American Institute of Mathematical Sciences

Advanced
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

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.  Google Scholar

[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.  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.  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.  doi: 10.1007/s00285-011-0479-4.  Google Scholar

[5]

C. Browne, Two Extensions of a Classical Virus Model,, PhD thesis, (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.  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.  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.  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.  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.  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.  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.  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.  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.  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.  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.  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.  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.  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.  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.  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.  doi: 10.1007/s11538-013-9820-y.  Google Scholar

show all references

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.  Google Scholar

[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.  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.  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.  doi: 10.1007/s00285-011-0479-4.  Google Scholar

[5]

C. Browne, Two Extensions of a Classical Virus Model,, PhD thesis, (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.  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.  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.  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.  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.  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.  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.  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.  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.  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.  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.  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.  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.  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.  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.  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.  doi: 10.1007/s11538-013-9820-y.  Google Scholar

[1]

Adam Sullivan, Folashade Agusto, Sharon Bewick, Chunlei Su, Suzanne Lenhart, Xiaopeng Zhao. A mathematical model for within-host Toxoplasma gondii invasion dynamics. Mathematical Biosciences & Engineering, 2012, 9 (3) : 647-662. doi: 10.3934/mbe.2012.9.647
[2]

Cameron J. Browne, Sergei S. Pilyugin. Global analysis of age-structured within-host virus model. Discrete & Continuous Dynamical Systems - B, 2013, 18 (8) : 1999-2017. doi: 10.3934/dcdsb.2013.18.1999
[3]

Yilong Li, Shigui Ruan, Dongmei Xiao. The Within-Host dynamics of malaria infection with immune response. Mathematical Biosciences & Engineering, 2011, 8 (4) : 999-1018. doi: 10.3934/mbe.2011.8.999
[4]

Andrei Korobeinikov, Conor Dempsey. A continuous phenotype space model of RNA virus evolution within a host. Mathematical Biosciences & Engineering, 2014, 11 (4) : 919-927. doi: 10.3934/mbe.2014.11.919
[5]

Shengqiang Liu, Lin Wang. Global stability of an HIV-1 model with distributed intracellular delays and a combination therapy. Mathematical Biosciences & Engineering, 2010, 7 (3) : 675-685. doi: 10.3934/mbe.2010.7.675
[6]

Urszula Foryś, Jan Poleszczuk. A delay-differential equation model of HIV related cancer--immune system dynamics. Mathematical Biosciences & Engineering, 2011, 8 (2) : 627-641. doi: 10.3934/mbe.2011.8.627
[7]

Tao Feng, Zhipeng Qiu, Xinzhu Meng. Dynamics of a stochastic hepatitis C virus system with host immunity. Discrete & Continuous Dynamical Systems - B, 2019, 24 (12) : 6367-6385. doi: 10.3934/dcdsb.2019143
[8]

Stephen Pankavich, Christian Parkinson. Mathematical analysis of an in-host model of viral dynamics with spatial heterogeneity. Discrete & Continuous Dynamical Systems - B, 2016, 21 (4) : 1237-1257. doi: 10.3934/dcdsb.2016.21.1237
[9]

Hossein Pourbashash, Sergei S. Pilyugin, Patrick De Leenheer, Connell McCluskey. Global analysis of within host virus models with cell-to-cell viral transmission. Discrete & Continuous Dynamical Systems - B, 2014, 19 (10) : 3341-3357. doi: 10.3934/dcdsb.2014.19.3341
[10]

Hui Cao, Yicang Zhou. The basic reproduction number of discrete SIR and SEIS models with periodic parameters. Discrete & Continuous Dynamical Systems - B, 2013, 18 (1) : 37-56. doi: 10.3934/dcdsb.2013.18.37
[11]

Yun Tian, Yu Bai, Pei Yu. Impact of delay on HIV-1 dynamics of fighting a virus with another virus. Mathematical Biosciences & Engineering, 2014, 11 (5) : 1181-1198. doi: 10.3934/mbe.2014.11.1181
[12]

Zhaohui Yuan, Xingfu Zou. Global threshold dynamics in an HIV virus model with nonlinear infection rate and distributed invasion and production delays. Mathematical Biosciences & Engineering, 2013, 10 (2) : 483-498. doi: 10.3934/mbe.2013.10.483
[13]

Alexander S. Bratus, Svetlana Yu. Kovalenko, Elena Fimmel. On viable therapy strategy for a mathematical spatial cancer model describing the dynamics of malignant and healthy cells. Mathematical Biosciences & Engineering, 2015, 12 (1) : 163-183. doi: 10.3934/mbe.2015.12.163
[14]

Nicolas Bacaër, Xamxinur Abdurahman, Jianli Ye, Pierre Auger. On the basic reproduction number $R_0$ in sexual activity models for HIV/AIDS epidemics: Example from Yunnan, China. Mathematical Biosciences & Engineering, 2007, 4 (4) : 595-607. doi: 10.3934/mbe.2007.4.595
[15]

Nirav Dalal, David Greenhalgh, Xuerong Mao. Mathematical modelling of internal HIV dynamics. Discrete & Continuous Dynamical Systems - B, 2009, 12 (2) : 305-321. doi: 10.3934/dcdsb.2009.12.305
[16]

Rachid Ouifki, Gareth Witten. A model of HIV-1 infection with HAART therapy and intracellular delays. Discrete & Continuous Dynamical Systems - B, 2007, 8 (1) : 229-240. doi: 10.3934/dcdsb.2007.8.229
[17]

Tianhui Yang, Lei Zhang. Remarks on basic reproduction ratios for periodic abstract functional differential equations. Discrete & Continuous Dynamical Systems - B, 2019, 24 (12) : 6771-6782. doi: 10.3934/dcdsb.2019166
[18]

Nikolay Pertsev, Konstantin Loginov, Gennady Bocharov. Nonlinear effects in the dynamics of HIV-1 infection predicted by mathematical model with multiple delays. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 0-0. doi: 10.3934/dcdss.2020141
[19]

Stephen Pankavich, Deborah Shutt. An in-host model of HIV incorporating latent infection and viral mutation. Conference Publications, 2015, 2015 (special) : 913-922. doi: 10.3934/proc.2015.0913
[20]

Bing Li, Yuming Chen, Xuejuan Lu, Shengqiang Liu. A delayed HIV-1 model with virus waning term. Mathematical Biosciences & Engineering, 2016, 13 (1) : 135-157. doi: 10.3934/mbe.2016.13.135

2018 Impact Factor: 1.008

Tools

Metrics

  • PDF downloads (28)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences