Backward Bifurcation of an Epidemiological Model with Saturated Incidence, Isolation and Treatment Functions

Abstract

This work considers an epidemic model with saturated incidence rate and saturated isolation and treatment functions. The incidence function (as well as the treatment and isolation functions) is of the Holling type II form, where, in the cases of treatment and isolation, these functions describes the effect of delayed treatment and isolation with large number of infected individuals in a population especially in situations where isolation facilities are available but fewer medical personnel. It is shown that the disease-free equilibrium (DFE) is locally-asymptotically stable whenever the effective reproduction number is less than unity. However, it is shown that the global-asymptotic stability of the disease-free equilibrium was largely dictated by the delay in treatment (of non-isolated infected individuals) and isolation of infected individuals; when such delay effect is weak, then the DFE is globally asymptotically stable when the effective reproduction number is less than unity. When the delay effect is strong, it is shown that there is the possibility of the existence of the backward bifurcation phenomenon whereby the DFE will co-exists with two endemic equilibria, when the effective reproduction number is less than unity. The backward bifurcation phenomenon existed when the delays occurred either singly or jointly. Mathematical analysis provides threshold conditions that allows for the global stability of the endemic equilibrium whenever the associated reproduction number is greater than unity. The results suggests that with timely isolation of infected individuals (for treatment under isolation) and treatment of non-isolated infectious cases, the ultimate goal of disease eradication is possible.