03-modreview.Rmd

# Modeling canine rabies virus transmission dynamics

*This chapter was originally published in:*

*__Rajeev M.__, Metcalf C.J.E., & Hampson K. Modeling canine rabies virus transmission dynamics. Chapter in Rabies, 4th Edition. May 2020.*

*The authors' pre-editorial version is included here.*

\newpage
\setlength{\parskip}{2em}

## Abstract {-}

Mathematical models of infectious disease are used to develop an
understanding of disease dynamics and aid in designing control
strategies. Modeling can also shed light on how dynamics, and therefore
intervention strategies, may change as control is implemented. In light
of the mounting evidence that elimination of canine rabies is a
realistic objective, the WHO has set a global target of zero human
deaths due to dog-mediated rabies by 2030. In this chapter, we focus on
how dynamic epidemiological modeling can guide efforts to achieve this
goal. We review existing modeling work and identify insights generated,
outstanding questions, and gaps in our knowledge. We further discuss the
role that modeling can play in the future to inform elimination.

**Key Words**: Canine rabies, Zero by 30, disease modeling, mass dog
vaccination, transmission dynamics

## Introduction

Models of disease dynamics are a powerful tool in the arsenal of disease
prevention and control efforts, and can be used to estimate key
epidemiological parameters, establish targets for control, and guide
policy [1]. Modeling can also identify counter-intuitive outcomes that
emerge as interventions are implemented, and challenges in the endgame
when disproportionate resources are necessary to reach the last mile of
elimination [2]. In light of the global goal to eliminate human deaths
due to dog-mediated rabies by 2030, models of rabies virus transmission
have potential to inform control efforts as countries progress towards
elimination.

### History of modeling rabies virus transmission dynamics

Modeling rabies in domestic dog populations is a relatively nascent
effort. In contrast, models of wildlife rabies guided early control
efforts [3]. Elimination of fox rabies in Europe was kick-started by
modeling studies that demonstrated the feasibility of control [4].
Surveillance of rabies in wildlife systems in Europe and North America
provided rich data sets to characterize dynamics, identifying the wave
front of outbreaks to target control geographically [5], establishing
that landscape features such as rivers act as barriers to disease
dispersal [6], and delineating how birth pulses shape seasonality in
transmission [7]. This work provides a foundation for modeling canine
rabies, but there are fundamental differences between wildlife and
domestic dog systems. Human populations, behavior, and culture structure
dog populations [8]. In addition, canine rabies persists in low- and
middle-income countries where surveillance capacity is limited and
representative disease data are lacking [9]. Beyond capturing core
infection biology, models of canine rabies must also encompass human
influences and be tractable to interpretation in data-sparse settings.

### The modeling backbone for canine rabies

Rabies can be modeled in an **SEIV** framework, with **Susceptible**,
**Exposed**, **Infectious,** and **Vaccinated** classes (Figure \@ref(fig:ch4-fig1)).
Dog demography governs the dynamics of the susceptible and vaccinated
classes. The **Susceptible** population is replenished by births and
depleted by mortality (both natural and disease-induced) and
vaccination. The **Vaccinated** population is governed by the rate of
vaccination, but depleted by natural mortality and waning of immunity
generated by vaccines (most high quality vaccines are protective for at
least 3 years, [10]. For canine rabies, evidence suggests that
domestic dogs are the reservoir host even in areas with complex wild
carnivore communities [11], [12]. While other wildlife hosts may
contribute to transmission, single-host models of rabies in the dog
population are likely sufficient to understanding and predicting
dynamics in most endemic areas [13].

Rabies virus is directly transmitted, typically via bites, in the saliva
of infectious animals. Transmission is on average low: most dogs do not
transmit or only infect one or two other dogs. However, there is also
substantial heterogeneity in transmission, and some dogs are capable of
biting upwards of 20 other dogs during their short infectious period
[14]. The incubation period is about 21 days but is highly variable.
Most exposed dogs become infectious within one month, but some
infections manifest months after initial exposure [14]-[16]. The
infectious period, on the other hand, is predictably short, and
infection results in death generally within 10 days of showing
neurological signs of infection [14], [17]. There is little evidence
that individuals can be infectious but sub-clinical (i.e. no carrier
class), and there is no recovered class, as exposure does not confer
immunity [18], and following onset of clinical signs, rabies is
invariably fatal.

Although transmission is mostly local (\< 1 km), rabies can cause
erratic and unpredictable behavior, with infected dogs able to run more
than 15 km, beyond the typical home range of most healthy dogs [14].
As a result, secondary cases often occur from disease-mediated
incursions spread from neighboring populations (e.g., nearby populated
settlements within the range of rabid dog movement). In addition,
long-distance human-mediated incursions of incubating dogs can result in
outbreaks being seeded from otherwise unconnected populations [19].


```{r ch4-fig1, out.width="90%", fig.cap = '(ref:ch4-fig1-cap)', fig.scap='The Susceptible-Exposed-Infectious-Vaccinated (SEIV) modeling framework for canine rabies.'}
knitr::include_graphics("figs/ch3/image1.jpeg")
```

(ref:ch4-fig1-cap) The Susceptible-Exposed-Infectious-Vaccinated (SEIV)
modeling framework for canine rabies: circles indicate epidemiological
classes, arrows linking circles indicate how individuals can move
between classes, insets describe underlying processes and influences.
**A) Host demography (i.e., the balance between births and deaths) and
vaccination** govern the susceptible and vaccinated population dynamics.
Following vaccination campaigns, vaccination coverage (y axis, inset)
first increases (vertical jumps) then wanes over time (x axis) as
vaccinated individuals die, susceptible individuals are born, or as
immunity conferred by vaccination wanes (in this example, campaigns
reach 70% of the population annually, but coverage wanes to
approximately 35% before the next annual campaign). **B) Transmission**
is on average low, but highly heterogeneous. Inset shows number of
secondary cases generated from a negative binomial distribution (n =
1000 draws, mean number of secondary cases = 1.2, red dashed line).
**C)** Individuals move from **exposed to infectious** on average after
22.3 days (inset, dashed line) but this is also highly variable with
some infections occurring months to years after exposure. **D)
Disease-induced mortality** is complete, and the infectious period is
short, on average 3.1 days (dashed line), with deaths due to infection
occurring within 10 days. **E) Introductions** from outside the
population modeled may seed cases within. Introductions may results from
**disease-mediated** movement of infectious dogs (sometimes upwards of
10 km; inset shows dispersal kernel, gamma distribution) and
**human-mediated** movements of incubating dogs (potentially on the
scale of 100s of km through movement along roads; the inset shows an
example of a major road network in Tanzania). All parameters used and
associated references are listed in Table \@ref(tab:ch4-tab1)


Table: (\#tab:ch4-tab1) Key parameter values associated with underlying processes illustrated in \@ref(fig:ch4-fig1).

