Estimating dengue transmission intensity from serological data: A comparative analysis using mixture and catalytic models.

BACKGROUND: Dengue virus (DENV) infection is a global health concern of increasing magnitude. To target intervention strategies, accurate estimates of the force of infection (FOI) are necessary. Catalytic models have been widely used to estimate DENV FOI and rely on a binary classification of serostatus as seropositive or seronegative, according to pre-defined antibody thresholds. Previous work has demonstrated the use of thresholds can cause serostatus misclassification and biased estimates. In contrast, mixture models do not rely on thresholds and use the full distribution of antibody titres. To date, there has been limited application of mixture models to estimate DENV FOI.
METHODS: We compare the application of mixture models and time-constant and time-varying catalytic models to simulated data and to serological data collected in Vietnam from 2004 to 2009 (N ≥ 2178) and Indonesia in 2014 (N = 3194).
RESULTS: The simulation study showed larger mean FOI estimate bias from the time-constant and time-varying catalytic models (-0.007 (95% Confidence Interval (CI): -0.069, 0.029) and -0.006 (95% CI -0.095, 0.043)) than from the mixture model (0.001 (95% CI -0.036, 0.065)). Coverage of the true FOI was > 95% for estimates from both the time-varying catalytic and mixture model, however the latter had reduced uncertainty. When applied to real data from Vietnam, the mixture model frequently produced higher FOI and seroprevalence estimates than the catalytic models.
CONCLUSIONS: Our results suggest mixture models represent valid, potentially less biased, alternatives to catalytic models, which could be particularly useful when estimating FOI from data with largely overlapping antibody titre distributions.

Introduction

Dengue fever is caused by infection with one or more of four antigenically distinct serotypes of dengue virus (DENV1-4), a Flavivirus carried by Aedes mosquitoes [1,2]. DENV infects approximately 105 million people each year [3], primarily in tropical and sub-tropical regions. The geographical range of DENV is increasing [1,4,5] and it is expected that the spread of dengue will be influenced by rising global temperatures and increasing urbanisation [1,6]. Intervention measures to date rely essentially on vector control due to the absence of antiviral treatment, challenges in the use of the first licensed dengue vaccine for widespread dengue prevention and control [7], as well as in the use of rapid diagnostic tests for screening [8]. The current and expected future burden of dengue on health-systems is therefore high, demonstrating a continuing need for increased understanding of DENV transmission.

Estimating epidemiological parameters such as the force of infection (FOI, the per capita rate at which a susceptible person is infected) and population seroprevalence (the proportion of people in a population exposed to a virus, as determined by the detection of antibodies in the blood) allow us to gain insights into the subsets of populations most at risk of infection and disease [9], to assess the predicted impact of an intervention strategy [10], and to inform public health policy [11,12].

Both the FOI and seroprevalence can be estimated using mathematical models calibrated to age-stratified serological data measuring IgG antibody levels (also called titres) from blood samples. IgG titres are obtained using Enzyme-Linked Immunosorbent Assays (ELISAs) and are often classified into qualitative, binary test results (seropositive or seronegative) based on the manufacturer’s threshold.

Catalytic models, first proposed in 1934, estimate disease FOI from age-stratified serological or case notification data [13]. In these models, large rates of increase in seroprevalence between individuals who are age a versus age a+1 are explained by high age-specific FOI (assuming the FOI is constant in time) or high time-specific FOI experienced by individuals of all ages during the period a to a+1 years ago [14]. Catalytic models have been used extensively for measles [15], rubella [16], Hepatitis A [17], Chagas disease [18], and DENV [12,14,19-21]. Whilst commonly used, previous work suggests that catalytic models risk generating biased estimates due to data-loss and/or misclassification [22-24]. For example, samples with titres greater than the seronegative threshold but lower than the seropositive threshold are classified as ‘equivocal’ and discarded from the analysis. Furthermore, titre levels of seropositive individuals in a given population may be affected by factors including host response, the degree of exposure to the pathogen and infection timing, which could lead to misclassification.

Mixture models are flexible statistical models that can be applied to continuous data from different clusters or populations, called components. Mixture models can therefore be applied to the absolute antibody titre values in serology datasets, rather than to the counts of titres in each of two classes (seropositive/seronegative) as is necessary for catalytic models [22]. The components’ distributions and their defining parameters (e.g., the mean titre of each component distribution) are inferred from a fitted mixture model which is used to estimate the FOI and population seroprevalence [22,25]. To date, mixture models have been applied to serological data to estimate the seroprevalence of infectious diseases such as parvovirus B19 and rubella in England [26,27], human papillomavirus in the Netherlands [23], measles in Italy [28], and a selection of arboviruses inlcuding DENV in Zambia [29]. In addition, mixture models have been used to develop frameworks capable of distinguishing between primary and post-primary DENV infections [30,31], and recent and historical influenza A infections [32]. Recently, DENV FOI was estimated using catalytic and mixture models applied to serological data collected in three locations in Vietnam (N > = 266) and in Chennai, India (N = 799) [31]. In this study, the estimates from mixture models were deemed more robust than those from binary catalytic models [31].

Here, we implement a simulation study to assess the ability of mixture and catalytic models to reconstruct the FOI value used for simulating the data. Furthermore, we add to the growing body of evidence exploring the use of mixture models by presenting a comparitative analysis of the DENV trasmission intensity estimates obtained from mixture and catalytic models applied to age-stratified serological datasets from Vietnam (N ≥ 2178, for years 2004–2009) and Indonesia (N = 3194, for 2014).

Methods

Ethics statement

Ethical approval for the secondary analysis of the age-stratified seroprevalence datasets was granted by the Imperial College Research Ethics Committee (Approval Reference 21IC7066).

Data

Age-stratified seroprevalence data

DENV IgG data were collected in Long Xuyen, Vietnam, during a prospective epidemiological study that was conducted to assess the suitability of the area for future CYD-TDV vaccine efficacy trials, as described previously [33]. Samples were collected from children under 11 years old in 2004 and then from children under 15 years old during September to February in 2004–2005, 2005–2006, 2006–2007, 2007–2008 and 2008–2009 (Datasets A-1 to A-6, Table 1). The titres were measured using in-house ELISA assays (Arbovirus Laboratory of Pasteur Institute, Ho Chi Minh City).

