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Impact of vaccine arrival on the optimal control of a newly emerging infectious disease: A theoretical study
1. | Department of Mathematics and Applications, University of Naples Federico II, via Cintia, I-80126 Naples, Italy |
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. |
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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