞ ¼ MbrðtÞ Exi

(44)

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where Mbr(t) is the number of bites in month t (Jan., Feb., .) for a person with relative exposure equal to 1. The relative exposure Exi is calculated as, Exi ¼ Exaðai ; si Þ$Exii ;

(45)

where Exa(ai,si) is the relative exposure of a person of age a and sex s, assumed to be zero at birth, to increase linearly with age between the ages of 0 and 20 years until a maximum of 1.0 for men and 0.7 for women, and to remain constant from 20 years onwards, and, Exii w Gamma(1.0,aExi) is the exposure index of person i. Exii is assumed to follow a gamma distribution with mean 1.0 and shape and rate equal to aExi. The exposure index of a person remains constant throughout their lifetime. For selected West African villages (within the OCP), estimated aExi values vary between 1.6 and 12.7 (for the simulations illustrated in this chapter values were 1.0 or 3.5; see Table 3 and Coffeng et al., 2014a). Mbr(t) values were obtained from six years of blackfly collections near the village of Asubende (Ghana) conducted between 1978 and 1985. In this site with perennial transmission, monthly biting rates of, on average, 2570 bites per person, varying from 1500 in March to 3750 in November had been found. For the actual biting rates (Mbr(t)) inside the village Asubende, these figures were multiplied by a factor (called the relative biting rate) of 0.95. (Since we have no measurements of biting rates actually experienced by villagers, we have arbitrarily defined a relative biting rate of 1.0, i.e., a mean Mbr ¼ 2750 as the biting rate that results in a geometric mean number of microfilariae (mf) per skin snip (ss) of 100 in a hypothetical village where all its inhabitants are permanently characterized by a relative exposure of 1.0.) Assuming the same seasonal pattern for other villages, relative biting rates have been estimated to vary from 0.4 to 0.9. 2.2.2.2 Parasite establishment

The monthly transmission potential (MTP) in ONCHOSIM is defined as mbr(t)  lr(t). When accumulated over a year, we get the ATP. If during a blood meal by a simuliid fly in month t, lr infective larvae are released on average, the force-of-infection foii(t), defined as the expected number of new adult parasites acquired by person i in month t, is calculated as, foii ðtÞ ¼ mbri ðtÞ lrðtÞ sr;

(46)

where sr is the success ratio, namely, the fraction of inoculated L3 larvae succeeding in developing to adult male or female worms, with value

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sr ¼ 0.0031. An average male:female sex ratio of 1:1 is assumed (SchulzKey, 1990). In month t, a person i is assumed to become infected according to a Poisson process with rate foii(t). ONCHOSIM assumes that there is no density dependence in this process. 2.2.2.3 Mating probability of female adult worms

Chance processes determine the number of male and female worms present in each human individual. The degree of parasite overdispersion in the human population depends on exposure heterogeneity assumptions. The reproductive lifespan of male and female parasites is a random variable, Tl w Weibull(muTl,aTl), with mean muTl ¼ 10 years and shape aTl ¼ 3.8. (For readers used to other commonly used parameterizations of the Weibull distribution in terms of shape k and scale l, shape k is aTl (as described in this document) and scale is l ¼ muTl/G(1 þ 1/aTl).) The microfilarial productivity r(a,t) of a female worm of age a in month t is calculated as, rða; tÞ ¼ RðaÞ fmðtÞ;

(47)

where R(a) is the potential microfilarial productivity of a female worm of age a (in years): R(a) ¼ 0 for 0  a < 1; R(a) ¼ 1 for 1  a < 6; R(a) ¼ 1  ((a  6)/15) for 6  a < 21; R(a) ¼ 0 for a > 21 and fm(t) mating factor at time t. Quantifying R(a) ¼ 0 for 0  a < 1 is equivalent to assuming an intrinsic latency period (i.e., the time needed for a parasite to become mature and be able to reproduce) of exactly 1 year for all male and female worms. This duration is user-specified in ONCHOSIM; other values can be given and it can also be specified as a continuous probability distribution. Similarly other values can be specified for the potential microfilarial production by female worms once the intrinsic latency period has elapsed. However, the intrinsic latency period is not necessarily equal to the prepatent period, as the latter is defined as the time from the moment of infection until patency (microfilarial infection) can be detected in the skin. The mating factor is defined as follows. To continue microfilarial production, a female worm must be inseminated each rc months (rc ¼ reproductive cycle ¼ 3 per year). If insemination takes place less than rc months

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ago, then fm(t) ¼ 1. Otherwise, the probability of insemination or reinsemination Pins(t) in month t is given by,

Pins ðtÞ ¼ Wm ðtÞ Wf ðtÞ if Wm < Wf ; (48) Pins ðtÞ ¼ 1 if otherwise where W(t) is the number of male (Wm) or female (Wf) parasites in the human host at time t. If no insemination takes place then fm(t) ¼ 0 and the female worm has a new opportunity in month t þ 1. If insemination occurs in month ti then fm(t) ¼ 1 during ti  t < ti þ rc. In ONCHOSIM there is one additional parameter influencing the mating probability Pins. This parameter is called male potential and is multiplied with the male:female worm sex ratio. Assigning a high value to this male potential (e.g., 100) implies that mating (if required) will always take place if there is at least one adult male worm. The skin microfilarial density sl(t) at time t is calculated by accumulating the microfilarial production of all female parasites over the past Tm months, slðtÞ ¼ cw$elðtÞ; elðtÞ ¼

ni X Tm  1 X rj aj  x; t  x ; Tm j¼1 x¼1

(49) (50)

where rj(aj,t) is the microfilarial productivity of a female worm j of age aj in month t el(t) is the effective parasite load at time t. This intermediate variable can be interpreted as the total number of female worms that have contributed to skin microfilarial counts at time t, weighted for their average microfilarial productivity over the past Tm months, cw is the average contribution of an inseminated female worm at peak fecundity (R ¼ 1) to the skin microfilarial density: cw ¼ 7.6 microfilariae/worm. (Instead of a linear relationship between sl and el, other functional relationships can be chosen, e.g., a saturating function.) Tm is the (fixed) microfilarial lifespan, with an assumed value Tm ¼ 9 months (Plaisier et al., 1995), and ni is the number of parasites alive during at least one of the months t  1,.,t  Tm. 2.2.2.4 Microfilarial counts in skin snips

