Vienna Yearbook of Population Research 2022 (Vol. 20), pp. 1–24

The mathematics of the reproduction number R
for Covid-19: A primer for demographers

Luis Rosero-Bixby1,∗ and Tim Miller2

Abstract

The reproduction number R is a key indicator used to monitor the dynamics of
Covid-19 and to assess the effects of infection control strategies that frequently have
high social and economic costs. Despite having an analog in demography’s “net
reproduction rate” that has been routinely computed for a century, demographers
may not be familiar with the concept and measurement of R in the context of
Covid-19. This article is intended to be a primer for understanding and estimating
R in demography. We show that R can be estimated as a ratio between the numbers
of new cases today divided by the weighted average of cases in previous days.
We present two alternative derivations for these weights based on how risks have
changed over time: constant vs. exponential decay. We then provide estimates of
these weights, and demonstrate their use in calculating R to trace the course of the
first pandemic year in 53 countries.

Keywords: Covid-19; reproductive number R; demographic methods; net reproduc-
tion rate

1 Introduction

Health professionals and world leaders are talking more and more about the
numbers R and R0 (R-naught), the basic reproduction number.

Angela Merkel, a rare head of state with a scientific background, explained the
trajectory of the Covid-19 pandemic on April 16, 2020, as follows:

“We are now at about a reproduction number of 1, so one person is
infecting another one. . . . If we get to the point where everybody infects

1University of Costa Rica, San Pedro, SJ, Costa Rica
2Department of Economic and Social Affairs, United Nations
∗Correspondence to: Luis Rosero-Bixby, Lrosero@mac.com

The views expressed in this paper are those of the authors and do not necessarily reflect the views of
the United Nations and the University of Costa Rica.

https://doi.org/10.1553/populationyearbook2022.res1.3

https://orcid.org/0000-0002-3063-3111


2 The mathematics of the reproduction number R for Covid-19

1.1 people, then by October we will reach the capacity of our health care
system with the assumed number of hospital beds. If we get to 1.2, so that
everyone is infecting 20% more – out of five people, one infects two and
the rest one. Then, we will reach the limits of our health care system in
July. And if it is up to 1.3 people, then in June we will reach the limits of
our health care system. So that’s where we can see how little the margin
is”. (The Guardian News, 2020).

The R was explicitly defined for the first time by the epidemiologist Klaus Dietz
in 19751 (Dietz, 1975) as the expected number of infections (secondary cases)
generated by a typical infected individual. If this occurs in a population in which
everyone is susceptible (that is, at the beginning of an epidemic; hence the
subscript zero), this number is R0, or the basic reproduction number. In later stages
of an epidemic, epidemiologists usually call the R the “effective” reproductive
number, which is often represented as R(t). This number can, in turn, be a cohort
(longitudinal) R, which is called in some texts the “case reproductive number;” or a
period (cross-sectional) indicator, which is sometimes called the “instantaneous R”
(Gostic et al., 2020). This article focuses on the instantaneous, effective reproductive
number, the R(t), which we usually refer to simply as R.

R is considered to be an important indicator for monitoring the Covid-19
pandemic, and particularly for assessing the effects of infection control measures
that frequently have high social and economic costs. R is also an important input
for projecting future scenarios of disease spread. Moreover, knowing R0 allows us
to identify the threshold for herd immunity: i.e., the proportion of individuals in
a completely susceptible population who need to become immune (naturally or by
vaccination) in order to stop the growth of the epidemic curve. This threshold occurs
at (R0 − 1)/R0 in homogeneous populations (Fine et al., 2011).

The demand for information about R for Covid-19 is so great that several websites
provide estimates of R at the national and subnational levels, as well as the tools
for producing estimates with user-provided data. The website https://shiny.dide.
imperial.ac.uk/epiestim/ is an example of the latter (Cori et al., 2013). A systematic
review of the Covid-19 literature up to September 2020 found 524 studies that
reported R estimates, including 49 that explained the method and the data they used
(Billah et al., 2020).

Although the concept of R is clear, the logic for its calculation in epidemiology
is not easy to follow, as it usually requires the use of mathematical models and
complex algorithms (Bettencourt and Ribeiro, 2008; Dietz, 1993; Nikbakht et al.,
2019; Wallinga and Lipsitch, 2007). In addition, the results may vary substantially
depending on the method used in the estimate (Billah et al., 2020). Hence, there is
a demand for transparent and reasonable estimates of R.

1 Earlier epidemiology in the field of malaria transmission used the concept of R in an effort to identify
critical thresholds of population densities of mosquitos per human for stopping the spread of infection
(Heesterbeek, 2002).

https://shiny.dide.imperial.ac.uk/epiestim/
https://shiny.dide.imperial.ac.uk/epiestim/


Luis Rosero-Bixby and Tim Miller 3

The purpose of this article is to use the toolbox of demographers to understand
R, and to provide a straightforward procedure for estimating it. We seek to
demystify the complexities of estimating this important indicator by following well-
known procedures in demography, a discipline in which an analog of R – the net
reproduction rate (NRR) – has been routinely computed for more than a century.
The approach to estimating R we present in this article is similar to an approach that
was recently developed in epidemiology by Cori and colleagues (Cori et al., 2013).

2 Simple (but not useful) formulas

In an ideal world in which we had access to perfect data, the reproduction number R
could be calculated for each generation of infected individuals as the simple average
of the number of infections generated by each member of the cohort. For example,
the cohort of the first two infected persons in Costa Rica (March 6, 2020) had an
R = 4.5, since, according to press reports, one case was a tourist who infected his
spouse and the other was a doctor who infected eight people: R = (1 + 8)/2 = 4.5.
However, this type of information is not available for the subsequent cohorts of
individuals who were infected in the days that followed. Moreover, this information
is not perfect, as it is possible that there were additional people who were infected
by these two initial cases, but whose infections were not reported.

