Reaction-diffusion on random spatial networks with scale-free jumping rates via effective medium theory

We study epidemic processes using a metapopulation approach on the line featuring random transport rates between arbitrarily distant sites. An average transport network is found using a recently developed variant of the effective medium approximation (EMA) that is capable of dealing with these long-range connections. Using a Feynman-Kac argument in the effective medium, we derive an estimate on the size of the infected domain, and reproduce the known result of its exponential growth in time. We hereby demonstrate the applicability of long-range EMA to dynamical processes on networks more intricate than simple diffusion.

Figure 1

Left panel: scheme of the one-dimensional metapopulation model. The dynamics is regulated by two different time scales, the one of diffusion, corresponding to the supopulation layer, and the reaction, governed by the SIS infection dynamics at the individual layer. Right panel: illustration of a sample metapopulation network consisting of $N=20$ subpopulations with symmetric transition rates $W_{x,y}$. The graph is constructed from a one-dimensional ring topology with all connections permitted, thus allowing the mapping with a two-dimensional space such as the earth geographical space. The edge color and size scales accordingly with the values of each transition rate.

Figure 2

Diameter of the infected population obtained from numerical integration of the reaction-diffusion equations and the EMA prediction given by the upper bound of Eq. (14). The epidemic is generated by a subpopulation SI reaction with $\beta =0.2$ in $N=4000$ subpopulations with Lévy exponent $\alpha =1.5$.

Figure 3

Prevalence curves (violet) for the SIS reaction with $\beta =0.2$ and $\mu =0.1$ of the $N=8000$ fully connected subpopulations with rates' Lévy exponent $\alpha =1.5$. The asymptotic value of the SIS steady state, the disease prevalence ${\rho }_x\left(\infty \right)=(\beta -\mu )/\beta$, is marked by the black dashed line while the concentration threshold $\overline{c}$ that defines the infection outbreak in each subpopulation is marked by the dash-dot blue line.

Figure 4

The extrapolated value of the Lévy exponent $\alpha$ with the corresponding error (blue bars) evaluated from the error propagation of the numerical fit error in $C$, shown in the inset, as a function of the basic reproductive number ${\mathcal{R}}_0=\beta /\mu$ at fixed $\alpha =1.5$ (a). Theoretical growth rate $C_{\text{the}}$ and the respective simulation fit value $C_{\text{fit}}$ for the SIS reaction with $\mu =0.1$ in $N=8000$ subpopulations as a function of the Lévy exponent $\alpha \in (1,2]$ at fixed $\beta =0.2$ (b).

Figure 5

Absolute value of the difference $\mathrm{\Delta }C:=|C_{\text{the}}-C_{\text{fit}}|$ between the theoretical $C_{\text{the}}=(\beta -\mu )/(1+\alpha )$ and the simulation fit value $C_{\text{fit}}$ for the SIS reaction with $\beta =0.2$ and $\mu =0.1$ as a function of the subpopulations number $N$. Different lines are for different Lévy exponents from dark to light in the range $\alpha \in (1,2)$. Inset: close-up in doubly logarithmic scale for $\alpha \in (1,1.5)$. For larger values of $\alpha$, the error fluctuates around 0.005 which is the numerically attainable accuracy.

Figure 6

Diameter of the infected population obtained from the simulation (light-blue scatter) of the SIS reaction in $N=8000$ subpopulations with $\beta =0.2$ and $\mu =0.1$. The theoretical prediction (dark-red line) is given by EMA for various Lévy exponent $\alpha \in (1,2]$.