Stochastic SIS And SIRS Epidemic Models With Time Delay

As we all know,epidemic comes with the progress of human civilization and have a profound and comprehensive impact on human civilization,such as economic development,human health,national security and so on.For a long time,although human beings have been fighting against various infectious diseases,and achieved success to a certain extent,the road to completely conquer epidemic is still long and tortuous.Just like the outbreak in COVID-19(Corona Virus Disease 2019)in late 2019,this epidemic has seriously affected human health and brought serious threats to economic development and national security.Therefore,studying epidemic models and their dynamic behaviors is an important method to prevent and control the epidemic of infectious diseases.This paper mainly studies the dynamic behavior of several types of epidemic models.Firstly,the paper considers a stochastic SIS(susceptible-infected-susceptible)epidemic models with time delay and a stochastic SIS epidemic models with non-monotone incidence rate and time delay.We first prove the existence and uniqueness,extinction of the system by using It^ lemma and Lyapunov function.Then we prove that it is able to make the disease die out by reducing the stochastic disturbance that is favorable to the spread of the disease,or increasing the stochastic disturbance that is beneficial to reduce the infection.Moreover,we study the factors that make the disease persistent for a stochastic SIS epidemic models with time delay.Finally,we use the euler method to simulate the model,and obtain the curves of the state variables,which proves the conclusion of this paper.Secondly,the paper considers a stochastic SIRS(susceptible-infected-recoveredsusceptible)epidemic models with double time delays.We study the dynamic behavior of the state variables of this model by using It^ lemma and Lyapunov function,that is,the existence and uniqueness,extinction of the system.we prove that under certain conditions,the control of the range of values of the stochastic disturbance can cause the death of the disease.Finally,the conclusions of this paper are summarized and prospected.