Backward bifurcation in a smoking cessation model with media campaigns

Nowadays, most of the public policies to manage the social issues are focusing on motivating people via media campaigns. In view of this, we propose and analyze a nonlinear mathematical model to study the effect of media campaigns on smoking cessation. The equilibria of the model have been obtained and their stability discussed. Using center manifold reduction theory, we reduce the proposed model to a system of lower dimension. The reduced system contains all the necessary information regarding the asymptotic behavior of small solutions of the original system. The analysis shows that on changing one parameter of the system (reproduction number, R, which depends on various other parameters), two different manifolds of fixed points cross each other and transcritical bifurcation occurs. Further, for large value of relapse rate the bifurcation is subcritical (backward). This shows that requirement R<1 is only necessary, but not sufficient, for smoking cessation. Numerical simulation also supports the analytically obtained results.