Research On Optimal Control Of Two Classes Of Stochastic Epidemic Models

Infectious diseases pose a serious threat to human health and affect social development.COVID19,which burst out in December 2019 and spread rapidly around the world,had a negative impact on people’s lives and economic development.Mathematical models can characterize the dynamic behavior of infectious diseases and help us establish prevention and control measures.Therefore,it is necessary to strengthen the qualitative and quantitative research on infectious disease models to provide a theoretical basis for predicting the spread of diseases and establishing optimal control strategies.In this paper,based on the complex phenomena of Markov switching,Levy noise,impulse and time delay,two classes of stochastic epidemic models are established.Some effective numerical analysis and optimal control strategies are proposed,the control theory of the stochastic susceptible-infectedvaccinated(SIV)and stochastic susceptible-exposure-infected-recovered(SEIR)systems are developed.The main research contents are as follows:(i)A stochastic SIV epidemic model with Markov switching is established based on the deterministic epidemic model.With the help of logarithmic truncation method and Euler-Maruyama(EM)numerical method,the logarithmic truncated EM scheme is established,which ensures positive numerical solutions and achieves order-1 convergence of the stochastic SIV epidemic model.The existence of an invariant measure is proved for the stochastic SIV epidemic model.The numerical analysis results show that the logarithmic truncation preserving scheme can achieve strong convergence of exact solutions.(ii)A class of SIV epidemic models are proposed including reaction-diffusion,impulse,latency and stochastic disturbances.Finite-time contraction stability of the unstable stochastic SIV epidemic system was studied by adding feedback control variables.With the help of the comparison principle,sufficient conditions of finite-time contraction stability of the stochastic SIV epidemic system are given.The effects of impulse,delay and noise on the finite-time contraction stability of the system are demonstrated by numerical simulations.(iii)Based on the impact of economic activities on the spread of infectious diseases,a stochastic susceptible-exposure-infective-recovered-economic(SEIRG)model is established.By using Hamiltonian function and maximum principle,the sufficient and necessary conditions for the near-optimal control of the SEIRG system are given,and the near optimal control strategy is obtained.In addition,the numerical results indicate that the level of economic development has an important positive feedback effect on epidemic control.(iv)A mean field stochastic SIV epidemic model is presented by applying the mean field principle whose parameters depend on the state process.The sufficient and necessary conditions of optimal control of the novel epidemic model are obtained by maximum principle.Numerical simulations show that vaccination has a significant effect on controlling the number of handfoot-and-mouth disease(HFMD)cases.(v)Relaxed controls for the stochastic SIV epidemic model with Markov switching are investigated under nonconvex assumptions.The Markov chain approximation method is used to give numerical approximations of the value function and the optimal control.The results show that the approximation schemes converge to the optimal strategy as continuous-time interpolation h goes to zero.