Optimal Control Of An HIV/AIDS Epidemic Model With Infective Immigration And Behavioral Change

Despite the very constraint ways to spread,HIV/AIDS disease continues to spread alarmingly.In order to find out the effect of human(sexual)behaviour change and immigration in spreading the HIV/AIDS,we have formulated a deterministic model of HIV/AIDS with infective immigration.At first,the analysis which related to the wild na-ture of HIV/AIDS and its prevalence across the world are given as a research background.And some works from mathematical field are analyzed to establish our own new findings.The second part is some basic knowledge.The third part mainly consists of following two mathematical models:First we established an HIV/AIDS model with infected immigration,analyzed ba-sic properties of the model,including non-negativity and boundedness of the solutions,existence together with both locally and global asymptotically stability of the endemic equilibrium.In accordance with earlier works,we have also computed the basic repro-duction number,R0,by using next generation method when there is no recruitment of either type of infectives.We have consulted the geometrical approach to obtain the glob-ally asymptotically stability of endemic equilibrium.Next the basic model is extended to include several control efforts aimed at reducing infection and changing behaviour.We use Pontriagin's maximum principle to derive the optimality system and solve the system numerically.Our numerical findings are illustrated through computer simulations using MATLAB,which show reliability of our model from the practical point of view.Based on the related studies and first model,second model is modified version that considering both standard incidence rate and time delay.At first some basic properties of the model are analyzed.It proved the non-negativity and boundedness of the solu-tions.Some conditions to the existence of positive equilibrium are given.The globally asymptotically stability of the endemic equilibrium are obtained by using suitable Lya-punov function.The delay model analyzed as an optimal control problem which sets behavioural change and screening as main control strategy.At the end numerical results of theoretical results of model are given,which shows correctness and effectiveness of the model.As it widely believed that mathematical models plays significant role to cope against the transmission disease like HIV/AIDS,especially in evaluation of intervention measures and predicting the outcome of such measures.So the model analyzed above can serve as an example of prediction of future prevalence of the disease.It can be concluded that persuading more people to practice safe sex and and screening are more efficient ways to prevent further prevalence of HIV/AIDS.These conclusions will undoubtedly help to realize the disease control goal.