A Wastewater-Based Epidemic Model for SARS-CoV-2 with Application to Three Canadian Cities

epidemic model ; COVID-19 ; environmental surveillance ; wastewater saved on 2021-07-19 15:54:19-04:00 1 / 43 . CC-BY-ND 4.0 International licenseIt 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 (which was not certified by peer review) The copyright holder for this preprint this version posted July 25, 2021. ; https://doi.org/10.1101/2021.07.19.21260773doi: medRxiv preprint NOTE: This preprint reports new research that has not been certified by peer review and should not be used to guide clinical practice. Wastewater-Based Epidemic Modelling 1 Abstract26 The COVID-19 pandemic has stimulated wastewater-based surveillance, allowing public health to track the epidemic by monitoring the concentration of the genetic fingerprints of28 SARS-CoV-2 shed in wastewater by infected individuals. Wastewater-based surveillance for COVID-19 is still in its infancy. In particular, the quantitative link between clinical30 cases observed through traditional surveillance and the signals from viral concentrations in wastewater is still developing and hampers interpretation of the data and actionable32 public-health decisions. We present a modelling framework that includes both SARS-CoV-2 transmission at the34 population level and the fate of SARS-CoV-2 RNA particles in the sewage system after faecal shedding by infected persons in the population.36 Using our mechanistic representation of the combined clinical/wastewater system, we per- form exploratory simulations to quantify the effect of surveillance effectiveness, public-38 health interventions and vaccination on the discordance between clinical and wastewater signals. We also apply our model to surveillance data from three Canadian cities to provide40 wastewater-informed estimates for the actual prevalence, the effective reproduction num- ber and incidence forecasts. We find that wastewater-based surveillance, paired with this42 model, can complement clinical surveillance by supporting the estimation of key epidemio- logical metrics and hence better triangulate the state of an epidemic using this alternative44 data source. saved on 2021-07-19 15:54:19-04:00 2 / 43 . CC-BY-ND 4.0 International licenseIt 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 (which was not certified by peer review) The copyright holder for this preprint this version posted July 25, 2021. ; https://doi.org/10.1101/2021.07.19.21260773doi: medRxiv preprint Wastewater-Based Epidemic Modelling 2 Introduction46 Wastewater has been used previously for monitoring of a wide range of behavioural, socio- economic and biological markers including: medical and illicit drugs [1, 2, 3]; antibiotic48 and antimicrobial resistance [4, 5, 6]; and industrial pollutant chemicals [7, 8]. Spatial and temporal screening of the wastewater collection system or “sewershed” can provide50 qualitative and quantitative information on the marker of interest within the population in a given sewer catchment contributing to the wastewater. The wastewater data when52 used as an index of disease burden can be incorporated into a clinical surveillance program that is purposeful, economical and action-oriented for public health [9]. Wastewater-based54 surveillance (WBS) has also proven to be a low-cost and non-invasive tool for the manage- ment of infectious disease pathogens such as norovirus [10, 11] and poliovirus [12, 13, 14]56 where viral concentration in wastewater served to supplement clinical surveillance. Since the start of the COVID-19 pandemic, SARS-CoV-2 RNA has been detected and quantified58 in sewage in many locations worldwide (as of June 2021, 55 countries have pilot programs for a wastewater surveillance system with 2,287 sampling sites [15]) and was employed60 successfully in correlating the concentration of SARS-CoV-2 in wastewater to clinical cases reported in the sewershed [16, 17, 18, 19, 20, 21, 22, 23, 24]. In some instances of targeted62 surveillance, the leading wastewater signal (measured as SARS-CoV-2 RNA concentration in wastewater) compared to the clinical reports provided an early sign for the introduction64 or resurgence of COVID-19 into a community [25, 26, 27, 28] enabling rapid deployment of public health response and mitigation efforts.66 Despite numerous successes with wastewater-based surveillance during the pandemic, uti- lizing wastewater surveillance data as a public-health tool for quick response remains chal-68 lenging for some jurisdictions, especially at the municipal level [29, 30]. A major hurdle is the lack of a quantitative framework to assess and interpret the wastewater data generated70 and to translate that into public health action [31, 32]. The common practice is to use the detection of SARS-CoV-2 in wastewater as a signal for COVID-19 (re)introduction in a72 community and/or perform trend analysis in parallel with clinical surveillance of COVID- 19. At the time of this manuscript, it is generally not recommended to use SARS-CoV-274 WBS for direct inference of key epidemiological indicators such as prevalence of active infections [31, 32, 16, 33].76 Public health response guided by SARS-CoV-2 levels in wastewater is currently hindered by a lack of structured interpretive criteria, which is at present obscured by the inherent78 complexity and variation imparted by diverse sewersheds and their contributing popula- tions [34, 35, 36]. Sources of data variability includes individual’s shedding dynamics, sam-80 pling frequency of wastewater, non-standardized laboratory methods, sewershed-specific viral degradation and signal attenuation during its journey from the site of faecal shedding82 (and potentially from urinary or sputum deposit [37, 38]) to the sampling point. Attenu- ation of RNA signal in wastewater involves several factors, such as dilution in municipal84 saved on 2021-07-19 15:54:19-04:00 3 / 43 . CC-BY-ND 4.0 International licenseIt 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 July 25, 2021. ; https://doi.org/10.1101/2021.07.19.21260773doi: medRxiv preprint Wastewater-Based Epidemic Modelling wastewater constituents (e.g., storm water effects in combined sewers and infiltration ef- fects in both combined and separated sewers), RNA degradation (e.g., due to household86 detergents and industrial wastewaters) and viral degeneration in the harsh wastewater en- vironment due to temperature, bioactive chemicals, pH, etc. Solids sedimentation and88 resuspension may also play a key role in the transportation and decay of SARS-CoV-2 RNA because of the hydrophobic characteristics of the viral envelope and its strong asso-90 ciations to solids [39, 33]. In addition, concentration methods for detection enhancement and minimization of inhibitory substances of molecular tests can result in some loss of the92 viral target [40, 41]. Here, we present a modelling framework that attempts to link quantified SARS-CoV-294 levels in wastewater with estimates of infections in the population within the sewershed, and to support policy decisions. The model incorporates both the viral transmission within96 the population via a standard epidemiological SEIR-like model (“Susceptible - Exposed - Infectious - Recovered”) [42] and the fate of SARS-CoV-2 in wastewater using a simplified98 hydrological transport framework. To illustrate potential applications, we fit our model to WBS data and traditional clinical reports gathered from six wastewater treatment plants100 (WWTPs) located in three Canadian cities (Edmonton, Ottawa and Toronto) and provide wastewater-informed estimates of key epidemiological metrics. We also perform exploratory102 simulations to investigate how the wastewater signal can be mechanistically associated with clinical surveillance of COVID-19.104 3 Methods We develop a mathematical model that mechanistically describes both the transmission at106 the population level (“above ground”) and the concentration of SARS-CoV-2 in wastewater as a result of faecal shedding from the infected individuals (“below ground”).108 3.1 Transmission between individuals To model SARS-CoV-2 transmission in the population, we use a SEIR-type epidemiolog-110 ical model. The disease progression of individuals is captured through several compart- ments that reflect their epidemiological states and disease outcomes (Table 1). Individuals112 can be susceptible (S ); exposed (infected but not yet infectious, E); symptomatically in- fected who will later become hospitalized ( J) or recovered without hospitalization during114 active COVID-19 ( I); asymptomatically infected (A); hospitalized (H ); those recovered and no longer infectious but still shedding virus in faeces ( Z); fully recovered and per-116 manently immune but not shedding anymore ( R) and deceased (D ). We ignore any mi- gration movements, so at any given time the total population is constant and equal to118 N = S + E + J + I + A + H + Z + R + D. Infection occurs at a time-dependent trans- mission rate βt between infectious (states I, J or A) and susceptible individuals (S ). Once120 saved on 2021-07-19 15:54:19-04:00 4 / 43 . CC-BY-ND 4.0 International licenseIt 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 July 25, 2021. ; https://doi.org/10.1101/2021.07.19.21260773doi: medRxiv preprint Wastewater-Based Epidemic Modelling infected, susceptible individuals enter the latent (non-infectious) state (E ) for an average duration of 1/ϵ days, where no faecal shedding occurs. A proportion α of all infections are122 asymptomatic. A fraction h of symptomatic individuals are hospitalized (H) for an average duration of 1/ℓ days and for those, the COVID-19-associatied mortality is δ. After their in-124 fectious period ends, patients enter the post-infection shedding state Z where SARS-CoV-2 faecal shedding still occurs for 1/η days on average. The exposed ( E), infectious (A, I and126 J) and post-infection shedding ( Z) states are modelled with a series of sub-compartments in order to have their respective sojourn time gamma-distributed [43, 44, 45]. Note that in128 our model, we make the simplifying assumption that hospitalized patients–assumed mostly bedridden and a small fraction of the shedding population–do not contribute significantly130 to faecal shedding. The transmission dynamics are represented by the system of differential equations 1a-1n132 and illustrated in Figure 1. ˙S = −βtS (A +I +J)/N (1a) ˙E1 =βtS (A +I +J)/N −nEεE1 (1b) ˙Ek =nEε(Ek−1 −Ek) 2 ≤k ≤nE (1c) ˙A1 =αεE −nAθA1 (1d) ˙Ak =nAθ(Ak−1 −Ak) 2 ≤k ≤nA (1e) ˙I1 = (1 −h) (1 −α)εE −nIνI1 (1f) ˙Ik =nIν(Ik−1 −Ik) 2 ≤k ≤nI (1g) ˙J1 =h(1 −α)εE −nJµJ1 (1h) ˙Jk =nJµ(Jk−1 −Jk) 2 ≤k ≤nJ (1i) ˙H =µnJJnJ −ℓH (1j) ˙Z1 =nIνInI +nAθAnA −nZηZ1 (1k) ˙Zk =nZη(Zk−1 −Zk) 2 ≤k ≤nZ (1l) ˙R =nZηZnZ + (1 −δ)ℓH (1m) ˙D =δℓH (1n) 134 where A = ξ ∑nA k=1 φkAk, I = ∑nI k=1 ψkIk and J = ∑nJ k=1 ψkJk. We use the dot notation to symbolize derivation with respect to time (e.g., ˙S = dS/dt).136 saved on 2021-07-19 15:54:19-04:00 5 / 43 . CC-BY-ND 4.0 International licenseIt 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 July 25, 2021. ; https://doi.org/10.1101/2021.07.19.21260773doi: medRxiv preprint Wastewater-Based Epidemic Modelling S E1:nE I1:nI J1:nJ A1:nA Z1:nZ H R D Faecal shedding in wastewater (W ∗) Reported SARS-CoV-2 concentration in wastewater ( W ) β h(1 −α) (1 −h)(1 −α) α 1− δ δ Deca y & Delay (g,κ,ℓ ww) Figure 1: Diagram of compartmental model. See main text for a description of the epidemiological states. The notation 1 : n• indicates a modelling using n• sub-compartments to obtain a gamma- distributed sojourn time in the associated epidemiological state. The parameters φ and ψ are multiplicative adjustments to the baseline transmission rate βt to represent the infectious profile during the course of infection. The values for φk and138 ψk were chosen to represent the best estimate of the temporal infectiousness profile given the different results published (see Appendix, Figure S2). The parameter ξ models the140 relative infectiousness of asymptomatic cases compared to symptomatic ones. The effective reproduction number of this model is (see Appendix for details on its calculation):142 Rt = βt ( αξ 1 θ + (1− h)(1− α) 1 ν + h(1− α) 1 µ ∑nJ k=1 ψk nI ) St N (2) 3.2 SARS-CoV-2 viral concentration in wastewater 3.2.1 Deposited Viral Concentration144 The daily concentration of SARS-CoV-2 in wastewater is directly calculated from the total number of individuals that are actively shedding into the sewage system. SARS-CoV-2146 faecal shedding varies according to the infected individual’s clinical state and disease out- comes. Depending on the disease progression, infected individuals shed a variable amount148 saved on 2021-07-19 15:54:19-04:00 6 / 43 . CC-BY-ND 4.0 International licenseIt 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 July 25, 2021. ; https://doi.org/10.1101/2021.07.19.21260773doi: medRxiv preprint Wastewater-Based Epidemic Modelling of SARS-CoV-2 while they are in the shedding states ( A, I, J and Z ). The total concen- tration of SARS-CoV-2 RNA entering the wastewater at time t is given by150 W ∗(t) = ω× ( nJ∑ k=1 λkJk(t) + nI∑ k=1 λkIk(t) + ξ nA∑ k=1 λA k Ak(t) + nZ∑ k=1 λZ kZk(t) ) (3) The parameters λk, λA k and λZ k represent SARS-CoV-2 faecal shedding dynamics per capita when the infected individual is in the epidemiological states I, J, A and Z respectively.152 Given the current lack of observational data, we used the same parameters λk for all epidemiological states (i.e., λk = λA k = λZ k). Values for the parameters λk were chosen154 to represent mid-range values of the different results published (see Appendix Figure S2). Note that we assume the same reduction in faecal shedding as in respiratory shedding for156 asymptomatic