2.1 Ethics approval for the study

The Federal Capital Territory Health Research Ethics Committee (FCT HREC) approved the study and was ratified by the University of Queensland Research Ethics Committee 2019001486 /FHREC/2019/01/04/21-01-19. No victim identifiers were present in this paper.

2.2 Data on victims of dog bites in Abuja

We performed a retrospective study of self-reported dog bite occurrence in all 14 secondary healthcare facilities in the FCT, one tertiary facility (Gwagwalada), three private facilities (Gwagwalada, AMAC and Bwari), and one Primary facility (AMAC) between January 2000 and December 2018.

To begin the study, Disease Surveillance and Notification Officers attached to the facility and student volunteers, reviewed all dog bite cases reported to the accident and emergency unit from the registers. Secondly, they retrieved all files and extracted the following information on the dog bite victim: age, gender, occupation, victims’ address, site of the bite, severity of the bite, reported use of PEP (number of vaccination doses taken by the victim). The PEP data were further classified into two categories for analysis (i.e., those who took at least one vaccine dose and those who failed to take any doses), the ownership and vaccination status of the offending dog, and the fate of the victim (whether or not the victim died showing clinical signs consistent with rabies as documented by the attending clinician (i.e. probable based on WHO classification) [18]. For example, all 49 victims who died had documented clinical notes. Of these, 9 received PEP while showing clinical signs of rabies. In addition, we attempted to confirm the fate of other victims using phone calls. However, due to logistic challenges and initial apprehension, we relied primarily on the clinical notes. Any victim who returned to the clinic for other health conditions, or responded to our phone calls, was deemed alive. We discarded information for victims who had no follow-up information and those for whom we could not ascertain the status of the victims via phone calls. Victims who died showing clinical signs consistent with rabies were classified as zero, while those alive were classified as one. All bite victims who showed clinical signs while taking PEP were considered unvaccinated. Confirmatory laboratory diagnosis for human rabies is unlikely across West Africa and Nigeria [5,19]. Hence, we based our outcome of interest on the victim’s binary fate.

For the offending dogs, only six had a confirmatory rabies diagnosis. For the ownership status of dogs, we relied on the history provided by the victims as documented by the physician. We classified dogs as free-roaming if the clinical notes indicated they were free-roaming or stray. It is likely that those dogs were owned who were allowed to roam freely, because a previous report indicated only a 3% minority of dogs are feral in Africa [20]. All information was entered manually into a standardised data abstraction tool. Finally, data were transferred into SPSS, exported to Microsoft Excel, and cleaned for analysis. We attempted to obtain data from the veterinary hospitals in the FCT and the Federal Ministry of Agriculture to supplement the data, because a study reported that Nigerian bite victims often report to the veterinary hospital [21]. However, no records were found from the clinics and the Federal Ministry of Agriculture, in agreement with a report of a study in Ogun State, Nigeria, indicated that record keeping in a veterinary hospital was poor [21]. We also had no information to evaluate the risk of rabies in the offending dog.

2.3 Environmental, socioeconomic, and epidemiological variables

Human rabies is associated with sociodemographic and ecological factors that bring people into close contact with infected dogs [22]. Previous studies showed that poverty, population density, literacy, distance to the road network, and residing in a rural area were associated with canine and human rabies [22,23]. For this study, we obtained several maps: a poverty raster map (approximately 1km resolution) of the proportion of people per grid square living in poverty in the FCT [20]; a literacy raster map (approximately 1km resolution) showing the proportion of men and women aged 15–49 per grid square as literate in the FCT [21]; an urban extent grid (v.1) showing the proportion of rural and urban areas in the LGA, extracted from the Global Rural-Urban Mapping Project (GRUMP v.1) [24]; and a population density map at 30 arc-second (~1 km at the equator) resolution from the World Pop project [25]. We used a 20-year average for land cover and population density. In contrast, for literacy and poverty, we used data from 2013, and the current distance to the veterinary clinic and hospital. We assumed no significant changes in the number of Government human and veterinary hospitals where dog victims were likely to report in the event of a dog bite, because the cost of care is relatively affordable in a government compared to a private hospital [26].

