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

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doi:10.3934/mbe.2018011

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2018, Volume 15, Issue 1: 255-273. Doi: 10.3934/mbe.2018011

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

^1,2,

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 23, 2016

Revised: March 02, 2017

Published: January 2018

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Abstract

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.

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Mathematics Subject Classification: Primary: 92B05, 92D30, 90B50, 91A80; Secondary: 34B60, 91A10, 90C31, 00A71.

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Figure 1. Global financing for the fight of HIV from Donor Governments: Commitments & Disbursements, 2002-2013. Taken from [15].

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Figure 2. Schematic diagram for global resource allocations.

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Figure 3. Model diagram that show flow of individuals between the compartments.

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Figure 4. 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

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Figure 5. 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

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Figure 6. 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

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Table 1. 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

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Table 2. 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

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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

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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

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Table 3. 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

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