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Impact of human mobility on the
periodicities and mechanisms underlying
measles dynamics
Ramona Marguta and Andrea Parisi
Research
Cite this article: Marguta R, Parisi A. 2015
Impact of human mobility on the periodicities
and mechanisms underlying measles
dynamics. J. R. Soc. Interface 12: 20141317.
http://dx.doi.org/10.1098/rsif.2014.1317
Received: 30 November 2014
Accepted: 19 January 2015
Subject Areas:
computational biology, biocomplexity
Keywords:
human mobility, measles, periodicity, SIR
Author for correspondence:
Andrea Parisi
e-mail: [email protected]
Electronic supplementary material is available
at http://dx.doi.org/10.1098/rsif.2014.1317 or
via http://rsif.royalsocietypublishing.org.
Centro de Fı́sica da Matéria Condensada, Biosystems and Integrative Sciences Institute and Departamento
de Fı́sica, Faculdade de Ciências da Universidade de Lisboa, Campo Grande Edificio C8,
1749-016 Lisboa, Portugal
Three main mechanisms determining the dynamics of measles have
been described in the literature: invasion in disease-free lands leading to
import-dependent outbreaks, switching between annual and biennial
attractors driven by seasonality, and amplification of stochastic fluctuations close to the endemic equilibrium. Here, we study the importance
of the three mechanisms using a detailed geographical description of
human mobility. We perform individual-based simulations of an SIR
model using a gridded description of human settlements on top of
which we implement human mobility according to the radiation model.
Parallel computation permits detailed simulations of large areas. Focusing
our research on the British Isles, we show that human mobility has an
impact on the periodicity of measles outbreaks. Depending on the level
of mobility, we observe at the global level multi-annual, annual or biennial
cycles. The periodicity observed globally, however, differs from the local
epidemic cycles: different locations show different mechanisms at work
depending on both population size and mobility. As a result, the periodicities observed locally depend on the interplay between the local
population size and human mobility.
1. Introduction
A large body of research in recent years has been devoted to the study of
the geographical spread of childhood diseases and, in particular, measles.
Early work was carried out by Cliff and co-workers on the geographical
spread of measles in Iceland, showing that the disease is not self-sustained,
but depends on the intensity of imports from outer countries [1,2]. Examining
measles incidence between 1901 and 1970, they showed [1] that pre-1938
measles in Iceland was characterized by isolated outbreaks separated by long
(up to 8 years) disease-free periods. Increased communication after 1945 led
to a reduction in the interepidemic periods: between 1955 and 1968, the epidemic peaks were separated by 4 years, during which the disease was again
extinct. These observations were consistent with the rate of imports being the
major determinant of the occurrence of epidemics in disease-free lands.
A series of studies on the incidence of measles in England and Wales [3– 6]
showed that the situation was different when large populated areas were investigated. In particular, measles shows complex nonlinear dynamics that can be
explained as a result of a switching between close-by attractors (annual and
biennial) caused by the strength of the seasonal forcing [5]. This mechanism
is shared by other childhood diseases; for instance, for rubella, the coexistence
of 1 and 5 year attractors was similarly found [5]. Further analysis of a large set
of administrative areas in England and Wales using wavelets demonstrated that
measles propagates from large cities to smaller populated areas, with large
populated areas synchronized by the seasonal forcing [6].
Recently, another approach has been suggested for the dynamics of childhood diseases conferring lifelong immunity. The idea comes from the
observation that in finite populations the stochastic fluctuations around the
& 2015 The Author(s) Published by the Royal Society. All rights reserved.
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for an SIR model parametrized for measles changes when
mobility among individuals is enhanced or reduced.
2. Computer model
Tij ¼ Ti
m i nj
,
(mi þ sij )(mi þ nj þ sij )
(2:1)
where sij is the total population within the circle of radius rij
around the source i, excluding both populations mi and nj.
P
Here, Ti ¼ j=iTij is the total flux of commuters leaving
location i and it is proportional to the population of the
source location. Hence,
Ti ¼ mi
Nc
,
N
(2:2)
where Nc is the total number of commuters and N is the total
population in the country. The fraction Nc/N, which represents the fraction of individuals in the population that
participate in the long-distance displacements, controls the
level of transmission to different cells. This fraction can be
estimated by existing data on human mobility. The model
has a few notable properties that are particularly advantageous to us. The flux between locations is controlled by
a single parameter Nc/N, which represents the fraction
of individuals in the population that participate in the
J. R. Soc. Interface 12: 20141317
The computer model used for our simulations combines three
level of descriptions: (i) a population level, providing a detailed
geographical distribution of human population; (ii) a mobility
level, detailing how individuals move around the geographical
area under study; and (iii) an epidemiological level, detailing
how the disease evolves.
