Main Article Content
Abstract
In this paper we consider a system of delay differential equations as a model for the dynamics of tumor-immune system interaction. We carry out a stability analysis of the proposed model. In particular, we show that the system can have up to two steady states: the tumor free steady state, which always exist, and the tumor persistent steady state, which exists only when the relative rate of increase of the tumor cells exceeds the ratio between the natural proliferation rate and the relative death rate of the effector cells. We also determine an upper bound for the delay, such that stability is preserved. Numerical simulations of the system under different parameter values are performed.
Keywords
Article Details
References
- Kuznetsov, V.A., Makalkin, I.A., Taylor, M. and Perelson, A.S. Nonlinear dynamics of immunogenic tumors: parameter estimation and global bifurcation analysis. Bulletin of Matematical Biology, 1994, 56, 295-321.
- Kuznetsov, V.A. and Volkenshtein, M.V. Dynamics of cellular immunological anti-tumor reactions II. Qualitative analysis of the model (in Russian), Mathematical Methods of Systems Theory, Frunze: Kirghiz State University, 1979,1, 72-100.
- Rihan, F.A., Abdel Rahman, D.H., Lakshmanan, S. and Alkhajeh, A.S. A time delay model of tumour-immune system interactions: Global dynamics, parameter estimation, sensitivity analysis. Applied Mathematics and Computation, 2014, 232, 606-623.
- Rihan, F.A., Hashish, A., Al-Maskari, F., Hussein, M.S., Ahmed, E., Riaz, M.B. and Yafia, R. Dynamics of tumor-immune system with fractional-order. Journal of Tumor Research, 2016, 2(1),109-115.
- Rihan, F.A. and Rihan, N.F. Dynamics of cancer immune system with external treatment and optimal control. Journal of Cancer Science and Therapy, 2016, 8(10), 257-261 .
- Galach, M. Dynamics of the tumor-immune system competition-the effect of time delay. International Journal of Applied Mathematics and Computer Science, 2003, 13(3), 395-406.
- Arciero, J.C., Jackson, T.L. and Kirschner, D.E. Mathematical model of tumor-immune evasion and siRNA treatment. Discrete and Continuous Dynamical Systems - Series B, 2004, 4(1),39-58.
- Kirschner, D. and Panetta, J.C. Modeling immunotherapy of the tumor-immune interaction. Journal of Mathematical Biology, 1998, 37, 235-252.
- Y. Kuang, 1993, Delay Differential Equations with Applications in Population Dynamics, London, Academic Press.
- Dionysiou, D.D. and Stamatakos, G.S. Applying a 4D multiscale in vivo tumor growth model to the exploration of radiotherapy scheduling: the effects of weekend treatment gaps and p53 gene status on the response of fast growing solid tumors. Cancer Inform, 2006, 2,113-121.
References
Kuznetsov, V.A., Makalkin, I.A., Taylor, M. and Perelson, A.S. Nonlinear dynamics of immunogenic tumors: parameter estimation and global bifurcation analysis. Bulletin of Matematical Biology, 1994, 56, 295-321.
Kuznetsov, V.A. and Volkenshtein, M.V. Dynamics of cellular immunological anti-tumor reactions II. Qualitative analysis of the model (in Russian), Mathematical Methods of Systems Theory, Frunze: Kirghiz State University, 1979,1, 72-100.
Rihan, F.A., Abdel Rahman, D.H., Lakshmanan, S. and Alkhajeh, A.S. A time delay model of tumour-immune system interactions: Global dynamics, parameter estimation, sensitivity analysis. Applied Mathematics and Computation, 2014, 232, 606-623.
Rihan, F.A., Hashish, A., Al-Maskari, F., Hussein, M.S., Ahmed, E., Riaz, M.B. and Yafia, R. Dynamics of tumor-immune system with fractional-order. Journal of Tumor Research, 2016, 2(1),109-115.
Rihan, F.A. and Rihan, N.F. Dynamics of cancer immune system with external treatment and optimal control. Journal of Cancer Science and Therapy, 2016, 8(10), 257-261 .
Galach, M. Dynamics of the tumor-immune system competition-the effect of time delay. International Journal of Applied Mathematics and Computer Science, 2003, 13(3), 395-406.
Arciero, J.C., Jackson, T.L. and Kirschner, D.E. Mathematical model of tumor-immune evasion and siRNA treatment. Discrete and Continuous Dynamical Systems - Series B, 2004, 4(1),39-58.
Kirschner, D. and Panetta, J.C. Modeling immunotherapy of the tumor-immune interaction. Journal of Mathematical Biology, 1998, 37, 235-252.
Y. Kuang, 1993, Delay Differential Equations with Applications in Population Dynamics, London, Academic Press.
Dionysiou, D.D. and Stamatakos, G.S. Applying a 4D multiscale in vivo tumor growth model to the exploration of radiotherapy scheduling: the effects of weekend treatment gaps and p53 gene status on the response of fast growing solid tumors. Cancer Inform, 2006, 2,113-121.