# Modelling and qualitative analysis of an illicit drugs model with saturated incidence rate and relapse

Volume 28, Issue 4, pp 373--392
Publication Date: July 26, 2022 Submission Date: September 26, 2021 Revision Date: February 11, 2022 Accteptance Date: May 20, 2022
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### Authors

K. P. Tan - Faculty of Ocean Engineering Technology and Informatics, Universiti Malaysia Terengganu, 21030 Kuala Nerus, Terengganu, Malaysia. A. A. Mat Daud - Faculty of Ocean Engineering Technology and Informatics, Universiti Malaysia Terengganu, 21030 Kuala Nerus, Terengganu, Malaysia.

### Abstract

Drug abuse is now regarded as a global issue that brings severe consequences on the health, social well-being, and economy. In this study, an illicit drugs model with saturated incidence rate and relapse of individuals who quit using drugs is proposed and analyzed qualitatively. This study aims to determine the behavior of a drug epidemic when the psychological or inhibitory effect and relapse are being considered as well as assist the policymakers in devising effective control measures. The basic reproduction number $R_0$ is derived and used as a threshold parameter in the global stability analysis. It is found that the drug-free equilibrium is globally asymptotically stable when $R_0\le 1$. This implies that we can eradicate a drug epidemic when the threshold $R_0$ is less than or equal to one, irrespective of the initial population size of drug users. On the other hand, the drug persistent equilibrium is globally asymptotically stable when $R_0>1$. This indicates that the phenomenon of drug use will remain in a community when the threshold $R_0$ is greater than one, irrespective of the initial population size of drug users. Next, the sensitivity analysis is performed and the results show that the effective contact rate should be targeted to reduce its value. The numerical simulations are also carried out to illustrate the analytical results and investigate the relationship between the measure of psychological or inhibitory effect and the number of drug users.

### Share and Cite

##### ISRP Style

K. P. Tan, A. A. Mat Daud, Modelling and qualitative analysis of an illicit drugs model with saturated incidence rate and relapse, Journal of Mathematics and Computer Science, 28 (2023), no. 4, 373--392

##### AMA Style

Tan K. P., Mat Daud A. A., Modelling and qualitative analysis of an illicit drugs model with saturated incidence rate and relapse. J Math Comput SCI-JM. (2023); 28(4):373--392

##### Chicago/Turabian Style

Tan, K. P., Mat Daud, A. A.. "Modelling and qualitative analysis of an illicit drugs model with saturated incidence rate and relapse." Journal of Mathematics and Computer Science, 28, no. 4 (2023): 373--392

### Keywords

• Illicit drugs model
• saturated incidence rate
• relapse
• global stability analysis
• sensitivity analysis

•  00A71
•  34C60
•  34A34