**Analysis of the effective reproduction number,** R*SD*

The reproduction number is differentiated into categories:

#### **Case 1: No intravenous drug users in the community**

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 given by

$$\mathcal{R}\_{0\_{\rm S}} = \frac{\beta\_{\rm n} c\_{\rm n}}{a\_{\rm 3}},$$

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 driving HIV/AIDS in male prisons.

#### **Increase in intravenous drug users**

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

$$\mathcal{R}\_{\mathbf{0}\_{\mathrm{D}}} = \frac{\beta\_d \mathbf{c}\_d + \beta\_{d\_1} \mathbf{c}\_{d\_1}}{a\_{\mathbf{3}}}.\tag{10}$$

(*a*) (*b*)

**<sup>0</sup> <sup>5</sup> <sup>10</sup> <sup>15</sup> <sup>20</sup> <sup>0</sup>**

**Time (years)**

**<sup>0</sup> <sup>5</sup> <sup>10</sup> <sup>15</sup> <sup>20</sup> <sup>0</sup>**

**Time (years)**

**50**

**An(t)**

**100**

**150**

**In(t)**

**200**

**250**

**300**

are in Table 1.

(*a*)

(*a*)

are in Table 1.

Fig. 2. Effects of varying the HIV infected-only initial conditions when R*SD* > 1. Parameter values used

HIV/AIDS Transmission Dynamics in Male Prisons 73

(*b*)

(*b*)

Fig. 3. Simulations showing the effects of varying the rate at which non-drug using prisoners become intravenous drug users on the population on drug using HIV-infected prisoners. Parameter values used

**<sup>0</sup> <sup>5</sup> <sup>10</sup> <sup>15</sup> <sup>20</sup> <sup>0</sup>**

**Time (years)**

**<sup>0</sup> <sup>5</sup> <sup>10</sup> <sup>15</sup> <sup>20</sup> <sup>0</sup>**

**Time (years)**

**Ad(t)**

**Id(t)**

R0*<sup>D</sup>* just like is R0*<sup>S</sup>* is a decreasing function of *ω*, meaning that use of open prison systems will be beneficial in the control of HIV among male prisoners. It is important to note that levels of sexual contact are higher among intravenous drug users than non-drug users, with it increased risk of contracting HIV, so *cnβ<sup>n</sup>* < *cdβ<sup>d</sup>* and this translates R0*<sup>S</sup>* < R0*<sup>D</sup>* . This suggest that intravenous drug use enhances HIV transmission in male prisons. Drug using prisoners are at an increased risk of HIV infection than their non-drug using counterparts. May be introducing drug substitution treatment together with introducing needle free exchange programmes will reduce the epidemic in prisons. A reduction in needle in sharing among prisoners result in HIV/AIDS prevalence as the sharing of unsterile needles in a major source of HIV transmission among male prisoners.

#### **4. Numerical simulations**

In this section, we carry out detailed numerical simulations using MatLab programming language to assess the effects of HIV transmission among male prisoners in the absence any interventional strategy which is more common in developing countries in Africa for different initial conditions. The parameter values that we use for numerical simulations are in Table 1. In Table 1, NPA denotes National Prison Administration (Zimbabwe). For influence of


Table 1. Model parameters and their interpretations.

peer pressure forces influencing one to become an IDU, we used values adapted from Bhunu et al. (4) which are peer pressure forces necessary for one to start smoking, for the sake of illustration. For influence of peer pressure forces influencing one to become an IDU, we used values adapted from Bhunu et al. (2010) (4) which are peer pressure forces necessary for one to start smoking.

Figure 2 is a graphical representation showing the effect of varying initial conditions when R*SD* > 1. In Figures 2(a) and 2(b) show the effects of varying the HIV-infected not yet showing sysmptoms on HIV-infected only and AIDS, respectively. Both graphs show a higher number of HIV-only and AIDS among intravenous drug using male prisoners than non-drug users. This tends to show intravenous drug using male prisoners are at increased risk of HIV infection due to sharing of unsterile needles and increased rates of homosexual sex habits.

