Periodic solutions in an SIRWS model with immune boosting and cross-immunity

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Tiffany Leung a, Barry D. Hughes a, Federico Frascoli b, James M. McCaw a, c, d,Periodic solutions in an SIRWS model with immune boosting and cross-immunityJournal of Theoretical Biology 410, December 2016, 55-64.


Incidence of whooping cough, an infection caused by Bordetella pertussis and Bordetella parapertussis, has been on the rise since the 1980s in many countries. Immunological interactions, such as immune boosting and cross-immunity between pathogens, have been hypothesised to be important drivers of epidemiological dynamics. We present a two-pathogen model of transmission which examines how immune boosting and cross-immunity can influence the timing and severity of epidemics. We use a combination of numerical simulations and bifurcation techniques to study the dynamical properties of the system, particularly the conditions under which stable periodic solutions are present. We derive analytic expressions for the steady state of the single-pathogen model, and give a condition for the presence of periodic solutions. A key result from our two-pathogen model is that, while studies have shown that immune boosting at relatively strong levels can independently generate periodic solutions, cross-immunity allows for the presence of periodic solutions even when the level of immune boosting is weak. Asymmetric cross-immunity can produce striking increases in the incidence and period. Our study underscores the importance of developing a better understanding of the immunological interactions between pathogens in order to improve model-based interpretations of epidemiological data.


a School of Mathematics and Statistics, University of Melbourne, Victoria 3010, Australia

b Department of Mathematics, Faculty of Science, Engineering and Technology, Swinburne University of Technology, Hawthorn, Victoria 3122, Australia

c Melbourne School of Population and Global Health, University of Melbourne, Victoria 3010, Australia

d Modelling and Simulation, Infection and Immunity Theme, Murdoch Childrens Research Institute, Royal Children’s Hospital, Parkville, Victoria 3052, Australia