A delayed synthetic drug transmission model with two stages of addiction and Holling Type-II functional response

Yougang Wang; Anwar Zeb; Ranjit Kumar Upadhyay; A Pratap; Yougang Wang; Anwar Zeb; Ranjit Kumar Upadhyay; A Pratap

doi:10.3934/math.2021001

AIMS Mathematics

2021, Volume 6, Issue 1: 1-22. doi: 10.3934/math.2021001

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Research article

A delayed synthetic drug transmission model with two stages of addiction and Holling Type-II functional response

1.
School of Management Science and Engineering, Anhui University of Finance and Economics, Bengbu, China
2.
Department of Mathematics, COMSATS University Islamabad, Abbottabad Campus, Abbottabad, 22060, Khyber Pakhtunkhwa, Pakistan
3.
Department of Mathematics & Computing, Indian Institute of Technology (Indian School of Mines), Dhanbad 826004, India
4.
Department of Mathematics, Alagappa University Alagappapuram, Karaikudi-630 004, India

Received: 24 July 2020 Accepted: 22 September 2020 Published: 28 September 2020
MSC : 34C23

This paper gropes the stability and Hopf bifurcation of a delayed synthetic drug transmission model with two stages of addiction and Holling Type-II functional response. The critical point at which a Hopf bifurcation occurs can be figured out by using the escalating time delay of psychologically addicts as a bifurcation parameter. Directly afterwards, properties of the Hopf bifurcation are explored with aid of the central manifold theorem and normal form theory. Specially, global stability of the model is proved by constructing a suitable Lyapunov function. To underline effectiveness of the obtained results and analyze influence of some influential parameters on dynamics of the model, some numerical simulations are ultimately addressed.
- delay,
- Hopf bifurcation,
- synthetic drugs model,
- stability,
- periodic solution
Citation: Yougang Wang, Anwar Zeb, Ranjit Kumar Upadhyay, A Pratap. A delayed synthetic drug transmission model with two stages of addiction and Holling Type-II functional response[J]. AIMS Mathematics, 2021, 6(1): 1-22. doi: 10.3934/math.2021001

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Abstract

This paper gropes the stability and Hopf bifurcation of a delayed synthetic drug transmission model with two stages of addiction and Holling Type-II functional response. The critical point at which a Hopf bifurcation occurs can be figured out by using the escalating time delay of psychologically addicts as a bifurcation parameter. Directly afterwards, properties of the Hopf bifurcation are explored with aid of the central manifold theorem and normal form theory. Specially, global stability of the model is proved by constructing a suitable Lyapunov function. To underline effectiveness of the obtained results and analyze influence of some influential parameters on dynamics of the model, some numerical simulations are ultimately addressed.

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