The expected number of microfilariae in a skin snip (of 2 mg) is given by, ssðtÞ ¼

ni Tm  X cw X dj rj aj  x; t  x ; Tm j¼1 x¼1

(51)

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where, dj is the dispersal factor of female parasite j. This is a random variable drawn for every ‘newborn’ worm and accounts for differences in the contribution of female worms to the microfilarial density at the standard site of the body where snips are taken (hips in Africa). We assume that dj follows an exponential distribution, dj w Expo(1.0). The actual (observed) number of microfilariae per skin snip (mf/ss) at examination time t, ssobs(t) follows a Poisson distribution, ssobs(t) w Poisson(ss(t)) e although other discrete probability functions (e.g., geometric) can be used. At each epidemiological survey two snips (or any other number as appropriate) are taken from all simulated persons. The results of such a survey are postprocessed to arrive at age- and sex-specific and standardized microfilarial prevalences. 2.2.2.5 Blindness and excess human mortality

The event of a person going blind at age a (in months) depends on the cumulative parasite load (elc) of a person, elcðaÞ ¼

a X

elðxÞ:

(52)

x¼0

Each person has a threshold level elc (denoted as Elc) at which a person goes blind. Elc follows a Weibull probability distribution, Elc w Weibull(muElc,aElc), with mean muElc ¼ 10,000 and shape aElc ¼ 2.0. Person i goes blind at age a when, elci ðaÞ  Elci > elci ða  1Þ:

(53)

At that moment the remaining lifespan at age a is reduced by a factor rl, which follows a uniform distribution on [0,1] (hence on average rl ¼ 0.5). (Any other probability distribution defined on [0,1] can be used, e.g., a beta distribution.) 2.2.3 Parasite population regulation in vectors 2.2.3.1 Parasite establishment

On the basis of fly-feeding experiments conducted in the OCP (analyzed by Bas
an ~ez et al., 1995 and summarized by Soumbey-Alley et al., 2004), the following expression for the relation between L1 uptake (lu) and skin microfilarial density in humans (sl) was derived by Plaisier et al. (1991b),   lu ¼ a$ 1 eb$sl $ 1þ ec$sl ; (54) with a ¼ 1.2, b ¼ 0.0213 and c ¼ 0.0861 (the initial slope of this relationship equals 2ab and, therefore, the maximum probability of parasite establishment is 0.0511 per microfilaria in a skin snip; the maximum number of larvae

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establishing within the thoracic muscles of the fly is 1.2). Other functional relationships can also be defined. For instance, we can set c ¼ 1.0 to simulate a situation with less pronounced negative density dependence in transmission (which may be reflective of the situation in forest areas). This alternative parameter value results in a less concave shape of the function, while the slope in the origin (which equals 2 ab) and the final saturation level (a) remain the same. In terms of L1 uptake, this means that uptake is up to 40% lower for skin microfilarial densities 40 mf/ss. Increases in c beyond 1.0 do not affect the shape of the function by much. The choice of setting the value c ¼ 1.0 was arbitrary and does not necessarily represent forest vectoreparasite complexes. The mean L1 uptake in the blackfly population per fly bite in month t is calculated as, , NðtÞ NðtÞ X X luðtÞ ¼ ðExi lui Þ Exi ; (55) i¼1

i¼1

where N(t) is the number of persons bitten in month t. It is assumed that a fixed proportion of the L1 (thoracic) larvae will develop to the L3 stage and be released during subsequent bites, lrðtÞ ¼ n luðtÞ;

(56)

where lr(t) is the mean number of L3 larvae released per bite in month t and v is the transmission probability from vector to humans, defined as the average probability that an L1 larva completes the extrinsic incubation period within the blackfly vector and is released as an infective, L3 larva. In calculating v we take into account the life history of the fly starting from her first blood meal. We assume that blood meals are taken at fixed hours during daytime, so that we can use 1 day time steps. Although we take into account differences in the length of the gonotrophic cycle between flies (the cycle may take between 2 and 5 days with a mean of 3.5 days), in the model we assume that a particular fly always has the same cycle length (which equals the time between two successive blood meals). We further explicitly account for variation in the duration of development from L1 to L3 (which is mainly determined by environmental temperature; Cheke et al., 2015). The basic assumption underlying the use of a fixed proportion v is that at any moment the fly population has a stable age distribution and that the number of bites per person is large enough to disregard the age of the biting flies.

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2.2.3.2 Transmission probability

For the current version of ONCHOSIM, transmission probability v has to be calculated outside the model and given as a parameter. This section describes the necessary calculations. Assume that a fly engorges one L1 larva (microfilariae are essentially L1 larvae as there is no moult between the microfilarial and the L1 stage) at her mth blood meal, then the probability to release an L3 larva n blood meals later is given by, i Prel ðnji; j; mÞ ¼ PL1/L3 $ð1 PL3/ Þi $PL3/L3 $PL3/ $Sðm; n$jÞ;

(57)

where Prel(nji,j,m) is the probability to release one L3 larva at the (m þ n)th blood meal if one L1 larva has been ingested during the mth blood meal, given that • a gonotrophic cycle takes j days, • between blood meals m and m þ n there have been i potentially infective blood meals (i.e., blood meals at which the L1 larvae have already developed to the L3 stage), PL1/L3 is the probability that an L1 larva develops to the L3 stage, given survival of the fly, with PL1/L3 ¼ 0.85 (Collins et al., 1977), PL3/L3 is the probability that an L3 larva, which is not released at a given blood meal survives to a next blood meal, given survival of the fly: PL3/L3 ¼ 0.90, PL3/ is the probability that an L3 larva is released at a blood meal: PL3/ ¼ 0.65 (Duke, 1973; Renz, 1987), and S(m,t) is the probability that a fly survives for t days until blood meal m. In order to arrive at a general solution for all possible values of i, we use the probability distribution of the number of potentially infective blood meals since the meal during which microfilariae (L1 larvae) were ingested and before the blood meal during which the infective L3 larvae are released, Prel ðnjj; mÞ ¼

n1 X ½Prel ðnji; j; mÞ$Pib ðijn; jÞ;

