Review of Stochastic Stability and Analysis Tumor-Immune Systems

Abstract

Abstract:In this paper we review and at the same time investigate some stochastic models for tumor-immune systems. To describe these models, we used a Wiener process, as the noise has a stabilization effect. Their dynamics are studied in terms of stochastic stability around the equilibrium points, by constructing the Lyapunov exponent, depending on the parameters that describe the model. Stochastic stability was also proved by constructing a Lyapunov function and the second order moments. We have studied and analyzed a Kuznetsov-Taylor like stochastic model and a Bell stochastic model for tumor-immune systems. These stochastic models are studied from stability point of view and they were graphically represented using the second order Euler scheme and Maple 12 software.

Stochastic modelling of slow-progressing tumors: Analysis and applications to the cell interplay and control of low grade gliomas

“…Important examples range from gene expression [48] , cellular differentiation [41] , to animal migration and infectious disease risk [3] . The interface of malignant cancer cells with the surrounding healthy tissue provides another relevant scenario for stochastic processes [4,15,19,23,24,30,31,47,63] . Disruption of the balance between early formed neoplastic cells and the normal cells results in the invasion and dissemination of the malignant cells and eventually a transition from a benign to a malignant tumor.…”

“…Thus, a considerable restructuration of the pre-existing healthy tissue occurs, typically resulting in the growth of a vascular network, which is often aberrant and dysfunctional, and in the development of necrotic regions [1] . The modification of the spatial environment re-defines the malignant-normal cell interaction that initially governed the system [18,56] . As a consequence, the conditions that were present during tumorigenesis change dramatically and may become masked by the complexity of a fast-evolving and multi-variable system.…”