Stability Analysis and Optimal Control Strategies of Giving Up Relapse Smoking Model with Bilinear and Harmonic Mean Type of Incidence Rates

Abstract

In this manuscript, a giving-up smoking model is develop using bilinear incidence rate, harmonic mean type of incidence rate and keeping in view the relapse factor associated to smoking. It is shown that the model' solution is bounded and positive for the appropriate initial data. The equilibria of the model are obtained, and it is proved that the smoking-free equilibrium is both locally and globally asymptotically stable for $R_0$ less than unity. It is shown that the model has a positive light smoker present equilibrium whenever $R_0$ is greater than one, and it is locally asymptotically stable if we have $1<R_0<1+2{\beta }_1\left(\mu +d+\delta \right)/\Lambda {\beta }_2$ . Conditions for the global stability of light smoker present equilibrium are rigorously investigated. Also, it is proved that the model has a smoking present equilibrium when $R_0$ satisfies some condition which is investigated both for local and global behavior. By considering a few control measures, optimal control strategies are achieved with the help of Pontryagin’s maximum principle. The analytical results are verified numerically, and effectiveness of the control program is presented.