Reconstructing influenza incidence by deconvolution ofdaily mortality time series
Edward Goldsteina,1 , Jonathan Dushoffb,c , Junling Mad , Joshua B. Plotkine , David J. D. Earnc,d , and Marc LipsitchaaCenter for Communicable Disease Dynamics, Department of Epidemiology, Harvard School of Public Health, Boston, MA 02115; b Department of Biology,
McMaster University, Hamilton, ON, Canada L8S 4K1; c M. G. Degroote Institute for Infectious Disease Research, McMaster University, Hamilton, ON, Canada L8N
3Z5; d Department of Mathematics and Statistics, McMaster University, Hamilton, ON, Canada L8S 4K1; and e Department of Biology, University of Pennsylvania,
Philadelphia, PA 19104

We propose a mathematically straightforward method to infer
the incidence curve of an epidemic from a recorded daily death
curve and time-to-death distribution; the method is based on the
Richardson–Lucy deconvolution scheme from optics. We apply themethod to reconstruct the incidence curves for the 1918 influenza
epidemic in Philadelphia and New York State. The incidence curves
are then used to estimate epidemiological quantities, such as daily
reproductive numbers and infectivity ratios. We found that during
a brief period before the official control measures were implemented in Philadelphia, the drop in the daily number of new infections due to an average infector was much larger than expectedfrom the depletion of susceptibles during that period; this finding was subjected to extensive sensitivity analysis. Combining thiswith recorded evidence about public behavior, we conclude thatpublic awareness and change in behavior is likely to have had amajor role in the slowdown of the epidemic even in a city whose
response to the 1918 influenza epidemic is considered to have beenamong the worst in the U.S.
1918 pandemic | incidence curve | death curve | Richardson–Lucy deconvolution | infectivity ratiosIn characterizing a newly emerging or historical epidemic, one
often has access to an epidemic curve that provides the number of persons who became ill on a certain day (the symptom
curve), or the number of cases that were reported each day(the report curve), or the number of dying that were reported
each day (the death curve). Of more direct interest, for the purposes of visualizing the spread of the epidemic and calculating
relevant quantities such as the daily reproductive number, isthe (usually unobserved) epidemic curve of the number of persons becoming infected on each day (the incidence curve). The
other three curves—the symptom, report, and death curves—
provide information about the incidence curve but are imperfectrepresentations of it, first because not all cases may appear inthese other curves—asymptomatics, unreported cases, or nonfatal
cases respectively will be missed—and perhaps more importantly
because the delay between infection and subsequent events—
symptom onset, report, or death—is a random variable that adds
horizontal variation or “smear” to the epidemic curve. For some
infections (e.g., HIV), diagnostic symptoms (i.e., AIDS-defining
illness) may occur years after infection, so the symptom curve isa poor reflection of the evolution of the epidemic; there is muchliterature on the problem of deconvolution to estimate the HIV
incidence curve from the symptom (AIDS-incidence) curve (1).
For acute infections, such as SARS and influenza, the incubation
period is relatively short compared with the growth rate of theepidemic; hence the symptom curve and the incidence curve aresimilar. The time to death for such infections, however, can be several weeks, equivalent to three or more disease generations, andis highly variable (2–4), making the death curve a poor surrogate
for the incidence curve.
In this paper, we propose a mathematically straightforward
method to infer the incidence curve from a death curve. The incidence curve is then used to estimate the basic epidemiologicalwww.pnas.org / cgi / doi / 10.1073 / pnas.0902958106

