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]

Math. Biosci., 163 (2000), 1-33. doi: 10.1016/S0025-5564(99)00047-4.  Google Scholar

[2]

Oxford University Press, Oxford, UK, 1992. Google Scholar

[3]

J. Theor. Biol., 253 (2003), 118-130. doi: 10.1016/j.jtbi.2008.02.026.  Google Scholar

[4]

Opt. Control Appl. Meth., 21 (2000), 269-285. doi: 10.1002/oca.678.  Google Scholar

[5]

Springer-Verlag, 2001.  Google Scholar

[6]

Math. Biosc. $&$ Eng., 7 (2010), 1-15. 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]

Journal of Optimization Theory and Applications, 19 (1976), 445-454. doi: 10.1007/BF00941486.  Google Scholar

[9]

Nonlinear Analysis, 47 (2001), 4753-4762. doi: 10.1016/S0362-546X(01)00587-9.  Google Scholar

[10]

in Mathematical Approaches for Emerging and Reemerging Infectious Diseases (eds., C. Castillo-Chavez, et al.), Springer-Verlag, IMA, 125 (2001), 153-163. 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]

J. Theor. Biol., 241 (2006), 193-204. doi: 10.1016/j.jtbi.2005.11.026.  Google Scholar

[13]

J. Roy. Soc. Interface, 4 (2007), 55-66. doi: 10.1098/rsif.2006.0161.  Google Scholar

[14]

J. Biol. Dynamics, 1 (2007), 307-393. doi: 10.1080/17513750701605515.  Google Scholar

[15]

Theoret. Popul. Biol., 46 (1994), 363394. doi: 10.1006/tpbi.1994.1032.  Google Scholar

[16]

Nature, 442 (2006), 448-452. doi: 10.1038/nature04795.  Google Scholar

[17]

SIAM Rev, 42 (2000), 599-653. doi: 10.1137/S0036144500371907.  Google Scholar

[18]

Journal of Mathematical Analysis and Applications, 243 (2000), 429-452. doi: 10.1006/jmaa.1999.6679.  Google Scholar

[19]

Operations Research, 15 (1967), 139-146. doi: 10.1287/opre.15.1.139.  Google Scholar

[20]

Mathematical Models and methods in Applied Sciences, 15 (2005), 1519-1531. doi: 10.1142/S0218202505000856.  Google Scholar

[21]

Amsterdam: North-Holland, 1991.  Google Scholar

[22]

J. Theor. Biol., 265 (2010), 136-150. doi: 10.1016/j.jtbi.2010.04.003.  Google Scholar

[23]

Chapman & Hall, CRC Mathematical and Computational Biology series, 2007.  Google Scholar

[24]

Optimization Methods & Software Archive, 22 (2007), 959-969. Google Scholar

[25]

Nature, 432 (2004), 904-906. doi: 10.1038/nature03063.  Google Scholar

[26]

International Journal of Tourism Policy, 3 (2010), 1-15. doi: 10.1504/IJTP.2010.031599.  Google Scholar

[27]

Springer, 2006.  Google Scholar

[28]

Theor. Pop. Biol., Elsevier, 71 (2007), 20-29. Google Scholar

[29]

Wiley, New Jersey, 1962.  Google Scholar

[30]

Bull. Math. Biol., 72 (2009), 1-33. doi: 10.1007/s11538-009-9435-5.  Google Scholar

[31]

Second Edition, Springer, 2000. Google Scholar

[32]

Acta Biotheoretica, Springer, 2010. Google Scholar

[33]

PLoS ONE, www.plosone.org, 5 (2010). doi: 10.1371/journal.pone.0009018.  Google Scholar

[34]

Math. and Computer Modelling, 40 (2004), 1491-1506. doi: 10.1016/j.mcm.2005.01.007.  Google Scholar

show all references

References:
[1]

Math. Biosci., 163 (2000), 1-33. doi: 10.1016/S0025-5564(99)00047-4.  Google Scholar

[2]

Oxford University Press, Oxford, UK, 1992. Google Scholar

[3]

J. Theor. Biol., 253 (2003), 118-130. doi: 10.1016/j.jtbi.2008.02.026.  Google Scholar

[4]

