A study on the stability behavior of an epidemic model with ratio-dependent incidence and saturated treatment

Abstract

In the present article, the dynamics of a novel combination of ratio-dependent incidence rate and saturated treatment rate in susceptible-infected-recovered disease compartmental model has been presented. The ratio-dependent incidence rate has been incorporated into the model to monitor the situation when ratio of the number of infectives to that of the susceptibles is getting higher. The saturated treatment rate of the infected population has been considered as Holling type II functional, which explains the limitation in treatment availability. From the mathematical analysis of the model, two types of equilibria of the model have been obtained, which are named as disease-free equilibrium (DFE) and endemic equilibrium (EE). The local stability behavior of equilibria has been investigated by the basic reproduction number $\left( {R_{0} } \right)$, center manifold theory and Routh–Hurwitz criterion. It has been investigated that the DFE is locally asymptotically stable when $R_{0} < 1$, and when $R_{0} = 1$, the DFE exhibits either a forward bifurcation or a backward bifurcation under some conditions. The local stability behavior of the EE has also been analyzed, and some conditions are obtained for the same. Finally, some numerical computations have been performed in support of our theoretical results.