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Models of Infectious Disease Transmission to Study the Effects of Waning and Boosting of Immunity on Infectious Disease Dynamics

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Models of Infectious Disease Transmission to Study the Effects of Waning and Boosting of Immunity on Infectious Disease Dynamics MODELS OF INFECTIOUS DISEASE TRANSMISSION TO EXPLORE THE EFFECTS OF IMMUNE BOOSTING Ngo Nam (Tiffany) LEUNG ORCID iD: 0000-0002-9075-4134 Doctor of Philosophy August 2019 SCHOOL OF MATHEMATICS AND STATISTICS THE UNIVERSITY OF MELBOURNE The thesis is being submitted in total fulfilment of the degree. The degree is not being completed under a jointly awarded degree. Copyright © 2019 Ngo Nam (Tiffany) LEUNG All rights reserved. No part of the publication may be reproduced in any form by print, photoprint, microfilm or any other means without written permission from the author. Abstract Despite advances in prevention and control, infectious diseases continue to be a burden to human health. Many factors, including the waning and boosting of immunity, are involved in the spread of disease. The link between immunological processes at an individual level and population-level immunity is complex and subtle. Mathematical models are a useful tool to understand the biological mechanisms behind the observed epidemiological patterns of an infectious disease. Here I construct deterministic, compartment models of infectious disease transmission to study the effects of waning and boosting of immunity on infectious disease dynamics. While waning immunity has been studied in many models, incorporation of immune boost- ing in models of transmission has largely been neglected. In my study, I look at three different aspects of immune boosting: (i) the influence of immune boosting on the criti- cal vaccination thresholds required for disease control; (ii) the effect of immune boosting and cross-immunity on infectious disease dynamics; and (iii) the influence of differentiat- ing vaccine-acquired immunity from natural (infection-acquired) immunity and different mechanisms of immune boosting on infection prevalence. Models can provide support for public health control measures in terms of critical vac- cination thresholds. There is a direct relationship, from mathematical theory, between the critical vaccination threshold and the basic reproduction number, R0. Key epidemiologi- cal quantities, such as R0, are measured from data, but the selection of the model used to infer these quantities matters. I show how the inferred values of R0—and thus, critical vaccination thresholds—can vary when immune boosting is taken into account. I also investigate the effects of interactions between immune boosting and cross- immunity on infectious disease dynamics, using a two-pathogen transmission model. iii Immunity to one pathogen that confers immunity to another pathogen, or to another strain of a given pathogen, is known as cross-immunity. Varying levels of susceptibility to infection conferred by cross-immunity are included in the model. Using a combination of numerical simulations and bifurcation analyses, I show that immune boosting at strong levels can lead to recurrent epidemics (or periodic solutions) independent of cross-immunity. Where immune boosting is weak, cross-immunity allows the model to generate periodic solutions. For some diseases, there are differences in infection-acquired immunity and vaccine- acquired immunity. I explore the effect of vaccination and immune boosting on epidemio- logical patterns of infectious disease. I construct and analyse a model that differentiates vaccine-acquired immunity from infection-acquired immunity in the form of duration of protection. The model also distinguishes between primary and secondary infections. I show that vaccination is effective at reducing primary infections but not necessarily secondary infections, which can maintain overall transmission. Two different mechanisms through which immune boosting provides protection are also explored. Whether immune boosting delays or bypasses a primary infection can determine whether primary or secondary infec- tions contribute most to transmission. iv Declaration This is to certify that 1. the thesis comprises only my original work towards the PhD except where indicated in the Preface, 2. due acknowledgement has been made in the text to all other material used, 3. the thesis is less than 100,000 words in length, exclusive of tables, maps, bibliogra- phies and appendices. Ngo Nam (Tiffany) LEUNG, August 2019 v Preface Chapter7 contains an article, “Periodic solutions in an SIRWS model with immune boosting and cross-immunity”, that was published by the Journal of Theoretical Biology in August 2016, authored by me (Tiffany Leung; TL), Barry D. Hughes (BH), Federico Frascoli (FF) and James M. McCaw (JM) [98]. I was the primary author of this publication and con- tributed to approximately 85% of the work. Regarding