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.  doi: 10.1016/S0025-5564(99)00047-4.  Google Scholar

[2]

R. M. Anderson and R. M. May, "Infectious Diseases of Humans: Dynamics and Control,", Oxford University Press, (1992).   Google Scholar

[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.  doi: 10.1016/j.jtbi.2008.02.026.  Google Scholar

[4]

H. Behncke, Optimal control of deterministic epidemics,, Opt. Control Appl. Meth., 21 (2000), 269.  doi: 10.1002/oca.678.  Google Scholar

[5]

F. Brauer and C. Castillo-Chavez, "Mathematical Models in Population Biology and Epidemiology,", Springer-Verlag, (2001).   Google Scholar

[6]

F. Brauer, Z. Feng, and C. Castillo-Chavez, Discrete epidemic models,, Math. Biosc. $&$ Eng., 7 (2010), 1.   Google Scholar

[7]

P. Brewer, Economic effects of pandemic flu in a recession, 2009,, http://www.wisebread.com/economic-effects-of-pandemic-flu-in-a-recession., ().   Google Scholar

[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.  doi: 10.1007/BF00941486.  Google Scholar

[9]

C. Castillo-Chavez and A-A. Yakubu, Discrete-time S-I-S models with complex dynamics,, Nonlinear Analysis, 47 (2001), 4753.  doi: 10.1016/S0362-546X(01)00587-9.  Google Scholar

[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., 125 (2001), 153.   Google Scholar

[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, ().   Google Scholar

[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.  doi: 10.1016/j.jtbi.2005.11.026.  Google Scholar

[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.  doi: 10.1098/rsif.2006.0161.  Google Scholar

[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.  doi: 10.1080/17513750701605515.  Google Scholar

[15]

R. Durrett and S. A. Levin, The importance of being discrete (and spatial),, Theoret. Popul. Biol., 46 (1994).  doi: 10.1006/tpbi.1994.1032.  Google Scholar

[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.  doi: 10.1038/nature04795.  Google Scholar

[17]

H. W. Hethcote, The mathematics of infectious diseases,, SIAM Rev, 42 (2000), 599.  doi: 10.1137/S0036144500371907.  Google Scholar

[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.  doi: 10.1006/jmaa.1999.6679.  Google Scholar

[19]

C. Hwang and L. Fan, A Discrete version of Pontryagin's maximum principle,, Operations Research, 15 (1967), 139.  doi: 10.1287/opre.15.1.139.  Google Scholar

[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.  doi: 10.1142/S0218202505000856.  Google Scholar

[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).   Google Scholar

[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.  doi: 10.1016/j.jtbi.2010.04.003.  Google Scholar

[23]

S. Lenhart and J. Workman, "Optimal Control Applied to Biological Models,", Chapman & Hall, (2007).   Google Scholar

[24]

B. Marinkovic, Optimality conditions for discrete optimal control problems,, Optimization Methods & Software Archive, 22 (2007), 959.   Google Scholar

[25]

C. E. Mills, J. M. Robins and M. Lipsitch, Transmissibility of 1918 pandemic influenza,, Nature, 432 (2004), 904.  doi: 10.1038/nature03063.  Google Scholar

[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.  doi: 10.1504/IJTP.2010.031599.  Google Scholar

[27]

J. Nocedal, "Numerical Optimization,", Springer, (2006).   Google Scholar

[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., 71 (2007), 20.   Google Scholar

[29]

L. S. Pontryagin, V. Boltyanskii, R. Gamkrelidze and E. Mishchenko, "The Mathematical Theory of Optimal Processes,", Wiley, (1962).   Google Scholar

[30]

Z. Qiu and Z. Feng, Transmission dynamics of an influenza model with vaccination and antiviral treatment,, Bull. Math. Biol., 72 (2009), 1.  doi: 10.1007/s11538-009-9435-5.  Google Scholar

[31]

S. P. Sethi and G. L. Thompson, "Optimal Control Theory: Applications to Management Science and Economics,", Second Edition, (2000).   Google Scholar

[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, (2010).   Google Scholar

[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, 5 (2010).  doi: 10.1371/journal.pone.0009018.  Google Scholar

[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.  doi: 10.1016/j.mcm.2005.01.007.  Google Scholar