Another way to estimate R is the approach that has been used in demography
since around 1880 (Lewes, 1984), and that was formally developed by Alfred Lotka,
the father of mathematical demography (Dublin and Lotka, 1925). Lotka defined
the NRR as the ratio of total births of daughters2 in two successive generations,
expressed as:

NRR = R =

∫ v

u
b(a)p(a) da (1)

Where b(a) is the fertility rate of women at age a and p(a) is the probability of
reaching this age alive (both variables refer only to females and female offspring),
and the limits of the integral include the reproductive age range of women, which
is, in practice, from u = 15 to v = 49 years.

If instead of applying the formula to population growth, we apply it to the
reproduction of an outbreak – that is, to a cohort of individuals infected on the same
date – the number of days elapsed since each cohort member was infected would be
represented by a (the “age”, defined as the days since infected); b(a) would become
the transmission rate of the infection at that “age” of a days, or the average number
of people infected on day a; and p(a) would become the probability of still being
able to spread the disease after a days. The limits of the integral would be from u,
or the first day when an individual achieves a sufficiently high viral load to become

2 Lotka originally defined the NRR for generations of men and sons. However, for practical reasons,
demographers compute it for women and daughters.



4 The mathematics of the reproduction number R for Covid-19

infectious; to the maximum number of days v that an infected person can still be
infectious. Hence, R becomes the NRR of infected individuals or the reproduction
number R in the lexicon of epidemiologists.

However, to use this formula as is customary in demography, it would be
necessary to have data on daily counts of new cases of infected persons tabulated by
the time-since-infection (duration of infection) of the person who infected them. The
newly recovered cases,3 as well as the deceased cases, should also be tabulated by
the duration of the infection. Given that these data usually do not exist, it is necessary
to make assumptions about the functional form of b(a) and p(a) to be able to
estimate the reproduction number R indirectly given the lack of data disaggregated
by duration a.

In the following sections, we present two approaches or models for estimating the
reproduction number R using widely available data. To simplify the presentation, we
assume no demographic change; i.e., a process with no births, deaths or migrants.
In the discussion section, we address the robustness of the method to violations of
these and other assumptions.4

3 A simple model with constant rates

Two heroic assumptions that can be used to simplify the estimation process are that
the effective transmission rates and the recovery rates (or, more broadly to include
deaths, the “removal rates”) are constant throughout a person’s infectious period;
that is, that the rates are invariant with respect to a, days since infection.

If b(a) is invariant with respect to a over the interval from u to v, then b(a) = b,
which can be placed outside of the integral:

R = b
∫ v

u
p(a) da (2)

Where b is the daily rate of effective transmission or the average number of people
infected per day.

The probability of continuing to be infectious – or survival function p(a) – is
driven by the removal rate g(a). In survival time analysis, this is the “hazard”,
“failure” or “mortality” rate. The following identity relates the survival function to
the failure rate, which, in turn, nicely simplifies into a negative-exponential function

3 Recovered cases are those of individuals who are no longer able to produce replication-competent
virus.
4 The acquired immunity of recovered individuals means that R declines over time because the
pool of susceptible individuals is depleted. This dynamic of epidemics does not occur in the NRR of
demography, as giving birth is a renewable process. The effect of a naturally declining R is, however,
nil on the few days that individuals are sick with Covid-19, and can thus be omitted from the models
used to estimate R in this article.



Luis Rosero-Bixby and Tim Miller 5

when the recovery rate g(a) is invariant with respect to a (Keyfitz, 1968):

p(a) = e−
∫ a

0 g(x) d(x) = e−ga (3)

The integral of p(a) is well-known in demography and in survival time analysis: it is
the area under the survival curve, which defines life expectancy; or, in this case, the
number of days lived while infectious during the interval between u and v, which
we call E.5 Here, “surviving” means to continue in the infectious state.

Recalling Equation (2), the equation for the reproduction number R therefore
simplifies into:

R = b · E (4)

b is the daily effective transmission rate of the infection (new infections per day),
and
E is the mean number of days infectious.

This simple identity is useful to show that the reproduction number R has two
components: the rate at which the infection is transmitted from one person to
another and the mean duration of the infectious period. For example, if the daily
transmission rate is b = 0.2 and the mean duration of the infectious period is E = 10
days, the reproduction number of the epidemic would be R = 2.0. Each case would
produce two infections on average, under the two assumptions of invariance noted
above.

4 Estimation of the effective transmission rate in a real
population

The expected length of the infectious period E, and the recovery rate from the
disease g that determines it, can reasonably be considered universal parameters
determined by the biology of the infectious agent, which, in practice, vary little over
time and from one population to another, at least as long as there is no treatment
to speed recovery. Early data for Covid-19 suggest that the virus has an average
infectious period of between eight and 15 days (Anastassopoulou et al., 2020; WHO,
2020; You et al., 2020). If an exogenous value of E is used, estimating R is a question
of determining the specific transmission rate b of the population at each time t. The
average transmission rate (under the aforementioned assumption of constancy over
the infectious period) can be estimated as:

b(t) = c(t)/A(t) (5)

c(t) is the number of new cases on day t, and

5 Solving the definite integral of p(a) in Equation (3) yields the expected number of days a person
remains infectious on average: E = [p(u) − p(v)]/g. If a person is infectious over the entire disease
period, E is simply the inverse of g.



