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Abstract
In this article, we consider the discretized classical Susceptible‐Infected‐Recovered (SIR) forced epidemic model to investigate the consequences of the introduction of different transmission rates and the effect of a constant vaccination strategy, providing new numerical and topological insights into the complex dynamics of recurrent diseases. Starting with a constant contact (or transmission) rate, the computation of the spectrum of Lyapunov exponents allows us to identify different chaotic regimes. Studying the evolution of the dynamical variables, a family of unimodal‐type iterated maps with a striking biological meaning is detected among those dynamical regimes of the densities of the susceptibles. Using the theory of symbolic dynamics, these iterated maps are characterized based on the computation of an important numerical invariant, the topological entropy. The introduction of a degree (or amplitude) of seasonality, ε, is responsible for inducing complexity into the population dynamics. The resulting dynamical behaviors are studied using some of the previous tools for particular values of the strength of the seasonality forcing, ε. Finally, we carry out a study of the discrete SIR epidemic model under a planned constant vaccination strategy. We examine what effect this vaccination regime will have on the periodic and chaotic dynamics originated by seasonally forced epidemics.
Highlights
We examine what effect this vaccination regime will have on the periodic and chaotic dynamics originated by seasonally forced epidemics
Understanding the dynamical behaviors of emergent infectious diseases in humans is viewed with increasing importance in epidemiology
Mathematical modeling has been used to provide useful insights to enhance our comprehension of complex processes associated with the pathogenesis of diseases, as well as to quantify the likely effects of different intervention/control strategies
Summary
Understanding the dynamical behaviors of emergent infectious diseases in humans is viewed with increasing importance in epidemiology. Of our study, the time step h is chosen as a bifurcation parameter to study different complex dynamical behaviors of model (5) in two regimes: epidemics without seasonality (ε = 0), taking β(t) = β0, (Section 2) and epidemics with seasonal contact rate (ε > 0), taking β(t) = β0(1 + εφ(t)), with φ(t) given by (3) (Section 3). By choosing h as a bifurcation parameter in the considered interval, [2.92, 3.4], we are able to exhibit a variety of dynamical behaviors from period 1 and period-doubling bifurcations, passing through interesting rich qualitative dynamics (flip bifurcation, Hopf bifurcation) to chaos This way, the comprehensive range of [2.92, 3.4] for values of h was chosen to fulfil the aim of displaying a window of global noticeable qualitative features of the dynamics of the studied discrete-time epidemic model.
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