cases (parameter ξ). The parameter ω implies that our model can only determine up to a constant the concentration of SARS-CoV-2 in wastewater [14], even if158 the limit of detection of the assay is known. This reflects our current inability to quantify the various complex processes that affect the concentration, from patients’ shedding to160 the concentration measured in laboratories ( e.g., frequency and timing of sampling, RNA degradation in the sewer system, recovery efficiency of assays).162 3.2.2 RNA transport and sampled viral concentration We use a simple advection-dispersion-decay model to simulate the fate of SARS-CoV-2164 along its journey in wastewater from the shedding points to the sampling site. This model is a combination of an exponential viral decay [46] and aτ-day dispersed plug-flow function,166 g(τ), representing all possible hydrodynamic processes ( e.g., dilution, sedimentation and resuspension) that leads to RNA degradation as well as decrease and delay of signal at the168 time of sampling. The dispersed plug-flow g(τ) acts as a transformation function, which reshapes the initial deposited concentration, W ∗, into a delayed viral distribution over τ170 days as a result of the transit of SARS-CoV-2 in the sewer system. Hence, we defined the sampled viral concentration at time t as:172 Wsamp(t) = ∫ t 0 W ∗(t− τ) g(τ) e−κτdτ, (4) where κ is the daily decay rate of SARS-CoV-2 due to the harsh, complex and bioactive environment of wastewater [46]. The SARS-CoV-2 RNA concentration entering the sewage174 system daily is modelled as a single hydrodynamic pulse per day and the plug-flow func- tion, g, is obtained by the analytical solution of the axial dispersed plug flow differential176 equation [47]. We then re-parametrize the analytical solution with the mean delay time ¯ τ and its standard deviation σ into a Gaussian distribution178 g(τ) = 1 √ 2πσ exp ( −(τ− ¯τ)2 2σ2 ) . (5) saved on 2021-07-19 15:54:19-04:00 7 / 43 . CC-BY-ND 4.0 International licenseIt 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 July 25, 2021. ; https://doi.org/10.1101/2021.07.19.21260773doi: medRxiv preprint Wastewater-Based Epidemic Modelling See the Appendix for more details on our advection-dispersion-decay model of the RNA transport.180 3.2.3 Wastewater reported sample The sample transportation, laboratory processing time and reporting lags, introduce re-182 porting delays of RNA concentration in wastewater. Hence, we define the reported wastew- ater concentration as184 W (t) = Wsamp(t− ℓww), (6) where ℓww is the reporting lag between wastewater sampling and concentration report after laboratory analysis. (Note that the reporting delay of the wastewater measurement is186 independent from the delay caused by the transport of RNA particles in the sewer system as defined in Equation 4).188 3.3 Clinical reported cases We also model surveillance data derived from laboratory confirmed and clinically diagnosed190 COVID-19 cases, acknowledging that instantaneous identification and complete reporting after initial infection is not possible. We assume that a fraction ρ of symptomatic incidence192 is reported with a lag of aℓclinical days from the time of infection. If i(t) is the total incidence at time t, we define the number of clinical cases reported at time t as:194 C(t) = ρ (1− α) i(t− ℓclinical), (7) 3.4 Wastewater and clinical surveillance data We apply our modelling framework to data sets from six wastewater sampling sites located196 in three Canadian cities: Edmonton (Alberta), Ottawa (Ontario) and Toronto (Ontario). Sampling sites are the following municipal WWTPs (abbreviation / approximate popula-198 tion served): Gold Bar in Edmonton (EGB / 900,000); Robert O. Pickard Environmental Centre in Ottawa (OTW / 1,000,000 [48]); Toronto Ashbridges Bay (TAB / 1,603,700 [49]);200 Toronto Humber (THU / 685,000 [50]); Toronto Highland Creek (THC / 533,000 [51]); and Toronto North Toronto (TNT / 252,530 [52]).202 3.4.1 Data collection Wastewater samples were collected approximately two (Edmonton and Toronto) to seven204 (Ottawa) times a week. The sampling location was at the influent of the wastewater treatment plant of each city. Wastewater samples were collected before de-gritting in206 Toronto, and after for Edmonton and Ottawa. saved on 2021-07-19 15:54:19-04:00 8 / 43 . CC-BY-ND 4.0 International licenseIt 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 July 25, 2021. ; https://doi.org/10.1101/2021.07.19.21260773doi: medRxiv preprint Wastewater-Based Epidemic Modelling Wastewater samples from Edmonton and Toronto were shipped to the National Micro-208 biology Laboratory in Winnipeg, Manitoba, where SARS-CoV-2 RNA concentration was measured. RNA from wastewater samples was purified using two methods. Prior to Febru-210 ary 12th 2021, 15 mL of clarified supernatant (after 4000 × g centrifugation for 20 min at 4 ◦C), was concentrated using an ultracentrifugal filter device (4000 × g for 35 min at212 4◦C) (Amicon Ultra-15, 10 kDa MWCO, Millipore-Sigma, St. Louis, MO, U.S.A). Total RNA was extracted from the resultant concentrate (∼200 µL) using the MagNA Pure 96214 DNA and Viral NA Large Volume Kit (Roche Diagnostics, Laval, QC) using the Plasma External Lysis 4.0 protocol as per manufacturer instructions. After February 12th 2021,216 the pellet resultant from clarifying (4000g for 20 minutes at 4 ◦C) 30 mL of wastewater was resuspended in 700 µL Qiagen Buffer RLT (Qiagen, Germantown, MD) containing 1%218 2-mercaptoethanol. To this, 200 µL of 0.5 mm zirconia-silica beads (Biospec, Bartlesville, OK) were added and the sample was processed with a Bead Mill 24 Homogenizer (Fisher220 Scientific, Ottawa, ON) using 4 × 30s pulses at 6 m/s, then clarified by centrifugation (12000× g, 3 min) and the resultant lysate used for RNA extraction using the MagNA222 Pure 96 instrument as described above. Viral RNA was quantified using RTq-PCR with the US-CDC N1 and N2 primers.224 For Ottawa, daily 24-hour composite primary sludge samples, consisting of four discrete samples collected at 6 hour intervals and subsequently mixed, were collected and trans-226 ported on ice to the University of Ottawa, where samples were analyzed within 24 hours of reception. Samples were concentrated by centrifugation at 10,000g for 45 minutes and228 RNA was extracted from a 250 mg portion of the resulting pellet using a modified ver- sion of the Qiagen RNeasy PowerMicrobiome kit [28]. Quantification was performed using230 singleplex probe-based RTq-PCR for the N1 and N2 gene regions of the virus. For Edmonton and Toronto, the SARS-CoV-2 concentrations in wastewater were normal-232 ized by the total solid suspension (TSS) measured on the day the sample was collected at the treatment plant. For Ottawa, they were normalized with the concentration of the234 Pepper mild mottle virus measured in the sample. For all cities, the reported viral con- centration used in the model ( W ) was the average normalized concentration across all236 technical replicates for both the N1 and N2 genes. We obtained clinical cases and hospital admissions (except for Toronto) for the catchment238 area of each of the six wastewater treatment plants. Hence, we were able to link clinical and wastewater surveillances. The data sets for the three cities are plotted in Figure 2240 Seroprevalence values at the city and province level were obtained from Canadian Blood Services (CBS) [53]. The wastewater and clinical surveillance data used in this study are242 available in Supplementary File S1. saved on 2021-07-19 15:54:19-04:00 9 / 43 . CC-BY-ND 4.0 International licenseIt 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 July 25, 2021. ; https://doi.org/10.1101/2021.07.19.21260773doi: medRxiv preprint Wastewater-Based Epidemic Modelling THC THU TNT EGB OTW TAB 1−Feb '20 1−Apr '20 1−Jun '20 1−Aug '20 1−Oct '20 1−Dec '20 1−Feb '21 1−Apr '21 1−Jun '21 1−Feb '20 1−Apr '20 1−Jun '20 1−Aug '20 1−Oct '20 1−Dec '20 1−Feb '21 1−Apr '21 1−Jun '21 1−Feb '20 1−Apr '20 1−Jun '20 1−Aug '20 1−Oct '20 1−Dec '20 1−Feb '21 1−Apr '21 1−Jun '21 Data type Case report Hospital admission Wastewater concentration Figure 2: Data sets used in this study for Edmonton, Ottawa and Toronto. Each horizontal panel is a city and colors represent the type of data (reported cases, hospital admissions and SARS-CoV-2 RNA concentration in wastewater). All curves were normalized to 1 (dividing by their respective maximum value) to plot them in one single panel to facilitate visual comparison. All data sets used in this study are available in Supplementary File S1. saved on 2021-07-19 15:54:19-04:00 10 / 43 . CC-BY-ND 4.0 International licenseIt 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 July 25, 2021. ; https://doi.org/10.1101/2021.07.19.21260773doi: medRxiv preprint Wastewater-Based Epidemic Modelling 3.4.2 Fit to data244 We use an Approximate Bayesian Computation (ABC) algorithm [54] to fit the unknown or unobserved model parameters to the available data. For each ABC prior iteration, the246 error function is defined as a weighted trajectory matching ei = wC(C− Cobs)2 + wH(H− Hobs)2 + wW (W− Wobs)2 (8) We use 50,000 prior ABC iterations and retain the 100 smallest errors to generate posterior248 distributions (acceptance ratio 2 × 10−3). The parameters fitted are the time-dependent transmission rate βt, ω, the hospitalization rate h and the mean transit time ¯ τ. More250 details about the fitting procedure is given in Appendix. We define three types of fitting-to-data procedures. “Clinical” when wC = wH = 1 and252 wW = 0, to use data from clinical sources only ; “WW” when wW = 1 and wH = wC = 0, to use wastewater data only; and finally “Combined” by choosing the weights wC, wH and254 wW such that the contribution of each error term (in Equation 8) are, on average, approx- imately equal. The “Combined” fitting procedure aims to have approximately the same256 contribution from clinical and wastewater data sources (despite the different observation frequencies).258 3.4.3 Inference of unobserved epidemiological quantities For a given location, once the model is fitted data, we can infer unobserved quantities of260 epidemiological importance by generating epidemic trajectories from the posterior samples. The posterior prevalence distribution (at each time point) is defined by simply adding the262 populations from the compartments representing active infection, that is prev(t) = E + nA∑ i=1 Ai + nI∑ i=1 Ii + nJ∑ i=1 Ji + H (9) The posterior cumulative incidence is obtained by summing Equation 1a until time t264 cuminc(t) =− t∑ i=1 ˙Si (10) The fitted model can also provide an estimate of the effective reproduction number from the different data sources (e.g., clinical and/or wastewater) using Equation 2.266 3.5 Simulations 3.5.1 Detection timing differential268 In order to explore wastewater-based surveillance as a leading indicator of infection in the community, the time when SARS-CoV-2 is first reported from wastewater is noted270 saved on 2021-07-19 15:54:19-04:00 11 / 43 . CC-BY-ND 4.0 International licenseIt 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 July 25, 2021. ; https://doi.org/10.1101/2021.07.19.21260773doi: medRxiv preprint Wastewater-Based Epidemic Modelling dww and defined as W (t = dww) = LOD where LOD is the limit of detection of the lab- oratory method. In addition, the time dclinical when COVID-19 is first reported is defined as272 C(t = dclinical) = 1. Finally, we define the reported detection differential ∆ =dww− dclinical. As a result, the wastewater signal can be classified as a leading indicator over traditional274 clinical surveillance when ∆ < 0. We assess how the reported detection differential ∆ can be impacted by varying model parameters that would typically differ from one community-276 sewer system to another. We select only three combinations of parameters (among many) to illustrate how ∆ can be affected, and most importantly how its sign can change indi-278 cating its transition between a leading and lagging indicator. We consider two levels of COVID-19 reporting, with ρ = 30% to reflect a relatively inefficient clinical surveillance280 system in the population, and ρ = 70% that represents a more efficient one. 3.5.2 Impact of vaccination282 Although the model presented here does not explicitly have a vaccination process, a mech- anism using the existing framework can be implemented to mimic the main effects of an284 infection-permissive vaccine (as this is the case for the COVID-19 vaccines currently avail- able). We model a simple scenario that rolls out an infection permissive vaccine by gradu-286 ally decreasing the transmission rate (β) by 70% over 50 days and increasing the proportion of asymptomatic infection (α ) from 30% to 90%. This reflects the growing protection of288 the population from severe outcomes of COVID-19 as well as decrease in transmissions as the vaccine is administered. To assess the differential impact of vaccination on clinical and290 wastewater observations, we consider the ratio of the level of SARS-CoV-2 in wastewater over the reported clinical cases, W (t)/C(t).292 4 Results We present our results in two sections. First, we apply our modelling framework to wastew-294 ater and clinical surveillance data from six sampling sites located in three Canadian cities (Edmonton, Ottawa and Toronto) and infer epidemiological parameters such as prevalence,296 effective reproduction number and incidence forecast. The second section is based on ex- ploratory simulations (not fitted to data of a specific location) that highlight important298 mechanistic aspects between clinical and wastewater surveillances. 