The generic model

We model transmission of disease in continuous time across a region . In principle, any disease could be modelled, whether of animals, plants or humans. However, the case study presented focusses on animal disease. We partition the region into a collection of patches, with each patch indexed by an integer () and associated with a given sub-region of . Within the region , the spatial distribution of the population at risk is uncertain. However, we assume that we have covariates available that can be used to model the variation in patch susceptibility/infectivity across the region. In practice, we choose covariates that are likely to explain at least some of the variation in host population risk profile across the region. The patches are the units of infection within the model framework. For each patch , there is a set of associated covariate values . In principle, the covariates may vary both temporally and spatially. However, for notational simplicity, we do not show explicit time dependence in the presentation that follows, and in all the applications of the methodology that we consider in this manuscript the covariates are constant in time.

The chosen size of the patches depends on a number of factors, including the spatial resolution of the covariates and the disease case data. If the covariates have a different spatial resolution to the case data then one or both data sets may be aggregated or de-aggregated to the chosen resolution. Patch size should be small enough to capture the apparent pattern of disease spread across the region.

For simplicity we consider modelling transmission according to a spatial SI model. That is, each patch is either susceptible to infection or has become infected/infectious. However, as shown in Case study, even where individual-host dynamics are not limited to susceptible and infectious compartments, patch-level SI dynamics can still provide a good approximation to the spatial transmission process. In addition, we note in general that more complex dynamics and a greater set of possible states could in principle be accounted for at the patch-scale if suitable data were available. However, such complexities are outside the scope of the current implementation. The SI assumption is most appropriate for the early stages of a disease outbreak, but can continue to be reasonable if the disease is persistent at the patch level on the timescales of the outbreak and this criterion should be considered when selecting an appropriate patch size for the analysis – for example, see Case study. The transmission of disease is modelled over some time interval via a continuous time discrete state space Markov process. The initial conditions comprise a list of the patches that are infected at time , with the remainder assumed susceptible.

Given the initial conditions, the transmission model is fully specified once we define the force of infection on a susceptible patch at any point in time. The force of infection

(1)

on a susceptible patch is comprised of two terms, the first term representing primary infection due to external transmission from outside the region under study, the second term representing transmission from infected patches within the region . Each patch has susceptibility and, when infected, infectivity which can be functions of spatially varying covariates – meaning that they vary between patches. The parameter controls the overall rate of background infection, while controls the overall rate of secondary transmission. The summation is over all infectious patches , is a so-called transmission kernel and is the distance between the centroids of patches and . Typically, we use a truncated power law kernel . Here, is an indicator function (equal to 1 if its argument is a true statement, 0 otherwise), is a normalisation constant defined in S1 Appendix (see also [27]), is a fixed maximum transmission distance, and is a parameter estimated from the outbreak data. If the patches are all squares of the same size then distance, as an input to the kernel function, is measured in units such that one unit is equal to the width of a patch, meaning that, e.g., precisely when patches and are nearest neighbours. The above expressions together represent what we call the constant-in-time model, although as noted above this can accommodate temporal variation in the covariates.

We also consider a variant of this model, which we term the varying-in-time model, in which the total force of infection is multiplied by a function of time . This function is intended to model the effects of disease control. For identifiability we assume that , with then typically decreasing following the application of control measures such as removal of hosts. However, we assume only that for all , allowing to both increase and decrease over time, so that such changes are estimated from the data. For simplicity we assume that is piecewise constant on pre-defined subintervals of the modelled time interval with for identifiability, then on the ’th subinterval (.

