Minimizing \mathcal R_0 for in-host virus model with periodic combination antiviral therapy

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December 2016, 21(10): 3315-3330. doi: 10.3934/dcdsb.2016099

Minimizing $\mathcal R_0$ for in-host virus model with periodic combination antiviral therapy

Cameron J. Browne ^1, and Sergei S. Pilyugin ^2,

1.	Department of Mathematics, University of Louisiana at Lafayette, Lafayette, LA 70504, United States
2.	Department of Mathematics, University of Florida, 1400 Stadium Road, Gainesville, FL 32611

Received December 2015 Revised May 2016 Published November 2016

Abstract
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The aim of this article is to provide formulas and properties of the basic reproduction number, $\mathcal R_0$, for a within-host virus model with periodic combination drug treatment. In particular, we extend and further results about how the phase difference between drug treatment can critically affect the asymptotic dynamics of model and corresponding $\mathcal R_0$. Our main theorem establishes that $\mathcal R_0$ is minimized for out-of-phase efficacies and maximized for in-phase efficacies in a special case where drug efficacies are ``bang-bang'' functions.

Keywords: optimization, within-host virus dynamics, HIV, Mathematical model, periodic differential equation system, reproduction number., combination antiviral therapy.

Mathematics Subject Classification: Primary: 34C60; Secondary: 92B0.

Citation: Cameron J. Browne, Sergei S. Pilyugin. Minimizing $\mathcal R_0$ for in-host virus model with periodic combination antiviral therapy. Discrete & Continuous Dynamical Systems - B, 2016, 21 (10) : 3315-3330. doi: 10.3934/dcdsb.2016099

References:

[1]	B. Adams, H. Banks, H. Kwon and H. Tran, Dynamic multidrug therapies for HIV: Optimal and STI control approaches,, Mathematical Biosciences and Engineering, 1 (2004), 223. doi: 10.3934/mbe.2004.1.223.
[2]	N. Bacaër and S. Guernaoui, The epidemic threshold of vector-borne diseases with seasonality,, Journal of Mathematical Biology, 53 (2006), 421. doi: 10.1007/s00285-006-0015-0.
[3]	N. Bacaër and R. Ouifki, Growth rate and basic reproduction number for population models with a simple periodic factor,, Mathematical Biosciences, 210 (2007), 647. doi: 10.1016/j.mbs.2007.07.005.
[4]	N. Bacaër et al., On the biological interpretation of a definition for the parameter r 0 in periodic population models,, Journal of Mathematical Biology, 65 (2012), 601. doi: 10.1007/s00285-011-0479-4.
[5]	C. Browne, Two Extensions of a Classical Virus Model,, PhD thesis, (2012).
[6]	C. J. Browne and S. S. Pilyugin, Periodic multidrug therapy in a within-host virus model,, Bulletin of Mathematical Biology, 74 (2012), 562. doi: 10.1007/s11538-011-9677-x.
[7]	C. J. Browne and S. S. Pilyugin, Global analysis of age-structured within-host virus model,, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), 1999. doi: 10.3934/dcdsb.2013.18.1999.
[8]	C. J. Browne, R. J. Smith and L. Bourouiba, From regional pulse vaccination to global disease eradication: Insights from a mathematical model of poliomyelitis,, Journal of Mathematical Biology, 71 (2015), 215. doi: 10.1007/s00285-014-0810-y.
[9]	J. M. Conway and A. S. Perelson, Post-treatment control of HIV infection,, Proceedings of the National Academy of Sciences, 112 (2015), 5467. doi: 10.1073/pnas.1419162112.
[10]	P. De Leenheer, Within-host virus models with periodic antiviral therapy,, Bulletin of Mathematical Biology, 71 (2009), 189. doi: 10.1007/s11538-008-9359-5.
[11]	P. De Leenheer and S. S. Pilyugin, Multistrain virus dynamics with mutations: A global analysis,, Mathematical Medicine and Biology, 25 (2008), 285. doi: 10.1093/imammb/dqn023.
[12]	N. M. Dixit and A. S. Perelson, Complex patterns of viral load decay under antiretroviral therapy: Influence of pharmacokinetics and intracellular delay,, Journal of Theoretical Biology, 226 (2004), 95. doi: 10.1016/j.jtbi.2003.09.002.
[13]	A. d'Onofrio, Periodically varying antiviral therapies: Conditions for global stability of the virus free state,, Applied Mathematics and Computation, 168 (2005), 945. doi: 10.1016/j.amc.2004.09.014.
[14]	M. A. Nowak and R. M. May, Virus Dynamics,, 2000., ().
[15]	A. S. Perelson and P. W. Nelson, Mathematical analysis of HIV-1 dynamics in vivo,, SIAM Review, 41 (1999), 3. doi: 10.1137/S0036144598335107.
[16]	L. Rong, Z. Feng and A. S. Perelson, Emergence of HIV-1 drug resistance during antiretroviral treatment,, Bulletin of Mathematical Biology, 69 (2007), 2027. doi: 10.1007/s11538-007-9203-3.
[17]	D. I. Rosenbloom, A. L. Hill, S. A. Rabi, R. F. Siliciano and M. A. Nowak, Antiretroviral dynamics determines hiv evolution and predicts therapy outcome,, Nature Medicine, 18 (2012), 1378. doi: 10.1038/nm.2892.
[18]	H. L. Smith and P. De Leenheer, Virus dynamics: a global analysis,, SIAM Journal on Applied Mathematics, 63 (2003), 1313. doi: 10.1137/S0036139902406905.
[19]	W. Wang and X.-Q. Zhao, Threshold dynamics for compartmental epidemic models in periodic environments,, Journal of Dynamics and Differential Equations, 20 (2008), 699. doi: 10.1007/s10884-008-9111-8.
[20]	X. Wang, X. Song, S. Tang and L. Rong, Dynamics of an HIV model with multiple infection stages and treatment with different drug classes,, Bulletin of Mathematical Biology, 78 (2016), 322. doi: 10.1007/s11538-016-0145-5.
[21]	Y. Wang, F. Brauer, J. Wu and J. M. Heffernan, A delay-dependent model with HIV drug resistance during therapy,, Journal of Mathematical Analysis and Applications, 414 (2014), 514. doi: 10.1016/j.jmaa.2013.12.064.
[22]	Z. Wang and X.-Q. Zhao, A within-host virus model with periodic multidrug therapy,, Bulletin of Mathematical Biology, 75 (2013), 543. doi: 10.1007/s11538-013-9820-y.

