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PLoS One. 2011; 6(1): e16591.

Published online 2011 January 31. doi: 10.1371/journal.pone.0016591

PMCID: PMC3031602

Paolo Bajardi,^{#}^{1,}^{2} Chiara Poletto,^{#}^{1} Jose J. Ramasco,^{3} Michele Tizzoni,^{1,}^{4} Vittoria Colizza,^{5,}^{6,}^{7,}^{*} and Alessandro Vespignani^{8,}^{9,}^{10,}^{*}

Matjaz Perc, Editor^{}

University of Maribor, Slovenia

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.

Received 2010 December 7; Accepted 2010 December 30.

Copyright Bajardi et al. This is an open-access article distributed under the
terms of the Creative Commons Attribution License, which permits unrestricted use,
distribution, and reproduction in any medium, provided the original author and
source are credited.

This article has been cited by other articles in PMC.

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 [1]. The target of these control measures is the decrease of travel to/from the areas affected by the epidemic outbreak and the corresponding decline of infected individuals reaching countries not yet affected by the epidemic. While the effects of slowing down the international propagation of an epidemic can be statistically evaluated based on available data and bootstrap techniques [2], the impossibility of disentangling the role played by travel from other contributing factors in the spread of an epidemic [3] has generated discussion about the appropriate strategy for mobility restrictions. In this context the only way to systematically gauge uncertainty and the effectiveness of competing control strategies is through data-driven modeling efforts [4]–[9]. Unfortunately, most previous works have focused on synthetic pandemic influenza scenarios and only a few empirical examples are available to validate models and evaluate the effectiveness of travel restrictions in general [10]–[12].

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 [1]. The aggregation of data on the H1N1pdm therefore represents an unprecedented opportunity to calibrate and validate a modeling approach to the global spread of epidemics that integrates detailed information on human mobility and travel. In the present work, we use the Global Epidemic and Mobility model (GLEaM) [13] that, fully integrating high resolution demographic and mobility data, allows the calibration to the H1N1pdm data of the invasion during the early stage of the epidemic and the exploration of hypothetical scenarios in which reductions in the international travel to/from Mexico with different timing and magnitude are considered. Interventions acting on mobility are found to be scarcely efficient in delaying the invasion process of the pandemic. This computational evidence can be explained within a simplified theoretical framework in terms of a phase transition between invasion and non-invasion dynamics of the metapopulation system, where the critical value is crucially affected by the topological fluctuations of the mobility network.

The Global Epidemic and Mobility model is based on a metapopulation scheme [4], [8], [9], [14]–[20] in which the world is
divided into geographical regions defining a subpopulation network where
connections among subpopulations represent the individual fluxes due to the
transportation and mobility infrastructure. GLEaM is composed of three different
layers [13]:
(i) the population layer that integrates census areas for a total of 3,362
subpopulations around major transportation hubs in 220 countries of the world
with a resolution up to ¼°×¼° [21]; (ii) the human mobility
layer that integrates both commuting flows collected from various sources in
more than 30 countries and airline traffic flows provided by the International
Air Transport Association (IATA) database [22]; and (iii) the
disease dynamics layer that implements a refined *SEIR*-like
model [23]
taking into account the specific etiology of the H1N1pdm [18].

The model simulates short-range mobility between subpopulations with a time scale
separation approach that defines the effective force of infections in connected
subpopulations [13], [18], [24], [25]. The airline mobility
from one subpopulation to another is modeled by an individual based stochastic
procedure in which the number of passengers of each compartment traveling from a
subpopulation *j* to a subpopulation *l* is an
integer random variable defined by the actual data from the airline
transportation database [8]. The infection dynamics takes place within each
subpopulation. We adopt a *SEIR*-like model [23] in which we consider
separate compartments for symptomatic traveling and not traveling, as well as
asymptomatic individuals in each subpopulation. More in detail, a susceptible
individual in contact with a symptomatic or asymptomatic infectious person
contracts the infection at rate *β* or
*r _{β}β*
[26], [27],
respectively, and enters the latent compartment where he is infected but not yet
infectious. After an average latency period

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 [18], [28], [29].
Infection parameters describing the transmission potential and the duration of
the stages of the disease are obtained through a maximum likelihood procedure
based on the empirical data of the H1N1 international seeding events (see Figure 1A). In particular, we
use the reproductive number
*R _{0}*=1.75 with the generation
interval set to 3.6 days (average latency period

