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Dynamics of synthetic drug transmission models

  • Shitao Liu and Liang Zhang EMAIL logo
Published/Copyright: March 29, 2021
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International Journal of Nonlinear Sciences and Numerical Simulation
From the journal International Journal of Nonlinear Sciences and Numerical Simulation Volume 23 Issue 3-4

Abstract

The deterministic and stochastic synthetic drug transmission models with relapse are formulated. For the deterministic model, the basic reproduction number R 0 is derived. We show that if R 0 < 1, the drug-free equilibrium is globally asymptotically stable and if R 0 > 1, there exists a unique drug-addition equilibrium which is globally asymptotically stable. For the stochastic model, we show there exists a unique global positive solution of the stochastic model for any positive initial value. Then by constructing some stochastic Lyapunov functions, we show that the solution of the stochastic model is going around each of the steady states of the corresponding deterministic model under certain parametric conditions. The sensitive analysis of the basic reproduction number R 0 indicates that it is helpful to reduce the relapse rate of people who have a history of drug abuse in the control of synthetic drug spreading. Numerical simulations are carried out and approve our results.

Keywords: globally asymptotical behavior; stochastic fluctuation; synthetic drug transmission model
2010 MSC: 34D23; 35B40; 37N25; 60H10
Corresponding author: Liang Zhang, College of Science, Northwest A&F University Yangling, Shaanxi 712100, PR China, E-mail:

Funding source: National Natural Science Foundation of China

Award Identifier / Grant number: 11601405

Acknowledgements

This work was partially supported by the CSC (201806305025), NSF of Shaanxi Province (2020JM-175), and NSF of China (No. 11601405). The authors also are grateful to the Reviewers and the Editor, and their valuable suggestions led to the improvement of our original manuscript.

  1. Author contribution: All the authors have accepted responsibility for the entire content of this submitted manuscript and approved submission.

  2. Research funding: None declared.

  3. Conflict of interest statement: The authors declare no conflicts of interest regarding this article.

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Received: 2019-01-27
Revised: 2020-05-11
Accepted: 2021-03-05
Published Online: 2021-03-29
Published in Print: 2022-06-25

© 2021 Walter de Gruyter GmbH, Berlin/Boston

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  1. Frontmatter
  2. Original Research Articles
  3. Dynamics of synthetic drug transmission models
  4. Collision-induced amplitude dynamics of pulses in linear waveguides with the generic nonlinear loss
  5. Global dissipativity of non-autonomous BAM neural networks with mixed time-varying delays and discontinuous activations
  6. On the inverse problem for nonlinear strongly damped wave equations with discrete random noise
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https://doi.org/10.1515/ijnsns-2019-0037
Keywords for this article
globally asymptotical behavior; stochastic fluctuation; synthetic drug transmission model

Articles in the same Issue

  1. Frontmatter
  2. Original Research Articles
  3. Dynamics of synthetic drug transmission models
  4. Collision-induced amplitude dynamics of pulses in linear waveguides with the generic nonlinear loss
  5. Global dissipativity of non-autonomous BAM neural networks with mixed time-varying delays and discontinuous activations
  6. On the inverse problem for nonlinear strongly damped wave equations with discrete random noise
  7. A second-order nonlocal regularized variational model for multiframe image super-resolution
  8. Algebro-geometric integration of a modified shallow wave hierarchy
  9. Singularity analysis of a 7-DOF spatial hybrid manipulator for medical surgery
  10. Propagation of diffusing pollutant by kinetic flux-vector splitting method
  11. Limit cycles in a tritrophic food chain model with general functional responses
  12. Local and parallel stabilized finite element methods based on full domain decomposition for the stationary Stokes equations
  13. Stress concentration effect on deflection and stress fields of a master leaf spring through domain decomposition and geometry updation technique
  14. Electrostatically actuated double walled piezoelectric nanoshell subjected to nonlinear van der Waals effect: nonclassical vibrations and stability analysis
  15. Traveling wave solutions of the generalized Rosenau–Kawahara-RLW equation via the sine–cosine method and a generalized auxiliary equation method
  16. A predictor–corrector compact finite difference scheme for a nonlinear partial integro-differential equation
  17. Parameter inference with analytical propagators for stochastic models of autoregulated gene expression
  18. DCSK performance analysis of a chaos-based communication using a newly designed chaotic system
  19. On successive linearization method for differential equations with nonlinear conditions
  20. Comparison of different time discretization schemes for solving the Allen–Cahn equation
  21. The homoclinic breather wave solution, rational wave and n-soliton solution to a nonlinear differential equation
  22. Diversity of interaction phenomenon, cross-kink wave, and the bright-dark solitons for the (3 + 1)-dimensional Kadomtsev–Petviashvili–Boussinesq-like equation
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