key: cord-0707294-ixv7ef09
authors: Proverbio, Daniele; Kemp, Françoise; Magni, Stefano; Ogorzaly, Leslie; Cauchie, Henry-Michel; Gonçalves, Jorge; Skupind, Alexander; Aalto, Atte
title: Model-based assessment of COVID-19 epidemic dynamics by wastewater analysis
date: 2022-03-01
journal: Sci Total Environ
DOI: 10.1016/j.scitotenv.2022.154235
sha: 72368a6d9843065009abe74610a901d8abc2d533
doc_id: 707294
cord_uid: ixv7ef09

Continuous surveillance of COVID-19 diffusion remains crucial to control its diffusion and to anticipate infection waves. Detecting viral RNA load in wastewater samples has been suggested as an effective approach for epidemic monitoring and the development of an effective warning system. However, its quantitative link to the epidemic status and the stages of outbreak is still elusive. Modelling is thus crucial to address these challenges. In this study, we present a novel mechanistic model-based approach to reconstruct the complete epidemic dynamics from SARS-CoV-2 viral load in wastewater. Our approach integrates noisy wastewater data and daily case numbers into a dynamical epidemiological model. As demonstrated for various regions and sampling protocols, it quantifies the case numbers, provides epidemic indicators and accurately infers future epidemic trends. Following its quantitative analysis, we also provide recommendations for wastewater data standards and for their use as warning indicators against new infection waves. In situations of reduced testing capacity, our modelling approach can enhance the surveillance of wastewater for early epidemic prediction and robust and cost-effective real-time monitoring of local COVID-19 dynamics.

SARS-CoV-2 viral load in wastewater. Our approach integrates noisy wastewater data and daily case numbers into a dynamical epidemiological model. As demonstrated for various regions and sampling protocols, it quantifies the case numbers, provides epidemic indicators and accurately infers future epidemic trends. Following its quantitative analysis, we also provide recommendations for wastewater data standards and for their use as warning indicators against new infection waves. In situations of reduced testing capacity, our modelling approach can enhance the surveillance of wastewater for early epidemic prediction and robust and cost-effective real-time monitoring of local COVID-19 dynamics.

demanded, along with other mathematical models presenting different structures that could decently capture the correlation between viral load in wastewater and the shedding/infected population. Recently, a number of studies have proposed methods to infer the size of the shedding population, based on the viral abundance sampled in wastewaters. Some are restricted to qualitative and semi-quantitative retrospective studies of lagged correlations (Nemudryi et al., 2020; Kumar et al., 2020) , others employ parametrical and non-parametrical regression models (Cao and Francis, 2021; Vallejo et al., 2021; Li et al., 2021; Hasan et al., 2021; Huisman et al., 2021) ; an alternative is to use an epidemiological SEIR (Susceptible-Exposed-Infectious-Recovered) model informed by estimates of individual viral trajectories (McMahan et al., 2021) . These models, tested on one specific country at a time, well showed the difficulty of estimating the true number of positive cases from wastewater samples, because of noisy data (including those of detected case numbers), uncertain ratios of detected and true positive cases, and the variety of individual infection periods.

In this article, we propose an alternative, automated and causal-based approach to address the aforementioned challenges. We develop a new method that couples a Susceptible-Exposed-Infectious-Recovered (SEIR) epidemiological model (Anderson and May, 1979) to the extended Kalman filter -EKF (Kalman, 1960) , a natural approach for combining noisy measurement data and modelling. The EKF deals effectively with the problem of producing model simulations that are representative of real system observations, which are in turn prone to uncertainties. It has been a standard tool in systems theory, with many applications ranging from automated control (Nagy Stovner et al., 2018) to finance (Davis and Leo, 2013) , bio-mechanics (Marchesseau et al., 2013) , time series analysis (Harvey, 1990) methods. This way, the model estimates population-level COVID-19 diffusion without the need to rely on individual viral trajectories. It is also very flexible with respect to sampling routines and considered regional areas. Hence, including empirical measurements and tuning the model can be done in a straightforward way. In addition, the underlying SEIR model allows interpretation of the inferred infection dynamics in terms of transmitting interactions and overcomes the extrapolation limitations of correlation-based statistical approaches (Cao and Francis, 2021; Li et al., 2021) . A Kalman filter was used for wastewater viral abundance data by Cluzel et al. (2022) and Courbariaux et al. (2022) , but they employed a linear first-order autoregressive model. This makes it a signal processing tool primarily aimed at reducing noise in the wastewater measurements, setting our approach apart.

