abundance-density-tl-SI.Rtex

%--------------------------------------------------------------------------------------------------------------------------------%
% Code for "Supplementary information: A mechanistic model to compare the importance of interrelated population measures: population size, population density and colony size"
% by Tim CD Lucas, Hilde Herbots and Kate Jones
%
%---------------------------------------------------------------------------------------------------------------------------------%



%%begin.rcode settings, echo = FALSE, cache = FALSE, message = FALSE, results = 'hide', eval = TRUE

# There are figures created in the data analysis which are not in the final chapter document.
#   If TRUE, they will be included in the output.
#   Use 'hide' to remove them.
extraFigs <- 'hide'

#knitr options
#  Set figure prefix as A so that no figures will be overwritten.
opts_chunk$set(cache.path = '.SICache/', fig.path = 'figure/')
source('misc/KnitrOptions.R')

%%end.rcode

%%begin.rcode libs, cache = FALSE

# ggplot2 theme.
source('misc/theme_tcdl.R')
theme_set(theme_grey() + theme_pub)

library(xtable)
library(palettetown)
library(binom)
library(ggplot2)
library(dplyr)

library(MetapopEpi)

library(broom)

%%end.rcode


\begin{table}[h!]
\captionsetup{width=\textwidth}
\centering
\small

\caption{A summary of all symbols used along with their units and default values.
The justifications for parameter values are given in the main manuscript.}

\begin{tabular}{@{}lp{7cm}p{2.9cm}r@{}}
\toprule
Symbol & Explanation & Units & Value\\
\midrule
$\rho$ & Number of pathogens && 2\\
$x, y$ & Group (colony) index &&\\
$p$ &  Pathogen index i.e.\ $p\in\{1,2\}$ for pathogens 1 and 2 & &\\
$q$ & Disease class i.e.\ $q\in\{1,2,12\}$&\\
$S_x$ & Number of susceptible individuals in group $x$ &&\\
$I_{qx}$ & Number of individuals infected with disease(s) $q \in \{1, 2, 12\}$ in group $x$ &&\\
$R_x$ & Number of individuals in group $x$ in the recovered with immunity class  &&\\
$N$ & Total Population size && 2,000 -- 32,000\\
$m$ & Number of groups && 5 -- 80\\
$n$ & Group size && 100 -- 1600\\
$a$ & Area & km\textsuperscript{2}& 2,500 -- 40,000\\
$d$ & Density & individuals.km$^{-2}$ &  0.2 -- 3.2\\
$\beta$ & Transmission rate &  & 0.1 -- 0.3\\
$\alpha$ & Coinfection adjustment factor  & & 0.1\\
$\gamma$ & Recovery rate & year$^{-1}.$individual$^{-1}$ & 1\\
$\xi$ & Dispersal rate & year$^{-1}.$individual$^{-1}$ & 0.01\\
$\Lambda$ & Birth rate & year$^{-1}.$individual$^{-1}$ & 0.05\\
$\mu$ & Death rate & year$^{-1}.$individual$^{-1}$ & 0.05\\
$g$  & Population growth rate &   & 1\\
$k_y$ & Degree of node $y$ (number of groups that individuals from group $y$ can disperse to) &&\\
$\delta$ & Waiting time until next event & years &\\
$e_i$ & The rate at which event $i$ occurs & year$^{-1}$&\\
$R_0$ & The (single subpopulation) basic reproduction number & infections &  \\
%$b_i$ & Regression coefficient & & \\
%$c$ & Regression intercept & & \\
%$\epsilon$ & Binomially distributed regression error term & & \\
\bottomrule
\end{tabular}

\label{t:params}
\end{table}

\clearpage

\section{Supplementary materials and methods: Stochastic simulations} \label{sup:stochmodel}



We examined the model using stochastic, continuous-time simulations implemented in \emph{R} \cite{R}.
The implementation is available as an \emph{R} package on GitHub \cite{metapopepi}.
The model can be written as a continuous-time Markov chain.
The Markov chain contains the random variables $((S_x)_{x = 1\ldots m}, (I_{x, q})_{x =1\ldots m,\:q \in \{1, 2, 12\}}, (R_x)_{x = 1\ldots m})$.
Here, $(S_x)_{x = 1\ldots m}$ is a length $m$ vector of the number of susceptibles in each group
$(I_{x, q})_{x =1\ldots m, q \in \{1, 2, 12\}}$ is a length $m \times 3$ vector describing the number of individuals of each disease class ($q \in \{1, 2, 12\}$) in each group.
Finally, $(R_x)_{x = 1\ldots m}$ is a length $m$ vector of the number of individuals in the recovered class.
The model is a Markov chain where extinction of either pathogen species and extinction of the host population are absorbing states.
The expected time for the host population to go extinct is much larger than the duration of the simulations.

