# American Institute of Mathematical Sciences

February  2018, 15(1): 255-273. doi: 10.3934/mbe.2018011

## Three-level global resource allocation model for HIV control: A hierarchical decision system approach

 1 Department of Mathematics and Statistical Sciences, Botswana International University of Science and Technology (BIUST), P/Bag 16, Palapye, Botswana 2 Department of Mathematics, Addis Ababa University, P.O.Box 1176, Addis Ababa, Ethiopia

Received  August 24, 2016 Revised  March 03, 2017 Published  May 2017

Funds from various global organizations, such as, The Global Fund, The World Bank, etc. are not directly distributed to the targeted risk groups. Especially in the so-called third-world-countries, the major part of the fund in HIV prevention programs comes from these global funding organizations. The allocations of these funds usually pass through several levels of decision making bodies that have their own specific parameters to control and specific objectives to achieve. However, these decisions are made mostly in a heuristic manner and this may lead to a non-optimal allocation of the scarce resources. In this paper, a hierarchical mathematical optimization model is proposed to solve such a problem. Combining existing epidemiological models with the kind of interventions being on practice, a 3-level hierarchical decision making model in optimally allocating such resources has been developed and analyzed. When the impact of antiretroviral therapy (ART) is included in the model, it has been shown that the objective function of the lower level decision making structure is a non-convex minimization problem in the allocation variables even if all the production functions for the intervention programs are assumed to be linear.