Previous studies in Nigeria showed that 27% of dog bite victims reported to a veterinary clinic, while the rest reported to a human hospital [5]. The distance to a veterinary clinic or hospital was used as proxy for PEP access. To derive a map of distances to the veterinary clinic, we obtained the coordinates of all H.F. in the FCT from the Federal Ministry of Health [27] and geolocated veterinary hospitals/clinics in Nigeria using Google Earth (https://earth.google.com/web/). We then calculated the Euclidean distance between human dog bite cases to a hospital/clinic and veterinary clinics in ArcGIS version 10.8. We also obtained a road network raster from DIVA-GIS (https://www.diva-gis.org free/gdata) and calculated the mean distance to the road network in meters. We scaled all continuous variables to unit variance by subtracting the mean and dividing by one standard deviation.

3.1 Model definition

Bayesian inference allows the estimation of a joint posterior distribution over unknown parameters for a statistical model, which is the product of the prior distribution and the likelihood of the data. Our analysis was conducted with R version 4.1.3, using the rjags package, which provides an interphase for JAGS (Just Another Gibbs Sampler) library to perform Bayesian analysis [28]. Our model was constructed considering that the observed vector of human deaths after a dog bite in Abuja was drawn from a Bernoulli distribution with unknown parameters p, the probability of human rabies death. We modelled p using a logit link function and linear predictor to estimate the additive effects of covariates on the probability of a human rabies death. The number of parameters to be estimated was 18, including the intercept and a vector of individual, sociodemographic, and landscape predictors:

logit(p(Y = 1)) = β0 + β1*Gender+ β2*Age + β3*Site of bite + β4*Bite type+ β5*Category of bite + β6*PEP + β7*Dog ownership+ β8*Dog vaccination status + β9*Distance to a hospital + β10*Distance to a veterinary hospital + β11*Urban/rural location + β12*Population density + β13*Poverty level + β14*Literacy level + β15*Distance to road + β16*Grass + β17*Shrubland, where Y is a binary indicator of human rabies deaths following dog bite (1 if death and 0 alive).

3.2 Accounting for missing data in the predictors of rabies deaths

For covariates with missing data, including the site of the bite (44%, 506/1,155) and vaccination status of the offending dogs (68.3%, 789/1,155), we used prior information/evidence from the literature on dog vaccination rates in the FCT and general rabies epidemiology in Nigeria to model missing values [5,29]. For dogs whose victims could not identify an owner, we termed an unknown dog with missing information on vaccination status. We drew binary values from a Bernoulli distribution, assuming that each dog had a 38% chance of being vaccinated, using evidence from a recent Abuja study that reported vaccination coverage for owned dogs [15]. We drew missing values for the site of bite from a categorical distribution, setting relative prior weights of 2-1-2 for a bite to the lower limb, head or multiple sites, because historical data and previous reports from Nigeria indicate victims are more likely to be bitten on the lower limb and multiple locations than the head [29]. An interaction term between the site of the bite and the victim’s age was initially included (assuming the site of the bite depended on the victim’s age). However, chains mixed poorly, and Markov Chain Monte Carlo (MCMC) diagnostics suggested the data were not informative to accurately estimate marginal and interaction effects. Prior to modelling, we carried out a pairwise correlation to measure our predictor variables using the cor() command in R and removed highly collinear variables in our model.

3.3 Accounting for sparse rabies death outcome data and multiple predictors

Due to the large number of candidate covariates and our prior assumption that most effects on a patient’s probability of death due to rabies will be near-zero and challenging to estimate with finite outcome rabies death data, we guarded against overfitting by trialling two regularisation priors for the β parameters. Regularisation aims to shrink uninformative coefficients towards zero to produce a sparser and more parsimonious model, while accounting for interdependencies between the regularised set of predictors. First, we fit a model that used a modified version of the regularised horseshoe prior [30] for the βs, which was defined as follows:

βj ∼ Normal(Mean = 0, SD = λ * τ2j)

λ ∼ half-Cauchy(Mean = 0, DF = 1)

τ2j ∼ Gamma(Shape = 0.1, Rate = 0.001),

A regularised horseshoe prior tends to shrink small coefficients towards zero by combination of a global shrinkage penalty (λ) and coefficient-specific penalties (τ2j). The prior density is symmetric about zero but incorporates fat tails and a large spike at zero, making it suitable for models with many regression coefficients where only a minority of these coefficients is expected to be non-zero. However, as the regularised horseshoe can strongly pull effects toward zero, it can be challenging to estimate effects with probability mass at small, non-zero values accurately. Therefore, we trialled a less informative regularising prior for the βs by placing a double exponential (i.e., Laplace) prior on these parameters. A Laplace prior is loosely regarded as the Bayesian equivalent of L1 regularisation (LASSO penalty), simultaneously allowing for many zero effects and some large non-zero effects and is defined by: y ~ Laplace (Mean = 0, Precision = 1).

