The Qualitative Analysis Of Two Types Of Epidemic Disease Transmission Models

Malaria is one of the most widely popular diseases in the world,mainly in Africa.In 2015,91 countries and areas had ongoing malaria transmission.Malaria is a kind of mosquito-borne diseases,due to the widespread application of antimalarial drugs,drug resistance had formed in parasites,which makes it more difficult to control malaria transmission,but the latest research shows that a special drug resistance cannot be spread by mosquitoes,this feature may be applied to control the spread of malaria.HIV,as a major global public health problem is also mainly in Africa,so the malaria and HIV co-infection is a problem that we have to focus on now.This paper is divided into four chapters.In chapter 1,we introduce the curren-t situation of malaria and HIV,the research background,the research status and related mathematical knowledge.In chapter 2,a model of the drug resistance about malaria cannot be spread by mosquitoes is introduced.By means of calculation,we obtain two threshold R0,R*.When R0<1,the disease-free equilibrium is locally asymptotically stable;when R0>1,the system has a unique positive equilibrium;when RO<1 and k3>k2+ k4,if R*>1,the system exists positive equilibrium,if R*<1,the system does not exist positive equilibrium;then,the paper proves the existence of backward bifurcation,analyzes the effect of resistance generating rate ?h and malaria patients' recovery rate ?i(i = 1,2)on the spread of malaria.Finally,some numerical simulations are performed to confirm our analytic results.In chapter 3,a model of malaria and HIV co-infection is introduced.By mean-s of calculation,we obtain several kinds of basic reproduction number R0,R0H,R0M,R0M(EH),R0H(Em).When R0<1,the disease-free equilibrium is a locally asymptotically stable;when R0H>1,the system has a unique HIV only endemic equilibrium EH and the equilibrium is locally asymptotically stable;when R0M>1,the system has a unique malaria only endemic equilibrium;when R0M<1 and h3>h2 + +h4,if R*? 1,the system exists positive equilibrium;when R0H>1,if R0M(EH)<1,the malaria-free equilibrium EH is stable,only HIV persists,if R0M(EH)>1,the malaria-free equilibrium EH is unstable,malaria can invade and persist in a population;when R0M>1,if R0H(EM)<1,the HIV-free equilibrium EM is stable,only malaria persists,if R0H(Em)>1,the HIV-free equilibrium EM is unstable,HIV can invade and persist in a population.Finally,by means of numerical simulation,we get that the malaria treatment to I2m has a positive effect on the treatment for HIV and in the case of malaria treatment,malaria has a suppressive effect on the transmission of HIV.In chapter 4,we summarize the main conclusions of this article and point out the deficiency of this article and the direction of this research in the future.