model_definition.md

CFA wastewater-informed hospital admissions forecasts

This document describes the data sources, model structure, and implementation details for the wastewater-informed hospital admissions forecasts submitted to the COVID-19 Forecast Hub under the name cfa-wwrenewal. The methods described will evolve as we continue to develop and improve the model. We use the model to produce forecasts for 53 jurisdictions: the 50 U.S. states, the District of Columbia, Puerto Rico, and the United States as a whole.

We welcome feedback on the model structure and implementation, please feel free to submit an issue directly on GitHub or reach out via this form.

For an example of how to fit the model to simulated data, we recommend the toy data vignette.

Data sources

We use two data sources as input to the model: viral genome concentrations in wastewater ("genome concentrations") and incident hospital admissions ("admissions"). We describe each of these data sources in detail below.

Viral genome concentration in wastewater data

We source viral genome concentration in wastewater from the CDC National Wastewater Surveillance System (NWSS) full analytic dataset. Concentrations are typically reported in units of estimated genome copies per unit volume wastewater or per unit mass of dry sludge. At present, we use only measurements per unit volume wastewater. The dataset is defined in the data dictionary and is described in plain text on the NWSS website. The full dataset is available to researchers upon request but a dashboard with summary statistics is publicly available.

We use the most recent data available at the time of submission to generate the forecast which is typically partly complete for the most recent week.

These data are complex and have many sources of variability and uncertainty. For example,

• Each jurisdiction (for our forecasts, typically a U.S. state or territory) has a different number of wastewater sampling sites (e.g., wastewater treatment plants) that each cover a catchment area. This means that different sites cover different populations, and when combining the sites across a jurisdiction, represent some subset of the jurisdiction's total population. The fraction of the total population covered by sampling sites varies significantly among the jurisdictions we consider.
• Different wastewater sites may use different sample collection methodologies. These methodologies may also vary through time at an individual site.
• Different sites sample at different cadences. Some sites sample nearly daily, while others sample less than once a week. Some sites report that data within the week while others have reporting latencies on the order of weeks.
• Different samples are processed in different labs that may use different methodologies for extraction, concentration, and quantification. These labs may also vary their methodologies through time.
• Not all of the jurisdictions we forecast submit genome concentrations to CDC's National Wastewater Surveillance System (NWSS).

These complexities mean that wastewater data are often difficult to interpret. For example, a change in genome concentration sampled at a site could reflect a change in the underlying number of infections, a change in the number of samples collected or processed, a change in the collection or processing methodologies, a change in the reporting cadence or latency, and/or a change in the viral shedding kinetics of the population (e.g., due to a new variant or a change in the age distribution of cases).

Details about data preprocessing, outlier detection, and aggregation are in the Appendix.

Hospital admissions data

We source hospital admissions data for each jurisdiction from healthdata.gov. As this data is updated on Friday afternoon with admissions up until the previous Saturday, and we submit forecasts on the following Monday, this means that we have a 9 day delay between the most recent data and when the model is fitted.

Details about data preprocessing and outlier detection are in the Appendix.

Model overview

We use three models to forecast COVID-19 hospital admissions. The models share a common structure and are fit using the same codebase. The models differ in the data they use as input, in whether or not they include a wastewater component, and the aggregation of the estimates they produce.

Model 1 is wastewater-informed, site-level infection dynamics model. It uses genome concentrations at the site level and admissions at the jurisdictional level as input and produces forecasts of admissions at the jurisdictional level. We use Model 1 to generate forecasts for jurisdictions with any wastewater data available. However, if a jurisdiction has not reported any genome concentration data in the past 21 days or if there are fewer than 5 data points per site in any site, we flag this in our metadata, as it is unlikely that the wastewater data is meaningfully informing the forecast in that jurisdiction.

Model 2 is a no-wastewater, jurisdiction-level infection dynamics model. It uses jurisdiction-level admissions as input and produces forecasts of jurisdiction-level admissions as output. We use Model 2 to generate forecasts for jurisdictions without any wastewater data.

Model 3 is a nationwide aggregated wastewater model. It uses nationally-aggregated genome concentrations and national-level admissions as input and produces forecasts of national-level admissions. We use Model 3 to generate nationwide forecasts for the entire United States. Of the forecasts being submitted to the COVID-19 Forecast Hub, only the nationwide (U.S.) forecasts use this observation model.

Model	Input WW data	Input hospitalization data	Infection dynamics	Predicted hospitalizations
Model 1	Site-level	Jurisdiction-level	Coupled site- and jurisdiction-level	Jurisdiction-level
Model 2	None	Jurisdiction-level	Jurisdiction-level	Jurisdiction-level
Model 3	Nationally aggregated	Nationally aggregated	National	National

Alongside each forecast, we publish a corresponding metadata table with jurisdiction-by-jurisdiction notes on wastewater data status and model used. If a jurisdiction has no wastewater data during the fitting period (the past 90 days), we fit Model 2 and indicate this in the metadata. If a jurisdiction has minimal wastewater data or no recent wastewater measurements, we still fit Model 1, but we indicate in the metadata that the wastewater data may be insufficient to inform the forecast.

