Dynamics Of An Age-structured Malaria Model With Sterile Mosquitoes

Malaria is one of the world's most important mosquito-born diseases, and it can be transmitted between humans by female mosquitoes. Now, about half of the world's pop-ulation are at risk of malaria, children under the age of five were at high risk, and about two-thirds of all the deaths from malaria were children under the age of five. An effective way to prevent the mosquito-born diseases is to control mosquitoes. Among the mosquito control measures, the sterile insect technique was proved as an effective way. The sterile mosquito can affect the mating rate between wild mosquitoes. In this paper, we present an age-structured malaria model with sterile mosquitoes, and the population was divided into two parts: the pre-school children and the remaining population, then research the effect on the spread of malaria.This paper consists of five chapters. In the first chapter, it introduced the background of malaria research, the purpose of this research, the research status and the research content.The background included three groups: the introduction of malaria, malaria in the world and in China. In the second chapter, we introduced some basic knowledge that will be used in this paper.The third chapter established an age-structured model with constant release of sterile mosquitoes and the mating rate was constant also. We got the basic reproduction number R0 by the next generation matrix method. R0 was consisted of two parts: the spread of the disease caused by the pre-school children and the remaining population, both of which were decreased with the increase of the releasing of sterile mosquitoes within a certain range, but the releasing of sterile mosquitoes had a greater impact on the remaining part.The stability of the disease-free equilibrium could be got by the R0; The more the number of sterile mosquitoes released to the environment the less the number of the sick that got to reach the balance in a certain range; Finally, we found an interval about the number of the sterile mosquitoes, in this interval, under certain conditions, wild mosquitoes still exist,but the disease will disappear.Taking into account that the producing of sterile mosquitoes cost time, labor and money, when the number of wild mosquitoes is very small , constant releasing of sterile mosquitoes will cause a waste. When the number of the wild mosquitoes is very small there would appeared mating difficult, so in the fourth chapter we established an age-structured model, with releasing of sterile mosquitoes proportional to wild mosquitoes,and the mating rate was considered as the Allee effect. Using the next generation matrix method we obtained the basic reproduction number, and the reproduction number was neg-atively correlated with k in a certain interval. The more the ratio of sterile mosquitoes to wild mosquitoes the less the number of the sick that got to reach the balance in a certain interval. The results showed that the impact of different age groups on the spread of malar-ia was different, and an interval of proportional k is given. In this interval, under certain conditions, wild mosquitoes still exist, but the disease will disappear.