# Impact of Intravenous Drug Use on HIV/AIDS among Women Prisoners: A Mathematical Modelling Approach

Authors: C. P. Bhunu; S. Mushayabasa
Journal: ISRN Computational Biology
 (2013)

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## Abstract

Intravenous drug use and tattooing remain one of the major routes of HIV/AIDS transmission among
prisoners. We formulate and analyze a deterministic model for the role of intravenous drug use in
HIV/AIDS transmission among women prisoners. With the aid of the Centre Manifold theory, the
endemic equilibrium is shown to be locally asymptotically stable when the corresponding reproduction number is greater than unity. Analysis of the reproduction number and numerical simulations
suggest that an increase in intravenous drug use among women prisoners as they fail to cope with
prison settings fuels the HIV/AIDS epidemic in women prisoners. Failure to control HIV/AIDS among
female prisoners may be a time bomb to their communities upon their release. Thus, it may be best
to consider free needle/syringe exchange and drug substitution treatment programmes in women
prisons as well as considering open prison systems for less serious crimes.


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## Body

## 1. Introduction

The imprisonment of large numbers of drug addicts has the potential to create environments within which social networks that enhance the transmission of infectious diseases form [1–6]. Prisons are high-risk settings for HIV transmission. However, HIV prevention and treatment are not adequately developed and implemented to respond to HIV in prisons [7]. There is evidence to show that health programmes for the particular needs of imprisoned drug users are not enough in USA and Canada [8, 9]. In Russia, a study of intravenous drug users demonstrated the critical role of prisons in the transmission of HIV through high levels of needle (syringes) sharing among the imprisoned [10].In many jurisdictions, a larger proportion of women than men are in prison for drug related offences [11]. Drug use and HIV infection are more prevalent among women in prison than among imprisoned men [12]. Women in prisons are more susceptible to self-harm through unsafe tattooing and injecting drug use [13]. In the absence of sterile injecting equipment, women, like men, will inject with used needles or home-made syringes, Doll [14] Like men, women get tattooed in prison [14]. In the absence of proper precautions and access to safe equipment tattooing can be a high-risk activity for the transmission of HIV [15, 16].Women constitute a very small proportion of the total prison population, globally accounting 5% of the total prison population [15]. Women prisoners present specific challenges for correctional authorities as most of the existing prison facilities for women inmates have all being developed for men, who have historically accounted the largest proportion of the prison population [15, 17–19]. Scholarly research has called attention to the presence of HIV-infected women in prison and their lack of care while under institutional supervision [20]. With increase in female inmate population in the USA, the prison system is left with a small but significant population who suffers as a result of both HIV infection upon entering the criminal justice system, and contracting the infection while under correctional custody [21, 22]. Quite recently [23], we employed a similar approach to assess the impact of educational campaigns on HCV control among women prisoners. However, the present study differs from the previous work [23] in that here we are dealing with HIV/AIDS whose stages of disease progression are different from those of HCV. This results in the two models studied having different forces of infection. Furthermore, the potential benefits of open prison systems are explored here. Some studies [24] did analyse a model to predict the effects of nondisclosure by HIV positive people on the spread of the virus. Although disclosure is not the subject of this study, HIV status disclosure plays a significant role in the control of the epidemic. This is probably the first model that attempts to look into transmission dynamics of HIV/AIDS among female prisoners.The rest of this paper is organized as follows. In the next section, the model and its basic properties are presented. In Section3, we determine stability analysis of the equilibria states. Numerical simultaions are presented in Section 4 and finally the last section concludes the paper.