Opt. Control Appl. Meth., 21 (2000), 269-285. doi: 10.1002/oca.678.  Google Scholar

[5]

Springer-Verlag, 2001.  Google Scholar

[6]

Math. Biosc. $&$ Eng., 7 (2010), 1-15. 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]

Journal of Optimization Theory and Applications, 19 (1976), 445-454. doi: 10.1007/BF00941486.  Google Scholar

[9]

Nonlinear Analysis, 47 (2001), 4753-4762. doi: 10.1016/S0362-546X(01)00587-9.  Google Scholar

[10]

in Mathematical Approaches for Emerging and Reemerging Infectious Diseases (eds., C. Castillo-Chavez, et al.), Springer-Verlag, IMA, 125 (2001), 153-163. 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]

J. Theor. Biol., 241 (2006), 193-204. doi: 10.1016/j.jtbi.2005.11.026.  Google Scholar

[13]

J. Roy. Soc. Interface, 4 (2007), 55-66. doi: 10.1098/rsif.2006.0161.  Google Scholar

[14]

J. Biol. Dynamics, 1 (2007), 307-393. doi: 10.1080/17513750701605515.  Google Scholar

[15]

Theoret. Popul. Biol., 46 (1994), 363394. doi: 10.1006/tpbi.1994.1032.  Google Scholar

[16]

Nature, 442 (2006), 448-452. doi: 10.1038/nature04795.  Google Scholar

[17]

SIAM Rev, 42 (2000), 599-653. doi: 10.1137/S0036144500371907.  Google Scholar

[18]

Journal of Mathematical Analysis and Applications, 243 (2000), 429-452. doi: 10.1006/jmaa.1999.6679.  Google Scholar

[19]

Operations Research, 15 (1967), 139-146. doi: 10.1287/opre.15.1.139.  Google Scholar

[20]

Mathematical Models and methods in Applied Sciences, 15 (2005), 1519-1531. doi: 10.1142/S0218202505000856.  Google Scholar

[21]

Amsterdam: North-Holland, 1991.  Google Scholar

[22]

J. Theor. Biol., 265 (2010), 136-150. doi: 10.1016/j.jtbi.2010.04.003.  Google Scholar

[23]

Chapman & Hall, CRC Mathematical and Computational Biology series, 2007.  Google Scholar

[24]

Optimization Methods & Software Archive, 22 (2007), 959-969. Google Scholar

[25]

Nature, 432 (2004), 904-906. doi: 10.1038/nature03063.  Google Scholar

[26]

International Journal of Tourism Policy, 3 (2010), 1-15. doi: 10.1504/IJTP.2010.031599.  Google Scholar

[27]

Springer, 2006.  Google Scholar

[28]

Theor. Pop. Biol., Elsevier, 71 (2007), 20-29. Google Scholar

[29]

Wiley, New Jersey, 1962.  Google Scholar

[30]

Bull. Math. Biol., 72 (2009), 1-33. doi: 10.1007/s11538-009-9435-5.  Google Scholar

[31]

Second Edition, Springer, 2000. Google Scholar

[32]

Acta Biotheoretica, Springer, 2010. Google Scholar

[33]

PLoS ONE, www.plosone.org, 5 (2010). doi: 10.1371/journal.pone.0009018.  Google Scholar

[34]

Math. and Computer Modelling, 40 (2004), 1491-1506. doi: 10.1016/j.mcm.2005.01.007.  Google Scholar

[1]

Andrea Signori. Penalisation of long treatment time and optimal control of a tumour growth model of Cahn–Hilliard type with singular potential. Discrete & Continuous Dynamical Systems, 2021, 41 (6) : 2519-2542. doi: 10.3934/dcds.2020373

[2]

Tobias Geiger, Daniel Wachsmuth, Gerd Wachsmuth. Optimal control of ODEs with state suprema. Mathematical Control & Related Fields, 2021  doi: 10.3934/mcrf.2021012

[3]

Diana Keller. Optimal control of a linear stochastic Schrödinger equation. Conference Publications, 2013, 2013 (special) : 437-446. doi: 10.3934/proc.2013.2013.437

[4]

Lorenzo Freddi. Optimal control of the transmission rate in compartmental epidemics. Mathematical Control & Related Fields, 2021  doi: 10.3934/mcrf.2021007

