Dynamics of delayed SIR model with quadratic treatment under the effect of pollution
We analyse epidemic model in this article, with incubation delay under the effect of pollution, where disease transmission follows saturated incidence rate and quadratic treatment has also been considered. Boundedness and presence of endemic equilibrium point ($Q$) as well as point of infection-free equilibrium ($Q^*$) for the model has been established. We investigate the stability analysis of the model around $Q$ and $Q^*$. $R_0$ as a basic reproduction number has been calculated. When $R_0<1$, the model is stable for $Q$, but when $R_0>1$, it shifts from a stable to an unstable environment for $\varrho \geq 0$. Using center manifold theory, it was shown that at $R_0 = 1$, the model exhibits backward bifurcation, when the bifurcating constant $a>0$, while if $a<0$, then the bifurcation is forward. We analyse the presence of Hopf bifurcation for endemic equilibrium point $Q^*$ by taking time delay as the critical parameter. Considering the normal form theory, we obtain the direction and the stability of Hopf Bifurcation. We establish the global stability of $Q^*$ using Lyapunov Lasalle theorem. Our results indicate that pollution affects the disease transmission dynamics of the model. We observe that in the diseased model with pollution the survival of the species is significantly less as compared when there is only disease and no pollution. Numerical simulations are performed to validate our logical findings.