key: cord-1049351-i2rmc37q
authors: Perez-Reche, Francisco; Strachan, Norval
title: Importance of untested infectious individuals for the suppression of COVID-19 epidemics
date: 2020-04-17
journal: nan
DOI: 10.1101/2020.04.13.20064022
sha: 84722adbd8ed8ede8fafd67029875202a63c12e9
doc_id: 1049351
cord_uid: i2rmc37q

A mathematical model which accounts for tested and untested infectious individuals is calibrated during the early stages of COVID-19 outbreaks in Germany, the Hubei province, Italy, Spain and the UK. The predicted percentage of untested infected individuals depends on the specific outbreak but we found that they typically represent 50% to 80% of the infected individuals. Even when unreported cases are taken into account, we estimate that less than 8% of the population would have been exposed to SARS-CoV-2 by 09/04/2020 in the analysed outbreaks. These levels are far from the 70-85% needed to ensure herd immunity and we predict a resurgence of infection if ongoing lockdowns in the analysed outbreaks are fully lifted. We propose that partially lifted lockdowns together with fast and thorough testing allowing for quick isolation of both symptomatic and asymptomatic cases could lead to suppression of secondary waves of COVID-19 epidemics.

Coronavirus disease produced by the SARS-CoV-2 virus emerged in Wuhan, China in December 2019 1 . The virus has spread at an unprecedented rate since then, leading to 1,521,252 confirmed cases and 92,798 deaths distributed in 213 countries as of 10 April 2020 2 . The worldwide burden of the disease is still growing despite significant efforts of many countries to suppress the spread of the virus. So far, efforts have focused on non-pharmaceutical interventions which range from handwashing or social distancing to more stringent measures such as isolation of infected individuals, banning of large gatherings or severe lockdowns 3, 4 .

Optimising interventions to mitigate or suppress the burden of COVID-19 remains a pressing global challenge due to significant uncertainties regarding the transmissibility of SARS-CoV-2 and other factors such as possibly a large proportion of undocumented infections as well as political, social and economic considerations 5, 6 . Underreporting of infections may depend on the testing ability of different countries and the presence of asymptomatic infected individuals [7] [8] [9] . There is no consensus on the proportion of unreported cases and their potential impact on the spread of SARS-CoV-2. For instance, a World Health Organization report in February suggested that "the proportion of truly asymptomatic infections is unclear but appears to be relatively rare and does not appear to be a major driver of transmission" 10 . Studies testing for SARS-CoV-2 infection in both symptomatic and asymptomatic individuals [7] [8] [9] , however, suggest that asymptomatic carriers can represent 50% or more of the cases. In many countries that mostly test individuals when they have symptoms, unreported infections are likely to include at least most of the asymptomatic individuals or those with mild symptoms. Such individuals can act as silent carriers for SARS-CoV-2 and have been suggested as a key factor promoting the rapid spread of the virus 11 , similar to what has been observed in other infectious diseases 12 . On the positive side, if recovery from infection leads to immunity, one could hope that untested positive individuals could significantly contribute to the build-up of herd immunity in the population 13, 14 . It is not clear to what extent this could be the case. The importance of silent carriers on interventions for mitigation and suppression 15 of the infection is not clear either.

Mathematical modelling has been very successful in epidemiology [16] [17] [18] and there is an ongoing effort to propose models to describe the dynamics of COVID-19 epidemics 3, 11, 14, 15, [19] [20] [21] [22] [23] [24] [25] . Unreported infectious individuals have been included in some models 3, 11, 22, 26 but their influence on control strategies has not been analysed.

Here, we use data from the outbreaks in Germany, Hubei (China), Italy, Spain and UK to calibrate a mathematical model that accounts for the force of infection associated with both tested (reported) and untested infectious individuals (see a scheme of the simplest version of the model in Figure 1 and more details in Methods). In order to compare outbreaks in different regions/countries, we fit the model independently to each outbreak. The calibrated model is used to study the effect of two suppression strategies: Interventions aiming for a reduction of transmission at the population level (representing, e.g., social distancing or a lockdown) and local interventions consisting in isolation of both tested and untested infectious individuals. Figure 1 . Simplest version of the model used to simulate the SARS-CoV-2 epidemic. At any given time, individuals can belong to one of 7 compartments: (susceptible to SARS-CoV-2), (exposed but not yet infectious), (infectious tested), (recovered from tested infected), (dead), (infected untested) and (recovered from untested infected). It is assumed that all deaths are associated with tested infectious individuals. Recovered individuals are assumed to be immune to the virus. The transition rates between compartments are indicated on the arrows. is the rate at which susceptible individuals become exposed. The force of infection, ( ), accounts for both tested and untested individuals. Here, is the size of the modelled population. The mean incubation period is −1 . The proportion of exposed individuals that become infected and are tested defines the testing rate, . Tested infectious individuals die or recover at a rate . The parameter