After calibration, our CoWWAn (COVID-19 Wastewater Analyser) method causally and quantitatively links the results of wastewater analysis with those of population-wide testing, also accounting for the "dark number" (ratio of real and detected case numbers) through meta-parameters. This way, we are able to quantify the goodness-of-fit to observed cases (what is empirically measured) and to justify the reconstruction of the shedding population. Then, CoWWAn causally infers the shedding population, estimates the effective reproduction number eff R , and provides projections of future epidemic trends. Quantifying these variables enables assessment of the epidemic status within a region and comparison between regions, and supports effective mitigation policy making. We also use this model to quantitatively assess the warning potentials of wastewater monitoring, combined with population testing or on its own. Originally, we applied CoWWAn in support to the Research Luxembourg COVID-19 taskforce of our local government. Moreover, to demonstrate its general applicability, we applied CoWWAn to public J o u r n a l P r e -p r o o f datasets from 12 regional areas from Europe and North America, associated with different population sizes and based on different wastewater data processing protocols.

Our pipeline requires at least three types of data to be calibrated: data about the COVID-19 RNA load in wastewater, detected cases associated to the area covered by the sewage system, and possibly the estimates about the ratio of detected versus true case numbers. In addition to its routine application to Luxembourg data, CoWWAn was tested on various datasets with different normalisation protocols for wastewater data, to show its general applicability after proper calibration. The dataset was constructed according to the following criteria. First, we employed the COVID19 Poops Dashboard (Naughton et al., 2021) 

Among the data collected, we observed some peculiarities. First, Raleigh county reported case numbers normalised to 10,000 inhabitants and rounded to an integer value; their subsequent up-scaling induces a further uncertainty. Second, the countrywide wastewater data for Netherlands are reported as averages over a week. To improve the temporal resolution of the data, we used instead the data from all communal treatment plants, averaging over samples from the same day. 

As a basis for the Extended Kalman filter to model the epidemic dynamics, we use a modified SEIR model, which has been shown to accurately describe COVID-19 epidemic dynamics He et al., 2020) . As we aim at estimating community incidence from noisy data, we choose a simple and descriptive model rather than a complex one, which is difficult to calibrate and could suffer from identifiability issues (Roda et al., 2020; Kemp et al., 2021) .

The classic, deterministic SEIR model considers Susceptible () St , Exposed () Et , Infectious () It and Removed () Rt compartments, and population flows governed by rate parameters. We follow the standard interpretation of E compartment as the set of individuals J o u r n a l P r e -p r o o f who have been exposed and infected, but who are not yet infectious due to incubation lag (Lai et al., 2020) . The mean incubation period 1   models the progression to becoming contagious (Kollepara et al., 2021) . The total community population is conserved, i.e.

(with constant N ) and we assume no possibility of re-infection during each period of infection transmission (no RS  ). The latter assumption is supported by waning immunity being estimated in a matter of months (Goldberg et al., 2021) . To model intrinsic stochasticity in transmission processes and viral shedding, we employ a stochastic version of this SEIR model, associating each transition between compartments with a random process:

where j w are mutually independent white noise processes. See Appendix A for details. The  -parameter is assumed to be time-varying, reflecting changes in social interaction, other mitigation measures (masks, vaccines, etc.) , and varying infectivity of emerging viral variants. () t  will as well be estimated by the Kalman filter.

