Efficient simulation of the spatial transmission dynamics of influenza.

Abstract

Early data from the 2009 H1N1 pandemic (H1N1pdm) suggest that previous studies over-estimated the within-country rate of spatial spread of pandemic influenza. As large spatially resolved data sets are constructed, the need for efficient simulation code with which to investigate the spatial patterns of the pandemic becomes clear. Here, we present a significant improvement to the efficiency of an individual-based stochastic disease simulation framework commonly used in multiple previous studies. We quantify the efficiency of the revised algorithm and present an alternative parameterization of the model in terms of the basic reproductive number. We apply the model to the population of Taiwan and demonstrate how the location of the initial seed can influence spatial incidence profiles and the overall spread of the epidemic. Differences in incidence are driven by the relative connectivity of alternate seed locations. The ability to perform efficient simulation allows us to run a batch of simulations and take account of their average in real time. The averaged data are stable and can be used to differentiate spreading patterns that are not readily seen by only conducting a few runs.

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Efficient Simulation of the Spatial Transmission
Dynamics of Influenza
Meng-Tsung Tsai1, Tsurng-Chen Chern1, Jen-Hsiang Chuang2, Chih-Wen Hsueh3, Hsu-Sung Kuo4, Churn-
Jung Liau1, Steven Riley5, Bing-Jie Shen6, Chih-Hao Shen7, Da-Wei Wang1, Tsan-Sheng Hsu1*
1Institute of Information Science, Academia Sinica, Taipei, Taiwan, 2Epidemic Intelligence Center, Centers for Disease Control, Taipei, Taiwan, 3Department of Computer
Science and Information Engineering, National Taiwan University, Taipei, Taiwan, 4Centers for Disease Control, Taipei, Taiwan, 5Department of Infectious Disease
Epidemiology, University of Hong Kong, Hong Kong, 6Department of Radiation Oncology, Far Eastern Memorial Hospital, Taipei, Taiwan, 7Department of Computer
Science, University of Virginia, Charlottesville, Virginia, United States of America
Abstract
Early data from the 2009 H1N1 pandemic (H1N1pdm) suggest that previous studies over-estimated the within-country rate
of spatial spread of pandemic influenza. As large spatially resolved data sets are constructed, the need for efficient
simulation code with which to investigate the spatial patterns of the pandemic becomes clear. Here, we present a
significant improvement to the efficiency of an individual-based stochastic disease simulation framework commonly used in
multiple previous studies. We quantify the efficiency of the revised algorithm and present an alternative parameterization of
the model in terms of the basic reproductive number. We apply the model to the population of Taiwan and demonstrate
how the location of the initial seed can influence spatial incidence profiles and the overall spread of the epidemic.
Differences in incidence are driven by the relative connectivity of alternate seed locations. The ability to perform efficient
simulation allows us to run a batch of simulations and take account of their average in real time. The averaged data are
stable and can be used to differentiate spreading patterns that are not readily seen by only conducting a few runs.
Citation: Tsai M-T, Chern T-C, Chuang J-H, Hsueh C-W, Kuo H-S, et al. (2010) Efficient Simulation of the Spatial Transmission Dynamics of Influenza. PLoS
ONE 5(11): e13292. doi:10.1371/journal.pone.0013292
Editor: Vladimir Brusic, Dana-Farber Cancer Institute, United States of America
Received May 13, 2010; Accepted September 2, 2010; Published November 4, 2010
Copyright: ? 2010 Tsai 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.