quantities of interest, such as the daily reproductive numbers andinfectivity ratios; the latter can be utilized to assess major changes
in the epidemic’s dynamics.
First, we describe a technique for deconvolution to estimate
the incidence curve from the death curve and the time-to-death
distribution. The technique, called the Richardson-Lucy (RL)
deconvolution, was originally developed for use in optics (5, 6),
and is adapted in a simple way to the slightly different setting of
the death-to-incidence deconvolution problem. The technique isillustrated for the 1918 influenza epidemic in Philadelphia and
New York State (2, 7, 8).
Second, we use the reconstructed incidence curves to perform
inference on daily reproductive numbers in that epidemic. One
level of difficulty arises from the fact that because of the rapid progression of the epidemic, there is saturation of susceptibles and apossible change in behavior during the course of infection for persons infected on a given day (9). To deal with this issue, we haverederived the Wallinga–Teunis estimator by using an approachsimilar to the one in ref. 10. The derivation uses daily infectivity ratios, a concept which essentially appeared before in refs. 10
and 11.
Third, we examine the spread of pandemic influenza in the cityof Philadelphia around the end of September and the beginningof October, 1918. On September 28, a 200,000-person Liberty
Loan Drive took place on the streets of Philadelphia against theadvice of medical professionals (12). Within 72 hours, “every single bed in the city’s 31 hospitals was filled” (ref. 13, p. 220). By
October 1, residents were often encouraged to stay home to stem
spread of the disease (14). On the evening of October 3, the closure of schools, churches, and places of public amusement was
adopted by the Philadelphia city council (ref. 15, p. 74). The deconvolved incidence curve, which peaks around October 1–2, shows
a drastic change in the growth patterns/infectivity ratios between
September 26 and October 3. In particular, the infectivity ratios,
representing the average number of infections by an infector on agiven day, dropped by more than half. We estimate, assuming thatcase-fatality ratio is at least 2% (16, 17), that during this period
depletion of susceptibles was at most 16% (here and elsewhere,
depletion is characterized in terms of the number of susceptibles
on September 26, which equals Philadelphia’s population minus
the number of infected by September 25). This difference suggests that depletion of susceptibles played only a modest role inslowing the growth of the epidemic during that period, which took
place before the closures went into effect. Our main conclusion,
which is not merely of historical interest, is that public awareness
Author contributions: E.G., J.D., J.M., J.B.P., D.J.D.E., and M.L. designed research; E.G., J.D.,
J.M., J.B.P., D.J.D.E., and M.L. performed research; E.G., J.D., J.M., J.B.P., D.J.D.E., and M.L.
analyzed data; and E.G. and M.L. wrote the paper.
The authors declare no conflict of interest.
This article is a PNAS Direct Submission.
Freely available online through the PNAS open access option.
1 To whom correspondence should be addressed. E-mail: egoldste@hsph.harvard.edu.

This article contains supporting information online at www.pnas.org/cgi/content/full/
0902958106/DCSupplemental.

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Edited by Burton H. Singer, Princeton University, Princeton, NJ, and approved October 23, 2009 (received for review March 17, 2009)

Fig. 1. The influenza epidemic in Philadelphia (Upper) and New York State(Lower) in 1918. Black is death curve. Blue is the initial condition for RL iterations (death curve, shifted back by nine days). Red is the estimated incidence(scaled by a factor of p, where p is the case fatality ratio), obtained from thefirst iteration for which the χ2 value from Eq. 3 is below 1 (sixth iteration for
Philadelphia and fourth iteration for New York State). Green are results of RL
after 30 iterations.

and changes in behavior are likely to have had a major role inthe slowdown of the epidemic before the official control measures
were in place.
In the SI Appendix we use simulations and sensitivity analyses totest the robustness of our methodology/conclusions. We examine
how well the deconvolved incidence curve represents the original
one, and what is the best way to estimate the growth rate and theinitial reproductive number of the original incidence curve. We
also study the epidemic progression in Philadelphia between September 26 and October 3, considering various “incidence” curves
deconvolved by using a collection of randomly generated timeto-death distributions and also allowing for time dependency ofthe case-fatality ratios due to potential changes in demographicsof the infected. Our general conclusion remains valid in all cases:
Within that period, a drastic change in the epidemic’s dynamics
took place, and depletion of susceptibles was probably insufficient
to explain that development.
Results
Reconstructing the Daily Incidence Curves for the 1918 Influenza Epidemic. We apply the procedure from the Methods section to the1918 influenza and pneumonia death curves in Philadelphia and
New York State (excluding the city) to obtain the daily incidence
in those places. Fig. 1 depicts the reconstructed incidence curves,which are scaled by a factor of p, where p is the case fatality ratio.
Notably, the peaks of the incidence curves are shifted backward
by approximately nine days, but the incidence curves have narrower peaks with steeper upward and downward trajectories as
one would expect when removing the smear induced by the timeto-death distribution, which is analogous to removing the blur due
to camera motion from a photographic image.
Estimates of the Daily Reproductive Numbers. The effective reproductive number Rt on day t measures the mean number of infections caused by persons who become infected on day t. Fig. 2 plots
the estimated reproductive numbers for certain periods of theepidemic’s progression, in both Philadelphia and New York State.
The reproductive numbers fluctuate initially because of the smallvalues for the deconvolved incidence. The reproductive numbers descend in the later part of the exponential growth period,
representing future infections which happen under saturation ofsusceptibles or behavior change.
21826

www.pnas.org / cgi / doi / 10.1073 / pnas.0902958106

Fig. 2. Epidemic progression in Philadelphia (Upper) and New York State(Lower), for certain time periods. Red is deconvolved incidence (scaled), andblack represents the estimated reproductive numbers.

