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Conceived and designed the experiments: PB CP JJR MT VC AV. Performed the experiments: PB CP JJR MT VC AV. Analyzed the data: PB CP JJR MT VC AV. Contributed reagents/materials/analysis tools: PB CP JJR MT VC AV. Wrote the paper: PB CP JJR MT VC AV.

The authors have declared that no competing interests exist.

After the emergence of the H1N1 influenza in 2009, some countries responded with travel-related controls during the early stage of the outbreak in an attempt to contain or slow down its international spread. These controls along with self-imposed travel limitations contributed to a decline of about 40% in international air traffic to/from Mexico following the international alert. However, no containment was achieved by such restrictions and the virus was able to reach pandemic proportions in a short time. When gauging the value and efficacy of mobility and travel restrictions it is crucial to rely on epidemic models that integrate the wide range of features characterizing human mobility and the many options available to public health organizations for responding to a pandemic. Here we present a comprehensive computational and theoretical study of the role of travel restrictions in halting and delaying pandemics by using a model that explicitly integrates air travel and short-range mobility data with high-resolution demographic data across the world and that is validated by the accumulation of data from the 2009 H1N1 pandemic. We explore alternative scenarios for the 2009 H1N1 pandemic by assessing the potential impact of mobility restrictions that vary with respect to their magnitude and their position in the pandemic timeline. We provide a quantitative discussion of the delay obtained by different mobility restrictions and the likelihood of containing outbreaks of infectious diseases at their source, confirming the limited value and feasibility of international travel restrictions. These results are rationalized in the theoretical framework characterizing the invasion dynamics of the epidemics at the metapopulation level.

The human mobility flows that determine the spreading of infectious diseases and the
control measures based on limiting or constraining human mobility are considered in
the contingency planning of several countries

In the recent 2009 H1N1 pandemic (H1N1pdm), control measures included travel bans
to/from Mexico, the screening of travelers on entry into airports, and travel
advisories against non-essential travel to Mexico

The Global Epidemic and Mobility model is based on a metapopulation scheme

The model simulates short-range mobility between subpopulations with a time scale
separation approach that defines the effective force of infections in connected
subpopulations _{β}β^{−1}, each latent individual becomes
infectious, entering the symptomatic compartments with probability
1−_{a} or becoming asymptomatic with
probability _{a}
_{t}) and those who would stop
traveling when ill (with probability 1−_{t})
_{min} during the summer
season to α_{max} during the winter season. Here we consider
α_{max} = 1.1, whereas α_{min}
assumes the best estimate value obtained from the calibration of the model to
the H1N1pdm invasion data (see next Subsection)

_{β}

The model is calibrated on the H1N1pdm data. The initial conditions of the
epidemic are set near La Gloria, Mexico, on 18 February 2009 in agreement with
the information published in official reports and with previous works _{0}^{−1} = 1.1 days
and average infectious period ^{−1}
= 2.5 days). Through a maximum likelihood approach, the
above estimates are obtained that best reproduce the actual chronology of newly
infected countries (additional details can be found in Ref. _{min} determines the strength
of the seasonality effect on the disease transmission. Here we consider
α_{min} in the range [0.6–0.7], that is the best
estimate obtained in Ref.

During the early stage of the outbreak, several countries implemented a variety
of travel-related interventions (see

It is important to stress that, contrary to previous approaches based on samples
of airline mobility data

The good agreement of the model with the actual data from the H1N1pdm allows us
to assess the effect of the observed decline in travel flows to/from Mexico by
comparing the results obtained in the reference scenario with a version of the
model in which no travel reduction is considered. Compartmentalization permits
tracking of the arrival of detectable (i.e. symptomatic) and non-detectable
(i.e. latent or asymptomatic) infected individuals in a given country. By
defining the arrival time as the date the first symptomatic case arrives in the
country under study, it is possible to quantify the delay in the spreading of
the epidemic. It is quite impressive to notice that the 40% drop in
travel flows observed in reality only led to an average delay in the arrival of
the infection in other countries (i.e. the first imported case) of less than 3
days (see

Travel measures imposing a reduction of

By considering the time at which the cumulative probability for the seeding from
Mexico has reached 90%, we can calculate the delay induced by larger
reductions in air travel.

The exponential increase of cases in the outbreak region explains the negligible
impact of travel restrictions over the course of the pandemic. Given two coupled
populations with deterministic infection dynamics, the delay

Another important question concerns the degree to which mobility restrictions are
able to achieve containment at the source of the pandemic, especially in
combination with timely mitigation policies in the country of origin. To this
end we consider a simplified modeling framework based on a metapopulation scheme
describing a network of subpopulations (nodes) coupled with mobility processes
(links, see _{ij}_{i}_{j}

_{*}_{*}_{0}_{0}_{*}

Disregarding the high-resolution details of numerical approaches, this synthetic
metapopulation model can now be analyzed, defining a new theoretical framework
that allows for the study of epidemic containment. Starting from a single
subpopulation infected at time _{*}_{0} at the individual level,
_{*}_{*}

The r.h.s. of Eq. (1) describes the contribution of the subpopulations of degree
_{a}_{t}_{*}

The effect of interventions like travel restrictions, mitigation, etc., are
unfortunately damped by the large topological fluctuations of human mobility
patterns. The function _{*}_{t}_{0}

Our analysis of the 2009 H1N1 pandemic shows that the observed decline in air travel to/from Mexico was of too small a magnitude to impact the international spread. Stricter regimes of travel reduction would have led to delays on the order of two weeks even in the optimistic case of early intervention. It is unlikely that given the ever-increasing mobility of people travel restrictions could be used effectively in a future pandemic event.

(PDF)

We are grateful to the International Air Transport Association for making the airline commercial flight database available to us. We would like to thank Marco Quaggiotto for his help in the visualization design.