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 R 0 (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.

New Investigation on the Generalized K-Fractional Integral Operators

“…This framework is considered to be a mind-boggling system, and the units that create the whole framework are viewed as the hubs of the intricate system. An attractive characteristic of this field is that there are numerous fractional operators, and this permits researchers to choose the most appropriate operator for the sake of modeling the problem under investigation (see [9][10][11][12][13]). Besides, because of its simplicity in application, researchers have been paying greater interest to recently introduced fractional operators without singular kernels [2,14,15], after which many articles considering these kinds of fractional operators have been presented.…”

Fractional optimal control in transmission dynamics of West Nile virus model with state and control time delay: a numerical approach

“…Since most of the fractional-order differential equations do not have exact analytic solutions, the approximation and numerical techniques must be used [26]. Delayed fractional differential equations (DEDEs) are also used to describe dynamical systems [32], for more details, see [33][34][35]. Recently, many papers have been devoted to DEDEs (see [36][37][38], and the references cited therein).…”

Numerical Simulations of a Delay Model for Immune System-Tumor Interaction

“…Mathematical models, based on ordinary differential equations, delay differential equations or partial differential equations, have proven to be useful tools in analysing and understanding the IS-tumor interactions. Several mathematical models have been suggested to describe the interactions between tumour and immune system [1][2][3][4][5][6][7][8][9].…”