On the Stochastic Dynamics of a Social Epidemics Model

Alcohol abuse is a major social problem, which has caused a lot of damages or hidden dangers to the individual and the society. In this paper, with random factors of alcoholism considered in mortality rate of compartment populations, we formulate a stochastic alcoholism model according to compartment theory of infectious disease. Based on this model, we investigate the long-term stochastic dynamics behaviors of two equilibria of the corresponding deterministic model and point out the effect of random disturbance on the stability of the system. We find that when R0 ≤ 1, we get the estimation between the trajectory of stochastic system andE0 = (Π/μs, 0, 0, 0) in the average in time with respect to the disturbance intensity, while whenR0 > 1, stochastic system is ergodic and has the unique stationary distribution. Finally, we carry out numerical simulations to support the corresponding theoretical results.


Introduction
Along with the rapid development of human civilization, the pressure of human life is becoming more and more large, especially in the spirit and emotion.In order to release the pressure, many people will lose their rationality and take some unwise actions; for example, they will take addictive drugs or indulge in long-term alcohol abuse or heavy smoking [1].These extreme behaviors not only seriously hurt the people's own health but also cause serious security risks to the total society; for example, drunk driving and drunk dangerous sexual behaviors have frequently occurred [2][3][4].Sociologists have noticed these undesirable negative phenomena, which are called social diseases with great destructive and infectivity due to mental or habitual addiction [5].In the above social phenomena, alcoholism is particularly serious, showing the trend of lowering ages and femininity [6].
Actually, similar to some typical infectious diseases, alcohol abuse also has a strong infectiousness, especially for the social crowd in frequent contact; therefore, it is often called social infectious diseases [5,7].In view of this, in the last ten years, many applied mathematicians and interdisciplinary workers are committed to formulating mathematical models to study the spreading behavior of alcohol abuse and predict the trend of alcoholism from a mathematical point of view [8][9][10][11][12][13][14].
Sanchez et al. [8] earlier proposed a simple and basic mathematical model with relapse to describe the spread and infection of alcohol abuse.In their model, according to the progress of alcoholism, the population is divided into three compartments, that is, normal persons () who do not drink or drink moderately; the alcoholics (), who will infect the people around to become new alcoholics; and the persons who temporarily quit drinking, denoted by ().The model in [8] is as follows: = Λ −  − ,   =  +  − ( + ) ,   =  −  − , (1) based on which, the authors calculated the basic reproduction number of alcoholism  0 and proved the existence and stability of the two equilibria by virtue of  0 and, simultaneously, analyzed the influence of the parameters on the stability of the system.However, this model is somewhat rough and simple.On the basis of this model, many authors have built new mathematical models from different angles to study the behavior of alcoholism.Taking the different stages of alcohol abuse into account, [9,10] proposed a twostage alcoholism model, while [10] also investigated different infection rates according to the two stages.References [12] considered the warning function of information publicity and education on persons against alcohol abuse.Besides the awareness of information, [14] also considered the time delay between the contact and infection during the course of alcohol abuse.To highlight the negative effects of alcohol abuse, [13] proposed a rather complex model to investigate the effect of alcohol on HIV infection.
Due to the fact that alcoholism can be prevented and treated [15], based on an earlier model of alcoholism [8], Wang et al. [16] proposed an SATR-type alcoholism model, and it is presented as follows: =  2 ()  − ( +  + ) , in which the persons in treatment are considered as a population compartment separately and put forward effective measures to control alcohol abuse.Specifically, in model (2), the total population is partitioned into four compartments: (), which refers to the persons who never drink or drink moderately without affecting the physical health; (), which refers to the persons who drink heavily; (), which refers to the persons being treated by taking pills or other medical interventions after alcoholism; (), which refers to the persons who recover from alcoholism after treatment and never drink again.Recently, Wang et al. [17] continue to consider model (2) with a distributed time delay between contact and infection during the course of alcoholism.Furthermore, in order to make the model have wider applicability in different environments, they generalize the incidence function from standard case to the abstract one.
Actually, in one hand, alcoholism is a social behavior; therefore, it is only infected and transmitted in a limited population.To depict this attribute, we will adopt saturation generating function ()()/(1 + ()) as [18] does.In the other hand, alcoholism will inevitably be affected by many uncertainties, such as emotions and environment.However, the models in the above-mentioned references did not take these random factors into account.In this paper, based on model (2), we will consider the random disturbances of the mortality in all populations due to the effect of alcohol abuse.To do that, we let (Ω, F, {F  } ≥0 , P) be a complete probability space with a filtration {F  } ≥0 satisfying the usual conditions (i.e., it is increasing and right continuous while F 0 contains all P-null sets) and let   (), 1 ≤  ≤ 4 be 4 independent standard Brownian motions.In practice, we usually estimate a parameter by an average value plus an error term.In this case, the parameters   ,   ,   , and   in (2) change to random variables μ , μ , μ , and μ , respectively, such that μ =   + error 1 , μ =   + error 2 , μ =   + error 3 , μ =   + error 4 .
(3) By the central limit theorem, the error terms may be approximated by normal distributions with mean zero and variance  2  , 1 ≤  ≤ 4, respectively, so we represent them as follows: With the above crucial factors considered, in this paper, based on model (2), we can get a new stochastic alcoholism dynamic model which is characterized by For the parameters in (5) and their explanations, please see Table 1.We will utilize stochastic analysis method to investigate the dynamic behavior of system (5).
This paper is arranged as follows.In Section 2, we put forward preliminaries including some tools for stochastic analysis and basic properties of deterministic model corresponding to (5) with   = 0,  = 1, 2, 3, 4. In Section 3, we discuss the existence and uniqueness of the positive solution of (5).In Section 4, we discuss the stochastic stability of alcohol-free equilibrium point  0 .In Section 5, we discuss the stochastic stability of the internal alcoholism equilibrium point  * .In Section 6, we carry out some simulations to support our theoretical results, In the last section, we give some conclusions to end this paper.The ratio of alcoholics in the treatment The ratio of the treatment population who are failed and return to be alcoholics