Funding: This work was supported in part by the following grants: Jen-Hsiang Chuang is supported in part by DOH98-DC-2036 from the Centers for Disease
Control, Department of Health, Taiwan, R.O.C.; Tsan-sheng Hsu and Bing-Jie Shen are supported in part by 97-2221-E-001-011-MY3 from National Science Council,
Taiwan, R.O.C.; Churn-Jung Liau is supported in part by 98-2221-E-001-013-MY3 from National Science Council, Taiwan, R.O.C.; Steven Riley is supported in part by
R01 TW008246-01 from Fogarty International Centre, RAPIDD program from Fogarty International Centre with the Science and Technology Directorate
Department of Homeland Security, and Research Fund for the Control of Infectious Disease of the Government of the Hong Kong SAR. The funders had no role in
study design, data collection and analysis, decision to publish, or preparation of the manuscript.
Competing Interests: The authors have declared that no competing interests exist.
* E-mail: tshsu@iis.sinica.edu.tw
Introduction
The current global spread of a novel influenza strain [1]
highlights gaps in our understanding of the spatial component of
disease transmission at national and regional scales. For example,
the early summer 2009 wave in the United States affected some
populations much moresothan others(CentersforDiseaseControl,
USA), even at similar latitudes. In addition, there was substantial
transmission in parts of southern England throughout the summer
of 2009, but very little in most of northern mainland Europe
(European Centre for Disease Prevention and Control). This slow
progression between national and regional level synchrony is not
obviously consistent with previous theoretical studies of the within-
country dynamics of pandemic influenza [2–4], in which census-
reported commuting patterns and airline flight data were used to
characterize very rapid spatial spread. Explaining these early
patterns of spatial spread for the 2009 pandemic will likely be an
active area of epidemiological research in the coming years.
Stochastic spatial transmission models, in which individuals or
small communities are represented explicitly in space, are an
extension of more traditional approaches and have been a
valuable tool in the study of infectious diseases in humans and
animals [5]. Traditionally, mathematical models of epidemics
often take the form of deterministic differential equations in which
the variables represent the expected number of individuals in
broad disease classes (e.g., susceptible, infected, or recovered) [6].
Although such models can be extended to model the geographic
spread of infectious diseases on patches [7], when it is not clear
which spatial scales are most important, it is difficult to use
compartmental approaches with confidence.
Here, we describe an algorithmic refinement of a spatial stochastic
model of individuals and their communities. This framework was
originally designed to investigate community interventions against
influenza in a generic sense [8]. It was later extended to examine the
optimalresponsetoabio-terroristsmallpoxattack[9]andtoexamine
the potential for the containment of influenza pandemic in large well-
mixedpopulations[10].Aspatialcomponentwasaddedtothemodel
to study the feasibility of containing an emergent influenza pandemic
inaruralsettinginSoutheastAsia[11].Initslastmajordevelopment,
the underlying algorithm was parallelized to allow it to run with a
population of 300 million, and used to predict the likely impact of
mitigation measures on an influenza pandemic in the United States
[2].Morerecently,thesameframeworkhasbeenused to describethe
likely fall wave transmission dynamics for H1N1pdm in Los Angeles
County [12], and to study the effects of school closure strategies in
Allegheny County, Pennsylvania [13].
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We have implemented a more efficient algorithm for this
popular disease transmission model. We demonstrate increased
computational efficiency compared with previous implementations
and we describe a parameterization scheme for the model using
the basic reproductive number, rather than the per contact
transmission potential. We illustrate the utility of the refined model
with simulation studies of seeding dynamics for a pandemic of
influenza in Taiwan.
