# American Institute of Mathematical Sciences

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

 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

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.
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), 223-241. doi: 10.3934/mbe.2004.1.223.  Google Scholar [2] N. Bacaër and S. Guernaoui, The epidemic threshold of vector-borne diseases with seasonality, Journal of Mathematical Biology, 53 (2006), 421-436. doi: 10.1007/s00285-006-0015-0.  Google Scholar [3] N. Bacaër and R. Ouifki, Growth rate and basic reproduction number for population models with a simple periodic factor, Mathematical Biosciences, 210 (2007), 647-658. doi: 10.1016/j.mbs.2007.07.005.  Google Scholar [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), 601-621. doi: 10.1007/s00285-011-0479-4.  Google Scholar [5] C. Browne, Two Extensions of a Classical Virus Model, PhD thesis, University of Florida, 2012. Google Scholar [6] C. J. Browne and S. S. Pilyugin, Periodic multidrug therapy in a within-host virus model, Bulletin of Mathematical Biology, 74 (2012), 562-589. doi: 10.1007/s11538-011-9677-x.  Google Scholar [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), 1999-2017. doi: 10.3934/dcdsb.2013.18.1999.  Google Scholar [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), 215-253. doi: 10.1007/s00285-014-0810-y.  Google Scholar [9] J. M. Conway and A. S. Perelson, Post-treatment control of HIV infection, Proceedings of the National Academy of Sciences, 112 (2015), 5467-5472. doi: 10.1073/pnas.1419162112.  Google Scholar [10] P. De Leenheer, Within-host virus models with periodic antiviral therapy, Bulletin of Mathematical Biology, 71 (2009), 189-210. doi: 10.1007/s11538-008-9359-5.  Google Scholar [11] P. De Leenheer and S. S. Pilyugin, Multistrain virus dynamics with mutations: A global analysis, Mathematical Medicine and Biology, 25 (2008), 285-322. doi: 10.1093/imammb/dqn023.  Google Scholar [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), 95-109. doi: 10.1016/j.jtbi.2003.09.002.  Google Scholar [13] A. d'Onofrio, Periodically varying antiviral therapies: Conditions for global stability of the virus free state, Applied Mathematics and Computation, 168 (2005), 945-953. doi: 10.1016/j.amc.2004.09.014.  Google Scholar [14] M. A. Nowak and R. M. May, Virus Dynamics,, 2000., ().   Google Scholar [15] A. S. Perelson and P. W. Nelson, Mathematical analysis of HIV-1 dynamics in vivo, SIAM Review, 41 (1999), 3-44. doi: 10.1137/S0036144598335107.  Google Scholar [16] L. Rong, Z. Feng and A. S. Perelson, Emergence of HIV-1 drug resistance during antiretroviral treatment, Bulletin of Mathematical Biology, 69 (2007), 2027-2060. doi: 10.1007/s11538-007-9203-3.  Google Scholar [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), 1378-1385. doi: 10.1038/nm.2892.  Google Scholar [18] H. L. Smith and P. De Leenheer, Virus dynamics: a global analysis, SIAM Journal on Applied Mathematics, 63 (2003), 1313-1327. doi: 10.1137/S0036139902406905.  Google Scholar [19] W. Wang and X.-Q. Zhao, Threshold dynamics for compartmental epidemic models in periodic environments, Journal of Dynamics and Differential Equations, 20 (2008), 699-717. doi: 10.1007/s10884-008-9111-8.  Google Scholar [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), 322-349. doi: 10.1007/s11538-016-0145-5.  Google Scholar [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), 514-531. doi: 10.1016/j.jmaa.2013.12.064.  Google Scholar [22] Z. Wang and X.-Q. Zhao, A within-host virus model with periodic multidrug therapy, Bulletin of Mathematical Biology, 75 (2013), 543-563. doi: 10.1007/s11538-013-9820-y.  Google Scholar

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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), 223-241. doi: 10.3934/mbe.2004.1.223.  Google Scholar [2] N. Bacaër and S. Guernaoui, The epidemic threshold of vector-borne diseases with seasonality, Journal of Mathematical Biology, 53 (2006), 421-436. doi: 10.1007/s00285-006-0015-0.  Google Scholar [3] N. Bacaër and R. Ouifki, Growth rate and basic reproduction number for population models with a simple periodic factor, Mathematical Biosciences, 210 (2007), 647-658. doi: 10.1016/j.mbs.2007.07.005.  Google Scholar [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), 601-621. doi: 10.1007/s00285-011-0479-4.  Google Scholar [5] C. Browne, Two Extensions of a Classical Virus Model, PhD thesis, University of Florida, 2012. Google Scholar [6] C. J. Browne and S. S. Pilyugin, Periodic multidrug therapy in a within-host virus model, Bulletin of Mathematical Biology, 74 (2012), 562-589. doi: 10.1007/s11538-011-9677-x.  Google Scholar [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), 1999-2017. doi: 10.3934/dcdsb.2013.18.1999.  Google Scholar [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), 215-253. doi: 10.1007/s00285-014-0810-y.  Google Scholar [9] J. M. Conway and A. S. Perelson, Post-treatment control of HIV infection, Proceedings of the National Academy of Sciences, 112 (2015), 5467-5472. doi: 10.1073/pnas.1419162112.  Google Scholar [10] P. De Leenheer, Within-host virus models with periodic antiviral therapy, Bulletin of Mathematical Biology, 71 (2009), 189-210. doi: 10.1007/s11538-008-9359-5.  Google Scholar [11] P. De Leenheer and S. S. Pilyugin, Multistrain virus dynamics with mutations: A global analysis, Mathematical Medicine and Biology, 25 (2008), 285-322. doi: 10.1093/imammb/dqn023.  Google Scholar [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), 95-109. doi: 10.1016/j.jtbi.2003.09.002.  Google Scholar [13] A. d'Onofrio, Periodically varying antiviral therapies: Conditions for global stability of the virus free state, Applied Mathematics and Computation, 168 (2005), 945-953. doi: 10.1016/j.amc.2004.09.014.  Google Scholar [14] M. A. Nowak and R. M. May, Virus Dynamics,, 2000., ().   Google Scholar [15] A. S. Perelson and P. W. Nelson, Mathematical analysis of HIV-1 dynamics in vivo, SIAM Review, 41 (1999), 3-44. doi: 10.1137/S0036144598335107.  Google Scholar [16] L. Rong, Z. Feng and A. S. Perelson, Emergence of HIV-1 drug resistance during antiretroviral treatment, Bulletin of Mathematical Biology, 69 (2007), 2027-2060. doi: 10.1007/s11538-007-9203-3.  Google Scholar [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), 1378-1385. doi: 10.1038/nm.2892.  Google Scholar [18] H. L. Smith and P. De Leenheer, Virus dynamics: a global analysis, SIAM Journal on Applied Mathematics, 63 (2003), 1313-1327. doi: 10.1137/S0036139902406905.  Google Scholar [19] W. Wang and X.-Q. Zhao, Threshold dynamics for compartmental epidemic models in periodic environments, Journal of Dynamics and Differential Equations, 20 (2008), 699-717. doi: 10.1007/s10884-008-9111-8.  Google Scholar [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), 322-349. doi: 10.1007/s11538-016-0145-5.  Google Scholar [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), 514-531. doi: 10.1016/j.jmaa.2013.12.064.  Google Scholar [22] Z. Wang and X.-Q. Zhao, A within-host virus model with periodic multidrug therapy, Bulletin of Mathematical Biology, 75 (2013), 543-563. doi: 10.1007/s11538-013-9820-y.  Google Scholar
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