4.1 Application to Canadian surveillance data300 In this section, we compare the inferences made on key epidemiological variables by fitting the model to different data sources. Our goal is to assess the added-value of the wastewater-302 based data stream. Hence, in the following, we present inferences by fitting the model to different data sources available (“Clinical”, “WW” and “Combined”) and comparing the304 outcomes. saved on 2021-07-19 15:54:19-04:00 12 / 43 . CC-BY-ND 4.0 International licenseIt 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 July 25, 2021. ; https://doi.org/10.1101/2021.07.19.21260773doi: medRxiv preprint Wastewater-Based Epidemic Modelling 4.1.1 Prevalence estimates306 Figure 3 shows, for selected locations, the SARS-CoV-2 prevalence estimated by sampling the posterior distributions fitted to the various data sources “Clinical”, “WW” and “Com-308 bined” and the evaluation of Equation 9. The right-most panel also displays estimates of cumulative incidence (Equation 10) compared to available SARS-CoV-2 seroprevalence lev-310 els estimated from surveys by the Canadian Blood Services performed on banks of blood donors in Edmonton, Ottawa and Toronto [53]. Note that our model was not fitted to312 seroprevalence data and this comparison acts as a crude check that prevalence estimates from the model are approximately consistent with other independent data sources. For all314 locations, as expected, wastewater-only prevalence estimates are close to the clinical-only ones when the levels of SARS-CoV-2 in wastewater mimic the COVID-19 trends in the316 population (Figure 2). For example, the prevalence estimated from wastewater-only and clinical-only are comparable for the December 2020 wave in Edmonton and April 2021 wave318 in Ottawa. However, when the clinical and wastewater signals are discordant, prevalence estimates can be significantly different. For example, wastewater-based prevalence esti-320 mates in January 2021 for Toronto Highland Creek (THC) do not show the peak seen from clinical observations. On the other hand, this January peak was captured in Toronto Ash-322 bridges Bay (TAB)–another part of the city–and the subsequent March-May 2021 wave in Toronto Highland Creek (THC) was identified by both wastewater and clinical surveillance.324 Finally, we note that, because of the larger variability of SARS-CoV-2 WBS and/or their lower sampling frequency as compared to daily clinical surveillance, credible intervals of our326 wastewater-only inferences are generally larger than the clinical-only ones (Figure 3). 4.1.2 Effective reproduction number328 The effective reproduction number (Rt) is a key epidemiological parameter that has gained recognition and application during the COVID-19 pandemic [55, 56]. Using the same ap-330 proach as for prevalence estimates, we inferred Rt from epidemic trajectories generated from posterior distributions fitted to the three different data sources ( i.e., “Clinical”,332 “WW” and “Combined”) and Equation 2. For comparison, we also calculate Rt using the popular R package EpiEstim (version 2.2) [57] from the reported clinical cases as334 a separate approach. Results shown in Figure 4 exhibit the same behaviour as for the prevalence estimates, that is, mean estimates of Rt are similar when trends of clinical336 and wastewater surveillance are comparable. Despite being based on a different modelling framework, estimates from EpiEstim are consistent with clinical-only estimates from our338 model. Finally, like for prevalence inferences, Rt estimates from wastewater-only data (Figure 4, blue solid line) tend to have broader uncertainty interval compared to Rt from340 clinical-only data. saved on 2021-07-19 15:54:19-04:00 13 / 43 . CC-BY-ND 4.0 International licenseIt 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 July 25, 2021. ; https://doi.org/10.1101/2021.07.19.21260773doi: medRxiv preprint Wastewater-Based Epidemic Modelling 0 10,000 20,000 30,000 Prevalence A 0 10,000 20,000 30,000 40,000 Apr '20 Jun '20 Aug '20 Oct '20 Dec '20 Feb '21 Apr '21 Jun '21 Credible interval width B EGB Apr '20 Jun '20 Aug '20 Oct '20 Dec '20 Feb '21 Apr '21 Jun '21 10 100 1,000 10,000 100,000 Cumulative incidence C 0 5,000 10,000 15,000 Prevalence D 0 5,000 10,000 15,000 20,000 25,000 Mar '20 May '20 Jul '20 Sep '20 Nov '20 Jan '21 Mar '21 May '21 Credible interval width E OTW Mar '20 May '20 Jul '20 Sep '20 Nov '20 Jan '21 Mar '21 May '21 10 100 1,000 10,000 100,000 Cumulative incidence F 0 10,000 20,000 30,000 40,000 Prevalence G 0 10,000 20,000 30,000 40,000 Feb '20 Apr '20 Jun '20 Aug '20 Oct '20 Dec '20 Feb '21 Apr '21 Jun '21 Credible interval width H TAB Feb '20 Apr '20 Jun '20 Aug '20 Oct '20 Dec '20 Feb '21 Apr '21 Jun '21 10 100 1,000 10,000 100,000 Cumulative incidence I 0 5,000 10,000 15,000 Prevalence J 0 5,000 10,000 15,000 Apr '20 Jun '20 Aug '20 Oct '20 Dec '20 Feb '21 Apr '21 Jun '21 Credible interval width K THC Apr '20 Jun '20 Aug '20 Oct '20 Dec '20 Feb '21 Apr '21 Jun '21 10 100 1,000 10,000 100,000 Cumulative incidence L sourcefit clin combined ww Figure 3: SARS-CoV-2 prevalence estimates. Each quadrant block represents a location. The top- left panel of each quadrant block (panels A,D,G,J) shows the estimates of SARS-CoV-2 prevalence time series. Each colour represents the different data sources used to fit the model (dark red: “Clinical”, pink: “Combined”, blue: “WW”) . The lines show the mean estimate of prevalence. The shaded ribbon indicates the 95% CrI for the estimate fitted on the “Combined” data set (CrIs for other data sources are omitted for clarity). The bottom-left panel of each quadrant block (panels B,E,H,K) represents the width of the 95% CrI for the estimates fitted on the different data sets (using the same colour code as the panel for prevalence). The right-most panel of each quadrant block (panels C,F,I,L) compares the cumulative incidence estimated by the model fitted on the “Combined” data set to seroprevalence levels reported by the Canadian Blood Services for each city (grey point indicates the mean, the vertical grey bars show the 95% confidence intervals). saved on 2021-07-19 15:54:19-04:00 14 / 43 . CC-BY-ND 4.0 International licenseIt 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 July 25, 2021. ; https://doi.org/10.1101/2021.07.19.21260773doi: medRxiv preprint Wastewater-Based Epidemic Modelling THC THU TNT EGB OTW TAB Aug '20 Dec '20 Apr '21 Aug '20 Dec '20 Apr '21 Aug '20 Dec '20 Apr '21 0.50 1.00 1.50 2.00 0.50 1.00 1.50 2.00 0.0 1.0 2.0 0.50 1.00 1.50 2.00 0.0 1.0 2.0 3.0 0.50 1.00 1.50 2.00 Effective reproduction number sourcefit clin EpiEstim combined ww Figure 4: Effective reproduction number. Each panel represents a wastewater treatment plant. Solid lines represent the mean of effective reproduction number estimated with different methods (model presented here and, in grey, EpiEstim) and data sources (colour-coded). The ribbon indicates the 95% credible interval for the “Clinical” data source. Note that wastewater data are available from early October 2020 for Edmonton and Toronto. The Rt curves were spline-smoothed, see Appendix for details. saved on 2021-07-19 15:54:19-04:00 15 / 43 . CC-BY-ND 4.0 International licenseIt 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 July 25, 2021. ; https://doi.org/10.1101/2021.07.19.21260773doi: medRxiv preprint Wastewater-Based Epidemic Modelling 0 1000 2000 3000 4000 Sep '20Oct '20Nov '20Dec '20Jan '21Feb '21Mar '21Apr '21May '21 reported clinical cases EGB 0 100 200 300 Sep '20 Oct '20 Nov '20 Dec '20 Jan '21 reported clinical cases THC 0 300 600 900 1200 Sep '20 Oct '20 Nov '20 Dec '20 Jan '21 reported clinical cases OTW Figure 5: Forecast examples for Edmonton (left panel), Toronto/Highland Creek (middle panel) and Ottawa (right panel). Filled points represent past data of reported clinical cases. Circles represent reported clinical cases not yet observed at the time of forecast. Colour represents the type of data the model was fitted to (blue, wastewater only; red, clinical data only). Dashed coloured lines indicate the fitted mean for reported cases. The thick solid line shows the 1-month-ahead mean forecast, and the shaded areas their respective 95%CrI. 4.1.3 Forecasts342 Another key advancement in our modelling framework is to generate forecasts based on clinical, hospital (if available) and wastewater data.344 In Figure 5 we show three 1-month-ahead forecasting examples for Edmonton, Toronto Highland Creek and Ottawa using either wastewater data only, or clinical data only. The346 Edmonton example (Figure 5, left panel) shows forecasts made as of April 1st, 2021. In this case, the forecasts are relatively similar because both the clinical reports and the348 wastewater signals are comparable and the model fits were similar using either wastewater or clinical data. The Toronto Highland Creek example forecasts as of November 30th,350 2020 (Figure 5, middle panel). For this location, the wastewater signal and clinical reports are discordant from December 2020 to February 2021. During this period the wastewater352 concentration is low and approximately flat whereas the clinical reports indicate a new wave of infections. As a result, the model fitted on these two data sources interprets the354 epidemic differently for this period and hence provides contrasting forecasts. The forecast for Ottawa as of December 15th, 2020 (Figure 5, right panel) illustrates the case when356 the wastewater forecast is more accurate than the one based on clinical surveillance only. At that time, the wastewater signal in Ottawa has picked up a resurgence earlier than358 clinical surveillance. This resurgence is then captured by the model fit and hence the wastewater-based forecast correctly projects the resurgence.360 saved on 2021-07-19 15:54:19-04:00 16 / 43 . CC-BY-ND 4.0 International licenseIt 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 July 25, 2021. ; https://doi.org/10.1101/2021.07.19.21260773doi: medRxiv preprint Wastewater-Based Epidemic Modelling 4.2 Simulations In this section, we report results from simulations that provide general insights in inter-362 preting WBS. 4.2.1 Leading signal and reported detection differential364 We vary the LOD of the wastewater assay across a broad, but realistic, range [58] and calculate ∆, the detection time difference, for each simulation for a given value of LOD366 and the clinical reporting rate ρ. Because SARS-CoV-2 RNA concentration in wastewater can only be determined up to a constant in our model, the LOD values chosen here are368 rescaled to the parameters used to run our simulations and cannot be directly interpreted as RNA copies per ml, the traditional unit for LOD. Panel A in Figure 6 shows that,370 depending on the LOD of the laboratory assay, the wastewater concentration of SARS- CoV-2 RNA can either be a leading (∆ < 0 for assays with low LODs) or a trailing372 indicator of cases (re)introduction when compared to reported clinical cases. This is the case whether the clinical surveillance system in the population is efficient or not (coloured374 curves, Figure 6A). We also vary the decay rate of RNA SARS-CoV-2 in wastewater within a broad realistic376 range [46, 59, 60], as well as the transit time of SARS-CoV-2 between the shedding and sampling sites. Figure 6B shows that , here again, the relative timing of (re)introduction378 detection by WBS compared to clinical surveillance can be affected by both the harshness of the wastewater (represented by the decay rate) and the transit time of SARS-CoV-2 in380 the sewer system. We note, as expected, that with a fast transit time (illustrated by a 1- day travel time in the left-most panel of Figure 6) the decay rate will not have a significant382 impact on ∆ clinical surveillance (efficient, ρ = 70%, or not, ρ = 30%), but as the transit time increases to 3 days (an upper bound considering strong sediment and recirculation384 effects) the effect of RNA decay becomes more important (increasing slope for the 1-day and 3-day transit times, Figure 6B).386 4.2.2 Assessing public health intervention effectiveness The ability to use wastewater signal to detect intervention effectiveness can also be explored388 using model simulations in order to explore if a public health intervention is more clearly observable in wastewater concentration measurements or in clinical reports. To do this,390 we choose to model an intervention such that the transmission rate β, constant until the intervention time, decreases linearly to a lower value β/3 during a period of time of392 Tinterv and remains constant afterwards (this aims to crudely simulate a lockdown). We consider the relative difference between the observed peak and 7 days later for COVID-394 19 by clinical surveillance, scl(t) = C(t + 7)/C(t)− 1 and the level of SARS-CoV-2 in wastewater, sww = W (t + 7)/W (t)− 1. The more negative the slope, the clearer the396 saved on 2021-07-19 15:54:19-04:00 17 / 43 . CC-BY-ND 4.0 International licenseIt 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 July 25, 2021. ; https://doi.org/10.1101/2021.07.19.21260773doi: medRxiv preprint Wastewater-Based Epidemic Modelling ww late ww early −15 −10 −5 0 5 10 15 500 1000 1500 2000 WW limit of detection Detection time difference (in days) Δ A ww late ww early ww late ww early Mean transit time = 1 day Mean transit time = 3 days 0.00 0.25 0.50 0.75 1.000.00 0.25 0.50 0.75 1.00 −10 0 10 Decay rate Detection time difference (in days) Δ B Reporting proportion 0.3 0.7 Figure 6: Simulations were run varying selected parameters to show their impact on the reported detection time differential ( ∆). Panel A: effect of the limit of detection of the quantification assay performed on wastewater. Values below the 0-intercept horizontal dashed line indicate a leading signal from wastewater concentrations than