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A note on the use of influenza vaccination strategies when supply is limited
A note on the use of optimal control on a discrete time model of influenza dynamics
1. | Program in Computational Science, The University of Texas at El Paso, El Paso, TX 79968-0514, United States |
2. | Mathematical, Computational and Modeling Sciences Center, School of Human Evolution and Social Change, Arizona State University, Tempe, AZ 85287 |
3. | Program in Computational Science, Department of Mathematical Sciences, The University of Texas at El Paso, El Paso, TX 79968-0514, United States |
4. | Mathematics, Computational and Modeling Sciences Center, Arizona State University, PO Box 871904, Tempe, AZ 85287 |
References:
[1] |
L. J. Allen and A. M. Burgin, Comparison of deterministic and stochastic SIS and SIR models in discrete time, Math. Biosci., 163 (2000), 1-33.
doi: 10.1016/S0025-5564(99)00047-4. |
[2] |
R. M. Anderson and R. M. May, "Infectious Diseases of Humans: Dynamics and Control," Oxford University Press, Oxford, UK, 1992. |
[3] |
J. Arino, F. Brauer, P. van den Driessche, J. Watmough and J. Wu, A model for influenza with vaccination and antiviral treatment, J. Theor. Biol., 253 (2003), 118-130.
doi: 10.1016/j.jtbi.2008.02.026. |
[4] |
H. Behncke, Optimal control of deterministic epidemics, Opt. Control Appl. Meth., 21 (2000), 269-285.
doi: 10.1002/oca.678. |
[5] |
F. Brauer and C. Castillo-Chavez, "Mathematical Models in Population Biology and Epidemiology," Springer-Verlag, 2001. |
[6] |
F. Brauer, Z. Feng, and C. Castillo-Chavez, Discrete epidemic models, Math. Biosc. $&$ Eng., 7 (2010), 1-15. |
[7] |
P. Brewer, Economic effects of pandemic flu in a recession, 2009,, http://www.wisebread.com/economic-effects-of-pandemic-flu-in-a-recession., ().
|
[8] |
C. A. Burdet and S. P. Sethi, On the maximum principle for a class of discrete dynamical systems with Lags, Journal of Optimization Theory and Applications, 19 (1976), 445-454.
doi: 10.1007/BF00941486. |
[9] |
C. Castillo-Chavez and A-A. Yakubu, Discrete-time S-I-S models with complex dynamics, Nonlinear Analysis, 47 (2001), 4753-4762.
doi: 10.1016/S0362-546X(01)00587-9. |
[10] |
C. Castillo-Chavez and A-A. Yakubu, Discrete-time S-I-S models with simple and complex population dynamics, in Mathematical Approaches for Emerging and Reemerging Infectious Diseases (eds., C. Castillo-Chavez, et al.), Springer-Verlag, IMA, 125 (2001), 153-163. |
[11] |
M. Chan, World now at the start of 2009 influenza pandemic, 11 Jun. 2009., http://who.int/mediacentre/news/statements/2009/h1n1_pandemic_phase6_20090611/en/index.html, ().
|
[12] |
G. Chowell, C. E. Ammon, N. W. Hengartner and J. M. Hyman, Transmission dynamics of the great influenza pandemic of 1918 in Geneva, Switzerland: Assessing the effects of hypothetical interventions, J. Theor. Biol., 241 (2006), 193-204.
doi: 10.1016/j.jtbi.2005.11.026. |
[13] |
G. Chowell, H. Nishiura and L. M. A. Bettencourt, Comparative estimation of the reproduction number for pandemic influenza from daily case notification data, J. Roy. Soc. Interface, 4 (2007), 55-66.
doi: 10.1098/rsif.2006.0161. |
[14] |
W. Ding, L. Gross, K. Langston, S. Lenhart and L. Real, Rabies in racoons: Optimal control for a discrete time model on a spatial grid, J. Biol. Dynamics, 1 (2007), 307-393.