 | Process              | Distribution                                | Parameters                 | Value | Source                | Inset |
|----------------------|---------------------------------------------|----------------------------|-------|-----------------------|-------|
| Birth rate           | --                                          | Mean annual rate (dogs/yr) | 0.5   |  [20]                 | A     |
| Death rate           | --                                          | Mean annual rate (dogs/yr) | 0.42  |  [20]                 | A     |
| Vaccine waning       | --                                          | Mean annual rate (dogs/yr) | 0.33  | [10]                  | A     |
| Secondary cases (R0) | Negative binomial, mean 1.2 secondary cases | Mean                       | 1.2   | Townsend et al., 2013 | B     |
|                      |                                             | Dispersion parameter (k)   | 1.3   |                       |       |
| Incubation period    | Gamma, mean 22.3 days                       | Shape                      | 1.15  | Hampson et al., 2009  | C     |
|                      |                                             | Rate                       | 0.04  |                       |       |
| Infectious period    | Gamma, mean 3.1 days                        | Shape                      | 2.9   | Hampson et al., 2009  | D     |
|                      |                                             | Rate                       | 1.01  |                       |       |
| Dispersal kernel     | Gamma, mean 0.88 km                         | Shape                      | 0.215 | Townsend et al., 2013 | E     |
|                      |                                             | Rate                       | 0.245 |                       |       |                         

## How to model rabies virus transmission?

There has been considerable debate about how to model rabies virus
transmission, which echoes a larger debate within the disease ecology
community [21]. Theory indicates that for diseases with
density-dependent transmission, i.e. when transmission scales with host
density, there exists a threshold density below which the disease cannot
persist [22]. However, there is no such threshold when transmission is
frequency-dependent, i.e. transmission rates are independent of host
density [21].

For canine rabies, the basic reproductive number (*R~o~*) or the average
number of secondary cases resulting from a single infection in a
completely susceptible population, is generally estimated as between 1-2
[14], [23]-[25]. Such consistently low estimates of R~0~ across a
range of dog densities suggest that rabies virus transmission is largely
frequency-dependent [14], [24], [26]-[28]. That is, rabid dogs
have on average the same number of infectious contacts regardless of the
density of dogs around them. As a result, reductions in population
densities are not likely to be effective in eliminating rabies. In
practice, although a common practice and one predicated on assumptions
of density-dependent transmission, indeterminate culling of dogs does
not curtail rabies transmission [29].

Despite evidence for frequency-dependent transmission, many modeling
studies formulate rabies transmission as density-dependent (Fig \@ref(fig:ch4-fig2)D). For a given R~0~, this assumption of density-dependent
transmission does not impact herd immunity thresholds; the critical
proportion that needs to be vaccinated, p~c~, is equal to 1 - 1/R~0~
regardless of the form of transmission [22]. However density-dependent
models predict reductions in transmission due to declining dog density
(e.g., via culling or disease-induced mortality) that are unlikely to
translate to the real world.

Models with frequency-dependent transmission are also not entirely
consistent with empirical observations. Frequency-dependent models that
assume homogeneous mixing (i.e. equal contact probabilities between all
individuals in a population, also referred to as 'mass action') result
in eventual population extinction for fatal pathogens like rabies
[30]. Only under very low transmission (1.01-1.02) and high population
growth can rabies persist in models with frequency-dependent
transmission. For models with density-dependent transmission, even with
R~0~ between 1.01 and 1.1, models of rabies show high annual incidence
(Figure \@ref(fig:figS1)), which is at odds with empirical evidence. Where
measured, rabies incidence is low (\< 1-2% annually) and consequently
has little demographic impact on dog populations [31]. Additional
model structure is therefore necessary to explain how rabies can persist
at such low incidence.

Transmission heterogeneity may be a potential mechanism to explain the
relatively low incidence of rabies. A high proportion of dead-end or
singleton transmissions result in negligible depletion of susceptibles,
while occasional super-spreaders may seed and maintain transmission. In
addition to unrealistic estimates of rabies incidence, if heterogeneity
in transmission is not captured, there is a risk that models may
generate biased estimates of control indicators, such as the time to
elimination and the threshold level of vaccination that this requires.

Accounting for the spatial scale of transmission could also explain how
rabies persists at low incidence. As most transmission occurs within a 1
km radius of infected animals, susceptible depletion at such fine scales
may limit transmission in a way that is not captured in mass action
models [32]. Phenomenological approximations may offer a solution to
this challenge [33], [34], but have yet to be thoroughly explored
for rabies. Spatially-explicit individual-based models implemented at
the scale at which most mixing occurs generate more realistic dynamics
[24], [35], but are computationally intensive and not analytically
tractable. Nonetheless, such models provide insights into underlying
mechanisms that could be simplified for more expedient models. Finally,
human behavior has also been implicated in curtailing epidemics, with
responses such as tying and killing infectious dogs and reactive
vaccination thought to scale with incidence [36].

There is limited data to disentangle these potential mechanisms, which
could reconcile empirical observations with modeling results. Further
work is necessary to ensure sufficient model realism to inform policy,
but balancing realism and complexity is a key challenge for any modeling
study [37]. Building in realism requires additional parameterization
and, often, additional assumptions. Robust epidemiological and
biological data are therefore key to improving our understanding of how
to model rabies transmission.


```{r ch4-fig2, out.width="90%", fig.cap = '(ref:ch4-fig2-cap)', fig.scap='Summary of studies with a dynamic model of canine rabies.'}
knitr::include_graphics("figs/ch3/image2.jpeg")
```

(ref:ch4-fig2-cap) Summary of studies with a dynamic model of canine
rabies. A total of 51 studies were included. A) Year of publication,
with most studies published after 2006; B) Countries where rabies
dynamics were modeled: studies were concentrated in China, Tanzania, and
Chad, but many also examined dynamics in hypothetical contexts, not
specific to any geographic situation. C) Estimates of R~0~: most studies
estimated R~0~ below 2 (10 studies, with 31 estimates; estimates of R~e~
(the effective reproduction number which accounts for ongoing
vaccination) and R~t~ (time-varying reproductive number) were excluded
(N = 3). D) Key features of models (N = 51): most assumed
density-dependent transmission (N = 27). Less than half were fit to data
(N = 20), stochastic (N = 20), or spatially-explicit (N = 19). 15/51
studies incorporated individual heterogeneity in transmission and 14/51
introductions from outside the population modeled. Only 10 included an
observation model in their analysis or accounted for under-reporting in
their inference. Full bibliography and metadata included in
Supplementary Table 1.

## Existing Modeling Studies

Two systematic reviews of rabies models recently examined the
effectiveness and cost-effectiveness of control and prevention
strategies. They concluded that estimates of R~0~ are consistently below
2 and dog vaccination is an effective strategy, but vaccination coverage
is critically influenced by dog demography [38]. Both mass dog
vaccination and provisioning of PEP to bite patients are cost-effective,
in contrast to dog culling which has rarely been identified as either
economically feasible or effective [39]. Building off these reviews,
we examined studies with a dynamic modeling component and synthesized
insights generated and data used to inform them. We searched for papers
that had the terms "rabies" AND ("domestic dog\*" OR "canine") AND
"model\*" on PubMed and Scopus, including all English language papers
published between January 1995 and July 2019 that incorporated a
transmission model of rabies virus in domestic dogs. Of the 547 unique
records retrieved, 51 papers fitted these inclusion criteria (Fig
\@ref(fig:ch4-fig3), Online Supplementary Table S1).

### Insights and limitations

Of studies that compared intervention strategies (generally: mass dog
vaccination, human PEP provisioning, and dog population control
including culling), the majority show that dog vaccination is most
effective, and essential to achieve elimination. Despite the potential
to maximize population-level immunity, synchronizing vaccination
campaigns geographically had little impact on probability of
elimination, at least for annual vaccination campaigns. In contrast,
spatial heterogeneity in vaccination coverage had a greater impact, with
even small contiguous coverage gaps reducing the probability of rabies
being eliminated [24], [35].

