Model overview

Our model aims at estimating the prevalence of infection at both host and tick levels, as well as the probability of pathogen transmission through direct host-to-tick transmission and indirect cofeeding interactions among ticks. Our model incorporates hierarchical structure to capture the nested nature of host-tick relationships, assuming that ticks share environmental exposure and immune responses on the same hosts. It also accounts for variations at the host level to capture the unobserved heterogeneity between birds in transmission risk to ticks.

Hereafter, our model assumes a ‘bird-tick interface’, with hosts designated as birds, because we use simulated and field data from this ecosystem as a case study (see ‘Simulation-based model assessments’ and ‘Model fitting to bird-tick pathogen data’, respectively). Importantly, for model fitting, we use datasets routinely collected as part of field surveys studying bird-tick-pathogen interactions, including the infection status and life stage (i.e., larva or nymph) of individual ticks, as well as the identity of the birds from which these ticks were collected (i.e., host membership). For tick life stages, our model considers larvae and nymphs but not adults, because they are rarely found on birds, as observed in our field survey datasets. Furthermore, additional details about individual hosts (e.g., host species and age) and ticks (e.g., engorgement level and cofeeding ticks) would enable statistical investigations into specific factors associated with host-to-tick transmission and cofeeding transmission, as illustrated with our bird-tick pathogen data (See ‘Model fitting to bird-tick pathogen data’).

Input data and parameters estimated are provided in Table 1. The minimum data requirements include, for each bird sampled, (i) the number of ticks identified and (ii) the number of ticks sampled and tested, and for each tick sampled and tested, its (a) life stage, (b) pathogen test results, and (c) the id of the bird from which it was sampled. Other data on other bird- and tick-level variables can be used to assess their impact on the respective transmission probabilities described below.

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Table 1. Input data and parameters estimated in the modelling framework within the bird-tick system.

https://doi.org/10.1371/journal.pcbi.1014146.t001

1. Bird-to-tick transmission probability

As the first condition for bird-to-tick transmission, a bird needs to be infected and infectious at the time of feeding. For bird k from which tick originates , its probability of being infectious, , is modelled as a logistic function of the bird’s characteristics:

(1)

is an intercept representing the baseline probability of a bird being infectious, while is a vector of bird-specific characteristics with being a vector of coefficients for these characteristics (i.e., bird species and age group). Whether bird is infectious or not, , is drawn from a Bernoulli distribution:

(2)

If bird is determined to be infectious, the probability of direct bird-to-tick transmission (i.e., the probability of an effective bird-to-tick contact), , becomes applicable for each individual tick i attached to that bird. is modelled as a logistic function of the tick’s characteristics.

(3)

is an intercept representing the baseline probability of an effective contact with an infectious bird, leading to bird-to-tick transmission. is a vector of tick-specific characteristics with being a vector of coefficients for these characteristics (i.e., tick life stage and engorgement level). As a result, the overall bird-to-tick transmission probability for tick i collected from bird becomes .

It should be noted that the framework does not consider the transmission from a tick to its host sampled together. Although this pathway is important, we hypothesise that the host must have been infected prior to being fed on by the sampled ticks. This is because the short feeding period of the tick would not provide sufficient time for the host to become infectious, while localised cofeeding transmission would remain possible [11,22].

2. Cofeeding transmission probability

We assume that cofeeding transmission involves two conditional probabilistic steps. As the first condition, a tick must be feeding on the same bird with at least one infected nymph, and then as the second condition, cofeeding transmission then must occur from infected nymph(s) to non-infected tick(s) with the probability, (i.e., the probability of an effective contact between cofeeding ticks).

Ideally, knowing the first condition would require detailed information on the infection status of all individual ticks sampled from the same bird. Although in practice this is not feasible, in most sampling situations, ticks are usually collected from easily accessible areas on birds, such as on the head around the eyes and beak, where ticks are more visible and often cofeed in groups. The infection status of these collected ticks can, therefore, serve as an approximate indicator of whether cofeeding transmission could occur among them. Following this logic, when fitting the model to the bird-tick pathogen data, we assume that cofeeding transmission occurs if there is at least one infected nymph among those feeding on the same bird, and is estimated separately (see ‘Model fitting to bird-tick pathogen data’ below).

