### AIMS Mathematics

2022, Issue 10: 18173-18193. doi: 10.3934/math.20221000
Research article Special Issues

# Analysis of illegal drug transmission model using fractional delay differential equations

• Received: 20 June 2022 Revised: 14 July 2022 Accepted: 17 July 2022 Published: 11 August 2022
• MSC : 00A71, 26A33, 92D30, 93C43

• The global burden of illegal drug-related death and disability continues to be a public health threat in developed and developing countries. Hence, a fractional-order mathematical modeling approach is presented in this study to examine the consequences of illegal drug usage in the community. Based on epidemiological principles, the transmission mechanism is the social interaction between susceptible and illegal drug users. A pandemic threshold value ($\Lambda$) is provided for the illegal drug-using profession, which determines the stability of the model. The Lyapunov function is employed to determine the stability conditions of illegal drug addiction equilibrium point of society. Finally, the proposed model has been extended to include time lag in the delayed illegal drug transmission model. The characteristic equation of the endemic equilibrium establishes a set of appropriate conditions for ensuring local stability and the development of a Hopf bifurcation of the model. Finally, numerical simulations are performed to support the analytical results.

Citation: Komal Bansal, Trilok Mathur, Narinderjit Singh Sawaran Singh, Shivi Agarwal. Analysis of illegal drug transmission model using fractional delay differential equations[J]. AIMS Mathematics, 2022, 7(10): 18173-18193. doi: 10.3934/math.20221000

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• The global burden of illegal drug-related death and disability continues to be a public health threat in developed and developing countries. Hence, a fractional-order mathematical modeling approach is presented in this study to examine the consequences of illegal drug usage in the community. Based on epidemiological principles, the transmission mechanism is the social interaction between susceptible and illegal drug users. A pandemic threshold value ($\Lambda$) is provided for the illegal drug-using profession, which determines the stability of the model. The Lyapunov function is employed to determine the stability conditions of illegal drug addiction equilibrium point of society. Finally, the proposed model has been extended to include time lag in the delayed illegal drug transmission model. The characteristic equation of the endemic equilibrium establishes a set of appropriate conditions for ensuring local stability and the development of a Hopf bifurcation of the model. Finally, numerical simulations are performed to support the analytical results.

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