The first layer is thus the population description: it
uses data from the Gridded Population of the World (GPW)
database [20], which provides estimates of the resident
population of a given geographical area on a regular grid
of cells of angular size 2.5 arc-minutes. This corresponds to
a longitudinal size of approximately 5 km at the Equator,
and around 3.5 km in northern Europe. The database also
provides maps with a 30 arc-second resolution, but we did
not use these highly detailed maps here. Because a single
cell is our basic geographical unit, we consider each of
these cells as a well-mixed population in which the disease
can evolve following a specified epidemiological model.
Hence, transmission within the cell occurs as in a wellmixed model; however, transmission between different cells
is also taken into consideration by simulating human mobility. We allow each individual to move to a different cell
with a given probability that depends on the propensity of
individuals to move to that cell. We assume that individuals
commute daily between different locations and, in order to
evaluate the probability of reaching a specific location, we
are interested in knowing what the daily flux is between
the home location and the destination location. The description of human mobility we use represents the second layer
of our model and is based on the recently introduced radiation
model [19], which was shown to accurately predict human
mobility patterns and overcomes a series of limitations of
the widely used gravity model [19]. The model gives the
flux of individuals Tij travelling from source i characterized
by a population mi to the destination j characterized by a
population nj, the two being separated by a distance rij. The
flux is given by the following formula:
2
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endemic equilibrium can become the source of recurrent
epidemics. McKane & Newmann [7] studied a simple
predator –prey model with fixed carrying capacity, and
showed that the sustained oscillations around the equilibrium
can be explained as a result of an amplification of stochastic fluctuations around a preferred frequency. The
whole system behaves as a forced oscillator, except that
the forcing is not provided by any external source: it is the
result of the intrinsic stochasticity of the system and its
nonlinear response. The same approach was demonstrated in a susceptible–infected –recovered (SIR) model with
demography [8]. In particular, it was shown that, when
parametrized for measles, the oscillations around the equilibrium have a preferred frequency of 2.1 years, in line with
the frequency of outbreaks reported in historical time series.
Further research showed that, as soon as spatial [9] or
temporal correlations [9,10] are taken into account, the
amplitude and coherence of these fluctuations increase
considerably, making them more regular.
The three different mechanisms described (importdependent spread, multi-stable dynamics driven by seasonality,
stochastic amplification) have been suggested as major determinants in the observed incidence data, but they have been studied
separately. Our aim is to explore their relative importance in a
more realistic geographical setting. The increase in computational power witnessed in recent years already permits
detailed simulations of geographical disease spread in both
humans and animals. An important example in which the geographical spread was studied in detail is that of foot-and-mouth
disease [11]: with detailed information of farm locations and
cattle movements, individual-based simulations provided accurate predictions on the geographical dynamics of the disease.
For human-related diseases, information about mobility is
much harder to obtain. Nevertheless, simulations of the
spread of influenza [12–15] at the global level and severe
acute respiratory syndrome (SARS) at local level [16] have
been performed in recent years. Our aim here is to perform
simulations on regional scales with a description of human
mobility as detailed as possible.
The main problem facing numerical work on regional geographical scales is the lack of detailed information on human
mobility at such scales. For instance, Riley et al. [16] used a
metapopulation model to describe the spread of SARS in
Hong Kong, where each metapopulation represents a different
district, but interaction among districts was based on their adjacency, with a fixed rate among neighbouring districts and a
different fixed rate among more distant districts independent
of their distance and population. Colizza et al. [13,14] instead
describe global spread using the large-scale aeroplane connection network, but mobility at lower scales needs additional
information on the mobility of individuals. Recent studies
however have shown that human mobility can be statistically
described on multiple scales using a truncated power law
[17,18]. Moreover, a model for human mobility was recently
developed [19] that depends only on a single parameter that
can be fitted to available datasets on mobility for a specific
country. The model overcomes several limitations of the traditionally used gravity model, and incorporates the scale
invariance of commuting patterns. This model permits a coherent description of mobility on multiple scales. In this study, we
present computer simulations of disease spread using a
detailed geographical description of human mobility on a
specific area, the British Isles, and explore how the dynamics
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(1)
(2)
(3)
(4)
b
transmission: SIj ! Ij I1
gL
recovery Ij ! I jþ1 (1 j , L)
gL
recovery IL ! R
m
death/birth {Ij , R} ! S:
We use the same rate m for death and birth ensuring, as is
commonly done, that the population size remains constant.
The parameter m does not change during a simulation, thus
we do not consider changes in birth and/or death rate.
We include seasonality by letting the parameter b be time
dependent—the form of the seasonality is assumed to be
term-time forcing [5]:
b(t) ¼ b0 [1 þ b1 term (t)],
(2:3)
3
J. R. Soc. Interface 12: 20141317
the amount of data we are dealing with, we parallelize the
algorithm by assigning chunks of the geographical map
under study to different computing nodes. To achieve this,
an algorithm is used that partitions the map into regions
that are compact in shape and support similar population
sizes. The algorithm performs a multi-level coarsening of the
gridded map in order to obtain a coarsed map that is composed by a small number of coarsed grid elements. An
optimal subdivision of the map is then rapidly found using
a simulated annealing algorithm. The map is then decoarsened
one level at a time, and, each time, the final part of a simulated
annealing algorithm is performed to adapt the optimal solution found at the coarser level to the new decoarsed level.
