Backward bifurcation and optimal control of a vector borne disease

This paper deals with a simple mathematical model for the transmission dynamics of a vector-borne disease that incorporates both direct and indirect transmission. The model is analyzed using dynamical systems techniques and it reveals the backward bifurcation to occur for some range of parameters. In such cases, the reproduction number does not describe the necessary elimination effort of disease rather the effort is described by the value of the critical parameter at the turning point. The model is extended to assess the impact of some control measures, by re-formulating the model as an optimal control problem with density-dependent demographic parameters. The optimality system is derived and solved numerically to investigate that there are cost effective control efforts in reducing the incidence of infectious hosts and vectors.