Received XXX; revised XXX; accepted for publication XXX

# A joint Bayesian hierarchical model for estimating SARS-CoV-2 diagnostic and
subgenomic RNA viral dynamics and seroconversion

Tracy Q. Dong1∗ and Elizabeth R. Brown1,2
1Vaccine and Infectious Disease Division, Fred Hutchinson Cancer Center
2Department of Biostatistics, University of Washington
<EMAIL_ADDRESS>
###### Abstract

Understanding the viral dynamics and immunizing antibodies of the severe acute
respiratory syndrome coronavirus 2 (SARS-CoV-2) is crucial for devising better
therapeutic and prevention strategies for COVID-19. Here, we present a
Bayesian hierarchical model that jointly estimates the diagnostic RNA viral
load reflecting genomic materials of SARS-CoV-2, the subgenomic RNAs (sgRNA)
viral load reflecting active viral replication, and the rate and timing of
seroconversion reflecting presence of antibodies. Our proposed method accounts
for the dynamical relationship and correlation structure between the two types
of viral load, allows for borrowing of information between viral load and
antibody data, and identifies potential correlates of viral load
characteristics and propensity for seroconversion. We demonstrate the features
of the joint model through application to the COVID-19 PEP study and conduct a
cross-validation exercise to illustrate the model’s ability to impute the
sgRNA viral trajectories for people who only had diagnostic viral load data.
Joint models; Bayesian hierarchical models; SARS-CoV-2; Correlates; Viral
load; Seroconversion.

00footnotetext: To whom correspondence should be addressed.

##  1 Introduction

The severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) has spread
rapidly around the world, resulting in more than 6 million deaths worldwide to
date (World Health Organization, 2022). Current knowledge regarding SARS-CoV-2
viral dynamics is built on diagnostic viral load detected via real-time
reverse transcription polymerase chain reaction (RT-PCR)(Cevik _and others_ ,
2021; Gastine _and others_ , 2021; Mellis _and others_ , 2022; Pajon _and
others_ , 2022), the most sensitive and widely used technique for the
diagnosis of coronavirus disease 2019 (COVID-19). However, the diagnostic RT-
PCR tests give limited insight into infectivity because the results may
reflect residual genomic material rather than replicating virus (Singanayagam
_and others_ , 2020). Viral culture and quantification by plaque assays
provide a gold standard for assessing infectivity but are not readily
available at large scale for a diagnostic assay. Therefore, detection of
subgenomic RNAs (sgRNA) in clinical samples has been suggested as an
additional diagnostic tool to track infectious virus because sgRNA transcripts
indicate active viral replication (Dagotto _and others_ , 2021; Bravo _and
others_ , 2022). Characterization of both SARS-CoV-2 diagnostic and sgRNA
viral dynamics is crucial for understanding the pathogenesis of the virus,
identifying patients most at risk , and optimizing the period of self-
isolation to reduce onward transmission while reducing unnecessary loss of
productivity.

Developing a model for SARS-CoV-2 viral load is of particular interest because
viral load is an important outcome of COVID-19 treatment, vaccine, and non-
vaccine prevention studies but viral load trajectories are rarely fully
observed (levine202a1initial; Gottlieb _and others_ , 2021; Goldberg _and
others_ , 2021; Chaccour _and others_ , 2021; Portal-Celhay _and others_ ,
2022; Levine-Tiefenbrun _and others_ , 2022; Eyre _and others_ , 2022). One
commonly used approach is compartmental models based on ordinary differential
equations (ODE) (Goyal _and others_ , 2020; Néant _and others_ , 2021; Ke _and
others_ , 2021; Challenger _and others_ , 2022). While the ODE-based models
are useful for capturing cell-level within-host viral kinetics and simulating
potential effects of antiviral therapies under various assumptions, the model
parameters cannot be easily translated into key viral trajectory
characteristics such as peak viral load and shedding duration. Therefore,
Bayesian mixed-effects models are a popular alternative for estimating
population- and individual-level viral load trajectories (Kissler _and others_
, 2021a, b; Stankiewicz Karita _and others_ , 2021; Jones _and others_ , 2021;
Singanayagam _and others_ , 2021; Deming _and others_ , 2022). This approach
has the advantage of being able to directly model viral load features and
incorporate covariates and correlation structures to explore potential
associations between viral dynamics and patient characteristics. So far, all
existing models estimate the diagnostic and sgRNA viral load separately
without considering any dependency between the two.

In addition to viral dynamics, serology characteristics of SARS-CoV-2
infection, especially the rate and timing of seroconversion defined by the
development of neutralizing antibodies, are also of interest. Not all persons
recovering from SARS-CoV-2 infection develop antibodies (Oved _and others_ ,
2020; Pathela _and others_ , 2021; Wellinghausen _and others_ , 2020).
Understanding the natural immunity acquired after SARS-CoV-2 infection and the
correlates of seroconversion is crucial for studying disease transmission,
vaccine effectiveness, and population-wide seroprevalence and immunity. Since
most serological studies were conducted among hospitalized people who
exhibited moderate to severe COVID-19 symptoms and were tracked in the later
course of the infection (Lou _and others_ , 2020; Yadav _and others_ , 2021;
Masiá _and others_ , 2021), little is known regarding seroconversion rate and
time in outpatients settings and among persons with very mild or asymptomatic
infection (Centers for Disease Control and Prevention, 2021).

