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2014, 11(5): 1045-1063. doi: 10.3934/mbe.2014.11.1045

Optimal control of vaccination dynamics during an influenza epidemic

1. 

Department of Physiology, McGill University, Montreal, Quebec, H3G 1Y6, Canada

2. 

Agent-Based Modelling Laboratory, York University, Toronto, Ontario, M3J 1P3, Canada

Received  July 2013 Revised  April 2014 Published  June 2014

For emerging diseases like pandemic influenza, several factors could impact the outcome of vaccination programs, including a delay in vaccine availability, imperfect vaccine-induced protection, and inadequate number of vaccines to sufficiently lower the susceptibility of the population by raising the level of herd immunity. We sought to investigate the effect of these factors in determining optimal vaccination strategies during an emerging influenza infection for which the population is entirely susceptible. We developed a population dynamical model of disease transmission and vaccination, and analyzed the control problem associated with an adaptive time-dependent vaccination strategy, in which the rate of vaccine distribution is optimally determined with time for minimizing the total number of infections (i.e., the epidemic final size). We simulated the model and compared the outcomes with a constant vaccination strategy in which the rate of vaccine distribution is time-independent. When vaccines are available at the onset of epidemic, our findings show that for a sufficiently high vaccine efficacy, the adaptive and constant vaccination strategies lead to comparable outcomes in terms of the epidemic final size. However, the adaptive vaccination requires a vaccine coverage higher than (or equivalent to) the constant vaccination regardless of the rate of vaccine distribution, suggesting that the latter is a more cost-effective strategy. When the vaccine efficacy is below a certain threshold, the adaptive vaccination could substantially outperform the constant vaccination, and the impact of adaptive strategy becomes more pronounced as the rate of vaccine distribution increases. We observed similar results when vaccines become available with a delay during the epidemic; however, the adaptive strategy may require a significantly higher vaccine coverage to outperform the constant vaccination strategy. The findings indicate that the vaccine efficacy is a key parameter that affects optimal control of vaccination dynamics during an epidemic, raising an important question on the trade-off between effectiveness and cost-effectiveness of vaccination policies in the context of limited vaccine quantities.
Citation: Majid Jaberi-Douraki, Seyed M. Moghadas. Optimal control of vaccination dynamics during an influenza epidemic. Mathematical Biosciences & Engineering, 2014, 11 (5) : 1045-1063. doi: 10.3934/mbe.2014.11.1045
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[2]

J. Arino, F. Brauer, P. van den Driessche, J. Watmough and J. Wu, A model for influenza with vaccination and antiviral treatment,, Journal of Theoretical Biology, 253 (2008), 118. doi: 10.1016/j.jtbi.2008.02.026.

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N. E. Basta, D. L. Chao, M. E. Halloran, L. Matrajt and I. M. Jr. Longini, Strategies for pandemic and seasonal influenza vaccination of schoolchildren in the United State,, American Journal of Epidemiology, 170 (2009), 679. doi: 10.1093/aje/kwp237.

[4]

C. T. Bauch, A. P. Galvani and D. J. Earn, Group interest versus self-interest in smallpox vaccination policy,, Proceedings of the National Academy of Sciences of the United States of America, 100 (2003), 10564. doi: 10.1073/pnas.1731324100.

[5]

C. S. Bowman, J. Arino and S. M. Moghadas, Evaluation of vaccination strategies during pandemic outbreaks,, Mathematical Biosciences and Engineering, 8 (2011), 113. doi: 10.3934/mbe.2011.8.113.

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R. M. Bush, Influenza evolution,, in Encyclopedia of Infectious Diseases: Modern Methodologies (ed. M. Tibayrenc, (2007), 199.

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M. Baguelin, M. Jit, E. Miller and W. J. Edmunds, Health and economic impact of the seasonal influenza vaccination programme in England,, Vaccine, 30 (2012), 3459. doi: 10.1016/j.vaccine.2012.03.019.

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N. M. Ferguson, S. Mallett, H. Jackson, N. Roberts and P. Ward, A population-dynamic model for evaluating the potential spread of drug-resistant influenza virus infections during community-based use of antivirals,, Journal of Antimicrobial Chemotherapy, 51 (2003), 977. doi: 10.1093/jac/dkg136.

