Qualitative analysis and optimal control of an SIR model with logistic growth, non-monotonic incidence and saturated treatment

This paper describes an SIR model with logistic growth rate of susceptible population, non-monotonic incidence and saturated treatment rate. The existence stability analysis equilibria have been investigated. It has shown that the disease free equilibrium point (DFE) is globally asymptotically stable if basic reproduction number less than unity transmission infection some threshold. system exhibits transcritical bifurcation at DFE respect to cure We also found condition for occurring backward bifurcation, which implies value not enough eradicate disease. Stability or instability different endemic analytically. experiences saddle-node Hopf bifurcation. Bogdanov-Takens (BT) co-dimension 2 investigated through numerical simulations. Here we used two control functions, one vaccination other control. solved optimal problem both analytically numerically. Finally, efficiency determine best strategy among treatment.