gives the proportion of infected tested that die. Untested infectious individuals recover at a rate . Figure 2 shows that the model accurately captures the early stages of the fitted outbreaks. Estimates for the parameters of each outbreak are given in Table 1 .

The obtained values for the testing rate reveal that during the early stage of outbreaks, Germany scored the highest in terms of testing for infection (median 53%). Hubei follows Germany in terms of testing, followed by Spain, Italy and the UK. Our prediction for Hubei is not far from the 65% reporting rate estimated by Li et al. 11 for China in the period considered here. The high testing rate predicted for Germany agrees with the known high testing capacity in this country 27 . Taking the confidence intervals into account, we estimate that for each infected individual tested in the UK, there could have been between 2 and 10 untested infected individuals. At the other end of the testing spectrum, we estimate that for each infected individual tested in Germany, between 0.2 and 2 individuals might have not been tested at the beginning of the epidemic. A higher testing rate for Germany is in qualitative agreement with estimates given elsewhere 22, 26 . Our estimates for the reporting rate, however, tend to be higher than those obtained by Jagodnik et al. 26 and the differences we found between countries are not as extreme as those given by Chicci et al. 22 Assuming that the testing rate remains constant during the course of epidemics and no control interventions are implemented, our model predicts that the number of tested and untested infected individuals would evolve in parallel in all the studied outbreaks which would last for around 12 weeks in all cases (see Figure 3 ). We see, however, that the epidemic in Germany would belong to a different class in the sense that the number of untested infected individuals remains smaller than the number of tested individuals during the whole epidemic. Italy, Spain and UK exhibit the opposite behaviour with more untested than tested individuals. For Hubei, we predict similar levels of untested and tested rates. Obviously, these predictions will not be fulfilled since control interventions are imposed in all these countries and testing strategies might change during the pandemic.

Bearing in mind that 50% of infected individuals might be asymptomatic 7,9 , we conclude that most of those that were infected but not tested in Germany were asymptomatic. In contrast, untested individuals in other countries might include a significant number of individuals with mild symptoms in addition to those that are asymptomatic. Table 2 ). Logarithmic scale is used in the vertical axis of each plot. Table 1 . Estimates of the model parameters given in terms of the 5% percentile, median and 95% percentile. is the transmission rate, is the proportion of tested infectious (in percentage), is the proportion of tested infectious that die (in percentage), is the rate of recovery of tested infectious individuals, is the rate of recovery of untested infectious individuals, (0) is the initial number of exposed individuals and ℛ 0 is the reproduction number. The proportion of tested infected individuals that die, , is smaller for the outbreak in Germany than for the other outbreaks. This might be a combined effect of the fact that infected individuals in this country were relatively young at the beginning of the outbreak 28 and the high testing rate. On the one hand, the COVID-19 fatality rate is lower for the younger than for the elderly 29 . On the other hand, the higher the testing rate, the more individuals with mild symptoms will be included in the tested infected compartment of our model. The lower death rate of individuals with mild symptoms will lead to an effectively lower death rate for the whole set of infected individuals in this compartment. We note that the specific value of for a country does not fully determine the expected fraction of deaths in the country. Indeed, we found that the predicted fraction of deaths by the end of unmitigated epidemics is not too different for different countries (medians are as follows: 0.4% for Germany, 0.9% for Hubei, 0.5% for Italy, 0.8% for Spain and 0.3% for UK).

