Complex dynamics in a seasonal infectious disease model

Abstract

In this thesis, we investigate the dynamical aspects of a Susceptible–Exposed–Infected–Recove- red–Susceptible (SEIRS) model with a time-dependent contact rate. Moreover, we explore

how vaccination strategies can be used to control chaotic dynamics and reduce the number of infected individuals. First, we introduce fundamental concepts of nonlinear dynamics, such as local stability, bifurcation, chaos, and Lyapunov exponents. Following this, we present key ideas from mathematical epidemiology by describing the compartmental models: Susceptible– Infected (SI), Susceptible–Infected–Susceptible (SIS), Susceptible–Infected–Recovered (SIR), and Susceptible–Exposed–Infected–Recovered (SEIR). As a practical application, we show an extension of SEIR model incorporating two vaccination doses. The next chapter focuses on the dynamical behavior of the SEIRS model. We first obtain the equilibrium solutions, namely the disease-free and endemic states, and analyze their stability. Subsequently, we introduce a time-dependent contact rate and examine the effects of its frequency on Lyapunov exponents. Our findings reveal that certain frequencies lead to multistable solutions, where chaotic and periodic attractors coexist. Further, we fix the contact rate frequency to one year and investigate the influence of other model parameters in the dynamics. Through bifurcation diagrams, we observe that almost all parameters affect the dynamics, often resulting in bistable regimes where periodic and chaotic attractors coexist. Additionally, we find that the inverse of the latent period is crucial for epidemic forecasting, as it is associated with tipping points where solutions change from periodic to chaotic attractors, thus compromising the forecast horizon. The final part of the thesis incorporates a constant vaccination campaign into the model. We first derive the stationary solutions for the unforced model and then extend these solutions to include seasonal forcing. This approach allows us to establish a vaccination threshold for disease eradication, which we validate through numerical simulations and found its limitations. For parameter sets exhibiting bistable solutions, we explore the effects of the vaccination campaign and uncover that constant vaccination can either suppress bistability or sustain it, depending on specific values. By analyzing the impact of constant immunization programs on Lyapunov exponents, we identify complex structures in parameter planes, including shrimps. Overall, the results presented in this thesis illustrate the intricate dynamical behavior of seasonal infectious diseases governed by SEIRS models, highlighting both the challenges and opportunities in controlling such systems.