Hemagglutinin sequence clusters and the antigenic
evolution of influenza A virus
Joshua B. Plotkin*†‡, Jonathan Dushoff*, and Simon A. Levin**Department of Ecology and Evolutionary Biology, Princeton University, Princeton, NJ 08540; and †Institute for Advanced Study, Princeton, NJ 08540
Contributed by Simon A. Levin, February 22, 2002

quasi species 兩 vaccination 兩 clustering 兩 H3N2

I

nfluenza A virus is a negative-stranded RNA virus that causes
significant human mortality and morbidity worldwide (1). The
virus is divided into subtypes based on major differences in thesurface proteins hemagglutinin (HA) and neuraminidase, which
are the most important targets for the human immune system.
Within each subtype of the influenza virus, gradual mutations to
the HA gene continually produce immunologically distinct
strains (referred to as drift variants); an influenza infection
brings lasting immunity to the infecting strain, but most people
are susceptible to re-infection by a new drift variant within a few
years. Over the past century, the annual epidemics associatedwith antigenic drift have had an even greater cumulative impact
than the three pandemics associated with major reassortment
events, known as antigenic shifts (2–4). Antigenic drift requiresthat vaccines be updated annually to correspond with thedominant epidemic strains of HA. Thus the prediction of HA’s
evolutionary course is of great practical importance to publichealth.
Recent developments in molecular biology and computation
have made possible remarkable phylogenetic reconstructions of
HA evolution (3, 5). Such studies reveal that modifications to
HA1, the immunogenic part of HA, accrue at a dramatic rate.
Those sites of HA1 involved in antigen determination exhibit
significantly more non-synonymous nucleotide substitutions
than synonymous substitutions (6, 7), whereas the remaining
sites show the more common pattern of primarily synonymous
variation. These observations demonstrate that HA is undergoing positive Darwinian selection for new antigenic variants (8).
Bush et al. (9) have identified 18 HA1 codon sites with significantly higher non-synonymous to synonymous ratios. Viewed
retrospectively, these 18 sites usually predict where the trunk of
the phylogeny will emerge: among the circulating sequences ina given influenza season, the one with the largest number ofamino acid replacements among these 18 sites is usually mostclosely related to future evolutionary lineages.
In this paper, we present an approach to analyzing and, tosome extent, predicting the course of influenza sequence evolution. Our approach is related and complementary to phylogenetic techniques, but we are less concerned with reconstructing the evolutionary relationships between HA1 sequences.
Instead, we identify natural scales at which HA1 amino acidwww.pnas.org兾cgi兾doi兾10.1073兾pnas.082110799

sequences aggregate into clusters, or ‘‘swarms,’’ and we study
their spatio-temporal patterns. We will focus on the relationships
between observed cluster structure, worldwide vaccination history, and the primary antibody-combining regions of the HA
protein.
Data and Methods
Data. This study uses 560 sequences, each 987 nucleotides long,of the H3 type HA1 gene isolated between 1968 and 2000 fromlocations around the globe. The sequences were obtained froma public database [ref. 10; Los Alamos National Laboratory,
Influenza Sequence Database (http:兾兾www.flu.lanl.gov兾)]. We
use the terms genotype and strain interchangeably to refer to anucleotide sequence of HA1. Viruses were isolated by either egg
or kidney cell cultures (3). All sequences were easily aligned
without gaps.
Each of the 560 sequences is associated with a calendar yearof isolation, in some cases inferred from the strain name. For 439
of the sequences, however, more detailed information is available, allowing them to be partitioned into influenza seasons,defined as 1 October through 30 September. For example, the‘‘94兾5 season’’ refers to those sequences collected between 1
October 1994 and 30 September 1995.
Most of the sequences were generated as part of the long-term
World Health Organization (WHO) influenza surveillance program. As we discuss below, only a small proportion of viruses
isolated by the WHO are also sequenced. Novel antigenic
isolates are preferentially sequenced by the WHO (11); as aresult, the database provides a biased approximation of worldwide strain frequencies.
Methods. To identify clusters of viral sequences, we must firstassign a distance between sequences. We define the distance
between two HA1 sequences as the sum of the pairwise distances
between their 329 composite amino acids. Several amino acidmetrics are possible. The simplest metric, called the Hamming
metric, equals zero or one depending on whether two aminoacids are identical. Alternative metrics weight the differences
between amino acids according to their stereochemical properties [e.g., the Miyata metric (12)], or their substitution frequencies in protein families (13). Here, we present results based on
the Hamming metric. Results based on the Miyata metric aresimilar.
Ideally, the distance between a pair of sequences shouldreflect the immunogenic properties of the corresponding viralproteins. Some steps have been taken in this direction. For
example, Lapedes and Farber (14) derived a distance measure
for HA from antibody binding assays, whereas Wilson and Cox(15) developed a metric based on changes in the solventaccessible surface of the folded HA molecule. Nonetheless,Abbreviations: HA, hemagglutinin; WHO, World Health Organization; HI, hemagglutination inhibition.
‡To whom reprint requests should be addressed. E-mail: plotkin@ias.edu.