Materials and Methods
Our model incorporates epidemiological attributes of viral
infection with computer generated mock population to simulate
the spatio-temporal spreading of pandemic influenza viruses. The
mock population is constructed according to national demograph-
ics and daily commuter (worker flow) statistics from Taiwan
Census 2000 Data (http://www.stat.gov.tw/) in order to retain
some population characteristics. The model is, effectively, a highly
connected network model representing the 23 million people living
in Taiwan. The connection between any two individuals indicates
the possibility of regular (daily) and relatively close contact that
could result in the successful transmission of the flu virus. A
contact group is a close association of individuals, where every
member is connected to all other members in the group. We
designate ten classes of such contact groups in our model:
community, neighborhood, household cluster, household, work
group, high school, middle school, elementary school, daycare
center, and playgroup. It is important to note that these contact
groups do not represent all people at any physical location such as
a workplace or school, but rather the groups of people who share
the same surrounding activities and sustain regular close contact
for potential viral infection. Furthermore, the entire population is
classified into five age groups: preschoolers (0–4 years old), school-
age children (5–18 years old), young adults (19–29 years old),
adults (30–64 years old), and elders (65+ years old). Each
individual is a member of one of the five age groups throughout
the simulation, and can belong to several contact groups
simultaneously at any time. The probability of any two individuals
staying in contact that could result in the successful transmission of
the flu virus is called the contact probability, and an empiric value
is assigned depending on the group where contact occurs and the
ages of both individuals. Age not only affects the probability of an
individual being infected, it also determines the individual’s
daytime contact groups: preschoolers stay either in daycare
centers or in playgroups; school-age children stay either in schools
or in households as dropouts; young adults and adults stay either in
work groups or in households if unemployed. Each simulation runs
in cycles of two 12-hour periods, daytime and nighttime, with each
cycle representing a day in the simulation. The simulation can
cover any specified duration of days; we usually operate in 180
days for typical influenza season, but there are times when 365
days duration is imperative for a slow progressing epidemic.
Contact occurs between individuals in each contact group every
day, there are no exceptions for weekends or holidays until we can
properly ascertain their effects. During nighttime, contact occurs
only in communities, neighborhoods, household clusters, and
households; whereas in the daytime, contact occurs in all contact
groups. Children do not go outside of their residential community
for daytime activities because the probabilities for such occasional
contacts are too low to be captured by any contact group. The
only inter-community transmission occurs when working adults
commute between household and work group as specified by
worker flow data. The implementation details of the base model
are provided in supporting text (Appendix S1); model parameters,
such as the full listing of contact probabilities, are given in the
supporting information of a study by Germann et al. [2]
The discrete-time simulation of infection events in individual-
based epidemic models can be reduced to the generation of
binomialdeviates.Withinanygivenmodel,therecanbemanytypes
of infectious individual and many types of susceptible individual.
For example, there can be many age groups and many stages in the
natural history of a disease. The set of all possible pairs in which the
first element is an infectious individual and the other element is a
susceptible individual (an I–S pair) defines the set from which
infectioneventscanbesimulatedatanypoint intime.Ifmanyofthe
pairs have exactly the same probability of generating an infection (S
Table 1. Algorithm 1: Naive algorithm.
foreach time period T do
foreach infected individual I do
update the status of I according to T
if I is infectious then
foreach individual S do
if S is susceptible then
foreach contact group G do
if I and S are in the same group G then
(1) calculate the probability pIS, that S is infected by I
(2) use a random number generator to decide whether S is
infected by I with a probability of pIS
if S is infected then
update the status of S
end
end
end
end
end
end
end
end
doi:10.1371/journal.pone.0013292.t001
Table 2. Algorithm 2: Our improved algorithm.
foreach time period T do
foreach infected individual I do
update the status of I according to T
if I is infectious then
foreach contact group G that I is in do
(1) calculate the infection probabilities pISbetween I and all
susceptible individual S in G
(2) use the Sieve algorithm below to decide all individuals in G to be
infected by I
(3) update the status of newly infected individuals
end
end
end
end
doi:10.1371/journal.pone.0013292.t002
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of exactly same type and I of the exactly the same type) then many
infection events can be generated with relatively few binomial
deviates. However, if the pairs are largely different, then many
binomial deviates need to be drawn to generate a similar number of
infections. The introduction of spatial dimensions into individual-
based formulations greatly increases the heterogeneity of the model
because every small group of individuals with a unique location
forms, effectively, their own risk group.
