Global asymptotic dynamics of a nonlinear illicit drug use system
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Global asymptotic dynamics of a nonlinear illicit drug use system John O. Akanni1 · Samson Olaniyi1
· Folake O. Akinpelu1,2
Received: 14 July 2020 / Revised: 11 August 2020 / Accepted: 14 August 2020 © Korean Society for Informatics and Computational Applied Mathematics 2020
Abstract In this paper, a nonlinear mathematical model of illicit drug use in a population is studied using dynamical system theory. The work is largely concerned with the analysis of asymptotic behaviour of solutions to a six-dimensional system of differential equations modeling the influence of illicit drug use in the population. The model is mathematically well-posed based on positivity and boundedness of solutions. A key threshold which measures the potential spread of the illicit drug use in the population is derived analytically. The model is shown to exhibit forward bifurcation property, implying the existence, uniqueness and local stability of an illicit drug-present equilibrium. Furthermore, the global asymptotic dynamics of the model around the illicit drug-free and drug-present equilibria are extensively investigated using appropriate Lyapunov functions. Numerical simulations are carried out to complement the obtained theoretical results, and to examine the effects of some parameters, such as influence rate, rehabilitation rates of drug users and relapse rate, on the dynamical spread of illicit drug use in the population. Measures to guide against the menace of the illicit drug use are suggested. Keywords Dynamical system · Nonlinear model · Lyapunov functionals · Bifurcation analysis · Simulations Mathematics Subject Classification 34A34 · 34D05 · 34D23 · 37N99
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Samson Olaniyi [email protected]
1
Department of Pure and Applied Mathematics, Ladoke Akintola University of Technology, Ogbomoso, Nigeria
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Department of Mathematics, University of Lagos, Lagos, Nigeria
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1 Introduction Illicit drug use has been one of the social and public health problems facing most countries of the world today. The catastrophic effects of illicit drug use on development of nations cannot be ignored. According to the World Health Organization (WHO) [1], it was reported that about 275 million people have used illicit drug at least once in the year 2016. As appeared in WHO [2], almost 31 million of people suffer from drug use disorders and leading to an estimated 3.3 million deaths each year. The problem of addiction to illicit drug is not limited to developing countries only, rather it spreads all over the world. Illicit drug use has now been reported in more than 140 countries, with an estimated number of 13.2 million injection drug users (IDUs) worldwide [3]. A drug can be defined as a substance either synthetic or natural which when taken could bring about biological, physiological, chemical and behavioural changes in the body. Thus, drugs are substances taken either by injecting, smoking, inhaling, snorting or swallowing which modifies the perceptions, cognition, mood, behaviour and general body functions [4].
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