Dynamic Analysis Of Two Stochastic SIQR Epidemic Models

Epidemic is a kind of contagious disease caused by pathogenic microorganisms,such as bacteria,viruses and parasites.Because of the persistence of microorganisms,epidemics always exist and spread in a wide range of ways.Therefore,the prevention and control of infectious diseases have always been the core concern of WHO and a key issue about the life and health of all mankind.From a practical perspective,there are three ways to prevent the transmission of disease involving controlling the sources of infection,cutting off the transmission routes and protecting susceptible population.In the view of biomathematics,based on the epidemic dynamics,we can analyse the transmission laws of infectious disease by establishing suitable mathematical model and find key factors which affect the spread of disease.Finally,we optimize control strategies of epidemic.In addition,at the beginning of the COVID-19 outbreak in2020,isolation is the best measure to control the spread of the disease,then it is necessary to consider the mathematical model with isolation,at the same time,stochastic factors disturb the diffusion of disease inevitably.Hence,in this thesis,in view of the influences of environmental noises and quarantine measures during the process of disease transmission,we mainly research the dynamic behaviors of two stochastic SIQR epidemic systems under different perturbations.The main contents are as follows:Firstly,we mainly illustrate the background and significance of epidemic dynamics,further with the development and current situations of epidemic models are introduced,then we show definitions,theorems and lemmas used in this article.Secondly,based on the deterministic model with constant input and standard incidence rate,we construct a stochastic SIQR epidemic model with a proportional relationship between environmental noises and system variables.By utilizing the stoptime concept,we obtain the existence and uniqueness of the global positive solution for stochastic system with any initial value.Next,by establishing suitable Lyapunov function and using Has’ minskii theory,we prove the existence of ergodic stationary distribution of system.Then,we gain the threshold condition to determine whether the disease is prevalent or not.Moreover,we verify the reasonability of the theoretical results with the help of the MATLAB simulation software.Finally,we assume the effect of environmental noises on the natural death rate satisfies the mean-reverting Ornstein-Uhlenbeck process and build the stochastic SIQR epidemic model in which only susceptibles have Logistic regenerative capacities.In this chapter,we state the existence and uniqueness of the global positive solution by the comparison theorem of stochastic differential equation.And then,by utilizing the It^o’s formula and Lyapunov function,we get the threshold conditions which represent the persistence and distinction of systems and epidemics respectively.In addition,the validity of theoretical results is verified by numerical simulations.What’s more,the sensitive analysis shows the effects of reversion speed and disturbance intensity on the transmission of disease during the mean-reverting process.Compared with a linear function of white noise,the mean-reverting process possess several important features,such as continuity,nonnegativity,practicality,possession of asymptotic distributions and so on.It is better to characterize environmental variabilities in biological systems.