The publication costs of this article were defrayed in part by page charge payment. This
article must therefore be hereby marked “advertisement” in accordance with 18 U.S.C.
§1734 solely to indicate this fact.

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Continual mutations to the hemagglutinin (HA) gene of influenza
A virus generate novel antigenic strains that cause annual epidemics. Using a database of 560 viral RNA sequences, we study the
structure and tempo of HA evolution over the past two decades.
We detect a critical length scale, in amino acid space, at which HA
sequences aggregate into clusters, or swarms. We investigate thespatio-temporal distribution of viral swarms and compare it to thetime series of the influenza vaccines recommended by the World
Health Organization. We introduce a method for predicting future
dominant HA amino acid sequences and discuss its potential
relevance to vaccine choice. We also investigate the relationshipbetween cluster structure and the primary antibody-combining
regions of the HA protein.

probably attributable to the WHO bias toward sequencing
antigenically novel strains. By focusing on the larger clusters, wefilter out these outliers.
Here we use ‘‘swarm’’ to denote a cluster of related viralgenotypes that to some degree operate as a selective unit
(19–21). Viral swarms, which are shaped by mutation andselection, are often called quasispecies (22), although this usage
differs from Eigen’s original usage to describe a collection ofmacromolecules generated by mutations from a fixed wild-type
configuration (23).

Fig. 1. The cluster-size curve for 560 sequences of HA1. This curve shows therelationship between the threshold distance d (at which to connect two
sequences into the same cluster) and the mean cluster size C(d), defined as thenormalized first moment of the resulting distribution of cluster sizes. Equivalently, C(d) is the probability that two randomly chosen sequences lie in thesame cluster. Plateaus in the cluster size curve correspond to stable lengthscales at which the sequences form nonrandom clusters. Random data would
not exhibit any plateaus except for C ⫽ 0 and C ⫽ 1 (18). The smooth cluster
size curve results from averaging over 100 probabilistic Gaussian draws foreach mean distance parameter d, with a 5% coefficient of variation (18). The
HA1 data exhibit two significant plateaus corresponding to clusterings at d ⫽2–3 and d ⫽ 4 –5. The long tail for d ⱖ 6 corresponds to the gradual accumulation of outlier sequences. When d ⫽ 2, there are 174 resulting clusters with
C (2) ⫽ 0.0614; at this scale, the expected size of the cluster containing arandomly chosen sequence is 560 ⫻ 0.0614 ⫽ 34.4 sequences. (The clustering
for d ⫽ 3 is extremely similar to d ⫽ 2, as the first plateau indicates.)

more detailed antibody-binding data and a better understandingof HA folding are required before these metrics can be comprehensively applied in our context.
Using their pairwise distances we identify a natural partitioning of the sequences into disjoint groups, or clusters, via asingle-linkage clustering algorithm (16). Traditional singlelinkage clustering produces a hierarchy of partitions starting witheach sequence in its own unique ‘‘cluster’’ and successively
merging clusters whose nearest neighbors are a minimal distanceapart, eventually grouping all sequences into one large cluster.
Cluster hierarchies have been used to generate phylogenetic
trees (17). In this paper, however, we do not consider thehierarchy of clusters, but rather inspect the clusters themselves
at a suitably chosen linkage distance. In this sense our analysis
is a logical complement to the phylogenetic approach.
Results
Identifying Natural Scales of Aggregation. To choose a suitable

