Stability of a delayed SIR epidemic model by introducing two explicit treatment classes along with nonlinear incidence rate and Holling type treatment

Abstract

In this article, we analyze the stability of a time-delayed susceptible–infected–recovered (S–I–R) epidemic model by introducing two explicit treatment classes (or compartments) along with nonlinear incidence rate. The treatment classes are named as a pre-treated class $$ \left( {T_{1} } \right) $$ and post-treated class $$ \left( {T_{2} } \right) $$. The pre-treatment and post-treatment rates are being considered as Holling type I and Holling type III, respectively. Long-term qualitative analysis has been carried out after incorporating incubation time delay $$ \left( \tau \right) $$ into the incidence rate. The model analysis shows that the model has two equilibrium points, named as disease-free equilibrium (DFE) and endemic equilibrium (EE). The disease-free equilibrium is locally asymptotically stable when the basic reproduction number ($$ R_{0} $$) is less than one and unstable when $$ R_{0} $$ is greater than one for time lag $$ \tau \ge 0 $$, and when $$ R_{0} = 1 $$ by Castillo-Chavez and Song theorem, the disease-free equilibrium changes its stability from stable to unstable and the model exhibits transcritical bifurcation. Furthermore, some conditions for stability of the endemic equilibrium are obtained. Finally, numerical simulations are presented to exemplify the analytical studies.