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, (2008), 12. Google Scholar

[2]

M. Bodnar, U. U. Foryś and Z. Szymańska, Model of AIDS-related tumor with time delay,, Appl. Math. (Warsaw), 36 (2009), 263. doi: 10.4064/am36-3-2. Google Scholar

[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. doi: 10.1002/cncr.20354. Google Scholar

[4]

C. Boshoff and R. Weiss, AIDS-related malignancies,, Nat. Rev. Cancer, 2 (2002), 373. doi: 10.1038/nrc797. Google Scholar

[5]

S. Bunimovich-Mendrazitsky, H. Byrne and L. Stone, Mathematical model of pulsed immunotherapy for superficial bladder cancer,, Bull. Math. Biol., 70 (2008), 2055. doi: 10.1007/s11538-008-9344-z. Google Scholar

[6]

L. Preziosi, "Cancer Modeling and Simulation,", Chapman & Hall, (2003). doi: 10.1201/9780203494899. Google Scholar

[7]

K. L. Cooke and P. van den Driessche, On zeros of some transcendental equations,, Funkcialaj Ekvacioj, 27 (1986), 77. Google Scholar

[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. doi: 10.1016/S0025-5564(00)00006-7. Google Scholar

[9]

C. DeLisi and A. Rescigno, Immune surveillance and neoplasia: A minimal mathematical model,, Bull. Mat. Biol., 39 (1977), 201. Google Scholar

[10]

P. J. Delves, D. J. Martin, D. R. Burton and I. M. Roitt, "Roitt's Essential Immunology,", 11$^{th}$ edition, (2006). Google Scholar

[11]

J. Gołąb, M. Jakóbisiak and W. Lasek (eds.), "Immunologia" (in Polish),, PWN, (2002). Google Scholar

[12]

D. Kirschner and J. C. Panetta, Modeling immunotherapy of the tumor-immune interaction,, J. Math. Biol., 37 (1998), 235. doi: 10.1007/s002850050127. Google Scholar

[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. Google Scholar

[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. Google Scholar

[15]

J. Lou and T. Ruggeri, A time delay model about AIDS-related cancer: Equilibria, cycles and chaotic behavior,, Ric. Mat., 56 (2007), 195. doi: 10.1007/s11587-007-0013-6. Google Scholar

[16]

J. D. Murray, "Mathematical Biology. An Introduction,", Springer Verlag, (2002). Google Scholar

[17]

P. W. Nelson, "Mathematical Models in Immunology and HIV Pathogenesis,", Ph.D thesis, (1998). Google Scholar

[18]

P. W. Nelson, J. D. Murray and A. S. Perelson, Delay model for the dynamics in HIV infection,, Math. Biosci., 163 (2000), 201. doi: 10.1016/S0025-5564(99)00055-3. Google Scholar

[19]

J. Palefsky, Human papillomavirus infection in HIV-infected persons,, Top HIV Med., 15 (2007), 130. Google Scholar

[20]

A. S. Perelson and P. W. Nelson, Mathematical models of HIV-1 dynamics in vivo,, SIAM Rev., 41 (1999), 3. doi: 10.1137/S0036144598335107. Google Scholar

[21]

T. E. Wheldon, "Mathematical Models in Cancer Research,", IOP Publishing Ltd., (1988). Google Scholar

show all references

References:
[1]

M. Bodnar, U. Foryś and Z. Szymańska, A model of AIDS-related tumor with time delay,, in, (2008), 12. Google Scholar

[2]

M. Bodnar, U. U. Foryś and Z. Szymańska, Model of AIDS-related tumor with time delay,, Appl. Math. (Warsaw), 36 (2009), 263. doi: 10.4064/am36-3-2. Google Scholar

[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. doi: 10.1002/cncr.20354. Google Scholar

[4]

C. Boshoff and R. Weiss, AIDS-related malignancies,, Nat. Rev. Cancer, 2 (2002), 373. doi: 10.1038/nrc797. Google Scholar

[5]

