Doubling time of infectious diseases (2024)

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J Theor Biol. 2022 Sep 13 : 111278.

doi:10.1016/j.jtbi.2022.111278 [Epub ahead of print]

PMCID: PMC9477213

PMID: 36113624

Asami Anzai and Hiroshi Nishiura^⁎

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Abstract

The concept of doubling time has beenincreasingly used since the onset of the coronavirus disease 2019(COVID-19) pandemic, but its characteristics are not well understood,especially as applied to infectious disease epidemiology. The presentstudy aims to be a practical guide to monitoring the doubling time ofinfectious diseases. Via simulation exercise, we clarify theepidemiological characteristics of doubling time, allowing possibleinterpretations. We show that the commonly believed relationship betweenthe doubling time and intrinsic growth rate in population ecology doesnot strictly apply to infectious diseases, and derive the correctrelationship between the two. We examined the impact of varying (i) thegrowth rate, (ii) the starting point of counting cumulative number ofcases, and (iii) the length of observation on statistical estimation ofdoubling time. It was difficult to recover values of growth rate fromdoubling time, especially when the growth rate was small. Starting timeperiod is critical when the statistical estimation of doubling timeoccurs during the course of an epidemic. The length of observation wascritical in determining the overall magnitude of doubling time, and whenonly the latest 1–2 weeks’ data were used, the resulting doubling timewas very short, regardless of the intrinsic growth rater. We suggest that doubling time estimates ofinfectious disease epidemics should at a minimum be accompanied bydescriptions of (i) the starting time at which the cumulative count isinitiated and (ii) the length of observation.

Keywords: exponential growth, intrinsic growth rate, mathematical model, severe acute respiratory syndrome coronavirus-2(SARS-CoV-2), transmission dynamics

1. Introduction

Doubling time, the time it takes for a numberof individuals to double, is classically used in the field of populationecology. Nowadays, doubling time is also used in the field of infectiousdisease epidemiology as a measurement of the spread of disease,representing the time required for the cumulative number of infections todouble during the course of an epidemic (University of Cambridge, 2021;Vynnycky and White, 2010). Usually, doubling time is estimated using thetime series data of (infected) individuals or from the growth rate ofindividuals, and the growth rate usually represents the rate at whichindividuals increase per unit time, i.e., a nearly opposing concept ofdoubling time (Gotelli, J, 2008). As a more widely accepted measurementof transmissibility of infectious diseases, the basic reproductionnumber, which is interpreted as the average number of secondary casesproduced by a single primary case in a fully susceptible population, is awell-defined dimensionless quantity. Nevertheless, the basic reproductionnumber requires us to know the length of the generation time in advanceof the estimation, especially to attain a real-time assessment. Theimportance of estimating an alternative epidemiological metric to assessthe transmissibility or the speed of growth of cases in the early stagesof the COVID-19 pandemic has been emphasized (Thompson et al., 2000).When the generation time has yet to be fully quantified, the speed ofepidemic growth must be measured in real time (Dushoff and Park, 2021, Ridenhour et al., 2014), and doubling time, T_d, is one of the available alternativemetrics, and intrinsic growth rate, r, is anotherpossible choice.

$C (t) = C (0) 2^{\frac{t}{T_{d}}},$

(1)

In the field of ecology, the relationshipbetween T_d and r has beenunderstood to be simple. LetC(t) be the populationsize at calendar time t, andC(0) be the initial value of the population. Asfor the concept of doubling, we have

and for the exponentially growing phase, we have

$C (t) = C (0) e x p (r t) .$

(2)

The right-hand sides of (1), (2) areequated, and we then obtain the relationship

$T_{d} = \frac{l n 2}{r},$

(3)

which is well known in the field of ecology(Gotelli, J, 2008). Nevertheless, the equation (2) is not strictly the case for thecumulative number of infectious diseases, and the estimation ofT_d using (3) is not applicable (seeMethods).

