Virus

In each experiment, a single animal (called the seeder) was infected with the PPRV Senegal 20-GP strain. The virus was isolated in BTS cells, a recombinant CV1 cell expressing the bovine SLAM protein (C. Adombi and A. Diallo, unpublished data), like the CHS cell which is a recombinant CV1 cell expressing the goat SLAM [34]. Both SLAM protein-based recombinant cells are efficient for PPRV in vitro isolation from pathological samples (C. Adombi and A. Diallo, unpublished data). This virus was grown in the BTS cells, and the collected virus suspension was aliquoted and stored in a 1.5 ml tube at −80°C. It was used at the 3rd passage in the BTS cells. The contents of three tubes were defrosted and titrated independently by the endpoint dilution assay on the BTS cells in 96-well flat-bottom plates for their cytopathic effect (CPE). The virus suspension titer was expressed as the 50% tissue culture infectious dose (TCID50), calculated according to the Spearmann-Kärber methodology as described by [35]. The mean value of the three titrations was 10^3.2 TCID50/ml. It belongs to the PPRV lineage IV according to its partial Np sequence data [36]. In preliminary animal pathogenicity testing (Diallo A., Diop. M, Sagna A. and M. Lo, unpublished data), all goats inoculated with this virus exhibited acute PPR symptoms: high fever up to 40–41°C, diarrhea, significant ocular and nasal discharges, and death in four out of ten inoculated animals.

Animals

A total of 82 goats, naïve and aged between six and twelve months, were used in the twelve experiments. They were purchased from different farms in Senegal by a team from the Laboratoire National d’Elevage et de Recherches Vétérinaires (LNERV) of the Institut Sénégalais de Recherches Agricoles (ISRA). Before purchase, all were proved negative for both PPRV and antibodies anti-PPRV by the ID.vet Innovative Diagnostic PPR pen-side test (ID Rapid® PPR Antigen) and the ID.vet Innovative Diagnostic ID Screen® PPR Competition respectively (See S1 File). The tests were performed by the ISRA laboratory team in the village of the farmers before purchasing the animals. After the purchase, the animals were brought to the ISRA experiment farm at Sangalkam (14.7955°N, 17.2289°W; WGS 84), about 30 km from Dakar. Upon arrival they were housed in a 300 m² barn for a minimum of two weeks for quarantine and acclimatization. During the first and second weeks, they were submitted to a second round of antibody anti-PPRV detection assay (ID.vet Innovative Diagnostic ID Screen® PPR) and the Reverse Transcription Polymerase Chain Reaction (RT-PCR) for PPRV nucleic acid detection according to the method of [37]. Only animals that tested negative again in both assays were included in the experiments. Any goat that was tested positive by one of the two tests was immediately removed from the group and excluded from the study.

Experiment facilities

At the Sangalkam experiment farm, the experimental infection protocol was conducted in a dedicated and specially equipped area of 40m² (see Fig S1, Section 1 in S1 File), located at about 50 meters from the quarantine barn. Within this area, paddock 2 was designated for controlled PPRV infection, while the animal-to-animal PPRV transmission experiments were conducted in paddock 1. Both open experimental barns were equipped with fenced pens designed to house each animal individually after the transmission experiments, to facilitate individual monitoring of PPR symptoms. The storage rooms were designated for storing animal feed and technical equipment, including syringes, blood collection tubes, gloves, coats, and containers for pathological sample collection. Outside the periods of experimental infection protocols, the animals were housed in the 300 m² barn. All barns, paddocks, and individual pens were equipped with troughs to provide animals with ad libitum access to feed and water. All the animal facilities were cleaned and disinfected before and after housing animals. During the experiment, the occupied rooms were cleaned daily.

Experiment design

Each experience was divided into three phases as shown in Fig 1:

Download:

• PNG
  larger image
• TIFF
  original image

Fig 1. Detailed Schema of the Experimental Infection Protocol for Studying PPR lineage IV Transmission Dynamics.

This diagram outlines the sequential stages of the experimental protocol designed to investigate the transmission of PPR among goats. The protocol includes three primary periods: the experimental infection period, the exposure period, and the monitoring period.

https://doi.org/10.1371/journal.pone.0358839.g001

1. Experimental Infection phase: infection of animal to be used for transmission of the PPRV to naïve animals by contact: the seeder.
2. Exposure Phase: The seeder is in contact with other naïve animals to allow transmission from animal-to-animal in a paddock for a period of variable duration.
3. Monitoring phase: exposed animals are isolated and monitored until they show PPR symptoms or any other PPR test positive results.

