Transmission dynamics of some epidemiological patch models

of a dissertation at the University of Miami. Dissertation supervised by Professor Shigui Ruan. No. of pages of text. (111) As we known, infectious diseases can be transmitted from one region to another due to extensive travel and migration. Meanwhile, different regions have different demographic and epidemiological characteristics. To capture these features, multi-patch epidemic models have been developed to study disease transmission in heterogeneous environments. These models are usually described by a system of differential equations with the dynamics of each patch coupled to that of other patches by population dispersal. Typically, a patch may represent a city, a village, or a biological habitat. A full understanding the effect of travel on the spatial spread of disease between patches can definitely improve disease control and prevention measures. In this thesis, we propose several epidemic models in a patchy environment to investigate some specific epidemiological problems. In the first chapter , a susceptible-infectious-susceptible patch model with nonconstant transmission coefficients is formulated to investigate the effect of media coverage and human movement on the spread of infectious diseases among patches. The basic reproduction number R0 is determined. It is shown that the disease-free equilibrium is globally asymptotically stable if R0 ≤ 1, and the disease is uniformly persistent and there exists at least one endemic equilibrium if R0 > 1. In particular, when the disease is nonfatal and the travel rates of susceptible and infectious individuals in each patch are the same, the endemic equilibrium is unique and is globally asymptotically stable as R0 > 1. Numerical calculations are performed to illustrate some results for the case with two patches. In chapter 2, we propose a multi-patch model to study the effects of population dispersal on the spatial spread of malaria between patches. The basic reproduction number R0 is derived and it is shown that the disease-free equilibrium is locally asymptotically stable if R0 1. Bounds on the disease-free equilibrium and R0 are given. A sufficient condition for the existence of an endemic equilibrium when R0 > 1 is obtained. For the two-patch submodel, the dependence of R0 on the movement of exposed, infectious, and recovered humans between the two patches is investigated. Numerical simulations indicate that travel can help the disease to become endemic in both patches, even though the disease dies out in each isolated patch. However, if travel rates are continuously increased, the disease may die out again in both patches. In Chapter 3, based on the classical Ross-Macdonald model, we propose a periodic malaria model to incorporate the effects of temporal and spatial heterogeneity in disease transmission. We define the basic reproduction number R0 and show that either the disease-free periodic solution or the positive periodic solution is globally asymptotically stable. Numerical simulations are conducted to confirm the analytical results. Chapter 4 is devoted to studying the spatial spread of Rift Valley fever in Egypt. We propose a three-patch model for the process that animals enter Egypt from Sudan are moved up the Nile, and then consumed at the feast. The basic reproduction number for each patch is introduced and then the threshold dynamics of the model is established. We simulate an interesting scenario showing possible explanation to the observed phenomenon in Egypt. Finally, we summarize the main results of this thesis and list some possible future research directions in Chapter 5.