Modeling And Studying Of Cholera And Infectious Diseases With Media Coverage

The dynamics of infectious diseases is an important method of studying the spread ofinfectious diseases qualitatively and quantitatively. It’s based on the specific property ofpopulation growth, to construct mathematical models reflecting the dynamic properties ofinfectious diseases, to analysis the dynamical behavior and to conduct numerical simulations.The research results are helpful to reveal the epidemic rules of infectious diseases, to predictthe developing tendency of infectious diseases, to seek the strategies of preventing andcontrolling the spread of infectious diseases. In this paper, we use the ordinary differentialequations to describe the two kinds of infectious diseases dynamical model. At the same time,some dynamical behaviors are discussed, including the positive invariant region of thesolutions, the existence and stability of equilibria, the persistence and extinction of dynamicalsystem, etc.. Besides, the biological significance is also discussed. Two aspects of maincontent are as follows:In chapter2, a cholera model with vaccination is investigated. A autonomous andfive-dimensional SVIR-B model is proposed. The control reproduction numberRv whichdetermines whether the diseases is extinct or not is permanent. When Rv1, it is proved thatthe disease-free equilibrium is local asymptotically stable by using the Routh-Hurwitzcriterion. Some sufficient conditions of global asymptotic stability of the disease-freeequilibrium are obtained, which means the cholera will tend to extinct; When Rv1, thesystem has a unique endemic equilibrium. Using the Routh-Hurwitz criterion, the localasymptotic stability of the endemic equilibrium is proved. The system is uniformly persistent.And some sufficient conditions are obtained by using compound matrices theory. Then underthe above conditions, the endemic equilibrium is global asymptotically stable and the cholerawill be permanent. Finally, we perform sensitivity analysis ofRv on the parameters, presentsome numerical simulations and propose that the cholera will go to extinction until both thevaccinated rate and the wanning rate reach some critical value simultaneously.In chapter3, a dynamic system with media coverage is investigated. We discuss a SIRSmodel with general contact function and obtain the basic reproduction numberR0. WhenR01, it is proved that the disease-free equilibrium is local asymptotically stable by using theRouth-Hurwitz criterion, then the infectious diseases will die out. WhenR01, using the zerotheorem, it is proved that a unique endemic equilibrium exists. Using the Routh-Hurwitzcriterion, the local asymptotic stability of the endemic equilibrium is proved. UsingBendixson criterion, we show some sufficient conditions of global asymptotic stability of theendemic equilibrium. Moreover, the infectious diseases will go to endemic. In the end, weconduct numerical simulations and indicate the impact of media coverage on the spread andcontrol of infectious diseases.