Some Epidemic Models With Nonlinear Incidence Rate And Treatment

In this paper, some epidemic models with nonlinear incidence rate and treatment are studied. In Chapter 1, an SIS epidemic model with staged treatment function is proposed where the disease-related death rate is negligible. The incidence rate of the model, which can include the bilinear incidence rate and the standard incidence rate, is a general nonlinear incidence rate. The global dynamics of the model are studied and then we can understand the effect of the capacity for treatment. It is found that a backward bifurcation occurs and there exist bistable endemic equilibria if the capacity is low. Mathematical results suggest that decreasing the basic reproduction number is insufficient for disease eradication and improving the efficiency and capacity of treatment is important for this end.In Chapter 2, an SIS epidemic model with staged treatment function and disease-related death rate is proposed. That is to say, we add a disease-related death rate to the model studied in the first chapter and the incidence rate is unchanged. The global analysis of this model becomes more complex. Through mathematical studies and simulations we find that there are still multiple endemic equilibria and the backward bifurcation in this model.In Chapter 3, an epidemic model with saturated incidence rate and saturated treatment function is studied. Here the treatment function adopts a continuous and differentiable function which can describe the effect of the infected being delayed for treatment when the number of infected individuals is getting larger and the medical condition is limited. The global dynamics of the model suggest that the basic reproduction number being one is a strict threshold for disease eradication when such effect is weak. However, it is shown that a backward bifurcation will take place when this delayed effect for treatment is strong. Therefore, it is not enough for us to drive the basic reproduction number below one to eradicate the disease. So a critical value at the turning point is deduced to be taken as a new threshold. The sufficient conditions for the disease-free equilibrium and the endemic equilibrium being globally asymptotically stable are also deduced. The results in this Chapter suggest that giving the patients timely treatment, improving the cure efficiency and decreasing the infective coefficient are all valid methods for the control of the disease.