The defining relationship betweenT_d and r has not beenwell formulated, nor have statistical methods to measure doubling time.Doubling time is estimated using the cumulative number of cases overtime, e.g., by using confirmed cases, and can therefore be used to detectrapid increases in the number of cases. In particular, it is attractivethat T_d can be measured even when intrinsiccharacteristics of an infectious disease, such as the generation time orincubation period, remain unknown (Pellis et al., 2021). For instance, during theepidemic of severe acute respiratory syndrome (SARS) from 2002–3,time-dependent changes in doubling time value were noted, and possiblefactors affecting doubling time have been discussed (Galvani et al., 2003).Doubling time has been also used for monitoring the epidemiologicaldynamics of coronavirus disease-2019 (COVID-19). In China, where COVID-19was first widespread, the doubling time in the early stages of theepidemic was estimated to be 1.4–3.1 days by province (Muniz-Rodriguez et al., 2020).Following China, the epidemic was seen in Italy (Remuzzi and Remuzzi, 2020; World HealthOrganization, 2020), and the initial doubling time in Italy was estimatedto be 3 days (Riccardo et al.,2020). Doubling time was also employed to measure thespread of the Omicron variant (B.1.1.529) in the United Kingdom duringits early phases (UK Health Security Agency, 2021). In addition tomonitoring the epidemiological dynamics, some studies have used doublingtime to evaluate interventions against the epidemic (Khosrawipour et al., 2020, Liang et al., 2021). In sum, in situations where an epidemicgrows rapidly, doubling time has been used to describe the rapidity ofincrease, and such an exercise has preceded an accurate statisticalestimation of the basic (or effective) reproduction number; additionally,no technical discussion took place as to (i) when and how to startcounting the cumulative number of cases, (ii) during which epidemicphases the measurement would be deemed useful, or (iii) how to interpretthe estimate of doubling time.

Doubling time has been increasingly usedsince the COVID-19 epidemic began, but its characteristics are not wellunderstood, especially in its application to infectious diseaseepidemiology. As the measurement relies on the cumulative number ofcases, it is vital to understand when and for how long the cumulativecount should be taken. Moreover, even provided that a given doubling timeis 3 days, we have not firmly understood how to interpret such a value.Considering that doubling time is easy to compute, this measurement willlikely continue to be employed as part of epidemiological monitoring. Itis vital to understand the epidemiological characteristics of doublingtime in advance of such use.

2.1. Modelling doubling time

Here we describe the analyticalrelationship. Doubling time T_d is estimated using the cumulative numberof infected cases at time t,C(t), i.e.,

$C (t) = C (0) 2^{\frac{t}{T_{d}}},$

(4)

where C(0) is theinitial value. It should be noted that C(0)cannot be zero forC(t)>0 fort >0, and thus doubling time does notassume that cases are counted from the very beginning of anepidemic.

Next, in the case of infectious diseases,not the cumulative number but the incidence of infection growsexponentially with the intrinsic growth rate r.The number of newly infected cases at time t,i(t), is expressed by

$i (t) = i (0) e x p (r t) .$

(5)

where i(0) is theinitial value. Taking the integral from time 0 to timet, we obtain the cumulative number ofinfected individuals,I(t):

$I (t) = \int_{0}^{t} i (s) d s = \frac{i (0)}{r} (\exp (r t) - 1) .$

(6)

It should be noted thatC(t) andI(t) cannot beimmediately equated; there must be a clock zero to start countingC(t), the cumulativenumber of cases, say t₀. At time t₀, we have the relationship:

$C (0) = i (t_{0}) = i (0) \exp (r t_{0}) .$

(7)

That is, the cumulative counting starts att₀ and C(0) is thesame as the incidence at time t₀,i(t₀). In particular, we assume thatC(0) was seen overt∈[t₀-1, t₀]. From the time t₀, the cumulative number of cases at timet₀+ $τ$ , following equation (4), is

$C (τ) = C (0) 2^{\frac{τ}{T_{d}}} = i (0) e x p (r t_{0}) 2^{\frac{τ}{T_{d}}} .$

(8)

It should be noted thatτ is the time elapsed since the start ofcounting the cumulative number of cases. As for the cumulative numberof cases following equation (6), we have

$I (t_{0}) = \frac{i (0)}{r} (\exp (r t_{0}) - 1),$

(9)

as the summation from time 0 tot₀. What corresponds to the quantity in theright-hand side of equation (8) is the integral ofi(t) from timet₀-1 to t₀+ $τ$ , thus,