## 2. Model Formulation

Consider a population consisting of the following subpopulations: susceptible female prisoners nondrug injectorsSn, HIV positive only female prisoners nondrug injectors Inh, AIDS cases among female prisoners nondrug injectors Anh, susceptible female prisoners who are drug injectors Sd, HIV positive only female prisoners who are drug injectors Idh, and AIDS cases among female prisoners who are drug injectors Adh; thus the total population is given by
(1)N=Nn+Nd,Nn=Sn+Inh+Anh,Nd=Sd+Idh+Adh,
where Nn and Nd are the total population of female nondrug using prisoners and total population of drug injecting female prisoners, respectively. Female prisoners are recruited into the prisons through committing different crimes at a constant rate Λ. We assume that fractions π0,π1,π2, and π3(∑n=03‍πn=1) are recruited into Sn,Inh,Sd, and Idh, respectively. Here, we further assume that AIDS patients are too sick to commit a crime as such there is no recruitment into the AIDS classes. So the AIDS cases we have in prisons are a direct result of progression to the AIDS stage while in prison. Due to peer pressure nondrug injecting female prisoners in SnandInh acquire drug injecting habits at rate ρ to move into Sd and Idh classes, respectively. We assume that AIDS patients Anh are now too sick to acquire drug injecting habits. Susceptible intravenous drug using female prisoners acquire HIV through sharing contaminated drug injection equipment at a rate
(2)λh=βh(Idh+θAdh)N,
where βh is the product of the probability of getting HIV infection per syringe sharing event and the number of people one shares it with; θ∈(0,1) accounts for a reduction in the rate of HIV transmission due to a decrease in the number of female prisoners willing to share drug injecting equipment with someone displaying AIDS symptoms. Natural mortality occurs in all classes at a rate μ, while individuals in the AIDS stage suffer an additional disease-induced mortality at a rate ν. Due to sickness drug injecting AIDS patients Adh are assumed to leave drug injecting habits at a rate α and move into Anh. Upon completion of their sentences female prisoners leave prisons at rate ω back into their communities. The model flow diagram is depicted in Figure 1.Figure 1
Model flow diagram.From the assumptions on the dynamics of the epidemic made above, the following are the model equations:(3)Sn′=Λπ0-(ρ+ω+μ)Sn,Inh′=Λπ1-(ρ+γ+ω+μ)Inh,Anh′=γInh+αAdh-(ω+μ+ν)Anh,Sd′=Λπ2+ρSn-(λh+ω+μ)Sd,Idh′=Λπ3+ρInh+λhSd-(ω+σ+μ)Idh,Adh′=σIdh-(ω+α+μ+ν)Adh.
The first octant of system (3) is positively invariant and attracting, and solutions starting in the first octant where all the variables are nonnegative stay there. Thus, system (3) will be analyzed in a suitable region 𝒟⊂ℝ+6 given by
(4)𝒟={(Sn,Inh,Anh,Sd,Idh,Adh)∈ℝ+6:N≤Λμ+ω},
which is positively invariant and attracting. Existence, uniqueness and continuation results for system (3) hold in this region.


## 3. Equilibrium States, Reproductive Number, and Stability

Model system (3) has a disease-free equilibrium (DFE) which occurs when π1=π3=0 given by
(5)𝒱0=(Sn0,Inh0,Anh0,Sd0,Idh0,Adh0)=(Λπ0ρ+ω+μ,0,0,Λ(ρ+(ω+μ)π2)(ω+μ)(ρ+ω+μ),0,0).
Following Van Den Driessche and Watmough [25], we have
(6)ℛ0f=βh(ρ+(μ+ω)π2)(μ+σ+ω)(μ+ω+ρ)(1+θσα+μ+ν+ω),
which is defined as the number of secondary HIV infections produced by one infected female individual through the sharing of nonsterile syringes/needles in a prison. Theorem 1 follows from Van Den Driessche and Watmough [25].Theorem 1.
The disease-free equilibrium𝒱0 is locally asymptotically stable when ℛ0f<1 and unstable otherwise.Analysis Reproduction Number
ℛ
0
f. Here we answer some issues pertaining to drug misuse in prisons by way of analysing the basic reproduction number ℛ0f in an effort to understand the dynamics of HIV infections in female prisons. First, let us consider the effect intravenous drug injections have on HIV transmission. This is done by partially differenting ℛ0f with respect to ρ to obtain
(7)∂ℛ0f∂ρ=βhμ+σ+ω(1+θσα+μ+ν+ω)×(1-π2)(μ+ω)(μ+ω+ρ)2>0.
The fact that (7) is positive suggests that an increase in females becoming intravenous drug users in prisons results in an increase of HIV related infections in women prisoners. Most women in prison resort to drug use and tattooing in prisons due to failure to cope with stressful prison conditions. Against this background it may be best to consider open prison systems for less serious crimes so that people may serve their sentences while coming from their homes. In this way they are surrounded by their families most of the time and hence are able to defeat the temptations to engage in intravenous drug use. Now consider
(8)limπ2→0ℛ0f=ℛ0fd=βhρ(μ+σ+ω)(μ+ω+ρ)×(1+σθα+μ+ν+ω).
Substracting ℛ0fd from ℛ0f we have
(9)ℛ0f-ℛ0fd=βhπ2(μ+ω)(μ+σ+ω)(μ+ω+ρ)×(1+σθα+μ+ν+ω)>0.
The issue of (9) being positive suggests that reducing the number of intravenous drug users in prisons reduces the spread of HIV in prisons. While theoretically this is feasible, in reality this is not feasible as drug users are more associated with crime than the general population. In light of that, perhaps, it is more feasible to have free needle exchange in prisons. This will assist in ensuring that female convicts do not share contaminated syringes. Furthermore, provision of drug substituting treatment will be of beneficial use to prisoners.