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M. E. Halloran and I. M. Jr. Longini, Community studies for vaccinating schoolchildren against influenza,, Science, 311 (2006), 615.

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E. Hansen and T. Day, Optimal control of epidemics with limited resources,, Journal of Mathematical Biology, 62 (2011), 423. doi: 10.1007/s00285-010-0341-0.

[23]

E. Hansen and T. Day, Optimal antiviral treatment strategies and the effects of resistance,, Proceedings of the Royal Society B, 278 (2011), 1082. doi: 10.1098/rspb.2010.1469.

[24]

M. Jaberi-Douraki, J. M. Heffernan, J. Wu and S. M. Moghadas, Optimal treatment profile during an influenza epidemic,, Differential Equations and Dynamical Systems, 21 (2013), 237. doi: 10.1007/s12591-012-0149-z.

[25]

M. Jaberi-Douraki and S. M. Moghadas, Optimality of a time-dependent treatment profile during an epidemic,, Journal of Biological Dynamics, 7 (2013), 133. doi: 10.1080/17513758.2013.816377.

[26]

C. D. Johnson, Singular solutions in problems of optimal control,, Automatic Control, 8 (1963), 4.

[27]

H. R. Joshi, Optimal control of an HIV immunology model,, Optimal Control Applications and Methods, 23 (2002), 199. doi: 10.1002/oca.710.

[28]

H. R. Joshi, S. Lenhart and H. Gaff, Optimal harvesting in an integro-difference population model,, Optimal Control Applications and Methods, 27 (2006), 61. doi: 10.1002/oca.763.

[29]

H. R. Joshi, S. Lenhart, H. Lou and H. Gaff, Harvesting control in an integro-difference population model with concave growth term,, Nonlinear Analysis: Hybrid Systems, 1 (2007), 417. doi: 10.1016/j.nahs.2006.10.010.

[30]

E. Jung, S. Lenhart and Z. Feng, Optimal control of treatments in a two-strain tuberculosis model,, Discrete and Continuous Dynamical Systems B, 2 (2002), 473. doi: 10.3934/dcdsb.2002.2.473.

[31]

E. Jung, Y. Takeuchi and T. C. Jo, Optimal control strategy for prevention of avian influenza pandemic,, Journal of Theoretical Biology, 260 (2009), 220. doi: 10.1016/j.jtbi.2009.05.031.

[32]

H. J. Kelley, R. E. Kopp and H. G. Moyer, Singular extremals,, in Topics in Optimization (ed. G. Leitmann), (1967), 63.

[33]

D. E. Kirk, Optimal Control Theory: An Introduction,, Dover Publications Inc., (2004).

[34]

R. E. Kopp and H. G. Moyer, Necessary conditions for singular extremals,, American Institute of Aeronautics and Astronautics (AIAA) Journal, 3 (1965), 1439. doi: 10.2514/3.3165.

[35]

E. G. Kyriakidis and A. Pavitsos, Optimal intervention policies for a multidimensional simple epidemic process,, Mathematical and Computer Modelling, 50 (2009), 1318. doi: 10.1016/j.mcm.2009.06.012.

[36]

M. Laskowski, et al., The impact of demographic variables on disease spread: Influenza in remote communities,, Scientific Reports: Nature, 1 (2011). doi: 10.1038/srep00105.

[37]

S. Lee, M. Golinski and G. Chowell, Modeling optimal age-specific vaccination strategies against pandemic influenza,, Bulletin of Mathematical Biology., 74 (2012), 958. doi: 10.1007/s11538-011-9704-y.

[38]

S. Lenhart and J. T. Workman, Optimal Control Applied to Biological Models,, Chapman & Hall, (2007).

[39]

D. L. Lukes, Differential Equations: Classical to Controlled,, Mathematics in Science and Engineering, (1982).

[40]

L. Matrajt and I. M. Jr. Longini, Optimizing vaccine allocation at different points in time during an epidemic,, PLoS ONE, 5 (2010). doi: 10.1371/journal.pone.0013767.

[41]

J. Medlock and A. P. Galvani, Optimizing influenza vaccine distribution,, Science, 325 (2009), 1705. doi: 10.1126/science.1175570.

