Bleaching and Withdrawal: A Modeling Approach for Assessing the Role of Bleaching and Individual Withdrawal on Controlling HIV among
Intravenous Drug Users 
Department of Mathematics, University of Zimbabwe, P.O. Box MP 167, Harare, Zimbabwe
S. Mushayabasa^{*}


Corresponding Author : 
S. Mushayabasa
University of Zimbabwe
Department of Mathematics
P.O. Box MP 167, Harare, Zimbabwe
Email: steadymushaya@gmail.com 

Received September 18, 2011; Accepted January 25, 2012; Published January 29, 2012 

Citation: Mushayabasa S (2012) Bleaching and Withdrawal: A Modeling Approach for Assessing the Role of Bleaching and Individual Withdrawal on Controlling HIV among Intravenous Drug Users. J AIDS Clinic Res S7:001.
doi: 10.4172/21556113.S7001


Copyright: © 2012 Mushayabasa S. This is an openaccess article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credit ed.


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Keywords 

HIV; Intravenous drug users; Bleaching; Withdrawal;
Reproductive number; Sensitivity analysis 

Introduction 


Injection of illicit drugs is a key pathway for the transmission
of HIV, and is the primary mode of transmission in certain regions,
such as Eastern Europe, Russia and south Asia. For instance, in Russia
more than 90% of new HIV infections are caused by this type of
exposure, and 37% of injecting drug users (IDUs) are believed to be
HIV positive [13]. Worldwide estimates suggest that as many as 3
million IDUs are HIV positive, [1] and that injection of illicit drugs
has contributed substantially to wider morbidity and mortality [2]. The
first recognizable HIV outbreak in China occurred among injecting
drug users (IDUs) in the city of Ruili, Yunnan province in 1989 [4],
following which the epidemic rapidly expanded throughout Yunnan
and neighboring provinces. By 2009, an estimated 740,000 people were
infected with HIV in China, including 105,000 with AIDS [5]. The
strategy of disinfecting syringes to prevent HIV emerged in California
in the 1980s. East Coast epidemics among IDUs (especially in New
York) made public health officials fear that HIV would be a major threat
to California IDUs [6,7]. Although, bleaching have been encourage
for more than twenty years, several studies are demonstrating high
HIV prevalence among IDUS [13]. In 2002, over a decade after the
epidemic commenced, needle and syringes programs (NSPs) were
initiated by various agencies throughout China as a harm reduction
strategy. Despite the large investment in NSPs, it is estimated that less
than 25% of IDUs in Yunnan obtain their injecting equipment through
NSPs [8,9] and rates of sharing of injecting equipment remains as high
as 45% [10]. NSPs have been shown to be a safe and effective means to
reduce HIV transmission in some developed and developing country
settings [1117]. 


If it is done carefully and thoroughly, disinfection can reduce the
amount of live HIV in a syringe [6]. However, the repeated use of
needles and syringes necessarily compromises their sterility and safety.
The aim of this paper is to use a simple mathematical model to assess
the role of bleaching and individual withdrawal on controlling HIV
among intravenous drug users. 

A Mathematical Framework and Approach 

Based on epidemiological status, the population of injection drug
users is subdivided into the following subgroups: Susceptible (S),
newlyinfected individuals in primary or acute seroconversion stage
(P) (less than 28 days of infection), symptomatic HIV carriers (I), and
AIDS stage (A). Thus, the total population is given by, N = S + P +
I + A. The susceptible population is increased by the recruitment of
individuals (assumed susceptible) into the population through peer
pressure, self motivation or any other reasons at rateΛ. Susceptible
individuals are infected with HIV through sharing contaminated
needles or syringes at rate ,where β is the
probability of getting infected whenever a susceptible individual uses a
contaminated needle, syringe or any other tools that might be used to
share intravenous drugs, 0<η<1 accounts for the relative infectiousness
of symptomatic HIV carriers (I) in relation to newlyinfected individuals
in acute seroconversion stage (P). The role of bleaching is modeled by
a factor of, (1 θ), with, 0 ≤ θ ≤ 1, θ = 0 it implies that bleaching is 0%
effective to prevent HIV infection among injecting drug users, while θ
= 1 implies that bleaching is 100% effective to prevent HIV infection
among injecting drug users. A fraction, q of symptomatic HIV carriers
is assumed to withdraw from intravenous drug injection activity. The
fraction q appears in both the numerator and the denominator of the
incidence term. Further, it is assumed that individuals progress from
the acute seroconversion stage (P) to the asymptomatic stage (I) at rate
γ. Progression from the asymptomatic stage (I) to the AIDS stage (A) occurs at rate γ. Natural mortality occurs in all classes at rate μ, and
AIDS individuals suffer an additional diseaseinduced death at rate
δ. Parameters in the model are summarized and explained in Table 1,
and the model is depicted in the transfer diagram in Figure 1. All the
parameters in the model are assumed to be nonnegative. 

