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- On the dynamical complexity of a seasonally forced discrete SIR epidemic model with a constant vaccination strategyPublication . Rashidinia, Jalil; Sajjadian, Mehri; Duarte, Jorge; Januário, Cristina; Martins, NunoIn 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 iteratedmaps 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.
- Controlling infectious diseases: the decisive phase effect on a seasonal vaccination strategyPublication . Duarte, Jorge; Januário, Cristina; Martins, Nuno; Seoane, Jesús M.; SANJUAN, MIGUEL A. F.The study of epidemiological systems has generated deep interest in exploring the dynamical complexity of common infectious diseases driven by seasonally varying contact rates. Mathematical modeling and field observations have shown that, under seasonal variation, the incidence rates of some endemic infectious diseases fluctuate dramatically and the dynamics is often characterized by chaotic oscillations in the absence of specific vaccination programs. In fact, the existence of chaotic behavior has been precisely stated in the literature as a noticeable feature in the dynamics of the classical Susceptible-Infected-Recovered (SIR) seasonally forced epidemic model. However, in the context of epidemiology, chaos is often regarded as an undesirable phenomenon associated with the unpredictability of infectious diseases. As a consequence, the problem of converting chaotic motions into regular motions becomes particularly relevant. In this article, we consider the so-called phase control method applied to the seasonally forced SIR epidemic model to suppress chaos. Interestingly, this method of controlling chaos has a clear meaning as a weak perturbation on a seasonal vaccination strategy. Numerical simulations show that the phase difference between the two periodic forces - contact rate and vaccination - plays a very important role in controlling chaos.