Table 1

Description of the datasets used in the analyses.

Summary statistics including notation, region, the assay used, the year of testing, the age range of the children participating to the study and the sample sizes.

Dataset	Region	Assay type	Year	Age Range	Sample Size
A-1	Long Xuyen, Vietnam	IgG ELISA	2004	3–10	2,178
A-2	Long Xuyen, Vietnam	IgG ELISA	2004–2005	3–13	3,681
A-3	Long Xuyen, Vietnam	IgG ELISA	2005–2006	3–14	3,727
A-4	Long Xuyen, Vietnam	IgG ELISA	2006–2007	3–14	3,651
A-5	Long Xuyen, Vietnam	IgG ELISA	2007–2008	4–14	2,959
A-6	Long Xuyen, Vietnam	IgG ELISA	2008–2009	5–14	2,249
B	30 urban subdistricts, Indonesia	IgG ELISA	2014	1–18	3,194
C	Simulated data	Simulated	N/A	1–18	3,194

DENV IgG data from 30 urban subdistricts in Jakarta, Indonesia were collected from 3,194 children under 18 years old as part of a cross-sectional seroprevalence survey in 2014 [34] (Dataset B). Given the small spatial scale of the range of data collection, we did not account for spatial differences when modelling. IgG titres were measured using the commercial Panbio Dengue IgG Indirect ELISA kit.

Simulated datasets

We simulated 540 antibody titre datasets (Dataset C), with the same age-distribution and sample size as the Indonesian seroprevalence survey data (Dataset B). For each simulation the distributions used for sampling seronegative and seropositive log(titres + 1) were selected from a normal, gamma or Weibull distribution. This gave 9 possible distribution pairs for seronegative and seropositive log(titre + 1) values, and we generated an equal number of simulations (N = 60) for each combination. Normal, gamma and Weibull distributions were chosen based on preliminary work on our antibody titre datasets showing that these mistures were most frequently selected among a wider set of distributions. Parameter values were randomly drawn from uniform distributions with limits as shown in S1 Table. The serostatus of each individual was drawn from a Bernoulli distribution with probability 1−e−λa, where a is the age of the individual and λ is the FOI (which is assumed to be constant with age and time), and therefore λa represents the cumulative FOI experienced by individuals over their lifetime. Log(titre + 1) values for each individual were subsequently randomly drawn from the respective component distributions. The analysis was conducted in the statistical programming language R [35].

Catalytic model

Data preparation

Catalytic models rely on data that are binarily classified as seropositive or seronegative. For Datasets A-1 to A-6, a background/control titre (t) was measured for each assay. An individual titre was classified as seronegative if ≤t and seropositive otherwise. For Dataset B, samples with titres ≤ 9 PanBio units were classified as seronegative and ≥11 as seropositive. Titres >9 and <11 were discarded (28 out of 3,194 samples). For simulated Dataset C, titres were classified as seronegative if they were ≤X and seropositive if they were ≥Y. X and Y are thresholds that were optimised using the ‘true’ simulated serostatuses: the optim function in R, using the Nelder Mead algorithm, was used to calulate the X and Y values per simulated dataset resulting in the fewest titre misclassifications. The optimisation process occassionally failed to estimate realistic classification thresholds (X < 25% quantile of seronegative titres or Y > 75% quantile of seropositive titres) and these simulations were excluded from analysis (N = 31). For the remaining 509 simulations, titres >X and

Parameter estimation

We used a catalytic model as previously described [14,36]. The yearly FOI, i.e., the per capita rate of infection experienced by individuals in a given year X−i, where X is the year the serosurvey was conducted, is assumed to be either constant in time (constant across the years) or time-varying (piecewise constant across the years). When we assumed a time-varying FOI, the number of FOI estimates is equal to the number of single year age groups available in the datasets (maximum age group A –minimum age group M). The proportion of seropositive individuals in age group a during year X (π), was estimated as in Eq 1. Here, the yearly FOI (λ) is summed over the lifetime of the individuals in age group a to give the cumulative FOI experienced by the individuals in this age group. If the minimum age group M does not equal 1, then we estimated an average FOI for the M years without age-specific data (X−M to X) denoted λ. When we assume a time-constant FOI over the whole study period, Eq 1 can be expressed as shown in Eq 2: A binomial log-likelihood was assumed for the FOI (Eq 3), where N is the total number of individuals and P is the number of seropositive individuals in age group a during year X [19]. The optim function in R was used to find the maximum likelihood estimate of the FOI using Eq 3. When we assumed a time-varying FOI, the λ values were averaged to produce a mean FOI experienced over the years in the study period (A−M) (Eq 4) which was compared to the FOI estimated by the time-constant catalytic model and the mixture model. We estimated 95% Confidence Intervals (CI) using a bootstrap method, where the titre data were sampled with replacement and the age-stratified proportion of seropositive indivuals was calculated, 500 times. The 95% CI was given by the 2.5% to 97.5% quantiles of the estimates from the catalytic models applied to the bootstrap samples.

Mixture model

Applying the mixture models to the titre distributions

Mixture models were applied to the bimodal distribution of individual antibody titres as described in Bollaerts et al., 2012 and Hens et al., 2012 [22, 25]. All individual antibody titre measurements were used in each dataset, which differs from the data used for the catalytic model where equivocal titres were discarded and titre measurements were classified as either seropositive or seronegative. The mixture model defines the distribution (z) of the log(titres + 1) as a mixture of two distinct distributions: one for susceptible individuals (seronegative, z) and one for individuals who have been previously infected (seropositive, z). The two-component mixture model is represented by: where f and f represent the probability density function of the seronegative and seropositive components, respectively, and where μ and σ represent the mean and standard deviation of each component, and π represents the age-specific seroprevalence during year X, when the serosurvey was conducted.

