Assessing the impact of homelessness on HIV/AIDS transmission dynamics

Figures & data

Figure 1. Structure of the model.

Table 1. Effects of lack of entertainment, poor nutrition, co-infections with other STIs, and reduced socio-economic status on HIV/ AIDS transmission dynamics. In each case, all parameters are equalized between the homeless and non-homeless, except those under the heading “Conditions”.

Case	Description	Conditions	${\mathcal{R}}_{0_{H_2}}-{\mathcal{R}}_{0_{H_1}}$
1	Lack of entertainment	$c_2=b_1{c}_1,b_1>1$	${\displaystyle \frac{(b_1-1)c_1{\mathit{\beta }}_1({\mathit{\rho }}_1+{\mathit{\varphi }}_1{g}_4)}{g_4{g}_7}>0}$
2	Poor nutrition	${\mathit{\rho }}_2=b_2{\mathit{\rho }}_1$, $b_2>1$	${\displaystyle \frac{c_1{\mathit{\beta }}_1(b_2-1)(\mathit{\mu }-{\mathit{\varphi }}_1{g}_4)}{g_4{g}_5{g}_7}>0,0<{\mathit{\varphi }}_1<\frac{\mathit{\mu }}{g_4}}$
3	Co-infections with other STIs	${\mathit{\beta }}_2=b_3{\mathit{\beta }}_1$, $b_3\ge 1$	${\displaystyle \frac{(b_3-1)c_1{\mathit{\beta }}_1({\mathit{\rho }}_1+{\mathit{\varphi }}_1{g}_4)}{g_4{g}_7}>0}$
4	Reduced socio-economic status	${\mathit{\nu }}_2=b_4{\mathit{\nu }}_1$, $b_4\ge 1$	${\displaystyle \frac{(b_4-1)c_1{\mathit{\beta }}_1{\mathit{\nu }}_1({\mathit{\varphi }}_1{\mathit{\nu }}_1-{\mathit{\rho }}_1)}{g_4{g}_6{g}_7}>0,\frac{{\mathit{\rho }}_1}{{\mathit{\nu }}_1}\le {\mathit{\varphi }}_1<1}$

Figure 2. Effects of lack of entertainment or co-infections with other STIs (solid line), poor nutrition (dashed line), and reduced socio-economic status (dotted line) noting that $c_2=b_1{c}_1,{\mathit{\rho }}_2=b_2{\mathit{\rho }}_1,{\mathit{\beta }}_2=b_3{\mathit{\beta }}_1$, and ${\mathit{\nu }}_2=b_4{\mathit{\nu }}_1$. Parameter values are as in Table 2.

Table 2. Default (baseline) model parameters used in the analysis and simulations. Here, ZIMSTAT stands for Zimbabwe National Statistics Agency (2012).

Definition	Symbol	Value (Range)	Source
Recruitment rate	$\mathrm{\Lambda }$	0.029 yr${}^{-1}$$\times 4.2\times {10}^6$	ZIMSTAT
Natural mortality rate	$\mathit{\mu }$	0.02 yr${}^{-1}$	ZIMSTAT
Modification parameters	${\mathit{\varphi }}_1,{\mathit{\varphi }}_2,\mathit{\theta }$	0.125 (0.1-1)	Bhunu and Mushayabasa (2012)
Rate of being put on treatment	${\mathit{\alpha }}_i,{\mathit{\alpha }}_a$	0.33(0.01-1) yr${}^{-1}$	Bhunu et al. (2009)
Product of effective contact rate
for HIV transmission and probability	${\mathit{\beta }}_i{c}_j(i,j=1,2)$	0.125(0.011-0.95) yr${}^{-1}$	Hyman, Li, and Stanley (1999)
of HIV transmission per sexual contact
Rate of progression to AIDS	${\mathit{\rho }}_1,{\mathit{\rho }}_{1_t}$	0.1(0.075?0.95) yr${}^{-1}$	Bhunu et al. (2009)
AIDS-related death (non-homeless)	${\mathit{\nu }}_1$	0.333 (0.3?0.75) yr${}^{-1}$	Bhunu et al. (2009)
Homegeneous mixing	$p_{11},p_{22}$	0.67	Assume
Heterogeneous mixing	$p_{12},p_{21}$	0.33	Assume
Modification parameter	$b_k(k=1,\cdots ,4)$	1.125($\ge$1)	Assume

Figure 3. Simulations showing the possible effects of antiretroviral therapy using parameter values in Table 2. The direction of the arrow shows an increase in the levels of treatment.

Figure 4. Simulations showing the possible effects of lacking entertainment using parameter values in Table 2 in the presence of antiretroviral therapy. The direction of the arrow shows a decrease in the levels of entertainment.

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Hyman, J. M., Li, J., & Stanley, E. A. (1999). The differential infectivity and staged progression models for the transmission of HIV. Mathematical Biosciences, 155, 77–109.

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