The procedure is repeated until decoarsening leads to the
original gridded map. This complex algorithm allows us to
partition large maps in very short times. The full procedure
is detailed in the electronic supplementary material, S1.
The third layer in our simulations is the epidemiological
model, which describes the disease dynamics in each cell.
The simulation program supports any compartmental
model provided we define the compartments and the set of
transitions with their corresponding transition rates. In this
work, we focused on the SIR model. In each cell, individuals
can be either susceptible (S), infected (I) or recovered (R).
When infected, they can transmit the disease to other
susceptibles, whereas recovered individuals have a lifelong
immunity to the disease. Demography acts by removing
infected and recovered individuals and replacing them with
new susceptible individuals, thus replenishing the susceptible class. We also include gamma-distributed recovery
profiles [22] by using multiple infective classes. The number
of infective classes L interpolates among an exponential
recovery for L ¼ 1 and a deterministic recovery for L ! 1.
Simulations with L ¼ 1 and L ¼ 2 show a remarkable qualitative and quantitative difference: this is understandable,
because the distribution of recovery times is exponential for
L ¼ 1 (thus corresponding to a constant rate of recovery),
whereas it is non-exponential with a maximum away from
0 for L . 1 (corresponding to a recovery rate which depends
on the time from infection), hence changing L from 1 to 2
introduces an important difference. No qualitative difference
appears instead among simulations with L ¼ 2 and simulations with higher values of L. In this work, our simulations
were performed with L ¼ 2: we did not explore any further
dependence on L, reserving an eventual exploration for
later studies.
The processes considered are thus described by the
following set of events:
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long-distance displacements and thus controls the level
of transmission to different cells. By modifying this parameter, the fluxes between locations can be varied with an
explicit meaning in terms of the number of moving individuals. Additionally, the fluxes are source limited: increasing
Nc/N the number of individuals leaving each cell never
exceeds the population of the cell, a property that does not
hold for the widely used gravity model.
Detailed studies on human mobility, based on the tracking
of a large number of mobile phone users [18], have shown that
individuals have a set of preferred locations they visit with
specific frequencies. In particular, an individual’s probability
of going to visit a specific location was generally found to
follow a Pareto law [18]; a detailed study on human mobility
within the city of Portland, OR, USA [21] showed it to
follow a power law. We mimic a similar behaviour by assigning to each individual a set of preferred locations: each
individual has its own specific set, and he will visit them
with a frequency that reflects his propensity to visit as dictated
by the radiation model. These preferred locations are chosen
randomly for each individual, but in such a way that the overall fluxes between locations correspond to those predicted by
the radiation model. In practice, we build the set of preferred
locations and their frequencies as follows: for individuals in
each cell i, we calculate the probability of visiting cell j. This
P
is simply given by pij ¼ Tij/mi. The probability pi ¼ j=ipij
corresponds to the probability that an individual leaves cell
i and through equation (2.2) it corresponds to pi ¼ Nc/N: it follows that 1 2 pi represents the probability of not leaving cell i.
Hence, the full set of choices for an individual are all the cells of
the map j (with j = i), each chosen with probability pij, and
location i with probability 1 2 pi. The associated probabilities
form a discrete distribution that can be used to extract preferred
locations. We sample this probability distribution M times,
obtaining a list of locations in which those with high probability will have been chosen more often. We can thus count
the number of times each distinct location has been chosen
and transform this count into a frequency by dividing it by
the size M of the sample. The result is a list of distinct locations
with an associated frequency of visit that can be used to decide
where the individual should be moved to. The size of the
sample M was chosen to be equal to 50 in all simulations:
values larger than one order of magnitude did not lead to
significantly different results even for low values of Nc/N.
Individuals are moved daily: at the start of each day, a
destination is chosen for each individual among his set of preferred locations according to their frequency of visit. The
individual is then moved to that preferred location and participates in the dynamics of that cell, thus permitting
transmission of the disease to different cells. At the end of
the day, the individual is moved back to his original location.
The use of a set of preferred locations is particularly advantageous for us, because it reduces the problem of choosing
a destination from a number of possible destinations as
large as the total number of grid cells with non-uniform probability, to that of choosing among a small set of preferred
locations. This is justified by the fact that most locations
will have a very low probability of visit. For the simulations
here presented, the average number of preferred locations per
individual was at most close to 15, compared with a total
number of cells close to 90 000.
As mentioned earlier, our aim is to study large geographical areas and thus large populations. In order to easily handle
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1
,
R0
j
Lg
m
1
,
1
R0
Lg þ m Lg þ m
mR0
1
R ¼ 1 1
:
R0
b
Ij ¼
0j,L
Note that the total number of infectives at equilibrium is
P
m
I ¼ L1
(R0 1) depending on the number of infecj¼0 Ij ¼
b
tive classes L only through R0. Individuals are initially
assigned one epidemiological class with a probability proportional to the corresponding endemic equilibrium prevalence,
thus building in each cell a random discrete realization close
to the endemic equilibrium. Owing to the mobility of individuals, the resulting set-up does not correspond to the local
endemic equilibrium, but it is sufficiently close to it: the
system settles after about 7–8 years of simulated time.