(58)

i¼0

Pib ðijn; jÞ ¼ FdL1/L3 ðjðn  iÞÞ  FdL1/L3 ðjðn  i  1ÞÞ;

(59)

where Pib(ijn,j) is the probability that before the nth blood meal since microfilarial intake, i blood meals have been potentially infective (L1 has become L3), given a cycle length of j days, and

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FdL1/L3(t) is the probability that the duration of development of L1 to L3 is equal to or less than t days (FdL1/L3(t) ¼ 0.0 for t  5; 0.07 for t ¼ 6; 0.86 for t ¼ 7; 1.0 for t  8 days). A general solution for all possible values of m can be obtained by incorporating the probability that a fly takes her mth blood meal, Prel ðnjjÞ ¼

mmax X

½Prel ðnjj; mÞ$Pb ðmjjÞ;

(60)

m¼1

, Pb ðmjjÞ ¼ Lðjðm  1ÞÞ

mmax X

ðjðm  1ÞÞ;

(61)

m¼1

where Pb(mjj) is the probability that a feeding fly takes her mth blood meal at a cycle length of j days and L(t) is the probability that a fly lives for at least t days. At present we assume an age- (and microfilarial load)-independent daily survival of 0.78. This is in rough agreement with a probability of daily survival of 0.81 at 25 C for S. damnosum s.s (Cheke et al., 2015). Generalizing for j can be achieved by summation, weighted by the probability distribution of the duration of the gonotrophic cycle, PðnÞ ¼

jmax X   Prel ðnjjÞ$Pgc ðjÞ ;

(62)

j¼jmin

where Pgc( j) is the probability that a gonotrophic cycle takes j days (i.e., j days between successive blood meals; Pgc( j) ¼ 0.0 for j  2; 0.2 for j ¼ 3; 0.6 for j ¼ 4; 0.2 for j ¼ 5; 0.0 for j  6 days). Using the following equality, Sðm; n$jÞ ¼ Lðjðmþ n  1ÞÞ=Lðjðm 1ÞÞ;

(63)

the average probability that an L1 larva taken from a human will develop to the L3 stage and be released to another human is given by, Prel ¼ PL1/L3 $PL3/ $ " ( # 9 8 mmax nmax n1 X X X 1 > i > > > P P ðjÞ$ ½ð1  P Þ$P  $ Lðjðm þ n  1ÞÞ$ $ > > gc L3/ L3/L3 m max > > > > > > m¼1 Lðjðm  1ÞÞ i¼0 m¼1 n¼1 > > = jP max < : > > j¼jmin > > ) > > > > > > P > > n1 > > ½ð1  PL3/ Þ$PL3/L3 i $½FdL1/L3 ðjðn  iÞÞ  FdL1/L3 ðjðn  i  1ÞÞ ; : i¼0

(64)

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In Eqs (58)e(61) mmax ¼

amax þ 1; j

truncated to integer

nmax ¼

amax  ðm$jÞ þ 1; j

truncated to integer

;

(65)

where amax is the maximum attainable age of the fly (i.e., age at which L(T) approaches zero). The transmission probability from vectors to humans v is now given by, v ¼ Prel $ð1  zÞ;

(66)

where z is the fraction of fly bites taken on nonhuman blood host (zoophagy). This value is highly dependent on local circumstances (e.g., human and nonhuman blood host density, blackfly density and blackfly species (Lamberton et al., 2016). In ONCHOSIM the value used is z ¼ 0.04, meaning that 96% of the blood meals are assumed to be taken on human hosts. This is close to the zoophagy index, z ¼ 0.08 recorded in the Beffa form of S. soubrense, in Ghana (Lamberton et al., 2016). Using the indicated quantifications, we have calculated a value for v of 0.073 released infective L3 larvae per L1 larva resulting from a given microfilarial uptake. Note that Eq (64) reduces to a much simpler form if we assume that each day a fraction S of the flies survive, that the gonotrophic cycle has a fixed duration of dgc days and that the number of blood meals needed to complete the development of L1 to L3 is fixed to n1 / 3, Prel ¼ PL1/L3 $PL3/ $

S nl/3$dgc 1  S dgc $ð1  PL3/ Þ$PL3/L3

(67)

2.2.3.3 Excess vector mortality

Excess mortality of infected flies is not considered in ONCHOSIM.

2.3 Comparison between EPIONCHO and ONCHOSIM The EPIONCHO and ONCHOSIM models were developed independently by research teams (respectively based at Imperial College London, London, and Erasmus Medical Center, Rotterdam) using distinct modelling approaches, on the basis of diverse datasets and with different initial purposes. EPIONCHO was built following the modelling tradition and methodology of Anderson and May (1982, 1985, 1991), and Dietz (1976, 1982) for infectious diseases in general and helminthic infections in

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Table 3 Definition and values of variables and parameters for ONCHOSIM Parameter Values and Units

Sources

Human demography

Human life table, F(a) Human fertility, R(t)

See Section 2.2.1 Human population demography See Section 2.2.1 Human population demography

United Nations (2013)

Gamma distribution with mean 1.0 and shape and rate equal to 1.0 or 3.5 Eq. (45)

Plaisier (1996); unpublished OCP data

104%, 91%, 58%, 75%, 75%, 66%, 102%, 133%, 117%, 128%, 146%, and 105% times the average monthly biting rate (JanuaryeDecember)

Alley et al. (1994)

United Nations (2013)

Exposure to simuliid vectors

Interindividual variation in exposure to fly bites (Exi) Variation in exposure to fly bites by age and sex (Exa) Seasonal variation in exposure to fly bites (mbr)

Plaisier (1996)

Life history and microfilarial productivity of the parasite in the human host

Worm longevity (Tl ) Pre-patent period Age (a)-dependent potential microfilarial production, R(a)

Plaisier et al. (1991a) Duke (1980) and Prost (1980) Albiez (1985) and Karam et al. (1987)

291

Weibull distribution with mean 10 and shape 3.8 (year) 1 year R(a) ¼ 0 for 0  a < 1 year R(a) ¼ 1 for 1  a < 6 years R(a) ¼ 1  ((a  6)/15) for 6  a < 21 years R(a) ¼ 0 for a > 21 years

(Continued)

