The Scaling of Human Contacts and Epidemic Processes in Metapopulation Networks

Abstract

We study the dynamics of reaction-diffusion processes on heterogeneous metapopulation networks where interaction rates scale with subpopulation sizes. We first present new empirical evidence, based on the analysis of the interactions of 13 million users on Twitter, that supports the scaling of human interactions with population size with an exponent γ ranging between 1.11 and 1.21, as observed in recent studies based on mobile phone data. We then integrate such observations into a reaction- diffusion metapopulation framework. We provide an explicit analytical expression for the global invasion threshold which sets a critical value of the diffusion rate below which a contagion process is not able to spread to a macroscopic fraction of the system. In particular, we consider the Susceptible-Infectious-Recovered epidemic model. Interestingly, the scaling of human contacts is found to facilitate the spreading dynamics. This behavior is enhanced by increasing heterogeneities in the mobility flows coupling the subpopulations. Our results show that the scaling properties of human interactions can significantly affect dynamical processes mediated by human contacts such as the spread of diseases, ideas and behaviors.

Figure 1

(A) Rescaled cumulative degree C_r against population N, measured between 13129406 Twitter users distributed across 2371 basins in 205 countries. (B) Rescaled cumulative degree against population, measured between 4606444 Twitter users in 1344 metropolitan areas in 31 countries. We normalized the values of C_r and N by their average to compare the results across different countries. Insets show the dependency of C_r on N restricted to the Twitter users in the US and Europe.

Figure 2

(A) Phase diagram defined by the threshold condition R_*(p, λ) = 1, corresponding to the solid lines, for η = 0 and η = 0.12. We consider uncorrelated scale-free networks of V = 10⁵ nodes, and P(k) ~ k^−2.1. We set θ = 0.5, , and μ = 0.3. (B) Simulated global attack rate D_∞/V as a function of the mobility rate p for different values of the contact scaling exponent η = 0, 0.06, 0.12 and λ = 0.35. Vertical lines indicate the critical threshold value p_c calculated by setting R_* = 1 in Eq. 5. Each point is averaged over at least 2 × 10³ simulations. Error bars correspond to the standard error of the mean.

Figure 3. Simulated global attack rate D_∞/V as a two-dimensional function of the mobility rate p and the transmissibility λ for different mobility network structures characterized by θ = 0.5 (A) and θ = −0.4 (B).

Black solid lines indicate the analytical predictions for the critical values of p and λ corresponding to R_* = 1. Here the network parameters are the same as in Fig. 2 and η = 0.12. Each point of the phase-space is averaged over 2 × 10³ simulations. To facilitate the visual comparison between the simulations and the analytical solutions we plot the z-axis considering the negative log₁₀ of D_∞/V.