key: cord-297161-ziwfr9dv
authors: Sauter, T.; Pires Pacheco, M.
title: TESTING INFORMED SIR BASED EPIDEMIOLOGICAL MODEL FOR COVID-19 IN LUXEMBOURG
date: 2020-07-25
journal: nan
DOI: 10.1101/2020.07.21.20159046
sha: 
doc_id: 297161
cord_uid: ziwfr9dv

The interpretation of the number of COVID-19 cases and deaths in a country or region is strongly dependent on the number of performed tests. We developed a novel SIR based epidemiological model (SIVRT) which allows the country-specific integration of testing information and other available data. The model thereby enables a dynamic inspection of the pandemic and allows estimating key figures, like the number of overall detected and undetected COVID-19 cases and the infection fatality rate. As proof of concept, the novel SIVRT model was used to simulate the first phase of the pandemic in Luxembourg. An overall number of infections of 13.000 and an infection fatality rate of 1,3% was estimated, which is in concordance with data from population-wide testing. Furthermore based on the data as of end of May 2020 and assuming a partial deconfinement, an increase of cases is predicted from mid of July 2020 on. This is consistent with the current observed rise and shows the predictive potential of the novel SIVRT model.

The pandemic disease COVID-19, caused by the coronavirus SARS-CoV-2, became in a few months one of the leading causes of death worldwide with now over 580.000 fatalities and 13 million reported cases (Dong et al., 2020; John Hopkins Dashboard July 16, 2020, n.d.) . The total number of cases and recovered patients is unknown as a fraction of the virus carriers do only show mild or no symptoms and hence, escape any diagnostics or could not get tested especially at the onset of the crisis due to a lack of infrastructure and test material. To a lesser extent, the number of cases is likely to be underestimated in countries that did not count deaths outside care facilities. Whereas other countries like Belgium included every fatality that has been tested positive for the virus regardless of the cause of death (https://www.politico.eu/article/why-is-belgiums-death-toll-so-high/). The lack of consistency among testing strategies and case counts prevents the reliable and comparable calculation of simple measures, such as the infection fatality rate (IFR) or the effective reproduction number

Rt_eff, which is required to better assess the virulence and spread of the disease. A unified large-scale testing strategy and a more rigorous integration of the testing information would enable more precise political decisions on measures beyond following the all-or-nothing example of China, which imposed a lockdown on their population to avoid a breakdown of the healthcare system due to a saturation of ICU beds by COVID-19 patients. Among the first affected countries by the virus, only the ones that had experienced the previous MERS-CoV outbreak such as South Korea and Singapore and hence had mitigation strategies in place or had established large test infrastructures like Iceland could avoid strict containment strategies.

Other countries like Sweden and England attempted to find a balance between lockdown and uncontrolled spread, by slowing the contagion to protect the healthcare system and the elderly people without facing the economic harm caused by a full lockdown.

In Luxembourg, the first case and death were reported on the 29 th of February and the 13 th of March 2020, respectively. On the 16 th of March, schools were closed and all non-crucial workers shifted to remote work or were furloughed. Two days later, the state of emergency was declared, in-person gatherings were prohibited, and restaurants and bars were closed. Over 70.000 workers were furloughed and 30.000 more took a leave for family reasons to homeschool their children. In parallel, within the CON-VINCE study serologic tests were performed to assess the presence of IgG and IgA in the plasma, as well as nose and mouth swabs on a random set of 1.800 habitants to assess the spread of the disease in the Luxembourgish population. Around 1,9% of the samples had antibodies and 5 people tested positive, indicating that Luxembourg was far away from herd immunity (Snoeck et al., 2020) .

Epidemic models such as compartment models have proven to be a useful tool in other outbreaks to access the efficiency of mitigation strategies and to plan the timing and strength of interventions. More specifically, SIR (Susceptible, Infected and Removed) and SEIR (Susceptible, Exposed, Infected, and Removed) models, formulated as Ordinary Differential

Equations (ODE), allow determining when social distancing, hand washing, testing, and voluntary remote working should be sufficient to prevent an exponential growth of the cases and when a significant portion of the population has to return into lockdown (Song et al., 2020; Tang et al., 2020; Wangping et al., 2020; Yang et al., 2020) . Adaptations and extensions of SIR and SEIR models were published for COVID-19 already (Giordano et al., 2020; Siwiak et al., 2020; Tang et al., 2020) . Such models allow describing the dynamics of mutually exclusive states such as Susceptible (S) which for COVID-19 is assumed to be the entire population of a country, a region or city, the number of Infected (I) and Removed (R) that often combines (deaths and recovered), as well as the number of Exposed (E) for SEIR models. The variables I and R, are often unknown as the number of cases and announced recovered patients only accounts for a fraction of the real values that is dependent on the testing performed within a country. Therefore the numbers of Susceptible and Exposed that equals the total population minus Infected and Removed in SEIR models are also undetermined. Several studies extended the number of considered states in such models to further differentiate between detected and undetected cases (Susceptible (S), Infected (I), Diagnosed (D), Ailing (A), Recognized (R), threatened (T), Healed (H) and Extinct (E)), (Gaeta, 2020; Giordano et al., 2020) or took the severity of the disease into account in relation to the age of the infected person (Balabdaoui & Mohr, 2020; Wu et al., 2020) . However, with an increase in the number of states and parameters describing the transitions between these states, more data is required to calibrate the model, i.e. to estimate the model parameters. Roda et al (Roda et al., 2020) showed that an SIR model seems to better represent data obtained from case reports than SEIR models.