The mixdist R package was used to fit the mixture models to the titre data by maximum likelihood using an Expectation Maximisation (EM) algorithm [37]. The package was adapted to allow fitting of different distributions for the seronegative and seropositive titre components: normal, gamma and Weibull distributions, giving 9 possible combinations. The best fitting mixture was chosen using the Akaike Information Criterion (AIC). For Dataset C, the estimated means (μ and μ) and standard deviations (σ and σ) for the seronegative and seropositive components of the best mixture were compared against the true parameter values used for simulating the data. We explored multiple parameterisations, including fixing the standard deviation of the two mixture components. For the Vietnamese datasets, we optimised the model having constrained the standard deviation of the seropositive component to multiple different values (for Dataset A-4 we set σ equal to all values from 0.02 to 0.08 in steps of 0.01, for the other five Datasets we set σ equal to all values from 0.05, to 0.15 in steps of 0.01). For the Indonesian dataset (Dataset B) the standard deviations of both components were constrained (σ was set equal to 0.10 to 0.15 in steps of 0.001, and σ was set equal to 0.15 to 0.3 in steps of 0.05).

The relationship between the age-dependent mean log(titre + 1) (μ), the age-specific seroprevalence (π) and the means of the mixture components (μ and μ) is described in Eq 6. We estimated μ by least-squares regression using a monotonically increasing P spline [22,25,38] using the mpspline.fit function from the serostat R package [39]. Equally spaced cubic polynomial segments (degree = 3) made up the spline. The optimal smoothing parameter (α) and number of segments (knots) were determined using the Bayesian Information Criterion, having explored combinations of α values (set equal to 0.001, 0.01, 0.1, 0.5, 1, 5, 10, 50, 100, 500) and knots (set equal to values in the sequence: 5 to the maximum number of x-axis age categories, step size = 1). The seroprevalence was calculated using Eq 7.

The time-varying FOI was derived from the age-specific seroprevalence as described in Eq 8 [22], where the rate of change in the seroprevalence between two sequential age groups (a−1 and a) is divided by the proportion of seronegative individuals in age group a, to give the FOI experienced in the year X−a (λ). Eq 8 can in turn be expressed as a function of the underlying antibody titre distribution as shown in Eq 9, where μ′ represents the derivative of the age-specific mean log(titre + 1). The μ′ terms were calculated by taking the gradient of the fitted μ spline at each age group a. The time-varying FOI can be averaged across the years in the study period to give the total FOI λ (Eq 4).

The 95% CI around the FOI and seroprevalence estimates were calculated using a boostrap method, where the titre data were sampled with replacement 5000 times. The 95% CI were given by the 2.5% to 97.5% quantiles of the estimates from the bootstrap samples.

Bias in the mixture and catalytic model estimates of FOI and seroprevalence for Dataset C was calculated as the estimated value minus the true simulated value of the parameter. Uncertainty was calculated as the width of the 95% CIs around the parameter estimates. Coverage was calculated as the percentage of simulations where the estimated 95% CIs contained the true parameter value. Code for the simulation study analysis is available at: https://github.com/Tori-Cox/Mixture-catalytic-models.

Results

Simulated data

The mixture model identified the correct distributions used to simulate both seropositive and seronegative titres in 76.1% (411/540) of simulations, one of the two distributions in 22.2% (120/540) of simulations and did not correctly identify either distribution in 1.7% (9/540) of the simulations. Whether the distributions were gamma, normal or Weibull did not influence the ability of the mixture model to correctly identify them (S2 Table). The estimated 95% CIs contained the true parameter values used to simulate the data in 88.1% (476/540), 86.9% (469/540), 86.5% (467/540) and 89.4% (483/540) of simulations for μ, μ, σ and σ, respectively (S1A Fig). Simulations where the seronegative titre distribution was Weibull distributed were over-represented in the simulations which produced outlying estimates of μ, μ, σ and σ (S1B Fig).

The mixture model coverage for the FOI was 95% of the total simulations (513/540) and 95% (485/509) of the simulations included in the catalytic model analysis, and for the seroprevalence was 88% (475/540) and 89% (451/509) respectively. The time-varying catalytic model coverage for the FOI was 96.7% (492/509) and for the seroprevalence was 78.8% (401/509). The time-constant catalytic model coverage for the FOI and seroprevalence was 38.9% (198/509) and 55.0% (280/509). It should be noted that the time-varying catalytic model produced wider CIs compared to when assuming a time-constant FOI or when using a mixture model (Fig 1). Average bias in the FOI estimates (0.001 (95% CI -0.036, 0.065), -0.007 (95% CI -0.069, 0.029) and -0.007 (95% CI -0.095, 0.043) for the mixture, time-constant and time-varying catalytic models, respectively) and the seropreavelance estimates (-0.003 (95% CI -0.144, 0.108), -0.007 (95% CI -0.244, 0.087) and -0.005 (95% CI -0.241, 0.100)) was smaller for the mixture model estimates (Fig 1). The increased negative bias in the catalytic model estimates compared to the mixture model estimates demonstrates that the catalytic models are more prone to underestimation of FOI and seroprevalence (Figs 1 and 2). High antibody titre misclassification error rates were positively associated with increased bias in the parameter estimates from the catalytic models (S3 Fig). As expected, model performance was improved when we fitted the catalytic models to the simulated ‘true serostatus’ (i.e., without classifying the titres using optimised thresholds): the coverage of the FOI was 99% (536/540) (95% CI: 98%, 100%) and 42% (228/540) (95% CI: 38%, 47%) for the time-varying and time-constant FOI catalytic models respectively, and the coverage of the seroprevalence was 100% for both models. The average bias in the FOI estimates was 0.007 (95% CI -0.020, 0.056) and -0.005 (95% CI -0.016, -0.477), and in the seroprevalence estimates was 0.000 (95% CI -0.001, 0.001) and 0.002 (95% CI -0.001, 0.001) for the time-varying and time-constant FOI catalytic models respectively.

Fig 1

Bias, coverage, and degree of uncertainty for seroprevalence and force of infection (FOI) estimates using catalytic and mixture models fitted to simulated datasets (Dataset C).

Bias is calculated as the estimated parameter value minus the true parameter value. Uncertainty is the width of the 95% Confidence intervals (CIs) around the central estimates, calculated using the bootstrap method. The coverage is the percentage of simulations where the estimated CIs contained the true values. The dashed line at 95% shows the threshold for the ideal coverage. For the bias and the uncertainty, the mean and 95% CI across the 509 simulations are given. For the coverage, the 95% exact binomial CI are given.

Fig 2

True versus estimated seroprevalence and force of infection (FOI) values from the mixture and catalytic models fitted to the simulated datasets (Dataset C).