The median of the recovery rate gives the following estimates for the time from reporting to recovery or death of tested infected individuals, −1 : 3 days for Spain, 3.3 days for Italy, 3.6 days for the UK, 4.2 days for Hubei and 7 days for Germany. The time for Hubei is consistent with the 3.48 days reported by Li et al. 11 for China. In general, the values we obtained are smaller than the infectious period (time from infection to death/recovery) reported elsewhere for COVID-19 3,29,30 . Our estimates thus probably reflect a reporting delay in all the studied outbreaks, in agreement with data on the onset of symptoms and reporting 28, 31, 32 . Our model predicts the smallest reporting delay for Germany. This is again in agreement with the high testing capacity of this country.

The recovery period for untested infected individuals, −1 , takes values of around 3 days for all the studied outbreaks. Comparison with the reporting-to-recovery period −1 and bearing in mind the reporting delays in all outbreaks, our estimates of −1 suggest that untested individuals remain infectious for a shorter time than tested individuals. This is in line with the lower infectivity of untested individuals proposed in a recent study that, instead of assuming different recovery rates for tested and untested individuals, assumed a lower transmission rate for unreported infectious individuals 11 .

We predict that the number of exposed individuals at the beginning of our simulations, (0), is of the order of several thousand for all the countries, in qualitative agreement with estimates of a previous study for China 11 .

We obtained similar values of the reproduction number ℛ 0 for all the studied outbreaks. To some extent this reflects our prior assumption that transmission of SARS-CoV-2 is intrinsically similar in different regions. The transmission rate, , was derived from the estimates of , , and ℛ 0 , using Eq. (4) (see Methods). Values of are statistically similar for all countries except Germany which features a smaller value. Bearing in mind that ℛ 0 and take similar values for all the countries, we conclude that a lower value of is a consequence of the higher testing rate and smaller recovery rate . CC-BY-NC-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.

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for this country. Indeed, according to Eq. (4), decreases with increasing and decreasing (or increasing period −1 ). Table 2 in Methods).

Interventions such as lockdowns or social distancing can be effectively studied by reducing the transmission rate in our model. As illustrated in Figure 4 (a), the outbreak can be significantly delayed if the transmission rate is reduced from early stages in the epidemic, in agreement with other works 15 .

In spite of that, the predicted number of deaths by the end of the epidemic only reduces significantly when is reduced by a factor close to = 1 − 1/ℛ 0 to ensure an early eradication of the infection 16 . Based on our estimated values for ℛ 0 , this requires reducing the transmission rate by more than 70% in all of the studied outbreaks. This is illustrated in Figure 4 (b) for the UK. As can be seen, the number of deaths would only reduce significantly if the number of contacts were reduced by approximately 80%.

Our model can be readily used to predict the effect of lockdowns of arbitrary duration and different exit strategies from such lockdowns. Figure 5 shows predictions for the lockdowns in Hubei, Italy and Spain. Predictions for Germany and the UK are also possible but a comparison with observed effects is uncertain at present since lockdowns were implemented more recently in these countries.

For Hubei, the model reproduces well the observed daily deaths assuming a 90% reduction of the transmission rate from 6 February 2020 (see Figure 5 (a)). The later is an effective date between the 23 January when the lockdown was implemented in Wuhan, the capital of the province, and 13 February when it was implemented in the whole province. Despite the spectacular reduction of daily deaths induced by the lockdown in Hubei, our model predicts that fully removing the lockdown after . CC-BY-NC-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 . https://doi.org/10.1101/2020.04. 13.20064022 doi: medRxiv preprint 60 days (i.e. approximately at the time of writing) would lead to a rapid resurge of the epidemic (see the marked increase predicted after week 12 in Figure 5 (a)). In contrast, an exit strategy in which the transmission is kept reduced by a 75% is predicted to keep the number of daily deaths below 100 for many weeks (see Figure 5 (c)).

We estimate that the lockdown ordered in Italy reduced the transmission by around 80%. Our prediction suggests that this will lead to a significant decrease of the number of deaths if it is kept for a long enough time. In particular, Figure 5 (b) shows a scenario in which the lockdown is kept at the same level for 90 days since its implementation on 11 March 2020. In this case, we predict around 69 [90% CI: (16, 568) ] daily deaths at the end of the lockdown. As for Hubei, a full removal of the lockdown leads to a fast resurge of the epidemic (Figure 5(b) ). For Italy, we estimate that an exit strategy from this lockdown should still keep the transmission at low values (~70% reduction) for the daily deaths to remain at a moderate value of around 100 (see Figure 5 (d)).