We now define the susceptibility and infectivity of a patch in terms of the covariates for that patch. There are many ways of defining the relationship between susceptibility/infectivity and the covariates. Here we suggest a general approach. Typically, covariates are either compositional or non-compositional [28]. Non-compositional covariates have unconstrained values, for example, average temperature and rainfall for the patch. Compositional covariates are constrained and dependent, for example, the proportion of the patch occupied by different land uses. These covariates are proportions that sum to one. Suppose that the set of covariates for patch comprises non-compositional covariates and compositional covariates , with being the proportion of patch on which a spatial categorical variable takes value (of distinct possible values). For example, these proportions may be obtained from a partition of each patch into distinct land uses. Then, a plausible model for the susceptibility of a patch is where the Greek letters are parameters to be estimated and each . We also require that for identifiability, bearing in mind that will be multiplied by the overall rate in the expression for the secondary force of infection. This general form of susceptibility function is analogous to the suitability function defined in [27] in the context of biological invasions. A similar expression is assumed for patch infectivity . Again, the Greek letters for and for are parameters to be estimated, with each and .

Each of the categories contributes additively to the susceptibility, with the contributions weighted by the covariates . Each non-compositional covariate contributes multiplicatively, with negative relationships possible for these covariates when . However, it is important to note that negative relationships are also possible for the compositional covariates, even though the are non-negative, due to the constraint that . For example, if the smallest component of is then the covariate will be negatively related to susceptibility in the sense that, if increases and the for decrease subject to the summation constraint, then the overall effect is that will decrease.

Bayesian inference and prediction based on disease case reports

This section describes details of the approach to estimating model parameters from data, how we compare between different model formulations, quantify uncertainty in model predictions and test performance of the iPAR framework. Some readers may wish to skip these details on initial reading.

Parameter and latent variable estimation.

Having defined the model, we now present methodology that allows fitting of the model to data including estimation of unobserved (latent) infection times. For a disease outbreak, case reports describe observed cases of the disease in individual animals or on individual farms. We assume that each case has a location and a detection time associated with it. Locations are taken to be specified at patch resolution or better. Detection times of disease cases are taken to be interval-censored, which allows for the common scenario whereby new case reports are provided at regular intervals (e.g., daily, or monthly), but can also accommodate irregular observation times. We denote these reporting times by where is the time interval over which case report data are available. We assume that interval-censored detection times should constrain the actual infection times of the individuals/farms within the patch, i.e., that a case occurring after time but before time is reported at time . In practice, we expect performance of this modelling framework to degrade if a significant number of infections occurring in an interval are unobserved at . To some extent this can be countered by extending these reporting time windows used in the analysis resulting in greater estimated uncertainty in inferred infection times (see below). When aggregating to the patch level, we define the infection time of a patch to be the earliest recorded infection time of the set of individuals/farms within it. We assume that all infected patches are detected; in this sense the system is taken to be perfectly observed, a common assumption in the literature, see, e.g., [19,29].

We use Bayesian techniques to infer the model parameters from outbreak data. The true infection time of each patch is not known with certainty so, as is common when fitting epidemic models, we use data augmented MCMC to facilitate the inference [7]. This general approach to inference is well established in the literature and has been applied in, for example, [29–31]. Let denote the set of model parameters and let denote the set of unobserved patch infection times. For each patch , let denote the ‘time category’ into which the patch falls in terms of reporting. So, if was already reported as infected at time and () if was first reported as infected at reporting time . Finally, if was not reported as infected at any point in the time interval , which we assume implies that remains susceptible at time . Finally, let denote the set of observed . These observed time categories are the data we use to infer the model parameters and the actual and unobserved or latent patch infection times . Bayes’ theorem states that

Here represents the joint prior distribution of the model parameters. The prior distribution is typically chosen to be non-informative (see S2 Appendix for details). The factor above is usually termed the complete data likelihood for the model parameters conditioned on the observed data and a given set of latent infection times. The transmission process is a continuous time discrete state space Markov process, so this likelihood can be computed using standard techniques as described in, for example, [30,32]. We are now in a position to draw samples from the joint posterior distribution . This is achieved by Markov Chain Monte Carlo (MCMC) methods. Full details of the likelihood calculation and sampling algorithm may be found in S3 Appendix. The computer code for running the simulations and inference was written in C, with R used for pre- and post-processing of data.

