Analysis of an epidemic model with bistable equilibria using the Poincaré index

摘要

In most disease transmission models, disease eradication depends on a certain threshold quantity (R0), known as the basic reproductive number, such that if R0<1, then the disease-free equilibrium is stable and the disease can be eradicated from the population. However, several recent studies have shown that reducing R0 to values less than unity is not sufficient to control the spread of a disease. These studies have considered epidemic models that have bistable equilibria, where both the disease-free equilibrium and an endemic equilibrium are locally asymptotically stable. This paper proposes a technique for analysing the models that exhibit such dynamics. It is shown that the stability of the equilibria for such models can be established using the Poincaré index of a piecewise Jordan curve defined as the boundary of a positively invariant region for the model. An application of this result to a vaccination model for the transmission dynamics of an infectious disease is given.