At any time, suppose the system is in state $((s_x), (i_{x,q}), (r_x))$.
At each step in the simulation we calculate the rate at which each possible event might occur.
One event is then randomly chosen, weighted by its rate
\begin{align}
  p(\text{event } i) = \frac{e_i}{\sum_j e_j},
\end{align}
where $e_i$ is the rate at which event $i$ occurs and $\sum_j e_j$ is the sum of the rates of all possible events.
Finally, the length of the time step, $\delta$, is drawn from an exponential distribution 
\begin{align}
  \delta \sim \operatorname{Exp}\left(\sum_j e_j  \right).
\end{align}


We can now write down the rates of all events. 
Assuming asexual reproduction, that all classes reproduce at the same rate and that individuals are born into the susceptible class we get
\begin{align}
  s_x \rightarrow s_x + 1 \;\;\;\text{at a rate of}\;\; \Lambda\left( s_{x}+\sum_q i_{qx} + r_{x}\right) 
\end{align}
where $s_x \rightarrow s_x + 1$ is the event that the number of susceptibles in group $x$ will increase by 1 (a single birth) and $\sum_q i_{qx}$ is the sum of all infection classes $q~\in~\{1, 2, 12\}$.
The rates of death, given a death rate $\mu$, and no increased mortality due to infection, are given by
\begin{align}
  s_x \rightarrow s_x-1  &\;\;\;\text{at a rate of}\;\; \mu s_x, \\
  i_{qx}  \rightarrow i_{qx}-1 &\;\;\text{at a rate of}\;\; \mu i_{qx},\\
  r_x  \rightarrow r_x-1 &\;\;\;\text{at a rate of}\;\; \mu r_x.
\end{align}



We modelled transmission as being density-dependent.
This assumption was more suitable than frequency-dependent transmission as we were modelling a disease transmitted by saliva or urine in highly dense populations confined to caves, buildings or potentially a small number of tree roosts.
We were notably not modelling a sexually transmitted disease (STD) as spillover of STDs from bats to humans is likely to be rare.
Infection of a susceptible with either Pathogen 1 or 2 is therefore given by
\begin{align}
  i_{1x} \rightarrow i_{1x}+1,\;\;\; s_x \rightarrow s_x-1 &\;\;\text{at a rate of}\;\; \beta s_x\left(i_{1x} + i_{12x}\right),\\
  i_{2x} \rightarrow i_{2x}+1,\;\;\; s_x \rightarrow s_x-1  &\;\;\text{at a rate of}\;\; \beta s_x\left(i_{2x} + i_{12x}\right),
\end{align}
while coinfection, given the coinfection adjustment factor $\alpha$, is given by
\begin{align}
  i_{12,x} \rightarrow i_{12,x}+1,\;\;\; i_{1x} \rightarrow i_{1x}-1 &\;\;\text{at a rate of}\;\; \alpha\beta i_{1x}\left(i_{2x} + i_{12x}\right),\\
  i_{12,x} \rightarrow i_{12,x}+1,\;\;\; i_{2x} \rightarrow i_{2x}-1 &\;\;\text{at a rate of}\;\; \alpha\beta i_{2x}\left(i_{1x} + i_{12x}\right).
\end{align}
Note that lower values of $\alpha$ give lower rates of coinfection as in \cite{castillo1989epidemiological}.



The rate of migration from group $y$ (with degree $k_y$) to group $x$, given a dispersal rate $\xi$ is given by
\begin{align}
  s_x \rightarrow s_x+1,\;\;\; s_y \rightarrow s_y-1 &\;\;\text{at a rate of}\;\; \frac{\xi s_y}{k_y},\\
  i_{qx} \rightarrow i_{qx}+1,\;\;\; i_{qy} \rightarrow i_{qy}-1 &\;\;\text{at a rate of}\;\; \frac{\xi i_{qy}}{k_y},\\
  r_x \rightarrow r_x+1,\;\;\; r_y \rightarrow r_y-1 &\;\;\text{at a rate of}\;\; \frac{\xi r_y}{k_y}.
\end{align}
Note that the dispersal rate does not change with infection.
As above, this is due to the low virulence of bat viruses.
Finally, recovery from any infectious class occurs at a rate $\gamma$
\begin{align}
  i_{qx} \rightarrow i_{qx}-1,\;\; r_x \rightarrow r_x+1  \;\;\text{at a rate of}\;\; \gamma i_{qx}.
\end{align}


\clearpage






\section{Supplementary materials and methods: Deterministic model}

% todo Align equal signs across groups
% http://tex.stackexchange.com/questions/11855/alignment-of-equals-sign-in-multiple-align-environments
%
% Should probably put whole, single patch model into sympy and see what I get out...