Detailed procedures for each experimental phase are provided in the S1 File, Section 1.

Data collection for indoor positioning

UWB solutions are used to measure the distance between a reference sensor (referred to as an anchor) and a sensor whose position is to be determined (referred to as a tag, equipped on the animal) using different metrics. For these experiments, paddock 1 was equipped with six anchors (UWB BeSpoon - STMicroelectronics, France) fixed to the ceiling, one at each corner and two near the center of the room. Each anchor was powered by a Power Over Ethernet cable connected to a dedicated 16-port switch (DES-1018MP; D-Link, Taipei – Taïwan). The tags (BeSpoon Industrial Tag; Length: 101 mm, Width: 52 mm, Thickness: 25 mm; BeSpoon - STMicroelectronics, France) were enclosed in a custom 3D-printed case to provide shock protection and ensure better attachment to the collar. The UWB system was controlled via a dedicated local server (a laptop with a Linux operating system), which provided a user interface for verifying the general system status and initiating data acquisitions. This experimental setup was designed to monitor and record, in near-real time, the precise indoor location of each individual.

The theoretical performance of the solution allows positioning an object with an accuracy of +/- 20 centimeters, but these performances depend notably on the tracking environment and the deployment conditions. Therefore, in order to evaluate the performance of this indoor positioning system under our experimental conditions, we designed and implemented a custom evaluation protocol using the different available tags (n = 7). Accuracy and precision were assessed using pairwise inter-sensor distance measurements across multiple geometric configurations:

• Sensors arranged in a straight line on the ground, with inter-sensor distances of 6, 10, 15, 30, and 50 cm, respectively.
• Sensors arranged in a straight line on a wooden bench at a height of 51.6 cm (approximating their actual monitoring height when equipped on the animals), with inter-sensor distances of 20 cm.
• Sensors arranged in a hexagon configuration with one sensor at the center and inter-sensor distances of 15, 30, and 100 cm, respectively.

Across all configurations, the system demonstrated a mean accuracy of 28.3 ± 6.2 cm and a mean precision of 22.2 ± 4.8 cm. These results were consistent with the manufacturer’s specifications and met our experimental requirements.

During the monitoring phase, the acquisition frequency was set to f = 1 Hz (one acquisition per second). Data were stored on the server as a csv file, with each line representing the estimated position of each tag on the x, y, and z axes. Distances between the infected individual and each susceptible animal were then computed from these data, using only the x and y axes. This frequency of indoor positioning provided a comprehensive temporal map of animal interactions. High-resolution proximity data enabled us to analyze not only the frequency of close contacts, but also the duration of these interactions at different distance thresholds.

Ethics statement

Animals were monitored daily for clinical signs. Because severe diarrhea is a frequent precursor to mortality in PPRV-infected animals, the onset of this clinical sign was established as the humane endpoint for euthanasia. Euthanasia was performed via intravenous injection of 25 mL of pentobarbital sodium (PENTOBARSOL™), a method that induces rapid and painless death, thereby minimizing distress. For biosecurity purposes, at the conclusion of each experiment, all remaining animals were humanely euthanized using this same protocol, and the facilities were thoroughly cleaned and disinfected.

All the animal experiments have been carried out by staff of the LNERV/ISRA at the ISRA experimental farm. LNERV/ISRA is the Senegalese national veterinary laboratory for diagnostics and research in animal health and production. One of the supervisors of the experiments, Mr. Adama Diallo, DVM and PhD, has about 40 years’ experience in animal viral disease vaccine development with testing in animals. The second, Mr. Moustapha Lo, head of the Animal Health Program at LNERV, at the time of the experiment, is responsible for biosecurity and biosafety at ISRA/LNERV. The procedures and protocols of the animal experiment which results have been reported here were submitted to and approved by the Scientific Direction of LNERV/ISRA, a competent body for animal research.