[5]

Marzia Bisi, Maria Groppi, Giorgio Martalò, Romina Travaglini. Optimal control of leachate recirculation for anaerobic processes in landfills. Discrete & Continuous Dynamical Systems - B, 2021, 26 (6) : 2957-2976. doi: 10.3934/dcdsb.2020215

[6]

Luke Finlay, Vladimir Gaitsgory, Ivan Lebedev. Linear programming solutions of periodic optimization problems: approximation of the optimal control. Journal of Industrial & Management Optimization, 2007, 3 (2) : 399-413. doi: 10.3934/jimo.2007.3.399

[7]

Xiaohong Li, Mingxin Sun, Zhaohua Gong, Enmin Feng. Multistage optimal control for microbial fed-batch fermentation process. Journal of Industrial & Management Optimization, 2021  doi: 10.3934/jimo.2021040

[8]

John T. Betts, Stephen Campbell, Claire Digirolamo. Examination of solving optimal control problems with delays using GPOPS-Ⅱ. Numerical Algebra, Control & Optimization, 2021, 11 (2) : 283-305. doi: 10.3934/naco.2020026

[9]

Livia Betz, Irwin Yousept. Optimal control of elliptic variational inequalities with bounded and unbounded operators. Mathematical Control & Related Fields, 2021  doi: 10.3934/mcrf.2021009

[10]

Christian Meyer, Stephan Walther. Optimal control of perfect plasticity part I: Stress tracking. Mathematical Control & Related Fields, 2021  doi: 10.3934/mcrf.2021022

[11]

Shi'an Wang, N. U. Ahmed. Optimal control and stabilization of building maintenance units based on minimum principle. Journal of Industrial & Management Optimization, 2021, 17 (4) : 1713-1727. doi: 10.3934/jimo.2020041

[12]

Changjun Yu, Lei Yuan, Shuxuan Su. A new gradient computational formula for optimal control problems with time-delay. Journal of Industrial & Management Optimization, 2021  doi: 10.3934/jimo.2021076

[13]

Jaouad Danane. Optimal control of viral infection model with saturated infection rate. Numerical Algebra, Control & Optimization, 2021, 11 (3) : 363-375. doi: 10.3934/naco.2020031

[14]

Vladimir Gaitsgory, Ilya Shvartsman. Linear programming estimates for Cesàro and Abel limits of optimal values in optimal control problems. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021102

[15]

Shanjian Tang, Fu Zhang. Path-dependent optimal stochastic control and viscosity solution of associated Bellman equations. Discrete & Continuous Dynamical Systems, 2015, 35 (11) : 5521-5553. doi: 10.3934/dcds.2015.35.5521

[16]

Fabio Camilli, Serikbolsyn Duisembay, Qing Tang. Approximation of an optimal control problem for the time-fractional Fokker-Planck equation. Journal of Dynamics & Games, 2021  doi: 10.3934/jdg.2021013

[17]

Marita Holtmannspötter, Arnd Rösch, Boris Vexler. A priori error estimates for the space-time finite element discretization of an optimal control problem governed by a coupled linear PDE-ODE system. Mathematical Control & Related Fields, 2021  doi: 10.3934/mcrf.2021014

[18]

Bernard Bonnard, Jérémy Rouot. Geometric optimal techniques to control the muscular force response to functional electrical stimulation using a non-isometric force-fatigue model. Journal of Geometric Mechanics, 2021, 13 (1) : 1-23. doi: 10.3934/jgm.2020032

[19]

Kehan Si, Zhenda Xu, Ka Fai Cedric Yiu, Xun Li. Open-loop solvability for mean-field stochastic linear quadratic optimal control problems of Markov regime-switching system. Journal of Industrial & Management Optimization, 2021  doi: 10.3934/jimo.2021074

[20]

Wenjuan Zhao, Shunfu Jin, Wuyi Yue. A stochastic model and social optimization of a blockchain system based on a general limited batch service queue. Journal of Industrial & Management Optimization, 2021, 17 (4) : 1845-1861. doi: 10.3934/jimo.2020049

2018 Impact Factor: 1.313

Metrics

  • PDF downloads (83)
  • HTML views (0)
  • Cited by (11)

[Back to Top]