the contributions of the authors: • Conceived the study: TL, BH, FF, JM. • Developed the model: TL, BH, FF, JM. • Analysed the model: TL, FF, with input from BH, FF, JM. • Drafted the manuscript: TL, with input from BH, FF, JM. Chapter8 contains an article, “Infection-acquired immunity versus vaccine-acquired immunity in an SIRWS model”, that was published by Infectious Disease Modelling in June 2018, authored by TL, Patricia T. Campbell (PC), BH, FF and JM [97]. I was the primary author of this publication and contributed to approximately 90% of the work. Regarding the contributions of the authors: • Conceived the study: TL, BH, JM. • Developed the model: TL, BH, PC, JM. • Analysed the model: TL, with input from BH, PC, FF, JM. • Drafted the manuscript: TL, with input from BH, PC, JM. vii All other work in the thesis is my own. None of the work in the thesis has been submitted for other qualifications. None of the work in the thesis was carried out prior to enrolment in the degree. No third party editorial assistance was provided in preparation of the thesis. I have been supported by a Melbourne International Research Scholarship. viii Acknowledgements First and foremost, I would like to thank my supervisors, James McCaw, Barry Hughes and Federico Frascoli, for giving me continuous support over my PhD research, for their time, patience and motivation. Thank you for encouraging me to publish and meticulously reading through the drafts of this work. Your professional and scientific guidance over the last four years has been instrumental to the development of this study. It has been an exceptional experience undertaking this study with you. I would like to thank my committee chair, James Osborne, for keeping me on track to complete my PhD. Thank you to the Modelling and Simulation group at the Melbourne School of Population and Global Health for listening to me talk about my research and providing advice. I would like to acknowledge the Centre for Research Excellence in Policy Relevant In- fectious diseases Simulation and Mathematical Modelling (PRISM2) for financial support to allow me to attend conferences. In particular, thank you to Alex Strich for administrative support. Thank you to the School of Mathematics and Statistics for financial support, and to Kirsten Hoak for assistance with all administrative questions since the start of this PhD. Thank you to everyone in the Mathematical and Computational Biology group, includ- ing those on our floor, for creating a supportive work environment and fostering a sense of belonging between staff and students. Thank you to my friends, in and out of the office, both past and present, for moral support, especially in the last phase of my PhD. A special men- tion to Michael Lydeamore, Claire Miller, Hilary Hunt and Dominic Maderazo, thank you for listening to my musings, rhapsodising about plants and taking walks with me. Thank you to friends in my board games group for being an excellent (and much needed) distrac- tion to my study. I appreciate those who took the time to check in on me, particularly ix Olivia Smith, Maryam Fanaeepour, Robin Wagner, Frank Wagner and my family. You have all contributed to my sense of community. To John Wagner, without whom I would not even have taken the first step into tertiary education, thank you for being there to celebrate every victory throughout the past decade, and for your support during stressful times. The light and beauty you have added to my life transcend the sum meaning of words. x Contents 1 Introduction 1 1.1 A brief history of immunity and vaccination.....................1 1.2 Mathematical models of infectious disease.....................4 1.3 Thesis outline.......................................6 2 Vaccine-preventable diseases with temporary immunity9 2.1 Temporary immunity..................................9 2.2 Cross-immunity..................................... 10 2.3 Imperfect vaccines.................................... 11 2.4 Summary......................................... 13 3 Mathematical background 15 3.1 Introduction........................................ 15 3.2 Dynamical systems................................... 15 3.2.1 Definitions.................................... 15 3.2.2 Equilibria and stability............................. 17 3.2.3 Bifurcations.................................... 20 3.3 The SIR model with demography........................... 24 3.3.1 Equilibria and stability of the SIR model with demography....... 27 3.3.2 Bifurcation in the SIR model with demography.............. 29 3.4 Epidemiological quantities of the SIR model with demography......... 30 3.4.1 The basic reproduction number........................ 30 3.4.2 Mean age of first infection........................... 32 3.5 Common assumptions of compartmental transmission models.......... 33 3.6 Summary......................................... 35 4 Existing compartmental models of transmission
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