While the critical vaccination threshold (*p~c~* or 1 - 1/*R~o~)* should
theoretically be much lower than 70% for a disease with the low range of
R~0~ estimated for rabies (Figure \@ref(fig:ch4-fig2)C), the coverage level recommended
by WHO reflects an empirical consensus [23], [40]. Models show that
due to high turnover in domestic dog populations, annual campaigns that
reach at least 70% of the population are necessary to maintain coverage
\> 20% throughout the year. Furthermore, heterogeneity in transmission
and frequent introductions of rabies cases increase both the vaccination
threshold necessary to interrupt transmission, and the probability of
observing small outbreaks even when vaccination coverage is high [14],
[41].

Most published models were deterministic (33/51) and did not incorporate
heterogeneities in transmission (36/51, Figure \@ref(fig:ch4-fig2)D). However, as R~0~
for rabies appears to be low, the interaction between stochasticity and
heterogeneity in transmission may be influential. In general, for
diseases with high transmissibility (i.e. measles), heterogeneities in
transmission can often be ignored as these complexities have little
impact on the emergent dynamics of infection [30]. However, for a
disease with lower transmission, heterogeneities may result in
unpredictable outbreaks [37]. Stochasticity is especially crucial in
the endgame, when elimination probabilities and incursion dynamics
depend on rare events.

Most studies model rabies virus transmission in a closed population,
that is without introductions from neighboring areas (Figure \@ref(fig:ch4-fig2)D).
While this is a reasonable approach in island settings such as in Bali,
Indonesia [24], recent modeling and phylogenetic work shows the
importance of incursions in less isolated populations in sustaining
rabies virus transmission (Bourhy et al., 2016; Zinsstag et al., 2017)
and that multiple strains co-circulate within a population [42],
[43]. Human behavior is also a key driver of transmission patterns,
facilitating as well as dampening transmission [44]. Multiple studies
have found signals of long distance transmission beyond the range of
disease-mediated dispersal, showing the role of human-mediated movement
of incubating dogs [44]-[46]. Road networks have been identified as
correlates of phylogenetic distance, indicating that human movement
could shape the spatial structure of canine rabies virus [44]-[46].
There is also strong pyhlogenetic evidence that historical
human-mediated long-distance movements underlie much of the contemporary
global distribution of canine rabies [47]. This work emphasizes the
need to understand how the size and connectivity of populations affects
the persistence of disease. Models have productively explored this
historically important question for childhood infections such as measles
[48], but for canine rabies, this remains an important challenge,
which may well define progress towards elimination.

A few studies look at how contact networks and movement behaviors could
drive transmission [49]-[51]. These studies simulated outbreaks on
contacts networks constructed using data from healthy domestic dogs.
They found that in general, targeting highly connected dogs or dogs with
larger home ranges for vaccination results in a higher probability of
disease elimination, but few predictors of connectivity of individuals
emerged. Broadly, these results are consistent with previous work on
transmission heterogeneity and could bring valuable benefits if it were
possible to *a priori* identify and target high-risk animals. However,
these traits are difficult to estimate in most endemic settings, where
there is limited data on dog populations, let alone individual dog
traits. Moreover, as rabies causes severe neurological symptoms, the
validity of these findings depends on how representative data from
healthy dogs are of movement and contact patterns of rabid dogs.

Dynamic models have been integrated with economic models to estimate
cost-effectiveness of interventions, demand for rabies PEP, and disease
burden. Early cost-effectiveness models critically lacked data on the
costs of PEP for those seeking care for non-rabid dog bites [27],
[52], [53]. Decision tree models have addressed these issues and
provide a framework to integrate field data on rabies exposures,
health-seeking, and access and adherence to PEP into estimates of burden
[54]-[56]. These more recent studies demonstrate that PEP is still a
very cost-effective intervention even when accounting for management of
patients bitten by non-rabid animals and emphasize the potential value
of administering rabies vaccine intradermally using the latest WHO
recommended abridged regimens [57]. However, they also highlight two
other critical points for policy. First, without strategies for more
judicious use, costs of PEP will remain high and continue to rise even
when dog rabies is controlled. Moreover, human rabies deaths will
continue to occur and the target of zero deaths by 2030 cannot be
achieved through PEP alone. A massive scaling up of dog vaccination is
required in most endemic countries. Support for human rabies vaccines
through Gavi, the Vaccine Alliance, is therefore a promising step
towards the 2030 goal [56], but more investment and commitment is
still needed.


```{r ch4-fig3, out.width="90%", fig.cap = '(ref:ch4-fig3-cap)', fig.scap='Rabies data reported in modeling studies (N = 25 studies reporting 30 unique data sources).'}
knitr::include_graphics("figs/ch3/image3.jpeg")
```

(ref:ch4-fig3-cap) Rabies data reported in modeling studies (N = 25
studies reporting 30 unique data sources). A) Type of data used. B)
The scales of temporal (x-axis) and spatial (colors) information
available and the duration (y axis). The size of the points is
proportional to the number of observations in each data set. Any rabies
data that was reported in studies were included (even if not used for
fitting purposes, only for qualitative comparison). If multiple data
sets were used, they were included as separate data sources, and if the
same data set was used in multiple studies it was only included once.

## The gap between models and data

Despite limited surveillance, few studies incorporated observation
models into their analyses or conducted sensitivity analyses on how
under-reporting might bias their inferences (Figure \@ref(fig:ch4-fig2)D). Developing
models of the observation process and integrating them into dynamic
models (often termed state-space modeling, [58]-[60] is essential
when fitting to incomplete data. But, these modeling frameworks can also
guide surveillance strategies across the elimination timeline by
estimating the minimum detection levels and time necessary to verify
elimination [61].

A major limitation of many existing modeling studies is a lack of data
to inform conclusions, with less than 40% of models fit to data (Fig \@ref(fig:ch4-fig2)D). For studies which did report incidence data, the scale and
quality of the data also varied greatly. Human deaths reported at the
national or regional level and numbers of clinical and laboratory
confirmed animal cases were the most commonly used data (Figure \@ref(fig:ch4-fig3)A).
The number of observations and length of the time series varied greatly,
from over 1000+ observations at a fine spatiotemporal scale over a
15-year period to annual cases reported for only 2 years (Figure \@ref(fig:ch4-fig3)B).
Ultimately, integrating data on rabies incidence and dog populations
into models of transmission is a critical step to moving modeling
efforts forward. Below we describe the various data sources that can be
used to fit and inform models and associated challenges and solutions to
collecting this data.

### Bite data

Bite data, (i.e. data on patients seeking care for animal bites) are
often used as a proxy for rabies exposure incidence. However, these data
often lack details on the status of the biting animal and are heavily
skewed by who has access to care, both geographically and
socioeconomically. Paradoxically, in settings where the direct cost of
PEP is charged to patients, bite records may be more reflective of
rabies exposures: people may be less likely to seek care when the
perceived risk is low (i.e. fewer people seek care for provoked bites by
known healthy and/or vaccinated animals) due to the associated costs
[62], [63]. In settings where PEP is provided for free and
indiscriminately, a higher proportion of reported bites may be due to
non-rabid animals [64]-[66], and many Category 1 exposures, i.e.
those for which PEP is not indicated [67], receive unnecessary PEP
[66], [68], [69].