Assessing the iPAR framework performance

We now assess the iPAR framework by applying it to simulated outbreak data. In Estimation and prediction for an illustrative simulated outbreak, we analyse data from a single simulated outbreak – simulated from the constant-in-time model with spatially varying susceptibility/infectivity – in order to illustrate the typical workflow and outputs obtained from the model fitting process. In Estimation of key epidemiological parameters, we conduct a systematic simulation study in which we generate outbreak data for a wider range of scenarios, though still for the constant-in-time model with spatially varying susceptibility/infectivity, and assess the ability of the inference to recover the ‘true’ parameters. In Estimation of temporal trends in transmission, we explore the ability of the inference to recover temporal trends in transmission, using simulated outbreak data from the varying-in-time model with spatially varying susceptibility/infectivity. Finally, in Benefits of modelling spatial variation in susceptibility and infectivity, we assess the benefits of modelling spatial variation in susceptibility and infectivity, one of the key aspects of the iPAR framework, both in terms of estimating key parameters and in terms of predicting held-out future simulated outbreak data. To do this, we generate simulated outbreak data from the constant-in-time model with spatially varying susceptibility/infectivity, but we fit either a model of similar form or a model which assumes spatially homogeneous susceptibility/infectivity.

The results of the assessments carried out here demonstrate that the iPAR framework is able to reliably infer epidemiologically relevant information from case-only reports across the range of scenarios considered. We found that the simulations run very quickly but the inference code can take between 1 and 5 hours to run for the scenarios presented here. The time taken increases with the number of patches and the number of patches that become infected, but in a non-straightforward manner due to optimisations implemented in the inference code. Computations were carried out using an Intel i7-8665u processor with 32GB memory. Five MCMC chains were run in parallel, with 100000 iterations per chain. Simulated trajectories were selected at random. Some readers may wish to skip the following details on initial reading and proceed directly to Case study, later returning to gain greater insights into those latter results.

Estimation and prediction for an illustrative simulated outbreak

Outbreak data are simulated using the constant-in-time iPAR model on the Estonian landscape, which is divided into 575 10km square patches, with model parameters similar to those estimated from the outbreak data for ASF in wild boar in Estonia (see Case study). As in the case study, land use data for Estonia are used as the basis for computing covariates used to model spatial variation in both susceptibility and infectivity. Four patches are initially seeded with infection, and the duration of the simulation is chosen such that a substantial proportion of patches become infected by the end of the simulation.

A single simulated outbreak can be summarised by plotting the number of infected patches over time (Fig 1A, black line). The constant-in-time iPAR model was fitted to the early-stage epidemic data up to month 12 from this single simulated outbreak, with simulated patch infection times interval censored into one-month long intervals. The resultant posterior distribution provides estimates for all model parameters. For example, we can visualise the posterior marginal distribution of the background transmission rate parameter (Fig 1B, with true value represented by a vertical line) by aggregating MCMC samples. We can also generate the 95% credible intervals for parameters and functions of the model parameters such as the transmission kernel (Fig 1C, true kernel in black, 95% credible interval in blue shading). Credible intervals for the components of the susceptibility parameter (Fig 1D, 95% credible intervals represented by blue lines) are often quite wide, but in general these intervals align well with the ‘true’ values.

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Fig 1. Estimated parameters and predictions from fitting the iPAR model to simulated outbreak data.

A: number of infected patches over time in the simulated outbreak data (black line). The model was fitted to the outbreak data up to month 12. Predicted number of infected patches (95% credible interval) beyond month 12 is shown in blue. B: marginal posterior distribution for the background transmission rate, with ‘true’ value indicated by the black vertical line. C: ‘true’ kernel function (black line) and estimated kernel function 95% credible interval (blue shading). D: estimated susceptibility parameter for each land use (95% credible interval, blue) and ‘true’ value (black circle).

https://doi.org/10.1371/journal.pcbi.1012622.g001

The model, fitted to data up to month 12, was then used to predict progression of the epidemic beyond the first 12 months. This was achieved by simulating 10000 times with parameters sampled from the posterior distribution and initial conditions specified by the state of the epidemic at month 12. These predictions may be summarised by 95% credible intervals for the number of infected patches over time (Fig 1A, blue shading). The intervals contain the ‘true’ epidemic curve in this case, suggesting satisfactory inference of the model parameters.

