Exact results for a simple epidemic model on a directed network: Explorations of a system in a nonequilibrium steady state

Motivated by fundamental issues in nonequilibrium statistical mechanics, we study the venerable susceptible-infected-susceptible (SIS) model of disease spreading in an idealized, simple setting. Using Monte Carlo and analytic techniques, we consider a fully connected, unidirectional network of odd number of nodes, each having an equal number of in- and out-degrees. With the standard SIS dynamics at high infection rates, this system settles into an active nonequilibrium steady state. We find the exact probability distribution and explore its implications for nonequilibrium statistical mechanics, such as the presence of persistent probability currents.

Figure 1

Comparison of the simulation results of the two systems with the exact solution (14). There are $N=7$ nodes in both the directed and undirected systems, set at $r=1$ and $\theta \approx 0.267$. (a) For each value of $k$, simulation data for both systems and the exact results coincide. (b)The relative error, $|1-P_{\text{simulations}}^{*}\left(k\right)/P_{\text{exact}}^{*}\left(k\right)|$, compared with the expected sampling error, $1/\sqrt{f_k}$, $f_k$ being the frequency we observe the system being in microstate $k$ during the run.

Figure 2

Probability currents, $K^{*}(k,k^{\prime })$, measured in simulations for the (a) directed and (b) undirected systems. Only a portion of all the $128\times 128$ currents are shown. Note the difference in the scales for the two cases.

Figure 3

Coarse-grained vorticity around the $\mathrm{i}\text{−}\mathrm{j}$ plaquette in a directed (a) and an undirected (b) network. (c) Adjacency matrix in our directed network.

Figure 4

Scaled plot of $\langle {\mathcal{L}}_t\rangle /N$, showing good data collapse for four cases: $N=41,81,161,401$. The unit of time here is MCS.

Figure 5

(a) Distributions of $\mathcal{L}$, $p\left(\mathcal{L}\right)$, obtained from histograms of observed values in a run of ${10}^8$ steps, for the undirected network ($\sigma =0$, green pluses) and the directed one ($\sigma =1$, red open circles). (b) Asymmetry in the distributions, defined as $\left[p\right(\mathcal{L})-p(-\mathcal{L}\left)\right]/\left[p\right(\mathcal{L})+p(-\mathcal{L}\left)\right]$, highlighting the different behaviors associated with the undirected network ($\sigma =0$, green pluses) and the directed one ($\sigma =1$, red open circles).