Stochastic dynamics of a nonlinear tumor-immune competitive system

The paper uses nine coupled ordinary differential equations (ODEs) to describe a tumor-immune competitive system. The model is then reduced to four nonlinear coupled ODEs, encompassing tumor cells, cytotoxic T-lymphocytes, macrophages, and dendritic cells by utilizing quasi-steady-state approximations. We explore the dynamics of biologically feasible steady states and their local stability analysis. We introduce stochastic fluctuation terms into the deterministic system to account for uncertainty and variability in the tumor-immune interaction system. The uniqueness and existence of our stochastic system is established by applying Ito's lemma. Additionally, it is demonstrated that the solution of the stochastic system is both stochastically ultimately bounded and permanent. We established the criteria for determining the extinction of tumor cell population, and conditions under which our stochastic system exhibits asymptotic stability in a mean square sense are derived. Numerical illustrations are performed to validate both deterministic and stochastic models under different intensities of population fluctuation. Our model explores that in the absence of intensity fluctuations, the deterministic model remains stable around a high tumor-presence steady state. Additionally, the tumor cell population quickly approaches zero for sufficiently small values of intensity fluctuation parameters. If the intensity value of population fluctuations is increased, then the cell populations are more fluctuated. Moreover, the stochastic mean solution confirms the influence of stochastic noise on the cell population.