𝜂
The ratio of successfully treated population and never drink hereafter

Preliminaries
Now, we give some criteria on the ergodic property.Denote In general, let  be a regular temporally homogeneous Markov process in   ⊂   + described by the stochastic differential equation with initial value ( 0 ) =  0 ∈   and   (), 1 ≤  ≤ , are standard Brownian motions defined on the above probability space.The diffusion matrix is defined as follows: Define the differential operator  associated with (7) by If  acts on a function  ∈  2,1 (  ×  + ; ), then where   = (/ Lemma 1 (see [19]).We assume that there exists a bounded domain  ⊂   with regular boundary, having the following properties:  [20, page 1163]).

Basic Properties of Deterministic Model.
To compare some results between the deterministic model and stochastic model, firstly, we let   = 0,  = 1, 2, 3, 4 to get the corresponding deterministic model as follows: We can derived the alcohol-free equilibrium  0 of the deterministic model corresponding to (13); that is,  0 = (Π/  , 0, 0, 0).By using the method of next generation matrix [21], we can calculate to get the fundamental reproduction number of alcoholism which is The biological meaning of  0 is as follows.Once an alcoholic is placed in the environment full of healthy persons with the initial population Π/  , during the course of alcohol transmission time 1/( 1 +   ), the alcoholic will infect the healthy persons. 0 denotes the number of healthy persons who are successfully infected.

The Existence and Uniqueness of the Positive Solution of (5)
For the sake of biological and practical meanings, in this section, we will investigate the long-term behaviors of system (5); that is, no matter how much the disturbance intensity is, it has a unique global positive solution.According to classical theory of the stochastic differential equations, in order to make sure the existence and uniqueness of the global solution of system (5) are under the given initial conditions (i.e., it will not blow up in a limited time), we generally require that the coefficient of stochastic system (5) satisfies the linear growth condition and the local Lipschitz condition [22].However, the coefficient of system (5) does not satisfy the linear growth condition; therefore, the solution is likely to blow up in a finite time.In the following, we will use the analysis method of Lyapunov functional to prove global existence of positive solution.