### 3.1. Endemic Equilibrium

In terms of the HIV and drug induction forces of infectionλh*andρ, the endemic equilibrium is given by 𝒱*=(Sn*,Inh*,Anh*,Sd*,Idh*,Adh*), where
(10)Sn*=Λπ0ρ+ω+μ,Inh*=Λπ0ρ+γ+ω+μ,Anh*=γΛπ0(ω+μ+ν)(ρ+γ+ω+μ)+ασΛ(ω+μ+ν)(ω+α+μ+ν)(ω+μ+σ)×((π1+π3)ρ+π3(γ+ω+μ)ρ+γ+ω+μ+λh*(π2(ρ+μ+ω)+π0ρ)(λh*+μ+ω)(ρ+μ+ω)),Sd*=Λ(π2(ρ+μ+ω)+π0ρ)(ρ+μ+ω)(λh*+μ+ω),Idh*=Λλh*(π2(ρ+μ+ω)+π0ρ)(ω+σ+μ)(ρ+μ+ω)(λh*+μ+ω)+Λ(π3(ρ+γ+ω+μ)+π1ρ)(ρ+γ+ω+μ)(ω+σ+μ),Adh*=σΛ(ω+α+μ+ν)(ω+μ+σ)×((π1+π3)ρ+π3(γ+ω+μ)ρ+γ+ω+μ+λh*(π2(ρ+μ+ω)+π0ρ)(λh*+μ+ω)(ρ+μ+ω)).
To establish the local asymptotic stability of this equilibrium point, we use the Centre Manifold theory [26] as illustrated by Castillo-Chavez and Song [27]. To apply the Centre Manifold theory we make the following change of variables: Sn=x1,Inh=x2,andAnh=x3,Sd=x4,Idh=x5,Anh=x6 so that ∑n=16‍xn. We now use the notation of the form X=(x1,x2,x3,x4,x5,x6)T. Then model system (3) can be written as dX/dt=F=(f1,f2,f3,f4,f5,f6)T, where
(11)x1′(t)=f1=Λπ0-(ρ+ω+μ)x1,x2′(t)=f2=Λπ1-(ρ+ω+γ+μ)x2,x3′(t)=f3=γx2+αx6-(μ+ω+ν)x3,x4′(t)=f4=Λπ2+ρx1-βh(x5+θx6)x4∑n=16xn-(ω+μ)x4,x5′(t)=f5=Λπ3+ρx2+βh(x5+θx6)x4∑n=16xn-(ω+σ+μ)x5,x6′(t)=f6=σx5-(μ+ω+α+ν)x6.
The Jacobian matrix of system (11) at 𝒱0 is given by(12)J(𝒱0)=[-A2000000-(A2+γ)00000γ-A300αρ00-A1-βh(ρ+π2A1)A2-βhθ(βd+π2A1)A20ρ00βh(ρ+π2A1)A2-A4βhθ(ρ+π2A1)A20000σ-(A3+α)], with A1=ω+μ, A2=A1+ρ, A3=A1+ν, and A4=A1+σ.It can be shown fromJ(𝒱0) in (12), the reproduction number of system (11) is
(13)ℛ0f=βh(ρ+(μ+ω)π2)(μ+σ+ω)(μ+ω+ρ)(1+θσα+μ+ν+ω).
If βh is taken as a bifurcation point and if we consider the case ℛ0f=1 and βdh we have
(14)βh=βh*=(μ+σ+ω)(μ+ω+ρ)(α+μ+ν+ω)(α+μ+ν+ω+σθ)(ρ+(μ+ω)π2).
Noting that the linearised system (11) with βh=βh* has simple zero eigenvalue, hence, the Centre Manifold theory [26] can be used to analyze the dynamics of system (11) near βh=βh*. It can be shown that the Jacobian of system (11) at βh=βh* has a right eigenvector with the zero eigenvalue given by u=[u1,u2,u3,u4,u5,u6]T, where