6 The mathematics of the reproduction number R for Covid-19

A(t) is the number of currently active cases (infected people who are still spreading
the disease) as of day t.

The number of new cases each day is a widely available statistic that is usually
published in a timely fashion. However, the number of currently active cases needs
to be estimated, which can be done using the data series of new cases in the previous
days. Borrowing a basic relationship in demography (Lotka, 1998), which defines
the size of a population based on the number of past births and the survival function,
the number of active cases in the infective period u to v can be estimated with:

A(t) =

∫ v

u
c(t − a)p(a) da (6)

Recalling from (2) that R = b
∫ v

u p(a) da, the reproduction number R at time t is:

R(t) =
c(t)∫ v

u c(t − a)p(a) da
·

∫ v

u
p(a) da

Dividing both the numerator and the denominator by
∫ v

u p(a) da gives an expression
with a clearer interpretation,

R(t) =
c(t)∫ v

u c(t − a)
[

p(a)∫ v
u p(a) da

]
da

(7)

The numerator of this quotient is the number of new cases counted on day t, while
the denominator is the weighted average of the cases reported during the previous
u to v days. The weights used to obtain this average are represented by the term
in square brackets, which we will call w(a).6 The weighting term is none other
than the distribution of the “survival” function for the infectious state; that is, the
proportion of people who continue to be infectious (t − a) days after they first
became infected. As previously shown in Equation (3), this is a simple negative
exponential distribution under the assumption that the recovery rate is independent
of the time elapsed since infected.

Moving on to the discrete version in which we solve the integral and simplify the
fraction, we arrive at the following handy formula for estimating R(t), which also
assumes a fixed lag of six days between the date the infection occurs and the date
the case is reported:

R(t − 6) = c(t)
/ a=v∑

a=u

c(t − a)w(a) (8)

The weights w(a) are the aforementioned distribution of the survival function p(a)
evaluated over the interval u to v, which is determined by the following formula

6 A quick and rough estimate of the denominator can be obtained by calculating the simple average –
without weighting – of the cases in a period of at least 14 previous days.



Luis Rosero-Bixby and Tim Miller 7

(see Footnote 5):
w(a) = ge−ga/(e−gu − e−gv) (9)

Plausible parameters for estimating these factors are:

• Infectious interval: u = 2 and v = 30 days, and
• Daily recovery rate g = 1/10, which implies:

◦ Mean duration of illness = 10 days and
◦ Mean duration of infectiousness E = 6 days.

We took these parameters from early reports of the epidemiology of Covid-19 as
observed mostly in the Hubei province in China (Anastassopoulou et al., 2020;
Park et al., 2020; WHO, 2020). As knowledge of this disease progresses, different
parameters may be favored in the future.

5 A (more realistic) model with exponential rates

Although epidemiology models of Covid-19 often assume that transmission and
recovery rates are constant during the illness period, it is useful to explore alternative
specifications of these two functions to better approximate the rates that have been
observed during the first few months of the pandemic.

Regarding the transmission rate b(a), initial data on the outbreak and measure-
ments of the viral load while infected with the disease suggest a distribution with an
early peak at two or three days followed by a sharp decline (He et al., 2020; Prakash,
2020). To keep the math simple, we assume a negative exponential function that
declines quickly from the peak day of infection, which is also assumed to be the
first day of infectiousness u:

b(a) = B0e−B1(a−u) (10)

B0 parameter representing the peak transmission rate on the initial day u, and
B1 parameter indicating the speed of the decline in the transmission rate.
Regarding the removal rate g(a), we did not find any estimates of its distribution

for the novel Covid-19 disease in the literature. However, it seems reasonable to
assume that the chance of recovery of an infected individual increases with time.
The Gompertz model is a well-known function (and is convenient for integration
purposes) for representing this behavior. It assumes that the rate of interest increases
with duration time at a constant speed, which is a pattern observed for failure rates
in most biological and mechanical entities (Keyfitz, 1968; Pollard, 1991):

g(a) = G0eG1a (11)

G0 parameter representing the recovery at the beginning of the disease, and
G1 parameter measuring the speed of increase in the recovery rate per unit of a.



8 The mathematics of the reproduction number R for Covid-19

The proportion of individuals who are still infectious after a days, or the survival
function, is obtained by solving the integral in the formula below, which results in a
double exponential function:

p(a) = e−
∫ a

0 g(x) d(x) = e[(G0/G1)(1−eG1a)] (12)

Determining the effective reproduction number R(t) with the functions b(a) and p(a)
would entail estimation at each time t of the parameters defining these functions;
most importantly, those of the transmission rate b(a). The data required to do this
are not available. Instead, we propose following a procedure that is well-known in
demographic analysis: indirect standardization (Shryock and Siegel, 1976). In the
first step of the procedure, we estimate the expected number of cases consistent with
a reproductive number R = 1 with plausible distributions of b(a) and p(a), given the
composition by duration a of active (currently infected) cases at time t.

The following relation estimates the expected number of cases given that R = 1:

c(t,R = 1) =

∫ v

u
c(t − a)[b(a)p(a)] da (13)

In a second step, the R(t) factor is estimated as a quotient between the observed and
the expected cases:

R(t) ≈
c(t)∫ v

u c(t − a)[b(a)p(a)] da
(14)

Note that the denominator is, like in the model of constant rates (Equation (7)),
a weighted average of the series of cases in the previous days, with the term in
rectangular brackets as the weighting factor we have called w(a).