Statistical analysis and models

We employed a Bayesian statistical framework to estimate the transmission rate, incubation period, and model transmission probability, a choice that was driven by the need to integrate high-resolution individual proximity data with batch-level and individual level infection outcomes. This probabilistic approach is particularly suited to experimental datasets with a limited number of independent batches and small sample size, as it avoids reliance on large-sample asymptotic assumptions and it allows for the rigorous quantification of uncertainty through posterior distributions. By employing a Bayesian structure, we defined model-specific priors to ensure biological plausibility. Depending on the particular quantity to estimate, we used different likelihoods to characterize the binary outcome of infection status and the duration of the incubation phase. In all cases we used minimally informative prior distributions to minimize potential bias and applied the Markov Chain Monte Carlo (MCMC) approach to estimate model parameters. Table 1 summarizes the four transmission probability models described in this section, detailing their respective outcomes, predictors, parameters, and epidemiological assumptions.

Download:

• PNG
  larger image
• TIFF
  original image

Table 1. Comparative roadmap of transmission probability models.

https://doi.org/10.1371/journal.pone.0358839.t001

Transmission rate.

We estimated the hourly transmission rate of the strain following the procedure of Dekker et al. [38]. The infection outcome for each susceptible individual in an interval of time Δt modeled as an independent and identically distributed Bernoulli random variable with infection probability p_t.

(1)

Where N_t is the total number of animals in the pen/paddock, the number of initially infected individuals (1 in all cases) and Δt is the duration of the exposure phase ranges from 1 hour to 48 hours (Δt = 1, 6, 24, 44, 48). Due to the variation in the length of the exposure phase, we estimated the transmission rate per hour. We estimated using the No-U-Turn Sampler (NUTS), fitting the model to serological data using a binomial likelihood.

Basic reproductive number.

In a continuous transmission model, considering an exponential process, the daily transmission rate can be estimated from the hourly transmission rate as:

(2)

Based on this, a first derived estimate of the basic reproductive number was calculated using the following formula derived from the SIR model definition:

(3)

where, is the daily transmission rate, and γ stands for the recovery rate (the inverse of the infectious period) commonly used in PPR modeling. The infection period could not be estimated from our experimental settings. Therefore, we calculated using two infectious period estimates from the literature (5 and 10 days).

Transmission probability.

We studied the effect of cluster dependence and within-experiment correlation on infectious status of the animals.

We used logistic regression model with a random effect, using the predicted variable as the individual infectious status and the exposure measure ( as a predictor of infectious status.

(4)

Exposure was considered to be the inverse of the distance from the source of infection at each time point, and we considered both the cumulative exposure and the duration of the experiment. We then performed a comparison between the different models using ANOVA test, to find if any difference existed between the different experiments.

Our analytical approach comprised four distinct modeling strategies for PPRV transmission in our study: (i) baseline model, (ii) segmented model, (iii) envelope model, and (iv) Stratified model. In our Bayesian statistical approach, we initially developed a baseline model that intentionally excluded proximity factors, focusing solely on temporal exposure patterns (see more details in Fig 2). In this approach we characterized transmission dynamics without spatial considerations. Subsequently, we progressively incorporated proximity data from UWB to systematically evaluate whether distance information significantly enhances transmission risk prediction. By comparing models with increasing spatial complexity, from a null temporal model to models integrating precise inter-animal distances, we aimed to empirically determine the statistical significance of proximity in PPRV transmission mechanisms. In all these models, we used Bernoulli likelihood to consider individual differences in contact patterns and activity.

Download:

• PNG
  larger image
• TIFF
  original image

Fig 2. Graphical representation of three different models (Baseline, Segmented, and Stratified) illustrating the spread of infection over time.

The infected seeder individual, denoted as S, interacts with susceptible individuals (1), (2), and (3). The arcs represent the varying distances between S and the susceptible individuals at three time points: t = 1, t = 2, and t = 3 (in minutes). The circles depict the dynamic radius of interaction, which either increases or decreases over time, showing the changing proximity and potential for infection spread in all orientations. The envelope model is a modification of Stratified model, where ranges are determined based on animals’ movement patterns.

https://doi.org/10.1371/journal.pone.0358839.g002

Baseline model.

Our baseline model, which serves as the null hypothesis, assumes a constant probability of transmission per unit of time, independent of proximity. The individual cumulative probability of infection for this model is expressed as follows:

(5)

where represents the probability of a successful transmission per unit time, and the total duration of the experiment. The simplicity of this model provided a fundamental benchmark against which more complex models could be compared.

Segmented model.