Model components

Our models are constructed from a set of generative components. These are:

• Infection component: A renewal model for the infection dynamics, which generates estimates of incident latent infections per capita.
• Hospital admissions component: A model for the expected number of hospital admissions given incident latent infections per capita.
• Viral genome concentration in wastewater: A model for the expected genome concentration given incident infections per capita.

Depending on the model, these components are implemented at different spatial scales and with different observation processes. In particular, the link to the observables depends on the model and the form of the observables, see below for detailed descriptions of each component and the following sections for model-specific details.

See the notation section for an overview of the mathematical notation we use to describe the model components, including how probability distributions are parameterized.

Infection component

Renewal process for incident infections

This component assumes that latent (unobserved) expected incident infections per capita $I(t)$ are generated from a renewal ¹²³ process described by: $$I(t) = \mathcal{R}(t) \sum_{\tau = 1}^{T_g} I(t-\tau) g(\tau)$$ Where $g(\tau)$ is the discrete generation interval, which describes the distribution of times from incident infection to secondary infection (i.e. infectiousness profile) and $\mathcal{R}(t)$ is the instantaneous reproduction number, representing the expected number of secondary infections occurring at time $t$, divided by the number of currently infected individuals, each scaled by their relative infectiousness at time $t$ ⁴. $T_g$ is the maximum generation interval, which is the maximum time from infection to secondary infection that we consider, and is set to 15 days.

This process is initialized by estimating an initial exponential growth² of infections for 50 days prior to the calibration start time $t_0$:

$$ I(t) = I_0\exp(rt) $$

where $I_0$ is the initial per capita infection incident infections and $r$ is the exponential growth rate.

Instantaneous reproduction number

We decompose the instantaneous reproduction number $\mathcal{R}(t)$ into two components: an unadjusted instantaneous reproduction number $\mathcal{R}^\mathrm{u}(t)$ and a damping term that accounts for the effect of recent infections on the instantaneous reproduction number⁵.

We assume that the unadjusted reproduction number $\mathcal{R}^\mathrm{u}(t)$ is a piecewise-constant function with weekly change points (i.e., if $t$ and $t'$ are days in the same week, then $\mathcal{R}^\mathrm{u}(t)$ = $\mathcal{R}^\mathrm{u}(t')$ ). To account for the dependence of the unadjusted reproduction number in a given week on the previous week, we use a differenced auto-regressive process for the log-scale reproduction number. A log-scale representation is used to ensure that the reproduction number is positive and so that week-to-week changes are multiplicative rather than additive.

$$ \log[\mathcal{R}^\mathrm{u}(t_3)] \sim \mathrm{Normal}\left(\log[\mathcal{R}^\mathrm{u}(t_2)] + \beta \left(\log[\mathcal{R}^\mathrm{u}(t_2)] - \log[\mathcal{R}^\mathrm{u}(t_1)]\right), \sigma_r \right) $$

where $t_1$, $t_2$, and $t_3$ are days in three successive weeks, $\beta$ is an autoregression coefficient which serves to make week-to-week changes correlated, and $\sigma_r$ determines the overall variation in week-to-week changes. We bound $\beta$ to be between 0 and 1 so that any changes in trend in $\mathcal{R}^\mathrm{u}(t)$ are damped over time to a degree determined by $\beta$.

The damping term we use is based on Asher et al. 2018⁵ but extended to be applicable to a renewal process. It assumes that the instantaneous reproduction number is damped by recent infections weighted by the generation interval. This is a simple way to account for the fact that the instantaneous reproduction number is likely to decrease when there are many infections in the population, due to factors such as immunity, behavioral changes, and public health interventions. The damping term is defined as:

$$ \mathcal{R}(t) = \mathcal{R}^\mathrm{u}(t) \exp \left( -\gamma \sum_{\tau = 1}^{T_f}I(t-\tau)g(\tau) \right) $$

where $\gamma$ is the infection feedback term controlling the strength of the damping on $\mathcal{R}(t)$, and the summation is analogous to the "force of infection." See Prior Distributions below for description of prior choice on $\gamma$.

Hospital admissions component

Following other semi-mechanistic renewal frameworks, we model the expected hospital admissions per capita $H(t)$ as a convolution of the expected latent incident infections per capita $I(t)$, and a discrete infection to hospitalization distribution $d(\tau)$, scaled by the probability of being hospitalized $p_\mathrm{hosp}(t)$.

