Mathematical modelling and analysis of alcohol-methamphetamine co-abuse in the Western Cape Province of South Africa

Figures & data

Figure 1. A compartmental representation of the epidemic of alcohol-methamphetamine co-use.

Table 1. Descriptions of co-abuse model parameters

Parameter	Description
$\mathrm{\Lambda }$	Recruitment rate for susceptible persons
$\mu$	Natural mortality rate
${\beta }_1$	Alcohol transmission rate
${\beta }_2$	Methamphetamine transmission rate
${\gamma }_1$	Relapse rate for alcohol abusers
${\gamma }_2$	Relapse rate for co-abuser
${\gamma }_3$	Relapse rate for methamphetamine abusers
${\gamma }_4$	Co-abusers’ relapse rate into alcohol only abuse
${\gamma }_5$	Co-abusers’ relapse rate into methamphetamine only abuse
${\delta }_1$	Alcohol-induced mortality rate
${\delta }_2$	Co-abuse-induced mortality rate
${\delta }_3$	Methamphetamine-induced mortality rate
${\sigma }_1$	Recovery rate for alcohol abusers
${\sigma }_2$	Recovery rate for co-abusers of alcohol and methamphetamine
${\sigma }_3$	Recovery rate for methamphetamine abusers
${\rho }_1$	Rate at which co-abusers revert to methamphetamine abuse only without treatment
${\rho }_2$	Rate at which co-abusers revert to alcoholism only without treatment
${\rho }_3$	Quitting rate for alcohol abusers without treatment
${\rho }_4$	Quitting rate for alcohol abusers upon success treatment
${\rho }_5$	Quitting rate for co-abusers upon successful treatment
${\rho }_6$	Quitting rate for methamphetamine abusers upon successful treatment
${\rho }_7$	Quitting rate for methamphetamine abusers without treatment
${\eta }_a$, ${\eta }_t$, ${ϵ}_1$, ${ϵ}_2$	Modification parameters
${\zeta }_1$, ${\zeta }_2$, ${\zeta }_3$, ${ϵ}_3$	Modification parameters

Table 2. Description of parameters of the co-abuse model, their values and their ranges

Symbol	Value	Range	Source	Symbol	Value	Range	Source
$\mathrm{\Lambda }$	0.03	0.02–0.04	(Day & Gray, 2008)	${\sigma }_1$	0.2		Estimated
$\mu$	0.02	0.02–0.03	(Farai, Musekwa, & Senelani, 2010)	${\sigma }_2$	0.3		Estimated
${\beta }_1$	0.25	0.1–0.4	(Bhunu & Mushayabasa, 2012b)	${\sigma }_3$	0.3	0.09–0.3	(Farai et al., 2010)
${\beta }_2$	0.2	0.1–0.4	(Farai et al., 2010)	${\rho }_1$	0.35	0–1	(Bhunu & Mushayabasa, 2012b)
${\gamma }_1$	0.01	0–0.1	(Bhunu & Mushayabasa, 2012b)	${\rho }_2$	0.5	0–1	(Sharomi & Gumel, 2008)
${\gamma }_2$	0.1	0.1–0.5	Estimated	${\rho }_3$	0.35	0.2–0.5	(Bhunu & Mushayabasa, 2012b)
${\gamma }_3$	0.0011	0–0.9	(Kalula & Nyabadza, 1998)	${\rho }_3$	0.25	0–1	(Bhunu & Mushayabasa, 2012b)
${\gamma }_4$	0.3	0.1–0.5	Estimated	${\rho }_5$	0.25		Estimated
${\gamma }_5$	0.3	0.1–0.5	Estimated	${\rho }_6$	0.8312	0–91	(Kalula & Nyabadza, 1998)
${\delta }_1$	0.03	0.01–0.05	(Farai et al., 2010)	${\rho }_7$	0.0095	0–0.3	(Kalula & Nyabadza, 1998)
${\delta }_2$	0.04	0.01–0.05	Estimated	${\eta }_a$,${\eta }_t$,${\zeta }_2$${ϵ}_2$	$1.05$	1–3	Estimated
${\delta }_3$	0.033	0.01–0.05	(Bhunu & Mushayabasa, 2012b)	${\zeta }_1$, ${\zeta }_3$, ${ϵ}_1$, ${ϵ}_3$	$0.7$	0–1	Estimated

Figure 2. Graphs showing the population dynamics of $S$, $U_a$, $R_t$, $U_t$, $R_t$, $U_{at}$ and $R_{at}$ for ${\mathcal{R}}_{at}0=0.6303<1$. Used parameter values are available in Table 2.