Building on the baseline approach, our segmented model has incorporated distance-based stratification to account for the potential influence of proximity on the virus transmission probability. This model uses the same fundamental equation as the baseline model:

(6)

However, in this context, represents the cumulative time spent within specific distance thresholds (1 m, 1.5 m, and 2 m). Separate models were fitted for each threshold, enabling us to discern how the probability of transmission might vary with different distances.

Stratified model.

Our stratified model represents a more subtle method of integrating transmission dynamics as a function of distance. The observed distances were divided into different ranges:

• t₁: cumulative time spent between 0.05 m and 0.5 m,
• t₂: cumulative time spent between 0.5 m and 1 m,
• t₃: cumulative time spent between 1 m and 2 m,
• t₄: cumulative time spent beyond 2 m.

The probability of infection for this model is given by:

(7)

where is the probability of success, are scale factors for different distances, and , represents the time spent in each range. This formulation allows for a differential weighting of transmission risk according to proximity. We studied five variants of this model, each emphasizing different distances:

• Stratified 1: j = 1, i = 2, 3, 4
• Stratified 2: j = 1, i = 2
• Stratified 3: j = 3, i = 4
• Stratified 4: j = 2, i = 3, 4
• Stratified 5: j = 1, i = 2,3

Envelope model.

Our most sophisticated approach, the envelope model, integrates both temporal and spatial proximity data clustering. We identified distinct envelop interaction groups for each individual and used the total duration of these groups along with their median distances to estimate infection probability. This model was applied with different distance filters (≤ 1 m, ≤ 1.5 m, and ≤ 2 m). The mathematical formulation is detailed in (S1 File, Section 3).

Exposure time estimation.

To estimate the minimum exposure duration required for a susceptible animal to become infected, we simulated various contact scenarios by analytically inverting the cumulative probability of infection functions derived from our best-fitting transmission models. The cumulative probability of infection after an exposure time (measured in minutes) was modeled under two main assumptions regarding animal movement and spatial density:

First, to represent a scenario where animals move freely and interact at varying distances without extended close contact (e.g., roaming herds), we used the Baseline model (see Eq. (5))

Second, to represent high-density environments where animals are maintained at relatively fixed, close distances (e.g., crowded livestock markets or transport), we used the Stratified models. For a susceptible animal kept at a constant distance within a specific spatial interval , the cumulative probability of infection simplifies to:

(8)

In this formulation, represents the reference transmission probability for the baseline distance interval and scaling factors .

For both scenarios, the minimum exposure time required to reach a specific probability of infection threshold (e.g., 50% and 95%) was calculated by solving the cumulative probability equations.

(9)

Where is the effective transmission probability per minute ( for the Baseline model, and for the Stratified models).

We used the posterior mean and 95% HDI estimates of the transmission probabilities () and scaling factors () obtained from the Bayesian inference of the models. We used the Baseline model to estimate exposure time for the unrestricted movement scenario. For fixed distances, we used the Stratified 1 model for the 0.05–0.5 m range, and the Stratified 4 model for the remaining distances (0.5–1 m, 1–2 m, and ≥ 2 m). Finally, we converted all exposure times from minutes to hours to make them easier to compare across the different density scenarios.

Parameter estimation

The choice of prior distributions for model parameters was guided by both biological plausibility and computational considerations. The specific parametrizations of the priors were selected based on preliminary analyses and expert knowledge of PPRV transmission dynamics. Since individuals could vary based on their activity patterns, we used a binomial likelihood to elicit the differences among individuals’ patterns.

We used the NUTS, an MCMC method. This approach facilitated efficient exploration of the parameter space and estimation of posterior distributions. Our sampling protocol was designed to ensure thorough exploration of the parameter space. We configured the sampler to draw 10,000 samples per chain, with a tuning phase of 1,000 iterations, and ran ten independent chains. This configuration allowed for comprehensive sampling and robust convergence diagnostics. We chose a high target acceptance rate of 0.95 to promote efficient exploration of parameter space, which was particularly important for our more complex models. Posterior distributions were obtained for each model parameter, and the Bayesian estimates were summarized using the posterior mean estimate and the 95% highest (posterior) density interval (HDI). The 95% HDI is the minimum Bayesian credible interval (BCI) that contains 95% of the posterior probability distribution. Convergence of the chains was confirmed by visual inspection of trace and density plots and, the Gelman—Rubin diagnostic (R̂) which compares the within-chain and between-chain variances [40]. If the chains have converged, the variances should be similar and results in an R̂ close to 1.

