key: cord-1054733-l1xgoz1z
authors: Aguiar, M.; Van-Dierdonck, J. B.; Stollenwerk, N.
title: Reproduction ratio and growth rates: measures for an unfolding pandemic
date: 2020-05-21
journal: nan
DOI: 10.1101/2020.05.18.20105528
sha: 14ec56c8a3eeab7a1be06656acf62b90b6c37298
doc_id: 1054733
cord_uid: l1xgoz1z

The initial exponential growth rate of an epidemic is an important measure that follows directly from data at hand, commonly used to infer the basic reproduction number. As the growth rates {lambda}(t) of tested positive COVID-19 cases have crossed the threshold in many countries, with negative numbers as surrogate for disease transmission deceleration, lockdowns lifting are linked to the behavior of the momentary reproduction numbers r(t), often called R0. Important to note that this concept alone can be easily misinterpreted as it is bound to many internal assumptions of the underlying model and significantly affected by the assumed recovery period. Here we present our experience, as part of the Basque Country Modeling Task Force (BMTF), in monitoring the development of the COVID-19 epidemic, by considering not only the behaviour of r(t) estimated for the new tested positive cases - significantly affected by the increased testing capacities, but also the the momentary growth rates for hospitalizations, ICU admissions, deceased and recovered cases, in assisting the Basque Health Managers and the Basque Government during the lockdown lifting measures. Two different data sets, collected and then refined during the COVID-19 responses, are used as an exercise to estimate the momentary growth rates and reproduction numbers over time in the Basque Country, and the implications of using those concepts to make decisions about easing lockdown and relaxing social distancing measures are discussed. These results are potentially helpful for task forces around the globe which are now struggling to provide real scientific advice for health managers and governments while the lockdown measures are relaxed.

responses, are used as an exercise to estimate the momentary growth rates and reproduction numbers over time in the Basque Country, and the implications of using those concepts to make decisions about easing lockdown and relaxing social distancing measures are discussed. These results are potentially helpful for task forces around the globe which are now struggling to provide real scientific advice for health managers and governments while the lockdown measures are relaxed. the number of secondary cases generated from a primary infected case during its infectiousness before recovering in a completely susceptible population [6] . Both concepts can be extended to larger compartmental models and into the phase when effects of the control measures become visible and parameters slowly change, leading to the so called momentary momentary growth rates λ(t), and momentary reproduction ratios r(t).

In the beginning of COVID-19 epidemics, the process of collecting data were often not yet well organized or pre-organized in the way that we could immediately use to feed models and extract accurate measurements for the momentary growth rates and the momentary reproduction numbers. To mitigate and suppress COVID-19 transmission, draconian intervention measures were rapidly implemented, crippling our economies as lockdowns were implemented. As research to develop an effective vaccine is ongoing, epidemiologists and public health workers are the frontline of this pandemic, focusing on the well known public health surveillance strategies of testing, isolation and contact tracing of infected COVID-19 individuals. Up to date, more than 4 million cases were confirmed with about 300 thousand deaths, and these numbers are still increasing [7] .

After several weeks of social distancing restrictions, lockdowns start now to be lifted and modeling task forces around the globe are struggling to apply the concept of r(t), often called R 0 , to decide whether social distancing relaxation decisions are taken in the right period of time,

i.e, when the outbreak is assumed to be controlled, with negative growth rates and a momentary reproduction numbers below 1. Although the absolute value of r(t) can vary, countries we should rather look at the threshold behavior as it is independent, when we use the growth rates primarily, of those modeling uncertainties and clearly indicates if the outbreak is under control or not when estimations are below or above 1. Complementary measures of growth rates for different variables such as hospitalization, intensive care unites (ICU) admissions and deceased, where data is also collected, should be evaluated when political decisions are taken.

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In this report we present the growth rates and reproduction numbers for the COVID-19 epidemic in the Basque Country, an autonomous community in northern Spain with 2.2 million inhabitants. For the reproduction number calculation we use a refined stochastic SHARUCDtype model -an extension of the well known simple SIR model that is frequently used to model different disease outbreaks [8, 9, 10] , developed within a multidisciplinary task force (so-called Basque Modelling Task Force, BMTF) created to assist the Basque Health managers and the Basque Government during the COVID-19 responses. The model is calibrated using the empirical data provided by the Basque Health Department and the Basque Health Service (Osakidetza), continually collected with specific inclusion and exclusion criteria. Able to describe well the incidences of disease for different variables of tested positive individuals, this framework is now used to monitor disease transmission, including estimations of the momentary growth rates and reproduction numbers, while the country lockdown is gradually lifted [11] .