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.  doi: 10.1016/S0025-5564(99)00047-4.  Google Scholar

[2]

R. M. Anderson and R. M. May, "Infectious Diseases of Humans: Dynamics and Control,", Oxford University Press, (1992).   Google Scholar

[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.  doi: 10.1016/j.jtbi.2008.02.026.  Google Scholar

[4]

H. Behncke, Optimal control of deterministic epidemics,, Opt. Control Appl. Meth., 21 (2000), 269.  doi: 10.1002/oca.678.  Google Scholar

[5]

F. Brauer and C. Castillo-Chavez, "Mathematical Models in Population Biology and Epidemiology,", Springer-Verlag, (2001).   Google Scholar

[6]

F. Brauer, Z. Feng, and C. Castillo-Chavez, Discrete epidemic models,, Math. Biosc. $&$ Eng., 7 (2010), 1.   Google Scholar

[7]

P. Brewer, Economic effects of pandemic flu in a recession, 2009,, http://www.wisebread.com/economic-effects-of-pandemic-flu-in-a-recession., ().   Google Scholar

[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.  doi: 10.1007/BF00941486.  Google Scholar

[9]

C. Castillo-Chavez and A-A. Yakubu, Discrete-time S-I-S models with complex dynamics,, Nonlinear Analysis, 47 (2001), 4753.  doi: 10.1016/S0362-546X(01)00587-9.  Google Scholar

[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., 125 (2001), 153.   Google Scholar

[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, ().   Google Scholar

[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.  doi: 10.1016/j.jtbi.2005.11.026.  Google Scholar

[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.  doi: 10.1098/rsif.2006.0161.  Google Scholar

[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.  doi: 10.1080/17513750701605515.  Google Scholar

[15]

R. Durrett and S. A. Levin, The importance of being discrete (and spatial),, Theoret. Popul. Biol., 46 (1994).  doi: 10.1006/tpbi.1994.1032.  Google Scholar

[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.  doi: 10.1038/nature04795.  Google Scholar

[17]

H. W. Hethcote, The mathematics of infectious diseases,, SIAM Rev, 42 (2000), 599.  doi: 10.1137/S0036144500371907.  Google Scholar

[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.  doi: 10.1006/jmaa.1999.6679.  Google Scholar

[19]

C. Hwang and L. Fan, A Discrete version of Pontryagin's maximum principle,, Operations Research, 15 (1967), 139.  doi: 10.1287/opre.15.1.139.  Google Scholar

[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.  doi: 10.1142/S0218202505000856.  Google Scholar

[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).   Google Scholar

[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.  doi: 10.1016/j.jtbi.2010.04.003.  Google Scholar

[23]

S. Lenhart and J. Workman, "Optimal Control Applied to Biological Models,", Chapman & Hall, (2007).   Google Scholar

[24]

B. Marinkovic, Optimality conditions for discrete optimal control problems,, Optimization Methods & Software Archive, 22 (2007), 959.   Google Scholar

[25]

C. E. Mills, J. M. Robins and M. Lipsitch, Transmissibility of 1918 pandemic influenza,, Nature, 432 (2004), 904.  doi: 10.1038/nature03063.  Google Scholar

[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.  doi: 10.1504/IJTP.2010.031599.  Google Scholar

[27]

J. Nocedal, "Numerical Optimization,", Springer, (2006).   Google Scholar

[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., 71 (2007), 20.   Google Scholar

[29]

L. S. Pontryagin, V. Boltyanskii, R. Gamkrelidze and E. Mishchenko, "The Mathematical Theory of Optimal Processes,", Wiley, (1962).   Google Scholar

[30]

Z. Qiu and Z. Feng, Transmission dynamics of an influenza model with vaccination and antiviral treatment,, Bull. Math. Biol., 72 (2009), 1.  doi: 10.1007/s11538-009-9435-5.  Google Scholar

[31]

S. P. Sethi and G. L. Thompson, "Optimal Control Theory: Applications to Management Science and Economics,", Second Edition, (2000).   Google Scholar

[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, (2010).   Google Scholar

[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, 5 (2010).  doi: 10.1371/journal.pone.0009018.  Google Scholar

[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.  doi: 10.1016/j.mcm.2005.01.007.  Google Scholar

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