Modeling canine rabies virus transmission dynamics

Chapter 20 in Rabies, 4th edition
Full text (author’s pre-editorial version) available here
Figures, tables, supplementary materials, and code

2020-06-26

Authors

Malavika Rajeev, C. Jessica E. Metcalf
Department of Ecology and Evolutionary Biology
Princeton University,
Princeton, New Jersey, USA

Katie Hampson
Institute of Biodiversity, Animal Health and Comparative Medicine
College of Medicine, Veterinary Medicine and Life Sciences
University of Glasgow
Glasgow G12 8QQ, UK

Abstract

Mathematical models of infectious disease are used to develop an understanding of disease dynamics and aid in the design of control strategies. Specifically, modeling can guide planning and implementation of interventions and improve our understanding of how disease dynamics, and therefore intervention strategies, may change as control measures are 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 contribute to guiding efforts to achieve this goal. We review existing modeling work and the insights generated from these studies, and identify outstanding questions and gaps in our knowledge. We further discuss the role that modeling can play in the future to address these questions and inform elimination.

Figure 20.1: SEIV framework for modeling rabies

## Project population
vacc_reconstruct <- function(t, y, parms){
  dS <- -parms["deaths"] * y["S"] + parms["births"] * (y["S"] + y["V"]) + parms["waning"] * y["V"]
  dV <- -parms["deaths"] * y["V"] - parms["waning"] * y["V"]
  dWaning <- -parms["waning"] * y["V"]
  dDeaths <- -parms["deaths"] * y["V"]
  dBirths <- -parms["births"] * (y["S"] + y["V"])
  return(list(c(dS, dV)))
} 

eventfun <- function(t, y, parms){
  with (as.list(y),{
    diff <- 0.7*(S + V) - V ## newly vaccinated
    V <- V + diff
    S <- S - diff
    return(c(S, V))
  })
}


## times in relevant timestep
times <- seq (1/52, 5, by = 1/52) ## weekly for 10 years

## initial pop
IS <- 70000
IV <- 0

## param
births <- 0.50
deaths <- 0.42
waning <- 0.33
parms <- c(deaths = deaths, births = births, waning = waning)

## run
calcsus <- lsoda(y = c(S = IS, V = IV), times = times, func = vacc_reconstruct, parms = parms, 
                 events = list(func = eventfun, time = seq(1/52, 5, by = 1)))
vacc_df <- data.frame(Year = calcsus[, "time"], propVacc = calcsus[,"V"]/(calcsus[,"V"] + calcsus[,"S"]))
vacc_a <- ggplot(data = vacc_df, aes(x = Year, y = propVacc)) +
  geom_line(color = "lightseagreen", size = 1) +
  xlab ("Year") +
  ylab("Coverage") + 
  ylim(c(0, 1)) +
  geom_hline(yintercept = 0.69, color = "aquamarine4", linetype = 2, size = 1) +
  newtheme +
  theme(text = element_text(color = "aquamarine4"), axis.text = element_text(color = "aquamarine4"), 
        axis.line = element_line(color = "aquamarine4"))
ggsave("../figs/fig1/vacc_a.jpeg", vacc_a, device = "jpeg", width = 5, height = 5)

### Secondary cases
## params
set.seed(123)
secondaries <- data.frame(secondaries = rnbinom(1000, mu = 1.2, size = 1.3))
sec_b <- ggplot(data = secondaries, aes(x = secondaries)) +
  geom_histogram(binwidth = 1, color = "lightgray", fill = "red", alpha = 0.5) +
  geom_vline(xintercept = 1.2, color = "red", linetype = 2, size = 1.1) +
  xlab("Secondary cases") +
  ylab("Frequency") +
  newtheme +
  theme(text = element_text(color = alpha("firebrick3", 1)),
        axis.text = element_text(color = alpha("firebrick3", 1)), 
        axis.line = element_line(color = alpha("firebrick3", 1)))
ggsave("../figs/fig1/sec_b.jpeg", sec_b, device = "jpeg", width = 5, height = 5)
# include_graphics("figs/fig1/secc_b.jpeg")

### Incubation period
incubation <- as.data.frame(list(Days = seq(0, 365, 1), 
                           Density = dgamma(seq(0, 365, 1), shape = 1.1518, 
                                    rate = 0.0429)))
inc_c <- ggplot(data = incubation, aes (x = Days, y = Density)) +
  geom_line(color = "darkred", size = 1.2) +
  xlab("Incubation Period \n (days)") +
  geom_vline(xintercept = 22.3, color = "darkred", linetype = 2, size = 1.1, alpha = 0.5) +
  newtheme +
  theme(text = element_text(color = "darkred"), axis.text = element_text(color = "darkred"), 
        axis.line = element_line(color = "darkred"))
ggsave("../figs/fig1/inc_c.jpeg", inc_c, device = "jpeg", width = 5, height = 5)

### Infectious period
infectious <- as.data.frame(list(Days = seq(0, 10, 0.1), 
                           Density = dgamma(seq(0, 10, 0.1), shape = 2.9012, 
                                    rate = 1.013)))
inf_d <- ggplot(data = infectious, aes (x = Days, y = Density)) +
  geom_line(color = "navy", size = 1.2) +
  xlab("Infectious Period \n (days)") +
  geom_vline(xintercept = 3.1, color = "navy", linetype = 2, size = 1.2, alpha = 0.5) +
  newtheme +
  theme(text = element_text(color = "navy"), axis.text = element_text(color = "navy"), 
        axis.line = element_line(color = "navy"))
ggsave("../figs/fig1/inf_d.jpeg", inf_d, device = "jpeg", width = 5, height = 5)


### Road network with scale
tz_roads <- readOGR("../data/Tanzania_Roads/Tanzania_Roads.shp")
tz_roads <- fortify(tz_roads)

map_e1 <- ggplot(data = tz_roads, aes(x = long, y = lat, group = group)) +
  geom_line(color = "mediumorchid4") +
  scalebar(tz_roads, location = "bottomleft", 
           dist = 200, dist_unit = "km", transform = TRUE, box.fill = c("mediumorchid4", "grey"),
           box.color =  c("mediumorchid4", "grey"),
           st.size = 5, st.dist = 0.05, st.color = "mediumorchid4", 
           height = 0.02, model = 'WGS84') +
  theme_map()

ggsave("../figs/fig1/map_e1.jpeg", map_e1, device = "jpeg", width = 5, height = 5)


### Dispersal kernel
## params
dispersal <- as.data.frame(list(km = seq(0, 10, 0.01), 
                           Density = dgamma(seq(0, 10, 0.01), shape = 0.215, 
                                    scale = 0.245)))
disp_e2 <- ggplot(data = dispersal, aes (x = km, y = Density)) +
  geom_line(color = "mediumorchid4", size = 1.2) +
  xlab("Distance to next bite \n (km)") +
  geom_vline(xintercept = 0.88, color = "mediumorchid4", linetype = 2, size = 1.2, alpha = 0.75) +
  newtheme +
  theme(text = element_text(color = "mediumorchid4"), axis.text = element_text(color = "mediumorchid4"), 
        axis.line = element_line(color = "mediumorchid4"))

ggsave("../figs/fig1/disp_e2.jpeg", disp_e2, device = "jpeg", width = 5, height = 5)

