Backward bifurcation in SIR endemic models : this thesis is presented in partial fulfillment of the requirements for the degree of Masters of Information Science in Mathematics at Massey University, Albany, Auckland, New Zealand

Abstract

In the well known SIR endemic model, the infection-free steady state is globally stable for R0 < 1 and unstable for R0 > 1. Hence, we have a forward bifurcation when R0 = 1. When R0 > 1, an asymptotically stable endemic steady state exists. The basic reproduction number R0 is the main threshold bifurcation parameter used to determine the stability of steady states of SIR endemic models. In this thesis we study extensions of the SIR endemic model for which a backward bifurcation may occur at R0 = 1. We investigate the biologically reasonable conditions for the change of stability. We also analyse the impact of di erent factors that lead to a backward bifurcation both numerically and analytically. A backward bifurcation leads to sub-critical endemic steady states and hysteresis. We also provide a general classi cation of such models, using a small amplitude expansion near the bifurcation. Additionally, we present a procedure for projecting three dimensional models onto two dimensional models by applying some linear algebraic techniques. The four extensions examined are: the SIR model with a susceptible recovered class; nonlinear transmission; exogenous infection; and with a carrier class. Numerous writers have mentioned that a nonlinear transmission function in relation to the infective class, can only lead to a system with an unstable endemic steady state. In spite of this we show that in a nonlinear transmission model, we have a function depending on the infectives and satisfying certain biological conditions, and leading to a sub-critical endemic equilibriums.