S. Bunimovich-Mendrazitsky, H. Byrne and L. Stone, Mathematical model of pulsed immunotherapy for superficial bladder cancer,, Bull. Math. Biol., 70 (2008), 2055. doi: 10.1007/s11538-008-9344-z. Google Scholar

[6]

L. Preziosi, "Cancer Modeling and Simulation,", Chapman & Hall, (2003). doi: 10.1201/9780203494899. Google Scholar

[7]

K. L. Cooke and P. van den Driessche, On zeros of some transcendental equations,, Funkcialaj Ekvacioj, 27 (1986), 77. Google Scholar

[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. doi: 10.1016/S0025-5564(00)00006-7. Google Scholar

[9]

C. DeLisi and A. Rescigno, Immune surveillance and neoplasia: A minimal mathematical model,, Bull. Mat. Biol., 39 (1977), 201. Google Scholar

[10]

P. J. Delves, D. J. Martin, D. R. Burton and I. M. Roitt, "Roitt's Essential Immunology,", 11$^{th}$ edition, (2006). Google Scholar

[11]

J. Gołąb, M. Jakóbisiak and W. Lasek (eds.), "Immunologia" (in Polish),, PWN, (2002). Google Scholar

[12]

D. Kirschner and J. C. Panetta, Modeling immunotherapy of the tumor-immune interaction,, J. Math. Biol., 37 (1998), 235. doi: 10.1007/s002850050127. Google Scholar

[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. Google Scholar

[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. Google Scholar

[15]

J. Lou and T. Ruggeri, A time delay model about AIDS-related cancer: Equilibria, cycles and chaotic behavior,, Ric. Mat., 56 (2007), 195. doi: 10.1007/s11587-007-0013-6. Google Scholar

[16]

J. D. Murray, "Mathematical Biology. An Introduction,", Springer Verlag, (2002). Google Scholar

[17]

P. W. Nelson, "Mathematical Models in Immunology and HIV Pathogenesis,", Ph.D thesis, (1998). Google Scholar

[18]

P. W. Nelson, J. D. Murray and A. S. Perelson, Delay model for the dynamics in HIV infection,, Math. Biosci., 163 (2000), 201. doi: 10.1016/S0025-5564(99)00055-3. Google Scholar

[19]

J. Palefsky, Human papillomavirus infection in HIV-infected persons,, Top HIV Med., 15 (2007), 130. Google Scholar

[20]

A. S. Perelson and P. W. Nelson, Mathematical models of HIV-1 dynamics in vivo,, SIAM Rev., 41 (1999), 3. doi: 10.1137/S0036144598335107. Google Scholar

[21]

T. E. Wheldon, "Mathematical Models in Cancer Research,", IOP Publishing Ltd., (1988). Google Scholar

[1]

Jinliang Wang, Lijuan Guan. Global stability for a HIV-1 infection model with cell-mediated immune response and intracellular delay. Discrete & Continuous Dynamical Systems - B, 2012, 17 (1) : 297-302. doi: 10.3934/dcdsb.2012.17.297

[2]

Songbai Guo, Wanbiao Ma. Global behavior of delay differential equations model of HIV infection with apoptosis. Discrete & Continuous Dynamical Systems - B, 2016, 21 (1) : 103-119. doi: 10.3934/dcdsb.2016.21.103

[3]

Arni S.R. Srinivasa Rao, Masayuki Kakehashi. Incubation-time distribution in back-calculation applied to HIV/AIDS data in India. Mathematical Biosciences & Engineering, 2005, 2 (2) : 263-277. doi: 10.3934/mbe.2005.2.263

[4]

Hui Miao, Zhidong Teng, Chengjun Kang. Stability and Hopf bifurcation of an HIV infection model with saturation incidence and two delays. Discrete & Continuous Dynamical Systems - B, 2017, 22 (6) : 2365-2387. doi: 10.3934/dcdsb.2017121

[5]

Yu Ji. Global stability of a multiple delayed viral infection model with general incidence rate and an application to HIV infection. Mathematical Biosciences & Engineering, 2015, 12 (3) : 525-536. doi: 10.3934/mbe.2015.12.525

[6]