[42]

G. N. Mercer, S. I. Barry and H. Kelly, Modelling the effect of seasonal influenza vaccination on the risk of pandemic influenza infection,, BMC Public Health, 11 (2011). doi: 10.1186/1471-2458-11-S1-S11.

[43]

S. M. Moghadas, Management of drug resistance in the population: Influenza as a case study,, Proceedings of the Royal Society B, 275 (2008), 1163. doi: 10.1098/rspb.2008.0016.

[44]

S. M. Moghadas, C. S. Bowman, G. Röst and J. Wu, Population-Wide Emergence of Antiviral Resistance during Pandemic Influenza,, PLoS ONE, 3 (2008). doi: 10.1371/journal.pone.0001839.

[45]

A. S. Monto, K. Hornbuckle and S. E. Ohmit, Influenza vaccine effectiveness among nursing home residents: A cohort study,, American Journal of Epidemiology, 154 (2001), 155. doi: 10.1093/aje/154.2.155.

[46]

K. L. Nichol, K. Tummers, A. Hoyer-Leitzel, J. Marsh, M. Moynihan and S. McKelvey, Modeling seasonal influenza outbreak in a closed college campus: Impact of pre-season vaccination, in-season vaccination and holidays/breaks,, PLoS ONE, 5 (2010). doi: 10.1371/journal.pone.0009548.

[47]

R. Patel, I. M. Jr. Longini and M. E. Halloran, Finding optimal vaccination strategies for pandemic inflenza using genetic algorithms,, Journal of Theoretical Biology, 234 (2005), 201. doi: 10.1016/j.jtbi.2004.11.032.

[48]

R. R. Regoes and S. Bonhoeffer, Emergence of drugresistance influenza virus: Population dynamical considerations,, Science, 312 (2006), 389. doi: 10.1126/science.1122947.

[49]

A. H. Reid, T. A. Janczewski, R. M. Lourens, A. J. Elliot, R. S. Daniels, C. L. Berry, J. S. Oxford and J. K. Taubenberger, 1918 influenza pandemic caused by highly conserved viruses with two receptor-binding variants,, Emerging Infectious Diseases, 9 (2003), 1249. doi: 10.3201/eid0910.020789.

[50]

L. B. Shaw and I. B. Schwartz, Enhanced vaccine control of epidemics in adaptive networks,, Physical Review E: Statistical, 81 (2010). doi: 10.1103/PhysRevE.81.046120.

[51]

H. J. Sussmann, A bang-bang theorem with bounds on the number of switchings,, SIAM Journal on Control and Optimization (SICON), 17 (1979), 629. doi: 10.1137/0317045.

[52]

R. Ullah, G. Zaman and S. Islam, Prevention of influenza pandemic by multiple control strategies,, Journal of Applied Mathematics, (2012).

[53]

E. Verriest, F. Delmotte and M. Egerstedt, Control of epidemics by vaccination,, Proceedings of the American Control Conference, 2 (2005), 985. doi: 10.1109/ACC.2005.1470088.

[54]

R. G. Webster, Influenza: An emerging disease,, Emerging Infectious Diseases, 4 (1998), 436. doi: 10.3201/eid0403.980325.

[55]

R. G. Webster, W. J. Bean, O. T. Gorman, T. M. Chambers and Y. Kawaoka, Evolution and ecology of influenza A viruses,, Microbiology and Molecular Biology Reviews, 56 (1992), 152.

[56]

D. Weycker, D. Weycker, J. Edelsberg, M. E. Halloran, I. M. Jr. Longini, A. Nizam, V. Ciuryla and G. Oster, Population-wide benefits of routine vaccination of children against influenza,, Vaccine, 23 (2005), 1284. doi: 10.1016/j.vaccine.2004.08.044.

[57]

K. Wickwire, Optimal immunization rules for an epidemic with recovery,, Journal of Optimization Theory and Applications, 27 (1979), 549. doi: 10.1007/BF00933440.

show all references

References:
[1]

J. Arino, C. S. Bowman and S. M. Moghadas, Antiviral resistance during pandemic influenza: Implications for stockpiling and drug use,, BMC Infectious Diseases, 9 (2009). doi: 10.1186/1471-2334-9-8.