(15)u1=u2=0,u3=αω+μ+νu6,u6>0,u5=μ+α+ω+νσu6,u4=-βh*(ρ+(ω+μ)π2)(μ+α+ω+ν+θσ)(ω+μ)(ω+μ+ρ)σu6.
The left eigenvector of J(𝒱0) is given by v=[v1,v2,v3,v4,v5,v6]T, where
(16)v1=v3=v4=0,v2=ρ(μ+α+ω+ν)(μ+ω+ρ)βh*θ(π2(μ+ω)+ρ)(ω+μ+γ+ρ)v6,v5=(μ+α+ω+ν)(μ+ω+ρ)βh*θ(π2(μ+ω)+ρ)v6,v6>0.
In order to establish the existence of a bifurcation, we use Theorem 4 in [27].Computation of
a
and
b
. For model system (11), the associated nonzero derivatives of F associated with b at 𝒱0 as noted in Theorem 4 in [27] are given by
(17)∂2f4∂βh*∂x5=-ρ+(ω+μ)π2ρ+ω+μ,∂2f4∂βh*∂x6=-θ(ρ+(ω+μ)π2)ρ+ω+μ,∂2f5∂βh*∂x5=ρ+(ω+μ)π2ρ+ω+μ,∂2f5∂βh*∂x6=-θ(ρ+(ω+μ)π2)ρ+ω+μ.
It follows from (15)–(17) that
(18)b=∑k,i=16vkui∂2fk∂xi∂βh*(𝒱0)=(μ+α+ω+ν)(μ+α+ω+ν+σθ)σθβh*u6v6>0.
The non-zero partial derivatives of F associated with a at 𝒱0 as noted in Theorem 4 in [27] are given by
(19)∂2f5∂x1∂x5=∂2f5∂x5∂x1=∂2f5∂x2∂x5=∂2f5∂x5∂x2=-βh*(ω+μ)(ρ+(ω+μ)π2)Λ(ρ+ω+μ),∂2f5∂x1∂x6=∂2f5∂x6∂x1=-βh*θ(ω+μ)(ρ+(ω+μ)π2)Λ(ρ+ω+μ),∂2f5∂x2∂x6=∂2f5∂x6∂x2=-βh*θ(ω+μ)(ρ+(ω+μ)π2)Λ(ρ+ω+μ),∂2f5∂x3∂x5=∂2f5∂x5∂x3=-βh*(ω+μ)(ρ+(ω+μ)π2)Λ(ρ+ω+μ),∂2f5∂x3∂x6=∂2f5∂x6∂x3=-βh*θ(ω+μ)(ρ+(ω+μ)π2)Λ(ρ+ω+μ),∂2f5∂x4∂x5=∂2f5∂x5∂x4=βh*π0(ω+μ)2Λ(ρ+ω+μ),∂2f5∂x4∂x6=∂2f5∂x6∂x4=βh*θπ0(ω+μ)2Λ(ρ+ω+μ),∂2f5∂x52=-2βh*(ω+μ)(ρ+(ω+μ)π2)Λ(ρ+ω+μ),∂2f5∂x62=-2βh*θ(ω+μ)(ρ+(ω+μ)π2)Λ(ρ+ω+μ),∂2f5∂x5∂x6=∂2f5∂x6∂x6=-βh*(1+θ)(ω+μ)(ρ+(ω+μ)π2)Λ(ρ+ω+μ).
It follows from (15), (16), and (19) that
(20)a=∑k,i,j=16‍vkuiuj∂2fk∂xi∂xj(𝒱0)=-2(ω+μ)(ω+μ+ν+σ)(ω+μ+ν+α)2σΛ(ω+μ+ν)×(1+ω+μ+ν+ασθ)u62v6-2(ω+μ+α+ν+θσ)2(ω+μ+α+ν)(μ+ω)βh*π0θσ2Λ(ω+μ+ρ)×u62v6<0.
It is worth noting from (20) that a is negative. Using Theorem 4 [27] item (iv) we establish the following result.Theorem 2.
The endemic equilibrium point𝒱* is locally asymptotically stable for ℛ0f>1 but close to 1.