The catalytic model was run under the assumption that the FOI was time-constant or time-varying. The 95% Confidence Intervals for the estimated values were calculated using a bootstrap method and are shown here as error bars; the point denotes the central estimate. The Pearson’s correlation coefficients (R) are shown. The dashed line represents the line y = x and shows where points would be located in a scenario with zero bias in the estimated values.

Long Xuyen, Vietnam data

When we applied the mixture model (Fig 3) to the data from Long Xuyen, Vietnam, the total population-level seroprevalence estimates ranged from 0.163 (95% CI 0.138–0.188) in 2004 to 0.376 (95% CI 0.249–0.403) in 2005–2006. The seroprevalence estimates from the time-constant and time-varying catalytic models were consistent with each other, with the latter ranging from 0.189 (95% CI 0.163–0.217) in 2006–2007 to 0.299 (95% CI 0.262–0.337) in 2008–2009. The seroprevalence estimates from all three models were consistent (as determined by the 95% CIs) for 4 out of 6 datasets (Fig 4, S3 Table). The general trend in the age-specific seroprevalence estimates, specifically for Datasets A-2:A-5, differed significantly between the mixture model and the catalytic models, with the mixture model estimating higher seroprevalence at the older ages (Fig 5).

Fig 3

Mixture model fitted to the Vietnamese (A1:A6) and Indonesian (B) datasets.

The distribution of log(titre+1) is shown in dark grey, the fitted mixture model is shown in blue, and the red dashed lines represent the mean antibody titre of each component of the fitted mixture model (μ and μ for the seronegative and seropositive components respectively). Note that the y-axis limits differ for each panel.

Fig 4

Force of infection (FOI) and total population level seroprevalence (SP) estimates from the mixture model and the catalytic models fitted to the observed data.

The catalytic model was run under the time-constant and time-varying FOI assumption. The 95% Confidence Intervals (CI) which were calculated by bootstrapping for all models are given as error bars. Note that the y-axis limits differ for each panel.

Fig 5

Age-specific seroprevalence estimates for the IgG data from Vietnam (Dataset A1:A6) and from Indonesia (Dataset B).

Mixture model estimates are in orange, catalytic model estimates are in green and blue when applied under the assumption that the FOI is time-constant or time-varying respectively. Shading represents the 95% Confidence Intervals (CI). The grey points show the observed seroprevalence per age group calculated from the binarily classified IgG data (seropositive individuals / tested individuals), with error bars indicating the 95% exact binomial CIs. The seroprevalence data and model estimates are overlayed for the purpose of comparison. However, it is important to note that the mixture model was not fitted to the data (grey points), as the former does not depend on the titre classification. The size of the grey data points represents the number of individuals tested in each age group.

The average FOI estimated by the mixture model ranged from 0.026 (95% CI 0.019–0.033) for the period 1993 to 2004, to 0.099 (95% CI 0.077–0.124) for 1990 to 2005. The average FOI estimated by the time-varying catalytic model ranged from 0.024 (95% CI 0.007–0.058) for the period 1991 to 2007, to 0.050 (95% CI 0.001–0.118) for 1990 to 2005. The FOI estimates from the mixture model versus the time-varying catalytic model were consistent for 6 out of 6 datasets, and versus the time-constant catalytic model they were consistent for 3 out of 6 datasets (Fig 4, S3 Table). There is a higher degree of uncertainty around the estimates from the catalytic model when assuming a time-varying FOI compared to the time-constant FOI assumption (Fig 4). We observe greater differences in the estimates from each model when comparing the year-specific FOI as opposed to the averaged total FOI (S4 Fig).

Indonesian data

The mixture and catalytic models fitted to the Indonesian data produced consistent FOI, total population seroprevalence and age-specific seroprevalence estimates. The FOI for the period 1996 to 2014 was estimated at 0.154 (95% CI: 0.106–0.213), 0.143 (95% CI 0.136–0.150) and 0.164 (95% CI 0.022–0.814), and the seroprevalence in 2014 was estimated at 0.718 (95% CI 0.694–0.741), 0.700 (95% CI 0.686–0.714) and 0.700 (95% CI 0.655–0.743) by the mixture model and the time-constant and time-varying catalytic models, respectively (Fig 4, S3 Table).

Discussion

In this analysis, we explored the accuracy and bias of FOI and seroprevalence estimates obtained from mixture and catalytic models applied to serological data. The catalytic models were applied assuming a time-constant or time-varying FOI. We performed a simulation study to compare the performance of each model with known parameter values used to generate the simulated data, and we observed significantly greater accuracy in FOI and seroprevalence estimates from the mixture and time-varying catalytic models than time-constant catalytic models. We observed reduced bias and uncertainty in estimates from the mixture compared to the time-varying catalytic model.

In our simulation study, larger bias in the catalytic model estimates of FOI and seroprevalence (Figs 1 and 2), was associated with increased serostatus misclassification (S3 Fig). Serostatus misclassification occurred more often in simulations where the difference between the mean log(titre + 1) for the susceptible/seronegative component and the mean log(titre + 1) for the infected/seropositive component was lower (S2 Fig), indicating greater overlap between the distributions of the two components. Our results are consistent with previous work which showed greater bias in seroprevalence estimates using methods which employ cut-off thresholds to classify simulated antibody data as opposed to mixture models, when there was high overlap in the underlying components [24].

Differences in the degree of overlap between components in real serological datasets are likely impacted by many factors, including differences in the ELISA tests used to measure antibody titres, the age groups sampled and the underlying age structure of the population as well as the transmission setting and spatiotemporal heterogeneities in the risk of infection at the local scale. In datasets where there is clear separation in the bimodal distribution of antibody titres, catalytic and mixture models are expected to produce more similar estimates of FOI and seroprevalence as fewer samples are misclassified during the binary classification of the data needed to calibrate catalytic models [22,24]. This is consistent with the results from our simulation study and with the reduced variability we observe in our FOI and seroprevalence estimates from each model when they were applied to serological data from Indonesia compared to Vietnam, where the former had higher separation of titre distributions (Figs 3 and 4, S3 Table).