In order to model viral flows into wastewater, we introduce another variable () At to model the effective number of active shedding cases producing virions to wastewater. Similarly to above, we incorporate stochastic processes. The dynamics of A are given by:

The A compartment is parallel to E , I , and R , that is, it still holds that J o u r n a l P r e -p r o o f

The influx to the A compartment is the same as that to the E compartment, while the outflux lumps together the dynamics of viral production (which is known to follow some kinetic trajectory in the hosts' body (Néant et al., 2021) ), the decay rate of SARS-CoV-2 RNA in water (Gundy et al., 2009; Sala-Comorera et al., 2021) , and inertia in abundance dynamics due to mixing in wastewater collecting pools. Since the parameter  lumps together properties of the virus and details on wastewater sampling, it is separately fitted for each region. We do not take into account delays associated with in-sewer travel time, as it was estimated to be significantly lower than the transmission time scales (median of 3.3h versus 1 day (Kapo et al., 2017) ). The A compartment thus allows to better follow the time evolution, including potential decaying inertia, and to consider explicitly the uncertainties associated to the shedding mechanism instead of the disease progression. Together, Eq. 1 and Eq. 2 form the combined SEIR-WW system.

The outputs from the model that are compared to the real-world measurements are the number of daily detected cases and the virion abundance in wastewater. The number of detected cases on day t  is assumed to be a share of people passing the incubation period on that day, that is, where  is a tuning parameter to reflect the incubation, production and shedding of viral load from infected people (Néant et al., 2021; Wölfel et al., 2020; Miura et al., 2021) and normalisation of the wastewater data. We do not consider explicit corrections linked to precipitations or other environmental factors, as previous studies evaluated them to be poorly correlated with RT-qPCR observations (Vallejo et al., 2021; Li et al., 2021) . An implicit tuning is nonetheless included in the fitting, cf. Eq. 5.

An Extended Kalman filter requires an underlying dynamical model (such as a SEIR-like one), its output and associated noise covariance matrices, and measurement data. The extended Kalman filter algorithm to estimate the state of the SEIR-WW system, based on different types of data, is presented in Algorithm 1. The inputs for the algorithm are the update function () fx (implementing Eq. 1), the observation matrices () Ct (for case numbers and wastewater data), the state noise Q (uncertainty on estimated variables), and the measurement error covariance () Ut (uncertainties on empirical data). Then, the method evaluates the set of variables of interest (

) and their associated uncertainty matrix P .

The algorithm is used to calculate three different estimates using only case number data, only wastewater data or using both case and wastewater data. These were then used to estimate the data that were not employed for the state estimation, initially for calibration and reconstruction of daily cases, and then to perform the desired predictions.

For details about the numerical implementation, the characterization of inputs and outputs and the discussion of each matrix introduced in Algorithm 1, we refer to Appendix B. Our current implementation is done with custom MATLAB 2019b code (see Code Availability section). 

Our model comes with a number of free parameters to be fixed from the data or with educated assumptions. As most time series data begin after the pandemic already diffused within a region, the initial sizes for the E and I compartments are automatically computed (cf. Appendix C for details).

Another parameter to be estimated is the average ratio of total and detected cases at day t , t  . This is necessary to link the measurements of population testing with those of wastewater analysis (ideally objective and insensitive to testing capacities 

This way, we minimise the error in estimating the case numbers by the EKF state estimate using only wastewater data. Model parameters, either fixed by literature or fitted from Eq. 5, are reported J o u r n a l P r e -p r o o f Journal Pre-proof in Tab. 1. Note that, due to different wastewater data normalisations,  parameters are not comparable between regions. Similarly,  parameters might depend on the used techniques. Data from different laboratories may contain significant differences .