Model evaluation

Model comparison was facilitated through the computation of WAIC [41] because it provides an estimate of predictive accuracy while accounting for model complexity, without requiring cross-validation or separate training/test datasets. WAIC estimates how well a model generalizes to unseen data by balancing goodness of fit (how well the model fits the observed data) and penalizing complexity (to avoid overfitting). A model with a lower WAIC value is preferable, as it indicates better predictive performance. However, models with close WAIC values can be considered as having similar performance, and other factors such as interpretability and model simplicity can be considered for the final selection.

We conducted a receiver operating characteristic (ROC) curve analysis and generated areas under the curve (AUCs) estimates using the individual transmission probability for each model to compare their binary classification performance. The ROC curve is a graphical representation that plots the True Positive Rate (TPR) against the False Positive Rate (FPR) across all possible thresholds. The AUC values range from 0 to 1, with the higher AUC values signifying superior performance of the predictive model.

We assessed the goodness of fit through posterior predictive checks using aggregated Bernoulli draws at the experiment level. This approach involves simulating replicated datasets by drawing from a Bernoulli distribution and then summing up these binary classifications to compute the total number of positive outcomes. We compared the mode of the simulated aggregates to the observed number of positive cases for each experiment graphically.

The Bayesian analyses were performed using the MCMC algorithm and were conducted in Python (version 3.9) using the PyMC language model (version 5.10.3) with BlackJax backend (version 1.1.0). WAIC computations were handled using the built-in function from Arviz library (version 0.17.0). The statistical tests were conducted using R (version 4.4.2).

Observed contact distribution and patterns

In our analysis, we considered only those experiments for which we had a complete dataset covering the entire duration of the experiment.

Fig S2-A (S1 File, Section 2) shows the distance distribution among animals for all experiments. From a visual inspection, the distance distribution appears different between experiments, which is confirmed by the Kruskal-Wallis test results ( = 1331181, p-value < 2.2e-16) indicating that some of the distributions are different, and the post-hoc pairwise Wilcox test, indicating that there is a significant difference among all the experiments (all p-value < 2.2e-16) except two (namely the experiments 1_2 and 2_2) for which the corresponding p-value = 0.19.

Fig S2-B (S1 File, Section 2) summarizes the distribution in the form of a boxplot. As we can notice, the median distance in each experiment varies. In particular, we notice that for five of them the median is in the range 1.5–2.5 m., while for the others it is between 3.3 and 4.5 m. The differences among contact patterns could indicate different behavior of the animals, due to external reasons, and could influence transmission of the virus. We analyzed the temporal network to identify possible eras, i.e., specific periods related to different collective behaviors that could indicate the presence of circadian activity. In Fig S3 (S1 File, Section 2), the median (solid line) and the minimum and maximum distances among animals are reported for each second of the exposure phase. The era identification analysis shows inconclusive results: a single large period was identified, lasting more than 90% of the time, together with some very short eras (of the order of the minutes).

Instead of focusing on the characteristics of the overall network, we extracted information about the contact pattern between the seeder and each susceptible animal (called proximity analysis). The proximity results presented in Fig S4 (S1 File, Section 2) shows the fraction of time susceptible animals spent in very close contact (less than the 0.5 m), medium range contact (0.5–1.0 m), long range (1.0–2.0 m) and distant contact (more than 2.0 m) The pattern of dynamic behavior is characterized by frequent, spontaneous fluctuations, where susceptible individuals intermittently reduce their proximity to the seeder before increasing the distance again. Specifically, susceptible individuals would briefly enter closer proximity zones (under 1 meter) before returning to the mostly distancing pattern of beyond 2 meters. Overall, individuals spend a very small fraction of time in close contact with the seeder, while most of the time they are at a large distance.

Clinical observations and serology results

In this section, we present the results of the tests and observations conducted to assess the animals’ health status and to determine whether they were infected (through RT-PCR), as well as their immune response to the exposure to the pathogen (through serological testing). The data collected from each experiment, including the monitoring period and the number of positive serological results, are summarized in Table 2. Overall, 18 positive cases based on positive serology were identified out of 70 susceptible individuals. A significant association between exposure time and infection status was found (Fisher Exact test, p-value < 0.001), indicating that exposure duration influences infection likelihood. Three of these (ID 32L, 42L and 81) were found to be positive during post-mortem analysis, while the others exhibited an immune response (positive serology) during the monitoring period. Only five animals (ID 104, 108, 99, 4 and 4D) were also positive in the RT-PCR test, and of these, only three (ID 104, 108 and 99) exhibited symptoms such as nasal and ocular discharges, diarrhea. At this last symptom, 4 animals were humanely euthanized (see Table 3).