To account for day-of-week effects in hospital reporting, we use an estimated weekday effect $\omega(t)$. If $t$ and $t'$ are the same day of the week, $\omega(t) = \omega(t')$. The seven values that $\omega(t)$ takes on are constrained to have mean 1.

$$H(t) = \omega(t) p_\mathrm{hosp}(t) \sum_{\tau = 0}^{T_d} d(\tau) I(t-\tau)$$

Where $T_d$ is the maximum delay from infection to hospitalization that we consider.

We define the discrete hospital admissions delay distribution $d(\tau)$ as a convolution of the incubation period distribution ⁶ and a separate estimate of the distribution of time from symptom onset to hospital admission (see Parameter section below for further details).

Infection-hospitalization rate

In the models that include fits to wastewater data, we allow the population-level infection-hospitalization rate (IHR) to change over time. An inferred change in the IHR could reflect either a true change in the rate at which infections result in hospital admissions (e.g. the age distribution of cases could shift, a more or less severe variant could emerge, or vaccine coverage could shift) or a change in the relationship between infections and genomes shed in wastewater $G$ (which we currently treat as fixed, but which could change in time if, for example, immunity reduces wastewater shedding without reducing transmission, or a variant emerges with a different per infection wastewater shedding profile).

Therefore, we model the proportion of infections that give rise to hospital admissions $p_\mathrm{hosp}(t)$ as a piecewise-constant function with weekly change points. If $t$ and $t'$ are two days in the same week, then $p_\mathrm{hosp}(t) = p_\mathrm{hosp}(t')$.

The values $p_\mathrm{hosp}(t)$ follow a logit-scale AR(1) process with linear scale median $\mu_{p_\mathrm{hosp}}$

$$ \mathrm{logit} (p_{\mathrm{hosp}}(t)) = \mathrm{logit} (\mu_{p_{\mathrm{hosp}}}(t)) + \delta_{H}(t)$$

where $\delta_{H}(t)$ is the time-varying site effect on $\mathrm{logit} (p_{\mathrm{hosp}}(t))$, modeled as,

$$\delta_{H}(t) = \varphi_H \delta_{H}(t-1) + \epsilon_{Ht}$$

where $0 < \varphi_H < 1$ and $\epsilon_{Ht} \sim \mathrm{Normal}(0, \sigma_{H\delta})$.

We chose a relatively strong prior of $\varphi_H \sim \mathrm{beta}(1,100)$ to impose minimal autocorrelation, and similarly chose a small prior on $\sigma_{H\delta} \sim \mathrm{Normal}(0, 0.01)$ to impose our believe that the IHR should only change in time if the data strongly suggests it. See Prior Distributions for the specified prior on $\mu_{p_\mathrm{hosp}}$.

In hospital admissions only models, we model the IHR as constant. We assign this constant IHR the same prior distribution that we assign $\mu_{p_\mathrm{hosp}}$ in the wastewater model.

Hospital admissions observation model

We model the observed hospital admission counts $h_t$ as:

$$h_t \sim \mathrm{NegBinom}(n H(t), \phi)$$

where the jurisdiction population size $n$ is used to convert from per-capita hospitalization rate $H(t)$ to hospitalization counts.

Currently, we do not explicitly model the delay from hospital admission to reporting of hospital admissions. In reality, corrections (upwards or downwards) in the admissions data after the report date are possible and do happen. See outlier detection and removal for further details.

Viral genome concentration in wastewater component

We model viral genome concentrations in wastewater $C(t)$ as a convolution of the expected latent incident infections per capita $I(t)$ and a normalized shedding kinetics function $s(\tau)$, multiplied by $G$ the number of genomes shed per infected individual over the course of their infection and divided by $\alpha$ the volume of wastewater generated per person per day:

$$C(t) = \frac{G}{\alpha} \sum_{\tau = 0}^{\tau_\mathrm{shed}} s(\tau) I(t-\tau)$$

where $\tau_\mathrm{shed}$ is the total duration of fecal shedding. Note there is no need to scale by wastewater catchment population size because $I(t)$ is measured as new infections per capita.

This approach assumes that $G$ and $\alpha$ are constant through time and across individuals. In fact, there is substantial inter-individual variability in shedding kinetics and total shedding. This approximation is more accurate when population sizes are large.