Value Source (0)) E Uncertainty of (0) E regional var( (0)) I Uncertainty of (0) I regional 2 ( (0) / 2) I sources. The parameter q  , controlling the allowed change of () t  in one day, is changed after 30 days. This is done to allow rapid changes in the beginning of the pandemic, when a strict lockdown quickly suppressed its propagation and to account for errors in initial (0)  . d stands for "days". When the source is not indicated, the parameter values is first initiated as an educated guess and then tested with sensitivity analysis (see Supplementary Fig. 18 ).

The sensitivity of the model performance on assumed parameter values is assessed in Supplementary Fig. 18 , which demonstrates the robustness of the model and justifies the current parameter choices. The sensitivity analysis was performed by varying the reference parameters up to 50%  of heir original value. The results are reported in Supplementary Fig. 18 

The workflow of our CoWWAn approach, integration of empirical data into a SEIR model through the Extended Kalman filter, is illustrated in Fig. 1 

After appropriate calibration to test cases with parameter fitting, CoWWAn quantitatively reconstructs the time evolution of observed cases from wastewater data (Fig. 2a) Thanks to the model structure, we could thus compare our results with the true number of detected cases (Fig. 2a, red and black lines) , before inferring the global magnitude of the shedding population (Fig. 2a , blue line, from mean estimates). The latter is an extrapolation from CoWWAn estimates and information about the "dark number" of undetected cases (provided as a model parameter); further independent studies to estimate this quantity help fine-tune the results.

We compared our results with a linear regression model (after data curation to reduce the noise, in a similar spirit to (Vallejo et al., 2021) ): CoWWAn's inferences achieve consistently higher correlation (Fig. 2b , blue and red sets), demonstrating the power of our mechanistic-based approach. These observations hold for all considered regions ( Fig. 2c and Supplementary Fig.   3-14) : the correlation coefficient  between inferred case numbers and true detected case numbers is typically in the range between 0.7 and 0.9 even for rather noisy data like Netherlands.

Frequent sampling improves the model calibration and the subsequent reconstruction performance, like for Luxembourg with = 0.91  for two probes/week and Milwaukee with = 0.95  for two (sometimes more) probes/week compared e.g. to Barcelona with = 0.70  with one probe/week (Fig. 2d) . The main discrepancies originate from either unnoticed changes in the share of detected cases or from changes in testing/sampling strategies (Supplementary Fig.   3-14) . In addition, we notice ( Fig. 2a and c) that the largest uncertainties come together with the highest case numbers, which are often associated to an augmented positivity rate (Ritchie et al., 2020) . Detecting such discrepancies can provide additional evidence about potential undertesting and could guide targeted scaling of population tests. Interpolating wastewater data points before

Journal Pre-proof the EKF estimation can improve the reconstruction (Fig. 2b , red and yellow sets), in particular for regions with low sampling frequency like for Barcelona Prat de Llobregat (PdL) and Kranj. In general, the Extended Kalman filter improves its predictions as new data points are available, so an adequate sampling rate is recommended to improve its performance.

CoWWAn allows estimation of the effective reproduction number eff R , an essential indicator for the trends of epidemic diffusion in a community (Huisman et al., 2021) , which depends on containment measures, infectivity of viral variants, population behavior and other factors. As exemplified for Luxembourg (Fig. 2a) Overall, for the different epidemic phases and all considered regions, the short-term J o u r n a l P r e -p r o o f Journal Pre-proof predictions compare well with the real case data and with the case-based predictions (see also Supplementary Fig. 3-14) . To quantify their performance, we determined the average standardised prediction error as the average discrepancy between predicted and actual case numbers in the corresponding time frame, normalised to case numbers and equivalent population Eq. (7). The performance of our wastewater-based pipeline is usually slightly lower, as it reconstruct the case numbers themselves before making the predictions, but remains similar with that of case-based predictions: all regional estimates lie within one standard deviation of the 1:1 (equal performance) line (Fig. 3b) . The only exceptions are estimates for Oshkosh, probably due to under-testing during late 2020 (refer to Supplementary Fig. 2 ) which induced discrepancies in the detected cases fraction, and Kranj, whose low case numbers are subject to larger uncertainties ( Supplementary   Fig. 5 ). In general, the largest discrepancies are observed when case numbers plateau or decline after a rapid increase, yielding a potential overshoot of the predictions (Fig. 3a and Supplementary   Fig. 3-14) . This effect is associated to large changes in social activities during epidemic waves and rapid implementations of stricter restrictions, which are not explicitly included in the model but implicitly learned from the epidemic curve by the EKF with some delay.

The standardised error grows quite linearly with increasingly long prediction horizons ( Fig. 3c) . There, wastewater predictions are more stable (their uncertainty grows slower for longer prediction horizons) than those based on case numbers as they are usually less susceptible to daily fluctuations (Supplementary Tab. 2). This aspect allows quantifying and comparing the precision for different horizons.

In addition to using one type of data at a time, CoWWAn's EKF-based approach enables integrating different types of data to further improve the quality of predictions. Including both wastewater and case data slightly but systematically improves the prediction accuracy compared outputs for single countries in Supplementary Fig. 17 ) for three inputs: case numbers, wastewater data, or both data combined. For all panels, "inh." stands for inhabitants.

Due to heterogeneous and evolving adaptations of population behavior and institutional measures, epidemic forecasts are typically only meaningful for relatively short time horizons. In fact, it is known that small uncertainties for short-term predictions are amplified over longer periods and the precision drops, similarly to what happens in weather forecasts (Petropoulos and Makridakis, 2020) . Nevertheless, long-term projections that assume no changes in infection dynamics can be useful for counterfactual analysis about the potential effects of current social or pharmaceutical measures and/or changed viral infectivity (Fig. 4a,b) . They can also be used to investigate plausible scenarios, by artificially modifying the model parameters.

J o u r n a l P r e -p r o o f

As for other models applied to complex systems, our projection uncertainties increase with longer time horizons (Fig. 4b) , reflecting the set of potential changes of conditions. Nonetheless, projections based on case numbers or on wastewater data are consistent with each other within error bounds, therefore supporting the possibility of using wastewater data for consistent what-if analysis. In addition, our mechanistic-based model allows assessing the changes in desired precision. Models applied in quickly changing conditions are known to be uncertain (Santosh, 2020) . Similarly, our projections' precision varies depending on whether they are conducted during a rapid increase of case numbers or during stable trends, calling for caution in interpreting these results as plausible projections rather than forecasts. Other examples are reported in Supplementary Fig. 15 . 

Among the purposes of this article is to investigate the utility of wastewater sampling to alert against new waves of infections and to inform its interpretation. It has been suggested by Cao and Francis (2021) wastewater-based predictions. In Fig. 5 and Supplementary Fig. 16 , we plot the predictions about the pandemic trends, obtained from wastewater data (red) and from detected case numbers (yellow), and compared with the true observed evolution (blue). We can then compare if and when the red and yellow curves correctly track the increasing trend of the blue curve. We observe that, overall, the prediction curves accurately increase when a new COVID-19 wave is observed in a region, but the timing might slightly differ depending, e.g., on the testing frequency. This analysis demonstrates the potential of wastewater data to detect incoming increasing trends and quantitatively verifies the recent calls by Bibby et al. (2021) for cautious interpretation: alerts based on wastewater analysis might be just-on-time or even lagging slightly behind the true infection waves. Nonetheless, they are often more advanced than reliable alerts based on case numbers alone, e.g. for Kitchener or Raleigh, despite counterexamples exist (e.g., Kranj). As a result, we suggest that wastewater-based monitoring could be an effective method to detect new waves of infection, but that the lead time should be carefully assessed case-by-case, according to the sampling frequency and other characteristics of the wastewater-analysis pipeline. In short, reliable warnings can be triggered, but it still remains to properly verify how "early". Supplementary Fig. 16 ). We compare 7-days projections from case numbers and from wastewater data with the true detected case numbers.