For data on bite patients to be more useful for modeling and
surveillance purposes, supplementary information for each bite beyond
the date reported and number of doses received is needed. Categorizing
the type of exposure per the WHO categories can help to exclude Category
1 exposures. Reporting clinical signs and the outcome of the biting
animal at each patient visit can identify probable rabies exposures and
trigger field investigations and sample collection to improve
surveillance. Finally, information on the geographical location where
the patient was bitten, for example to the finest scale administrative
unit identifiable, could be used to understand spatial patterns of
transmission, estimate demand for PEP, and identify determinants of
health seeking behavior.

### Laboratory confirmed case data

Laboratory confirmed case data are considered a gold standard due to the
high sensitivity and specificity of diagnostic tests for rabies, but
represent the tip of the iceberg in terms of true incidence [9],
[61]. Diagnostic confirmation of rabies cases is often lacking in many
endemic settings due to limited laboratory and field capacity. Even with
strong laboratory resources in country, collecting a brain sample from a
suspected rabid animal or human case can be challenging. Lack of cold
chain and accessibility to communities, limited veterinary capacity and
training in euthanasia and sampling methods, and low reporting of
suspected cases are all significant barriers to case confirmation. For
humans, nuchal samples can be collected non-invasively (from nape of the
neck) to confirm a rabies case ante-mortem [70]. However, confirmation
of a human case first requires a person to seek care, and rabies deaths
are most common in populations with the least access to health care
[71]. For animal cases, field sample collection methods, like the
straw method of sampling brain tissue that does not require the
submission of the whole head, and alternative forms of sample storage
and testing, such as rapid diagnostic tests and filter papers, have
potential to address some of these challenges [72]. While these
alternative tests may not be appropriate for guiding patient treatment,
they could greatly improve surveillance and understanding of rabies
virus transmission if implemented more routinely.

Even with the gold-standard diagnostic test, using laboratory
confirmation to guide administration of PEP in endemic settings may be
impractical, due to delays in sampling and testing. Integrated bite case
management (IBCM, see Chapter 18) programs, which combine risk
assessments, field investigations, animal observation/quarantine, and
sampling of suspected cases, are a promising method of improving rabies
surveillance and PEP provisioning. IBCM can increase both detection of
and confirmation of clinically suspect animal cases and guide referrals
for PEP , as well as limit further exposures by euthanizing rabid
animals once detected [73]. However, IBCM relies on coordination
between human and animal health practitioners and resources to support
clinical rabies diagnosis and field sample collection, which is still
lacking in most low-income countries.

### Sequence data

Sequence data can be used to make inferences about transmission
processes, particularly when linked with epidemiological data
[43]-[45], [60], [74], [75]. Recent studies have demonstrated
the added value of whole-genome sequencing (WGS) for understanding finer
scale transmission dynamics of canine rabies [44], [74], but WGS has
yet to be routinely generated for canine rabies. Sequencing capacity is
even more limited than general laboratory capacity in rabies-endemic
countries and exporting samples for sequencing is costly. Advances in
portable, real-time sequencing could help to tackle these limitations in
the field (ARTICnetwork, <http://artic.network/index.html>). Portable
sequencers such as the MinION could support rapid generation and
dissemination of sequence data. Methods to sequence from alternative
sample types, such as rapid diagnostic tests and filter papers, could
also help to overcome obstacles in field sample collection and transport
[72]. Bioinformatic pipelines and open sharing of sequences, such as
those developed for other viral pathogens [76], could greatly
facilitate our understanding of rabies dynamics at a regional and global
scale. In general, low-cost, high-throughput sequencing methods should
be developed to increase the timely availability of representative
sequence data from endemic settings.

### Dog population and vaccination data

Data on the dog population is necessary to further understand how the
distribution, density, and connectivity of the host population drives
transmission [74]. Estimates of vaccination coverage and other
intervention efforts facilitate inference of the mechanisms driving
transmission and the impact of interventions, helping to predict future
outcomes given different control strategies [75]. In most endemic
countries, limited systematic data is collected on dog populations. If
integrated into more routine census or demographic surveys (i.e. the
Demographic and Health Surveys, <https://dhsprogram.com>), questions on
dog ownership and vaccination status at the household level could be a
potential way to get this data where the majority of the dog population
is owned. However, if conducted as standalone surveys, these can be
resource intensive and difficult to implement in a representative way,
particularly in more rural/remote areas. Alternatively, integrating
post-vaccination coverage surveys into campaigns has been shown to be a
cost-effective way to generate coverage and population estimates, and
only requires temporary marking of vaccinated dogs [77]-[79]. As
spatial heterogeneity in coverage is likely a key factor driving the
success of vaccination campaigns, such coverage estimates at the scale
at which campaigns are implemented could be critical to understanding
rabies persistence and elimination probabilities.

## Conclusions

Modeling studies, in combination with decades of empirical evidence,
have demonstrated that dog vaccination is the optimal intervention
strategy for controlling canine rabies. As global momentum for
implementing national rabies control programs grows, models should move
beyond comparing vaccination and other strategies in idealized
populations towards linking models with field data to identify
refinements to intervention strategies. To date, most work has focused
on studying control efforts and identifying drivers of dynamics (often
without using data), and studies of the impact of control have rarely
been linked to analyses grounded in empirical data (i.e. studies that
explained observed patterns or estimated key parameters, see
Figure \@ref(fig:figS2) for an overview of existing studies). Models
should aim to integrate these questions and test specific vaccination
strategies, such as ring vaccination or establishment of control
corridors based on geographic barriers as implemented for wildlife
rabies in Europe.

Key parameters to estimate from models and data include transmission
heterogeneity (captured in the distribution of secondary cases), the
dispersal kernel, and introduction rates (including how to differentiate
ongoing local transmission from imported cases). Integrating models of
surveillance into dynamic models can further establish surveillance
requirements necessary to verify freedom from disease and inform policy
decisions regarding the cessation and scaling back of control efforts.
Importantly, models can predict how these requirements might change over
the elimination timeline. Given the challenges in generating
high-quality surveillance data for canine rabies, these models can also
be used to account for under-reporting and determine the minimum level of
detection necessary for robust inference. Phylodynamic approaches, which
combine both epidemiological and genetic data, are a promising avenue to
tackle many of these questions. Critically, progress in this area will
require strong surveillance systems and representative data from a range
of populations.

Countries have made varying progress towards elimination, ranging from
some that lack a realized national control policy and others in the
end-game stages of elimination. Now, we are tasked with building
flexible models that can capture rabies dynamics and the impacts of
control across the elimination timeline. Identifying where and how
implementation of control efforts needs improving and delivering such
improvements will require a much closer collaboration between
scientists, practitioners and policymakers.

## Data and code availability

All data and code used to generate figures and supplemental files, as
well as the bibliography for the literature review are available online
at [https://github.com/mrajeev08/ModelingChapter](https://github.com/mrajeev08/ModelingChapter).

## References
\setlength{\parskip}{1em}

[1]	H. Heesterbeek, R. M. Anderson, V. Andreasen, S. Bansal, D. De Angelis, C. Dye, K. T. D. Eames, W. J. Edmunds, S. D. W. Frost, S. Funk, T. D. Hollingsworth, T. House, V. Isham, P. Klepac, J. Lessler, J. O. Lloyd-Smith, C. J. E. Metcalf, D. Mollison, L. Pellis, J. R. C. Pulliam, M. G. Roberts, C. Viboud, Isaac Newton Institute IDD Collaboration, “Modeling infectious disease dynamics in the complex landscape of global health,” Science, vol. 347, no. 6227, pp. aaa4339–aaa4339, Mar. 2015.

