A mathematical analysis of Zika virus transmission with optimal control strategies

In this paper, we formulate and analyze a system of ordinary differential equations of Zika virus disease by considering standard incidence type interaction for the human to human transmission.Primarily, the Zika virus is transmitted to humans through the bite of infected Aedes mosquitoes. This vector-borne disease can be spread through sexual transmission and blood transfusions also. The proposed model is an SEIR type model for the human population and an SEI type model for mosquito populations. The equilibria of the proposed model are found and the basic reproduction number $R_0$ is computed. The local and global stability of the disease is investigated theoretically in detail. The backward bifurcation is presented for $R_0<1$, which suggests that the disease is possible to eliminate from the population. A sensitivity analysis of the parameters of the basic reproduction number for our model is established. Incorporating three types of time-dependent control parameters in the system to reduce the transmission. Electronic devices, insecticide-treated bed nets, and mosquito repulsive lotions are used to reduce mosquito biting rates. Considering that fact, we find the appropriate optimal control strategies to eliminate the virus from the tropical area. To exhibit the optimal control strategies we used the Pontryagin's Maximum Principle. It was noticed that the optimal control gives better result than without the optimal control model. Numerical simulation is presented to support our mathematical findings.