6 Will-be-set-by-IN-TECH

R0*<sup>D</sup>* just like is R0*<sup>S</sup>* is a decreasing function of *ω*, meaning that use of open prison systems will be beneficial in the control of HIV among male prisoners. It is important to note that levels of sexual contact are higher among intravenous drug users than non-drug users, with it increased risk of contracting HIV, so *cnβ<sup>n</sup>* < *cdβ<sup>d</sup>* and this translates R0*<sup>S</sup>* < R0*<sup>D</sup>* . This suggest that intravenous drug use enhances HIV transmission in male prisons. Drug using prisoners are at an increased risk of HIV infection than their non-drug using counterparts. May be introducing drug substitution treatment together with introducing needle free exchange programmes will reduce the epidemic in prisons. A reduction in needle in sharing among prisoners result in HIV/AIDS prevalence as the sharing of unsterile needles in a major source

In this section, we carry out detailed numerical simulations using MatLab programming language to assess the effects of HIV transmission among male prisoners in the absence any interventional strategy which is more common in developing countries in Africa for different initial conditions. The parameter values that we use for numerical simulations are in Table 1. In Table 1, NPA denotes National Prison Administration (Zimbabwe). For influence of

Parameter Symbol Value Source Recruitment rate <sup>Λ</sup> 0.00163yr−<sup>1</sup> <sup>∗</sup> 3000000 NPA Natural mortality rate *μ* 0.02yr−<sup>1</sup> (5) Natural rate of progression to AIDS *γ* 0.1yr−<sup>1</sup> (5) Rate of leaving prison due to AIDS related sickness *φ* 0.25 Assumed AIDS related death rate *ν* 0.4yr−<sup>1</sup> (5)

per drug injection *cd*<sup>2</sup> *<sup>β</sup>d*<sup>2</sup> 0.562yr−<sup>1</sup> (16) Probability of HIV transmission per sexual contact *βn*, *β<sup>d</sup>* 0.125 (0.01-0.95)yr−<sup>1</sup> (5) Sexual acquistion rate *cn*, *cd* 3yr−<sup>1</sup> (5)

drug misuser per contact with a misuser *cd*<sup>1</sup> *<sup>β</sup>d*<sup>1</sup> 0.4yr−<sup>1</sup> (4) Rate of quitting drug misuse of sickness *α* 0.3yr−<sup>1</sup> (4) Rate of release from prison *ω* 0.25yr−<sup>1</sup> Assumed Proportion recruited into *Sn*, *Sd*, *In*, *Id* classes *π*1, *π*2, *π*3, *π*<sup>4</sup> 0.375,0.375,0.125,0.125 Assumed

peer pressure forces influencing one to become an IDU, we used values adapted from Bhunu et al. (4) which are peer pressure forces necessary for one to start smoking, for the sake of illustration. For influence of peer pressure forces influencing one to become an IDU, we used values adapted from Bhunu et al. (2010) (4) which are peer pressure forces necessary for one

Figure 2 is a graphical representation showing the effect of varying initial conditions when R*SD* > 1. In Figures 2(a) and 2(b) show the effects of varying the HIV-infected not yet showing sysmptoms on HIV-infected only and AIDS, respectively. Both graphs show a higher number of HIV-only and AIDS among intravenous drug using male prisoners than non-drug users. This tends to show intravenous drug using male prisoners are at increased risk of HIV infection due to sharing of unsterile needles and increased rates of homosexual sex habits.

of HIV transmission among male prisoners.

**4. Numerical simulations**

Product of effective contact rate for HIV infection and probability of HIV transmission

Product of effective contact rate for becoming a drug user and probability of becoming a

to start smoking.

Table 1. Model parameters and their interpretations.

Fig. 2. Effects of varying the HIV infected-only initial conditions when R*SD* > 1. Parameter values used are in Table 1.

Fig. 3. Simulations showing the effects of varying the rate at which non-drug using prisoners become intravenous drug users on the population on drug using HIV-infected prisoners. Parameter values used are in Table 1.

