2.1.1 Antibody kinetics model.

In the most general case, the antibody kinetics model, for an individual i, we define k immunological stimuli, which occur at times , such that <for all i. These immunological stimuli could represent exposure to an infection or vaccination. If the functions represent the antibody kinetics following immunological stimulus for biomarker b, then, given a serological sample at time , we estimate the biomarker value (e.g., log neutralising titre) at this time,, in the model as:

(3)

That is, the antibody titre is governed by an individual’s most recent exposure and the function for this response is conditional on the time since this exposure, the biomarker level at that time, and any exposure-specific parameters. Given that our analysis focuses on inferring infections, we assume that for a subset of individuals, the immunological stimuli, , results in an infection, and the timing of this infection . We refer to this framework as an antibody kinetics model as, in this study, we are measuring titre values to antibody levels, but note the same method can be applied to any measured immunological biomarker (e.g., memory B cell concentrations).

2.1.2 Observational model.

The observation model defines the likelihood function that links the model-estimated biomarker value at the sampled time to the measured value at that time. In many applications, particularly when serological measurements are log-transformed, it is common to assume a normally distributed error structure for expected log assay observations, alongside censoring where measurements are discrete and bounded (e.g., dilution thresholds). This assumption can be appropriate for assays such as ELISA, where measured values are continuous, bounded away from zero, and approximately homoscedastic on the log scale. However, this is not universally applicable: for example, PRNT and HAI assays often exhibit scale-dependent variance and discretisation effects due to dilution steps. Users of serojump are therefore encouraged to tailor the likelihood function to their specific assay, as the R package allows full flexibility to define custom likelihoods (e.g., interval-censored, overdispersed, or non-normal). In our analysis, we used a normal likelihood, on log-transformed ELISA values, which we found to be well calibrated and consistent with the distributional properties of the underlying measurements [25].

2.1.3 Infection model.

The prior for the infection time, , for each individual can be estimated on the population-level from existing serological surveys or surveillance data to identify likely periods of infection. For example, surveillance data can still indicate the distribution of infection burden over time, if not the true cumulative number of infections. A key feature of our framework is that can be specified to reflect a shared force of infection (FOI) across individuals, ensuring that infection times are conditionally dependent given the population-level FOI. This avoids the naive assumption that values are fully independent for each individual. If this prior is not specified in the model, or there is no known data to infer the epidemic curve, the model will assume a uniform risk of infection over the study period (Section Ad in S1 Methods).

2.1.4 Prior distributions.

The prior distribution on the parameters describing the observational model and antibody kinetics model, , can be defined by the user and are assumed to be independent. We note that, the prior distributions chosen for the simulated and empirical models described below reflect practical considerations rooted in both biological plausibility and computational stability. For instance, the decay rate parameter, , reflects empirically observed values for SARS-CoV-2 and other viral infections, while the observational noise prior is calibrated to the expected scale of log-transformed ELISA values. While these priors may appear narrow, they are not required by the RJ-MCMC algorithm per se; rather, they help stabilise inference under transdimensional sampling, where high-dimensional or weakly identified models may suffer from poor mixing or identifiability issues. The prior distribution on the infection status binary vector, , as with similar inference problems, [25] has an implicit prior resulting from the definition of the reversible jump algorithm. This is Bin(0.5, N), or 50% of the population infected on average. To remove this implicit prior and instead assign a uniform distribution to the total number of infected individuals during the epidemic, we choose the combinatorial prior where ||Z||₁ is the number of infected individuals:

(4)

If prior knowledge on the number of missed infected individuals is known (e.g., from random community testing regardless of symptoms), these can be added to this prior (Sections Ad-Ae in S1 Methods). To support model robustness and transparency, we conduct prior predictive checks for both the antibody kinetics functions and the observational models to ensure they produce biologically plausible outputs before fitting to data. These checks are described in Section Af in S1 Methods. We strongly encourage users of serojump to follow similar workflows and adapt the priors to reflect assay characteristics, study populations, and research questions. The R package supports fully custom prior specifications for all components of the model, including hierarchical or unbounded alternatives.

