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
References:
[1]

Avert, Funding for HIV and AIDS, 2016. Available from URL http://www.avert.org/node/353/pdfGoogle Scholar

[2]

M. L. BrandeauG. S. Zaric and A. Richter, Resource allocation for control of infectious diseases in multiple independent populations: Beyond cost-effectiveness analysis, Journal of Health Economics, 22 (2003), 575-598. Google Scholar

[3]

M. L. BrandeauG. S. Zaric and V. De Angelis, Improved allocation of HIV prevention resources: Using information about prevention program production functions, Health Care Management Science, 8 (2005), 19-28. Google Scholar

[4]

Centers for Disease Control and Prevention (CDC), Achievements in public health, reduction in perinatal transmission of HIV infection -United States, 1985 -2005, MMWR Morb Mortal Wkly Rep, 55 (2006), 592-597.Google Scholar

[5]

D. DonnellJ. M. BaetenJ. KiarieK. K. ThomasW. StevensC. R. CohenJ. MclntyreJ. R. Lingappa and C. Celum, Heterosexual HIV-1 transmission after initiation of antiretroviral therapy: A prospective cohort analysis, Lancet, 375 (2010), 2092-2098. doi: 10.1016/S0140-6736(10)60705-2. Google Scholar

[6]

M. Drummond, B. O'Brien, G. L. Stoddart and G. J. Torrance (Eds. ), Methods for the Economic Evaluation of Health Care Programs, Oxford University Press, New York, 2000.Google Scholar

[7]

S. R. EarnshawK. HicksA. Richter and A. Honeycutt, A linear programming model for allocating HIV prevention funds with state agencies: A pilot study, Health Care Manage Sci, 10 (2007), 239-252. doi: 10.1007/s10729-007-9017-8. Google Scholar

[8]

S. Flessa, Where efficiency saves lives: A linear programme for the optimal allocation of health care resources in developing countries, Health Care Management Science, 3 (2000), 249-267. Google Scholar

[9]

A. M. Kassa and S. M. Kassa, A multi-parametric programming algorithm for special classes of non-convex multilevel optimization problems, An International Journal of Optimization and Control: Theories & Applications (IJOCTA), 3 (2013), 133-144. doi: 10.11121/ijocta.01.2013.00156. Google Scholar

[10]

A. M. Kassa and S. M. Kassa, A branch-and-bound multi-parametric programming approach for non-convex multilevel optimization with polyhedral constraints, Journal of Global Optimization, 64 (2016), 745-764. doi: 10.1007/s10898-015-0341-0. Google Scholar

[11]

A. M. Kassa and S. M. Kassa, Deterministic solution approach for some classes of nonlinear multilevel programs with multiple followers, J Glob Optim, (2017), 1-19. doi: 10.1007/s10898-017-0502-4. Google Scholar

[12]

S. M. Kassa and A. Ouhinou, The impact of self-protective measures in the optimal interventions for controlling infectious diseases of human population, Journal of Mathematical Biology, 70 (2015), 213-236. doi: 10.1007/s00285-014-0761-3. Google Scholar

[13]

S. M. Kassa and A. Ouhinou, Epidemiological models with prevalence dependent endogenous self-protection measure, Mathematical Biosciences, 229 (2011), 41-49. doi: 10.1016/j.mbs.2010.10.007. Google Scholar

[14]

J. Kates, J. A. Izazola and E. Lief, Financing the response to HIV in low-and middle-income countries: , International assistance from donor governments in 2015,2015. Available from: http://files.kff.org/attachment/Financing-the-Response-to-HIV-in-Low-and-Middle-Income-Countries-International-Assistance-from-Donor-Governments-in-2015Google Scholar

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[16]

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A. P. KourtisC. H. SchmidD. J. Jamieson and J. Lau, Use of Antiretroviral therapy in HIV-infected pregnant women and the risk of premature delivery: A meta-analysis, AIDS, 21 (2007), 607-615. doi: 10.1097/QAD.0b013e32802ef2f6. Google Scholar

[18]

A. LasryG. S. Zaric and M. W. Carter, Multi-level resource allocation for HIV prevention: A model for developing countries, European Journal of Operational Research, 180 (2007), 786-799. doi: 10.1016/j.ejor.2006.02.043. Google Scholar

[19]