In this paper, we develop a Bayesian hierarchical model that jointly analyzes
SARS-CoV-2 diagnostic and sgRNA viral load data and interval-censored
seroconversion data to estimate viral trajectories, their association with
patient characteristics, and the correlates, rate, and time of seroconversion.
Our proposed model has several advantages compared to the existing models:
first, it takes into account the dynamical relationship between the diagnostic
PCR and sgRNA viral trajectories and can be used to impute sgRNA viral
trajectories for patients who only had diagnostic viral load data. In
addition, by jointly modeling viral load and antibody measurements, our model
allows the borrowing of information across different data types to better
estimate key virological and serological parameters. Our model also uses a
pair of indicator variables for the observed and true viral shedding status to
allow direct modeling of false positive and false negative cases. Finally, our
model explicitly accounts for the increased uncertainty in RT-PCR viral load
measurements below the test’s limit of quantification (LoQ).

In Section 2, we introduce a motivating example of the COVID-19 post-exposure
prophylaxis (PEP) study (Barnabas _and others_ , 2021). Section 3 describes
the model formulation and inference procedure. In Section 4, we apply our
methods to analyze data collected in the COVID-19 PEP study and summarize the
results. We end the paper with a discussion in Section 5.

##  2 A motivating example: the COVID-19 PEP Study

The COVID-19 PEP Study was a double-blinded, household-randomized controlled
trial comparing hydroxychloroquine to placebo-like control for prevention of
SARS-CoV-2 infection (Barnabas _and others_ , 2021). The study was remotely
conducted between March and August 2020 before COVID-19 vaccines were
available and enrolled 829 initially asymptomatic household contacts and
healthcare workers recently exposed ($<$96 hours) to persons with laboratory-
confirmed SARS-CoV-2 infection from 41 U.S. states. The study found no
differences between hydroxychloroquine and placebo-like control arms in SARS-
CoV-2 acquisition.

Participants provided demographic information via enrollment surveys. They
also completed daily symptom questionnaires and self-collected daily mid-
turbinate swabs for 14 consecutive days. Swabs were tested for SARS-CoV-2 RNA
at the University of Washington (UW) Virology Laboratory using RT-PCR
targeting nucleocapsid genes N1 and N2. Samples with detectable SARS-CoV-2 RNA
via RT-PCR were further tested for sgRNA targeting sgE (Bruce _and others_ ,
2021).

RT-PCR for both N and sgE were performed to quantify the viral titer via the
cycle threshold (Ct), defined as the number of thermal cycles needed to
amplify sampled viral RNA to a detectable level. Quantification standards were
run with both assays to convert Ct value to viral load as copies/ml during
assay validation. Both tests used a Ct cutoff of 40, corresponding to a lower
limit of detection (LoD) of approximately $10^{2.4}$ copies/ml. In addition,
the diagnostic RT-PCR had a limit of quantification (LoQ), defined as the
lowest amount of viral RNA in a sample that can be quantitatively determined
with acceptable precision accuracy (Forootan _and others_ , 2017), of
approximately $10^{4.9}$ copies/ml, corresponding to a Ct value of 31.
Therefore, the conversion from the diagnostic Ct values to viral
concentrations suffers higher uncertainty for nasal swabs with observed Ct
values between the LoD and LoQ.

Dried blood spot (DBS) samples were voluntarily provided by a subset of
participants on study day 1, 14 and 28. The DBS samples were assayed to detect
Immunoglobulin G (IgG) antibodies to SARS-CoV-2 spike protein at the UW
Department of Laboratory Medicine & Pathology. For brevity, we will use
“seroconversion” to refer to anti-spike IgG seroconversion for the rest of the
paper. However, the model can be generalized to accommodate other types of
seroonversion as well. More information can be found in the Discussion
section.

##  3 Methods

We assume that the diagnostic and sgRNA viral load trajectories consist of a
proliferation phase with exponential growth in viral RNA concentration,
followed by a clearance phase characterized by exponential decay in viral RNA
concentration. On the scale of log10 copies/ml, this roughly corresponds to a
linear increase to a peak followed by a linear decrease. Figure 1 shows a
graphical illustration of the latent viral RNA concentration trends overlay
with the observed diagnostic and sgRNA viral load measurements over time for
an individual participant. We describe the details of the model in the
following sections.

###  3.1 Modeling the diagnostic RNA viral load

Let $i$ be index for participants and $j$ be index for nasal swabs. For person
$i$, let $w_{a,i}$ represent time from shedding onset to peak viral load,
$w_{b,i}$ represent time from peak to viral clearance, and $v_{p,i}$ represent
the magnitude of peak viral load with respect to LoD (Figure 1). We use the
latent peak as a reference point and let $s$ represent time since the latent
peak viral load. As such, we have $s=0$ at the latent peak viral load,
$s=-w_{a,i}$ at shedding onset, and $s=w_{b,i}$ at viral clearance. For the
$j$th swab collected from person $i$, $s_{ij}$ is the time from peak viral
load to swab collection, and the latent diagnostic viral load at $s_{ij}$,
denoted by $\mu_{y}(s_{ij})$, is

$\displaystyle\mu_{y}(s_{ij})=\mbox{LoD}+v_{p,i}+\left(\frac{v_{p,i}}{w_{a,i}}\right)\times
s_{ij}\times\mathbbm{1}\left[{s_{ij}\leq
0}\right]+\left(-\frac{v_{p,i}}{w_{b,i}}\right)\times
s_{ij}\times\mathbbm{1}\left[{s_{ij}>0}\right],$

where $\mathbbm{1}\left[{\cdot}\right]$ is an indicator function. In
particular, $\left(\frac{v_{p,i}}{w_{a,i}}\right)$ and
$\left(-\frac{v_{p,i}}{w_{b,i}}\right)$ represent the upward and downward
slopes respectively for the linear change in diagnostic viral trajectory over
time. To model the person-level variability in $w_{a,i}$, $w_{b,i}$ and
$v_{p,i}$ while accounting for potential correlation among these parameters,
we use a multivariate normal distribution for the random effects on the log
scale:

