# American Institute of Mathematical Sciences

November  2020, 25(11): 4449-4477. doi: 10.3934/dcdsb.2020107

## 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

Received  December 2018 Revised  December 2019 Published  March 2020

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.

Citation: Abdennasser Chekroun, Mohammed Nor Frioui, Toshikazu Kuniya, Tarik Mohammed Touaoula. Mathematical analysis of an age structured heroin-cocaine epidemic model. Discrete & Continuous Dynamical Systems - B, 2020, 25 (11) : 4449-4477. doi: 10.3934/dcdsb.2020107
##### References:

show all references

##### References:
Transfer diagram for model (1)
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 $\theta_2$
The evolution of solution $S$ and $R$ with respect to time $t$ are drawn. The case of disease-free equilibrium with $r(\Psi'[0]) < 1$
The evolution of solutions $i_1$ and $i_2$ with respect to time $t$ and age $a$. The case of disease-free equilibrium with $r(\Psi'[0]) < 1$
The evolution of solution $S$ and $R$ with respect to time $t$ are drawn. The case of endemic equilibrium with $r(\Psi'[0]) > 1$
The evolution of solutions $i_1$ and $i_2$ with respect to time $t$ and age $a$. The case of endemic equilibrium with $r(\Psi'[0]) > 1$
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
 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
 [1] Linlin Li, Bedreddine Ainseba. Large-time behavior of matured population in an age-structured model. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2561-2580. doi: 10.3934/dcdsb.2020195 [2] Rafael Luís, Sandra Mendonça. A note on global stability in the periodic logistic map. Discrete & Continuous Dynamical Systems - B, 2020, 25 (11) : 4211-4220. doi: 10.3934/dcdsb.2020094 [3] Lakmi Niwanthi Wadippuli, Ivan Gudoshnikov, Oleg Makarenkov. Global asymptotic stability of nonconvex sweeping processes. Discrete & Continuous Dynamical Systems - B, 2020, 25 (3) : 1129-1139. doi: 10.3934/dcdsb.2019212 [4] Jing Feng, Bin-Guo Wang. An almost periodic Dengue transmission model with age structure and time-delayed input of vector in a patchy environment. Discrete & Continuous Dynamical Systems - B, 2021, 26 (6) : 3069-3096. doi: 10.3934/dcdsb.2020220 [5] Raimund Bürger, Christophe Chalons, Rafael Ordoñez, Luis Miguel Villada. A multiclass Lighthill-Whitham-Richards traffic model with a discontinuous velocity function. Networks & Heterogeneous Media, 2021, 16 (2) : 187-219. doi: 10.3934/nhm.2021004 [6] Simão Correia, Mário Figueira. A generalized complex Ginzburg-Landau equation: Global existence and stability results. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021056 [7] Michael Grinfeld, Amy Novick-Cohen. Some remarks on stability for a phase field model with memory. Discrete & Continuous Dynamical Systems, 2006, 15 (4) : 1089-1117. doi: 10.3934/dcds.2006.15.1089 [8] Rui Hu, Yuan Yuan. Stability, bifurcation analysis in a neural network model with delay and diffusion. Conference Publications, 2009, 2009 (Special) : 367-376. doi: 10.3934/proc.2009.2009.367 [9] Miroslav Bulíček, Victoria Patel, Endre Süli, Yasemin Şengül. Existence of large-data global weak solutions to a model of a strain-limiting viscoelastic body. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021053 [10] Lu Xu, Chunlai Mu, Qiao Xin. Global boundedness of solutions to the two-dimensional forager-exploiter model with logistic source. Discrete & Continuous Dynamical Systems, 2021, 41 (7) : 3031-3043. doi: 10.3934/dcds.2020396 [11] Harumi Hattori, Aesha Lagha. Global existence and decay rates of the solutions for a chemotaxis system with Lotka-Volterra type model for chemoattractant and repellent. Discrete & Continuous Dynamical Systems, 2021  doi: 10.3934/dcds.2021071 [12] Azeddine Elmajidi, Elhoussine Elmazoudi, Jamila Elalami, Noureddine Elalami. Dependent delay stability characterization for a polynomial T-S Carbon Dioxide model. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021035 [13] Jonathan DeWitt. Local Lyapunov spectrum rigidity of nilmanifold automorphisms. Journal of Modern Dynamics, 2021, 17: 65-109. doi: 10.3934/jmd.2021003 [14] Wenmeng Geng, Kai Tao. Lyapunov exponents of discrete quasi-periodic gevrey Schrödinger equations. Discrete & Continuous Dynamical Systems - B, 2021, 26 (6) : 2977-2996. doi: 10.3934/dcdsb.2020216 [15] Jianfeng Lv, Yan Gao, Na Zhao. The viability of switched nonlinear systems with piecewise smooth Lyapunov functions. Journal of Industrial & Management Optimization, 2021, 17 (4) : 1825-1843. doi: 10.3934/jimo.2020048 [16] Sara Munday. On the derivative of the $\alpha$-Farey-Minkowski function. Discrete & Continuous Dynamical Systems, 2014, 34 (2) : 709-732. doi: 10.3934/dcds.2014.34.709 [17] Ralf Hielscher, Michael Quellmalz. Reconstructing a function on the sphere from its means along vertical slices. Inverse Problems & Imaging, 2016, 10 (3) : 711-739. doi: 10.3934/ipi.2016018 [18] Skyler Simmons. Stability of Broucke's isosceles orbit. Discrete & Continuous Dynamical Systems, 2021, 41 (8) : 3759-3779. doi: 10.3934/dcds.2021015 [19] Charles Fulton, David Pearson, Steven Pruess. Characterization of the spectral density function for a one-sided tridiagonal Jacobi matrix operator. Conference Publications, 2013, 2013 (special) : 247-257. doi: 10.3934/proc.2013.2013.247 [20] Dayalal Suthar, Sunil Dutt Purohit, Haile Habenom, Jagdev Singh. Class of integrals and applications of fractional kinetic equation with the generalized multi-index Bessel function. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021019

2019 Impact Factor: 1.27