The estimates for Indonesia from each model were consistent with each other and with previously published FOI estimates from catalytic models fitted to case-notification data from 2008–2017 in Jakarta, Indonesia (0.130, 95% CI: 0.129–0.131) [12], and seroprevalence estimates from time-constant catalytic models applied to the same serology dataset (Dataset B) [34,40]. Our results show that the mixture and catalytic models do not significantly differ in their FOI and seroprevalence estimates in this setting. In contrast, the mixture model applied to the six datasets from Vietnam produced more variable estimates (FOI range = 0.026–0.099, seroprevalence range = 0.16.3–0.376) than the catalytic models (FOI range = 0.023–0.037 and 0.024–0.050, seroprevalence range = 0.190–0.300 and 0.189–0.299 under the assumption of a time-constant or time-varying FOI respectively). The variance was even greater in the age-specific seroprevalence and yearly FOI estimates (Figs 5 and S4). As expected, the time-varying catalytic model and the mixture model (which implicitly models FOI as time-varying), were better able to capture the age-specific seroprevalence than the time-constant catalytic model. The estimates from the mixture model tended to exceed those obtained from the catalytic models (Fig 4, S3 Table). Given the greater negative bias observed for the catalytic models in our simulation study, we expect the higher mixture model estimates to be more accurate for the Vietnamese setting. Lam et al., similarly observed higher FOI estimates when applying mixture models compared to catalytic models to serological data from Vietnam, for example 0.12 (95% CI: 0.11–0.14) compared to 0.07 (95% CI: 0.06–0.09), in Ho Chi Minh City [31].

A major advantage of the mixture model is the comparative ease with which it can be applied to serological data to estimate transmission intensity without the need to use thresholds to process the data. Furthermore, to generate robust estimates, there are fewer data requirements for mixture models than for catalytic models: in the former, the data are pooled, and age is used only to calculate the age-specific mean log(titre + 1) using a spline, meaning that there are no constraints on the number of participants per age category. However, it is important to consider the bias that will be introduced if the mixture distributions fit the titre data poorly [24]. In this study we accounted for this by using an information criterion to select the best fitting models from a range of options. In the future we will explore implementing the models in a Bayesian framework [22] which would allow us to perform posterior predictive checks to more robustly assess model fit. It would also be interesting to explore the FOI estimates obtained when applying a mixture model with more than two mixture distributions, which may better account for the complex immunity profiles observed in areas where multiple DENV serotypes circulate. For example, Biggs et al. and Lam et al. fit three-component mixture distributions to DENV antibody titre data in the Philippines and Vietnam respectively, to develop frameworks capable of distinguishing between post and primary DENV infection [30,31] by specifying mixture components for seronegative, seropositive with a primary infection and seropositive with post-primary infections.

In summary, our results suggest that mixture models represent a good alternative to catalytic models to quantify DENV time-varying FOI and seroprevalence from age-stratified serological data, with potentially less bias and less uncertainty. They may be particularly useful when estimating FOI from data where there is high overlap between the component distributions, where the risk of serostatus misclassification and bias introduction when using cut-off threshold methods is greater (S2 and S3 Figs).

We have provided code to run the simulation study to encourage further exploration and comparison of the different methodologies. Critically, further investigation of the use of mixture models depends on the availability of raw antibody titre data. For these reasons, we would encourage current and future seroprevalence studies on DENV, as well as other infectious diseases, to publish anonymised individual-level antibody titre data where it is possible to do so.

Parameter values used for generating 540 simulated datasets.

(DOCX)

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Number of simulations where the mixture model correctly specified the distributions of the seronegative and/or seropositive component of the simulated antibody titre datasets (Dataset C).

Here, n represents the number of simulated datasets out of 540.

(DOCX)

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The observed data is serology data collected in Vietnam (Datasets A-1:A-6) and Indonesia (Dataset B). 95% Confidence Intervals (CI) were calculated by the bootstrap method.

(DOCX)

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(A) True versus estimated parameter values from the mixture model fitted to the simulated datasets (Dataset C). The estimated parameters are the mean log(titre + 1) value of the seronegative/susceptible (S) and seropositive/infected (I) components (μs and μI respectively) and the corresponding standard deviations (σs and σI). Red indicates the estimates where the true parameter value was not captured by the estimates (i.e., the 95% Confidence Interval of the estimate did not contain the true value). Note that the axes limits differ for each panel. (B) The percentage of parameter outliers after fitting the mixture model to Dataset C, per seronegative and seropositive titre family distributions. The percentage of the total number of outliers of μs, μI, σS and σI (red in panel A) per distribution combination on the x-axis, where the two letters represent the seronegative (first letter) and the seropositive (second letter) distribution pair (N = normal, G = gamma and W = Weibull).

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Association between the true component mean titre values in Dataset C versus the serostatus misclassification error.

The x-axis shows the difference between the true mean log(titre + 1) value of the seronegative (μ) and the seropositive component (μ) for each realisation over 509 simulated datasets. The titres are classified as seropositive or seronegative using realisation-specific optimised titre thresholds. The loess regression line and corresponding 95% Confidence Intervals are shown.

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Serostatus misclassification versus catalytic model estimate bias.

The bias in the estimates from the time-constant and time-varying catalytic models for the realisations over 509 simulated datasets are plotted against the serostatus misclassification error rate. The serostatuses of the titres in each of the 509 simulated datasets are classified as seropositive or seronegative using realisation-specific optimised titre thresholds. The serostatus misclassification error rate is calculated as the percentage of titres in each dataset that are misclassified. Absolute bias is calculated as the absolute value of the estimated value–true value for the force of infection (FOI) and seroprevalence. The linear regression lines and corresponding 95% Confidence Intervals are shown, as well as the Pearson’s correlation coefficients (R). Three outliers with FOI estimates > 0.4 were removed from the time-varying catalytic model panel and corresponding regression line estimation.

(TIF)

Click here for additional data file.

Force of infection (FOI) estimates across time.

Yearly FOI estimates from the catalytic and mixture models. 95% Confidence Intervals were calculated by bootstrapping.

(TIF)

Click here for additional data file.