A high-level description of a naive algorithm for the basic model
is presented in Algorithm 1 (Table 1). The basic idea is to
substantiate viral transmission to every susceptible individual in
every contact group of every infectious individual during every 12-
hours period of the simulation.
The Sieve algorithm we have developed greatly improves the
efficiency with which infection events can be generated across
large numbers of similar risk pairs. Here, we briefly describe the
key features of the algorithm as it relates to the efficient simulation
of spatial epidemics. The methods are described in more details
elsewhere [14]. In essence, the approach is to use lazy evaluation
for large groups of pairs with similar probabilities of an infection
event. For example, one infectious individual a in community A
has a certain maximum probability of infecting members of
community B, based on the flow of workers between those two
communities. The precise probability of infection for each
member of community B will depend on their age and other
risk variables. However, the maximum probability for any
individual in group B, pmax, may be very small if the worker
flow between A and B is small. Working with the Sieve algorithm,
our first step is to generate a random variable for the provisional
number of infection events that occur by assuming that all pairs
have the same probability of an infection occurring. This however
generates too many infections, and the second step is to select
Table 3. Algorithm 3: Sieve algorithm.
(1) let pmax~maxfpISg for all susceptible individual S in G
(2) let N be the number of susceptible individuals in G
(3) decide a tight bound K that is the upper bound of possible infected persons according to a binomial distribution with an inclusion probability pmaxand N trials
(4) randomly pick K candidates from the group of susceptible individuals in G
foreach picked candidate b do
use a random number generator to decide whether b is infected by I with a probability of pIb=pmax
end
doi:10.1371/journal.pone.0013292.t003
Figure 1. The computation of the probability that individual j will be infected by individual i according to the natural history
model.
doi:10.1371/journal.pone.0013292.g001
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specific pairs at random and either accept or reject provisional
infections using the precise probability of infection between
individual a and each individual b (in the provisional set of
infections in community B). We define the precise probability to
be pb. If we accept each provisional infection event with
probability pb=pmax, it is clear that the overall probability of
individual b being infected is equal to pb. Therefore, our method
reiterates the same stochastic process as if we evaluated each
individual pbseparately, and is not an approximation.
A high-level description of our improved algorithm is available
in Tables 2 and 3.
We are able to prove that the statistical behaviors of the Sieve
algorithm are the same as the naive algorithm where each
candidate is decided one by one, sequentially. The proof of this
equivalence is given in [14]. Note that our Sieve algorithm decides
a set of candidates in a batch. One of the reasons that our
algorithm can run faster is because in practice, pmaxis very small.
Thus, the size of the candidates K selected in the Sieve algorithm
is much smaller than N, the pool of people to be considered.
By treating the model explicitly as a network, we calculate the
average number of secondary cases a priori, rather than using semi-
empirical methods to calibrate the model. The basic reproductive
number R0 is the expected number of secondary infections
generated by a single typically infectious individual in an otherwise
susceptible population [15]. R0 is a threshold parameter that
determines whether an infectious disease will spread through a
population. Strictly, for models with multiple types of infectious
individuals, R0 should be defined in terms of a next generation
matrix and an eigenvector for the exponential phase of growth. The
eigenvector is important in that it defines what is typical during the
exponential phase. Often, a typical type of infectious individual will
be different from a randomly chosen individual. For network models
of infectious disease, the formal approach presents some problems
because every individual is, essentially, a different type. Therefore,
we follow many previous network models and use the average
number of secondary cases per randomly chosen individual as R0.
Based on the influenza model and parameters, we compute the
probability that infectious individual i will infect susceptible
individual j, namely wij, as follows. First, the infection probability
resulting from i and j’s contact in group k is defined as
pijk~ptrans|ck, where ptransis the disease-dependent transmission
probability and ck is the group-dependent contact probability.