from clinical reports. Panel B: effect of the SARS- CoV-2 RNA decay rate in wastewater for different transit times between the shedding and sampling site. The colour of the curves represents the proportion of clinical cases reported ( ρ) out of the total symptomatic incidence. signal. Using our baseline parameters (Table 2), we simulate an intervention that reduces the398 contact rate to a third of its pre-intervention value at different time of the simulations, ranging from 20 to 90 days after the introduction of the index case. Figure 7 shows400 that, overall, the effect of an intervention that significantly reduces transmission yields a larger relative decrease in number of COVID-19 cases than the level of SARS-CoV-2 in402 wastewater because the post-peak slope from clinical surveillance ( scl) is consistently more negative than the one from WBS ( sww). The difference is more pronounced as the change404 in transmission is more sudden (Tinterv small). Hence, given the observation noise typically encountered, we expect that the effect of a sudden change in transmission rate would be406 more clearly observable from clinical surveillance than from WBS. 4.2.3 Differential impact of vaccination408 In Figure 8 shows how W (t)/C(t), the ratio of reported wastewater concentration over reported cases, increases following vaccination with a infection-permissive vaccine. Indeed,410 while an infection-permissive vaccine does reduce transmission, it still allows for infections to occur (mostly asymptomatic) and in particular, faecal shedding. Hence, a smaller pro-412 portion of infections are reported (because most of them are asymptomatic) but faecal saved on 2021-07-19 15:54:19-04:00 18 / 43 . CC-BY-ND 4.0 International licenseIt 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 July 25, 2021. ; https://doi.org/10.1101/2021.07.19.21260773doi: medRxiv preprint Wastewater-Based Epidemic Modelling 50 60 70 80 90 simulation time normalized value A T_interv = 3 days T_interv = 10 days T_interv = 20 days 20 40 60 80 20 40 60 80 20 40 60 80 20% 40% 60% start time of intervention relative reduction after post−intervention peak B data type clinical wastewater Figure 7: Detectability of a sharp transmission reduction. Panel A: example of how the post-peak relative changes are calculated. The colour-coded dashed lines represent the time series of reported clinical cases and SARS-CoV-2 RNA concentration in wastewater. The shaded area indicates when the transmission rate linearly decreases to a third of its baseline value (here, Tinterv = 10 days). The segment illustrates the relative change between the peak value and 7 days later ( sww and scl). Panel B: the horizontal axis represents the time (since the start of the epidemic) when starts the intervention that reduces transmission to a third of its value. The vertical axis represents the post- peak relative changes from clinical reports (s cl, red lines) or wastewater ( sww, blue lines). Each panel indicates a different value (3, 10 and 20 days) for Tinterv, the time it takes to reduces the transmission rate to a third of its initial value. saved on 2021-07-19 15:54:19-04:00 19 / 43 . CC-BY-ND 4.0 International licenseIt 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 July 25, 2021. ; https://doi.org/10.1101/2021.07.19.21260773doi: medRxiv preprint Wastewater-Based Epidemic Modelling shedding is less affected by this reporting bias and level of SARS-CoV-2 in wastewater414 decreases less steadily than COVID-19 surveillance. In this simulation, the approximately constant ratio before the start of vaccination (Figure 8B) indicates that reports from clin-416 ical or wastewater data sources provide a similar picture of the epidemic for that period. However, once vaccination is implemented, the increasing ratio highlights a discordance418 between the two data sources. 5 Discussion420 Surveillance through detection and quantification of targeted pathogens in wastewater has been a noteworthy tool for public health across the world [61, 16, 18, 10, 11, 12, 13, 14].422 While pathogen surveillance in wastewater is not new, the scale and urgency of scientific development for WBS are witnessed during the unprecedented COVID-19 pandemic. Be-424 cause of the novelty of SARS-CoV-2-related WBS and the lack of quantitative tools for analysis, the interpretation of levels SARS-CoV-2 in wastewater and their translation into426 actionable public health measures is still challenging [31, 32, 16, 33]. Here, we have provided a modelling framework to improve the understanding of the mech-428 anisms at play between the viral transmission in the population and viral concentration shed in wastewater. This model can also provide estimates of key unobserved epidemiolog-430 ical parameters. We demonstrated the applicability of our model by fitting it to data from three Canadian cities and made wastewater-informed inferences of important epidemio-432 logical metrics (prevalence, effective reproduction number and forecasted incidence). Our estimates for cumulative incidence were approximately in line with seroprevalence levels434 from cohorts of blood donors independently observed in the three cities. Importantly, we observed that estimates based on wastewater-only data usually provide a436 similar picture of the epidemic trajectory (Figure 3) but discordant signals can occur and lead to drastically different interpretations. This was the case, for example, in January438 2021 in Toronto Highland Creek where the wastewater signal did not indicate a resurgence of infections, despite the wave observed from the reported clinical cases. We believe this440 muted peak in wastewater signal was not caused by a laboratory issue, but rather from undetermined events in this particular sewershed at that specific time that need to be442 further investigated. Similarly, the effective reproduction numbers inferred from wastewater data only are consis-444 tent with more traditional methods, such as using clinical reports with the popular software EpiEstim (Figure 4). The fact that wastewater data can potentially act as a substitute446 for clinical surveillance (albeit with more uncertainty) to provide critical epidemiological metrics is encouraging, although more realistically, it will likely act as a complementary448 data source. The ability to estimate epidemiological metrics using wastewater surveillance represents a step forward in demonstrating the use of wastewater data for actionable pub-450 saved on 2021-07-19 15:54:19-04:00 20 / 43 . CC-BY-ND 4.0 International licenseIt 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 July 25, 2021. ; https://doi.org/10.1101/2021.07.19.21260773doi: medRxiv preprint Wastewater-Based Epidemic Modelling Vaccination Baseline 25 50 75 100 125 rescaled value (log scale) Outcome Clinical cases (C) Wastewater concentration (W) A Vaccination starts 1 3 10 25 50 75 100 125 simulation time log( W / C ) Scenario Baseline Vaccination B Figure 8: Infection-permissive vaccination. Panel A shows the trajectories of reported clinical cases and SARS-CoV-2 concentration in wastewater under a scenario using an infection-permissive vaccine (“Vaccination”), or not (“Baseline”). In the vaccination scenario, the reported clinical cases decrease more rapidly than the lavel of SARS-CoV-2 in wastewater because sub-clinical infections tend to be less reported whereas faecal shedding continues. Panel B highlights this difference showing W (t)/C(t), the ratio of reported wastewater concentration over reported cases, for the baseline / no- vaccination (pink) and the vaccination (green) scenarios. The ratio is normalized to have a starting value at 1 to make it easier to quantify the increase visually. The vertical dotted line indicates when vaccination starts (at time 70). saved on 2021-07-19 15:54:19-04:00 21 / 43 . CC-BY-ND 4.0 International licenseIt 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 July 25, 2021. ; https://doi.org/10.1101/2021.07.19.21260773doi: medRxiv preprint Wastewater-Based Epidemic Modelling lic health metrics, if they are available to public health in a timely manner. In addition, the ability to triangulate the state of an epidemic using alternative data sources helps en-452 sure additional confidence in the estimation of relevant parameters or forecasting. Indeed, the COVID-19 pandemic has consumed public-health resources at levels that are probably454 not sustainable for long-term surveillance of this pathogen. However, the current wastew- ater surveillance performed in many communities can probably be continued as long as456 necessary given its relative low cost [9]. Our modelling framework provides a more principled alternative to simpler smoothing tech-458 niques (e.g., moving averages, polynomial interpolations) that have been used to support the interpretation of WBS [28]. However, we note that less complex modelling options460 are possible if the focus is on specific epidemiological metrics ([62, 63]). We also note recent efforts to use machine learning techniques and artificial neural network that incor-462 porate WBS [64]. While those methods are promising, they cannot–by design–explain the epidemiological mechanisms at play.464 Our model enables in silico experiments on the epidemic/wastewater system to identify key parameters and processes that can play an important role for the epidemiological inter-466 pretation. We showed, using simulations, that the relative timing of the wastewater signal (whether it is leading or not) compared to traditional clinical surveillance is actually influ-468 enced by the characteristics of both systems (Figure 6). On the one hand, the laboratory analysis of wastewater samples may not detect the presence of SARS-CoV-2 because, for470 example, its limit of detection is too high, or prevalence of infection in the community is very small, or the viral RNA has degraded before reaching the sampling site. Shipment472 time of wastewater samples can also be significant (e.g., several days) for remote sampling locations without any laboratory capacity. On the other hand, the delay in clinical cases474 reports is usually caused by the incubation period and the reporting time of an infection by the health system (turnaround time for contact tracing and/or laboratory results of in-476 dividuals’ swab) or availability of testing. Some communities would typically have a longer lag for clinical reporting than for wastewater surveillance [20, 22, 28, 24], while others may478 have the opposite (for example when a very effective contact tracing system is in place [65] or rapid testing is implemented). Moreover, a community may experience both situa-480 tions, that is a period when clinical surveillance is extremely efficient at detecting cases so rapidly that it leads wastewater surveillance while, at other times, it can lag (for example482 when incidence is high, overwhelming contact-tracing and clinical testing capacities). The model presented here allows to quantify how various factors can impacts the relative timing484 between clinical and wastewater surveillances. It can be tempting to monitor the effect of public health interventions using changes in the486 levels of SARS-CoV-2 in wastewater given its non-invasive nature. Indeed, WBS should be less affected by sampling bias than clinical surveillance (for example the latter may488 miss most of the subclinical infections). However, our simulations showed that wastewater saved on 2021-07-19 15:54:19-04:00 22 / 43 . CC-BY-ND 4.0 International licenseIt 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 July 25, 2021. ; https://doi.org/10.1101/2021.07.19.21260773doi: medRxiv preprint Wastewater-Based Epidemic Modelling surveillance may be inferior to clinical surveillance to identify sharp declines in transmis-490 sion, as typically seen after a lockdown is implemented (Figure 7). The long period of faecal shedding creates a lag in comparison to the sudden drop of incidence caused by the492 public health intervention, inhibiting a prompt signal in wastewater. This effect is visible on the Canadian data sets presented here (Figure 2).494 We also highlight the potential for an infection-permissive vaccine to generate discordant signals between wastewater and clinical surveillances (Figure 8). Indeed, vaccination im-496 plies a larger proportion of asymptomatic infections which are less likely to be detected by clinical surveillance, but still picked up in wastewater because of continued faecal shedding.498 Note that we use our model to highlight this potential effect, whereas detecting it in real data is probably challenging without studies purposely designed to detect this.500 Here, we used a mixed approach regarding the normalization of the levels of SARS-CoV- 2 in wastewater, with Ottawa using PMMV-normalization versus TSS-normalization for502 Toronto and Edmonton. It is likely that some normalization is necessary to discount the variations of the total faecal mass shedded and viral degradation, but it is still not clear504 which normalization is the most appropriate for a given sewershed. We note that we considered separately each WWTP in Toronto to provide examples of application at the506 sub-municipal level. Our modelling approach has several weaknesses. We did not precisely model the transport508 and fate of SARS-CoV-2 in municipal sewer systems. The lack of data about flow dynamics and particles binding of SARS-CoV-2 in wastewater hampered a more detailed approach.510 Hence, we took a simple approach to model the below-ground component and assumed the flow dynamic followed a low-dispersion plug flow model with a plausible fixed decay rate512 [46] and varied the mean transit time (from shedding to sampling sites) within a range of possible values. As