However, this approach complicates simulation-based model assessment, as simulating the infection status of individual ticks requires information on the presence of infected nymphs feeding on the same bird, leading to a circular dependency in the data structure. Therefore, for general model assessment purposes, we propose two alternative approaches. In the first approach, the probability that tick i is feeding with infected nymphs depends on the estimated number of ticks on its bird at the time of collection. That is, it assumes that the probability increases with the number of ticks feeding on the bird (Cofeeding probability assumption 1). We quantify this relationship by fitting a logistic regression model that assesses the association between the presence of at least one infected nymph and the estimated number of ticks in the bird-tick pathogen data. We use the predicted values from this logistic regression model as the probability parameter in a Bernoulli distribution, . This allows us to estimate whether tick i and other ticks feeding on bird are in proximity to infected nymphs, :

(4)

(5)

In the second approach, we assume that the probability that tick i is feeding with infected nymphs is defined by the proportion of hosts with at least one infected nymph, using a Bernoulli distribution (Cofeeding probability assumption 2), and we define :

(6)

(7)

The implications of these alternative approaches are tested and discussed through simulation-based model assessment (see ‘Simulation-based model assessments’ below).

Finally, if tick i is determined to be feeding with infected nymphs based on either of the above approaches, the probability of cofeeding transmission, , is applicable, modelled as a logistic function of the tick’s characteristics.

(8)

is an intercept representing a baseline cofeeding transmission probability, while is a vector of tick-specific characteristics with being a vector of coefficients for these characteristics (i.e., tick life stage and engorgement level). As a result, the overall cofeeding transmission probability for tick i collected from host becomes .

3. Combined force of infection on ticks while attached on the sampled host

We model the combined force of infection as follows:

(9)

(10)

represents the probability that tick i on bird becomes infected through either bird-to-tick transmission, cofeeding transmission, or both, accounting for potential non-independence between the two transmission pathways and heterogeneity in transmissibility at the bird level with and , respectively, while does not account for these factors. represents a covariance effect between host-to-tick transmission and cofeeding transmission. That is, these two transmission routes are assumed not independent. represents a random effect for each host , accounting for unobserved variability in transmissibility among hosts. was sampled from a Normal distribution with the mean of 0 and the standard deviation, σ, estimated through model fitting.

4. Adjusted infection probability, accounting for ticks being infected prior to feeding the sampled host

Finally, the combined force of infection, , is adjusted to account for the probability that tick i is already infected prior to feeding on host k, :

(11)

(12)

The prior-infection probability is modelled separately for larvae ( and nymphs (. For example, if prior infection is assumed only for nymphs, in Eq. 11 becomes for larvae and remains for nymphs. > 0 is an offset parameter ensuring . That is, nymphs are assumed to have a higher probability than larvae, considering that they have a blood meal at their larval stage, whereas larvae can only be infected prior to feeding through transovarial transmission. Previously interrupted blood meals are also possible, but nymphs would still, on average, have more opportunities for such events and therefore a higher prior-infection probability than larvae.

Finally, the infection status of tick i from bird is drawn from a Bernoulli distribution, with as the probability parameter:

(13)

Therefore, the log-likelihood of the observed tick infection data, given the fixed parameters (Table 1), the random effects , and the latent bird infection status , becomes:

(14)

