Stochastic Dynamics of Tumor-Immune System

Abstract

In this work, a very well known tumor-immune system model from the literature, Kuznetsov et al.'s model, is converted into two different Stochastic Differential Equation (SDE) models and an It\^{o} formalism of the models are obtained. Furthermore, the models are made discrete by using Euler-Maruyama scheme, simulation results of the corresponding models are investigated and compared.

Keywords

• Ito Calculus,Stochastic Model,Markov Chain,Euler-Maruyama Method,Tumor-Immune System

References

1. J. A. Adam, N. Bellomo, A Survey of Models for Tumor-Immune System Dynamics. Birkh\"{a}user, Boston, MA, 1996.
2. E. Allen, Modeling with Ito Stochastic Differential Equations, Springer, 2007.
3. E. J. Allen, L. J. S. Allen, A. Arciniega, P. Greenwood, Construction of equivalent stochastic differential equation models. Stoch. Anal. Appl., 26, pages: 274-297, 2008.
4. L. J. S. Allen, An Introduction to Stochastic Processes with Applications to Biology, CRC Press, 2011.
5. J. C. Arciero, T. L. Jackson, D. Kirschner, A Mathematical Model of Tumor-Immune Evasion and siRNA Treatment, Discrete and Continuous Dynamical Systems- Series B, Volume 4, Number 1, Pages:39-58, 2004.
6. N. Bellomo, M. Delitala, From the mathematical kinetic, and stochastic game theory to modelling mutations, onset, progression and immune competition of cancer cells, Physics of Life Reviews, Volume 5, Issue 4, Pages 183-206, December 2008.
7. J. Cresson, S. Sonner, A note on a derivation method for SDE models: Application in biology and viability criteria, Stochastic Analysis and Applications, Vol. 36, No. 2, pages 224-239, 2018.
8. A. Eladdadi, P. Kim, D. Mallet, Mathematical Models of Tumor-Immune System Dynamics, Springer, 2014.

9. R. A. Gatenby, T. L. Vincent, Application of quantitative models from population biology andevolutionary game theory to tumor therapeutic strategies, Mol Cancer Ther., 2(9):919-27, 2003.
10. D. Kirschner, J.C. Panetta. Modeling immunotherapy of the tumor-immune interaction. J Math Biol., 37(3):235-52, 1998.
11. V. A. Kuznetsov, I. A. Makalkin, M. A. Taylor, A. S. Perelson. Nonlinear dynamics of immunogenic tumors: Parameter estimation and global bifurcations analysis. Bull Math Biol., 56(2):295-321, 1994.
12. D. G. Mallet, L. G. De Pillis, A cellular automata model of tumor–immune system interactions, Journal of Theoretical Biology, Volume 239, Issue 3, Pages 334-350, 2006.
13. L. G. de Pillis, A. E. Radunskaya, C. L. Wiseman, A Validated Mathematical Model of Cell-Mediated Immune Response to Tumor Growth, Cancer Res., 65(17):7950-8, 2005.
14. M. Robertson-Tessi, A. El-Kareh, A. Goriely. A mathematical model of tumor-immune interactions. J Theor Biol., 294:56-73, 2012.