Role of Epidemic Model to Control Drinking Problem

discussed. The local asymptotical stability of equilibrium is verified by analyzing the eigenvalues and using the Routh-Hurwitz criterion. Also discuss the global asymptotical stability of the drinking-free equilibrium by using LaSalle’s invariance principle and endemic equilibrium by autonomous convergence theorem. The stability analysis of the model shows that the system is locally asymptotically stable at drinking-free equilibrium 0 E when 0 1 R  . When 0 1 R  , endemic equilibrium * E exists and the system becomes locally asymptotically stable at * E and 0 E becomes unstable. Finally, numerical findings by using actual

© 2018, IJSRMSS All Rights Reserved 325 programmes usually offer counselling and therapy to discuss alcoholism and its effects, mental health support, medical care etc. There are two major forms of intervention policy of alcohol abuse: (i) prevention initiation into alcohol abuse and (ii) rehabilitation of alcohol abusers. Among the many problems confronting these programmes, the most important is the very high rates of relapse after treatment. The National Institute on Alcohol Abuse and Alcoholism estimates that up to 70% of treated alcohol abusers relapse after treatment which is indeed a big problem. Therefore prevention and control efforts that include treatment and education (Awareness) programmes should be improved so that the rate of relapse from treatment can be reduced. It is obvious that alcohol abuse and alcoholism not only causes health problems but also has great social and economic impacts on the countries. Therefore, it is very important to understand the dynamics of alcoholism spread among the populations and identify the parameters of greater importance which will help the policy-makers in targeting prevention and treatment resources for maximum effectiveness. Although drinking is a problem of significant public health importance, not much has been analyzed in terms of using mathematical modelling to gain insight into its transmission dynamics at population level. Most of the existing works on alcohol abuse and alcoholism are of clinical aspects.
Mathematical models could mimic the process of drinking and provided useful tools to analyze the spread and control of drinking behaviour. Several different mathematical models for drinking had been formulated and studied [2,4,5,6,7,8,9,10]. Giuseppe and Brian [4] developed a two-stage (four components) model for youths with serious drinking problems and their treatment. The stability of all the equilibria was analyzed. Mubayi et al. [2] introduced a simple framework where drinking was modeled as a socially contagious process in low and high-risk connected environments. Lee et al. [5] introduced a mathematical model of drinking that incorporated the impact of relapse, and was analyzed primarily under the impact of two time-dependent controls put in place over a infinite time horizon. Xiang et al. [6] presented a quit drinking model taking into account of permanent quit drinker compartment and relapse, and global stability of equilibria was obtained. Huo and Song [7] introduced a more realistic two-stage model for binge drinking problem, where the youths with alcohol problems were divided into those who admitted the problem and those who did not admit. Mathematical analyses established that the global dynamics of the model were determined by the basic reproduction number. Swarnali Sharma [8] and Isaac Kwasi Adu [9] introduce four compartmental models with treatment and relapse. For the other mathematical models for drinking, we referred to [10] and the references therein.
The organization of the paper is as follows. In section-III, 3.1 describe the development of model and give a detail of model assumptions, 3.2 presented basic properties of the system, and 3.3 presented the basic reproduction and give a detail of sensitivity analysis of reproduction in 3.4. The detail of equilibrium points presented in 3.5. The local stability of the system presented in 3.6 and the global stability of the system presented in 3.7. A numerical simulation experiment has been presented in Section-IV. Conclusion and future scope presented in the section-V.

II. RELATED WORK
Motivated by the work of Swarnali Sharma [8] and Isaac Kwasi Adu [9], in this paper, an SHTR epidemic model included with the effect of saturated incidence rate () 1

SH g S H S
    concerned. This rate was proposed by Anderson and May [1]. The purpose of this paper is to show that the effect of treatment and awareness parameter to control drinking problem.