1. Parameter identifiability

The first assessment focuses on the model’s ability to recover true parameter values by fitting it to synthetic datasets and comparing the posterior parameter estimates with the true parameter values used to generate those datasets. We generated 100 synthetic datasets. The number of birds is assumed to be the same as the number of birds in the bird-tick pathogen empirical data (see “Model fitting to bird-tick pathogen data” below). For each bird, the number of ticks is simulated using a negative binomial distribution, with the mean (µ = 7.130), overdispersion (k = 1.217), and larvae-to-nymph ratio (0.472) informed by the bird-tick pathogen data. Other parameter values were randomly selected from the joint posterior distribution of the baseline model fitted to the B. garinii dataset among the bird-tick pathogen data. Based on the parameter values selected, the infection status of each tick was then determined by in Eq. 13. Note that we did not draw parameters values independently and directly from priors, because doing so can produce unrealistic combinations of parameter values and infection scenarios; instead, drawing from the joint posterior allows parameter recovery to be evaluated within empirically plausible ranges while preserving posterior dependencies among parameters. We used the B. garinii posterior because its underlying dataset showed a significant association between the presence of at least one infected nymph and the estimated number of ticks in the bird-tick pathogen data, enabling us to assess parameter recovery under the more complex model assumption (Cofeeding probability assumption 1, in Eq. 4). The impact of assuming a constant cofeeding probability (Cofeeding probability assumption 2, in Eq. 6) is evaluated in the subsequent assessment steps.

The model is subsequently fitted to each synthetic dataset using Gibbs Markov chain Monte Carlo (MCMC) sampling algorithm in JAGS [26] in R 4.3.2. Across the model fits, the following model performance metrics are obtained for each parameter: (i) the distribution of bias, defined as the difference between the median posterior estimate and the true parameter value, (ii) 95% highest density intervals (HDI) coverage, defined as the percentage of the synthetic datasets in which the true parameter value lies within the 95% HDI, and (iii) the distribution of precision, defined as the inverse of the 95% HDI width.

2. Impact of sample size on parameter estimation

The second assessment examines the impact of sample size on the model performance metrics. In most sampling situations where hosts are captured first and ticks are collected from these hosts, it is realistic to treat the number of hosts sampled as the primary design target because the number of ticks collected depends primarily on the number of birds captured. Accordingly, we define sample size as the number of birds sampled, with the number of ticks subsequently determined by the assumed per-bird tick-burden distribution (see the next section, “Impact of tick overdispersion on parameter estimation”). We then assess the effect of sample size on model estimates across bird sample sizes ranging from 10 to 250, using 100 synthetic datasets for each sample size.

For the same reason as in the assessment on parameter identifiability, we generate synthetic datasets using posterior parameter estimates from the model fitted to the B. garinii dataset (see “Model fitting to bird-tick pathogen data” below), rather than relying on random samples from weakly informed prior distributions.

3. Impact of tick overdispersion on parameter estimation

Finally, we examine how heterogeneity in tick burdens across birds affects estimation performance, because it changes both the amount and distribution of information in the data and can therefore influence model fitting. To do this, we fit the model to synthetic datasets generated under different scenarios for how ticks are distributed across birds: (i) overdispersed tick burdens following a negative binomial distribution, as observed in the bird-tick pathogen data (baseline assumption, µ = 7.130, overdispersion parameter k = 1.217); (ii) increased overdispersion, also under a negative binomial distribution (µ = 7.130, k = 0.561); and (iii) non-overdispersed tick burdens following a Poisson distribution (µ = 7.130). Each scenario is combined with the two assumptions about the probability of cofeeding with infected nymphs described above: constant across birds or varying with observed tick burden.

1. Bird-tick pathogen data

The bird-tick pathogen data include the infection and demographic information of 187 larvae and 209 nymphs as well as 26 blackbirds (Turdus merula), 30 song thrushes (Turdus philomelos), and 12 robins (Erythacus rubecula) from which these ticks were collected. The birds and ticks were sampled in the Forêt Domaniale de Haye (48.6779°N, 06.1001°E), Northeast France, from June 14 to July 4 in 2023. Bird capture and blood sampling were conducted under licence from the Centre de Recherches sur la Biologie des Populations d’Oiseaux (CRBPO, BF permit no. 1073). Bird hosts were captured using mist nets. T. merula and T. philomelos in the Turdidae family, and E. rubecula in the Muscicapidae family, were selected for sampling, given that they are known to exhibit the highest tick prevalence among common forest-dwelling passerines [9]. Each captured bird was checked visually for ticks, with particular attention on the head, where ticks are most found [23]. The number of ticks detected on each bird was recorded, and ticks were removed using tick-removal tools or fine pointed forceps, then immediately placed in microtubes filled with 70% ethanol for later laboratory analysis. When tick removal was too tricky, ticks were not collected to avoid injuring the birds. In cases where more than 10 ticks were detected on a bird, only 10 ticks were collected and analysed for pathogen presence; engorged ticks were prioritised to maximise the probability of pathogen detection within the available budget. This resulted in testing not all the ticks identified in 35.3% of the captured birds. From each captured T. merula bird, 300–400µl of blood was drawn from the brachial vein using sterile 27G needles, collected into 100µl heparinised tubes, and deposited in duplicate onto FTA cards for laboratory analysis. Birds were released after confirming that bleeding had stopped.