We model the shedding kinetics $s(\tau)$ as a discretized, scaled triangular distribution⁷:

$$\log_{10}[s^\mathrm{cont}(\tau)] = \begin{cases} V_\mathrm{peak} \frac{\tau}{\tau_\mathrm{peak}} & \tau \leq \tau_\mathrm{peak} \\\ V_\mathrm{peak} \left( 1 - \frac{\tau - \tau_\mathrm{peak}}{\tau_\mathrm{shed} - \tau_\mathrm{peak}} \right) & \tau_\mathrm{peak} < \tau \leq \tau_\mathrm{shed} \\\ 0 & \tau > \tau_\mathrm{shed} \end{cases}$$

where $V_\mathrm{peak}$ is the peak number or viral genomes shed on any day, $\tau_\mathrm{peak}$ is the time from infection to peak shedding, and $\tau_\mathrm{shed}$ is the total duration of shedding. Then:

We model the log observed genome concentrations as Normally distributed:

$$ \log[c_t] \sim \mathrm{Normal}(C(t), \sigma_c) $$

This component does not mechanistically simulate each step involved in sample collection, processing, and reporting. Instead, it aims to to account for these processes, at a summary level. Future iterations of this model will evaluate the utility of mechanistic modeling of wastewater collection and processing.

Model 1: Site-level infection dynamics

In this model, we represent hospital admissions at the jurisdictional level and viral genome concentrations at the site level. We use the components described above but divide the jurisdiction's total population into subpopulations representing sampled wastewater sites' catchment populations, with an additional subpopulation to represent individuals who do not contribute to sampled wastewater.

We model infection dynamics in these subpopulations hierarchically: subpopulation infection dynamics are distributed about a central jurisdiction-level infection dynamic, and jurisdiction's total infections are simply the sum of the subpopulation-level infections.

Subpopulation definition

In Model 1, a jurisdiction consists of $K_\mathrm{total}$ subpopulations $k$ with corresponding population sizes $n_k$. We associate one subpopulation to each of the $K_\mathrm{sites}$ wastewater sampling sites in the jurisdiction and assign that subpopulation a population size $n_k$ equal to the wastewater catchment population size reported to NWSS for that site.

Whenever the sum of the wastewater catchment population sizes $\sum\nolimits_{k=1}^{K_\mathrm{sites}} n_k$ is less than the total jurisdiction population size $n$, we use an additional subpopulation of size $n - \sum\nolimits_{k=1}^{K_\mathrm{sites}} n_k$ to model individuals in the jurisdiction who are not covered by wastewater sampling.

The total number of subpopulations is then $K_\mathrm{total} = K_\mathrm{sites} + 1$: the $K_\mathrm{sites}$ subpopulations with sampled wastewater, and the final subpopulation to account for individuals not covered by wastewater sampling.

This amounts to modeling the wastewater catchments populations as approximately non-overlapping; every infected individual either does not contribute to measured wastewater or contributes principally to one wastewater catchment. This approximation is reasonable because we only use samples taken from primary wastewaster treatment plants, which avoids the possibility that an individual might be sampled once in a sample taken upstream and then sampled again in a more aggregated sample taken further downstream; see data filtering for further details.

If the sum of the wastewater site catchment populations meets or exceeds the reported jurisdiction population ($\sum\nolimits_{k=1}^{K_\mathrm{sites}} n_k \ge n$) we do not use a final subpopulation without sampled wastewater. In that case, the total number of subpopulations $K_\mathrm{total} = K_\mathrm{sites}$. In our data, this is true only for the District of Columbia.

When converting from predicted per capita incident hospital admissions $H(t)$ to predicted hospitalization counts, we use the jurisdiction population size $n$, even in the case of the District of Columbia where $\sum n_k &gt; n$.

This amounts to making two key additional modeling assumptions:

• Any individuals who contribute to wastewaster measurements but are not part of the jurisdiction population are distributed among the catchment populations approximately proportional to catchment population size.
• Whenever $\sum n_k \ge n$, the fraction of individuals in the jurisdiction not covered by wastewater is small enough to have minimal impact on the jurisdiction-wide per capita infection dynamics.

Subpopulation-level infection dynamics

We couple the subpopulation- and jurisdiction-level infection dynamics at the level of the un-damped instantaneous reproduction number $\mathcal{R}^\mathrm{u}(t)$.

We model the subpopulations as having infection dynamics that are similar to one another but can differ from the overall jurisdiction-level dynamic.

We represent this with a hierarchical model where we first model a jurisdiction-level un-damped effective reproductive number $\mathcal{R}^\mathrm{u}(t)$, but then allow individual subpopulations $k$ to have individual subpopulation values of $\mathcal{R}^\mathrm{u}_{k}(t)$

The jurisdiction-level model for the undamped instantaneous reproductive number $\mathcal{R}^\mathrm{u}(t)$ follows the time-evolution described above. Subpopulation deviations from the jurisdiction-level reproduction number are modeled via a log-scale AR(1) process. Specifically, for subpopulation $k$:

$$ \log[\mathcal{R}^\mathrm{u}_{k}(t)] = \log[\mathcal{R}^\mathrm{u}(t)] + \delta_k(t) $$

where $\delta_k(t)$ is the time-varying subpopulation effect on $\mathcal{R}(t)$, modeled as,

$$\delta_k(t) = \varphi_{R(t)} \delta_k(t-1) + \epsilon_{kt}$$

where $0 &lt; \varphi_{R(t)} &lt; 1$ and $\epsilon_{kt} \sim \mathrm{Normal}(0, \sigma_{R(t)\delta})$.