1 Dec 2021

Dear Ms Cox,

Thank you very much for submitting your manuscript "Estimating dengue transmission intensity from serological data: a comparative analysis using mixture and catalytic models." for consideration at PLOS Neglected Tropical Diseases. As with all papers reviewed by the journal, your manuscript was reviewed by members of the editorial board and by several independent reviewers. In light of the reviews (below this email), we would like to invite the resubmission of a significantly-revised version that takes into account the reviewers' comments.

We cannot make any decision about publication until we have seen the revised manuscript and your response to the reviewers' comments. Your revised manuscript is also likely to be sent to reviewers for further evaluation.

When you are ready to resubmit, please upload the following:

[1] A letter containing a detailed list of your responses to the review comments and a description of the changes you have made in the manuscript. Please note while forming your response, if your article is accepted, you may have the opportunity to make the peer review history publicly available. The record will include editor decision letters (with reviews) and your responses to reviewer comments. If eligible, we will contact you to opt in or out.

[2] Two versions of the revised manuscript: one with either highlights or tracked changes denoting where the text has been changed; the other a clean version (uploaded as the manuscript file).

Important additional instructions are given below your reviewer comments.

Please prepare and submit your revised manuscript within 60 days. If you anticipate any delay, please let us know the expected resubmission date by replying to this email. Please note that revised manuscripts received after the 60-day due date may require evaluation and peer review similar to newly submitted manuscripts.

Thank you again for your submission. We hope that our editorial process has been constructive so far, and we welcome your feedback at any time. Please don't hesitate to contact us if you have any questions or comments.

Sincerely,

Kendall McKenzie

Staff

PLOS Neglected Tropical Diseases

Joseph Wu

Deputy Editor

PLOS Neglected Tropical Diseases

***********************

Reviewer's Responses to Questions

Key Review Criteria Required for Acceptance?

As you describe the new analyses required for acceptance, please consider the following:

Methods

-Are the objectives of the study clearly articulated with a clear testable hypothesis stated?

-Is the study design appropriate to address the stated objectives?

-Is the population clearly described and appropriate for the hypothesis being tested?

-Is the sample size sufficient to ensure adequate power to address the hypothesis being tested?

-Were correct statistical analysis used to support conclusions?

-Are there concerns about ethical or regulatory requirements being met?

Reviewer #1: (No Response)

Reviewer #2: This paper by Cox et al. compares different methods to estimate the force of infection and the seroprevalence of DENV from cross-sectional seroprevalence surveys. They show that serocatalytic models induce a bias in the parameter estimations compared to a mixture model. I think the authors do a good job at advocating the use of quantitative antibody titers. I appreciated the simulation study, and the statistical analysis is done correctly.

--------------------

Results

-Does the analysis presented match the analysis plan?

-Are the results clearly and completely presented?

-Are the figures (Tables, Images) of sufficient quality for clarity?

Reviewer #1: (No Response)

Reviewer #2: (No Response)

--------------------

Conclusions

-Are the conclusions supported by the data presented?

-Are the limitations of analysis clearly described?

-Do the authors discuss how these data can be helpful to advance our understanding of the topic under study?

-Is public health relevance addressed?

Reviewer #1: (No Response)

Reviewer #2: The authors show well the importance of using quantitative serological data to estimate key epidemiological parameters. I have however some concerns about the novelty of the work. Moreover, some of the points raised in the discussion did not convince me.

--------------------

Editorial and Data Presentation Modifications?

Use this section for editorial suggestions as well as relatively minor modifications of existing data that would enhance clarity. If the only modifications needed are minor and/or editorial, you may wish to recommend “Minor Revision” or “Accept”.

Reviewer #1: (No Response)

Reviewer #2: Minor points

I would suggest minor changes to the figures.

- Figure 1: the y axis need to be consistent, for instance the coverage has to go between 0 and 100.

- Figure 5, it would be better to have the same x axis in all panels.

- The labels of the supplementary figures are not consistent (they start at Figure S5 in the main text, and at S1 in the supplementary material)

Reference 22: Bollaerts et al., part of the title is missing

--------------------

Summary and General Comments

Use this section to provide overall comments, discuss strengths/weaknesses of the study, novelty, significance, general execution and scholarship. You may also include additional comments for the author, including concerns about dual publication, research ethics, or publication ethics. If requesting major revision, please articulate the new experiments that are needed.

Reviewer #1: (No Response)

Reviewer #2: 1/ Using mixture models to extract epidemiological parameters from serological studies is quite common now. The authors cite a few papers that use mixture models to estimate the FOI and claim line 97 that they are the first to use these methods to estimate DENV FOI. I’m not yet convinced that applying these methods to estimate DENV FOI is a big improvement to what was done in the various cited papers that used mixture models. Unless accounting for specificities of this virus (like primary/post-primary infections) or for instance specificities in the type of serological data, I don’t see a big novelty brought by the fact that the virus studied here is DENV. It is thus important to show more clearly the novelty of the work in the introduction.

2/ In the author summary, the authors write that “in high transmission areas catalytic and mixture models produced consistent estimates”. I think this is a bit misleading and hard to understand when compared to Figure 2 that show that the difference between the mixture and catalytic models increase with the FOI. The claim in the summary seems to be not general but true for the Indonesian dataset – which is obtained in a context of high circulation, and where catalytic and mixture models produced consistent estimates – but it seems that the main reason for this consistency is the little overlap in the negative and positive titer distribution.

If Figure 2 was generated with the values of the seronegative and seropositive components estimated for dataset B (the distributions and values of muS/muI), would they observe a similar difference between the mixture models and catalytic estimates? I expect no impact of the high value of the seroprevalence.

3/ Line 330 : “Differences in the degree of overlap between components in real serological datasets (S8 Figure) reflect differences in transmission intensity and variable degrees of spatiotemporal heterogeneities in the risk of infection (3,37). “

The titer distributions in the datasets of Vietnam vs Indonesia are so different, that I doubt that they are simply reflecting differences in the exposure to the virus. Since it seems that different ELISA tests were used for datasets A and B, isn’t it the most likely reason for the different titer distributions?

“In datasets collected from areas which experience hyperendemic DENV circulation, one expects greater separation between the titre components because most seropositive individuals likely have had multiple infections, translating to higher antibody titres”

I don’t believe this is true. If it were true, the way I interpret the Indonesian dataset based on this claim is that some individuals have never been exposed to the virus, whereas the other ones have been exposed multiple times. Is it realistic, and is there a way to check this? I suggest to break down by age groups the titer distribution of the seropositive components. Younger kids are expected to having been fewer infections than older ones.

