Two Types Of Infectious Disease Model Stability And Branch Of Research

Epidemic dynamics is to study disease spread, predict the trends of the disease, fnd outthe key factors in the process of the spread, and prevent the disease by the best methods.In this paper, we take into account diferent factors and create two SIR models and an SIVepidemic model.Firstly, we introduce an epidemic model with the population growth rate of Logisticsfunction and the treatment function of piecewise function. It is assumed that treatment rateis proportional to the number of infectives below the capacity and is a constant when thenumber of infectives is greater than the capacity. First, we provide the basic reproductionnumber and fnd out the conditions of the existence of endemic equilibria; Secondly, wediscuss the local and global asymptotic stability of disease-free equilibrium; thirdly, we provethe local stability of endemic equilibrium, and use the Lyapunov function obtained the globalstability; Finally, we perform the numerical simulation.Also，we investigate the stability and the bifurcation curve of an SIV model for diseasetransmission with vaccination. In particular, we provide conditions for the existence ofmultiple endemic equilibria and backward bifurcations. We extend the results to includemodels in which the parameters may depend on the level of infection. The global stabilityof the equilibrium are proved by constructing appropriate Lyapunov function.Finally，an SIR epidemic model with saturated incidence rate and saturated treatmentfunction is investigated. The stability and the bifurcation curve of which are also obtained.Here continuous and diferentiable functions are employed as the incidence rate and treat-ment function, which can give rise to backward bifurcation. The global dynamics of themodel suggest that the basic reproduction number being the unity is a strict threshold fordisease eradication when such efect is weak. However, it is shown that a backward bifurca-tion will take place when this delayed efect for treatment is strong. Thus, driving the basicreproduction number below the unity is not enough to eradicate the disease. In addition, acritical value at the turning point is deduced as a new threshold. Some sufcient conditions for the disease-free equilibrium and the endemic equilibrium being globally asymptoticallystable are also obtained.