[2]	P. Klepac, C. J. E. Metcalf, A. R. McLean, and K. Hampson, “Towards the endgame and beyond: complexities and challenges for the elimination of infectious diseases,” Philosophical Transactions of the Royal Society B: Biological Sciences, vol. 368, no. 1623, pp. 20120137–20120137, Jun. 2013.

[3]	V. G. Panjeti and L. A. Real, “Mathematical models for rabies.,” Adv. Virus Res., vol. 79, pp. 377–395, 2011.

[4]	R. M. Anderson, H. C. Jackson, R. M. May, and A. M. Smith, “Population dynamics of fox rabies in Europe,” Nature, vol. 289, no. 5800, pp. 765–771, Feb. 1981.

[5]	J. D. Murray, E. A. Stanley, and D. L. Brown, “On the spatial spread of rabies among foxes,” Proceedings of the Royal Society of London. Series B. Biological Sciences, vol. 229, no. 1255, pp. 111–150, Nov. 1986.

[6]	D. L. Smith, B. Lucey, L. A. Waller, J. E. Childs, and L. A. Real, “Predicting the spatial dynamics of rabies epidemics on heterogeneous landscapes.,” Proc Natl Acad Sci USA, vol. 99, no. 6, pp. 3668–3672, Mar. 2002.

[7]	S. M. Duke-Sylvester, L. Bolzoni, and L. A. Real, “Strong seasonality produces spatial asynchrony in the outbreak of infectious diseases,” Journal of The Royal Society Interface, vol. 8, no. 59, pp. 817–825, Jan. 2011.

[8]	S. Cleaveland, H. Beyer, K. Hampson, D. Haydon, F. Lankester, T. Lembo, F.-X. Meslin, M. Morters, Z. Mtema, M. Sambo, and S. Townsend, “The changing landscape of rabies epidemiology and control.,” Onderstepoort J. Vet. Res., vol. 81, no. 2, pp. E1–8, Apr. 2014.

[9]	T. P. Scott, A. Coetzer, A. S. Fahrion, and L. H. Nel, “Addressing the Disconnect between the Estimated, Reported, and True Rabies Data: The Development of a Regional African Rabies Bulletin,” Front Vet Sci, vol. 4, no. 4, pp. 1–6, Feb. 2017.

[10]	N. Lakshmanan, T. C. Gore, K. L. Duncan, M. J. Coyne, M. A. Lum, and F. J. Sterner, “Three-year rabies duration of immunity in dogs following vaccination with a core combination vaccine against canine distemper virus, canine adenovirus type-1, canine parvovirus, and rabies virus.,” Vet. Ther., vol. 7, no. 3, pp. 223–231, 2006.

[11]	T. Lembo, D. T. Haydon, A. Velasco-Villa, C. E. Rupprecht, C. Packer, P. E. Brandão, I. V. Kuzmin, A. R. Fooks, J. Barrat, and S. Cleaveland, “Molecular epidemiology identifies only a single rabies virus variant circulating in complex carnivore communities of the Serengeti.,” Proceedings of the Royal Society B: Biological Sciences, vol. 274, no. 1622, pp. 2123–2130, Sep. 2007.

[12]	T. Lembo, K. Hampson, D. T. Haydon, M. Craft, A. Dobson, J. Dushoff, E. Ernest, R. Hoare, M. Kaare, T. Mlengeya, C. Mentzel, and S. Cleaveland, “Exploring reservoir dynamics: a case study of rabies in the Serengeti ecosystem.,” J Appl Ecol, vol. 45, no. 4, pp. 1246–1257, Aug. 2008.

[13]	S. Cleaveland, F. Lankester, S. Townsend, T. Lembo, and K. Hampson, “Rabies control and elimination: a test case for One Health.,” Vet. Rec., vol. 175, no. 8, pp. 188–193, Aug. 2014.

[14]	K. Hampson, J. Dushoff, S. Cleaveland, D. T. Haydon, M. Kaare, C. Packer, and A. Dobson, “Transmission dynamics and prospects for the elimination of canine rabies.,” PLOS Biol, vol. 7, no. 3, p. e53, Mar. 2009.

[15]	C. M. Foggin, “Rabies and rabies-related viruses in Zimbabwe: Historical, virological and ecological aspects,” Jan. 1988.

[16]	T. Hemachudha, J. Laothamatas, and C. E. Rupprecht, “Human rabies: a disease of complex neuropathogenetic mechanisms and diagnostic challenges,” Lancet Neur, vol. 1, no. 2, pp. 101–109, Jun. 2002.

[17]	V. Tepsumethanon, H. Wilde, and F. X. Meslin, “Six criteria for rabies diagnosis in living dogs.,” J Med Assoc Thai, vol. 88, no. 3, pp. 419–422, Mar. 2005.

[18]	Y.-Z. Zhang, Z. F. Fu, D.-M. Wang, J.-Z. Zhou, Z.-X. Wang, T.-F. Lv, C.-L. Xiong, Y. Zou, W.-R. Yao, M.-H. Li, G.-M. Dong, G.-L. Xu, M. Niezgoda, I. V. Kuzmin, and C. E. Rupprecht, “Investigation of the Role of Healthy Dogs as Potential Carriers of Rabies Virus,” Vector-Borne and Zoonotic Diseases, vol. 8, no. 3, pp. 313–320, Jun. 2008.

[19]	K. Brunker, K. Hampson, D. L. Horton, and R. Biek, “Integrating the landscape epidemiology and genetics of RNA viruses: rabies in domestic dogs as a model.,” Parasitology, vol. 139, no. 14, pp. 1899–1913, Dec. 2012.

[20]	A. M. Czupryna, J. S. Brown, M. A. Bigambo, C. J. Whelan, S. D. Mehta, R. M. Santymire, F. J. Lankester, and L. J. Faust, “Ecology and Demography of Free-Roaming Domestic Dogs in Rural Villages near Serengeti National Park in Tanzania,” PLoS ONE, vol. 11, no. 11, p. e0167092, Nov. 2016.

[21]	J. O. Lloyd-Smith, P. C. Cross, C. J. Briggs, M. Daugherty, W. M. Getz, J. Latto, M. S. Sánchez, A. B. Smith, and A. Swei, “Should we expect population thresholds for wildlife disease?,” Trends in Ecology & Evolution, vol. 20, no. 9, pp. 511–519, Sep. 2005.

[22]	H. McCallum, N. Barlow, and J. Hone, “How should pathogen transmission be modelled?,” Trends in Ecology & Evolution, vol. 16, no. 6, pp. 295–300, Jun. 2001.

[23]	P. G. Coleman and C. Dye, “Immunization coverage required to prevent outbreaks of dog rabies,” Vaccine, 1996.

[24]	S. E. Townsend, I. P. Sumantra, Pudjiatmoko, G. N. Bagus, E. Brum, S. Cleaveland, S. Crafter, A. P. M. Dewi, D. M. N. Dharma, J. Dushoff, J. Girardi, I. K. Gunata, E. F. Hiby, C. Kalalo, D. L. Knobel, I. W. Mardiana, A. A. G. Putra, L. Schoonman, H. Scott-Orr, M. Shand, I. W. Sukanadi, P. P. Suseno, D. T. Haydon, and K. Hampson, “Designing programs for eliminating canine rabies from islands: Bali, Indonesia as a case study.,” PLoS Negl Trop Dis, vol. 7, no. 8, p. e2372, 2013.