Notably, SIR models captured a link between the transmission rate β and the case-infection ratio that was missed by SEIR models. The underestimation of the Infected, Deaths, and

Removed, due to not considering country specific testing information, causes SIR models to predict IFR and effective reproduction number (Rt_eff ) values that vary drastically across countries with different testing and might often be overestimated. To overcome this issue, we

propose an extended SIR model (SIVRT) which is informed by the number of performed tests and also takes the number of hospitalizations into account to parametrize the model. This allows for a better prediction of the evolution of the disease and the estimation of key pandemic parameters, as well as the analysis of different deconfinement and testing strategies.

The novel SIVRT model ( and Detected Death cases (D D ). All these transitions are modeled with first-order laws with rate constants kIR, kIV, kVR, kVD, kIDR, kIDVD, kVDR, and kVDD, respectively. The rate constants for detected and non-detected states are assumed to be equal. Regarding the testing, it is presumed that (i) Severe cases S are tested with high probability compared to asymptomatic cases, as it is more likely that the severe cases will be spotted within the population; (ii) Susceptible S and Recovered people RCum (=R+RD) are tested with the same probability.

Testing of Severe cases S is modelled as a first-order term as well (kVVD), and the remaining performed tests are distributed among Infected (I) and the sum of Susceptible and Recovered (S+RCum). The ratio between these two groups is adjusted with parameter kTIvsS which is also subjected to optimization. 

For Luxembourg data on people tested positive, death cases, hospitalisation and performed tests were obtained from the website of the Luxembourgish government https://coronavirus.gouvernement.lu/en.html and are summarized in Appendix 2.

The model was implemented in the IQM toolbox (Sunnåker & Schmidt, 2016) 

For the predicted no lockdown at all scenarios (Figure 3 ), the lockdown event on Day 17 was removed. For the predicted light lockdown scenario (Figure 4) , the lockdown event on day 17 was kept, but the infection rate parameter (kSI) was increased by 10% of the difference of its value during the full lockdown compared to its value before the lockdown. For the predicted partial lifting of the lockdown scenario as of end of May ( Figure 5 ), the infection rate parameter (kSI) was increased on Day 85 of the simulation by 20% of the difference of its value during the full lockdown compared to its value before the lockdown. The testing rate was kept constant.

For the predicted lifting of the lockdown scenario as of end of May with increased testing approximately matching the Luxembourgish strategy of testing (Figure 6 ), on Day 85 of the simulation the infection rate parameter (kSI) was set to its value before the lockdown and the testing rate was increased to 5.000 tests per day.

As the number of performed tests strongly influences the dynamic analysis of the COVID-19 pandemic in a country or region, we developed a novel SIR based epidemiological model (SIVRT, Figure 1 ) which allows the integration of this key information. The model consists of two layers describing the undetected and detected cases whereby the transition between these layers is realized by testing. The model distinguishes severe from non/less symptomatic cases .

The probability for severe cases to get tested is assumed to be higher. The model consists of 9 states and has been implemented in the IQM toolbox within Matlab (Methods & Appendix 1).

Importantly it allows the fitting to epidemiological data, among others to detected cases, deaths, March 25, 2020, which had one of the largest numbers of cases and tests performed at the onset of the pandemic (Kim et al., 2020) and more importantly with the estimated IFR after adjusting for delay from confirmation to death obtained on the Diamond Princess Cruise Ship (Russell et al., 2020 are predicted to have occurred in Luxembourg with only around 5.000 deaths being detected and assigned to the pandemic ( Figure 3 ). As the same number of tests in this simulation was assumed as performed in reality, a high number of deaths would not have been detected.

Whereas, a lighter lockdown could have led already to a second infection wave as shown in an example simulation (Figure 4 ). Thus obviously also the model supports the huge necessity of the performed lockdown, in comparison with alternative scenarios. Example simulation showing a reduced risk of second infection wave arising around mid of July (Day 135), compared to lower testing as shown in Figure 5 . Legend as in Figure 2 .

In summary, the novel testing informed SIVRT model structure allows to describe and analyze the COVID-19 pandemic data of Luxembourg in dependency of the number of performed tests.

This enables the estimation of the overall and recovered cases, including detected and nondetected cases and thereby the estimation of the infection fatality rate (IFR). It is furthermore possible to perform predictions on past and future scenarios of combinations of lockdown lifting and testing.

Simulations of the novel SIVRT model with parameters estimated from the data of the early pandemic in Luxembourg give a full dynamic picture including detected and non-detected cases. In particular, the overall number of cases until end of May and the IFR is estimated at around 13.000, representing 2,1% of the population and 1,3%, respectively. This is in line with 1,9% of volunteers in the CON-VINCE study that had IgG antibodies against SARS-CoV-2 in their plasma and the estimated IFR rate on the Diamond Princess cruise ship 1,3 (95% CI: 0.38-3.6) (Russell et al., 2020) . The SIVRT model also allowed predicting the appearance of a second wave in a time frame of 50 days after a partial lifting of the lockdown. This is in concordance with the rise in cases see in Luxembourg as of mid of July. 

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Estimating the infection and case fatality ratio for coronavirus disease (COVID-19) using age-adjusted data from the outbreak on the Diamond Princess cruise ship

From a Single Host to Global Spread. The Global Mobility Based Modelling of the COVID-19 Pandemic Implies Higher Infection and Lower Detection Rates than Current Estimates

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