Bifurcation Thresholds and Optimal Control in Transmission Dynamics of\n Arboviral Diseases

Abstract

In this paper, we derive and analyse a model for the control of arboviral\ndiseases which takes into account an imperfect vaccine combined with some other\nmechanisms of control already studied in the literature. We begin by analyse\nthe basic model without controls. We prove the existence of two disease-free\nequilibrium points and the possible existence of up to two endemic equilibrium\npoints (where the disease persists in the population). We show the existence of\na transcritical bifurcation and a possible saddle-node bifurcation and\nexplicitly derive threshold conditions for both, including defining the basic\nreproduction number, R 0 , which determines whether the disease can persist in\nthe population or not. The epidemiological consequence of saddle-node\nbifurcation (backward bifurcation) is that the classical requirement of having\nthe reproduction number less than unity, while necessary, is no longer\nsufficient for disease elimination from the population. It is further shown\nthat in the absence of disease--induced death, the model does not exhibit this\nphenomenon. We perform the sensitivity analysis to determine the model\nrobustness to parameter values. That is to help us to know the parameters that\nare most influential in determining disease dynamics. The model is extended by\nreformulating the model as an optimal control problem, with the use of five\ntime dependent controls, to assess the impact of vaccination combined with\ntreatment, individual protection and vector control strategies (killing adult\nvectors, reduction of eggs and larvae). By using optimal control theory, we\nestablish optimal conditions under which the disease can be eradicated and we\nexamine the impact of a possible combined control tools on the disease\ntransmission. The Pontryagin's maximum principle is used to characterize the\noptimal control. Numerical simulations, efficiency analysis and cost\neffectiveness analysis show that, vaccination combined with other control\nmechanisms, would reduce the spread of the disease appreciably, and this at low\ncost.\n