Spatiotemporal fluctuation-induced transition in a tumor model with immune surveillance

We report on a simple model of spatially extended antitumor system with a fluctuation in growth rate, which can undergo a nonequilibrium phase transition. Three states as excited, subexcited and nonexcited states of a tumor are defined to describe its growth. The multiplicative noise is found to have opposite effects: The positive effect on a nonexcited tumor and the negative effect on an excited tumor.

Figure 1

The solution, $m$, of the self-consistency equation is the intersection point between $F\left(m\right)=m$ and $F\left(m\right)=y\left(m\right)$ for noise level ${\sigma }^2=8.0\times {10}^{-3}$.

Figure 2

$m$ as a function of ${\sigma }^2$ given by Eq. (7). The points correspond to the intersection of curves in Fig. 1. The critical immune coefficients are ${\beta }_{c1}=2.156$ and ${\beta }_{c2}=2.209$, respectively, which divide the state of a tumor into three levels: excited $\left(E\right)$, subexcited and $\left(S\right)$, nonexcited $\left(N\right)$.

Figure 3

Stationary probability distributions of average population of tumor cells for different noise intensities and immune coefficients. The parameters are (a) $\beta =2.12$, ${\sigma }^2=0.01$, (b) $\beta =2.30$, ${\sigma }^2=0.01$, (c) $\beta =2.12$, ${\sigma }^2=0.40$, (d) $\beta =2.30$, ${\sigma }^2=0.40$.

Figure 4

$m$ as a function of ${\sigma }^2$ given by Eq. (7). The points are obtained by a method which is the same as for Figs. 1, 2.

Figure 5

One-dimensional simulation for the relationship between $m$ and ${\sigma }^2$. $L$ is the lattice size. The parameters are the same as for Fig. 4.