# American Institute of Mathematical Sciences

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

 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

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.
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.

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##### 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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