Optimal Control And Dynamic Behaviors Of Stochastic Epidemic Model

Infections are diseases that can be transmitted among human beings,animals,or humans and animals in a short period of time.And they are harmful for our lives,the society and the development of the economic.In order to prevent the diseases from spreading effectively,constructing a mathematical model to discuss the rules of spreading and study their dynamic behaviors,and then formulate some reasonable and effective control strategies,which has become one of the main methods for the study of infectious diseases.Therefore,the main contents of this thesis are as follows:1.We discuss the dynamic behavior of the stochastic SIRS epidemic model with me-dia coverage.It mainly studies on two aspects:On the one hand,using Markov semigroup theory to prove that the basic reproductive number R0s can control the dynamic behavior of stochastic systems.If R0s<1,the stochastic system has a disease-free equilibrium which implies that the disease is extinct;If R0s>1,under the mild extra condition,the stochastic differential system has an endemic equilibrium that is globally asymptotically stable,which means the persistence of the disease.On the other hand,we have demonstrated that environ-mental fluctuations can suppress the outbreak of disease.That is to say,for a deterministic system,the disease is persistent when R0>1,however,if R0s<1(even under the condition R0>1),the disease still dies out with probability one for the stochastic model.Finally,numerical examples were given to illustrate our results,and explained media coverage can reduce the number of infected people.2.We study the problem of near-optimal control for a stochastic SIRS epidemic mod-el.The near-optimal control is more flexible and easier to be implemented than the exact optimal control.Therefore,we mainly discuss the near-optimal problem of the epidemic model and keep the treatment cost to a minimum.We obtained the sufficient and necessary conditions for the near-optimality.And showed that any near-optimal control can be solved by using the Hamiltonian function to approximate the cost function in some integral sense.According to the adjoint equations,we derived the estimate for the error bound of the near-optimality.Numerical simulations are presented to illustrate the effect of treatment control on the disease.