American Institute of Mathematical Sciences

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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

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
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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

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[29]

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[30]

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

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