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

Understanding and forecasting the progression of disease epidemics is possible through the study of nonlinear epidemic biochemical models that describe the relationship among susceptible, infected, and immune individuals in a population. In this paper, by determining the algebraic invariant planes and studying the Hopf bifurcation on these invariant planes, we study the stability of the Hopf bifurcation in the infection-free and endemic states of the SIR and SIRS epidemic models with bilinear incidence rate. We analyze the stability of the limit cycles of the bilinear incidence SIR and SIRS models at the steady state point where infection vanishes and at the endemic steady state point where the system behaves in an oscillatory manner. We demonstrate the algebraic results by numerical simulations for parameter values that satisfy the conditions for both free and endemic states.