This paper considers the Susceptible-Infected-Vaccinated-Recovered (SIRV) deterministic model with a non linear force of infection and treatment, where individual humans that are vaccinated losses their vaccination after some time and become vulnerable to infections. The basic reproduction number R 0 obtained from the model system is an epidemic threshold that determines if a disease will continue to ravage the human population or not. The model state equations considered in this paper possess two steady-state solutions such that if R 0 < 1, the infection-absent steady-state solutions are locally and globally asymptotically stable. Also, if R 0 > 1, a unique infection-persistent steady-state solutions are established, which is also locally and globally asymptotically stable. Thus, it leads to the persistence of infections in the human host population. Finally, numerical simulations were carried out to validate our theoretical results.

In this paper, a deterministic optimal control problem involving a Susceptible - Vaccinated - Infected - Recovered (SVIR) epidemic model is considered. The optimal control problem is characterized using the Pontryagin’s maximum principle involving three control strategies namely, social mobilization, screening and sanitation. The derived optimality system is numerically solved using the forward - backward Runge - Kutta fourth order method via the computational software matlab. The numerical simulations depict that each of the control strategy has its significance in minimizing the spread of diseases, but the optimal combination of these controls are more effective in stemming the emergence and spread of an epidemic.