Critical behavior of the susceptible-infected-recovered model on a square lattice

By means of numerical simulations and epidemic analysis, the transition point of the stochastic asynchronous susceptible-infected-recovered model on a square lattice is found to be $c_0=0.1765005\left(10\right)$, where $c$ is the probability a chosen infected site spontaneously recovers rather than tries to infect one neighbor. This point corresponds to an infection/recovery rate of ${\lambda }_c=\left(1-c_0\right)/c_0=4.66571\left(3\right)$ and a net transmissibility of $\left(1-c_0\right)/\left(1+3c_0\right)=0.538410\left(2\right)$, which falls between the rigorous bounds of the site and bond thresholds. The critical behavior of the model is consistent with the two-dimensional percolation universality class, but local growth probabilities differ from those of dynamic percolation cluster growth, as is demonstrated explicitly.

Figure 1

(Color online) Plot of $s^{\tau -2}{P}_{\ge s}$ vs $s^{\sigma }$ for the asynchronous SIR model, using dynamic percolation values $\tau =187/91$ and $\sigma =36/91$, for (top to bottom for large $s$): $c=0.176490$, $0.176495$, $0.176500$, $0.176505$, and $0.176510$. The number of samples are, respectively, $335000$, $10500000$, $440000$, $11000000$, and $360000$.

Figure 2

Slope of the linear part of the three central curves of Fig. 1, measured for $s^{\sigma }\ge 125$. The intercept of the line on the horizontal axis gives an estimate for $c_0$.

Figure 3

Plot of $\text{ln}{P}_{\ge s}$ vs $\text{ln}s$ for the SIR model, simulated on a $16384\times 16384$ lattice, with a cutoff of $s=2097152$. The equation represents a linear fit of the points for $s>100$, where $x$ represents $\text{ln}s$ and $y$ represents $\text{ln}{P}_{\ge s}$.

Figure 4

(Color online) Example of a completed bond percolation cluster of $51456$ wetted sites, generated at the threshold $p_c=1/2$. The red (dark) dot marks the place where the cluster growth began.

Figure 5

(Color online) Example of a completed SIR cluster of $51034$ recovered sites, generated at the threshold $c_0=0.1765$. The red (dark) dot shows the location of origin of the infection.