Exponential Stability Of The Trival Solution Of Two Stochastic Epidemic Models On Networks

Epidemics will seriously threaten human life and property security.According to the transmission mechanism of epidemics,it is important to establish the reasonable epidemic models on complex networks.And then,one can predict the transmission law of epidemics by analyzing dynamical behaviors of these models,which can provide the theoretical base and scientific foundation for the control of epidemics.Therefore,qualitative research on the dynamics of epidemics on complex networks has become a research hotspot for scholars at home and abroad.Considering the heterogeneity of individuals in the process of disease transmission and the impact of stochastic impact from environment,stochastic epidemic models can more reasonably reflect the transmission mechanism of epidemics.Based on the above factors,this paper mainly studies the pth moment exponential stability of the trival solution of two stochastic models on complex networks.The specific content is arranged as follows.Chapter 1 mainly analyzes the necessity of establishing epidemic models on complex networks and analyzing their dynamic properties.Then we introduce the current research status of corresponding epidemic models and give the main work as well as preliminary knowledge of this paper.Chapter 2 establishes a stochastic SIS model on complex networks as follows(?) where k=1,2,…,N.Firstly,the existence and uniqueness of the global positive solution are given by constructing Lyapunov function and using Ito's formula.Secondly,combining Kirchhoff matrix-tree theorem in graph theory,we give the sufficient conditions for the pth moment exponential stability of the trival solution of the model.Finally,the theoretical results are verified by numerical experiment.In chapter 3,we establish a stochastic SAIS model on complex networks as following(?) We first verify that there exists a unique global positive solution of the model by constructing Lyapunov function and using Ito's formula.And then,the sufficient conditions for the pth moment exponential stability of the trival solution of the model are provided by using Kirchhoff matrix-tree theorem,Jensen inequality and Young inequality.In the end,numerical simulations are presented to validate main results.Chapter 4 summarizes the contents of chapter 2 and chapter 3 and presents the short-comings of the model and its research results as well as the direction of future improvement.