Using two different available data sets for the Basque Country, collected from March 4 to May 9, 2020, the data was revised as variable definition for positive cases was changed in respect to the diagnostic test used, we present results obtained for the momentary growth rates and reproduction ratios during the ongoing COVID-19 epidemic in the Basque Country and discuss the implications of using those concepts during an unfolding pandemic.

For the Basque Country we use the cumulative data for the following variables defined as: i) total tested positive patients (I cum ) in yellow which are recorded in categories for ii) hospital admissions (C H ), in red, iii) intensive care units admissions (C U ) in purple, iv) recovered (C R ) in green and v) deceased (D) in black. At the beginning of the outbreak, only patients with severe symptoms admitted to a hospital were tested using the PCR (polymerase chain reaction) method. As testing capacities increased, including also antibody tests used mainly as screen-4 . CC-BY-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review)

The copyright holder for this preprint this version posted May 21, 2020. . https://doi.org/10.1101/2020.05.18.20105528 doi: medRxiv preprint ing tool in nursing homes, less severe symptomatic cases started to be tested, contributing to enhance the number of confirmed positive cases in the population. This data collection (named "Data set A") includes, for each category or variable, patients tested with both PCR and rapid antibody tests. "Data set A" has now being revised to include patiences, in all categories, who were tested positive with PCR method only (named "Data set B"). Using the data for the all positive cases, the momentary growth rates (λ) and the momentary reproduction numbers (r(t)) are calculated for both data sets, A and B, and results are compared.

We consider primarily SHARUCD model versions as stochastic processes in order to compare with the available data which are often noisy and to include population fluctuations, since at x := (x 1 , ..., x 10 ) tr , giving the dynamics for the probabilities p(x, t) as

with n = 10 different transitions w j (x), as described by the mechanisms above, and small deviation from state x as ∆x j := 1 N · r j . For the basic SHARUCD model we have explicitly the 5 . CC-BY-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review)

The copyright holder for this preprint this version posted May 21, 2020. . following transitions w j (x) and its shifting vectors r j given by

With these w j (x) and r j specified we also can express the mean field ODE system The deterministic version of the refined model is given by a differential equation system for all classes, including the recording classes of cumulative cases C H , C A , C R and C U by

Model parameters and initial conditions are shown in Table 1 , where β is the infection 6 . CC-BY-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.

The copyright holder for this preprint this version posted May 21, 2020. . rate and φ is the ratio describing the asymptomatic/mild infections contribution to the force of infection. γ is the recovery rate, µ is the disease induced death rate and ν is the ratio of hospitalized going to the ICU. η is the proportion of susceptible being infected, develop sever symptoms and being hospitalized whereas 1 − η is the proportion of susceptible becoming infected and developing mild disease or asymptomatic. ξ is the ratio of detected, via testing, mild/asymptomatic infect individuals. is the import rate needed to describe the introductory phase of the epidemics and for the present study, we assume to be much smaller than the other additive terms of the force of infection, given the strong observational insecurities on the data collected at the beginning of the outbreak.

For completeness of the system and to be able to describe the initial introductory phase of the epidemic, an import term should be also included into the force of infection. For the present study, we assume to be much smaller than the other additive terms of the force of infection, given the strong observational insecurities on the data collected at the beginning of the outbreak, when would matter most.

After an introductory phase, the epidemic entered into an exponential growth phase, which started in the Basque Country around the March 10, 2020 and due to the effects of the imposed control measures has left to a slower growth around March 27, 2020 [11] . This exponential growth phase is typical for any outbreak with disease spreading in a completely susceptible population, as observed already in the SIR-system, from the dynamics of the infected dI dt = β S N − γ · I when S(t) ≈ N , such that a linear differential equation dI dt = (β − γ) · I =: λ · I with an exponential growth factor λ is obtained. This growth factor then can be measured again from disease data via λ = 1 I · dI dt = d dt ln(I) giving a straight line in a semi-logarithmic plot of the data.

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Description

severe disease and hospitalized 54.0 A(t 0 ) mild disease and asymptomatic 80.0

proportion of mild/asymptomatic 0.4 ρ import parameter − 

now including disease induced transition to death via the mortality rate µ and transition to ICU admission with admission rate ν. For an epidemic in its initial phase, i.e. S(t) ≈ N , we now

for entries into the disease classes and is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.

