Global Stability Of Two Classes Of Dengue Epidemic Models

Dengue is an acute infectious disease caused by the dengue virus, mainlyspread by aedes, and has become the second harmful disease after malaria. Since the20th century, dengue disease transmission has three peaks, besides, the geograp-hical distribution is wide, and the last peak occurs about1995. Nowadays, thegovernment departments around the world has paid great attention on the disease,and took effective prevention and control measures, such as large scale anti-mosquito, laboratory-based surveillance, and the vaccine invention, etc. However, itcannot be completely wiped out. Therefore, the study of dengue model and thecontrol of the spread are imperative.In this paper, two classes of the global stability of dengue infection are studied.Based on some reasonable assumption conditions, the first part of this article istalking about dengue disease transmission model with diffusion. Using the theory ofpartial differential equations, such as positivity lemma and the comparison principle,positive and uniformly bounded of the solution of model are proved. By Lyapunovmethod, the method of graph theory and the second Green’s formula, sufficientconditions for global asymptotical stability of the disease-free equilibrium and theendemic equilibrium are given. Finally, several simulation figures are illustrated toshow the justifications of the theoretical conclusions.In the second part of this article, a dengue discrete numerical transmissionmodel with nonlinear incidences is constructed by applying nonstandard finitedifference methods. Firstly, it is proved that numerical solution is unconditionallypositive and uniformly bounded. Then, by the knowledge of graph theory andLyapunov method, its global stability is studied. When the basic reproductivenumber is less than or equal to1, the disease-free equilibrium is the uniqueequilibrium and it is globally asymptotic stable; otherwise, when it is greater than1,the positive endemic equilibrium of system exists and it is globally asymptoticallystable while the disease free equilibrium is unstable, and the global stability is notrestricted by the time step. It implies that the discrete model can maintain somedynamic behaviors of the original continuous system, such as positive, uniformlybounded and the global stability. At last, numerical simulation is also presented toverify the obtained conclusions.