Model assumptions
The following assumptions were made in the model: The drinking epidemic occurs in a closed environment.
(ii) Farmers, labors and sex workers taking wine every evening for body pain relief, mental peace and for sleep.
(iii) Social status and race do not affect the probability of becoming a heavy drinker.
(iv) Heavy drinking is transmitted to non-drinkers when they are in contact with heavy drinkers.
(v) Members mix homogeneously (have the same interaction to the same degree).
(vi) Drinkers in treatment may only become heavy drinkers again after passing through the recovery and susceptible compartments respectively.
(vii) Drinkers who have stopped drinking enter into recovery compartment.
Under the above assumptions the SIR epidemic model takes the following form: with the initial conditions (0) 0 Rt  for all t ≥ 0. Proof. Since the right hand side of system (1) is completely continuous and locally Lipschitzian on C (space of continuous functions), the solution ( ( ), ( ), ( ), ( )) S t H t T t R t of (1) with initial conditions (2) exists and is unique on [0, ) Similarly, from the forth equation of (1) have Finally, it follows from the first equation of the system (1) that, Rt   t ≥ 0. This completes the proof.

Invariant region
Adding the equations of the system (1), obtain The solution () Nt of the differential equation (3) has the following property, where, (0) N represents the sum of the initial values of the variables.
, then the solution ( ( ), ( ), ( ), ( )) S t H t T t R t enters  or approach it asymptotically. Hence it is positively invariant under the flow induced by the system (1). Thus in  , the model (1) is well-posed epidemiologically and mathematically. Hence it is sufficient to study the dynamics of the model in  .

The basic reproduction number 0
R Basic reproduction number [3] for drinking epidemic model is defined as the number of heavy drinkers produced when a single heavy drinker is introduced into susceptible (non-drinkers) population, i.e., (1) is given by 0

Sensitivity analysis of 0 R
The basic reproduction number 0 R of system (1) depends on five parameters, namely, the transmission coefficient from susceptible to heavy drinkers  , drinking related death rate of heavy drinkers 1  , the pro-portion of drinkers who enter into treatment  ,  awareness effect parameter and the natural death rate of population  . Among those parameters, we cannot control the natural death rate of population  . Therefore, to examine the sensitivity of 0 R to the parameters  , 1  ,  and  , normalized forward sensitivity index with respect to each of those parameters are computed as follows: From the above discussion it is clear that the basic reproduction number 0 R is most sensitive to changes in  , the transmission coefficient from susceptible population to heavy drinkers. If  will increase 0 R will increase in same proportion and if  will decrease 0 R will also decrease in same proportion. On the other hand 1  ,  and  have an inversely proportional relationship with 0 R , i.e., an increase in any of them will cause a decrease in 0 R and a decrease in any of them will cause an increase in 0 R . But the increase in 1  , the drinking related death rate of the heavy drinkers not in treatment, is neither ethical nor practical. So, it is better to focus either on  , the transmission rate from susceptible population to heavy drinker,  awareness parameter or  , the proportion of drinkers who enter into treatment. As 0 R is more sensitive to changes in  than © 2018, IJSRMSS All Rights Reserved 329  and  , it seems sensible to focus on the reduction of  and  is increase to control the alcohol abuse. This sensitivity analysis tells us that efforts to increase prevention are more effective in controlling the spread of alcohol abuse in population than efforts to increase the numbers of heavy drinkers accessing treatment.

Equilibrium points and existence
In this section, the drinking-free (problem free) and the endemic equilibrium points of system (1) find and analyze their existence.
The equilibrium points of the model system (1) are: 1. Drinking-free (problem free) equilibrium: 0 ( , 0, 0, 0) b E  , 2. Endemic equilibrium: * * * * * ( , , , ) E S H T R , Here use the term "drinking-free equilibrium" to describe the state where a drinking culture does not exist, i.e. the equilibrium points of system (1) at the origin (0, 0). On the other hand, "endemic equilibrium" stands for the state where a drinking culture exists, i.e. the non-trivial positive solution of system (1).