Second, Dijis the set of i and j’s contact groups in the daytime, and
Nij is the set of i and j’s contact groups during the night. The
intersection of Dijand Nijcan be either empty or nonempty. Third,
when the infectious individuals are incubating or asymptomatic, the
infection probability is reduced by a factor of r, where rw1. For
clarity, we define hijk~pijk=r. In our model, the current setting of r
istwo,asin[2].Thus,inconjunctionwith all daytime andnighttime
contacts, the daily infection probability is calculated by
Pij~1{
P
k[Dij
(1{pijk)
"#
P
k[Nij
(1{pijk)
"#
,
Hij~1{
P
k[Dij
(1{hijk)
"#
P
k[Nij
(1{hijk)
"#
,
where Pij is the daily infection probability when individual i is
symptomatic, and Hij is the daily infection probability when
individual i is incubating or asymptomatic. Finally, by adopting the
Table 4. Comparison of R0.
ptrans
Theoretical R0
Sample R0
Simulated R0
0.071.1141.114 (1.133E-03) 1.147 (1.811E-04)
0.081.2691.270 (1.407E-03)1.262 (6.985E-05)
0.09 1.424 1.424 (1.468E-03)1.379 (6.777E-05)
0.101.577 1.576 (1.558E-03)1.500 (8.114E-05)
0.11 1.730 1.730 (1.796E-03)1.622 (7.376E-05)
0.12 1.8821.882 (2.101E-03) 1.745 (7.623E-05)
0.132.033 2.033 (2.154E-03)1.868 (9.316E-05)
0.14 2.1832.184 (2.577E-03) 1.990 (9.551E-05)
0.15 2.3332.333 (2.538E-03) 2.111 (1.011E-04)
0.16 2.4822.481 (2.808E-03)2.231 (1.188E-04)
0.172.6302.631 (2.916E-03)2.349 (1.240E-04)
0.182.7772.777 (2.664E-03) 2.466 (1.202E-04)
List of R0, calculated by three different methods, for the selected range of ptrans.
Theoretical R0is the average number of expected secondary infections per
individual in the entire population. Sample R0is the average of R0derived from
100 samples of &2,000 initial infectious case; the 95% confidence interval (CI) is
listed in parentheses. Simulated R0is the average of R0estimations derived
from 100 baseline simulations; the 95% CI is listed in parentheses.
doi:10.1371/journal.pone.0013292.t004
Figure 2. The precision and efficiency of the Sieve algorithm, as applied to a model of pandemic influenza transmission in Taiwan.
(A) Demonstrates the correct implementation of the Sieve algorithm such that the attack rates from both algorithms stay nearly identical throughout
the selected range of ptrans. (B) Shows the speedup of the Sieve algorithm for the selected range of ptrans. Speedup is defined as the ratio of the
average computation time for the naive algorithm over the Sieve algorithm.
doi:10.1371/journal.pone.0013292.g002
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naturalhistorymodel, wijcanbecalculated astheweighted sumofall
branches in Figure 1. The expected number of people infected by
individual i isP
an equal chance of being the initial infectious case, we calculate the
expected number of secondary infections for everyone in the entire
population; and by definition, the Theoretical R0is the average of all
suchsecondaryinfections.Table4liststhevalueofTheoreticalR0for
a selected range of ptrans, along with two R0estimations derived from
alternative methods. The first method samples, stratified by age
group, &2,000 people as the index cases and calculates R0for the
sample group. We then define the Sample R0as the average R0from
100 such sample groups. We find that even with a small sample size,
the Sample R0approximates the Theoretical R0closely if we take
sufficient samples. In addition, since the model population remains
unchanged throughout the simulations, we can estimate R0based on
the prevalence of infections at the point of endemic equilibrium [16].
The second method is to average the estimated R0from 100 baseline
simulations for each ptrans, we callit the Simulated R0. The estimated
R0for each simulation resultis calculated using the following formula
jwij, when i is the single infectious case in the
otherwise susceptible population. Assuming that each individual has
R~A
N,
R0&{ln(1{R)
R
,
where N is the number of people in the population, A is the number
of people who experience the event (become infected), and R is the
proportion of the population who become infected, also known as the
infection attack rate.
