Qualitative Analysis Of An Epidemic Mathematical Model

On the basis of classical SIR compartmental model, this thesis focuses on the situation where there are two different infectious compartments – asymptomatic and symptomatic infectious individuals, both of whom are able to infect susceptible individuals. Therefore we divide the population into four compartments –susceptible individuals, asymptomatic infectious individuals, symptomatic infectious individuals and recovered individuals, with the assumption that recovered individuals don't have permanent immunity. The four compartments are denoted by S(t), Ia(t), Is(t), R(t), respectively, and then a four-dimensional system of ordinary differential equations is built to describe the dynamics of diseases.The infection rate ?, coefficient of the ratio between the infections caused by one infectious individual and the number of total susceptible individuals, is very important in the model. Since the infection rate of some diseases vary from season to season while others don't, we will study two cases where ? is a constant(Chapter 3) or a periodic function ?(t) of t(Chapter 4). We calculated the basic reproduction number R0, and found it bigger than that of classical SIR model.We also proved that R0= 1 is a threshold value that decides whether the disease persists or not. To be specific, if ? is a constant, then the disease-free equilibrium is globally asymptotically stable when R0< 1, and unstable when R0> 1; when R0> 1, the model has a positive endemic equilibrium which is locally asymptotically stable, and a sufficient condition of its global asymptotical stability is given.If ? = ?(t) is a periodic function of t, then the disease-free constant solution is globally attractive when R0< 1, and unstable when R0> 1; when R0> 1, the disease is uniformly persistent. We also illustrated the dynamics of the model by numerical simulations, and showed the influence of asymptomatic infectious individuals on the spread of disease. The results can provide theoretical basis for the decision-making process of public health authorities.