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2012, 9(3): 539-552. doi: 10.3934/mbe.2012.9.539

Impact of vaccine arrival on the optimal control of a newly emerging infectious disease: A theoretical study

Bruno Buonomo 1, and  Eleonora Messina 1,

1. 

Department of Mathematics and Applications, University of Naples Federico II, via Cintia, I-80126 Naples, Italy

Received  April 2011 Revised  March 2012 Published  July 2012

When a newly emerging human infectious disease spreads through a host population, it may be that public health authorities must begin facing the outbreaks and planning an intervention campaign when not all intervention tools are readily available. In such cases, the problem of finding optimal intervention strategies to minimize both the disease burden and the intervention costs may be addressed by considering multiple intervention regimes. In this paper, we consider the scenario in which authorities may rely initially only on non-pharmaceutical interventions at the beginning of the campaign, knowing that a vaccine will later be available, at an exogenous and known switching time. We use a two-stage optimal control problem over a finite time horizon to analyze the optimal intervention strategies during the whole campaign, and to assess the effects of the new intervention tool on the preceding stage of the campaign. We obtain the optimality systems of two connected optimal control problems, and show the solution profiles through numerical simulations.
Keywords: Two-stage, epidemic models, vaccination., optimal control, non-pharmaceutical intervention.
Mathematics Subject Classification: Primary: 92D30, 49J1.
Citation: Bruno Buonomo, Eleonora Messina. Impact of vaccine arrival on the optimal control of a newly emerging infectious disease: A theoretical study. Mathematical Biosciences & Engineering, 2012, 9 (3) : 539-552. doi: 10.3934/mbe.2012.9.539
References:
[1]

S. Anita, V. Arnăutu and V. Capasso, "An Introduction to Optimal Control Problems in Life Sciences and Economics," Modeling and Simulation in Science, Engineering and Technology, Birkhäuser/Springer, New York, 2011.

[2]

E. Asano, L. J. Gross, S. Lenhart and L. A. Real, Optimal control of vaccine distribution in a rabies metapopulation model, Math. Biosci. Engineering, 5 (2008), 219-238.

[3]

H. Behncke, Optimal control of deterministic epidemics, Optimal Control Appl. Methods, 21 (2000), 269-285. doi: 10.1002/oca.678.

[4]

K. W. Blayneh, A. B. Gumel, S. Lenhart and T. Clayton, Backward bifurcation and optimal control in transmission dynamics of West Nile Virus, Bull. Math. Biol., 72 (2010), 1006-1028. doi: 10.1007/s11538-009-9480-0.

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R. Boucekkine, J. B. Krawczyk and T. Vall\'ee, Environmental quality versus economic performance: A dynamic game approach, Optimal Control Appl. Methods, 32 (2011), 29-46. doi: 10.1002/oca.927.

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R. Boucekkine, C. Saglam and T. Vall\'ee, Technology adoption under embodiment: A two-stage optimal control approach, Macroeconomic Dynamics, 8 (2004), 250-271.

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R. Bulirsch, E. Nerz, H. J. Pesch and O. von Stryk, Combining direct and indirect methods in optimal control: Range maximization of a hang glider, in "Optimal Control" (Freiburg, 1991), Internat. Ser. Numer. Math., 111, Birkh\"auser, Basel, (1993), 273-288.

[8]

B. Buonomo, A simple analysis of vaccination strategies for rubella, Math. Biosci. Engineering, 8 (2011), 677-687.

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B. Buonomo, On the optimal vaccination strategies for horizontally and vertically transmitted infectious diseases, J. Biol. Sys., 19 (2011), 263-279. doi: 10.1142/S0218339011003853.

[10]

C. W. Clark, "Mathematical Bioeconomics: The Optimal Management of Renewable Resources," Third edition, Pure and Applied Mathematics, A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 2010.

[11]

A. d'Onofrio and P. Manfredi, Information-related changes in contact patterns may trigger oscillations in the endemic prevalence of infectious disease, J. Theor. Biol., 256 (2009), 473-478. doi: 10.1016/j.jtbi.2008.10.005.

[12]

Z. Feng, Y. Yang, D. Xu, P. Zhang, M. M. Mc Cauley and J. W. Glasser, Timely identification of optimal control strategies for emerging infectious diseases, J. Theor. Biol., 259 (2009), 165-171. doi: 10.1016/j.jtbi.2009.03.006.

