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A Risk-Structured Model for Understanding the Spread of Drug Abuse

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Abstract

Drug abuse is an issue of considerable concern due to its correlation with negative effects such as delinquency, unemployment, divorce and health problems. Understanding the dynamics of drug abuse is important in developing effective prevention programs. We formulate a mathematical model for the spread of drug abuse using nonlinear ordinary differential equations. Susceptibility to drug use varies, due to differences in behavioral, social and environmental factors. Risk structure is included before initiation and after recovery to help differentiate those more likely (high risk) to abuse drugs and those less likely (low risk) to abuse drugs. The model allows back and forth transition between risk groups. It is shown that the model has multiple equilibria and using the center manifold theory, the model exhibits the phenomenon of backward bifurcation whose implications to rehabilitation are discussed. An epidemic threshold value, \({\mathcal {R}}_a\), termed the abuse reproduction number, is proposed and defined herein in the drug-using context. Sensitivity analysis of the abuse reproduction number and numerical simulations were performed. The results show that education about effective coping response and/or skills to deal with the risky situation may better equip individuals to stand against initiating or re-initiating into drug abuse.

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Acknowledgements

J. Mushanyu acknowledge, with thanks, the support of the Department of Mathematics, University of Zimbabwe. F. Nyabadza acknowledges with gratitude the support from National Research Foundation and Stellenbosch University for the production of this manuscript.

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Authors and Affiliations

  1. Department of Mathematics, University of Zimbabwe, Box MP 167, Mount Pleasant, Harare, Zimbabwe

    J. Mushanyu

  2. Department of Mathematical Sciences, Stellenbosch University, P. Bag X1, Matieland, 7602, South Africa

    F. Nyabadza

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  1. J. Mushanyu
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  2. F. Nyabadza
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Correspondence to J. Mushanyu.

Appendices

Appendix 1: Coefficients of Polynomial (23)

$$\begin{aligned} \xi _0= & {} \mu ^2g_1g_2(\mu +\varepsilon _1+\varepsilon _2)(\mu +\omega _1+\omega _2)(1-{\mathcal {R}}_a),\\ \xi _1= & {} -\,\beta _1\varLambda g_2(\beta _1 \eta _1 \mu (\mu +\epsilon _1+\epsilon _2)-\beta _2(\mu +\eta _2 (\mu +\epsilon _1)+\epsilon _2) (\eta _1 (\mu (p-1)-\omega _1)\\&+\,\mu (-p)-\omega _2))\\&+\,\mu g_2(\beta _2 (\mu +\omega _1+\omega _2)(\mu +\eta _2 (\mu +\epsilon _1)+\epsilon _2)\\&+\,\beta _1(\mu +\epsilon _1+\epsilon _2)(\eta _1(\mu +\omega _1)+\mu +\omega _2))g_1-\beta _2 \mu (\mu +\omega _1+\omega _2)\\&\times (\gamma _h (\eta _2 (\mu \rho _l+\epsilon _1 (\rho _l+\sigma ))+\epsilon _2(\rho _l+\sigma )+\mu \sigma )+\rho _h (\mu +\eta _2 \epsilon _1+\epsilon _2) g_2\\&+\,(\eta _2 (\mu +\epsilon _1)+\epsilon _2) (\gamma _l (\rho _l+\sigma )+\mu \rho _l)),\\ \xi _2= & {} -\,\beta _1 \beta _2 \varLambda g_2 (\beta _1 \eta _1 (\mu +\eta _2(\mu +\epsilon _1)+\epsilon _2)+\beta _2 \eta _2(\eta _1 (\mu +\mu (-p)+\omega _1)+\mu p+\omega _2))\\&+\,g_2(\beta _2^2 \eta _2 \mu (\mu +\omega _1+\omega _2)+\beta _2 \beta _1(\eta _1(\mu +\omega _1)+\mu +\omega _2)\\&\quad (\mu +\eta _2 (\mu +\epsilon _1)+\epsilon _2)+\beta _1^2 \eta _1 \mu \\&\times (\mu +\epsilon _1+\epsilon _2))g_1-\beta _2(\beta _2 \eta _2 \mu (\mu +\omega _1+\omega _2)(\sigma (\gamma _h+\gamma _l)+\rho _lg_2)\\&+\,\beta _1(\eta _1 (\mu +\omega _1)+\mu +\omega _2)\\&\times (\gamma _h (\eta _2(\mu \rho _l+\epsilon _1(\rho _l+\sigma ))+\epsilon _2(\rho _l+\sigma )+\mu \sigma )\\&+\,(\eta _2 (\mu +\epsilon _1)+\epsilon _2)(\gamma _l(\rho _l+\sigma )+\mu \rho _l)) \\&+\,\rho _hg_2(\beta _2 \eta _2 \mu (\mu +\omega _1+\omega _2)+\beta _1(\eta _1(\mu +\omega _1)+\mu +\omega _2)(\mu +\eta _2 \epsilon _1+\epsilon _2))),\\ \xi _3= & {} \beta _1 \beta _2 (\beta _2 \eta _2 (\eta _1(\mu +\omega _1)+\mu +\omega _2)(\mu (\delta +\mu +\sigma )+(\delta +\mu )(\gamma _h+\gamma _l))\\&+\,\beta _1 \eta _1(\mu ^2 (\delta +\mu +\sigma )\\&+\,\delta \mu \epsilon _2+\delta \mu \gamma _h+\delta \epsilon _2 \gamma _h+\mu ^2 \gamma _h+\mu \epsilon _2 \gamma _h+\eta _2(\mu (\delta +\mu +\sigma )(\gamma _h+\mu )-\beta _2 \varLambda g_2\\&+\,\epsilon _1(\mu (\delta +\mu +\sigma )+(\delta +\mu )(\gamma _h+\gamma _l))+\mu \rho _h g_2+\mu (\delta +\mu ) \gamma _l)+\mu \rho _lg_2+\delta \mu \gamma _l\\&+\,\delta \epsilon _2 \gamma _l+\mu ^2 \gamma _l+\mu \sigma \gamma _l+\mu \epsilon _2 \gamma _l+\mu ^2 \epsilon _2+\mu \sigma \epsilon _2)),\\ \xi _4= & {} \beta _1^2 \beta _2^2 \eta _1 \eta _2 \left( \mu (\delta +\mu +\sigma )+(\delta +\mu ) \left( \gamma _h+\gamma _l\right) \right) . \end{aligned}$$

