Asymptotic Behavior Analysis Of An Improved Epidemic Model

It is not easy for scientists to study infectious diseases by experiments and so weneed to theoretically analyze how epidemic spread in a community and how could wemake effective policies to prevent and control epidemic spread. In order to study andforecast the behavior of diseases reasonably based on data collected from infected peo-ple, we have to construct suitable epidemic models. In this paper, we first improve anepidemic model with stochastic disease transmission with the purpose of well describingsome fatal diseases of human being such as HIV. And then we demonstrate the existenceand uniqueness of the solution to our improved model and the boundedness as well byLyapunov function theory and Ito formula. With the help of Fokker Planck Equation(FPE), we could carry out complete parameters analysis for asymptotic behaviors of in-fectious diseases. Our model analysis shows that the probability of vertical transmissionof diseases, P = 1/2 +α/2B, is the threshold value of disease outbreak in a commu-nity to a large extent. However, P = 1/2 is much more meaningful in that it might beconsidered as a realistic cordon of precautions and controls of disease outbreak.In addition, we also add environmental perturbations to the increase of mortality dueto diseases while considering the probability of disease transmission affected by environ-mental ?uctuations. That is to say, we employ an epidemic model with double stochasticdisturbance. In order to carry out complete parameter analysis for our new model, westill apply FPE to it. As a result, model analysis displays asymptotic behaviors of in-fectious diseases in accordance with circumstances which these two independent coeffi-cients of environmental disturbances intensity satisfy. Furthermore, one thing worthy ofmentioning is that big outbreak of diseases in a community would be suppressed if twoindependent stochastic perturbations are taken into account.From our model analysis, we could find that it is definitely necessary to improve theoriginal model as we do and it is also of significance for the department of disease pre-caution and control to ponder those new and interesting behaviors of infectious diseaseswhile making related policies. In fact, our model analysis would not only provide policiesof disease precautions and controls with realistic theoretical foundation but make thosepolicies effective and easily handled.