From the descriptions and assumptions on the dynamics of the
epidemic made above, the following are the model equations. 

(1) 

Since system (1) monitors human population, it is assumed that all
the state variables are nonnegative for t ≥ 0. Hence; we consider system
(1) to be biologicallyfeasible in the region. 



It can be shown that all solutions of the system starting in D remain
in D for all t ≥ 0. Thus, D is positivelyinvariant and sufficient to consider
the dynamics of the flow generated by (1) in this positively invariant
domain D. It can be shown that unique solutions exist in D for all
positive time. Thus, the model is epidemiologically and mathematically
well posed (see [20] for further discussion). 

Reproductive Number 

The basic reproductive number (or, R_{0}) is defined as the number
of secondary cases generated by a primary case when the virus
is introduced in a population of fully susceptible individuals at a
demographic steady state [21]. That is, R_{0} measures the power of a
disease to invade a population under conditions that facilitate maximal
growth. If, R_{0} >1 then the epidemic progresses. If, R_{0}>1, the epidemic
dies out. The higher the reproductive number, the faster the infecting agent runs out of susceptible individuals (i.e., the faster it decreases).
For system (1) the basic reproductive number is given by the unitless
expression: 

(2) 

The threshold quantity, R_{0} measures the average number of new
secondary HIV cases generated by a single HIV infective intravenous
drug user, when introduced in a susceptible population in the presence
of aforementioned control measures are in place. The effectiveness of
interventions (medical or behavioural) is evaluated by the ability of the
program to reduce, R_{0}. Ideally, one would like to bring the system to the
point where, R_{0} < 1. 

Analysis of the Reproductive Number 

In this section, I will analyse the impact of bleaching and individual
withdrawal, as means of controlling HIV among intravenous drug users
in the absence of any antiretroviral therapy. 

Case 1: (No bleaching and withdrawal). In the absence of bleaching
(θ = 0) and withdrawal (q = 0), the reproductive number reduces to, R_{1} which is given by 

(3) 

Case 2: (Bleaching only). In the absence of withdrawal (q = 0), the reproductive number reduces to, R_{2} which is given by 

(4) 

Case 3: (Withdrawal, only). In the absence of bleaching (θ = 0), the
reproductive number reduces to, R3 which is given by 

(7) 

Comparing equations (35), one can easily observe that ,suggesting that bleaching and individual withdrawal
of symptomatic HIV carriers from drug injection misuse have a positive
impact on controlling the spread of HIV among intravenous drug
users. Further analysis on these reproductive numbers suggests that
implementing both strategies may have a more influence on reducing
the disease among intravenous drug users. 

Uncertainty and Sensitivity Analyses 

The uncertainty analyses were used to assess the variability in the
empirical, R_{0} distribution generated from the variability in parameter
estimates. The sensitivity analyses assess the amount and type of change
inherent in the model as captured by the terms which define, R_{0}. If,
R_{0} is very sensitive to a particular parameter, then a perturbation of
the conditions that connect the dynamics to such a parameter may
prove useful in identifying policies that reduce the transmission of
HIV among injection drug users. Sensitivity and uncertainty analyses
are common in the study of the role of variability in tippingpoint
phenomena [2226]. 

Partial rank correlation coefficients (PRCC) were calculated to
estimate the correlation between values of, R_{0} and the seven model
parameters across one thousand (1000) random draws from the
empirical distribution of, R_{0} and its associated parameters. A large PRCC is indicative of high sensitivity to parameter estimates, while a
small PRCC reflects low sensitivity [2226]. The signs of PRCC values
determine whether a parameter is correlated directly or inversely to the
reproductive number (R_{0}). 

