**HIV/AIDS Transmission Dynamics in Male Prisons**

#### C.P. Bhunu\*and S. Mushayabasa

*Department of Applied Mathematics, Modelling Biomedical Systems Research Group National University of Science and Technology, Bulawayo Zimbabwe*

#### **1. Introduction**

66 Global View of HIV Infection

Xu, J. (2009). Prospective cohort study to the incidence of HIV/STIs among FSWs in

Yu, M. and Vajdy, M. (2010). Mucosal HIV transmission and vaccination strategies through

and Prevention.

10(8): 1181-1195.

Kaiyuan City. PhD [dissertation].Beijing, China: China Center for Disease Control

oral compared with vaginal and rectal routes. *Expert Opinion on Biological Therapy*,

The imprisonment of large numbers of drug addicts has the potential to create environments within which social networks that enhances the transmission of infectious diseases form (7–11; 14). About 668,000 men and women are incarcerated in sub-Saharan Africa with South Africa having the highest prison population with 157,402 people behind bars in the region and 335 prisoners per 100,000 of the national population; it has the ninth largest prison population in the world (21). International data show that HIV prevalence among prisoners is between six to fifty times higher than that of the general adult population. For example, in the USA the ratio is 6:1, in France it is 10:1; in Switzerland 27:1 and in Mauritius 50:1 (17). On a global scale, the prison population is growing rapidly, with high incarceration rates leading to overcrowding, which largely stems from national law and criminal justice policies. In most countries, overcrowding and poor physical conditions prevail (20). This phenomenon poses significant health concerns with regard to control of infectious diseases-and HIV prevention and care most of all (21). Prisons are high risk settings for HIV transmission. However, HIV prevention, treatment are not adequately developed and implemented to respond to HIV in prisons (13). There is evidence to show that health programmes for the particular needs of imprisoned drug users are not enough in USA and Canada (15; 22). 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 (23).

Prison populations are predominantly male and most prisons are male-only institutions, including the prison staff. In such a gender exclusive environment, male-to-male sexual activity (prisoner-to-prisoner and guard-to-prisoner) is frequent (18). While much of the sex among men in prisons is consensual, rape and sexual abuse are often used to exercise dominance in the culture of violence that is typical of prison life (19). Inmate rape, including male rape, is considered one of the most ignored crimes. Sexual and physical abuse in custody remains a tremendous human rights problem (1). Intravenous drug use, tattooing and the following aspects of man-to-man sexual activity in prison make it a high risk for HIV transmission: anal intercourse, rape and the presence of sexually transmitted infections (STIs). Related problems in prisons across Southern Africa include overcrowding, shortages, corruption, and the presence of juveniles alongside adult prisoners. The potential for the spread of HIV is also increased by a lack of information and education, and a lack of proper medical care. STIs, if left untreated, can greatly increase a person's vulnerability to HIV

<sup>\*</sup>Visiting Fellow, Clare Hall College, University of Cambridge

with *βi*, *i* = (*n*, *d*) is probabbility on individual being infected with HIV by an individual from the n- or d-class per sexual contact; *cj*, *j* = (*n*, *d*) are the number of sexual partners an individual acquires per year (partner acquistion rates); *βd*<sup>2</sup> is the probability an intravenous drug user getting HIV infection through sharing non-sterile needles during drug injections

HIV/AIDS Transmission Dynamics in Male Prisons 69

It is assumed people are recruited into prison at rate Λ through committing various crimes and the following proportions *π*1, *π*2, *π*<sup>3</sup> and *π*<sup>4</sup> recruited enter the classes *Sn*(*t*), *Sd*(*t*), *In*(*t*) and *Id*(*t*), respectively. We further assume that AIDS cases are too sick to commit a crime, so there are no recruitment of prisoners already in the AIDS stage of disease. Furthermore, it is assumed that intravenous drug using prisoners showing AIDS symptoms are nolonger able to exert peer pressure strong enough to make one become a drug user. Individuals in *Sn*(*t*) and *In*(*t*) acquire drug misusing habits at rate *λd*(*t*) due to peer pressure