$\displaystyle\log\left([v_{p,i},w_{a,i},w_{b,i}]^{\intercal}\right)\sim$
$\displaystyle~{}\mbox{MVN}\left(\bm{\mu}_{log,i},\bm{\Sigma}_{log}\right)$
$\displaystyle\bm{\mu}_{log,i}[1]=$
$\displaystyle~{}\mu_{lvp}+\bm{\beta}_{vp}^{\intercal}\bm{X}_{vp,i}$
$\displaystyle\bm{\mu}_{log,i}[2]=$
$\displaystyle~{}\mu_{lwa}+\bm{\beta}_{wa}^{\intercal}\bm{X}_{wa,i}$
$\displaystyle\bm{\mu}_{log,i}[3]=$
$\displaystyle~{}\mu_{lwb}+\bm{\beta}_{wb}^{\intercal}\bm{X}_{wb,i}$

where $\bm{X}_{vp,i}$, $\bm{X}_{wa,i}$ and $\bm{X}_{wb,i}$ are vectors of
person-level covariates for peak viral load, time from shedding onset to peak,
and time from peak to viral clearance, respectively; and $\bm{\beta}_{vp}$,
$\bm{\beta}_{wa}$ and $\bm{\beta}_{wb}$ are the corresponding vectors of
coefficients for these covariates.

We use a pair of indicator variables to model the true positive rate (TPR) and
true negative rate (TNR) for the observed diagnostic viral detection. Let
$y_{ij}$ denote the observed viral RNA concentration in log10 copies/ml for
swab $j$ of person $i$, $B_{ij}$ be the fully observed indicator variable of
whether the swab had detectable diagnostic RNA via RT-PCR (i.e.,
$B_{ij}=\mathbbm{1}\left[{y_{ij}>\mbox{LoD}}\right]$), and let $S_{ij}$ be the
latent indicator variable of whether person $i$ was truly experiencing
diagnostic viral shedding when swab $j$ was collected (i.e.,
$S_{ij}=\mathbbm{1}\left[{-w_{a,i}\leq s_{ij}\leq w_{b,i}}\right]$). We
specify the following model:

$\displaystyle B_{ij}|S_{ij}\sim$
$\displaystyle~{}\mbox{Bernoulli}\left(\mbox{expit}(\alpha_{0}+\alpha_{1}S_{ij})\right).$

In particular, the true positive rate (TPR), defined as the probability of
detecting diagnostic RNA from a nasal swab when a person was experiencing
viral shedding, is

$\displaystyle\mbox{TPR}=\mathbbm{E}[B_{ij}=1|S_{ij}=1]=\mbox{expit}(\alpha_{0}+\alpha_{1});$

and the true negative rate (TNR), defined as the probability of not detecting
viral RNA from a nasal swab when a person was clear of viral RNA, is

$\displaystyle\mbox{TNR}=\mathbbm{E}[B_{ij}=0|S_{ij}=0]=1-\mbox{expit}(\alpha_{0}).$

The distribution of a swab’s observed diagnostic viral load depends on whether
the test result is a true/false positive/negative. When the $j$th swab
collected from person $i$ has no detectable diagnostic RNA, i.e., $B_{ij}=0$,
the observed viral load equals to the LoD. When a swab does have detectable
viral RNA, i.e., $B_{ij}=1$, we further consider two scenarios: if the result
is false positive, i.e., $S_{ij}=0$, we use a normal distribution centered
around some viral load level above the LOD with a moderate standard deviation:

$\displaystyle
y_{ij}|B_{ij}=1,S_{ij}=0\sim\mbox{Normal}\left(\mbox{LoD}^{+},0.5\right).$

If the result is true positive, i.e., $S_{ij}=1$, we use a normal distribution
centered around the latent viral trajectory and a standard deviation that
depends on LoQ:

$\displaystyle
y_{ij}|B_{ij}=1,S_{ij}=1\sim\mbox{Normal}\left(\mu_{y}(s_{ij}),\sigma_{y}\left(Q_{ij}\right)\right),$

where

$\displaystyle\sigma_{y}^{2}\left(Q_{ij}\right)=\sigma^{2}_{yy}\times\left(1+Q_{ij}\times\delta_{Q}\right).$

Our specification the variance $\sigma_{y}^{2}$ aims to account for the
increased uncertainty in viral load measurements for swabs with diagnostic RNA
concentration below the LoQ. Specifically, we let $\sigma^{2}_{yy}$ be the
base variance term denoting the unmeasured variability in the observations,
and $Q_{ij}$ be an indicator variable of whether the observed viral
concentration from swab $j$ lies between LoD and LoQ (i.e.,
$Q_{ij}=\mathbbm{1}\left[{\mbox{LoD}<y_{ij}<\mbox{LoQ}}\right]$). When a swab
has an observed diagnostic RNA viral load between LoD and LoQ, i.e.,
$Q_{ij}=1$, the variance term will be higher than the base level by a factor
of $\delta_{Q}$ to account for the extra uncertainty in these observations.

Finally, since the latent peak viral load is not observable, we estimate the
time from the observed peak to the latent peak for person $i$, denoted
$t_{p,i}$. If person $i$’s observed peak viral load was recorded within the
first two days of follow-up, we will assume the latent peak happens at the
latest one day after the observed peak:

$\displaystyle
t_{p,i}\sim\mbox{Normal}\left(\mu_{tpL},\sigma_{tpL}\right)\mbox{truncated
}[-\infty,1],$

where $\mu_{tpL}$ is negative. If person $i$’s observed peak viral load was
recorded during the last two days of follow-up, we will assume the latent peak
happens at the earliest one day before the observed peak:

$\displaystyle
t_{p,i}\sim\mbox{Normal}\left(\mu_{tpR},\sigma_{tpR}\right)\mbox{truncated
}[-1,\infty],$

where $\mu_{tpR}$ is positive. Otherwise, we assume the latent peak happens
around the time of the observed peak:

$\displaystyle t_{p,i}\sim\mbox{Normal}\left(0,\sigma_{tp}\right).$

We can specify $\sigma_{tpL}$, $\sigma_{tpR}$, and $\sigma_{tp}$ to reflect
our prior belief in how variable the time differences $t_{p,i}$ are among the
participants.

