James Watmough - A simple SIS epidemic model with a backward bifurcation

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James Watmough - A simple SIS epidemic model with a backward bifurcation

JAMES WATMOUGH, University of Victoria, Victoria, British Columbia V8W 3P4

A simple SIS epidemic model with a backward bifurcation

Classical disease transmission models with constant contact rates typically have only a single stable equilibrium. There is a threshold level of the reproduction number below which the disease dies out and above which the disease approaches an endemic level.

In this talk I will formulate an nonlinear Volterra integral equation to model the dynamics of a simple disease. The model always possesses a disease free equilibrium (zero infectives) which loses stability as a parameter (the basic reproduction number) is increased through a threshold. In classical models, this bifurcation is always super critical. However, with the introduction of a non-constant contact rate, the bifurcation may be sub-critical, implying the existence of multiple equilibrium and hysterisis.

Let I(t) and S(t)=1-I(t) be the fraction of the population in each of the two disjoint classes, infective and susceptible, at time $t\ge 0$ . Susceptible individuals become infective at a rate $\lambda\bigl(I(t)\bigr)I(t)S(t)$ . The integral equation studied has the form

$\begin{displaymath}I(t) = I_o(t) + \int_0^t \lambda\bigl(I(u)\bigr)I(u)\bigl(1-I(u)\bigr) P(t-u)e^{-b(t-u)}\,du, \end{displaymath}$

(1)

where S(t) has been replaced by 1-I(t). Briefly, P(t-u)e^-b(t-u) is the probability that an individual infected at time u is still infectious at time t. The birth and death rate of individuals are both equal to the parameter b. The integral sums the individuals that entered the infective class at time $u\ge 0$ and have remained infective through to time t.

We determine the existence and stability of equilibria of Equation () as a function of the reproduction number

$\begin{displaymath}R_o = \lambda(0)\int_0^\infty P(u)e^{-bu} \,du, \end{displaymath}$

(2)

which is the expected number of infectives produced by a single infective during its lifetime. There are two thresholds of the reproduction number, $R_o^m\ge R_o^c>0$ , such that the disease free equilibrium is the only equilibrium solution and is globally asymptotically stable for R_o<R_o^c, and there is a single, globally asymptotically stable endemic (positive) equilibrium for R_o>R_o^m. Sufficient conditions for the existence of multiple stable equilibria are given by the following result.

Theorem 1 For the model of Equation (

), with the assumptions of the previous section and $\lambda(0)>0$ , there is a transcritical bifurcation at R_o=1, $\bar{I}=0$ . This bifurcation is in the forward direction if $\lambda'(0)<\lambda(0)$ and in the backward direction if $\lambda'(0)>\lambda(0)$ . Further, if the model has a backward bifurcation, then R_o^c<1 and there are multiple stable equilibria for R_o^c<R_o<1.

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