doi: 10.1080/17513750701605515. |
[15] |
R. Durrett and S. A. Levin, The importance of being discrete (and spatial), Theoret. Popul. Biol., 46 (1994), 363394.
doi: 10.1006/tpbi.1994.1032. |
[16] |
N. M. Ferguson, D. A. T. Cumminangs, C. Fraser, J. C. Cajika, P. C. Cooley and D. S. Burke, Strategies for mitigating an influenza pandemic, Nature, 442 (2006), 448-452.
doi: 10.1038/nature04795. |
[17] |
H. W. Hethcote, The mathematics of infectious diseases, SIAM Rev, 42 (2000), 599-653.
doi: 10.1137/S0036144500371907. |
[18] |
R. Hilschera and V. Zeidanb, Discrete optimal control: The accessory problem and necessary optimality conditions, Journal of Mathematical Analysis and Applications, 243 (2000), 429-452.
doi: 10.1006/jmaa.1999.6679. |
[19] |
C. Hwang and L. Fan, A Discrete version of Pontryagin's maximum principle, Operations Research, 15 (1967), 139-146.
doi: 10.1287/opre.15.1.139. |
[20] |
E. Jung, S. Lenhart, V. Protopopescu and C. F. Babbs, Optimal control theory applied to a difference equation model for cardiopulmonary resuscitation, Mathematical Models and methods in Applied Sciences, 15 (2005), 1519-1531.
doi: 10.1142/S0218202505000856. |
[21] |
M. I. Kamien and N. L. Schwarz, "Dynamic Optimization. The Calculus of Variations and Optimal Control in Economics And Management," Amsterdam: North-Holland, 1991. |
[22] |
S. Lee, G. Chowell and C. Castillo-Chavez, Optimal control for pandemic influenza: The role of limited antiviral treatment and isolation, J. Theor. Biol., 265 (2010), 136-150.
doi: 10.1016/j.jtbi.2010.04.003. |
[23] |
S. Lenhart and J. Workman, "Optimal Control Applied to Biological Models," Chapman & Hall, CRC Mathematical and Computational Biology series, 2007. |
[24] |
B. Marinkovic, Optimality conditions for discrete optimal control problems, Optimization Methods & Software Archive, 22 (2007), 959-969. |
[25] |
C. E. Mills, J. M. Robins and M. Lipsitch, Transmissibility of 1918 pandemic influenza, Nature, 432 (2004), 904-906.
doi: 10.1038/nature03063. |
[26] |
J. C. Monterrubio, Short-term economic impacts of influenza A(H1N1) and government reaction on the Mexican tourism industry: an analysis of the media, International Journal of Tourism Policy, 3 (2010), 1-15.
doi: 10.1504/IJTP.2010.031599. |
[27] | |
[28] |
M. Nuno, G. Chowell, X. Wang and C. Castillo-Chavez, On the role of cross-immunity and vaccines on the survival of less fit flu-strains, Theor. Pop. Biol., Elsevier, 71 (2007), 20-29. |
[29] |
L. S. Pontryagin, V. Boltyanskii, R. Gamkrelidze and E. Mishchenko, "The Mathematical Theory of Optimal Processes," Wiley, New Jersey, 1962. |
[30] |
Z. Qiu and Z. Feng, Transmission dynamics of an influenza model with vaccination and antiviral treatment, Bull. Math. Biol., 72 (2009), 1-33.
doi: 10.1007/s11538-009-9435-5. |
[31] |
S. P. Sethi and G. L. Thompson, "Optimal Control Theory: Applications to Management Science and Economics," Second Edition, Springer, 2000. |
[32] |
J. M. Tchuenche, S. A. Kamis, F. B. Agusto and S. C. Mpesche, "Optimal Control and Sensitivity Analysis of an Influenza Model with Treatment and Vaccination," Acta Biotheoretica, Springer, 2010. |
[33] |
S. M. Tracht, S. Del Valle and J. Hyman, Mathematical modeling of the effectiveness of facemasks in reducing the spread of novel influenza A (H1N1) PLoS ONE, www.plosone.org, 5 (2010).