more research focuses on the fate of SARS-CoV-2 in wastewater, the514 transport module of our model can be enhanced. The comparison with observed seroprevalence level must be done with caution, because we516 do not model seroreversion (patients who were infected but subsequently test seronegative because of loss of immunity or antibodies falling to undetectable levels). Our model does518 not model vaccination explicitly. We made this choice to keep the first version of our model relatively simple. However, we believe that we can appropriately approximate the520 effects of infection-permissive vaccination by reducing the transmission rate and increasing the proportion of asymptomatic infections. As the proportion of vaccinated individuals522 increases, modelling an explicit vaccination process is necessary. We note that for the Canadian cities studied here, the vaccination coverage was either null or low during the524 study period. We model SARS-CoV-2 as a single-strain pathogen which is an oversimplification of reality,526 given the numerous variants circulating in Canada since late 2020 [66]. However, it is not clear how (or if) multi-variants modelling would affect our results, given that the difference528 of viral shedding (respiratory and faecal) between variants is still not fully understood saved on 2021-07-19 15:54:19-04:00 23 / 43 . CC-BY-ND 4.0 International licenseIt 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 July 25, 2021. ; https://doi.org/10.1101/2021.07.19.21260773doi: medRxiv preprint Wastewater-Based Epidemic Modelling [67, 68].530 Because of ordinary differential equations, this model is not well adapted to either small communities or very low prevalence settings. While its epidemiological structure (Figure 1)532 would still be valid for such environments, a more advanced statistical modelling would be preferable to handle low incidence counts and observation uncertainty [69, 70].534 A further limitation of our model is the use of a scaling coefficient for the amount of SARS- CoV-2 shedded in the wastewater by the infected population (parameter ω in Equation 3).536 This scaling coefficient embeds all the uncertainties associated with sampling strategy and laboratory analysis, such as assay recovery efficiency, limit of detection, and total faecal538 mass normalization. Most of those processes are currently poorly known for SARS-CoV-2 and, as long as more observational data is not available, will constrain modelling (note540 that this limitation has already been identified for polio models [14]). An ultimate goal of wastewater surveillance may be to measure all the components of the scaling coefficient542 (here, ω) in order to estimate infection prevalence in a community directly from viral concentration readings.544 To conclude, the model presented here–built upon previous similar approach for other pathogens [71, 14, 72]–is a first step to better understand the mechanistic relationships546 between the COVID-19 epidemic spreading in a community and the SARS-CoV-2 RNA concentration in wastewater caused by faecal shedding of infected individuals (and poten-548 tially from urinary or sputum shedding). Future developments should explicitly incorporate vaccination and multiple variants/strains given the ability of new assays to detect variants550 from wastewater samples [73, 74, 75]. This model can be the basis of quantitative tools to support public health decision making that embraces wastewater-based epidemiology.552 Beyond the SARS-CoV-2/COVID-19 pandemic, WBS coupled with the type of model pre- sented here could be leveraged and applied to other transmissible pathogens where urinary554 or faecal shedding occurs, such as other respiratory diseases ( e.g., influenza, respiratory syncytial virus, adenovirus) and some enteric diseases (e.g., norovirus, rotavirus, shigel-556 losis). saved on 2021-07-19 15:54:19-04:00 24 / 43 . CC-BY-ND 4.0 International licenseIt 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 July 25, 2021. ; https://doi.org/10.1101/2021.07.19.21260773doi: medRxiv preprint Wastewater-Based Epidemic Modelling 6 Acknowledgments558 • Wastewater samples from Edmonton and Toronto were provided to the NML through a collaboration with Statistics Canada’s Canadian Wastewater Survey560 • Dave Spreitzer, Ravinder Lidder, Codey Dueck, Quinn Wonitowy, Umar Mohammed and Graham Cox for their contribution in wastewater sample processing and technical562 assistance. • Dana Al-Bargash provided data and local expertise for the City of Toronto.564 • The wastewater treatment plant Gold Bar, EPCOR Water Services Inc., Edmonton, Alberta, Canada for providing sewage samples in this study for the city of Edmonton.566 • Peter Vanrolleghen’s group at the University of Laval (QC, Canada) for insightful discussions on viral transport and fate in wastewater568 7 Funding Robert Delatolla acknowledges funding by Ontario Ministry of Environment Conserva-570 tion and Parks Wastewater Surveillance Initiative Transfer Payment Agreement 2020-11- 1-1463261970.572 8 Competing interests All authors declare they do not have any competing interests.574 saved on 2021-07-19 15:54:19-04:00 25 / 43 . CC-BY-ND 4.0 International licenseIt 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 July 25, 2021. ; https://doi.org/10.1101/2021.07.19.21260773doi: medRxiv preprint Wastewater-Based Epidemic Modelling 9 Tables Table 1: Description of the model’s compartments and parameters for the SARS-CoV-2 RNA transmission and disease outcome. Symbole Definition S susceptibles E exposed susceptibles but not infectious Ak asymptomatic infectious cases in k th subcompartment Ik symptomatic infectious cases in k th subcompartment Jk symptomatic infectious cases in k th subcompartment who later admits to hospital Zk non-infectious cases but fecal shedding SARS-CoV-2 RNA in k th subcompartment H hospitalized patients R recovered cases D deceased cases βI,k transmission rate in k th subcompartment among symptomatics (per contact) βA,k transmission rate in k th subcompartment among asymptomatics (per contact) 1/ε ave. latency time (days) 1/νk ave. duration in k th subcompartment among symptomatics (days) 1/µk ave. duration in k th subcompartment among symptomatics goes to hospital (days) 1/θk ave. duration in k th subcompartment among asymptomatics (days) 1/ηk ave. duration in k th subcompartment of shedding after infectiousness (days) 1/ℓ ave. length of stay in a hospital (days) nI total number of subcompartments in I state nJ total number of subcompartments in J state nA total number of subcompartments in A state nZ total number of subcompartments in Z state α proportion of exposed cases that are asymptomatic h proportion of symptomatic cases that need hospital admission δ proportion of deceased individuals among hospitalized patients saved on 2021-07-19 15:54:19-04:00 26 / 43 . CC-BY-ND 4.0 International licenseIt 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 July 25, 2021. ; https://doi.org/10.1101/2021.07.19.21260773doi: medRxiv preprint Wastewater-Based Epidemic Modelling T able 2: Description of fixed parameters used in this model and their sources sym bol description value unit source ξ relativ e infectiousness of asymptomatic compared to symptomatic states 0.8 none [76, 77, 78, 79, 80, 81] 1/ϵ Latent mean duration 2 day [45, 44] 1/ν Infectiousness duration for symptomatic individual 12 day [82, 83, 84, 85] 1/µ Infectiousness duration for symptomatic individual before admis- sion to hospital 8 day [86, 44] 1/θ Infectiousness duration for asymptomatic individual 10 day [82, 83, 84, 85] 1/η fecal sheding duration after infectious period 24 day [87] 1/ℓ Length of hospital stay 10 day assumed α asymptomatic proportion 0.316 none [88, 89, 90] δ proportion of death from hospitalized 0.19 none Report of Cana- dian Institute for Health Information https://wwwcihica/en/covid- 19-hospitalization-and- emergency-department- statistics ℓww reporting lag between sampling date and reporting date in days 2 day assumed κ decay rate of RNA in ww 0.18 none [46] τ mean transit time between shedding and sampling sites (in days) 1 day assumed σ std dev transit time between shedding and sampling sites (in days) 0.3 day assumed saved on 2021-07-19 15:54:19-04:00 27 / 43 . CC-BY-ND 4.0 International licenseIt 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 July 25, 2021. ; https://doi.org/10.1101/2021.07.19.21260773doi: medRxiv preprint Wastewater-Based Epidemic Modelling References576 [1] Lizhou Feng, Wei Zhang, and Xiqing Li. Monitoring of regional drug abuse through wastewater-based epi- demiology? A critical review (article). Science China Earth Sciences , 61(3):239–255, 2018.578 [2] E. Zuccato, C. Chiabrando, S. Castiglioni, R. Bagnati, and R. Fanelli. Estimating community drug abuse by wastewater analysis. Environ Health Perspect, 116(8):1027–1032, Aug 2008.580 [3] E. Zuccato, C. Chiabrando, S. Castiglioni, D. Calamari, R. Bagnati, S. Schiarea, and R. Fanelli. Cocaine in surface waters: a new evidence-based tool to monitor community drug abuse. Environ Health, 4:14, Aug 2005.582 [4] A. Christou, A. Aguera, J. M. Bayona, E. Cytryn, V. Fotopoulos, D. Lambropoulou, C. M. Manaia, C. Michael, M. Revitt, P. Schr¨ oder, and D. Fatta-Kassinos. The potential implications of reclaimed wastewater reuse for584 irrigation on the agricultural environment: The knowns and unknowns of the fate of antibiotics and antibiotic resistant bacteria and resistance genes - A review. Water Res, 123:448–467, 10 2017.586 [5] L. Rizzo, C. Manaia, C. Merlin, T. Schwartz, C. Dagot, M. C. Ploy, I. Michael, and D. Fatta-Kassinos. Urban wastewater treatment plants as hotspots for antibiotic resistant bacteria and genes spread into the environment:588 a review. Sci Total Environ , 447:345–360, Mar 2013. [6] M. Laht, A. Karkman, V. Voolaid, C. Ritz, T. Tenson, M. Virta, and V. Kisand. Abundances of tetracy-590 cline, sulphonamide and beta-lactam antibiotic resistance genes in conventional wastewater treatment plants (WWTPs) with different waste load. PLoS One, 9(8):e103705, 2014.592 [7] N. I. Rousis, E. Zuccato, and S. Castiglioni. Wastewater-based epidemiology to assess human exposure to pyrethroid pesticides. Environ Int, 99:213–220, Feb 2017.594 [8] N. I. Rousis, E. Zuccato, and S. Castiglioni. Monitoring population exposure to pesticides based on liquid chromatography-tandem mass spectrometry measurement of their urinary metabolites in urban wastewater:596 A novel biomonitoring approach. Sci Total Environ , 571:1349–1357, Nov 2016. [9] BM Gawlik, i S Tavazz, G Mariani, H Skejo, M Sponar, T Higgins, G Medema, and T Wintgens. Sars-cov-2598 surveillance employing sewage. towards a sentinel system. Technical report, European Union, 2021. [10] J. H. Lun, J. Hewitt, A. Sitabkhan, J. S. Eden, D. Enosi Tuipulotu, N. E. Netzler, L. Morrell, J. Merif, R. Jones,600 B. Huang, D. Warrilow, K. A. Ressler, M. J. Ferson, D. E. Dwyer, J. Kok, W. D. Rawlinson, D. Deere, N. D. Crosbie, and P. A. White. Emerging recombinant noroviruses identified by clinical and waste water screening.602 Emerg Microbes Infect, 7(1):50, Mar 2018. [11] J. M. Fioretti, T. M. Fumian, M. S. Rocha, I. A. L. Dos Santos, F. A. Carvalho-Costa, M. R. de Assis, J. S.604 Rodrigues, J. P. G. Leite, and M. P. Miagostovich. Surveillance of Noroviruses in Rio De Janeiro, Brazil: Occurrence of New GIV Genotype in Clinical and Wastewater Samples. Food Environ Virol , 10(1):1–6, 03606 2018. [12] H. Asghar, O. M. Diop, G. Weldegebriel, F. Malik, S. Shetty, L. El Bassioni, A. O. Akande, E. Al Maamoun,608 S. Zaidi, A. J. Adeniji, C. C. Burns, J. Deshpande, M. S. Oberste, and S. A. Lowther. Environmental surveillance for polioviruses in the Global Polio Eradication Initiative. J Infect Dis, 210 Suppl 1:294–303, Nov610 2014. [13] R. J. Duintjer Tebbens, M. Zimmermann, M. A. Pallansch, and K. M. Thompson. Insights from a Systematic612 Search for Information on Designs, Costs, and Effectiveness of Poliovirus Environmental Surveillance Systems. Food Environ Virol, 9(4):361–382, 12 2017.614 [14] Andrew F. Brouwer, Joseph N. S. Eisenberg, Connor D. Pomeroy, Lester M. Shulman, Musa Hindiyeh, Yossi Manor, Itamar Grotto, James S. Koopman, and Marisa C. Eisenberg. Epidemiology of the silent polio outbreak616 in rahat, israel, based on modeling of environmental surveillance data. Proceedings of the National Academy of Sciences, 115(45):E10625–E10633, 2018.618 [15] C.C. Naughton. COVIDPoops19 Summary of Global SARS-CoV-2 Wastewater Monitoring Efforts by UC Merced Researchers. ArcGIS Online Dashboard, 2020.620 saved on 2021-07-19 15:54:19-04:00 28 / 43 . CC-BY-ND 4.0 International licenseIt 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 July 25, 2021. ; https://doi.org/10.1101/2021.07.19.21260773doi: medRxiv preprint Wastewater-Based Epidemic Modelling [16] Gertjan Medema, Leo Heijnen, Goffe Elsinga, Ronald Italiaander, and Anke Brouwer. Presence of sars- coronavirus-2 rna in sewage and correlation with reported covid-19 prevalence in the early stage of the epidemic622 in the netherlands. Environmental Science & Technology Letters , 7(7):511–516, 2020. [17] W. Ahmed, N. Angel, J. Edson, K. Bibby, A. Bivins, J. W. O’Brien, P. M. Choi, M. Kitajima, S. L. Simpson,624 J. Li, B. Tscharke, R. Verhagen, W. J. M. Smith, J. Zaugg, L. Dierens, P. Hugenholtz, K. V. Thomas, and J. F. Mueller. First confirmed detection of SARS-CoV-2 in untreated wastewater in Australia: A proof of626 concept for the wastewater surveillance of COVID-19 in the community. Sci Total Environ , 728:138764, Aug 2020.628 [18] S. Wurtzer, V. Marechal, J. M. Mouchel, Y. Maday, R. Teyssou, E. Richard, J. L. Almayrac, and L. Moulin. Evaluation of lockdown effect on SARS-CoV-2 dynamics through viral genome quantification in waste water,630 Greater Paris, France, 5 March to 23 April 2020. Euro Surveill, 25(50), 12 2020. [19] J. Peccia, A. Zulli, D. E. Brackney, N. D. Grubaugh, E. H. Kaplan, A. Casanovas-Massana, A. I. Ko, A. A.632 Malik, D. Wang, M. Wang, J. L. Warren, D. M. Weinberger, W. Arnold, and S. B. Omer. Measurement of SARS-CoV-2 RNA in wastewater tracks community infection dynamics. Nat Biotechnol, 38(10):1164–1167, 10634 2020. [20] G. La Rosa, M. Iaconelli, P. Mancini, G. Bonanno Ferraro, C. Veneri, L. Bonadonna, L. Lucentini, and636 E. Suffredini. First detection of SARS-CoV-2 in untreated wastewaters in Italy. Sci Total Environ, 736:139652, Sep 2020.638 [21] D. A. Larsen and K. R. Wigginton. Tracking COVID-19 with wastewater. Nat Biotechnol, 38(10):1151–1153, 10 2020.640 [22] W. Randazzo, P. Truchado, E. Cuevas-Ferrando, P. Simon, A. Allende, and G. Sanchez. SARS-CoV-2 RNA in wastewater anticipated COVID-19 occurrence in a low prevalence area. Water Res, 181:115942, Aug 2020.642 [23] P. M. D’Aoust, E. Mercier, D. Montpetit, J. J. Jia, I. Alexandrov, N. Neault, A. T. Baig, J. Mayne, X. Zhang, T. Alain, M. A. Langlois, M. R. Servos, M. MacKenzie, D. Figeys, A. E. MacKenzie, T. E. Graber, and644 R. Delatolla. Quantitative analysis of SARS-CoV-2 RNA from wastewater solids in communities with low COVID-19 incidence and prevalence. Water Res, 188:116560, Jan 2021.646 [24] A. Hata, H. Hara-Yamamura, Y. Meuchi, S. Imai, and R. Honda. Detection of SARS-CoV-2 in wastewater in Japan during a COVID-19 outbreak. Sci Total Environ , 758:143578, Mar 2021.648 [25] C. Gibas, K. Lambirth, N. Mittal, M. A. I. Juel, V. B. Barua, L. Roppolo Brazell, K. Hinton, J. Lontai, N. Stark, I. Young, C. Quach, M. Russ, J. Kauer, B. Nicolosi, D. Chen, S. Akella, W. Tang, J. Schlueter,650 and M. Munir. Implementing building-level SARS-CoV-2 wastewater surveillance on a university campus. Sci Total Environ, 782:146749, Aug 2021.652 [26] Jaclyn Peiser. University of Arizona used wastewater testing to detect cases of coronavirus in a dorm. The Washington Post, Aug 2020.654 [27] Denise Paglinawan. University of Guelph testing campus residences’ wastewater to detect COVID-19. CBC News - The Canadian Press , Oct 2020.656 [28] P. M. D’Aoust, T. E. Graber, E. Mercier, D. Montpetit, I. Alexandrov, N. Neault, A. T. Baig, J. Mayne, X. Zhang, T. Alain, M. R. Servos, N. Srikanthan, M. MacKenzie, D. Figeys, D. Manuel, P. Juni, A. E.658 MacKenzie, and R. Delatolla. Sci Total Environ Catching a resurgence: Increase in SARS-CoV-2 viral RNA identified in wastewater 48 h before COVID-19 clinical tests and 96 h before hospitalizations.Sci Total Environ,660 770:145319, May 2021. [29] National Institute for Public Health Netherlands, Welfare the Environment, Ministry of Health, and Sport.662 Coronavirus monitoring in sewage research. Amsterdam: National Institute for Public Health and the Envi- ronment, Mar 2021.664 [30] Public Health Ottawa. Wastewater COVID-19 surveillance. Ottawa Public Health, Ontario, Canada, 2021. [31] Public Health Ontario. Ontario Agency for Health Protection and Promotion, Wastewater surveillance of666 COVID-19,Toronto, ON. Queen ’s Printer for Ontario , Apr 2021. saved on 2021-07-19 15:54:19-04:00 29 / 43 . CC-BY-ND 4.0 International licenseIt 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 July 25, 2021. ; https://doi.org/10.1101/2021.07.19.21260773doi: medRxiv preprint Wastewater-Based Epidemic Modelling [32] WHO. World Health Organization, Status of environmental surveillance for SARS-CoV-2 virus: scientific brief.668 World Health Organization , 2020. [33] Paola Foladori, Francesca Cutrupi, Nicola Segata, Serena Manara, Federica Pinto, Francesca Malpei, Laura670 Bruni, and Giuseppina La Rosa. Sars-cov-2 from faeces to wastewater treatment: What do we know? a review. Science of The Total Environment , 743:140444, 2020.672 [34] X. Li, S. Zhang, J. Shi, S. P. Luby, and G. Jiang. Uncertainties in estimating SARS-CoV-2 prevalence by wastewater-based epidemiology. Chem Eng J, 415:129039, Jul 2021.674 [35] Y. Zhu, W. Oishi, C. Maruo, M. Saito, R. Chen, M. Kitajima, and D. Sano. Early warning of COVID-19 via wastewater-based epidemiology: potential and bottlenecks. Sci Total Environ , 767:145124, May 2021.676 [36] Mia Rabson. Researchers looking for clues to the COVID-19 pandemic in city sewers. Global News, The Canadian Press, Apr 2021.678 [37] David L. Jones, Marcos Quintela Baluja, David W. Graham, Alexander Corbishley, James E. McDonald, Shelagh K. Malham, Luke S. Hillary, Thomas R. Connor, William H. Gaze, Ines B. Moura, Mark H. Wilcox, and680 Kata Farkas. Shedding of sars-cov-2 in feces and urine and its potential role in person-to-person transmission and the environment-based spread of covid-19. Science of The Total Environment , 749:141364, 2020.682 [38] Kun Wang, Xin Zhang, Jiaxing Sun, Jia Ye, Feilong Wang, Jing Hua, Huayu Zhang, Ting Shi, Qiang Li, and Xiaodong Wu. Differences of severe acute respiratory syndrome coronavirus 2 shedding duration in sputum and684 nasopharyngeal swab specimens among adult inpatients with coronavirus disease 2019. Chest, 158(5):1876– 1884, 2020.686 [39] P. M. Gundy, C. P. Gerba, and I. L. Pepper. Survival of Coronaviruses in Water and Wastewater.Food Environ Virol, 2009.688 [40] M. Kitajima, W. Ahmed, K. Bibby, A. Carducci, C. P. Gerba, K. A. Hamilton, E. Haramoto, and J. B. Rose. Sci Total EnvironSARS-CoV-2 in wastewater: State of the knowledge and research needs. Sci Total Environ ,690 739:139076, Oct 2020. [41] I. Michael-Kordatou, P. Karaolia, and D. Fatta-Kassinos. J Environ Chem EngSewage analysis as a tool for692 the COVID-19 pandemic response and management: the urgent need for optimised protocols for SARS-CoV-2 detection and quantification. J Environ Chem Eng , 8(5):104306, Oct 2020.694 [42] Roy M Anderson and Robert M May. Infectious Diseases of Humans - Dynamics and Control . Oxford University Press, 1991.696 [43] Helen J Wearing, Pejman Rohani, and Matt J Keeling. Appropriate models for the management of infectious diseases. PLoS Medicine, 2(7):e174, 2005.698 [44] X. He, E. H. Y. Lau, P. Wu, X. Deng, J. Wang, X. Hao, Y. C. Lau, J. Y. Wong, Y. Guan, X. Tan, X. Mo, Y. Chen, B. Liao, W. Chen, F. Hu, Q. Zhang, M. Zhong, Y. Wu, L. Zhao, F. Zhang, B. J. Cowling, F. Li, and700 G. M. Leung. Temporal dynamics in viral shedding and transmissibility of COVID-19.Nat Med, 26(5):672–675, 05 2020.702 [45] Q. Li, X. Guan, P. Wu, X. Wang, L. Zhou, Y. Tong, R. Ren, K. S. M. Leung, E. H. Y. Lau, J. Y. Wong, X. Xing, N. Xiang, Y. Wu, C. Li, Q. Chen, D. Li, T. Liu, J. Zhao, M. Liu, W. Tu, C. Chen, L. Jin, R. Yang,704 Q. Wang, S. Zhou, R. Wang, H. Liu, Y. Luo, Y. Liu, G. Shao, H. Li, Z. Tao, Y. Yang, Z. Deng, B. Liu, Z. Ma, Y. Zhang, G. Shi, T. T. Y. Lam, J. T. Wu, G. F. Gao, B. J. Cowling, B. Yang, G. M. Leung, and Z. Feng.706 Early Transmission Dynamics in Wuhan, China, of Novel Coronavirus-Infected Pneumonia. N Engl J Med , 382(13):1199–1207, 03 2020.708 [46] Warish Ahmed, Paul M. Bertsch, Kyle Bibby, Eiji Haramoto, Joanne Hewitt, Flavia Huygens, Pradip Gyawali, Asja Korajkic, Shane Riddell, Samendra P. Sherchan, Stuart L. Simpson, Kwanrawee Sirikanchana, Erin M.710 Symonds, Rory Verhagen, Seshadri S. Vasan, Masaaki Kitajima, and Aaron Bivins. Decay of sars-cov-2 and surrogate murine hepatitis virus rna in untreated wastewater to inform application in wastewater-based712 epidemiology. Environmental Research, 191:110092, 2020. [47] A. Kayode Coker. Chapter eight - residence time distributions in flow reactors. pages 663–761, 2001.714 saved on 2021-07-19 15:54:19-04:00 30 / 43 . CC-BY-ND 4.0 International licenseIt 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 July 25, 2021. ; https://doi.org/10.1101/2021.07.19.21260773doi: medRxiv preprint Wastewater-Based Epidemic Modelling [48] City of Ottawa. Wastewater collection and treatment, Robert O. Pickard Environmental Centre (ROPEC). Online, 2021.716 [49] Toronto Water. City of Toronto, Ashbridges Bay wastewater treatment plant annual report of 2020. Online, 2021.718 [50] Toronto Water. City of Toronto, Humber wastewater treatment plant annual report of 2020. Online, 2021. [51] Toronto Water. City of Toronto, Highland Creek wastewater treatment plant annual report of 2020. Online,720 2021. [52] Statistics Canada. 2016 Census of Canada via Toronto Public Health. 2016.722 [53] Canadian Blood Services. Covid-19 public seroprevalence report: April 2020 to january 2021. Technical report, Canadian Blood Services, 2021.724 [54] Mark A Beaumont, Wenyang Zhang, and David J Balding. Approximate Bayesian Computation in Population Genetics. Genetics, 162(4):2025–2035, 12 2002.726 [55] C. Hollingsworth D. Keeling M. Vegvari C. Baggaley R. Maddren R. Anderson, R. Donnelly. Reproduction number (R) and growth rate (r) of the COVID-19 epidemic in the UK: methods of estimation, data sources,728 causes of heterogeneity, and use as a guide in policy formulation. The Royal Society , 2020. [56] J. M. Brauner, S. Mindermann, M. Sharma, D. Johnston, J. Salvatier, T. Gaven?iak, A. B. Stephenson,730 G. Leech, G. Altman, V. Mikulik, A. J. Norman, J. T. Monrad, T. Besiroglu, H. Ge, M. A. Hartwick, Y. W. Teh, L. Chindelevitch, Y. Gal, and J. Kulveit. ScienceInferring the effectiveness of government interventions732 against COVID-19. Science, 371(6531), 02 2021. [57] A Cori, N M Ferguson, C. Fraser, and S Cauchemez. A New Framework and Software to Estimate Time-734 Varying Reproduction Numbers During Epidemics. American Journal of Epidemiology, 178(9):1505–1512, October 2013.736 [58] Brian M Pecson, Emily Darby, Charles N Haas, Yamrot M Amha, Mitchel Bartolo, Richard Danielson, Yeggie Dearborn, George Di Giovanni, Christobel Ferguson, Stephanie Fevig, et al. Reproducibility and sensitivity738 of 36 methods to quantify the sars-cov-2 genetic signal in raw wastewater: findings from an interlaboratory

evaluation in the us. Environmental Science: Water Research & Technology, 7(3):504–520, 2021.740 [59] Aaron Bivins, Justin Greaves, Robert Fischer, Kwe Claude Yinda, Warish Ahmed, Masaaki Kitajima, Vin- cent J. Munster, and Kyle Bibby. Persistence of sars-cov-2 in water and wastewater. Environmental Science742 & Technology Letters, 7(12):937–942, 2020. [60] P. Mandal, A. K. Gupta, and B. K. Dubey. A review on presence, survival, disinfection/removal methods of744 coronavirus in wastewater and progress of wastewater-based epidemiology. J Environ Chem Eng, 8(5):104317, Oct 2020.746 [61] N. Sims and B. Kasprzyk-Hordern. Environ IntFuture perspectives of wastewater-based epidemiology: Moni- toring infectious disease spread and resistance to the community level. Environ Int, 139:105689, 06 2020.748 [62] Jana S. Huisman, J´ er´ emie Scire, Lea Caduff, Xavier Fernandez-Cassi, Pravin Ganesanandamoorthy, Anina Kull, Andreas Scheidegger, Elyse Stachler, Alexandria B. Boehm, Bridgette Hughes, Alisha Knudson, Aaron750 Topol, Krista R. Wigginton, Marlene K. Wolfe, Tamar Kohn, Christoph Ort, Tanja Stadler, and Timothy R. Julian. Wastewater-based estimation of the effective reproductive number of sars-cov-2. medRxiv, 2021.752 [63] A. Xiao, F. Wu, M. Bushman, J. Zhang, M. Imakaev, P. R. Chai, C. Duvallet, N. Endo, T. B. Erickson, F. Armas, B. Arnold, H. Chen, F. Chandra, N. Ghaeli, X. Gu, W. P. Hanage, W. L. Lee, M. Matus, K. A.754 McElroy, K. Moniz, S. F. Rhode, J. Thompson, and E. J. Alm. medRxivMetrics to relate COVID-19 wastewater data to clinical testing dynamics. medRxiv, Jun 2021.756 [64] X. Li, J. Kulandaivelu, S. Zhang, J. Shi, M. Sivakumar, J. Mueller, S. Luby, W. Ahmed, L. Coin, and G. Jiang. Sci Total EnvironData-driven estimation of COVID-19 community prevalence through wastewater-758 based epidemiology. Sci Total Environ , 789:147947, May 2021. saved on 2021-07-19 15:54:19-04:00 31 / 43 . CC-BY-ND 4.0 International licenseIt 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 July 25, 2021. ; https://doi.org/10.1101/2021.07.19.21260773doi: medRxiv preprint Wastewater-Based Epidemic Modelling [65] Sofia K. Mettler, Jihoo Kim, and Marloes H. Maathuis. Diagnostic serial interval as a novel indicator for760 contact tracing effectiveness exemplified with the sars-cov-2/covid-19 outbreak in south korea. International Journal of Infectious Diseases, 99:346–351, 2020.762 [66] Angela McLaughlin, Vincent Montoya, Rachel L. Miller, Gideon J. Mordecai, Michael Worobey, Art F. Y. Poon, and Jeffrey B. Joy. Early and ongoing importations of sars-cov-2 in canada. medRxiv, 2021.764 [67] Michael Kidd, Alex Richter, Angus Best, Nicola Cumley, Jeremy Mirza, Benita Percival, Megan Mayhew, Oliver Megram, Fiona Ashford, Thomas White, Emma Moles-Garcia, Liam Crawford, Andrew Bosworth,766 Sowsan F Atabani, Tim Plant, and Alan McNally. S-Variant SARS-CoV-2 Lineage B1.1.7 Is Associated With Significantly Higher Viral Load in Samples Tested by TaqPath Polymerase Chain Reaction. The Journal of768 Infectious Diseases, 223(10):1666–1670, 02 2021. [68] Stephen M. Kissler, Joseph R. Fauver, Christina Mack, Caroline G. Tai, Mallery I. Breban, Anne E. Watkins,770 Radhika M. Samant, Deverick J. Anderson, David D. Ho, Jessica Metti, Gaurav Khullar, Rachel Baits, Matthew MacKay, Daisy Salgado, Tim Baker, Joel T. Dudley, Christopher E. Mason, Nathan D. Grubaugh,772 and Yonatan H. Grad. Densely sampled viral trajectories for sars-cov-2 variants alpha (b.1.1.7) and epsilon (b.1.429). medRxiv, 2021.774 [69] A A King, M Domenech de Celles, F M G Magpantay, and P. Rohani. Avoidable errors in the modelling of outbreaks of emerging pathogens, with special reference to Ebola. Proceedings of the Royal Society B:776 Biological Sciences, 282(1806):20150347–20150347, April 2015. [70] Michael Li, Jonathan Dushoff, and Benjamin M Bolker. Fitting mechanistic epidemic models to data: A com-778 parison