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In this paper, we derive and analyse a model for the control of arboviraldiseases which takes into account an imperfect vaccine combined with some othermechanisms of control already studied in the literature. We begin by analysethe basic model without controls. We prove the existence of two disease-freeequilibrium points and the possible existence of up to two endemic equilibriumpoints (where the disease persists in the population). We show the existence ofa transcritical bifurcation and a possible saddle-node bifurcation andexplicitly derive threshold conditions for both, including defining the basicreproduction number, R 0 , which determines whether the disease can persist inthe population or not. The epidemiological consequence of saddle-nodebifurcation (backward bifurcation) is that the classical requirement of havingthe reproduction number less than unity, while necessary, is no longersufficient for disease elimination from the population. It is further shownthat in the absence of disease--induced death, the model does not exhibit thisphenomenon. We perform the sensitivity analysis to determine the modelrobustness to parameter values. That is to help us to know the parameters thatare most influential in determining disease dynamics. The model is extended byreformulating the model as an optimal control problem, with the use of fivetime dependent controls, to assess the impact of vaccination combined withtreatment, individual protection and vector control strategies (killing adultvectors, reduction of eggs and larvae). By using optimal control theory, weestablish optimal conditions under which the disease can be eradicated and weexamine the impact of a possible combined control tools on the diseasetransmission. The Pontryagin's maximum principle is used to characterize theoptimal control. Numerical simulations, efficiency analysis and costeffectiveness analysis show that, vaccination combined with other controlmechanisms, would reduce the spread of the disease appreciably, and this at lowcost.

Bifurcation Thresholds and Optimal Control in Transmission Dynamics of Arboviral Diseases