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ISSN:2164-6457 (print)
ISSN:2164-6473 (online)
Journal of Applied Nonlinear Dynamics
Miguel A. F. Sanjuan (editor), Albert C.J. Luo (editor)
Miguel A. F. Sanjuan (editor)

Department of Physics, Universidad Rey Juan Carlos, 28933 Mostoles, Madrid, Spain

Email: miguel.sanjuan@urjc.es

Albert C.J. Luo (editor)

Department of Mechanical and Industrial Engineering, Southern Illinois University Ed-wardsville, IL 62026-1805, USA

Fax: +1 618 650 2555 Email: aluo@siue.edu

Age-Structured Heroin Transmission Model with Delay

Journal of Applied Nonlinear Dynamics 14(2) (2025) 271--284 | DOI:10.5890/JAND.2025.06.004

Imane Hana Zirar$^{1}$, Abdelkader Lakmeche$^{1}$, Nacera Helal$^{1,2}$, Salih Djilali$^3$, Anwar Zeb$^4$

$^1$ Biomathematics Laboratory, Djillali Liabes University of Sidi Bel-Abbes, 22000 Algeria

$^2$ Ecole sup'{e}rieure en informatique 08 Mai 1945 Sidi Bel-Abb`{e}s, Algeria

$^3$ Department of Mathematics, Faculty of Exact Sciences and informatics, Hassiba Benbouali University, Chlef 02000, Algeria

$^4$ Department of Mathematics, COMSATS University Islamabad, Abbottabad, 22060, Pakistan

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Abstract

This study investigates the outbreak of heroin addiction using a new model for heroin transmission with delay and a finite treatment period. The investigated model considers two different repulsion mechanisms, namely, repulsion from treatment to heroin addiction during treatment, and repulsion from treatment to potentially addicted individuals after treatment. These two repulsions make investigating the suggested age-structured model challenging. Indeed, we establish that the drug-free (resp. drug spread) equilibrium is locally asymptotically stable if the incidence rate $\beta$ is less than a certain threshold value $\beta^*$, with additional assumptions on the parameters of the model. Moreover, we have demonstrated the global asymptotic stability of the drug-free equilibrium when $\beta$ is less than another threshold value $\beta^{}$ (where $\beta^{**} < \beta^*$). Some numerical investigations of the model are conducted to identify effective measures for containing the epidemic.

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