doi: 10.1371/journal.pone.0009018. |
[34] |
Y. Zhou, Z. Ma and F. Brauer, A discrete epidemic model for SARS transmission and control in China, Math. and Computer Modelling, 40 (2004), 1491-1506.
doi: 10.1016/j.mcm.2005.01.007. |
show all references
References:
[1] |
L. J. Allen and A. M. Burgin, Comparison of deterministic and stochastic SIS and SIR models in discrete time, Math. Biosci., 163 (2000), 1-33.
doi: 10.1016/S0025-5564(99)00047-4. |
[2] |
R. M. Anderson and R. M. May, "Infectious Diseases of Humans: Dynamics and Control," Oxford University Press, Oxford, UK, 1992. |
[3] |
J. Arino, F. Brauer, P. van den Driessche, J. Watmough and J. Wu, A model for influenza with vaccination and antiviral treatment, J. Theor. Biol., 253 (2003), 118-130.
doi: 10.1016/j.jtbi.2008.02.026. |
[4] |
H. Behncke, Optimal control of deterministic epidemics, Opt. Control Appl. Meth., 21 (2000), 269-285.
doi: 10.1002/oca.678. |
[5] |
F. Brauer and C. Castillo-Chavez, "Mathematical Models in Population Biology and Epidemiology," Springer-Verlag, 2001. |
[6] |
F. Brauer, Z. Feng, and C. Castillo-Chavez, Discrete epidemic models, Math. Biosc. $&$ Eng., 7 (2010), 1-15. |
[7] |
P. Brewer, Economic effects of pandemic flu in a recession, 2009,, http://www.wisebread.com/economic-effects-of-pandemic-flu-in-a-recession., ().
|
[8] |
C. A. Burdet and S. P. Sethi, On the maximum principle for a class of discrete dynamical systems with Lags, Journal of Optimization Theory and Applications, 19 (1976), 445-454.
doi: 10.1007/BF00941486. |
[9] |
C. Castillo-Chavez and A-A. Yakubu, Discrete-time S-I-S models with complex dynamics, Nonlinear Analysis, 47 (2001), 4753-4762.
doi: 10.1016/S0362-546X(01)00587-9. |
[10] |
C. Castillo-Chavez and A-A. Yakubu, Discrete-time S-I-S models with simple and complex population dynamics, in Mathematical Approaches for Emerging and Reemerging Infectious Diseases (eds., C. Castillo-Chavez, et al.), Springer-Verlag, IMA, 125 (2001), 153-163. |
[11] |
M. Chan, World now at the start of 2009 influenza pandemic, 11 Jun. 2009., http://who.int/mediacentre/news/statements/2009/h1n1_pandemic_phase6_20090611/en/index.html, ().
|
[12] |
G. Chowell, C. E. Ammon, N. W. Hengartner and J. M. Hyman, Transmission dynamics of the great influenza pandemic of 1918 in Geneva, Switzerland: Assessing the effects of hypothetical interventions, J. Theor. Biol., 241 (2006), 193-204.
doi: 10.1016/j.jtbi.2005.11.026. |
[13] |
G. Chowell, H. Nishiura and L. M. A. Bettencourt, Comparative estimation of the reproduction number for pandemic influenza from daily case notification data, J. Roy. Soc. Interface, 4 (2007), 55-66.
doi: 10.1098/rsif.2006.0161. |
[14] |
W. Ding, L. Gross, K. Langston, S. Lenhart and L. Real, Rabies in racoons: Optimal control for a discrete time model on a spatial grid, J. Biol. Dynamics, 1 (2007), 307-393.
doi: 10.1080/17513750701605515. |
[15] |
R. Durrett and S. A. Levin, The importance of being discrete (and spatial), Theoret. Popul. Biol., 46 (1994), 363394.
doi: 10.1006/tpbi.1994.1032. |
[16] |
N. M. Ferguson, D. A. T. Cumminangs, C. Fraser, J. C. Cajika, P. C. Cooley and D. S. Burke, Strategies for mitigating an influenza pandemic, Nature, 442 (2006), 448-452.