The copyright holder for this preprint this version posted May 21, 2020. . https://doi.org/10.1101/2020.05.18.20105528 doi: medRxiv preprint d dt x = Jx and its solution

with matrix exponential including the eigenvalue matrix Λ and the transformation matrix T from the eigenvectors of matrix J. The eigenvalues of the matrix J are given by

with the parameter dependent trace tr = (η + φ(1 − η)) · β − (2γ + µ + ν) and determinant 

with constants K H and K A depending on parameters and initial conditions (from H(t) = K H,1 · e λ 1 (t−t 0 ) + K H,2 · e λ 2 (t−t 0 ) etc.). In the limiting case of a simple SIR-type model (with φ ≈ 1 and µ, ν γ) we obtain tr = (η + φ(1 − η)) · β − (2γ + µ + ν) ≈ β − 2γ and det = γ(γ + µ + ν) − ((γ + µ + ν)φ(1 − η) + γη) · β ≈ γ 2 − γβ and hence λ 1 ≈ β − γ and λ 2 ≈ −γ

The concept of the growth rate can be extended into the phase when effects of the control measures become visible and parameters slowly change, such that for short times the above analysis holds as for constant parameters. The momentary growth rates are analyzed below.

Another measure of the spreading of the disease in its initial phase is the basic reproduction number (R 0 ), the number of secondary cases I s from a primary case I p during its infectiveness before recovering in a completely susceptible population.

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In its simplest version for SIR models a primary case, I p (t 0 ) = 1, recovers via dIp dt = −γI p , hence I p (t) = I p (t 0 )e −γ(t−t 0 ) . The number of secondary cases from the primary case is given by dIs dt = β S N I p (t) with I s (t 0 ) = 0, a simple inhomogeneous linear differential equation in case of a entirely susceptible population S(t) = N . The solution is I s (t) = β γ · I p (t 0 ) 1 − e −γ(t−t 0 ) + I s (t 0 ) and gives the total number of secondary cases from a primary case as the long time limit as I s (t → ∞) = β γ , hence the basic reproduction number is simply R 0 = β γ . So we have the relation between R 0 and the growth rate λ here as R 0 = β γ = 1 + λ γ . Generalized, the reproduction ratio is then given by r = I s (t → ∞)/I p (t 0 ) = β γ as the ratio of secondary cases produced by primary cases during their infectiousness.

This concept can be also generalized for larger compartmental models, with the notions of matrices B and G as introduced above. For any primary cases H p or A p we have with

, using again the matrix exponential. For secondary cases H s and A s we have the dynamics of x s = (H s , A s ) tr given by d dt x s = B · x p (t) with solution analogously to the SIR case as x s (t) − x s (t 0 ) = BG −1 1| − e −G(t−t 0 ) x p (t 0 ) with F = BG −1 the next generation matrix, since x s (t → ∞) = F x p (t 0 ) or from generation x n to generation x n+1 the discrete iteration x n+1 = F cot x n . For the present case we have the next generation matrix given as

with its dominant eigenvalue for the basic SHARUCD-model

and the other one being zero. In the limiting case of a simple SIR-type model (with φ ≈ 1 and µ, ν γ) we obtain again r 1 = β/γ as can be easily seen.

This concept of the reproduction ratio can be extended into the phase when effects of the control measures become visible and parameters slowly change. The momentary reproduction 10 . CC-BY-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.

The copyright holder for this preprint this version posted May 21, 2020. . https://doi.org/10.1101/2020.05.18.20105528 doi: medRxiv preprint ratios (r) can be analyzed, as frequently done for the COVID-19 epidemics, but often called "basic reproduction number". While the momentary growth rate follows directly from the time continuous data at hand, the momentary reproduction ratio depends on the notion of a generation time γ −1 .

we use smoothing of the differences of logarithmic positive cases with τ = 7 d and ∆t = 5 d

To obtain the momentary growth rates from data directly we use λ = d dt ln(I) at first applied to the cumulative tested positive cases I cum (t) obtaining, via a smoothing window, the new cases after time τ as

and hence, the growth rate

From the growth rate, the reproduction ratio is calculated with the recovery period γ −1 obtained from our underlying model and recent literature about SARS-CoV-2 interaction with human hosts [12, 13, 14, 15, 16, 17] . Assuming the recovery period to be of 10 days, we use 7 days smoothing of the differences of logarithmic positive cases to include all possible fluctuations during the data collection process such as "weekend effects", for example, when we often observe a consistent low number of cases reported that are then adjusted shortly after. For this first exercise, both data sets show negative growth rates from April 1st, 2020, confirming a decrease in disease transmission. Nevertheless, when looking at the long term results for the "Data set A" (see Fig. 2 a-b) ), an increase of the growth rate over time is estimated, with values crossing the threshold and becoming positive from April 23 to May 1, 2020, whereas "Data set B" (see . CC-BY-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.