Estimation of key epidemiological parameters

We systematically tested the constant-in-time model over a wide range of parameter combinations using the sensitivity analysis approach described in Testing the iPAR modelling approach in Methods. The simulation model set-up was identical to that in the previous section except for the use of different sets of model parameters. The data from most of the simulated outbreak (up to month 52) was used to fit the model. Plots of true versus inferred parameters are available in Fig 2 for , , and in S6 Appendix for the remaining parameters. Key model parameters such as the kernel parameter and the background transmission rate are well recovered from the simulated outbreak data, though the overall transmission rate tends to be somewhat overestimated. Reasons for the latter phenomenon will be discussed later. The susceptibility parameters tend to be well recovered, while the infectivity parameters seem more difficult to recover, which is consistent with the results in [36].

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Fig 2. Reliable parameter inference for the constant-in-time iPAR model.

Each panel corresponds to a parameter of the model. The ‘true’ value of the parameter is plotted against the posterior median (cross) and posterior 95% credible interval (vertical line). For comparison purposes we also superimpose the diagonal line representing perfect agreement between inferred and true values. This figure includes panels for the kernel parameter , the overall transmission rate and the background transmission rate . Analogous figures for the other parameters may be found in S6 Appendix.

https://doi.org/10.1371/journal.pcbi.1012622.g002

Estimation of temporal trends in transmission

Next, we focus on the varying-in-time iPAR model with parameters similar to those obtained from fitting this variant of the model to the Estonian ASF outbreak data (see Case study) as a baseline parameterisation. We then vary the form of the temporal trend function according to three pre-chosen scenarios: (1) decreasing linearly (perhaps representing increasing levels of disease control), (2) constant (no disease control), and (3) initially constant then suddenly drops to low levels (corresponding to rapid strong disease control after some delay). The changepoints of the function are fixed and are kept equal to those used in the case study. The function from the first scenario is closest to what was actually estimated from the data in the case study. Note that, in the case study, was seen to initially increase before decreasing, which is why we have also allowed for an initial increase in in the first and third scenarios. We assess the extent to which these temporal trends can be recovered by inference from data simulated under each of the above three scenarios. We also fit the constant-in-time model to the same sets of simulated outbreak data, which allows us to compare model fits (constant-in-time versus varying-in-time) using as discussed in Model selection in Methods.

Fig 3 shows the simulated ‘true’ incidence curve for each of the three temporal trends. Overlaid on this ‘true’ incidence curve is a region representing 95% credible intervals for incidence, obtained from simulations of the fitted varying-in-time model with initial conditions the same as those of the simulated outbreak data used for fitting the model. For all three temporal trends, the agreement between the credible intervals and the ‘true’ incidence curve shows that the algorithm is able to reliably recover parameters giving temporal trends in incidence that agree with the ‘true’ incidence curve. Fig 3 also shows the 95% credible interval for in the same set of three simulations. Despite the striking fact that the form of this function is not at all obvious if presented with a single outbreak incidence curve, the trend function is inferred correctly from single outbreaks, though with some overestimation. We also computed values for each model fit (see S4 Appendix for actual values obtained). For scenarios (1) and (3), in which , the indicated a strong preference for the varying-in-time model. For scenario (2), in which , there was a mild preference for the constant-in-time model. These results were all as we would expect.

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Fig 3. Reliable recovery of temporal trends in the time varying simulation study.