[25]	A. Kurosawa, K. Tojinbara, H. Kadowaki, K. Hampson, A. Yamada, and K. Makita, “The rise and fall of rabies in Japan: A quantitative history of rabies epidemics in Osaka Prefecture, 1914–1933,” PLoS Negl Trop Dis, vol. 11, no. 3, pp. e0005435–19, Mar. 2017.

[26]	M. C. Fitzpatrick, K. Hampson, S. Cleaveland, L. A. Meyers, J. P. Townsend, and A. P. Galvani, “Potential for Rabies Control through Dog Vaccination in Wildlife-Abundant Communities of Tanzania,” PLoS Negl Trop Dis, vol. 6, no. 8, pp. e1796–6, Aug. 2012.

[27]	J. Zinsstag, S. Durr, M. A. Penny, R. Mindekem, F. Roth, S. M. Gonzalez, S. Naissengar, and J. Hattendorf, “Transmission dynamics and economics of rabies control in dogs and humans in an African city,” Proc Natl Acad Sci USA, vol. 106, no. 35, pp. 1–22, Aug. 2009.

[28]	H. Tian, Y. Feng, B. Vrancken, B. Cazelles, H. Tan, M. S. Gill, Q. Yang, Y. Li, W. Yang, Y. Zhang, Y. Zhang, P. Lemey, O. G. Pybus, N. C. Stenseth, H. Zhang, and S. Dellicour, “Transmission dynamics of re-emerging rabies in domestic dogs of rural China,” PLoS Pathog, vol. 14, no. 12, pp. e1007392–18, Dec. 2018.

[29]	M. K. Morters, O. Restif, K. Hampson, S. Cleaveland, J. L. N. Wood, and A. J. K. Conlan, “Evidence-based control of canine rabies: a critical review of population density reduction,” J Anim Ecol, vol. 82, no. 1, pp. 6–14, Sep. 2012.

[30]	M. J. Keeling and P. Rohani, Modeling infectious diseases in humans and animals. 2008.

[31]	K. Hampson, B. Abela-Ridder, K. Brunker, S. T. M. Bucheli, M. Carvalho, E. Caldas, J. Changalucha, S. Cleaveland, J. Dushoff, V. Gutierrez, A. R. Fooks, K. Hotopp, D. T. Haydon, A. Lugelo, K. Lushasi, R. Mancy, D. Marston, Z. Mtema, M. Rajeev, L. R. M. P. Dourado, J. F. G. Roldan, K. Rysava, S. M. Rocha, M. Sambo, L. Sikana, M. Vigilato, and V. Del Rio Vilas, “Surveillance to Establish Elimination of Transmission and Freedom from Dog-mediated Rabies,” bioRxiv, p. 096883, Dec. 2016.

[32]	M. J. Ferrari, S. E. Perkins, L. W. Pomeroy, and O. N. Bjørnstad, “Pathogens, Social Networks, and the Paradox of Transmission Scaling,” Interdisciplinary Perspectives on Infectious Diseases, vol. 2011, no. 2, pp. 1–10, 2011.

[33]	J. P. Aparicio and M. Pascual, “Building epidemiological models from R0: an implicit treatment of transmission in networks,” Proceedings of the Royal Society B: Biological Sciences, vol. 274, no. 1609, pp. 505–512, Feb. 2007.

[34]	M. Pascual, M. Roy, and K. Laneri, “Simple models for complex systems: exploiting the relationship between local and global densities,” Theor Ecol, vol. 4, no. 2, pp. 211–222, Mar. 2011.

[35]	E. A. Ferguson, K. Hampson, S. Cleaveland, R. Consunji, R. Deray, J. Friar, D. T. Haydon, J. Jimenez, M. Pancipane, and S. E. Townsend, “Heterogeneity in the spread and control of infectious disease: consequences for the elimination of canine rabies,” Nature Publishing Group, vol. 5, no. 1, pp. 1–13, Dec. 2015.

[36]	K. Hampson, J. Dushoff, J. Bingham, G. Brückner, Y. H. Ali, and A. Dobson, “Synchronous cycles of domestic dog rabies in sub-Saharan Africa and the impact of control efforts.,” Proc Natl Acad Sci USA, vol. 104, no. 18, pp. 7717–7722, May 2007.

[37]	N. C. Grassly and C. Fraser, “Mathematical models of infectious disease transmission,” Nat Rev Micro, vol. 180, pp. 1–12, May 2008.

[38]	W. Rattanavipapong, M. Thavorncharoensap, S. Youngkong, A. J. Genuino, T. Anothaisintawee, U. Chaikledkaew, and A. Meeyai, “The impact of transmission dynamics of rabies control: Systematic review.,” Vaccine, Dec. 2018.

[39]	T. Anothaisintawee, A. Julienne Genuino, M. Thavorncharoensap, S. Youngkong, W. Rattanavipapong, A. Meeyai, and U. Chaikledkaew, “Cost-effectiveness modelling studies of all preventive measures against rabies: A systematic review.,” Vaccine, Dec. 2018.

[40]	World Health Organization, “WHO Expert Consultation on Rabies. Second report.,” World Health Organ Tech Rep Ser, no. 982, pp. 1–139– back cover, 2013.

[41]	J. O. Lloyd-Smith, S. J. Schreiber, P. E. Kopp, and W. M. Getz, “Superspreading and the effect of individual variation on disease emergence,” Nature, vol. 438, no. 7066, pp. 355–359, Nov. 2005.

[42]	M. Laager, M. Léchenne, K. Naissengar, R. Mindekem, A. Oussiguere, J. Zinsstag, and N. Chitnis, “A metapopulation model of dog rabies transmission in N'Djamena, Chad,” Journal of Theoretical Biology, vol. 462, pp. 408–417, Feb. 2019.

[43]	H. Bourhy, E. Nakouné, M. Hall, P. Nouvellet, A. Lepelletier, C. Talbi, L. Watier, E. C. Holmes, S. Cauchemez, P. Lemey, C. A. Donnelly, and A. Rambaut, “Revealing the Micro-scale Signature of Endemic Zoonotic Disease Transmission in an African Urban Setting,” PLoS Pathog, vol. 12, no. 4, pp. e1005525–15, Apr. 2016.

[44]	K. Brunker, D. A. Marston, D. L. Horton, S. Cleaveland, A. R. Fooks, R. Kazwala, C. Ngeleja, T. Lembo, M. Sambo, Z. J. Mtema, L. Sikana, G. Wilkie, R. Biek, and K. Hampson, “Elucidating the phylodynamics of endemic rabies virus in eastern Africa using whole-genome sequencing.,” Virus Evol, vol. 1, no. 1, pp. vev011–11, 2015.

[45]	C. Talbi, P. Lemey, M. A. Suchard, E. Abdelatif, M. Elharrak, N. Jalal, A. Faouzi, J. E. Echevarría, S. Vazquez Morón, A. Rambaut, N. Campiz, A. J. Tatem, E. C. Holmes, and H. Bourhy, “Phylodynamics and Human-Mediated Dispersal of a Zoonotic Virus,” PLoS Pathog, vol. 6, no. 10, pp. e1001166–10, Oct. 2010.