Stochastic Stability of Alcohol-Free Equilibrium Point 𝐸 0
Seen from Theorem 4, for deterministic system (13), as  0 < 1, alcohol-free equilibrium point  0 = (Π/  , 0, 0, 0) is globally asymptotically stable, which means the alcohol population will disappear in the appropriate conditions finally, and the alcohol behavior will be effectively controlled.However, there is not any equilibrium point in the stochastic system (5); therefore, in this section, we will discuss the disturbance behavior of system (5) by investigating the stability and the ergodic property of  0 .
Remark 7. Besides,  0 becomes the disease-free equilibrium of system (1) as  1 = 0.In the proof, we see Thus, the solution of system is stochastically asymptotically stable in the large [22] as the conditions   /2 >  2 , ( +   )/2 >  3 are satisfied.

Stochastic Stability of the Internal
Equilibrium Point  * In this section, we will investigate the stochastic behavior of (5) around the internal equilibrium point  * in deterministic model (13) from two aspects; that is, one is concerned with the stability of its distribution, the other is concerned with ergodic property of equilibrium point  * .
Theorem 8.If the following conditions are satisfied, where then ( 5) is ergodic and positive recurrent.
Proof.Construct Lyapunov function  1 as and calculate to get (53) Next, we let and calculate to get Then, it is easy to know where the last inequality is derived from  ≥ log  + 1,  > 0.
we calculate to get similar to the technique in computing  1 , we calculate to get Considering the linear combination of  3 and  4 as then Next, we define then We let it is easy to prove that We consider the linear combination of then where   ,  = 1, 2, 3, 4 is represented as then  ≤ − for some  > 0. The proof is complete.
Next, we will discuss the mean convergence of system (5) based on the condition in Theorem 6.
Therefore, if we define  = /(1 + ) or  or  or , respectively, we can prove the corresponding convergence.By the use of ( 5), we can get The others conclusions can be similarly proved.The proof is completed.
Synthesizing and comparing the information in Figures 1 and 2, we can conclude that when  0 < 1, regardless of the size of the disturbance intensity, the solution of stochastic system (5) will randomly tend to the alcohol-free equilibrium  0 = (Π/  , 0, 0, 0) of deterministic system (13).Specifically speaking, if we can take some effective measures, for example, controlling the population recruitment rate Π or reducing the infection rate , the population in (), (), () will eventually tend to zero while the population in () will eventually tend to a constant Π/  .Figures 1 and 2 agree with the conclusions in Theorems 6 and 9. (2) We choose population recruitment rate as Π = 0.2 and alcohol infection rate as  = 0.002 to make  0 = 1.59 > 1.
Similarly, by synthesizing and comparing the information in Figures 3 and 4, we can conclude that, in one hand, when  0 > 1, regardless of the size of the disturbance intensity, the solution of stochastic system (5) will randomly tend to the internal equilibrium point  * = ( * ,  * ,  * ,  * ) of deterministic system corresponding to   = 0 ( = 1, 2, 3, 4) in (5).Specifically speaking, if the population recruitment rate Π or the infection rate  is rather large, the population in (), (), () will increase with time going and eventually tends to a fixed level, while the population in () will decrease with time going and eventually tends to a constant.In the other hand, the disturbance intensity of the system will affect the speed of the trend; specifically, the greater the disturbance, the slower this trend, and vice versa.(3) To testify the ergodic property of system (5), using the same parameters as case (2), we draw Figures 5 and 6   follows only with relatively large disturbance intensity   = 0.05,  = 1, 2, 3, 4.
Figures 5 and 6 reveal the fact that if system parameters meet the conditions to make  0 > 1, then the model is ergodic and has a uniqueness stationary distribution which illustrates the persistence.This agrees with the conclusions in Theorem 9.
Figures 3  and 4  agree with the conclusions in Theorems 8 and 9. as