Figure 20.1. 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 20.1

Table 20.1: Key parameter values associated with underlying processes illustrated in Figure 20.1

table1 <- read.csv("../tables/maintable_20.1.csv")
kable(table1, rownames = FALSE) %>%
  kable_styling(bootstrap_options = c("striped", "hover")) %>%
  collapse_rows()
Process Distribution Parameters Value Source Inset
Birth rate – Mean annual rate (dogs/yr) 0.500 (Czupryna et al. 2016) A
Death rate 0.420
Vaccine waning 0.330 (Lakshmanan et al. 2006)
Secondary cases (R0) Negative binomial, mean 1.2 secondary cases Mean 1.200 (S. E. Townsend, Sumantra, et al. 2013) B
Dispersion parameter (k) 1.300
Incubation period Gamma, mean 22.3 days Shape 1.150 (Hampson et al. 2009) C
Rate 0.040
Infectious period Gamma, mean 3.1 days Shape 2.900 D
Rate 1.010
Dispersal kernel Gamma, mean 0.88 km Shape 0.215 (S. E. Townsend, Sumantra, et al. 2013) E

Figure 20.2: Summarizing studies

studies <- read.csv("../data/modeling_studies.csv")

## A. Studies by Year
year_A <- ggplot(data = studies, aes(x = Year)) +
  geom_bar(fill = "grey50") +
  scale_y_continuous(labels = scales::number_format(accuracy = 1)) +
  ylab("Number of studies") +
  labs(tag = "A") +
  newtheme +
  theme(axis.text.x = element_text(angle = 45, hjust = 1))

## B. Studies by country
## refactor countries first
studies %>%
  mutate(Country = fct_recode(Country, Other = "India", Other = "Israel", Other = "Kenya")) -> studies
## then plot   
country_B <- ggplot(data = studies, aes(x = fct_relevel(reorder(Country, Country, function(x)-length(x)), 
                                            "Multiple (Africa)", "Multiple (global)", 
                                           "Other", "Hypothetical", after = 10))) + 
  geom_bar(fill = "grey50") +
  scale_y_continuous(breaks = seq(0, 10, by = 2), labels = scales::number_format(accuracy = 1)) +
  ylab("Number of studies") +
  xlab("") +
  labs(tag = "B") +
  newtheme +
  theme(axis.text.x = element_text(angle = 45, hjust = 1))

## C. R0 plot
r_ests <- read.csv("../data/r_ests.csv")
R0_C <- ggplot(data = filter(r_ests, R_type == "R0"), aes(x = Estimate)) +
  geom_histogram(binwidth = 0.25, color = "white", fill = "grey50") +
  scale_y_continuous(labels = scales::number_format(accuracy = 1)) +
  labs(tag = "C") +
  xlab("R0 estimate") +
  ylab("Number of studies") +
  newtheme

## D. Key modeling decisions and features
studies %>%
  select(14:20) %>%
  gather() %>%
  filter(value %in% "Y") %>%
  group_by(key) %>%
  summarize(proportion = n()/51) -> summary
features_D <- ggplot(data = summary, aes(x = reorder(key, -proportion), y = proportion)) +
  geom_col(fill = "grey50") +
  scale_x_discrete(labels = c("DDT", 
                              "Fit to data", "Stochastic", "Spatial", 
                              "Heterogeneity", "Introductions", 
                              "Underreporting")) +
  ylab("Proportion of studies") +
  ylim(c(0, 0.8)) +
  xlab("") +
  labs(tag = "D") +
  newtheme +
  theme(axis.text.x = element_text(angle = 45, hjust = 1))


fig2 <- year_A + R0_C + country_B + features_D + plot_layout(ncol = 2, nrow = 2)
ggsave("../figs/fig2.jpeg", fig2, device = "jpeg", width = 8, height = 8)
include_graphics("../figs/fig2.jpeg")

Figure 20.2. 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 R0: most studies estimated R0 below 2 (10 studies, with 31 estimates; estimates of Re (the effective reproduction number which accounts for ongoing vaccination) and Rt (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 underreporting in their inference. Full bibliography and metadata included in Supplementary Table 1.)

Figure 20.3: summary of data used in modeling studies

## Reported data
repdata <- read.csv("../data/reported_data.csv")
repdata %>%
  mutate(Nobs_maxed = case_when(Nobs < 250 ~ as.numeric(Nobs), Nobs >= 250 ~ 250), 
         Spatial.scale = fct_relevel(Spatial.scale, "National", after = Inf), 
         Temporal.scale = fct_relevel(Temporal.scale, "Month", after = 2)) -> repdata

data_A <- ggplot(data = repdata, aes(x = reorder(Type.of.data, Type.of.data,function(x)-length(x)))) + 
  geom_bar() +
  newtheme +
  scale_x_discrete(labels = c("Clinical and lab \n confirmed animal cases", 
                              "Reported human \n deaths", 
                              "Sequence data", "Lab confirmed \n animal cases", 
                              "Lab confirmed \n human exposures", "Reported \n animal bites")) +
  xlab("Type of data") +
  ylab("Number of data \n sources") +
  labs(tag = "A") +
  coord_flip()

data_B <- ggplot(data = repdata, aes(x = Temporal.scale, y = Length, fill = Spatial.scale, size = Nobs_maxed)) +
  ggbeeswarm::geom_beeswarm(cex = 3.5, alpha = 0.75, shape = 21, color = "grey50", stroke = 1.5) +
  scale_fill_manual(values = c("Darkred", "Red", "Pink", "Grey50"), name = "Spatial scale") +
  scale_size_area(name = "# of Observations", breaks = c(10, 100, 150, 250), 
                       labels = c("10", "100", "150", "250+")) +
  ylab("Length of time \n series (years)") +
  xlab("Temporal scale") +
  newtheme +
  labs(tag = "B") +
  theme(axis.text.x = element_text(angle = 30, hjust = 1)) +
  guides(fill = guide_legend(override.aes = list(size = 3)))
fig3 <- data_A + data_B + plot_layout(ncol = 2, widths = c(1, 1))
ggsave("../figs/fig3.jpeg", plot = fig3, device = "jpeg", height = 8, width = 12)
include_graphics("../figs/fig3.jpeg")

Figure 20.3. 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.

