An efficient computational analysis for stochastic fractional heroin model with artificial decay term

Feliz Minhós; Ali Raza; Umar Shafique; Feliz Minhós; Ali Raza; Umar Shafique

doi:10.3934/math.2025278

AIMS Mathematics

2025, Volume 10, Issue 3: 6102-6127. doi: 10.3934/math.2025278

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An efficient computational analysis for stochastic fractional heroin model with artificial decay term

Feliz Minhós ^{1,2
,
,},
Ali Raza ^{2,3
,
, ,},
Umar Shafique ^{4
,}

1.
Department of Mathematics, School of Science and Technology, University of Évora, Rua Romão Ramalho, 59, Évora, 7000-671, Portugal
2.
Center for Research in Mathematics and Applications (CIMA), Institute for Advanced Studies and Research (IIFA), University of Évora, Rua Romão Ramalho, 59, Évora, 7000-671, Portugal
3.
Department of Physical Sciences, The University of Chenab, Gujrat, 50700, Pakistan; ali@phs.uchenab.edu.pk
4.
Department of Mathematics, National College of Business Administration and Economics, Lahore, 54660, Pakistan

Received: 08 September 2024 Revised: 28 November 2024 Accepted: 03 December 2024 Published: 19 March 2025
MSC : 65M06, 65M12, 35K15, 35K55, 35K57

Heroin addiction is a continuously progressing phenomenon that represents a major problem for world's the public health; this indicates that the development of new methodologies to address the issue at the international level is a crucial priority. To study its transmission dynamics, a new stochastic fractional delayed heroin model based on stochastic fractional delay differential equations (SFDDEs) was developed to focus on the positive aspects of randomness and memory effects. The positive, boundedness, existence, and uniqueness of the model were studied rigorously. The equilibria (i.e., heroin-free equilibrium and the present equilibrium, which gives a clue about both eradication and persistence cases), reproduction number, and sensitivity of parameters were analyzed. The local and global stability of the new model was studied around its steady states. Also, well-known theorems are presented to investigate the extinction and persistence of heroin. The Grunwald-Letnikove non-standard finite difference (GL-NSFD) method was used for the efficient computational analysis of the stochastic fractional delayed model. For the dynamical consistency of the model, the positivity and boundedness of an efficient method were studied rigorously. The given study focuses on delay strategies and fractional calculus that could be useful in formulating specific measures for regulating addiction. Moreover, the simulated results support the theoretical analysis of the model and validate it.
- heroin model,
- stochastic fractional delay differential equations (SFDDEs),
- existence and uniqueness,
- stability results,
- extinction and persistence,
- GL-NSFD,
- results
Citation: Feliz Minhós, Ali Raza, Umar Shafique. An efficient computational analysis for stochastic fractional heroin model with artificial decay term[J]. AIMS Mathematics, 2025, 10(3): 6102-6127. doi: 10.3934/math.2025278

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Abstract

Heroin addiction is a continuously progressing phenomenon that represents a major problem for world's the public health; this indicates that the development of new methodologies to address the issue at the international level is a crucial priority. To study its transmission dynamics, a new stochastic fractional delayed heroin model based on stochastic fractional delay differential equations (SFDDEs) was developed to focus on the positive aspects of randomness and memory effects. The positive, boundedness, existence, and uniqueness of the model were studied rigorously. The equilibria (i.e., heroin-free equilibrium and the present equilibrium, which gives a clue about both eradication and persistence cases), reproduction number, and sensitivity of parameters were analyzed. The local and global stability of the new model was studied around its steady states. Also, well-known theorems are presented to investigate the extinction and persistence of heroin. The Grunwald-Letnikove non-standard finite difference (GL-NSFD) method was used for the efficient computational analysis of the stochastic fractional delayed model. For the dynamical consistency of the model, the positivity and boundedness of an efficient method were studied rigorously. The given study focuses on delay strategies and fractional calculus that could be useful in formulating specific measures for regulating addiction. Moreover, the simulated results support the theoretical analysis of the model and validate it.

References

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