Stability analysis of a delayed social epidemics model with general contact rate and its optimal control

Xun-Yang  Wang; Khalid  Hattaf; Hai-Feng  Huo; Hong  Xiang

doi:10.3934/jimo.2016.12.1267

Article Contents

2016, Volume 12, Issue 4: 1267-1285. Doi: 10.3934/jimo.2016.12.1267

This issue Previous Article Improved approximating $2$-CatSP for $\sigma\geq 0.50$ with an unbalanced rounding matrix Next Article A necessary condition for mean-field type stochastic differential equations with correlated state and observation noises

Stability analysis of a delayed social epidemics model with general contact rate and its optimal control

1.
College of Electrical and Information engineering, Lanzhou University of Technology, Lanzhou, Gansu, 730050
2.
Department of Mathematics and Computer Science, Faculty of Sciences Ben M'sik, Hassan II University, P.O. Box 7955, Sidi Othman, Casablanca
3.
Department of Applied Mathematics, Lanzhou University of Technology, Lanzhou, Gansu, 730050

Received: April 30, 2014

Revised: September 30, 2015

Published: September 2016

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Abstract

In this paper, we formulate an alcohol quitting model in which we consider the impact of distributed time delay between contact and infection process by characterizing dynamic nature of alcoholism behaviours, and we generalize the infection rate to the general case, simultaneously, we consider two different control strategies. Next, we discuss the qualities on the model, the existence and boundedness as well as positivity of the equilibrium are involved. Then, under certain proper conditions, we construct appropriate Lyapunov functionals to prove the global stability of alcohol free equilibrium point $E_{0}$ and alcoholism equilibrium $E^{*}$ respectively. Furthermore, the optimal control strategies are derived by proposing an objective functional and using classic Pontryagin's Maximum Principle. Numerical simulations are conducted to support our theoretical results derived in optimal control.

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Mathematics Subject Classification: Primary: 92D25.

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