Given the assumed functions for b(a) and p(a), and with the aforementioned lag
of six days between infection and diagnosis, we arrive at the following formula in
discrete terms for computing an estimate of the effective reproduction number R(t)
under the model we call “exponential rates”:

R(t − 6) = c(t)
/ a=v∑

a=u

c(t − a)w(a) (15)

This is the same formula as the one with the constant rates model (Equation (8)),
but with a different set of weighting factors w(a):

w(a) = B0e[−B1(a−u)+(G0/G1)(1−eG1a)] (16)

These weighting factors w(a) are the distributions derived by multiplying b(a) times
p(a), starting with the day a = u when infectiousness begins, which we are also
assuming is the peak day of Covid-19 infectiousness.

Plausible parameters for estimating the set of weighting factors are:

• Infectious interval: u = 2 and v = 30 days (however, the upper limit is
irrelevant, since the weighting factors reach zero by day 22);



Luis Rosero-Bixby and Tim Miller 9

• Parameters for the survival function p(a) chosen to conveniently reproduce a
10-day mean duration of illness:
G0 = 0.0169 and
G1 = 0.220;
• Parameters for the effective transmission function b(a) chosen to reproduce,

in conjunction with p(a), a convenient reproduction number R = 1:
B0 = 0.157 and
B1 = 0.0508.

As before, we chose the parameters on the basis of early knowledge of the Covid-19
epidemiology, mostly from the Chinese province of Hubei (Anastassopoulou et al.,
2020; He et al., 2020; Park et al., 2020; Prakash, 2020; WHO, 2020).

6 Weighting factors, generation time and growth

With two different sets of assumptions, we have arrived at the same relationship for
estimating R(t) as the quotient between the numbers of new cases in day t divided
by the weighted average of cases in the previous days. Therefore, the choice of the
correct set of weighting factors w(a) becomes a key issue in estimating R. Figure 1
compares the w(a) distributions in the previously presented constant and exponential
models (the functions b(a) and p(a) behind the weighting factors are shown in
Figure A.1 in the Appendix).

The constant rates model gives more weight to cases that occurred farther back in
the past, while the exponential rates model gives more importance to more recent
cases. If the number of new cases has changed little in the past, the R(t) estimated
with the two models will be similar. Remembering that these factors are in the
denominator of the R(t) formula, the constant rates model will result in higher R(t)
when the number of daily cases is increasing. The reverse will happen in later stages
of the epidemic, when the number of daily cases is declining: i.e., the R(t) estimates
with the constant model will be lower. Therefore, the constant rates model and, in
general, wider distributions will exaggerate extreme values of R(t) estimates.

The two models can be considered archetypes for the choice of a weighting
distribution for the indirect estimation of the reproduction number R(t). Choosing a
narrow distribution, as in the exponential model, gives more weight to recent cases,
while a wider distribution, as in the constant model, gives more weight to older
cases.

The shape of the w(a) distribution is mostly driven by the shape of the transmis-
sion rate curve b(a). To understand the transmission pattern of Covid-19, it is useful
to look to evidence from recent outbreaks of other respiratory infections, such as:
(1) the seasonal influenza curve with a high and narrow concentration in the first
few days of illness; and (2) the SARS-2003 coronavirus outbreak with a wider and
later distribution, which is somewhat similar to our rectangle of constant b(a) (see
Figure A.1 in the Appendix). Emerging data and estimates for the novel Covid-
19 virus suggest that its transmission pattern resembles that of seasonal influenza,



10 The mathematics of the reproduction number R for Covid-19

Figure 1:
Weighting factors distributions

0

.04

.08

.12

.16

W
ei

gh
tin

g 
fa

ct
or

s 
w

(a
)

0 5 10 15 20 25 30
Days `a´ since contagion

Exponential
Constant

Model:

rather than of SARS, with a high concentration in days two to four (He et al., 2020);
as in our exponential model.

The generation time7 or length is an important indicator that epidemiologists
often use to summarize the time it takes for an infected person to pass on the
infection to others. It is a key input element in many epidemiological models that
estimate the reproduction number R. This indicator is the mean duration a in our
w(a) distribution of weighting factors, which we call T:

T =

∫ v

u
aw(a) da (17)

7 The epidemiologic literature often uses the “serial interval” as an estimate of the “generation time”.
The generation time is the interval between the onset of infection for the “parent-child” cases. The
serial interval is the observed period of the onset of symptoms between the infector and the infectee.
The onset of infection and the onset of symptoms are separated by the “incubation period”.



Luis Rosero-Bixby and Tim Miller 11

Since the integral in the exponential model does not have a simple analytical
solution, we use numerical integration to derive the generation time (see Table A.1
in the Appendix) with:

T = 10.20 days in the model of constant rates, and
T = 6.06 days in the model of exponential rates.
Four review papers have identified nearly 40 articles on Covid-19 with estimates

of T ranging from four to eight days (Billah et al., 2020; Griffin et al., 2020; Hussein
et al., 2021; Park et al., 2020; Rai et al., 2021). As the estimates of our exponential
model fall in the middle of this range, it appears that this model better represents the
current state of knowledge about Covid-19 transmission than our constant model.
An example of a set of R(t) estimates with a shorter generation time of 3.6 days is
from the Centre for the Mathematical Modeling of Infectious Diseases (CMMID)
at the London School of Hygiene and Tropical Medicine (Abbott et al., 2020). As
expected, these estimates result in smaller extreme figures; or, in other words, the
estimates are very close to R(t) = 1 at all times.