Results
The Sieve algorithm shows significant improvement over the
naive algorithm when applied to a real-world application. For a
simulation involving population of 23 million people (approxi-
mately the size of Taiwan’s population), we calibrated the strength
of transmission to have an infection attack rate of 60% (a severe
pandemic) and let the infectious period of an infector be, on
average, three days. Even with a coarse half-day time step, the
naive algorithm would still need to evaluate an order of 1,015
interactions (providing every infectious individual has a non-zero
probability of infecting any susceptible host). By using the re-
sampling approach of the Sieve algorithm, the execution time is
drastically reduced (Figure 2B) without any notable loss of
precision (Figure 2A).
These performance data were derived from groups of 32 runs of
the baseline simulation for each ptransand algorithm combination.
On a server with dual Intel Xeon W5580, quad-cores, 3.20 GHz
CPUs and 48GB DDR3 memory, and 16 simulations running
concurrently, the Sieve algorithm finishes ptrans~0:20 baseline
simulation in just under three minutes (Figure 3A); in contrast: the
naive algorithm takes about three hours and twelve minutes.
Figure 3A illustrates the average simulation time of the Sieve
Figure 3. Average computation time for various 180-day baseline simulations. (A) Simulation time on a mock Taiwan population for the
selected range of ptrans. (B) Simulation time on multiples of Taiwan population for ptrans=0.10.
doi:10.1371/journal.pone.0013292.g003
Figure 4. Statistical properties of simulation results. (A) Histogram and the estimated normal distribution for the average day of the 1,000-th
symptomatic case. (B) Quantile-quantile (q-q) plot of the observed distribution with the theoretical normal distribution.
doi:10.1371/journal.pone.0013292.g004
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algorithm, including 20 seconds for generating the mock popula-
tion. The simulation time remains relatively low up to a threshold
value (ptrans&0:06), after which both the simulation time and
cumulative number of infections (attack rate in Figure 2A) increase
substantially. Figure 3B shows the time required for the simulation
of multiples of Taiwan’s population for ptrans~0:10, here we
perform a single simulation for each population size due to
memory limitations.
Stochastic convergence
The stochastic process, by nature, involves non-deterministic
trials evolving through time and abiding by miscellaneous
characteristics with probability distributions. This means that
even if all conditions are known in advance, there will be
numerous possible outcomes, while some are more probable than
others. With all trials guided by the same set of characteristics and
probability distributions, the sequence of essentially random events
is expected to settle into a pattern. Multiple realizations of the
same scenario are necessary to elucidate this underlying pattern. A
fast realization tool for the stochastic process is especially beneficial
in dealing with various aspects of the model itself, such as
sensitivity analysis.
Next, we describe experiments conducted to assess the
variability of the simulation results. First, we randomly picked a
mock population and simulated 2,000 baseline realizations with
constant transmission parameters. For each of the 2,000
realizations, we extracted information on important properties,
such as the day of the 1,000-th (10,000-th, …) symptomatic case
and the final number of infected people. We then treated the
statistics from all 2,000 results as if they were the real sample
space and assumed that the parameters of the real unknown
sample space were comparable. Thus, each production run is
merely a sample derived from the 2,000-run sample space
(2KSS). First, we observe that the histograms of the important
properties are all bell-shaped. We use a maximum likelihood
heuristic to estimate the most likely normal distribution to match
the histogram, as shown in Figure 4A. Next, we then compare the
observed distribution with the theoretical normal distribution in a
quantile-quantile (q-q) plot. The q-q plot is a graphical technique
for determining if two data sets come from populations with a
common distribution. It is a plot of the quantiles of the first data
set against the quantiles of the second data set. By a quantile, we
mean the fraction (or percentage) of points below a given value. A
45-degree reference line is also plotted. If the two sets come from
a population with the same distribution, the points should fall
approximately along this reference line [17]. As illustrated in
Figure 4, the normality of the observed distribution is not only
visually correlated on the left, and also statistically verifiable on
the right.
