Mathematical Analysis of Vector-Host Diseases Models

Abstract

The emphasis of this dissertation lies on the theoretical study of different models of vectorborne diseases in order to get better understanding of the transmission and spread of these diseases. The patterns of infection in the host population can be understood more precisely if we comprehend those factors that influence the transmission of the disease. Five mathematical models are presented in this dissertation. Four of these explore the dynamics of the disease in relation to human population and mosquitoes. One model is dedicated to pine wilt disease in which hosts are pine trees and vectors are bark beetles. The dynamics of vector-borne diseases are explored on three scales. First, various mathematical models are constructed by using ordinary differential equations. These models are developed by considering bilinear contact rates, nonlinear incidence rates and standard incidence rates. The models explore direct as well as vector mediated transmission. In mathematical model of pine wilt disease, it is considered that susceptible beetles (vectors of pine wilt disease) receive infection directly from infectious ones through mating. Next, the global behavior of equilibria of models are analyzed. The analytical expressions for the basic reproduction number R0 are obtained and global dynamics of the models are completely described by this number. Using Lyapunov functional theory it is proved that the disease-free equilibria are globally asymptotically stable whenever R0 ≤ 1. The geometric approach is utilized to study the global stabilities of endemic equilibria whenever the basic reproduction number exceeds unity. Finally, in order to assess the effectiveness of disease control measures, the sensitivity analysis of the basic reproductive number R0 and the endemic proportions with respect to epidemiological and demographic parameters is provided. This sensitivity analysis provide an aid to design effective control strategies. It may be an important tool in the decision support system.