Numerical Schemes On Two Stochastic Epidemic Models

The emergence of infectious diseases has caused a large number of deaths and economic losses,which has seriously endanger human health and social stability.Confronted with these grim facts,it is extremely important to use mathematical theories and methods to prevent and control infectious diseases.Due to the influence of environmental noise,the parameters of traditional deterministic infectious disease models may be subject to some degree of random disturbance(such as birth rate,mortality,transmission coefficient and so on).Moreover,age structure is one of the important factors for the spread and prevention of infectious diseases.Therefore,this paper considers the environ-mental noise and age structure into the epidemic models,and then studies its numerical schemes and properties.The main contents of the paper are shown as follows:(1)The numerical scheme of a stochastic Susceptible-Infected-Quarantined-Susceptible(SIQS)epidemic model with intervention environmental white noise is investigated.Using the ex-plicit Euler-Maruyama(EM)scheme,we can obtain a numerical approximate solution of the stochastic SIQS epidemic model.We prove that the EM approximate solution will converge to the exact solution of the model.To preserve positivity of numerical scheme,we give a balanced implicit numerical method of model by constructing control function.Then,the convergence property of the balanced implicit numerical method is established.Numerical simulations are carried out to verify the theoretical results and explore the influence of integration interval and time stepsize for the balanced implicit numerical method.(2)We study numerical scheme for an age-structured Susceptible-Infected-Removed(SIR)epi-demic model with environmental white noise.Using the semigroup theory,the existence and uniqueness of global positive solution for the model is discussed.We then define a truncat-ed function and establish a partially truncated EM numerical scheme to the stochastic age-structured SIR epidemic model.We present the pth moment boundedness of the partially truncated EM numerical approximate solutions.Furthermore,the strong Lq convergence is established for the condition of 2 ? q<p of the partially truncated EM scheme.Final-ly,numerical simulations and examples are provided to demonstrate theoretical results and to illustrate validity of the partially truncated EM scheme.