Improving the Estimation of the COVID-19 Effective Reproduction Number using Nowcasting
IImproving the Estimation of the COVID-19 Basic ReproductionNumber using Nowcasting
Joaqu´ın Salas ∗ Abstract
As the interactions between people increases, the impending menace of COVID-19 outbreaksmaterialize, and there is an inclination to apply lockdowns. In this context, it is essential tohave easy-to-use indicators for people to use as a reference. The basic reproduction number ofconfirmed positives, R t , fulfill such a role. This document proposes a data-driven approach tonowcast R t based on previous observations’ statistical behavior. As more information arrives,the method naturally becomes more precise about the final count of confirmed positives. Ourmethod’s strength is that it is based on the self-reported onset of symptoms, in contrast to othermethods that use the daily report’s count to infer this quantity. We show that our approachmay be the foundation for determining useful epidemy tracking indicators. After a period of confinement due to the presence of COVID-19 and facing economic and socialpressures, societies start to open up, seeking to return to productive, sport, and recreational activi-ties. As the interactions between people increase, the impending menace of outbreaks materializes.Naturally, there is a tendency to apply once again lockdowns, in what has been called the hammerand the dance [22]. In this context, it is essential to have easy to apply indicators for people to useas a reference. The basic reproduction number, R t , fulfill such a role [15]. When R t is higher thanone, the number of infected people grows exponentially, i.e. , their number will double in a shortperiod. When R t is less than one, the epidemic will tend to disappear. However, estimating R t accurately at the required level of geospatial resolution is a complex problem.Although applicable to any country, let us take the case of Mexico, as an example. The recordsgenerated by the epidemiological surveillance system contain information that includes, amongother predictors, the number of confirmed positives, deaths, and suspects. Daily, the Ministryof Health informs the public about the status of its records [26]. However, the data it disclosesupdates records of events that occurred in the past, sometimes as far as 50 or 60 days ago. Atother times, with a significant frequency, the records that were previously released are discarded.Although publishers often drop these erroneous entries overnight, there have been cases of recordseliminated after more than 50 days.Besides the integrity of the information, there are other difficulties in tracking the epidemyinherent to the pandemic and interesting for researchers, decision-makers, and the general public.SARS-CoV-2 is an airborne virus [4], which infects some people without causing symptoms [19].On a significant number of occasions, people begin to spread COVID-19 before they start to feelsick [13]. Also, each infected person reacts differently and will have, if anything, a different latency ∗ We thank Carlo Tomasi, Duke University, for providing the fundamental idea for our approach. Send yourcorrespondence to Joaquin Salas, [email protected], Instituto Polit´ecnico Nacional a r X i v : . [ q - b i o . P E ] J u l nd incubation period [12]. People will have a different contagious period, manifested with inequalintensity during that time [7]. Although the symptoms are known, one may reveal them differently.People will require different types of medical attention, which may or may not require hospital-ization [11]. In some cases, someone ill may need or not a ventilator [18]. Eventually, a givenperson may recover, possibly with sequels, or will pass away [24]. About the whole process, webegin to have some statistical knowledge on which we can develop models. In this paper, propose adata-driven approach that leverage experience to create a simple, yet effective nowcasting methodfor R t that can be used by policy-makers as well by the general public. Our main contribution isan approach to use past observations to generate plausible sequences of estimates for the number ofconfirmed positive cases that could have possibly occurred in the recent days to compute variationsof the basic reproduction number.We base our method on the statistical behavior of previous observations. As more informationarrives, the estimation naturally becomes more precise about the final count of confirmed positives.In the next section, we review the literature about related methods. Then, in §
3, we discuss theintrinsic delay in information flow that exists in the process of detecting a COVID-19 confirmedpositive and detail our approach to estimate plausible sequences for the number of infected people.In §
4, we review the underlying method we employ to estimate the basic reproduction number outof the possible sequences. We show our implementation of the nowcasting method for R t in § Though recent, COVID-19 has kickstarted some novel ideas to track it reliably. The researcheffort to nowcast the basic reproduction number can be classified in either mechanistic approaches,Bayesian approaches, or a hybrid combination of both.