> *<sup>λ</sup>dd* <sup>=</sup> *<sup>β</sup>d*<sup>1</sup> *cd*<sup>1</sup> (*Sd* <sup>+</sup> *Id*) *Nd*

where *βd*<sup>1</sup> is the probability of becoming an intravenous drug user (IDU) following contact with an IDU and *cd*<sup>1</sup> are the number of contacts necessary for one to become an IDU (partner acquistion rate). Individuals in *Sn*(*t*) class acquire HIV infection at a rate *λnh* (*t*) to move into *In*(*t*). Individuals in *Sd*(*t*) class acquire HIV infection at a rate *λdh* (*t*) to move *Id*(*t*) class. Individuals infected with HIV-only not yet displaying symptoms (*In*(*t*), *Id*(*t*)) progress to the AIDS stage ((*An*(*t*), *Ad*(*t*)) at a rate *γ*. Individuals in *Ad*(*t*) leave the intravenous drug using habits at a rate *α* to get into *An*(*t*) class. Individuals in all classes experience natural death at a rate *μ* and those in AIDS stage of the disease experience an additional disease induced death at a rate *ν*. Individuals in all classes leave the prison at rate *ω* upon completion of their sentences. Individuals in the AIDS stage of the disease (final terminal stages) are further released from prison due to sickness at rate *φ*. The model flow diagram is shown in Figure 1. Based on these assumptions the following system of differential equations describe the model.

*<sup>n</sup>*(*t*) = *π*1Λ − (*λ<sup>d</sup>* + *λnh* )*Sn* − (*μ* + *ω*)*Sn*,

*<sup>n</sup>*(*t*) = *γIn* + *αAd* − (*μ* + *ω* + *φ* + *ν*)*An*,

*<sup>d</sup>*(*t*) = *π*2Λ + *λdSn* − *λdhSd* − (*μ* + *ω*)*Sd*,

*<sup>d</sup>*(*t*) = *π*4Λ + *λdhSd* + *λ<sup>d</sup> In* − (*μ* + *ω* + *γ*)*Id*,

In this section, we study the basic results of solutions of model system (5), which are essential

*<sup>d</sup>*(*t*) = *γId* − (*μ* + *α* + *ω* + *φ* + *ν*)*An*.

*<sup>n</sup>*(*t*) = *π*3Λ + *λnhSn* − *λ<sup>d</sup> In* − (*μ* + *ω* + *γ*)*In*,

, (4)

(5)

and *cd*<sup>2</sup> are the number drug sharing partners an individual acquires.

and move into *Sd*(*t*) and *Id*(*t*, respectively with

*S*�

*I* �

*A*�

*S*�

*I* �

*A*�

**2.1 Model basic properties**

in the proofs of stability results.

through sexual contact, UNAIDS noted (26). Men get tattooed in prison (12). In the absence of proper precautions and access to safe equipment tattooing can be a high-risk activity for the transmission of HIV (24; 25).

The literature and development of mathematical epidemiology is well documented (2; 3; 6). This paper seeks to use mathematical models to gain insights on transimission of HIV among male prisoners while in prison in the context of homosexuality and intravenous drug use. The rest of this paper is organized as follows. In the next section, the model and its basic properties are presented. In Section 3, we determine stability analysis of the equilibria states. Numerical simualtions are presented in Section 4 and finally the last section concludes the paper.

#### **2. Model description**

The model sub-divides the total male prisoner population into the following sub-populations of susceptible intravenous drug users *Sd*(*t*), susceptible non-drug users *Sn*(*t*), intravenous drug using HIV-only infected people not yet showing AIDS symptoms *In*(*t*), non-drug using HIV-only infected people not yet showing AIDS symptoms *Id*(*t*), intravenous drug using AIDS cases *Ad*(*t*) and non-drug using AIDS cases *Ad*(*t*). There is sexual interaction between intravenous drug users and non-drug users making HIV transmission across different these two distinct distinct groups possible. The population is patterns is heterogeneous mixing with regard to sexual behaviour. The total population is given by;

$$N(t) = N\_d(t) + N\_n(t),\ N\_d(t) = S\_d(t) + I\_d(t) + A\_d(t),\ N\_n(t) = S\_n(t) + I\_n(t) + A\_n(t),\tag{1}$$

with *Nn*(*t*) and *Nd*(*t*) being the total number of non-drug using and intravenous drug using male prisoners (intravenous drug users-IDU), respectively. The group *j* members make *cj*, *j* = (*d*, *n*) sexual contacts per unit time, and that a fraction of the contacts made by a member of group *j* is with a member of group *i* is *pji* , *i* = (*d*, *n*). Then *pnn* + *pnd* = *pdd* + *pdn* = 1. The total number of sexual contacts per unit time by members of group 'n' (non-drug users) with members of group 'd' (intravenous drug users) is *cn pnd Nn* and because this must be equal to the number of contacts made by members of group 'd' with members of group 'n', we have a balance relation

$$\frac{p\_{\text{n}\_d}c\_{\text{n}}}{N\_d} = \frac{p\_{d\_{\text{n}}}c\_d}{N\_{\text{n}}}.\tag{2}$$

