Many differential equations arising in engineering and the physical sciences, particularly in the study of vibratory or oscillatory phenomena, have the form 



(1)

in which αn, αn-1,…, αl, αo and f are given functions of x. Such an equation is known as a linear ordinary differential equation of order n. In this chapter it will sometimes be convenient to write a linear equation in the abbreviated form 



(2)

where L denotes a linear differential operator: 



The linearity property of L is expressed by the easily confirmed result 



where c1 and c2 are any constants and y and z any functions of x possessing derivatives up to the nth order.

Ordinary Differential Equations of Order > 1

The aim of the present study is to consider a heroin epidemic model with age-structure only in the active heroin users. The model was formulated with the help of available literature on heroin epidemic. Instead of treatment as a class, we incorporated recovered population and considered treatment as a control variable and thus a control problem is presented for further analysis. The techniques of weak derivatives and sensitivities were used for obtaining the adjoint equations. The maximum principle of Pontryagin’ type was used for obtaining the optimal value of the control variable. Sample simulations are presented at the end of the study in order to show the effectiveness of the treatment.

/pdf/controlling-heroin-addiction-via-age-structured-modeling-2bs5yflbi3.pdf

Controlling heroin addiction via age-structured modeling

Analysis of an age-structured multi-group heroin epidemic model

In this paper, a synthetic drugs transmission model with treatment is formulated based on the principles of mathematical epidemiology. The model considers that relapse can occur among those individuals who have a history of drug abuse and we distinguish the addiction rates of susceptible individuals who have a history of drug abuse and those who have not. The global dynamics of this model are determined by the basic reproduction number, $$R_{0}$$
 , under certain conditions. If $$R_{0}<1$$
 , the drug-free equilibrium is globally exponentially stable for a special case and the exponential convergence rate can be unveiled, and if $$R_{0}>1$$
 , the drug-addiction equilibrium is globally asymptotically stable under certain conditions. Sensitivity analysis is performed to seek for effective control measures for drug abuse. Numerical simulations are also carried out to confirm the analytical results.

Modelling and stability of a synthetic drugs transmission model with relapse and treatment

Synthetic drugs are replacing traditional ones and becoming the main popular ones gradually, which have given rise to serious social issues in recent years. In this paper, a synthetic drugs transmission model with psychological addicts and general contact rate is proposed. The local and global stabilities are decided by the basic reproduction number R 0 . By analyzing the sensitivity of parameters, we obtain that controlling psychological addiction is better than drugs treatment. These results are verified by numerical simulations.

Dynamics of synthetic drugs transmission model with psychological addicts and general incidence rate

Dependence on methamphetamine remains one of the major health and social problem in the Western Cape province of South Africa. We consider a mathematical model that takes into account two forms of rehabilitation, namely; inpatient and outpatient. We examine the trends of these two types of rehabilitation. We also seek to investigate the global dynamics of the developed methamphetamine epidemic model. The model is designed by likening the initiation process to an infection that spreads in a community through interactions between methamphetamine users and non-users. We make use of Lyapunov functions obtained from a suitable combination of common quadratic and Volterra-type functions to establish the global stability of the methamphetamine-persistent steady state. The least squares curve fit routine (lsqcurvefit) in Matlab with optimization is used to estimate the parameter values. The model analysis shows that the model has two equilibria, the methamphetamine free equilibrium and the methamphetamine persistent equilibrium, that are both globally stable when the threshold $$\mathcal {R}_a<1$$
 and $$\mathcal {R}_a>1$$
 , respectively. Upon fitting the model to data on drug users under rehabilitation, parameter values that give the best fit were obtained. The projections carried out the long term trends of these forms of rehabilitation. The results suggest that inpatient rehabilitation programs have an increased potential of enhancing the chances of recovery for methamphetamine addicts.

/pdf/modelling-the-trends-of-inpatient-and-outpatient-wlshmptf4m.pdf

Modelling the trends of inpatient and outpatient rehabilitation for methamphetamine in the Western Cape province of South Africa

Adolescence methamphetamine use is an issue of considerable concern due to its correlation with later delinquency, divorce, unemployment and health problems. Understanding how adolescents initiate methamphetamine abuse is important in developing effective prevention programs. We formulate a mathematical model for the spread of methamphetamine abuse using nonlinear ordinary differential equations. It is assumed that susceptibles are recruited into methamphetamine use through imitation. An epidemic threshold value, $${\mathcal {R}}_a$$
 , termed the abuse reproduction number, is proposed and defined herein in the drug-using context. The model is shown to exhibit the phenomenon of backward bifurcation. This means that methamphetamine problems may persist in the population even if $${\mathcal {R}}_a$$
 is less than one. Sensitivity analysis of $${\mathcal {R}}_a$$
 was performed to determine the relative importance of different parameters in methamphetamine abuse initiation. The model is then fitted to data on methamphetamine users less than 20 years old reporting methamphetamine as their primary substance of abuse in the treatment centres of Cape Town and parameter values that give the best fit are chosen. Results show that the proportion of methamphetamine users less than 20 years old reporting methamphetamine as their primary substance of abuse will continue to decrease in Cape Town of South Africa. The results suggest that intervention programs targeted at reducing adolescence methamphetamine abuse, are positively impacting methamphetamine abuse.

On the Role of Imitation on Adolescence Methamphetamine Abuse Dynamics.

Modelling Drug Abuse Epidemics in the Presence of Limited Rehabilitation Capacity

Drug dependence is a ‘chronic disease’ treatable through rehabilitation. Many drug addicts progress through a series of rehabilitation and relapsing episodes. In this paper, we formulate a mathematical model with n-alternate stages of rehabilitation and relapsing. The dynamics of drug abuse are treated as an infectious disease that spreads through a population. The model analysis shows that the model has two equilibria, the drug free equilibrium and the drug persistent equilibrium, that are both globally stable when the threshold $${\mathcal {R}}_{0}<1$$
 and $${\mathcal {R}}_{0}>1$$
 respectively. The model is fitted to data on individuals under repeated rehabilitation and parameter values that give the best fit chosen. The projections carried out the long term trends of proportions for repeated rehabilitants. The relative impact for each subgroup is determined to find out which population subgroup is responsible for a disproportionate number of initiations. The results have huge implications to designing policies aligned to rehabilitation processes.

A. G. R. Stewart

Papers

Modelling the trends of inpatient and outpatient rehabilitation for methamphetamine in the Western Cape province of South Africa

On the Role of Imitation on Adolescence Methamphetamine Abuse Dynamics.

Modelling Drug Abuse Epidemics in the Presence of Limited Rehabilitation Capacity

Modelling multiple relapses in drug epidemics