Analysis of stochastic vector-host epidemic models

Abstract

In this dissertation, deterministic and stochastic mathematical models are proposed to study vector-host epidemic models with direct transmission. The total population of the host and the vector is divided into different compartments as susceptible hosts, infected hosts, susceptible vectors and infected vectors. In the first chapter, we model and study the deterministic vector-host epidemics with direct transmission using a nonlinear system of differential equations. First we obtain the disease-free equilibrium point E_0 and the endemic equilibrium point E_1. After that we derive the basic reproductive number R_0, and study the local and global stabilities of E_0 and E_1 in relation to R_0: Using the perturbation of fixed point estimation, we investigate the sensitivity of the basic reproductive number in relation to the parameters used in the model. Next by adding environmental fluctuations to the deterministic model, we obtain a nonlinear system of stochastic differential equation that describes the dynamics of the stochastic vector-host epidemic model. By defining a stochastic Lyapunov function, we prove the existence of a unique non negative global solution to the stochastic model. Moreover, we show that the solution of the stochastic model is stochastically ultimately bounded and stochastically permanent. Similar to the deterministic case, we obtain the basic reproductive number for the stochastic model R_0^s and we show that the infection will die out or persist depending on the value of R_0^s: In particular, we show that random effects may lead to extinction in the stochastic case while the deterministic model predicts persistence. We also present necessary conditions for the infection to be persistent in the stochastic model. Finally we present a stochastic vector-host epidemic model with direct transmission in random environment, governed by a system of stochastic differential equations with regime-switching diffusion. We first examine the existence and uniqueness of a non negative global solution. Then we investigate stability properties of the solution, including almost sure and p^th moment exponential stability and stochastic asymptotic stability. Moreover, we provide conditions for the existence and uniqueness of a stationary distribution. In all the chapters, we provide numerical simulations and examples to illustrate some of the theoretical results.