Sources

Longevity of microfilariae (Tm) Worm contribution to the skin microfilarial load (cw) Variability in microfilariae per skin snip (2 mg) Dispersal factor for worm contribution to skin snip (d ) Mating cycle (rc)

9 months 7.6 mf/worm

Plaisier et al. (1995) Plaisier (1996)

Poisson distribution with mean ss(t), Eq. (51) Exponential distribution with mean 1

Habbema et al. (1996a)

3 months

Male potential

100 female worms

Schulz-Key (1990) and Schulz-Key and Karam (1986) Habbema et al. (1996a)

292

Table 3 Definition and values of variables and parameters for ONCHOSIMdcont'd Parameter Values and Units

Habbema et al. (1996a)

Vision loss

Blindness threshold (Elc)

Weibull distribution with mean 10,000 and shape 2.0 50%

Coffeng et al. (2013a)

Fly survival, L(t)

0.78 day1

Probability of gonotrophic cycle duration, Pgc( j )

Pgc( j ) ¼ 0.0 Pgc( j ) ¼ 0.2 Pgc( j ) ¼ 0.6 Pgc( j ) ¼ 0.2 Pgc( j ) ¼ 0.0

Habbema et al. (1996a) and Cheke et al. (2015) Habbema et al. (1996a); expert opinion (OCP entomologists)

Reduction in remaining life expectancy due to blindness

Coffeng et al. (2013a), Kirkwood et al. (1983) and Plaisier et al. (1990)

Parasite in simuliid vector

j  2 days j ¼ 3 days j ¼ 4 days j ¼ 5 days j  6 days

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for for for for for

Probability of duration of larval development (from L1 to L3), FdL1/L3(t) Larval survival (L1 / L3) L3 survival (L3 / L3) Larval release (L3) Success ratio (sr)

0.04

Habbema et al. (1996a); expert opinion (OCP entomologists)

Eq. (54); a ¼ 1.2, b ¼ 0.0213, and c ¼ 0.0861 (main analysis); a ¼ 1.2, b ¼ 0.0213, and c ¼ 1.0 (sensitivity analysis) FdL1/L3(t) ¼ 0 for t  5 days FdL1/L3(t) ¼ 0.07 for t ¼ 6 days FdL1/L3(t) ¼ 0.86 for t ¼ 7 days FdL1/L3(t) ¼ 1.0 for t  8 days 0.85 0.90 0.65 0.0031

Plaisier et al. (1991b)

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Zoophagy index, proportion of blood meals taken on nonhumans blood hosts (z) Microfilarial uptake, lu

Habbema et al. (1996a); expert opinion (OCP entomologists)

Plaisier et al. (1996)

Mass treatment coverage and adherence

Coverage, Cw Age- and sex-specific adherence cr(k,s) Individual adherence index (co)

User-defined See page s13 of Supplementary File S1 Text of Coffeng et al. (2014a) Uniform distribution [0,1]

Unpublished OCP data Habbema et al. (1996a)

Ivermectin

Microfilaricidal efficacy (assumption sets 1 and 2)

100%

Plaisier et al. (1995)

Weibull distribution with mean 1 and shape 2

Plaisier et al. (1995)

Assumption set 1 293

Relative effectiveness (v)

(Continued)

Duration of embryostatic effect (Tr, s) Per dose (cumulative) reduction in worm fecundity, d Macrofilaricidal efficacy

Sources

294

Table 3 Definition and values of variables and parameters for ONCHOSIMdcont'd Parameter Values and Units

11 months 35% 0%

Assumption set 2

Embryostatic effect (tau) Macrofilaricidal efficacy (on male worms) Macrofilaricidal efficacy (on female worms)

Exponential distribution with mean 3.5 (years) Beta distribution with mean 12.3% and sample size 50 Beta distribution with mean 6% and sample size 50

Coffeng et al. (2014a)

User-defined User-defined

Plaisier et al. (1997)

Larviciding

Timing Coverage

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particular. The precursors of EPIONCHO (Bas
an ~ez and Boussinesq, 1999; Bas
an ~ez and Ric
ardez Esquinca, 2001; Bas
an ~ez et al., 2002, 2007), based on (deterministic) differential equations, were developed with the primary objective of understanding the impact of parasite population regulatory processes on the population biology of O. volvulus and the transmission dynamics of onchocerciasis. These precursors were subsequently extended to explore the influence of parasite overdispersion (Churcher et al., 2005, 2006), to include human host age- and sex-structure (Filipe et al., 2005), to account for parasite population genetic structure with the aim to investigate the spread of anthelmintic resistance (Churcher and Bas
an ~ez, 2008, 2009) and to add realism regarding treatment coverage and adherence patterns (Turner et al., 2013a), thus evolving into a full-transmission and control model capable of supporting decision-making by intervention programmes. ONCHOSIM is an individual-based model for simulating onchocerciasis transmission and control in a dynamic human population, based on the technique of stochastic microsimulation (Alley, 1992; Habbema et al., 1996b). The underlying generalized modelling framework has formed the basis of similar models for other helminthic diseases (De Vlas et al., 1996; Plaisier et al., 1998). ONCHOSIM was conceived from the start with the purpose of informing control strategies (Remme, 2004a), formerly during the OCP (Habbema et al., 1996a; Plaisier et al., 1990,b, 1997; Remme et al., 1986; Remme et al., 1990a), and more recently to support APOC’s elimination efforts and to quantify its overall health impact (Coffeng et al., 2013a, 2014a). A formal comparison of EPIONCHO and ONCHOSIM regarding the required duration of mass ivermectin treatment for onchocerciasis elimination in Africa has been presented by Stolk et al. (2015). In this paper, the two models were ‘docked’ as much as possible in terms of parameter values (see Table 2 of Stolk et al., 2015), and the focus was on comparing, contrasting and understanding the similarities and differences in projected elimination outcomes (under annual or biannual ivermectin distribution) for a number of epidemiological scenarios (defined in terms of initial endemicity, microfilarial prevalence, CMFL and vector biting rates), ranging from mesoendemic to holoendemic onchocerciasis, and a number of programmatic scenarios (defined by therapeutic coverage and treatment adherence). This comparison revealed several differences in model predictions, despite harmonization of key parameters. The remainder of this section discusses some of the convergences and divergences identified.