3.4 Parameter estimation

We used a MCMC Gibbs sampling algorithm to sample from the joint posterior of unknown model parameters for each of the above prior specifications. Estimation was conducted in the software JAGS using the R interface rjags [28]. A total of four MCMC chains were run for 5,000 iterations for burn-in, discarded—visual inspection of the posterior density and trace plots was used to note convergence after 100,000 post burn-in iterations. Following convergence, we stored the posterior distributions from model parameters for inference and calculated the summaries of the parameters, including posterior means and 95% credible intervals (Crl).

Posterior predictive checking is designed to monitor the quality of a statistical model by detecting any inherent difference between the model-simulated data and the observed data through the generation of replicated data sets from the posterior distribution and comparing the observed data for a feature of interest [9]. To conduct predictive checks for each of the above models, we simulated Bernoulli draws from the model’s estimated probability of rabies death for each individual and compared these simulations to the observed data to check for systematic inconsistencies between the observations and model assumptions. A Bayesian p-value was calculated as the proportion of MCMC iterations in which the sum of the simulated Bernoulli vector (the model’s estimated total number of deaths) was larger than the sum of the corresponding observed vector (the true total number of deaths). A model with a Bayesian p-value close to 0.5 indicates a lack of systematic discrepancy between the observed data and the model’s assumed data generating process. We also compared model fits using the penalised deviance information criterion, which is estimated by adding the effective number of parameters (Pd) to the expected deviance [31]. Superior model fitting was evidenced by the lower deviance information criterion (DIC).

3.5 Posterior visualisation of key parameters

We used marginal posterior simulations to understand how the probability of human death due to rabies was expected to change across different magnitudes/levels for modifiable covariates. We simulated how much the probability of death was expected to change when a dog bite victim took at least one dose of vaccine (or none) and when the dog bite victim took varying vaccine doses (and none). We simulated how much the probability of death was expected to change when the offending dogs were vaccinated vs. unvaccinated and when owned vs. free roaming.

3.6 Residual spatial autocorrelation

To gain insight into any evidence for spatial dependence left unexplained by the model, we calculated randomised quantile (Dunn-Smyth) residuals for each posterior simulation and inspected these residuals for possible spatial autocorrelation. Briefly, we computed randomised quantile residuals, which samples variability in the parameter estimates by inverting the fitted distribution function for each response value and finding the equivalent standard normal [9]. We fit semi-variograms to each posterior draw of these residuals. A semi-variogram is described by nugget effect, range, and partial sill [32]. Nugget represents a small-scale data variability. The range is the distance the data are no longer correlated. The partial sill is the semivariance due to spatial dependence. The nugget plus the partial sill is also known as the total variance or sill [32]. We calculated the nugget:sill ratio for each posterior semi-variogram, revealing the overall variance at a distance smaller than the smallest lag interval and how much variance was accounted for in the model. A nugget:sill ratio of <0.25 indicates strong residual spatial dependence, 0.25–0.75 shows moderate spatial dependence, and >0.75 indicates weak spatial dependency [33].

3.7 Sensitivity analysis

Sensitivity analysis is a way of checking the final model result based on the reference prior (original) concerning the results obtained using different priors to aid in interpreting results transparently [34]. To examine the effect of our prior imputations for dog vaccination with large numbers of missing values, we conducted a sensitivity analysis by building four models that made different assumptions about vaccination probability for missing values. In the first model, we adjusted the probability of dog vaccination to 35% to account for 3% of the free-roaming dog population out of the initial 38% because a previous report indicated that only 3% of African dogs are strays [18]. In the second model, we allowed the probability of vaccination to vary (i.e., between 20% and 45%) in each MCMC iteration. In the third model, we incorporated slightly more realism by allowing owned and free-roaming dogs to have different vaccination probabilities, with values between 35% and 60% for owned dogs and 30% for free-roaming dogs. For the last model, we varied the probabilities for both owned and free-roaming dogs, with values between 25% and 75% for owned dogs and between 20%- and 45% for free-roaming dogs. We compared posterior contrast distributions from each model to see how these prior choices impacted our resulting inferences.