Download:

• PNG
  larger image
• TIFF
  original image

Table 2. Summary of experimental infection results by experiment session and duration.

https://doi.org/10.1371/journal.pone.0358839.t002

Download:

• PNG
  larger image
• TIFF
  original image

Table 3. Day of symptom onset, RT-PCR positivity, serology positivity, and death/humanely euthanized for the confirmed cases of PPR by experiment and ID, as the number of days elapsed since the end of the exposure phase.

https://doi.org/10.1371/journal.pone.0358839.t003

Fig 3 visually depicts the proportions of infected versus non-infected individuals across different exposure time categories, emphasizing that no infections were recorded for exposure times under 24 hours. We observed that experiments with longer durations of exposure resulted in a higher number of infections, indicating that the length of the exposure period may impact the probability of transmission.

Download:

• PNG
  larger image
• TIFF
  original image

Fig 3. Proportions of infection by exposure time categories.

Outputs of the experimental infections are grouped based on the duration of the exposure phase. Red corresponds to the fraction of positive in each category, Cyan to the negative ones.

https://doi.org/10.1371/journal.pone.0358839.g003

Transmission rate estimation

The transmission rates were initially estimated on an hourly basis. However, for reporting purposes, these values were multiplied by 24 to convert them into daily rates. The estimated value of is 0.86 (95% highest density interval (HDI): 0.62–0.94).

The basic reproductive number is estimated to be 4.3 (95% HDI: 3.1–4.8) when considering an infectious period of 5 days as used by El Arbi et al. [42] and 8.6 (95% HDI: 6.2–9.6) when considering an infectious period of 10 days, as reported by Fournié et al. (2018) [15] and Herzog et al. (2024) [17].

Incubation period estimation

Table 3 shows, for each positive confirmed animal, the day at which it showed symptoms (symptoms), was positive to RT-PCR or serological tests, and in case of deaths, the day of death. For the survival analysis, we considered the first day on which one of these factors was registered. As the table shows, very few positive animals (only 5) showed symptoms or were positive to RT-PCR during the follow-up period, indicating that the virus may circulate sub-clinically in a herd. This occurred an average of 11 days after exposure, with a range of 7–17 days. The same animals tested seropositive around two days later on average (2.3 days). Animals that did not exhibit symptoms became seropositive within an interval of 9–37 days after exposure phase.

Preliminary survival analysis was done to check if there was a difference between exposure duration and length of the incubation period. The log-rank test showed no significant difference between the groups (Chi-square 0.1, p-value 0.8). Because of this, we conducted survival analysis without taking into consideration the duration of the exposure phase. Among the three parametric models, the LogNormal one outperformed the others (Table 4). The average incubation period is around 14.7 days (95% CI 11.6–20.1).

Download:

• PNG
  larger image
• TIFF
  original image

Table 4. Estimation of the incubation period using three different parametric distributions (Weibull, Gamma, and LogNormal) showing the median of the parameter estimates, the 95% Credible Interval and Watanabe Akaike Information Criterion (WAIC).

https://doi.org/10.1371/journal.pone.0358839.t004

Fig 4 shows the empirical Kaplan-Meier curve, and the theoretical one obtained using a lognormal distribution. During the first four days after exposure, none of the animals can secret the virus. From the fifth day till the 20^th day, the probability of not developing symptoms decreases. Around the 14^th day, half of the population is no longer incubating, and animals either show symptoms or are tested positive. After the 22^nd day, only a small percentage of animals still develop symptoms.

Download:

• PNG
  larger image
• TIFF
  original image

Fig 4. Kaplan-Meier curve for the empirical (black solid line) and the estimated one (blue).

The shaded area corresponds to the 95% C.I. of the survival plot. The dashed line indicates the time (in days) half of the population has become infectious.

https://doi.org/10.1371/journal.pone.0358839.g004

Transmission probability estimates

The preliminary analysis using logistic model results implies a significant effect of the group on the infectious status indicating that there could be actually a difference in the outcomes that could depend on the exposure/contact pattern (See S1 File, Section 6).