296

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EPIONCHO is a deterministic, population-based model; ONCHOSIM is a stochastic individual-based model. Although there are some similarities, the models also differ in important aspects, e.g., on the extent to which heterogeneities in the human population (e.g., in exposure to blackfly bites) and density dependencies in various processes are captured (e.g., in parasite establishment rate within humans and excess mortality of infected flies, as described above). More specifically, and in the context of the ability of both models to predict elimination, EPIONCHO does not account for the possibility of chance elimination of the parasite population (stochastic fadeout), which becomes increasingly likely at very low intensities of infection, especially for small settings (villages) with a couple of hundred inhabitants (as assumed by ONCHOSIM). Secondly, the models differ with respect to assumptions about density dependence in the various processes involved in transmission dynamics (Table 4), which may also be important for elimination prospects (Duerr et al., 2005, 2011). In particular, EPIONCHO includes a (negative) density-dependent relationship between the annual transmission potential and the parasite establishment rate (Section 2.1.1.1); ONCHOSIM does not capture this mechanism, which makes the model more optimistic regarding elimination prospects. Thirdly, the assumed distribution of adult worm and microfilarial survival times and assumptions regarding microfilarial productivity in relation to worm age may play a role. The current version of EPIONCHO assumes an exponential distribution of worm survival times with a long right tail, implying that worm mortality rates are independent of worm age (an implicit assumption of the exponential model). ONCHOSIM assumes a Weibull distribution (Plaisier et al., 1991a), a more symmetrical distribution with the same mean survival time but a shorter right tail, implying age dependency of worm mortality rates (Section 2.2.2.3). Therefore, it takes considerably longer for the parasite population to die out naturally in EPIONCHO than in ONCHOSIM. In addition to this, ONCHOSIM assumes that the microfilarial production rate declines in older worms (Section 2.2.2.3), so that the relatively old worm population remaining after long-term ivermectin mass treatment has a relatively low microfilarial production. Such a process is not considered by EPIONCHO. EPIONCHO models microfilarial prevalence as a function of mean microfilarial density assuming an underlying negative binomial distribution whose overdispersion parameter is a function of the mean (such that the distribution becomes increasingly aggregated as the intensity of infection decreases). In ONCHOSIM individual outputs are aggregated to obtain information on the microfilarial prevalence

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Table 4 Overview of the main characteristics of the EPIONCHO and ONCHOSIM models Characteristics EPIONCHO ONCHOSIM Basic model structure

Modelling approach

Number and type of spatial locations modelled Way of representing infection in hosts

Interventions considered in previous publications

Deterministic, population meanbased Single population

Stochastic, individualbased (excepting the vector component) Single population

Mean density in population subgroups (e.g., age, sex, treatment adherence group). Prevalence as a function of mean density assuming an underlying negative binomial distribution Mass treatment

Presence and density at individual level

Mass treatment, selective treatment (test-and-treat), vector control

Features included in the model

Human population demographics Heterogeneities in the human population

Birth and death rate; age and sex composition Age, sex, life expectancy, level of exposure to blackflies, adherence with MDA

Blackfly population density

Fixed input as annual biting rate (ABR); seasonality in biting rates can be included Heterogeneous (dependent on age and sex)

Exposure to blackfly vectors

Birth and death rate; age and sex composition Age, sex, life expectancy, level of exposure to blackflies, adherence with MDA, treatment efficacy Fixed input as annual biting rate (ABR); seasonal monthly biting rates Heterogeneous (dependent on age, sex, personal attractiveness to blackflies) (Continued)

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Table 4 Overview of the main characteristics of the EPIONCHO and ONCHOSIM modelsdcont'd Characteristics EPIONCHO ONCHOSIM

Uptake of infection by blackfly vectors

Infection in blackfly vectors Excess mortality of infected flies Parasite acquisition in humans

Infection in humans

Diagnostic outcomes

Elimination end-points

Varying non-linearly (density-dependent) with infection intensity in human hosts Density (average L3 load per fly) Yes

Varying non-linearly (density-dependent) with infection intensity in human hosts Density (average L3 load per fly) No

Nonlinearly (densitydependent) related to rate of exposure to L3 larvae

Linearly proportional to mean number of L3 larvae inoculated, denoted by the success ratio Density (immature or mature worms, mf per skin snip) Microfilarial count sampling to relate model predictions to data

Density (nonfertile and fertile worms, mf per mg of skin) Sampling process and diagnostic performance of skin snipping not yet included pOTTIS and transmission breakpoints; does not account for stochastic fade-out

pOTTIS and probability of elimination; stochastic fade-out possible

Adapted from Stolk, W.A., Walker, M., Coffeng, L.E., Bas
an ~ez, M.G., de Vlas, S.J., 2015. Required duration of mass ivermectin treatment for onchocerciasis elimination in Africa: a comparative modelling analysis. Parasit. Vectors 8, 552.

(proportion of individuals with a positive microfilarial count in either of two skin snips) and intensity. Lastly, the distribution of adult worms among the human population will play a role again through its influence on the mating probability. This assumed distribution is explicit in EPIONCHO (Section 2.1.1.2) and implicit in ONCHOSIM, driven by between-host heterogeneities in exposure and adherence with treatment (Section 2.2.2.3) (Stolk et al., 2015). Table 4 gives an overview of structural similarities and differences between the two models; Table 5 compares parametric assumptions by the two models.