[46]	K. Tohma, M. Saito, C. S. Demetria, D. L. Manalo, B. P. Quiambao, T. Kamigaki, and H. Oshitani, “Molecular and mathematical modeling analyses of inter-island transmission of rabies into a previously rabies-free island in the Philippines,” INFECTION, GENETICS AND EVOLUTION, vol. 38, no. C, pp. 22–28, Mar. 2016.

[47]	A. A. King, A. R. Fooks, M. Aubert, and A. I. Wandeler, Historical perspective of rabies in Europe and the Mediterranean Basin. 2004.

[48]	O. N. Bjørnstad and B. T. Grenfell, “Hazards, spatial transmission and timing of outbreaks in epidemic metapopulations,” Environ Ecol Stat, vol. 15, no. 3, pp. 265–277, Dec. 2008.

[49]	M. Laager, C. Mbilo, E. A. Madaye, A. Naminou, M. Léchenne, A. Tschopp, S. K. Naïssengar, T. Smieszek, J. Zinsstag, and N. Chitnis, “The importance of dog population contact network structures in rabies transmission,” PLoS Negl Trop Dis, vol. 12, no. 8, p. e0006680, Aug. 2018.

[50]	E. G. Hudson, V. J. Brookes, M. P. Ward, and S. Dürr, “Using roaming behaviours of dogs to estimate contact rates: the predicted effect on rabies spread,” Epidemiol. Infect., vol. 147, p. e135, Jan. 2019.

[51]	J. K. Wilson-Aggarwal, L. Ozella, M. Tizzoni, C. Cattuto, G. J. F. Swan, T. Moundai, M. J. Silk, J. A. Zingeser, and R. A. McDonald, “High-resolution contact networks of free-ranging domestic dogs Canis familiaris and implications for transmission of infection.,” PLoS Negl Trop Dis, vol. 13, no. 7, p. e0007565, Jul. 2019.

[52]	K. Hampson, S. Cleaveland, and D. Briggs, “Evaluation of Cost-Effective Strategies for Rabies Post-Exposure Vaccination in Low-Income Countries,” PLoS Negl Trop Dis, vol. 5, no. 3, p. e982, Mar. 2011.

[53]	M. C. Fitzpatrick, K. Hampson, S. Cleaveland, I. Mzimbiri, F. Lankester, T. Lembo, L. A. Meyers, A. D. Paltiel, and A. P. Galvani, “Cost-effectiveness of canine vaccination to prevent human rabies in rural Tanzania.,” Ann. Intern. Med., vol. 160, no. 2, pp. 91–100, Jan. 2014.

[54]	D. L. Knobel, S. Cleaveland, P. G. Coleman, E. M. Fèvre, M. I. Meltzer, M. E. G. Miranda, A. Shaw, J. Zinsstag, and F.-X. Meslin, “Re-evaluating the burden of rabies in Africa and Asia.,” Bull. World Health Organ., vol. 83, no. 5, pp. 360–368, May 2005.

[55]	K. Hampson, L. Coudeville, T. Lembo, M. Sambo, A. Kieffer, M. Attlan, J. Barrat, J. D. Blanton, D. J. Briggs, S. Cleaveland, P. Costa, C. M. Freuling, E. Hiby, L. Knopf, F. Leanes, F.-X. Meslin, A. Metlin, M. E. Miranda, T. Müller, L. H. Nel, S. Recuenco, C. E. Rupprecht, C. Schumacher, L. Taylor, M. A. N. Vigilato, J. Zinsstag, J. Dushoff, Global Alliance for Rabies Control Partners for Rabies Prevention, “Estimating the global burden of endemic canine rabies.,” PLoS Negl Trop Dis, vol. 9, no. 4, p. e0003709, Apr. 2015.

[56]	WHO Rabies Modelling Consortium, “The potential impact of improved provision of rabies post-exposure prophylaxis in Gavi-eligible countries: a modelling study,” Lancet Infect Dis, vol. 19, no. 1, pp. 102–111, Jan. 2019.

[57]	A. Tarantola, S. Ly, M. Chan, S. In, Y. Peng, C. Hing, C. N. Taing, C. Phoen, S. Ly, S. Cauchemez, P. Buchy, P. Dussart, H. Bourhy, and J.-Y. Mary, “Intradermal rabies post-exposure prophylaxis can be abridged with no measurable impact on clinical outcome in Cambodia, 2003-2014.,” Vaccine, Nov. 2018.

[58]	H. L. Beyer, K. Hampson, T. Lembo, S. Cleaveland, M. Kaare, and D. T. Haydon, “Metapopulation dynamics of rabies and the efficacy of vaccination,” Proceedings of the Royal Society B: Biological Sciences, vol. 278, no. 1715, pp. 2182–2190, Dec. 2010.

[59]	N. Mollentze, L. H. Nel, S. Townsend, K. le Roux, K. Hampson, D. T. Haydon, and S. Soubeyrand, “A Bayesian approach for inferring the dynamics of partially observed endemic infectious diseases from space-time-genetic data.,” Proc. Biol. Sci., vol. 281, no. 1782, pp. 20133251–20133251, May 2014.

[60]	A. Cori, P. Nouvellet, T. Garske, H. Bourhy, E. Nakouné, and T. Jombart, “A graph-based evidence synthesis approach to detecting outbreak clusters: An application to dog rabies,” PLoS Comput Biol, vol. 14, no. 12, pp. e1006554–22, Dec. 2018.

[61]	S. E. Townsend, T. Lembo, S. Cleaveland, F. X. Meslin, M. E. Miranda, A. A. G. Putra, D. T. Haydon, and K. Hampson, “Surveillance guidelines for disease elimination: A case study of canine rabies,” “Comparative Immunology, Microbiology and Infectious Diseases,” vol. 36, no. 3, pp. 249–261, May 2013.

[62]	J. Changalucha, R. Steenson, E. Grieve, S. Cleaveland, T. Lembo, K. Lushasi, G. Mchau, Z. Mtema, M. Sambo, A. Nanai, N. J. Govella, A. Dilip, L. Sikana, F. Ventura, and K. Hampson, “The need to improve access to rabies post-exposure vaccines: Lessons from Tanzania.,” Vaccine, Oct. 2018.

[63]	K. Hampson, A. Dobson, M. Kaare, J. Dushoff, M. Magoto, E. Sindoya, and S. Cleaveland, “Rabies Exposures, Post-Exposure Prophylaxis and Deaths in a Region of Endemic Canine Rabies,” PLoS Negl Trop Dis, vol. 2, no. 11, pp. 1–9, Nov. 2008.

[64]	K. Rysava, M. E. Miranda, R. Zapatos, S. Lapiz, P. Rances, L. M. Miranda, M. C. Roces, J. Friar, S. E. Townsend, and K. Hampson, “On the path to rabies elimination: The need for risk assessments to improve administration of post-exposure prophylaxis,” Vaccine, pp. 1–9, Dec. 2018.

[65]	R. M. Wallace, H. Reses, R. Franka, P. Dilius, N. Fenelon, L. Orciari, M. Etheart, A. Destine, K. Crowdis, J. D. Blanton, C. Francisco, F. Ludder, V. Del Rio Vilas, J. Haim, and M. Millien, “Establishment of a Canine Rabies Burden in Haiti through the Implementation of a Novel Surveillance Program,” PLoS Negl Trop Dis, vol. 9, no. 11, pp. e0004245–15, Nov. 2015.