Supplementary Figure S1: comparing models

# models
SEI.rabies <- function(nu = 0.50, mu = 0.42, R0 = 1.2, sigma = (1/22.3*365),
                       gamma = (1/3.1*365), K = 20, START.S = 50000, 
                       START.E = 0, START.I = 2,
                       START.S.density = 15, START.I.density = 0.01, 
                       START.E.density = 0.00, years = 20, steps = 1/52, model = "frequency",...){
  
  ### Deterministic skeleton of models for rabies
  
  require(deSolve)
  
  times <- seq(from = 0, to = years, by = steps)
  
  ## Simple FDT model with complete disease-induced mortality
  if (model == "frequency"){
    
    START.N <- START.S + START.E + START.I
    
    beta = (R0*gamma*(sigma + mu))/sigma
      
      # This function models a time step for the SIR:
      dx.dt.SIR <- function(t, y, parms) {
        N <- y["S"] + y["E"] + y["I"] 
        S <- y["S"]
        E <- y["E"]
        I <- y["I"]
        
        # Calculate the change in Susceptible
        dS <-  parms["nu"]*(S + E) - parms["mu"]*S - parms["beta"]*S*I/N
        
        #Calculate the change in Exposed
        dE <- parms["beta"]*S*I/N - parms["mu"]*E - parms["sigma"]*E
        
        # Calculate the change in Infected
        dI <- parms["sigma"]*E - parms["gamma"]*I
        
        # Return a list with the changes in S, E, I, R at the current time step
        return(list(c(dS, dE, dI)))
      }
      
      # Create the parameter vector
      parms <- c(nu = nu, mu = mu, beta = beta, sigma = sigma, gamma = gamma)
      inits <- c(S = START.S, E = START.E, I = START.I)
  }
  
  ## Classical fox rabies model (widely used for dog rabies as well, adapted from Anderson et al. 1981)
  if (model == "density"){
    
    START.N.density <- START.S.density + START.I.density + START.E.density
    
    # the old school rabies model
    # This function models a time step for the SIR:
    dx.dt.SIR <- function(t, y, parms) {
      N <- y["S"] + y["I"] + y["E"]
      S <- y["S"]
      E <- y["E"]
      I <- y["I"]

      ## Calculate dmort and beta here from parms and plug them in
      dmort <- (parms["nu"] - parms["mu"])/parms["K"]
      beta <- (parms["R0"]*(parms["sigma"] + parms["nu"])*(parms["gamma"] + parms ["nu"]))/(parms["sigma"]*parms["K"])
      
      # Calculate the change in Susceptible
      dS <-  parms["nu"]*N - parms["mu"]*S - dmort*S - beta*S*I
      
      #Calculate the change in Exposed
      dE <- beta*S*I - parms["mu"]*E - dmort*N*E - parms["sigma"]*E
      
      # Calculate the change in Infected
      dI <- parms["sigma"]*E - parms["mu"]*I - dmort*N*I - parms["gamma"]*I
      
      # Return a list with the changes in S, E, I, R at the current time step
      return(list(c(dS, dE, dI)))
    }
    
    # Create the parameter vector
    parms <- c(nu = nu, mu = mu, R0 = R0, K = K, sigma = sigma, gamma = gamma)
    inits <- c(S = START.S.density, E = START.E.density, I = START.I.density)
  }
  
  # Run the ODE solver
  SIR.output <- lsoda(y = inits, 
                      times = times, 
                      func = dx.dt.SIR, 
                      parms = parms)
  SIR.output <- as.data.frame(SIR.output)
  SIR.output$N <- SIR.output$S + SIR.output$E + SIR.output$I
  return (SIR.output)
}

## Density dependent
ddt_1.01 <- SEI.rabies(R0 = 1.01, years = 20, model = "density")
ddt_1.01 <- data.frame(infected = unname(tapply(ddt_1.01$I, (seq_along(ddt_1.01$I)-1) %/% 4, sum)[1:260]),
                       pop = ddt_1.01$N[seq(1, 20*52, 4)], R0 = 1.01, trans = "Density")
ddt_1.1 <- SEI.rabies(R0 = 1.1, years = 20, model = "density")
ddt_1.1 <- data.frame(infected = unname(tapply(ddt_1.1$I, (seq_along(ddt_1.1$I)-1) %/% 4, sum)[1:260]),
                       pop = ddt_1.1$N[seq(1, 20*52, 4)], R0 = 1.1, trans = "Density")
ddt_1.05 <- SEI.rabies(R0 = 1.05, years = 20, model = "density")
ddt_1.05 <- data.frame(infected = unname(tapply(ddt_1.05$I, (seq_along(ddt_1.05$I)-1) %/% 4, sum)[1:260]),
                      pop = ddt_1.05$N[seq(1, 20*52, 4)], R0 = 1.05, trans = "Density")

ddt <- bind_rows(ddt_1.01, ddt_1.1, ddt_1.05)
ddt$time <- 1:260

ddt_plot <- ggplot(data = ddt, aes(x = time, y = infected/pop, group = R0, color = as.factor(R0))) + 
  geom_line(size = 1) +
  scale_color_manual(values = c("lightcoral", "red", "darkred"), name = "R0") +
  xlab("Months") +
  ylab("Incidence (monthly proportion \n of population infected)") +
  newtheme +
  labs(tag = "B", subtitle = "Density")

## Frequency dependent
fdt_1.01 <- SEI.rabies(R0 = 1.01, years = 20, model = "frequency")
fdt_1.01 <- data.frame(infected = unname(tapply(fdt_1.01$I, (seq_along(fdt_1.01$I)-1) %/% 4, sum)[1:260]),
                       pop = fdt_1.01$N[seq(1, 20*52, 4)], R0 = 1.01, trans = "Frequency")
fdt_1.1 <- SEI.rabies(R0 = 1.1, years = 20, model = "frequency")
fdt_1.1 <- data.frame(infected = unname(tapply(fdt_1.1$I, (seq_along(fdt_1.1$I)-1) %/% 4, sum)[1:260]),
                      pop = fdt_1.1$N[seq(1, 20*52, 4)], R0 = 1.1, trans = "Frequency")
fdt_1.05 <- SEI.rabies(R0 = 1.05, years = 20, model = "frequency")
fdt_1.05 <- data.frame(infected = unname(tapply(fdt_1.05$I, (seq_along(fdt_1.05$I)-1) %/% 4, sum)[1:260]),
                       pop = fdt_1.05$N[seq(1, 20*52, 4)], R0 = 1.05, trans = "Frequency")
fdt <- bind_rows(fdt_1.01, fdt_1.1, fdt_1.05)
fdt$time <- 1:260

fdt_plot <- ggplot(data = fdt, aes(x = time, y = infected/pop, group = R0, color = as.factor(R0))) + 
  geom_line(size = 1) +
  scale_color_manual(values = c("lightcoral", "red", "darkred"), name = "R0") +
  xlab("Months") +
  ylab("Incidence (monthly proportion \n of population infected)") +
  guides(colour = "none") +
  newtheme +
  labs(tag = "A", subtitle = "Frequency")
figS1 <- fdt_plot / ddt_plot
ggsave("../figs/figS1.jpeg", plot = figS1, device = "jpeg", height = 7, width = 7)
include_graphics("../figs/figS1.jpeg")

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 20.1 (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 km2 , 0.01 infectious dogs per km2, and carrying capacity of 29 dogs per km2.

