Front propagation in a spatial system of weakly interacting networks

We consider the spread of epidemic in a spatial metapopulation system consisting of weakly interacting patches. Each local patch is represented by a network with a certain node degree distribution and individuals can migrate between neighboring patches. Stochastic particle simulations of the SIR model show that after a short transient, the spatial spread of epidemic has a form of a propagating front. A theoretical analysis shows that the speed of front propagation depends on the effective diffusion coefficient and on the local proliferation rate similarly to fronts described by the Fisher-Kolmogorov equation. To determine the speed of front propagation, first, the early-time dynamics in a local patch is computed analytically by employing degree based approximation for the case of a constant disease duration. The resulting delay differential equation is solved for early times to obtain the local growth exponent. Next, the reaction diffusion equation is derived from the effective master equation and the effective diffusion coefficient and the overall proliferation rate are determined. Finally, the fourth order derivative in the reaction diffusion equation is taken into account to obtain the discrete correction to the front propagation speed. The analytical results are in a good agreement with the results of stochastic particle simulations.

Figure 1

Front profiles at early and late times. Shown is the fraction of recovered nodes [panel (a)], the fraction of susceptible nodes [panel (b), solid lines], and the fraction of infected nodes [panel (b), dashed-dotted lines]. Each local network consists of $N=10000$ nodes; the transmission rate is $\gamma =0.01$ day${}^{-1}$, the disease duration is $T=15$ days, and the strength of interaction between neighboring patches $D=0.03$ day${}^{-1}$. To illustrate the stochastic nature of simulations, the profiles are shown for five different runs both for $t_{in}=200$ days and $t_f=700$ days.

Figure 2

Histogram of front speeds for three values of local network sizes, $N=1000$ (black squares), $N=5000$ (blue circles), and $N=15000$ (green triangles). The transmission rate is $\gamma =0.01$ day${}^{-1}$, the disease duration is $T=15$ days, and the strength of interaction between neighboring patches $D=0.06$ day${}^{-1}$. The total number of runs for each value of $N$ is 500.

Figure 3

Average front speed $\langle c\rangle$ and the width of speed distribution $\sigma$ as a function of network size $N$. The upper panel shows that the average speed saturates at large values of $N$. Here the circles denote the simulation results, while the dashed line represents the power-law dependence (see text). The red dotted line shows the theoretical result; see text. The inset shows that the width of the speed distribution slowly decreases (circles). The power law decay of the average speed shift is clearly seen in the lower panel. The parameters are as follows: the transmission rate is $\gamma =0.01$ day${}^{-1}$, the disease duration is $T=15$ days, and the strength of interaction between neighboring hubs $D=0.03$ day${}^{-1}$. The results were averaged over 500 runs for each data point.

Figure 4

Speed of front propagation as a function of $D$—the strength of interaction between neighboring patches. The theoretical curve (blue dashed line) is given by Eq. (11), while the red circles denote the simulation results. The parameters are as follows: each local network consists of $N=5000$ nodes, the transmission rate is $\gamma =0.01$ day${}^{-1}$, the disease duration is $T=15$ days, and the simulation results were averaged over 100 runs.