In this case the sexual contact rates (partner acquistion rates) *cd* and *cn* are saturating terms for the total population and the mixing proportions may change with time. It is worth mentioning here that intravenous drug users are more likely to have more sexual partners than the general population. Therefore, *cd* = B*cn*, B ≥ 1. We assume that male prisoners in AIDS stage of the disease are nolonger sexually active as they are nolonger capable of attracting sexual mates among prisoners. Also drug using AIDS patients nolonger share their needles with others as other prisoners do not like sharing needles with someone whose AIDS symptoms are visible. The forces of HIV infection for intravenous drug users and non-drug users in the male prison are:

$$\begin{aligned} \lambda\_{d\_h} &= \frac{p\_{d\_d} c\_d \beta\_d I\_d}{N\_d} + \frac{p\_{d\_n} c\_d \beta\_n I\_n}{N\_n} + \frac{c\_{d\_2} \beta\_{d\_2} I\_d}{N\_d}, \\\\ \text{and } \lambda\_{n\_h} &= \frac{p\_{n\_n} c\_n \beta\_{n} I\_n}{N\_n} + \frac{p\_{n\_d} c\_n \beta\_d I\_d}{N\_d}, \text{ respectively} \end{aligned} \tag{3}$$

2 Will-be-set-by-IN-TECH

through sexual contact, UNAIDS noted (26). Men get tattooed in prison (12). In the absence of proper precautions and access to safe equipment tattooing can be a high-risk activity for the

The literature and development of mathematical epidemiology is well documented (2; 3; 6). This paper seeks to use mathematical models to gain insights on transimission of HIV among male prisoners while in prison in the context of homosexuality and intravenous drug use. The rest of this paper is organized as follows. In the next section, the model and its basic properties are presented. In Section 3, we determine stability analysis of the equilibria states. Numerical simualtions are presented in Section 4 and finally the last section concludes the paper.

The model sub-divides the total male prisoner population into the following sub-populations of susceptible intravenous drug users *Sd*(*t*), susceptible non-drug users *Sn*(*t*), intravenous drug using HIV-only infected people not yet showing AIDS symptoms *In*(*t*), non-drug using HIV-only infected people not yet showing AIDS symptoms *Id*(*t*), intravenous drug using AIDS cases *Ad*(*t*) and non-drug using AIDS cases *Ad*(*t*). There is sexual interaction between intravenous drug users and non-drug users making HIV transmission across different these two distinct distinct groups possible. The population is patterns is heterogeneous mixing with

*N*(*t*) = *Nd*(*t*) + *Nn*(*t*), *Nd*(*t*) = *Sd*(*t*) + *Id*(*t*) + *Ad*(*t*), *Nn*(*t*) = *Sn*(*t*) + *In*(*t*) + *An*(*t*), (1)

, *i* = (*d*, *n*). Then *pnn* + *pnd* = *pdd* + *pdn* = 1. The

. (2)

(3)

with *Nn*(*t*) and *Nd*(*t*) being the total number of non-drug using and intravenous drug using male prisoners (intravenous drug users-IDU), respectively. The group *j* members make *cj*, *j* = (*d*, *n*) sexual contacts per unit time, and that a fraction of the contacts made by a member of

total number of sexual contacts per unit time by members of group 'n' (non-drug users) with members of group 'd' (intravenous drug users) is *cn pnd Nn* and because this must be equal to the number of contacts made by members of group 'd' with members of group 'n', we have a

In this case the sexual contact rates (partner acquistion rates) *cd* and *cn* are saturating terms for the total population and the mixing proportions may change with time. It is worth mentioning here that intravenous drug users are more likely to have more sexual partners than the general population. Therefore, *cd* = B*cn*, B ≥ 1. We assume that male prisoners in AIDS stage of the disease are nolonger sexually active as they are nolonger capable of attracting sexual mates among prisoners. Also drug using AIDS patients nolonger share their needles with others as other prisoners do not like sharing needles with someone whose AIDS symptoms are visible. The forces of HIV infection for intravenous drug users and non-drug users in the male prison