[4] Bhunu CP, Mushayabasa S, Tchuenche JM. A theoretical assessment of the effects of smoking on the transmission dynamics of tuberculosis. *Bulletin of Mathematical Biology* DOI 10.1007/s11538-010-9568-6, 2010(a). Available at

HIV/AIDS Transmission Dynamics in Male Prisons 75

[5] Bhunu CP, Garira W, Mukandavire Z. Modelling HIV/AIDS and tuberculosis

[6] Brauer F, C. Castillo-Chavez. Mathematical Models in Population Biology and

[7] Buavirat A, Page-Shafer K, van Griensven GJP, Mandel JS, Evans J, Chuaratanaphong J, Chiamwongpat S, Sacks R, Moss A. Risk of prevalent HIV infection associated with incarceration among injecting drug users in Bangkok, Thailand: case-control study. *BMJ*

[8] Centers for Disease Control and Prevention (CDC). HIV transmission among male inmates in a state prison system-Georgia, 1992-2005. *MMWR Morb Mortal Wkly Rep*

[9] Centers for Disease Control and Prevention (CDC). National Center for HIV/AIDS, Viral Hepatitis, STD, and TB Prevention. Prevention and control of tuberculosis in correctional and detention facilities: recommendations from CDC, endorsed by the Advisory Council for the Elimination of Tuberculosis, the National Commission on Correctional Health Care, and the American Correctional Association. *MMWR Recomm Rep* 55(RR-9): 1–44,

[10] Centers for Disease Control and Prevention. Tuberculosis outbreaks in prison housing units for HIV-infected inmatesUCalifornia, 1995–1996. ˚ *MMWR Morb Mortal Wkly Rep*

[11] Crofts N, Stewart T, Hearne P, Ping XY, Breschkin AM, Locarnini SA. Spread of bloodborne viruses among Australian prison entrants. *BMJ* 310(6975): 285–288, 1995.

[13] Dolan J, Kite B, Aceijas C, Stimson GV. HIV in prison in low income and middle-income

[14] Dolan KA, Wodak A. HIV transmission in a prison system in an Australian state. *Med J*

[15] Flanigan TP, Rich JD, Spaulding A. HIV care among incarcerated persons: a missed

[16] Green D, Al-Fwzan W. An improved optimistic three-stage model for the spread of HIV amongst injecting intravenous drug users. *J Discrete Contin Dyn Sys* (Supplement 2009):

[18] Human Rights Watch. World Report; Special Programs and CampaignsUPrisons, 2002. ˚

[20] International Centre for Prison Studies. The World Female Imprisonment List. KingŠs

[21] International Centre for Prison Studies. The World Prison Population List. KingŠs

[22] Martin RE, Gold F, Murphy W, Remple V, Berkowitz J, Money D. Drug use and risk of bloodborne infections: a survey of female prisoners in British Columbia. *Can J Public*

[12] Doll D. Tattooing in prison and HIV infection. *The Lancet* 2(9): 66-67, 1988.

countries. *Lancet Infectious Diseases* 7: 32-43, 2007.

[17] Human Rights Watch. HIV/AIDS in prisons, 2006.

[19] Human Rights Watch. No Escape: Male Rape in USA Prisons, 1991.

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Figure 3 shows the effects of intravenous drug use on HIV/AIDS transmission dynamics among male prisoners is illustrated by varying the rate a male prisoner becoming an intravenous drug user while in prison. It shows an increase in the number of HIV/AIDS cases among drug users cases with increase in drug use while the opposite will be happening among non-drug users. This suggest that effective control of HIV among male prisoners to some extent on the control of intravenous drug use.