Wang et al. [27] developed a hybrid model to complement the dynamics of the SIR (Suspected,Infectious, Recovered) model with spatiotemporal analysis. The space-time component is modeled,at the start, with a Poisson distribution to describe rare events. Then, they complemented it witha negative binomial random model during over-dispersion. Balabdaoui and Mohr [5] propose anage-stratified discrete compartment model as an alternative to SIR type models. Their approachfollows the trajectory of individuals that includes the exposed, the asymptomatic, the symptomaticinfectious, the symptomatic in self-isolation, the patients in the intermediate care unit, and thepatients in the intensive care unit. Masjedi et al. [16] compares phenomenologic and mechanisticmodels. The former based on generalized Richards models [23] (an extension of sigmoid func-tions) and the latter on a modified SEIR (Susceptible, Exposed, Infectious, and Recovered) model.They fit the models with observed data to forecast the next month. They observe that althoughphenomenologic models fit the data, they are not reliable for decision-making. In contrast, SEIRmodels predicted the phenomena better. Contaldi [9] presents SIRFH, an extension of the SIRmodel that tracks hospitalizations and hospital-based fatalities introducing additional differentialequations. The estimation for the basic reproduction number derives from the solution to this ex-tended model. Finally, Annan and Hargreaves [3] produce a nowcasting method based on the SEIRmodel. To calibrate the parameters, they use observational data and a Bayesian approach. Annanand Hargreaves’ analysis includes the uncertainties associated with deaths’ stochastic nature, thereporting errors, and the model itself. 2a) (b)Figure 1: Delays in reporting. The vertical axis shows the day of onset. The vertical axis indicatesthe number of patients confirmed positives. Each layer is an update to registers in the past.
Altmejd et al. [2] present a model based on the removal method [21], where one extracts batches ofa fixed population. Their models deal with lags arising from the calendar patterns, where eventsreported during the weekends are less. Their Bayesian approach uses a likelihood that considers thenumber of reports by day of the week, and priors with improper uniform distribution. Their modelprovides better estimates than seven days averages. Schneble et al. [25] present a nowcasting modelbased on the number of deaths, as quantifying their correct number is more reliable than for infectedpeople. Their epidemic spread model considers region and age-specific Poisson distributions, wherethey consider lag to report. They model the effect of age, gender, weekday, and location as a quasiPoisson distribution. Then, they infer a posterior using a Gaussian prior. For nowcasting, theymodel the delay as a random variable which will provide death counts. They distribute these deathcounts as a quasi-binomial distribution. Chitwood et al. [8] propose to use a Bayesian frameworkfor nowcasting. They take into account delayed and incomplete reporting. They assume that onecan understand the COVID-19 complex spread system by examining the individual components.In that model, they consider the uncertainty that results from available diagnosis and delays inthe estimation of disease progression and reporting systems. Lastly, Abbot et al. [1] employ aquasipoisson regression model to estimate the spread rate. Interestingly, they base their analysison the reported dates for the confirmed positives and infer the symptom onset through statisticalmodeling.
In our approach, we characterize the frequency at which the counting updates of COVID-19 con-firmed positives occur. In this section, we analyze the origin of such delays and describe the formwe model them.
Declaring a person confirmed positive involves a complex process that may take days, even nowa-days, when it is of paramount importance to achieve certainty for decision-making. Just considerthe case of a person showing symptoms related to COVID-19 [17] that decides to visit the physician.3igure 2: Confirmed positives. (a) As the days pass, updates eventually level off to a final count fora given day C t ( D ). (b) and (c) show the normalized daily number of reported confirmed positiveand the accumulated number of cases. Our method relies on the assumption that it is possible tomodel the daily variations with a data distribution.After an interview to collect some necessary clinic information, the physician chooses to take eithera sample from the nasopharynx using a long swab [20] or a CT (Computer Tomography) [14]. Insome places, the sample can be analyzed via the RT-PCR(reverse-transcription polymerase chainreaction) [28] in situ with results on the same day but frequently it may take a week or longer tobe processed. Afterward, the results will be uploaded in computer systems and summarized foranalysis.In Figure 1, we illustrate the effect of delays in reporting using the data set made public bythe Mexican Health Ministery [26]. The horizontal and vertical axes show the day of onset and thenumber of confirmed positive cases. Each layer corresponds to the number of cases added to a priordate. Although the number of updates may be significant for a given day, they eventually convergeto the total number of confirmed positives for that day, C t ( D ), for D large, and where t expressesthe day of interest (see Figure 2(a)). If we divide the daily accumulated of confirmed positives C t ( δ ), for a given day δ , by C t ( D ), the cumulated distribution will tend to one. We illustrate thisin Figure 2(b)-(c), where we show both, the rate of daily change and the cumulative change. Ourapproach aims to characterize the variations we observe in these distributions to develop a modelfor nowcasting. We aim to estimate the number of confirmed positive cases C t ( D ) for the day t using the following δ days of reports available. In principle, we would learn about C t ( D ) when D is cosiderably large.But in practice, D can be as short as one month and a half of daily updates. Given the number ofconfirmed positives δ days after day t , C t ( δ ), the number of confirmed positives on day C t ( δ + 1)can be expressed as C t ( δ + 1) = C t ( δ )(1 + ρ t ( δ )) , (1)where ρ t ( δ ) is the rate of change from one day δ to the next δ + 1, for reference day t . If we solvethe recursion, we will have the expression C t ( D ) = C t (0) D − (cid:89) δ =0 (1 + ρ t ( δ )) , (2)where one assumes that the daily rate changes over time. In the cases we are studying, the curvesexpressing the rate of change of the number of confirmed positive relative to the day before, for adifferent starting day, seem to be somewhat consistent over the samples. In our case, we model ρ t as4 random variable, for which we may be able to fit some standard distribution to the experimentalsamples. Then, on the day t + δ , the best-guess prediction for the number of confirmed positive, C t ( δ ), is C t ( D ) = C t ( δ )(1 + ρ Dt ( δ )) , (3)where our newly defined random variable ρ Dt ( δ ) expresses the rate of change from day δ + t to day D . In our approach, we model ρ Dt ( δ ) as a random variable with different parameters for each day δ , for more fine-grained or longer-term prediction. One may find the relationship between ρ Dt ( δ )and ρ t ( δ ) by noting that (2 and (3) solve for C ( t D ) as C t (0) D − (cid:89) δ =0 (1 + ρ t ( δ )) = C t ( δ )(1 + ρ Dt ( δ )) . (4)Expanding C t ( δ ) using the recurrence relationship in (1), we have C t (0) D − (cid:89) δ =0 (1 + ρ t ( δ )) = C t (0) δ − (cid:89) d =0 (1 + ρ t ( d ))(1 + ρ Dt ( δ )) , (5)from where, after eliminating for the common factors, solving for ρ Dt ( δ ) results in ρ Dt ( δ ) = D − (cid:89) d = δ (1 + ρ t ( d )) − . (6) R t Given a particular sequence of the observed number of infected people { C ( δ ) , C ( δ − . . . , C t (0) } ,and the argument of the number of days the report has been updated, we aim to nowcast thebasic reproduction number R t , i.e. , given the distribution of the rate of change ρ Dt ( δ ), we generateensembles of sequences aiming to estimate { C ( D ) , C ( D ) . . . , C t ( D ) } before proceeding to calculate R t . We first review EpiEstim , a method proposed by Cori et al. [10], to estimate R t from theobserved number of cases.Cori et al. [10] proposed a Bayesian framework to compute R t , where the number of infectedpeople observed at day t , C t , follows a Poisson process. In a simplification, they assume that thedaily observations of infected people are independent. Thus, one may express the likelihood ofobserving a sequence of infected people between day t − τ − t as [10] P ( C t − τ +1 , . . . , C t | C , . . . , C t − , w , R t,τ ) = t (cid:89) s = t − τ +1 ( R t,τ Λ s ) C s e − R t,τ Λ s C t ! , (7)where the transmisibility R t,τ is assumed to be constant over the period [ t − τ + 1 , t ], Λ t = (cid:80) ts =1 C t − s w s is the total infectiousness of infected people at time t , and w = ( w , . . . , w t ) T isa mass density probability profile of infectivity profile for an individual. Cori et al. [10] assumethat the basic reproduction number R t,τ is a random variable which probability follows a Gammadistribution as [10] P ( R t,τ ) = R a − t,τ Γ( a ) b a e − R t,τ /b , (8)5here a and b are the parameters of shape and scale. Since the Poisson and Gamma probabil-ity distributions are conjugate, one can express the posterior in closed form, again as a Gammadistribution, as [10] (9) P ( C t − τ +1 , . . . , C t , R t,τ | C , . . . , C t − τ , w ) ∝ R α − t,τ e R t,τ /β t (cid:89) s = t − τ +1 Λ C s s C s ! , from where the mean α and standard deviation β are given by α = a + t (cid:88) s = t − τ +1 C s and β = 1 t (cid:88) s = t − τ +1 Λ s + 1 b . (10)Given C t ( δ ), the information about the number of infected people δ days after the day of interest t , and the model for the probability function for ρ Dt ( δ ), we produce N random samples which willcorrespond to the number of people infected that day. We then compute R t for each of the sequencesusing the model proposed by Cori et al. [10]. Finally, we calculate the mean and standard deviationfor R t to provide the most likely value and uncertainty at one standard deviation. To take intoaccount the difference between the accepted values for the average incubation (five days) [12] andlatency periods (three days) [13], we represent them two days before t . We took the data set for COVID-19 cases provided by the Mexican Health Ministery correspondingto July 11, 2020. The data set contains 723,668 records, out of which 295,268 correspond toconfirmed positives. As time passes by, the number of confirmed positives for a given day t isupdated. In Figure 1, we illustrate how each day the updates stack up a layer of updated registerstoward the past. As we accumulate the number of confirmed positive updates, we observe that thetotal quantity levels off and reaches a maximum at C t ( D ) (see Figure 2). About 98% of reportsare filled out by day 33. When we divide the daily updates for the day t by C t ( D ), we obtain thenormalized updates by day and accumulated registers illustrated in Figure 2.We then proceed to fit Gamma distributions to the variation of ρ Dt ( δ ). We show illustrationsof this fit for δ = 1 , , , ,