[2]

J. Arino, F. Brauer, P. van den Driessche, J. Watmough and J. Wu, A model for influenza with vaccination and antiviral treatment,, Journal of Theoretical Biology, 253 (2008), 118. doi: 10.1016/j.jtbi.2008.02.026.

[3]

N. E. Basta, D. L. Chao, M. E. Halloran, L. Matrajt and I. M. Jr. Longini, Strategies for pandemic and seasonal influenza vaccination of schoolchildren in the United State,, American Journal of Epidemiology, 170 (2009), 679. doi: 10.1093/aje/kwp237.

[4]

C. T. Bauch, A. P. Galvani and D. J. Earn, Group interest versus self-interest in smallpox vaccination policy,, Proceedings of the National Academy of Sciences of the United States of America, 100 (2003), 10564. doi: 10.1073/pnas.1731324100.

[5]

C. S. Bowman, J. Arino and S. M. Moghadas, Evaluation of vaccination strategies during pandemic outbreaks,, Mathematical Biosciences and Engineering, 8 (2011), 113. doi: 10.3934/mbe.2011.8.113.

[6]

R. M. Bush, Influenza evolution,, in Encyclopedia of Infectious Diseases: Modern Methodologies (ed. M. Tibayrenc, (2007), 199.

[7]

M. Baguelin, M. Jit, E. Miller and W. J. Edmunds, Health and economic impact of the seasonal influenza vaccination programme in England,, Vaccine, 30 (2012), 3459. doi: 10.1016/j.vaccine.2012.03.019.

[8]

A. E. Jr. Bryson and Y. C. Ho, Applied Optimal Control: Optimization, Estimation and Control,, Taylor & Francis, (1975).

[9]

M. G. Cojocaru, C. T. Bauch and M. D. Johnston, Dynamics of vaccination strategies via projected dynamical systems,, Bulletin of Mathematical Biology, 69 (2007), 1453. doi: 10.1007/s11538-006-9173-x.

[10]

R. B. Couch, et al., Respiratory viral infections in immunocompetent and immunocompromised persons,, The American Journal of Medicine, 102 (1997), 2.

[11]

N. J. Cox and K. Subbarao, Influenza,, Lancet, 354 (1999), 1277.

[12]

O. Diekmann and J. A. P. Heesterbeek, Mathematical Epidemiology of Infectious Diseases: Model Building, Analysis and Interpretation,, John Wiley & Sons, (2000).

[13]

J. Dushoff, et al., Vaccinating to protect a vulnerable subpopulation,, {PLOS Medicine}, 4 (2007). doi: 10.1371/journal.pmed.0040174.

[14]

N. M. Ferguson, S. Mallett, H. Jackson, N. Roberts and P. Ward, A population-dynamic model for evaluating the potential spread of drug-resistant influenza virus infections during community-based use of antivirals,, Journal of Antimicrobial Chemotherapy, 51 (2003), 977. doi: 10.1093/jac/dkg136.

[15]

N. M. Ferguson, D. A. T. Cummings, S. Cauchemez, C. Fraser, S. Riley, A. Meeyai, S. Iamsirithaworn and D. S. Burke, Strategies for containing an emerging influenza pandemic in Southeast Asia,, Nature, 437 (2005), 209. doi: 10.1038/nature04017.

[16]

K. R. Fister and J. C. Panetta, Optimal control applied to cell-cycle-specific cancer chemotherapy,, SIAM Journal on Applied Mathematics (SIAP), 60 (2000), 1059. doi: 10.1137/S0036139998338509.

[17]

W. H. Fleming and R. W. Rishel, Deterministic and Stochastic Optimal Control,, Springer-Verlag, (1975).

[18]

H. Gaff and E. Schaefer, Optimal control applied to vaccination and treatment strategies for various epidemiological models,, Mathematical Biosciences and Engineering, 6 (2009), 469. doi: 10.3934/mbe.2009.6.469.

[19]

M. E. Halloran, F. G. Hayden, Y. Yang, I. M. Jr. Longini and A. S. Monto, Antiviral effects on influenza viral transmission and pathogenicity: observations from household-based trials,, American Journal of Epidemiology, 165 (2006), 212. doi: 10.1093/aje/kwj362.

