A Stochastic Model For AIDS Transmission And Its Risk Analysis

The AIDS epidemic belongs to a kind of disease which threats the life and health of people seriously. As AIDS spreads all over the world, to research its transmission dynamics and then render some strategies to decrease its transmission risk plays a very important role in risk management and control for AIDS. Until now, most of the work focuses on ordinary differential equation (ODE) on modeling AIDS transmission dynamics without consideration of the effect of stochastic disturbance on its transmission. Consequently, accurate prediction for its spread can not be obtained. And in such a way, effective way to decrease its transmission risk can not be offered. This thesis aims to research AIDS transmission with application of stochastic model and predict the number of people infected with HIV in the future more accurately, then analyze the critical attributes affecting its course of transmission and study some strategy to decrease its transmission risk in order to render some decision support for the risk management and control of its social risk behavior.This thesis can be separated into two main parts. The result in the first part covers the effect of the environmental disturbance on AIDS transmission. When the disturbance of the environmental noise is taken into account, the disease transmission is changed from a constant to a stochastic process. In such case, we obtain the following results: Consdering the effect of the enviorment on AIDS transmission, the stochastic differential equation (SDE) model for AIDS transmission is extended and formulated, based on its orginal model. The deterministic and stochastic situation is compared. With application of numerical solution, the future proportion of the population infected with HIV against total population is forecasted in next few years in China with actual data, and then the distribution trend for the proportion of the population infected with HIV against total population is obtained. The effect of the disease transmission on AIDS transmission is analyzed and some strategies are presented to reach a certain control by means of adjusting the transmission rate control index, and the corresponding risk analysis is given meanwhile. Based on this, the existence and uniqueness for the solution is proved, and it is concluded that the solution is bounded and belongs to the interval [0, 1]. And then the asymptotic behavior is analyzed and the condition for the stability is obtained, and the p-th moment stability is also attained. Next, by simulation, a critical value for disease transmission rate is demonstrated with the help of actual data and some condition for the speed for AIDS spread to fall and then finally for AIDS to die out is obtained. And, the effect of AIDS's broken out on economics is also analysed.Actually, the government takes action to prevent AIDS from spreading. They are dedicated to develop the vaccine and new medicine to treat AIDS. They also pay attention to assemble the patients and treat them. When the treatment rate is considered into the model, the property of the more complex model is analyzed and the main contributes in the second part includes: With consideration of treatment rate, a stochastic model with treatment rate is formulated to describe AIDS transmission. A condition for the asymptotic behavior of the solution is obtained as to the extended model with application of Ito's formula, and the asymptotic behavior is analyzed and the condition for the stability is obtained, and the p-th moment stability is also attained. Then a critical value is illustrated for the treatment rate to make the proportion of the population infected with HIV against total population fall and then for AIDS to die out by simulation when other parameters is unchanged. Furthermore, the input-output model for treatment of patients is formulated based on the model. Then the risk of social behavior leading to AIDS is analysed. And, by simulation, the risk probability is computed of the proportion of the population infected with HIV against total population exceeding the given control value based on the stochastic model with treatment rate. Based on this, the effect is analyzed of the knowledge, attitudes and action of people infected with HIV on the model, and the Fokker-Plank forward equation is obtained for the probability distribution of the proportion of the population infected with HIV against total population and this result is illustrated by simulation. The changing trend of the proportion of the population infected with HIV against total population in the future is compared among different situations as far as the knowledge, attitudes, behavior of the infected person are concerned. And a conclusion can be made that the speed for AIDS transmission can be decresed to the largest degree only if the infected person can be confronted with HIV/AIDS very bravely and is willing to get treatment positively. Finally, further analysis is given for the risk of AIDS' social behavior based on the analysis of the model.