Abstract
This paper investigates the long time behavior of tumor cells evolution in a tumor-immune system competition model perturbed by environmental noise. Sufficient conditions for extinction, stochastic persistence, and strong persistence in the mean of tumor cells are derived by constructing Lyapunov functions. The study results show that environmental noise can accelerate the extinction of tumor cells under immune surveillance of effector cells, which means that noise is favorable for the extinction of tumor in this condition. Finally, numerical simulations are introduced to support our results.
Highlights
Cancer is becoming the leading cause of death around the world, but our cognition of its causes, methods of prevention and cure are still in its infancy
The goal of this paper is to explore the long time behavior of tumor and effector cells in the tumor-immune system competition model perturbed by environmental noise
It is obvious that the mean time to extinction (MTE) in the stochastic model is less than that in the deterministic model when the values of ε are the same, which shows that environmental noise can accelerate the extinction of tumor cells
Summary
Cancer is becoming the leading cause of death around the world, but our cognition of its causes, methods of prevention and cure are still in its infancy. Applying Itô’s formula to the second equation of model ( ), we get d ln y = (α – αβy – x) dt Integrating this from to t and dividing by t on both sides, we have ln y(t) – ln y = α – αβ y(t) – x(t) t. Applying Itô’s formula again to exp(t)V (y) gives d exp(t)V (y) = exp(t)V (y) dt + exp(t) dV (y) = exp(t)yq dt + exp(t) qαyq – qαβyq+ – qxyq dt = exp(t) –qαβyq+ + (qα + )yq – qxyq dt ≤ exp(t)K, where K is a positive constant Integrating this inequality and taking expectations on both sides, one can see that. Consider that both x(t) and y(t) are bounded, we have lim inf y(t) ≥ δα – ε δα – ε αβδ + ωα αβδ + ωα αβδ + ωα we can derive that if δα > ε, y(t) ∗ > a.s
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