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Table 5 Overview of current parametric assumptions by EPIONCHO and ONCHOSIM Parametric assumption EPIONCHO ONCHOSIM

Human population size

Large

Demography of the human host population

Following survivorship function for Cameroon; good agreement with OCP reference population Blindness and excess mortality are functions of microfilarial load; the latter is a sigmoid function, with a higher mortality risk, for a given microfilarial load, for those aged 20 years or less Yes

Blindness and excess mortality of the human host associated with onchocerciasis

Parameterized for O. volvulusesavannah species of Simulium damnosum s.l. Parameterized for O. volvuluseforest and forestesavannah mosaic species of Simulium damnosum s.l. Proportion of blood meals taken by vectors on humans

Life expectancy of adult worms Distribution of adult worm survival times Parasite establishment probability within human hosts

Small, maximum population size of 440 people Reflecting survivorship function for subSaharan Africa

Blindness is a function of cumulative parasite load and blind individuals have their remaining life expectancy shortened by 50% on average

Yes

No

No

h, varied between 0.1 and 0.99; set at 0.3 for Cameroon; set to 0.96 for formal model comparison 10 years

1  z ¼ 0.96 (where z is the zoophagy index, set at 0.04)

Exponential

Weibull

dH, a (negative) densitydependent function of the number of L3 larvae received per year (following a

sr, a densityindependent constant set at 0.0031

10 years

(Continued)

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Table 5 Overview of current parametric assumptions by EPIONCHO and ONCHOSIMdcont'd Parametric assumption EPIONCHO ONCHOSIM

Contribution to skin microfilariae per female worm Life expectancy of microfilariae

non-linear relationship between microfilarial load and annual transmission potential) (Eq. (2)), varying between a maximum value of 0.0854 and a minimum of 0.00299 2.4 to 5.8 microfilariae/ worm 15 months, assuming exponential distribution of survival times

Skin microfilarial load

M, the mean number of microfilariae per mg of skin; EPIONCHO has used a snip weight of 1.7e2.84 mg

Prophylactic efficacy Microfilaricidal efficacy

Not included 98%e99%, 2 mo. after treatment Fertile worms exposed to ivermectin decrease their microfilarial production according to the dynamics presented in Bas
an ~ez et al. (2008) and would fully recover if further untreated

Embryostatic effect

Macrofilaricidal efficacy

0%

cw ¼ 7.6 microfilariae/ worm Fixed at 9 months, assuming that all microfilariae live for 9 months and then die sl, the number of microfilariae per skin snip (ss); ONCHOSIM assumes a weight of 2 mg per snip Not included 100%, instantaneous reduction All female worms temporarily stop producing microfilariae but resume production gradually, reaching maximum production capacity 11 months post-treatment on average (Plaisier et al., 1995) 0% or 12.3% for male worms and 6% for female worms; set to 0% for formal model comparison

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Table 5 Overview of current parametric assumptions by EPIONCHO and ONCHOSIMdcont'd Parametric assumption EPIONCHO ONCHOSIM

Cumulative, per dose reduction in female worm fertility Parasite establishment probability within vector hosts

0%, 7%, 30%, set to 35% for formal model comparison dV, a (negative) densitydependent function of microfilarial load per mg (following a hyperbolic relationship between L1 uptake and microfilarial load, Eq. (6)), varying between a maximum of 0.0207 and zero

Maximum number of L1 larvae establishing within the fly as microfilarial load becomes infinitely large Mortality rate of vectors

1.4

Background probability of daily survival by vectors Probability that an infective, L3 larva is shed, per bite Survival probability of L3 larvae (if not shed)

35%

A (negative) densitydependent function of microfilarial load per ss (following an exponential relationship between L1 uptake and microfilarial load, Eq. (54)), varying between 0.0511 and zero. (Note that converting ss into mg, the maximum probability would be 0.0256) 1.2

Increases linearly with microfilarial load, Eq. (8) 0.87

Density independent

0.54e0.8

0.65

0.75

0.90

0.78

As a consequence of the above, and for the comparison presented in Stolk et al. (2015), EPIONCHO predicted (under either annual or biannual mass ivermectin distribution) a faster initial decline in both microfilarial prevalence and intensity for all the epidemiological scenarios explored,

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but the decline levelled off and the two infection indicators tended to move towards a new equilibrium, possibly as a consequence of (negative) density dependence. In ONCHOSIM, the initial decline in infection levels was less pronounced and so was the levelling-off in the long term. Eventually, the infection indicators reached zero faster in ONCHOSIM than in EPIONCHO. The difference between the two models was more pronounced for the microfilarial prevalence than for the mean microfilarial intensity, prompting modifications to the way that EPIONCHO models prevalence (Walker et al., unpublished data).

3. MODEL VALIDATION 3.1 EPIONCHO EPIONCHO outputs have been compared against published (Bas
an ~ez and Boussinesq, 1999) precontrol data on microfilarial prevalence (the gold standard measure of infection prevalence) and annual vector biting rates from Burkina Faso, Cameroon and C^ ote d’Ivoire (Fig. 5). EPIONCHO matched the pattern in the data very closely by modifying the human blood index (h, the proportion of blood meals taken on humans) of the blackfly vectors. This modification was motivated by recent (molecular-based) estimates obtained from field samples collected in Ghana (Lamberton et al., 2012, 2016). The proportion of blood meals taken on humans was increased

Figure 5 Validation of EPIONCHO against precontrol microfilarial prevalence data from ^te d’Ivoire (Bas
an ~ez and Boussinesq, 1999). The solid line Burkina Faso, Cameroon and Co represents the predicted endemic equilibrium prevalence of skin microfilariae in the human population for different values of the annual biting rate (the number of vector bites per person per year) assuming a human blood index of 67% for the savannah members of the Simulium damnosum s.l. complex.

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303

Figure 6 Validation of EPIONCHO and ONCHOSIM against longitudinal microfilarial prevalence data from foci where onchocerciasis has been eliminated with mass distribution of ivermectin. The solid and dashed lines are projections from EPIONCHO and ONCHOSIM, respectively. Data points are from mesoendemic (left panel) and hyperendemic (right panel) endemic communities in the river Gambia, Senegal (Diawara et al., 2009). Treatment was yearly in 1988 and 1989, and 6-monthly from 1990 onwards. Models are fitted to the precontrol microfilarial prevalence by modifying the annual biting rate of blackfly vectors. The cumulative antifecundity effect of ivermectin was set to 35% per dose for both models (Plaisier et al., 1995).

from one-third to two-thirds (for the savannah members of the S. damnosum species complex). In addition, EPIONCHO has been validated (and compared with corresponding ONCHOSIM projections) against longitudinal data on microfilarial loads (the gold standard measure of infection intensity) from sentinel communities in Mali and Senegal where annual and biannual treatments with ivermectin were sufficient to eliminate the infection in approximately 15 years (see Fig. 6 for Senegal). EPIONCHO captured well the long-term longitudinal trends in the data and matched closely the projections of ONCHOSIM when a similar assumption was implemented on the severity of the cumulative effect of ivermectin on the fertility of adult female O. volvulus. Formerly, a 7% cumulative reduction on adult female worm fertility had been used as nominal value and varied in sensitivity analysis (Turner et al., 2014a,c). By increasing this magnitude to 35%, results are closer to those of ONCHOSIM, indicating that this is a crucial assumption in the models, which needs to be scrutinized further (Stolk et al., 2015).

