Bifurcation Analysis of a Discrete Epidemic Model with Vaccination

This paper explores the dynamical characteristics of a discrete epidemic model incorporating vaccination. The model is vital for understanding how vaccination impacts the spread and control of infectious diseases, especially in settings, where population interactions are best described by discrete processes. The analysis reveals that epidemic model possesses a disease-free fixed point for all model’s parameters, while an endemic fixed point emerges under specific parametric condition(s). Through linear stability theory, the local dynamics at these fixed points are elucidated, offering insights into the stability and potential oscillatory behaviors of the disease spread. Furthermore, in order to understand the complex dynamics of the understudied model, we also explore the existence of possible bifurcations by the bifurcation theory and center manifold theorem. More specifically, our bifurcation analysis identifies that at the disease-free fixed point, no flip bifurcation occurs, whereas Neimark–Sacker and flip bifurcations are present at the endemic fixed point. These bifurcations suggest transitions to complex dynamical behaviors such as periodic outbreaks and chaos. The application of OGY and Hybrid control strategies demonstrates methods for mitigating chaotic dynamics, thereby enhancing epidemic management strategies. The theoretical findings are confirmed by numerical simulations.