> <sup>+</sup> *pdn cdβ<sup>n</sup> In Nn*

+

<sup>+</sup> *pnd cnβ<sup>d</sup> Id Nd*

*cd*<sup>2</sup> *βd*<sup>2</sup> *Id Nd*

,

, respectively

<sup>=</sup> *pdn cd Nn*

*pnd cn Nd*

regard to sexual behaviour. The total population is given by;

*<sup>λ</sup>dh* <sup>=</sup> *pdd cdβ<sup>d</sup> Id Nd*

and *<sup>λ</sup>nh* <sup>=</sup> *pnn cnβ<sup>n</sup> In*

*Nn*

group *j* is with a member of group *i* is *pji*

transmission of HIV (24; 25).

**2. Model description**

balance relation

are:

with *βi*, *i* = (*n*, *d*) is probabbility on individual being infected with HIV by an individual from the n- or d-class per sexual contact; *cj*, *j* = (*n*, *d*) are the number of sexual partners an individual acquires per year (partner acquistion rates); *βd*<sup>2</sup> is the probability an intravenous drug user getting HIV infection through sharing non-sterile needles during drug injections and *cd*<sup>2</sup> are the number drug sharing partners an individual acquires.

It is assumed people are recruited into prison at rate Λ through committing various crimes and the following proportions *π*1, *π*2, *π*<sup>3</sup> and *π*<sup>4</sup> recruited enter the classes *Sn*(*t*), *Sd*(*t*), *In*(*t*) and *Id*(*t*), respectively. We further assume that AIDS cases are too sick to commit a crime, so there are no recruitment of prisoners already in the AIDS stage of disease. Furthermore, it is assumed that intravenous drug using prisoners showing AIDS symptoms are nolonger able to exert peer pressure strong enough to make one become a drug user. Individuals in *Sn*(*t*) and *In*(*t*) acquire drug misusing habits at rate *λd*(*t*) due to peer pressure and move into *Sd*(*t*) and *Id*(*t*, respectively with

$$
\lambda\_{d\_d} = \frac{\beta\_{d\_1} c\_{d\_1} (S\_d + I\_d)}{N\_d},
\tag{4}
$$

where *βd*<sup>1</sup> is the probability of becoming an intravenous drug user (IDU) following contact with an IDU and *cd*<sup>1</sup> are the number of contacts necessary for one to become an IDU (partner acquistion rate). Individuals in *Sn*(*t*) class acquire HIV infection at a rate *λnh* (*t*) to move into *In*(*t*). Individuals in *Sd*(*t*) class acquire HIV infection at a rate *λdh* (*t*) to move *Id*(*t*) class. Individuals infected with HIV-only not yet displaying symptoms (*In*(*t*), *Id*(*t*)) progress to the AIDS stage ((*An*(*t*), *Ad*(*t*)) at a rate *γ*. Individuals in *Ad*(*t*) leave the intravenous drug using habits at a rate *α* to get into *An*(*t*) class. Individuals in all classes experience natural death at a rate *μ* and those in AIDS stage of the disease experience an additional disease induced death at a rate *ν*. Individuals in all classes leave the prison at rate *ω* upon completion of their sentences. Individuals in the AIDS stage of the disease (final terminal stages) are further released from prison due to sickness at rate *φ*. The model flow diagram is shown in Figure 1. Based on these assumptions the following system of differential equations describe the model.

$$\begin{aligned} S'\_{\eta}(t) &= \pi\_1 \Lambda - (\lambda\_d + \lambda\_{\eta\_k}) S\_{\eta} - (\mu + \omega) S\_{\eta \nu} \\\\ I'\_{\eta}(t) &= \pi\_3 \Lambda + \lambda\_{\eta\_k} S\_{\eta} - \lambda\_d I\_{\eta} - (\mu + \omega + \gamma) I\_{\nu \nu} \\\\ A'\_{\eta}(t) &= \gamma I\_{\mathfrak{n}} + \alpha A\_d - (\mu + \omega + \phi + \nu) A\_{\mathfrak{n} \nu} \\\\ S'\_d(t) &= \pi\_2 \Lambda + \lambda\_d S\_{\mathfrak{n}} - \lambda\_{d\_h} S\_d - (\mu + \omega) S\_{d \nu} \\\\ I'\_d(t) &= \pi\_4 \Lambda + \lambda\_{d\_h} S\_d + \lambda\_d I\_{\mathfrak{n}} - (\mu + \omega + \gamma) I\_{d \nu} \\\\ A'\_d(t) &= \gamma I\_d - (\mu + \alpha + \omega + \phi + \nu) A\_{\mathfrak{n}} \end{aligned} \tag{5}$$

#### **2.1 Model basic properties**

In this section, we study the basic results of solutions of model system (5), which are essential in the proofs of stability results.