We can study the model analytically if we restrict the population to a single subpopulation.
We first study the endemic pathogen. 
We then have a typical SIR model with vital dynamics \cite{Kermack1932}.

\begin{align}
\frac{dS}{dt} & =  \Lambda n - \mu S -  \beta S I_1,  \label{smodel2} \\
\frac{dI_1}{dt} & = \beta S I_1 - (\gamma + \mu) I_1, \\
\frac{dR}{dt} & = \gamma I_1 - \mu R.
\end{align}

\subsubsection{Equilibrium values of single pathogen model}


To find the equilibrium number of susceptibles, $S^\ast$, we set $\frac{dI_1}{dt} = 0$.
We then have the trival steady state of $S^* = n$ and  $I_1^* = 0$.
To find the endemic steady state ($I_1^* \ne 0$) we write

\begin{align}
\beta S^\ast I_1^\ast &= (\gamma + \mu) I_1^\ast,\\
S^\ast &= \frac{\gamma + \mu}{\beta} \label{sstar}.
\end{align}

To find $I^\ast$ we substitute Equation \ref{sstar} into Equation \ref{smodel2} and set $\frac{dI_1}{dt} = 0$ giving

\begin{align}
0 = & \Lambda n - \mu S^\ast -  \beta S^\ast I_1^\ast , \\
0 = & \Lambda n - \frac{\mu \left(\gamma + \mu\right)}{\beta} -   \left(\gamma + \mu\right)I_1^\ast , \\
I_1^\ast = & \frac{\Lambda n}{\gamma + \mu} - \frac{\mu }{\beta} \label{istar}.
\end{align}

\subsubsection{Second pathogen}

We assume that Pathogen 2 is introduced when the endemic pathogen is at equilibrium.
The introduction of Pathogen 2 will cause the system to shift to a new equilibrium.
However, we continue to use $S^*$ and $I_1^*$ to mean the values above.
Initially, the dynamics of Pathogen 2 are governed by

\begin{align}
\frac{dI_2}{dt} & = \beta S^\ast I_2 + \alpha \beta I_1^\ast I_2 - (\gamma + \mu) I_2 \label{path2I}
\end{align}

where $\beta S^\ast I_2$ is the number of infections per unit time,  $ \alpha \beta I_1^\ast I_2$ is the number of coinfections per unit time and $(\gamma + \mu) I_2 $ is the number of deaths and recoveries per unit time. 
$I_2$ is therefore the number of individuals infected with Pathogen 2 regardless of whether they are infected with Pathogen 1.
In this sense it is the combined size of classes $I_2$ and $I_{12}$  as used in the stochastic model (Section \ref{sup:stochmodel}).
For simplicity here we simply label this group $I_2$.
Substituting in Equations \ref{sstar} and \ref{istar}, and noting that $R_0 = \frac{\beta n }{\gamma + \mu}$  we get 

\begin{align}
\frac{dI_2}{dt} & =  (\gamma + \mu) I_2 + \alpha \beta \left( \frac{\Lambda n}{\gamma + \mu} - \frac{\mu}{\beta} \right) I_2 - (\gamma + \mu) I_2, \\
\frac{dI_2}{dt} & = \alpha\left( \frac{\beta \Lambda n}{\gamma + \mu} - \mu \right) I_2, \\
\frac{dI_2}{dt} & = \alpha\left( \Lambda R_0 - \mu \right) I_2.
\end{align}

This is greater than zero when 

\begin{align}
\alpha\left( \Lambda R_0 - \mu \right) I_2 > 0
\end{align}

As $I_2$ is positive, this is greater than zero when $ \Lambda R_0 - \mu > 0$ and $\alpha > 0$. 
As $\Lambda = \mu$ due to the assumption of stable population size, $ \Lambda R_0 - \mu > 0$ is true as long as $R_0 > 1$. 
$R_0$ is greater than one (as we have assumed that Pathogen 1 is endemic), 
Therefore, $\frac{dI_2}{dt}$ is greater than zero provided $\alpha$ is greater than zero.