## 3.1. Endemic Equilibrium

In terms of the HIV and drug induction forces of infectionλh*andρ, the endemic equilibrium is given by 𝒱*=(Sn*,Inh*,Anh*,Sd*,Idh*,Adh*), where
(10)Sn*=Λπ0ρ+ω+μ,Inh*=Λπ0ρ+γ+ω+μ,Anh*=γΛπ0(ω+μ+ν)(ρ+γ+ω+μ)+ασΛ(ω+μ+ν)(ω+α+μ+ν)(ω+μ+σ)×((π1+π3)ρ+π3(γ+ω+μ)ρ+γ+ω+μ+λh*(π2(ρ+μ+ω)+π0ρ)(λh*+μ+ω)(ρ+μ+ω)),Sd*=Λ(π2(ρ+μ+ω)+π0ρ)(ρ+μ+ω)(λh*+μ+ω),Idh*=Λλh*(π2(ρ+μ+ω)+π0ρ)(ω+σ+μ)(ρ+μ+ω)(λh*+μ+ω)+Λ(π3(ρ+γ+ω+μ)+π1ρ)(ρ+γ+ω+μ)(ω+σ+μ),Adh*=σΛ(ω+α+μ+ν)(ω+μ+σ)×((π1+π3)ρ+π3(γ+ω+μ)ρ+γ+ω+μ+λh*(π2(ρ+μ+ω)+π0ρ)(λh*+μ+ω)(ρ+μ+ω)).
To establish the local asymptotic stability of this equilibrium point, we use the Centre Manifold theory [26] as illustrated by Castillo-Chavez and Song [27]. To apply the Centre Manifold theory we make the following change of variables: Sn=x1,Inh=x2,andAnh=x3,Sd=x4,Idh=x5,Anh=x6 so that ∑n=16‍xn. We now use the notation of the form X=(x1,x2,x3,x4,x5,x6)T. Then model system (3) can be written as dX/dt=F=(f1,f2,f3,f4,f5,f6)T, where
(11)x1′(t)=f1=Λπ0-(ρ+ω+μ)x1,x2′(t)=f2=Λπ1-(ρ+ω+γ+μ)x2,x3′(t)=f3=γx2+αx6-(μ+ω+ν)x3,x4′(t)=f4=Λπ2+ρx1-βh(x5+θx6)x4∑n=16xn-(ω+μ)x4,x5′(t)=f5=Λπ3+ρx2+βh(x5+θx6)x4∑n=16xn-(ω+σ+μ)x5,x6′(t)=f6=σx5-(μ+ω+α+ν)x6.
The Jacobian matrix of system (11) at 𝒱0 is given by(12)J(𝒱0)=[-A2000000-(A2+γ)00000γ-A300αρ00-A1-βh(ρ+π2A1)A2-βhθ(βd+π2A1)A20ρ00βh(ρ+π2A1)A2-A4βhθ(ρ+π2A1)A20000σ-(A3+α)], with A1=ω+μ, A2=A1+ρ, A3=A1+ν, and A4=A1+σ.It can be shown fromJ(𝒱0) in (12), the reproduction number of system (11) is
(13)ℛ0f=βh(ρ+(μ+ω)π2)(μ+σ+ω)(μ+ω+ρ)(1+θσα+μ+ν+ω).
If βh is taken as a bifurcation point and if we consider the case ℛ0f=1 and βdh we have
(14)βh=βh*=(μ+σ+ω)(μ+ω+ρ)(α+μ+ν+ω)(α+μ+ν+ω+σθ)(ρ+(μ+ω)π2).
Noting that the linearised system (11) with βh=βh* has simple zero eigenvalue, hence, the Centre Manifold theory [26] can be used to analyze the dynamics of system (11) near βh=βh*. It can be shown that the Jacobian of system (11) at βh=βh* has a right eigenvector with the zero eigenvalue given by u=[u1,u2,u3,u4,u5,u6]T, where
