Estimating seroconversion rates accounting for repeated infections by approximate Bayesian computation

Abstract

This study presents a novel approach for inferring the incidence of infections by employing a quantitative model of the serum antibody response. Current methodologies often overlook the cumulative effect of an individual's infection history, making it challenging to obtain a marginal distribution for antibody concentrations. Our proposed approach leverages approximate Bayesian computation to simulate cross-sectional antibody responses and compare these to observed data, factoring in the impact of repeated infections. We then assess the empirical distribution functions of the simulated and observed antibody data utilizing Kolmogorov deviance, thereby incorporating a goodness-of-fit check. This new method not only matches the computational efficiency of preceding likelihood-based analyses but also facilitates the joint estimation of antibody noise parameters. The results affirm that the predictions generated by our within-host model closely align with the observed distributions from cross-sectional samples of a well-characterized population. Our findings mirror those of likelihood-based methodologies in scenarios of low infection pressure, such as the transmission of pertussis in Europe. However, our simulations reveal that in settings of higher infection pressure, likelihood-based approaches tend to underestimate the force of infection. Thus, our novel methodology presents significant advancements in estimating infection incidence, thereby enhancing our understanding of disease dynamics in the field of epidemiology.

Keywords: approximate Bayesian computation; empirical distribution function; reinfection; seroincidence.

Fig. A1:

Predicted seroresponse for the longitudinal model fitted to pertussis antibody data.

Fig. A2:

Posterior predictive density estimates of the peak antibody level and the time to peak, calculated from the estimated parameters $\left({\mu }_0,{\mu }_1,c^{\mathrm{*}}\right)$ and $y_0$.

Fig. A3:

(a) Relation between serum antibody baseline $y_0$ and subsequent peak level $y_1=f\left(y_0\right)$. (b) Distribution of baseline threshold for a subsequent “small jump” in seroresponse.

Fig. A4:

Simulated seroresponse of a hypothetical subject, from birth to age 80 (years), infections occurring as a Poisson process with rate 0.2/yr (a,b) and 1/yr (c,d). Longitudinal parameters fitted to pertussis data (a,c): parameters and baseline chosen at random at each new infection. This corresponds with published analyses. (b,d): parameters $\left({\mu }_0,{\mu }_1,c,\alpha ,r\right)$ and baseline $y_0$ chosen at birth and kept fixed. Baseline antibody level $y_0$ for subsequent infections carried over from the prior episode. Triangles indicate symptomatic (“large jump”: red) or asymptomatic seroconversions (“small jump”: blue).

Fig. B1:

Prior (gray) and posterior (black) densities for simulated cross–sectional data. Estimated seroconversion rate $\lambda$, and the two noise parameters $(\mu ,\sigma )$. (a), (b), (c): ${\lambda }_{\mathrm{s}\mathrm{i}\mathrm{m}}=0.03(1/\mathrm{y}\mathrm{r})$; (d), (e), (f): ${\lambda }_{\mathrm{s}\mathrm{i}\mathrm{m}}=0.013(1/\mathrm{y}\mathrm{r})$. The prior for $\mathrm{l}\mathrm{o}\mathrm{g}(\lambda )$ was $N(\mathrm{l}\mathrm{o}\mathrm{g}(0.1),5.0)$; for $\mu$ this was $N(0.1,0.5)$ and for $\log \left(\sigma \right)N(\mathrm{l}\mathrm{o}\mathrm{g}(0.6),0.5)$.

Fig. B2:

Prior (gray) and posterior (black) densities for cross–sectional data from the Netherlands (ESEN study). Estimated seroconversion rate $\lambda$, and the two noise parameters $(\mu ,\sigma )$. (a), (b), (c): ages 5 – 10yr; (d), (e), (f): ages 55 – 60 yr. The prior for $\mathrm{l}\mathrm{o}\mathrm{g}(\lambda )$ was $N(\mathrm{l}\mathrm{o}\mathrm{g}(1.0),2.0)$; for $\mu$ this was $N(1.0,1.0)$ and for $\log \left(\sigma \right)N(\mathrm{l}\mathrm{o}\mathrm{g}(1.0),1.0)$.

