American Institute of Mathematical Sciences

Advanced
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

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.  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

[1]

Eunha Shim. Optimal strategies of social distancing and vaccination against seasonal influenza. Mathematical Biosciences & Engineering, 2013, 10 (5&6) : 1615-1634. doi: 10.3934/mbe.2013.10.1615
[2]

Ellina Grigorieva, Evgenii Khailov, Andrei Korobeinikov. An optimal control problem in HIV treatment. Conference Publications, 2013, 2013 (special) : 311-322. doi: 10.3934/proc.2013.2013.311
[3]

Majid Jaberi-Douraki, Seyed M. Moghadas. Optimal control of vaccination dynamics during an influenza epidemic. Mathematical Biosciences & Engineering, 2014, 11 (5) : 1045-1063. doi: 10.3934/mbe.2014.11.1045
[4]

Maria do Rosário de Pinho, Helmut Maurer, Hasnaa Zidani. Optimal control of normalized SIMR models with vaccination and treatment. Discrete & Continuous Dynamical Systems - B, 2018, 23 (1) : 79-99. doi: 10.3934/dcdsb.2018006
[5]

Joaquim P. Mateus, Paulo Rebelo, Silvério Rosa, César M. Silva, Delfim F. M. Torres. Optimal control of non-autonomous SEIRS models with vaccination and treatment. Discrete & Continuous Dynamical Systems - S, 2018, 11 (6) : 1179-1199. doi: 10.3934/dcdss.2018067
[6]

Cristiana J. Silva, Delfim F. M. Torres. A TB-HIV/AIDS coinfection model and optimal control treatment. Discrete & Continuous Dynamical Systems - A, 2015, 35 (9) : 4639-4663. doi: 10.3934/dcds.2015.35.4639
[7]

Urszula Ledzewicz, Mohammad Naghnaeian, Heinz Schättler. Dynamics of tumor-immune interaction under treatment as an optimal control problem. Conference Publications, 2011, 2011 (Special) : 971-980. doi: 10.3934/proc.2011.2011.971
[8]

Yali Yang, Sanyi Tang, Xiaohong Ren, Huiwen Zhao, Chenping Guo. Global stability and optimal control for a tuberculosis model with vaccination and treatment. Discrete & Continuous Dynamical Systems - B, 2016, 21 (3) : 1009-1022. doi: 10.3934/dcdsb.2016.21.1009
[9]

Cristiana J. Silva, Delfim F. M. Torres. Optimal control strategies for tuberculosis treatment: A case study in Angola. Numerical Algebra, Control & Optimization, 2012, 2 (3) : 601-617. doi: 10.3934/naco.2012.2.601
[10]

Sanjukta Hota, Folashade Agusto, Hem Raj Joshi, Suzanne Lenhart. Optimal control and stability analysis of an epidemic model with education campaign and treatment. Conference Publications, 2015, 2015 (special) : 621-634. doi: 10.3934/proc.2015.0621
[11]

Djamila Moulay, M. A. Aziz-Alaoui, Hee-Dae Kwon. Optimal control of chikungunya disease: Larvae reduction, treatment and prevention. Mathematical Biosciences & Engineering, 2012, 9 (2) : 369-392. doi: 10.3934/mbe.2012.9.369
[12]

Holly Gaff, Elsa Schaefer. Optimal control applied to vaccination and treatment strategies for various epidemiological models. Mathematical Biosciences & Engineering, 2009, 6 (3) : 469-492. doi: 10.3934/mbe.2009.6.469
[13]

Kbenesh Blayneh, Yanzhao Cao, Hee-Dae Kwon. Optimal control of vector-borne diseases: Treatment and prevention. Discrete & Continuous Dynamical Systems - B, 2009, 11 (3) : 587-611. doi: 10.3934/dcdsb.2009.11.587
[14]

Xun-Yang Wang, Khalid Hattaf, Hai-Feng Huo, Hong Xiang. Stability analysis of a delayed social epidemics model with general contact rate and its optimal control. Journal of Industrial & Management Optimization, 2016, 12 (4) : 1267-1285. doi: 10.3934/jimo.2016.12.1267
[15]

Maciej Leszczyński, Urszula Ledzewicz, Heinz Schättler. Optimal control for a mathematical model for anti-angiogenic treatment with Michaelis-Menten pharmacodynamics. Discrete & Continuous Dynamical Systems - B, 2019, 24 (5) : 2315-2334. doi: 10.3934/dcdsb.2019097
[16]

Marco Arieli Herrera-Valdez, Maytee Cruz-Aponte, Carlos Castillo-Chavez. Multiple outbreaks for the same pandemic: Local transportation and social distancing explain the different "waves" of A-H1N1pdm cases observed in México during 2009. Mathematical Biosciences & Engineering, 2011, 8 (1) : 21-48. doi: 10.3934/mbe.2011.8.21
[17]

Nasser Sweilam, Fathalla Rihan, Seham AL-Mekhlafi. A fractional-order delay differential model with optimal control for cancer treatment based on synergy between anti-angiogenic and immune cell therapies. Discrete & Continuous Dynamical Systems - S, 2020, 13 (9) : 2403-2424. doi: 10.3934/dcdss.2020120
[18]

Pierre Degond, Gadi Fibich, Benedetto Piccoli, Eitan Tadmor. Special issue on modeling and control in social dynamics. Networks & Heterogeneous Media, 2015, 10 (3) : i-ii. doi: 10.3934/nhm.2015.10.3i
[19]

Elena Fimmel, Yury S. Semenov, Alexander S. Bratus. On optimal and suboptimal treatment strategies for a mathematical model of leukemia. Mathematical Biosciences & Engineering, 2013, 10 (1) : 151-165. doi: 10.3934/mbe.2013.10.151
[20]

Mudassar Imran, Hal L. Smith. A model of optimal dosing of antibiotic treatment in biofilm. Mathematical Biosciences & Engineering, 2014, 11 (3) : 547-571. doi: 10.3934/mbe.2014.11.547

2018 Impact Factor: 1.313

Tools

Metrics

  • PDF downloads (76)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences