The dynamical consequences of seasonal forcing, immune boosting and demographic change in a model of disease transmission

• Immune boosting induces cyclical behaviour in a model of infectious disease dynamics.
• Seasonal forcing of transmission also induces cyclical behaviour in this system.
• The birth rate, waning rate and forcing interact to generate complex dynamics.
• Periodic cycles in the forced system are related to unforced limit cycle dynamics.
• The “demographic transition” may lead to new dynamical regimes for certain diseases.

The impact of seasonal effects on the time course of an infectious disease can be dramatic. Seasonal fluctuations in the transmission rate for an infectious disease are known mathematically to induce cyclical behaviour and drive the onset of multistable and chaotic dynamics. These properties of forced dynamical systems have previously been used to explain observed changes in the period of outbreaks of infections such as measles, varicella (chickenpox), rubella and pertussis (whooping cough). Here, we examine in detail the dynamical properties of a seasonally forced extension of a model of infection previously used to study pertussis. The model is novel in that it includes a non-linear feedback term capturing the interaction between exposure and the duration of protection against re-infection. We show that the presence of limit cycles and multistability in the unforced system give rise to complex and intricate behaviour as seasonal forcing is introduced. Through a mixture of numerical simulation and bifurcation analysis, we identify and explain the origins of chaotic regions of parameter space. Furthermore, we identify regions where saddle node lines and period-doubling cascades of different orbital periods overlap, suggesting that the system is particularly sensitive to small perturbations in its parameters and prone to multistable behaviour. From a public health point of view — framed through the ‘demographic transition’ whereby a population׳s birth rate drops over time (and life-expectancy commensurately increases) — we argue that even weak levels of seasonal-forcing and immune boosting may contribute to the myriad of complex and unexpected epidemiological behaviours observed for diseases such as pertussis. Our approach helps to contextualise these epidemiological observations and provides guidance on how to consider the potential impact of vaccination programs.