3. Methods and data
For the investigation described in this work, our area of study
corresponds to that of the British Isles. The choice comes
from the wealth of data available for England and Wales
for childhood diseases, and the fact that the British Isles can
be considered to some extent an isolated unit, neglecting
imports from outer regions. The population count maps are
obtained by combining the GPW maps for the UK, Ireland
and the Isle of Man, thus excluding any other surrounding
country. It is of course possible to extend the study to include
the influence of neighbouring countries, but we did not
pursue this goal in this study.
Disease parameters for measles were chosen following
[5]. We set the transmission rate to b0 ¼ 1.175 d21; the recovery rate was set to g ¼ 1/13 d21 and the death/birth rate to
m ¼ 5.5 1025 d21. The seasonal forcing was set to b1 ¼
0.25 following [5], except when we were interested in
4. Results
Before illustrating results obtained through simulations, it is
worth discussing briefly two extreme cases. The first case
occurs when Nc/N is set to zero: this corresponds to lack of
mobility of individuals who are constrained to their residence
cell. Thus, the model describes a collection of isolated wellmixed populations. In the map under consideration, all
cells have a population below the critical community size
(the population size of the inhabited cells varies between 1
and 137 646 individuals, the latter corresponding to a cell
located in the area of London), and because no interaction
between cells is possible, they will experience permanent disease extinction. On the other hand, for values of Nc/N close
to 1, the model describes a set of cells where all individuals
travel to other cells: hence, the dynamics is more wellmixed, but still it does not correspond to that of a wellmixed population. In fact, each individual will most probably
interact with individuals having the same preferred location,
hence the probability of interaction with any other individual
is not uniform. This should not be surprising, because it
reflects a realistic description of human mobility where
4
J. R. Soc. Interface 12: 20141317
S ¼
comparing with the non-forced model; in that case, we
used b0 ¼ 0.
The only remaining parameter needed to perform simulations of mobility is the ratio Nc/N that controls human
mobility. We did not consider vaccination in this work.
Hence, the value of this ratio should be referred to the interval of years corresponding to that of available pre-vaccination
data that, for measles, range between 1944 and 1966 [26];
however, there is no available information about mobility
for those years. As a consequence, we used mobility as a
proxy to investigate its influence on the dynamics. First, we
estimated the mobility ratio using recent census data available from the Office of National Statistics of the United
Kingdom [27]. We were able to obtain a value of Nc/N ¼
0.176 + 0.001 for our simulations of the UK by comparing
the distribution of the average flux of commuters, with
respect to distance covered as predicted by our radiation
model based on gridded maps, with that derived from
census data of the UK. The procedure used for this estimate
is detailed in the electronic supplementary material, S2, and
takes into account the specific set-up of our model. We then
assumed this value to be valid for the whole area under
investigation, and used it as a reference for the mobility
ratio. We explored different values of Nc/N to investigate
the influence of this ratio on the dynamics, varying it between
0 (for which no mobility occurs) and 0.2.
Incidence data for measles in England and Wales were
made available by the authors of [26]. Data provide biweekly
incidence for 60 cities in England and Wales between 1944
and 1966, as well as average population size of each of the
cities. The original unaggregated weekly data come from
the Office of Population Censuses and Surveys [26,28].
Under-reporting was estimated to be around 50% [26,29]:
the correct infective incidence is estimated in [4] through a
time-series analysis that takes into account the variation in
birth rates during those years. Because we assume constant
birth rates, here we use a simplified approach, and we
assume that the correct number of infective cases is twice
the reported number.
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where term (t) is a function taking values þ1 during school
time and – 1 otherwise [5,23,24], and b1 is a parameter that
controls the amplitude of the forcing.
The dynamics is simulated in each cell soon after moving
individuals to new locations chosen from their sets of preferred locations according to their frequency of visit as
detailed earlier in this section. Each cell thus will be reduced
in population owing to the individuals who leave the cell for
other destinations, and will increase owing to individuals
coming from other cells. Individuals who, after the mobility
phase, are in the cell (hence residents who did not leave the
cell and visitors coming from other cells) can interact for
that day. Dynamics is simulated using the Gillespie eventselection algorithm for 1 day [25], and disease can hence be
transmitted between current residents and visitors of the
cell. At the end of the day, individuals are moved back to
their home cell, and then the whole procedure is repeated.
The system is initially placed close to endemic equilibrium.
