Abstract
This paper describes an SIR model with logistic growth rate of susceptible population, non-monotonic incidence rate and saturated treatment rate. The existence and stability analysis of equilibria have been investigated. It has been shown that the disease free equilibrium point (DFE) is globally asymptotically stable if the basic reproduction number is less than unity and the transmission rate of infection less than some threshold. The system exhibits the transcritical bifurcation at DFE with respect to the cure rate. We have also found the condition for occurring the backward bifurcation, which implies the value of basic reproduction number less than unity is not enough to eradicate the disease. Stability or instability of different endemic equilibria has been shown analytically. The system also experiences the saddle-node and Hopf bifurcation. The existence of Bogdanov-Takens bifurcation (BT) of co-dimension 2 has been investigated which has also been shown through numerical simulations. Here we have used two control functions, one is vaccination control and other is treatment control. We have solved the optimal control problem both analytically and numerically. Finally, the efficiency analysis has been used to determine the best control strategy among vaccination and treatment.
Highlights
Mathematical modelling has been increasingly recognized as an important tool for understanding the transmission processes of different infectious diseases
If R0 = 1, the disease free equilibrium point (DFE) is non-hyperbolic equilibrium point and it is shown that it is stable by using Centre Manifold Theory
It is shown that DFE is globally asymptotically stable if R0 < 1 and the transmission rate of infection less than some quantity, which signifies that the disease will die out for low transmission rate along with basic reproduction number R0 < 1
Summary
Mathematical modelling has been increasingly recognized as an important tool for understanding the transmission processes of different infectious diseases. We have considered an SIR model with logistic growth rate of susceptible population, a nonmonotonic incidence rate of the form βSI 1+αI 2 and a saturated treatment function of the form T (I) =. The treatment control parameter have considered two controls, one is vaccination control u1 and other is treatment control u2 These two controls have been used to address the question of how to optimally combine the vaccination and treatment strategies to minimize the susceptible and infected population as well as the cost of implementation of these two interventions.
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