We chose a prior of $\varphi_{R(t)} \sim \mathrm{beta}(2,40)$ to impose limited autocorrelation in the week-by-week deviations. We set a weakly informative prior $\sigma_{R(t)\delta} \sim \mathrm{Normal}(0, 0.3)$ to allow for either limited or substantial site-site heterogeneity in $\mathcal{R}(t)$, with the degree of heterogeneity inferred from the data.

The subpopulation $\mathcal{R}_{k}(t)$ is subject to the infection feedback described above such that:

$$\mathcal{R}_k(t) = \mathcal{R}^\mathrm{u}_k(t) \exp \left(-\gamma \sum_{\tau = 1}^{T_f} I_k(t-\tau) g(\tau) \right)$$

From $\mathcal{R}_{k}(t)$, we generate values of the supopulation-level expected latent incident infections per capita $I_k(t)$ using the renewal process described in the infection component.

To obtain the number of infections per capita $I(t)$ in the jurisdiction as a whole, we sum the $K_\mathrm{total}$ subpopulation per capita infection counts $I_k(t)$ weighted by their population sizes:

$$I(t) = \frac{1}{\sum\nolimits_{k=1}^{K_\mathrm{total}} n_k} \sum_{k=1}^{K_\mathrm{total}} n_k I_k(t)$$

We infer the site level initial per capita incidence $I_k(0)$ hierarchically. Specifically, we treat $\mathrm{logit}[I_k(0)]$ as Normally distributed about the corresponding jurisdiction-level value $\mathrm{logit}[I(0)]$, with an estimated standard deviation $\sigma_{i0}$:

$$ \mathrm{logit}[I_k(0)] \sim \mathrm{Normal}(\mathrm{logit}[I(0)], \sigma_{k0}) $$

Viral genome concentration in wastewater

We model site-specific viral genome concentrations in wastewater $C_i(t)$ independently for each site $i$ using the same model as described in the wastewater component. The latent incident infections in subpopulation $k$ are mapped to the corresponding site $i$.

Genome concentration measurements can vary between sites, and even within a site through time, because of differences in sample collection and lab processing methods. To account for this variability, we add a scaling term $M_{ij}$ and a variablity term $\sigma_{cij}$ that vary across sites $i$ and also within sites across labs $j$:

$$\log[c_{ijt}] \sim \mathrm{Normal}(\log[M_{ij} C_i(t)], \sigma_{cij})$$

Both $M_{ij}$ and $\sigma_{cij}$ are modeled as site-level random effects.

In the rare cases when a site submits multiple concentrations for a single date and lab method, we treat each record as an independent observation.

Censoring of wastewater observations below the limit of detection

Lab processing methods have a finite limit of detection (LOD), such that not all wastewater measurements can be modeled using the log-normal approach above. This limit of detection varies across sites, between methods, and potentially also over time.

If an observed value $c_{ijt}$ is above the corresponding LOD, then the likelihood is:

$$ f_\mathrm{Normal}(\log[c_{ijt}]; \log[M_{ij} C_i(t)], \sigma_{cij}) $$

where $f_\mathrm{Normal}(x; \mu, \sigma)$ is the probability density function of the Normal distribution. When the observed value is below the LOD, we use a censored likelihood:

$$\int_{-\infty}^{\log [\mathrm{LOD}_{ijt}]} f_\mathrm{Normal}(x; \log[M_{ij} C_i(t)], \sigma_{cij}) \mathrm{d}x$$

(This is mathematically equivalent to integrating the probability density function of the log-normal distribution from zero to the LOD.)

If a sample is flagged in the NWSS data as below the LOD (field pcr_target_below_lod) but is missing a reported LOD (field lod_sewage), the 95th percentile of LOD values across the entire data is used as the integral's upper limit.

If a sample has a reported concentration (field pcr_target_avg_conc) above the corresponding reported LOD, but the sample is nevertheless flagged as below the LOD (field pct_target_below_lod), we assume the flag takes precedence and treat the sample as below LOD for the purposes of censoring.

Model 2: no wastewater

The no wastewater, jurisdiction-level infection dynamics model is the simplest model, because it does not include wastewater data. Each jurisdiction is modeled as a separate population according to the infection component and hospitalization component described above. We use this model when no wastewater data is available for a jurisdiction. This model also serves as a comparative baseline for evaluating wastewater-informed models.