###  3.2 Modeling the sgRNA viral load

We model the sgRNA viral load in relation to the diagnostic viral trajectory.
For clarity, we define a parallel set of notations for the sgRNA viral load by
adding ′ after the previously defined notations. We assume that the diagnostic
and sgRNA viral shedding starts at the same time. In addition, we know that
sgRNA viral clearance occurs no later than the diagnostic RNA viral clearance
(Figure 1).

Let $t_{d,i}$ denote time from the latent peak diagnostic viral load to the
latent peak sgRNA viral load for person $i$. We use the following truncated
normal distribution for $t^{\prime}_{d,i}$:

$\displaystyle t^{\prime}_{d,i}\sim$
$\displaystyle~{}\mbox{Normal}\left(\mu_{td},2\right)\mbox{ truncated
}[-w_{a,i},w_{b,i}]$

The time from sgRNA shedding onset to peak viral load for person $i$, denoted
$w^{\prime}_{a,i}$, can be calculated as

$\displaystyle w^{\prime}_{a,i}=w_{a,i}+t^{\prime}_{d,i}.$

We use a truncated normal distribution to model the log-transformed time from
sgRNA viral clearance to diagnostic RNA rival clearance, denoted
$w^{\prime}_{d,i}$:

$\displaystyle\log\left(w^{\prime}_{d,i}\right)\sim\mbox{Normal}\left(\mu_{lwd},\sigma_{lwd}\right)\mbox{
truncated }\left[0,\log\left(w_{b,i}-t^{\prime}_{d,i}\right)\right].$

The time from peak sgRNA viral load to viral clearance for person $i$, denoted
$w^{\prime}_{b,i}$, can be calculated as

$\displaystyle
w^{\prime}_{b,i}=w_{a,i}+w_{b,i}-w^{\prime}_{a,i}-w^{\prime}_{d,i}.$

As for the magnitude of the peak sgRNA viral load $v^{\prime}_{p,i}$, we
parameterize it as the product of the diagnostic peak viral load and a
multiplier $q_{i}$, i.e., $v^{\prime}_{p,i}=q_{i}\times v_{p,i}$. Since the
peak sgRNA viral load is always lower than the peak diagnostic RNA viral load,
we model the multiplier $q_{i}$ using a beta distribution:

$\displaystyle q_{i}\sim\mbox{Beta}\left(\gamma_{1},\gamma_{2}\right)$

In the PEP study, only the swabs with detectable diagnostic RNA were tested
for sgRNA. Therefore, we model the TPR and TNR of sgRNA detection conditional
on the observed diagnostic viral load measurement:

$\displaystyle B^{\prime}_{ij}|S^{\prime}_{ij},B_{ij}\sim$
$\displaystyle~{}\mbox{Bernoulli}\left(B_{ij}\times\left[\mbox{expit}(\alpha^{\prime}_{0}+\alpha^{\prime}_{1}S^{\prime}_{ij})\right]\
\right).$

Specifically, when a swab was negative for diagnostic RNA, i.e., $B_{ij}=0$,
it would also be negative for sgRNA, i.e., $B^{\prime}_{ij}=0$. Otherwise, the
probability of detecting sgRNA from a nasal swab when a person was
experiencing sgRNA viral shedding is

$\displaystyle\mbox{TPR}^{\prime}=\mathbbm{E}[B^{\prime}_{ij}=1|S^{\prime}_{ij}=1,B_{ij}=1]=\mbox{expit}(\alpha_{0}^{\prime}+\alpha_{1}^{\prime});$

and the probability of not detecting sgRNA from a nasal swab when a person was
clear of sgRNA is

$\displaystyle\mbox{TNR}^{\prime}=\mathbbm{E}[B^{\prime}_{ij}=0|S^{\prime}_{ij}=0,B_{ij}=1]=1-\mbox{expit}(\alpha_{0}^{\prime}).$

The distribution of the observed sgRNA viral load follows a similar structure
as the diagnostic viral load and depends on whether the sgRNA test result is a
true/false positive/negative. When the $j$th swab collected from person $i$
has no detectable sgRNA, i.e., $B^{\prime}_{ij}=0$, the observed viral load
equals to the LoD. When a swab does have detectable sgRNA and the result is
false positive, we use a normal distribution centered around some viral load
level above the LOD with a moderate standard deviation:

$\displaystyle
y^{\prime}_{ij}|B^{\prime}_{ij}=1,S^{\prime}_{ij}=0\sim\mbox{Normal}\left(\mbox{LoD}^{+},0.5\right).$

Finally, if the sgRNA viral load detected is true positive, we use a normal
distribution:

$\displaystyle
y^{\prime}_{ij}|B^{\prime}_{ij}=1,S^{\prime}_{ij}=1\sim\mbox{Normal}\left(\mu_{y}^{\prime}(s^{\prime}_{ij}),\sigma^{\prime}_{y}\right),$

where

$\displaystyle\mu^{\prime}_{y}(s^{\prime}_{ij})=\mbox{LoD}^{\prime}+v^{\prime}_{p,i}+\left(\frac{v^{\prime}_{p,i}}{w^{\prime}_{a,i}}\right)\times
s^{\prime}_{ij}\times\mathbbm{1}\left[{s^{\prime}_{ij}\leq
0}\right]+\left(-\frac{v^{\prime}_{p,i}}{w^{\prime}_{b,i}}\right)\times
s^{\prime}_{ij}\times\mathbbm{1}\left[{s^{\prime}_{ij}>0}\right].$