CoWWAn combines two powerful approaches to process wastewater data in an automated and mechanistic-based manner: an epidemiological SEIR model and an extended Kalman filter, to fit the model parameters adequately and to provide predictions of epidemic trends. This allows for new avenues for wastewater-based epidemic monitoring. In situations of reduced population testing, our approach allows to enhance the performances and robustness of real-time surveillance in a cost-effective manner. Our model can support the reconstruction of the infection curves from wastewater data and allows projections of future trends, in particular close to epidemic resurgence.

Since hospital admission is downstream of the susceptible-exposed-infectious flow , healthcare management as well (D'Aoust et al., 2021; Saguti et al., 2021) can obtain crucial information from an early detection of increasing case numbers supported by quantitative models that account for noise. We recall that our approach provides information on a community level but does not single out the infected individuals, hence it does not enable contact tracing nor does it reveal detailed information like age distribution of cases or infection clusters. Nonetheless, as already proven in our applications for the Luxembourg government, it proves useful to track the evolution of the pandemic, as a complement or even supplement to widespread testing. As a consequence, our results can enhance the SWEEP (Surveillance of Wastewater for Early Epidemic Prediction) framework recently proposed by (Tiwari et al. (2021) However, we acknowledge the limitations of our approach, to be further improved in future studies. To begin with, the reconstruction of case numbers depend on the mean-field SEIR approximation: although meaningful when concentrating on average epidemic trends (Kollepara et al., 2021) , it might yield uncertainties in case of heterogeneous behaviors like clusters. In addition, we observe that tailoring region-specific model parameters is recommended to fine-tune the performance and reduce the uncertainties over the estimates. The parameters can usually be estimated with independent methods or educated prior information, in particular concerning seroprevalence, so we acknowledge that the current set of proposed parameters might not be complete for all countries. As observed in the sensitivity analysis, the projections are somewhat sensitive to the ratio of total and detected cases. This is a shortcoming of every model-based projection. Long-term projections are more influenced by the choice of this parameter, due to potential errors in the estimated level of natural immunity in the population. Short-term projections are less sensitive, since any error in the estimated size of the susceptible population is compensated by the infectivity parameter estimate. In particular, we recommend using reliable estimates for the ratio of true versus detected cases during the model calibration. The best J o u r n a l P r e -p r o o f Journal Pre-proof estimates originate from seroprevalence studies that are able to distinguish between antibodies from previous infection and vaccination, that is, studies that detect antibodies against other parts of the virus than just the spike protein (Suhandynata et al., 2021) . Future discrepancies between wastewater-based estimates and detected cases might be used as indications about changes in the share of detected cases and could be used to trigger a warning against potential undertesting. As for what concern predictions, we observe a close relationship between data and prediction quality:

escalating the sampling precision and rates is essential to improve the model estimations. Finally, although we have paved the way for the assessment of on-line warnings of epidemic waves using model-based predictions, we suggest that future studies further quantify the lead time and the precision/recall. Future analysis may also concentrate on optimising the desired performance and the costs associated, or might focus on expanding the current methodology to other epidemic contexts.

Sampling and analysing SARS-CoV-2 fluxes in wastewater has been suggested as an efficient, non-invasive and cost-effective complement or alternative to testing routines. Our study leverages the potential of wastewater analysis to provide quantitative information for monitoring, alerting and decision-making. By introducing an effective coupling of causal-based models and wastewater sampling, our approach goes beyond statistical methods. In fact, it allows immediate interpretation of its outputs and enables counterfactual analysis, to estimate plausible epidemic scenarios. Overall, the flexibility of our freely available approach, its ease of implementation and its performance make it an important tool for long-term monitoring and support of epidemic mitigation.

The wastewater and case numbers data that support the findings of this study are available from the websites listed in Section 2. 