For each tick, the infection status of B. garinii, B. valaisiana, and A. phagocytophilum was determined using high-throughput microfluidic real-time PCR, performed on a BioMar real-time PCR system (Standard BioTools, USA). Primers and probes sequences are detailed in Melis, Batisti Biffignandi [24]. This technique uses the 48.48 Dynamic Arra (Standard BioTools, USA), which features two sets of 48 inlets, allowing simultaneous loading of 48 samples and 48 primers and probe (assay) mixtures. All 2,304 sample-assay combinations are processed and tested simultaneously by the automated microfluidic system, requiring only few total microliters of each sample and assay. The results were confirmed for each pathogen by conducting conventional PCR on around 20% of the positive samples; all showed >99% sequence homology at the species level (see Melis, Batisti Biffignandi [24] and Michelet, Delannoy [25] for details).

The life stage (larva or nymph) and engorgement level (mild, moderate, or full) of the ticks, and the id of the bird from which it was collected were recorded.

Additionally, prevalence estimates of B. garinii, B. valaisiana, and A. phagocytophilum in questing nymphs collected from the same Forêt Domaniale de Haye area in late June to early July from 2021 to 2023 were used to evaluate whether the model’s estimated probability of a nymph having been infected prior to feeding, , was comparable to general infection levels in questing nymphs. Here, questing nymphs were collected by dragging a cotton sheet over 200m² area in two different locations within the forest and tested by PCR as done for ticks collected from birds. For each bird, its species (T. merula, T. philomelos, and E. rubecula) and age (≤2-year-old or >2-year-old) were recorded. The molecular testing results for A. phagocytophilum were also available for the blood of birds from which ticks were collected. However, this information was not used for the model fitting itself but only to assess whether the model’s estimated probability of a bird being infected, , was comparable to actual molecular testing result.

2. Model fitting to observed bird-tick pathogen data

All models are fitted to bird-tick B. garinii, B. valaisiana, and A. phagocytophilum data separately using Gibbs MCMC sampling algorithm in JAGS [26] in R 4.3.2, with random initial values given to four chains. After discarding the first 30,000 posterior samples as burn-in, another 30,000 were retained per chain, and convergence was assessed by visual inspection of MCMC trace plots and Gelman-Rubin convergence diagnostic (<1.01) [27].

In the full model, (i) the probability of a bird being infectious (), (ii) the probability of direct bird-to-tick transmission (), and (iii) the probability of cofeeding transmission () were modelled as logistic functions of all available bird- and tick-level information. Bird species and bird age were included as predictors for , while tick life stage and engorgement level were used as predictors for and . Importantly, was valid for tick i on bird only when there was at least one infected nymph among those sampled from the same bird, i.e., in for the overall cofeeding transmission probability.

After model fitting, a posterior predictive check was performed using the proportion of ticks that tested positive to each pathogen, differentiated by bird species and tick life stage, as the predicted measure. For each predictor, its effect size was evaluated relative to a reference group, and its statistical significance was assessed by comparing model fits between the full model and the model without the predictor based on deviance information criterion (DIC), obtained with the dic.sample fuction of the rjags package [26]. A decrease in DIC greater than 5 units was considered to indicate a statistically significantly better model fit.

1. Results of simulation-based model assessment

The comparison between posterior parameter estimates and true parameter values indicates that the model could recover true parameter values, with no systematic bias observed in posterior parameter estimates and acceptable 95% highest density intervals (HDI) coverages ranging between 93.0% and 97.0% (Fig A in S1 Text).

