# 2014 Ebola Outbreak: Worldwide Air-Transportation, Relative Import Risk and Most Probable Spreading Routes

## An interactive network analysis

#### August 4th, 2014

Dirk Brockmann^{1,2,3,*}, Lars Schaade^{1}, Luzie Verbeek^{1}

^{1}Robert Koch-Institute, Berlin, Germany

^{2}Institute for Theoretical Biology, Berlin, Germany

^{3}Northwestern Institute on Complex Systems, Northwestern University, Evanston, USA

^{*}Contact: dirk.brockmann@hu-berlin.de

### Ebola

### Challenges

**interactive network analysis**we provide here (see image on the right) displays computationally estimated relative import risks. Before you use the tool, please make sure you understand what these quantities mean and how to interpret them.

#### What is relative import risk?

P(Y|X).

In words: Given an infected individual boards a plane at X, P(Y|X) is the probably that the individual exits the system at location Y. This conditional probability may involve multiple flight legs and multiple possible routes that can be taken from X to Y.

#### What is absolute import risk?

P(Y,X) = P(Y|X) x P(X)

This is the conditional probability P(Y|X) multiplied by the probability that an infected individual boards a plane at X.

Because the probability **P(X) is usually much much smaller** than P(Y|X) the actual import probability is also much smaller than P(Y|X). For example if only 1 in 1000 passengers that travel from X to Y is infected than the absolute import risk is 1000 times smaller than the relative import risk.

### How to interpret and use relative and absolute risk?

P(A|X) = 10%

P(B|X)=1%

The import probability to A is thus 10 times larger than to B. This says nothing about the absolute import probabilities P(A,X) and P(B,X) because these require knowledge of P(X), the probability that an infected individual actually boards a plane at X. P(X) is very difficult to assess as it depends on a number of parameters that we do not know, and some things that are impossible to measure.

On can, however, get a rough estimate of the order of magnitude of P(X) in the context of the Ebola outbreak. Say we have a country with a population N of approx. 10 Mio. (like Guinea for instance). Further assume we have between I = 10-100 infected, asymptomatic individuals that can potentially board a plane. Let's assume that approx. 1000 individuals leave the country every day on average. The probability that at least on infected person is among these is approx:

**P(X) ~ 1 - (1 - I/N)^N ~ 0.1 - 1 %**

This number is based on the assumption that infected individuals are distributed uniformly in the local population, which they are not. We can expect that 0.1-1% / day is an overestimation of P(X). Nevertheless, if we assume this as an upper bound for P(X) we obtain for the actual daily import probability for A and B would be

P(A|X) = 0.01-0.1% / day

P(B|X)=0.001-0.01% / day

Relative probabilities are mostly determined by properties of global mobility and can be estimated based on data on the worldwide air-transportation network. Absolute probabilities are determined by factors at the outbreak site and are subject to strong variability.

#### Why is relative risk a useful quantity?

### A model for the global spread of infectious diseases during early phases of the outbreak - Stochastic Estimates of Relative Risks

However, particularly during the early phase of an outbreak, when specifics of the situations are still unclear, unknown or poorly understood, it is difficult to perform detailed computer simulations because they require fixing a multitude of parameters, or, if parameters are unknown, systematic parameter scans. These are computationally resource and time demanding. Furthermore, models that are useful in one context may not be useful in another because substantial differences may exist between e.g. disease specific mechanisms or parameters.

In addition to developing more fine-tuned models, each suitable for specific contexts, it is thus also important to develop techniques that focus on features that different diseases share and extract information that will be valid (within limits) irrespective of disease specifics and can be practically used during the early phases of an outbreak, inform about spreading aspects early on, and may guide decision processes and help establish an intuition about global risks.

In a recent study (D. Brockmann & D. Helbing, Science (2013)) we showed that valuable information about potential, global spreading patterns can be obtained using a geometric approach to the problem and leveraging methods from complex network theory. In this paper, we showed that contagion phenomena in complex, strongly heterogeneous transportation networks are dominated by the most probable pathways a disease can take through the network. This allowed the introduction of a novel type of effective distance that is a reliable predictor of epidemic arrival times.

To estimate risk for the 2014 Ebola outbreak we used insights from this study and devised a stochastic model for the probabilities of ensembles of paths an infected individual that enters an source node in the network (e.g. one of the airports in the outbreak region) can take. The details of this model are currently being prepared for publication. In a nutshell, for every path from X to Y, potentially via a sequence of intermediate locations, we compute two quantities, the probabilities for every step along the path, and the probability that the current location is the destination. Both types of quantities are computed from traffic flux across the worldwide air transportation, a network of approx. 4000 airports and approx. 25000 direct links. Every simulated individual is going to reach a destination somewhere in the network. For two locations X and Y we integrate over all possible paths that contribute to the individual exiting the system at Y.

In addition to the overall relative import risks this method estimates, it provides topological information of the most likely spreading pathways and the roles different airports play in the global distribution of risk. For instance, from the perspective of a chosen location in the outbreak region, the most likely spread to other locations is equivalent to a tree structure in which different airports have different numbers of branches. If an airport has many branches, it plays a more important role (imposes a larger risk) of further disseminating the disease. A clear example of this can be seen by comparing the perspective of the potential Ebola spread from Sierra Leona on one hand, and Guinea on the other. Whereas in the former UK airports play a dominant role in terms of distribution propensity, in the latter case Paris, CDG does.

### Interactive Risk Assessment

For example, the tool illustrates the quantitative and topological impact that air traffic reduction may have on the spread of Ebola across the network, including a few examples of the impact of single connection removal in the network.

The tool will open in a separate window if you click on the image on the top right. The usage of the tool is fairly self-explanatory. A few directions are provided on the bottom right in the tool window: When you hover over a node, node information will be displayed, clicking on a country will highlight the country's nodes in the network and cumulative information will be displayed. In the top left of the tool window, you can pick from different perspectives, i.e. different root or reference nodes.

The tool only depict a reduced network of the 1227 largest airports to allow better rendering and avoid clutter.