## Results: 5

1.

2.

Black crosses indicate the location of the 17 HCFs that have been designated for ARV rollout by the South African government, and the spatial distribution of communities distinguished by the number of individuals infected with HIV (by both size and color). Durban (represented by the large red diamond) is the capital city of the province and has more individuals with HIV than any other community. Pietermaritzburg and Newcastle (represented by orange diamonds) have the next greatest numbers of individuals with HIV.

3.

Box plots of the percentage of infected people that obtain treatment per community for the three different sets of HCFs simulated in our analysis for ARV rollout, namely, using the 17 HCFs likely to be used, the 27 HCFs suggested by the South African government as potential implementation points, and all of the 54 hospitals in the KwaZulu–Natal province. These cases are represented for each of the three catchment region sizes we considered (with radii of 20 km, 40 km, or 60 km) and referenced against the ideal fraction treated (dotted blue line) under perfect conditions of egalitarian distribution, given the limited ARV supply. The red crosses indicate the median percentage of people with HIV that obtain treatment per community.

4.

The three strategies considered are as follows: allocation of ARVs according to the results of minimizing our objective function (first row) allocation of ARVs only to one HCF in Durban (second row), allocation of ARVs equally to each of the 17 HCFs (third row). The proportion of ARVs allocated by these strategies to the 17 different HCFs is indicated in (A), (D), and (G), with each HCF represented by a different color. The spatial allocation of ARVs is shown in (B), (E), and (H), respectively. The respective percentage of infected people that are treated throughout the KwaZulu–Natal province is simulated in (C), (F), and (I). Here, the

*x–y*plane represents spatial location, and the shaded color at a location refers to the proportion of individuals with HIV that are treated at the specified location. The plots were obtained by generating an interpolating surface where the*z*-ordinate, colored by magnitude, represents the proportion of treated individuals, and then orientating the view of the surface normal to the*x–y*plane. We performed surface data interpolation using the method of translates [32].5.

(A) A histogram indicating heterogeneity in the distance from communities in KwaZulu–Natal to the closest HCF. The treatment accessibility function used in our model is a Gaussian distribution, exp(−

(B) The catchment region is shown with an effective radius of 20 km for coverage from each HCF (

(C) The catchment region is shown with an effective radius of 40 km for coverage from each HCF (

(D) The catchment region is shown with an effective radius of 60 km for coverage from each HCF (

In each case, the red dots indicate the location of the HCF, the green circles represent the locations where treatment accessibility has been reduced to 50% relative to someone located at the HCF, and the blue circles represent the locations where treatment accessibility has been reduced to 1% relative to someone located at the HCF. The locations of communities are presented as black diamonds. The large black diamonds denote large communities (with population greater than 10,000 people), and the small black diamonds denote small communities (with population less than 10,000 people). Substantially more area of the province is covered if HCFs have catchment regions of 60-km radius, relative to catchment regions of 40-km radius, and substantially less area of the province is covered if HCFs have a catchment region of only 20-km radius. However, the proportion of people with access does not differ greatly between the different catchment sizes because of the great spatial heterogeneity in the prevalence of people with HIV.

*kd*^{2}), indicating that accessibility is strongly related to distance*(d),*and*k*is a dispersal length scale parameter.(B) The catchment region is shown with an effective radius of 20 km for coverage from each HCF (

*k =*0.0151).(C) The catchment region is shown with an effective radius of 40 km for coverage from each HCF (

*k***=**0.003786).(D) The catchment region is shown with an effective radius of 60 km for coverage from each HCF (

*k*= 0.00168).In each case, the red dots indicate the location of the HCF, the green circles represent the locations where treatment accessibility has been reduced to 50% relative to someone located at the HCF, and the blue circles represent the locations where treatment accessibility has been reduced to 1% relative to someone located at the HCF. The locations of communities are presented as black diamonds. The large black diamonds denote large communities (with population greater than 10,000 people), and the small black diamonds denote small communities (with population less than 10,000 people). Substantially more area of the province is covered if HCFs have catchment regions of 60-km radius, relative to catchment regions of 40-km radius, and substantially less area of the province is covered if HCFs have a catchment region of only 20-km radius. However, the proportion of people with access does not differ greatly between the different catchment sizes because of the great spatial heterogeneity in the prevalence of people with HIV.