Supplementary Figure S2: types of studies

## Study category
studies %>%
  mutate_at(vars(starts_with("Category")), 
            function(x) ifelse(x == "N" | is.na(x), 0, 1)) %>%
  select(Reference, starts_with("Category")) -> categories
names(categories)
names(categories) <- c("Reference", "Explain patterns",
                       "Estimate parameters", "Study control",
                       "Identify drivers", "Predict trends")

## Upset plot
jpeg("../figs/figS2.jpeg", width = 1500, height = 1500)
upset(categories, nsets = 5, nintersects = NA,
      sets.x.label = "Type of study", mainbar.y.label = "Number of studies", text.scale = 5)
dev.off()

## Bar plot
studies %>%
  select(Reference, starts_with("Category")) %>%
  gather() %>%
  filter(value %in% "Y") -> cat_summary
table(cat_summary$key)

Types of modeling studies (categories adapted 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 Ro, 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).

Supplementary Table 1: Modeling studies included in literature review

Table S1

Table_S1 <- read_excel("../tables/supptable_S1.xlsx")
Table_S1 %>%
  dplyr::select(-Citation, -`Cite Key`) %>%
  datatable(rownames = FALSE)

Key for columns

cols_S1 <- read_excel("../tables/supptable_S1.xlsx", sheet = "DefineColumns")
kable(cols_S1, rownames = FALSE) %>%
  kable_styling(bootstrap_options = c("striped", "hover")) %>%
  collapse_rows()
Column Definition Notes
Reference Reference key
Year Year published
Country Country which the modeling analysis refers to
Scale of analysis The scale at which the modeling analysis was conducted
Model type The type of modeling analysis used in the study (analysis components which pertain to canine rabies dynamics)
Aims The aims of the study
Category: explain observed patterns Studies which develop models to match the dynamics observed in data (Lloyd-Smith et al. 2009)
Category: estimate key parameters Studies which use models and data to estimate key epidemiological paramemeters such as reproductive numbers, pathogen dispersal, etc.
Category: study control measures Studies which use models to simulate/compare the impacts of different control and surveillance measures
Category: predict future trends Studies that use models to predict incidence and/or burden of rabies in the future given current estimates and data
Category: identify drivers of dynamics Studies which use models to examine potential drivers of dynamics (separate category from explain observed patterns because do these do not use data or observations but use models as hypotheticals)
Reported rabies data Did the study report any rabies data?
Data description Description of the data reported
Fit to rabies data Was the data reported used to fit models/estimate parameters?
Observation/Underreporting Did the modeling analysis include an observation model or account for underreporting in their analysis (i.e. through sensitivity analysis)?
Introductions Did the model include introductions from outside of the population explicitly modeled?
Heterogeneity in transmission Did the model include individual heterogeneity in transmission?
Density-dependent transmission Did the model use the density-dependent formulation of transmission?
Stochastic Was the model stochastic?
Spatial Was the model spatially explicit?
Citation Full citation
Notes Notes on the analysis, etc.

Key for values in columns

cols_S1 <- read_excel("../tables/supptable_S1.xlsx", sheet = "DefineTerms")
kable(cols_S1, rownames = FALSE) %>%
  kable_styling(bootstrap_options = c("striped", "hover"))
Term Definitions
ODE Ordinary Differential Equation
DE Difference Equation
ORV Oral Rabies Vaccine
PEP Post-Exposure Prophylaxis
ML Maximum likelihood
IBM Individual-Based Model
R0 Reproductive number (the average number of secondary cases arising from a single infection in a completely susceptible population)
Rt Time-varying reproductive number
Reff Effective-reproductive number (accounting for proportion susceptible in a population, i.e. with ongoing vaccination)
Pc Critical vaccination threshold (1- 1/R0, the threshold level of vaccination necessary to interrupt transmission/eliminate infection in a population)