3.2 ONCHOSIM ONCHOSIM was first published in 1990 (Plaisier et al., 1990) and since then model predictions have frequently been compared with observed data for model fitting and validation. The model has been shown to mimic adequately trends in infection prevalence, infection intensity and

304

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an

microfilarial count frequency distribution during vector control (Plaisier et al., 1991a,b), changes in infection intensity for a cohort of people following a single treatment or repeated annual treatments with ivermectin (Plaisier et al., 1995) and average trends in microfilarial prevalence during long-term ivermectin mass treatment (Coffeng et al., 2014a; Tekle et al., 2016). The model also adequately captures the precontrol association between community-level microfilarial prevalence and prevalence of blindness as well as trends in blindness prevalence during vector control (Coffeng et al., 2013a). A more formal validation of EPIONCHO and ONCHOSIM regarding their ability to model microfilarial prevalence temporal trends and subsequent elimination in the Mali and Senegal foci where this has been successfully achieved (Traore et al., 2012) is underway and will be presented elsewhere (Walker et al., unpublished data).

4. MODELLING CURRENT TREATMENT STRATEGIES In this section we turn our attention to the modelling of current, mass treatment, strategies based on annual (and biannual) ivermectin distribution. Biannual ivermectin treatment has been included among the alternative intervention strategies, particularly for those foci and projects that are not progressing well under the annual strategy e but are not coendemic with loiasis e or have not yet started/are starting large-scale implementation. However, we include biannual treatment in this section because some countries (e.g., Ghana, Ethiopia) have already switched to 6-monthly ivermectin distribution (Carter Center, 2013; Turner et al., 2013b; Frempong et al., 2016), or have been successful in achieving elimination in some foci by doing so (e.g., Sudan, Uganda) (Higazi et al., 2013; Katabarwa et al., 2012; Katabarwa and Richards, 2014). The same rationale applies to the increasing of therapeutic coverage and reduction of nonadherence to treatment, the impact of which is also discussed in this section. Therefore, this section presents our attempts to tackle the first question posed by the NTD Modelling Consortium, namely, ‘under which epidemiological scenarios do models predict that onchocerciasis can be eliminated with current strategies?’ We restrict ourselves to discussing epidemiological scenarios where species of the S. damnosum s.l. are the main vectors (i.e., we do not consider S. neavei-transmitted onchocerciasis as in those, East African, foci where this species prevails, onchocerciasis elimination has been aided by

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elimination of the vector (Garms et al., 2009; Lakwo et al., 2013), which is facilitated by its very particular breeding site ecology in phoretic association with freshwater Potamonautes crabs). We commence by defining elimination endpoints, i.e., what are the targets to be achieved and how do the models quantify time to elimination and programme duration.

4.1 Defining elimination endpoints Most recent applications of EPIONCHO and ONCHOSIM e particularly since the switch from onchocerciasis morbidity control to elimination e have focused on exploring the feasibility of MDA-based interventions in achieving elimination endpoints within the time horizons defined by the WHO’s 2020 and 2025 elimination targets described in Section 1.2.4 above (Turner et al., 2014c; Coffeng et al., 2014a). The (positive) density-dependent transmission processes built into EPIONCHO (e.g., the mating probability) ensure a theoretical ‘breakpoint’ parasite density below which transmission would be unsustainable and the parasite population would decline terminally until (local) elimination (Anderson and May, 1985). Such breakpoints are multidimensional, depending on the density of each parasite stage, but particularly the longest-lived adult stage (Duerr et al., 2011). Under equilibrium conditions, or by applying simplifying assumptions on the dynamical responses elicited by interventions, breakpoints can be explored analytically (Bas
an ~ez et al., 2009; Duerr et al., 2011). More generally, they must be assessed from numerical outputs, evaluating the parasite population projections after cessation of a simulated intervention. Many of the population processes that ensure a deterministic breakpoint in EPIONCHO also apply when elimination is achieved in ONCHOSIM, albeit with the crucial difference that the stochastic nature of ONCHOSIM permits elimination by chance, socalled ‘stochastic fade-out’. The importance of stochastic fade-out compared to the influence of breakpoints inherent to the transmission process will be presented elsewhere but has been briefly discussed by Stolk et al. (2015). The somewhat hypothetical and elusive nature of parasite breakpoints has motivated the use by EPIONCHO of the more pragmatic and programmatically relevant provisional operational (prevalence) thresholds for treatment interruption and commencement of surveillance that were proposed by APOC (2010) in its conceptual and operational framework for onchocerciasis elimination with ivermectin treatment. These

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thresholds, referred to as pOTTIS are (1) a microfilarial prevalence (by skin snipping) of 1 would still be necessary for both the unstable and stable equilibria to exist). Attempts to quantify R0 for O. volvulus in African savannah villages with different vector biting rates were made by Dietz (1982) and Bas
an ~ez and Boussinesq (1999). Figures ranged from 1.5 (for an ABR of 1000 per person per year and endemic microfilarial prevalence of 50%) to 260 (for 175,000 bites per person per year and 98% microfilarial prevalence). The values in Bas
an ~ez and Boussinesq (1999) obtained following the principles of stability analysis (May, 1974) pertain to the nonage structured version of EPIONCHO and are highly influenced by the vector biting rate (to which the used formulation of R0 is linearly proportional). Using the host age- and sex-structured version of EPIONCHO, Filipe et al. (2005) calculated average R0 values across a number of villages in Cameroonian (savannah), Guatemalan (coffee plantation) and Venezuelan (forest) endemic settings as, respectively, 7.7 (95% CI 7.0e8.4, for an average ABR of w40,000); 7.3 (95% CI 5.6e8.9, for an ABR of w200,000) and 5.3 (95% CI 4.1e 6.5, for an ABR of w60,000). Other attempts to calculate R0 assumed a (hypothetical) nonlinear relationship between the basic reproduction ratio and the annual biting rate and yielded values ranging from 2 to 160e170 (Bas
an ~ez et al., 2007). Following the effective reproduction ratio approach and for onchocerciasis in