of simple Markov chain Monte Carlo approaches. Statistical Methods in Medical Research , 27(7):1956– 1967, 2018.780 [71] C. S. McMahan, S. Self, L. Rennert, C. Kalbaugh, D. Kriebel, D. Graves, J. A. Deaver, S. Popat, T. Karanfil, and D. L. Freedman. medRxivCOVID-19 Wastewater Epidemiology: A Model to Estimate Infected Popula-782 tions. medRxiv, Nov 2020. [72] A. N. M. Kraay, A. F. Brouwer, N. Lin, P. A. Collender, J. V. Remais, and J. N. S. Eisenberg. Proc Natl Acad784 Sci U S AModeling environmentally mediated rotavirus transmission: The role of temperature and hydrologic factors. Proc Natl Acad Sci U S A , 115(12):E2782–E2790, 03 2018.786 [73] G. La Rosa, P. Mancini, G. Bonanno Ferraro, C. Veneri, M. Iaconelli, L. Lucentini, L. Bonadonna, S. Brusaferro, D. Brandtner, A. Fasanella, L. Pace, A. Parisi, D. Galante, and E. Suffredini. Water ResRapid screening for788 SARS-CoV-2 variants of concern in clinical and environmental samples using nested RT-PCR assays targeting key mutations of the spike protein. Water Res, 197:117104, Jun 2021.790 [74] S. Agrawal, L. Orschler, S. Schubert, K. Zachmann, L. Heijnen, S. Tavazzi, B. M. Gawlik, M. de Graaf, G. Medema, and S. Lackner. A pan-European study of SARS-CoV-2 variants in wastewater under the EU792 Sewage Sentinel System. medRxiv, Jun 2021. [75] Katharina Jahn, David Dreifuss, Ivan Topolsky, Anina Kull, Pravin Ganesanandamoorthy, Xavier Fernandez-794 Cassi, Carola B¨ anziger, Elyse Stachler, Lara Fuhrmann, Kim Philipp Jablonski, Chaoran Chen, Catharine Aquino, Tanja Stadler, Christoph Ort, Tamar Kohn, Timothy R. Julian, and Niko Beerenwinkel. Detection of796 sars-cov-2 variants in switzerland by genomic analysis of wastewater samples. medRxiv, 2021. [76] E. T. Landaas, M. L. Storm, M. C. Tollanes, R. Barlinn, A. B. Kran, K. Bragstad, A. Christensen, and798 T. Andreassen. Diagnostic performance of a SARS-CoV-2 rapid antigen test in a large, Norwegian cohort. J Clin Virol, 137:104789, 04 2021.800 [77] Di Tian, Zhen Lin, Ellie M. Kriner, Dalton J. Esneault, Jonathan Tran, Julia C. DeVoto, Naima Okami, Rachel Greenberg, Sarah Yanofsky, Swarnamala Ratnayaka, Nicholas Tran, Maeghan Livaccari, Marla Lampp, Noel802 Wang, Scott Tim, Patrick Norton, John Scott, Tony Y. Hu, Robert Garry, Patrice Delafontaine, Lee Hamm, and Xiao-Ming Yin. Sars-cov-2 load does not predict transmissibility in college students. medRxiv, 2021.804 [78] M. D. Folgueira, J. Luczkowiak, F. Lasala, A. Perez-Rivilla, and R. Delgado. Prolonged SARS-CoV-2 cell culture replication in respiratory samples from patients with severe COVID-19. Clin Microbiol Infect , Feb806 2021. saved on 2021-07-19 15:54:19-04:00 32 / 43 . CC-BY-ND 4.0 International licenseIt 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 July 25, 2021. ; https://doi.org/10.1101/2021.07.19.21260773doi: medRxiv preprint Wastewater-Based Epidemic Modelling [79] A. A. Sayampanathan, C. S. Heng, P. H. Pin, J. Pang, T. Y. Leong, and V. J. Lee. Infectivity of asymptomatic808 versus symptomatic COVID-19. Lancet, 397(10269):93–94, 01 2021. [80] L. K. Kociolek, W. J. Muller, R. Yee, J. Dien Bard, C. A. Brown, P. A. Revell, H. Wardell, T. J. Savage, S. Jung,810 S. Dominguez, B. A. Parikh, R. C. Jerris, S. C. Kehl, A. Campigotto, J. M. Bender, X. Zheng, E. Muscat, M. Linam, L. Abuogi, C. Smith, K. Graff, A. Hernandez-Leyva, D. Williams, and N. R. Pollock. Comparison812 of Upper Respiratory Viral Load Distributions in Asymptomatic and Symptomatic Children Diagnosed with SARS-CoV-2 Infection in Pediatric Hospital Testing Programs. J Clin Microbiol, 59(1), 12 2020.814 [81] Stephen M. Kissler, Joseph R. Fauver, Christina Mack, Scott W. Olesen, Caroline Tai, Kristin Y. Shiue, Chaney C. Kalinich, Sarah Jednak, Isabel M. Ott, Chantal B.F. Vogels, Jay Wohlgemuth, James Weisberger,816 John DiFiori, Deverick J. Anderson, Jimmie Mancell, David D. Ho, Nathan D. Grubaugh, and Yonatan H. Grad. Sars-cov-2 viral dynamics in acute infections. medRxiv, 2020.818 [82] Daniel Owusu, Mary A Pomeroy, Nathaniel M Lewis, Ashutosh Wadhwa, Anna R Yousaf, Brett Whitaker, Elizabeth Dietrich, Aron J Hall, Victoria Chu, Natalie Thornburg, Kimberly Christensen, Tair Kiphibane,820 Sarah Willardson, Ryan Westergaard, Trivikram Dasu, Ian W Pray, Sanjib Bhattacharyya, Angela Dunn, Jacqueline E Tate, Hannah L Kirking, Almea Matanock, and Household Transmission Study Team. Persistent822 SARS-CoV-2 RNA Shedding without Evidence of Infectiousness: A Cohort Study of Individuals with COVID- 19. The Journal of Infectious Diseases, 02 2021. jiab107.824 [83] Sukbin Jang, Ji-Young Rhee, Yu Mi Wi, and Bo Kyeung Jung. Viral kinetics of sars-cov-2 over the preclinical, clinical, and postclinical period. International Journal of Infectious Diseases, 102:561–565, 2021.826 [84] Nad` ege N´ eant, Guillaume Lingas, Quentin Le Hingrat, Jade Ghosn, Ilka Engelmann, Quentin Lepiller, Alexan- dre Gaymard, Virginie Ferr´ e, C´ edric Hartard, Jean-Christophe Plantier, Vincent Thibault, Julien Marlet,828 Brigitte Montes, Kevin Bouiller, Fran¸ cois-Xavier Lescure, Jean-Fran¸ cois Timsit, Emmanuel Faure, Julien Poissy, Christian Chidiac, Fran¸ cois Raffi, Antoine Kimmoun, Manuel Etienne, Jean-Christophe Richard, Pierre830 Tattevin, Denis Garot, Vincent Le Moing, Delphine Bachelet, Coralie Tardivon, Xavier Duval, Yazdan Yaz- danpanah, France Mentr´ e, C´ edric Laou´ enan, Benoit Visseaux, and J´ er´ emie and Guedj. Modeling sars-cov-2832 viral kinetics and association with mortality in hospitalized patients from the french covid cohort. Proceedings of the National Academy of Sciences , 118(8), 2021.834 [85] Jared Bullard, Kerry Dust, Duane Funk, James E Strong, David Alexander, Lauren Garnett, Carl Boodman, Alexander Bello, Adam Hedley, Zachary Schiffman, Kaylie Doan, Nathalie Bastien, Yan Li, Paul G Van Cae-836 seele, and Guillaume Poliquin. Predicting Infectious Severe Acute Respiratory Syndrome Coronavirus 2 From Diagnostic Samples. Clinical Infectious Diseases, 71(10):2663–2666, 05 2020.838 [86] C. Faes, S. Abrams, D. Van Beckhoven, G. Meyfroidt, E. Vlieghe, N. Hens, A. S. Aouachria, K. Bafort, L. Belkhir, N. Bossuyt, V. Colombie, N. Dauby, P. De Munter, J. Deblonde, D. Delmarcelle, M. Del-840 vallee, R. Demeester, T. Dugernier, X. Holemans, B. Kerzmann, P. Y. Machurot, P. Minette, J. M. Minon, S. Mokrane, C. Nachtergal, S. Noirhomme, D. Pierard, C. Rossi, C. Schirvel, E. Sermijn, F. Staelens, F. Triest,842 N. Van Goethem, J. Van Praet, A. Vanhoenacker, S. Cooreman, E. Willems, and C. Wyndham-Thomas. Time between Symptom Onset, Hospitalisation and Recovery or Death: Statistical Analysis of Belgian COVID-19844 Patients. Int J Environ Res Public Health , 17(20), 10 2020. [87] Till Hoffmann and Justin Alsing. Faecal shedding models for sars-cov-2 rna amongst hospitalised patients and846 implications for wastewater-based epidemiology. medRxiv, 2021. [88] K. Mizumoto, K. Kagaya, A. Zarebski, and G. Chowell. Estimating the asymptomatic proportion of coronavirus848 disease 2019 (COVID-19) cases on board the Diamond Princess cruise ship, Yokohama, Japan, 2020. Euro Surveill, 25(10), 03 2020.850 [89] H. Nishiura, T. Kobayashi, T. Miyama, A. Suzuki, S. M. Jung, K. Hayashi, R. Kinoshita, Y. Yang, B. Yuan, A. R. Akhmetzhanov, and N. M. Linton. Estimation of the asymptomatic ratio of novel coronavirus infections852 (COVID-19). Int J Infect Dis, 94:154–155, 05 2020. [90] D. Buitrago-Garcia, D. Egli-Gany, M. J. Counotte, S. Hossmann, H. Imeri, A. M. Ipekci, G. Salanti, and854 N. Low. Occurrence and transmission potential of asymptomatic and presymptomatic SARS-CoV-2 infections: A living systematic review and meta-analysis. PLoS Med, 17(9):e1003346, 09 2020.856 saved on 2021-07-19 15:54:19-04:00 33 / 43 . CC-BY-ND 4.0 International licenseIt 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 July 25, 2021. ; https://doi.org/10.1101/2021.07.19.21260773doi: medRxiv preprint Wastewater-Based Epidemic Modelling [91] Muge Cevik, Matthew Tate, Ollie Lloyd, Alberto Enrico Maraolo, Jenna Schafers, and Antonia Ho. Sars-cov-2, sars-cov, and mers-cov viral load dynamics, duration of viral shedding, and infectiousness: a systematic review858 and meta-analysis. The Lancet Microbe, 2(1):e13–e22, 2021. [92] Rita Jaafar, Sarah Aherfi, Nathalie Wurtz, Clio Grimaldier, Thuan Van Hoang, Philippe Colson, Didier Raoult,860 and Bernard La Scola. Correlation Between 3790 Quantitative Polymerase Chain Reaction–Positives Samples and Positive Cell Cultures, Including 1941 Severe Acute Respiratory Syndrome Coronavirus 2 Isolates. Clinical862 Infectious Diseases, 09 2020. ciaa1491. [93] Dorothy Anderson and Ray Watson. On the spread of a disease with gamma distributed latent and infectious864 periods. Biometrika, 67(1):191–198, 1980. [94] Anne Weiss, Mads Jellingso, and Morten Otto Alexander Sommer. Spatial and temporal dynamics of sars-cov-2866 in covid-19 patients: A systematic review and meta-analysis. EBioMedicine, 58:102916, 2020. [95] Amy E Benefield, Laura A Skrip, Andrea Clement, Rachel A Althouse, Stewart Chang, and Benjamin Muir868 Althouse. Sars-cov-2 viral load peaks prior to symptom onset: a systematic review and individual-pooled analysis of coronavirus viral load from 66 studies. medRxiv, 2020.870 [96] Mathilde Bellon, Stephanie Baggio, Frederique Jacquerioz Bausch, Herv´ e Spechbach, Julien Salamun, Camille Genecand, Aglae Tardin, Laurent Kaiser, Arnaud G L’Huillier, and Isabella Eckerle. SARS-CoV-2 viral load872 kinetics in symptomatic children, adolescents and adults. Clinical Infectious Diseases, 05 2021. [97] Fuminari Miura, Masaaki Kitajima, and Ryosuke Omori. Duration of sars-cov-2 viral shedding in faeces as a874 parameter for wastewater-based epidemiology: Re-analysis of patient data using a shedding dynamics model. Science of The Total Environment , 769:144549, 2021.876 [98] Wen-Cheng Liu and Wei-Cher Huang. Modeling the transport and distribution of fecal coliform in a tidal estuary. Science of The Total Environment , 431:1–8, 2012.878 [99] Christopher Staley, Kenneth H. Reckhow, Jerzy Lukasik, and Valerie J. Harwood. Assessment of sources of human pathogens and fecal contamination in a florida freshwater lake. Water Research, 46(17):5799–5812,880 2012. [100] Rob Jamieson, Doug M. Joy, Hung Lee, Ray Kostaschuk, and Robert Gordon. Transport and deposition of882 sediment-associated escherichia coli in natural streams. Water Research, 39(12):2665–2675, 2005. [101] Bhuban Ghimire and Zhiqiang Deng. Hydrograph-based approach to modeling bacterial fate and transport in884 rivers. Water Research, 47(3):1329–1343, 2013. [102] Sen Bai and Wu-Seng Lung. Modeling sediment impact on the transport of fecal bacteria. Water Research,886 39(20):5232–5240, 2005. [103] Kyung Hwa Cho, Yakov A. Pachepsky, David M. Oliver, Richard W. Muirhead, Yongeun Park, Richard S. Quil-888 liam, and Daniel R. Shelton. Modeling fate and transport of fecally-derived microorganisms at the watershed scale: State of the science and future opportunities. Water Research, 100:38–56, 2016.890 [104] Chris R. Rehmann and Michelle L. Soupir. Importance of interactions between the water column and the sediment for microbial concentrations in streams. Water Research, 43(18):4579–4589, 2009.892 [105] Octave Levenspiel. The Dispersion Model , pages 47–70. Springer New York, New York, NY, 2012. [106] A McDonald, D Kay, and A Jenkins. Generation of fecal and total coliform surges by stream flow manipulation894 in the absence of normal hydrometeorological stimuli. Applied and Environmental Microbiology, 44(2):292–300, 1982.896 [107] J. W. Nagels, R. J. Davies-Colley, A. M. Donnison, and R. W. Muirhead. Faecal contamination over flood events in a pastoral agricultural stream in New Zealand. Water Sci Technol , 45(12):45–52, 2002.898 saved on 2021-07-19 15:54:19-04:00 34 / 43 . CC-BY-ND 4.0 International licenseIt 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 July 25, 2021. ; https://doi.org/10.1101/2021.07.19.21260773doi: medRxiv preprint Wastewater-Based Epidemic Modelling Appendix900 saved on 2021-07-19 15:54:19-04:00 35 / 43 . CC-BY-ND 4.0 International licenseIt 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 July 25, 2021. ; https://doi.org/10.1101/2021.07.19.21260773doi: medRxiv preprint Wastewater-Based Epidemic Modelling time transmission rate β(b1) β(b2) β(b3) β(b4) Figure S1: Illustration of the piecewise linear function to model the transmission rate βt. The break dates b1, b2, . . . are chosen manually by visual inspection of the time series of interest, and the values β(bi) are fitted with an ABC algorithm. A-1 Fitting procedure To fit the model to the clinical and/or wastewater surveillance data, we model the transmis-902 sion rate, βt as a piecewise linear function. We parsimoniously choose the times b1, b2, . . . (“break times”) defining each segment by visual inspection of the time series (i.e., incidence904 of new COVID-19 cases and/or hospitalizations for clinical surveillance; SARS-CoV-2 con- centration for wastewater surveillance). Those times should correspond to changes in the906 transmission dynamics. The value of the transmission rate at a break date, β(bi) is fit- ted using an Approximate Bayesian Computation (ABC). We chose not to fit the break908 times bi because the fitting algorithm would require a computation time that would not be practical. See Figure S1.910 The mean transit time ¯τ and the scaling factor ω are also fitted to data. For those parame- ters, the goal is primarily to allow for uncertainty rather than infer a posterior distribution912 as they are essentially not identifiable. Finally, the hospitalization rate is also fitted to the hospitalization data, when available.914 saved on 2021-07-19 15:54:19-04:00 36 / 43 . CC-BY-ND 4.0 International licenseIt 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 