In Table 5, an overview of the prior distributions used for each parameter across all models is presented. For the transmission probability parameters, we employed Beta distributions to constrain values between 0 and 1, reflecting the probabilistic nature of these parameters. The decay parameter in the envelope model was assigned a Gamma distribution to ensure positivity while allowing for a range of plausible values. For scaling factors in the Stratified model, we used a uniform prior over a range that encompassed biologically reasonable values.

Download:

• PNG
  larger image
• TIFF
  original image

Table 5. Prior distributions for model parameters.

https://doi.org/10.1371/journal.pone.0358839.t005

The estimated probability parametersfrom MCMC, differ across models depending on how they incorporate distance. The results of the inference through calibration are shown in Table 6. Probability estimates for the Baseline case are the lowest. This is mainly due to the fact that in the model, individuals are considered always to be exposed independently of the distance.

Download:

• PNG
  larger image
• TIFF
  original image

Table 6. Summary of posterior distributions for all models.

https://doi.org/10.1371/journal.pone.0358839.t006

For the other models, spatial information was incorporated post hoc, by applying distance-based filters to the proximity data after it was collected; no constraints were imposed during data generation or experimentation. Segmented models used simple distance filters (e.g., allowing transmission only within 1 m, 1.5 m, or 2 m) produced progressively lowerestimates (p = 0.0063, 0.0028, and 0.0014, respectively), showing that increasing the spatial window reduces the estimated per-individual transmission probability. Stratified models further refined this approach by dividing the spatial range into intervals (e.g., 0.05–0.5 m, 0.5−1 m, 1–1.5 m, etc.), and we noticed that the more models cover close-contact (0.05–0.5 m) the more the probability estimates increase (e.g., Stratified 2, Stratified 5 and Stratified 1), and for the case when models integrated distances above 2 m, theestimates decrease (e.g., Stratified 3). In particular, we noticed that for Stratified 2 model the probability of transmission at very close distance (less than 0.5 m) is 39 times larger than in the baseline case. This result indicates that spatial density strongly impacts the diffusion of pathogens.

Model evaluation results

The ROC curve analysis was performed to assess the diagnostic performance of our models, with the mean AUC and its 95% HDI presented in Table 7. Fig 5 illustrates the average ROC curves for these models, showcasing their sensitivity and specificity across various thresholds. The analysis of AUC values provided critical insights into the predictive capabilities of the evaluated models. Notably, the baseline model and the Stratified 4 model exhibited the highest AUC values, indicating superior predictive accuracy, with the baseline model achieving an AUC of 0.84 and the Stratified 4 model closely following at 0.82. These findings suggest that both models are highly effective in distinguishing between different classes. Additionally, the Stratified 3 and Stratified 1 models demonstrated similar AUC values of 0.81 each, reflecting similar levels of predictive efficiency. In contrast, the segmented and envelope models yielded lower AUC values overall; however, within their respective categories, the segmented model and the envelope model performed relatively better at distances below 2 meters, achieving AUC values of 0.73 and 0.72, respectively. These results underscore the robustness of both basic and Stratified models in predictive tasks while highlighting potential limitations associated with segmented and envelope models.

Download:

• PNG
  larger image
• TIFF
  original image

Table 7. Comparison of AUCs for different models.

https://doi.org/10.1371/journal.pone.0358839.t007

Download:

• PNG
  larger image
• TIFF
  original image

Fig 5. Mean ROC curves of best models.

Colors correspond to the best models identified in each category and compared to the baseline case when no contact information is considered. The average ROC and AUC are estimated from the average over a parameter sample of 100 000 transmission probability values.

https://doi.org/10.1371/journal.pone.0358839.g005

In Table 8, we summarized the performance of the models based on the WAIC results. In the table “elpd_waic”: This is the expected log pointwise predictive density (ELPD). High values indicate better predictive accuracy. “p_waic”: This is the effective number of parameters in the model that provides an estimate of model complexity; “elpd_diff”: This is the difference in elpd_waic between the current model and the best model. Smaller differences indicate that the models have similar predictive performance; “se”: The standard error of ELPD estimate. “rank” indicates the ranking of models based on their WAIC values. A low rank indicates a better model.