Citations

library(DT)
library(readxl)
Table_S1 <- read_excel("../tables/supptable_S1.xlsx")
Table_S1 %>%
  dplyr::select(`Cite Key`, Citation) %>%
  kable(rownames = FALSE) %>%
  kable_styling(bootstrap_options = c("striped", "hover")) %>%
  scroll_box(height = "1000px")
Cite Key Citation
Cleaveland and Dye (1995) Cleaveland, S, and C Dye. “Maintenance of a Microparasite Infecting Several Host Species: Rabies in the Serengeti.” Parasitology 111.S1 (1995): S33–S47. Web.
Coleman and Dye (1996) Coleman, P G, and C Dye. “Immunization Coverage Required to Prevent Outbreaks of Dog Rabies.” Vaccine (1996): n. pag. Print.
Bohrer et al. (2002) Bohrer, Gil et al. “The Effectiveness of Various Rabies Spatial Vaccination Patterns in a Simulated Host Population with Clumped Distribution.” Ecological Modelling 152.2-3 (2002): 205–211. Print.
Kitala et al. (2002) Kitala, P M et al. “Comparison of Vaccination Strategies for the Control of Dog Rabies in Machakos District, Kenya.” Epidemiology and Infection 129.1 (2002): 215–222. Web.
Hampson et al. (2007) Hampson, Katie, Jonathan Dushoff, John Bingham, et al. “Synchronous Cycles of Domestic Dog Rabies in Sub-Saharan Africa and the Impact of Control Efforts..” Proceedings of the National Academy of Sciences 104.18 (2007): 7717–7722. Web.
Wang and Lou (2008) WANG, XIAOWEI, and JIE LOU. “Two Dynamic Models About Rabies Between Dogs and Human.” Journal of Biological Systems 16.04 (2008): 519–529. Web.
Lembo et al. (2008) Lembo, Tiziana et al. “Exploring Reservoir Dynamics: a Case Study of Rabies in the Serengeti Ecosystem..” The Journal of applied ecology 45.4 (2008): 1246–1257. Web.
Hampson et al. (2009) Hampson, Katie, Jonathan Dushoff, Sarah Cleaveland, et al. “Transmission Dynamics and Prospects for the Elimination of Canine Rabies..” PLOS Biol 7.3 (2009): e53. Web.
Zinsstag et al. (2009) Zinsstag, J, S Durr, et al. “Transmission Dynamics and Economics of Rabies Control in Dogs and Humans in an African City.” Proceedings of the National Academy of Sciences 106.35 (2009): 1–22. Web.
Carroll et al. (2010) Carroll, Matthew J et al. “The Use of Immunocontraception to Improve Rabies Eradication in Urban Dog Populations.” Wildlife Research 37.8 (2010): 676–12. Web.
Talbi et al. (2010) Talbi, Chiraz et al. “Phylodynamics and Human-Mediated Dispersal of a Zoonotic Virus.” Ed. Michael Emerman. PLoS Pathogens 6.10 (2010): e1001166–10. Web.
Beyer et al. (2010) Beyer, H L, 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 278.1715 (2010): 2182–2190. Web.
Zhang et al. (2011) Zhang, Juan, Zhen Jin, Gui-Quan Sun, Tao Zhou, et al. “Analysis of Rabies in China: Transmission Dynamics and Control.” Ed. Pere-Joan Cardona. PLoS ONE 6.7 (2011): e20891. Web.
Bhunu (2011) BHUNU, C P. “Impact of Culling Stray Dogs and Vaccination on the Control of Human Rabies: a Mathematical Modeling Approach.” International Journal of Biomathematics 04.04 (2011): 379–397. Web.
J Zhang et al. (2012) Zhang, J, Z Jin, G Q Sun, and X Sun. “Spatial Spread of Rabies in China.” Journal of Applied Analysis and Computation (2012): n. pag. Print.
Fitzpatrick et al. (2012) Fitzpatrick, Meagan C, Katie Hampson, Sarah Cleaveland, Lauren Ancel Meyers, et al. “Potential for Rabies Control Through Dog Vaccination in Wildlife-Abundant Communities of Tanzania.” Ed. Richard Reithinger. PLoS Neglected Tropical Diseases 6.8 (2012): e1796–6. Web.
Hou, Jin, and Ruan (2012) Hou, Qiang, Zhen Jin, and Shigui Ruan. “Dynamics of Rabies Epidemics and the Impact of Control Efforts in Guangdong Province, China.” Journal of Theoretical Biology 300.C (2012): 39–47. Web.
Beyer et al. (2012) Beyer, Hawthorne L, Katie Hampson, Tiziana Lembo, Sarah Cleaveland, Magai Kaare, and Daniel T Haydon. “The Implications of Metapopulation Dynamics on the Design of Vaccination Campaigns.” Vaccine 30.6 (2012): 1014–1022. Web.
Juan Zhang et al. (2012) Zhang, Juan, Zhen Jin, Gui-Quan Sun, Xiang-Dong Sun, et al. “Modeling Seasonal Rabies Epidemics in China.” Bulletin of Mathematical Biology 74.5 (2012): 1226–1251. Web.
S. E. Townsend, Sumantra, et al. (2013) Townsend, Sunny E, Tiziana Lembo, et al. “Surveillance Guidelines for Disease Elimination: a Case Study of Canine Rabies.” “Comparative Immunology, Microbiology and Infectious Diseases” 36.3 (2013): 249–261. Web.
S. E. Townsend, Lembo, et al. (2013) Townsend, Sunny E, I Putu Sumantra, et al. “Designing Programs for Eliminating Canine Rabies From Islands: Bali, Indonesia as a Case Study..” Ed. Charles E Rupprecht. PLoS Neglected Tropical Diseases 7.8 (2013): e2372. Web.
Mollentze et al. (2014) Mollentze, Nardus et al. “A Bayesian Approach for Inferring the Dynamics of Partially Observed Endemic Infectious Diseases From Space-Time-Genetic Data..” Proceedings. Biological sciences / The Royal Society 281.1782 (2014): 20133251–20133251. Web.
Fitzpatrick et al. (2014) Fitzpatrick, Meagan C, Katie Hampson, Sarah Cleaveland, Imam Mzimbiri, et al. “Cost-Effectiveness of Canine Vaccination to Prevent Human Rabies in Rural Tanzania..” Annals of internal medicine 160.2 (2014): 91–100. Web.
Ferguson et al. (2015) Ferguson, Elaine A et al. “Heterogeneity in the Spread and Control of Infectious Disease: Consequences for the Elimination of Canine Rabies.” Nature Publishing Group 5.1 (2015): 1–13. Web.
Dürr and Ward (2015) Dürr, Salome, and Michael P Ward. “Development of a Novel Rabies Simulation Model for Application in a Non-Endemic Environment.” Ed. Jakob Zinsstag. PLoS Neglected Tropical Diseases 9.6 (2015): e0003876–22. Web.
Wiraningsih et al. (2015) Wiraningsih, E D et al. Stability Analysis of Rabies Model with Vaccination and Culling Effect on Dogs. Applied Mathematical …, 2015. Web.
Chen et al. (2015) Chen, Jing et al. “Modeling the Geographic Spread of Rabies in China.” Ed. Charles E Rupprecht. PLoS Neglected Tropical Diseases 9.5 (2015): e0003772–18. Web.
Bourhy et al. (2016) Bourhy, Hervé et al. “Revealing the Micro-Scale Signature of Endemic Zoonotic Disease Transmission in an African Urban Setting.” Ed. Colin Parrish. PLoS Pathogens 12.4 (2016): e1005525–15. Web.
Fitzpatrick et al. (2016) Fitzpatrick, Meagan C, Hiral A Shah, et al. “One Health Approach to Cost-Effective Rabies Control in India.” Proceedings of the National Academy of Sciences 113.51 (2016): 14574–14581. Web.
Wera et al. (2016) Wera, E et al. “Cost-Effectiveness of Mass Dog Vaccination Campaigns Against Rabies in Flores Island, Indonesia.” Transboundary and Emerging Diseases 64.6 (2016): 1918–1928. Web.
Bilinski et al. (2016) Bilinski, Alyssa M et al. “Optimal Frequency of Rabies Vaccination Campaigns in Sub-Saharan Africa..” Proceedings. Biological sciences / The Royal Society 283.1842 (2016): 20161211–9. Web.
Sparkes et al. (2016) Sparkes, Jessica et al. “Rabies Disease Dynamics in Naïve Dog Populations in Australia..” Preventive Veterinary Medicine 131 (2016): 127–136. Web.
Chapwanya, Lubuma, and Terefe (2016) Chapwanya, M, J M S Lubuma, and Y A Terefe. “Analysis and Dynamically Consistent Nonstandard Discretization for a Rabies Model in Humans and Dogs.” Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas 110.2 (2016): 783–798. Web.
Tohma et al. (2016) Tohma, Kentaro et al. “Molecular and Mathematical Modeling Analyses of Inter-Island Transmission of Rabies Into a Previously Rabies-Free Island in the Philippines.” INFECTION, GENETICS AND EVOLUTION 38.C (2016): 22–28. Web.
Kurosawa et al. (2017) Kurosawa, Aiko et al. “The Rise and Fall of Rabies in Japan: a Quantitative History of Rabies Epidemics in Osaka Prefecture, 1914–1933.” Ed. Charles E Rupprecht. PLoS Neglected Tropical Diseases 11.3 (2017): e0005435–19. Web.
Leung and Davis (2017) Leung, Tiffany, and Stephen A Davis. “Rabies Vaccination Targets for Stray Dog Populations.” Frontiers in veterinary science 4.5 (2017): e0003786–10. Web.
Liu, Jia, and Zhang (2017) Liu, Junli, Ying Jia, and Tailei Zhang. “Analysis of a Rabies Transmission Model with Population Dispersal.” Nonlinear Analysis: Real World Applications 35 (2017): 1–22. Web.
Asamoah et al. (2017) Asamoah, Joshua Kiddy K et al. “Modelling of Rabies Transmission Dynamics Using Optimal Control Analysis.” Journal of Applied Mathematics 2017.12 (2017): 1–23. Web.
Mengistu Tulu (2017) Mengistu Tulu, Aberu. “The Impact of Infective Immigrants on the Spread of Dog Rabies.” American Journal of Applied Mathematics 5.3 (2017): 68–10. Web.
Zinsstag et al. (2017) Zinsstag, Jakob, Monique Léchenne, et al. “Vaccination of Dogs in an African City Interrupts Rabies Transmission and Reduces Human Exposure..” Science translational medicine 9.421 (2017): eaaf6984. Web.
Cori et al. (2018) Cori, Anne et al. “A Graph-Based Evidence Synthesis Approach to Detecting Outbreak Clusters: an Application to Dog Rabies.” Ed. Mercedes Pascual. PLoS computational biology 14.12 (2018): e1006554–22. Web.
Huang et al. (2018) Huang, Jicai et al. “Modeling the Transmission Dynamics of Rabies for Dog, Chinese Ferret Badger and Human Interactions in Zhejiang Province, China.” Bulletin of Mathematical Biology 81.4 (2018): 939–962. Web.
WHO Rabies Modelling Consortium (2019) Hampson, Katie, Francesco Ventura, et al. “The Potential Effect of Improved Provision of Rabies Post-Exposure Prophylaxis in Gavi-Eligible Countries: a Modelling Study.” The Lancet. Infectious diseases 19.1 (2018): 102–111. Web.
Tian et al. (2018) Tian, Huaiyu et al. “Transmission Dynamics of Re-Emerging Rabies in Domestic Dogs of Rural China.” Ed. Matthias Johannes Schnell. PLoS Pathogens 14.12 (2018): e1007392–18. Web.
Borse et al. (2018) Borse, Rebekah H et al. “Cost-Effectiveness of Dog Rabies Vaccination Programs in East Africa.” Ed. Jakob Zinsstag. PLoS Neglected Tropical Diseases 12.5 (2018): e0006490–21. Web.
Kadowaki et al. (2018) Kadowaki, H et al. “The Risk of Rabies Spread in Japan: a Mathematical Modelling Assessment.” Epidemiology and Infection 146.10 (2018): 1245–1252. Web.
Laager et al. (2018) Laager, Mirjam, Céline Mbilo, et al. “The Importance of Dog Population Contact Network Structures in Rabies Transmission.” Ed. Charles E Rupprecht. PLoS Neglected Tropical Diseases 12.8 (2018): e0006680. Web.
Hudson et al. (2019) Hudson, Emily G et al. “Using Roaming Behaviours of Dogs to Estimate Contact Rates: the Predicted Effect on Rabies Spread.” Epidemiology and Infection 147 (2019): e135. Web.
Anderson et al. (2019) Anderson, Aaron et al. “A Bioeconomic Model for the Optimization of Local Canine Rabies Control.” Ed. Monique Léchenne. PLoS Neglected Tropical Diseases 13.5 (2019): e0007377–24. Web.
Laager et al. (2019) Laager, Mirjam, Monique Léchenne, et al. “A Metapopulation Model of Dog Rabies Transmission in N•Djamena, Chad.” Journal of Theoretical Biology 462 (2019): 408–417. Web.
Wilson-Aggarwal et al. (2019) Wilson-Aggarwal, Jared K et al. “High-Resolution Contact Networks of Free-Ranging Domestic Dogs Canis Familiaris and Implications for Transmission of Infection.” Ed. Sergio Recuenco. PLoS Neglected Tropical Diseases 13.7 (2019): e0007565. Web.