doi: 10.1038/nature04795. |
[17] |
H. W. Hethcote, The mathematics of infectious diseases, SIAM Rev, 42 (2000), 599-653.
doi: 10.1137/S0036144500371907. |
[18] |
R. Hilschera and V. Zeidanb, Discrete optimal control: The accessory problem and necessary optimality conditions, Journal of Mathematical Analysis and Applications, 243 (2000), 429-452.
doi: 10.1006/jmaa.1999.6679. |
[19] |
C. Hwang and L. Fan, A Discrete version of Pontryagin's maximum principle, Operations Research, 15 (1967), 139-146.
doi: 10.1287/opre.15.1.139. |
[20] |
E. Jung, S. Lenhart, V. Protopopescu and C. F. Babbs, Optimal control theory applied to a difference equation model for cardiopulmonary resuscitation, Mathematical Models and methods in Applied Sciences, 15 (2005), 1519-1531.
doi: 10.1142/S0218202505000856. |
[21] |
M. I. Kamien and N. L. Schwarz, "Dynamic Optimization. The Calculus of Variations and Optimal Control in Economics And Management," Amsterdam: North-Holland, 1991. |
[22] |
S. Lee, G. Chowell and C. Castillo-Chavez, Optimal control for pandemic influenza: The role of limited antiviral treatment and isolation, J. Theor. Biol., 265 (2010), 136-150.
doi: 10.1016/j.jtbi.2010.04.003. |
[23] |
S. Lenhart and J. Workman, "Optimal Control Applied to Biological Models," Chapman & Hall, CRC Mathematical and Computational Biology series, 2007. |
[24] |
B. Marinkovic, Optimality conditions for discrete optimal control problems, Optimization Methods & Software Archive, 22 (2007), 959-969. |
[25] |
C. E. Mills, J. M. Robins and M. Lipsitch, Transmissibility of 1918 pandemic influenza, Nature, 432 (2004), 904-906.
doi: 10.1038/nature03063. |
[26] |
J. C. Monterrubio, Short-term economic impacts of influenza A(H1N1) and government reaction on the Mexican tourism industry: an analysis of the media, International Journal of Tourism Policy, 3 (2010), 1-15.
doi: 10.1504/IJTP.2010.031599. |
[27] | |
[28] |
M. Nuno, G. Chowell, X. Wang and C. Castillo-Chavez, On the role of cross-immunity and vaccines on the survival of less fit flu-strains, Theor. Pop. Biol., Elsevier, 71 (2007), 20-29. |
[29] |
L. S. Pontryagin, V. Boltyanskii, R. Gamkrelidze and E. Mishchenko, "The Mathematical Theory of Optimal Processes," Wiley, New Jersey, 1962. |
[30] |
Z. Qiu and Z. Feng, Transmission dynamics of an influenza model with vaccination and antiviral treatment, Bull. Math. Biol., 72 (2009), 1-33.
doi: 10.1007/s11538-009-9435-5. |
[31] |
S. P. Sethi and G. L. Thompson, "Optimal Control Theory: Applications to Management Science and Economics," Second Edition, Springer, 2000. |
[32] |
J. M. Tchuenche, S. A. Kamis, F. B. Agusto and S. C. Mpesche, "Optimal Control and Sensitivity Analysis of an Influenza Model with Treatment and Vaccination," Acta Biotheoretica, Springer, 2010. |
[33] |
S. M. Tracht, S. Del Valle and J. Hyman, Mathematical modeling of the effectiveness of facemasks in reducing the spread of novel influenza A (H1N1) PLoS ONE, www.plosone.org, 5 (2010).
doi: 10.1371/journal.pone.0009018. |
[34] |
Y. Zhou, Z. Ma and F. Brauer, A discrete epidemic model for SARS transmission and control in China, Math. and Computer Modelling, 40 (2004), 1491-1506.
doi: 10.1016/j.mcm.2005.01.007. |
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