Jinhu Xu, Yicang Zhou. Bifurcation analysis of HIV-1 infection model with cell-to-cell transmission and immune response delay. Mathematical Biosciences & Engineering, 2016, 13 (2) : 343-367. doi: 10.3934/mbe.2015006

[7]

Huiyan Zhu, Xingfu Zou. Dynamics of a HIV-1 Infection model with cell-mediated immune response and intracellular delay. Discrete & Continuous Dynamical Systems - B, 2009, 12 (2) : 511-524. doi: 10.3934/dcdsb.2009.12.511

[8]

Bao-Zhu Guo, Li-Ming Cai. A note for the global stability of a delay differential equation of hepatitis B virus infection. Mathematical Biosciences & Engineering, 2011, 8 (3) : 689-694. doi: 10.3934/mbe.2011.8.689

[9]

Miguel Atencia, Esther García-Garaluz, Gonzalo Joya. The ratio of hidden HIV infection in Cuba. Mathematical Biosciences & Engineering, 2013, 10 (4) : 959-977. doi: 10.3934/mbe.2013.10.959

[10]

Cristiana J. Silva, Delfim F. M. Torres. A TB-HIV/AIDS coinfection model and optimal control treatment. Discrete & Continuous Dynamical Systems - A, 2015, 35 (9) : 4639-4663. doi: 10.3934/dcds.2015.35.4639

[11]

Cristiana J. Silva, Delfim F. M. Torres. Modeling and optimal control of HIV/AIDS prevention through PrEP. Discrete & Continuous Dynamical Systems - S, 2018, 11 (1) : 119-141. doi: 10.3934/dcdss.2018008

[12]

Christopher M. Kribs-Zaleta, Melanie Lee, Christine Román, Shari Wiley, Carlos M. Hernández-Suárez. The Effect of the HIV/AIDS Epidemic on Africa's Truck Drivers. Mathematical Biosciences & Engineering, 2005, 2 (4) : 771-788. doi: 10.3934/mbe.2005.2.771

[13]

Brandy Rapatski, Petra Klepac, Stephen Dueck, Maoxing Liu, Leda Ivic Weiss. Mathematical epidemiology of HIV/AIDS in cuba during the period 1986-2000. Mathematical Biosciences & Engineering, 2006, 3 (3) : 545-556. doi: 10.3934/mbe.2006.3.545

[14]

Moatlhodi Kgosimore, Edward M. Lungu. The Effects of Vertical Transmission on the Spread of HIV/AIDS in the Presence of Treatment. Mathematical Biosciences & Engineering, 2006, 3 (2) : 297-312. doi: 10.3934/mbe.2006.3.297

[15]

Wouter Rogiest, Koen De Turck, Koenraad Laevens, Dieter Fiems, Sabine Wittevrongel, Herwig Bruneel. On the optimality of packet-oriented scheduling in photonic switches with delay lines. Numerical Algebra, Control & Optimization, 2011, 1 (4) : 727-747. doi: 10.3934/naco.2011.1.727

[16]

Alexander Rezounenko. Stability of a viral infection model with state-dependent delay, CTL and antibody immune responses. Discrete & Continuous Dynamical Systems - B, 2017, 22 (4) : 1547-1563. doi: 10.3934/dcdsb.2017074

[17]

Alexander Rezounenko. Viral infection model with diffusion and state-dependent delay: Stability of classical solutions. Discrete & Continuous Dynamical Systems - B, 2018, 23 (3) : 1091-1105. doi: 10.3934/dcdsb.2018143

[18]

Jinliang Wang, Xiu Dong. Analysis of an HIV infection model incorporating latency age and infection age. Mathematical Biosciences & Engineering, 2018, 15 (3) : 569-594. doi: 10.3934/mbe.2018026

[19]

Chengjun Guo, Chengxian Guo, Sameed Ahmed, Xinfeng Liu. Moment stability for nonlinear stochastic growth kinetics of breast cancer stem cells with time-delays. Discrete & Continuous Dynamical Systems - B, 2016, 21 (8) : 2473-2489. doi: 10.3934/dcdsb.2016056

[20]

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

2018 Impact Factor: 1.313

Metrics

  • PDF downloads (5)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]