\clearpage




%%begin.rcode rawDataTables, cache = FALSE
dens1 <- read.csv('Data/DensSims.csv') %>% dplyr::select(-X)

dens1Means <- dens1 %>%
                #filter(nExtantDis > 0) %>%
                group_by(transmission, colonySize, colonyNumber) %>%
                summarise(`Area \\tiny{($\\times 1000$ km$^2$)}` = mean(area / 1000),
                          `$N$ \\tiny{($\\times 1000$)}` = mean(as.numeric(as.character(pop)) / 1000) , 
                          `Density \\tiny{(km$^{-2}$)}`  = mean(as.numeric(as.character(pop))) / mean(as.numeric(as.character(area))),
                          Invasions = sum(nExtantDis == 2)) %>%
                rename(`$\\beta$` = transmission, `$n$` = colonySize, `$m$` = colonyNumber)


# Constant density, altered colony number

dens2 <- read.csv('Data/Dens2Sims.csv') %>% dplyr::select(-X)

dens2Means <- dens2 %>%
                #filter(nExtantDis > 0) %>%
                group_by(transmission, colonySize, colonyNumber) %>%
                summarise(`Area \\tiny{($\\times 1000$ km$^2$)}` = mean(area / 1000),
                          `$N$ \\tiny{($\\times 1000$)}` = mean(as.numeric(as.character(pop)) / 1000) , 
                          `Density \\tiny{(km$^{-2}$)}`  = mean(as.numeric(as.character(pop))) / mean(as.numeric(as.character(area))),
                          Invasions = sum(nExtantDis == 2)) %>%
                rename(`$\\beta$` = transmission, `$n$` = colonySize, `$m$` = colonyNumber)

# Constant population, altered area

pop <- read.csv('Data/PopSims.csv') %>% dplyr::select(-X)

popMeans <- pop %>%
                #filter(nExtantDis > 0) %>%
                group_by(transmission, area) %>%
                summarise(`$\\beta$` = mean(transmission),
                          `$n$` = mean(colonySize),
                          `$m$` = mean(colonyNumber),
                          `Area \\tiny{($\\times 1000$ km$^2$)}` = mean(area / 1000),
                          `$N$ \\tiny{($\\times 1000$)}` = mean(as.numeric(as.character(pop)) / 1000) , 
                          `Density \\tiny{(km$^{-2}$)}` = mean(as.numeric(as.character(pop))) / mean(as.numeric(as.character(area))),
                          Invasions = sum(nExtantDis == 2)) %>%
                ungroup %>%
                dplyr::select(-area, -transmission)
%%end.rcode






% ----------------------------------------------- %
% Print tables
% ----------------------------------------------- %



%%begin.rcode pop, results = 'asis'


popTitle <- "
Raw data for geographic range size simulations
  "

popCapt <- "
Raw data for geographic range size simulations.
The population parameters are shown along with the number of invasions.
$\\beta$ is the transmission rate, $n$ is group size, $m$ is the number of groups and $N$ is the total population size.
"


print(xtable(popMeans,
             caption = c(popCapt, popTitle), 
             label = "C-pop",
             digits = c(0, 1, 0, 0, 1, 0, 1, 0),
             align = c('r', '@{}r', 'r', 'r', 'r', 'r', 'r', 'r@{}')), 
      size = "small", #Change size; useful for bigger tables
      include.rownames = FALSE, #Don't print rownames
      include.colnames = TRUE,
      sanitize.colnames.function = function(x){x},
      caption.placement = "top",
      booktabs = TRUE
    )

%%end.rcode


%%begin.rcode dens1, results = 'asis'


dens1Title <- "
Raw data for group size simulations
  "

dens1Capt <- "
Raw data for group size simulations. 
The population parameters are shown along with the number of invasions.
$\\beta$ is the transmission rate, $n$ is group size, $m$ is the number of groups and $N$ is the total population size.
"


print(xtable(dens1Means,
             caption = c(dens1Capt, dens1Title), 
             label = "C-dens1",
             digits = c(0, 1, 0, 0, 1, 0, 1, 0),
             align = c('r', '@{}r', 'r', 'r', 'r', 'r', 'r', 'r@{}')), 
      size = "small", #Change size; useful for bigger tables
      include.rownames = FALSE, #Don't print rownames
      include.colnames = TRUE,
      sanitize.colnames.function = function(x){x},
      caption.placement = "top",
      booktabs = TRUE
    )

