Modelling the factors affecting the probability for local rabies elimination by strategic control

Johann L. Kotzé; John Duncan Grewar; Aaron Anderson

doi:10.1371/journal.pntd.0009236

Abstract

Dog rabies has been recognized from ancient times and remains widespread across the developing world with an estimated 59,000 people dying annually from the disease. In 2011 a tri-partite alliance consisting of the OIE, the WHO and the FAO committed to globally eliminating dog-mediated human rabies by 2030. Regardless of global support, the responsibility remains with local program managers to implement successful elimination programs. It is well known that vaccination programs have a high probability of successful elimination if they achieve a population-coverage of 70%. It is often quoted that reducing population turnover (typically through sterilizations) raises the probability for local elimination by maintaining herd immunity for longer. Besides this, other factors that affect rabies elimination are rarely mentioned. This paper investigates the probability for local elimination as it relates to immunity, fecundity, dog population size, infectivity (bite rates), in-migration of immune-naïve dogs, and the initial incidence. To achieve this, an individual-based, stochastic, transmission model was manipulated to create a dataset covering combinations of factors that may affect elimination. The results thereof were analysed using a logistic regression model with elimination as the dependent variable. Our results suggest that smaller dog populations, lower infectivity and lower incidence (such as when epidemics start with single introductions) strongly increased the probability for elimination at wide ranges of vaccination levels. Lower fecundity and lower in-migration had weak effects. We discuss the importance of these findings in terms of their impact and their practical application in the design of dog-mediated rabies control programs.

Author summary

Most guidelines for rabies control call for at least 70% vaccination coverage of dogs. This level of immunity has a very high probability for the local elimination of rabies, but it is often not an achievable ideal due to resource constraints. Campaign managers can be strategic on how they allocate their resources. Lower infectivity rates are present in areas with more restricted dog movements and have higher probabilities for elimination at lower vaccination rates. Smaller sub-populations have higher probabilities for elimination at the same vaccination coverage levels compared to larger sub-populations. Vaccinating immune corridors can divide meta-populations into smaller sub-populations that are likely to result in elimination either due to their small size or due to the local low infectivity. Areas already free of rabies require lower vaccination levels to maintain freedom compared to endemic areas. Where donors do not specifically require sterilization campaigns, funds meant for rabies control should not be diverted to sterilizations.

Citation: Kotzé JL, Duncan Grewar J, Anderson A (2021) Modelling the factors affecting the probability for local rabies elimination by strategic control. PLoS Negl Trop Dis 15(3): e0009236. https://doi.org/10.1371/journal.pntd.0009236

Editor: Amy T. Gilbert, US Department of Agriculture, UNITED STATES

Received: February 18, 2020; Accepted: February 11, 2021; Published: March 4, 2021

This is an open access article, free of all copyright, and may be freely reproduced, distributed, transmitted, modified, built upon, or otherwise used by anyone for any lawful purpose. The work is made available under the Creative Commons CC0 public domain dedication.

Data Availability: Supporting information and additional intermediate data can be sourced at https://doi.org/10.5281/zenodo.4362497.

Funding: The author(s) received no specific funding for this work.

Competing interests: The authors have declared that no competing interests exist.

Introduction

Dogs are responsible for up to 99% of all human deaths attributed to rabies, transmitting the virus primarily through bites [1]. Exposure to virus does not invariably lead to disease, but when it does the case fatality rate approaches 100% [2]. Rabies has persisted since ancient times, being the oldest known infectious disease to man [3]. Whereas notable successes have been achieved to control dog-transmitted rabies in many countries, most of the globe is still in the endemic state. Despite its widespread prevalence, rabies remains rare (relative to other infectious diseases) with typical annual incidence rates in endemic countries below 0.2% [4]. This is a result of basic reproductive numbers mostly below 1.5 [5].

Good quality rabies vaccines are available, leaving the only remaining barrier to exit the endemic state the ability to get vaccine into dogs and achieve adequate herd immunity [6,7]. It is widely quoted that campaign managers should aim to achieve 70% vaccination coverage [8–10]. However, calculating the required coverage through the basic reproductive number, R₀ [11] gives much lower required coverages. Typically, these are between 20% and 40%. Factoring in the waning of vaccine induced immunity and the loss of immune individuals through rapid population turnover rates, the required annual coverage again approaches 70% [4,12].