A: the black line shows the incidence curve from a single simulation of the “linear decrease” scenario. The blue area shows the 95% credible intervals for incidence, defined as newly infected patches each month, obtained from the posterior predictive distribution of the fitted varying-in-time model, given the data from that single simulation. B: true and estimated for the same scenario. Black line represents true value, blue area shows the 95% credible interval. C,D: same plots for the “constant over time” scenario. E,F: same plots for the “sudden drop off” scenario (see text for details).

https://doi.org/10.1371/journal.pcbi.1012622.g003

Benefits of modelling spatial variation in susceptibility and infectivity

We assess the importance of modelling spatial variation in susceptibility and infectivity in the context of the constant-in-time iPAR model with land use covariates but no non-compositional covariates. The spatial variation in the patch susceptibilities and infectivities of the model arises as a result of spatial variation in the covariates, combined with differences in the susceptibility and infectivity parameters between land use categories. Our approach here is to simulate outbreak data incorporating such spatial variation and then fit two model variants to the simulated outbreak data. First, we fit a heterogenous model which accounts for this spatial variation and has the same form as the model used for simulation. Second, we fit a simpler homogeneous model in which there are no covariates and patch susceptibility/infectivity is spatially constant. Comparing the two model fits and their associated predictions enables us to assess the importance of taking into account variation in susceptibility and infectivity.

We consider model fitting given two levels of data availability. First, we fit models to data from the early stages of an outbreak. Second, we fit models to data from an entire outbreak. At both levels, we consider four different parameter combinations for the simulated outbreak data. This is less than the number of parameter combinations considered in Estimation of key epidemiological parameters, however the chosen parameter combinations cover a range of common transmission patterns. We use either a wide or a moderately wide transmission kernel. Narrow kernels, such as those approximating nearest neighbour kernels, are avoided because we intuitively expect there to be little difference between the predictive performance of the heterogenous and homogeneous models in this case – see Case study and associated discussion for more on this topic. In the simulations, we assume that there is a single land use with high susceptibility (all other land uses are assigned equal low susceptibilities), and the same applies for infectivity. Broadleaf woodland is taken to have high susceptibility in all scenarios, while the high infectivity land use alternates between broadleaf woodland and arable land. Precise details of the scenarios are provided in S8 Appendix.

We repeat the model fitting and prediction for 20 replicate simulations per parameter combination, in order to give a fuller picture of parameter recovery and predictive performance in each case. Mean (over all 20 replicates) AUC and mean TPR at 5% FPR measure discriminatory power. We compute the posterior predictive distribution of , the predicted number of infected patches divided by the true number of infected patches, and its reliability which is the % of replicates in which the 95% CrI for includes . We also compute the mean bias (posterior median of minus , averaged over all 20 replicates). Corresponding quantities are computed for the kernel parameter . For the susceptibility and infectivity parameters we adopt a different approach. We focus on estimation of the large susceptibility parameter and the large infectivity parameter. These parameters are likely more difficult to estimate than the other susceptibility/infectivity parameters because they are far from the main mass of the prior distribution on the simplex. So, we measure the difference between the posterior and prior medians (averaged across all 20 replicates), which captures the extent that the posterior has moved away from the prior. If the difference is very small then there is little information in the data regarding that parameter. For the given simulation set up, a posterior median that is 0.73 higher than the prior median corresponds to perfect estimation of the large susceptibility/infectivity parameter.

Using only early-stage data, predictive performance assessed using discrimination metrics was consistently better for the heterogeneous model, if not by a very large margin (Table 1). Ability to predict was good for both the heterogenous and homogeneous models. In terms of parameter recovery using only early stages data, we focus on the transmission kernel parameter, the susceptibility parameter associated with the high susceptibility land use class and the infectivity parameter associated with the high infectivity land use class. The kernel parameter was typically well estimated (Table 2), though the homogeneous model tended to somewhat overestimate this parameter, a phenomenon that was also observed by [27]. However, susceptibility and infectivity parameters were imperfectly recovered (Table 2). Comparison of the posterior and prior medians suggests that the early-stage data used for fitting provides some information on susceptibility but is less informative on infectivity.