Citation: Semu Mitiku Kassa. Three-level global resource allocation model for HIV control: A hierarchical decision system approach. Mathematical Biosciences & Engineering, 2018, 15 (1) : 255-273. doi: 10.3934/mbe.2018011
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].">Figure 1.  Global financing for the fight of HIV from Donor Governments: Commitments & Disbursements, 2002-2013. Taken from [15].
Schematic diagram for global resource allocations.
Model diagram that show flow of individuals between the compartments.
Prevalence graphs for Countries in Region 1. The figures to the left indicate the prevalence for Low Risk population groups while those in the right indicate the prevalence for High Risk population groups with in the same country. The broken lines indicate the prevalence if the optimal resource is invested in the planned 10 years period for each of the community groups
Prevalence graphs for Countries in Region 2. The broken lines indicate the prevalence if the optimal resource is invested in the planned 10 years period for each of the community groups
Prevalence graphs for Countries in Region 3. The broken lines indicate the prevalence if the optimal resource is invested in the planned 10 years period for each of the community groups
Disease parameters - Assumed to be Constant across regions and countries
 Parameter value Description $\delta_V = 0.15$ mortality rate for infected children $\delta_A = 0.2$ additional death rate due to AIDS $\sigma = 0.1$ rate of progression to AIDS, if not treated $\gamma = 1.3$ preferential rate of recruitment for children to receive ART $\beta = 0.12$ transmission probability $\epsilon_1 = 0.08$ factor of reduction on rate of disease transmission due to ART $\epsilon_2 = 1/6$ factor of reduction on rate of MTCT due to treatment
 Parameter value Description $\delta_V = 0.15$ mortality rate for infected children $\delta_A = 0.2$ additional death rate due to AIDS $\sigma = 0.1$ rate of progression to AIDS, if not treated $\gamma = 1.3$ preferential rate of recruitment for children to receive ART $\beta = 0.12$ transmission probability $\epsilon_1 = 0.08$ factor of reduction on rate of disease transmission due to ART $\epsilon_2 = 1/6$ factor of reduction on rate of MTCT due to treatment
Parameters that are assumed to vary from region to region or from risk groups to risk groups
 Parameter values Description (each is for the 6 countries) $\pi = [0.032, 0.036, 0.028, 0.021, 0.015, 0.012]$ Birth rates for the six countries, respectively $\mu = [0.029, 0.025, 0.025, 0.019, 0.013, 0.010]$ Death rates for the six countries, respectively $\lambda_H = [0.16, 0.31, 0.13, 0.12, 0.12, 0.11]$ Initial unsafe contact rates for high risk groups $\lambda_L = [0.11, 0.14, 0.09, 0.075, 0.07, 0.05]$ Initial unsafe contact rates for low risk groups $m_H = [0.31, 0.37, 0.28, 0.25, 0.22, 0.20]$ Initial rate of MTCT for high risk groups $m_L = [0.19, 0.22, 0.15, 0.14, 0.12, 0.11]$ Initial rate of MTCT for low risk groups $\alpha_A = [0.15, 0.32, 0.11, 0.12, 0.11, 0.10]$ Initial rate of defaulting in the use of ART-assumed to be the same for both risk groups in each of the countries $\rho_H = [0.15, 0.15, 0.20, 0.21, 0.23, 0.24]$ Initial rate of recruitment for ART in High risk groups $\rho_L = [0.075, 0.075, 0.10, 0.11, 0.15, 0.15]$ Initial rate of recruitment for ART in Low risk groups
 Parameter values Description (each is for the 6 countries) $\pi = [0.032, 0.036, 0.028, 0.021, 0.015, 0.012]$ Birth rates for the six countries, respectively $\mu = [0.029, 0.025, 0.025, 0.019, 0.013, 0.010]$ Death rates for the six countries, respectively $\lambda_H = [0.16, 0.31, 0.13, 0.12, 0.12, 0.11]$ Initial unsafe contact rates for high risk groups $\lambda_L = [0.11, 0.14, 0.09, 0.075, 0.07, 0.05]$ Initial unsafe contact rates for low risk groups $m_H = [0.31, 0.37, 0.28, 0.25, 0.22, 0.20]$ Initial rate of MTCT for high risk groups $m_L = [0.19, 0.22, 0.15, 0.14, 0.12, 0.11]$ Initial rate of MTCT for low risk groups $\alpha_A = [0.15, 0.32, 0.11, 0.12, 0.11, 0.10]$ Initial rate of defaulting in the use of ART-assumed to be the same for both risk groups in each of the countries $\rho_H = [0.15, 0.15, 0.20, 0.21, 0.23, 0.24]$ Initial rate of recruitment for ART in High risk groups $\rho_L = [0.075, 0.075, 0.10, 0.11, 0.15, 0.15]$ Initial rate of recruitment for ART in Low risk groups
 Pop. Size R1C1 R1C2 R2C1 R2C2 R3C1 R3C2 S(0) 15652504 870885.6 35863200 7341600 28122000 16815000 , V(0) 87640 12854.4 652800 151200 548250 285000 I(0) 1016624 107120 1632000 361200 1322250 627000 T(0) 438200 47132.8 1836000 411600 1451250 855000 A(0) 333032 33207.2 816000 134400 806250 418000
 Pop. Size R1C1 R1C2 R2C1 R2C2 R3C1 R3C2 S(0) 15652504 870885.6 35863200 7341600 28122000 16815000 , V(0) 87640 12854.4 652800 151200 548250 285000 I(0) 1016624 107120 1632000 361200 1322250 627000 T(0) 438200 47132.8 1836000 411600 1451250 855000 A(0) 333032 33207.2 816000 134400 806250 418000
 Pop. Size R1C1 R1C2 R2C1 R2C2 R3C1 R3C2 S(0) 11706200 611078.4 13785600 2656800 7998000 4512000 V(0) 110176 25708.8 499200 57600 193500 90000 I(0) 1170620 208636.8 2515200 417600 1236250 666000 T(0) 454476 84048 1632000 342000 1053500 600000 A(0) 330528 59328 768000 126000 268750 132000
 Pop. Size R1C1 R1C2 R2C1 R2C2 R3C1 R3C2 S(0) 11706200 611078.4 13785600 2656800 7998000 4512000 V(0) 110176 25708.8 499200 57600 193500 90000 I(0) 1170620 208636.8 2515200 417600 1236250 666000 T(0) 454476 84048 1632000 342000 1053500 600000 A(0) 330528 59328 768000 126000 268750 132000
Solution of the three level resource allocation
 $v_1 = 2,056.9$ $v_2 = 2,461.6$ $v_3 = 3,452.1$ $x_{11}=617.1$ $x_{12}=1,439.7$ $x_{21}= 1,723$ $x_{22}=738.5$ $x_{31}=1,187.9$ $x_{32}=2,264.1$ Low Risk Com. Grp $y^1$ 125.46 234.63 310.64 107.62 172.46 423.69 $y^2$ 89.14 193.46 154.29 98.947 193.42 244.36 $y^3$ 97.432 102.46 196.98 69.442 207.16 219.80 $y^4$ 49.864 299.87 130.26 124.67 154.32 501.64 High Risk Com. Grp $y^1$ 28.808 125.01 120.09 77.004 124.50 142.29 $y^2$ 65.128 166.29 276.41 85.677 103.49 321.64 $y^3$ 56.836 257.30 233.75 115.18 89.809 346.20 $y^4$ 104.41 60.049 300.46 59.954 142.64 64.384
 $v_1 = 2,056.9$ $v_2 = 2,461.6$ $v_3 = 3,452.1$ $x_{11}=617.1$ $x_{12}=1,439.7$ $x_{21}= 1,723$ $x_{22}=738.5$ $x_{31}=1,187.9$ $x_{32}=2,264.1$ Low Risk Com. Grp $y^1$ 125.46 234.63 310.64 107.62 172.46 423.69 $y^2$ 89.14 193.46 154.29 98.947 193.42 244.36 $y^3$ 97.432 102.46 196.98 69.442 207.16 219.80 $y^4$ 49.864 299.87 130.26 124.67 154.32 501.64 High Risk Com. Grp $y^1$ 28.808 125.01 120.09 77.004 124.50 142.29 $y^2$ 65.128 166.29 276.41 85.677 103.49 321.64 $y^3$ 56.836 257.30 233.75 115.18 89.809 346.20 $y^4$ 104.41 60.049 300.46 59.954 142.64 64.384
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