Stability and bifurcation analysis of a contaminated sir model with saturated type incidence rate and Holling type-III treatment function

Abstract

The paper deals with an epidemic model which is contaminated. Logistic growth input of vulnerable is taken in our SIR model. Saturated type incidence rate alongside Holling type III treatment function and time delay in growth component has been considered. The stability analysis of equilibrium points is done and the basic reproduction number (R 0 ) is obtained. Using (R 0 ), Local Stability around disease-free equilibrium point has been acquired. When R 0 is less than one, it is locally stable which means disease doesn’t exist in the environment anymore and when R0 is greater than one, it is unstable which means disease still exists in the environment. We have shown that at R 0 =1 the model exhibits forward bifurcation. The conditions for the local stability of endemic equilibrium point have also been obtained. Conditions for the existence of Hopf bifurcation along with its direction and stability has been determined. Numerical simulations have been performed utilizing MATLAB to confirm the logical outcomes acquired. We have seen that to diminish the effect of illness, contamination ought to be decreased. Also, suitable strategies should be chosen to expand the capacity of treatment to decrease the effect of infection.