The copyright holder for this preprint this version posted May 21, 2020. . https://doi.org/10.1101/2020.05.18.20105528 doi: medRxiv preprint transmission. The momentary reproduction numbers follow the same trends for both data sets respectively, depending on the data set used. The observed signal from "Data set A" would significantly impact decisions on lockdown lifting, as the national plan for lifting the restrictions imposed during the state of alarm, called "Plan for the Transition towards a new normality", was announced on April 28, 2020 [18] . Taking place over 4 phases, with a gradual de-escalation to "a new normality", the plan is dependent on the on-going progress of COVID-19 epidemic's control across the different regions of Spain. However, results obtained by "Data set B" would, alternatively, support the already started lockdown lifting with its "Phase Zero" initiated on May 4, 2020. When assuming a short recovery period of γ = 4 days, see Fig. 4 , similar results are observed between the different data sets, only with variation on the absolute values.

The momentary growth rates for the various variables are also calculated to verify and support the interpretation of the estimated r(t) threshold behaviour since for any assumed recovery period γ −1 , results obtained for the various variables are the same, changing only when considering the different data sets. Figure 4 shows the behavior of three variables that are synchronized in the Basque Country, I cum , C H , and C U . They also cross the threshold to a negative growth rate on April 1st, 2020, confirming the observed r(t) trend obtained by looking at data on I cum alone. Recovered and deceased cases, shown in Fig. 4 b) , follow 1-2 weeks later, due to the delay between onset of symptoms, hospitalization and eventually death, reaching negative growth rate on April 7 and April 11, 2020, respectively. Besides the observed deviation of I, for the "Data set A", the other variables are kept below the threshold, constantly negative until May 9, 2020, supporting the political decision of starting lifting the lockdown measures rather sooner than later in time. So which measure should be considered to guide political decisions? Here, the answer is simple. When the available data is consistently collected and defined, the momentary growth rates for different variables, I, H, U , R and D, measured directly from the data at hand, should also be considered as complementary investigation.

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The copyright holder for this preprint this version posted May 21, 2020. . https://doi.org/10.1101/2020.05.18.20105528 doi: medRxiv preprint Figure 1 : Ensemble of stochastic realizations of the SHARUCD-type model. a) Cumulative tested positive cases I cum (t). From April 6, 2020, we note an increase of reported positive cases as the testing capacities were increasing. In b) cumulative hospitalized cases C H (t), c) cumulative ICU admission C U (t), d) cumulative deceased cases D(t).

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The copyright holder for this preprint this version posted May 21, 2020. 14 . CC-BY-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.

The copyright holder for this preprint this version posted May 21, 2020. 

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The copyright holder for this preprint this version posted May 21, 2020. Figure 4 : Using data on PCR + rapid tests we plot the momentary growth rates estimation from the data on positive tested infected cases (yellow), hospitalizations (red) and ICU admission (purple) are plotted in a) and recovered (green) and deceased cases (black) in b). Using data on PCR tests only we plot the momentary growth rates estimation from the data on positive tested infected cases (yellow), hospitalizations (red) and ICU admission (purple) are plotted in c) and recovered (green) and deceased cases (black) in d).

As the concept of R 0 used alone can be easily misinterpreted, specially now when testing capacity is increasing and consequently the number of new notified cases, the BMTF now monitors the development of the COVID-19 epidemic in the Basque Country by considering not only the behaviour of the momentary growth rates λ(t) and momentary reproduction numbers r(t) for the positive cases I cum (t), but also the λ(t) for hospitalizations (C H ), ICU admissions (C U ), deceased (D) and recovered cases (C R ), assisting the Basque Health Managers and the Basque Government with results that are obtained by the model framework, based on available 16 . CC-BY-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.

The copyright holder for this preprint this version posted May 21, 2020. . data and evidence as scientific advise. Without interfering in any political decision, we now use "Data set B", with a clearer definition of tested positive cases I cum and all other variables that follow, C H , C U , C R and D, and recovery period of γ = 4, shown in Figure 3 d) and Figures 4   c-d) . At the moment, the reproduction ratio r is estimated to be below the threshold behavior of r = 1, but still close to 1, meaning that although the number of new cases reported in the Basque Country are decelerating, the outbreak is still in its linear phase and careful monitoring of the development of the dynamics of the new cases from all categories, based on new information and data, to support the upcoming political decisions that will change the current life situation of millions of people is required.

Using the available data for the Basque Country, a small community with short path for data collection and validation, we developed a modeling framework able to predict the course of the epidemic, from introduction to control measure response, potentially helpful for task forces around the globe which are now struggling to provide real scientific advice for health managers and governments while the lockdown measures are relaxed.. 

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The copyright holder for this preprint this version posted May 21, 2020. 

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