$C (τ) = I (t_{0} + τ) - I (t_{0} - 1) .$

(10)

Notably, C(0) inequation (7) wasdealt with as the discrete quantity, and therefore the integral fromt₀ to t₀+ $τ$ was calculated as the difference between $I (t_{0} - 1)$ and $I (t_{0} + τ)$ . From equation (10), we obtain

$i (0) \exp (r t_{0}) 2^{\frac{τ}{T_{d}}} = \frac{i (0)}{r} (\exp (r {(t}_{0} + τ)) - 1) - \frac{i (0)}{r} (\exp (r {(t}_{0} - 1)) - 1),$

(11)

which can be reduced to

$2^{\frac{τ}{T_{d}}} = \frac{\exp (r τ) - \exp (- r)}{r},$

(12)

In the end, we obtain

2.2. Computational exercise

While the abovementioned equation(13) providesthe analytical relationship between T_d and r, thefinding is restricted to exponential growth phase, and moreover, wehave not secured a stable interpretation of T_d. What if the starting time to countcumulative number of cases is varied? What if the length ofobservation τ is very short? We tackle thesepoints via numerical simulations.

There have been several different methodsfor estimating the doubling time. The first is to use equation(3). As wediscuss with equation (13), equation (3) may not be directly applicable toinfectious disease. The second method is to use two data points:supposing that the cumulative number of cases at timet₁ and time t₂ wereC(t₁) andC(t₂), respectively, the doubling time iscalculated as

$T_{d} = (t_{2} - t_{1}) \frac{l n 2}{l n \frac{C (t_{2})}{C (t_{1})} .}$

(14)

The equation (14) is preferred when the calculationmust be kept simple (e.g., when using a spreadsheet program).Nevertheless, this method forces us to select two specific time pointsarbitrarily, and such choice leads to biased estimation when thenumber of cases remains small and stochasticity cannot be ignored. Thethird method is to fit equation (4), i.e., the equation of the cumulative number ofcases, to the empirical data. This method uses an additional number ofdata points and can thereby avoid potential inflation ofT_d. Here we estimated doubling timeT_d from simulated epidemic data usingequation (4),assuming that the variations in cases are captured by Gaussiandistribution and employing a maximum likelihood method. In the presentstudy, doubling time is defined as the time that the cumulative numberof cases from the estimated start time doubles. The doubling timewould always take a positive value, because the cumulative number ofcases increases even when the growth rate is negative and theincidence is decreasing.

2.2.1. Varying interpretations duringexponential growth

First, we explored howT_d varies by varying (i) the growthrate, (ii) the starting point, and (iii) the time period used inthe calculation on the doubling time (i.e., the length ofobservation τ) during the exponential growthphase. Consider an epidemic in which the number of new infectedcases i(t) at time $t$ is described by $i (t) = i (0) e^{rt}$ . The initial value is not influential and is thereforeset at $i (0) = 1$ . Then, we varied $r f r o m - 0.1, - 0.01, 0, 0.01 t o 0.1$ per day, and the length of observation $τ$ was also varied from 0 to 50 days. We examined how theestimate of T_d differed when doubling time wascalculated for each given growth rate for different periods oftime, and when doubling time was calculated for the latest 7, 14,and 21 days. In the case of an infectious disease epidemic with asingle peak, all datasets from day zero would be used to calculatethe doubling time. However, in the case of COVID-19 with multipleepidemic waves, it has been practically the case that only thedatasets of the latest few weeks were subject for analysis. Theintrinsic growth rate was calculated from the doubling time usingequation (13), and we examined how well the estimateddoubling time reflected the growth of cases.

2.2.2. Doubling time during the course of anepidemic

Second, we conducted simulations usingthe susceptible-infectious-recovered/removed (SIR) model written byordinary differential equations :

$\frac{dS (t)}{dt} = - β S (t) I (t),$

$\frac{dI (t)}{dt} = β S (t) I (t) - γ I (t),$

$\frac{dR (t)}{dt} = γ I (t),$

(15)

where β is thetransmission coefficient and γ is the rateof recovery. Using the numerical solution of the incidence fromthis model, we examined how the estimates ofT_d varied depending on how the startingpoint was taken during the course of epidemic (e.g., during theincreasing phase, near the peak, or during the decreasing phase)and what period of the epidemic we used as the length ofobservation $τ$ . When the SIR model was employed, we performedsimulations with $r = 0.1$ and the mean generation time $T_{g} = 5$ days (assumed to be identical to the mean infectiousperiod), thus in the SIR model written by three ordinarydifferential equations, $R_{0} = 1 + r T_{g} = 2$ . Similar to the simulation with the exponential growth,T_d was estimated by varying (i) thestarting point and (ii) the time period used in the calculation ofthe doubling time.

