Rabies is a preventable infectious disease in warm-blooded animals that causes acute encephalitis and death. The etiological agent is a virus belonging to the genus Lyssavirus. Canine rabies is the form carried by domestic dogs that is overwhelmingly responsible for approximately 59,000 human deaths per year (1) where transmission of the virus occurs via a dog bite.

Despite the presence of rabies control programs, rabies remains endemic in over 80 countries (2). The main component of these programs is vaccination, where the long-held World Health Organization (WHO) recommendation of a dog vaccination target of 70% (2, 3) is most often the aim. What we now increasingly appreciate is that differences in the local ecology of the dog population (4), and the dogs’ relationships with the humans they live with (5), can determine the outcome of a control program (6, 7).

The efficacy of a vaccination program is expected to depend on vaccinated dogs living long enough for herd immunity to build up. A relatively rapid turnover in a dog population (where a population is largely made up of stray dogs, for example) ought to decrease the efficacy of a vaccination program (8, 9). Field studies that have measured turnover rates found they are higher in regions with higher dog density, and they are especially high in areas where the stray dog population is high (10). Such thinking and observations have led to the Animal Birth Control program implemented in India, for example, to sterilize the female dogs that were captured and vaccinated, conscious that this would improve the survival of these dogs (no costs of reproduction), reduce population turnover, and increase the impact of the vaccination effort (11, 12).

The proportion of the dog population that is stray (ownerless) varies considerably from one country to another, ranging from 5% in Tanzania (13), 19% in Sri Lanka (6), to as high as 60% in India (12). Stray dogs are typically wary of humans such that they cannot be held, and so vaccination of these dogs needs to be achieved either by physically capturing the dogs or by distributing oral baits (2, 14). In either case, reaching a sufficiently high vaccination coverage of the whole dog population becomes a more difficult and costly endeavor.

Another contributing factor to the success of vaccination programs is the relationships that dogs share with the humans they live with. Some owned dogs are fully confined, and others partially free-roaming (6). The degree of care humans provide for dogs ranges from none at all, to supplying food and/or shelter only (but not claiming ownership), to full adoption of the dog as a pet where the dog would be provided food, shelter, and health care (5). By contrast, some claim ownership but provide no care at all (15). These differences can have a considerable effect on the dog’s health and exposure to diseases.

A widely used epidemiological measurement of infectious disease transmission is the basic reproduction number, or R₀, which describes the average number of secondary cases produced by a typical infectious individual in a completely susceptible population (16). It can be interpreted as the initial growth rate of the spread of an infectious disease. It follows that when R₀ > 1, the disease will cause an epidemic in the population, and when R₀ < 1, the disease will die out. Theoretical literature shows how R₀ is calculated and then applied to public health, such that a vaccination coverage of 1 − 1/R₀ is expected to lead to the elimination of a disease (16, 17). However, this assumes that coverage occurs homogeneously across the population. In the context of dog rabies, this may not be the case since dog rabies vaccination campaigns typically target specific types of dogs, such as strays (11, 18). When control targets a single type of host then the required effort can be measured by the type-reproduction number (19).

The type-reproduction number, T₁, is an epidemiological quantity introduced by Roberts and Heesterbeek (19) that measures the effort required to prevent outbreaks when control targets a single type of host. For homogeneous populations, its threshold coincides with R₀. In a heterogeneous population, say, with three host types, the concept of T₁ supposes there is a single infectious type 1 host in an otherwise fully susceptible population; T₁ is the average number of type 1 infections caused by the primary case, either directly or via any chain of infection involving host types 2 and 3 (each chain starts with the initial infected type 1 host and ends in a second type 1 infection). Similar to R₀, the critical fraction of host type 1 vaccination coverage required to prevent outbreaks in the entire population is 1 − 1/T₁. We use this relationship to present results on vaccination targets for stray dogs.

The key to interpreting T₁ and how it relates to vaccination coverage targets is that there are three cases to consider. The first is that T₁ > 1. This implies that R₀ > 1 (19) and that control efforts that target type 1 have the potential to bring an outbreak under control, or prevent outbreaks from occurring, provided a vaccination coverage of 1 − 1/T₁ can be achieved. The second is where T₁ < 1 which implies that R₀ < 1 (20) so no control efforts need be applied to type 1 (or indeed to any type). The third and final case is where transmission between the other host types (types 2 and 3, for example) occurs enough such that even if every type 1 host were vaccinated or removed, outbreaks would still occur among the other types. Mathematically, this coincides with a single infected stray dog causing an infinite number of stray dog infections because the chains of infection in the owned dog host types never stop.

In this paper, we consider the control of canine rabies in a mixed dog population consisting of strays, owned free-roaming dogs, and owned confined dogs. In particular, we show how T₁ can be calculated from a differential equation model for rabies transmission, and how it is related to R₀. Our goal is to understand how vaccination targets for strays, determined by the quantity T₁, are affected by the composition of the dog population, the numbers of dogs (dog density), the vaccination coverage of the owned dog population, and the mortality rates of the stray population.

The mathematics we use is presented in discrete boxes and contained as figures. The reader will be able to “skip” the mathematics if they wish to do so and still follow the methods, results, and conclusions of the paper, provided they understand the epidemiological quantities, T₁ and R₀.

Motivated by the studies of dog populations by Matter et al. (6) and Massei et al. (5), who both observed differences in dog ownership styles, our modeling approach divides the dog population into three ownership types: stray (type 1), owned free-roaming (type 2), and owned confined (type 3). In our mathematical model of dog rabies transmission, we define stray dogs to be ownerless, hence have no health care, and must forage for their own food. Owned free-roaming dogs are defined to be regularly fed by the community but are provided little to no health care. Owned confined dogs are regularly fed and have access to health care through their owners. Under these differences, stray, owned free-roaming, and owned confined dogs have short, medium, and long life spans, respectively.