F. J. PalellaK. M. DelaneyA. C. MoormanM. O. LovelessJ. Fuhrer and G. A. Satten, Declining morbidity and mortality among patients with advanced human immunodeficiency virus infection, The New England Journal of Medicine, 338 (1998), 853-860. doi: 10.1056/NEJM199803263381301. Google Scholar

[20]

L. PalombiM. C. MarazziA. Voetberg and N. A. Magid, Treatment acceleration program and the experience of the DREAM program in prevention of mother-to-child transmission of HIV, AIDS, 21 (2007), S65-S71. doi: 10.1097/01.aids.0000279708.09180.f5. Google Scholar

[21]

A. PrendergastG. Tudor-WilliamsS. Burchett and P. Goulder, International perspectives, progress, and future challenges of paediatric HIV infection, Lancet, 370 (2007), 68-80. doi: 10.1016/S0140-6736(07)61051-4. Google Scholar

[22]

N. Siegfried, L. van der Merwe, P. Brocklehurst and T. T. Sint, Antiretrovirals for reducing the risk of mother-to-child transmission of HIV infection, Cochrane Database of Systematic Reviews 2011, 7 (2011), Art. CD003510. doi: 10.1002/14651858.CD003510.pub3. Google Scholar

[23]

J. A. C. SterneM. A. HenánB. LedergerberK. TillingR. Weber and P. Sendi, Long-term effectiveness of potent antiretroviral therapy in preventing AIDS and death: A prospective cohort study, Lancet, 366 (2005), 378-384. doi: 10.1016/S0140-6736(05)67022-5. Google Scholar

[24]

UNAIDS, AIDS by the numbers, 2016. Available from: http://www.unaids.org/sites/default/files/media_asset/AIDS-by-the-numbers-2016_en.pdf.Google Scholar

[25]

UNAIDS, Fast-track update on investments needed in the AIDS response, UNAIDS Reference, 2016. Available from: http://www.unaids.org/sites/default/files/media_asset/UNAIDS_Reference_FastTrack_Update_on_investments_en.pdfGoogle Scholar

[26]

R. Vardavas and S. Blower, The emergence of HIV transmitted resistance in Botswana: When will the WHO detection threshold be exceeded? PLoS ONE, 2 (2007), e152. doi: 10.1371/journal.pone.0000152. Google Scholar

[27]

M. C. Weinstein, From cos-effectiveness ratios to resource allocation: where to draw the line?, in Valuing Healthcare: Costs, Benefits, Effectiveness of phramaceuticals and other medical technologies (eds. F. A. Sloan), Cambridge University Press, New York (1995), 77-97.Google Scholar

[28]

World Health Organization, Towards Universal Access: Scaling up Priority HIV/AIDS Interventions in the Health Sector: Progress Report 2009, WHO, 2009.Google Scholar

[29]

A. T. Woldemariam and S. M. Kassa, Systematic evolutionary algorithm for general multilevel Stackelberg problems with bounded decision variables (SEAMSP), Annals of Operations Research, 229 (2015), 771-790. doi: 10.1007/s10479-015-1842-4. Google Scholar

[30]

G. S. Zaric and M. L. Brandeau, Resource allocation for epidemic control over short time horizons, Mathematical Biosciences, 171 (2001), 33-58. doi: 10.1016/S0025-5564(01)00050-5. Google Scholar

[31]

G. S. Zaric and M. L. Brandeau, A little planning goes a long way: Multilevel allocation of HIV prevention resources, Medical Decision Making, 27 (2007), 71-81. doi: 10.1177/0272989X06297395. Google Scholar

show all references

References:
[1]

Avert, Funding for HIV and AIDS, 2016. Available from URL http://www.avert.org/node/353/pdfGoogle Scholar

[2]

M. L. BrandeauG. S. Zaric and A. Richter, Resource allocation for control of infectious diseases in multiple independent populations: Beyond cost-effectiveness analysis, Journal of Health Economics, 22 (2003), 575-598. Google Scholar

[3]

M. L. BrandeauG. S. Zaric and V. De Angelis, Improved allocation of HIV prevention resources: Using information about prevention program production functions, Health Care Management Science, 8 (2005), 19-28. Google Scholar

[4]