[13]

K. R. Fister, S. Lenhart and J. S. McNally, Optimizing chemotherapy in an HIV model, Electron. J. Differential Equations, 1998, 12 pp.

[14]

K. R. Fister and J. C. Panetta, Optimal control applied to cell-cycle-specific cancer chemotherapy, SIAM J. Appl. Math., 60 (2000), 1059-1072. doi: 10.1137/S0036139998338509.

[15]

K. R. Fister and J. C. Panetta, Optimal control applied to competing chemotherapeutic cell-kill strategies, SIAM J. Appl. Math., 63 (2003), 1954-1971. doi: 10.1137/S0036139902413489.

[16]

W. H. Fleming and R. W. Rishel, "Deterministic and Stochastic Optimal Control," Applications of Mathematics, No. 1, Springer-Verlag, Berlin-New York, 1975.

[17]

H. Gaff and E. Schaefer, Optimal control applied to vaccination and treatment strategies for various epidemiological models, Math. Biosci. Engineering, 6 (2009), 469-492.

[18]

D. Grass, J. P. Caulkins, G. Feichtinger, G. Tragler and D. A. Behrens, "Optimal Control of Nonlinear Processes. With Applications in Drugs, Corruption, and Terror," Springer-Verlag, Berlin, 2008.

[19]

A. Huhtala, A post-consumer waste management model for determining optimal levels of recycling and landfilling, Environ. Resour. Econom., 10 (1997), 301-314. doi: 10.1023/A:1026475208718.

[20]

Italian Ministry of Health, "Nuova Influenza." Available from: http://www.nuovainfluenza.salute.gov.it/nuovainfluenza/nuovaInfluenza.jsp.

[21]

Italian Ministry of Health, "FluNews," no. 28, May 3-9, 2010 (18th week). Available from: http://www.nuovainfluenza.salute.gov.it/imgs/C_17_notiziario_63_allegato.pdf.

[22]

H. R. Joshi, Optimal control of an HIV immunology model, Optimal Control Appl. Methods, 23 (2002), 199-213. doi: 10.1002/oca.710.

[23]

E. Jung, S. Iwami, Y. Takeuchi and T.-C. Jo, Optimal control strategy for prevention of avian influenza pandemic, J. Theor. Biol., 260 (2009), 220-229. doi: 10.1016/j.jtbi.2009.05.031.

[24]

E. Jung, S. Lenhart and Z. Feng, Optimal control of treatments in a two-strain tuberculosis model, Discrete Contin. Dyn. Syst. Ser. B, 2 (2002), 473-482.

[25]

S. Lee, G. Chowell and C. Castillo-Chavez, Optimal control for pandemic influenza: The role of limited antiviraltreatment and isolation, J. Theor. Biol., 265 (2010), 136-150. doi: 10.1016/j.jtbi.2010.04.003.

[26]

S. Lee, R. Morales and C. Castillo-Chavez, A note on the use of influenza vaccination strategies when supply is limited, Math. Biosci. Engineering, 8 (2011), 171-182.

[27]

S. Lenhart and J. T. Workman, "Optimal Control Applied to Biological Models," Chapman & Hall/CRC Mathematical and Computational Biology Series, Chapman & Hall/CRC, Boca Raton, FL, 2007.

[28]

F. Lin, K. Muthuraman and M. Lawley, An optimal control theory approach to non-pharmaceutical interventions, BMC Infect. Dis., 10 (2010), 32-45. doi: 10.1186/1471-2334-10-32.

[29]

, MATLAB© , "Matlab Release 12,", The Mathworks Inc., (2000). 

[30]

R. L. Miller Neilan, E. Schaefer, H. Gaff, K. R. Fister and S. Lenhart, Modeling optimal intervention strategies for cholera, Bull. Math. Biol., 72 (2010), 2004-2018. doi: 10.1007/s11538-010-9521-8.

[31]

L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze and E. F. Mishchenko, "The Mathematical Theory of Optimal Processes," Interscience Publishers John Wiley & Sons, Inc. New York-London, 1962.