The basic reproductive number and the equilibrium values for
the number of susceptible, infective and recovered individuals
in a well-mixed population can be calculated from the deterministic approximation. For an SIR model with L infective classes,
the basic reproductive number depends on L and is given by
R0 ¼ b/m[1 2 (Lg/Lg þ m)L]. The equilibria can be expressed
in terms of R0 as
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(a)
42 500
40 000
10–3
10–2
Nc/N
(b)
(c)
200 000
200 000
100 000
100 000
30
40
50
30
60
(d)
I(t)
10–1
40
50
60
50
60
(e)
200 000
200 000
100 000
100 000
30
40
50
t
60
30
40
t
Figure 1. (a) Daily average number of infectives for the whole British Isles as a function of the mobility ratio Nc/N: circles correspond to simulations with seasonality
and squares to simulations without seasonality. The dashed line corresponds to the endemic equilibrium for a well-mixed population. All values are close to the
endemic equilibrium; simulations for Nc/N ¼ 0.0005 and lower values drop to zero owing to extinction events. The other plots show the time series of seasonally
forced simulations for the different values of the mobility ratio: (b) Nc/N ¼ 0.001, (c) Nc/N ¼ 0.008, (d) Nc/N ¼ 0.05, (e) Nc/N ¼ 0.2. Simulations with seasonality
were performed with parameters as described in the main text, resulting in kb(t)l ¼ 1:32. Simulations without seasonality were performed with
b0 ¼ kb(t)l ¼ 1:32. (Online version in colour.)
individuals interact most intensively with those visiting the
same preferred locations, be it for work, shopping, etc. As a
result, this model does not interpolate between isolated and
well-mixed populations, but rather it approximates realistic
human mobility at given values of the mobility ratio, with
the two extremes of Nc/N ¼ 0 and Nc/N ¼ 1 both describing
unrealistic situations.
First, we are interested in the global behaviour in the area
under study for different values Nc/N. Because the model
does not describe a well-mixed population for any value of the
mobility ratio, it becomes important to understand whether
the endemic equilibrium is maintained in these conditions, and
how the dynamics around the endemic equilibrium change. In
our simulations, we observe extinction for levels of Nc/N
lower than 0.0005: this value suggests that persistence of the disease can occur even with extremely low mobility levels; however,
the specific threshold is influenced by the size of the grid cells
and could be different if a population not artificially partitioned
into gridded cells was considered. For higher values of Nc/N, the
average infective incidence achieves a level close to that of the
endemic equilibrium (figure 1a). However, the overall dynamics
for low and high mobility differ, as shown in figure 1b–e. The
low-mobility simulation shows a remarkably different epidemic
cycle characterized by less frequent and more temporally wide
outbreaks (figure 1b). For mobility ratios of Nc/N ¼ 0.008
(figure 1c) and 0.05 (figure 1d), the frequency of the outbreaks
is annual, whereas it appears to be biennial for Nc/N ¼ 0.2
(figure 1e). These results suggest that the mobility ratio of individuals between different locations is a major determinant in
the overall dynamics of the disease in the area under study, controlling the frequency at which outbreaks occur.
If mobility influences disease dynamics globally, how
does it influence disease dynamics locally, at the city level?
We obtain time series for the cities by considering the aggregated incidence data of the cells within the area occupied by
the city. We simplify the shape of the city to a circle centred
on the geographical centre of the city, covering a surface
equivalent to the surface area of the city itself. The radius
of the circle is fine-tuned to get a population close to that
reported in census data. We specify in the caption of the
figures the resulting population used for simulation data as
well as the historical city population to which the measles
datasets refer.
Figure 2 shows the disease incidence in the area of
London for two values of the mobility ratio: Nc/N ¼ 0.05
(figure 2a) and Nc/N ¼ 0.2 (figure 2b). In both cases, the
plot shows a regular series of biennial epidemics, compatible
with the data available for the area of London (figure 2d):
except for the incidence level, the frequency of outbreaks
seems unaffected by this difference in mobility ratios.
This is remarkably in contrast with the global time series,
especially when considering that the area of London accounts
for about 12% of the whole population in the area under consideration. This suggests that for London the dynamics are
fundamentally internal: imports from other locations do not
play a relevant role. The different behaviour of the global
time series for Nc/N ¼ 0.05 must hence be due to dynamics
within lower populated areas. Of the three mechanisms we
are interested in, seasonality for London plays the major
role. For such large locations, the sequence of biennial outbreaks must be a consequence of the seasonal forcing. This
is illustrated in figure 2c where the incidence for London is
shown in the absence of seasonality (b1 ¼ 0). In this case,
the incidence level fluctuates around the endemic equilibrium, and the resulting time series differ remarkably from
the behaviour observed in datasets for London. Note that
J. R. Soc. Interface 12: 20141317
I(t)
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I
5
45 000
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(a)
(b)
6
0.005
35
40
45
50
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1950
1955
t
1960
1965
J. R. Soc. Interface 12: 20141317
0.004
0.003
I
0.002
0.001
0
30
(c)
35
40
45
50
30
(d)
0.005
0.004
0.003
I
0.002
0.001
0
30
35
40
45
50
1945
t
Figure 2. Biweekly incidence as a function of time for the city of London. City population: simulated 7 779 671 individuals; data 3 249 440. (a) Nc/N ¼ 0.05,
b1 ¼ 0.25. (b) Nc/N ¼ 0.2, b1 ¼ 0.25. (c) Nc/N ¼ 0.05, b1 ¼ 0. (d ) Historical time series. All other disease parameters are as described in the main text.