Centers for Disease Control and Prevention (CDC), Achievements in public health, reduction in perinatal transmission of HIV infection -United States, 1985 -2005, MMWR Morb Mortal Wkly Rep, 55 (2006), 592-597.Google Scholar

[5]

D. DonnellJ. M. BaetenJ. KiarieK. K. ThomasW. StevensC. R. CohenJ. MclntyreJ. R. Lingappa and C. Celum, Heterosexual HIV-1 transmission after initiation of antiretroviral therapy: A prospective cohort analysis, Lancet, 375 (2010), 2092-2098. doi: 10.1016/S0140-6736(10)60705-2. Google Scholar

[6]

M. Drummond, B. O'Brien, G. L. Stoddart and G. J. Torrance (Eds. ), Methods for the Economic Evaluation of Health Care Programs, Oxford University Press, New York, 2000.Google Scholar

[7]

S. R. EarnshawK. HicksA. Richter and A. Honeycutt, A linear programming model for allocating HIV prevention funds with state agencies: A pilot study, Health Care Manage Sci, 10 (2007), 239-252. doi: 10.1007/s10729-007-9017-8. Google Scholar

[8]

S. Flessa, Where efficiency saves lives: A linear programme for the optimal allocation of health care resources in developing countries, Health Care Management Science, 3 (2000), 249-267. Google Scholar

[9]

A. M. Kassa and S. M. Kassa, A multi-parametric programming algorithm for special classes of non-convex multilevel optimization problems, An International Journal of Optimization and Control: Theories & Applications (IJOCTA), 3 (2013), 133-144. doi: 10.11121/ijocta.01.2013.00156. Google Scholar

[10]

A. M. Kassa and S. M. Kassa, A branch-and-bound multi-parametric programming approach for non-convex multilevel optimization with polyhedral constraints, Journal of Global Optimization, 64 (2016), 745-764. doi: 10.1007/s10898-015-0341-0. Google Scholar

[11]

A. M. Kassa and S. M. Kassa, Deterministic solution approach for some classes of nonlinear multilevel programs with multiple followers, J Glob Optim, (2017), 1-19. doi: 10.1007/s10898-017-0502-4. Google Scholar

[12]

S. M. Kassa and A. Ouhinou, The impact of self-protective measures in the optimal interventions for controlling infectious diseases of human population, Journal of Mathematical Biology, 70 (2015), 213-236. doi: 10.1007/s00285-014-0761-3. Google Scholar

[13]

S. M. Kassa and A. Ouhinou, Epidemiological models with prevalence dependent endogenous self-protection measure, Mathematical Biosciences, 229 (2011), 41-49. doi: 10.1016/j.mbs.2010.10.007. Google Scholar

[14]

J. Kates, J. A. Izazola and E. Lief, Financing the response to HIV in low-and middle-income countries: , International assistance from donor governments in 2015,2015. Available from: http://files.kff.org/attachment/Financing-the-Response-to-HIV-in-Low-and-Middle-Income-Countries-International-Assistance-from-Donor-Governments-in-2015Google Scholar

[15]

J. Kates, A. Wexler and E. Lief, Financing the response to HIV in low-and middle-income countries: International assistance from donor governments in 2013, UNAIDS Report, July 2014. Available from: https://kaiserfamilyfoundation.files.wordpress.com/2014/07/7347-10-financing-the-response-to-hiv-in-low-and-middle-income-countries.pdfGoogle Scholar

[16]

J. M. KilbyH. Y. LeeJ. D. HazelwoodA. BansalR. P. BucyM. S. SaagG. M. ShawE. P. AcostaV. A. JohnsonA. S. Perelson and P. A. Goepfert, Treatment response in acute/early infection versus advanced AIDS: Equivalent first and second phase of HIV RNA decline, AIDS, 22 (2008), 957-962. doi: 10.1097/QAD.0b013e3282fbd1da. Google Scholar

[17]

A. P. KourtisC. H. SchmidD. J. Jamieson and J. Lau, Use of Antiretroviral therapy in HIV-infected pregnant women and the risk of premature delivery: A meta-analysis, AIDS, 21 (2007), 607-615. doi: 10.1097/QAD.0b013e32802ef2f6. Google Scholar

[18]