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Table 1. Early-stage data predictive performance.

https://doi.org/10.1371/journal.pcbi.1012622.t001

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Table 2. Early-stage data parameter recovery.

https://doi.org/10.1371/journal.pcbi.1012622.t002

In the ‘entire outbreak’ data availability setting, it is no longer meaningful to consider predictive performance on held-out future outbreak data. Parameter recovery is assessed for the same set of parameter combinations as for the ‘early stages’ setting. Reliability of inference for the transmission kernel parameter is excellent for the heterogenous model but less good for the homogenous model (Table 3). This parameter is consistently overestimated but especially by the homogeneous model, which again is consistent with what was observed in [27]. The posterior medians of the (large) susceptibility and infectivity parameters have shifted closer to their true values when compared to Table 2. This is as we would expect given that we have more data available to fit the model in this setting. In particular, there is now clearer evidence of a shift in the infectivity parameter posterior distribution away from the prior distribution.

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Table 3. Full outbreak parameter recovery.

https://doi.org/10.1371/journal.pcbi.1012622.t003

Finally, to put these results into context, we take a closer look at the spatial variation in the Estonian landscape, which we have used for these simulations. At the 10km resolution, the Estonian landscape is not particularly heterogeneous in terms of land use (Fig 4). Most of the 575 10km square patches comprise a mixture of land uses, with very few patches consisting mostly of a single land use. The low heterogeneity in land use is one factor leading to limited variation in estimated patch susceptibilities (Fig 5). In general, we might expect that increased heterogeneity in land use, increased differences in susceptibility/infectivity between land uses, a wide kernel and higher numbers of available case reports are all factors that should increase the benefits of modelling spatial variation in susceptibility and infectivity.

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Fig 4. Landscape heterogeneity in Estonia.

A: ternary diagram showing the land use composition of every 10km square patch covering Estonia (land use amalgamated into three categories – forest, agricultural and other – for ease of visualisation). B: histogram showing maximum patch land use proportion for every 10km square patch in Estonia, with six land uses as in the model covariates.

https://doi.org/10.1371/journal.pcbi.1012622.g004

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Fig 5. Typical distribution of inferred patch susceptibilities.

Typical distribution of (inferred) posterior mean patch susceptibilities given data from a simulated disease outbreak in Estonia. There is some variation but few patches with very high or low susceptibilities.

https://doi.org/10.1371/journal.pcbi.1012622.g005

Background/approach

Here, we used all ASF cases in wild boar populations reported in Estonia between 2014 and 2019 to the World Organisation for Animal Health (WOAH). The Estonian ASF outbreak had very serious impacts for both the pig industry and wildlife. African Swine Fever leads to high mortality, and its spread is difficult to control. It continues to pose a major ongoing threat in many regions of Europe [38–40]. In this study we focus on better quantifying risk of transmission and spread of ASF in the Estonia wild boar population. Knowledge of the wild boar population in this region is incomplete, though estimated wild boar density maps have been published [39]. Within Estonia, there have been extensive efforts to control the ASF outbreak, partly by reducing the density of wild boar. These efforts are reflected in density maps showing an apparent reduction in the population over time [39]. Case reports of ASF in dead wild boar are available at monthly time intervals, with detailed location information. Therefore, given the uncertainty in knowledge of the Estonian wild boar population described above, we apply the iPAR framework to this outbreak.

To apply the framework to the ASF outbreak in Estonian wild boar, we must first define a grid of patches covering Estonia, as well as covariates likely to be related to wild boar density and hence to patch susceptibility and infectivity. Land use is one of the key factors likely to determine wild boar density [41], and hence we use high resolution CORINE land use data available for Estonia [42]. The case report data for wild boar in Estonia also have high spatial resolution. However, if the patches used to define the infectious units of the model are too small then (a) fitting the model will be very computationally intensive, and (b) the persistence assumption inherent in our assumed SI framework is less likely to be reasonable. As a compromise we initially opted for square patches of size 10km by 10km, though smaller patches would also be feasible. This choice of patch size was informed by a persistence of infection analysis, which revealed that, at the 10km resolution, ASF in wild boar in Estonia typically persists for long periods of time, often multiple years (S9 Appendix). Any of the 10km patches composed wholly of sea or inland water were removed from the set of patches. In the case of Estonia, there are 575 patches covering the country. Fig 6 illustrates the spatial spread of ASF through the wild boar population in Estonia at the 10km patch resolution.