July 25, 2021. ; https://doi.org/10.1101/2021.07.19.21260773doi: medRxiv preprint Wastewater-Based Epidemic Modelling Over the study period (from early March 2020 to June 1st, 2021), we have about a dozen break times bi for each city to reflect the various interventions (e.g., lockdowns) or behaviour916 (e.g., change in contact rate when schools re-opened in September 2020). Hence, the parameter space to explore by the ABC algorithm is relatively large. To avoid unpractical918 computational times, we defined relative strong priors on all parameters, that is normal distributions with a mean close to the expected value (explored manually) and a relatively920 broad standard deviation (corresponding approximately to a coefficient of variation of 0.5). The normal distribution was censored to positive values. The outputs of the fitting922 procedure for all locations are shown in supplementary file File S2. A-2 Relative infectiousness924 In Equation 1a, the force of infection from symptomatic cases is β ∑nI k=1 ψkIk. For conve- nience, the ψk are chosen such that926 nI∑ k=1 ψk = nI . (A.1) Hence, when the infectiousness profile is constant (λ k = 1, for k = 1, . . . , nI), the force of infection is β ∑nI k=1 Ik. Using this normalization allows to keep a single baseline param-928 eter β. Only the relative values of the ψk affect the infectiousness profile. Similarly for asymptomatic infections, the parameters φk are chosen such that ∑nA k=1 φk = nA.930 We assume the infectious period for symptomatic infections is 12 days on average [91, 92, 85, 82] and divide this period into nI = 6 sub-compartments I1, . . . , I6 where infected932 individuals will stay, on average, 2 days in each of them. Note that with this representation, the duration of infectiousness has an Erlang distribution [93] with shape nI (and mean934 12 days). The parameter ψk represents the relative infectiousness of sub-compartment Ik. We assume infectiousness, that is the probability of transmission given contact, is936 proportional to the log viral load measured from respiratory samples in clinical studies [83, 84] and choose ˜ψ = (3, 6, 5, 4, 3, 2) and then normalize according to Equation A.1 with938 ψk = nI ˜ψk/ ∑ k ˜ψk. Similarly, we assume a shorter infectious period for asymptomatic infections of 10 days on940 average [94, 91], divided into nA = 5 sub-compartments with an average stay of 2 days each.942 A-3 Respiratory and faecal Viral kinetics Our model explicitly accounts for the temporal profile of respiratory shedding via the multi-944 ple sub-compartments for the infectious states (A, I and J) combined with the parameters φ and ψ. Similarly, it explicitly accounts for the faecal shedding kinetics via the shedding946 states (A, I, J and Z) and the parameters λ. saved on 2021-07-19 15:54:19-04:00 37 / 43 . CC-BY-ND 4.0 International licenseIt 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 July 25, 2021. ; https://doi.org/10.1101/2021.07.19.21260773doi: medRxiv preprint Wastewater-Based Epidemic Modelling 0 2 4 6 8 0 10 20 30 40 Days from symptom onset Log viral load source Bellon Benefield Jang Neant This study SARS−COV−2 kinetics − Respiratory 2 4 6 0 10 20 30 Days from symptom onset Log viral load source Benefield Hoffmann Miura This study SARS−COV−2 kinetics − Faecal Figure S2: SARS-CoV-2 viral kinetics. The values used in our model are represented by the black curve and the green curves show estimates from the literature. The full reference of each study can be found in the bibliography: Benefield [95], Bellon [96], Jang [83], Neant [84], Hoffmann [87], Miura [97]. We parameterize our model such that the SARS-CoV-2 viral kinetics reflect with levels948 reported in the literature. The black line with points in Figure S2 shows the values used for respiratory (top panel) and faecal (bottom panel) shedding, and how they compare to950 observational studies. A-4 Effective reproduction number952 As a first step, we establish the basic reproduction number,R0, for the model defined by equations 1a-1n. To deriveR0, we follow the methodology presented in [42].954 Asymptomatic individuals are infectious for an average duration of 1 /θ, and the ratio of their infectiousness compared to all the other infectious states (I and J) is ξ. Asymptomatic956 incidence is a proportion α of the overall incidence. Hence, the contribution to the basic reproduction number from the asymptomatically infected individuals is958 RA 0 = βαξ/θ (A.2) Infectious individuals that are symptomatic and that will recover without hospitalization (I) are infectious for an average duration of 1 /ν. Their proportion of the overall incidence960 is calculated by simply stating they are not hospitalized (1 − h) and not asymptomatic (1− α), hence the proportion is (1− h)(1− α). The contribution to the basic reproduction962 number from the symptomatically infected individuals that will not require hospitalization saved on 2021-07-19 15:54:19-04:00 38 / 43 . CC-BY-ND 4.0 International licenseIt 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 July 25, 2021. ; https://doi.org/10.1101/2021.07.19.21260773doi: medRxiv preprint Wastewater-Based Epidemic Modelling is964 RI 0 = β(1− h)(1− α)/ν (A.3) For the symptomatically infected persons that will require hospitalization ( J), the same logic applies to the I sub-group, except that we need to take into account their relative966 infectiousness compared to I, which is ∑nJ k=1 ψk/ ∑nI k=1 ψk. Note that the denominator is simply nI thanks to the normalization defined in Equation A.1. The subgroup in state968 J are hospitalized (h) and not asymptomatic (1 − α) so their contribution to the basic reproduction number is970 RJ 0 = ∑nJ k=1 ψk nI βh(1− α)/µ (A.4) We obtain the (overall) basic reproduction number by summing the respective contributions from all infectious epidemiological states972 R0 =RA 0 +RI 0 +RJ 0 (A.5) Finally, the effective reproduction numberRt is simply defined as Rt = St NR0 which gives Equation 2 in the main text.974 A-5 Fate and RNA transport in wastewater Upon fecal deposition into the wastewater, viral RNA undergoes various hydrodynamic pro-976 cesses and degrades during its journey from the shedding point to the sampling site [16, 33]. This degradation mainly comes from RNA dilution in municipal wastewater constitutes978 (e.g., hygiene products, household detergents, industrial wastewater and storm waters) and RNA decay resulting from harsh wastewater environment (e.g., temperature, bioac-980 tive chemicals, solids, pH, etc.). As a common practice, complex hydraulic models simulate the in-fluid transportation of the water contaminants (endemic viruses and fecal microor-982 ganisms) via solving a set of physics-based differential equations describing the flow and transport mechanisms [98, 99, 100, 101, 102, 103, 104].984 These hydrodynamical models are advection and dispersion mechanisms, which describe the microorganisms’ transportation by the flow velocity along the longitudinal axis and986 its diluting process into the surrounding fluid [103]. Once a mass concentration enters the stream, it gradually disperses due to many physical factors such as dissolving pro-988 cess, velocity profile, turbulent mixing, molecular diffusion, etc. [105]. Moreover, sediment association plays a significant role in the transport models. Several studies indicated at-990 tachments of microorganisms to sediments and the impact of this associations on their delayed transportation [104, 102, 100]. Due to solid mass and subsequent gravitational992 pull, sediment’s velocity differs from the flow velocity and results in solid settlement at the bottom of the stream. These microorganism-attached sediments, later, re-suspend during994 overflow periods, induced by heavy rainfall and industrial discharges, and act as a reservoir saved on 2021-07-19 15:54:19-04:00 39 / 43 . CC-BY-ND 4.0 International licenseIt 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 July 25, 2021. ; https://doi.org/10.1101/2021.07.19.21260773doi: medRxiv preprint Wastewater-Based Epidemic Modelling τ = 7−day delay τ = 3−day delay 40 60 80 100 0.0e+00 5.0e+05 1.0e+06 1.5e+06 2.0e+06 2.5e+06 0.0e+00 5.0e+05 1.0e+06 1.5e+06 2.0e+06 2.5e+06 Simulation time RNA concentration (gc/mL) per capita RNA concen. deposited sampled Figure S3: Impact of advection-dispersion-decay model (Eq. A.8). sampled RNA concentration (Wsamp) may have different viral distribution compared to deposited density (W ∗) due to delayed viral genetic materials resulting from hydrodynamical phenomenon in sewer system. and increase in-stream microorganism’s concentration irrespective of new input from the996 shedding source [106, 107]. Enveloped viruses, such as SARS-CoV-2, dissolve less in water and tends to attach to solids as a result of their hydrophobic envelope [39, 33], and there-998 fore, settlement/resuspension may play a key role in transportation of the SARS-CoV-2 genetic materials in wastewater.1000 Here, we used a simple advection-dispersion-decay model to simulate the fate of SARS- CoV-2 RNA along their journey from shedding points to the sampling site. We assumed the1002 deposited RNA concentration in the sewage system is a one-time pulse input per day and described the transfer process by a dispersed plug-flow model. As the plug concentration of1004 viral RNA enters into the flow with velocity u, it gradually disperses along the longitudinal axis (flow direction) with dispersion coefficient D (m2/s) within the sewage pathway. The1006 length from the shedding location to the sampling site is L. The total RNA mass arrives at the sampling point gradually over time, such that the daily pulse input concentration1008 is delayed. Assuming a small deviation from plug flow ( D/uL≤ 0.01), we can use the analytical solution of the 1-dimensional axial dispersed plug flow differential equation [47,1010 105] as a transfer function for viral RNA in wastewater. For low diffusion limit, the transfer function is approximately symmetrical and defined by a Gaussian distribution g(τ), which1012 represents the fraction of the deposited concentration at the sampling site τ days after its saved on 2021-07-19 15:54:19-04:00 40 / 43 . CC-BY-ND 4.0 International licenseIt 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 July 25, 2021. ; https://doi.org/10.1101/2021.07.19.21260773doi: medRxiv preprint Wastewater-Based Epidemic Modelling

to the sewage system1014 g(τ) = √ u3 4πDL exp ( −(L− uτ)2 4DL/u ) . (A.6) Defining the mean transit time ¯τ = L/u and its standard deviation σ = √ 2DL/u3, we can re-parametrize Equation A.6 based on the first two moments of the transit time:1016 g(τ) = 1√ 2πσ exp ( −(τ− ¯τ)2 2σ2 ) , (A.7) In addition to delay, the genetic materials of SARS-CoV-2 degrades exponentially at a daily rate κ due to the complex environment of wastewater [46]. As a result, W (t), the1018 sampled concentration at time t, is a combination of both delayed viral materials and degradation1020 Wsamp(t) = ∫ t 0 W ∗(t− τ) g(τ) e−κτdτ, (A.8) where W ∗ is the initial daily deposited concentration. A-6 Rt posterior estimates1022 The effective reproduction number Rt is estimated by fitting the model to different data sources (see main text). We also estimateRt using the R library EpiEstim [57] on reported1024 cases. The function βt is modelled as piecewise-linear. As a result, Rt from Eq. 2 is also a1026 piecewise linear function. We interpolated Rt with a polynomial curve (by smooth spline function in R) for Figure 4. For validity purpose, we compared the interpolated and the1028 piecewise linear functionRt and checked the smoothed and actual curves are similar. See Figure S5.1030 saved on 2021-07-19 15:54:19-04:00 41 / 43 . CC-BY-ND 4.0 International licenseIt 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 July 25, 2021. ; https://doi.org/10.1101/2021.07.19.21260773doi: medRxiv preprint Wastewater-Based Epidemic Modelling THC THU TNT 0.50 1.00 1.50 2.00 0.50 1.00 1.50 2.00 0.0 1.0 2.0 0.50 1.00 1.50 2.00 EGB OTW TAB Aug '20 Dec '20 Apr '21 Aug '20 Dec '20 Apr '21 Aug '20 Dec '20 Apr '21 0.0 1.0 2.0 3.0 0.50 1.00 1.50 2.00Effective reproduction number sourcefit clin clin.EpiEstim clinww ww A THC THU TNT EGB OTW TAB Aug '20 Dec '20 Apr '21 Aug '20 Dec '20 Apr '21 Aug '20 Dec '20 Apr '21 0.00 0.50 1.00 1.50 2.00 2.50 0.00 0.50 1.00 1.50 2.00 2.50Width of 95% CI for Rt B Figure S4: Panel A: Effective reproduction number inferred from model, utilizing different data sources, for all WWTP’s locations. Rt inferred by fitted model to clinical reported cases (clin), RNA wastewater concentration (ww), and both clinical reported cases and viral load in wastewater (combined clinww). Panel B: Associated 95% credible interval widths. saved on 2021-07-19 15:54:19-04:00 42 / 43 . CC-BY-ND 4.0 International licenseIt 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 July 25, 2021. ; https://doi.org/10.1101/2021.07.19.21260773doi: medRxiv preprint Wastewater-Based Epidemic Modelling EGB OTW TAB THU clinclinwwww Aug '20 Dec '20 Apr '21 Aug '20 Dec '20 Apr '21 Aug '20 Dec '20 Apr '21 Aug '20 Dec '20 Apr '21 0 1 2 3 0 1 2 3 0 1 2 3 4Effective reproduction number spl.smooth No Y es Figure S5: The effective reproduction number and its 95% credible interval plotted for different sources (clin, ww and combined clinww) of fitting in wastewater sampling locations; Edmonton (EGB), Ottawa (OTW), Toronto Ashbridges Bay (TAB) and Toronto Humber (THU). Colours representRt obtained from model and Eq. 2 (pink) and its interpolated smoothed spline (green). saved on 2021-07-19 15:54:19-04:00 43 / 43 . CC-BY-ND 4.0 International licenseIt 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 July 25, 2021. ; https://doi.org/10.1101/2021.07.19.21260773doi: medRxiv preprint