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Guatemala (transmitted by S. ochraceum s.l., with armed cibarium), Bas
an ~ez et al. (2009) presented values of Rmax (for mated female worms) ranging from 1.1 (for an ABR of w9000 and microfilarial prevalence of 34%) to 12.7 (for an ABR of 550,000 and microfilarial prevalence of 90%). The great variation of the values presented above highlights the difficulties in obtaining robust estimates of the basic and effective reproduction ratios for onchocerciasis. None of these estimates (with the exception of those presented by Filipe et al., 2005) account for heterogeneities in exposure and transmission and none have been obtained using the nextgeneration matrix approach. According to this approach, and for anthroponotic vector-borne infections such as onchocerciasis, R0 values would be proportional not to the product of the R0 from humans to vectors times the R0 from vectors to humans but to the square root of this product (Dobson, 2009). A better understanding of the relationship between endemic infection prevalence and/or intensity and R0, Reff or Rmax would be helpful for assessing the (relative) feasibility of elimination (a small, absolute, value of R0 does not necessarily imply that elimination is easy, but a scale of values calculated using the same method may provide a useful comparator) and identifying optimal combinations of interventions, according to setting, that may help to bring the effective reproduction ratio below 1 in the vicinity of the unstable transmission breakpoint.

7.3 Modelling the diagnostic performance of the skin snip method and serological assays in near-elimination scenarios Bottomley et al. (2016) have modelled the sensitivity of skin snips in nearelimination settings under a range of assumptions regarding the cumulative effect of ivermectin on female worm fertility based on other modelling studies (0% reduction, Bottomley et al., 2008; 7% per dose reduction, Turner et al., 2014a; and 35% per dose reduction, Plaisier et al., 1995). This study concluded that (1) the proposed surveillance period of 3e5 years after stopping treatment is adequate as the sensitivity of this parasitological method improves with time after the last treatment, and (2) taking four snips instead of two (as routinely used) would substantially increase sensitivity, particularly under the assumption of no cumulative effect of ivermectin on female worm fertility (Bottomley et al., 2008). Under the current protocols of two iliac skin snips, an observed microfilarial prevalence of 1% (a component of the pOTTIS) would correspond to a predicted (underlying) prevalence of 3e6% one year after treatment and of 1e2% 5 years after

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treatment depending on the above mentioned assumptions, highlighting that residual infection would be greater than what can be measured by skin snipping. Future refinements to the EPIONCHO model will incorporate the sensitivity of skin snips according to this study. The new WHO (2016) guidelines for stopping ivermectin MDA (World Health Organization, 2016), although endorsing the use of skin snips for monitoring progress during the treatment phase, do not recommend their use to verify elimination, and instead focus on the use of serological assays for the detection of human IgG4 to the O. volvulus antigen Ov16 (Lobos et al., 1991; Lipner et al., 2006). For verification of elimination, the guidelines suggest a seroprevalence threshold of 0.1% in a sample of 2000 children aged less than 10 years. In addition to the feasibility of sampling the required number of children in the stipulated age range in endemic communities, the ability of serological assays to measure this threshold will depend on the tests having, essentially, 100% specificity. Ongoing studies are being conducted to evaluate the diagnostic performance of Ov16 serology under a range of epidemiological settings, including Ov16 ELISA and the commercially available SD BIOLINE Onchocerciasis IgG4 rapid diagnostic test developed by PATH (http://sites.path.org/dx/ntd/onchocerciasis-thedisease/onchocerciasis-point-of-care-test/). However, since the ability of the skin snip method to act as the gold standard for any of these evaluations declines when very low intensities of microfilarial infection are reached, appropriate statistical modelling approaches should be used to analyze the data in the absence of a gold standard in order to estimate sensitivity and specificity as well as other test properties (Joseph et al., 1995; Dendukuri et al., 2010). Additionally, the same questions investigated regarding the pOTTIS apply here, namely, (1) what is the relationship between the proposed serological threshold, transmission breakpoints and the probability that elimination has been reached?, (2) what is the impact on the magnitude of the operational thresholds (parasitological, serological or otherwise) of the initial (precontrol) endemicity level and other epidemiological variables? These questions are particularly relevant when ivermectin treatment programmes have been conducted in the absence of vector control (as is the case in the majority of APOC projects), because original (entomological and ecological) transmission conditions may still prevail despite profound suppression of microfilarial densities (Duerr et al., 2011). ONCHOSIM has been used to address some of these questions based on assumptions regarding the parasite stage that elicits the IgG4 response to the Ov16 antigen, and the sensitivity and specificity of the serological assays (unpublished

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data). It is planned that EPIONCHO will use force-of-infection models (Muench, 1959; Hens et al., 2010) to relate incidence (rate of acquisition of incoming worms) in near-elimination scenarios to expected age seroprevalence profiles with which to investigate size of appropriate serological thresholds and age ranges that should be sampled to adequately power field studies seeking to verify elimination.

7.4 Modelling hypoendemic onchocerciasis Achieving elimination from the African continent will require the incorporation of hypoendemic onchocerciasis foci into ivermectin treatment programmes, since such foci were not included for priority ivermectin treatment when APOC focused on the elimination of the disease as a public health problem (concentrated in areas of meso- and hyperendemicity, Prost et al., 1979). The identification of optimal strategies to treat these areas has posed a number of questions that modelling can help to address as well as a number of challenges. Some hypoendemic areas, initially identified by the REMO methodology (Ngoumou and Walsh, 1993; Noma et al., 2002), have revealed to include meso- and hyperendemic communities when in-depth epidemiological evaluations have been conducted (World Health Organization/ African Programme for Onchocerciasis Control, 2015). These foci of higher endemicity could act as a source of infection throughout the area if not adequately delineated and treated, but the converse could also apply, with hypoendemic areas acting as a source of infection to controlled areas. The conjecture has been examined that hypoendemic areas may not support autochthonous transmission but rather are maintained as a result of spillover transmission from adjacent areas of higher endemicity (Katabarwa et al., 2010), with the conclusion that transmission is likely to be ongoing in some of these areas (defined as those with nodule prevalence