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Fig 6. Spread of ASF in Estonian wild boar.

Spatially aggregated African Swine Fever wild boar cases reported in Estonia. Snapshots at: November 2014, September 2015, July 2016. Bottom right panel shows the outbreak incidence curve, generated using spatially aggregated wild boar case reports. Map base layer taken from https://gadm.org/maps.html.

https://doi.org/10.1371/journal.pcbi.1012622.g006

We took CORINE land cover data [42] at 100 metre resolution and spatially aggregated it to the 10km patches to obtain proportions of each land use category for each patch. We also aggregated some of the CORINE land cover categories, resulting in a total of 6 categories: 1 = urban, 2 = agriculture, 3 = broadleaf/mixed forest, 4 = coniferous forest, 5 = semi-natural, 6 = wetlands/water bodies excluding sea. We defined the susceptibility of a patch to be where the are parameters to be estimated and is the proportion of land in patch occupied by land use category . Similarly, consistent with the notation in Methods, the infectivity of a patch is defined as .

The latent period of ASF in wild boar is very short relative to the scale of the spreading process, so we have chosen to ignore this aspect of disease dynamics, which is implicit in the SI assumption. In addition, as discussed above the SI assumption represents the patch-scale dynamic which may differ from disease progression in an individual host. A case report from the outbreak is associated with a specific month. So, the time of the case report is considered to be interval-censored, with the time interval equal to the relevant month. The model can now be fitted to the outbreak data as described in Methods. We could have taken into account other covariates but, for simplicity, we chose to use land use data only. This case study is intended as a proof of principle, rather than a definitive analysis of ASF transmission in wild boar in Estonia.

Parameter estimates

The constant-in-time and varying-in-time heterogeneous models were fitted to the full set of Estonian outbreak data, with the estimated parameters as in Table 4. The metric was computed for both model fits, and was found to strongly favour the varying-in-time model (details in S4 Appendix). This is not surprising given the strong decreasing trend in that was identified (Table 4). Generally, parameter estimates were consistent between the two model variants. There is much uncertainty, but broadleaf woodland seems to have the highest susceptibility, while agricultural land has the highest infectivity. External transmission occurs at very low levels, and the estimated kernel is relatively narrow. The estimates of the overall rate for the two models are not easily comparable because, in the varying-in-time model, is multiplied by the function which is estimated as less than one for most of the observation time interval. So it is not surprising that the estimated value of is higher for the varying-in-time model.

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Table 4. Parameter estimates obtained from fitting the iPAR model to data from the entire Estonian outbreak.

https://doi.org/10.1371/journal.pcbi.1012622.t004

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Fig 7. Model prediction and validation. Left column: Predicted numbers of infected 10km patches from models fitted to the early stages of the ASF outbreak in Estonian wild boar. Two constant-in-time models (heterogenous, homogeneous) were fitted to outbreak data up to September 2015. The black lines above show the true cumulative incidence curve at 10km patch resolution. A: the blue line and region show the posterior predictive median and 95% credible interval for the total number of infected patches under the homogeneous model. The red line and region show the corresponding quantities under the heterogenous model. B: the red line and region show the posterior predictive median and 95% credible interval for the total number of infected patches under the heterogeneous model. The grey line and region show the corresponding quantities under a varying-in-time heterogeneous model fitted to the full set of outbreak data. Right column: External validation via hunting bag data from [40]. C: estimated wild boar density in the east of Estonia. D: estimated wild boar density in the west of Estonia. E: estimated overall spread rate from the varying-in-time model fitted to the entire ASF outbreak in Estonia. The black line represents the posterior median, the blue area shows the 95% credible interval.

https://doi.org/10.1371/journal.pcbi.1012622.g007

Model predictions

Finally, we assess the predictive performance of the model fitted to the Estonian data. We fit the constant-in-time model to the early stages of the outbreak in Estonia, holding out data from later stages of the outbreak in order to test the model’s predictions. Both the heterogeneous and homogeneous variants of the model were fitted.

