title: Dynamical crises, multistability and the influence of the duration of immunity in a seasonally-forced model of disease transmission.
abstract: BACKGROUND
  Highly successful strategies to make populations more resilient to infectious diseases, such as childhood vaccinations programs, may nonetheless lead to unpredictable outcomes due to the interplay between seasonal variations in transmission and a population's immune status.
  
  
  METHODS
  Motivated by the study of diseases such as pertussis we introduce a seasonally-forced susceptible-infectious-recovered model of disease transmission with waning and boosting of immunity. We study the system's dynamical properties using a combination of numerical simulations and bifurcation techniques, paying particular attention to the properties of the initial condition space.
  
  
  RESULTS
  We find that highly unpredictable behaviour can be triggered by changes in biologically relevant model parameters such as the duration of immunity. In the particular system we analyse--used in the literature to study pertussis dynamics--we identify the presence of an initial-condition landscape containing three coexisting attractors. The system's response to interventions which perturb population immunity (e.g. vaccination "catch-up" campaigns) is therefore difficult to predict.
  
  
  CONCLUSIONS
  Given the increasing use of models to inform policy decisions regarding vaccine introduction and scheduling and infectious diseases intervention policy more generally, our findings highlight the importance of thoroughly investigating the dynamical properties of those models to identify key areas of uncertainty. Our findings suggest that the often stated tension between capturing biological complexity and utilising mathematically simple models is perhaps more nuanced than generally suggested. Simple dynamical models, particularly those which include forcing terms, can give rise to incredibly complex behaviour.
authors.forename: Mathew P
authors.initials: MP
authors.lastname: Dafilis
authors.forename: Federico
authors.initials: F
authors.lastname: Frascoli
authors.forename: Jodie
authors.initials: J
authors.lastname: McVernon
authors.forename: Jane M
authors.initials: JM
authors.lastname: Heffernan
authors.affiliation: Centre for Epidemiology and Biostatistics, Melbourne School of Population and Global Health, The University of Melbourne, Melbourne VIC, Australia. jamesm@unimelb.edu.au.
authors.forename: James M
authors.initials: JM
authors.lastname: McCaw
mesh_terms: D003141:Communicable Diseases / Q000276:immunology* / Q000635:transmission*; D006801:Humans; D007109:Immunity; D018448:Models, Immunological; D017711:Nonlinear Dynamics; D012621:Seasons; D049489:Stroboscopy; D013997:Time Factors