Appendix 2. Number of positive roots of polynomial (23)

See Tables 2 and 3.

Table 2 Number of positive roots
Full size table
Table 3 Number of positive roots
Full size table

Appendix 3. Associated non-zero partial derivatives of F at the drug-free equilibrium

$$\begin{aligned} \frac{\partial ^2 f_1}{\partial x_1\partial x_3}= & {} \frac{\partial ^2 f_1}{\partial x_3\partial x_1}=\frac{-\beta ^* _1 \mu \left( \mu (1-p)+\omega _1\right) }{\varLambda \left( \mu +\omega _1+\omega _2\right) },\quad \frac{\partial ^2 f_1}{\partial x_2\partial x_3}=\frac{\partial ^2 f_1}{\partial x_3\partial x_2}=\frac{\beta ^* _1 \mu \left( \mu p+\omega _2\right) }{\varLambda \left( \mu +\omega _1+\omega _2\right) },\\ \frac{\partial ^2 f_1}{\partial x^2_3}= & {} \frac{2 \beta ^* _1 \mu \left( \mu p+\omega _2\right) }{\varLambda \left( \mu +\omega _1+\omega _2\right) },\quad \frac{\partial ^2 f_1}{\partial x_3\partial x_4}=\frac{\partial ^2 f_1}{\partial x_4\partial x_3}=\frac{\beta ^* _1 \mu \left( \mu p+\omega _2\right) }{\varLambda \left( \mu +\omega _1+\omega _2\right) },\\ \frac{\partial ^2 f_1}{\partial x_3\partial x_5}= & {} \frac{\partial ^2 f_1}{\partial x_5\partial x_3}=\frac{\beta ^* _1 \mu \left( \mu p+\omega _2\right) }{\varLambda \left( \mu +\omega _1+\omega _2\right) },\quad \frac{\partial ^2 f_1}{\partial x_3\partial x_6}=\frac{\partial ^2 f_1}{\partial x_6\partial x_3}=\frac{\beta ^* _1 \mu \left( \mu p+\omega _2\right) }{\varLambda \left( \mu +\omega _1+\omega _2\right) },\\ \frac{\partial ^2 f_2}{\partial x_1\partial x_3}= & {} \frac{\partial ^2 f_2}{\partial x_3\partial x_1}=\frac{\beta ^* _1 \eta _1 \mu \left( \mu +\mu (-p)+\omega _1\right) }{\varLambda \left( \mu +\omega _1+\omega _2\right) },\quad \frac{\partial ^2 f_2}{\partial x_2\partial x_3}=\frac{\partial ^2 f_2}{\partial x_3\partial x_2}\\= & {} -\,\frac{\beta ^* _1 \eta _1 \mu \left( \mu p+\omega _2\right) }{\varLambda \left( \mu +\omega _1+\omega _2\right) },\\ \frac{\partial ^2 f_2}{\partial x^2_3}= & {} \frac{2 \beta ^* _1 \eta _1 \mu \left( \mu +\mu (-p)+\omega _1\right) }{\varLambda \left( \mu +\omega _1+\omega _2\right) },\quad \frac{\partial ^2 f_2}{\partial x_3\partial x_4}=\frac{\partial ^2 f_2}{\partial x_4\partial x_3}\\= & {} \frac{\beta ^* _1 \eta _1 \mu \left( \mu +\mu (-p)+\omega _1\right) }{\varLambda \left( \mu +\omega _1+\omega _2\right) },\\ \frac{\partial ^2 f_2}{\partial x_3\partial x_5}= & {} \frac{\partial ^2 f_2}{\partial x_5\partial x_3}=\frac{\beta ^* _1 \eta _1 \mu \left( \mu (1-p)+\omega _1\right) }{\varLambda \left( \mu +\omega _1+\omega _2\right) },\quad \frac{\partial ^2 f_2}{\partial x_6\partial x_3}=\frac{\partial ^2 f_2}{\partial x_3\partial x_6}\\= & {} \frac{\beta ^* _1 \eta _1 \mu \left( \mu +\mu (-p)+\omega _1\right) }{\varLambda \left( \mu +\omega _1+\omega _2\right) }, \end{aligned}$$
