The Research On The Neutral Stochastic Functional Differential Equations And The Stochastic Epidemic Models

Stochastic functional differential equation plays an important role in various fields in which time delay phenomena exist,for instance,in life science,mechanical engineering,social sciences,and the financial field.In the nature,almost all of the systems are disturbed by different types of random disturbance.So,the stochastic biomathematic models become closer to the nature.In this paper,we will use methods and theories of stochastic differential equations,to aim at two topics bellow.One is the study of the neutral stochastic functional differential equations,in which we will show the existence and uniqueness of the solution,the continuous dependence on initial value,error estimate between the approximate solution and accurate solution.Another is to explore the solutions of the stochastic epidemic models,in which we will focus on the effect of random disturbances.Firstly,we choose space Ch as the phase space and study the existence and uniqueness of the solution to neutral stochastic functional differential equations with infinite delay(INSFDEs).Under uniformly Lipschitz condition,weakened linear growth condition and contractive condition,the existence and uniqueness of the solution to INSFDEs is obtained by means of the Picard iteration,the Doob's martingale inequalities and the Gronwall's inequality.Furthermore,we discuss the continuous dependence of the solution on the initial value to INSFDEs,and then we show the error estimation between thep approximate solution and accurate solution.Secondly,we consider the dynamical behavior of the stochastic SIRS epidemic model with environmental disturbance and parameters disturbance.By defining the stopping time and using Lyapunov function,the unique global positive solution to the stochastic SIRS model is derived.Then we analyze the asymptotic behaviors of the stochastic SIRS model around the disease-free equilibrium and the endemic equilibrium.Under certain conditions,the stochastic SIRS model has a unique stationary distribution with ergodicity,and the solution asymptotically follows a 3-dimensional normal distribution.Some numerical simulations are carried out to clarify our results.Finally,a stochastic SIQS epidemic model with saturated incidence is discussed.We introduce the random disturbance into our model,stochastic perturbations are directly proportional to the differences S(t)-S*,I*-I(t),Q*-Q(t),where S*,I*,Q*are the steady-state values,respectively.By constructing a suitable Lyapunov function and using Ito formula,we obtain the global existence and uniqueness of the positive solution,stochastically ultimate boundedness and stochastic permanence.Numerical simulations are given out to verify the results.In this paper,the research results suggest that,the impacts on the corresponding deterministic system are different by introducing different random disturbances.In other words,if the disturbances are controlled in a suitable range,the systems still maintain the original properties.Otherwise,if the disturbances lie out of this range,the original properties of the systems will be destroyed.So far,we have built the mathematical models for the infectious diseases,and analyzed the dynamical characteristics of the models.And,with the assistance of computer,the numerical simulations are carried out.Our new results will provide the best posiibility of controlling the spread of the infectious diseases,and prevent the outbreak of the disease in a long run.