The equivalent of the generation time in demographic analysis is the “mean inter-
val between two consecutive generations,” which Alfred Lotka, in his 1934 book
Analytical Theory of Biological Associations, used to identify a relationship between
the net reproduction rate R and a key indicator of the multiplication capacity of a
population: the “intrinsic rate of growth” (Lotka, 1969). The relationship is:

R = eρT or ρ = ln(R)/T (18)

In the context of Covid-19, ρ is the “intrinsic” or underlying rate of growth of the
number of infectious individuals. Note that this growth rate may differ from the
observed or real rate usually represented by lowercase r. In Lotka’s words: “the ρ
exposes the fundamental capacity of multiplication . . .while the r does not give us
the true measure of that capacity since it is influenced by past factors we could call
adventitious. The ρ is an asymptotic value to which the observed r will approach
when those fundamental conditions remain the same” (Lotka, 1969, pp. 126–127).
The observed growth r of Covid-19 cases is determined by both the fundamental
conditions of its infectiousness and the momentum in the pool of individuals who
are the source of infection. The intrinsic ρ is a rate free of momentum effects.

It is worth noting that several epidemiological studies have developed estimation
procedures of R that start from this relationship and use observed growth rates as
input and borrow T from models.8 However, those studies usually do not make the
distinction between the observed little r and the intrinsic ρ.

8 Indeed, estimating the intrinsic growth rate directly from observed population data is a well-known
approach in demography. In stable populations, births, deaths and population numbers are all growing
at the intrinsic growth rate. In non-stable populations, Preston has shown that the growth rate of the
population segment below the mean length of a generation is a good approximation of the intrinsic
growth rate (Preston, 1986). Ediev, in generalizing the work of Fisher on reproductive value, has
provided a method for estimating the intrinsic growth rate based on the dynamics of the population age
structure (Ediev, 2007).



12 The mathematics of the reproduction number R for Covid-19

7 Estimates of R(t) for Covid-19 in the real world

In this section, we analyze our estimates of R(t) during the first year of the pandemic
for 53 European and Latin American countries.9 Figure 2 shows the results for Chile
and Costa Rica, two Latin American countries known for maintaining good-quality
health statistics. The figure illustrates the effect on R estimates of using the two
different weighting distributions w(a) corresponding to our constant and exponential
assumptions. The figure also shows the relationship between the behavior of R(t)
and the epidemic curve of incidence over time.

The two proposed models produce approximately similar time-trend curves. They
tell similar stories about when the reproduction number in each country is ascending,
declining and crossing the R = 1 threshold; and about the speed of change in this
indicator. However, at specific points in time, the level of the estimate may differ
substantially, especially at extremely high or low levels. As expected, the model
assuming constant rates exaggerates extreme values, tending toward higher values
at high levels and lower values at low levels. This is in part because the mean
generation time in the constant model is wider (10 days vs. six days). However,
it is also because new cases tend to be increasing when R > 1 and to be decreasing
when R < 1 (see the epidemic curves in the lower part of the figure), which, as we
explained above, pulls the estimate up or down due to the greater weight assigned
to older cases in the constant rates model.

As we noted in the previous section, our model of choice is the one that
assumes exponential rates of removal and transmission of the Covid-19 disease. The
“constant model” was developed for didactic purposes only.

Figure 2 also illustrates the relationship between R(t) and the epidemic curve of
incidence. In periods when R > 1, the epidemic curve increases; and in periods when
R < 1, the curve declines. When R is hovering around one, the number of new cases
plateaus. This can occur at high levels, such as in Costa Rica from September to
December; or at moderate levels, such as in Chile from August to November.

The points in time when R(t) falls below the threshold of one are approximately
the peak times of the pandemic waves: i.e., early July and early January in Chile
and mid-September and January 1, 2021, in Costa Rica. R(t) also shows the distinct
phases or waves of the epidemic, delimited by the red vertical lines of Figure 2.

The R(t) curves observed in these countries demonstrate the importance of taking
aggressive action to contain the pandemic in its very early stages. Costa Rica

9 We used the daily national series of confirmed Covid-19 cases from the “Our World in Data” website
(Ritchie, 2020), accessed on March 10, 2021. The raw curves of cumulative cases were first smoothed
out with local regression as implemented in the Stata software, command “lowess” (StataCorp, 2017).
Clean daily numbers of cases were obtained by the difference in the smoothed cumulative curve, and
were used as the input data in the estimation. Countries with populations of less than one million or
unreliable data were excluded, along with the period before there were 100 accumulated cases. Our
final analytical data file for Figures 2 and 3 is included as supplementary material in Excel and Stata-17
formats (available at https://doi.org/10.1553/populationyearbook2022.res1.3).

https://doi.org/10.1553/populationyearbook2022.res1.3


Luis Rosero-Bixby and Tim Miller 13

Figure 2:
R(t) and the incidence curve during the first year of the Covid-19 pandemic in Chile
and Costa Rica

0.5

0.7

1.0

1.5

2.0

3.0

4.0

R
(t

) 
fa

ct
or

Apr−01 Jul−01 Oct−01 Jan−01 Apr−01

Constant

Exponential

R(t) with model:

Chile

0

50

100

150

200

250

300

350

400

450

D
ai

ly
 c

as
es

 p
er

 m
ill

on
 p

eo
pl

e

Apr−01 Jul−01 Oct−01 Jan−01 Apr−01

0.5

0.7

1.0

1.5

2.0

3.0

4.0

R
(t

) 
fa

ct
or

Apr−01 Jul−01 Oct−01 Jan−01 Apr−01

Constant

Exponential

R(t) with model:

Costa Rica

0

50

100

150

200

250

300

350

400

450

D
ai

ly
 c

as
es

 p
er

 m
ill

on
 p

eo
pl

e

Apr−01 Jul−01 Oct−01 Jan−01 Apr−01

`Daily cases´ are weekly averages

Source: Daily national series of confirmed Covid-19 cases from the website “Our World in Data” (Ritchie, 2020),
accessed on March 10, 2021.

employed that strategy by implementing aggressive contact tracing and testing
programs, as well as drastic lockdown measures that essentially paralyzed the
country from March 15 to April 15 (Rosero-Bixby and Jiménez-Fontana, 2021).
Consequently, in Costa Rica, the R(t) factor fell well below one, and the number of
infections was contained at levels close to zero. In contrast, Chile did not reduce
its R to the threshold of one or lower in April, and paid dearly for this failure with
a devastating surge in infections in the following period. After the first month of
the pandemic, both countries had rising R, but because the increase started at very
different baselines, the results were vastly different. By June 15, the pandemic was
exploding in Chile, at 260 daily cases per million population; whereas in Costa Rica,
just 20 daily cases per million population were being reported.



14 The mathematics of the reproduction number R for Covid-19

The effects of Costa Rica’s initial success in containing the virus were still
apparent as long as one year after the start of the pandemic. As of March 10, 2021,
the cumulative mortality caused by Covid-19 was 561 deaths per million residents
in Costa Rica, compared to 1,117 deaths per million residents in Chile.

In general, subtle differences in the trajectory of the R(t) resulted in two
substantially different epidemic curves of incidence in Chile and Costa Rica. This
is an obvious point from a demographic perspective: the absolute increase in
population size is driven by both the reproduction rate and the initial population
size. By the same token, both the R factor and the number of actively contagious
individuals drive the incidence curve.

Broadening the scope of our analysis to 18 countries in Latin America and 35
countries in Europe, Figure 3 shows the results of our R estimates (exponential
model), with weekly boxes displaying the distribution of countries by R. The box’s
hinges indicate the interquartile interval, and each box’s central line indicates the
median value of R for that week.

Epidemiologists pay special attention to the R0 factor – the basic reproduction
number – to characterize and model epidemic outbreaks. The level of R(t) – the
effective reproduction number – in the first days of an outbreak is an approximation
of this basic R0. The first boxes in the figure thus suggest that Covid-19 R0 was in
the interquartile range of 1.9 to 2.8 in European populations, whereas it was in the
range of 2.3 to 2.5 in Latin American populations.

On both continents, the initial R declined sharply in the first few weeks, though
more so in Europe than in America. In the European countries, R leveled out at
around R = 0.8 in May, while in the Latin American countries, R leveled out at
around R = 1.15. This means that in Europe, the first pandemic wave peaked (R
crossed one) in early April, with the incidence of Covid-19 falling sharply thereafter.
By contrast, in Latin America as a whole, the peak (R = 1) of the first pandemic
wave seems to have occurred much later, in early August.

In Latin America, R hovered around R = 1 from August to December. Thus, the
first wave did not really end, but instead plateaued at high levels of incidence.

In Europe, the Covid-19 pandemic has followed a trajectory of three well-defined
waves: the initial wave peaked in April 2020; the second wave peaked in November
2020; and the third wave had not yet peaked by March 5, 2021.

One year after the start of the pandemic, the described trajectories of R(t) resulted
in a mortality toll that was 16% higher in Latin America, with 1,325 deaths per
million people, than it was in Europe, with 1,139 deaths per million people.

The data from the 18 Latin American countries confirm our previous observation
that the very early containment of R correlates with a less severe pandemic in the
following months. In these countries, the correlation coefficient between the national
level of R two weeks after case 100 was diagnosed and the death toll in the first
year of the pandemic is strong, at 81%. However, this association is not observed in
Europe, where the correlation coefficient is weak, at 5%. Figure A.2 in the Appendix
shows the scatter plots behind these correlations.



Luis Rosero-Bixby and Tim Miller 15

Figure 3:
Weekly distribution by R(t) of countries in Europe and Latin America

.5

.6

.8

1

1.2

1.5

2

3

R
(t

) 
fa

ct
or

. . . Apr
−0

3

. . . M
ay

−0
1

. . . . Ju
n−

05

. . . Ju
l−0

3

. . . . Aug
−0

7

. . . Sep
−0

4

. . . Oct−
02

. . . . Nov
−0

6

. . . Dec
−0

4

. . . Ja
n−

01

. . . . Feb
−0

5

. . . M
ar

−0
5

.

Europe

.5

.6

.8

1

1.2

1.5

2

3

R
(t

) 
fa

ct
or

. .

Apr
−0

3 . . .

M
ay

−0
1 . . . .

Ju
n−

05
. . .

Ju
l−0

3 . . . .

Aug
−0

7 . . .

Sep
−0

4 . . .

Oct−
02

. . . .

Nov
−0

6 . . .

Dec
−0

4 . . .

Ja
n−

01
. . . .

Feb
−0

5 . . .

M
ar

−0
5 .

Latin America

Source: Daily national series of confirmed Covid-19 cases from the website “Our World in Data” (Ritchie, 2020),
accessed on March 10, 2021.