[32]

O. Prosper, O. Saucedo, D. Thompson, G. Torres-Garcia, X. Wang and C. Castillo-Chavez, Modeling control strategies for concurrent epidemics of seasonal and pandemic H1N1 influenza, Math. Biosci. Engineering, 8 (2011), 141-170.

[33]

R. J. Rossana, Delivery lags and buffer stocks in the theory of investment by the firm, J. Econ. Dyn. Control, 9 (1985), 135-193.

[34]

W. H. Schmidt, Numerical methods for optimal control problems with ODE or integral equations, in "Large-Scale Scientific Computing," Lecture Notes in Comput. Sci., 3743, Springer, Berlin, 2006, 255-262.

[35]

K. Tomiyama, Two-stage optimal control problems and optimality conditions, J. Econ. Dyn. Control, 9 (1985), 317-337.

show all references

References:
[1]

S. Anita, V. Arnăutu and V. Capasso, "An Introduction to Optimal Control Problems in Life Sciences and Economics," Modeling and Simulation in Science, Engineering and Technology, Birkhäuser/Springer, New York, 2011.

[2]

E. Asano, L. J. Gross, S. Lenhart and L. A. Real, Optimal control of vaccine distribution in a rabies metapopulation model, Math. Biosci. Engineering, 5 (2008), 219-238.

[3]

H. Behncke, Optimal control of deterministic epidemics, Optimal Control Appl. Methods, 21 (2000), 269-285. doi: 10.1002/oca.678.

[4]

K. W. Blayneh, A. B. Gumel, S. Lenhart and T. Clayton, Backward bifurcation and optimal control in transmission dynamics of West Nile Virus, Bull. Math. Biol., 72 (2010), 1006-1028. doi: 10.1007/s11538-009-9480-0.

[5]

R. Boucekkine, J. B. Krawczyk and T. Vall\'ee, Environmental quality versus economic performance: A dynamic game approach, Optimal Control Appl. Methods, 32 (2011), 29-46. doi: 10.1002/oca.927.

[6]

R. Boucekkine, C. Saglam and T. Vall\'ee, Technology adoption under embodiment: A two-stage optimal control approach, Macroeconomic Dynamics, 8 (2004), 250-271.

[7]

R. Bulirsch, E. Nerz, H. J. Pesch and O. von Stryk, Combining direct and indirect methods in optimal control: Range maximization of a hang glider, in "Optimal Control" (Freiburg, 1991), Internat. Ser. Numer. Math., 111, Birkh\"auser, Basel, (1993), 273-288.

[8]

B. Buonomo, A simple analysis of vaccination strategies for rubella, Math. Biosci. Engineering, 8 (2011), 677-687.

[9]

B. Buonomo, On the optimal vaccination strategies for horizontally and vertically transmitted infectious diseases, J. Biol. Sys., 19 (2011), 263-279. doi: 10.1142/S0218339011003853.

[10]

C. W. Clark, "Mathematical Bioeconomics: The Optimal Management of Renewable Resources," Third edition, Pure and Applied Mathematics, A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 2010.

[11]

A. d'Onofrio and P. Manfredi, Information-related changes in contact patterns may trigger oscillations in the endemic prevalence of infectious disease, J. Theor. Biol., 256 (2009), 473-478. doi: 10.1016/j.jtbi.2008.10.005.

[12]

Z. Feng, Y. Yang, D. Xu, P. Zhang, M. M. Mc Cauley and J. W. Glasser, Timely identification of optimal control strategies for emerging infectious diseases, J. Theor. Biol., 259 (2009), 165-171. doi: 10.1016/j.jtbi.2009.03.006.

[13]

K. R. Fister, S. Lenhart and J. S. McNally, Optimizing chemotherapy in an HIV model, Electron. J. Differential Equations, 1998, 12 pp.

[14]

K. R. Fister and J. C. Panetta, Optimal control applied to cell-cycle-specific cancer chemotherapy, SIAM J. Appl. Math., 60 (2000), 1059-1072. doi: 10.1137/S0036139998338509.

[15]

K. R. Fister and J. C. Panetta, Optimal control applied to competing chemotherapeutic cell-kill strategies, SIAM J. Appl. Math., 63 (2003), 1954-1971. doi: 10.1137/S0036139902413489.