Model 3: nationwide wastewater

In this model, the entire United States is treated as a single population. This model uses the general infection, hospitalization, and wastewater components described above, but with the following modifications:

We first generate a thresholded population-weighted average viral genome concentration in wastewater:

1. For each week and site, if the site has more than one sample in that week, compute the mean genome concentration across those samples. This associates each site and week with a single genome concentration.
2. For each week, compute the weighted mean genome concentration across all sites. The weights are the site population, or 300,000, whichever is lower. The aim of this thresholding is to prevent large sites from dominating the average concentration.

This approach is similar to the algorithm used by Biobot Analytics to derive regional and national aggregate genome concentration. Future iterations of the model will evaluate the validity of this approach.

We then use this thresholded population-weighted average concentration as the input to the wastewater viral concentration component.

Model parameters

Prior distributions

We use informative priors for parameters that have been well characterized in the literature and weakly informative priors for parameters that have been less well characterized.

Parameter	Prior distribution	Source
Initial hospitalization probability	$\mathrm{logit}[p_{\mathrm{hosp}}(t_0)] \sim \mathrm{Normal}(\mathrm{logit}[0.01], 0.3)$	Perez-Guzman et al. 2023 8
Time to peak fecal shedding	$\tau_\mathrm{peak} \sim \mathrm{Normal}(5 \text{ days}, 1 \text{ day})$	Russell et al. 2023 9, Huisman et al. 2022 10, Cavany et al. 2022 11
Peak viral shedding $V_\mathrm{peak}$	$\log_{10}[V_\mathrm{peak}] \sim \mathrm{Normal}(5.1, 0.5)$	Miura et al. 2021 12
Duration of shedding	$\tau_\mathrm{shed} \sim \mathrm{Normal}(17 \text{ days}, 3 \text{ days})$	Cevik et al. 2021 13, Russell et al. 2023 9
Total genomes shed per infected individual	$\log_{10}[G] \sim \mathrm{Normal}(9, 2)$	Watson et al 202314
Initial infections per capita $I_0$	$I_0 \sim \mathrm{Beta}(1 + k i_\mathrm{est}, 1 + k (1-i_\mathrm{est}))$	where $i_\mathrm{est}$ is the sum of the last 7 days of hospital admissions, divided by jurisdiction population, and divided by the prior mode for $p_\mathrm{hosp}$, and $k = 5$ is a parameter governing the informativeness ("certainty") of the Beta distribution
Initial exponential growth rate	$r \sim \mathrm{Normal}(0, 0.01)$	Chosen to assume flat dynamics prior to observations
Infection feedback term	$\gamma \sim \mathrm{logNormal}(6.37, 0.4)$	Weakly informative prior chosen to have a mode of 500 in natural scale, based on posterior estimates of peaks from prior seasons in a few jurisdictions

Scalar parameters

Parameter	Value	Source
Maximum generation interval	$T_g = 15$ days
Maximum infection to hospital admissions delay	$T_d = 55$ days
Wastewater produced per person-day	$\alpha=$ 378,500 mL per person-day	Ortiz 202415

Distributional parameters

The discrete generation interval probability mass function $g(\tau)$ approximates a log-normal distribution⁶ with log-mean 2.9 and log-standard deviation of 1.64. To approximate the double censoring process necessary to discretize the continuous log-normal distribution, we use a simulation-based approach as recommended by Park et al.¹⁶. This assumes that the primary event is uniformly distributed (this ignores the influence of the growth rate within the primary interval but is a good approximation in most settings). The secondary event is then a sum of this primary interval and the continuous distribution and is observed within a day (see Figure 9 in ¹⁶). As the renewal process is not defined if there is probability mass on day zero we further left truncate this distribution. For more details refer to ¹⁶.

We derive the distribution $\delta(\tau)$ of the delay from infection to hospital admission as the sum of the incubation period (delay from infection to symptom onset) and the period from symptom onset to hospital admission.

We model the incubation period with a discretized, modified Weibull distribution⁶ with probability mass function $\delta(\tau)$:

$$ \delta^\mathrm{cont}(\tau) = \exp[0.15\tau] f_\mathrm{Weibull}(\tau; \mathrm{shape}=1.5, \mathrm{scale}=3.6) $$

$$\delta(\tau) = \begin{cases} \delta^\mathrm{cont}(\tau) / \left( {\sum}_{\tau'=0}^{23} g^\mathrm{cont}(\delta') \right) & 0 \leq \tau \leq 23 \\\ 0 & \text{otherwise} \end{cases}$$

We model the symptom onset to hospital admission delay distribution with a Negative Binomial distribution with probability mass function $\gamma(\tau)$ fit to line list patient data from Dananché et al. 2022¹⁷.

$$ \gamma(\tau) = f_\mathrm{NegBin}(\tau; 6.99 \text{ days}, 2.49 \text{ days}) $$

The infection-to-hospitalization delay distribution $d(\tau)$ is the convolution:

$$ d(\tau) = \sum_{x=0}^\tau \delta(x) \gamma(\tau - x) $$

This resulting infection to hospital admission delay distribution has a mean of 12.2 days and a standard deviation of 5.67 days.