Despite the common recommendation of 70% coverage, the actual required coverage will vary considerably between locations as a function of the local dog ecology [5,13,14]. Furthermore, budgetary resources at the local level are often not enough to achieve a 70% coverage [7]. Since most of the costs of vaccination campaigns are logistic in nature, achieving a uniform coverage across large meta-populations carries the greatest cost. Instead, campaign managers typically achieve patchy, irregular coverage based on convenience or as a response to reported cases. The ability to strategically select specific, spatially separated sub-populations for campaigns could achieve better long-term results but campaign managers are currently not equipped with enough knowledge to make evidence-based choices. There is therefore a need for a better quantitative understanding of the impact various factors have on the required vaccination coverage at the sub-population level. There is also limited literature on the impact of factors other than vaccination, dog densities and fecundity [15].

Field experiments would be ideal to answer these questions but conducting them may be unethical or impractical. This study attempts to address these needs through an individual-based, bioeconomic, stochastic disease model. Bio-economic models allow for flexibility in both disease parameters as well as management interventions, thereby allowing for their optimization [16]. Stochastic methods allow parameters to take on a constantly changing value based on a probability distribution rather than a fixed value. This is important for diseases with high demographic variability such as rabies [17]. For example, the number of secondary contacts from a rabid dog approaches a negative binomial distribution with mean 2.2 and variance 5.6 [4]. Ignoring such frequency distributions in a mathematical model can have substantial impacts on the outcome [17].

Our hope with this paper is to assist campaign managers to allocate their resources strategically to geographically separate sub-populations based on their relative importance to the sustainability of rabies in the larger meta-population.

Methods

Seven parameters of a published rabies transmission model were selected to be investigated for their impact on the probability for dog rabies elimination [16] (Table 1). The selected parameters represent factors that are either known or suspected to affect the local elimination of rabies. A range of possible values were chosen for each parameter reflecting the most likely field values. The specified values of the seven parameters created 103,125 unique combinations. Due to the model’s stochastic nature, we ran five iterations for each combination. The results of the transmission model were captured in a dataset containing 425,802 successful eliminations and 88,940 sustained outbreaks, each linked to a specific set of values for the seven parameters. We excluded from the analysis 883 iterations where the dog population became extinct. The values of the different parameters that had extinction events are presented in S1 Table and the calculations are shown in S1 Script.

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Table 1. Model parameters included in the transmission model.

https://doi.org/10.1371/journal.pntd.0009236.t001

This dataset was then analysed using a logistic regression model to identify the conditions that are most likely to result in rabies elimination. The prediction error of the regression model was minimized while controlling for complexity using stepwise regression, based on the Akaike Information Criterion (AIC). Since the aim of the study was to evaluate the impact on the required coverage, all factors were examined for their interaction with vaccination. The transmission model and the regression model were both computed using R [18]. Finally, two examples were calculated comparing the different contact rates (infectivity) in N’Djaména, Chad and comparing investment in sterilizations versus vaccinations in Hluvukani, South Africa.

The disease transmission model

We made use of a peer-reviewed, publicly available, individual-based, stochastic model (http://github.com/anderaa/BioEcon) [16]. We retained the values for all parameters of the published form of the model except for our seven chosen study variables that were manipulated from a chosen minimum to a maximum (Table 1). The adapted R script used in this study is reported in S2 Script. The model used daily time steps and tracked the age, sex, reproductive status, immunity and disease status of individual dogs. Each of the variables took one of five different values within the chosen limits, apart from immunity and initial incidence. Immunity had eleven levels because a high level of precision was desired for interpretation over all possible values from 0 to 100% herd immunity. Initial incidence took on only three values because we wanted to compare the difference during maximum size outbreaks found under hyperendemic conditions versus very low incidences found when reintroductions of infective dogs occur in previously rabies-free areas.

In the model, the rate of effective bites from an infective dog varied stochastically following an evidence-based, negative binomial distribution that results in frequency-dependent transmission of disease [4]. Density-dependent transmission is often used instead, but there is empirical evidence in favour of frequency-dependent transmission. The issue remains controversial [8,14,19–21]. If the true transmission rate is indeed a function of density, it would be an important factor determining epidemic outcome by increasing the number of bites from each rabid dog [15]. However, other local ecology factors also directly impact the bite rate and includes fences, dog restraining devices and home ranges. It is therefore simpler to model the bite rate which varies as a function of the total dog ecology that includes dog density. Dog density is further confounded by the fact that interventions that reduce density also impact dog turnover rate, which in turn affects the decline of herd immunity following a vaccination pulse. We included a multiplier of the bite rate as a variable and called it infectivity.

Fecundity is defined as the reproductive potential of an animal and can be controlled through sterilizations. Reduced fecundity equates to reduced population turnover rates, meaning that herd immunity through vaccination is sustained for longer [22]. Although frequently applied as an adjunct to vaccination campaigns, its efficacy has been questioned [1,20,22]. Fecundity was also varied by using a multiplier affecting both the litter size and the number of litters.

