An SIRS Pulse Vaccination Model with Nonlinear Incidence Rate and Time Delay

In this paper, we investigate the effectiveness of pulse vaccination as a control strategy for a time-delayed SIRS epidemic model with varying population size. The dynamics of the infectious disease are closely tied to the basic reproduction number, denoted as R₀ . Traditional epidemic models evaluate R₀ using the next generation matrix, but this approach is unsuitable for non-autonomous systems. As our study focuses on pulse vaccination strategies, our system naturally falls into the non-autonomous category. To address this, we adopt a general approach that derives R₀ in terms of spectral radii of Poincaré maps. Furthermore, we demonstrate the existence of an infectious-free periodic solution and establish its global attractiveness for R₀ < 1 while highlighting the persistence of the infectious disease for R₀ > 1. Lastly, we conduct a comprehensive sensitivity analysis for R₀ under the framework of the Holling type II functional response.