Existence of epidemic equilibrium * * * * * ( , , , ) E S H T R
At an endemic equilibrium, drinkers are present and the following conditions hold: By applying the Descarte's rule of signs if 0 1 R  , one positive equilibria exists and if 0 1 R  , system has no positive equilibrium. Summarizing the previous discussions come to the following result: Theorem 3.5. 1 The system (1) has a drinking-free equilibrium 0 ( , 0, 0, 0) b E  , which exists for all parameter values. If 0 1 R  , the system (1) also admits a unique endemic equilibrium * * * * * ( , , , ) E S H T R . If 0 1 R  , then the system has no endemic equilibrium and if 0 1 R  , system has a positive equilibrium.

Local stability of drinking-free (problem free) equilibrium
In this section find the local stability of the system (1) at drinking-free equilibrium 0 E . Let system (1) as:

HS S S HS J S H T R S S
The Jacobian matrix of system (1) at 0 ( , 0, 0, 0 Therefore, eigenvalues of the characteristic equation Here, 1  , 3  and 4  are clearly real and negative. Now, 0

HS
 Written as the above equation: Using the Routh-Hurwitz criterion [6]. It can be seen that all eigen values of the characteristic equation has negative real part if and only if Theorem 3.6.2 The endemic equilibrium * E of the model system (1) is locally asymptotically stable if and only if inequalities (5) are satisfied.

Global stability of drinking-free equilibrium
In this section discuss about the global stability of the drinking free equilibrium 0 E when * 0 1 Consider the Lyapunov function as follows: The derivative of ( , , ) : 0 , is the singleton 0 {} E . LaSalle's invariance principle [6] implies that 0 E is globally asymptotically stable in  when      then the drinking free equilibrium (DFE) 0 E of model system (1) is globally asymptotically stable.

Global stability of endemic equilibrium
Determine the global stability of the endemic equilibrium in this section.
Reduce the system (1), by using ( ) to eliminate () Rt from the first equation of system (1), which leads to the following three dimensional model: Tt  . Now, use the method of Li and Muldowney [7], the geometric approach method, for the global stability of an endemic equilibrium. We find the sufficient conditions for which the endemic equilibrium is globally asymptotically stable. We first briefly explain the geometric approach method. Consider () where :, nn f D R D R  is an open set and is simply connected and 1 () Let * x be the solution of (6) i.e. * ( ) 0 fx  . Assume that the following hypotheses hold. (H1) There exists a compact absorbing set KD  . (H2) Equation (7) has a unique equilibrium * x in D . The basic idea of this method is that if the equilibrium * x is locally stable, then the stability is assured provided that (H1) and (H2) hold and no non constant periodic solution of (7) exists. Therefore, sufficient conditions on f capable of precluding the existence of such solutions have to be detected. Suppose that assumptions (H1) and (H2) hold. Assume that (7) satisfies a Bendixson criterion that is robust under 1 C local perturbations of f at all non-equilibrium non-wandering points for (7). The * x is globally stable in D provided it is stable.  (8) where the matrix and the matrix [2] J is the second additive compound matrix of the Jacobian matrix J , that is, ( )  (11) It is proved in [7] that if (H1) and (H2) hold, condition It is shown in [7] that, if D is simply connected, the condition 0 q  rules out the presence of any orbit that gives rise to a simple closed rectifiable curve that is invariant for (7), such as periodic orbits, homoclinic orbits, and heteroclinic cycles. Moreover, it is robust under 1 C local perturbations of f near any non equilibrium point that is non-wandering. In particular, the following global-stability result is proved in Li and Muldowney [7]. x of (7) is globally stable in D if 0 q  . Now, study the global stability of the endemic equilibrium * E and obtain.
Theorem 3.7.2 If 0 1 R  then the endemic equilibrium * E of the system (6) is globally stable.   .This means that the model at 0 1 R  . This also indicates that the drinking free equilibrium 0 E is asymptotically stable (Figure. 2).

V. CONCLUSION and Future Scope
In this chapter, the model shows that, drinking epidemic cannot only be controlled by reducing the contact rate between the nondrinkers and heavy drinkers but also increasing the number of drinkers that go into treatment and aware (educating) drinkers to refrain from drinking can be useful in combating the epidemic.