Global analysis of within host virus models with cell-to-cell viral transmission

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December 2014, 19(10): 3341-3357. doi: 10.3934/dcdsb.2014.19.3341

Global analysis of within host virus models with cell-to-cell viral transmission

Hossein Pourbashash ^1, , Sergei S. Pilyugin ^1, , Patrick De Leenheer ^1, and Connell McCluskey ^2,

1.	Department of Mathematics, University of Florida, 1400 Stadium Road, Gainesville, FL 32611, United States, United States, United States
2.	Department of Mathematics, Wilfrid Laurier University, 75 University Avenue West, Waterloo, ON, N2L 3C5, Canada

Received February 2013 Revised April 2013 Published October 2014

Abstract
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Recent experimental studies have shown that HIV can be transmitted directly from cell to cell when structures called virological synapses form during interactions between T cells. In this article, we describe a new within-host model of HIV infection that incorporates two mechanisms: infection by free virions and the direct cell-to-cell transmission. We conduct the local and global stability analysis of the model. We show that if the basic reproduction number ${\mathcal R}_0\leq 1$, the virus is cleared and the disease dies out; if ${\mathcal R}_0>1$, the virus persists in the host. We also prove that the unique positive equilibrium attracts all positive solutions under additional assumptions on the parameters. Finally, a multi strain model incorporating cell-to-cell viral transmission is proposed and shown to exhibit a competitive exclusion principle.

Keywords: Lyapunov function, global stability, Within-host virus model, second compound matrices., cell-to-cell transmission.

Mathematics Subject Classification: Primary: 37N25, 97M60; Secondary: 34A34, 34D2.

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

References:

[1]	W. A. Coppel, Stability and Asymptotic Behavior of Differential Equations, Heath and Co., Boston, 1965.
[2]	P. De Leenheer and H. L. Smith, Virus dynamics: A global analysis, SIAM J. Appl. Math., 63 (2003), 1313-1327. doi: 10.1137/S0036139902406905.
[3]	P. De Leenheer and S. S. Pilyugin, Multistrain virus dynamics with mutations: A global analysis, Math. Med. Biol., 25 (2008), 285-322.
[4]	N. Dixit and A. Perelson, Multiplicity of human immunodeficiency virus infections in lymphoid tissue, J. Virol., 78 (2004), 8942-8945. doi: 10.1128/JVI.78.16.8942-8945.2004.
[5]	M. Fiedler, Additive compound matrices and inequality for eigenvalues of stochastic matrices, Czech. Math. J., 24 (1974), 392-402.
[6]	H. I. Freedman, M. X. Tang and S. G. Ruan, Uniform persistence and flows near a closed positively invariant set, J. Dynam. Differential Equations, 6 (1994), 583-600. doi: 10.1007/BF02218848.
[7]	H. K. Khalil, Nonlinear Systems, 3rd Edition, Prentice Hall, 2002.
[8]	A. Korobeinikov, Global properties of SIR and SEIR epidemic models with multiple parallel infectious stages, Bull. Math. Biol., 71 (2009), 75-83. doi: 10.1007/s11538-008-9352-z.
[9]	M. Y. Li, J. R. Graef, L. Wang and J. Karsai, Global dynamics of a SEIR model with varying total population size, Math. Biosci., 160 (1999), 191-213. doi: 10.1016/S0025-5564(99)00030-9.
[10]	M. Y. Li and J. S. Muldowney, A geometric approach to the global-stability problems, SIAM J. Math. Anal., 27 (1996), 1070-1083. doi: 10.1137/S0036141094266449.
[11]	M. Y. Li and J. S. Muldowney, Global stability for the SEIR model in epidemiology, Math. Biosci., 125 (1995), 155-164. doi: 10.1016/0025-5564(95)92756-5.
[12]	R. H. Jr. Martin, Logarithmic norms and projections applied to linear differential systems, J. Math. Anal. Appl., 45 (1974), 432-454. doi: 10.1016/0022-247X(74)90084-5.
[13]	D. Mazurov, A. Ilinskaya, G. Heidecker, P. Lloyd and D. Derse, Quantitative comparison of HTLV-1 and HIV-1 Cell-to- Cell infection with new replication dependent vectors, PLoS Pathogens, 6 (2010), e1000788. doi: 10.1371/journal.ppat.1000788.
[14]	B. Monel, E. Beaumont, D. Vendrame, O. Schwartz, D. Brand and F. Mammano, HIV cell-to-cell transmission requires the production of infectious virus particles and does not proceed through Env-mediated fusion pores, J. Virol., 86 (2012), 3924-3933. doi: 10.1128/JVI.06478-11.
[15]	J. S. Muldowney, Compound matrices and ordinary differential equations, Rocky Mount. J. Math., 20 (1990), 857-872. doi: 10.1216/rmjm/1181073047.
[16]	M. A. Nowak and R. M. May, Virus Dynamics, Oxford University press, New York, 2000.
[17]	A. S. Perelson and P. W. Nelson, Mathematical analysis of HIV-1 dynamics in vivo, SIAM Rev., 41 (1999), 3-44. doi: 10.1137/S0036144598335107.
[18]	V. Piguet and Q. Sattentau, Dangerous liaisons at the virological synapse, J. Clin. Invest., 114 (2004), 605-610. doi: 10.1172/JCI200422812.
[19]	O. Schwartz, Immunological and virological aspects of HIV cell-to-cell transfer, Retrovirology, 6 (2009), I16. doi: 10.1186/1742-4690-6-S2-I16.
[20]	H. L. Smith and P. Waltman, Perturbation of a globally stable steady state, Proc. Am. Math. Soc., 127 (1999), 447-453. doi: 10.1090/S0002-9939-99-04768-1.
[21]	M. Sourisseau, N. Sol-Foulon, F. Porrot, F. Blanchet and O. Schwartz, Inefficient human immunodeficiency virus replication in mobile lymphocytes, J. Virol., 81 (2007), 1000-1012. doi: 10.1128/JVI.01629-06.
[22]	P. van den Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosci., 180 (2002), 29-48. doi: 10.1016/S0025-5564(02)00108-6.
[23]	L. Wang and S. Ellermeyer, HIV infection and $CD4^+$ T cell dynamics, Discrete Contin. Dyn. Syst. Ser. B, 6 (2006), 1417-1430. doi: 10.3934/dcdsb.2006.6.1417.
[24]	L. Wang and M. Y. Li, Mathematical analysis of the global dynamics of a model for HIV infection of $CD4^{+}$ T cells, Math. Biosci., 200 (2006), 44-57. doi: 10.1016/j.mbs.2005.12.026.

show all references

References:

[1]	W. A. Coppel, Stability and Asymptotic Behavior of Differential Equations, Heath and Co., Boston, 1965.
[2]	P. De Leenheer and H. L. Smith, Virus dynamics: A global analysis, SIAM J. Appl. Math., 63 (2003), 1313-1327. doi: 10.1137/S0036139902406905.
[3]	P. De Leenheer and S. S. Pilyugin, Multistrain virus dynamics with mutations: A global analysis, Math. Med. Biol., 25 (2008), 285-322.
[4]	N. Dixit and A. Perelson, Multiplicity of human immunodeficiency virus infections in lymphoid tissue, J. Virol., 78 (2004), 8942-8945. doi: 10.1128/JVI.78.16.8942-8945.2004.
[5]	M. Fiedler, Additive compound matrices and inequality for eigenvalues of stochastic matrices, Czech. Math. J., 24 (1974), 392-402.
[6]	H. I. Freedman, M. X. Tang and S. G. Ruan, Uniform persistence and flows near a closed positively invariant set, J. Dynam. Differential Equations, 6 (1994), 583-600. doi: 10.1007/BF02218848.
[7]	H. K. Khalil, Nonlinear Systems, 3rd Edition, Prentice Hall, 2002.
[8]	A. Korobeinikov, Global properties of SIR and SEIR epidemic models with multiple parallel infectious stages, Bull. Math. Biol., 71 (2009), 75-83. doi: 10.1007/s11538-008-9352-z.
[9]	M. Y. Li, J. R. Graef, L. Wang and J. Karsai, Global dynamics of a SEIR model with varying total population size, Math. Biosci., 160 (1999), 191-213. doi: 10.1016/S0025-5564(99)00030-9.
[10]	M. Y. Li and J. S. Muldowney, A geometric approach to the global-stability problems, SIAM J. Math. Anal., 27 (1996), 1070-1083. doi: 10.1137/S0036141094266449.
[11]	M. Y. Li and J. S. Muldowney, Global stability for the SEIR model in epidemiology, Math. Biosci., 125 (1995), 155-164. doi: 10.1016/0025-5564(95)92756-5.
[12]	R. H. Jr. Martin, Logarithmic norms and projections applied to linear differential systems, J. Math. Anal. Appl., 45 (1974), 432-454. doi: 10.1016/0022-247X(74)90084-5.
[13]	D. Mazurov, A. Ilinskaya, G. Heidecker, P. Lloyd and D. Derse, Quantitative comparison of HTLV-1 and HIV-1 Cell-to- Cell infection with new replication dependent vectors, PLoS Pathogens, 6 (2010), e1000788. doi: 10.1371/journal.ppat.1000788.
[14]	B. Monel, E. Beaumont, D. Vendrame, O. Schwartz, D. Brand and F. Mammano, HIV cell-to-cell transmission requires the production of infectious virus particles and does not proceed through Env-mediated fusion pores, J. Virol., 86 (2012), 3924-3933. doi: 10.1128/JVI.06478-11.
[15]	J. S. Muldowney, Compound matrices and ordinary differential equations, Rocky Mount. J. Math., 20 (1990), 857-872. doi: 10.1216/rmjm/1181073047.
[16]	M. A. Nowak and R. M. May, Virus Dynamics, Oxford University press, New York, 2000.
[17]	A. S. Perelson and P. W. Nelson, Mathematical analysis of HIV-1 dynamics in vivo, SIAM Rev., 41 (1999), 3-44. doi: 10.1137/S0036144598335107.
[18]	V. Piguet and Q. Sattentau, Dangerous liaisons at the virological synapse, J. Clin. Invest., 114 (2004), 605-610. doi: 10.1172/JCI200422812.
[19]	O. Schwartz, Immunological and virological aspects of HIV cell-to-cell transfer, Retrovirology, 6 (2009), I16. doi: 10.1186/1742-4690-6-S2-I16.
[20]	H. L. Smith and P. Waltman, Perturbation of a globally stable steady state, Proc. Am. Math. Soc., 127 (1999), 447-453. doi: 10.1090/S0002-9939-99-04768-1.
[21]	M. Sourisseau, N. Sol-Foulon, F. Porrot, F. Blanchet and O. Schwartz, Inefficient human immunodeficiency virus replication in mobile lymphocytes, J. Virol., 81 (2007), 1000-1012. doi: 10.1128/JVI.01629-06.
[22]	P. van den Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosci., 180 (2002), 29-48. doi: 10.1016/S0025-5564(02)00108-6.
[23]	L. Wang and S. Ellermeyer, HIV infection and $CD4^+$ T cell dynamics, Discrete Contin. Dyn. Syst. Ser. B, 6 (2006), 1417-1430. doi: 10.3934/dcdsb.2006.6.1417.
[24]	L. Wang and M. Y. Li, Mathematical analysis of the global dynamics of a model for HIV infection of $CD4^{+}$ T cells, Math. Biosci., 200 (2006), 44-57. doi: 10.1016/j.mbs.2005.12.026.

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