Properties Analysis Of Two Malaria Transmission Models

Malaria is the world’s most important tropical parasitic disease currently, and it kills much more people than any other communicable disease except tuberculosis. Today, malaria is a public health problem. It involves more than ninety countries, 2.4 billion people around the world to 40 percent ratio. Every year,350 to 500 million cases of malaria occur worldwide,and one million people died, most of them are children in sub-Saharan African countries. At the same time, past of Asia, Latin America, Middle East, and European countries have also been affected by malaria. Faced with the severe epidemic situation, the World Health Organization and various countries have taken appropriate measures to control the spread of malaria. Currently, there are many models of malaria transmission been studied. But, vertical transmission is a neglected areas. Many reports say, the proportion of new born babies suffering from malaria is increasing. In this thesis, according to the model thought of epidemic dynamics method, the mechanism of the spread of the malaria, and other factors, two malaria dynamics transmission models were built. We make some qualitative analysis of the mathematical models and give some suggestions to control malaria.This article is divided into four chapters. The first chapter introduces malaria, the epidemic of malaria in China, the basic concepts of infectious disease dynamics, and the way to calculate the basic reproductive number.In chapter two, we establish and study malaria model with vertical transmis-sion. By constructing Lyapunov function, we get the result that the disease-free equilibrium is globally asymptotically stable when the basic reproductive number is less than unity. When the basic reproductive number is greater than unity, the disease-free equilibrium is unstable and only one endemic equilibrium exists. Use security number and the stability theorem of fourth-order real matrix partial, we prove the stability of the endemic equilibrium. The results of numerical simulation and the actual situation is consistent. through numerical simulation results with the basic reproduction number newborns born to infected increased rate increases. According to the theoretical results, we discuss some strategies for malaria control.In chapter three, a congenital malaria dynamic model with age structure is proposed, the people of childbearing age, and the remaining part. We get the basic reproductive number by the method provided by Van den Driessche and Watmough. Using Routh-Hurwitz criterion, we prove that the disease free equilibrium point is local stability. By numerical simulation, the existence and stability of the endemic equilibrium are proved, and when the ratio of the newborns are susceptible and infected is very low, the infected mosquitos and childbearing is positively correlated. When the ratio of the newborns are susceptible and infected increases, the proportion of child-bearing age who are ill will be greater than before the proportion of the remaining portion of the infected.In chapter four, we arrive at a conclusion by summarizing the current work and setting the research targets afterwards.