During the early stage of the outbreak, several countries implemented a variety
of travel-related interventions (see Text S1 for country-specific measures and
implementation details). Such measures, in addition to a spontaneous reaction of
individuals to the health emergency, led to a reduction in the international
traffic to/from Mexico of about 40% observed during the month of May,
followed by smaller reductions in the following months, and resulting in a slow
return to normality in about 3 months [31] (see Text S1).
Here we consider as a *reference scenario* the one produced by
the best estimates able to reproduce the initial chronology of newly infected
countries (i.e. the *baseline scenario*), where in addition we
take into account the empirically observed drop in air traffic, following the
data reported in Table 2 of Text S1. The reference scenario is then
compared to a set of hypothetical scenarios in which increasingly larger
restrictions in individual mobility are considered, as well as different
starting dates for the implementation of such restrictions. In addition, we also
test scenarios in which country-specific air travel bans are applied, and
scenarios in which ground mobility along the border between Mexico and the US is
restricted (see Text S1).

It is important to stress that, contrary to previous approaches based on samples of airline mobility data [4], [9], GLEaM simulations take into account the full air travel database and the role of intra-country mobility as well as border commuting flows (e.g. across the US-Mexico border [18]). GLEaM allows for the detailed simulation of the time evolution of the spreading pattern by reproducing the infection dynamics and computing the number of travelers in each compartment. It is therefore possible to track the movement of H1N1 cases and analyze the statistics associated with arrival times, case importation, and local transmission based on many realizations that incorporate the relevant stochastic effects. The efficacy of travel-related measures is therefore measured on the timing of seeding events and resulting delays.

Figure 2 summarizes the
simulation's accurate reproduction of the observed relative magnitude of
imported cases in the local epidemics of newly-affected countries that validate
the model. Panels A, B show cases in the United Kingdom and Germany,
respectively, during the early phase of the outbreak when case-based
surveillance was deployed in order to detect imported H1N1 cases and monitor
local H1N1 transmission [32], [33]. Computer simulations also allow us to explore the
level of stochasticity associated with the importation of infectious
individuals. We keep track for each time step of each realization of the
contribution of imported cases to the total prevalence in the country defined as
the ratio *Q* of imported cases versus the total number of
infectious individuals in the country. Since at the early stage of the epidemic
there are usually large fluctuations in the number of imported local
transmission cases, we measure the probability in time of observing a given
ratio *Q* by averaging over 2,000 realization of the global
simulation. Panels 2C, 2D show the time behavior of the probability distribution
*P(Q)* clearly illustrating that the importation of cases
dominates the initial phase of the epidemic in each country, which is soon
followed by a sustained local transmission. The contribution of imported cases
is observed at 100% with a finite probability only during the months of
April-May, after which the probability distribution progressively shrinks around
small values of *Q*, showing how the local H1N1 transmission
starts to dominate the epidemic.

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 Text S1 for more details). We then test whether an additional decrease in travel flows of magnitudes larger than the observed 40% would have provided an additional benefit in slowing down the propagation of the H1N1 virus across the world. We consider drops in the air travel flows connecting Mexico with the rest of the world starting on April 25 following the international alert, optimistically assuming a prompt implementation by authorities with no further delays. We also assume that the reduction is kept constant across time, differently from the empirically observed decline that successively decreased to become negligible in about 3 months.

Figure 3 shows changes induced by travel restrictions on the simulated chronology with respect to the reference case by tracking the arrival time probability distribution. Results are reported in panels A, B of Figure 3, where application of the interventions is shown to reduce the probability values right after the peak of the distribution, with almost no change in the date of the peak. If we focus on the first arrival from Mexico, considering all possible seeding events (i.e. latent, asymptomatic, and symptomatic), we observe similar reductions in the rate of increase in the cumulative probability distribution of the seeding event, pointing to a slower rate of importation (see Figure 3C, D). However, the resulting change is not able to halt the spread.

