Research On The Impact Of The Epidemic Model Of Random Noise

Infectious diseases have serious efects on human health and life safety, and the socialstability and economic development. By establishing a mathematical model to study theepidemic is an important research direction of epidemiology. In recent years, the internationalresearch progress on the dynamics of infectious diseases is rapid development, and a largenumber of epidemic models were used to analyze various infectious diseases. The epidemicmodel which is commonly proposed is a deterministic model, and rarely considers the efectof random factors. In fact, in the process of the spread of infectious diseases, it is inevitablyafected by random factors. Therefore, stochastic epidemic model is more accord with thereality of infectious diseases.In the second chapter, a stochastic SI epidemic model with double noises was proposed.With the stochastic averaging method and nonlinear dynamic theory, the SI epidemic modelwas simplifed. According to the Lyapunov exponent, we got the local stochastic stabilityconditions; according to the singular boundary theory, we got the global stochastic stabilityconditions. By dint of the Lyapunov exponent of invariant measure and the stationaryprobability density, the stochastic bifurcation of the model was explored and also analysis ofthe location and probability which the stochastic system occurred stochastic Hopf bifurcationunder the diferent parameters. Results show that under the efect of random factors theconditions of the stability of a deterministic system were changed and the system becomesmore sensitive and more unstable.In the third chapter, a stochastic SI epidemic model in the complex networks wasproposed. We show that this model has a unique global positive solution. Then we considerthe asymptotic behavior around the disease-free equilibrium of the deterministic system whenR0≤1. Furthermore, we consider the asymptotic behavior around the endemic equilibriumof the deterministic system when R0>1. Finally, a series of numerical simulations arepresented to illustrate our mathematical fndings.