Table 5. Selected Simulation Properties.
Simulation
Property
k~20
k~30
k~40
Day of
the 104-th case
0.98 (0.13)0.79 (0.08) 0.68 (0.05)
Day of
the 105-th case
1.06 (0.14) 0.84 (0.08)0.72 (0.05)
Day of
the 106-th case
1.06 (0.15)0.85 (0.09)0.73 (0.06)
Day of
the 107-th case
1.42 (0.18)1.14 (0.11)0.98 (0.07)
Number of
infected people
2,890 (420)2,330 (270) 2,000 (180)
The relationship between the number of simulation runs (k) and 95% CI for
several simulation properties. The mean and standard deviation, in parentheses,
of 95% CI per 1,000 groups of k simulation runs are shown.
doi:10.1371/journal.pone.0013292.t005
Figure 5. Simulation peaks distribution (with averages) for Taipei and Changhua scenarios when outbreaks occurs with one index
case in 1,000 simulation runs. The y axis shows the maximum daily new symptomatic cases of each simulated outbreaks (Taipei scenario, 95% CI
192,722–194,729; Changhua scenario, 95% CI 234,307–236,560), the x axis shows the day that peak occurs (Taipei scenario, 95% CI 129–131;
Changhua scenario, 95% CI 130–132).
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We then assess the variability among groups of simulation
results and attempt to establish an acceptable number of
simulation runs that would represent all possible outcomes with
high confidence. We calculate the 95% CI for selected important
properties in groups of k simulation runs, where k ranges from 2 to
100. For each value of k, we conduct 1,000 experiments by
sampling k instances out of 2KSS, and calculate the corresponding
95% CIs for each experiment. We then calculate the mean and
standard deviation of 95% CIs among 1,000 experiments for each
k. In Table 5, we summarize the mean and standard deviation of
95% CIs from experiments of 20, 30 and 40 simulation runs.
Based on these numbers, it is safe to say that a sensible decision is
to repeat each simulation at least 30 times.
Practical use of efficient simulations
To demonstrate the practical use of the model, we simulate a
severe flu pandemic, R0&1:6, in Taiwan. We design two scenarios
to best describe typical epidemic outbreaks: (1) An imported
infectious case by seeding one index case in Taipei, which is at the
northern end of the island and is densely populated with over 2.6
million people in the city and over 5 million in the greater
metropolitan area; as the political, economic, and cultural center
of the nation, Taipei is the most likely first stop for all international
travelers. (2) An endemic outbreak by seeding one index case in a
mid-latitude, less connected, remote farming town in Changhua
county, which has 1.3 million residents and the highest
concentration of chicken livestock in the country. We run each
scenario 1,000 times. For the Taipei scenario, the single index case
causes an outbreak in 513 out of 1,000 runs; while for the
Changhua scenario, 543 runs result in outbreaks. We plot the
averages of these outbreaks and observe that the epidemic
progresses more rapidly from Taipei to other areas, resulting in
a more synchronized epidemic; that is, the number of incidences is
similar in quite distant locations during the middle part of the
epidemic. In contrast, Changhua is less well connected, and the
epidemic takes longer to spread to other parts of the population.
Hence, the number of incidences in the mid-latitude area close to
the seed is higher than in other areas. This results in a slightly
slower epidemic (in terms of growth), but the peak is more
pronounced for the Changhua scenario. These simulation runs
illustrate the general principal that when epidemics fail to
synchronize spatially, the overall incidence is less peaked.
