Dynamical behaviour of the stochastic tumor-immune interaction model

Cancer is one of the major causes of death worldwide, despite there are many cancer treatments, such as surgery, radiation, chemotherapy, etc. This is the reason why immunotherapy has gained more attention and become approved for the treatment of many types of cancers. The development of the tumor itself is very complex and multifactorial process, which may differ from one person to the other due to his/hers state of immune system and environmental conditions. Motivated by this fact, in this paper we consider the stochastic tumor-immune interaction model, which dynamics is described by the three-dimensional system of stochastic differential equations. The model is obtained by incorporating white noise into deterministic tumor-immune interaction model, which is of predator-prey type. For our stochastic model, we verify that the environmental noise provides a solution that is positive, global and bounded. Also, we obtain conditions under which our model has an ergodic stationary distribution, which is important due to the fact that under these conditions tumor cells and immune cells are weakly persistent in mean, as well as, the conditions which lead to non-persistence in mean. We close the paper by presenting numerical simulations to verify our theoretical results. For that purpose we use reliable data for growth of the highly malignant B Lymphoma/Leukemic cells (BCL1) in the spleen of chimeric mice. Both theoretical and numerical results indicate that the random perturbations may make the model more realistic than its deterministic analogue.