Session info

sessionInfo()
## R version 4.0.0 (2020-04-24)
## Platform: x86_64-apple-darwin17.0 (64-bit)
## Running under: macOS Mojave 10.14.6
## 
## Matrix products: default
## BLAS:   /Library/Frameworks/R.framework/Versions/4.0/Resources/lib/libRblas.dylib
## LAPACK: /Library/Frameworks/R.framework/Versions/4.0/Resources/lib/libRlapack.dylib
## 
## locale:
## [1] en_US.UTF-8/en_US.UTF-8/en_US.UTF-8/C/en_US.UTF-8/en_US.UTF-8
## 
## attached base packages:
## [1] grid      stats     graphics  grDevices utils     datasets  methods  
## [8] base     
## 
## other attached packages:
##  [1] readxl_1.3.1     DT_0.13          UpSetR_1.4.0     patchwork_1.0.0 
##  [5] forcats_0.5.0    stringr_1.4.0    dplyr_0.8.5      purrr_0.3.4     
##  [9] readr_1.3.1      tidyr_1.1.0      tibble_3.0.1     tidyverse_1.3.0 
## [13] ggsn_0.5.0       ggthemes_4.2.0   rgdal_1.4-8      sp_1.4-2        
## [17] deSolve_1.28     ggplot2_3.3.0    kableExtra_1.1.0 knitr_1.28      
## 
## loaded via a namespace (and not attached):
##  [1] nlme_3.1-147        bitops_1.0-6        fs_1.4.1           
##  [4] sf_0.9-3            lubridate_1.7.8     webshot_0.5.2      
##  [7] httr_1.4.1          tools_4.0.0         backports_1.1.7    
## [10] R6_2.4.1            KernSmooth_2.23-17  vipor_0.4.5        
## [13] DBI_1.1.0           colorspace_1.4-1    withr_2.2.0        
## [16] tidyselect_1.1.0    gridExtra_2.3       compiler_4.0.0     
## [19] cli_2.0.2           rvest_0.3.5         xml2_1.3.2         
## [22] labeling_0.3        scales_1.1.1        classInt_0.4-3     
## [25] digest_0.6.25       foreign_0.8-79      rmarkdown_2.1      
## [28] jpeg_0.1-8.1        pkgconfig_2.0.3     htmltools_0.4.0    
## [31] dbplyr_1.4.3        highr_0.8           htmlwidgets_1.5.1  
## [34] rlang_0.4.6         rstudioapi_0.11     generics_0.0.2     
## [37] farver_2.0.3        jsonlite_1.6.1      crosstalk_1.1.0.1  
## [40] magrittr_1.5        Rcpp_1.0.4.6        ggbeeswarm_0.6.0   
## [43] munsell_0.5.0       fansi_0.4.1         lifecycle_0.2.0    
## [46] stringi_1.4.6       yaml_2.2.1          plyr_1.8.6         
## [49] maptools_1.0-1      crayon_1.3.4        lattice_0.20-41    
## [52] haven_2.2.0         hms_0.5.3           pillar_1.4.4       
## [55] rjson_0.2.20        reprex_0.3.0        glue_1.4.1         
## [58] evaluate_0.14       modelr_0.1.8        png_0.1-7          
## [61] vctrs_0.3.0         selectr_0.4-2       RgoogleMaps_1.4.5.3
## [64] cellranger_1.1.0    gtable_0.3.0        assertthat_0.2.1   
## [67] xfun_0.14           broom_0.5.6         e1071_1.7-3        
## [70] class_7.3-17        viridisLite_0.3.0   beeswarm_0.2.3     
## [73] units_0.6-6         ggmap_3.0.0         ellipsis_0.3.1

References

Anderson, Aaron, Johann Kotzé, Stephanie A Shwiff, Brody Hatch, Chris Slootmaker, Anne Conan, Darryn Knobel, and Louis H Nel. 2019. “A bioeconomic model for the optimization of local canine rabies control.” PLoS Neglected Tropical Diseases.

