Dynamics of an age-structured heroin transmission model with vaccination and treatment

Xi-Chao Duan; Xue-Zhi Li; Maia Martcheva; Xi-Chao Duan; Xue-Zhi Li; Maia Martcheva

doi:10.3934/mbe.2019019

Mathematical Biosciences and Engineering

2019, Volume 16, Issue 1: 397-420. doi: 10.3934/mbe.2019019

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Research article

Dynamics of an age-structured heroin transmission model with vaccination and treatment

1.
College of Science, University of Shanghai for Science and Technology, Shanghai 200093, China, School of Mathematical Sciences, Tongji University, Shanghai 200092, China
2.
College of Mathematics and Information Science, Henan Normal University, Xinxiang 453007, China
3.
Department of Mathematics, University of Florida, 358 Little Hall, PO Box 118105, Gainesville, FL 32611–8105, United States

Received: 04 April 2018 Accepted: 04 September 2018 Published: 14 December 2018

Based on the development of heroin vaccine, in this paper, we propose an age structured heroin transmission model with treatment and vaccination. The model allows the drug reuse rate of the individuals in treatment to depend on a treatment-age and the vaccine waning rate of the vaccinated to depend on a vaccination age. Meanwhile, the model allows that the heroin vaccine provides an imperfect protection (i.e., the vaccinated individuals can also become drug addicted). We derive the basic reproduction number which dependents on vaccination. The basic reproduction number completely determines the persistence and extinction of heroin spread, i.e., if the basic reproduction number is less than one the drug-free steady state is globally asymptotically stable (i.e., the heroin spread dies out), if the basic reproduction number is larger than one, there exists an unique positive steady state and it is locally and globally stable in some special cases. Finally, some numerical simulations are carried out to illustrate the stability of the positive steady state.
- heroin transmission model,
- age of vaccination,
- basic reproduction number,
- drug-free steady state,
- drug spread steady state
Citation: Xi-Chao Duan, Xue-Zhi Li, Maia Martcheva. Dynamics of an age-structured heroin transmission model with vaccination and treatment[J]. Mathematical Biosciences and Engineering, 2019, 16(1): 397-420. doi: 10.3934/mbe.2019019

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Abstract

Based on the development of heroin vaccine, in this paper, we propose an age structured heroin transmission model with treatment and vaccination. The model allows the drug reuse rate of the individuals in treatment to depend on a treatment-age and the vaccine waning rate of the vaccinated to depend on a vaccination age. Meanwhile, the model allows that the heroin vaccine provides an imperfect protection (i.e., the vaccinated individuals can also become drug addicted). We derive the basic reproduction number which dependents on vaccination. The basic reproduction number completely determines the persistence and extinction of heroin spread, i.e., if the basic reproduction number is less than one the drug-free steady state is globally asymptotically stable (i.e., the heroin spread dies out), if the basic reproduction number is larger than one, there exists an unique positive steady state and it is locally and globally stable in some special cases. Finally, some numerical simulations are carried out to illustrate the stability of the positive steady state.

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