4.1 Demographic and health characteristics of dog bite victims

A total of 1,155 dog bite victims were reported in 19 H.F. across the FCT during 2000–2018, of which 4.2% (49/1,155) were reported to have died of rabies. Most victims reported to a secondary H.F. (93.2%, 1,076/1,155). Victims were predominantly male (66%, 756/1,155), between the age of 0–15 (40.3%, 465/1,155), and primarily students (44%, 519/1,155). Only a minority of biting dogs were vaccinated against rabies (14%, 166/1,155). The majority were free-roaming dogs, and most dog bite incidents were due to provoked bites by a person (85%, 980/1,155). Table 1 compares individual-level information distribution across victims who died or were alive after a dog bite.

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Table 1. Distribution of variables among victims of dog bite who died or were alive following a dog bite in Abuja between 2000 and 2018.

https://doi.org/10.1371/journal.pntd.0011147.t001

4.2 Model selection and performance

Regarding the relative fits of the models to our observed data, most coefficients were drawn towards zero in model 1 (regularised horseshoe prior), though mixing of chains was poor for some parameters. This model also displayed a higher DIC (38086) and an inferior Bayesian p-value = 0.42. In contrast, model 2 (Laplace prior) showed good mixing of chains, reasonable Rhat split diagnostics, S1 File a lower DIC (38064), and a Bayesian p-value nearer to 0.5 (p = 0.46). We used model 2 for inference and subsequent analyses based on these results. Rhat refers to the potential scale reduction statistic, also known as the Gelman-Rubin statistic, which is the ratio of the variance of a parameter when the data are pooled across all the chains to the within-chain variance. Values of Rhat close to 1 indicate good performance of the sampler [35].

4.3 Factors associated with the probability of death of dog bite victims

Dog vaccination status, dog ownership, and victims who took rabies vaccination were estimated to have regression coefficients with a probability density predominately away from zero. Precisely, the probability of human rabies death was estimated to decline when comparing bites by owned vs. free-roaming dogs (Fig 1), for bites from vaccinated vs. unvaccinated dogs (Fig 2), and for victims who had PEP vs. no PEP (Fig 3). Other variables, including mean distance to the hospital, veterinary facility (proximity to a veterinary facility), population density, land covers, literacy level, poverty, count of rabies cases per ward, site of the bite and dog bite victim age, displayed weaker effects that had a substantial probability density near zero (S1 File). In contrast, the type of bite and urban locations were centred at zero (and considered non-significant herein, S1 File). Our results showed that individuals bitten by owned dogs were less likely to die of rabies (OR = 0.27, 95% CrI: 0.01–0.44) than those bitten by free-roaming dogs. Individuals bitten by vaccinated dogs were less likely to die from rabies (OR = 0.48, 95% CrI: 0.01–1.34) than those bitten by unvaccinated dogs. Individuals who received PEP were less likely (OR = 0.002, 95% CrI: 0.01–0.01) to die from rabies than those who had no PEP (Table 2).

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Fig 1. Posterior marginal contrast distribution demonstrating the estimated posterior probability of deaths was predicted to change when comparing bites from owned dogs to bites from free-roaming dogs.

https://doi.org/10.1371/journal.pntd.0011147.g001

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Fig 2. Posterior marginal contrast distribution demonstrating how the estimated posterior probability of deaths was predicted to change when comparing bites from unvaccinated dogs to bites from vaccinated dogs.

https://doi.org/10.1371/journal.pntd.0011147.g002

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Fig 3. Posterior marginal contrast distribution demonstrating the estimated posterior probability of deaths was predicted to change when comparing victims who received PEP and those who did not.

https://doi.org/10.1371/journal.pntd.0011147.g003

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Table 2. Posterior mean, standard deviation, credible intervals, and convergence diagnostics for estimated effects on the logit (probability of deaths) due to rabies after a dog bite.