(15)u1=u2=0,u3=αω+μ+νu6,u6>0,u5=μ+α+ω+νσu6,u4=-βh*(ρ+(ω+μ)π2)(μ+α+ω+ν+θσ)(ω+μ)(ω+μ+ρ)σu6.
The left eigenvector of J(𝒱0) is given by v=[v1,v2,v3,v4,v5,v6]T, where
(16)v1=v3=v4=0,v2=ρ(μ+α+ω+ν)(μ+ω+ρ)βh*θ(π2(μ+ω)+ρ)(ω+μ+γ+ρ)v6,v5=(μ+α+ω+ν)(μ+ω+ρ)βh*θ(π2(μ+ω)+ρ)v6,v6>0.
In order to establish the existence of a bifurcation, we use Theorem 4 in [27].Computation of
a
and
b
. For model system (11), the associated nonzero derivatives of F associated with b at 𝒱0 as noted in Theorem 4 in [27] are given by
(17)∂2f4∂βh*∂x5=-ρ+(ω+μ)π2ρ+ω+μ,∂2f4∂βh*∂x6=-θ(ρ+(ω+μ)π2)ρ+ω+μ,∂2f5∂βh*∂x5=ρ+(ω+μ)π2ρ+ω+μ,∂2f5∂βh*∂x6=-θ(ρ+(ω+μ)π2)ρ+ω+μ.
It follows from (15)–(17) that
(18)b=∑k,i=16vkui∂2fk∂xi∂βh*(𝒱0)=(μ+α+ω+ν)(μ+α+ω+ν+σθ)σθβh*u6v6>0.
The non-zero partial derivatives of F associated with a at 𝒱0 as noted in Theorem 4 in [27] are given by
(19)∂2f5∂x1∂x5=∂2f5∂x5∂x1=∂2f5∂x2∂x5=∂2f5∂x5∂x2=-βh*(ω+μ)(ρ+(ω+μ)π2)Λ(ρ+ω+μ),∂2f5∂x1∂x6=∂2f5∂x6∂x1=-βh*θ(ω+μ)(ρ+(ω+μ)π2)Λ(ρ+ω+μ),∂2f5∂x2∂x6=∂2f5∂x6∂x2=-βh*θ(ω+μ)(ρ+(ω+μ)π2)Λ(ρ+ω+μ),∂2f5∂x3∂x5=∂2f5∂x5∂x3=-βh*(ω+μ)(ρ+(ω+μ)π2)Λ(ρ+ω+μ),∂2f5∂x3∂x6=∂2f5∂x6∂x3=-βh*θ(ω+μ)(ρ+(ω+μ)π2)Λ(ρ+ω+μ),∂2f5∂x4∂x5=∂2f5∂x5∂x4=βh*π0(ω+μ)2Λ(ρ+ω+μ),∂2f5∂x4∂x6=∂2f5∂x6∂x4=βh*θπ0(ω+μ)2Λ(ρ+ω+μ),∂2f5∂x52=-2βh*(ω+μ)(ρ+(ω+μ)π2)Λ(ρ+ω+μ),∂2f5∂x62=-2βh*θ(ω+μ)(ρ+(ω+μ)π2)Λ(ρ+ω+μ),∂2f5∂x5∂x6=∂2f5∂x6∂x6=-βh*(1+θ)(ω+μ)(ρ+(ω+μ)π2)Λ(ρ+ω+μ).
It follows from (15), (16), and (19) that
(20)a=∑k,i,j=16‍vkuiuj∂2fk∂xi∂xj(𝒱0)=-2(ω+μ)(ω+μ+ν+σ)(ω+μ+ν+α)2σΛ(ω+μ+ν)×(1+ω+μ+ν+ασθ)u62v6-2(ω+μ+α+ν+θσ)2(ω+μ+α+ν)(μ+ω)βh*π0θσ2Λ(ω+μ+ρ)×u62v6<0.
It is worth noting from (20) that a is negative. Using Theorem 4 [27] item (iv) we establish the following result.Theorem 2.
The endemic equilibrium point𝒱* is locally asymptotically stable for ℛ0f>1 but close to 1.