[66]	M. Rajeev, G. Edosoa, C. Hanitriniaina, S. F. Andriamandimby, H. Guis, R. Ramiandrasoa, R. Ratovoson, L. Randrianasolo, M. Andriamananjara, J.-M. Heraud, L. Baril, C. J. E. Metcalf, and K. Hampson, “Healthcare utilization, provisioning of post-exposure prophylaxis, and estimation of human rabies burden in Madagascar,” Vaccine, pp. 1–10, Nov. 2018.

[67]	“Rabies vaccines: WHO position paper,” WHO Wkly Epidemiol Rec, 2018.

[68]	Tenzin, N. K. Dhand, and M. P. Ward, “Human rabies post exposure prophylaxis in Bhutan, 2005-2008: trends and risk factors.,” Vaccine, vol. 29, no. 24, pp. 4094–4101, May 2011.

[69]	V. Duong, A. Tarantola, S. Ong, C. Mey, R. Choeung, S. Ly, H. Bourhy, P. Dussart, and P. Buchy, “Laboratory diagnostics in dog-mediated rabies: an overview of performance and a proposed strategy for various settings,” Int J Infect Dis, vol. 46, pp. 107–114, May 2016.

[70]	L. Dacheux, J.-M. Reynes, P. Buchy, O. Sivuth, B. M. Diop, D. Rousset, C. Rathat, N. Jolly, J. B. Dufourcq, C. Nareth, S. Diop, C. Iehlé, R. Rajerison, C. Sadorge, and H. Bourhy, “A Reliable Diagnosis of Human Rabies Based on Analysis of Skin Biopsy Specimens,” Clinical Infectious Diseases, vol. 47, no. 11, pp. 1410–1417, Dec. 2008.

[71]	D. Wentworth, K. Hampson, S. M. Thumbi, A. Mwatondo, G. Wambura, and N. R. Chng, “A social justice perspective on access to human rabies vaccines,” Vaccine, pp. 1–3, Apr. 2019.

[72]	M. Léchenne, K. Naissengar, A. Lepelletier, I. O. Alfaroukh, H. Bourhy, J. Zinsstag, and L. Dacheux, “Validation of a Rapid Rabies Diagnostic Tool for Field Surveillance in Developing Countries,” PLoS Negl Trop Dis, vol. 10, no. 10, pp. e0005010–16, Oct. 2016.

[73]	E. A. Undurraga, M. I. Meltzer, C. H. Tran, C. Y. Atkins, M. D. Etheart, M. F. Millien, P. Adrien, and R. M. Wallace, “Cost-Effectiveness Evaluation of a Novel Integrated Bite Case Management Program for the Control of Human Rabies, Haiti 2014–2015,” Am J Trop Med Hyg, vol. 96, no. 6, pp. 1307–1317, Jun. 2017.

[74]	K. Brunker, P. Lemey, D. A. Marston, A. R. Fooks, A. Lugelo, C. Ngeleja, K. Hampson, and R. Biek, “Landscape attributes governing local transmission of an endemic zoonosis: Rabies virus in domestic dogs,” Molecular Ecology, vol. 27, no. 3, pp. 773–788, Feb. 2018.

[75]	J. Zinsstag, M. Léchenne, M. Laager, R. Mindekem, S. Naïssengar, A. Oussiguere, K. Bidjeh, G. Rives, J. Tessier, S. Madjaninan, M. Ouagal, D. D. Moto, I. O. Alfaroukh, Y. Muthiani, A. Traoré, J. Hattendorf, A. Lepelletier, L. Kergoat, H. Bourhy, L. Dacheux, T. Stadler, and N. Chitnis, “Vaccination of dogs in an African city interrupts rabies transmission and reduces human exposure.,” Sci Transl Med, vol. 9, no. 421, p. eaaf6984, Dec. 2017.

[76]	J. Hadfield, C. Megill, S. M. Bell, J. Huddleston, B. Potter, C. Callender, P. Sagulenko, T. Bedford, and R. A. Neher, “Nextstrain: real-time tracking of pathogen evolution | Bioinformatics | Oxford Academic,” Bioinformatics, vol. 34, no. 23, pp. 4121–4123, 2018.

[77]	A. D. Gibson, P. Ohal, K. Shervell, I. G. Handel, B. M. Bronsvoort, R. J. Mellanby, and L. Gamble, “Vaccinate-assess-move method of mass canine rabies vaccination utilising mobile technology data collection in Ranchi, India,” BMC Infect. Dis., pp. 1–10, Dec. 2015.

[78]	M. Sambo, P. C. D. Johnson, K. Hotopp, J. Changalucha, S. Cleaveland, R. Kazwala, T. Lembo, A. Lugelo, K. Lushasi, M. Maziku, E. Mbunda, Z. Mtema, L. Sikana, S. E. Townsend, and K. Hampson, “Comparing Methods of Assessing Dog Rabies Vaccination Coverage in Rural and Urban Communities in Tanzania,” Front Vet Sci, vol. 4, no. 8, pp. e0003709–12, Mar. 2017.

[79]	T. Tenzin, J. S. McKenzie, R. Vanderstichel, B. D. Rai, K. Rinzin, Y. Tshering, R. Pem, C. Tshering, N. Dahal, K. Dukpa, S. Dorjee, S. Wangchuk, P. D. Jolly, R. Morris, and M. P. Ward, “Comparison of mark-resight methods to estimate abundance and rabies vaccination coverage of free-roaming dogs in two urban areas of south Bhutan.,” Preventive Veterinary Medicine, vol. 118, no. 4, pp. 436–448, Mar. 2015.


## Supplementary Figures

\beginsupplement

```{r figS1, fig.cap = '(ref:figS1-cap)', fig.scap='Density vs. frequency-dependent transmission.'}
knitr::include_graphics("figs/ch3/image4.jpeg")
```

(ref:figS1-cap) Density vs. frequency-dependent transmission.
Monthly incidence (the proportion of the population
infected, and thus removed (as a result of mortality)) from mass-action
models of rabies with A) frequency and B) density dependent
transmission. Even in low transmission scenarios (R0 = 1.01 - 1.1),
incidence peaks at between 1.5 -- 2.0% per month for models with
density-dependent transmission and between 0.01 - 30% for
frequency-dependent transmission, compared with the 1 - 2% max annual
incidence observed empirically. Demographic and transmission parameters
are listed in Table \@ref(tab:ch4-tab1) (mean incubation and infectious periods were
input as annual rates). Frequency-dependent model is a SEI model with
starting dog population of 50,000 and seeded with 2 infectious
individuals. Density-dependent model is adapted from Anderson et al.
1981, with starting population density of 15 dogs per km^2^ , 0.01
infectious dogs per km^2^, and carrying capacity of 29 dogs per km^2^.


```{r figS2, fig.cap = '(ref:figS2-cap)', fig.scap= "Types of canine rabies modeling studies."}
knitr::include_graphics("figs/ch3/image5.jpeg")
```

(ref:figS2-cap) Types of modeling studies. Categories areadapted from Lloyd-Smith et al. 2009: 1) Predict future trends based on
currently available data and model projections; 2) Study control
measures (using models to estimate/simulate the impacts of control
efforts and compare intervention strategies); 3) Estimate key parameters
such as R~o~, the incubation period, the dispersal kernel; we also
differentiate between studies which 4) Identify drivers of dynamics
(that is look at hypothetical factors which may drive transmission*
*without comparing or fitting to data) and studies which 5) Explain
observed patterns (use models and data to determine likely drivers of
observed patterns).