With the fluctuation of the reproductive numbers, one may wonder what is the best way to estimate R0 , the mean number of new
infections caused by an infected individual during the early, exponential growth stage of the actual epidemic curve in Philadelphia.
We address this via simulations in the SI Appendix. Our estimate
is R0 = 2.14, with a standard error of 0.13.
Epidemic Peak in Philadelphia. In this section, we examine the peakof the Philadelphia epidemic in more detail. We first compute thedaily infectivity ratios, representing the number of infections on
a given day caused by a weighted average of previously infectedindividuals (Eq. 4). Fig. 3 plots the estimated infectivity ratios forthe period of September 14–October 19.
The infectivity ratio IRt on a given day t presents a basic assessment of the epidemic’s state on day t; in a mass action model,
the infectivity ratio is proportional to the number of susceptibles
left on that day. The reproductive number Rt on day t measures
the number of infections caused during an infectious period ofan average person who was infected on day t; it may be harder
to relate Rt to the epidemic’s state on day t because during that
infectious period further depletion of susceptibles may occur, andconditions related to the epidemic’s progression, such as an introduction of control measures and an increase in public awareness,
may take place.
The estimated infectivity ratios dropped by more than half
between September 26 and October 3. At the same time, depletionof susceptibles was much smaller. Philadelphia had a population of
1.7 million, with another 300, 000 added by the war industry (15).
As explained in Methods, the estimated incidence curve plottedin Figure 1 is scaled by a factor of p, where p is the case fatality
ratio. Various estimates of p, according to location, age, gender andFig. 3. Estimated infectivity ratios (black) vs. (scaled) incidence (red) in
Philadephia, September 14–October 19, 1918.

Goldstein et al.

Discussion
We have described and demonstrated the usefulness of a RL-type
deconvolution approach to reconstruct incidence curves from epidemic curves showing the times of death in an epidemic. Applied
to the 1918 influenza data, this approach reconstructs an epidemic
curve that is more sharply peaked than the death curve—as one
would expect given the smear introduced by the large variance inthe time-to-death distribution. Moreover, this reconstructed curve
allows the direct estimation of a daily value for the reproductive number Rt , bypassing the more-cumbersome methods usedpreviously, which involved extensive simulations and allowed theestimation of only a single value for R (4), or a parametric estimation of Rt (18). The estimate of Rt is given by the Wallinga–Teunis
formula (9). Our derivation of this estimate relies on the fact thatthe number of infected individuals is large hence the notion of an
average-infectiousness profile makes sense; this is different fromthe original setting of an emerging epidemic in ref. 9, where it
is assumed that all infected individuals had the same infectiousness profile. Finally, the deconvolved incidence curve was used
to show that public awareness and change in behavior is likely tohave played a major role in the slowdown of the epidemic evenin Philadelphia, a city whose response to the 1918 influenza epidemic is considered to have been among the worst in the U.S.
(13, 19). A related point regarding the belatedness of official
control measures in Philadelphia was made in ref. 18.
The latter qualitative conclusion was subjected to extensive
sensitivity analysis. The population of Philadelphia was not homogeneous, with case-fatality ratios, time-to-death distributions,
infectivity and susceptibility to infection varying by age and evengender. Although we cannot recover the complexity of the population network in Philadelphia, we have tested our observation thatthe drop in infectivity ratios during the week between September
26 and October 3 greatly exceeded the depletion of susceptibles under various scenarios; we have examined the sensitivity ofthat conclusion with respect to the time-to-death distribution, andallowed for certain forms of time dependency for the case-fatality
ratios, reflecting upon potential changes in the demographics of
the infected as the epidemic progressed. The main conclusion
persisted with remarkable stability in the simulations we have performed. Furthermore, visual inspection of the growth rate of thedeath curve (Fig. 4) shows a major change in its slope by October 5–6, which was likely predated by major changes in incidence
patterns. Combining this with contemporary journalistic evidence(14) suggests that our hypothesis about the change in behavior
before the official control measures were implemented is likely tobe real.
We have not considered asymptomatic infections in our analysis.
Data from the 1957 and the 1968 influenza epidemics yielded estimates of 20–42% for the rate of asymptomatic infections (20, 21).
A review of several volunteer challenge studies (22) gives an estimate of 33% for the rate of asymptomatic infections. There is noGoldstein et al.