4/ The titer values lines 142 and in Figure 3 for dataset B are not consistent.

5/ lines 138-140: I’d like to see the titer t used as a threshold. For instance as a vertical line in Figure 3.

--------------------

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Reviewer #1: Yes: Luc E. Coffeng

Reviewer #2: No

Figure Files:

While revising your submission, please upload your figure files to the Preflight Analysis and Conversion Engine (PACE) digital diagnostic tool, https://pacev2.apexcovantage.com. PACE helps ensure that figures meet PLOS requirements. To use PACE, you must first register as a user. Then, login and navigate to the UPLOAD tab, where you will find detailed instructions on how to use the tool. If you encounter any issues or have any questions when using PACE, please email us at figures@plos.org.

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Reproducibility:

To enhance the reproducibility of your results, we recommend that you deposit your laboratory protocols in protocols.io, where a protocol can be assigned its own identifier (DOI) such that it can be cited independently in the future. Additionally, PLOS ONE offers an option to publish peer-reviewed clinical study protocols. Read more information on sharing protocols at https://plos.org/protocols?utm_medium=editorial-email&utm_source=authorletters&utm_campaign=protocols

Submitted filename: PNTD-D-21-01566 comments LC.docx

Click here for additional data file.

28 Jan 2022

Submitted filename: Reviewer_feedback_response_letter.docx

Click here for additional data file.

11 May 2022

Dear Ms Cox,

Thank you very much for submitting your manuscript "Estimating dengue transmission intensity from serological data: a comparative analysis using mixture and catalytic models." for consideration at PLOS Neglected Tropical Diseases. As with all papers reviewed by the journal, your manuscript was reviewed by members of the editorial board and by several independent reviewers. The reviewers appreciated the attention to an important topic. Based on the reviews, we are likely to accept this manuscript for publication, providing that you modify the manuscript according to the review recommendations.

Please prepare and submit your revised manuscript within 30 days. If you anticipate any delay, please let us know the expected resubmission date by replying to this email.

When you are ready to resubmit, please upload the following:

[1] A letter containing a detailed list of your responses to all review comments, and a description of the changes you have made in the manuscript.

Please note while forming your response, if your article is accepted, you may have the opportunity to make the peer review history publicly available. The record will include editor decision letters (with reviews) and your responses to reviewer comments. If eligible, we will contact you to opt in or out

[2] Two versions of the revised manuscript: one with either highlights or tracked changes denoting where the text has been changed; the other a clean version (uploaded as the manuscript file).

Important additional instructions are given below your reviewer comments.

Thank you again for your submission to our journal. We hope that our editorial process has been constructive so far, and we welcome your feedback at any time. Please don't hesitate to contact us if you have any questions or comments.

Sincerely,

Joseph T. Wu

Deputy Editor

PLOS Neglected Tropical Diseases

Joseph Wu

Deputy Editor

PLOS Neglected Tropical Diseases

***********************

Reviewer's Responses to Questions

Key Review Criteria Required for Acceptance?

As you describe the new analyses required for acceptance, please consider the following:

Methods

-Are the objectives of the study clearly articulated with a clear testable hypothesis stated?

-Is the study design appropriate to address the stated objectives?

-Is the population clearly described and appropriate for the hypothesis being tested?

-Is the sample size sufficient to ensure adequate power to address the hypothesis being tested?

-Were correct statistical analysis used to support conclusions?

-Are there concerns about ethical or regulatory requirements being met?

Reviewer #1: (No Response)

Reviewer #2: No additional comments.

--------------------

Results

-Does the analysis presented match the analysis plan?

-Are the results clearly and completely presented?

-Are the figures (Tables, Images) of sufficient quality for clarity?

Reviewer #1: (No Response)

Reviewer #2: No additional comments.

--------------------

Conclusions

-Are the conclusions supported by the data presented?

-Are the limitations of analysis clearly described?

-Do the authors discuss how these data can be helpful to advance our understanding of the topic under study?

-Is public health relevance addressed?

Reviewer #1: (No Response)

Reviewer #2: No additional comments.

--------------------

Editorial and Data Presentation Modifications?

Use this section for editorial suggestions as well as relatively minor modifications of existing data that would enhance clarity. If the only modifications needed are minor and/or editorial, you may wish to recommend “Minor Revision” or “Accept”.

Reviewer #1: (No Response)

Reviewer #2: In this revision, Cox et al. respond adequately to the requests of the reviewers and I found the article much clearer in this version.

1/ I have one comment about the equations in the Methods section: they are part of the text and the punctuation should therefore be made accordingly.

2/ Are the datasets publicly available? This seems to be true for the Vietnam dataset and not for the Indonesian one. Please give more details about data availability.

--------------------

Summary and General Comments

Use this section to provide overall comments, discuss strengths/weaknesses of the study, novelty, significance, general execution and scholarship. You may also include additional comments for the author, including concerns about dual publication, research ethics, or publication ethics. If requesting major revision, please articulate the new experiments that are needed.

Reviewer #1: The author's efforts to address the reviewer comments are appreciated, in particular the clarification on what the models' underlying assumptions and what they are supposed to represent. This clarification has, however, raised a concern about the appropriateness of how the catalytic model was implemented in terms of maths. Equation 1 seems incorrect, or at least does not match the assumptions and model purpose described in the introduction. For details, please see the attachment.

Reviewer #2: No additional comments.

--------------------

PLOS authors have the option to publish the peer review history of their article (what does this mean?). If published, this will include your full peer review and any attached files.

If you choose “no”, your identity will remain anonymous but your review may still be made public.

Do you want your identity to be public for this peer review? For information about this choice, including consent withdrawal, please see our Privacy Policy.

Reviewer #1: Yes: Luc E. Coffeng

Reviewer #2: No

Figure Files:

While revising your submission, please upload your figure files to the Preflight Analysis and Conversion Engine (PACE) digital diagnostic tool, https://pacev2.apexcovantage.com. PACE helps ensure that figures meet PLOS requirements. To use PACE, you must first register as a user. Then, login and navigate to the UPLOAD tab, where you will find detailed instructions on how to use the tool. If you encounter any issues or have any questions when using PACE, please email us at figures@plos.org.