threshold distance d at which to stop the single-linkage algorithm, we inspect the ‘‘cluster-size curve’’ (18) for all 560
sequences in the data set (Fig. 1). When d ⫽ 0, there are 468
separate clusters, indicating the number of unique amino acidsequences in the data set. When d ⫽ 18, all 560 sequences fall
into a single cluster. Furthermore, Fig. 1 shows that there is anatural, nonrandom partitioning of the sequences into disjoint
clusters at a threshold distance of d ⫽ 2 amino acid changes. In
other words, d ⫽ 2 provides the finest natural scale at which thesequences aggregate. (There is another nonrandom partition
corresponding to d ⫽ 4, at which most sequences fall into threelarge clusters.) The strain name and corresponding cluster foreach of the 560 sequences are provided in supporting information, which is published on the PNAS web site (www.pnas.org).
The existence of a natural scale of non-random aggregationsuggests that HA1 sequences form viral swarms. There are 174
clusters corresponding to d ⫽ 2, but 134 of these clusters consistof a single outlier sequence. The abundance of outliers is6264 兩 www.pnas.org兾cgi兾doi兾10.1073兾pnas.082110799

Spatio-Temporal Evolution of Sequence Clusters. The above methodfor decomposing HA1 sequences into natural clusters provides
a revealing perspective on the evolution of influenza virus. In
Fig. 2 we plot the number of sequences in each cluster as afunction of isolation year. Even though we did not use any
information about isolation year when clustering the data, Fig.
2 shows that the resulting viral clusters are localized in time.
There is no cluster that has members spanning more than sevencollection years. Instead, dominant clusters of viral sequences
tend to replace one another every 2–5 years, in agreement withthe timescale of dominant antigenic replacements (24). Someclusters are significantly more long-lived than others, suggestingthat HA swarms do not evolve at a constant rate.
Note that the swarm evolution seen in Fig. 2 is not periodic
within the time-span of two decades. Once HA evolves awayfrom a given region of sequence space, it does not later revisit
that region. This result, seen here in terms of cluster structure,
is consistent with the one-trunk phylogenetic reconstructions of
HA1 (3). Such behavior is in sharp contrast to viruses with
distinct co-circulating serotypes [e.g., avian influenza (2)], orwith distinct serotypes that may cycle in time.
Host-mediated mutations acquired during viral culturing
could potentially affect the structure of sequence clusters.
However we believe this to be a secondary effect and find no
signature of it, e.g., no cluster that spans all collection years, inthe time series in Fig. 2.
It is generally believed that novel influenza A subtypes (e.g.,
the subtype H3N2) usually originate in Asia, especially in China.
Common wisdom also holds that novel strains within eachsubtype (i.e., drift variants) also originate in China (3). We canuse the three largest sequence clusters in Fig. 3 (those spanningseasons 87兾8–93兾4, 91兾2–93兾4, and 93兾4–97兾8) to test thishypothesis. We find that among the sequences within each ofthese large clusters, those sequences isolated in China or Hong
Kong are found preferentially in the first half of the cluster’s
lifetime (␹2 ⫽ 13.0, 9.0, 8.26; P ⬍ 0.005 for all). These results
support the hypothesis that dominant viral swarms tend to
originate in Asia and thereafter spread across the globe.
Cluster Structure and Vaccination Choice. Each February, the World
Health Organization recommends two strains of influenza A
virus (one of the H1N1 subtype, and one H3N2) and one strain
of influenza B virus to be used as the basis for the trivalentvaccine in the northern hemisphere influenza season (25).
Choices are based on the antigenic properties of circulatingstrains and the immunogenic properties of vaccine candidates.
The lead time is necessary for vaccine preparation. Case studies
indicate that in general vaccination is extremely effective (up to68%) for prevention (26) and for reduction of morbidity, evenamong unvaccinated individuals in close contact with vaccinees(27). Nevertheless, the antigenic plasticity of HA complicatesthe precise prediction of future dominant strains and vaccine
choice (24).
Ideally, the strain used as the basis for a vaccine each season
should correspond to the dominant antigenic strain that season.
Unfortunately, the standard hemagglutination inhibition (HI)assay (28) offers relatively poor resolution for comparing antiPlotkin et al.