Figure 3. Graphs showing the population dynamics of $S$, $U_a$, $R_t$, $U_t$, $R_t$, $U_{at}$ and $R_{at}$ for ${\mathcal{R}}_{at}0=1.1254>1$. Used parameter values are available in Table 2.

Figure 4. PRCCs for the full range of parameters from Table 2.

Figure 5. Graphs (a), (b), (c) and (d) show the Monte Carlo simulations for the four parameters with the greatest PRCC magnitude in the alcohol model. Parameter values in Table 2 and 1,000 simulations per run were used.

Figure 6. Graphs (a), (b), (c) and (d) show the Monte Carlo simulations for the four parameters with the greatest PRCC magnitude in the methamphetamine (MA) model. Used parameter values can be found in Table (2). One thousand simulations per run were used.

Table 3. Primary Methamphetamine-abuse for the period 1997a to 2016b in % (Source: (Siphokazi et al., 2017)). The letters a and b represent the first 6 months (January–June) and the second 6 months (July–December) of the year, respectively

Year	1997a	1997b	1998a	1998b	1999a	1999b	2000a	2000b	2001a
% Meth users use	0	0.1	0	0.1	0.1	0.1	0.2	0.1	0.1
Year	2001b	2002a	2002b	2003a	2003b	2004a	2004b	2005a	2005b
% Meth users	0.3	0.3	0.8	2.3	2.3	10.7	19.3	26.1	34.7
Year	2006a	2006b	2007a	2007b	2008a	2008b	2009a	2009b	2010a
% Meth users	37.2	42.3	40.7	36.1	35.8	35.1	40.6	35.5	33.6
Year	2010b	2011a	2011b	2012a	2012b	2013a	2013b	2014a	2014b
% Meth users	35.1	35.3	38.8	34	33	28	33	33	35
Year	2015a	2015b	2016a	2016b
% Meth users	35	37	32	29

Table 4. Primary Alcohol abuse: by 6-month period (from 1997a to 2016b in % (Source: (Siphokazi et al., 2017))

Year	1997a	1997b	1998a	1998b	1999a	1999b	2000a	2000b	2001a
% Alcohol users	82.0	78.0	74.0	64.0	56.0	50.0	48.0	51.0	46.0
Year	2001b	2002a	2002b	2003a	2003b	2004a	2004b	2005a	2005b
% Alcohol users	46.0	48.0	47.0	43.6	39.4	38.3	33.7	34.4	25.1
Year	2006a	2006b	2007a	2007b	2008a	2008b	2009a	2009b	2010a
% Alcohol users	30.2	26.4	29.5	29.7	30.0	27.6	26.8	29.4	29.8
Year	2010b	2011a	2011b	2012a	2012b	2013a	2013b	2014a	2014b
% Alcohol users	27.5	27.5	23.7	24	22	20	22	20	22
Year	2015a	2015b	2016a	2016b
% Alcohol users	21	20	22	21

Figure 7. Shows methamphetamine model (37) fitted to data for addicts seeking treatment for methamphetamine as a primary substance of abuse in the Western Cape Province of South Africa. Parameter values are in Table (2).

Figure 8. Shows alcohol-model (15) fitted to data for individuals seeking treatment for alcohol abuse in Cape town and Western Cape province of South Africa. Parameter values are in Table (2).

Figure 9. Graphs showing the projected population of methamphetamine (MA), (a) alcohol (b) abuse patients seeking treatment services in Western Cape province, South Africa.

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