The effectiveness of the lockdown imposed in Spain is predicted to be similar to the one in Italy ( Figure  5 (c)): The current lockdown managed to reduce the transmission by ~80% and resurge of infection is predicted to occur if the lockdown is completely removed after 90 days. At the end of the initial 90 days lockdown, we predict around 99 [90% CI: (22, 501) ] daily deaths. The number of daily deaths is predicted to remain at this level if the lockdown is partially lifted to a situation in which transmission is kept to a 72% reduced level.

Irrespective of the effectiveness of the lockdown, our model predicts that epidemics will resurge after relatively extended lockdowns in Hubei, Italy and Spain. The same is likely to occur in other countries. In fact, resurgence of the disease is predicted even for much longer lockdowns. This is due to the fact that an early lockdown delays the spread but does not lead to herd immunity. Assuming that recovered individuals are immune to SARS-CoV-2, herd immunity is only achieved when a proportion 1 − 1/ℛ 0 of the susceptible population has been infected and died or recovered. Even if the number of untested individuals that may have recovered are taken into account, we estimate that, as of 09/04/2020, the the proportion of the susceptible population that has been exposed to the virus (i.e. the attack rate, see Methods) is 0.65% [90% CI: 0.46%-1.18%] for Germany, 0.5% [0.3%-1.0%] for Hubei, 4.6% [3.3%-7.2%] for Italy, 3.7% [2.0%-6.4%] for Spain and 4.4% [2.7%-6.9%] for the UK. These proportions are small compared to the 70-85% needed to ensure herd immunity for these epidemics. Our conclusion is in qualitative agreement with the results by Flaxman et al. 3 despite the fact that their estimates for the attack rate tend to be higher than ours. Our conclusions, however, disagree with Lourenço et al. 14 that predicted much higher attack rates that would be close to herd immunity. . CC-BY-NC-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.

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Prompt isolation of infected individuals is regarded as an effective strategy to reduce the transmission of infection and significantly reduce the size of epidemics 33 . Identification of silent carriers is difficult, however, and this makes the implementation of this strategy challenging for SARS-CoV-2.

In order to study the effect of isolating tested and untested individuals, we extended the model shown in Figure 1 to include compartments for isolated tested and untested infected individuals (see Model 2 in Methods). Interventions are parametrised by the fraction of tested (untested) infected individuals that are randomly selected to be isolated, ( ), and the rate at which they are isolated after testing/reporting. Isolation strategies leading to eradication of infection satisfy the condition ℛ 0 < 1, where ℛ 0 is the reproduction number for Model 2 (see Eq. (6) in Methods). Given the significant reporting delays we found, isolation has to be fast after testing positive. In particular, we found that isolation after an average time of −1 = 1 day could only lead to eradication in Germany; for other countries, ℛ 0 remains larger than 1 for any and . Following this, in Figure 6 we show results for a scenario in which individuals are isolated after an average time of −1 = 1/2 days. In this case, eradication is possible in all the studied countries if both and are large enough, i.e. if enough tested and untested cases are isolated. The estimated boundaries separating the eradication region (ℛ 0 < 1) from the epidemic region (ℛ 0 > 1) are different for different countries but differences are statistically less marked if confidence intervals are taken into account.

Interventions that only isolate tested infected individuals (i.e. with = 0) are predicted to have a minor effect on the final fraction of deaths even if they manage to isolate all tested individuals (see Figure 6 (b)). Isolation of tested individuals is not effective due to the underlying transmission associated with silent carriers that are not isolated and keep ℛ 0 > 1.

For instance, an intervention in the UK in which 70% of tested infected individuals were isolated in = 1/2 days could drastically reduce the number of deaths if 40% of untested infected were isolated at the same rate. For Germany, the percentage of untested individuals that should be isolated to ensure eradication in this situation is around 15% since testing seems to be already faster and more effective than in other countries.

Isolating infected individuals after an average time of half a day since testing is likely to be difficult to implement in practice. This time could indeed be extended by rapid identification of carriers of the virus. This highlights the importance of fast identification of infected individuals. Identification of asymptomatic cases is expected to be challenging. However, we believe that efficient tracing of the contacts of symptomatic individuals and fast testing of such contacts could facilitate the identification of asymptomatic cases. 

