Global Dynamics of Some Malaria Models in Heterogeneous Environments

Malaria is one of the most important parasitic infections in humans and more than two billion people are at risk every year. There were an estimated 247 million malaria cases in 2006, causing nearly a million deaths. Currently, malaria is still endemic in 109 countries. Human malaria is caused by protozoan parasites of the genus Plasmodium, transmitted from human-to-human by the female Anopheles mosquito. Over the past century, considerable work has been invested in the study of malaria transmission. However, only a few studies with malaria consider the spatial and temporal heterogeneities of this disease. Hence, there is an essential need for more information on the spatial and temporal patterns of disease burden, distribution and control strategies. The aim of this thesis is to study the malaria transmission in heterogeneous environments.;Chapter 2 is devoted to the study of global dynamics of a periodic susceptible-infected-susceptible compartmental model with maturation delay. We first obtain sufficient conditions for the single population growth equation to admit a globally attractive positive periodic solution. Then we introduce the basic reproduction ratio R0 for the epidemic model, and show that the disease dies out when R0 < 1, and the disease remains endemic when R0 > 1. Numerical simulations are also provided to confirm our analytic results. The study in this chapter also enables us to consider time-delayed and periodic malaria models.;In chapter 3, we present a malaria transmission model with periodic birth rate and age structure for the vector population. We first introduce the basic reproduction ratio for this model and then show that there exists at least one positive periodic state and that the disease persists when R0 > 1. It is also shown that the disease will die out if R0 < 1, provided that the invasion intensity is not strong. We further use these analytic results to study the malaria transmission cases in KwaZulu-Natal Province, South Africa. Some sensitivity analysis of R0 is performed, and in particular, the potential impact of climate change on seasonal transmission and populations at risk of the disease is analyzed.;Based on the classical Ross-Macdonald model, we propose in chapter 4 a periodic model with diffusion and advection to study the possible impact of the mobility of humans and mosquitoes on malaria transmission. We establish the existence of the leftward and rightward spreading speeds and their coincidence with the minimum wave speeds in the left and right directions, respectively. For the model in a bounded domain, we obtain a threshold result on the global attractivity of either zero or a positive periodic solution.;We begin with a brief introduction of mathematical background for this thesis in chapter 1. We shall provide some mathematical terminologies and theorems related to the theories of monotone dynamical systems, uniform persistence, basic reproduction ratio, spreading speeds and traveling waves.;To understand how the spatial heterogeneity and extrinsic incubation period (EIP) of the parasite within the mosquito affect the dynamics of malaria epidemiology, we formulate a nonlocal and time-delayed reaction-diffusion model in chapter 5. We then define the basic reproduction ratio R0 and show that R0 serves as a threshold parameter that predicts whether malaria will spread. Furthermore, a sufficient condition is obtained to guarantee that the disease will stabilize at a positive steady state eventually in the case where all the parameters are spatially independent. Numerically, we show that the use of the spatially averaged system may highly underestimate the malaria risk. The spatially heterogeneous framework in this chapter can be used to design the spatial allocation of control resources.;At last, we summarize the results in this thesis, and also point out some problems for future research in chapter 6.