# American Institute of Mathematical Sciences

2011, 8(2): 627-641. doi: 10.3934/mbe.2011.8.627

## A delay-differential equation model of HIV related cancer--immune system dynamics

 1 University of Warsaw, Faculty of Mathematics, Informatics and Mechanics, Institute of Applied Mathematics and Mechanics, Banacha 2, 02-097 Warsaw

Received  March 2010 Revised  November 2010 Published  April 2011

In the human body, the appearance of tumor cells usually turns on the defensive immune mechanisms. It is therefore of great importance to understand links between HIV related immunosuppression and cancer prognosis. In the paper we present a simple model of HIV related cancer - immune system interactions in vivo which takes into account a delay describing the time needed by CD$4^+$ T lymphocyte to regenerate after eliminating a cancer cell. The model assumes also the linear response of immune system to tumor presence. We perform a mathematical analysis of the steady states stability and discuss the biological meanings of these steady states. Numerical simulations are also presented to illustrate the predictions of the model.
Citation: 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
##### References:
 [1] M. Bodnar, U. Foryś and Z. Szymańska, A model of AIDS-related tumor with time delay, in "Proceedings of the Fourteenth National Conference on Application of Mathematics in Biology and Medicine" (eds. M. Bodnar and U. Foryś), University of Warsaw, (2008), 12-17. [2] M. Bodnar, U. U. Foryś and Z. Szymańska, Model of AIDS-related tumor with time delay, Appl. Math. (Warsaw), 36 (2009), 263-278. doi: 10.4064/am36-3-2. [3] F. Bonnet, C. Lewden, T. May, L. Heripret, E. Jougla, S. Bevilacqua, D. Costagliola, D. Salmon, G. Chêne and P. Morlat, Malignancy-related causes of death in human immunodeficiency virus-infected patients in the era of highly active antiretroviral therapy, Cancer, 101 (2004), 317-324. doi: 10.1002/cncr.20354. [4] C. Boshoff and R. Weiss, AIDS-related malignancies, Nat. Rev. Cancer, 2 (2002), 373-382. doi: 10.1038/nrc797. [5] S. Bunimovich-Mendrazitsky, H. Byrne and L. Stone, Mathematical model of pulsed immunotherapy for superficial bladder cancer, Bull. Math. Biol., 70 (2008), 2055-2076. doi: 10.1007/s11538-008-9344-z. [6] L. Preziosi, "Cancer Modeling and Simulation," Chapman & Hall, 2003. doi: 10.1201/9780203494899. [7] K. L. Cooke and P. van den Driessche, On zeros of some transcendental equations, Funkcialaj Ekvacioj, 27 (1986), 77-90. [8] R. V. Culshaw and S. Ruan, A delay-differential equation model of HIV infection of CD$4^+$ T-Cells, Math. Biosci., 165 (2000), 27-39. doi: 10.1016/S0025-5564(00)00006-7. [9] C. DeLisi and A. Rescigno, Immune surveillance and neoplasia: A minimal mathematical model, Bull. Mat. Biol., 39 (1977), 201-221. [10] P. J. Delves, D. J. Martin, D. R. Burton and I. M. Roitt, "Roitt's Essential Immunology," 11th edition, Blackwell Science, Oxford, 2006. [11] J. Gołąb, M. Jakóbisiak and W. Lasek (eds.), "Immunologia" (in Polish), PWN, Warszawa, 2002. [12] D. Kirschner and J. C. Panetta, Modeling immunotherapy of the tumor-immune interaction, J. Math. Biol., 37 (1998), 235-252. doi: 10.1007/s002850050127. [13] V. A. Kuznetsov, I. A. Makalkin, M. A. Taylor and A. S. Perelson, Nonlinear dynamics of immunogenic tumors: Parameter estimation and global bifurcation analysis, Bull. Math. Biol., 56 (1994), 295-321. [14] J. Lou, T. Ruggeri and C. Tebaldi, Modeling cancer in HIV-1 infected individuals: Equilibria, cycles and chaotic behavior, Math. Biosci. and Eng., 3 (2006), 313-324. [15] J. Lou and T. Ruggeri, A time delay model about AIDS-related cancer: Equilibria, cycles and chaotic behavior, Ric. Mat., 56 (2007), 195-208. doi: 10.1007/s11587-007-0013-6. [16] J. D. Murray, "Mathematical Biology. An Introduction," Springer Verlag, New York, 2002. [17] P. W. Nelson, "Mathematical Models in Immunology and HIV Pathogenesis," Ph.D thesis, Department of Applied Mathematics, University of Washington, Seattle WA, 1998. [18] P. W. Nelson, J. D. Murray and A. S. Perelson, Delay model for the dynamics in HIV infection, Math. Biosci., 163 (2000), 201-215. doi: 10.1016/S0025-5564(99)00055-3. [19] J. Palefsky, Human papillomavirus infection in HIV-infected persons, Top HIV Med., 15 (2007), 130-133. [20] A. S. Perelson and P. W. Nelson, Mathematical models of HIV-1 dynamics in vivo, SIAM Rev., 41 (1999), 3-44. doi: 10.1137/S0036144598335107. [21] T. E. Wheldon, "Mathematical Models in Cancer Research," IOP Publishing Ltd., Bristol, 1988.

