Analysis and Optimal control of deterministic Vector-Borne diseases model

Abstract

In this dissertation, two systems of deterministic differential equations are introduced to study the transmission of vector-borne diseases between a host and a vector. The total population of the host and the vector are divided into different compartments. The total population of host is divided into susceptible , exposed, infected, and treated groups. The total population of vector is divided into susceptible, exposed, and infected. In chapter 2, we introduce a model to study vertically transmitted vector-borne diseases with nonlinear system of differential equations. We analyze the model by finding the disease free equilibrium point E0 and deriving the basic reproductive number R0 by using the next generation matrix method. We study the local and global stability of E0 and how the stability is related to R0. We study the sensitivity of R0 using the normalized forward sensitivity index and find the relation to the parameters in the model. We have numerical simulations to show the result we get from the analysis based on the dengue virus. In chapter 3, we introduce a model to study optimal control to find the best way to control viruses. We introduce two optimal controls, the prevention of contact between host and vector u1 and the treatment of host u2 in the model given in the chapter 2. We consider a cost functional related to the cost of the prevention and the treatment. We try to minimize the number of exposed and infected host groups and maximize the number of susceptible and treated host groups. We show the existence of u1 and u2 by using Carathodory’s existence theorem. We find the explicit formula of u1 and u2 with the status variables and the adjoint variables from Hamiltonian by using Pontryagin’s maximum principle. We find the numerical values of u1 and u2 by solving the given status system and the adjoint system derived from Hamiltonian. We use the forward-backward sweep method and the Runge-Kutta method in 3-dimension to solve the status system and the adjoint system. In the numerical simulation, we compare the result between the controlled case and uncontrolled case for each host and vector groups. Also we see which control among u1 and u2 is more effective to control the virus. In appendices, we show the Matlab code used in the numerical simulation.