Data Requirements:

Please note that, as a condition of publication, PLOS' data policy requires that you make available all data used to draw the conclusions outlined in your manuscript. Data must be deposited in an appropriate repository, included within the body of the manuscript, or uploaded as supporting information. This includes all numerical values that were used to generate graphs, histograms etc.. For an example see here: http://www.plosbiology.org/article/info%3Adoi%2F10.1371%2Fjournal.pbio.1001908#s5.

Reproducibility:

To enhance the reproducibility of your results, we recommend that you deposit your laboratory protocols in protocols.io, where a protocol can be assigned its own identifier (DOI) such that it can be cited independently in the future. Additionally, PLOS ONE offers an option to publish peer-reviewed clinical study protocols. Read more information on sharing protocols at https://plos.org/protocols?utm_medium=editorial-email&utm_source=authorletters&utm_campaign=protocols

References

Please review your reference list to ensure that it is complete and correct. If you have cited papers that have been retracted, please include the rationale for doing so in the manuscript text, or remove these references and replace them with relevant current references. Any changes to the reference list should be mentioned in the rebuttal letter that accompanies your revised manuscript. If you need to cite a retracted article, indicate the article's retracted status in the References list and also include a citation and full reference for the retraction notice.

Submitted filename: PNTD-D-21-01566_R1 comments LC.docx

Click here for additional data file.

27 May 2022

Submitted filename: Reviewer_feedback_response_letter.docx

Click here for additional data file.

16 Jun 2022

Dear Ms Cox,

We are pleased to inform you that your manuscript 'Estimating dengue transmission intensity from serological data: a comparative analysis using mixture and catalytic models.' has been provisionally accepted for publication in PLOS Neglected Tropical Diseases.

Before your manuscript can be formally accepted you will need to complete some formatting changes, which you will receive in a follow up email. A member of our team will be in touch with a set of requests.

Please note that your manuscript will not be scheduled for publication until you have made the required changes, so a swift response is appreciated.

IMPORTANT: The editorial review process is now complete. PLOS will only permit corrections to spelling, formatting or significant scientific errors from this point onwards. Requests for major changes, or any which affect the scientific understanding of your work, will cause delays to the publication date of your manuscript.

Should you, your institution's press office or the journal office choose to press release your paper, you will automatically be opted out of early publication. We ask that you notify us now if you or your institution is planning to press release the article. All press must be co-ordinated with PLOS.

Thank you again for supporting Open Access publishing; we are looking forward to publishing your work in PLOS Neglected Tropical Diseases.

Best regards,

Joseph T. Wu

Deputy Editor

PLOS Neglected Tropical Diseases

Joseph Wu

Deputy Editor

PLOS Neglected Tropical Diseases

***********************************************************

Reviewer's Responses to Questions

Key Review Criteria Required for Acceptance?

As you describe the new analyses required for acceptance, please consider the following:

Methods

-Are the objectives of the study clearly articulated with a clear testable hypothesis stated?

-Is the study design appropriate to address the stated objectives?

-Is the population clearly described and appropriate for the hypothesis being tested?

-Is the sample size sufficient to ensure adequate power to address the hypothesis being tested?

-Were correct statistical analysis used to support conclusions?

-Are there concerns about ethical or regulatory requirements being met?

Reviewer #1: (No Response)

Reviewer #2: (No Response)

**********

Results

-Does the analysis presented match the analysis plan?

-Are the results clearly and completely presented?

-Are the figures (Tables, Images) of sufficient quality for clarity?

Reviewer #1: (No Response)

Reviewer #2: (No Response)

**********

Conclusions

-Are the conclusions supported by the data presented?

-Are the limitations of analysis clearly described?

-Do the authors discuss how these data can be helpful to advance our understanding of the topic under study?

-Is public health relevance addressed?

Reviewer #1: (No Response)

Reviewer #2: (No Response)

**********

Editorial and Data Presentation Modifications?

Use this section for editorial suggestions as well as relatively minor modifications of existing data that would enhance clarity. If the only modifications needed are minor and/or editorial, you may wish to recommend “Minor Revision” or “Accept”.

Reviewer #1: (No Response)

Reviewer #2: (No Response)

**********

Summary and General Comments

Use this section to provide overall comments, discuss strengths/weaknesses of the study, novelty, significance, general execution and scholarship. You may also include additional comments for the author, including concerns about dual publication, research ethics, or publication ethics. If requesting major revision, please articulate the new experiments that are needed.

Reviewer #1: (No Response)

Reviewer #2: The authors answered the questions of the reviewers and I find no additional comment to add to my review. I therefore recommend this paper for publication.

**********

PLOS authors have the option to publish the peer review history of their article (what does this mean?). If published, this will include your full peer review and any attached files.

If you choose “no”, your identity will remain anonymous but your review may still be made public.

Do you want your identity to be public for this peer review? For information about this choice, including consent withdrawal, please see our Privacy Policy.

Reviewer #1: Yes: Luc E. Coffeng

Reviewer #2: No

6 Jul 2022

Dear Ms Cox,

We are delighted to inform you that your manuscript, "Estimating dengue transmission intensity from serological data: a comparative analysis using mixture and catalytic models.," has been formally accepted for publication in PLOS Neglected Tropical Diseases.

We have now passed your article onto the PLOS Production Department who will complete the rest of the publication process. All authors will receive a confirmation email upon publication.

The corresponding author will soon be receiving a typeset proof for review, to ensure errors have not been introduced during production. Please review the PDF proof of your manuscript carefully, as this is the last chance to correct any scientific or type-setting errors. Please note that major changes, or those which affect the scientific understanding of the work, will likely cause delays to the publication date of your manuscript. Note: Proofs for Front Matter articles (Editorial, Viewpoint, Symposium, Review, etc...) are generated on a different schedule and may not be made available as quickly.

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Thank you again for supporting open-access publishing; we are looking forward to publishing your work in PLOS Neglected Tropical Diseases.

Best regards,

Shaden Kamhawi

co-Editor-in-Chief

PLOS Neglected Tropical Diseases

Paul Brindley

co-Editor-in-Chief

PLOS Neglected Tropical Diseases