#### **5. Discussion**

A mathematical model have been presented in attempt to understand the transmission dynamics of HIV/AIDS among male prisoners. Male prisoners are infected with HIV while in prison through intravenous drug use using unsterile needles (syringes) and homosexual tendencies. Intravenous drug use in male prisons act in two way: (i) sharing of unsterile needles/ syringes enhance the transmission of HIV; (ii) flashing blood that drawing is blood from someone who would have injected himself with a drug and inject it into one self which on its own exposes the injector to the HIV infection. Also intravenous drug using prisoners are more likely to engage in homosexual relations with other male prisoners and with it increased risky of HIV transmission. Analysis of the reproduction number have shown that (i) a reduction in drug use results in a decrease of HIV/ AIDS prevalence among male prisoners, (ii) release of prisoners may also act in reducing the concentration HIV/AIDS cases in prisons. The later fact is not feasible, but perhaps implementing opening prison systems where prisoners of less serious crimes are allowed to save their crimes while staying with their families enables male prisoners to cope up with stressful prison conditions. Open prison systems will reduce the influence of peer pressure among prisoners as they will have moral and pyschological support from the family which does not exist in enclosed prison systems. Numerical simulations carried also support the analytic results that increase in drug use and tattooing increases HIV/ AIDS prevalence among male prisoners. The result of this study have 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 (8) and this might be the case world wide. HIV infected men in prison pose a risk to their communities upon release from prison, especially in Africa where partners in marriage rarely discuss safe sex so in the absence open prison systems, it may be best to have mandatory HIV/ AIDS screening and specific educational programmes for prisoners. This will reduce the prevalence of high-risk behaviours and lower HIV transmission in male prisons, thus reducing post-release 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 as a way of keeping in check HIV transmission in men prisons. Additionally provision of condoms might also help given the high levels of homosexuality in male prisons.

#### **6. References**


8 Will-be-set-by-IN-TECH

Figure 3 shows the effects of intravenous drug use on HIV/AIDS transmission dynamics among male prisoners is illustrated by varying the rate a male prisoner becoming an intravenous drug user while in prison. It shows an increase in the number of HIV/AIDS cases among drug users cases with increase in drug use while the opposite will be happening among non-drug users. This suggest that effective control of HIV among male prisoners to

A mathematical model have been presented in attempt to understand the transmission dynamics of HIV/AIDS among male prisoners. Male prisoners are infected with HIV while in prison through intravenous drug use using unsterile needles (syringes) and homosexual tendencies. Intravenous drug use in male prisons act in two way: (i) sharing of unsterile needles/ syringes enhance the transmission of HIV; (ii) flashing blood that drawing is blood from someone who would have injected himself with a drug and inject it into one self which on its own exposes the injector to the HIV infection. Also intravenous drug using prisoners are more likely to engage in homosexual relations with other male prisoners and with it increased risky of HIV transmission. Analysis of the reproduction number have shown that (i) a reduction in drug use results in a decrease of HIV/ AIDS prevalence among male prisoners, (ii) release of prisoners may also act in reducing the concentration HIV/AIDS cases in prisons. The later fact is not feasible, but perhaps implementing opening prison systems where prisoners of less serious crimes are allowed to save their crimes while staying with their families enables male prisoners to cope up with stressful prison conditions. Open prison systems will reduce the influence of peer pressure among prisoners as they will have moral and pyschological support from the family which does not exist in enclosed prison systems. Numerical simulations carried also support the analytic results that increase in drug use and tattooing increases HIV/ AIDS prevalence among male prisoners. The result of this study have 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 (8) and this might be the case world wide. HIV infected men in prison pose a risk to their communities upon release from prison, especially in Africa where partners in marriage rarely discuss safe sex so in the absence open prison systems, it may be best to have mandatory HIV/ AIDS screening and specific educational programmes for prisoners. This will reduce the prevalence of high-risk behaviours and lower HIV transmission in male prisons, thus reducing post-release 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 as a way of keeping in check HIV transmission in men prisons. Additionally provision of condoms might also help given the

[1] Amnesty International. Abuse of Women in Custody: Sexual Misconduct and Shackling

[2] Anderson, May RM. Infectious Diseases of Humans, Dynamics and Control, Oxford

[3] Bailey N, The Mathematical Theory of Infectious Diseases, Charles Griffin, 1975.

some extent on the control of intravenous drug use.

high levels of homosexuality in male prisons.

of Pregnant Women, 2001.

University Press, 1991.

**6. References**

**5. Discussion**


**Part 3** 

**Prevention and Treatment of** 

**AIDS-Related Diseases** 