3. Results

Figure 1 shows the effect of exponential growthrate on the statistical estimate of doubling time during the exponentialgrowth phase. The simulation was conducted for a total of 50 days, andthe doubling time was measured in four different ways, i.e., using datafrom time 0 to time 50 (entire period), using the data for the latest 7days (Day 44–50), the latest 14 days (Day 37–50), and the latest 21 days(Day 30–50). In the all-50-days case, T_d was estimated to range from 4.4 to 12.5days depending on the growth rate. Similarly, when the latest 7-day datawere used, T_d was estimated to range from 1.7 to 2.1days; when the latest 14-day or 21-day data were used,T_d was estimated to range from 2.5 to 3.5 daysand 3.1 to 5.5 days, respectively. That is, as the length of observation $τ$ was shortened, the resulting doubling time was estimated tobe shorter. Even when r was negative (i.e., in thedecreasing phase), values of T_d close to other growth rates were obtained.Especially when the latest 7-day or 14-day data only were used, theresulting doubling time values were close to each other and tended to bevery short (i.e., on the order of a few days).

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Fig. 1

Impact of varying growth rates and length of observation ondoubling time. Estimates of doubling time by differentexponential growth rates. The horizontal axis represents the length ofobservation used for estimation of doubling time. The simulation wasconducted for 50 days, and the doubling time was measured in fourdifferent ways, i.e., using data from time 0 to time 50 (all period),using the data for the latest 7 days (Day 44–50), the latest 14 days (Day37–50), and the latest 21 days (Day 30–50).

Table 1 summarizes the estimate of doubling timeduring the exponential growth phase from a different angle fromFigure 1. InTable 1, thesimulation was conducted for the total of 60 days, and thus, using thedata for the latest 7 days, 14 days and 21 days represent Days 54–60,Days 47–60 and Days 40–60, respectively. Qualitatively, similar patternsto Figure 1 wereobserved. In addition to Figure1, neither the estimate and the uncertainty bounds(i.e., 95% confidence intervals) were very sensitive to the time at whichestimation was conducted. That is, the results of estimation using 0–20days, 0–40 days and 0–60 days were not very variable. When the length ofobservation was short (e.g., 7 days), the impact of variations inr was again minimal.

Table 1

Estimates of doubling time duringexponential growth phase.

Period used to estimate doubling time	All observed days used (60 days)	Number of recent days used
Period used to estimate doubling time	All observed days used (60 days)	7 days	14 days	21 days
	Growth rate = -0.01
0 – 20		2.04 (1.85,2.33)	3.23 (3.02,3.52)	4.31 (4.08,4.63)
0 – 40	7.18 (6.89,7.54)	2.04 (1.85,2.32)	3.22 (3.02,3.52)	4.32 (4.09,4.63)
0 – 60	9.88 (9.54,10.30)	2.03 (1.85,2.32)	3.24 (3.03,3.53)	4.31 (4.08,4.63)
	Growth rate = 0.01
0 – 20		1.98 (1.81,2.25)	3.09 (2.90,3.35)	4.08 (3.88,4.34)
0 – 40	6.55 (6.34,6.82)	1.98 (1.81,2.25)	3.09 (2.91,3.35)	4.08 (3.88,4.34)
0 – 60	8.74 (8.51,9.01)	1.99 (1.81,2.25)	3.10 (2.91,3.35)	4.08 (3.88,4.34)
	Growth rate = 0.1
0 – 20		1.76 (1.63,1.94)	2.55 (2.44,2.69)	3.15 (3.06,3.26)
0 – 40	4.27 (4.22,4.32)	1.76 (1.63,1.94)	2.55 (2.45,2.69)	3.15 (3.06,3.26)
0 – 60	4.91 (4.88,4.93)	1.76 (1.63,1.94)	2.55 (2.45,2.69)	3.15 (3.06,3.26)

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Numbers in parenthesis represent 95%confidence intervals as computed via the profile likelihood method.During the maximum likelihood estimation, daily cases were assumed tofollow Gaussian distribution.

