Dynamical Behaviors of the Tumor-Immune System in a Stochastic Environment

Abstract

This paper investigates dynamic behaviors of the tumor-immune system perturbed by environmental noise. The model describes the response of the cytotoxic T lymphocyte (CTL) to the growth of an immunogenic tumour. The main methods are stochastic Lyapunov analysis, comparison theorem for stochastic differential equations (SDEs) and strong ergodicity theorem. Firstly, we prove the existence and uniqueness of the global positive solution for the tumor-immune system. Then we go a further step to study the boundaries of moments for tumor cells and effector cells and the asymptotic behavior in the boundary equilibrium points. Furthermore, we discuss the existence and uniqueness of stationary distribution and stochastic permanence of the tumor-immune system. Finally, we give several examples and numerical simulations to verify our results.

Polynomial ergodicity of an SIRS epidemic model with density‐dependent demographics

“…In view of (31), we easily obtain that there exists a η = η(𝜀, 𝜂) ∈ (0, 1) sufficiently small such that ln 𝐼(𝑡 0 ) + (𝑇 * 3 − 𝑡 0 ) < 0, for any 𝐼(𝑡 0 ) ∈ (0, 𝐶 4 η]. As a result of (22), it follows that…”

“…Cai et al 4 investigated a stochastic SIRS epidemic model and discovered that random perturbations could suppress disease outbreaks. We also refer the readers to Nguyen et al, 20 Dieu et al, 21 Li et al, 22 Tan et al, 23 Privault and Wang, 24 Zhou et al, 25 and the references therein for some related investigations.…”

Dynamic Properties for a Second-Order Stochastic SEIR Model with Infectivity in Incubation Period and Homestead-Isolation of the Susceptible Population

“…Mao et al have claimed that the incorporation of a small amount of random fluctuation in the deterministic model of infectious disease can actively prevent outbreaks in the latent population [5]. Recently, several authors have considered the random fluctuation factor into biological mathematical models of infectious diseases and environmental pollution [6][7][8][9][10]. In [11], in view of nonlinear stochastic disturbance, Mu et al discussed the extinction and persistence for a stochastic micro-organism flocculation model.…”

Stabilization of Chaos in a Cancer Model: The Effect of Oncotripsy

“…By considering the variability in cellular reproduction, death and the fluctuation of chemotherapy effect, the deterministic differential equation model has been extended to the stochastic one to analyze the dynamics of tumour cells and immune cells under chemotherapy in [16]. The culling rate of effector cells and the intrinsic growth rate of tumour cells have been modeled as stochastic processes and the effect of environmental noise on the dynamic behaviors of the tumour-immune model has been studied in [17]. A stochastic tumour-immune system with a combination of immunotherapy and chemotherapy has been modeled in [18] and the evolution of tumours has been analyzed in the presence of environmental noise and chemotherapeutic dose.…”