# American Institute of Mathematical Sciences

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

 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

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.
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 & 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., (1965). Google Scholar [2] P. De Leenheer and H. L. Smith, Virus dynamics: A global analysis,, SIAM J. Appl. Math., 63 (2003), 1313. doi: 10.1137/S0036139902406905. Google Scholar [3] P. De Leenheer and S. S. Pilyugin, Multistrain virus dynamics with mutations: A global analysis,, Math. Med. Biol., 25 (2008), 285. Google Scholar [4] N. Dixit and A. Perelson, Multiplicity of human immunodeficiency virus infections in lymphoid tissue,, J. Virol., 78 (2004), 8942. doi: 10.1128/JVI.78.16.8942-8945.2004. Google Scholar [5] M. Fiedler, Additive compound matrices and inequality for eigenvalues of stochastic matrices,, Czech. Math. J., 24 (1974), 392. Google Scholar [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. doi: 10.1007/BF02218848. Google Scholar [7] H. K. Khalil, Nonlinear Systems,, 3rd Edition, (2002). Google Scholar [8] A. Korobeinikov, Global properties of SIR and SEIR epidemic models with multiple parallel infectious stages,, Bull. Math. Biol., 71 (2009), 75. doi: 10.1007/s11538-008-9352-z. Google Scholar [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. doi: 10.1016/S0025-5564(99)00030-9. Google Scholar [10] M. Y. Li and J. S. Muldowney, A geometric approach to the global-stability problems,, SIAM J. Math. Anal., 27 (1996), 1070. doi: 10.1137/S0036141094266449. Google Scholar [11] M. Y. Li and J. S. Muldowney, Global stability for the SEIR model in epidemiology,, Math. Biosci., 125 (1995), 155. doi: 10.1016/0025-5564(95)92756-5. Google Scholar [12] R. H. Jr. Martin, Logarithmic norms and projections applied to linear differential systems,, J. Math. Anal. Appl., 45 (1974), 432. doi: 10.1016/0022-247X(74)90084-5. Google Scholar [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). doi: 10.1371/journal.ppat.1000788. Google Scholar [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. doi: 10.1128/JVI.06478-11. Google Scholar [15] J. S. Muldowney, Compound matrices and ordinary differential equations,, Rocky Mount. J. Math., 20 (1990), 857. doi: 10.1216/rmjm/1181073047. Google Scholar [16] M. A. Nowak and R. M. May, Virus Dynamics,, Oxford University press, (2000). Google Scholar [17] A. S. Perelson and P. W. Nelson, Mathematical analysis of HIV-1 dynamics in vivo,, SIAM Rev., 41 (1999), 3. doi: 10.1137/S0036144598335107. Google Scholar [18] V. Piguet and Q. Sattentau, Dangerous liaisons at the virological synapse,, J. Clin. Invest., 114 (2004), 605. doi: 10.1172/JCI200422812. Google Scholar [19] O. Schwartz, Immunological and virological aspects of HIV cell-to-cell transfer,, Retrovirology, 6 (2009). doi: 10.1186/1742-4690-6-S2-I16. Google Scholar [20] H. L. Smith and P. Waltman, Perturbation of a globally stable steady state,, Proc. Am. Math. Soc., 127 (1999), 447. doi: 10.1090/S0002-9939-99-04768-1. Google Scholar [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. doi: 10.1128/JVI.01629-06. Google Scholar [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. doi: 10.1016/S0025-5564(02)00108-6. Google Scholar [23] L. Wang and S. Ellermeyer, HIV infection and $CD4^+$ T cell dynamics,, Discrete Contin. Dyn. Syst. Ser. B, 6 (2006), 1417. doi: 10.3934/dcdsb.2006.6.1417. Google Scholar [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. doi: 10.1016/j.mbs.2005.12.026. Google Scholar

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##### References:
 [1] W. A. Coppel, Stability and Asymptotic Behavior of Differential Equations,, Heath and Co., (1965). Google Scholar [2] P. De Leenheer and H. L. Smith, Virus dynamics: A global analysis,, SIAM J. Appl. Math., 63 (2003), 1313. doi: 10.1137/S0036139902406905. Google Scholar [3] P. De Leenheer and S. S. Pilyugin, Multistrain virus dynamics with mutations: A global analysis,, Math. Med. Biol., 25 (2008), 285. Google Scholar [4] N. Dixit and A. Perelson, Multiplicity of human immunodeficiency virus infections in lymphoid tissue,, J. Virol., 78 (2004), 8942. doi: 10.1128/JVI.78.16.8942-8945.2004. Google Scholar [5] M. Fiedler, Additive compound matrices and inequality for eigenvalues of stochastic matrices,, Czech. Math. J., 24 (1974), 392. Google Scholar [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. doi: 10.1007/BF02218848. Google Scholar [7] H. K. Khalil, Nonlinear Systems,, 3rd Edition, (2002). Google Scholar [8] A. Korobeinikov, Global properties of SIR and SEIR epidemic models with multiple parallel infectious stages,, Bull. Math. Biol., 71 (2009), 75. doi: 10.1007/s11538-008-9352-z. Google Scholar [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. doi: 10.1016/S0025-5564(99)00030-9. Google Scholar [10] M. Y. Li and J. S. Muldowney, A geometric approach to the global-stability problems,, SIAM J. Math. Anal., 27 (1996), 1070. doi: 10.1137/S0036141094266449. Google Scholar [11] M. Y. Li and J. S. Muldowney, Global stability for the SEIR model in epidemiology,, Math. Biosci., 125 (1995), 155. doi: 10.1016/0025-5564(95)92756-5. Google Scholar [12] R. H. Jr. Martin, Logarithmic norms and projections applied to linear differential systems,, J. Math. Anal. Appl., 45 (1974), 432. doi: 10.1016/0022-247X(74)90084-5. Google Scholar [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). doi: 10.1371/journal.ppat.1000788. Google Scholar [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. doi: 10.1128/JVI.06478-11. Google Scholar [15] J. S. Muldowney, Compound matrices and ordinary differential equations,, Rocky Mount. J. Math., 20 (1990), 857. doi: 10.1216/rmjm/1181073047. Google Scholar [16] M. A. Nowak and R. M. May, Virus Dynamics,, Oxford University press, (2000). Google Scholar [17] A. S. Perelson and P. W. Nelson, Mathematical analysis of HIV-1 dynamics in vivo,, SIAM Rev., 41 (1999), 3. doi: 10.1137/S0036144598335107. Google Scholar [18] V. Piguet and Q. Sattentau, Dangerous liaisons at the virological synapse,, J. Clin. Invest., 114 (2004), 605. doi: 10.1172/JCI200422812. Google Scholar [19] O. Schwartz, Immunological and virological aspects of HIV cell-to-cell transfer,, Retrovirology, 6 (2009). doi: 10.1186/1742-4690-6-S2-I16. Google Scholar [20] H. L. Smith and P. Waltman, Perturbation of a globally stable steady state,, Proc. Am. Math. Soc., 127 (1999), 447. doi: 10.1090/S0002-9939-99-04768-1. Google Scholar [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. doi: 10.1128/JVI.01629-06. Google Scholar [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. doi: 10.1016/S0025-5564(02)00108-6. Google Scholar [23] L. Wang and S. Ellermeyer, HIV infection and $CD4^+$ T cell dynamics,, Discrete Contin. Dyn. Syst. Ser. B, 6 (2006), 1417. doi: 10.3934/dcdsb.2006.6.1417. Google Scholar [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. doi: 10.1016/j.mbs.2005.12.026. Google Scholar
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