To embed the SEIR dynamical system in the Extended Kalman filter, we formulate a time-discretised state-space version of the dynamical system Eq. 1 by explicit Euler method:

To obtain the number of daily new infections from the model on a given day, an additional auxiliary state variable () Dt is defined, whose dynamics are given by ( ) = 0, for ,

Dt is the differential counterpart of () 

As argued in the previous section, the state noise () wt can be well approximated as normally (1) corresponds to its respective reaction () j rx. The coefficient  is used to account for modelling errors. In particular, the SEIR model implicitly assumes a homogeneous and perfectly mixed population. This assumption leads to a rather small uncertainty. The coefficient  can also be interpreted as a sensitivity tuning parameter. Lower  leads to higher sensitivity but noisy estimates. Higher  decreases sensitivity but increases robustness against noise. The parameter  has no dynamics through () fx, but it is updated by the Kalman filter. The matrix Q  is otherwise zero, except for the element (6,6) being q  , which acts as a tuning parameter controlling the magnitude of change of () t  in one day.

The measurements from the model are either detected cases on a given day and/or wastewater sampling. To this end, we define possible observation matrices: 

where the sub-indices refer to cases ( c ), wastewater ( w ), and both ( b ). We recall that t c is the share of detected cases on a day t . It is a coefficient that reflects the testing strategy, which often depends on the day (reduced testing on weekends and on public holidays). The empirical measurements are assumed to be noisy, with an additive, normally distributed noise with mean 

where  and  are the transition rates EI  and IR  , respectively, whose inverses are the average duration an infected person remains in E and I compartments.

t on public non-weekend holidays, t c is reduced by a factor of 4 from the value given by Eq. C.4 to account for reduced testing. In case the weekly rhythm is not regular, manual tuning could help improving performance (or estimating t c based on number of performed tests, for example).

The variance of the wastewater measurements w U is estimated from data by where each i t is the time point when wastewater sampling is done. The scaling factor K is either 1/10 when wastewater data is used alone and =1 K when both case and wastewater data are used, as well as for the outlier detection. In the plots of wastewater data reconstruction, =1 K is used for plotting the uncertainty envelope.

Supplementary material (figures and tables) for this article can be found in the accompanying file.

Wastewater surveillance demonstrates high predictive value for COVID-19 infection on board repatriation flights to Australia

Estimating the reproduction number of Ebola virus (EBOV) during the 2014 outbreak in West Africa

Population biology of infectious diseases: Part I

Making waves: Plausible lead time for wastewater based epidemiology as an early warning system for COVID-19

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Modeling SARS-CoV-2 viral kinetics and association with mortality

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Measurement of SARS-CoV-2 RNA in wastewater tracks community infection dynamics

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Dynamical SPQEIR model assesses the effectiveness of non-pharmaceutical interventions against COVID-19 epidemic outbreaks

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High-resolution within-sewer SARS-CoV-2 surveillance facilitates informed intervention

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Surveillance of wastewater revealed peaks of SARS-CoV-2 preceding those of hospitalized patients with COVID-19

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Modeling the number of people infected with SARS-COV-2 from wastewater viral load in northwest Spain

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Virological assessment of hospitalized patients with COVID-2019

Evaluation of lockdown effect on SARS-CoV-2 dynamics through viral genome quantification in waste water

☐ The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work

☒ The authors declare the following financial interests/personal relationships which may be considered as potential competing interests

Daniele Proverbio reports financial support was provided by Luxembourg National Research Fund (FNR)

Francoise Kemp reports financial support was provided by Luxembourg National Research Fund (FNR)

Stefano Magni reports financial support was provided by Luxembourg National Research Fund (FNR)

Leslie Ogorzaly reports financial support was provided by Luxembourg National Research Fund (FNR)

Cauchie reports financial support was provided by Luxembourg National Research Fund

Jorge Goncalves reports financial support was provided by 111 Project on Computational Intelligence and Intelligent Control

The SEIR model is based on the assumption that each susceptible person has probability (Gillespie, 2000) , the binomial distribution can be well approximated by the normal distribution with mean Note that the procedure for the first five weeks is not causal, but some data is anyway needed for