Fig. 4. Natural logarithm of the recorded daily number of deaths frompneumonia and influenza in Philadelphia, 1918.

data for the rate of asymptomatic infections for the 1918 influenza
pandemic. Given its virulence, that rate could have been lower
than the rates above. In our simulations, the drop in infectivity
ratios between September 26 and October 3 surpasses depletionof susceptibles by a factor of at least 2.5. Thus, even if a third of
all infections were asymptomatic, depletion of susceptibles duringthat period cannot explain the drop in infectivity ratios.
Our approach in principle gives a general method for reconstructing an incidence curve from the death curve. In practice,
to have reliable estimates one needs the standard errors of thecounts to be small as compared with the means—say, if the (Poisson) counts reach into the hundreds. We recommend performing
simulations following the protocol in the SI Appendix to see how
well the deconvolution process works.
We note that estimated infectivity ratios in Philadelphia (Fig. 3)
first drop on September 27, one day before the notorious Liberty
Loan parade, and seven days before the implementation of official control measures. Although reductions in transmission in theabsence of official control measures have been inferred from the
dynamics of influenza in U.S. cities by a previous study (18), thisfinding for Philadelphia is surprising, given that the parade is often
interpreted as a sign that Philadelphia residents had not yet recognized the seriousness of the pandemic, and, moreover, that theparade is often seen as an opportunity for large-scale transmission. If we have accurately estimated the timing of the decline in
infectivity ratios, then the parade was probably not in fact a major
venue for transmission, perhaps because it took place in the open
air, and we must infer that behavior changes made a contribution
to the slowing of the epidemic prior to official control measures.
We have considered the possibility that the timing is estimatedincorrectly. For example, it is possible that the time-to-death distribution we have employed is too long, resulting in artificially
early changes in the incidence curve to reflect changes in thedeath curve. We believe that a shorter time-to-death distribution
is unlikely because we are unaware of any reliable data suggesting that the time-to-death distribution is substantially shorter thanthe one we have adopted from ref. 2; in fact, the distribution weused is shorter than the one obtained from military data and usedpreviously in refs. 4 and 18 (see SI Appendix). A likely source oferror in estimating the exact timing of declines in infectivity ratios
might be the nature of the deconvolution process, as seen in simulations in the SI Appendix. Briefly, deconvolved incidence curves
have an exponential growth period ending about two days before
the original one.
One may wonder if it is possible to use the death curve directly toinfer epidemiological parameters of interest, such as R0 . One cantry the exponential growth rate approach (see the SI Appendix).
In theory, given a long exponential growth period for incidence(long in comparison to the time-to-death distribution), the death
curve would have the same exponential growth rate. However
for the Philadelphia data, deconvolution shows that the death
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race, are given in refs. 16 and 17. There is considerable geographic
variation in case-fatality ratios, and for the surveyed communities
in the Northeastern United States (from Baltimore, MD to New
London, CT) case-fatality ratios were above 2%, topping 3% insome places. Assuming P ≥ 0.02 for Philadelphia and by using
the deconvolved incidence curve, we estimate that at most 16%of the susceptible population was depleted between September
26 and October 3, which cannot account for the over 50% dropin the infectivity ratios during that period; the latter finding was
subjected to extensive sensitivity analysis with regard to the timeto-death distribution and other parameters (see the SI Appendix
for more details). At the same time, we have evidence from refs. 12
and 14 about public awareness of the epidemic prior to October 3.
We conclude that public awareness and change in behavior likely
played a major role in the slowdown of the epidemic in Philadelphia before the official control measures were implemented on
October 4.

