Stéphane Pelleau

Tom Woudenberg

Jason Rosado

Françoise Donnadieu

Laura Garcia

Thomas Obadia

Soazic Gardais

Yasmine Elgharbawy

Aurelie Velay

Maria Gonzalez

Jacques Yves Nizou

Nizar Khelil

Konstantinos Zannis

Charlotte Cockram

Sarah Hélène Merkling

Annalisa Meola

Solen Kerneis

Benjamin Terrier

Jerome De Seze

Delphine Planas

Olivier Schwartz

François Dejardin

Stéphane Petres

Cassandre Von Platen

Laurence Arowas

Louise Perrin De Facci

Darragh Duffy

Clíona Ní Cheallaigh

Niall Conlon

Liam Townsend

Heidi Auerswald

Marija Backovic

Bruno Hoen

Arnaud Fontanet

Dr Michael White
email: michael.white@pasteur.fr

Serological reconstruction of COVID-19 epidemics through analysis of antibody kinetics to SARS-CoV-2 proteins

Introduction

Severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2), causing coronavirus disease 2019 (COVID- 19), has led to widespread morbidity and mortality since its emergence. The response to the SARS-CoV-2 pandemic is critically dependent on surveillance data, most notably numbers of COVID-19 associated hospital admissions and deaths recorded through health systems surveillance, as well as numbers of cases confirmed SARS-CoV-2 positive by PCR-based testing [START_REF] Salje | Estimating the burden of SARS-CoV-2 in France[END_REF] . Other tools are providing crucial complementary information, for example genomic surveillance has been key to tracking the emergence of novel SARS-CoV-2 variants [START_REF] Priesemann | An action plan for pan-European defence against new SARS-CoV-2 variants[END_REF] . Serology, based on the detection of antibodies induced by previous infection with SARS-CoV-2, represents another category of surveillance information [START_REF] Vu | Prevalence of SARS-CoV-2 antibodies in France: results from nationwide serological surveillance[END_REF][START_REF] Ward | Declining prevalence of antibody positivity to SARS-CoV-2: a community study of 365,000 adults[END_REF] . Appropriately designed sero-prevalence studies can provide estimates of the proportion of a population who have been previously infected. Although no substitute for health systems surveillance, sero-prevalence studies have the advantage of accounting for asymptomatic cases, and symptomatic individuals who do not present to health systems. Sero-prevalence studies also provide information on the status of SARS-CoV-2 epidemics in situations where record keeping by health systems is not possible [START_REF] Kagucia | Seroprevalence of anti-SARS-CoV-2 IgG antibodies among truck drivers and assistants in Kenya[END_REF] .

Infection with SARS-CoV-2 induces diverse humoral and cellular immune responses [START_REF] Dan | Immunological memory to SARS-CoV-2 assessed for up to 8 months after infection[END_REF] . Humoral immunity includes antibodies of several immunoglobulin isotypes targeting SARS-CoV-2 proteins, most notably Spike (S) and Nucleocapsid (N). The concentration of antibodies in blood varies substantially between individuals, and with time since infection [START_REF] Dan | Immunological memory to SARS-CoV-2 assessed for up to 8 months after infection[END_REF][START_REF] To | Temporal profiles of viral load in posterior oropharyngeal saliva samples and serum antibody responses during infection by SARS-CoV-2: an observational cohort study[END_REF][START_REF] Iyer | Persistence and decay of human antibody responses to the receptor binding domain of SARS-CoV-2 spike protein in COVID-19 patients[END_REF][START_REF] Isho | Persistence of serum and saliva antibody responses to SARS-CoV-2 spike antigens in COVID-19 patients[END_REF][START_REF] Röltgen | Defining the features and duration of antibody responses to SARS-CoV-2 infection associated with disease severity and outcome[END_REF][START_REF] Seow | Longitudinal observation and decline of neutralizing antibody responses in the three months following SARS-CoV-2 infection in humans[END_REF] . Studies of the duration of immunity to a range of coronaviruses demonstrated that antibodies remain detectable six years after infection, but continue to wane [START_REF] Tang | Lack of peripheral memory B cell responses in recovered patients with severe acute respiratory syndrome: a six-year follow-up study[END_REF] . Longitudinal follow-up of individuals infected with SARS-CoV-2 indicates a pattern of waning of antibody responses consistent with other coronaviruses [START_REF] Callow | The time course of the immune response to experimental coronavirus infection of man[END_REF][START_REF] Choe | MERS-CoV antibody responses 1 year after symptom onset, South Korea, 2015[END_REF] . Within the first three months, antibody levels boost sharply and wane rapidly. Over a longer interval of eight months, antibody levels wane more slowly [START_REF] Dan | Immunological memory to SARS-CoV-2 assessed for up to 8 months after infection[END_REF] . These observations can be explained by the bi-phasic nature of antibody kinetics [START_REF] Slifka | Humoral immunity due to long-lived plasma cells[END_REF] . In the first three months, antibodies are predominantly generated by short-lived plasma cells in secondary lymphoid organs. The long-term response is dominated by antibodies from long-lived plasma cells in the bone marrow. This pattern of bi-phasic waning is observed for infection-and vaccine-induced antibody responses to a wide range of pathogens [START_REF] Teunis | Linking the seroresponse to infection to within-host heterogeneity in antibody production[END_REF][START_REF] Andraud | Living on three time scales: the dynamics of plasma cell and antibody populations illustrated for hepatitis A virus[END_REF][START_REF] White | Immunogenicity of the RTS,S/AS01 malaria vaccine and implications for duration of vaccine efficacy: secondary analysis of data from a phase 3 randomised controlled trial[END_REF][START_REF] White | Antibody kinetics following vaccination with MenAfriVac: an analysis of serological data from randomised trials[END_REF] .

Initial concerns that waning antibody responses would lead to reduced sensitivity of SARS-CoV-2 serological diagnostics over time have unfortunately been confirmed [START_REF] Rosado | Multiplex assays for the identification of serological signatures of SARS-CoV-2 infection: an antibody-based diagnostic and machine learning study[END_REF] . Using self-administered lateral flow diagnostic tests, a sero-prevalence study of 365,000 adults after the first epidemic wave in the UK observed significantly declining sero-prevalence during July to September 2020 [START_REF] Ward | Declining prevalence of antibody positivity to SARS-CoV-2: a community study of 365,000 adults[END_REF] . Using a laboratory-based assay for measuring anti-N IgG antibodies, a sero-prevalence survey of blood donors from the Brazilian city of Manaus estimated that 26% of the population had previously been infected [START_REF] Buss | Three-quarters attack rate of SARS-CoV-2 in the Brazilian Amazon during a largely unmitigated epidemic[END_REF] .

Application of a statistical adjustment to account for antibody waning led to an increased estimate of 76% previously infected. As the pandemic progresses, this problem of declining sensitivity of serological diagnostics is likely to get worse, potentially undermining the utility of sero-prevalence studies.

Serological diagnostics typically classify a sample as positive if a measured antibody level is greater than a defined cutoff.

Analysis of quantitative rather than binary antibody levels provides additional information, for example antibody levels are associated with time since infection, symptom severity, and sex [START_REF] Scepanovic | Human genetic variants and age are the strongest predictors of humoral immune responses to common pathogens and vaccines[END_REF] . However, large variation in antibody levels between individuals prevents this from having predictive value: detected antibodies could be from a recent infection, or due to immunological memory of an infection that cleared a year ago. This limitation has recently been overcome for a range of pathogens through the combination of multiplex assays and classification algorithms. Using machine learning algorithms to analyse quantitative antibody responses to multiple antigens, the time since previous infection can be estimated for Plasmodium falciparum malaria [START_REF] Helb | Novel serologic biomarkers provide accurate estimates of recent Plasmodium falciparum exposure for individuals and communities[END_REF][START_REF] Yman | Distinct kinetics in antibody responses to 111 Plasmodium falciparum antigens identifies novel serological markers of recent malaria exposure[END_REF] , P. vivax malaria [START_REF] Longley | Development and validation of serological markers for detecting recent exposure to Plasmodium vivax infection[END_REF] , and cholera [START_REF] Azman | Estimating cholera incidence with cross-sectional serology[END_REF] .

In this study, we use multiplex assays to measure antibodies to SARS-CoV-2 in health care workers and hospitalized patients followed for up to eleven months after infection, and apply mathematical models to characterize antibody kinetics in the first year following infection. Classification algorithms were developed that minimize the reduction in the sensitivity of serological tests over time, in addition to estimating time since previous infection from a single blood sample.

Finally, we present a method for serological reconstruction of past SARS-CoV-2 transmission using samples from a single cross-sectional survey.

Methods

Samples

A panel of 407 negative control serum or plasma samples was assembled from pre-pandemic cohorts (before December, 2019) with ethical approval for broad antibody testing (Table 1). This included 258 serum samples from healthy adult French blood donors, 81 serum samples from Peruvian healthy adult donors, and 69 plasma samples from healthy adult donors from the Thai Red Cross. A panel of 407 positive control serum samples was assembled from individuals with recent SARS-CoV-2 infection. This included 72 samples from patients in Paris hospitals [START_REF] Lescure | Clinical and virological data of the first cases of COVID-19 in Europe: a case series[END_REF][START_REF] Hadjadj | Impaired type I interferon activity and inflammatory responses in severe COVID-19 patients[END_REF] , 161 samples from Strasbourg healthcare workers, all confirmed positive by PCR-based testing [START_REF] Fafi-Kremer | Serologic responses to SARS-CoV-2 infection among hospital staff with mild disease in eastern France[END_REF] . Also included were 174 samples from community members of Crépy-en-Valois, France, confirmed seropositive by flow cytometry based testing [START_REF] Grzelak | A comparison of four serological assays for detecting anti-SARS-CoV-2 antibodies in human serum samples from different populations[END_REF][START_REF] Dufloo | Asymptomatic and symptomatic SARS-CoV-2 infections elicit polyfunctional antibodies[END_REF] .

The duration of antibody responses following SARS-CoV-2 infection was studied in longitudinal cohorts of hospitalized patients and healthcare workers. 213 serum samples from 194 patients in Dublin hospitals were collected, with date post symptom onset extending to four months. 724 serum samples from 347 healthcare workers in Strasbourg hospitals were collected, with date post symptom onset extending to nine months [START_REF] Planas | Sensitivity of infectious SARS-CoV-2 B.1.1.7 and B.1.351 variants to neutralizing antibodies[END_REF] .

In April 2020, our team implemented a study of the sero-prevalence of SARS-CoV-2 in healthcare workers from Institut Mutualiste Montsouris, a hospital in Paris. Serum samples were collected from 769 healthcare workers, and tested with our multiplex assay (Appendix Figure 1). Healthcare workers who tested sero-positive in April 2020 were invited to present a second sample in January 2021. In total we obtained follow-up samples from 29 healthcare workers. Plates were read using a Luminex® MAGPIX® system and the median fluorescence intensity (MFI) was used for analysis. A 5-parameter logistic curve was used to convert MFI to relative antibody unit (RAU), relative to the standard curve performed on the same plate to account for inter-assay variations.

The data from our multiplex assay was compared against data from two different neutralization assays with live virus using a subset of serum samples. Firstly, we implemented an S-Fuse assay as described elsewhere [START_REF] Planas | Sensitivity of infectious SARS-CoV-2 B.1.1.7 and B.1.351 variants to neutralizing antibodies[END_REF] . Neutralization activity was measured as the reciprocal dilution required to obtain a 50% reduction in neutralization (IC50). Secondly, we implemented a foci reduction neutralization test (FRNT) based on the detection of neutralizing antibodies directed against SARS-CoV-2.

This assay was performed under BSL-3 conditions as it facilitates infection of African green monkey kidney cells (VeroE6;

ATCC CRL-1586) with live-virus of a Cambodian SARS-CoV-2 isolate (GISAID: EPI_ISL_411902). Infection is visualized 14-16h after inoculation by staining of infected cells with a SARS-CoV-2 specific antibody (# ABIN1030641), targeting the S2 subunit of the viral spike protein. Neutralizing antibody titers are expressed as the reciprocal serum/plasma dilution that induces 50% reduction of infection (FRNT50) and is calculated by log probit regression analysis.

Mathematical model of antibody kinetics

SARS-CoV-2 antibody kinetics are described using a previously published mathematical model of the immunological processes underlying the generation and waning of antibody responses following infection [START_REF] White | Dynamics of the antibody response to Plasmodium falciparum infection in African children[END_REF] .

𝑑𝑃 𝑠 𝑑𝑡 = 𝜌𝛽 -𝑐 𝑠 𝑃 𝑠 , 𝑑𝑃 𝑙 𝑑𝑡 = (1 -𝜌)𝛽 -𝑐 𝑙 𝑃 𝑙 , 𝑑𝐴 𝑑𝑡 = 𝑔𝑃 𝑠 + 𝑔𝑃 𝑙 -𝑟𝐴
β denotes the boost in antibody secreting plasma cells. It is assumed that a proportion ρ of plasma cells are short-lived (Ps) waning at rate cs, and a proportion 1 -ρ are long-lived (Pl) waning at rate cl. Plasma cells generate antibodies (IgG, IgM or IgA) at rate g, and r is the rate of decay of antibody molecules.

Statistical inference was implemented within a mixed-effects framework allowing for characterisation of the kinetics within all individuals while also describing the population-level patterns (Statistical Appendix). On the population level, both the mean and variation in antibody kinetics are accounted for. Models were fitted in a Bayesian framework using Markov chain Monte Carlo methods with priors informed by estimates from long-term studies of antibody kinetics following infection with other coronaviruses [START_REF] Rosado | Multiplex assays for the identification of serological signatures of SARS-CoV-2 infection: an antibody-based diagnostic and machine learning study[END_REF] .

Classification algorithms

Measurements of antibodies of three isotypes (IgG, IgM, IgA) to multiple SARS-CoV-2 antigens were used to create a training dataset. Samples where the time post symptom onset was ≤ 14 days or unknown were excluded. In total we had 407 samples from pre-pandemic negative controls and 1402 positive samples. In a first step, a Random Forests binary classification algorithm was developed. A threshold corresponding to 99% specificity was selected. In a second step, a Random Forests multiway classification algorithm was developed for categorizing samples into four classes: (i) negative;

(ii) infected ≤ 3 months ago; (iii) infected 3 -6 months ago; and (iv) infected 6 -12 months ago. The algorithm was calibrated to have 99% specificity for correctly classifying negative samples. Positive samples were then classified according to the maximum number of votes. Uncertainty in classification performance was assessed via 1000-fold crossvalidation with a training set comprising two thirds of the data and a disjoint testing set comprising a third of the data.

Classification algorithms were implemented in R (version 3.4.3).

Statistical methods for serological surveillance

Imperfect diagnostic sensitivity causes a downwards bias in sero-prevalence estimates, whereas imperfect specificity causes an upwards bias in seroprevalence estimates. This bias can be corrected for using a Rogan-Gladen estimator, with incorporation of cross-validated uncertainty [START_REF] Gelman | Bayesian analysis of tests with unknown specificity and sensitivity[END_REF] . Multiway classification algorithms provide estimates of the proportion of a population infected during different time intervals. These estimates are subject to biases which can be adjusted for using a multivariate extension of the Rogan-Gladen estimator described in the Statistical Appendix.

Results

SARS-CoV

Modelled SARS-CoV-2 antibody kinetics

Our data on measured SARS-CoV-2 antibody responses was supplemented with data from six other longitudinal studies of the SARS-CoV-2 antibody response, and one longitudinal study of the SARS-CoV-1 antibody response. Appendix Figure 5 provides a comparison of the measured antibody responses between the eight studies. The data have been rescaled so that the average antibody response for each study equals one at day 14 after symptom onset. In addition to the large inter-individual variation, there is notable variation in antibody levels between studies. For example, the large dynamic range in the study by Iyer et al [START_REF] Iyer | Persistence and decay of human antibody responses to the receptor binding domain of SARS-CoV-2 spike protein in COVID-19 patients[END_REF] , reveals the boosting and subsequent decay of anti-S IgG antibodies in the first two months post infection, a phenomenon that is missed by assays without a comparably high upper limit of detection.

A mathematical model of antibody kinetics was simultaneously fitted to data from all eight studies. Appendix Figure 6 provides an overview of the fit of the model to the data. Figure 2a provides example fits to data from one individual from each of the eight studies. Figure 2b provides model predictions of the percentage antibody level remaining over the first two years post infection, where 100% is assumed to be the antibody response at day 14 following symptom onset. There are considerable differences in the pattern of waning between isotypes and between antigens. Table 2 summarises the duration of the antibody response. One year following symptom onset, we estimate that 36% (95% range: 11%, 94%) anti-S IgG remains, 31% (9%, 89%) anti-RBD IgG remains, and 7% (1%, 31%) anti-N IgG remains. The uncertainty represents the considerable degree of inter-individual variation in the duration of the antibody responses. Antibodies of the IgM isotype waned more rapidly, with 6% (0%, 27%) anti-S IgM remaining after one year, 9% (2%, 32%) anti-RBD IgM remaining after one year, and 15% (4%, 50%) anti-N IgM remaining after one year. Antibodies of the IgA isotype also waned rapidly, with 18% (4%, 67%) anti-S IgA remaining after one year, 10% (3%, 38%) anti-RBD IgA remaining after one year, and 3% (0%, 13%) anti-N IgA remaining after one year. We also observed comparable reductions in titres for viral neutralization over time (Appendix Figure 4), although the small number of samples prevented application of our model.

Estimating time since previous SARS-CoV-2 infection

Using a dataset on measured IgG, IgM and IgA antibody levels from pre-pandemic negative controls and samples from individuals with SARS-CoV-2 infection confirmed by PCR-based testing and followed for up to eleven months after symptom onset, Random Forests binary classification algorithms were trained to identify individuals with previous SARS-CoV-2 infection (Appendix Figures 7 and8). Next, a Random Forests multi-way classification was trained to simultaneously identify previous infection and estimate the time since infection (Figure 3). The diagnostic test identified samples from individuals with previous SARS-CoV-2 infection with 99% specificity and 98% (95% confidence interval: 94%, 99%) sensitivity (Figure 3a). The diagnostic test classified samples from individuals infected within the previous 3 months (Figure 3b). Notably, it was easier to distinguish recent infections (<3 months) from older infections (6 -12 months), compared to infections that occurred 3 -6 months ago. There was limited statistical signal to distinguish between infections that occurred 3 -6 months ago, and older infections occurring more than 6 months ago (Figure 3c). A breakdown of classification performance by time since infection is provided in Figure 3D. The diagnostic test accurately classifies samples of all categories, with the exception of samples from individuals infected 3 -6 months ago. Many of these samples were incorrectly classified in the neighbouring infection categories of 0 -3 months or 6 -12 months.

Serological reconstruction of COVID-19 epidemics

When applied to samples from a cross-sectional sero-prevalence survey, a diagnostic test capable of estimating time since infection can provide a serological reconstruction of past SARS-CoV-2 epidemics. Figure 4 presents four simulated epidemics where 1,000 individuals were sampled, and 40% were previously infected. A range of scenarios were simulated, including a recent epidemic wave, an older epidemic wave, a double wave epidemic, and constant transmission. The algorithm was able to accurately reconstruct all epidemic profiles. For example, in the recent wave scenario 60% of the population were negative and the model estimated 59.8% (95% CI: 59.3%, 61.4%); 35.8% were infected 0 -3 months ago and the model estimated 35.5% (95% CI: 25.7%, 40.3%); 4.2% were infected 3 -6 months ago and the model estimated 4.2% (95% CI: 0.0%, 14.4%); and 0% were infected 6 -12 months ago and the model estimated 0% (95% CI: 0.0%, 3.3%).

Discussion

Infection with SARS-CoV-2 induces a complex, diverse immune response, that varies by orders of magnitude between individuals, and changes over time. This diversity is a challenge to immunologists and vaccinologists, but presents an opportunity to diagnostic developers armed with multiplex assays and machine learning algorithms. By quantifying the kinetics of different components of the humoral immune response, it is possible to provide classification of previous infection that minimizes the reduction of diagnostic sensitivity over time, and also allows estimation of the time since infection.

Based on data from a range of studies with up to eleven months follow up after symptom onset, we estimate that 31% (95% CrI: 9%, 89%) of anti-RBD IgG antibody levels remain one year after infection. Antibodies of other isotypes waned more rapidly, with 9% (95% CrI: 2%, 32%) of anti-RBD IgM antibody levels remaining after one year, and 10% (95% CrI: 3%, 38%) of anti-RBD IgA antibody levels remaining after one year. There was considerable variation in kinetic profiles between different SARS-CoV-2 antigens, with 7% (95% CrI: 1%, 31%) of anti-N IgG antibody levels remaining after one year. Although the determinants of the duration of antigen-specific antibody responses remain poorly understood [START_REF] Scepanovic | Human genetic variants and age are the strongest predictors of humoral immune responses to common pathogens and vaccines[END_REF] , the diversity in patterns of antibody kinetics can be quantified using epidemiological studies, yielding valuable information for serological diagnostics.

The majority of commercially available serological tests classify individuals as having previous SARS-CoV-2 infection if a measured antibody response is greater than a defined cutoff [START_REF]National SARS-CoV-2 Serology Assay Evaluation Group. Performance characteristics of five immunoassays for SARS-CoV-2: a head-to-head benchmark comparison[END_REF] . Instead of reducing a complex antibody response to a binary data point, a mode detailed serological signature based on quantitative measurements of multiple antibody responses provides two notable advantages [START_REF] Rosado | Multiplex assays for the identification of serological signatures of SARS-CoV-2 infection: an antibody-based diagnostic and machine learning study[END_REF] . Firstly, the reduction in diagnostic sensitivity associated with waning antibodies is minimized -no reduction in diagnostic sensitivity over the eleven months of follow up was observed.

Secondly, the time since previous infection can be estimated providing valuable additional epidemiological information.

For the current assay, previous infections were categorised into intervals of 0 -3 months, 3 -6 months, and 6 -12 months.

More precise classification is possible in theory, but this must be balanced against the statistical signal. For example, there was little statistical signal to discriminate between infections that occurred 6 -9 months ago, versus infections that occurred 9 -12 months ago. It is anticipated that further improvements to the assay such as the incorporation of new antigens, more training samples with a range of time since infection, and better algorithms will lead to improvements in accuracy.

Existing sero-prevalence studies estimate the proportion of a population previously infected with SARS-CoV-2. The addition of a diagnostic tool capable of estimating time since infection allows for the serological reconstruction of past transmission trends. Thus, using samples from a sero-prevalence study collected at a single time point, we can discriminate between a scenario of constant SARS-CoV-2 transmission and a scenario where transmission occurs in distinct epidemic waves. This diagnostic technology has a range of possible applications. For countries that experienced double wave SARS-CoV-2 epidemics, it has been challenging to quantify the relative magnitude of these waves due to the time required to scale up PCR-based diagnostic testing [START_REF] Pullano | Underdetection of cases of COVID-19 in France threatens epidemic control[END_REF] . Furthermore, there are many epidemiological settings where it is unknown if SARS-CoV-2 transmission was constant over time, or occurred as distinct epidemic waves [START_REF] Kagucia | Seroprevalence of anti-SARS-CoV-2 IgG antibodies among truck drivers and assistants in Kenya[END_REF] .

The widespread introduction of SARS-CoV-2 vaccines will lead to vaccine-induced immunity in many individuals. The majority of vaccines against SARS-CoV-2 target the Spike protein's RBD. As our diagnostic assay and algorithms include Spike and RBD antigens, serum samples from vaccinated individuals would likely be classified as positive. However, by measuring antibody responses to Nucleocapsid, Membrane, Envelope, and other viral proteins [START_REF] Hachim | ORF8 and ORF3b antibodies are accurate serological markers of early and late SARS-CoV-2 infection[END_REF] it will be possible to modify the diagnostic test to provide three-way classification: negative; previously infected; or vaccinated. Identification of individuals who have been both infected and vaccinated will be challenging. Having classified an individual as previously infected or vaccinated, the diagnostic algorithm can then be used to estimate time since infection or time since vaccination. Such an approach could contribute to efforts to measure population-level immunity to SARS-CoV-2, whether induced by infection or vaccination.

There are several limitations to this study. Estimates of the duration of antibody responses are based on data from multiple studies, each using a unique immunoassay [START_REF] Dan | Immunological memory to SARS-CoV-2 assessed for up to 8 months after infection[END_REF][START_REF] To | Temporal profiles of viral load in posterior oropharyngeal saliva samples and serum antibody responses during infection by SARS-CoV-2: an observational cohort study[END_REF][START_REF] Iyer | Persistence and decay of human antibody responses to the receptor binding domain of SARS-CoV-2 spike protein in COVID-19 patients[END_REF][START_REF] Isho | Persistence of serum and saliva antibody responses to SARS-CoV-2 spike antigens in COVID-19 patients[END_REF][START_REF] Röltgen | Defining the features and duration of antibody responses to SARS-CoV-2 infection associated with disease severity and outcome[END_REF][START_REF] Seow | Longitudinal observation and decline of neutralizing antibody responses in the three months following SARS-CoV-2 infection in humans[END_REF][START_REF] Tang | Lack of peripheral memory B cell responses in recovered patients with severe acute respiratory syndrome: a six-year follow-up study[END_REF] . Every immunoassay may differ in terms of background reactivity, cross-reactivity with other pathogens, protein formulation, dynamic range and reproducibility. We believe that the benefit of drawing on multiple data sources outweighs the benefit of having a smaller more homogeneous database, especially since the mathematical model of antibody kinetics is sufficiently flexible to incorporate data from multiple assays. Our selection of a mechanistic mathematical model of antibody kinetics is a potential limitation. The model is based on a mechanistic understanding of the immunological processes underlying the generation and persistence of antibodies, and imposes a flexible functional form on how antibody levels change over time. Although this approach has been validated in a range of applications [START_REF] Teunis | Linking the seroresponse to infection to within-host heterogeneity in antibody production[END_REF][START_REF] Andraud | Living on three time scales: the dynamics of plasma cell and antibody populations illustrated for hepatitis A virus[END_REF][START_REF] White | Immunogenicity of the RTS,S/AS01 malaria vaccine and implications for duration of vaccine efficacy: secondary analysis of data from a phase 3 randomised controlled trial[END_REF][START_REF] White | Antibody kinetics following vaccination with MenAfriVac: an analysis of serological data from randomised trials[END_REF] , there will be instances where the model fails to capture the true pattern of antibody kinetics, for example in immune-deficient individuals. An advantage of a mechanistic model versus a non-parametric statistical model is the ability to make projections forward in time. We have provided predictions up to two years following infection, for example by estimating that 16% (5%, 48%) of anti-RBD IgG antibodies remain after two years. There is a risk to providing predictions beyond the timescale of the data, but these predictions can be easily falsified via continued longitudinal studies.

The diagnostic assay is not exempt from the challenges of antibody waning. Although no reductions in diagnostic sensitivity over time were observed, reduced sensitivity will likely be observed as we analyse additional samples over longer durations of follow up. There are substantial challenges in providing estimates of time since infection. Although the approach can reliably reconstruct the distribution of infection times across a population, there will be substantial uncertainty in estimates of the time since previous infection for individual samples.

Sero-prevalence studies are playing a critical role in monitoring the progress of the SARS-CoV-2 pandemic. In the early stages of the pandemic, immunoassays had the advantage of measuring high antibody levels in the initial months after infection. As the pandemic progresses, sero-prevalence will become more challenging to accurately measure due to waning antibody responses and increased vaccine-induced immunity. Multiplex assays and algorithms accounting for how antibody levels change over time may be an important tool for ensuring the ongoing utility of sero-surveillance.

parameters 𝜃 𝑛,𝑚 = (𝐴 𝑏𝑔 𝑚,𝑛 , 𝛽 𝑚,𝑛 , 𝛿

Mixed effects likelihood

As described above, for each individual there are seven parameters to be estimated. These individual-level parameters are drawn from population-level distributions. Some parameters are assumed to depend on the assay in each study (the antibody kinetic parameters: 𝛿 𝑚,𝑛 , 𝑑 𝑏 𝑚,𝑛 , 𝑑 𝑠 𝑚,𝑛 , 𝑑 𝑙 𝑚,𝑛 , 𝑑 𝑎 𝑚,𝑛 , 𝜌 𝑚,𝑛 ). Each of these parameters are assumed to be drawn from the same distribution across all studies. Some parameters are assay dependent, notably 𝐴 𝑏𝑔 𝑚,𝑛 , 𝛽 𝑚,𝑛 . These parameters are drawn from distributions specific for each study. The mixed effects likelihood can be written as follows:

𝐿 𝑚𝑖𝑥 𝑚,𝑛 (𝜃 As the proportion of the ASCs that are long-lived must be bounded by 0 and 1, the individual-level parameters ρ m,n are assumed to be drawn from logit-Normal distributions.

Total model likelihood

Denote 𝐷 = {𝐷 1 , … , 𝐷 𝑁 } to be the vector of data for all N participants from all studies. We denote

)

where 𝜇 𝜌,0 and 𝜏 𝜌,0 parameterise the Normal prior distribution on the mean, and k and θ parameterise the Gamma prior distribution on the precision (inverse standard deviation).

Observational variance parameter Metropolis-Hastings Update.

 For each study m, update the observational variance parameter 𝜎 𝑜𝑏𝑠,𝑚 ′  Calculate updated total likelihood 𝐿 𝑡𝑜𝑡𝑎𝑙 (𝜃′|𝐷)and the updated prior probability density 𝑃(𝜃′)  Accept the parameter update with probability 𝑚𝑖𝑛 (1, 𝐿 𝑡𝑜𝑡𝑎𝑙 (𝜃′|𝐷)𝑃(𝜃′) 𝐿 𝑡𝑜𝑡𝑎𝑙 (𝜃|𝐷)𝑃(𝜃) )

The MCMC algorithm was implemented in C++ complied in Microsoft Visual Studio. The covariance of the multivariate-Normal proposal distributions for Metropolis-Hastings updates were adaptively tuned using the estimated posterior distributions during a burn-in phase of 1 million MCMC iterations. The magnitude of the proposed step size was calibrated using a Robbins-Munro algorithm to achieve an acceptance rate of approximately 23%. The total number of MCMC iterations was 10,000,000. The effective number of iterations was calculated using the effectiveSize routine in the R library coda and the effective size was checked to be > 1,000 for all parameters. The statistical inference procedure was repeated twice to allow for assessment of chain convergence using Gelman-Rubin convergence diagnostics Rc. For all populationlevel chains, we ensure Rc < 1.05. For all individual-level chains, we ensure Rc < 1.1.

Serological surveillance algorithms

A ROC curve obtained from a training data set consisting of positive and negative samples is described by a sequence of estimated sensitivities and specificities{𝐸(𝑠𝑒 𝑖 , 𝑠𝑝 𝑖 )}, where E denotes an estimator. N-fold cross-validation generates samples of sensitivity {𝑠𝑒 𝑖1 , … , 𝑠𝑒 𝑖𝑁 } for each 𝐸(𝑠𝑝 𝑖 ) and samples of specificity {𝑠𝑝 𝑖1 , … , 𝑠𝑝 𝑖𝑁 } for each E(𝑠𝑒 𝑖 ). Following a previously outlined approach [START_REF] Gelman | Bayesian analysis of tests with unknown specificity and sensitivity[END_REF] , for each pair i of sensitivity and specificity, we obtain N estimates of the measured seroprevalence Min in a scenario with true seroprevalence T as follows:

𝑀 𝑖𝑛 = 𝑇𝑠𝑒 𝑖𝑛 + (1 -𝑇)(1 -𝑠𝑝 𝑖𝑛 )
Following the Rogan-Gladen estimator approach, this equation can be rearranged to give an adjusted estimate of true seroprevalence:

  -2 antibodies over time Antibody responses of three isotypes (IgG, IgM, IgA) to nine coronavirus antigens were measured in 407 pre-pandemic serum samples, and 1402 serum samples from individuals with previous SARS-CoV-2 infection. 961/1402 of the positive samples were from individuals with SARS-CoV-2 infection confirmed by PCR-based testing with available data on time post symptoms. Figure 1 presents the IgG, IgM and IgA antibody responses to SARS-CoV-2 S, RBD, and N measured over time.Appendix Figures2 and 3presents the antibody responses to S2, ME, and the Spike proteins of the four human seasonal coronaviruses. Notably there is substantial inter-individual variation in antibody responses, with antibody levels varying by orders of magnitude between individuals. As a measure of functional immunity, we applied live virus neutralizing assays to a subset of samples. Substantial variation in neutralizing activity between individuals and over time (Appendix Figure4).

Figure 1 :

 1 Figure 1: Antibody kinetics in the first year following infection with SARS-CoV-2. A bead-based multiplex Luminex assay used to measure antibodies of multiple isotypes (IgG, IgM, IgA) to multiple antigens in serum samples from individuals with PCR-positive SARS-CoV-2 infection and pre-pandemic negative controls.

Figure 2 :

 2 Figure 2: Modelled SARS-CoV-2 antibody kinetics. A mathematical model of SARS-CoV-2 antibody kinetics was simultaneously fitted to data from seven studies of SARS-CoV-2 and one study of SARS-CoV-1. (a -top row) Examples of the model fit to the data for one individual from each study. Data are represented as points, posterior median model prediction as lines, and 95% credible intervals as shaded areas. (b -middle and bottom rows) Model predicted duration of antibodies within the first 2 years following infection. Antibody levels are shown relative to the expected antibody level at day 14 following symptom onset. Each point represents the prediction from an individual at 6, 12, 18 and 24 months post symptoms. The median predictions for each of the eight studies are presented as lines.

Figure 3 :

 3 Figure 3: Classification of time since previous SARS-CoV-2 infection. A cross-validated multi-way classification algorithm was trained to estimate time since infection. (A) The algorithm can differentiate between positive and negative samples. (B) The algorithm can classify individuals infected within the previous 3 months. (C) There is limited diagnostic power to distinguish between infections that occurred 3-6 months ago versus 6-12 months ago. (D) Breakdown of classification performance according to time since previous infection. Colours represent model predicted classification. >99% of negative samples are correctly classified as negative (blue). For the positive samples, the distribution shows the time since previous infection. Samples with time since infection <3 months are mostly classified in the 0 -3 month category (red). Samples with time since infection >6 months ago are mostly classified in the 6 -12 month category (purple). There is a substantial degree of misclassification of samples with time since infection 3 -6 months ago. This is due to the temporal imbalance in the training data.

Figure 4 :

 4 Figure 4: Serological reconstruction of previous COVID-19 epidemics.The grey histograms represent simulated data from a range of SARS-CoV-2 epidemic scenarios. Data were simulated from 1000 samples from a cross-sectional sero-prevalence study. It was assumed that 600 samples were negative, with 400 positive samples with varying time since infection. For previously infected individuals, the histograms represent the time since infection. Based on antibody levels sampled at a single time point, our computational method was used to provide a serological reconstruction of previous COVID-19 epidemics. The solid lines represent the model predictions for the categories of negative; infected 0 -3 months ago; infected 3 -6 months ago; and infected 6 -12 months ago. Shaded regions represent 95% confidence intervals.

Table 1 : Panels of samples. Age and days post symptom onset are presented as median and ranges. cohort status participants samples PCR positive age (years) gender (% male) days post symptoms

 1

	Établissement Français	pre-pandemic	45	45		> 18
	du Sang 1	negative controls
	Établissement Français	pre-pandemic	213	213		42 (18,81)	40%
	du Sang 2	negative controls
	Thai Red Cross	pre-pandemic	68	68		> 18
		negative controls
	Peruvian donors	pre-pandemic	81	81		> 18
		negative controls
	Hôpital Bichat	hospitalized patients	2	8	2	31 (30, 32) 100%	14 (8, 24)
	(Paris, France)
	Hôpital Cochin	hospitalized patients	64	64	64	55 (25, 79) 76%	17 (10, 28)
	(Paris, France)
	Strasbourg hospitals 1	infected healthcare workers	161	161	161	32 (20, 62) 31%	24 (13, 39)
	Crépy-en-Valois	infected community members	154	174	0	17 (15, 56) 34%
	community	(flow cytometry positive)
	Dublin hospitals	hospitalized patients	194	213	194	55 (21, 92) 47%	13 (1, 126)
	Strasbourg hospitals 2	follow-up of infected	347	724	347	41 (21,74)	23%	132 (11, 284)
		healthcare workers
	Institut Mutualiste	sero-prevalence survey in	769	769	20	41 (18, 72) 27%
	Montsouris	healthcare workers
	(Paris, France)	(unknown status)
	Institut Mutualiste	follow-up of sero-positive	29	29	12	37 (24, 63) 41%	304 (285, 336)
	Montsouris	healthcare workers
	(Paris, France)

Table 2 : Estimated duration of antibody responses following SARS-CoV-2 infection.

 2 The percentage antibody level remaining over time is compared to the measured antibody level 14 days after symptom onset. Estimates are presented as the population median, with the 95% range due to inter-individual variation.

		6 months	12 months	24 months
	Spike IgG	55% (16%, 100%)	36% (11%, 94%)	16% (5%, 55%)
	RBD IgG	43% (13%, 100%)	31% (9%, 89%)	16% (5%, 48%)
	Nucleocapsid IgG	30% (8%, 92%)	7% (1%, 31%)	0.8% (0%, 7%)
	Spike IgM	12% (1%, 52%)	6% (0%, 27%)	2% (0%, 9%)
	RBD IgM	16% (4%, 51%)	9% (2%, 32%)	4% (1%, 16%)
	Nucleocapsid IgM 23% (6%, 75%)	15% (4%, 50%)	7% (2%, 24%)
	Spike IgA	21% (4%, 82%)	18% (4%, 67%)	12% (3%, 47%)
	RBD IgA	12% (4%, 49%)	10% (3%, 38%)	6% (2%, 24%)
	Nucleocapsid IgA	6% (1%, 30%)	3% (0%, 13%)	0.6% (0%, 4%)

  𝑚,𝑛 , 𝑑 𝑠 𝑚,𝑛 , 𝑑 𝑙 𝑚,𝑛 , 𝑑 𝑎 𝑚,𝑛 , 𝜌 𝑚,𝑛 ) are estimated. The model predicted antibody levels will be {𝐴(𝑡 1 ), 𝐴(𝑡 2 ), … , 𝐴(𝑡 𝐽 )}. We assume log-Normally distributed measurement error such that the difference between 𝑙𝑜𝑔(𝑎 𝑗 ) and 𝑙𝑜𝑔 (𝐴(𝑡 𝑗 )) is Normally distributed with variance 𝜎 𝑜𝑏𝑠,𝑚 2. The parameters for observational variance are specific for each study due to differences in the assays utilised. For model predicted antibody levels 𝐴(𝑡 𝑗 ) the data likelihood for individual n from study m is given by

				2
	𝐿 𝑚𝑜𝑑 𝑚,𝑛 (𝜃 𝑚,𝑛 |𝐷 𝑚,𝑛 ) = ∏ 𝑗∈𝐽	𝑒	-	(𝑙𝑜𝑔(𝑎 𝑗 )-𝑙𝑜𝑔(𝐴(𝑡 𝑗 ))) 2 2𝜎 𝑜𝑏𝑠,𝑚 𝑎 𝑗 𝜎 𝑜𝑏𝑠,𝑚 √2𝜋

  𝑚,𝑛 |𝐷 𝑚,𝑛 ) =

				𝑒	-	(𝑙𝑜𝑔(𝐴 𝑏𝑔 𝑚,𝑛 )-𝜇 𝐴,𝑚 ) 2𝜎 𝐴,𝑚 2	2	𝑒	-	(𝑙𝑜𝑔(𝛽 𝑚,𝑛 )-𝜇 𝛽,𝑚 ) 2 2𝜎 𝛽,𝑚 2	𝑒	-	(𝑙𝑜𝑔(𝛿 𝑚,𝑛 )-𝜇 𝛿 ) 2 2𝜎 𝛿 2
				(		√2𝜋𝐴 𝑏𝑔 𝑚,𝑛 𝜎 𝐴,𝑚	) (		√2𝜋𝛽 𝑚,𝑛 𝜎 𝛽,𝑚	)	(	√2𝜋𝛿 𝑚,𝑛 𝜎 𝛿	)
	𝑒	-(𝑙𝑜𝑔(𝑑 𝑠 𝑚,𝑛 )-𝜇 𝑠 ) 2 2𝜎 𝑠 2 √2𝜋𝑑 𝑠 𝑚,𝑛 𝜎 𝑠	𝑒	-(𝑙𝑜𝑔(𝑑 𝑙 𝑚,𝑛 )-𝜇 𝑙 ) 2 2𝜎 𝑙 2 √2𝜋𝑑 𝑙 𝑚,𝑛 𝜎 𝑙	𝑒	-(𝑙𝑜𝑔(𝑑 𝑎 𝑚,𝑛 )-𝜇 𝑎 ) 2 2𝜎 𝑎 2 √2𝜋𝑑 𝑎 𝑚,𝑛 𝜎 𝑎	𝑒 √2𝜋𝜌 𝑚,𝑛 (1 -𝜌 𝑚,𝑛 )𝜎 𝜌 (𝑙𝑜𝑔( 2 𝜌 𝑚,𝑛 1-𝜌 𝑚,𝑛 )-𝜇 𝜌 ) -2 2𝜎 𝜌	𝐿 𝑚𝑜𝑑 𝑚,𝑛 (𝜃 𝑚,𝑛 |𝐷 𝑚,𝑛 )
	(	) (				) (					) (	)

  𝜃 𝑚,𝑛 = (𝜇 𝐴,𝑚 , 𝜎 𝐴,𝑚 , 𝜇 𝛽,𝑚 , 𝜎 𝛽,𝑚 , 𝜇 𝛿 , 𝜎 𝛿 , 𝜇 𝑠 , 𝜎 𝑠 , 𝜇 𝑙 , 𝜎 𝑙 , 𝜇 𝑎 , 𝜎 𝑎 , 𝜇 𝜌 , 𝜎 𝜌 , 𝜃 1 , … , 𝜃 𝑁 ) to be the combined vector of population-The model was fitted to the data using Markov Chain Monte Carlo (MCMC) methods. A three stage parameter update regimen was utilised with a Metropolis-within-Gibbs sampler with sequential updating of individual-level parameters, population-level parameters, and observational variance parameters. A′ indicates an attempted update. For each of the 𝑖 ∈ {𝛿, 𝑠, 𝑙, 𝑎} we obtain new estimates of the population level parameters 𝜇 𝑖 ′ and 𝜏 𝑖 ′ =where 𝜇 𝑖,0 and 𝜏 𝑖,0 parameterise the Normal prior distribution on the mean, and k and θ parameterise the Gamma prior distribution on the precision (inverse standard deviation). 𝑦 𝑖 𝑛 is the individual-level parameter i in individual n. For ρ we obtain new estimates of the population level parameters 𝜇 𝜌 ′ and 𝜏 𝜌 ′ = 1 𝜎 𝜌 ′ as follows:𝜇 𝜌 ′ ~𝑁 ( 𝜏 𝜌,0 𝜇 𝜌,0 + 𝜏 𝜌 ∑ ∑ 𝑙𝑜𝑔𝑖𝑡(𝜌 𝑛 )

		𝐿 𝑡𝑜𝑡𝑎𝑙 (𝜃|𝐷) = ∏ 𝐿 𝑚𝑖𝑥 𝑚,𝑛	(𝜃 𝑛 |𝐷 𝑛 )
							𝑛∈𝑁 𝜏 𝜌,0 + 𝜏 𝜌 𝑁 𝑁 𝑚 𝑛 𝑀 𝑚	,	1 √ 𝜏 𝜌,0 + 𝜏 𝜌 𝑁	)
	Markov Chain Monte Carlo parameter update 𝜏 𝜌 ′ ~Γ ( 𝑁 2	+ 𝑘, ( 𝜃 1	+	1 2	𝑛 ∑ ∑ (𝑙𝑜𝑔𝑖𝑡(𝜌 𝑛 ) -𝜇 𝜌 ′) 𝑚 2 𝑁 𝑚 𝑀	) -1
	1. Individual-level parameter Metropolis-Hastings update. For each participant n in study m:
	 Update local parameters: 𝜃 𝑚,𝑛 ′ = (𝐴 𝑏𝑔 𝑚,𝑛 ′ , 𝛽 𝑚,𝑛 ′, 𝛿 𝑚,𝑛 ′, 𝑑 𝑠 𝑚,𝑛 ′, 𝑑 𝑙 𝑚,𝑛 ′, 𝑑 𝑎 𝑚,𝑛 ′, 𝜌 𝑚,𝑛 ′)
	 Calculate updated mixed effects likelihood 𝐿 𝑚𝑖𝑥 𝑚,𝑛 (𝜃 𝑚,𝑛 ′|𝐷 𝑚,𝑛 )
	 Accept the parameter update with probability 𝑚𝑖𝑛 (1,	𝐿 𝑚𝑖𝑥 𝑚,𝑛 (𝜃 𝑚,𝑛 ′ |𝐷 𝑚,𝑛 ) 𝐿 𝑚𝑖𝑥 𝑚,𝑛 (𝜃 𝑚,𝑛 |𝐷 𝑚,𝑛 ) )
	2. Population-level parameter Gibbs update.
									𝑖 ′ =
	1 𝜎 𝑖 ′	as follows:
		𝜇 𝑖 ′ ~𝑁 (	𝜏 𝑖,0 𝜇 𝑖,0 + 𝜏 𝑖 ∑ 𝑙𝑜𝑔(𝑦 𝑖 𝑛 ) 𝑁 𝑚 𝜏 𝑖,0 + 𝜏 𝑖 𝑁 𝑚	,	1 √ 𝜏 𝑖,0 + 𝜏 𝑖 𝑁 𝑚	)
		𝜏 𝑖 ′ ~Γ ( 𝑁 𝑚 2	+ 𝑘, (	1 𝜃	+	2 1	∑ (𝑙𝑜𝑔(𝑦 𝑖 𝑁 𝑚 𝑛 ) -𝜇 𝑖 ′) 2	)	-1
	1 𝜎 𝑖 ′ as follows:
		𝜇 𝑖 ′ ~𝑁 (	𝜏 𝑖,0 𝜇 𝑖,0 + 𝜏 𝑖 ∑ ∑ 𝑙𝑜𝑔(𝑦 𝑖 𝑛 ) 𝑁 𝑚 𝑛 𝑀 𝑚 𝜏 𝑖,0 + 𝜏 𝑖 𝑁	,	1 √ 𝜏 𝑖,0 + 𝜏 𝑖 𝑁	)
	level parameters and individual-level parameters to be estimated. The total likelihood is obtained by multiplying the 𝜏 𝑖 ′ ~Γ ( 𝑁 2 + 𝑘, ( 1 𝜃 + 1 2 ∑ ∑ (𝑙𝑜𝑔(𝑦 𝑖 𝑛 𝑚 𝑛 ) -𝜇 𝑖 ′) 2 𝑀 -1 𝑁 𝑚 ) )
	likelihood for each participant

 For each of the 𝑖 ∈ {𝐴 𝑏𝑔 𝑚 , 𝛽 𝑚 } we obtain new estimates of the population level parameters 𝜇 𝑖 ′ and 𝜏 )

where 𝜇 𝑖,0 and 𝜏 𝑖,0 parameterise the Normal prior distribution on the mean, and k and θ parameterise the Gamma prior distribution on the precision (inverse standard deviation). 𝑦 𝑖 𝑛 is the individual-level parameter i in individual n.



Acknowledgements

The French COVID cohort is supported by the REACTing consortium and by the French Directorate General for Health.

Jessica Vanhomwegen (Institut Pasteur) is thanked for provision of a MAGPIX machine. Jérôme Hadjadj and Laura Barnabei (Institut Imagine, Paris) are thanked for their work on the Hôpital Cochin study. Dionicia Gamboa (Cayetano Heredia University, Lima) is thanked for sharing negative control samples from Peru. Jetsumon Sattabongkot (Mahidol University, Bangkok) is thanked for sharing negative control samples from Thailand. Marie-Noelle Ungeheuer and Blanca Liliana Perlaza are thanked for processing samples at the Clinical Investigation and Access to BioResources platform in Institut Pasteur. We thank all patients and health-care workers who kindly agreed for samples to be used for medical research purposes.

Funding

This work was supported by the European Research Council (MultiSeroSurv ERC Starting Grant 852373; MW), l'Agence Nationale de la Recherche and Fondation pour la Recherche Médicale (CorPopImm; MW), and the Institut Pasteur International Network (CoronaSeroSurv; MW). JR was supported by the Pasteur Paris University (PPU) International PhD Program. CC was supported by the European Research Council 771813. MB and AL were supported by the « URGENCE COVID-19 » fundraising campaign of Institut Pasteur. NC and CNC are part-funded by a Science Foundation Ireland (SFI) grant, Grant Code 20/SPP/3685. LT has been awarded the Irish Clinical Academic Training (ICAT) Programme, supported by the Wellcome Trust and the Health Research Board (Grant Number 203930/B/16/Z), the Health Service Executive, National Doctors Training and Planning and the Health and Social Care, Research and Development Division, Northern Ireland (https://icatprogramme.org/). MB and AM were supported by Institut Pasteur TaskForce funding (TooLab project). SFK lab is funded by Strasbourg University Hospitals (SeroCoV-HUS; PRI 7782), the Agence Nationale de la Recherche (ANR-18-CE17-0028), Laboratoire d'Excellence TRANSPLANTEX (ANR-11-LABX-0070_TRANSPLANTEX), and Institut National de la Santé et de la Recherche Médicale (UMR_S 1109).

Data availability

Data and code will be made available online following peer review of the article. In the meantime, requests for data or code should be made to the corresponding author.

Ethics

Declaration of interests

Appendix Figures

Appendix Figure 1

Statistical Appendix Data on longitudinal antibody responses to SARS-CoV-2 proteins

Several teams have conducted studies of the duration of the antibody response against SARS-CoV-2 with varying durations of follow up. Our data analyses the duration of the SARS-CoV-2 antibody response over approximately eleven months. To better characterize the antibody kinetics over different time scales, we supplemented our database with data from six published studies on SARS-CoV-2 antibody kinetics. These data are summarized in Appendix Table 1. In addition to the studies on SARS-CoV-2 antibody kinetics, we also included data from one study of SARS-CoV-1 antibody kinetics in individuals with six years follow-up [START_REF] Tang | Lack of peripheral memory B cell responses in recovered patients with severe acute respiratory syndrome: a six-year follow-up study[END_REF] .

Appendix

Mathematical model of antibody kinetics

SARS-CoV-2 antibody kinetics are described using a previously published mathematical model of the immunological processes underlying the generation and waning of antibody responses following infection or vaccination [START_REF] Rosado | Multiplex assays for the identification of serological signatures of SARS-CoV-2 infection: an antibody-based diagnostic and machine learning study[END_REF][START_REF] White | Dynamics of the antibody response to Plasmodium falciparum infection in African children[END_REF] . For antibodies of three classes (IgG, IgM and IgA) targeting the studied antigens, the antibody kinetics are described by the following equations:

where β denotes the boost in antibody secreting plasma cells. It is assumed that a proportion ρ of plasma cells are shortlived (Ps) waning at rate cs, and a proportion 1 -ρ are long-lived (Pl) waning at rate cl. Plasma cells generate antibodies (IgG, IgM or IgA) at rate g, and r is the rate of decay of antibody molecules. Assuming Ps(0) = Pl(0) = 0 and A(0) = Abg, these equations can be solved analytically to give:

δ is the time after symptom onset when antibody levels start to increase. B0 is the number of B cells, and g is the rate at which they secrete antibodies. As g and B0 are not both identifiable without detailed and invasive experiments (e.g. bone marrow aspirates to measure antigen-specific plasma cells), we estimate 𝛽 = 𝑔𝐵 0 . If r is the decay rate of antibody molecules, then we define 𝑑 𝑎 = .

Note that this model of antibody kinetics follows the formulation outlined in White et al. [START_REF] White | Dynamics of the antibody response to Plasmodium falciparum infection in African children[END_REF] and not that outlined in Rosado et al. [START_REF] Rosado | Multiplex assays for the identification of serological signatures of SARS-CoV-2 infection: an antibody-based diagnostic and machine learning study[END_REF] which included an additional equation for modelling the proliferation of memory B cells. Following experiments on model identifiability with simulated data, it was found that there was limited ability to recover model parameters describing memory B cell proliferation.

Methodology for statistical inference

The model was fitted to longitudinal antibody level measurements from all participants. Mixed effects methods were used to capture the natural variation in antibody kinetics between individual participants, whilst estimating the average value and variance of the immune parameters across the entire population of individuals. The models were fitted in a Bayesian framework using Markov Chain Monte Carlo (MCMC) methods. Mixed effects methods allow individual-level parameters to be estimated for each participant separately, with these individual-level (or mixed effects) parameters being drawn from global distributions.

We consider a scenario where we have data from M studies, each of which has data from Nm individuals. For example, for each participant n in study m the half-life of the short-lived ASCs may be estimated as 𝑑 𝑠 𝑚,𝑛 (an individual-level parameter).

The 𝑁 = ∑ 𝑁 𝑚 𝑀 𝑚=1

estimates of the local parameters 𝑑 𝑠 𝑚,𝑛 will be drawn from a probability distribution. A log-Normal distribution is suitable as it has positive support on [0, ∞). Thus we have 𝑙𝑜𝑔(𝑑 𝑠 𝑚,𝑛 )~𝑁(𝜇 𝑠 , 𝜎 𝑠 2 ). The mean ds and the variance Σ 𝑠 2 of the estimates of 𝑑 𝑠 𝑚,𝑛 are given by 𝑑 𝑠 = 𝑒 𝜇 𝑠 + 𝜎 𝑠 2 2 and Σ 𝑠 2 = (𝑒 𝜎 𝑠 2 -1)𝑒 2𝜇 𝑠 +𝜎 𝑠 2 .

Model likelihood

For individual n from study m we have data on observed antibody levels 𝐴 𝑚,𝑛 = {𝑎 1 , … , 𝑎 𝐽 } at times 𝑇 𝑚,𝑛 = {𝑡 1 , … , 𝑡 𝐽 }.

We denote 𝐷 𝑚,𝑛 = (𝐴 𝑚,𝑛 , 𝑇 𝑚,𝑛 ) to be the vector of data for individual n from study m. For individual n from study m, the 𝐸(𝑇 𝑖𝑛 ) = 𝑀 𝑖𝑛 + 𝐸(𝑠𝑝 𝑖 ) -1 𝐸(𝑠𝑒 𝑖 ) + 𝐸(𝑠𝑝 𝑖 ) -1 with 𝐸(𝑇 𝑖𝑛 ) = 0 if 𝑀 𝑖𝑛 < 1 -𝐸(𝑠𝑒 𝑖 ). Both 𝑀 𝑖 and 𝐸(𝑇 𝑖 ) are summarized as medians with 95% ranges. Define α K as a K-way classification algorithm that can classify a sample into one of K categories. Let C0 be the category for a sample from an individual not previously infected with SARS-CoV-2. Let Ck be the category for individuals previously infected with SARS-CoV-2 in one of K -1 time intervals defined by (0, t1, …, tK-1). α K can be any of a range of algorithms from Random Forests, Support Vector Machines, or ordinal logistic regression. α K can be trained on a data set, with classification performance assessed using N-fold cross-validation. For a serum sample, let Pk denote the classification score (analogous to a probability) that this sample belongs to category k.

Given a vector of classification scores Pk, we can assign a serum sample to a category. A frequently used approach is to select k according to the maximum of Pk. In order to retain better control over diagnostic specificity, we select an alternative two step approach:  if P0 ≥ γ99 then the sample is classified as negative  if P0 < γ99 then the sample is classified into category k according to the maximum of P1, …, PK-1 Here γ99 is selected based on the training data to ensure a target specificity of 99%. The classification performance of this approach can be assessed using a confusion matrix Cij defined by the proportion of samples of category i classified as category j. For the example with K = 4 explored in more detail here, we define three intervals: 0 -3 months; 3 -6 months; 6 -12 months.

Following the approach of the Rogan-Gladen estimator, define M to be the vector of measured prevalences in each of the K categories. An estimator of the true sero-prevalence is provided by E(T) = C -1 * M where C -1 is the inverse of the confusion matrix. Depending on the values of the confusion matrix, this approach may have Tk < 0. In this instance we assign Tk = 0 and transfer the predictions proportionally to the other categories.