Some Epidemiological Models With Control Measures

The aim of this work is to construct several mathematical models in epidemiology and to analyze the asymptotic behavior of these models. In the first part, some epidemic models with nonlinear incidence rate are studied to consider effects of protection measures and intervention policies. In the first two sections, we consider SIRS epidemiological models with nonlinear incidence rates (kIS)/(1+α(S+I)) and (kIS)/(1+α₁S+α₂I). A threshold for the disease to be epidemic or to be eradicated is established. The behavior of solution and the threshold conditions of existence and locally asymptotic stability are obtained. The sufficient conditions on the global asymptotic stability of the disease-free equilibrium and the endemic equilibrium. In the third section, a species mechanics epidemical model with nonlinear rate was considered. The sufficient conditions on the global asymptotic stability of the extinct equilibrium and the survival equilibrium are obtained. In the fourth section, we study the global dynamics of an epidemic model with a complete form of the nonmonotonic incidence rate of [41]. by carrying out the qualitative analysis of the model, it is shown that the model admits rich dynamical behaviors. It has saddle node bifurcations of endemic equilibria, Hopf bifurcations, the saddle node bifurcation of nonhyperbolic periodic orbit, which implies the existence of two limit cycles and homoclinic bifurcations. We have also found that the model admits bistable steady states such that the outcome of disease evolution depends upon initial positions.In the second part of this paper, we consider a kind of epidemic model with disease transmission in the predator-prey system. A predator-prey model modified to include an SIS parasitic infection in the prey with infected prey being more vulnerable to predation is studied. This means that in ecology it is predator-prey system, in the mechanism of disease transmission it is an SIS epidemiological model. we improve (3.1.2) in two-folds. First, we replace the standard incidence by a bilinear incidence. Secondly, we also consider the case whereα₂ <α₁, which essentially means that predators prefer to healthy prey. This induces interesting biological phenomena. An epidemic model with disease transmission in prey population is studied to find how predators regulate the evolution of disease in the prey population. To have further insights into the effect of biological control of predators to the spread of epidemic disease, we present the example. Conditions for the extinction of disease and conditions for the persistence of disease are established, and effects of the preference of predators to prey individuals are analyzed.