curve growth rate is somewhat smaller than the incidence curve
growth rate, and the estimate of R0 resulting from the death curve
growth rate ( 2 − 2.02) is biased downward. Alternatively, one
can consider the Wallinga and Teunis approach for estimating the
reproductive numbers by using the “death-to-death” distribution.
The latter can be understood as follows: For people who diedon day t, one can look at the times of death of the individuals
they’ve infected. However this distribution depends on the stateof the epidemic around time t, and thus changes with t. Moreover,assuming that the serial interval distribution is independent fordifferent pairs of cases, the death-to-death distribution will not be
independent for successive pairs of cases (e.g., A who infects B
who infects C), violating the assumptions of the approach. At thesame time, we found that having information about the incidence
curve can be used for a number of purposes besides estimating R0 .
Our estimates of the daily reproductive numbers and assessmentof the rapid decline of infectivity ratios around the peak of theepidemic in Philadelphia may not be accessible directly from thedeath curve.
It is also worthwhile to compare the approach to estimating Rt
here to an approach which assumes that future opportunities for
infection for each individual, regardless of when they are infected,
will follow a fixed distribution going forward in time. One might
expect that the relation between the reproductive numbers, theserial interval (wi ) (see Methods), and new infections is that foran average person infected on day t, the expected number Et,i ofpeople s/he will infect on day t + i is
Et,i = Rt · wi .

[1]

The relation above is known to hold during periods when there islittle to no depletion of susceptibles and behavior change (23)—during that period, Rt is in fact constant. However this may beuntenable for the whole duration of an epidemic, where the number of susceptibles may decline over the course of a single individual’s infectious period (24); similarly, measures to control theepidemic and behavior change may take place during that period.
Et,iTo put the issue a bit more precisely, consider the ratio E 1 . If
wt,i2

Eq. 1 was correct, this ratio would equal wi1 and would be indei2
pendent of the day t of the infection. However, when the epidemic
progresses very rapidly, populations of susceptibles (and their possible behavior) can be quite different on days t +i1 and t +i2 . Thus,
it is unreasonable to expect that this ratio is independent of t. The
formulation proposed here, and in ref. 9, avoids this pitfall.
Methods
Richardson–Lucy-Type Deconvolution. The deconvolution problem is toassess the daily incidence curve (It ) given the daily death curve (D1 , . . . , DN )and time from-infection-to-death distribution (d1 , . . . , dl ). Here, Dj is thenumber of deaths recorded on day j; and dk represents the probability that an
infected person who will eventually die, will die on day k after his/her infection. The daily death curves for Philadelphia and New York State (excludingcity) are recorded in refs. 7 and 8. The time from-infection-to-death distribution is obtained as a convolution of two distributions: time from symptom
onset to death, taken from ref. 2, based on 599 hospital autopsy reports
and the influenza latent period distribution, taken from ref. 25. It is plottedin Fig. 5.
To assess incidence It on day t, let p be the probability that an infectedindividual will die from influenza and pneumonia; in the SI Appendix, weallow for time dependence of p due to potential shifts in demographics of
the infected. The number of people who got infected on day t and later
died is binomial Bin(It , p); because It is large and p is small, it is well approximated by a Poisson variable Pois(λt ) with (an unknown) mean λt = p · It .
Those Pois(λt ) deaths are binned over the subsequent days according to thetime-to-death distribution.
We wish to estimate the daily incidence curve during some time period(t1 , . . . , t2 ), which overlaps with the period (1, . . . , N) for which the daily
counts for the number of deaths are available (N = 92 for Philadelphia andN = 75 for New York State). Because over 95% of deaths happen within three
weeks of infection, we want to estimate incidence from day t1 = −20 (three
weeks prior to the available death data); we also take t2 = N − 2 as d1 = 0 for21828

www.pnas.org / cgi / doi / 10.1073 / pnas.0902958106

Fig. 5. Assumed proportions of deaths on each day since infection. Basedon the Cook County Hospital data from ref. 2 for time from symptom onset
to death and the latent period distribution from ref. 25.
our time-to-death distribution. Thus, the number of unknown parameters is
N + 19, exceeding the number N of observations.
To assess the vector of unknown parameters (λt1 , . . . , λt2 ) (thus estimating
the incidence curve up to a multiple), we iterate in the space of parameters byusing the expectation maximization algorithm (26); this procedure is called
the RL iteration. The initial guess λ0 = (λ0t , . . . , λ0t ) for the parameters is1
2
the death curve, shifted back by nine days, as the time-to-death distribution(d1 , . . . , d31 ) peaks on day nine after infection (see Fig. 4). RL iterations produce a sequence λn = (λnt , . . . , λnt ) of the Poisson parameters. Explicitly, let
12qj = −j+1≤i≤N−j di be the probability that a death resulting from incidence
on day j will be observed during the interval
1, . . . , N; here −20 ≤ j ≤ N − 2and di = 0 for i ≤ 0 or i ≥ 32. Let Din = j<i di−j λnj be the expected numberof deaths to occur on day i, conditional on the parameters λn . Then,
λn+1
=
j