Asamoah, Joshua Kiddy K, Francis T Oduro, Ebenezer Bonyah, and Baba Seidu. 2017. “Modelling of Rabies Transmission Dynamics Using Optimal Control Analysis.” Journal of Applied Mathematics.

Beyer, Hawthorne L, Katie Hampson, Tiziana Lembo, Sarah Cleaveland, Magai Kaare, and Daniel T Haydon. 2012. “The implications of metapopulation dynamics on the design of vaccination campaigns.” Vaccine.

Beyer, H L, K Hampson, T Lembo, S Cleaveland, M Kaare, and D T Haydon. 2010. “Metapopulation dynamics of rabies and the efficacy of vaccination.” Proceedings of the Royal Society B: Biological Sciences.

Bhunu, C P. 2011. “Impacts of culling stray dogs and vaccination on the control of human rabies: a mathematical modeling approach.” International Journal of Biomathematics.

Bilinski, Alyssa M, Meagan C Fitzpatrick, Charles E Rupprecht, A David Paltiel, and Alison P Galvani. 2016. “Optimal frequency of rabies vaccination campaigns in Sub-Saharan Africa.” Proceedings. Biological Sciences / the Royal Society.

Bohrer, Gil, Shachar Shem-Tov, Eric Summer, Keren Or, and David Saltz. 2002. “The effectiveness of various rabies spatial vaccination patterns in a simulated host population with clumped distribution.” Ecological Modelling.

Borse, Rebekah H, Charisma Y Atkins, Manoj Gambhir, Eduardo A Undurraga, Jesse D Blanton, Emily B Kahn, Jessie L Dyer, Charles E Rupprecht, and Martin I Meltzer. 2018. “Cost-effectiveness of dog rabies vaccination programs in East Africa.” PLoS Neglected Tropical Diseases.

Bourhy, Hervé, Emmanuel Nakouné, Matthew Hall, Pierre Nouvellet, Anthony Lepelletier, Chiraz Talbi, Laurence Watier, et al. 2016. “Revealing the Micro-scale Signature of Endemic Zoonotic Disease Transmission in an African Urban Setting.” PLoS Pathogens.

Carroll, Matthew J, Alexander Singer, Graham C Smith, Dave P Cowan, and Giovanna Massei. 2010. “The use of immunocontraception to improve rabies eradication in urban dog populations.” Wildlife Research.

Chapwanya, M, J M S Lubuma, and Y A Terefe. 2016. “Analysis and dynamically consistent nonstandard discretization for a rabies model in humans and dogs.” Revista de La Real Academia de Ciencias Exactas, Fı́sicas Y Naturales. Serie A. Matemáticas.

Chen, Jing, Lan Zou, Zhen Jin, and Shigui Ruan. 2015. “Modeling the Geographic Spread of Rabies in China.” PLoS Neglected Tropical Diseases.

Cleaveland, S, and C Dye. 1995. “Maintenance of a microparasite infecting several host species: rabies in the Serengeti.” Parasitology.

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

Cori, Anne, Pierre Nouvellet, Tini Garske, Hervé Bourhy, Emmanuel Nakouné, and Thibaut Jombart. 2018. “A graph-based evidence synthesis approach to detecting outbreak clusters: An application to dog rabies.” PLoS Computational Biology.

Czupryna, Anna M, Joel S Brown, Machunde A Bigambo, Christopher J Whelan, Supriya D Mehta, Rachel M Santymire, Felix J Lankester, and Lisa J Faust. 2016. “Ecology and Demography of Free-Roaming Domestic Dogs in Rural Villages Near Serengeti National Park in Tanzania.” PLoS ONE.

Dürr, Salome, and Michael P Ward. 2015. “Development of a Novel Rabies Simulation Model for Application in a Non-endemic Environment.” PLoS Neglected Tropical Diseases.

Ferguson, Elaine A, Katie Hampson, Sarah Cleaveland, Ramona Consunji, Raffy Deray, John Friar, Daniel T Haydon, Joji Jimenez, Marlon Pancipane, and Sunny E Townsend. 2015. “Heterogeneity in the spread and control of infectious disease: consequences for the elimination of canine rabies.” Nature Publishing Group.

Fitzpatrick, Meagan C, Katie Hampson, Sarah Cleaveland, Lauren Ancel Meyers, Jeffrey P Townsend, and Alison P Galvani. 2012. “Potential for Rabies Control through Dog Vaccination in Wildlife-Abundant Communities of Tanzania.” PLoS Neglected Tropical Diseases.

Fitzpatrick, Meagan C, Katie Hampson, Sarah Cleaveland, Imam Mzimbiri, Felix Lankester, Tiziana Lembo, Lauren A Meyers, A David Paltiel, and Alison P Galvani. 2014. “Cost-effectiveness of canine vaccination to prevent human rabies in rural Tanzania.” Annals of Internal Medicine.

Fitzpatrick, Meagan C, Hiral A Shah, Abhishek Pandey, Alyssa M Bilinski, Manish Kakkar, Andrew D Clark, Jeffrey P Townsend, Syed Shahid Abbas, and Alison P Galvani. 2016. “One Health approach to cost-effective rabies control in India.” Proceedings of the National Academy of Sciences.

Hampson, Katie, Jonathan Dushoff, John Bingham, Gideon Brückner, Y H Ali, and Andy Dobson. 2007. “Synchronous cycles of domestic dog rabies in sub-Saharan Africa and the impact of control efforts.” Proceedings of the National Academy of Sciences.

Hampson, Katie, Jonathan Dushoff, Sarah Cleaveland, Daniel T Haydon, Magai Kaare, Craig Packer, and Andy Dobson. 2009. “Transmission dynamics and prospects for the elimination of canine rabies.” PLOS Biol.

Hou, Qiang, Zhen Jin, and Shigui Ruan. 2012. “Dynamics of rabies epidemics and the impact of control efforts in Guangdong Province, China.” Journal of Theoretical Biology.

Huang, Jicai, Shigui Ruan, Yaqin Shu, and Xiao Wu. 2018. “Modeling the Transmission Dynamics of Rabies for Dog, Chinese Ferret Badger and Human Interactions in Zhejiang Province, China.” Bulletin of Mathematical Biology.

Hudson, Emily G, Victoria J Brookes, Michael P Ward, and Salome Dürr. 2019. “Using roaming behaviours of dogs to estimate contact rates: the predicted effect on rabies spread.” Epidemiology and Infection.

Kadowaki, H, K Hampson, K Tojinbara, A Yamada, and K Makita. 2018. “The risk of rabies spread in Japan: a mathematical modelling assessment.” Epidemiology and Infection.