A. LasryG. S. Zaric and M. W. Carter, Multi-level resource allocation for HIV prevention: A model for developing countries, European Journal of Operational Research, 180 (2007), 786-799. doi: 10.1016/j.ejor.2006.02.043. Google Scholar

[19]

F. J. PalellaK. M. DelaneyA. C. MoormanM. O. LovelessJ. Fuhrer and G. A. Satten, Declining morbidity and mortality among patients with advanced human immunodeficiency virus infection, The New England Journal of Medicine, 338 (1998), 853-860. doi: 10.1056/NEJM199803263381301. Google Scholar

[20]

L. PalombiM. C. MarazziA. Voetberg and N. A. Magid, Treatment acceleration program and the experience of the DREAM program in prevention of mother-to-child transmission of HIV, AIDS, 21 (2007), S65-S71. doi: 10.1097/01.aids.0000279708.09180.f5. Google Scholar

[21]

A. PrendergastG. Tudor-WilliamsS. Burchett and P. Goulder, International perspectives, progress, and future challenges of paediatric HIV infection, Lancet, 370 (2007), 68-80. doi: 10.1016/S0140-6736(07)61051-4. Google Scholar

[22]

N. Siegfried, L. van der Merwe, P. Brocklehurst and T. T. Sint, Antiretrovirals for reducing the risk of mother-to-child transmission of HIV infection, Cochrane Database of Systematic Reviews 2011, 7 (2011), Art. CD003510. doi: 10.1002/14651858.CD003510.pub3. Google Scholar

[23]

J. A. C. SterneM. A. HenánB. LedergerberK. TillingR. Weber and P. Sendi, Long-term effectiveness of potent antiretroviral therapy in preventing AIDS and death: A prospective cohort study, Lancet, 366 (2005), 378-384. doi: 10.1016/S0140-6736(05)67022-5. Google Scholar

[24]

UNAIDS, AIDS by the numbers, 2016. Available from: http://www.unaids.org/sites/default/files/media_asset/AIDS-by-the-numbers-2016_en.pdf.Google Scholar

[25]

UNAIDS, Fast-track update on investments needed in the AIDS response, UNAIDS Reference, 2016. Available from: http://www.unaids.org/sites/default/files/media_asset/UNAIDS_Reference_FastTrack_Update_on_investments_en.pdfGoogle Scholar

[26]

R. Vardavas and S. Blower, The emergence of HIV transmitted resistance in Botswana: When will the WHO detection threshold be exceeded? PLoS ONE, 2 (2007), e152. doi: 10.1371/journal.pone.0000152. Google Scholar

[27]

M. C. Weinstein, From cos-effectiveness ratios to resource allocation: where to draw the line?, in Valuing Healthcare: Costs, Benefits, Effectiveness of phramaceuticals and other medical technologies (eds. F. A. Sloan), Cambridge University Press, New York (1995), 77-97.Google Scholar

[28]

World Health Organization, Towards Universal Access: Scaling up Priority HIV/AIDS Interventions in the Health Sector: Progress Report 2009, WHO, 2009.Google Scholar

[29]

A. T. Woldemariam and S. M. Kassa, Systematic evolutionary algorithm for general multilevel Stackelberg problems with bounded decision variables (SEAMSP), Annals of Operations Research, 229 (2015), 771-790. doi: 10.1007/s10479-015-1842-4. Google Scholar

[30]

G. S. Zaric and M. L. Brandeau, Resource allocation for epidemic control over short time horizons, Mathematical Biosciences, 171 (2001), 33-58. doi: 10.1016/S0025-5564(01)00050-5. Google Scholar

[31]

G. S. Zaric and M. L. Brandeau, A little planning goes a long way: Multilevel allocation of HIV prevention resources, Medical Decision Making, 27 (2007), 71-81. doi: 10.1177/0272989X06297395. Google Scholar

Figure 1.  Global financing for the fight of HIV from Donor Governments: Commitments & Disbursements, 2002-2013. Taken from [15].
Figure 2.  Schematic diagram for global resource allocations.
Figure 3.  Model diagram that show flow of individuals between the compartments.
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
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
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
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
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
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
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
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
$v_1 = 2,056.9$ $v_2 = 2,461.6$ $v_3 = 3,452.1$
$x_{11}=617.1$ $x_{12}=1,439.7$ $x_{21}= 1,723$ $x_{22}=738.5$ $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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