We included a variable for the probability of a dog to immune-convert rather than develop disease following an effective exposure. Exposure acquired immunity is a critical determinant of epidemic outcomes in high-recovery diseases such as smallpox [23]. Some rabies challenge experiments have reported high rates of survival of non-vaccinated dogs [24,25]. However, field observations by Tepsumethanon et al. of large sets of biting but non-rabid dogs as well as true rabid dogs have failed to confirm this occurrence [26]. We therefore assumed that, if present under field conditions, the rate would be lower than the sensitivity of the field observations by Tepsumethanon et al. (one in a thousand). We wanted to investigate if such events, even at low rates, impact the effectiveness of other control measures. We selected a low maximum probability to survive rabies challenge of 0.001. All survivors were non-infective and subsequently became immune.

The persistence of rabies has been reported to be unstable and reliant on frequent reintroductions from larger meta-populations [27–29]. Recently, a different stochastic model has found population size to be a highly significant predictor of elimination events [30]. Therefore, we included the initial population size and allowed it to vary over a large range (200 to 16,200). While rabies is often endemic in rural and remote regions where village sizes are smaller, very small population sizes are likely to result in mortality related population extinctions [19].

Substantial migration rates have been reported for dogs in Africa, where the typical in-migrant is not vaccinated [12]. We therefore classified all in-migrants as immune-naïve. We set the maximum in-migration at 10% per annum to preserve most of the original population characteristics, although higher rates have sometimes been reported in the field [12].

Migration is often associated with (re-)introductions of rabies [29]. This would represent the start of an epidemic with minimal incidence initially (before the epidemic gathers momentum). In mature epidemics with no interventions, maximal incidence can be expected. We varied initial incidence such to represent both these scenarios (as well as an intermediate value). At the lowest incidence value a population of 5,400 had two rabid dogs introduced for two consecutive months (all smaller populations had only one introduction for two months). At the highest incidence, a population of 5,400 had 14 rabid dogs introduced for two consecutive months.

Finally, as a seventh factor to examine, we varied initial herd immunity of the population between 0 and 1.

Each unique combination of parameter values (covariate pattern) was simulated 5 times each, producing a total of 515,625 iterations. From the authors’ previous work using Anderson et al.’s model this was considered a large enough dataset to achieve adequate convergence. The first year of the simulation presents the highest probability for elimination due to the natural decline in immunity with time. At the end of one year, if no infectious or exposed individuals were present in the population at the simulation end-point, an elimination success (i.e. 1) was recorded. If the population of dogs itself had become extinct then that simulation result was discarded, thereby excluding results that were not necessarily related to virus extinction. Excluded data are reported in S1 Table. Such an event occurred at a rate of 0.009% over all parameter values, reaching 2.15% when the population size was set at the minimum value, 3.1% when the infectivity was set at a maximum and 9.4% when there was no herd immunity. Small population sizes, high infectivity rates and low herd immunities pre-disposed the model to extinction events and are over-presented in the data excluded from the analysis. No extinction events occurred when initial immunity was set at 0.1 or higher.

The regression models

We calculated for each unique parameter combination (covariate pattern), the proportion of the iterations that resulted in successful elimination. The R script to do this is made available in S3 Script. We then modelled this proportion using a fractional logistic regression model and included the seven targeted parameters as covariates in a linear model (Model 1). Simple scatterplots of several of the independent variables revealed curvilinear relationships with elimination (S1 Fig). To account for potential non-linear relationships, a quadratic term was added for each independent variable (Model 2). Some of the extrema of the estimated quadratic equations fell just inside the parameter space, so we also investigated the inclusion of cubic terms (Model 3). A fourth alternative model was investigated adding first order interaction terms for each of the independent variables with polynomial terms (Model 4). The four models were compared by their residual deviance and AIC. (Table 2). It was clear from the reductions in the residuals and drop in the AIC that quadratic terms, cubic terms and interactions all contributed to improving the fit of the model whilst controlling for complexity. From this we sought the optimal model by backward and forward stepwise regression of the maximum model using the AIC as the inclusion criterion. We chose AIC as the evaluation metric because it rewards superior fit and penalizes model complexity, which minimizes the risk of overfitting [31]. The model selection process was implemented with the stepAIC() function from the package MASS [32]. A script executable in R to define the regression models is available in S4 Script. The R summaries of all intermediate models (including each step of the stepAIC function) are available in S1 Text.

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Table 2. Derivation of the optimal regression model.

https://doi.org/10.1371/journal.pntd.0009236.t002

Each independent variable was plotted against the probability for stochastic elimination while holding all other variables (except for immunity) to their default values (refer to Table 1). Immunity was varied using 10% increments producing multiple plots. The predicted values from the regression model (predicted) were shown as well as the actual fraction of elimination events from the transmission model (observed). (Figs 1–6).