16 The mathematics of the reproduction number R for Covid-19

8 Discussion

The reproduction number R is a key indicator that has been used to characterize
the dynamics of the Covid-19 pandemic, and to assess the effects of pandemic-
related policy interventions. Unfortunately, the available statistics do not allow us
to calculate this factor unequivocally. Instead, R must be estimated using indirect
methods based on theoretical models and assumptions about the behavior of this
novel disease. This article provides an approach for estimating R using methods
and models developed a century ago in demography. The strengths of the proposed
approach are the transparency of the assumptions from the point of view of
demographers and the simplicity of the procedure.

The simple relationship used to estimate R(t) on a daily basis is a quotient
between the current number of new cases divided by a weighted average of
the number of cases in the previous 20 or 30 days. We suggest using a set of
weighting factors derived from assuming that: (1) the transmission rate of an
infected individual declines sharply from a peak at day 2 of the illness following
a negative exponential function; and (2) the recovery rate from the disease follows
the Gompertz law of exponential growth with disease duration. A mean generation
time of six days summarizes this suggested set of weighting factors. Early estimates
of this interval, mostly for outbreaks in China’s provinces, range from four to eight
days. A weighting factors distribution with shorter generation times will result in
R(t) values that are closer to one; i.e., with less extreme values. We have shown
that during stages of the outbreak when the number of new cases is increasing,
shorter generation times (narrower distributions) result in lower R(t) estimates;
whereas during stages of the outbreak when the number of new cases is decreasing,
shorter generation times result in higher (closer to one) estimates. In spite of these
differences, the general time trend in R(t) does not change meaningfully when
different distributions are chosen. As our knowledge about this novel coronavirus
improves, researchers will have more information that will enable them to make
better informed choices about the distribution of the weighting factors used to
estimate R(t).

The strategy proposed in this article for estimating R is not new in epidemiology.
A similar equation was proposed by Wallinga and Lipsitch (2007, Equation 4.2), and
was implemented through web-based tools by Cori et al. (2013). The distribution
w(a), or the set of weighting factors of cases that occurred in previous days t − a, is
called the “infectivity profile” by these authors, which is also the distribution of the
generation time. Epidemiology studies assume a mathematical function for the w(a)
distribution, with the gamma function being the most commonly used (Knight and
Mishra, 2020).

Using the computer tool provided by Cori et al. (2013), we were able to reproduce
very closely our R estimates with the gamma function for a mean generation time of
six days and a standard deviation of three. One study has recommended using the
Cori et al. approach to estimate R after comparing it with two other epidemiological



Luis Rosero-Bixby and Tim Miller 17

methods applied to a simulated Covid-19 epidemic in which the true R is known
(Gostic et al., 2020).

The main contribution of this article is that we demonstrated how the problem
of estimating R can be approached with demographic thinking. The key set of
weighting factors w(a) is seen here not as a black box of a mathematical function,
but as the product of two well-known demographic concepts: a survival function
and a birth function, which could be defined analytically or with discrete observed
distributions.

Our model assumes the absence of demographic change, meaning that births,
deaths and migrations do not exist. Given that the time horizon involved in R(t)
estimates is short (30 days or less), including or excluding demographic change
is unlikely to change the results in a meaningful way. Potential exceptions to this
general observation are the arrival of imported cases of Covid-19 and mortality
caused by Covid-19 itself.

Imported cases should not be counted in the numerator if the information is
available, even though they must be included in the denominator. However, imported
cases are statistically important only when the outbreak is at very low levels, and is
in its initial stages.

Covid-19 deaths can be included by broadening the concept of the recovery rate
g(a) to a “removal rate” that would include both recovery and death as means
of exiting the population of the infected. However, this correction would change
the estimates very little, since the case fatality rate of Covid-19 has an order of
magnitude of 0.01 (Worldometer, 2020), which, along with a mean period of illness
of 15 days, is equivalent to a daily mortality rate of less than 0.001. Given that the
mean daily recovery rate of Covid-19 is around 0.1, the correction would thus be
about 1%. Such a small correction may well be omitted.

A weakness in all of the estimates on the numbers of reported cases is that this
statistic is just the tip of the iceberg of all Covid-19 infections. But this does not
necessarily invalidate the estimate. The estimated R would be valid insofar as these
known observations are representative of the whole. Regardless of what proportion
of cases is known and what proportion of cases is unknown, the important thing is
that the known cases reflect the characteristics of the whole, and that this proportion
does not change rapidly on the scale we are using to measure R. It is worth noting
that given this weakness in the available input data, it might be pointless to use more
intricate models to estimate R, which would seem to support the use of the simple
approach this article proposes.

The R number is probably the best indicator for monitoring the dynamics in
the propagation of an epidemic, and for taking action to contain it. It is like the
speedometer in a car that tells us how quickly an epidemic is moving, and it does
so in a more timely manner and with less contamination than its cousins; i.e., the
rates of variation in the curves of incidence, hospitalizations or deaths. For example,
in late January and early February 2021 in Costa Rica, the epidemic curve of
incidence was declining, whereas R was clearly increasing (Figure 2). Thus, the



18 The mathematics of the reproduction number R for Covid-19

former indicator was misleading, while the R estimates reinforced the need to keep
public health restrictions in place.

However, the R number tells only a partial story of an epidemic and its drivers.
It does not, for example, tell us about the severity of an outbreak, which is better
described by the incidence of diagnoses, the prevalence of hospitalizations or the
mortality rate. In addition, because it is just an average, R can miss several important
dimensions of reproduction, particularly in heterogeneous populations. For example,
the existence of super-spreader individuals or clusters, which can be crucial in
an outbreak, is totally hidden in this average. As a long tradition of demographic
research has shown us, estimating the reproduction rate and assessing its meaning
is just a first step in an ongoing quest to grasp the complexities of human behavior
and the conditions that drive it.