However, the results presented here do not describe local
incidences of infection, which would be more ‘‘peaky’’. The
animation in Movie S1 (which is published as supporting
information) demonstrates the spatial epidemiology of infectious
disease for both scenarios. The simulation cases presented in this
movie were selected to approximate, as closely as possible, the
Figure 6. County level spatio-temporal spreading patterns for Taipei and Changhua scenarios.
doi:10.1371/journal.pone.0013292.g006
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calculated average of all 500+ simulation runs for each scenario.
We also prepared another movie (Movie S2, which is published as
supporting information) by selecting simulation runs that were
farther away from the average behavior of each scenario to show
the unpredictable nature of the stochastic process.
In Movie S3 (which is published as supporting information), we
use a different representation to demonstrate the county level
spread of infectious disease, where each rectangular bar represents
a county or major city in Taiwan; hence, their geographical
relationships are also presented in these diagrams. The height of
each bar indicates the number of new symptomatic cases daily;
hence, we can easily observe the epidemic’s critical level for each
location.
If we use the peak of new cases and its date as an indicator of
each outbreak and plot the distribution of all simulation runs for
both scenarios, we find that although they are reasonably scattered
in a disk area, the two disks have a non-trivial overlay (Figure 5).
Such observations may not be possible if only a few simulations are
conducted. Figure 6 shows the spatio-temporal spreading patterns
for the Taipei and Changhua scenarios. In each figure, the whole
island of Taiwan is plotted as a rectangle. The day that an area
reports the first symptomatic case is plotted on the left; and the day
that the peak occurs in an area is plotted on the right. We observe
that Changhua has a less uniform spatio-temporal spreading
pattern. The Taipei scenario exhibits more coordinated behavior.
Discussion
We have described the application of a general re-sampling
algorithm to a widely used spatial model of infectious disease
transmission [8]. The resulting epidemic simulation tool achieves
substantial speedups compared with our own implementation of a
naive algorithm for the same model. Although derived indepen-
dently, the resulting simulation algorithm is similar to those used to
investigate the properties of the re-emergence of smallpox in the
UK [18], and the pandemic influenza in Thailand [19], the
United Kingdom and the United States [4].
We believe that further research on the underlying algorithms
for the model presented here and similar models is warranted. For
example, there are many ecological questions about the spatial
properties of the current H1N1pdm — not least the need to
explain the high degree of spatio-temporal variability observed on
a continental scale. More generally, on any scale, improved
computational efficiency of epidemic models, similar to that
demonstrated here, will substantially increase their utility as tools
for theoretical investigation.
Supporting Information
Appendix S1
Found at: doi:10.1371/journal.pone.0013292.s001 (0.16 MB
PDF)
Supporting text with implementation details.
Movie S1
patterns of an influenza epidemic in Taiwan with index case
seeding in two distinct locales. The daily prevalence of
symptomatic cases in each community is presented as an epidemic
alert level on a logarithmic color scale, with red indicating the
most critical situation when 3% or more of the population become
symptomatic.
Found at: doi:10.1371/journal.pone.0013292.s002 (11.02 MB
AVI)
Visualization of typical spatio-temporal spreading
Movie S2
epidemic in Taiwan with index case seeding in two distinct locales.
Found at: doi:10.1371/journal.pone.0013292.s003 (11.04 MB
AVI)
Spatio-temporal spreading patterns of a rare influenza
Movie S3
simulations in Taiwan.
Found at: doi:10.1371/journal.pone.0013292.s004 (7.28 MB AVI)
County level visualization of influenza epidemic
Author Contributions
Conceived and designed the experiments: MTT TCMC JHC HSK CJL
SR BJS CHS DWW TSH. Performed the experiments: MTT TCMC JHC
CWH HSK CJL SR BJS CHS DWW TSH. Analyzed the data: MTT
TCMC JHC CWH HSK CJL SR BJS CHS DWW TSH. Contributed
reagents/materials/analysis tools: MTT TCMC JHC CWH HSK CJL SR
BJS CHS DWW TSH. Wrote the paper: MTT TCMC SR BJS CHS
DWW TSH.
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