$$\begin{aligned} \frac{\partial ^2 f_3}{\partial x_1\partial x_3}= & {} \frac{\partial ^2 f_3}{\partial x_3\partial x_1}=\frac{\beta ^* _1 (1-\eta _1) \mu \left( \mu (1-p)+\omega _1\right) }{\varLambda \left( \mu +\omega _1+\omega _2\right) },\quad \frac{\partial ^2 f_3}{\partial x_2\partial x_3} =\frac{\partial ^2 f_3}{\partial x_3\partial x_2}\\= & {} \frac{-\beta ^* _1 \left( 1-\eta _1\right) \mu \left( \mu p+\omega _2\right) }{\varLambda \left( \mu +\omega _1+\omega _2\right) },\\ \frac{\partial ^2 f_3}{\partial x^2_3}= & {} \frac{-2 \beta ^* _1 \mu \left( \eta _1 \left( \mu (1-p)+\omega _1\right) +\mu p+\omega _2\right) }{\varLambda \left( \mu +\omega _1+\omega _2\right) },\quad \frac{\partial ^2 f_3}{\partial x_3\partial x_4}=\frac{\partial ^2 f_3}{\partial x_4\partial x_3}\\= & {} \frac{-\beta ^* _1 \mu \left( \eta _1 \left( \mu (1-p)+\omega _1\right) +\mu p+\omega _2\right) }{\varLambda \left( \mu +\omega _1+\omega _2\right) },\\ \frac{\partial ^2 f_3}{\partial x_3\partial x_5}= & {} \frac{\partial ^2 f_3}{\partial x_5\partial x_3}=\frac{\mu \left( \beta _2 \left( \mu +\omega _1+\omega _2\right) +\beta ^* _1 \left( \eta _1 \left( \mu (p-1)-\omega _1\right) +\mu (-p)-\omega _2\right) \right) }{\varLambda \left( \mu +\omega _1+\omega _2\right) },\\ \frac{\partial ^2 f_3}{\partial x_3\partial x_6}= & {} \frac{\partial ^2 f_3}{\partial x_6\partial x_3}=\frac{\mu \left( \beta _2 \eta _2 \left( \mu +\omega _1+\omega _2\right) +\beta ^* _1 \left( \eta _1 \left( \mu (p-1)-\omega _1\right) +\mu (-p)-\omega _2\right) \right) }{\varLambda \left( \mu +\omega _1+\omega _2\right) },\\ \frac{\partial ^2 f_5}{\partial x_3\partial x_5}= & {} \frac{\partial ^2 f_5}{\partial x_5\partial x_3}=-\,\frac{\beta _2 \mu }{\varLambda },\quad \frac{\partial ^2 f_6}{\partial x_3\partial x_6}=\frac{\partial ^2 f_6}{\partial x_6\partial x_3}=-\,\frac{\beta _2 \eta _2 \mu }{\varLambda },\\ \frac{\partial ^2 f_1}{\partial x_3\partial \beta ^*_1}= & {} -\frac{\mu p+\omega _2}{\mu +\omega _1+\omega _2},\quad \frac{\partial ^2 f_2}{\partial x_3\partial \beta ^*_1}=\frac{\eta _1 \left( \mu (p-1)-\omega _1\right) }{\mu +\omega _1+\omega _2},\\ \frac{\partial ^2 f_3}{\partial x_3\partial \beta ^*_1}= & {} \frac{\eta _1 \left( \mu +\mu (-p)+\omega _1\right) +\mu p+\omega _2}{\mu +\omega _1+\omega _2}. \end{aligned}$$

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Mushanyu, J., Nyabadza, F. A Risk-Structured Model for Understanding the Spread of Drug Abuse. Int. J. Appl. Comput. Math 4, 60 (2018). https://doi.org/10.1007/s40819-018-0495-9

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