Supplementary material

Available online at https://doi.org/10.1553/populationyearbook2022.res1.3

Supplementary file 1. Data in Excel format
Supplementary file 2. Data in Stata 17 format

ORCID

Luis Rosero-Bixby https://orcid.org/0000-0002-3063-3111

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Luis Rosero-Bixby and Tim Miller 21

Appendix

Figure A.1:
Transmission rate b(a) and “survival” function p(a) in the constant and exponential
rates models

0

.05

.1

.15

.2

b(
a)

 d
ai

ly
 r

at
e

0 5 10 15 20 25 30
Days `a´ since contagion

Model:

Exponential

Constant

Transmission rate b(a)

0

.2

.4

.6

.8

1

S
ur

vi
vi

ng
 p

ro
po

rt
io

n 
p(

a)

0 5 10 15 20 25 30
Days `a´ since contagion

Model:

Exponential

Constant

Survival function p(a)



22 The mathematics of the reproduction number R for Covid-19

Figure A.2:
Correlation between the early level of R and the Covid-19 crude death rate in the first
year

ALB

AUT

BEL

BGRBIH

BLR

CHE

CZE

DEU

DNK

ESP

EST

FIN

FRA

GBR

GRC

HRV

HUN

IRL

ITA

LTU

LVA

MDA

MKD

NLD

NOR

POL

PRT

ROU

RUS
SRB

SVK

SVN

SWE

UKR

ARG

BOL

BRA

CHL
COL

CRI

CUB

DOM

ECU

GTM
HND

JAM

MEX
PAN

PER

PRY

SLV
URY

0

.5

1

1.5

2

2.5

.5 1 1.5 2 2.5 3 .5 1 1.5 2 2.5 3

Europe R=0.05 Latin America R=0.81

C
ru

de
 d

ea
th

 r
at

e 
pe

r 
1.

00
0

R two weeks after case 100

Note: Countries are identified by their ISO alpha-3 codes. United Nations Statistics Division, “Standard Country or
Area Codes for Statistical Use” (M49 standard) https://unstats.un.org/unsd/methodology/m49/ Accessed on March
15, 2021.
Source: Daily national series of confirmed Covid-19 cases and deaths from the website “Our World in Data”
(Ritchie, 2020), accessed on March 10, 2021.

https://unstats.un.org/unsd/methodology/m49/


Luis Rosero-Bixby and Tim Miller 23

Table A.1:
Weighting factors w(a) to estimate R(t) with two models

Days Constant rates model Exponential rates model
a g(a) p(a) w(a) g(a) p(a) b(a) w(a)
0.5 0 0.9512 0 0.0189 0.9911 0 0
1.5 0 0.8607 0 0.0235 0.9704 0 0
2.5 0.10 0.7788 0.1013 0.0293 0.9452 0.1531 0.1746
3.5 0.10 0.7047 0.0917 0.0365 0.9147 0.1455 0.1529
4.5 0.10 0.6376 0.0830 0.0455 0.8781 0.1383 0.1328
5.5 0.10 0.5769 0.0751 0.0568 0.8345 0.1315 0.1142
6.5 0.10 0.5220 0.0679 0.0708 0.7831 0.1249 0.0970
7.5 0.10 0.4724 0.0615 0.0882 0.7235 0.1188 0.0811
8.5 0.10 0.4274 0.0556 0.1099 0.6555 0.1129 0.0665
9.5 0.10 0.3867 0.0503 0.1370 0.5797 0.1073 0.0532
10.5 0.10 0.3499 0.0455 0.1708 0.4973 0.1020 0.0413
11.5 0.10 0.3166 0.0412 0.2129 0.4108 0.0969 0.0309
12.5 0.10 0.2865 0.0373 0.2654 0.3237 0.0921 0.0220
13.5 0.10 0.2592 0.0337 0.3308 0.2406 0.0876 0.0148
14.5 0.10 0.2346 0.0305 0.4123 0.1662 0.0832 0.0092
15.5 0.10 0.2122 0.0276 0.5139 0.1048 0.0791 0.0053
16.5 0.10 0.1920 0.0250 0.6405 0.0590 0.0752 0.0027
17.5 0.10 0.1738 0.0226 0.7984 0.0288 0.0715 0.0012
18.5 0.10 0.1572 0.0205 0.9951 0.0118 0.0679 0.0004
19.5 0.10 0.1423 0.0185 1.2404 0.0039 0.0645 0.0001
20.5 0.10 0.1287 0.0167 1.5461 0.0010 0.0614 0.0000
21.5 0.10 0.1165 0.0152 1.9271 0.0002 0.0583 0.0000
22.5 0.10 0.1054 0.0137 2.4020 0.0000 0.0554 0.0000
23.5 0.10 0.0954 0.0124 2.9940 0.0000 0.0527 0.0000
24.5 0.10 0.0863 0.0112 3.7319 0.0000 0.0501 0.0000
25.5 0.10 0.0781 0.0102 4.6516 0.0000 0.0476 0.0000
26.5 0.10 0.0707 0.0092 5.7980 0.0000 0.0452 0.0000
27.5 0.10 0.0639 0.0083 7.2269 0.0000 0.0430 0.0000
28.5 0.10 0.0578 0.0075 9.0080 0.0000 0.0409 0.0000
29.5 0.10 0.0523 0.0068 11.2280 0.0000 0.0388 0.0000

Sum 1.0000 1.0000
T 10.20 6.06



24 The mathematics of the reproduction number R for Covid-19

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