2.1.5. Post-processing of posterior distributions and correlates of risk.

After fitting the model with the serojump algorithm, we obtain posterior distributions for the set . We calculate the posterior distribution of the post-exposure antibody trajectories for exposure e and biomarker b, . To assess the posterior distribution of the number of infected individuals across the population, we calculate (the number of infected individual) for each sample of the posterior distribution. We also calculate for each posterior sample, the “reference titre value” for each individual and biomarker. The reference titre value for those infected is the titre at which individuals become infected () for each individual and biomarker . For those not infected, we sample a random value from their titre trajectories across the study weighted by the prior probability of exposure timing , where Using the reference titre value, we assess the relationship between the biomarker value (e.g., antibody titre) and infection risk, to estimate a correlate of risk (COR). A COR represents either a direct mediator of immunity or a proxy for immune protection. To estimate this relationship from serojump, we fit a logistic curve with parameters k (gradient), x₀ (mid-point) and L (asymptote, representing the exposure rate), to describe how reference biomarker values relate to the infection risk, given by:

From this, we define the conditional protection curve (CPC) as the fitted probability curve, representing the direct relationship between biomarker levels and protection from infection. It can be derived from the fitted R(x) as:

The relative risk (RR), is a rescaled version of the CPC that normalises protection relative to a reference titre level (in this case the lower limit of detection. It can also be derived from fitted R(x) as

These estimates allow us to quantify the protective effect of immune markers and assess how biomarker levels modulate infection risk across a population (Sections Af-Ag in S1 Methods).

2.2.1 Description of the simulated data.

To test whether it is possible to correctly recover epidemiological and immunological dynamics from common serological cohort structures, we first simulate a serological dataset using the serosim R package [24] to test the serojump framework. We simulate continuous epidemic serosurveillance (CES) cohort data, which represents a serostudy in which individuals are followed over an epidemic wave and sampled at multiple random time points throughout. The simulated data includes N = 200 individuals with serological samples taken within the first seven days of the study’s starting and a sample within the last seven days of the study’s ending. These individuals also had three samples taken randomly throughout the study during an 120-day epidemic wave.

We model the epidemic as a stochastic transmission process, where individuals can be either susceptible, exposed, or infected. The probability of exposure is set at 60% per individual, meaning that individuals coming into contact with an infectious person have this probability of becoming exposed, assuming no prior immunity. Among those exposed, the probability of progressing to infection is 30%, meaning that some individuals remain uninfected despite exposure. This corresponds to a scenario where a pathogen spreads within a population that starts with partial immunity. Over the study timeframe we assume that individuals can have a maximum of one exposure. To model a symmetric epidemic peak, we simulate the exposure time from a normal distribution, N(60, 20) days. To assess the impact of biomarker levels on infection risk, we simulate two different relationships between the measured biomarker and the probability of infection given exposure: (1) No COP model (Model A), where infection occurs with a fixed probability of 50% regardless of biomarker titre, representing a non-titre-dependent correlate of protection; and (2) With COP model (Model B), where infection probability follows a logistic function of biomarker titre, a commonly used correlate of protection model [26,27]. Explicitly, the simulated probability of infection, given a titre value, x, is given by

Following infection, the antibody kinetics are assumed to follow a linear rise to a peak at 14 days, followed by an exponential decay to a set-point value [28]. The formula for this biphasic trajectory is given by

(5)

Biologically, Equation 5 models a biphasic antibody response characterised by a rapid linear rise to a peak (typically within ~2 weeks post-infection), followed by an exponential decay to a stable plateau. This reflects typical antibody dynamics observed after acute viral infections such as influenza. SARS-CoV-2, and dengue, where antibodies first expand quickly (activation phase), then contract and stabilise (memory phase). In this parameterisation, a controls the peak magnitude of the response, b the rate of exponential decay, and the long-term plateau value. This simple yet biologically interpretable form has been shown to effectively reproduce observed kinetics in controlled infection studies [28]. The simulated values we use are a = 1.5, b = 0.3, c = 1 and t, is the number of days post-infection. If an individual is not infected over the period, their titre remains unchanged and, therefore, equal to their start titre throughout. We assume all individuals have the same antibody kinetic response following infection but we add observational noise into our model, assuming a normal distribution with standard deviation . In the base case, we assume . We tested the robustness of the simulation recovery to noise by fitting the serojump algorithm to simulated data with increasing observational error. Specifically, we simulated using serosim, serological data with standard deviations on the normally distributed error for 11 values, (0.01, 0.05, 0.1, 0.15,.., 0.45, 0.5), and evaluated the capacity for serojump to recover i) the infection status of individuals, ii) the timing of the epidemic and iii) the recovered population-level antibody kinetics.

2.2.2 Model specification for serojump.

We assume each individual’s serological profile is characterised by the IgG biomarker (b={IgG}) and that this biomarker is measured using ELISA, with the units given by IU/mL. Two immunological events are considered: , the pre-infection state and , the infected state. The structural assumptions on the kinetics of the pre-infection and infection state are; for the pre-infection state, we assume the prior on the antibody levels Y_i are modeled as a linear wane

with a waning rate following the uniform distribution with a prior . For the post-infection dynamics, we assume that antibody-kinetics boost according to the functional definition defined as (Equation 5) with the priors, a ~ N(2, 2), b ~ N(0.3, 0.05), c ~ U(0, 4).