Figure 2 shows estimates ofT_d during the course of an epidemic. As wasexamined for the exponential growth model, we varied starting time tocount the cumulative number of cases and the length of observation duringthe course of epidemic as described by the SIR model. Doubling timeduring the increasing phase (from the beginning of the epidemic to thepeak, i.e., Day 22 to 42) yielded the shortest estimate, and the valuecalculated after the peak (Day 49–75) yielded the longestT_d. As the length of observation was shortenedfrom 21 days to 7 days, T_d became shorter.

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Fig. 2

Impact of varying period of observation and length ofobservation on doubling time during the course of anepidemic. The doubling time T_d was estimated during the course of anepidemic, simulated by the SIR model, varying the period of observationand the length of observation. Doubling times were estimated in sixdifferent time periods, including those at the start of the exponentialphase, near the epidemic peak, and after the peak of the epidemic. Thepeak incidence was observed on Day 42.

Table 2 shows the estimated doubling time duringthe course of an epidemic using the SIR model. Using the nonlinearepidemic model, Table2 shows the impact of involving datasets duringsub-exponential and decreasing phases on the estimate ofT_d, which can practically take place. Thelonger the time period used for estimation of T_d, the longer the doubling time estimateT_d would be. This is theoreticallyunderstandable because the highest growth rate is attained during theinitial exponential growth phase, and the growth rate monotonicallydecreases subsequently. Using shorter lengths of observation, shorterestimates of T_d were obtained.

Table 2

Doubling time by the time period andlength of observation during the course of an epidemic based on SIRepidemic model.

Period used to estimate doubling time	All observed days used	Number of recent days used
Period used to estimate doubling time	All observed days used	7 days	14 days	21 days
0 – 25	3.04 (3.02,3.06)	1.55 (1.46,1.67)	2.09 (2.03,2.15)	2.40 (2.37,2.44)
0 – 30	3.13 (3.12,3.15)	1.58 (1.48,1.71)	2.11 (2.05,2.18)	2.43 (2.39,2.47)
0 – 35	3.23 (3.22,3.25)	1.64 (1.53,1.79)	2.18 (2.11,2.27)	2.49 (2.45,2.54)
0 – 40	3.37 (3.35,3.40)	1.77 (1.63,1.97)	2.33 (2.24,2.46)	2.63 (2.57,2.70)
0 – 45	3.57 (3.54,3.61)	1.97 (1.79,2.26)	2.62 (2.48,2.82)	2.89 (2.80,3.02)
0 – 50	3.82 (3.78,3.88)	2.19 (1.96,2.56)	3.06 (2.85,3.36)	3.34 (3.18,3.55)
0 – 55	4.12 (4.05,4.19)	2.32 (2.06,2.74)	3.54 (3.25,3.97)	3.98 (3.74,4.31)
0 – 60	4.43 (4.36,4.53)	2.37 (2.11,2.81)	3.87 (3.54,4.39)	4.72 (4.39,5.22)
0 – 65		2.38 (2.11,2.83)	4.02 (3.67,4.58)	5.34 (4.92,5.96)
0 – 70		2.38 (2.11,2.82)	4.06 (3.70,4.63)	5.66 (5.20,6.34)

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Table 3 shows the results of calculating theintrinsic (exponential) growth rate from the estimated doubling timeusing equation (13), exploring whether the growth rate can be recovered.The uncertainty bound of r informs the uncertaintyof T_d. This was examined only during theexponential growth phase (for the total of 75 days). Notably, the growthrate was not successfully recovered when the length of observation wasshort or when the growth rate was small. Especially in cases of negativegrowth rate it became too difficult to recover the growth rate fromdoubling time. When r>0, a longer time periodof observation was required to obtain a value close to the originalexponential growth rate. Figure3compares equations (3), (13),attempting to recover the estimate of the intrinsic growth rate (set at0.01 per day) from doubling time. The estimated intrinsic growth rateusing equation (13)was closer to the original value compared with those recovered from thewidely used estimator (3).

