Dynamics of a model of tumor-immune interaction with time delay and noise

Abstract

We propose a model of tumor-immune interaction with time delay in immune reaction and noise in tumor cell reproduction. Immune response is modeled as a non-monotonic function of tumor burden, for which the tumor is immunogenic at nascent stage but starts inhibiting immune system as it grows large. Without time delay and noise, this system demonstrates bistability. The effects of response time of the immune system and uncertainty in the tumor innate proliferation rate are studied by including delay and noise in the appropriate model terms. Stability, persistence and extinction of the tumor are analyzed. We find that delay and noise can both induce the transition from low tumor burden equilibrium to high tumor equilibrium. Moreover, our result suggests that the elimination of cancer depends on the basal level of the immune system rather than on its response speed to tumor growth.

Global Dynamics of the model of tumor-immune interaction

“…This leads to the attempt of anti-tumor treatment by activating the immune system. Current studies (see [8,9] and reference therein) have shown that the immune system has specific and nonspecific responses to tumors, and its mechanism is very complex, involving a variety of immune cells and their secretory products. They interact, regulate and complement each other, and jointly complete the immune surveillance function.…”

“…In [6], the author proposed an ODEs model about tumor and immune effector. And then, the authors in [9] gave a special case of the model as follows…”

“…where ρ = ρ/μ, K = K/ √ κ, γ = (αγ)/(μ 2 √ κ) and β = ( β√ κ)/α are positive. It is shown in [9] that tumor-free equilibrium E 0 (0, β) is stable if ρ < γβ and unstable if ρ > γβ, respectively. Besides, if ρ > γβ, system (2) with three interior equilibria (namely tumor equilibria) demonstrates bistability.…”

“…They found that time delay and noise can both influence the tumor growth. Moreover, the results in [9] suggested that the basal level of the immune system plays an important part in the elimination of tumors.…”

“…Ser. S 13:2347-2363, 2020. We give the necessary and sufficient conditions for the model to have no tumor equilibrium, one, two or three tumor equilibria respectively.…”