For the observational model, we assume that antibody measurements follow the normal distribution probability density function ;

with the prior ~ U(0, 4). For the prior on the risk of infection over time, we assume this is uniform across the whole time period for all i. Finally, the distribution for the number of infection events, , is constrained by a combinatorial factor reflecting the arrangement of events across the population as described in Equation 4. A summary for the method specification for serojump can be found in the Section B in S1 Methods.

2.3.1 Description of the real-world serological data on SARS-CoV-2 from The Gambia.

To test serojump on real-world serological data, we used longitudinal serological data collected before and after the Delta wave [29] in a household cohort study of 349 individuals in The Gambia. Quantitative ancestral anti-spike and nucleocapsid (NCP) IgG data were generated using previously validated ELISA assays [10], calibrated to the WHO International Standard for anti-SARS-CoV-2 immunoglobulin (cat no NIBSC 20/136). In addition to serological data, SARS-CoV-2 PCR testing was conducted weekly regardless of symptoms and additionally for those who presented with influenza-like illness symptoms during this period. In addition to PCR-positive diagnoses, dates of SARS-CoV-2 vaccination and PCR-positive pre-Delta infections were also known. At the start of the study, which was just prior to the Delta variant circulating, 56.7% of people were seropositive for SARS-CoV-2 [29]. Over the study period, the Delta variant predominantly circulated with 99 PCR-confirmed cases, with an additional 21 PCR confirmed SARS-CoV-2 infections with pre-Delta variants, and 49 individuals who received a dose of vaccine. In this analysis, we explicitly model vaccination as a separate immunological stimulus with a known date of administration, which is provided as an input to the model. This ensures that vaccine-induced boosts in antibody levels—particularly to the spike protein—are not misclassified as infections. When vaccination dates are unknown or uncertain, such boosts could in principle be misattributed; however, the inclusion of multiple biomarkers (spike and nucleocapsid IgG) allows the model to disambiguate immunological responses. A description of the timing of the sample collection, and infections for each individual is given in Section C in S1 Methods.

2.3.2 Model specification for serojump.

Each individual’s serological profile is characterised by two biomarkers: IgG to SARS-CoV-2 ancestral spike and NCP protein (b = {spike, NCP}). We model four immunological events: , the pre-infection state, , the pre-Delta infection state, , the Delta-infected state, and the vaccination state. For the pre-infection state, we assume the prior on the antibody levels Y_i for biomarker b are modeled as a linear wane:

with a waning rate, w, with a prior w ~ U(0, 0.1), estimated separately for each biomarker. For post-infection (pre-Delta and Delta) and vaccination, we assume that antibody kinetics follow Equation 6 (below), which describes a heterogeneous antibody response incorporating variation in response magnitude, time-to-peak, and decay rate. This functional form is adapted from Teunis et al. (2009) [30], who demonstrated its utility in seroepidemiological inference across multiple pathogens. For SARS-CoV-2, recent work (e.g., Srivastava et al., 2024 [28]) has shown that antibody responses often peak around 14–21 days and then decline heterogeneously, with substantial inter-individual variation. Thus, parameters , and describe the peak titre and time to peak, respectively, governs the decay rate, and the observational uncertainty. This flexible form allows us to model the diverse immune responses to both infection and vaccination with SARS-CoV-2, and is well-suited to longitudinal data with multiple exposures and biomarkers. In addition, titre-dependent boosting of SARS-CoV-2 has been observed in previous studies [28,31], and we defined as the parameter which drives the degree to which titre-value at exposure attenuates the boosting effect. Therefore, we assume that antibody kinetics follow a boost of , defined by:

(6)

and

(7)

where , and the priors governing the boosting response are given by: , , , and . The value of is fixed to 0.001, which corresponds to a slow, long-term waning of the antibodies coming from the long-lasting memory B cell response. Thus, in the posterior distribution, each immunological event (pre-Delta infection, Delta infection, vaccination) and each biomarker has distinct parameter values describing its kinetics, but they share the same prior distributions. For the observational model, we assume that antibody measurements follow the normal distribution probability density ;

with prior ~ Exponential [1]. We choose an empirical distribution based on the PCR-positive data for the prior on the risk of infection over time. Finally, the distribution of the number of infection events, , is constrained by a combinatorial factor reflecting the arrangement of events across the population as described in Equation 4. A summary for the method specification for serojump can be found in the Section C in S1 Methods.

2.3.3 Benchmarking against standard serological heuristics.

We evaluate how well serojump and other serological approaches can infer true infection status (as defined by PCR-positivity) using serological data alone, without incorporating PCR results into the inference process. Specifically, we compare five different infection classification strategies based on serology: serojump, a four-fold rise in spike or NCP, and seropositivity thresholds for spike and NCP (IC50 values of 6.39 and 3.00) [29]. To assess the performance of these classification approaches, we compute and compare their sensitivity in detecting PCR-confirmed infections.

3.1.1. Simulation recovery for the base case simulated dataset.

For the default simulated dataset with low observational noise (), the individual-level antibody kinetics show good agreement between simulated and recovered kinetics under the With COP and No COP simulated data (Fig 1A–B) with the observed data points (grey dots) close to the median posterior distributions of the model-fitted individual-level antibody trajectories (red lines). We recover the infection status of each individual with 100% sensitivity and specificity, and the differences between simulated infection days and model-predicted infection days (Fig 1C–D) are small (within 14 days) for almost all individuals. This results in a close alignment between the simulated and the recovered epidemic curves (Fig 1E–F). These observations suggest that for the default simulated dataset, both the With COP and No COP data are well recovered, as evidenced by closer alignment of posterior medians to observed data, with minor errors in infection time predictions and an accurate reconstruction of the infection time distribution.

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Fig 1. Comparison of simulation recovery for With COP and No COP models with observational error (0.1).

(A, B) Individual-level antibody kinetics for a subset of individuals, showing observed data points (grey dots), and individual-level posterior medians (red lines) for With COP data and No COP data. (C, D) Model error (mean with 95% credible intervals) in recovering infection times, represented as the difference between simulated and model-predicted infection days for infected individuals With COP data and No COP data. (E, F) Distribution of infection times, comparing simulated infection times with recovered infection times.

https://doi.org/10.1371/journal.pcbi.1013467.g001

3.1.2. Stability of simulation recovery for increasing observational noise.

We assess predictive performance by measuring the Continuous Ranked Probability Score (CRPS, [35]) a metric that evaluates the accuracy of probabilistic predictions by comparing the predicted cumulative distribution function to the observed value. We find the functional form of the antibody kinetics for both No COP and With COP models is well recovered (CRPS < 0.35) across all levels of observational noise; however, decreasing in accuracy as observational noise increases (Fig 2A). In contrast, the recovery of the epidemic curve is adequate at low levels of uncertainty but quickly becomes inaccurate (CRPS > 0.7) at a moderate level of noise (Fig 2B). This suggests that for datasets with a high degree of observational error, the recovery of the epidemic timing may be inaccurate when a uniform prior on infection timing is used in serojump.

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Fig 2. Evaluation of model performance under varying levels of observational uncertainty.

(A) Accuracy in recovering antibody kinetics over various simulated observational errors, measured as the mean CRPS across all individuals of the observational model (B) Accuracy in recovering the epidemic curve over various simulated observational errors, assessed using the mean CRPS across all infection times (C, D) ROC curves for infection status predictions under simulated data with COP (C) and without COP (D) under different observational error (colour) (E) F1-scores for recovering infection status across different observational errors and model types (No COP and With COP). (F, G) Posterior means of the COP curves across various observational uncertainty values (blue colours) compared to the simulated function (dashed red line) (H) Accuracy in recovering the functional form for the COP, measured by Mean CRPS, across various levels of observational uncertainty for both model types (With COP and No COP).

https://doi.org/10.1371/journal.pcbi.1013467.g002

For recovery infection status, we find the model perfectly recovers the infection status of each individual (F1-score = 1) up until an observational error of 0.15, after which there is a decrease in F1-score as observational error increases, suggesting the sensitivity and specificity of the recovery of infection status is decreasing (Fig 2C–E). This decrease is driven by the increase in the false positive rate, with the true positive rate remaining consistent across all observational noise, suggesting that the model correctly identifies all infections but may incorrectly detect infections at high levels of noise. We find that the functional form of the COP is well recovered in both the With COP and No COP datasets when observational noise is below 0.3, with CRPS values less than 0.05 (Fig 2F–H). In high-noise settings, we observe that underestimation of the correlate of protection (COP) curve can lead to elevated false positive rates, as individuals with relatively high antibody titres may still be incorrectly inferred as infected. This interplay between inferred protection and infection classification underscores the difficulty of simultaneously recovering both infection dynamics and immunity correlates, particularly when data are noisy or sparse.

3.2.1. Testing sensitivity of serological detection methods.

We evaluate the sensitivity of various serological detection heuristics in identifying infection status. Using each of the five infection classification strategies based on serology (serojump, a four-fold rise in spike or NCP, and seropositivity thresholds for spike and NCP), we classified each of the PCR-confirmed individuals by their derived infection status and then calculated the sensitivity of each method (Fig 3). The serojump method demonstrates the highest sensitivity, outperforming the other four heuristics. The four-fold spike rise achieves the second-highest sensitivity, closely followed by the four-fold NCP rise. Both seropositive threshold methods (spike and NCP) perform less well, highlighting the limitations of using simple thresholds for infection detection. These observations highlight that leveraging dynamic changes in antibody kinetics can outperform static thresholds for serological detection of infection, with the serojump method demonstrating superior sensitivity, indicating its potential as a preferred approach for infection inference.

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Fig 3. Sensitivity of serological detection heuristics in identifying infection status for 99 PCR-positive individuals using serological data only.

Posterior probabilities of recovery for individuals using five detection methods: (A) Four-fold spike rise, (B) Four-fold NCP rise, (C) Seropositive threshold spike, (D) Seropositive threshold NCP, and (E) serojump. Green bars represent individuals inferred as true positives, while red points indicate false negatives.

https://doi.org/10.1371/journal.pcbi.1013467.g003

3.2.2. Combining PCR and serological data to infer epidemiological dynamics of the Delta wave.

We estimate the dynamics of antibody responses, epidemic patterns, and infection risk in the context of SARS-CoV-2 infection and vaccination during the Delta wave in The Gambia. This uses the same serological dataset described in Section 3.1 but includes PCR-confirmed infections, anchoring the infection status and infection time for 99 individuals. We infer antibody kinetics trajectories for the biomarkers, spike and NCP, for individuals infected prior to the Delta variant, during the Delta wave, and following vaccination (Fig 4A). We find that infections with Delta and infections with variants before Delta stimulate both spike and NCP responses, with spike responses seeing more sustained responses compared to NCP responses (AUC: 395 for spike and 257 for NCP for Delta infection). The spike responses remain above a four-fold rise up until day 46 for infection with the Delta variant. Following vaccination, we find robust spike responses but very low NCP responses (AUC: 607 vs 67 for spike and NCP, respectively). These observations are consistent with the spike-based composition of the vaccine used. We also summarise the total number of infections inferred from the serojump framework and compare this to the known PCR-confirmed infections (Fig 4B and 4C). Considering only PCR-confirmed infections, we find that the attack rate is 30%, increasing to 52% when we use serojump, implying analysing changes in serological kinetics finds many infections which are missed through virological surveillance alone. Using our informed prior ensures that the inferred timings of these missed infections agree with the known epidemic wave. Finally, we compared the relationship between titre levels at infection and the individual-level posterior probability of infection, serving as a correlate of risk (COR) (Fig 4D). For NCP and spike biomarkers, a negative relationship is observed between titre levels at infection and the probability of infection, suggesting that higher antibody titres are associated with reduced infection risk. When we consider the CPC after normalising the titre scales, spike has a slightly steeper gradient compared to NCP, suggesting a stronger correlate of protection (Fig 4E). In addition, high titre values providing only 60% protection suggest the Ancestral spike and NCP proteins correlate with protection despite being antigenically distant from the infecting variant (Delta), suggesting cross-immunity with the analysed proteins. The RR shows that having a log spike titre of 3.2 provides 50% more protection against infection compared to those with no measurable titre (Fig 4F).

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Fig 4. Antibody kinetics, recovery of infection timings, and correlates of protection from empirical data with PCR information.

(A) Fitted antibody kinetic trajectories for individuals with infection prior to the Delta variant (left), infection during the Delta variant wave (center), and following vaccination (right), showing fold-rise in antibody titre over time for NCP (blue) and spike (orange) biomarkers. Shaded regions represent 95% credible intervals. (B) Inferred epidemic wave during the Delta wave, illustrating the temporal distribution of PCR-confirmed cases (black bars) and total inferred infections (pink shaded curve), with shaded uncertainty. (C) The estimated attack rate during the Delta wave, partitioned into PCR-confirmed cases (dark grey) and total inferred infections (pink). Error bars indicate 95% credible intervals. (D) The fitted logistic curve to the infection risk, showing the posterior probability of infection as a function of antibody titre at infection for NCP (blue) and spike (orange) biomarkers, with shaded regions representing 95% credible intervals. (E) The CPC from the fitted infection risk curve and (F) shows the relative risk for the same biomarkers.

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