Implementation

Our framework is an extension of the widely used ^{18 19}, semi-mechanistic renewal framework {EpiNow2} ²⁰², using a Bayesian latent variable approach implemented in the probabilistic programming language Stan ²¹ using ²² to interface with R. For submission to the COVID-19 Forecast Hub, the model is run on Saturday to generate forecasts each Monday.

For each jurisdiction, we run 4 chains for 750 warm-up iterations and 500 sampling iterations, with a target average acceptance probability of 0.95 and a maximum tree depth of 12.

To generate forecasts per the hub submission guidelines, we calculate the necessary quantiles from the 2,000 draws from the posterior of the expected observed hospital admissions 28 days ahead of the Monday forecast date.

Appendix: Notation

The notation $X \sim \mathrm{Distribution}$ indicates that a random variable $X$ is distributed according to a given distribution.

We parameterize Normal distributions in terms of their mean and standard deviation: $\mathrm{Normal}(\mathrm{mean, standard\ deviation})$.

We parameterize Beta distributions in terms of their two standard shape parameters $\alpha$ and $\beta$, which can be interpreted in terms of the counts of observed successes and failures, respectively, in a Binomial experiment to infer a probability: $\mathrm{Beta}(\alpha, \beta)$.

We parameterize Negative Binomial distributions in terms of their mean and their positive-constrained dispersion parameter (often denoted $\phi$): $\mathrm{NegBinom}(\mathrm{mean, dispersion})$. As the dispersion parameter goes to 0, a Negative Binomial distribution becomes increasingly over-dispersed. As it goes to positive infinity, the Negative Binomial approximates a Poisson distribution with the same mean.

We write $\mathrm{logit}(x)$ to refer to the logistic transform: $\mathrm{logit}(x) \equiv \log(x) - \log(1 - x)$.

Observed data are labeled by data source: $c$ for wastewater concentrations, $h$ for hospital admissions. Hospitalization data are indexed by day $t$ (i.e., $h_t$). Wastewater data are indexed by site $i$, wastewater testing lab $j$, and day $t$ (e.g., $c_{ijt}$).

Appendix: Wastewater data pre-processing

Field names are references to fields in the analytic dataset.

Data filtering

• Primary wastewater treatment plants only (field sample_location equals wwtp)
• SARS-CoV-2 amplification targets only (field pcr_target equals sars-cov-2)
• No solid samples (field sample_matrix is not primary_sludge, and field pcr_target_units is not copies/g dry sludge)
• No samples flagged for quality issues (field quality_flag is not yes)
• Outliers removed (see below)

Viral genome concentration in wastewater outlier detection and removal

We identify potential outlier genome concentrations for each unique site and lab pair with an approach based on $z$-scores.

Briefly, we compute $z$-scores for the concentrations and their finite differences and remove any observations above a threshold values for either metric. In detail:

1. For purposes of outlier detection, exclude wastewater observations below the LOD.
2. For purposes of outlier detection, exclude observations more than 90 days before the forecast date.
3. For each site $i$, compute the change per unit time between successive observations $t$ and $t'$: $(\log[c_{it'}] - \log[c_{it}])/(t' - t)$.
4. Compute $z$-scores for $\log[c_{it}]$ across all sites $i$ and timepoints $t$. Flag values with $z$-scores over 3 as outliers and remove them from model calibration.
5. Compute $z$-scores for the change per unit time values across all sites and pairs of timepoints. For values with $z$-scores over 2, flag the corresponding wastewater concentrations $c_{it}$ as outliers and remove them from model calibration.

The $z$-score thresholds were chosen by visual inspection of the data.

Other data pre-processing

• For consistency, the reported viral genome concentration in wastewater (field pcr_target_avg_conc) and LOD (field lod_sewage) are converted to genome copies per mL wastewater using the reported measurement units (field pcr_target_units).
• For wastewater catchment population $n_{it}$, we use field population_served.
• We identify the unique combinations of sites and labs and add this as a column to our pre-processed dataset.

Hospital admissions pre-processing

We visually inspect the hospital admissions data for each jurisdiction before producing a forecast, identifying anomalies that seem implausible. We then remove these observations from the model inference, treating them as missing data (i.e. as NA values in the model). Often these implausible observations are due to reporting errors, such as a hospital reporting a large number of admissions on a single day that should have been spread out over multiple days and are later corrected. When this happens, we add the corrected data back into the model inference when it gets updated.