(a)
(b)
0.010
0.008
0.006
I
0.004
0.002
30
35
40
45
50
(c)
30
35
40
45
50
1950
1955
t
1960
1965
(d )
0.010
0.008
I
0.006
0.004
0.002
30
35
40
t
45
50
1945
Figure 3. Biweekly incidence as a function of time for the city of Sheffield. City population: simulated 473 005 individuals; data 499 505. (a) Simulation for Nc/N ¼
0.1, b1 ¼ 0.25; (b) simulation for Nc/N ¼ 0.05, b1 ¼ 0.25; (c) simulation for Nc/N ¼ 0.008, b1 ¼ 0.25; (d ) observed datasets. For all simulations, all other
parameters are as described in the main text.
since the definition of the extent of the area of London
changed in 1965, in our simulation, we use the current definition while datasets (figure 2d) are based on the previous
definition and thus refer to a smaller population size.
In figure 3, we show the time series for Sheffield, a city
with a population of about 500 000 individuals, considerably
smaller than London, but still beyond the critical community
size. The mobility among the high-populated cells of the area
leads to persistence and to a biennial cycle for Nc/N ¼ 0.1
(figure 3a). For Nc/N ¼ 0.05 (figure 3b), the interepidemic
period is slightly larger although no extinction occurs in the
area: the level of imports affects the local dynamics causing this light shift. For lower values of the mobility ratio
(figure 3c), the decrease in import leads to extinctions and
to a larger change in frequency of the outbreaks, being
more distant in time.
The situation is different when we look at low-populated
areas. Figure 4a–d shows the infective incidence for the city
of Chester. This is a city with a population of 57 000 residents, about 30 km from the larger city of Liverpool, and
60 km from Manchester. For Nc/N ¼ 0.1 (figure 4b), the
time series of measles incidence are quite irregular, but a
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0.006
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1955
t
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Figure 4. Biweekly incidence as a function of time for the city of Chester. City population: simulated 52 255 individuals; data 57 180. (a) Simulation for Nc/N ¼ 0.05,
b1 ¼ 0.25; (b) simulation for Nc/N ¼ 0.1, b1 ¼ 0.25; (c) simulation for Nc/N ¼ 0.05, b1 ¼ 0; (d) observed datasets. For all simulations, all other parameters are as
described in the main text. Data with and without seasonality show limited differences in terms of frequency of outbreaks.
09–12
12–12
02–13
04–13
07–13
09–13
1
13
105 785
Figure 5. Snapshots of a simulation of the spread of measles for Nc/N ¼ 0.1.
The month and year refer to simulation time. Our simulations extend over 68
years (25 000 days): here, we show a few frames taken during the 12th and
13th year. The colour gradient represents the number of infected individuals in
each cell. (Online version in colour.)
detailed analysis shows several extinction events. For lower
values of the mobility ratio, extinction events combined
with lower mobility become increasingly important: figure
4a shows the time series for Nc/N ¼ 0.05. The lower
number of imports results in more spaced outbreaks, with
a frequency for the main peaks roughly triennial. Here, seasonality has a limited effect; the amplitude and frequency in
the time series in (figure 4a) and (figure 4c) barely differ. This
suggests that, in contrast with the case of London, the frequency at which outbreaks are observed in datasets (figure
4d) for small populated areas is a direct consequence of the
level of mobility.
The irregularity of the time series might result from the
proximity of Chester to larger cities. In the electronic supplementary material, S3, we include a movie for Nc/N ¼ 0.1
showing how the disease spreads in the area we are studying.
Travelling waves originating from high-populated areas can
be observed: these propagating waves can avoid extinction
during the epidemic troughs and can trigger recurrent
epidemics in low-populated areas. Some of these lowerpopulated areas are subject to the occurrence of multiple
waves leading to irregular time series. The movie in the electronic supplementary material shows that the region of
Chester is exposed to waves propagating from the neighbouring large cities and to waves coming from the south and west.
Figure 5 shows some snapshots taken from this movie.
Besides the two cases of large and small populated cities,
a third case occurs when a city sustains a set of epidemic
outbreaks with rare extinction events. This typically occurs
for cities with intermediate population sizes: in the literature,
the city of York is often used to illustrate this case. In our
simulations, we fine-tune the city size to obtain a population
of about 100 000 individuals, in line with the population
reported in historical datasets for measles outbreaks [26].