Under various scenarios for the probability of cofeeding with infected nymphs and tick distributions across hosts, 95%HDI coverages remain robust across all sample sizes, with approximately 95% of true parameter values falling within these intervals across simulations (Fig B in S1 Text). Precision and bias improve noticeably with larger sample sizes when the number of ticks per bird is overdispersed following a negative binomial distribution (overdispersion parameter k = 1.217 or 0.561), compared to a Poisson distribution, and when cofeeding probability is allowed to vary with tick numbers (, blue circles and diamonds, Figs 1 and 2). In contrast, assuming a constant cofeeding probability across ticks leads to consistently poor precision for parameters related to bird-to-tick and cofeeding transmission, with minimal improvement observed with larger sample sizes (, red points, Fig 1A, 1B, and 1C). Similarly, assuming a non-overdispersed (i.e., Poisson) tick distribution, even with , results in a poor precision for the probability of cofeeding transmission () across all sample sizes (blue squares, Fig 1C). These trends in precision are also reflected in bias for and for the covariance parameter (), which captures the relationship between host-to-tick and cofeeding transmission (blue squares, Fig 2C and 2G).

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Fig 1. Precision of posterior parameter estimates across different sample sizes and model assumptions.

Point shapes indicate precision under assumptions about the distribution of tick numbers on birds, while point colours indicate precision under assumptions about the probability of cofeeding with infected nymphs (: blue points; : red points). Blue circles represent precision under the baseline assumptions. Vertical lines the number of birds in the bird-tick pathogen data.

https://doi.org/10.1371/journal.pcbi.1014146.g001

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Fig 2. Bias of posterior parameter estimates across different sample sizes and model assumptions.

Point shapes indicate bias under assumptions about the distribution of tick numbers on birds, while point colours indicate bias under assumptions about the probability of cofeeding with infected nymphs (: blue points; : red points). Blue circles represent bias under the baseline assumptions. Vertical lines the number of birds in the bird-tick pathogen data.

https://doi.org/10.1371/journal.pcbi.1014146.g002

2. Results of model fitting to observed bird-tick pathogen data

For all three pathogens studied (B. garinii, B. valaisiana, and A. phagocytophilum), the full model accurately predicted the empirical proportion of infected ticks by bird species and tick life stage (Fig 3). The full model provided a significantly better fit to the data compared to the baseline model, which assumed no effect of bird- and tick-level predictors (Table 2). Also, the model assuming cofeeding transmission yielded a significantly better fit to the data than the model assuming no cofeeding transmission (vs. Model 1.7, Table 2). For A. phagocytophilum, incorporating engorgement level into the probability of bird-to-tick transmission also improved the model fit (vs. Model 1.4, Table 2), whereas incorporating engorgement level into the probability of cofeeding transmission did not (vs. Model 1.6, Table 2). Other assumptions did not explain the data significantly better than the full model (DIC < 5, Table 2). However, to facilitate comparison across the three pathogens and to present the full set of transmission pathways that can be considered for tick-borne disease transmission in a host-tick system, the results presented in this study are based on the full model.

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Fig 3. Full model fit.

For each pathogen, the predicted distribution of the proportion of infected ticks was compared with the observed proportion by bird species and tick life stage. Circles and vertical lines represent the median and 95% percentile intervals of the predicted proportions, while diamonds indicate the observed proportions.

https://doi.org/10.1371/journal.pcbi.1014146.g003

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Table 2. Model comparison for Borrelia garinii.

https://doi.org/10.1371/journal.pcbi.1014146.t002

For all three pathogens, the median posterior probability of a bird being infectious ranged between 0.06 and 0.35 across different bird species and age groups, although no significant differences were observed between these groups (Figs 4A, 5A, and 6A). For A. phagocytophilum, the posterior probability of a T. merula bird being infectious was estimated to closely match the proportion of the T. merula birds that tested positive for this pathogen, which was used as a value only for comparison but not used for model fitting (Fig 6A).

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Fig 4. Posterior parameter distribution from the full model fitted to the B. garinii dataset.

