Dynamics Studies And Optimal Control Of Two Types Of HIV/AIDS Models

At present,AIDS has spread around the world.According to the available data reports,new trends are emerging in new infections——total number of HIV infections among young and elderly is on the rise.At the same time,because the probability of infection in men who have sex with men is nine times as high as heterosexual sex.The number of infected men is more than women.In this paper,two types of HIV/AIDS models are developed for age structure and sex structure,taking into account the transmission characteristics of AIDS.First of all,the knowledge about HIV/AIDS and some preparatory knowledge are explained in detail,and some domestic and foreign scholar’s research status and relevant conclusions about HIV/AIDS are introduced.In the second part of the parper,based on the different characteristics of HIV transmission in different age stages,we establish a three-stages HIV/AIDS epidemic model that includes unreported infection.We calculate the basic reproduction number R₀by the next generation matrix method.If R₀<1,the disease-free equilibrium is globally asymptotically stable.If R₀>1,we prove that the disease is uniform persistence and the model has an endemic equilibrium.We control the spread of disease by taking relative measures for different age groups,and we use the Pontryagin maximum principle to find the optimal control strategy.Numerical results indicate that increases in condom use is an effective way to reduce infection cases.Data fitting indicates that the number of HIV/AIDS infection cases in China will continue to rise in the future years.In the third part of the parper,we establish a HIV/AIDS epidemic model with sex structure which including homosexual contacts and heterosexual contacts.We derive the explicit expression for the basic reproduction number through the next generation matrix approach.If R₀<1,the disease-free equilibrium is globally asymptotically stable.If R₀>1,the disease is uniform persistence.In particular,numerical simulations are presented to reinforce these analytical results.We also formulate an optimal control problem and solve it using Pontryagins Maximum Principle.The model can be fitted to real data,which can predict the growth trend of HIV/AIDS patients in the future years.Finally,we summarize the existing research results and give out the prospect.