## 4. Numerical Simulations

In this section, we carry out detailed numerical simulations using Matlab programming language to assess the transmission dynamics of HIV in prisons housing female prisoners. The parameter values that we use for numerical simulations are in Table1.Table 1
Model parameters and their interpretations.
Parameter
Symbol
Value
Source
Recruitment rate
Λ
0.00163 yr−1*3*106
NPA
Natural mortality rate
μ
0.02 yr−1
b
*
Natural rate of progression to AIDS
γ
,
σ
0.1 yr−1
b
*
Modification parameter
θ
0.25
Assumed
AIDS related death rate
ν
0.4 yr−1
b
*
Product of effective contact rate for HIV infection and probability of HIV transmission per drug injection
β
h
0.562 yr−1
a
*
Rate of becoming a drug user
ρ
0.4 yr−1
c
*
Rate of quitting drug misuse of sickness
α
0.3 yr−1
c
*
Rate of release from prison
ω
0.25 yr−1
Assumed
Proportion recruited intoSn, Inh, Sd, and Idh classes
π
0
,
π
1
,
π
2
,
π
3
0.375, 0.125, 0.375, 0.125
AssumedIn Table1, NPA denotes National Prison Administration (Zimbabwe); a*, b*, and c* denote parameter values adapted from Green and Al-Fwzan [28], Bhunu et al. [29], and Bhunu et al. [30], respectively. For influence of peer pressure forces influencing one to become an IDU, we used values adapted from Bhunu et al. [30], which are peer pressure forces necessary for one to start smoking, for the sake of illustration.Figure2 is a graphical representation showing the effects of intravenous drug use on HIV and AIDS cases in female prisons. Figures 2(a) and 2(b) show that total HIV-only and AIDS cases increase with increase in the percentage of intravenous drug users, respectively. This suggests that decreasing the levels of intravenous drug use will also result in a decrease in levels of HIV/AIDS prevalence in prisons. Theoretically, this is feasible but in the real world this is not possible to achieve. Currently, while the problem of intravenous drug use exists in female prisons, authorities have not yet acknowledged its existence. Perhaps the answer to the problem of intravenous drug use among female prisons is accepting that it exists and then offering free needle exchange and/or drug substitution treatment.Simulations of model system (3) showing the effects of varying the percentage of intravenous drug users on the population of HIV positive people and AIDS cases in female prisoners. Parameter values used are in Table 1.
(a)
(b)Figure3 is a graphical representation showing how HIV and AIDS fare in 100% and 0% of intravenous drug using female prisoner populations. In Figure 3 the HIV and AIDS population is depicted for various initial conditions. The HIV and AIDS cases converge to their distinct endemic equilibrium states depending on the level of intravenous drug use in the female prison population as shown in Figures 3(a) and 3(b), respectively. The higher the percentage of intravenous drug users, the larger the numbers of the HIV-only and AIDS cases at the endemic equilibrium point. This suggests that control of drug use in female prisons may provide an answer to the problem of HIV/AIDS among female prisoners.Time series evolution of the HIV-only infected individuals and those displaying symptoms of AIDS for various initial conditions when all female prisoners are made of nondrug users and when they are all intravenous users. Parameter values used are in Table1.
(a)
(b)


## 5. Discussion

A mathematical model has been presented in attempt to understand the transmission dynamics of HIV/AIDS among female prisoners with particular reference to female prisoners in Zimbabwe. Drug use in female prisons is two way: (i) sharing of unsterile needles/syringes enhances the transmission of HIV and (ii) flashing blood that is drawing blood from someone who has injected herself with a drug and inject the blood into herself on its own exposes the injector to the HIV infection. Analysis of the reproduction number has shown that (i) a reduction in drug use results in a decrease of HIV/AIDS prevalence among female prisoners and (ii) release of prisoners may also act in reducing the concentration HIV/AIDS cases in prisons. The latter fact is not feasible, but perhaps implementing opening prison systems where prisoners of less serious crimes are allowed to serve their sentences while staying with their families enables female prisoners to cope with stressful prison conditions. Open prison systems will reduce the influence of peer pressure among prisoners as they will have moral and psychological support from the family which does not exist in enclosed female prison systems. Numerical simulations carried out also support the analytic results that increase in drug use and tattooing increases HIV/AIDS prevalence among women prisoners. The result of this study has a public health implication considering high rates of syringe lending and borrowing in prisons. This might explain why there are more HIV cases in prisons than the general population in the case of the USA [2, 3] and this might be the case world wide. HIV-infected women in prison pose a risk to their communities upon release from prison; so in the absence of open prison systems, it may be best to have mandatory HIV/AIDS screening and gender-specific educational programmes. This will reduce the prevalence of high-risk behaviours and lower HIV transmission in female prisons, thus reducing postrelease public health threat. Given the high levels of HIV in prisons, about three and half times higher among prisoners than the general population, it may be best to consider the introduction of needle/syringe free exchange programme and drug substitution treatment ways of keeping in check with HIV transmission in women prisons. The study is presented here is not exhaustive; it can be extended to include the contribution of male prison staff into the dynamics of HIV transmission among female prisoners as the male staff are the most likely people who sexually abuse female prisoners while incarcerated.



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