Fig. B3:

Densities for population samples of data of the Netherlands from the ESEN study, and densities from samples fitted by matching EDFs.

Fig. B4:

ESEN data, (a) Estimated seroconversion rates ${\lambda }_{\text{est}}$ by (5 yr) age categories. Two fitting methods are shown. ML: maximum likelihood seroincidence with B–noise fixed $\nu =3.0\mathrm{I}\mathrm{U}/\mathrm{m}\mathrm{l}$. ML adj: maximum likelihood seroincidence with B–noise adjusted to the 95th percentile of the distribution estimated by EDF. KS: EDF method simulating vaccination at age 2, using $p_{\mathrm{K}\mathrm{S}}$, fitted by ABC jointly estimating $\lambda$ and the two noise parameters. (b) Estimated log mean $\mu$ of B–noise. (c) Estimated $\log \mathrm{s}\mathrm{d}\sigma$ of B–noise.

Fig. B5:

ESEN data, (a) Estimated seroconversion rates ${\lambda }_{\text{est}}$ by (5 yr) age categories. Two fitting methods are shown. ML: maximum likelihood seroincidence with B–noise fixed $\nu =3.0\mathrm{I}\mathrm{U}/\mathrm{m}\mathrm{l}$. ML adj: maximum likelihood seroincidence with B–noise adjusted to the 95th percentile of the distribution estimated by EDF. KS: EDF method simulating vaccination at age 2, using $p_{\mathrm{K}\mathrm{S}}$, fitted by ABC jointly estimating $\lambda$ and the two noise parameters. (b) Estimated log mean $\mu$ of B–noise. (c) Estimated $\log \mathrm{s}\mathrm{d}\sigma$ of B–noise.

Fig. B6:

Kolmogorov probabilities for EDFs of antibody samples generated from posterior parameter estimates in approximate Bayesian computation. Simulated cross–sectional data without (a) and with (b) vaccination at age 2. Observed cross–sectional population data for the Netherlands from the ESEN study for pertussis, analysed in 10 yr age categories, results in Figures 9 and B5.

Fig. B7:

(a) Variation in $\lambda$. A subject sampled at age 10 was exposed to an outbreak 5 years ago, causing $\lambda$ to increase from a baseline 0.05 (1/yr) to a peak infection rate of 10 (1/yr). Duration of the outbreak was 465 days, approximately. (b) sample of intervals for this nonhomogeneous Poisson process $(\mathrm{n}=25)$.

Fig. B8:

Estimated ${\lambda }_{\text{est}}$ and B–noise parameters $\left({\mu }_{\text{est}},{\sigma }_{\text{est}}\right)$ for an outbreak 2, 5, 10, 20 and 50 years ago $(\mathrm{\Delta })$ at time of sampling, for subjects aged 0 – 80 years (a – d) or 0 – 10 years (e–h) (uniform age distributions). Baseline infection rate ${\lambda }_0=0.05(1/\mathrm{y}\mathrm{r})$, during the outbreak a peak rate of ${\lambda }_1=10(1/\mathrm{y}\mathrm{r})$ is reached. Simulated B–noise distribution parameters: $\left({\mu }_{\mathrm{s}\mathrm{i}\mathrm{m}},{\sigma }_{\mathrm{s}\mathrm{i}\mathrm{m}}\right)=(0.1,0.6)$. ML: maximum likelihood estimation; KS: EDF based method. Note how recent changes in $\lambda$ cause decreased $p_{\mathrm{K}\mathrm{S}}(\mathrm{d},\mathrm{h})$.

Figure 1:

Simulated seroresponse of a hypothetical subject, from birth to age 80 (years), infections occurring as a Poisson process with rate 0.2/yr. Longitudinal parameters fitted to pertussis data: $(\left.{\mu }_0,{\mu }_1,c,\alpha ,r\right)$ chosen at birth and kept fixed. The baseline antibody level $y_0$ is low at birth. After the first infection $y_0$ is carried over from each prior episode for any further infections. Triangles indicate symptomatic (“large jump”: red) or asymptomatic seroconversions (“small jump”: blue).

Figure 2:

Fitting $\lambda$: output for simulated data for subjects 0–80 yrs of age with baseline distribution parameters: (0.1, 0.6) and seroconversion rate ${\lambda }_{\text{sim}}=0.001$ and 0.05 (1/yr). (a) Probability density of (simulated) observed serum antibody distribution and best fitting (minimum $D_{\mathrm{K}\mathrm{S}}$ or $D_{\mathrm{K}\mathrm{L}}$) simulated distributions. (b) Scaled deviates as a function ${\lambda }_{\text{est}}$ of the simulated sample of antibody concentrations. (c) and (d): corresponding KS dev (Kolmogorov–Smirnov deviate $D_{\mathrm{K}\mathrm{S}}$); $\mathrm{K}\mathrm{L}$ div (Kullback–Leibler divergence $D_{\mathrm{K}\mathrm{L}}$); AD dev (Anderson–Darling deviate $A^2$); KS prob (Kolmogorov probability $\left.p_{\mathrm{K}\mathrm{S}}\right)$ as a function of $\lambda$.

Figure 3:

Fitting B–noise distribution parameters: output for simulated data for subjects 0 – 80yrs of age with baseline distribution parameters: (0.1, 0.6) and seroconversion rate ${\lambda }_{\mathrm{s}\mathrm{i}\mathrm{m}}=0.001$ and 0.05 (1/yr). (a) Scaled deviates as a function of the B–noise log mean ${\mu }_{\mathrm{e}\mathrm{s}\mathrm{t}}$ at ${\lambda }_{\mathrm{s}\mathrm{i}\mathrm{m}}=0.001(1/\mathrm{y}\mathrm{r})$. (b) Scaled deviates as a function of the B–noise $\mathrm{l}\mathrm{o}\mathrm{g}\mathrm{s}\mathrm{d}{\sigma }_{\text{est}}$ at ${\lambda }_{\text{sim}}=0.001(1/\mathrm{y}\mathrm{r})$. (c) and (d) Same at ${\lambda }_{\text{sim}}=0.05(1/\mathrm{y}\mathrm{r})$. KS dev: Kolmogorov–Smirnov deviate $D_{\mathrm{K}\mathrm{S}}$; $\mathrm{K}\mathrm{L}$ div: Kullback–Leibler divergence $D_{\mathrm{K}\mathrm{L}}$; AD dev: Anderson–Darling deviate $A^2$; KS prob: Kolmogorov probability $p_{\mathrm{K}\mathrm{S}}$.

Figure 4:

Estimated ${\lambda }_{\text{est}}$ and B–noise parameters $\left({\mu }_{\text{est}},{\sigma }_{\text{est}}\right)$ for a range of simulated ${\lambda }_{\text{sim}}$ ranging from 0.001 to 10 (1/yr) in subjects 0 – 80yrs of age and B–noise distribution parameters: $\left({\mu }_{\text{sim}},{\sigma }_{\text{sim}}\right)=(0.1,0.6)$. Baseline $y_0=y\left(t_{\text{inf}}\right)$ from previous infections, generating seroresponses as in Figure 1. (a) Two methods for estimation of $\lambda$ are compared: ML: maximum likelihood using the published seroincidence method with fixed B–noise parameter adjusted to the 95 th percentile of simulated noise $(\nu =2.97\mathrm{I}\mathrm{U}/\mathrm{m}\mathrm{l})$, and $\mathrm{K}\mathrm{S}$ : EDF based method using $p_{\mathrm{K}\mathrm{S}}$ to jointly estimate $(\lambda ,\mu ,\sigma )$. (b) B–noise log mean ${\mu }_{\text{est}}$ and (c) $\mathrm{l}\mathrm{o}\mathrm{g}\mathrm{s}\mathrm{d}{\sigma }_{\text{est}}$, estimated jointly with ${\lambda }_{\text{est}}$ using ABC. Dashed lines indicate simulated values: ${\lambda }_{\text{est}}={\lambda }_{\text{sim}},$ ${\mu }_{\text{est}}={\mu }_{\text{sim}}=0.1,$ ${\sigma }_{\text{est}}={\sigma }_{\text{sim}}=0.6$.

Figure 5:

Bias in ${\lambda }_{\text{est}}$ due to baseline “memory”. (a) Simulated cross–sectional sample generated with $y_0=y\left(t_{\text{inf}}\right)$; ${\lambda }_{\text{est}}$ calculated by ML and EDF fitting (KS) with fixed baseline $\left(y_0=y(0)\right)$ and instantaneous seroconversion $\left(t_1=0\right)$. (b) Simulated cross–sectional sample generated with fixed $y_0=y(0)$ and instantaneous seroconversion $\left(t_1=0\right);{\lambda }_{\text{est}}$ calculated by ML and EDF fitting (KS) with infection history $\left(y_0=y\left(t_{\text{inf}}\right)\right)$ and seroconversion $\left(t_1>0\right)$.

Figure 6:

Estimated ${\lambda }_{\text{est}}$ and B–noise parameters $\left({\mu }_{\text{est}},{\sigma }_{\text{est}}\right)$ for a simulated population vaccinated at age 2. a–c: ${\lambda }_{\text{est}}$ and B–noise parameters $\left({\mu }_{\text{est}},{\sigma }_{\text{est}}\right)$ estimated using a model including vaccination (KS). d–f: same parameters estimated using a model without vaccination at age 2 (KS). For comparison, likelihood based estimated are also shown (ML).

Figure 7:

ESEN data from the Netherlands: estimates of $\lambda$ and probability density of antibody levels. Age 35–40 yr (a): $D_{\mathrm{K}\mathrm{S}}$ (KS dev) and associated probability $p_{\mathrm{K}\mathrm{S}}$ (KS prob) as a function of ${\lambda }_{\text{est}}$. Also shown $D_{\mathrm{K}\mathrm{L}}$ ($\mathrm{K}\mathrm{L}$ div) and Anderson–Darling $A^2$ (AD dev). (b): probability densities of observed antibody levels and minimum $D_{\mathrm{K}\mathrm{S}}$ (maximum $p_{\mathrm{K}\mathrm{S}}$) sample, and minimum $D_{\mathrm{K}\mathrm{L}}$ sample.

Figure 8:

ESEN data, (a) Estimated seroconversion rates ${\lambda }_{\text{est}}$ by (5yr) age categories. Two fitting methods are shown. ML: maximum likelihood seroincidence with B–noise fixed $\nu =3.0\mathrm{I}\mathrm{U}/\mathrm{m}\mathrm{l}$. ML adj: maximum likelihood seroincidence with B–noise adjusted to the 95th percentile of the distribution estimated by EDF. KS: EDF method using $p_{\mathrm{K}\mathrm{S}}$, fitted by ABC jointly estimating $\lambda$ and the two noise parameters. (b) Estimated log mean $\mu$ of B–noise. (c) Estimated $\log \mathrm{s}\mathrm{d}\sigma$ of B–noise.

Figure 9:

ESEN data, (a) Estimated seroconversion rates ${\lambda }_{\text{est}}$ by (5 yr) age categories. Two fitting methods are shown. ML: maximum likelihood seroincidence with B–noise fixed $\nu =3.0\mathrm{I}\mathrm{U}/\mathrm{m}\mathrm{l}$. ML adj: maximum likelihood seroincidence with B–noise adjusted to the 95th percentile of the distribution estimated by EDF. KS: EDF method using $p_{\mathrm{K}\mathrm{S}}$, fitted by ABC jointly estimating $\lambda$ and the two noise parameters. (b) Estimated log mean $\mu$ of B–noise. (c) Estimated $\log \mathrm{s}\mathrm{d}\sigma$ of B–noise.