Download:

• PNG
  larger image
• TIFF
  original image

Table 8. WAIC comparison of the different models to assess their performance and complexity.

https://doi.org/10.1371/journal.pone.0358839.t008

We compared multiple models using WAIC. The baseline model had the highest ELPD (elpd_waic = –27.93) and thus the best out-of-sample predictive performance. The Stratified 3, Stratified 4, and Stratified 1 models showed minor decreases in ELPD relative to the baseline (elpd_diff = 0.84, 0.95, and 1.02 respectively), with standard errors that suggest these differences are not substantial (dse = 1.76, 1.86, and 1.9 respectively). All of these models had similar complexity, with p_waic values between 0.88 and 1.3.

The remaining models Segmented, Envelope and some Stratified models had notably lower ELPD indicating some problems in predictive performance. The elpd_diff values for these models were larger than the associated standard errors (se), supporting the conclusion that these models predict less well than the baseline and top Stratified models.

Goodness of fit

We conducted Bernoulli simulations using the estimated probability valuesderived from the best-ranked models: Baseline, Stratified 4, Stratified 3, and Stratified 1, to generate binary outcomes (0 or 1) for each scenario. These simulations were grouped by experiments, and the total number of positive cases per experiment was calculated by summing the binary outcomes. The results are summarized in Fig 6, which compares the predicted mode number of positive cases against the real observed values. The results demonstrate that the four models are nearly juxtaposed, indicating that their performance in terms of fitting and prediction is almost identical. This suggests a high level of consistency across the models in capturing the underlying patterns of the data. Additionally, the predicted curves closely follow the trends of the real observations, further validating the models’ accuracy. However, an exception is noted in experiment 2_24, where the predicted outcomes deviate slightly from the observed trends.

Download:

• PNG
  larger image
• TIFF
  original image

Fig 6. Comparison of model-predicted and experimentally observed PPR positive cases.

Predictions are displayed as mode values, with error bars indicating the standard deviation, and real observations are presented as blue dots.

https://doi.org/10.1371/journal.pone.0358839.g006

Exposure time estimation

To estimate the minimum exposure time, we simulated various exposure scenarios. In Fig 7 we report the results of the estimation of the probability for an individual to get infected while in contact with an infected animal after a certain amount of time. We considered several scenarios, based on assumptions about contact patterns. We initially used the estimated values of the probability of success from the baseline to determine the minimum exposure time, measured in hours, needed for successful contamination. In this example, we assumed that animals could move freely and that the chance of infection is uniform. If one of them is infected, there is a 50% chance for a naïve animal to catch the infection after being exposed for about 38.50 hours corresponding to one day and a half. On the other hand, if the probability of infection is 0.95, it is estimated that the average exposure time is around 166.40 hours (almost one week). These estimations are based on the fact that animals can move freely and interact at short distances rarely. However, livestock markets are very crowded places, where thousands of animals spend 6–8 hours in very close contact. To estimate the exposure times in these conditions, we considered simulations using the parameters for close contacts estimated by our models.

Download:

• PNG
  larger image
• TIFF
  original image

Fig 7. Estimating the minimal exposure duration required to achieve contamination.

The solid lines represent the cumulative probability of an animal becoming infected after being exposed for minutes. Colors indicate different scenarios: the baseline case without contact patterns, and hypothetical cases where animals remain at fixed distances, Stratified 1 was used for the first distance segment 0.05-0.5 m, and Stratified 4 for the remaining segments. Dashed lines mark the thresholds corresponding to 50% and 90% infection probability.

https://doi.org/10.1371/journal.pone.0358839.g007

The analysis of simulation using the Stratified 1 and Stratified 4 models revealed a consistent positive relationship between exposure time and the inter-individual distance. We used Stratified 1 model to estimate the exposure time related to a fixed distance within the first range of 0.05–0.5 m and Stratified 4 for the rest of distance segments. The exposure duration increased with distance to reach the probability thresholds of 0.5 and 0.95. To reach a transmission probability of 0.5, the mean exposure durations were 2.51 hours, 7.21 hours, 17.60 hours, and 31.83 hours for the distance segments 0.05–0.5 m, 0.5–1 m, 1–2 m, and ≥ 2 m, respectively. To reach a probability of 0.95, the required durations increased substantially, with corresponding values of 10.83 hours, 31.18 hours, 76.09 hours, and 137.58 hours, respectively, across the same distance intervals. These estimates indicate a substantial increase in the average exposure time required to achieve higher probabilities of infection, highlighting the variability and escalation of exposure duration necessary as the risk of infection increases.