Black points and vertical lines represent the posterior median and 95%HDI estimates. Panel A shows the probability that a bird was infected and infectious by bird species and age group. Panel B shows the probability that a tick was infected from an infectious bird, by life stage and engorgement level. Panel C shows the probability that a tick was infected via cofeeding with other infected nymphs, by life stage and engorgement level. Panel D shows the probability that a tick was already infected before feeding, by life stage. Red triangles and dashed lines represent the proportion of questing nymphs that tested positive to B. garinii in the same Forêt Domaniale de Haye region in Northeast France, which was not used for model fitting.

https://doi.org/10.1371/journal.pcbi.1014146.g004

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Fig 5. Posterior parameter distribution from the full model fitted to the B. valaisiana dataset.

Black points and vertical lines represent the posterior median and 95%HDI estimates. Panel A shows the probability that a bird was infected and infectious by bird species and age group. Panel B shows the probability that a tick was infected from an infectious bird, by life stage and engorgement level. Panel C shows the probability that a tick was infected via cofeeding with other infected nymphs, by life stage and engorgement level. Panel D shows the probability that a tick was already infected before feeding, by life stage. Red triangles and dashed lines represent the proportion of questing nymphs that tested positive to B. valaisiana in the same Forêt Domaniale de Haye region in Northeast France, which was not used for model fitting.

https://doi.org/10.1371/journal.pcbi.1014146.g005

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Fig 6. Posterior parameter distribution from the full model fitted to the A. phagocytophilum dataset.

Black points and vertical lines represent the posterior median and 95%HDI estimates. Panel A shows the probability that a bird was infected and infectious by bird species and age group. Panel B shows the probability that a tick was infected from an infectious bird, by life stage and engorgement level. Panel C shows the probability that a tick was infected via cofeeding with other infected nymphs, by life stage and engorgement level. Panel D shows the probability that a tick was already infected before feeding, by life stage. Red triangles and dashed lines represent the proportion of T. merula birds that tested positive to A. phagocytophilum, from which the ticks in the A. phagocytophilum dataset were collected. This T. merula infection status data was not used for model fitting.

https://doi.org/10.1371/journal.pcbi.1014146.g006

The median posterior probability of a tick becoming infected from an infectious bird was highest for fully engorged ticks, particularly for A. phagocytophilum (0.66, 95%HDI: 0.22 to 0.95), compared to B. valaisiana (0.21, 95%HDI: 0.01 to 0.74) and B. garinii (0.48, 95%HDI: 0.10 to 0.86) (Figs 4B, 5B, and 6B). Notably, the association between engorgement level and infection probability was particularly strong for A. phagocytophilum. Fully engorged ticks were estimated to have 9.67 (95%HDI: 1.52 to 53.90) times the odds of becoming infected with A. phagocytophilum from an infectious bird compared to mildly engorged ticks (Set 2 in Figs C and D in S1 Text).

The median posterior probability of cofeeding transmission in the presence of at least one infected nymph was estimated to be higher for B. garinii (0.58 to 0.73, Fig 4C) than B. valaisiana (0.17 to 0.40, Fig 5C) and A. phagocytophilum (0.24 to 0.58, Fig 6C) across ticks at different life stages and varying engorgement levels. While cofeeding transmission probability tended to increase with higher engorgement levels, these trends were not statistically significant for all three pathogens (Set 3 in Figs C and D in S1 Text).

The median posterior probability of a tick being already infected prior to feeding the sampled host was estimated to be the highest for B. garinii (0.014 for larvae and 0.022 for nymphs, Fig 4D), with B. valaisiana (0.002 for larvae and 0.005 for nymphs, Fig 5D) and A. phagocytophilum (0.002 for larvae and 0.005 for nymphs, Fig 6D) having similar levels. The probability was estimated to be lower in larvae than in nymphs, with no significant difference observed for a model that did not assume prior infection at the larval stage (Model 1.8, no in Eq. 11) compared with the full model that assumed it ( in Eq. 11) (Table 2). Finally, for all three pathogens, the probability of a nymph being infected prior to feeding the sampled host was estimated to fall within the same range observed from empirical data collected independently from the present study in the same forest region in northeast France (red triangles and vertical dashed lines in Figs 4D, 5D, and 6D).