genic properties of circulating strains; HI often characterizes up
to 95% of strains as identical (29). More refined antigenic assays
are possible, but more time consuming. Fortunately, the aminoacid sequence can be more sensitive (30) and is generally
correlated (29) with the results of refined HI assays.
The decomposition of HA1 sequences into disjoint clusters
affords a retrospective and predictive tool for analyzing influenza vaccine choices. Fig. 3 shows the sizes of the primary HA1clusters, each in a different color, as a function of influenza
season. Fig. 3 also shows the ‘‘color’’ of the WHO-recommended
vaccine in each season, that is, the color of the cluster corresponding to the strain on which each vaccine was based. Insofar
as the color of a vaccine indicates its antigenic properties—
which, we have argued, is a good first approximation—Fig. 3
provides a wealth of information about worldwide vaccination
strategy. Fig. 4 lists some representative strain names that are
members of the clusters in Fig. 3.
According to Fig. 3, a dominant cluster of influenza virus
can go extinct even while the recommended vaccine is chosenfrom outside of the cluster (e.g., the blue cluster, 94兾5). In otherwords, large viral swarms seem to drive themselves to extinction,
presumably by stimulating immunity in the human population,even without the pressure of vaccination. This behavior corroborates the common wisdom that, although WHO’s vaccination
recommendations have greatly reduced human mortality, vaccination has not significantly altered the evolutionary course of
HA (3). This conclusion is consistent with the constancy of HA’srate of evolution before and after the advent of widespread
vaccination (2, 31).
We emphasize that the HA1 sequences used in Fig. 3 were notavailable at the time when the WHO recommended the indicatedvaccines. Only recently has it become possible to sequence anisolate’s HA1 gene within the same season of the virus’s
isolation. This technological advance in sequencing speed may
allow HA1 sequences to play an expanded role, complementing
HI assays, in informing vaccine selection.
Plotkin et al.

With these provisos in mind, we now consider a sequencebased algorithm for choosing vaccine strains. Given the limited
amount of available data, we examine the simplest of suchalgorithms. First, we cluster all of the sequences collected during
or before the current influenza season (using d ⫽ 2 amino acidchanges). Then, we choose the HA sequence on which to basenext season’s vaccine as the most recent sequence in the current
season’s most dominant cluster. Fig. 3 shows the vaccines thatwould have been specified by this algorithm. The algorithm
would have agreed with the actual choices made by the WHO innine of the last 16 flu seasons, and it would have specified
different choices in seven seasons.
Our interpretation of Fig. 3 depends on the assumption thatthe sequence database reflects worldwide strain frequencies
each season. But our database is limited and potentially biased;
only a small proportion of viruses characterized by HI assays are
actually sequenced. By choosing the appropriate scale of aggregation (Fig. 1), we have tried to mitigate the influence of outlier
sequences. Nevertheless, the bias toward novelty in the sequence
database may cause sequence clusters to peak before the corresponding actual antigenic types do. (Conversely, the coarseness of the HI assay and the need to group types in real time may
cause the Weekly Epidemiological Record identification to lag
slightly behind actual antigenic changes.)
We emphasize that Fig. 3 provides only an approximate
indication of vaccine suitability. Vaccines should ideally matchthe dominant antigenic properties of circulating influenza sequences each season. The extent of the vaccine’s correlation withthe dominant amino acid sequence of influenza strains, indicatedby the colors in Fig. 3, is related but not identical to antigenic
correlation. The Hamming metric on sequences has been used as
an effective model of antigenic distances for B-cells in general(32) and for influenza in particular (33). Nevertheless, comparison of sequence data to direct immunological assays should be
used to quantify the correlation between amino acid composition and antigenic properties (14). The algorithm above cannot
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Fig. 2. The number of HA1 sequences within each cluster plotted as a function of calendar year of isolation. The clustering shown here corresponds to d ⫽ 2amino acids (see Fig. 1). Each cluster is indicated by a different color, with the eight largest clusters shown in bold. The dashed line indicates the total number
of isolates in the data set each year. The dominant sequence clusters tend to replace each other every 2–5 years. The dominant cluster in each year accounts formore than 25% of the sequences isolated that year. (The number of sequences each year does not reflect the severity of infections, but rather the temporal biases
in the sequence data set.)Fig. 3. The number of HA1 sequences within each cluster plotted as a function of influenza season. The graph shows the eight largest clusters as well as anyother clusters that contain sequences used in a WHO vaccine. The tiles denoted ‘‘WHO vaccine’’ indicate the ‘color’ of the WHO-recommended vaccine in eachseason, e.g., the color of the cluster corresponding to the strain on which each vaccine was based. The tiles denoted ‘‘Algorithmic vaccine’’ indicate the colorof vaccine prescribed each season by the algorithm proposed in the main text (the dominant cluster from the previous season). The tiles denoted ‘‘WER strain’’
indicate the color of the dominant antigenic type, based on HI assays, as reported by the WHO in its Weekly Epidemiological Record (40 –56). (Note that one of
the three strains reported in WER for 1999 –2000 is missing from the Los Alamos sequence database.) Both vaccines tend to match the WER strains well; in someseasons the WHO vaccine matches better, and in some seasons the algorithmic vaccine matches better.