###  3.3 Modeling seroconversion rate and time

Since not everyone infected with SARS-CoV-2 would eventually develop
neutralizing antibodies, we use an indicator variable $C_{i}$ to model whether
person $i$ would ever seroconvert:

$\displaystyle
C_{i}\sim\mbox{Bernoulli}\left(\mbox{expit}\left(\beta_{C0}+\bm{\beta}_{C}^{\intercal}\bm{X}_{C,i}\right)\right),$

where $\bm{X}_{C,i}$ are vectors of potential correlates of seroconversion. If
a person developed antibodies after infection, then the time from viral
shedding onset to seroconversion, denoted $w_{s,i}$, is modeled as

$\displaystyle w_{s,i}|C_{i}=1\sim$
$\displaystyle~{}\mbox{Gamma}\left(\kappa_{1},\kappa_{2}\right)$

If a person never seroconverts, $w_{s,i}=\infty$.

In our motivating example of the COVID-19 PEP study, DBS samples were only
collected on day 1, 14 and/or 28 of the study. Therefore, the servoconversion
time is censored. Specifically, if a participant only had negative IgG
samples, his/her seroconversion time is right-censored at the last negative
antibody test; if a participant only had positive IgG samples, his/her
seroconversion time is left-censored at the first positive antibody test; if a
participant had both negative and positive IgG samples, his/her seroconversion
time is interval-censored between the last negative test and the first
positive test. Finally, since not all participants had provided DBS samples
with confirmed results, only those who had antibody data available will
contribute to the likelihood of the model.

###  3.4 Prior elucidation

Inference for all model parameters can be conducted under a Bayesian
framework. The prior distributions for virological parameters
$\left\\{\mu_{lvp},\mu_{lwa},\mu_{lwb},\mu_{lwd}\right\\}$ can be specified
using normal distributions with means equal to the estimated diagnostic and
sgRNA peak viral load, time from shedding onset to peak and time from peak to
shedding cessation from previous literature. For variance and covariance
parameters
$\left\\{\sigma_{yy}^{2},\sigma_{y}^{{}^{\prime}2},\bm{\Sigma}_{log}\right\\}$,
we can use weakly informative priors such as the Cauchy distribution and
inverse wishart distribution. Multivariate normal distributions with zero
means can be used as priors for the coefficients of potential correlates of
viral dynamics and seroconversion
$\left\\{\bm{\beta}_{vp},\bm{\beta}_{wa},\bm{\beta}_{wb},\bm{\beta}_{C}\right\\}$.
For the model parameters related to TPR and TNR of diagnostic and sgRNA
detection
$\left\\{\alpha_{0},\alpha_{1},\alpha_{0}^{\prime},\alpha_{1}^{\prime}\right\\}$,
we can use relatively tight normal priors because we expect TPR and TNR to be
high. Similarly, we can specify a moderately informative beta prior for
$\delta_{Q}$ since we do not expect the extra uncertainty in the observed
diagnostic viral load below LoQ to be huge. As for the positive-valued
parameters $\left\\{\gamma_{1},\gamma_{2},\kappa_{1},\kappa_{2}\right\\}$, we
can use gamma distributions as priors such that the resulting prior
distributions for $q_{i}$ and $w_{i}$ have reasonable values as suggested by
previous literature.

###  3.5 sgRNA viral load imputation

In a resource-constrained setting, it is common that everyone in a study was
tested for diagnostic RNA but only a subset were tested for sgRNA. In this
case, our model can be used to impute the sgRNA viral trajectories for people
who only had diagnostic RNA viral load and covariates data by treating their
sgRNA measurements as missing data. Specifically, we can combine the people
without sgRNA samples with the people who had complete data and fit the model
to everyone to estimate the posterior distributions of
$\left\\{v^{\prime}_{p,i},w^{\prime}_{a,i},w^{\prime}_{b,i}\right\\}$ by
borrowing information from the rest of the parameters.

##  4 Application to the COVID-19 PEP study

###  4.1 Model implementation

We applied our methods to analyze data collected from the 80 COVID-19 PEP
study participants who had at least 2 positive sgRNA samples during the first
14 days of follow-up (Figures S1-2 of the Supplement). We chose this subset of
participants because we are interested in examining the viral dynamics and
seroconversion among SARS-CoV-2-infected people who had sustained viral
shedding. Among them, 33 participants had at least 1 DBS sample with confirmed
results, hence their anti-spike IgG antibody data contributed to the
seroconversion part of the model. For our primary model, the following
covariates were included as correlates for peak diagnostic viral load
($\bm{X}_{vp}$): age, sex at birth, treatment arm (hydroxychloroquine vs.
placebo), and presence of COVID-19 symptoms defined based on the case
definition approved by the U.S. Centers for Disease Control and Prevention
(CDC) (Centers for Disease Control and Prevention, 2020). The peak diagnostic
viral load, $v_{p}$, was included as a correlate for seroconversion
probability ($\bm{X}_{C}$). All analysis was implemented in R 4.2.1 (R Core
Team, 2022) using JAGS 4.3.0 (Plummer _and others_ , 2003). More details,
including prior specification, Markov Chain Monte Carlo (MCMC) setup, and
model diagnostics, can be found in Section S1 of the Supplement.