[16]

W. H. Fleming and R. W. Rishel, "Deterministic and Stochastic Optimal Control," Applications of Mathematics, No. 1, Springer-Verlag, Berlin-New York, 1975.

[17]

H. Gaff and E. Schaefer, Optimal control applied to vaccination and treatment strategies for various epidemiological models, Math. Biosci. Engineering, 6 (2009), 469-492.

[18]

D. Grass, J. P. Caulkins, G. Feichtinger, G. Tragler and D. A. Behrens, "Optimal Control of Nonlinear Processes. With Applications in Drugs, Corruption, and Terror," Springer-Verlag, Berlin, 2008.

[19]

A. Huhtala, A post-consumer waste management model for determining optimal levels of recycling and landfilling, Environ. Resour. Econom., 10 (1997), 301-314. doi: 10.1023/A:1026475208718.

[20]

Italian Ministry of Health, "Nuova Influenza." Available from: http://www.nuovainfluenza.salute.gov.it/nuovainfluenza/nuovaInfluenza.jsp.

[21]

Italian Ministry of Health, "FluNews," no. 28, May 3-9, 2010 (18th week). Available from: http://www.nuovainfluenza.salute.gov.it/imgs/C_17_notiziario_63_allegato.pdf.

[22]

H. R. Joshi, Optimal control of an HIV immunology model, Optimal Control Appl. Methods, 23 (2002), 199-213. doi: 10.1002/oca.710.

[23]

E. Jung, S. Iwami, Y. Takeuchi and T.-C. Jo, Optimal control strategy for prevention of avian influenza pandemic, J. Theor. Biol., 260 (2009), 220-229. doi: 10.1016/j.jtbi.2009.05.031.

[24]

E. Jung, S. Lenhart and Z. Feng, Optimal control of treatments in a two-strain tuberculosis model, Discrete Contin. Dyn. Syst. Ser. B, 2 (2002), 473-482.

[25]

S. Lee, G. Chowell and C. Castillo-Chavez, Optimal control for pandemic influenza: The role of limited antiviraltreatment and isolation, J. Theor. Biol., 265 (2010), 136-150. doi: 10.1016/j.jtbi.2010.04.003.

[26]

S. Lee, R. Morales and C. Castillo-Chavez, A note on the use of influenza vaccination strategies when supply is limited, Math. Biosci. Engineering, 8 (2011), 171-182.

[27]

S. Lenhart and J. T. Workman, "Optimal Control Applied to Biological Models," Chapman & Hall/CRC Mathematical and Computational Biology Series, Chapman & Hall/CRC, Boca Raton, FL, 2007.

[28]

F. Lin, K. Muthuraman and M. Lawley, An optimal control theory approach to non-pharmaceutical interventions, BMC Infect. Dis., 10 (2010), 32-45. doi: 10.1186/1471-2334-10-32.

[29]

, MATLAB© , "Matlab Release 12,", The Mathworks Inc., (2000). 

[30]

R. L. Miller Neilan, E. Schaefer, H. Gaff, K. R. Fister and S. Lenhart, Modeling optimal intervention strategies for cholera, Bull. Math. Biol., 72 (2010), 2004-2018. doi: 10.1007/s11538-010-9521-8.

[31]

L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze and E. F. Mishchenko, "The Mathematical Theory of Optimal Processes," Interscience Publishers John Wiley & Sons, Inc. New York-London, 1962.

[32]

O. Prosper, O. Saucedo, D. Thompson, G. Torres-Garcia, X. Wang and C. Castillo-Chavez, Modeling control strategies for concurrent epidemics of seasonal and pandemic H1N1 influenza, Math. Biosci. Engineering, 8 (2011), 141-170.

[33]

R. J. Rossana, Delivery lags and buffer stocks in the theory of investment by the firm, J. Econ. Dyn. Control, 9 (1985), 135-193.

[34]

W. H. Schmidt, Numerical methods for optimal control problems with ODE or integral equations, in "Large-Scale Scientific Computing," Lecture Notes in Comput. Sci., 3743, Springer, Berlin, 2006, 255-262.

[35]

K. Tomiyama, Two-stage optimal control problems and optimality conditions, J. Econ. Dyn. Control, 9 (1985), 317-337.

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