Kitala, P M, J J McDermott, P G Coleman, and C Dye. 2002. “Comparison of vaccination strategies for the control of dog rabies in Machakos District, Kenya.” Epidemiology and Infection.

Kurosawa, Aiko, Kageaki Tojinbara, Hazumu Kadowaki, Katie Hampson, Akio Yamada, and Kohei Makita. 2017. “The rise and fall of rabies in Japan: A quantitative history of rabies epidemics in Osaka Prefecture, 19141933.” PLoS Neglected Tropical Diseases.

Laager, Mirjam, Monique Léchenne, Kemdongarti Naissengar, Rolande Mindekem, Assandi Oussiguere, Jakob Zinsstag, and Nakul Chitnis. 2019. “A metapopulation model of dog rabies transmission in N’Djamena, Chad.” Journal of Theoretical Biology.

Laager, Mirjam, Céline Mbilo, Enos Abdelaziz Madaye, Abakar Naminou, Monique Léchenne, Aurélie Tschopp, Service Kemdongarti Naı̈ssengar, Timo Smieszek, Jakob Zinsstag, and Nakul Chitnis. 2018. “The importance of dog population contact network structures in rabies transmission.” PLoS Neglected Tropical Diseases.

Lakshmanan, Nallakannu, Thomas C Gore, Karen L Duncan, Michael J Coyne, Melissa A Lum, and Frank J Sterner. 2006. “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.” Veterinary Therapeutics : Research in Applied Veterinary Medicine.

Lembo, Tiziana, Katie Hampson, Daniel T Haydon, Meggan Craft, Andy Dobson, Jonathan Dushoff, Eblate Ernest, et al. 2008. “Exploring reservoir dynamics: a case study of rabies in the Serengeti ecosystem.” The Journal of Applied Ecology.

Leung, Tiffany, and Stephen A Davis. 2017. “Rabies Vaccination Targets for Stray Dog Populations.” Frontiers in Veterinary Science.

Liu, Junli, Ying Jia, and Tailei Zhang. 2017. “Analysis of a rabies transmission model with population dispersal.” Nonlinear Analysis: Real World Applications.

Lloyd-Smith, J O, D George, K M Pepin, V E Pitzer, J R C Pulliam, A P Dobson, P J Hudson, and B T Grenfell. 2009. “Epidemic Dynamics at the Human-Animal Interface.” Science.

Mengistu Tulu, Aberu. 2017. “The Impact of Infective Immigrants on the Spread of Dog Rabies.” American Journal of Applied Mathematics.

Mollentze, Nardus, Louis H Nel, Sunny Townsend, Kevin le Roux, Katie Hampson, Daniel T Haydon, and Samuel Soubeyrand. 2014. “A Bayesian approach for inferring the dynamics of partially observed endemic infectious diseases from space-time-genetic data.” Proceedings. Biological Sciences / the Royal Society.

Sparkes, Jessica, Steven McLeod, Guy Ballard, Peter J S Fleming, Gerhard Körtner, and Wendy Y Brown. 2016. “Rabies disease dynamics in naı̈ve dog populations in Australia.” Preventive Veterinary Medicine.

Talbi, Chiraz, Philippe Lemey, Marc A Suchard, Elbia Abdelatif, Mehdi Elharrak, Nourlil Jalal, Abdellah Faouzi, et al. 2010. “Phylodynamics and Human-Mediated Dispersal of a Zoonotic Virus.” PLoS Pathogens.

Tian, Huaiyu, Yun Feng, Bram Vrancken, Bernard Cazelles, Hua Tan, Mandev S Gill, Qiqi Yang, et al. 2018. “Transmission dynamics of re-emerging rabies in domestic dogs of rural China.” PLoS Pathogens.

Tohma, Kentaro, Mariko Saito, Catalino S Demetria, Daria L Manalo, Beatriz P Quiambao, Taro Kamigaki, and Hitoshi Oshitani. 2016. “Molecular and mathematical modeling analyses of inter-island transmission of rabies into a previously rabies-free island in the Philippines.” INFECTION, GENETICS AND EVOLUTION.

Townsend, Sunny E, Tiziana Lembo, Sarah Cleaveland, François X Meslin, Mary Elizabeth Miranda, Anak Agung Gde Putra, Daniel T Haydon, and Katie Hampson. 2013. “Surveillance guidelines for disease elimination: A case study of canine rabies.” "Comparative Immunology, Microbiology and Infectious Diseases".

Townsend, Sunny E, I Putu Sumantra, Pudjiatmoko, Gusti Ngurah Bagus, Eric Brum, Sarah Cleaveland, Sally Crafter, et al. 2013. “Designing programs for eliminating canine rabies from islands: Bali, Indonesia as a case study.” PLoS Neglected Tropical Diseases.

Wang, Xiaowei, and Jie Lou. 2008. “Two dynamic models about rabies between dogs and humans.” Journal of Biological Systems.

Wera, E, M C M Mourits, M M Siko, and H Hogeveen. 2016. “Cost-Effectiveness of Mass Dog Vaccination Campaigns against Rabies in Flores Island, Indonesia.” Transboundary and Emerging Diseases.

WHO Rabies Modelling Consortium. 2019. “The potential impact of improved provision of rabies post-exposure prophylaxis in Gavi-eligible countries: a modelling study.” The Lancet Infectious Diseases.

Wilson-Aggarwal, Jared K, Laura Ozella, Michele Tizzoni, Ciro Cattuto, George J F Swan, Tchonfienet Moundai, Matthew J Silk, James A Zingeser, and Robbie A McDonald. 2019. “High-resolution contact networks of free-ranging domestic dogs Canis familiaris and implications for transmission of infection.” PLoS Neglected Tropical Diseases.

Wiraningsih, E D, F Agusto, L Aryati, S Lenhart, and S Toaha. 2015. Stability analysis of rabies model with vaccination and culling effect on dogs. Applied Mathematical ….

Zhang, J, Z Jin, G Q Sun, and X Sun. 2012. “Spatial spread of rabies in China.” Journal of Applied Analysis and Computation.

Zhang, Juan, Zhen Jin, Gui-Quan Sun, Xiang-Dong Sun, and Shigui Ruan. 2012. “Modeling Seasonal Rabies Epidemics in China.” Bulletin of Mathematical Biology.

Zhang, Juan, Zhen Jin, Gui-Quan Sun, Tao Zhou, and Shigui Ruan. 2011. “Analysis of Rabies in China: Transmission Dynamics and Control.” PLoS ONE.

Zinsstag, Jakob, Monique Léchenne, Mirjam Laager, Rolande Mindekem, Service Naı̈ssengar, Assandi Oussiguere, Kebkiba Bidjeh, et al. 2017. “Vaccination of dogs in an African city interrupts rabies transmission and reduces human exposure.” Science Translational Medicine.

Zinsstag, J, S Durr, M A Penny, R Mindekem, F Roth, S M Gonzalez, S Naissengar, and J Hattendorf. 2009. “Transmission dynamics and economics of rabies control in dogs and humans in an African city.” Proceedings of the National Academy of Sciences.