be considered for use as a vaccine protocol until direct antigenic
data are incorporated. Our methodology for evaluating vaccine
candidates would be relatively easy to generalize using a combination HI- and sequence-derived metric.
Cluster Structure and Antibody-Combining Sites. The underlying
mechanism of influenza A’s antigenic plasticity, that is, how thevirus continually evades immunity by producing variant strains,remains an outstanding evolutionary problem with obvious
practical implications. To address this question, we inspect thestructure of the identified sequence clusters with regards to the
known epitopes (antibody-combining regions) of HA.

Fig. 4. Strain names of representative members from each of the clusters
seen in Fig. 3. Note that strains considered as antigenically distinct by the WHO(using HI assay) can fall in the same cluster.
6266 兩 www.pnas.org兾cgi兾doi兾10.1073兾pnas.082110799

Of the 329 codon sites in the HA1 gene, 131 sites lie in or near
the five main epitopes of the HA trimer, labeled A through E(15). Epitopic sites have been shown to exhibit greater variability, higher ratios of replacement to silent mutations, and greater
correlation with future phylogenetic trajectory (9)—the hallmarks of divergent selection. Having partitioned the HA sequences into clusters we may now ask specifically: (i) whether
different clusters are unusually homogenous with respect todifferent epitopes, and (ii) whether different epitopes change
each time influenza ‘‘jumps’’ from one cluster to the next.
For each of the epitopes, Fig. 5 shows the within-cluster
variation and the size of the ‘‘jumps’’ between the eight largest
clusters in our data set. We see some intriguing interactions
between epitopes and viral swarms: four of the five epitopes
account for the greatest amount of variation in at least one
cluster. Many of the epitopes also show interesting temporal
patterns. For example, the within-cluster variation in epitope B
declines sharply from the early clusters to the late clusters; thewithin-cluster variation of epitope D is prominent in the middle
clusters and then later declines.
Even more striking is the pattern of epitope changes when
jumping from one cluster to the next. Each jump is dominatedby a different epitope than the previous jump. For example, thedifference between the 1985 and 1987 clusters is greatest along
epitope B, whereas the difference from 1987 to 1990 is greatest
along epitope A. This suggests that influenza must jump in adifferent direction, i.e., along a different epitopic axis, every 2–5
years to escape from the immune system. The pattern ofinter-cluster jumps quantifies and verifies the criteria of Wilson
and Cox (15) that new drift variants require more than fouramino acid changes across two or more antigenic sites.
Plotkin et al.

indicate that this epitope is a prime immune target for strains inthat cluster, and is thus unable to vary from a fixed adaptive
sequence; or does it indicate that epitope A is under less immunepressure than other epitopes, and thus does not need to vary?

Moreover, the ‘‘non-epitopic’’ sites of HA1 show increasing
intra-cluster variability (Fig. 5a) and dramatically larger jumps
between clusters (Fig. 5b) in recent years. These trends suggestthat the locations of epitopic sites on the HA trimer may have
evolved since their original characterization (9, 15).
Furthermore, we find a strong correlation between withincluster variation and jump distance; in five cases the epitope oflargest within-cluster variation jumps the furthest (and in theother two cases the epitope of second largest variation jumps thefurthest). In other words, large evolutionary changes in an
epitope are possible only when a cluster features sufficientvariation on which selection may act.
We suspect that substantially more data, including antibodybinding or protein-folding information, will be needed to teaseapart all of the patterns in Fig. 5. We also need a clearer
understanding of how immune reactions shape swarm structureand epitopic sequence variation. In particular, what does convergence or divergence of a particular epitope in a cluster imply
about immuno-dominance? For example, does the small amountof within-cluster variation seen in epitope A in the 1985 cluster
Plotkin et al.