We observe that a moderate mobility ratio permits sustained
epidemics. Figure 6a,b shows the time series for Nc/N ¼ 0.1
and Nc/N ¼ 0.05. The results in figure 6a show a biennial cycle for major outbreaks with no extinctions. For
Nc/N ¼ 0.05 instead, the time series (figure 6b) shows a
slightly larger cycle close to 2.5 years: the increase of the
cycle period might be related to the occurrence of extinction
events occurring owing to the lower mobility. Figure 6c
shows the time series for the city of York for a non-seasonal
simulation (b1 ¼ 0) and a mobility ratio Nc/N ¼ 0.1. The frequency of outbreaks remains close to 2 years: this is in line
with the fact that the stochastic fluctuations have a periodicity
of 2 years for measles.
J. R. Soc. Interface 12: 20141317
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J. R. Soc. Interface 12: 20141317
0.008
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Figure 6. Biweekly incidence rates as a function of time for the city of York. City population: simulations 111 254 individuals; data 105 150. (a) Simulation for
Nc/N ¼ 0.05, b1 ¼ 0.25; (b) simulation for Nc/N ¼ 0.1, b1 ¼ 0.25; (c) simulation for Nc/N ¼ 0.05, b1 ¼ 0; (d ) observed datasets. For all simulations, all other
parameters are as described in the main text.
(a)
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Figure 7. Power spectra of the simulated time series for the city of York
for different values of the mobility ratio as a function of the frequency:
(a) Nc/N ¼ 0.1, (b) Nc/N ¼ 0.08 and (c) Nc/N ¼ 0.05. The continuous line
shows the non-seasonal power spectrum for simulations with b1 ¼ 0, whereas
the dashed line shows the power spectrum for simulations with b1 ¼ 0.25.
For all simulations, kb(t)l ¼ 1:32. Power spectra were obtained from an average over 100 simulations in each case, after discarding the first 10 simulated
years. (Online version in colour.)
Figure 7 shows a comparison of the power spectra of the
time series for Nc/N ¼ 0.1, 0.08 and 0.05 for the seasonal and
non-seasonal case as a function of the frequency. The
seasonal simulations show a peak at the frequency corresponding to a cycle of 1 year. Simulations for Nc/N ¼ 0.1 and 0.08
show a stochastic peak close to 0.5, corresponding to a biennial
cycle; for Nc/N ¼ 0.05, the stochastic peak corresponds to a
larger cycle of 2.5 years. Comparing seasonal and non-seasonal
simulations with the same Nc/N, we can observe that the stochastic peaks are close in amplitude. For Nc/N ¼ 0.05, the
period of outbreaks
P( f )
20
London
Sheffield
York
Chester
global
6
4
2
0
10–3
10–2
Nc/N
10–1
Figure 8. Periodicity of disease outbreaks for different values of the mobility
ratio. The plot shows periodicities observed in simulations for the four cities
discussed in the text and for the global time series. (Online version in colour.)
seasonal forcing does not affect the stochastic peak, whereas
a mild dependence is observed for larger values.
In figure 8, we report a detailed plot of the observed dependence of the period of outbreaks versus the mobility ratio Nc/N
for the cities discussed in the text as well as the global
time series. We can distinguish four behaviours depending
on Nc/N. For the highest values used (Nc =N . 0:15), the
period of outbreaks converges to a biennial cycle for all
cases. Below this level of mobility instead the period observed
for cities and that observed for the global time series differ. For
intermediate values of the mobility ratio (0:025 , Nc =N , 0:10),
the global time series show annual outbreaks. For these same
values, London shows its characteristic biennial cycle, in line
with our observations discussed earlier. Smaller cities show
instead a period of fluctuations that depends on Nc/N and
increases when mobility is decreased. Chester has a more pronounced dependence than Sheffield or York owing to its small
population size, which leads to local extinctions and a strong
dependence on imported cases from nearby locations for
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A large body of research has been devoted to measles in
order to understand its dynamics. Three mechanisms have
been identified as most relevant: import-dependent spread,
which permits propagation in areas where the disease went
extinct; multi-stable dynamics driven by seasonal forcing;
and stochastic amplification. These mechanisms have been
studied in detail at the level of cities on models where the
population is well mixed. Recently, some attempts have
been proposed to investigate how some of these mechanisms
interact on a larger collection of cities. Here, we have explored
a more detailed description of mobility using a model that
simulates propagation on a detailed scale, reaching large
cities as well as low-populated areas of the British Isles.
The new ingredient that we have introduced is a realistic
description of mobility at a level of detail and on an area
which extends to the whole British Isles. This level of description has two relevant characteristics: first, it provides the
possibility to investigate in silico the role of high-populated
as well as low-populated areas; second, it uses a description
of mobility that goes beyond the well-mixed modelling of
mobility used in recent publications.