There are two main implications from the models which are directly relevant for policy in dealing with the outbreak. The first, involves the existence of a significant proportion of cases that are not tested and may act as silent carriers of the infection. We found that the predicted percentage of untested infected individuals may represent 50% to 80% of the cases in Germany, Hubei, Italy, Spain and the UK. The specific percentage depends on the country and we found the lowest proportion of unreported cases in Germany. Based on studies in Iceland 9 and the Diamond Princess cruise 7 , we conclude that asymptomatic infected individuals are likely to be the main contribution to the untested cases in all analysed outbreaks but a fraction of cases with mild symptoms are also likely to be untested. Even when unreported cases are taken into account, we estimate that less than 8% of the population would have been exposed to SARS-CoV-2 by 09/04/2020 in the analysed outbreaks. In policy terms, our results demonstrate that the current suppression strategies being employed in Germany, Hubei, Italy, Spain and the UK will not facilitate sufficient levels of herd immunity in the population that would control and eventually eradicate the virus. This leaves the risk of re-emergence of the virus once suppression strategies are lifted, similar to second waves of infection observed in 1918 influenza epidemics 34 . We predict, however, that partial relaxation of ongoing lockdowns could keep the number of daily deaths to potentially tolerable levels.

The second implication involves the finding that unreported cases play an important role in the control of COVID-19 epidemics. In particular, unreported cases act as silent carriers and control strategies would need to account for them or be prone to the risk of re-emergence or ineffective suppression of

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The copyright holder for this preprint . https://doi.org/10.1101/2020.04.13.20064022 doi: medRxiv preprint spread. For instance, we predict that isolation of infected individuals can have a limited impact on the suppression of spread unless it includes silent carriers that are currently missed by most countries. In line with previous suggestions 14, 35 , we suggest that, thorough testing combined with contact tracing 20,21 , isolation of infected individuals and social distancing can be more effective to suppress SARS-CoV-2 spread than severe lockdowns. At present, however, lockdowns are probably the most effective way to delay epidemics until more effective pharmaceutical (e.g. a vaccine) or nonpharmaceutical interventions (e.g. fast and thorough testing) become feasible.

The main aim of our models is to contribute to the understanding of the epidemiological patterns of SARS-Cov-2 rather than to provide exact predictions. Hence the models should be viewed as a general guide of how the outbreak and interventions may play out rather than as an exact representation of COVID-19 epidemics 17 . We made several simplifying assumptions in the models. In particular, spatial and age heterogeneities were ignored. However, it is known that there is both spatial heterogeneity within the populations 36 (e.g. cities and rural areas) as well as differences in both susceptibility and mortality across different age and vulnerability groups 29, 37 . Another limitation of our approach is that it assumes that the transition times between compartments are exponentially distributed. This memory-less assumption is usual for classical compartmental models 17 . For COVID-19, however, transitions between compartments are better described in terms of gamma distributions 3,29 and using models with memory would provide a more precise description of the dynamics 3, 17, 38 . In addition, our model is limited in terms of the specific ways to make interventions operational. For instance, reductions in transmission are treated at a generic level without specifying if they could be achieved by enhanced social distancing, school closure, etc. Accounting for such details would require using individual-based simulations 15 .

The models assume immunity after recovery from infection but whether or not this is the case is still unclear 39 . Our model could easily be extended to account for re-infections and predictions might significantly change. There is also a growing body of evidence for pre-symptomatic transmission [40] [41] [42] [43] [44] . There is the potential to incorporate this in our models by varying the incubation rate parameter. Finally it is unknown what percentage of the population is actually susceptible. Here it is assumed all of the population are susceptible but it may be that a proportion are not for genetic reasons 45 or due to cross immunity 46 .

Data on numbers of infected and dead individuals by country or region were obtained from the Wolfram Data Repository 47 . Models were calibrated by considering data from the first available day in which the number of deaths is non-zero, as listed in Table 2 . The date when lockdowns were ordered in each of the countries/regions is also given in Table 2 . Table 2 . Details on the first day used to calibrate the models, number of deaths by that day, date when a lockdown was ordered in each of the countries/region analysed, and population of each country/region.