4 Discussion

Table 3

Recovery of exponential growth rateby the time period and length of observation during the course of anepidemic using equation (13).

Period used to estimate doubling time	Exponential growth rate (/day)
Period used to estimate doubling time	-0.1	-0.01	0	0.01	0.1	0.15
Most recent 7days(69 - 75)	0.000991	0.0922	0.102	0.113	0.193	0.243
Most recent 14days(62 - 75)	-0.0356	0.0429	0.052	0.0612	0.141	0.186
Most recent 21days(55 - 75)	-0.0475	0.0278	0.035	0.0438	0.124	0.17
0 – 25	-0.052	0.0209	0.0291	0.0372	0.117	0.164
0 – 30	-0.0548	0.0161	0.0244	0.0331	0.113	0.161
0 – 35	-0.0568	0.0131	0.0214	0.0297	0.11	0.158
0 – 40	-0.0582	0.0106	0.0189	0.0273	0.108	0.156
0 – 45	-0.059	0.00885	0.0169	0.0253	0.107	0.155
0 – 50	-0.0598	0.00729	0.0155	0.0238	0.106	0.154
0 – 55	-0.0603	0.00606	0.0141	0.0225	0.105	0.154
0 – 60	-0.0607	0.00495	0.0131	0.0213	0.104	0.154
0 – 65	-0.061	0.00408	0.0121	0.0204	0.104	0.153
0 – 70	-0.0612	0.00329	0.0113	0.0197	0.103	0.153
0 – 75	-0.0614	0.00257	0.0107	0.0189	0.103	0.153

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Fig. 3

Recovery of intrinsic growth ratefrom doubling time. The intrinsic growth rate rwas calculated using two types of methods (equations (3), (13) in themain text, represented by unfilled and filled bars) using doubling time.The growth rate was assumed to be 0.01 per day (horizontal dashedline).

The present study characterized the doublingtime of infectious diseases, frequently used as an epidemiologicalmeasurement to quantify the speed of epidemic growth. Through a shortanalytical exercise, we have shown that the commonly believedrelationship in population ecology (i.e., equation (3)) is not strictly the casefor infectious diseases, and instead equation (13) should be used to describe therelationship between T_d and r. In addition,we examined the impact of varying (i) the growth rate, (ii) the startingpoint of counting the cumulative number of cases, and (iii) the length ofobservation $τ$ on the statistical estimate of T_d. The growth rate did not easily recoverfrom T_d, especially when rwas small (e.g., negative). The starting time period is critical when thestatistical estimation of T_d is undertaken during the course of anepidemic. The length of observation $τ$ was critical in determining the overall magnitude ofT_d, and when only the data from the latest 1–2weeks were used, the resulting T_d appeared to be very short, regardless ofthe intrinsic growth rate r. Without accountingfor these findings, our simulations indicated that it was fairlydifficult to objectively interpret empirically estimated values ofT_d in the epidemiology of infectious diseases.Compared with T_d, the intrinsic growth rate may be regardedas a less biased metric to describe the increase and decrease in theincidence.

To our knowledge, the present study is thefirst to have derived the relationship between T_d and r in epidemicdata and to have clarified that T_d estimated from an identicalr value may yield a completely different valueif the empirical settings of observation are varied. These findings stemfrom the fact that doubling time is not calculated in a single uniqueway. If these issues persist, it is very difficult for an epidemiologistto judge whether a T_d estimate of 2–3 days is an alarming signalof epidemic growth.

To resolve this issue, we propose somefriendly guidance for estimating T_d. Three tips from our exercise cancontribute to it: First, whenever doubling time is presented, thestarting time at which the cumulative count is initiated must bedescribed (and the interpretation should take particular care on thispoint). It must also be remembered that the use of data from the latest1–2 weeks alone tends to yield very short T_d estimates, as shown in Fig. 1, Fig. 2.Second, the optimal length of time to assess T_d should be discussed in relation to theintrinsic transmission dynamics (or the natural history) of an infectiousdisease. Table 1indicates that T_d could be extended, if the entire epidemiccurve is used and the length of observation is extended. Third, it isvery difficult to translate the doubling time value to the intrinsicgrowth rate, especially when the growth rate is small. Of course, thelarge intrinsic growth rate still has a potential to be recovered fromthe doubling time, as shown in Table 3, and therefore the estimation ofr from T_d would be still sound when the actual growthof cases is fairly fast. In summary, we suggest that anyT_d estimates of an infectious disease epidemicshould at a minimum be accompanied by descriptions of the time at whichthe cumulative count is initiated and the length ofobservation.

