Role of Epidemic Model to Control Drinking Problem

Ankit Agrawal¹ , Abha Tenguria² , Geeta Modi³

Section:Research Paper, Product Type: Isroset-Journal
Vol.5 , Issue.4 , pp.324-337, Aug-2018

CrossRef-DOI:   https://doi.org/10.26438/ijsrmss/v5i4.324337

Online published on Aug 31, 2018

Copyright © Ankit Agrawal, Abha Tenguria, Geeta Modi . This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
 

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Abstract :
In this paper, we have developed a non-linear mathematical model of alcohol abuse with non-linear incidence rate proposed by Anderson and May [1] which consists of four compartments corresponding to four population classes, namely, non- drinkers , heavy drinkers , drinkers in treatment and temporarily recovered class . The basic properties and sensitivity analysis of the system are discussed. Next, Basic reproduction number is calculated. The local and global stability of the drinking-free (problem free) equilibrium and the endemic equilibrium of the model are 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 when . When , endemic equilibrium exists and the system becomes locally asymptotically stable at and becomes unstable. Finally, numerical findings by using actual data of my village are illustrated through computer simulations using MATLAB software, which show the reducing the contact rate between the non-drinkers and heavy drinkers, increasing the number of drinkers that go into treatment and awareness combating the drinking epidemic.

Key-Words / Index Term :
epidemic, treatment, awareness effect, drinking-free and endemic equilibrium, local and global stability

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