Bifurcation thresholds and optimal control in transmission dynamics of arboviral diseases

In this paper, we derive and analyse a model for the control of arboviral diseases which takes into account an imperfect vaccine combined with some other control measures already studied in the literature. We begin by analysing the basic model without control. We prove the existence of two disease-free equilibrium points and the possible existence of up to two endemic equilibrium points (where the disease persists in the population). We show the existence of a transcritical bifurcation and a possible saddle-node bifurcation and explicitly derive threshold conditions for both, including defining the basic reproduction number, , which provides whether the disease can persist in the population or not. The epidemiological consequence of saddle-node bifurcation is that the classical requirement of having the reproduction number less than unity, while necessary, is no longer sufficient for disease elimination from the population. It is further shown that in the absence of disease-induced death, the model does not exhibit this phenomenon. The model is extended by reformulating the model as an optimal control problem, with the use of five time dependent controls, to assess the impact of vaccination combined with treatment, individual protection and two vector control strategies (killing adult vectors and reduction of eggs and larvae). By using optimal control theory, we establish conditions under which the spread of disease can be stopped, and we examine the impact of combined control tools on the transmission dynamic of disease. The Pontryagin's maximum principle is used to characterize the optimal control. Numerical simulations and efficiency analysis show that, vaccination combined with other control mechanisms, would reduce the spread of the disease appreciably.