Qualitative Analysis For Several Stochastic Epidemic Models

It is well known that infectious diseases seriously threaten human health and life.For a long time,human beings have been fighting against various infectious diseases,such as smallpox,AIDS,influenza,etc.Some infectious diseases(such as smallpox)have been eliminated,however more infectious diseases(such as mixed influenza A and B,COVID-19)still plague humans.In addition,with the acceleration of globalization,the convenient transportation and the frequent movement of population have accelerated the spread of infectious diseases.An important method to study the spread of infectious diseases is to establish a dynamic model.By analyzing the dynamic characteristics of the model,we can reveal the law of infectious diseases,predict its development trend,and seek the strategies to prevent and cure the disease.Therefore,it is necessary to establish and analyze the mathematical model reflecting the dynamic characteristics of infectious diseases.In the real world,infectious diseases are inevitably affected by environmental noise,which makes the related parameters(such as contact rate,mortality rate,recovery rate,etc.)show random fluctuations.In many cases,the influence of random interference on disease transmission can not be ignored.It is not always ideal to describe and predict the development process and transmission law of disease with deterministic epidemic model.Thus it is more realistic to study the dynamic properties of stochastic epidemic model.Several stochastic epidemic models are considered in the paper.By means of stochastic differential equation and stochastic process,we analyze the stochastic dynamic behavior of the models(including the existence and uniqueness of the global positive solution of the initial value problem,stochastic extinction and persistence of the disease,stability and stationary distribution,etc.).Some theoretical results are obtained to provide scientific theoretical basis for the prevention and control of infectious diseases.The main contents are as follows:In Chapter 1,we introduce the research background and significance of epidemic dynamics model,summarize its research status at home and abroad,and give the related definitions and lemmas used in the paper.In Chapter 2,we investigate a stochastic SIRS model with nonlinear incidence and transfer from infectious to susceptible.Firstly,the existence and uniqueness of global positive solution of the model with any positive initial value are proved by Lyapunov analysis method.Next,sufficient conditions for extinction and persistence of the disease are established.By using stochastic stability theory,we discuss stochastic asymptotic stability of disease-free equilibrium of the model.Then,we obtain almost surely exponential stability of disease-free equilibrium,which implies that noises can lead to extinction of disease.By the Lyapunov method,we give conditions to ensure that the solution of the model fluctuates around endemic equilibrium of the corresponding deterministic model in average time.Finally,the theoretical results are verified by numerical simulations.In Chapter 3,we propose and analyze a stochastic SIRI epidemic model with relapse and media coverage.First,the existence and uniqueness of the global positive solution of the model are proved.Next,the existence of a stationary distribution is obtained by using the Has' minskii theory.Then sufficient conditions for extinction of the disease are established.By constructing the suitable Lyapunov functions,the dynamic properties of the solution around both the disease-free equilibrium and the endemic equilibrium of the corresponding deterministic model are studied.Finally,numerical simulation is carried out.In addition,the numerical simulation shows that increasing the media coverage can reduce the number of infected individuals,so media coverage can reduce the spread of infectious diseases.In Chapter 4,a stochastic SIS epidemic model with transport-related infection is proposed to investigate the dynamics of disease propagation between two cities.First,we investigate the asymptotic behaviors of positive solution to the stochastic model.Especially,by constructing a Lyapunov function and stopping times,we show that the difference between susceptible populations or infected populations in two cities will tend to 0 with probability one.Then,exponential extinction and persistence in mean of the disease are discussed.At last,numerical simulations are presented.The theoretical results show that noise can suppress the outbreak of disease.In addition,the numerical simulation shows that transport-related infection will cause an endemic disease more seriously on spreading disease.In the presence of the disease,the time mean of the number of susceptible in both cities decreases with the increase of individual migration rate,whereas the time mean of the number of infected in both cities increases with the increase of individual migration rate.Further,the time mean of the number of individuals in both cities decreases with the increase of individual migration rate.In Chapter 5,we investigate a stochastic SIQS epidemic model with saturated incidence and Markovian switching.First,the existence and uniqueness of the global positive solution of the model are proved.Then by using the ergodic property of Markov chains,the sufficient conditions of extinction and persistence in the mean of the disease are obtained.Finally,some numerical simulations are introduced to demonstrate the analytical results.The results show that if one of the subsystems is extinct,another is stochastically persistent,then the hybrid system may be either stochastically extinct or persistent,and the result depends on the probability that the Markov chain stays in each state.