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. Figure 4A shows the delays obtained for a selection of countries. Even given the unlikely assumption of a 90% travel reduction, the resulting delay would be on the order of 2 weeks, confirming results from previous studies [4], [5], [8], [9]. This time could be used to finalize the response by the public health infrastructure of unaffected countries following the international alert, thus gaining time to enhance surveillance systems and allocate resources. Unfortunately, this timescale is insufficient to develop and distribute a vaccine. Anticipation of travel reductions following local epidemiological alerts in Mexico or the onset of symptoms from the first case in the US would lead to similar results (see panel B of Figure 4).

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
Δ*t* is a logarithmic function of the
applied travel reduction of magnitude ,
, where is the timescale
of the epidemic's exponential growth in the seed population [34], [35]. The
exponential increase of cases in the outbreak region is therefore responsible
for the relatively limited delay induced by strong and lasting travel
reductions. When ,
or the corresponding
delays become approximately 1, 1.6, and 3 times, respectively, the timescale
that is typically on the order of a few days. The
logarithmic relation also explains more realistic situations in which the
epidemic origin is characterized by spatial heterogeneity and intra-region
mobility that is not subject to travel restrictions (see Text S1 for
the complete analytic treatment in this case). This is the case of the H1N1
pandemic, which initially diffused within Mexico before reaching international
hubs and propagating internationally.

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 Figure 5A) whose
features reproduce the topological and mobility properties of real-world
transportation systems [36], [37]. We assume: (i) the large-scale heterogeneity found
in the airline transportation network where the number of connections
*k* departing from each airport (i.e. the degree of the node)
follows a power-law distribution ; and (ii) that the
observed correlations between topology and traffic, relating the number of
passengers *w _{ij}* traveling from airport

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 , it is possible to
describe the invasion dynamics at the subpopulation level in a Levins-type
approach by considering the microscopic dynamics of infection and of individual
travel [37].
The system is characterized by a subpopulation reproductive number
*R _{*}*. Analogous to the reproductive number

(1)

The r.h.s. of Eq. (1) describes the contribution of the subpopulations of degree
*k*' at generation *n*-1 to the infection
of subpopulations with degree *k* at generation
*n*. Each of the has
(*k*'−1) possible connections along which the
infection can proceed (−1 takes into account the link through which each
of those subpopulations received the infection). In order to infect a
subpopulation of degree *k*, three conditions need to occur: (i)
the connections departing from nodes with degree *k*' point
to subpopulations of degree *k*, as indicated by the conditional
probability ; (ii) the reached subpopulations are not yet infected,
as indicated by the probability , where
is the total number of subpopulations with degree
*k*; (iii) the outbreak seeded by
infectious individuals traveling from
*k*' to *k* takes place, and the probability
for this event to happen is given by
[41]. The
latter term is the one that relates the microscopic dynamics of the local
infection occurring within a subpopulation to the coarse-grained view that
describes the disease invasion at the metapopulation level. It depends on the
details of the diffusion process of individuals as well as the individual travel
behavior and its interplay with the disease stages. The expression of
for the compartmental model here considered is derived
in Text
S1; by plugging it in to Eq. (1) we can derive the expression for the
global invasion threshold (more details are reported in Text
S1):

(2)

where is a function that
depends on the reproductive number only; is a combination of
the infection parameters ,
, *p _{a}* and

The effect of interventions like travel restrictions, mitigation, etc., are
unfortunately damped by the large topological fluctuations of human mobility
patterns. The function is expressed by
(Text S1), so that the topological
heterogeneities encoded in lead to very large
values of the ratio , which suppresses
reduction in the travel flows in , leading to values
of *R _{*}* well above the threshold at 1 as shown by
the 3D plot reported in Figure
5B. Similar conclusions apply for entry screening at the airports
modeled by a reduction in the traveling probability

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.

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.

**Competing Interests: **The authors have declared that no competing interests exist.

**Funding: **This work has been partially funded by the National Institutes of Health
R21-DA024259 award, the Lilly Endowment grant 2008 1639-000 and the
DTRA-1-0910039 award to AV; the EC-ICT contract no. 231807 (EPIWORK) to AV and
VC; the EC-FET contract no. 233847 (DYNANETS) to AV, VC, and JJR; the ERC Ideas
contract n.ERC-2007-Stg204863 (EPIFOR) to VC, PB, CP, and MT. The funders had no
role in study design, data collection and analysis, decision to publish, or
preparation of the manuscript.

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