We are grateful for valuable discussions with Viggo Andreasen, Freddy
Christiansen, David Earn, Juan Lin, Ellis McKenzie, Alan Perelson, Tom
Reichert, Derek Smith, Lone Simonsen, and Martin Weigert. We
especially thank Walter Fitch, Peter Palese, Robin Bush, and Nancy Cox
for critical readings of the manuscript, but we do not imply that theynecessarily endorse our interpretation of the results. We thank Los
Alamos National Laboratories and all who have contributed to the
Influenza Sequence Database, including those who have contributed
unpublished sequences: N. Komadina, A. Hapson, N. Cox, C. Bender, O.
Hungnes, and M. Brytting. This work was supported by a grant from the
National Institutes of Health, no. 1-R01-GM60729-01 to S.A.L. We also
thank the Alfred P. Sloan Foundation, The Ambrose Monell Foundation, The Florence Gould Foundation, and the J. Seward Johnson Trust.
J.B.P. acknowledges support from the National Science Foundation, theTeresa and H. John Heinz III Foundation, and the Burroughs Wellcome
Fund.
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Fig. 5. Within-cluster variation (a) and between-cluster distances (b), by
epitope, for the eight largest clusters in our data set. Within-cluster variation
is calculated as the mean pairwise Hamming distance, restricted to sites in agiven epitope, among sequences in a cluster. The abscissa shows the mean of
the calendar years for each cluster’s sequences. Note that the amount ofvariation among the 198 nonepitopic sites is of roughly the same magnitude
as variation in each of the epitopes. In b, the distance between successive
clusters is calculated as the distance between the cluster centroids in Hamming
space (using the Manhattan metric). The abscissa shows the temporal midpoint of the two clusters being compared. Note that the epitope with thelargest inter-cluster change is never repeated in two successive ‘‘jumps.’’

Discussion
The study of viral sequence evolution, and influenza virus
evolution in particular, has traditionally relied on phylogenetic
techniques. Here we have presented formal cluster-based techniques that can complement traditional methodologies by detecting natural scales of sequence aggregation. The resulting
partition of viral sequences into clusters yields new perspectives
on the structure of their genomic evolution.
Although phylogenetic algorithms can estimate the evolutionary relationships among sequences, influenza phylogenies, eventhose using extensive databases, are often plagued by poorbootstrap values and instabilities of tree topology, which havebeen systematically studied by only a few authors (3, 11, 34). The
problem of resolving an evolutionary time series to the level ofindividual sequences is thus difficult, but perhaps unnecessary.
Considerable evidence (20–22), suggests that rapidly evolving
RNA viruses effectively experience selection as swarms, rather
than as individuals. The techniques developed here allow us to
inspect viral evolution at the scale at which selection acts.
In this paper we have demonstrated that HA1 sequences
cluster in a non-random fashion, that clusters replace one
another every 2–5 years, that the persistence of clusters can beused to predict the next season’s influenza sequences, and thatclusters demonstrate interesting interactions with the five main
antibody-combining regions of hemagglutinin. All of these results rely intrinsically upon the quasispecies [see Domingo et al.
(22)] nature of the influenza A virus.
Recently, there has been increasing theoretical interest in theecology and evolution of influenza and other diseases with antigenically distinct, interacting strains (35–39). Our results provide
empirical grounding for that work, and identify the viral swarm asa ‘‘fundamental particle’’ for modeling. The dynamics of influenza
A viral evolution thus will be driven by selection pressures uponswarms, as mediated by patterns of cross-reactivity among them.
Although the approaches developed here offer an importantcomplement to phylogenetic techniques, our results perhaps raise
as many questions as they answer. Our method of predicting future
dominant influenza sequences still requires antibody-binding assays
before use as a vaccine protocol. Similarly, the observed clusterepitope interactions may indicate important directions for influenza modeling, but their immunological interpretation is not yet
clear. Despite these remaining questions, our results emphasize theimportance of an integrated approach to the ecology and evolutionof influenza virus. A focus on the cluster as the core element ofinfluenza A dynamics not only provides a framework for understanding the evolutionary history of the virus; it also helps to informthe prediction of future outbreaks.

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