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In this paper, we analyse population dynamics of liquor habit. Liquor free and Liquor endemic equilibrium are computed. The local and global stabilities of the proposed problem are established. The numerical simulation is given to validate the transmission of population indifferent compartment using state-space model.

The Ayurvedas depicted that the alcohol behave as a medicine if it is taken for the purpose of meditation otherwise it behave as a poison if it is taken in addiction manner for the humans. The liquor habits reflect person’s social status and cultural prestige. It is observed that transmission of liquoring spreads frequently these days due to availability of the liquor in the market. There is a similarity between spread of infectious disease and liquor habits. In other words, liquor habit can be treated as a virus which transmits among the compartments by social pressure like parties with friends, peers and executive meetings.

In this paper, we analyze quantitative model of a liquor habit transmission in a population similar to SEIR- model. The notations are described in Section 2. The mathematical model and basic reproduction number are formulated in Section 3. The local and global stability are derived in Sections 3.2.1 and 3.2.2 respectively. Numerical simulations are illustrated in Section 4 using State-Space model. Discussions and conclusions are given in Section 5.

The model is derived using following notations.

Anybody in the population is susceptible to liquor. In general, one starts taking pegs and increases it gradually. The consumption of two pegs is considered as a normal which falls in E-compartment. The work stress, financial stress, and many more factors to go for more than two pegs which fall under I-compartment. Certain fraction of population from E- and I-compartments may be removed. It is taken as R-compartment. Thus, it resembles toSEIR-model.

Let us call

From

Since, an epidemic model occurs in a short time period, we ignore moving portion of removal from liquor. So, we will analyze the first three equations forming new reduced system

Adding all these three equations, we have

Which gives,

Therefore, the feasible region for (2) is

Now, the basic reproduction number

Let

where

F and V are

So

where V is non-singular matrix, so that

Hence, the basic reproduction number

Next, we need to discuss equilibrium of the liquor habit system.

The liquor free equilibrium is locally asymptotically stable if all the eigenvalues of the matrix have positive real values (Al-Amoudi et al. (2014) [

Theorem 1 (Johnson (2004) [

Proof Let

Thus

Since

Then, we have

Finally, since

Thus,

Hence,

Similarly, it follows that

Û

Û

Hence,

It follows that

The liquor free equilibrium is stable if all the eigenvalues of the Jacobian matrix of the system (1) have negative real parts. For this, the Jacobian of the system (1) at

Here

The liquor free equilibrium is globally stable if

where

In this section, we perform numerical simulation of the system (1) using with the state-space model.

State-Space ModelState equation

Output equation

The general state-space description for a linear time invariant, continuous time dynamical system is

where

in

We carry out the simulation. The results are shown in

From

that more than two pegs are taken immediately after 3^{rd} week and increases exponentially. Almost 15% gets chain liquor (those who can’t survive without liquor). The removal compartment (

In this paper, a nonlinear mathematical model for liquor transmission is analysed. The local and global stability of the liquor free equilibrium point are established. It is proved that the free equilibrium is locally asymptotically stable when basic reproduction number

This model can be extended by educating youth for non-liquoring via advertisement, rehabilitation centre etc.

The first author thanks DST-FIST file # MSI-097 for technical support.

Nita H.Shah,Bijal M.Yeolekar,Nehal J.Shukla, (2015) Liquor Habit Transmission Model. Applied Mathematics,06,1208-1213. doi: 10.4236/am.2015.68112