Qualitative Analysis Of Three Malaria Transmission Models

Malaria, a mosquito-born disease, caused a wide attention all over the world, it hurts our health and distructs our life seriously. Malaria is transmitted by blood, mainly by blood-feeding Mosquitoes. At present, the ways of controlling malaria focuses on controlling the quantity of mosquitoes. But because of the resistance to drugs, the traditional method of use pesticides to kill mosquitoes appears ineffec-tively to control the spread of malaria. Moreover, widely spraying insecticides may cause environmental pollution, which will lead to serious environmental problems. So more effective and safe method to control the malaria should to be established. In this paper, three mathematical models are established based on the dynamics of the malaria transmission and considering the influence of treatment, vaccines and the environmental capacity of mosquitoes. In this paper, we make some qualitative analysis of the mathematical models and give some suggestions to control malaria.This article is divided into six chapters. The first chapter introduces the preva-lence of malaria in the world and in China. Recent research works on the mathe-matical models of malaria are also introduced in this chapter. The second chapter provides some preliminary knowledge, which is the theoretical foundation for re-searching the malaria transmission in the study.In the third chapter, we set up and studied the transmission model of malaria with constant immigration rate and incomplete treatment. By calculation we get the basic reproductive number R0. Lyapunov functions are also constructed to get a critical value Ra. When Ra is less than one, the disease-free equilibrium is globally asymptotically stable. When the basic reproductive number is larger than one, the system has an endemic equilibrium. By calculation, we know the model may have two equilibria, one equilibrium or no equilibrium when the basic reproductive number is less than one. So we can get the existence conditions of backward bifurcation in this model. We also analyzes the influence of the treatment r in the malaria transmission model. When r is large enough, we can make R0 smaller than one and Ra smaller than one. By this way the prevalence of malaria can be controlled. At present, in the most area of Africa and southeast Asia, there is a large proportion of people suffering from malaria and did not get effective treatment, our work shows that the treatment can greatly effect the spread of malaria.In the fourth chapter, we consider a malaria transmission model with immune class. Assume that the malaria vaccine got promotion, on the basis of the origi-nal model of we joined the immune class and get a new model. By constructing Lyapunov function, it is concluded that when Rb is less than one, the disease-free equilibrium is globally asymptotically stable. When the basic reproductive number is greater than one, there exists a positive equilibrium. Through the numerical sim-ulation we find that when the basic reproductive number is less than one, there are likely to be positive equilibrium point of the model. In addition, we analyzed how the coverage and the effectiveness of the vaccine effect the basic reproductive num-ber, and we found that the vaccine is always play a positive role. Although there is no effective malaria vaccine currently, our work shows the importance of vaccine in malaria control. Therefore, develop an effective vaccine is still an important task in malaria control.In the fifth chapter, we discussed a malaria transmission model with logistic growth mosquitoes. Assume that part of the new newborn babies got vaccined, we established a new model of malaria. By constructing Lyapunov function, we get that when Rc is less than one, the disease-free equilibrium is globally asymptotically stable. When the basic reproductive number is greater than one, there exists a posi-tive equilibrium. By numerical simulation, we find that when the basic reproductive number is less than one, there may be positive equilibrium point of the model. In addition, we also analyzed the influence of the coverage of the new born vaccina-tion, we found that the higher the coverage is, the less basic reproductive number is and the higher the efficient of the vaccine is, the smaller the basic reproductive number will be. In our model, the basic reproductive number is positive correlate with the environmental capacity of mosquitoes k, so improving the vaccination rates and reducing the environmental capacity of mosquitoes, will be effective measures to control the disease.In the sixth chapter, we discuss the main conclusions of this paper, and give some suggestions to control the malaria. We also point out the deficiency of this paper and the research works will be carried out in the future.