Figure 2 illustrates that reproductive number (R_{0}) is most sensitive
(inversely) to bleaching, and directly to the rate of progression to
AIDS stage. Comparing voluntary withdrawal of symptomatic HIV
carriers and bleaching, I observe that the reproductive number is more
sensitive to bleaching than individual withdrawal; this suggests that
effective bleaching may be more important on controlling HIV among
injection drug users compared to voluntary withdrawal of symptomatic
HIV carriers. By effective bleaching herein, I mean following current
bleach policy, as recommended by the health experts. Although,
voluntary withdrawal of symptomatic HIV carriers is not most sensitive
(inversely) to the reproductive it should not be discouraged, since it has show that it contributes to the reduction of the magnitude of R_{0}.
Results on Figure 2 further suggests that, the use of antiviral drugs can
substantially reduce the rate at which individuals progress to full blown
AIDS, thereby increasing the life expectancy of individuals within the
community. 

Figure 3, below illustrates the effect of bleaching on controlling
HIV among intravenous drug users. The results suggest that an increase
on the level of bleaching results in a decrease on the reproductive
number. Thus, bleaching will be an important intervention strategy
for controlling HIV among intravenous drug users, (in the absence of
antiviral drugs). If the level of bleaching can be 50% effective of all the
time or more, then the disease will be controlled. 

Discussion 

Injecting drug use has been associated with severe health and social harms. High rates of disease, death, crime, and the accompanying costs
are drugrelated harms experienced throughout the world. Injecting
drug use has also been identified as a key risk characteristic for HIV
infection in many countries around the world. Explosive epidemics of
HIV have emerged in various settings, demonstrating that HIV can
spread rapidly once established within communities of people who
inject drugs. The dynamics of injecting drug usedriven HIV epidemics
present unique challenges, giving policy makers and health authorities
little time to respond in an effort to contain outbreaks of HIV infection.
Here, a simple deterministic mathematical model for assessing the role of
bleaching and individual withdrawal of symptomatic HIV carriers from
drug injection misuse activity is presented. A threshold dimensionless
quantity known as the reproductive number (R_{0}) has been derived.
It (R_{0}) has been used to assess the role of bleaching and withdrawal
on controlling HIV among injecting drug users. The reproductive
number (R_{0}) measures the average number of new secondary HIV cases
generated by a single IDU in a population where the aforementioned
control measures are in place. If R_{0} ≤ 1, then HIV will not persist, while
if R_{0}>1 the disease will persist. At best this study suggests that bleaching
is more effective compared to withdrawal on controlling HIV among
intravenous drug users. The study further suggests that antiviral drug
may be use on controlling HIV among intravenous drug users. 

Acknowledgments 

The author is grateful to the anonymous referee and the handling editor for
their valuable comments and suggestions. 