25 and 35 in Figure 3. Note that δ = 0 is not present as generally thenumber of reported confirmed positive for C t (0) = 0 causing ρ t (0) to be undefined. Once we havethe models for ρ t ( δ ), we may proceed to generate estimates for the number of confirmed positivesfor C t ( D ) using (3). The mean and standard deviation statistics will provide us with the mostlikely value and an estimate for the uncertainty. We use the same set of randomly generated valuesto obtain sequences, which we evaluate using the method proposed by Cori et al. [10] to obtainthe instantaneous R t . Our implementation considers the pre-symptomatic transmission, i.e. , theincubation period, or the time it takes for an infected person to start showing symptoms, is greaterthan the latent period, or the time from which an infected person can spread to others. FollowingBar-On et al. [6], we assume that the latent period lasts for three days and the incubation periodfor five days.We compare the performance of our nowcasting with the proposed by Abbott et al. [1] (seeFigure 4). In their case, the nowcasting tends to closely follow the number of reported confirmedpositives, which gives the undesirable effect of resulting in a descending R t , when it is not. Ourproposal, on the other hand, increases its certainty naturally as more information is available.6a) (b) (c)(d) (e) (f)Figure 3: Distribution Fit. We fit the data to a Gamma distribution. Here, we show examples for t = 1 , , , , , δ in our nowcasting exercise, there is a tendency to observe fewer cases, becausethere has not been enough time for the information to arrive. We set a dynamic threshold tostop the nowcasting estimation when for a particular day of analysis, δ , the number of confirmedpositive cases is less than 30. Also, we have observed that as the number of confirmed positive isless, the normalized cumulative curves tend to be noisier. In Figure 5, we illustrate what happensfor entities in Mexico where the number of confirmed positive cases is 59,667 (Mexico City), 44,114(State of Mexico), 15,909 (Tabasco), and 2,667 (Quer´etaro). We believe that our method worksbest when the number of positive cases is beyond 2,600 for the observed interval of 90 days. Usingthis threshold, there are still currently 30 States (out of 32) and 32 Municipios (out of 2450) inMexico subject to our analysis.To foster further research, allowing other researchers to verify our results and serve as a step-ping stone, we make our code publicly available at . Conclusion
In this document, we have presented a nowcasting method to estimate the number of confirmedpositives. We have shown that this may be the foundation to generate plausible sequences outof which one may determine useful epidemy tracking indicators, such as the basic reproductionnumber. Our method naturally expresses uncertainty due to the lack of information but eventuallygains certainty as more data accumulates.Our method’s strength is that it is based on the self-reported onset of symptoms, in contrastto other methods that use the number of confirmed positives cases accumulated by the report’sday to infer this quantity. A potential drawback of our approach is that it relies on a regularity ofthe update cycle. As researchers implement more sophisticated systems for testing and reporting,the statistics may change. To remedy this potential effect, one may eliminate old observations and7a) Nowcasting the infected (b) R t = 1 . ± . et al. algorithm to compute R t . We fed the data to the method provided by Abbott et al. [1], which seems to follow the reported daily cases. Finally, we illustrate the output of ourapproach, including its area of uncertainty. Thanks to Dagoberto Pulido for implementing Abbott et al. [1] to generate (c).update the distributions for ρ t regularly. Due to the difference between the incubation and latentperiods, and delays in the detection and reporting cycle, our model estimates R t up to several daysin the past. We decided to take no further assumptions about the progression of the epidemy.Although potentially some form of state estimation may be possible to implement to fill the gap.We believe that it is crucial to continue developing solutions to quickly, robustly, and reliablyestimate indicators such as the basic reproduction number. A possible direction for future researchmay be to determine the disaggregation level to continue to generate a reliable indicator. The re-sulting nowcasting methods should compensate for the delays inherent in producing and processinginformation about this critical, global, and urgent problem. Also, we are planning to study theextend at which our model can be incorporated into dynamics-based models. This enhancementcould offer improved nowcasting. 8a) 1.41 ± ± ± ± R t for some states of Mexico and corresponding cumulative normalizeddistribution for Mexico City (a)-(b), Mexico State (c)-(d), Tabasco (e)-(f), and Queretaro (g)-(h). References [1] Sam Abbott, Joel Hellewell, Robin Thompson, Katharine Sherratt, Hamish Gibbs, NikosBosse, James Munday, Sophie Meakin, Emma Doughty, and June Young Chun. Estimatingthe time-varying reproduction number of SARS-CoV-2 using national and subnational casecounts.
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