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##### References:
 [1] M. Bodnar, U. Foryś and Z. Szymańska, A model of AIDS-related tumor with time delay, in "Proceedings of the Fourteenth National Conference on Application of Mathematics in Biology and Medicine" (eds. M. Bodnar and U. Foryś), University of Warsaw, (2008), 12-17. [2] M. Bodnar, U. U. Foryś and Z. Szymańska, Model of AIDS-related tumor with time delay, Appl. Math. (Warsaw), 36 (2009), 263-278. doi: 10.4064/am36-3-2. [3] F. Bonnet, C. Lewden, T. May, L. Heripret, E. Jougla, S. Bevilacqua, D. Costagliola, D. Salmon, G. Chêne and P. Morlat, Malignancy-related causes of death in human immunodeficiency virus-infected patients in the era of highly active antiretroviral therapy, Cancer, 101 (2004), 317-324. doi: 10.1002/cncr.20354. [4] C. Boshoff and R. Weiss, AIDS-related malignancies, Nat. Rev. Cancer, 2 (2002), 373-382. doi: 10.1038/nrc797. [5] S. Bunimovich-Mendrazitsky, H. Byrne and L. Stone, Mathematical model of pulsed immunotherapy for superficial bladder cancer, Bull. Math. Biol., 70 (2008), 2055-2076. doi: 10.1007/s11538-008-9344-z. [6] L. Preziosi, "Cancer Modeling and Simulation," Chapman & Hall, 2003. doi: 10.1201/9780203494899. [7] K. L. Cooke and P. van den Driessche, On zeros of some transcendental equations, Funkcialaj Ekvacioj, 27 (1986), 77-90. [8] R. V. Culshaw and S. Ruan, A delay-differential equation model of HIV infection of CD$4^+$ T-Cells, Math. Biosci., 165 (2000), 27-39. doi: 10.1016/S0025-5564(00)00006-7. [9] C. DeLisi and A. Rescigno, Immune surveillance and neoplasia: A minimal mathematical model, Bull. Mat. Biol., 39 (1977), 201-221. [10] P. J. Delves, D. J. Martin, D. R. Burton and I. M. Roitt, "Roitt's Essential Immunology," 11th edition, Blackwell Science, Oxford, 2006. [11] J. Gołąb, M. Jakóbisiak and W. Lasek (eds.), "Immunologia" (in Polish), PWN, Warszawa, 2002. [12] D. Kirschner and J. C. Panetta, Modeling immunotherapy of the tumor-immune interaction, J. Math. Biol., 37 (1998), 235-252. doi: 10.1007/s002850050127. [13] V. A. Kuznetsov, I. A. Makalkin, M. A. Taylor and A. S. Perelson, Nonlinear dynamics of immunogenic tumors: Parameter estimation and global bifurcation analysis, Bull. Math. Biol., 56 (1994), 295-321. [14] J. Lou, T. Ruggeri and C. Tebaldi, Modeling cancer in HIV-1 infected individuals: Equilibria, cycles and chaotic behavior, Math. Biosci. and Eng., 3 (2006), 313-324. [15] J. Lou and T. Ruggeri, A time delay model about AIDS-related cancer: Equilibria, cycles and chaotic behavior, Ric. Mat., 56 (2007), 195-208. doi: 10.1007/s11587-007-0013-6. [16] J. D. Murray, "Mathematical Biology. An Introduction," Springer Verlag, New York, 2002. [17] P. W. Nelson, "Mathematical Models in Immunology and HIV Pathogenesis," Ph.D thesis, Department of Applied Mathematics, University of Washington, Seattle WA, 1998. [18] P. W. Nelson, J. D. Murray and A. S. Perelson, Delay model for the dynamics in HIV infection, Math. Biosci., 163 (2000), 201-215. doi: 10.1016/S0025-5564(99)00055-3. [19] J. Palefsky, Human papillomavirus infection in HIV-infected persons, Top HIV Med., 15 (2007), 130-133. [20] A. S. Perelson and P. W. Nelson, Mathematical models of HIV-1 dynamics in vivo, SIAM Rev., 41 (1999), 3-44. doi: 10.1137/S0036144598335107. [21] T. E. Wheldon, "Mathematical Models in Cancer Research," IOP Publishing Ltd., Bristol, 1988.
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