References 

 Mathers BM, Degenhardt L, Phillips B, Wiessing L, Hickman M, et al. (2008) Global epidemiology of injecting drug use and HIV among people who inject drugs: a systematic review. Lancet 372: 17331745.
 Wolfe D, MalinowskaSempruch K (2004) Illicit drug policies and the global HIV epidemic: effects of UN and national government approaches. New York: Open Society Institute.
 Werb D, Mills EJ, Montaner JS, Wood E (2010) Risk of resistance to highly active antiretroviral therapy among HIVpositive injecting drug users: a metaanalysis. Lancet Infect Dis 10: 464469.
 Ma Y, Li ZZ, Zhang K (1990) HIV was first discovered among injection drug users in China. Chinese Journal of Epidemiology 11: 184185.
 Lu F, Wang N, Wu Z, Sun X, Rehnstrom J, et al. (2006) Estimating the number of people at risk for and living with HIV in China in 2005: methods and results. Sex Transm Infect 82: iii8791.
 CDC (2004) Syringe disinfection for injecting drug users.
 Markus B, Kirsten M, Michael VZ, Dieter E (2001) Treatment of hepatitis C infection in IDUs. Hepatology 34: 188193.
 Cheng F, Chen H, Li JH, Zhang LL, Hu H, et al. (2003) SASH survey on high risk behaviors of IDUs in four cities of Yunnan and Sichuan. Chinese Journal of Drug Dependence 12: 294298.
 Sha LI, Yunzhao LI, Yanyan ZHU, et al. (2008) A sampling survey on syringe exchange and methadone maintenance treatment among drug abusers in Yunnan province. Chinese Journal of AIDS/STD 14: 238239.
 Chu TX, Levy JA (2005) Injection drug use and HIV/AIDS transmission in China. Cell Res 15: 865869.
 Kwon JA, Iversen J, Maher L, Law MG, Wilson DP (2009) The impact of needle and syringe programs on HIV and HCV transmissions in injecting drug users in Australia: a modelbased analysis. J Acquir Immune Defic Syndr 51: 462469.
 Vickerman P, Kumaranayake L, Balakireva O, Guinness L, Artyukh O, et al. (2006) The costeffectiveness of expanding harm reduction activities for injecting drug users in Odessa, Ukraine. Sex Transm Dis 33: S89102.
 Jenkins C, Rahman H, Saidel T, Jana S, Hussain AM (2001) Measuring the impact of needle exchange programs among injecting drug users through the National Behavioural Surveillance in Bangladesh. AIDS Educ Prev 13: 452461.
 Wodak A, Cooney A (2006) Do needle syringe programs reduce HIV infection among injecting drug users: a comprehensive review of the international evidence. Subst Use Misuse 41: 777813.
 Bastos FI, Strathdee SA (2000) Evaluating effectiveness of syringe exchange programmes: current issues and future prospects. Soc Sci Med 51: 17711782.
 Wodak A (2006) Lessons from the first international review of the evidence for needle syringe programs: the band still plays on. Subst Use Misuse 41: 837839.
 Zhang L, Yap L, Xun Z, Wu Z, Wilson DP (2011) Needle and syringe programs in Yunnan, China yield health and financial return. BMC Public Health 11: 250.
 Mushayabasa S, Tchuenche JM, Bhunu CP, NgarakanaGwasira E (2011) Modeling gonorrhea and HIV cointeraction. Biosystems 103: 2737.
 Podder CN, Sharomi O, Gumel AB, Strawbridge E (2011) Mathematical analysis of a model for assessing the impact of antiretroviral therapy, voluntary tesing and condom use in curtailing HIV. Differ Equ Dyn Syst 19: 283302.
 Hethcote HW (2000) The mathematics of infectious diseases. SIAM Rev 42; 599653.
 Diekmann O, Heesterbeek JA (2000) Mathematical Epidemiology of Infectious Diseases: Model Building, Analysis and Interpretation. John Wiley and Sons, The Netherlands.
 Blower SM, Dowlatabadi H (1994) Sensitivity and uncertainty analysis of complex models of disease transmission: an HIV model, as an example. Int Stat Rev 62: 229243.
 Mushayabasa S, Bhunu CP, Smith RJ (2012) Assessing the impact of educational campaigns on controlling HCV among women in prison settings. Commun Nonlinear Sci Numer Simulat 17: 17141724.
 Helton JC (1993) Uncertainty and sensitivity analysis techniques for use in performance assessment for radioactive waste disposal. Reliability Eng Syst Saf 42: 327367.
 Saltelli A, Chan K, Scott EM (2000) Sensitivity Analysis. Chichester: John Wiley and Sons.
 Mubayi A, Greenwood P, Wang X, CastilloChávez C, Gorman DM, et al. (2011) Types of drinkers and drinking settings: an application of a mathematical model. Addiction 106: 749758.