###  4.2 Model results

On average, the diagnostic viral load reaches a peak of $8.0$ (95% credible
interval, CI: $\left[7.8,8.1\right]$) log10 copies/ml after 3.8 (95% CI:
$\left[3.2,4.7\right]$) days and then clears after 10.5 (95% CI:
$\left[10.1,10.9\right]$) days. The sgRNA viral load reaches a peak of $6.0$
(95% CI: $\left[5.9,6.2\right]$) log10 copies/ml approximately 0.6 (95% CI:
$\left[0.3,0.8\right]$) day after the peak diagnostic viral load and then
clears after only 5.7 (95% CI: $\left[5.3,6.1\right]$) days, approximately 4.3
(95% CI: $\left[4.0,4.6\right]$) days before the diagnostic viral clearance
(Table 1 and Figure 2). The estimated infectious period, as approximated by
the total duration of sgRNA viral shedding, is 10.1 days (95% CI:
$\left[9.4,11.0\right]$). The posterior means and 95% CIs of individual viral
load trajectories are shown in Figures S1-2 in the Supplement.

We estimated the multiplicative factors associated with various patient
characteristics for an increase in peak diagnostic viral load on the log10
copies/ml scale (Figure 3). The posterior means of the parameters indicated
that being older, male, in the hydroxychloroquine arm, and reporting COVID-19
symptoms were positively associated with peak viral load in our data. However,
all the posterior 95% CIs were wide and overlapped with 1, indicating that
there was high uncertainty associated with these estimates.

To identify specific symptoms that might potentially be associated with peak
diagnostic viral load, we fitted additional models by replacing the COVID-19
symptoms indicator with other symptom indicators defined based on the Flu-PRO
patient-reported outcome instrument (Powers _and others_ , 2015), including
fever, change in taste or smell, and chest/throat/nose/body/gastrointestinal
symptoms. Figure 4 shows the posterior means and 95% CIs of the multiplicative
factors associated with various symptoms for an increase in peak diagnostic
viral load on the log10 copies/ml scale. Among all symptoms investigated,
having body symptoms (chills, myalgia, headache, or fatigue) was most strongly
associated with a higher peak diagnostic viral load (multiplicative factor =
1.25, 95% CI: $\left[1.07,1.50\right]$). That is, on the log10 copies/ml
scale, the peak diagnostic RNA viral load was estimated to be 25% (95% CI:
$\left[7\%,50\%\right]$) higher among SARS-CoV-2 patients who experienced
chills, myalgia, headache, or fatigue than those who did not experience these
symptoms. The rest of the symptoms either had close-to-zero or positive
estimated associations, but all their 95% CIs overlapped with 1.

The overall seroconversion rate was 75.0% (95% CI:
$\left[62.5\%,85.0\%\right]$). The peak diagnostic viral load was estimated to
be positively associated with the probability of seroconversion (odds ratio =
1.24, 95% CI: $\left[1.00,1.54\right]$, for every 10-fold increase in
diagnostic peak viral load). Among those who seroconverted, the mean time from
viral shedding onset to the presence of detectable IgG antibodies is 14.4 days
(95% CI: $\left[12.5,16.6\right]$) (Table 1 and Figure 2).

To identify any symptoms that might potentially be associated with
seroconversion rate, we fitted additional models by replacing peak diagnostic
viral load with symptom indicators. Figure 5 shows the posterior means and 95%
CIs of the odds ratio of seroconversion associated with various symptoms. All
symptoms investigated were estimated to be positively associated with the
probability of seroconversion but all posterior 95% CIs overlapped with 1.

The probabilities that a sample collected during viral shedding would be
tested positive for diagnostic and sgRNA (TPR) are 91.8% (95% CI:
$\left[90.1\%,93.3\%\right]$) and 89.6% (95% CI: $\left[87.4\%,91.6\%\right]$)
respectively; the probabilities that a sample collected outside the viral
shedding periods would be tested negative for diagnostic and sgRNA (TNR) are
99.5% (95% CI: $\left[99.4\%,99.6\%\right]$) and 99.6% (95% CI:
$\left[99.5\%,99.7\%\right]$) respectively (Table 1). Note that these values
account for not only the properties inherent to the RT-PCR tests such as
sensitivity and specificity, but also the potential errors that occurred due
to misplaced swabs, contaminated samples, or improper swabbing that were
common in outpatient household studies.

Our model results indicated that there were no significant correlations among
peak diagnostic viral load, time from shedding onset to peak, and time from
peak to diagnostic viral clearance (Table S1 in the Supplement). As for the
extra uncertainty in diagnostic viral load measurements below LoQ (104.9
copies/ml, corresponding to a Ct value of 31), our model estimated that the
variance in the observed diagnostic viral load ($\sigma^{2}_{y}$) was
approximately 25% (95% CI: $\left[13\%,38\%\right]$) higher if the observed
values were below LoQ as compared to those above LoQ.

###  4.3 Cross-validation for sgRNA viral load imputation and seroconversion
probability estimation

We conducted an 10-fold cross-validation exercise to examine our model’s
ability to impute sgRNA viral load and seroconversion for people who only have
diagnostic viral load and covariate data. Specifically, we randomly divided
the 80 participants into 10 groups of 8 people. For each iteration, we removed
the sgRNA viral load data (and antibody data if available) from one group of
participants and fitted the model using the rest of the observed data. In
figures S3-4 in the Supplement, we present the observed and imputed sgRNA
viral load trajectories and the estimated probability of seroconversion from
all iterations. The posterior means of the sgRNA viral trajectories were very
close to the masked sgRNA viral load measurements. As expected, the imputed
sgRNA viral load had wider posterior 95% CIs than those of the diagnostic
viral trajectories. The estimated seroconversion probabilities ranged from 64%
to 89%, a reasonable range given the masked antibody data.

