The Stability Analysis Of The Mathematical Models On Alcoholism And Optimal Control Strategies

With the rapid development of economy and society,people's life,work and the pressure of the spirit are also unprecedented high,leading to all kinds of social problems in an endless stream,in which the phenomenon of excessive drinking and the related social problems caused by excessive drinking are particularly prominent.Alcoholism has been medically identified as a chronic infectious disease by WHO and other authorities.In this dissertation,based on social drinking status and fundamental medical hypotheses,on the one hand,from the angle of infectious diseases,the theory of the population compartment is utilized to formulate several different mathematical models adapted to different environments to analyze qualitatively,and thus,the trend of the drinking behavior in the future is predicted;on the other hand,the method of prevention,media publicity,medication,policy orientation and other means are used to control drinking behaviors.Specifically,based on the mathematical model established,double objective functions are proposed in which both abstinence effect and abstinence cost are considered.Pontryagin minimum principle is utilized to solve the optimal control related problems and obtain the optimal control strategies.Centered on these two basic issues,this dissertation systematically studies the following contents.Firstly,a four dimensional SAT Q model of alcoholism and alcohol abstinence with standard contact infection rate is established,and two control measures including prevention and treatment are also considered.The population involved in this model is divided into four compartments,i.e.,the healthy persons who are suspected alcoholism,denoted by S(t);the alcoholics denoted by A(t);the persons taking part in treatment,denoted by T(t)and the persons who are quitting drinking for ever,denoted by Q(t).Obviously,the model proposed in this dissertation is more objective and detailed compared with the existed SAQ one,it is the foundation of the follow-up models in this dissertation.The existence and positivity of the alcohol-free equilibrium point and internal alcoholism equilibrium point are proved.Then,the basic reproduction number of alcoholic drinks is obtained,and the global stability of alcohol-free equilibrium point is proved by the comparison principle.Furthermore,the appropriate Lyapunov function is constructed to prove the global asymptotic stability of the internal equilibrium point.Finally,the objective function is proposed,and the optimal control problem is considered.The existence and uniqueness of the optimal control are solved,and the characterization of the optimal control is also given.The results show that the optimal control performs better than other control methods,such as full control,semi-control and single control.Based on the model and objective function,in the comparison of single controls,the fact is found that the effect of treatment is better than that of prevention,which is basically consistent with the objective facts.Secondly,with the introduction of the abstract contact infection function and the distribution of alcohol infection delay,a general four dimensional SAT Q model of alcoholism and alcohol abstinence model is proposed,which extends the one formulated in chapter two.In this chapter,the existence of the equilibrium point of alcohol abuse is discussed,and the expression of the basic reproductive number is also given.Because of the abstraction of the model and the existence of the distributed time delay,the stability of the equilibria are rather difficult.The Lyapunov function is constructed by using the Huang-T akeuchi method to prove the global asymptotic stability of the two kinds of equilibrium points.An objective functional is proposed in which both the control effect and the control cost are taking into consideration,and then the existence of the optimal control using delay optimal control theory and method is proved,and the mathematical representation of optimal control is finally derived.The results show that the stability of the equilibrium point can not be destroyed by the time delay.However,the delay will affect the optimal control,that is,the greater the delay,the worse the control effect.Thirdly,a five-dimension complicated model with multiple control strategies on the interaction between alcoholism and HIV virus infection is established.It is proved that the existence,positivity and boundedness of the solution of the model.By the use of variable classification,global stability of the HIV disease free equilibrium is proved,indicating that the HIV virus can eventually become extinct even though under the conditions of alcoholism.Taking the three control measures into consideration,namely,the treatment of the alcoholics and the prevention of the suspected from being infected as well as the publicity and education on the suspected population of HIV virus.This dissertation also proposes the double-effect objective function with discounted coefficient to research on related problems of optimal control to derive the optimal control strategies which satisfy the double-effect purpose.The sensitivity analysis of the model parameters showes that alcohol abuse acts as the role of adding fuel to the flames on the spread of HIV virus infection.Fourth,considering the drinking environment diversity and the differences on alcoholics behavior of individuals which can be embodied in the rate of alcohol contact infection,and also the effect of drinking behavior on the population mortality rate,then gradually establish a stochastic model with single factor random disturbance and another one with multi-disturbance respectively.Based on these two stochastic model,firstly,the existence,uniqueness and non-blowup of the solution of the models are respectively investigated.Further,under the appropriate conditions,compared to the corresponding deterministic model,the mean asymptotic stability and ergodic behaviors of the stochastic model is proved.In the proof of the random behavior of the equilibria of the multi-factor model,due to the nonlinearity of the disturbance term,it is difficult to determine the sign of many nonlinear terms including high-order moments.In this paper,the method of constructing multiple sub functions and then fitting them in a proper linear combination is used to eliminate the cross terms.The results show that,on the one hand,when there is no external disturbance,the solutions of the determined system are bounded,while the solutions of perturbed system are ergodic(unbounded);on the other hand,the greater the intensity of disturbance,the faster the vanishment of the alcoholics,which in a sense means that disturbance helps to control drinking behavior.