Phys Rev E. 2017 Jan;95(1-1):012313. doi: 10.1103/PhysRevE.95.012313.

Effective distances for epidemics spreading on complex networks.

Iannelli F1, Koher A2, Brockmann D3,4, Hövel P2, Sokolov IM1.
1 Institute for Physics, Humboldt-University of Berlin, Newtonstraße 15, 12489 Berlin, Germany.
2 Institute for Theoretical Physics, Technische Universität Berlin, Hardenbergstraße 36, 10623 Berlin, Germany.
3 Robert Koch-Institute, Nordufer 20, 13353 Berlin, Germany.
4 Institute for Theoretical Biology and Integrative Research Institute of Life Sciences, Humboldt-University of Berlin, Philippstraße 13, Haus 4, 10115 Berlin, Germany.


We show that the recently introduced logarithmic metrics used to predict disease arrival times on complex networks are approximations of more general network-based measures derived from random walks theory. Using the daily air-traffic transportation data we perform numerical experiments to compare the infection arrival time with this alternative metric that is obtained by accounting for multiple walks instead of only the most probable path. The comparison with direct simulations reveals a higher correlation compared to the shortest-path approach used previously. In addition our method allows to connect fundamental observables in epidemic spreading with the cumulant-generating function of the hitting time for a Markov chain. Our results provides a general and computationally efficient approach using only algebraic methods.




Forecast and control of global epidemic outbreaks such as SARS (2003), H1N1 (2009) and Ebola (2014) remains a major challenge to public health institutions. One of the most relevant information for containment strategies is the time until the first infected individual arrives at a specific place. Due to the complex nature of human mobility, however, the spreading patterns of infectious diseases in geographical space appears erratic and unpredictable.

Here, we provide a mathematical framework that allows to estimate infection arrival times, based only on the topology of the network. In particular, we derive a quantity that is based on the mean hitting time. That is the expected number of steps until a random walker, which starts at the outbreak location, first arrives at a target node.

The derivation uncovers the close connection between a simple diffusion process and epidemics spreading on complex networks. Detailed numerical simulations on the global air-traffic network support our results.

The excellent performance of the effective distance shows that it can be used for containment strategies and risk assessments in real epidemic scenarios.



Figure 1: The global mobility network used in the simulations consisting of V = 3865 airports and E = 51 440 flights [1]



Figure 2: Correlation of the infection arrival times in days obtained from direct simulation of the metapopulation infection model with the two relevant effective distance approaches defined using only the network topology. The random-walk approach (blue) improves even further the dominant path approach (red) proposed in [2] and [3]. The rescaling allows to estimate with great precision the simulations infection arrival times. The source infected node is São Paulo Guarulhos International Airport, and each point in the scatter plot corresponds to a destination airport in the global mobility network, with size proportional to its out-traffic . The Pearson correlation coefficient for the random walk approach is r=0.99 indicating an almost perfectly linear relation between infection arrival time and effective distance.




[1] Global mobility data of air traffic was provided by OAG (

[2] A. Gautreau, A. Barrat, and M. Barthelemy, J. Theor. Biol. 251,509 (2008)

[3] D. Brockmann and D. Helbing, Science 342, 1337 (2013)