A difference in life span between groups of dogs creates a slight epidemiological difference (12, 21) because high natural mortality or additional density-dependent mortality implies that an exposed stray could die of natural means before becoming rabid, whereas this is much more unlikely for an owned dog. The stronger epidemiological differences between the host types are in terms of exposure, and therefore biting rates and transmission to other dogs. Our premise is that an owned confined dog, unless it is momentarily free to roam during hunting or some other activity, will not normally be free to come across or attack other dogs, and hence, is assumed not to wander beyond their owner’s property.

We first visualize the results of solving the differential equation model given in Figure 1 while varying the initial composition of the dog population and the initial size of the dog population. The initial proportions of dogs that are stray, free-roaming, and confined are varied but always sum to 1. This means that the effects of the composition of the dog population on R₀; the number of infected dogs at equilibrium; and rabies prevalence can all be visualized as ternary plots (see Figure 5). The corners of each of the pictured triangles represent dog populations at the extremes where there are only strays, or only owned free-roaming dogs, or only owned confined dogs. The sides of the triangle represent when the dog population is a mix of two dog types, and the region inside the triangle represents populations that are a mix of all three.

We have used a differential equation model and the epidemiological quantity, T₁, introduced by Roberts and Heesterbeek (19), to explore how vaccination targets for stray dogs might be expected to depend on dog population size and composition, stray dog demography, and on the vaccination rate that is achieved in the owned dog population. Our study shows that an increase in public knowledge around rabies (such as proper disposal of garbage) can be effective in reducing the required vaccination targets for stray dogs. We have found that the required stray dog vaccination rate is sensitive to the proportion of the total dog population that is owned and free-roaming. Indeed, a dog population consisting wholly of free-roaming owned dogs is predicted to require a higher vaccination coverage to ensure it is rabies-free, compared to any other mix of types, and even compared to a population consisting entirely of stray dogs.

This is not to say that populations of owned free-roaming dogs are more challenging for rabies control because owned dogs can be handled and are therefore easier to vaccinate. With this consideration, populations of stray dogs are clearly the most problematic. However, for example, our findings do underline concerns about rabies entering Australia’s northern communities where dog populations consist entirely of free-roaming dogs (30). There are typically no stray dogs in this population, but there are contacts with wild dogs (4). The insights we have presented here would predict that this mix of wild dogs and free-roaming domestic dogs could be highly vulnerable to a rabies incursion.

We emphasize that our modeling assumes that stray and owned free-roaming dogs have similar roaming behavior and hence similar exposure to infected dogs. The bottom right-hand corners of the ternary plots represent populations where a high proportion of the dog population is confined and these always correspond to the “safest” situations because the transmission rates of confined dogs are assumed to be much lower. The effect of higher proportions of stray dogs, which can be observed by traveling along the border of the ternary plot, is to subtly reduce R₀ (Figures 5A–C). This is because higher demographic turnover means that an exposed stray dog is slightly less likely to survive long enough to become rabid.

In general, the three types would likely exist to some degree in all populations of dogs, though for specific regions and countries one or two of the types might be in negligible numbers. Defining these three types of dogs is a simplification of the varied and complex relationships between dogs and humans. Whether an owned dog is confined or allowed to roam, and even whether a dog has an owner or not, will have answers that vary from yes to no to everything in between. We therefore acknowledge that in reality the composition of a dog population and the validity of allocating every member to one of three types will vary from one region to another. Similarly, it is clear that dog densities vary from location to location, and contact rates too, so while we can endeavor to be in the right “ball park” it is not easy to be more precise than we have been unless we parameterize the model for a particular region where dog density, dog composition, and R₀ are all known. Despite these caveats it is insightful to explore the consequences for rabies vaccination programs if there were groups of dogs that vary in their exposure (to bites from rabid dogs) as well as their freedom to infect other dogs should they become rabid.

The critical assumptions of this study are that dogs mix randomly at the population level, do not change from one type to another over their lifetime, and that every dog clearly belongs to one and only one type. The degree to which this is a simplification of reality, and what effect introducing further, more realistic, heterogeneity in the dog population would have on the epidemiological quantities that determine vaccination targets is unknown. Another core assumption is that the transmission rate between dogs has a linear relationship with density. This is a common feature of many rabies modeling studies (3, 23, 31); replacing the density-dependence with frequency-dependent contact in a standard SEIR compartmental model gives behavior that is discordant with observed dynamics (the dog population becomes extinct) (32), but there is conflicting evidence for density-dependent transmission and hence for the population thresholds one would expect to see as a result.

Our study highlights the importance of developing a better understanding of the dog ecology, such as dog population densities and degree of contact between the dogs, to guide future model development, improve model-based interpretation, and provide guidance to the design of vaccination programs. Our study provides a firmer scientific basis for vaccination programs aimed at stray or neighborhood dogs that cannot be handled and therefore cannot be brought to vaccination points. If the key dog ecology parameters can be estimated, then the theory provides definite vaccination targets, which can give greater confidence that a control program represents sufficient effort to eliminate canine rabies.

TL and SD together defined the research questions of the paper and designed the approach. TL performed much of the analytical work and obtained numerical solutions for the differential equations, created the figures, and helped write the manuscript. SD recovered many of the analytic results found by TL and led the writing of the manuscript.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.