The predicted number of infected patches under the heterogeneous and homogeneous models tends to be overestimated, though the predictions of the heterogeneous model lie a little closer to the actual data (Fig 7A). The predicted number of infected patches from simulation of a heterogeneous varying-in-time model fitted to the entire outbreak is much closer to the actual data (Fig 7B). This indicates that the varying-in-time model is sufficiently flexible to capture the key trends in the observed cumulative incidence curve in Estonia.

External validation

Many aspects of the fitted model are difficult to validate externally. However, one aspect of interest is the time varying function , which was estimated to initially increase from the baseline value of 1 in 2015 to a maximum of 1.71 in 2016, before dropping to 0.06 by the end of the outbreak (Table 4). The function is intended to model the effects of disease control measures, and in Estonia a key control measure was reduction of the wild boar population. The trend seen in the estimated time varying function is broadly consistent with reported trends in wild boar population density across Estonia based on information from hunters (see Fig 5 in [40]). This comparison provides some evidence for the validity of the fitted varying-in-time model (see Fig 7C–E).

Quantitative risk assessment for disease control

In addition to the retrospective analysis described above which shows the impact of the control measures that were adopted, the iPAR framework also provides information relevant to the within-outbreak targeting of disease control activities. For example, Fig 8A illustrates the estimated susceptibilities of patches in Estonia for the constant-in-time model, identifying which locations are more likely to be susceptible to disease incursion. Although there are some areas with greater susceptibilities there does not seem to be a strong trend in susceptibility across the country. However, Table 4 also provides potentially useful information to inform disease control efforts by identifying the land use classes (broadleaved and agricultural) that contribute most strongly to susceptibility and infectivity. Such information may be useful in guiding development of control policies and programmes.

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Fig 8. Quantifying disease risk across Estonia.

(A) Susceptibility map for Estonia obtained from fitting the constant-in-time model to the Estonian ASF wild boar outbreak data. (B) Risk map illustrating predictions from the constant-in-time heterogeneous model fitted to the early stages of the ASF outbreak in wild boar in Estonia. Grey patches are recorded as infected at the end of the early stage of the outbreak. Red colours indicate probability of infection four months beyond this end point. The black dots show patches that did in fact become infected within this time horizon. Alternative summaries of the same predictions may be found in Table 5 and Fig 7A. Map base layer taken from https://gadm.org/maps.html.

https://doi.org/10.1371/journal.pcbi.1012622.g008

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Table 5.

Predictive performance of the fitted constant-in-time model.

https://doi.org/10.1371/journal.pcbi.1012622.t005

In addition, risk maps can be helpful for visualising spatial predictions of which areas will become infected over defined scenarios and specified time horizons. Based on observation of the early stage epidemic, up to September 2015 (grey patches), Fig 8B shows model predictions of the probability of future infection. Such information can aid decisions about where to focus limited resources for disease control as the outbreak unfolds in real time. This prediction is over a specified time horizon under the assumption that rates of spread remain fixed. However, we also note that such use of this information means that actual spread will ideally be different from that predicted. This is due to the impact of disease controls as illustrated in Fig 7C–E which suggests a substantial benefit of disease control onwards from 2016.

By comparing spatial risk prediction to actual case reports over a limited time horizon we see the heterogeneous model identifies future infected patches slightly better than the homogeneous model (Table 5). However, the differences between the two models’ predictions are not substantial. One reason for this observation is likely to be that the landscape of Estonia is relatively homogeneous in the sense that there is little spatial variation in susceptibility at large scales (see Benefits of modelling spatial variation in susceptibility and infectivity in Results). Another explanation may be the narrowness of the estimated kernel, which is much narrower than the kernels used in Benefits of modelling spatial variation in susceptibility and infectivity in Results. Further simulations (S10 Appendix) similar to those in Benefits of modelling spatial variation in infectivity and susceptibility in Results but with a narrow kernel give predictive performance similar to that in Table 5.