A note on the use of optimal control on a discrete time model of influenza dynamics

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2011, 8(1): 183-197. doi: 10.3934/mbe.2011.8.183

A note on the use of optimal control on a discrete time model of influenza dynamics

Paula A. González-Parra ^1, , Sunmi Lee ^2, , Leticia Velázquez ^3, and Carlos Castillo-Chavez ^4,

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

Received June 2010 Revised September 2010 Published January 2011

Abstract
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Under a Creative Commons license

A discrete time Susceptible - Asymptomatic - Infectious - Treated - Recovered (SAITR) model is introduced in the context of influenza transmission. We evaluate the potential effect of control measures such as social distancing and antiviral treatment on the dynamics of a single outbreak. Optimal control theory is applied to identify the best way of reducing morbidity and mortality at a minimal cost. The problem is solved by using a discrete version of Pontryagin's maximum principle. Numerical results show that dual strategies have stronger impact in the reduction of the final epidemic size.

Keywords: antiviral treatment., Influenza, optimal control, social distancing.

Mathematics Subject Classification: Primary: 92B05, 49K21, 93C55; Secondary: 92D4.

Citation: Paula A. González-Parra, Sunmi Lee, Leticia Velázquez, Carlos Castillo-Chavez. A note on the use of optimal control on a discrete time model of influenza dynamics. Mathematical Biosciences & Engineering, 2011, 8 (1) : 183-197. doi: 10.3934/mbe.2011.8.183

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]	J. Nocedal, "Numerical Optimization," Springer, 2006.
[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]	J. Nocedal, "Numerical Optimization," Springer, 2006.
[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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