Mathematical modeling and optimal control of a vector borne disease

Abstract

This dissertation is concerned with mathematical modeling and optimal control of a vector borne disease. We derive and rigorously analyze mathematical models to better understand the transmission and spread of vector borne diseases. First, a mathematical model is formulated to evaluate the impact of biological control of a vector borne disease "malaria" by considering larvivorous fish as a sustainable larval control method. To evaluate the potential impacts of this biological control measure on malaria transmission, we investigate the model describing the linked dynamics between the predator-prey interaction and the host-vector interaction. The dynamical behavior with all possible equilibria of the model is presented. The model also exhibits backward bifurcation phenomenon which give rise to the existence of multiple endemic equilibria. The backward bifurcation phenomenon suggests that the reproductive number R0 < 1 is not enough to eliminate the disease from the population under consideration. So an accurate estimation of parameters and level of control measures is important to reduce the infection prevalence of malaria in an endemic region. Our control techniques for elimination of malaria in a community suggest that the introduction of larvivorous fish can in principle have important consequence for the control of malaria but also indicate that it would require a strong predator on larval mosquitoes. Then, a new epidemic model of a vector-bornediseasewhichhasbothdirectandthevectormediatedtransmissionsis considered. The model incorporates bilinear contact rates between the mosquitoes vector and the humans host populations. Using Lyapunov function theory some sufficient conditions for global stability of both the disease-free equilibrium and the endemic equilibrium are obtained. We derive the basic reproduction number ℜ0 and establish that the global dynamics are completely determined by the values of ℜ0. For the basic reproductive numberℜ0 < 1, the disease free equilibrium is globally asymptotically stable, while for ℜ0 > 1, a unique endemic equilibrium exists and is globally asymptotically stable. The model is extended to assess the impact of some control measures, by using an optimal control theory. In order to do this, first we show the existence of the control problem and then use both analytical and numerical techniques to investigate that there are cost effective control efforts for prevention of direct and indirect transmission of disease. Finally, we present complete characterization and numerical simulations of the optimal control problem. In order to illustrate the overall picture of the epidemic, individuals under the optimal control and without control are shown in figures. Our theoretical results areconfirmedbynumericalsimulationsandsuggestapromisingwayforthecontrol of a vector borne disease.