##  5 Discussion

In this paper, we presented a joint Bayesian hierarchical model that allows
for borrowing of information between viral load and antibody data to make
inferences on key characteristics of diagnostic and sgRNA viral dynamics and
seroconversion. Our method can be used to identify potential correlates of
viral load trajectories and propensity for seroconversion, to estimate the
seroconversion time after a SARS-CoV-2 infection, and to impute sgRNA viral
load for people who only have diagnostic RNA viral load data. By jointly
modeling the diagnostic and sgRNA viral trajectories, our method is able to
take into account the dynamical relationship and correlation structure between
the two types of viral load. Our model also uses indicator variables to
represent latent and observed viral shedding status separately, accounting for
false positive and false negative viral detection due to not only the RT-PCR
test properties but also the potential sample collection errors in outpatient
study settings. As a demonstrating example, we implemented our proposed
methods using JAGS in R and analyzed data collected from 80 participants of
the COVID-19 PEP study who had at least 2 positive sgRNA samples during the
first 14 days of follow-up. A cross-validation exercise was also conducted to
examine the model’s ability to impute sgRNA viral load trajectories.

Based on the COVID-19 PEP data, our model-estimated population-level viral
load trajectories were characterized by a rapid rise to reach viral peak and
subsequent slower decline, similar to what has been demonstrated in previous
literature (Goyal _and others_ , 2020; Néant _and others_ , 2021; Ke _and
others_ , 2021; Kissler _and others_ , 2021a, b; Stankiewicz Karita _and
others_ , 2021; Jones _and others_ , 2021; Singanayagam _and others_ , 2021;
Deming _and others_ , 2022). The model results on diagnostic and sgRNA viral
shedding duration were consistent with previous findings that SARS-CoV-2
infectivity usually terminates after 10 days (van Kampen _and others_ , 2021;
Wölfel _and others_ , 2020; Bullard _and others_ , 2020; Arons _and others_ ,
2020; The COVID-19 Investigation Team, 2020; Buder _and others_ , 2022) while
SARS-CoV-2 RNA is detectable in respiratory samples for an average of 17 days
(Cevik _and others_ , 2021). In addition, our analysis confirms that anti-
spike IgG seroconversion is positively associated with viral load (Liu _and
others_ , 2021; Masiá _and others_ , 2021). To our knowledge, our model is the
first that quantifies the association between peak SARS-CoV-2 viral load and
IgG seroconversion rate and the time from SARS-CoV-2 infection to IgG
seroconversion.

One limitation of our paper is the small sample size of the COVID-19 PEP data
especially the limited number of people with seroconversion data. Repeating
the analysis with data from more participants and more frequent antibody
sampling schedule may increase the precision of the parameter estimates. In
addition, we only focused on the people with sustained viral shedding, which
we subset based on the criteria of having at least 2 positive sgRNA samples
during the first 14 days. Extending our method to include people who only had
one positive sample is possible. However, adjustment needs to be made to
adequately account for the possible cases of intermittent viral detection at
the very end of viral shedding, clearance of the virus by innate immunity due
to low viral inoculum, or/and limited infections attributed to immune priming
by prior seasonal coronaviruses (Stankiewicz Karita _and others_ , 2021).

In addition to identifying correlates of peak diagnostic viral load, as
demonstrated in the analysis of PEP data, our model can be used to investigate
factors associated with other viral dynamics characteristics as well. For
example, we can estimate the potential association between vaccination and
viral shedding duration by using vaccination status as a covariate for time
from viral shedding onset to peak ($\bm{X}_{wa,i}$) and/or time from peak to
viral clearance ($\bm{X}_{wb,i}$). Although we only focused on anti-spike IgG
seroconversion in this paper, our model can be generalized to study other
immune responses, such as the development of anti-nucleocapsid IgG and
Immunoglobulin M (IgM) antibodies, if relevant data is available. Given the
flexible structure of the joint hierarchical model, it is straight forward to
extend our methods to model multiple immune markers simultaneously.

Our model will be a valuable tool for future SARS-CoV-2 vaccine studies that
focus on assessing the vaccine effect on transmission (VET)(Kennedy-Shaffer
_and others_ , 2021; Follmann and Fay, 2022). Since detection of sgRNA
indicates active viral replication and tracks infectious virus, our methods
can be used to impute sgRNA viral load and better estimate proxies of VET. Our
model can also be used to better understand the potential impact of
interventions that reduce viral load on seroconversion rate. For example,
previous vaccine studies have shown that vaccinated people tended to have
lower viral load at diagnosis and have lower rate of seroconversion as
compared to the control group (Pajon _and others_ , 2022; Follmann _and
others_ , 2022). Models like ours could help explain how much of the decrease
in seroconversion rates is due to vaccination and how much is due to the
vaccination effect on viral load. Finally, our proposed model can facilitate
analysis of household transmission studies by providing estimated viral load
trajectories for each individual in a household to inform the direction of
transmission.

##  6 Software

Software in the form of R code and JAGS code is available online at
https://github.com/dq0708/joint_vl_sero.

##  7 Supplementary Material

Supplementary material is available online at
http://biostatistics.oxfordjournals.org.

## Acknowledgments

Funding for the project was provided by Bill & Melinda Gates Foundation. The
authors thank the Hydroxychloroquine COVID-19 PEP Study Team and participants
for providing scientific insights and data. Special thanks to Dr. Ruanne
Barnabas and Dr. Anna Bershteyn for reviewing the manuscript drafts and
providing valuable suggestions. We also thank Dr. Leigh Fisher for additional
advice regarding RT-PCR viral load quantification.

Conflict of Interest: None declared.

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Figure 1: Graphical illustration of the joint model. Along the x-axis is time
since the latent peak diagnostic RNA viral load in day. Along the y-axis is
the magnitude of viral load in log10 copies/ml. The limit of detection (LoD)
for both diagnostic and sgRNA RT-PCR tests were approximately $10^{2.4}$
copies/ml. In addition, the diagnostic RT-PCR had a limit of quantification
(LoQ) of approximately $10^{4.9}$ copies/ml. Black dots represent the observed
diagnostic RNA viral load from RT-PCR tests, and blue segments represent the
latent diagnostic RNA viral load trajectory. Red dots represent the observed
sgRNA viral load from RT-PCR tests, and maroon segments represent the latent
sgRNA viral load trajectory. The black and red dots were actual data from one
participant from the COVID-19 PEP study. $w_{a}$ and $w_{a}^{\prime}$
represent time from shedding onset to peak for diagnostic and sgRNA viral load
respectively. $w_{b}$ and $w_{b}^{\prime}$ represent time from peak to viral
clearance for diagnostic and sgRNA viral load respectively. $w_{d}^{\prime}$
represents the time from sgRNA viral clearance to diagnostic RNA viral
clearance. $t_{p}$ represents time from the observed peak diagnostic RNA viral
load to the latent peak diagnostic RNA viral load, and $t_{d}^{\prime}$
represents time from the latent peak diagnostic RNA viral load to the latent
peak sgRNA viral load. Finally, $v_{p}$ represents the magnitude of peak
diagnostic RNA viral load with respect to LoD, and $q$ is the multiplicative
factor of peak sgRNA viral load relative to peak diagnostic RNA viral load.
Figure 2: The posterior means and 95% credible intervals (CI) of the
population-level diagnostic and sgRNA viral load trajectories and time of
anti-spike IgG seroconversion (among those who seroconverted).  Figure 3: The
posterior means and 95% credible intervals of the multiplicative factors
associated with various patient characteristics for an increase in peak
diagnostic viral load on the log10 copies/ml scale.  Figure 4: The posterior
means and 95% credible intervals of the multiplicative factors associated with
various symptoms for an increase in peak diagnostic viral load on the log10
copies/ml scale. Specifically, “COVID-19” symptoms were defined based on the
case definition approved by the U.S. Centers for Disease Control and
Prevention (Centers for Disease Control and Prevention, 2020); the rest were
defined based on the Flu-PRO patient-reported outcome instrument (Powers _and
others_ , 2015): “Chest” symptoms include cough and shortness of breath;
“Taste or Smell” symptoms include olfactory and taste disorders; “Nose”
symptoms include congestion, runny nose, and olfactory disorder; “Body”
symptoms include chills, myalgia, headache, and fatigue; “GI” symptoms include
diarrhea, nausea, and vomit; “Throat” symptoms include sore throat and taste
disorder.  Figure 5: The posterior means and 95% credible intervals of the
odds ratio of seroconversion associated with various symptoms. Specifically,
“COVID-19” symptoms were defined based on the case definition approved by the
U.S. Centers for Disease Control and Prevention (Centers for Disease Control
and Prevention, 2020); “Chest” symptoms include cough and shortness of breath;
“Taste or Smell” symptoms include olfactory and taste disorders; “Nose”
symptoms include congestion, runny nose, and olfactory disorder; “Body”
symptoms include chills, myalgia, headache, and fatigue; “GI” symptoms include
diarrhea, nausea, and vomit; “Throat” symptoms include sore throat and taste
disorder.

Table 1: The posterior means and 95% credible intervals (CI) for selected quantities of interest. In particular, $n=80$ is the total number of people with diagnostic RNA viral load data.  Type | Quantity of interest | Estimand | Posterior mean [95% CI]
---|---|---|---
Diagnostic RNA | Mean time from shedding onset to peak viral load (days) | $\sum_{i=1}^{n}\hat{w}_{a,i}/n$ | $3.8\left[3.2-4.7\right]$
viral load | Mean time from peak viral load to viral clearance (days) | $\sum_{i=1}^{n}\hat{w}_{b,i}/n$ | $10.5\left[10.1-10.9\right]$
| Mean peak viral load (log10 copies/ml) | $\sum_{i=1}^{n}\hat{v}_{p,i}/n+\text{LoD}$ | $8.0\left[7.8-8.1\right]$
| True Positive Rate | $\mbox{expit}(\hat{\alpha}_{0}+\hat{\alpha}_{1})$ | $91.8\%\left[90.1\%-93.3\%\right]$
| True Negative Rate | $1-\mbox{expit}(\hat{\alpha}_{0})$ | $99.5\%\left[99.4\%-99.6\%\right]$
sgRNA | Mean time from shedding onset to peak viral load (days) | $\sum_{i=1}^{n}\hat{w}^{\prime}_{a,i}/n$ | $4.4\left[3.8-5.3\right]$
viral load | Mean time from peak viral load to viral clearance (days) | $\sum_{i=1}^{n}\hat{w}^{\prime}_{b,i}/n$ | $5.7\left[5.3-6.1\right]$
| Mean peak viral load (log10 copies/ml) | $\sum_{i=1}^{n}\hat{v}^{\prime}_{p,i}/n+\text{LoD}^{\prime}$ | $6.0\left[5.9-6.2\right]$
| True Positive Rate | $\mbox{expit}(\hat{\alpha}^{\prime}_{0}+\hat{\alpha}^{\prime}_{1})$ | $89.6\%\left[87.4\%-91.6\%\right]$
| True Negative Rate | $1-\mbox{expit}(\hat{\alpha}^{\prime}_{0})$ | $99.6\%\left[99.5\%-99.7\%\right]$
Anti-spike IgG | The overall seroconversion rate | $\sum_{i=1}^{n}\hat{C}_{i}/n$ | $75.0\%\left[62.5\%-85.0\%\right]$
Seroconversion | The odds ratio of seroconversion associated with | $\exp\left(\hat{\bm{\beta}}_{C}[2]\right)$ | $1.24\left[1.00-1.54\right]$
|  every 10-fold increase in peak diagnostic viral copies |  |
| Mean time from infection to seroconversion (days) | $\hat{\kappa_{1}}/\hat{\kappa_{2}}$ | $14.4\left[12.5-16.6\right]$