**3. Disease-free equilibrium and stability analysis**

<sup>E</sup><sup>0</sup> <sup>=</sup> *S*0 *<sup>n</sup>*, *I* 0 *<sup>n</sup>*, *<sup>A</sup>*<sup>0</sup> *<sup>n</sup>*, *<sup>S</sup>*<sup>0</sup> *d*, *I* 0 *<sup>d</sup>* , *<sup>A</sup>*<sup>0</sup> *d* =

+

Watmough (27).

given by

system (5) is given as

<sup>R</sup>*SD* <sup>=</sup> *cd*<sup>2</sup> *<sup>β</sup>d*2+*cd pdd <sup>β</sup><sup>d</sup>*

*cd*<sup>2</sup> *<sup>β</sup>d*2+*cd pdd <sup>β</sup><sup>d</sup>*

R*SD* < 1 *and unstable otherwise.*

driving HIV/AIDS in male prisons. **Increase in intravenous drug users**

**Analysis of the effective reproduction number,** R*SD* The reproduction number is differentiated into categories: **Case 1: No intravenous drug users in the community**

<sup>2</sup>*a*<sup>3</sup> + *cn*

<sup>2</sup>*a*<sup>3</sup> + *cn*

The disease free equilibrium of model system (5), <sup>E</sup><sup>0</sup> is given by

*π*1Λ

*<sup>π</sup>*<sup>1</sup> *pnd <sup>a</sup>*1*cd*<sup>1</sup> *<sup>β</sup>d*<sup>1</sup> *<sup>β</sup><sup>d</sup>*

*<sup>π</sup>*<sup>1</sup> *pnd <sup>a</sup>*1*cd*<sup>1</sup> *<sup>β</sup>d*<sup>1</sup> *<sup>β</sup><sup>d</sup>*

*μ* + *ω* + *βd*<sup>1</sup> *cd*<sup>1</sup>

<sup>2</sup>*a*<sup>3</sup> *<sup>a</sup>*<sup>4</sup> *<sup>a</sup>*<sup>6</sup> <sup>+</sup> *pnn <sup>β</sup><sup>n</sup>*

<sup>2</sup>*a*<sup>3</sup> *<sup>a</sup>*<sup>4</sup> *<sup>a</sup>*<sup>6</sup> <sup>+</sup> *pnn <sup>β</sup><sup>n</sup>*

with *a*<sup>1</sup> = *μ* + *ω*, *a*<sup>2</sup> = *μ* + *ω* + *φ* + *ν*, *a*<sup>3</sup> = *μ* + *ω* + *γ*, *a*<sup>4</sup> = *μ* + *ω* + *γ* + *cd*<sup>1</sup> *βd*<sup>1</sup> ,

Following van den Driessche and Watmough (27), the effective reproduction number of model

HIV/AIDS Transmission Dynamics in Male Prisons 71

2*a*<sup>4</sup> 

<sup>2</sup>

2*a*<sup>4</sup>

*a*<sup>5</sup> = *μ* + *ω* + *φ* + *ν*, *a*<sup>6</sup> = *π*2(*μ* + *ω*) + *cd*<sup>1</sup> *βd*<sup>1</sup> throughout the manuscript. The reproduction number R*SD* is defined as the number of secondary HIV infections produced by one HIV infected individual during his/ her entire infectious period in a mixed population of non-drug users and intravenous drug male prisoners. Theorem 2 follows from van den Driessche and

**Theorem 2.** *The disease free equilibrium* <sup>E</sup><sup>0</sup> *of model system (5) is locally asymptotically stable if*

In this case *β<sup>d</sup>* = *βdc cdc* = *βdcd* = *pnd* = *pdn* = 0, *pnn* = 1 so that R*SD* becomes R0*<sup>S</sup>* which is