References

Footnotes

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2. Abbott, S. et al. EpiNow2: Estimate real-time case counts and time-varying epidemiological parameters. https://doi.org/10.5281/zenodo.3957489 ↩ ↩² ↩³

3. Fraser, C. (2007). Estimating individual and household reproduction numbers in an emerging epidemic. PLoS One, 2(8), e758 (2007). https://doi.org/10.1371/journal.pone.0000758 ↩

4. Gostic, K.M. et al. Practical Considerations for Measuring the Effective Reproductive Number, Rt. PLoS Comput Biol. 16(12) (2020). https://doi.org/10.1371/journal.pcbi.1008409 ↩

5. Asher, J. Forecasting Ebola with a regression transmission model. Epidemics. 22, 50-55 (2018). https://doi.org/10.1016/j.epidem.2017.02.009 ↩ ↩²

6. Park, S.W. et al. Inferring the differences in incubation-period and generation-interval distributions of the Delta and Omicron variants of SARS-CoV-2. Proc Natl Acad Sci U S A. 120(22):e2221887120 (2023). https://doi.org/10.1073/pnas.2221887120 ↩ ↩² ↩³

7. Larremore, D.B. et al. Test sensitivity is secondary to frequency and turnaround time for COVID-19 screening. Science Advances (2021). https://doi.org/10.1126/sciadv.abd5393 ↩

8. Perez-Guzman, P.N. et al. Epidemiological drivers of transmissibility and severity of SARS-CoV-2 in England. Nat Commun 14, 4279 (2023). https://doi.org/10.1038/s41467-023-39661-5 ↩

9. Russell, T.W. et al. Within-host SARS-CoV-2 viral kinetics informed by complex life course exposures reveals different intrinsic properties of Omicron and Delta variants. medRxiv (2023). https://doi.org/10.1101/2023.05.17.23290105 ↩ ↩²

10. Huisman, J.S. et al. Estimation and worldwide monitoring of the effective reproductive number of SARS-CoV-2 eLife 11:e71345 (2022). https://doi.org/10.7554/eLife.71345 ↩

11. Cavany S, et al. Inferring SARS-CoV-2 RNA shedding into wastewater relative to the time of infection. Epidemiology and Infection 150:e21 (2022). https://doi.org/10.1017/S0950268821002752 ↩

12. Miura F, Kitajima M, Omori R. Duration of SARS-CoV-2 viral shedding in faeces as a parameter for wastewater-based epidemiology: Re-analysis of patient data using a shedding dynamics model. Sci Total Environ 769:144549 (2021). https://doi.org/10.1016/j.scitotenv.2020.144549 ↩

13. Cevik, M. et al. SARS-CoV-2, SARS-CoV, and MERS-CoV viral load dynamics, duration of viral shedding, and infectiousness: a systematic review and meta-analysis. Lancet Microbe 2(1),e13-e22 (2021). https://doi.org/10.1016/S2666-5247(20)30172-5 ↩

14. Leighton, M. et al. Improving estimates of epidemiological quantities by combining reported cases with wastewater data: a statistical framework with applications to COVID-19 in Aotearoa New Zealand. medRxiv (2023). https://doi.org/10.1101/2023.08.14.23294060 ↩

15. Ortiz, P. Wastewater facts - statistics and household data in 2024. https://housegrail.com/wastewater-facts-statistics/ ↩

16. Park, S.W. et al. Estimating epidemiological delay distributions for infectious diseases. medRxiv (2024). https://doi.org/10.1101/2024.01.12.24301247 ↩ ↩² ↩³

17. Danaché, C. et al. Baseline clinical features of COVID-19 patients, delay of hospital admission and clinical outcome: A complex relationship. PLoS One 17(1):e0261428 (2022). https://doi.org/10.1371/journal.pone.0261428 ↩

18. US Centers for Disease Control and Prevention. Current Epidemic Growth Status (Based on Rt) for States and Territories. https://www.cdc.gov/forecast-outbreak-analytics/about/rt-estimates.html (2024). ↩

19. US Centers for Disease Control and Prevention. Technical Blog: Improving CDC’s Tools for Assessing Epidemic Growth (2024). https://www.cdc.gov/forecast-outbreak-analytics/about/technical-blog-rt.html ↩

20. Abbott, S. et al. Estimating the time-varying reproduction number of SARS-CoV-2 using national and subnational case counts. Wellcome Open Res. 5:112 (2020). https://doi.org/10.12688/wellcomeopenres.16006.2 ↩

21. Stan Development Team. Stan Modeling Language Users Guide and Reference Manual. (2023). https://mc-stan.org ↩

22. CmdStanR: the R interface to CmdStan. (2024). https://mc-stan.org/cmdstanr/index.html ↩