Numerical Analysis And Global Stability Of Two Kinds Of Epidemic Models

Vaccination is one of the commonly used measures in preventing and controlling thespread of some communicable diseases. Dengue fever, an endemic in many countries, hascaused great harm to the human health. As a result, it has been listed in the appendix of theinternational health regulations, which was revised in 2005. Various forms of epidemicmodel with vaccination and dengue fever epidemic model have been constructed andstudied. This paper makes a study on the global dynamic behaviors of discretized andmultigroup form of two typical kinds of models.Discretized epidemic models not only produce good approximation to exact solutionof continuous system, but also make better use of the infection statistics. However, thedynamic behavior of discretized system being very complex, the choice of the numericalschemes which can guarantee the global asymptotical stability for the endemic equilib-rium of the models remains a very important topic. But, when the endemic equilibriumof continuous system is globally asymptotically stable, the discretized model obtainedby various methods is persistent, and few results show that the endemic equilibrium isglobally asymptotically stable. There is less locally asymptotical stability.Firstly, a type of discretized vaccination system is constructed with non-standarddifference method and it is shown that the endemic equilibrium is globally asymptoticallystable. The dynamic behavior of discretized system is in keeping with the continuousthrough adaptive step size. The numerical simulation illustrates that the positivity ofnumerical solution may be destroyed and exhibits period two. Moreover, when the stepsize reaches a certain degree, a loss of stability is likely to occur. In order to overcomethis defect, we gain a new discretized model by transforming the form of nonstandarddifference. We obtain that the discretized system is unconditionally positive. In addition,for any step size, we produce that when the basic reproductive number is greater than 1,the endemic equilibrium is globally asymptotically stable by Lyapunov function.Secondly, with Lyapunov method, the global stability of fifth dimensional denguemodel is obtained. After that, we construct an unconditionally positive discretized systemby nonstandard di?erence method and prove that when the basic reproductive numberis less than 1, the disease free equilibrium is globally asymptotically stable, and whenthe basic reproductive number is greater than 1, the discretized system is persistent. A second-order discretized system is obtained by finite di?erence. It is proved that for anystep size, when the basic reproductive number is less than 1, the disease free equilibriumis locally asymptotically stable, which overcomes the defect that the discretized systemproduced by traditional second-order methods will lose stability with the growth of stepsize. However, numerical simulation shows that this second-order discretized system isnot unconditionally positive. Furthermore, a second-order nonstandard di?erence dis-cretized system is given by combining of nonstandard di?erence and finite di?erence.Numerical experiment demonstrates that through adjustment of the relevant parameters,the numerical solution can be guaranteed unconditionally positive.On the other hand, the study of multigroup infectious disease model is of great ne-cessity based on the spread of infection mechanism. However, due to the large scale andcomplexity of multigroup models, progress in the mathematical analysis of their globaldynamics is slow, and especially when the basic reproduction number is greater than 1,the uniqueness and global stability of the endemic equilibrium poses a problem. In thispaper, we also study multigroup form of the above two kinds of models. With the methodof Lyapunov and LaSalle invariant theory, we show that when the threshold value is lessthan 1 (or 0 ), the system has only one disease free equilibrium which is globally asymp-totically stable; when the threshold value is greater than 1 (or 0 ), besides disease freeequilibrium, the system also has one endemic equilibrium. The disease-free equilibriumis unstable and the endemic equilibrium is globally asymptotically stable. Therefore, wederive su?cient conditions that the multi-group form keep dynamics behavior of thesetwo kinds of models.