[20]

M. E. Halloran and I. M. Jr. Longini, Community studies for vaccinating schoolchildren against influenza,, Science, 311 (2006), 615.

[21]

E. Hansen, Applications of Optimal Control Theory to Infectious Disease Modeling,, Ph.D. Thesis, (2011).

[22]

E. Hansen and T. Day, Optimal control of epidemics with limited resources,, Journal of Mathematical Biology, 62 (2011), 423. doi: 10.1007/s00285-010-0341-0.

[23]

E. Hansen and T. Day, Optimal antiviral treatment strategies and the effects of resistance,, Proceedings of the Royal Society B, 278 (2011), 1082. doi: 10.1098/rspb.2010.1469.

[24]

M. Jaberi-Douraki, J. M. Heffernan, J. Wu and S. M. Moghadas, Optimal treatment profile during an influenza epidemic,, Differential Equations and Dynamical Systems, 21 (2013), 237. doi: 10.1007/s12591-012-0149-z.

[25]

M. Jaberi-Douraki and S. M. Moghadas, Optimality of a time-dependent treatment profile during an epidemic,, Journal of Biological Dynamics, 7 (2013), 133. doi: 10.1080/17513758.2013.816377.

[26]

C. D. Johnson, Singular solutions in problems of optimal control,, Automatic Control, 8 (1963), 4.

[27]

H. R. Joshi, Optimal control of an HIV immunology model,, Optimal Control Applications and Methods, 23 (2002), 199. doi: 10.1002/oca.710.

[28]

H. R. Joshi, S. Lenhart and H. Gaff, Optimal harvesting in an integro-difference population model,, Optimal Control Applications and Methods, 27 (2006), 61. doi: 10.1002/oca.763.

[29]

H. R. Joshi, S. Lenhart, H. Lou and H. Gaff, Harvesting control in an integro-difference population model with concave growth term,, Nonlinear Analysis: Hybrid Systems, 1 (2007), 417. doi: 10.1016/j.nahs.2006.10.010.

[30]

E. Jung, S. Lenhart and Z. Feng, Optimal control of treatments in a two-strain tuberculosis model,, Discrete and Continuous Dynamical Systems B, 2 (2002), 473. doi: 10.3934/dcdsb.2002.2.473.

[31]

E. Jung, Y. Takeuchi and T. C. Jo, Optimal control strategy for prevention of avian influenza pandemic,, Journal of Theoretical Biology, 260 (2009), 220. doi: 10.1016/j.jtbi.2009.05.031.

[32]

H. J. Kelley, R. E. Kopp and H. G. Moyer, Singular extremals,, in Topics in Optimization (ed. G. Leitmann), (1967), 63.

[33]

D. E. Kirk, Optimal Control Theory: An Introduction,, Dover Publications Inc., (2004).

[34]

R. E. Kopp and H. G. Moyer, Necessary conditions for singular extremals,, American Institute of Aeronautics and Astronautics (AIAA) Journal, 3 (1965), 1439. doi: 10.2514/3.3165.

[35]

E. G. Kyriakidis and A. Pavitsos, Optimal intervention policies for a multidimensional simple epidemic process,, Mathematical and Computer Modelling, 50 (2009), 1318. doi: 10.1016/j.mcm.2009.06.012.

[36]

M. Laskowski, et al., The impact of demographic variables on disease spread: Influenza in remote communities,, Scientific Reports: Nature, 1 (2011). doi: 10.1038/srep00105.

[37]

S. Lee, M. Golinski and G. Chowell, Modeling optimal age-specific vaccination strategies against pandemic influenza,, Bulletin of Mathematical Biology., 74 (2012), 958. doi: 10.1007/s11538-011-9704-y.

[38]

S. Lenhart and J. T. Workman, Optimal Control Applied to Biological Models,, Chapman & Hall, (2007).

[39]

D. L. Lukes, Differential Equations: Classical to Controlled,, Mathematics in Science and Engineering, (1982).

[40]

L. Matrajt and I. M. Jr. Longini, Optimizing vaccine allocation at different points in time during an epidemic,, PLoS ONE, 5 (2010). doi: 10.1371/journal.pone.0013767.

