### Mathematical Biosciences and Engineering

2022, Issue 11: 11195-11216. doi: 10.3934/mbe.2022522
Research article Special Issues

# A two-dimensional discrete delay-differential system model of viremia

• Received: 29 April 2022 Revised: 16 July 2022 Accepted: 24 July 2022 Published: 04 August 2022
• A deterministic model is proposed to describe the interaction between an immune system and an invading virus whose target cells circulate in the blood. The model is a system of two ordinary first order quadratic delay-differential equations with stipulated initial conditions, whose coefficients are eventually constant, so that the system becomes autonomous. The long-term behavior of the solution is investigated with some success. In particular, we find two simple functions of the parameters of the model, whose signs often, but not always, determine whether the virus persists above a nonzero threshold in the circulation or heads toward extinction.

Citation: Joseph E. Carroll. A two-dimensional discrete delay-differential system model of viremia[J]. Mathematical Biosciences and Engineering, 2022, 19(11): 11195-11216. doi: 10.3934/mbe.2022522

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• A deterministic model is proposed to describe the interaction between an immune system and an invading virus whose target cells circulate in the blood. The model is a system of two ordinary first order quadratic delay-differential equations with stipulated initial conditions, whose coefficients are eventually constant, so that the system becomes autonomous. The long-term behavior of the solution is investigated with some success. In particular, we find two simple functions of the parameters of the model, whose signs often, but not always, determine whether the virus persists above a nonzero threshold in the circulation or heads toward extinction.

 [1] G. I. Marchuk, Mathematical Models in Immunology, Optimization Software, Inc., Publications Division, Springer-Verlag, New York, 1983. [2] V. Bernhauerová, B. Lisowski, V. Rezelj, M. Vignuzzi, Mathematical modelling of SARS-CoV-2 infection of human and animal host cells reveals differences in the infection rates and delays in viral particle production by infected cell, J. Theor. Biol. , 531 (2021), 110895. https://doi.org/10.1016/j.jtbi.2021.110895 doi: 10.1016/j.jtbi.2021.110895 [3] F. A. Rihan, V. Gandhi, Dynamics and sensitivity of fractional-order delay differential model for Coronavirus (COVID-19) infection, Prog. Fract. Diff. Appl., 7 (2021), 43–61. https://doi.org/10.18576/pfda/070105 doi: 10.18576/pfda/070105 [4] L. N. Cooper, Theory of an immune system retrovirus, PNAS, 83 (1986), 9159–9163. https://doi.org/10.1073/pnas.83.23.9159 doi: 10.1073/pnas.83.23.9159 [5] N. Intrator, G. P. Deocampo, L. N. Cooper, Analysis of immune system retrovirus equations, in Theoretical Immunology, Part 2, A. S. Perelson, Ed., Addison-Wesley, Redwood City, Calif., (1988), 85–100. [6] A. McLean, HIV infection from an ecological viewpoint, in Theoretical Immunology, Part 2, A. S. Perelson, Ed., Addison-Wesley, Redwood City, Calif., (1988), 77–84. [7] S. Merrill, AIDS: Background and the dynamics of the decline of immunocompetence, in Theoretical Immunology, Part 2, A. S. Perelson, Ed., Addison-Wesley, Redwood City, Calif., (1988), 59–75. [8] A. S. Perelson, Modeling the interaction of the immune system with HIV, in Mathematical and Statistical Approaches to AIDS Epidemiology (Lecture Notes Biomath., Vol. 83), Ed., C. Castillo-Chavez, Springer-Verlag, New York, (1989), 350–370. https://doi.org/10.1007/978-3-642-93454-4_17 [9] A. S. Perelson, D. E. Kirschner, R. De Boer, Dynamics of HIV infection of CD4+ T cells, Math. Biosci. , 114 (1993), 81–125. https://doi.org/10.1016/0025-5564(93)90043-A doi: 10.1016/0025-5564(93)90043-A [10] A. S. Perelson, R. M. Ribiero, Modeling the within host dynamics of HIV infection, BMC Biol. , 11 (2013), 96. https://doi.org/10.1186/1741-7007-11-96 doi: 10.1186/1741-7007-11-96 [11] C. Rajivganthi, F. A. Rihan, Global dynamics of a stochastic viral infection model with latently infected cells, Appl. Sci. , 11 (2021), 10484. https://doi.org/10.3390/app112110484 doi: 10.3390/app112110484 [12] X. Zhou, X. Song, X. Shi, A differential equation model of HIV infection of CD4+ T-cells with cure rate, J. Math. Anal. App. , 342 (2008), 1342–1355. https://doi.org/10.1016/j.jmaa.2008.01.008 doi: 10.1016/j.jmaa.2008.01.008 [13] A. V. M. Herz, S. Bonhoeffer, R. M. Anderson, R. M. May, M. A. Novak, Viral dynamics in vivo: limitations on estimates of intracellular delay and virus decay, PNAS, 93 (1996), 7247–7251. https://doi.org/10.1073/pnas.93.14.7247 doi: 10.1073/pnas.93.14.7247 [14] D. Li, W. Ma, Asymptotic properties of a HIV-1 infection model with time delay, J. Math. Ana. App., 335 (2007), 683–691. https://doi.org/10.1016/j.jmaa.2007.02.006 doi: 10.1016/j.jmaa.2007.02.006 [15] P. W. Nelson, A. S. Perelson, Mathematical analysis of delay differential equation models of HIV-1 infection, Math. Biosci. , 179 (2002), 73–94. https://doi.org/10.1016/S0025-5564(02)00099-8 doi: 10.1016/S0025-5564(02)00099-8 [16] J. Yang, X. Wang, F. Zhang, A differential equation model of HIV infection of CD4+ T-cells with delay, Discr. Dyn. Nat. Soc. , 2008 (2008). https://doi.org/10.1155/2008/903678 doi: 10.1155/2008/903678 [17] K. Hattaf, N. Yousfi, Qualitative analysis of a generalized virus dynamics model with both modes of transmission and distributed delays, Int. J. Differ. Equations, 2018 (2018). https://doi.org/10.1155/2018/9818372 doi: 10.1155/2018/9818372 [18] M. Maziane, K. Hattaf, N. Yousfi, Spatiotemporal dynamics of an HIV infection model with delay in immune response activation, Int. J. Differ. Equations, 2018 (2018). https://doi.org/10.1155/2018/3294268 doi: 10.1155/2018/3294268 [19] D. Kirschner, Using mathematics to understand HIV immune dynamics, Not. AMS, 43 (1996), 191–202. [20] R. Bellman, K. L. Cooke, Differential-Difference Equations, Academic Press, New York, 1963. https://doi.org/10.1063/1.3050672 [21] K. L. Cooke, P. van den Driessche, On zeroes of some transcendental equations, Funkcialaj Ekvacioj, 29 (1986), 77–90.
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