<sup>R</sup>0*<sup>S</sup>* <sup>=</sup> *<sup>β</sup>ncn a*3

which is the number of secondary HIV infections produced by one HIV infected individual through homosexual tendencies in a male prison. It is important to note R0*<sup>S</sup>* is a decreasing function of *ω*, suggesting that increasing the number of prisoners leaving the prison reduces the concentration of HIV cases in prison. Theoretically this is feasible, in reality this begs more questions than answers as sentences communicated cannot be reversed because of HIV. Perhaps, it may be necessary to consider the use of open prison systems where prisoners with less serious crimes can serve their sentences while staying at their homes. This has a further advantage of reducing the high levels of raping of man by man in prisons and the homosexual tendencies which male prisoners resort to in enclosed prisons which is one of the major forces

> <sup>R</sup>0*<sup>D</sup>* <sup>=</sup> *<sup>β</sup>dcd* <sup>+</sup> *<sup>β</sup>d*<sup>1</sup> *cd*<sup>1</sup> *a*3

In this case (*ppp* , *βd*<sup>1</sup> *cd*<sup>1</sup> ) → (1, ∞) so that R*SD* becomes R0*<sup>D</sup>* which is given by

, 0, 0,

Λ(*π*2(*μ* + *ω*) + *βd*<sup>1</sup> *cd*<sup>1</sup> ) (*μ* + *ω*)(*μ* + *ω* + *βd*<sup>1</sup> *cd*<sup>1</sup> )

<sup>−</sup> *cn <sup>β</sup>n*(*cd*<sup>2</sup> *pnn <sup>β</sup>d*2+*cdβ<sup>d</sup>* (*pnn pdd*−*pnd pdn* )) *a*<sup>3</sup> *a*<sup>4</sup>

, (9)

. (10)

, 0, 0

. (7)

(8)

Fig. 1. Structure of the model.

**Lemma 1.** *The equations preserve positivity of solutions.*

*Proof.* The vector field given by the right hand side of (5) points inward on the boundary of **R**<sup>6</sup> <sup>+</sup> \ {0}. For example, if *Sn* = 0 then *S*� *<sup>n</sup>* = *π*1Λ ≥ 0. All the other components are similar.

**Lemma 2.** *Each non-negative solution is bounded in L*1*-norm by* max {*N*(0), <sup>Λ</sup>/*μ*}*.*

*Proof.* The norm *L*<sup>1</sup> norm of each non-negative solution is *N* and it satisfies the inequality *N*� ≤ Λ − *μN*. Solutions to the equation *M*� = Λ − *μM* are monotone increasing and bounded by Λ/*μ* if *M*(0) < Λ/*μ*. They are monotone decreasing and bounded above if *M*(0) ≥ Λ/*μ*. Since *N*� ≤ *M*� the claim follows.

**Corollary 1.** *The region*

$$\Phi = \left\{ (\mathbb{S}\_{\mathbb{II}} \, \_{\mathbb{II}} A\_{\mathbb{II}} \, \_{\mathbb{II}} \mathbb{S}\_{d \prime} I\_d \, A\_d) \in \mathbb{R}\_+^6 : N \le \frac{\Lambda}{\mu} \right\}.\tag{6}$$

*is invariant and attracting for system (5).*

**Theorem 1.** *For every non-zero, non-negative initial value, solutions of model system (5) exist for all times*

*Proof.* Local existence of solutions follow from standard arguments since the right hand side of (5) is locally Lipschitz. Global existence follows from the a-priori bounds.

4 Will-be-set-by-IN-TECH

*Proof.* The vector field given by the right hand side of (5) points inward on the boundary of

*Proof.* The norm *L*<sup>1</sup> norm of each non-negative solution is *N* and it satisfies the inequality *N*� ≤ Λ − *μN*. Solutions to the equation *M*� = Λ − *μM* are monotone increasing and bounded by Λ/*μ* if *M*(0) < Λ/*μ*. They are monotone decreasing and bounded above if *M*(0) ≥ Λ/*μ*.

(*Sn*, *In*, *An*, *Sd*, *Id*, *Ad*) <sup>∈</sup> **<sup>R</sup>**<sup>6</sup>

**Theorem 1.** *For every non-zero, non-negative initial value, solutions of model system (5) exist for all*

*Proof.* Local existence of solutions follow from standard arguments since the right hand side

of (5) is locally Lipschitz. Global existence follows from the a-priori bounds.