Whereas we have shown that calculatingdoubling time can result in considerably different values by varyingempirical estimation settings, T_d is very easy to calculate for monitoringinfectious disease epidemics, especially when the natural history of thedisease, including the generation time, is yet to be known. However,because of this easy-to-calculate feature, it is necessary to clearlyspecify the settings when using doubling time. Perhaps these pointsraised can be better understood if we imagine measuring the growth ofcases by exponential growth rate (rather than T_d): we would have to specify (i) for whattime we have used the data and (ii) how long the exponential growth ratewas assumed to continue. The same applies to T_d. Of course, it is still useful tocontinuously monitor the same geographic area with a rapid increase incases with a T_d of 2–3 days (as with COVID-19) to measurethe increase using reasonable calculations. However, once the increase inthe number of cases slows down, the use of T_d becomes complicated:T_d estimates based on long-term data may notadequately represent the epidemiological situation, and therefore onlythe latest data may be used, but the use of the latest 1–2 weeks’ dataalone tends to result in smaller T_d estimates than those based on longerobservation.

There are number of technical limitationsthat should be discussed. First, our study rests on a simulation studywith limited parameter space, and it should be noted that the extent ofbias depends on intrinsic transmission dynamics. Second, we have not beenable to explicitly account for heterogeneity in measuring the doublingtime. Alternative metrics other than a single growth rate or a singledoubling time would be merited when the incidence is structured (e.g., byage group). Third, doubling time depends on cases, and when enormousgrowth is observed, the empirical data are influenced by ascertainmentbias; e.g., once an epidemic is recognized, ascertainment of casesabruptly improves and the growth rate may be estimated as very large(i.e., doubling time is estimated as very short) because of this improvedascertainment over time. Without additional data, ascertainment biascannot be addressed.

5 Conclusions

The present study characterized the doublingtime of infectious diseases, frequently used as an epidemiologicalmeasurement to quantify the speed of epidemic growth. Cautions must beexercised when handling empirical data and providing estimate of arepresentative epidemiological metric. Equation (13) should be used to describe therelationship between T_d and r for infectiousdiseases. Doubling time should be interpreted with caution, especiallywhen estimated using data from the most recent few weeks. We suggest thatT_d estimates of infectious disease epidemicshould be accompanied at a minimum by descriptions of the starting timeat which cumulative count is initiated and the length ofobservation.

Uncited references

Gotelli and N., , 2008, Thompson et al., 2020, xxxx, Vynnycky and White, 2010, World Health Organization, 2020.

Declaration of CompetingInterest

The authors declare that they have no knowncompeting financial interests or personal relationships that could haveappeared to influence the work reported in this paper.

Acknowledgments

Acknowledgments

A.A. received funding from the Kyoto UniversityMedical Student Support-Fund and JSPS KAKENHI (22J14304). H.N. receivedfunding from Health and Labour Sciences Research Grants (20CA2024, 20HA2007,21HB1002, and 21HA2016), the Japan Agency for Medical Research andDevelopment (JP20fk0108140 and JP20fk0108535), the JSPS KAKENHI (21H03198),the Environment Research and Technology Development Fund (JPMEERF20S11804)of the Environmental Restoration and Conservation Agency of Japan, the JapanScience and Technology Agency CREST program (JPMJCR1413), and the SICORPprogram (JPMJSC20U3 and JPMJSC2105). We thank the local governments, publichealth centers, and institutes for surveillance, laboratory testing,epidemiological investigations, and data collection. We also thank JohnDaniel from Edanz (https://jp.edanz.com/ac) for editing a draft of thismanuscript. The funders had no role in the study design, data collection andanalysis, decision to publish, or preparation of the manuscript.

Data availability

No data was used for the research described inthe article.

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