λnj  di−j Di
·
.
qj
Din

[2]

i>j

The probability of observations D1 , . . . , DN , conditional on the Poisson parameters λn and the time-to-death distribution, increases with each iteration.
While iterating, we do not seek convergence to a maximum likelihood solution; rather, we use a criterion to end iterations and produce our estimateof thePoisson parameters. To understand this, for the nth iteration, let
n
n
Ein =
j<i λj di−j be the expected number of deaths on day i. If λ werenthe true parameters, Di would be Poisson distributed with mean Ei ; thus,
the expectation E((E n −Di )2i) would be 1. This suggests to iterate until theEninormalized χ2 statistic

χ2 =1  (Ein − Di )2
N
Ein

[3]

i

descends below 1 for the first time.
Bootsrapping simulations in the SI Appendix address the question of how
closely a deconvolved curve resembles the original incidence curve.

Estimating Reproductive Numbers and Infectivity Ratios. In thissection, we show how to estimate the daily reproductive numbers for the1918 influenza from incidence curves and the infectiousness profile distribution. We remark that the words “infectiousness profile” and “serial interval”
are often used interchangeably, though this is not quite accurate (see ref. 27,
where “infectiousness profile distribution” is called “infectious contact distribution”). We prefer the term “infectiousness profile distribution” because it
reflects upon average individual infectivity, which depends only on the timesince one became infected, unlike the proportion of infectious contacts oneach day after infection or the time between one’s infection and the infectionof one’s infector.
We set a cutoff of 10 days for the infectivity process. The infectiousness
profile distribution (w1 , . . . , w10 ), essentially taken from ref. 25, is plottedin the SI Appendix. Here each number wi represents the proportion of the
cumulative infectiousness which falls between days (i − 0.5, i + 0.5) for an
average person. Note that, conveniently, the latent period in ref. 25 has anoffset of 0.5 days.
By using the infectiousness profile distribution and the incidence curve, we
can describe some key epidemiological parameters of interest. The infectivity
ratio IRt on day t measures the number of people infected by an “average”
infector on day t:
IRt = It
i<t Ii · wt−i.

[4]Goldstein et al.

where it is assumed that all the infected have the same infectiousness profile.
The two derivations help to clarify the notion that the reproductive number
Rt is forward-looking, taking account of all the events that may happen after
time t that will affect how many secondary cases an individual infected att will infect. By contrast, the infectivity ratio looks backward from the dayon which infection occurs. In the case of exponential growth (where opportunities for transmission remain constant over time), the two measures areequivalent.

This is the Wallinga–Teunis formula, and its derivation essentially follows
ref. 11. Note that this derivation relies on the fact that the number of infected
is large, hence the notion of an average infectiousness profile makes sense;
this is different from the original setting of an emerging epidemic in ref. 9,

ACKNOWLEDGMENTS. M.L. and E.G. were funded by the U.S. National Institutes of Health Models of Infectious Disease Agent Study Cooperative Agreements 5U01GM076497 and 1U54GM088588 to M.L. D.J.D.E., J.M., and J.D.
were supported by the Canadian Institutes of Health Research and the Natural Sciences and Engineering Research Council of Canada. D.E. and J.B.P.
were supported by the J. S. McDonnell Foundation. J.B.P. was supported by
the Burroughs Wellcome Fund, the David and Lucille Packard Foundation, andNational Institutes of Health Grant 3U54AI057168-06S1. The authors would
like to thank the Banff International Research Station for hosting focusedresearch group meetings at which this project was begun.

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Rt =


i>0

Goldstein et al.

IRt+i · wi =

i>0

wi It+i.
i−1
l=i−10 wi−l It+l

[5]

PNAS

December 22, 2009

vol. 106

no. 5121829

MEDICAL SCIENCES

This concept was already used in ref. 10, where it was denoted by φ1 (t), and inref. 11, where it was denoted by Rt . If the population of susceptibles and their
behavior does not change much over a reasonable interval (covering the timeof one individual’s infectiousness history), the infectivity ratio would equal
the reproductive number. This is not the case for the epidemics in question.
However, because the number of infected is large, we can compute the reproductive number Rt on day t in a “forward-looking” way, by adding up thenumbers of infections caused in subsequent days by an average person who
got infected on day t: