Statistical identifiability and sample size calculations for serial seroepidemiology

Abstract

Inference on disease dynamics is typically performed using case reporting time series of symptomatic disease. The inferred dynamics will vary depending on the reporting patterns and surveillance system for the disease in question, and the inference will miss mild or underreported epidemics. To eliminate the variation introduced by differing reporting patterns and to capture asymptomatic or subclinical infection, inferential methods can be applied to serological data sets instead of case reporting data. To reconstruct complete disease dynamics, one would need to collect a serological time series. In the statistical analysis presented here, we consider a particular kind of serological time series with repeated, periodic collections of population-representative serum. We refer to this study design as a serial seroepidemiology (SSE) design, and we base the analysis on our epidemiological knowledge of influenza. We consider a study duration of three to four years, during which a single antigenic type of influenza would be circulating, and we evaluate our ability to reconstruct disease dynamics based on serological data alone. We show that the processes of reinfection, antibody generation, and antibody waning confound each other and are not always statistically identifiable, especially when dynamics resemble a non-oscillating endemic equilibrium behavior. We introduce some constraints to partially resolve this confounding, and we show that transmission rates and basic reproduction numbers can be accurately estimated in SSE study designs. Seasonal forcing is more difficult to identify as serology-based studies only detect oscillations in antibody titers of recovered individuals, and these oscillations are typically weaker than those observed for infected individuals. To accurately estimate the magnitude and timing of seasonal forcing, serum samples should be collected every two months and 200 or more samples should be included in each collection; this sample size estimate is sensitive to the antibody waning rate and the assumed level of seasonal forcing.

Keywords: Antibody waning; Complete disease dynamics; Influenza; Maximum likelihood; Serial seroepidemiology; Seroepidemiology; Statistical identifiability.

Fig. 1

Class diagram for model (1). Population classes are exposed individuals (E), infected individuals (I), recently recovered individuals who have not yet mounted a specific antibody response (P), and recovered individuals with antibody titer i (R_i). Using constraints (2) and (3), hosts inside the dashed box are completely refractory to reinfection.

Fig. 2

(A) Susceptibility reduction (ɛ_i) in the different R-classes. Three values of γ are shown. (B) Five different antibody waning rates explored in this analysis. From left to right the five lines are for c = 0.2, 0.4, 0.53, 0.625, 0.7.

Fig. 3

Difference in log-likelihood values between the true parameter combination Θ₀ (used to generate data sets X_A) and false parameter combinations Θ_j that generate similar dynamics. Panel A shows free parameter sets Θ_j as defined by (9), and panel B shows constrained parameters sets ${\mathrm{\Theta }}_j^{\prime }$ as defined by (10). Each point corresponds to one of 100 false parameter combinations, and a negative number on the ordinate indicates that a particular Θ_j had lower log-likelihood than Θ₀. When the log-likelihood difference is greater than five, no point is plotted; instead the number of points for which the log-likelihood difference is greater than five is indicated above the tick marks on the horizontal axis. The amplitude A is shown on the horizontal axis. Likelihoods are computed given eleven different data sets X_A with different seasonal forcing amplitudes. As seasonal forcing increases, statistical identifiability improves.

Fig. 4

Confidence intervals for parameters β, A, and φ as a function of the sample size N, under a scenario of fast antibody waning (1.8 years to immune loss). Amplitudes of A = 0.2 (top row) and A = 0.1 (bottom row) are used. Gray lines correspond to a three-year data set and black lines to a four-year data set, and the dashed lines indicate the true parameter values. Data are simulated with system (1) and constraints (2)–(4), and inference is performed with the reduced model (5). Parameters are $\gamma =0.6,\mu =2.0,w=50,c=0.2$. Confidence intervals shown are medians over ten simulated data sets, and the square on each line is the mean maximum-likelihood estimate.

Fig. 5

Confidence intervals for parameters β, A, and φ as a function of the sample size N, under a scenario of moderate antibody waning (3.3 years to immune loss). Amplitudes of A = 0.2 (top row) and A = 0.1 (bottom row) are used. Gray lines correspond to a three-year data set and black lines to a four-year data set, and the dashed lines indicate the true parameter values. Data are simulated with system (1) and constraints (2)–(4), and inference is performed with the reduced model (5). Parameters are $\gamma =0.6,\mu =2.0,w=50,c=0.4$. Confidence intervals shown are medians over ten simulated data sets, and the square on each line is the mean maximum-likelihood estimate.

Fig. 6

Effect of antibody waning on our ability to perform inference on seasonality parameters. Plots show equilibrium dynamics (left and middle columns) and likelihood surfaces (right column) for five different rates of antibody waning; the c parameter and time to immune loss are shown on the left. The first column of figures shows the dynamics of I(t) relative to its long-term mean value, and the second column shows the dynamics of ∑R_i(t) relative to its long-term mean value. As the rate of antibody waning slows down, the relative magnitude of oscillations in the recovered classes decreases, making these oscillations more difficult to detect. The right column shows the 95% confidence region for A and φ, with the darker regions having highest likelihood. When antibody waning is slow (c > 0.5), detecting the phase φ may be impossible. Simulations and inference were performed with system (1) with constraints (2)–(4). Parameters used in the simulation were A = 0.1, φ = π, γ = 0.6, μ = 2.0, β = 0.3, and $w=50$. Inference was performed on a 4-year data set with 200 samples collected every two months.