Dynamical properties of a stochastic tumor–immune model with comprehensive pulsed therapy

Abstract

In this paper, a stochastic tumor–immune model with comprehensive pulsed therapy is established by taking stochastic perturbation and pulsed effect into account. Some properties of the model solutions are given in the form of the Theorems. Firstly, we obtain the equivalent solutions of the tumor–immune system by through three auxiliary equations, and prove the system solutions are existent, positive and unique. Secondly, a Lyapunov function is constructed to prove the global attraction in the mean sense for the system solution, and the boundness of the solutions’ expectation is proved by the comparison theorem of the impulsive differential equations. Next, the sufficient conditions for the extinction and non-mean persistence of tumor cells, hunting T-cells and helper T-cells, as well as the weak persistence and stochastic persistence of the tumor, are obtained by way of combining It$\hat{o}$’s differential rule and strong law of large numbers, respectively. The results pass the confirmation by numerical Milsteins method. The results show that when the noise intensity gradually increases, the tumor state changes from the weak persistence to the extinction, it demonstrates that the effect of stochastic perturbations on tumor cells is very prominent. In addition, by adjusting the value of $a\left(nP\right)$ to simulate different medication doses, the results show that the killing rate of the medication to the tumor cells is the dominant factor in the long-term evolution of the tumor, and the bigger killing rate can lead to a rapid decrease in the number of tumor cells. Increasing the frequency of pulse therapy has also significant effects on tumor regression. The conclusion is consistent with the clinical observation of tumor treatment.