Mathematical analysis of an age structured heroin-cocaine epidemic model

Abdennasser Chekroun; Mohammed Nor Frioui; Toshikazu Kuniya; Tarik Mohammed Touaoula

doi:10.3934/dcdsb.2020107

Article Contents

2020, Volume 25, Issue 11: 4449-4477. Doi: 10.3934/dcdsb.2020107

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Mathematical analysis of an age structured heroin-cocaine epidemic model

1.
Laboratoire d'Analyse Nonlinéaire et Mathématiques Appliquées, University of Tlemcen, Tlemcen 13000, Algeria
2.
Graduate School of System Informatics, Kobe University, 1-1 Rokkodai-cho, Nada-ku, Kobe 657-8501, Japan

^* Corresponding author
^* Corresponding author

Received: November 30, 2018

Revised: November 30, 2019

Early access: March 2020

Published: November 2020

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Abstract

This paper is devoted to studying the dynamics of a certain age structured heroin-cocaine epidemic model. More precisely, this model takes into account the following unknown variables: susceptible individuals, heroin users, cocaine users and recovered individuals. Each one of these classes can change or interact with others. In this paper, firstly, we give some results on the existence, uniqueness and positivity of solutions. Next, we obtain a threshold value $ r(\Psi'[0]) $ such that an endemic equilibrium exists if $ r(\Psi'[0]) > 1 $. We then show that if $ r(\Psi'[0]) < 1 $, then the disease-free equilibrium is globally asymptotically stable, whereas if $ r(\Psi'[0]) > 1 $, then the system is uniformly persistent. Moreover, for $ r(\Psi'[0]) > 1 $, we show that the endemic equilibrium is globally asymptotically stable under an additional assumption that epidemic parameters for heroin users and cocaine users are same. Finally, some numerical simulations are presented to illustrate our theoretical results.

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Mathematics Subject Classification: Primary: 35Q92, 37N25; Secondary: 92D30.

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Figure 1. Transfer diagram for model (1)

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Figure 2. The functions $ \beta $, $ \theta_1 $ and $ \theta_2 $ with respect to age $ a $. (left) $ \theta_1 \equiv \theta_2 $ as in (40), (right) $ \theta_1 $ greater than $ \theta_2 $

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Figure 3. The evolution of solution $ S $ and $ R $ with respect to time $ t $ are drawn. The case of disease-free equilibrium with $ r(\Psi'[0]) < 1 $

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Figure 4. The evolution of solutions $ i_1 $ and $ i_2 $ with respect to time $ t $ and age $ a $. The case of disease-free equilibrium with $ r(\Psi'[0]) < 1 $

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Figure 5. The evolution of solution $ S $ and $ R $ with respect to time $ t $ are drawn. The case of endemic equilibrium with $ r(\Psi'[0]) > 1 $

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Figure 6. The evolution of solutions $ i_1 $ and $ i_2 $ with respect to time $ t $ and age $ a $. The case of endemic equilibrium with $ r(\Psi'[0]) > 1 $

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Table 1. Description of each symbol in model (1)

Symbol	Description
$ S(t) $	Density of susceptible individuals at time $ t $
$ i_1(t, \xi_1) $	Density of heroin users at time $ t $ and age $ \xi_1 $
$ i_2(t, \xi_2) $	Density of cocaine users at time $ t $ and age $ \xi_2 $
$ R(t) $	Density of recovered individuals at time $ t $
$ A $	Number of all newborns per unit time
$ \mu $	Natural death rate per capita and unit time
$ \beta(\xi_i) $ $ (i=1, 2) $	Transmission rate for drug users with age $ \xi_i $ $ (i=1, 2) $
$ \theta_1(\xi_1) $	Recovery rate from the consumption of heroin at age $ \xi_1 $
$ \theta_2(\xi_2) $	Recovery rate from the consumption of cocaine at age $ \xi_2 $
$ k_1 $	Rate at which an individual recovered from the consumption
	of heroin becomes a cocaine user
$ k_2 $	Rate at which an individual recovered from the consumption
	of cocaine becomes a heroin user
$ \delta_1 $	Rate at which a recovered individual becomes a heroin user
$ \delta_2 $	Rate at which a recovered individual becomes a cocaine user

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