First day considered

Date when lockdown ordered 3, 4 Population, N 

We used extensions of the SEIR model 17 to include two types of infected individuals described by the compartments and (see Figure 1 ). The SEIR model with a single compartment for infectious individuals has already been used to describe the COVID-19 outbreak in China 19,23 and a model with two compartments for infected individuals analogous to those proposed here was used by Li et al. 11 

The model shown in Figure 1 is run with deterministic, continuous-time dynamics given by the following differential equations: (1)

Here,

is the force of infection. is the population size.

The attack rate at a given time is defined as the fraction of individuals in a population of size that have been exposed to the disease by that time. For the model described by Eqs. (1), we calculate the attack rate as follows:

Attack rate( ) = ( ) + ( ) + ( ) + ( ) + ( ) + ( ) .

The reproductive number corresponding to this model can be analytically calculated using the next generation method 18 and is given by

Isolation of infectious individuals is modelled by adding two more compartments, and , which contain isolated tested and untested infectious individuals, respectively (see Figure 7 ). The fraction of tested and untested infectious individuals are denoted as and , respectively. Both types of infectious individuals are assumed to become isolated with the same rate, . The force of infection in this model is

One can again use the next generation method 18 to obtain the following expression for the reproduction number: Figure 7 . Extension of the basic epidemic model of Figure 1 to incorporate a compartment for isolation of a tested infectious individual and a compartment for isolation of untested infectious individuals.

We fitted the Model 1 to data. Values for the incubation rate was set to 37 = 1/5.2 days -1 . The free parameters in our fits were the rate of transmission, , proportion of infectious that were tested, , proportion of tested infectious that die, , rate to recovery of tested infectious individuals, , rate of recovery of untested infectious individuals, , and initial number of exposed individuals, (0). We denote the free parameters by a vector = { , , , , , (0)}. The model was fit to the time series for the number of daily reported infected individuals and cumulative deaths, obs = { , } =1 , in a period of days in the early stages of epidemics (here, is used to denote discrete time in days). In particular, we used = 15 days since the first data point with a positive number of deaths (see Table 2 ). We used data at early stage of each outbreak to minimise the influence of suppression strategies on our parameter estimates.

Using data on deaths is important to obtain reliable descriptions of COVID-19 epidemics since data on deaths is more accurate than data on infected and recovered individuals 3, 14, 26, 48 . In addition to deaths, we can use data on infected individuals which is represented by the tested infectious compartment, , in our models.

Our fitting procedure aims at calculating the posterior probability density function for the parameters given the data, ( | obs ). To this end, we use an approximate Bayesian algorithm which follows the same steps as the minimum distance method proposed by Perez-Reche et al. 49 ; the only difference being that here we use a likelihood function to quantify the similarity of simulated and observed time series instead of a quadratic distance.

The posterior ( | obs ) is approximated by the empirical distribution of a set of 500 point estimates ̂ of the model parameters. A point estimate ̂ is obtained by simulating = 3000 epidemics with parameters sampled from a prior probability density ̂( ). In each realization, a simulation of Model 1 produces deterministic evolution functions ( ) and ( ) for the number of tested cases and cumulative deaths. The functions ( ) and ( ) are used to build a random daily time series sim ( ) = { sim ( ), sim ( )} =1 , where sim and sim are, respectively, the number of tested infected and deaths predicted at day . We assume that sim ( )~ Pois( ( )) and sim ( )~ Pois( ( )), i.e. the predicted number of tested infected and deaths are described as random variables obeying a Poisson distribution with mean ( ) and ( ), respectively. The point estimate ̂ is defined as the parameter vector corresponding to the realization that gives the closest prediction, sim , to the observations, obs . More explicitly, the point estimate for the model parameters is given by ̂= argmax { ℒ( obs | sim ( ))} ,

where ℒ( obs | sim ( )) is a loglikelihood function defined as ℒ( obs | sim ( )) = ∑ ln(pois( | sim ( )) pois( | sim ( ))) =1 .

Here,

is the Poisson probability mass function.

The prior probability density is defined as the product of priors for each parameter:

The priors used in our fits are summarized in Table 3 . Normally distributed informative priors were used when prior information on the parameter was available. In particular, since the reproductive number has been more thoroughly studied in previous works than the transmission rate, we set an informative prior for ℛ 0 and derive from Eq. (4) using the priors of ℛ 0 , , and . Derived from other parameters using Eq. (4) Number of exposed at the first data point, (0) ln (0)~ (8,0.5 2 )

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