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
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Date
2008
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Massey University
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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.
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Mathematical biology, Linear algebra, Nonlinear transmission, Exogenous infection