Biochemical models of SIR and SIRS: Effects of bilinear incidence rate on infection-free and endemic states.

This article focuses on the mathematical analysis of SIR (Susceptible-Infected-Recovered) and SIRS (Susceptible-Infected-Recovered-Susceptible) epidemic models incorporating a bilinear incidence rate, which describes the interaction between susceptible and infected individuals. It investigates the local stability and existence of Hopf bifurcations—transitions leading to oscillatory dynamics—at both infection-free and endemic equilibrium points by determining algebraic invariant planes and employing Lyapunov function methods. The study provides conditions under which stable or oscillatory epidemic states arise, supported by numerical simulations and computational algebra tools such as MATLAB and MATHEMATICA. These results contribute to understanding the qualitative behavior of epidemic outbreaks modeled by these nonlinear biochemical systems.