Dynamics of Tumor-Immune System with Fractional-Order

Abstract

Most of biological systems have long-range temporal memory. Modeling of such systems by fractional-order (or arbitrary-order) models provides the systems with long-time memory and gains them extra degrees of freedom. Herein, we suggest a simple fractional-order model to describe the dynamics of tumor-immune interactions. Two effector cells are considered, in the model, with a Holling function response of type-III. The model is extended to include treatment terms which represent an external source of the effectors cells by ACI and an external input of IL- 2. Asymptotic stabilities of tumour-free steady state and persistent- tumour steady state are studied. The threshold parameter R0 (average number of newly infected cells produced by a single councerous cell) is deduced. The numerical simulations show that the fractional-order derivative enriches the dynamics of the system and increases the complexity of the observed behaviours, which confirms that the fractional-order may play the role of memory in the system.