Our results show that the level of mobility of individuals
influences disease incidence both locally and globally. At the
global level, the time series change from showing multiannual cycles for very low mobility levels, to annual cycles,
to biennial cycles for increasing mobility ratio. Locally, the
change in disease incidence depends on the population size
of the areas under study. In particular, we observe different
mechanisms at work: in the large populated areas, the seasonal forcing drives the system out of equilibrium creating the
typical biennial outbreaks observed in datasets for most
values of the mobility ratio. Low-populated areas instead
have the frequency of outbreaks dependent on the level of
mobility owing to the extinction following each outbreak,
until the mobility ratio is high enough to sustain the epidemic. When this occurs, the time series is still heavily
influenced by the local mobility, with waves propagating
from the surrounding areas that make the time series strongly
irregular. An intermediate regime occurs for populations large
enough to be less influenced by the surrounding areas, for
which extinction is a rare event. In this situation, with an
adequate level of mobility, the population sustains the epidemic and, owing to the low population size, amplifies the
stochastic fluctuations of the disease.
The results summed up in figure 8 show the complex
dependence of the period of outbreaks on the mobility
ratio. Two behaviours are particularly interesting. First, the
response of the city of London to variations of the level of
human mobility is unique when compared with that of smaller locations. While we have discussed the cause of the
9
J. R. Soc. Interface 12: 20141317
5. Discussion and conclusions
biennial cycles observed for intermediate values of Nc/N,
we have no clear explanation for the annual cycles observed
for low levels of mobility. Second, the global time series show
biennial cycles for high levels of mobility, whereas intermediate and low levels of mobility lead to annual cycles, in
contrast with what occurs at the local level. The origin of
this discrepancy might be linked to the contribution coming
from locations which happen to be out of synchrony with
respect to other locations. For low levels of Nc/N, the range
of periodicities observed at different locations might concur
to a similar result. The analysis discussed here is by no
mean exhaustive. For instance, the movie in the electronic
supplementary material, S3, shows an area in the north
of England, which includes Liverpool, Manchester and
Leeds, which behaves as the beating heart of the region (see
also [6]), not dissimilarly from what happens in the area of
London. However, the origin of such behaviour is probably
related to synchronization effects between these cities and
the high-density populated surrounding area, and requires
further studies.
A few observations are required at this point. First, the
level of mobility reported in this work should not be compared too closely with the level of mobility taken from
census data. The latter refers to the level of commuting for
work as measured from census data for 2011. Here, we
study a disease which typically affects children, and as such
the mobility patterns for these young individuals might
differ from those of adult individuals. Our model does not
have any age structure, therefore no distinction is attempted
among younger and older individuals. We do not have
data for mobility between 1944 and 1966, hence we have
used the value of Nc/N obtained for census data only as a
reference value. The mobility level that we use should be
regarded as an effective mobility that weights the mobility
for different age classes in a way similar to a single contact
parameter SIR when compared with an age-structured SIR.
Second, our simulations have been performed using an SIR
model with two infective classes. We have not explored
higher values of the number of infective classes nor SEIR
(susceptible, exposed, infectious, recovered) simulations. We
selected seasonality and disease parameter values as used
in previous simulation works for measles, but did not perform any sensitivity analysis on them. It has also been
shown that the inclusion of non-constant birth/death rates
leads to improved agreement with the historical incidence
data. Using the variation in birth rates in England and Wales,
it was possible to reconstruct the dynamics of the susceptibles
and thus explore the effects of variation in birth rates on the
time series both at global level [4] and at the level of cities
[26]. This could explain for instance the pre-1950 annual
cycles observed in time series [4]. There are thus several aspects
of the epidemiological modelling that require further investigation: we will eventually report results on these issues in a
separate publication.
On a more computational ground, we presented a computer model that enables the study of the spread of infectious
diseases in geographically detailed areas, using a statistical
description of mobility based on the radiation model. Our
simulation model uses a simulated annealing technique
based on coarse-graining of base areal units to partition the
geographical map of interest into regions that can be assigned
to different computing nodes for parallel computation, thus
permitting unlimited complexity. Epidemiological models
rsif.royalsocietypublishing.org
most values of Nc/N. For low values of the mobility ratio
(0:001 , Nc =N ,
0:01), the city of London joins the global
time series in showing annual cycles, whereas smaller cities
maintain their strong dependence on Nc/N. Finally, for
very low values of mobility (Nc =N ≃ 0:001), the global time
series become multi-annual, and most cities follow the same
periodicity. Interestingly, London shows a smaller period of
fluctuations, once again deviating from the behaviour of the
other cities.
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individual) unmodified. One possibility could be to estimate
such fluxes from data of mobile phone tracking: a recent
paper has shown that the radiation model and mobiletracking data can provide complementary results on the
study of the spatial spread of infectious diseases [30].
Acknowledgement. The authors acknowledge funding from the Fundação para a Ciência e a Tecnologia (FCT) under contract no. PTDC/
SAU-EPI/112179/2009.
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References
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other than the SIR can be easily implemented by modifying
the number of classes simulated and the corresponding list
of transition events, while maintaining the underlying structure describing human mobility unmodified. On the other
hand, the radiation model was used by us to infer the probability for an individual to move to a different location.
Hence, it is always possible to use a different model for
human mobility that can provide an alternative description
of the fluxes between locations, while maintaining other features (for instance, the use of a list of preferred locations per