[41]

J. Medlock and A. P. Galvani, Optimizing influenza vaccine distribution,, Science, 325 (2009), 1705. doi: 10.1126/science.1175570.

[42]

G. N. Mercer, S. I. Barry and H. Kelly, Modelling the effect of seasonal influenza vaccination on the risk of pandemic influenza infection,, BMC Public Health, 11 (2011). doi: 10.1186/1471-2458-11-S1-S11.

[43]

S. M. Moghadas, Management of drug resistance in the population: Influenza as a case study,, Proceedings of the Royal Society B, 275 (2008), 1163. doi: 10.1098/rspb.2008.0016.

[44]

S. M. Moghadas, C. S. Bowman, G. Röst and J. Wu, Population-Wide Emergence of Antiviral Resistance during Pandemic Influenza,, PLoS ONE, 3 (2008). doi: 10.1371/journal.pone.0001839.

[45]

A. S. Monto, K. Hornbuckle and S. E. Ohmit, Influenza vaccine effectiveness among nursing home residents: A cohort study,, American Journal of Epidemiology, 154 (2001), 155. doi: 10.1093/aje/154.2.155.

[46]

K. L. Nichol, K. Tummers, A. Hoyer-Leitzel, J. Marsh, M. Moynihan and S. McKelvey, Modeling seasonal influenza outbreak in a closed college campus: Impact of pre-season vaccination, in-season vaccination and holidays/breaks,, PLoS ONE, 5 (2010). doi: 10.1371/journal.pone.0009548.

[47]

R. Patel, I. M. Jr. Longini and M. E. Halloran, Finding optimal vaccination strategies for pandemic inflenza using genetic algorithms,, Journal of Theoretical Biology, 234 (2005), 201. doi: 10.1016/j.jtbi.2004.11.032.

[48]

R. R. Regoes and S. Bonhoeffer, Emergence of drugresistance influenza virus: Population dynamical considerations,, Science, 312 (2006), 389. doi: 10.1126/science.1122947.

[49]

A. H. Reid, T. A. Janczewski, R. M. Lourens, A. J. Elliot, R. S. Daniels, C. L. Berry, J. S. Oxford and J. K. Taubenberger, 1918 influenza pandemic caused by highly conserved viruses with two receptor-binding variants,, Emerging Infectious Diseases, 9 (2003), 1249. doi: 10.3201/eid0910.020789.

[50]

L. B. Shaw and I. B. Schwartz, Enhanced vaccine control of epidemics in adaptive networks,, Physical Review E: Statistical, 81 (2010). doi: 10.1103/PhysRevE.81.046120.

[51]

H. J. Sussmann, A bang-bang theorem with bounds on the number of switchings,, SIAM Journal on Control and Optimization (SICON), 17 (1979), 629. doi: 10.1137/0317045.

[52]

R. Ullah, G. Zaman and S. Islam, Prevention of influenza pandemic by multiple control strategies,, Journal of Applied Mathematics, (2012).

[53]

E. Verriest, F. Delmotte and M. Egerstedt, Control of epidemics by vaccination,, Proceedings of the American Control Conference, 2 (2005), 985. doi: 10.1109/ACC.2005.1470088.

[54]

R. G. Webster, Influenza: An emerging disease,, Emerging Infectious Diseases, 4 (1998), 436. doi: 10.3201/eid0403.980325.

[55]

R. G. Webster, W. J. Bean, O. T. Gorman, T. M. Chambers and Y. Kawaoka, Evolution and ecology of influenza A viruses,, Microbiology and Molecular Biology Reviews, 56 (1992), 152.

[56]

D. Weycker, D. Weycker, J. Edelsberg, M. E. Halloran, I. M. Jr. Longini, A. Nizam, V. Ciuryla and G. Oster, Population-wide benefits of routine vaccination of children against influenza,, Vaccine, 23 (2005), 1284. doi: 10.1016/j.vaccine.2004.08.044.

[57]

K. Wickwire, Optimal immunization rules for an epidemic with recovery,, Journal of Optimization Theory and Applications, 27 (1979), 549. doi: 10.1007/BF00933440.

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