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References:

[1]	B. Adams, H. Banks, H. Kwon and H. Tran, Dynamic multidrug therapies for HIV: Optimal and STI control approaches,, Mathematical Biosciences and Engineering, 1 (2004), 223. doi: 10.3934/mbe.2004.1.223.
[2]	N. Bacaër and S. Guernaoui, The epidemic threshold of vector-borne diseases with seasonality,, Journal of Mathematical Biology, 53 (2006), 421. doi: 10.1007/s00285-006-0015-0.
[3]	N. Bacaër and R. Ouifki, Growth rate and basic reproduction number for population models with a simple periodic factor,, Mathematical Biosciences, 210 (2007), 647. doi: 10.1016/j.mbs.2007.07.005.
[4]	N. Bacaër et al., On the biological interpretation of a definition for the parameter r 0 in periodic population models,, Journal of Mathematical Biology, 65 (2012), 601. doi: 10.1007/s00285-011-0479-4.
[5]	C. Browne, Two Extensions of a Classical Virus Model,, PhD thesis, (2012).
[6]	C. J. Browne and S. S. Pilyugin, Periodic multidrug therapy in a within-host virus model,, Bulletin of Mathematical Biology, 74 (2012), 562. doi: 10.1007/s11538-011-9677-x.
[7]	C. J. Browne and S. S. Pilyugin, Global analysis of age-structured within-host virus model,, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), 1999. doi: 10.3934/dcdsb.2013.18.1999.
[8]	C. J. Browne, R. J. Smith and L. Bourouiba, From regional pulse vaccination to global disease eradication: Insights from a mathematical model of poliomyelitis,, Journal of Mathematical Biology, 71 (2015), 215. doi: 10.1007/s00285-014-0810-y.
[9]	J. M. Conway and A. S. Perelson, Post-treatment control of HIV infection,, Proceedings of the National Academy of Sciences, 112 (2015), 5467. doi: 10.1073/pnas.1419162112.
[10]	P. De Leenheer, Within-host virus models with periodic antiviral therapy,, Bulletin of Mathematical Biology, 71 (2009), 189. doi: 10.1007/s11538-008-9359-5.
[11]	P. De Leenheer and S. S. Pilyugin, Multistrain virus dynamics with mutations: A global analysis,, Mathematical Medicine and Biology, 25 (2008), 285. doi: 10.1093/imammb/dqn023.
[12]	N. M. Dixit and A. S. Perelson, Complex patterns of viral load decay under antiretroviral therapy: Influence of pharmacokinetics and intracellular delay,, Journal of Theoretical Biology, 226 (2004), 95. doi: 10.1016/j.jtbi.2003.09.002.
[13]	A. d'Onofrio, Periodically varying antiviral therapies: Conditions for global stability of the virus free state,, Applied Mathematics and Computation, 168 (2005), 945. doi: 10.1016/j.amc.2004.09.014.
[14]	M. A. Nowak and R. M. May, Virus Dynamics,, 2000., ().
[15]	A. S. Perelson and P. W. Nelson, Mathematical analysis of HIV-1 dynamics in vivo,, SIAM Review, 41 (1999), 3. doi: 10.1137/S0036144598335107.
[16]	L. Rong, Z. Feng and A. S. Perelson, Emergence of HIV-1 drug resistance during antiretroviral treatment,, Bulletin of Mathematical Biology, 69 (2007), 2027. doi: 10.1007/s11538-007-9203-3.
[17]	D. I. Rosenbloom, A. L. Hill, S. A. Rabi, R. F. Siliciano and M. A. Nowak, Antiretroviral dynamics determines hiv evolution and predicts therapy outcome,, Nature Medicine, 18 (2012), 1378. doi: 10.1038/nm.2892.
[18]	H. L. Smith and P. De Leenheer, Virus dynamics: a global analysis,, SIAM Journal on Applied Mathematics, 63 (2003), 1313. doi: 10.1137/S0036139902406905.
[19]	W. Wang and X.-Q. Zhao, Threshold dynamics for compartmental epidemic models in periodic environments,, Journal of Dynamics and Differential Equations, 20 (2008), 699. doi: 10.1007/s10884-008-9111-8.
[20]	X. Wang, X. Song, S. Tang and L. Rong, Dynamics of an HIV model with multiple infection stages and treatment with different drug classes,, Bulletin of Mathematical Biology, 78 (2016), 322. doi: 10.1007/s11538-016-0145-5.
[21]	Y. Wang, F. Brauer, J. Wu and J. M. Heffernan, A delay-dependent model with HIV drug resistance during therapy,, Journal of Mathematical Analysis and Applications, 414 (2014), 514. doi: 10.1016/j.jmaa.2013.12.064.
[22]	Z. Wang and X.-Q. Zhao, A within-host virus model with periodic multidrug therapy,, Bulletin of Mathematical Biology, 75 (2013), 543. doi: 10.1007/s11538-013-9820-y.

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