References

 Mathers BM, Degenhardt L, Phillips B, Wiessing L, Hickman M, et al. (2008) Global epidemiology of injecting drug use and HIV among people who inject drugs: a systematic review. Lancet 372: 17331745.
 Wolfe D, MalinowskaSempruch K (2004) Illicit drug policies and the global HIV epidemic: effects of UN and national government approaches. New York: Open Society Institute.
 Werb D, Mills EJ, Montaner JS, Wood E (2010) Risk of resistance to highly active antiretroviral therapy among HIVpositive injecting drug users: a metaanalysis. Lancet Infect Dis 10: 464469.
 Ma Y, Li ZZ, Zhang K (1990) HIV was first discovered among injection drug users in China. Chinese Journal of Epidemiology 11: 184185.
 Lu F, Wang N, Wu Z, Sun X, Rehnstrom J, et al. (2006) Estimating the number of people at risk for and living with HIV in China in 2005: methods and results. Sex Transm Infect 82: iii8791.
 CDC (2004) Syringe disinfection for injecting drug users.
 Markus B, Kirsten M, Michael VZ, Dieter E (2001) Treatment of hepatitis C infection in IDUs. Hepatology 34: 188193.
 Cheng F, Chen H, Li JH, Zhang LL, Hu H, et al. (2003) SASH survey on high risk behaviors of IDUs in four cities of Yunnan and Sichuan. Chinese Journal of Drug Dependence 12: 294298.
 Sha LI, Yunzhao LI, Yanyan ZHU, et al. (2008) A sampling survey on syringe exchange and methadone maintenance treatment among drug abusers in Yunnan province. Chinese Journal of AIDS/STD 14: 238239.
 Chu TX, Levy JA (2005) Injection drug use and HIV/AIDS transmission in China. Cell Res 15: 865869.
 Kwon JA, Iversen J, Maher L, Law MG, Wilson DP (2009) The impact of needle and syringe programs on HIV and HCV transmissions in injecting drug users in Australia: a modelbased analysis. J Acquir Immune Defic Syndr 51: 462469.
 Vickerman P, Kumaranayake L, Balakireva O, Guinness L, Artyukh O, et al. (2006) The costeffectiveness of expanding harm reduction activities for injecting drug users in Odessa, Ukraine. Sex Transm Dis 33: S89102.
 Jenkins C, Rahman H, Saidel T, Jana S, Hussain AM (2001) Measuring the impact of needle exchange programs among injecting drug users through the National Behavioural Surveillance in Bangladesh. AIDS Educ Prev 13: 452461.
 Wodak A, Cooney A (2006) Do needle syringe programs reduce HIV infection among injecting drug users: a comprehensive review of the international evidence. Subst Use Misuse 41: 777813.
 Bastos FI, Strathdee SA (2000) Evaluating effectiveness of syringe exchange programmes: current issues and future prospects. Soc Sci Med 51: 17711782.
 Wodak A (2006) Lessons from the first international review of the evidence for needle syringe programs: the band still plays on. Subst Use Misuse 41: 837839.
 Zhang L, Yap L, Xun Z, Wu Z, Wilson DP (2011) Needle and syringe programs in Yunnan, China yield health and financial return. BMC Public Health 11: 250.
 Mushayabasa S, Tchuenche JM, Bhunu CP, NgarakanaGwasira E (2011) Modeling gonorrhea and HIV cointeraction. Biosystems 103: 2737.
 Podder CN, Sharomi O, Gumel AB, Strawbridge E (2011) Mathematical analysis of a model for assessing the impact of antiretroviral therapy, voluntary tesing and condom use in curtailing HIV. Differ Equ Dyn Syst 19: 283302.
 Hethcote HW (2000) The mathematics of infectious diseases. SIAM Rev 42; 599653.
 Diekmann O, Heesterbeek JA (2000) Mathematical Epidemiology of Infectious Diseases: Model Building, Analysis and Interpretation. John Wiley and Sons, The Netherlands.
 Blower SM, Dowlatabadi H (1994) Sensitivity and uncertainty analysis of complex models of disease transmission: an HIV model, as an example. Int Stat Rev 62: 229243.
 Mushayabasa S, Bhunu CP, Smith RJ (2012) Assessing the impact of educational campaigns on controlling HCV among women in prison settings. Commun Nonlinear Sci Numer Simulat 17: 17141724.
 Helton JC (1993) Uncertainty and sensitivity analysis techniques for use in performance assessment for radioactive waste disposal. Reliability Eng Syst Saf 42: 327367.
 Saltelli A, Chan K, Scott EM (2000) Sensitivity Analysis. Chichester: John Wiley and Sons.
 Mubayi A, Greenwood P, Wang X, CastilloChávez C, Gorman DM, et al. (2011) Types of drinkers and drinking settings: an application of a mathematical model. Addiction 106: 749758.

Tables at a glance

Table 1 
Figures at a glance
 
 

Figure 1  
Figure 2  
Figure 3 


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