https://doi.org/10.1371/journal.pntd.0011147.t002

Posterior marginal contrast distributions highlight the non-normality of expected changes in the probability of human rabies deaths following potential interventions. In particular, our model estimated strongly left-skewed decreases in the probability of human rabies deaths when comparing bites from owned vs. free-roaming (Fig 1) and when comparing bites from unvaccinated vs. vaccinated dogs (Fig 2), suggesting decreases of up to 20% in probability were not considered likely, but also not implausible by the model. Our model also estimated a significant left-skewed decrease in the probability of human rabies deaths when comparing victims who received PEP and those who failed to receive PEP (Fig 3).

4.4 Residual spatial dependence

The posterior mean nugget:sill ratio was 0.824, suggesting there was weak evidence of remaining spatial dependency—that is, there was little spatial dependence in the data unaccounted for the variables in the model (i.e., that estimates of coefficient standard errors and effect sizes are unlikely biased by non-independence).

4.5 Sensitivity analysis

After visual inspection of the posterior marginal plots, our results indicated non-normality of the expected changes in the probability of human rabies deaths following prior imputations at different vaccination and ownership probabilities with consistent results with the initial assumption. Our model estimated a strongly right-skewed decrease in the probability of human rabies deaths when comparing bites from owned vs free-roaming dogs at different vaccination probabilities between the ranges of 10% and 20% across all imputations, with a slightly more decrease when the probability for owned dogs was varied between 35% and 60% and 30% for free-roaming dogs (bottom left) and when we varied the probabilities between 25% - 75% for owned and 20%- 45% for free-roaming dogs (bottom right) (Fig 4).

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Fig 4. Posterior marginal contrast distribution demonstrating how the estimated probability of deaths was predicted to change, comparing bites from owned to free-roaming dogs at 35% vaccination probability to account for 3% of free-roaming (Top left), when the probability of vaccination was varied between 20% and 45% in each MCMC iteration(Top right), when the probability for owned dogs was varied between 35% and 60% and 30% for free-roaming dogs(Bottom left) and when we varied the probabilities between 25% - 75% for owned and 20%- 45% for free-roaming dogs(Bottom right).

https://doi.org/10.1371/journal.pntd.0011147.g004

When comparing bites from vaccinated vs unvaccinated dogs at different prior imputations, our models estimated a left-skewed decrease in the probability of human rabies across all prior imputations without much difference (Fig 5). Similarly, our models estimated a strong skewed decrease in the probability of human rabies deaths when comparing victims who received PEP vs those who did not, about 50% decrease, especially when the probability of vaccination was varied between 20% and 45% in each MCMC iteration (top right), when the probability for owned dogs was varied between 35% and 60% and 30% for free-roaming dogs (bottom left) and when we varied the probabilities between 25% - 75% for owned and 20%- 45% for free-roaming dogs (bottom right) (Fig 6). The consistency of our results across different prior configurations likely indicates that the observations with missing vaccination data were less informative for the likelihood overall.

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Fig 5. Posterior marginal contrast distribution demonstrating how the estimated probability of deaths was predicted to change when comparing bites from vaccinated to unvaccinated dogs, when the vaccination probability was adjusted to 35% to account for 3% of free-roaming dogs (Top left) when the probability of vaccination was varied between 20% and 45% in each MCMC iteration(Top right) when the probability for owned and free-roaming dogs were varied between 35% and 60% for owned dogs and 30% for free-roaming dogs(Button left) and we varied the probabilities between 25% - 75% for owned and 20%- 45% for free-roaming dogs(Button right).

https://doi.org/10.1371/journal.pntd.0011147.g005

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Fig 6. Posterior marginal contrast demonstrating how the estimated posterior probability of deaths was predicted to change when comparing victims who received PEP and those who did not at 35% vaccination probability to account for 3% of free-roaming dogs (Top left) when the probability of vaccination was varied between 20% and 45% in each MCMC iteration (Top right) when the probability for owned dogs was varied between 35% and 60% and 30% for free-roaming dogs (Button left) and when we varied the probabilities between 25% - 75% for owned and 20%- 45% for free-roaming dogs(button right).

https://doi.org/10.1371/journal.pntd.0011147.g006