**Lemma 2.** *Each non-negative solution is bounded in L*1*-norm by* max {*N*(0), <sup>Λ</sup>/*μ*}*.*

*<sup>n</sup>* = *π*1Λ ≥ 0. All the other components are similar.

<sup>+</sup> : *N* ≤

Λ *μ* 

. (6)

Fig. 1. Structure of the model.

**R**<sup>6</sup>

*times*

**Lemma 1.** *The equations preserve positivity of solutions.*

Φ = 

<sup>+</sup> \ {0}. For example, if *Sn* = 0 then *S*�

Since *N*� ≤ *M*� the claim follows.

*is invariant and attracting for system (5).*

**Corollary 1.** *The region*

#### **3. Disease-free equilibrium and stability analysis**

The disease free equilibrium of model system (5), <sup>E</sup><sup>0</sup> is given by

$$\mathcal{L}^{0} = \left( S\_{\nu\nu}^{0} I\_{\nu\nu}^{0} A\_{\nu\nu}^{0} S\_{d\prime}^{0} I\_{d\prime}^{0} A\_{d}^{0} \right) = \left( \frac{\pi\_{1} \Lambda}{\mu + \omega + \beta\_{d1} \mathfrak{c}\_{d\_{1}}}, 0, 0, \frac{\Lambda (\pi\_{2}(\mu + \omega) + \beta\_{d\_{1}} \mathfrak{c}\_{d\_{1}})}{(\mu + \omega)(\mu + \omega + \beta\_{d\_{1}} \mathfrak{c}\_{d\_{1}})}, 0, 0 \right) . \tag{7}$$

Following van den Driessche and Watmough (27), the effective reproduction number of model system (5) is given as

$$\begin{split} \mathcal{R}\_{SD} &= \frac{c\_{\sharp f}\delta\_{\sharp} + c\_{\sharp}\mu\_{\sharp}\delta\_{\sharp}}{2a\_{3}} + c\_{\mathrm{n}} \left( \frac{\pi\_{1}p\_{u\sharp}a\_{1}c\_{\sharp d\_{1}}\mathfrak{f}\_{d\_{1}}\mathfrak{f}\_{d}}{2a\_{3}a\_{4}a\_{6}} + \frac{p\_{u\sharp}\mathfrak{f}\_{u}}{2a\_{4}} \right) \\ &+ \sqrt{\left( \frac{c\_{\sharp}\mathfrak{f}\_{2}\mathfrak{f}\_{d\_{2}} + c\_{\sharp}p\_{d\_{4}}\mathfrak{f}\_{d}}{2a\_{3}} + c\_{\mathrm{n}} \left( \frac{\pi\_{1}p\_{u\sharp}a\_{1}c\_{\sharp d\_{1}}\mathfrak{f}\_{d\_{1}}\mathfrak{f}\_{d}}{2a\_{3}a\_{4}a\_{6}} + \frac{p\_{u\sharp}\mathfrak{f}\_{u}}{2a\_{4}} \right) \right)^{2} - \frac{c\_{\sharp}\mathfrak{f}\_{\sharp}\left(c\_{\sharp}\mathfrak{f}\_{d}p\_{u\sharp}\mathfrak{f}\_{d} + c\_{\sharp}\mathfrak{f}\_{d}(p\_{u\sharp}p\_{d\_{4}} - p\_{u\sharp}p\_{d\_{4}})\right)}{a\_{3}a\_{4}}} \end{split} \tag{8}$$

with *a*<sup>1</sup> = *μ* + *ω*, *a*<sup>2</sup> = *μ* + *ω* + *φ* + *ν*, *a*<sup>3</sup> = *μ* + *ω* + *γ*, *a*<sup>4</sup> = *μ* + *ω* + *γ* + *cd*<sup>1</sup> *βd*<sup>1</sup> , *a*<sup>5</sup> = *μ* + *ω* + *φ* + *ν*, *a*<sup>6</sup> = *π*2(*μ* + *ω*) + *cd*<sup>1</sup> *βd*<sup>1</sup> throughout the manuscript. The reproduction number R*SD* is defined as the number of secondary HIV infections produced by one HIV infected individual during his/ her entire infectious period in a mixed population of non-drug users and intravenous drug male prisoners. Theorem 2 follows from van den Driessche and Watmough (27).

**Theorem 2.** *The disease free equilibrium* <sup>E</sup><sup>0</sup> *of model system (5) is locally asymptotically stable if* R*SD* < 1 *and unstable otherwise.*
