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From the guest editors: "Delay Differential Equations: Theory, Applications and New Trends"
September  2020, 13(9): 2347-2363. doi: 10.3934/dcdss.2020140

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

 School of Mathematical and Statistical Sciences, Arizona state University, Tempe, AZ 85287-1804, USA

* Corresponding author: Yang Kuang

Received  January 2019 Published  September 2020 Early access  November 2019

Fund Project: The authors are supported by NSF grant 161587 and NIH grant 1R01GM131405-01

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.

Citation: Lifeng Han, Changhan He, Yang Kuang. Dynamics of a model of tumor-immune interaction with time delay and noise. Discrete & Continuous Dynamical Systems - S, 2020, 13 (9) : 2347-2363. doi: 10.3934/dcdss.2020140
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##### References:
Nullclines of (4). Open circles denote unstable fixed points and filled circle denotes stable fixed points. Parameter values used: $\rho = 2.5, {\beta = 0.02, \gamma = 5, K = 10}$
Time course of $u$ and $v$ with and without delay. There is a stability switch as $\tau$ increases and eventually the solution settles down to high tumor equilibrium. The observation indicates that responsiveness of the immune system is important to contain the tumor in its nascent size. If there is long time delay in immune response, the tumor can grow in oscillatory fashion and eventually escape the control of the immune system. Parameter values used to generate the plots: $\rho = 2.5, {\beta = 0.02, \gamma = 5, K = 10}$
">Figure 3.  Computer simulated sample paths of the stochastic system (12) in comparison with its deterministic version (4). (a) tumor extinction in small noise regime; (b) tumor extinction in big noise regime; (c) monostable fluctuation; (d) bistable switching. Parameter values used here are summarized in Table 1
stationary distribution of (15) with $\tau = 0, 0.5, 1, 2$. The histogram is formed by 5000 samples of $u(1000)$
Parameter values used in Figure 3
 small noise induced extinction (a) big noise induced extinction (b) monostable (c) bistable (d) $\rho$ 0.12 2.5 1.5 2.5 $\beta$ 0.02 0.02 0.02 0.02 $\gamma$ 5 5 5 5 $K$ 10 10 10 10 $\sigma$ 0.3 5.66 0.65 0.65
 small noise induced extinction (a) big noise induced extinction (b) monostable (c) bistable (d) $\rho$ 0.12 2.5 1.5 2.5 $\beta$ 0.02 0.02 0.02 0.02 $\gamma$ 5 5 5 5 $K$ 10 10 10 10 $\sigma$ 0.3 5.66 0.65 0.65
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