The Study Of Stability Of Infectious Disease Dynamics Model

This paper mainly studies the stability of the SEIRS infectious disease models. Byconsidering more comprehensive factors of affecting the development of the diseases,we establish three kinds of models which could reflect the pathogenesis of the diseasesand the transmission law of the diseases better. Then we analyze the stability of thesemodels to research the development of the diseases. The main contents as follows:1. According to the latent period and immune period of the diseases, thetransmission of the diseases, we establish an SEIRS epidemic model with two timedelays. Firstly we get the basic reproductive number of the model and the conditions ofthe disease free equilibrium and the endemic equilibrium. Then by applying Hurwitzdiscrimination, the judgment theorem of the asymptotical stability, we obtain the localasymptotical stability of the disease free equilibrium and the endemic equilibrium.2. According to the disease feature in latent and infected period, the mortality ratebecause of the illness, we establish an SEIRS epidemic model with infection in latentperiod and infectious period. Firstly by using of the monotonicity of the function andthe theorem of the existence of the zero, we get the basic reproductive number of themodel and the conditions of the disease free equilibrium and the endemic equilibrium.Then by applying Lyapunov stability theorem, LaSalle invariance principle, Routh-Hurwitz criterion, we obtain the global asymptotical stability of the disease freeequilibrium and the local asymptotical of the endemic equilibrium.3. According to the periodic of the diseases and the bilinear incident rate, weestablish an SEIRS epidemic model in a periodic environment. Firstly we obtain thebasic reproductive number of the model. Secondly by applying the comparison theorem,disease persistence theorem, Poincaré map, we get the stability condition of the diseasefree periodic solution and the endemic periodic solution. Finally we apply the numericalsimulation to verify the validity of the result.