Natural Human Mobility Patterns and Spatial Spread of Infectious Diseases

We investigate a model for spatial epidemics explicitly taking into account bidirectional movements between base and destination locations on individual mobility networks. We provide a systematic analysis of generic dynamical features of the model on regular and complex metapopulation network topologies and show that significant dynamical differences exist to ordinary reaction-diffusion and effective force of infection models. On a lattice we calculate an expression for the velocity of the propagating epidemic front and find that, in contrast to the diffusive systems, our model predicts a saturation of the velocity with an increasing traveling rate. Furthermore, we show that a fully stochastic system exhibits a novel threshold for the attack ratio of an outbreak that is absent in diffusion and force of infection models. These insights not only capture natural features of human mobility relevant for the geographical epidemic spread, they may serve as a starting point for modeling important dynamical processes in human and animal epidemiology, population ecology, biology, and evolution.

Popular Summary

Planning containment strategies for emergent epidemics, as epitomized by the recent H1N1 pandemic, requires efficient forecasts with answers to three basic questions: How many people will be infected, where, and when? To answer the last two questions requires the knowledge of the effective speed of a spreading epidemic. Physical models can relate that speed to key parameters of the underlying processes. A class of frequently used models are the so-called reaction-diffusion models, where “reaction” refers to infection and where the motion of people is assumed to be “diffusion—a type of random motion.” These models typically predict that the speed increases with the magnitude of the diffusion. Human mobility, however, is strikingly different from the assumed diffusion. This fact challenges predictions of these models and puts their universal features into question.

The main approach described in this paper replaces the diffusion model by a more realistic one for human mobility patterns. In the new model, individuals have their own home bases and typically frequent only a limited number of places from those bases—a very different mobility pattern from diffusion. This more realistic description leads to a number of predictions fundamentally different from those of the reaction-diffusion models: One, there is an upper bound on the speed of a spreading epidemic no matter of how high the overall mobility in the system of moving/residing individuals is. This means that the reaction-diffusion models may overestimate the spreading speed considerably. Two, there exists a new type of outbreak threshold in how frequently individuals travel between different places. Both of these effects show up robustly even when the specifics are varied about how different places or populations are connected. These insights are not only important for the development of containment strategies, but also lay the foundation for improved computational models designed to forecast future epidemics. And, beyond human epidemiology, the work should also find potential applications in a wide range of scientific problems in human or animal ecology, population dynamics, and evolution.

Figure 1

Metapopulational mobility models. Patches and arrows represent individual populations and travel. (a) Diffusive dispersal: indistinguishable individuals travel randomly between different locations governed by the set of transition rates ${\omega }_{nm}$. (b) Dispersal capturing individual mobility patterns: individuals with label $k$ possess travel rates ${\omega }_{mk}^k$ and ${\omega }_{km}^k$ at which they travel from their base location $k$ to connected locations $m$ and back.

Figure 2

Front velocity $c(\omega )$ of the model defined by Eq. (6) as a function of mobility rate $\omega$ in comparison to ordinary reaction-diffusion dynamics [Eq. (2)]. For stochastic simulations we used a lattice with ${10}^2$ sites and $N={10}^4$ individuals/site. Mean velocities are indicated by red (bidirectional mobility) and blue (reaction-diffusion) symbols. Analytical results, i.e., Eqs. (9, 4) are shown by dashed and dash-dotted lines, respectively.

Figure 3

Results of stochastic simulations. Attack ratio $\rho ({\omega }^{-}/N)$ as a function of the return travel rate ${\omega }^{-}$ for SIR epidemics on a lattice with ${10}^2$ sites with $N$ agents/site. Epidemic parameters: $\alpha =1$, $\beta =0.1$. Insets: (a) $\rho ({\omega }^{-}/N)$ for a scale-free network ($\gamma =1.5$) of ${10}^3$ nodes populated uniformly with $⟨N⟩=250/\mathrm{\text{site}}$, with $k_{\min }=5$ and $k_{\max }=50$; (b) $\rho ({\omega }^{-}/N)$ for an Erdős-Rényi network with 500 nodes and $\overline{k}=10$. Results were averaged over 50 realizations. Interlocation flux rate was $⟨\omega ⟩=1$.