MATH COLLOQUIUM

Department of Mathematical Sciences

University of Wisconsin-Milwaukee

Forward bifurcation and backward bifurcation of equilibria in epidemic models

Dr. Maia Martcheva

Polytechnic University

Abstract

The use of mathematical modeling as a tool for disease control has historically relied on threshold results, in which a certain factor related to disease transmission must be changed beyond a given level in order to eradicate the disease. The most well-known threshold criterion is the basic reproductive number of a disease, typically denoted ${\cal R}_0$, which represents the average number of secondary infections caused by one infective in a pool of susceptibles. In simple epidemic models when ${\cal R}_0 <1$ the disease-free equilibrium is often globally stable which suggests that it is sufficient to reduce ${\cal R}_0$ below one and the disease will disappear from the population. I consider a simple ODE model of hepatitis C as an illustration. However, several recent studies have shown that the ${\cal R}_0 <1$ criterion is not always sufficient to control the spread of a disease. Dynamically it is possible that the transcritical bifurcation that occurs at ${\cal R}_0=1$ may change directions, creating what has become known in the literature as a ``backward'' bifurcation, in which the endemic equilibrium arises from the disease-free equilibrium for ${\cal R}_0 <1$ rather than for ${\cal R}_0\gt 1$ as in the simplest cases. From a control point of view when backward bifurcation occurs it is not sufficient to lower ${\cal R}_0$ below one but below another threshold value which is the leftmost point on the bifurcation curve for which an endemic equilibrium exists. I consider a structured model for a disease with a progressing and a quiescent exposed class and variable susceptibility to super-infection. The model exhibits backward bifurcations under certain conditions, which allow for both stable and unstable endemic states when the basic reproduction number is smaller than one.

Friday, March 1, 2002

4:00 pm, EMS E495

Refreshments: 3:30 pm, EMS E495B