%%end.rcode




%%begin.rcode dens2, results = 'asis'


dens2Title <- "
Raw data for number of groups simulations
  "

dens2Capt <- "
Raw data for number of groups simulations.
The population parameters are shown along with the number of invasions.
$\\beta$ is the transmission rate, $n$ is group size, $m$ is the number of groups and $N$ is the total population size.
"


print(xtable(dens2Means,
             caption = c(dens2Capt, dens2Title), 
             label = "C-dens2",
             digits = c(0, 1, 0, 0, 1, 0, 1, 0),
             align = c('r', '@{}r', 'r', 'r', 'r', 'r', 'r', 'r@{}')), 
      size = "small", #Change size; useful for bigger tables
      include.rownames = FALSE, #Don't print rownames
      include.colnames = TRUE,
      sanitize.colnames.function = function(x){x},
      caption.placement = "top",
      booktabs = TRUE
    )

%%end.rcode



%%begin.rcode colonyNetCaptions

colonyNetTitle <- 'Relationship between geographic range size and metapopulation network structure'

colonyNetCapt <- '
The relationship between geographic range size and metapopulation network structure.
Groups are shown by circles.
Groups that are close enough for animals to disperse between (i.e.\\  within 100 km of each other) are joined by a line.
Groups are placed randomly in spaces of various sizes (grey dashed lines).
A and C) the default geographic range size (10000 km\\textsuperscript{2}).
B and D) the largest geographic range size (40000 km\\textsuperscript{2}).
A and B) the smallest number of groups (five).
C and D) the default number of groups (20).
The mean number of connections per subpopulation, $\\bar{k}$, is shown for each metapopulation.
'


%%end.rcode

\clearpage



%%begin.rcode colonyNetworkPlots, fig.height = 5, fig.cap = colonyNetCapt, fig.scap = colonyNetTitle, out.width = '0.7\\textwidth', fig.showtext = TRUE


set.seed(3)

  #   Area and meanColonySize are the actal input arguments
  pop <- c(2000, 4000, 8000, 16000, 32000)
  colonySize <- 400
  colonyNumber <- pop / colonySize

  area <- pop / 0.8
  side <- sqrt(area)


# Make the population.
p1 <- makePop(space = side[3], 
             transmission = 0.1, 
             meanColonySize = colonySize, 
             nColonies = colonyNumber[1], 
             model = 'SIR', 
             events = 20,
             sample = 2,
             maxDistance = 100)
k1 <- sum(p1$adjacency != 0 )/p1$parameters['nColonies']


# Make the population.
p2 <- makePop(space = side[5], 
             transmission = 0.1, 
             meanColonySize = colonySize, 
             nColonies = colonyNumber[3], 
             model = 'SIR', 
             events = 20,
             sample = 2,
             maxDistance = 100)
k2 <- sum(p2$adjacency != 0 )/p2$parameters['nColonies']

# Make the population.
p3 <- makePop(space = side[3], 
             transmission = 0.1, 
             meanColonySize = colonySize, 
             nColonies = colonyNumber[3], 
             model = 'SIR', 
             events = 20,
             sample = 2,
             maxDistance = 100)
k3 <- sum(p3$adjacency != 0 )/p3$parameters['nColonies']

# Make the population.
p4 <- makePop(space = side[5], 
             transmission = 0.1, 
             meanColonySize = colonySize, 
             nColonies = colonyNumber[1], 
             model = 'SIR', 
             events = 20,
             sample = 2,
             maxDistance = 100)
k4 <- sum(p4$adjacency != 0 )/p4$parameters['nColonies']


labSz <- 1.1
bigy <- 8
bigx <- 6
titleSz <- 3
axSz <- 2
points <- 0.35

par(mfrow = c(2,2))


plotColonyNet(p1, area = side[5], cex = points, mar = c(bigx, bigy, 2, 2), alpha = 1, lowgrey = 0.2, highgrey = 0.7, col = pokepal('Charizard')[c(4, 14)])
#axis(2, col = 'grey', cex.axis = axSz, at = c(0, 50, 100, 150, 200), labels = c('0', '', '100', '', '200'), col.axis = '#a4a4a4')
axis(1)
#title(main = bquote(italic(bar(k)) ~ '='~ .(k1)), cex.main = titleSz, line = -1.18)
mtext(side = 2,  line = 4, text = 'Latitude', cex = labSz, col  =  "#8B8B8B")
lines(x = c(0, 100, 100), y = c(100, 100, 0), lty = 2, col = 'grey')
mtext(side = 3,  line = -0.3, text = 'A)', cex = 1, at = -62)

plotColonyNet(p4, area = side[5], cex = points, mar = c(bigx, 6, 2, 2), alpha = 1, lowgrey = 0.2, highgrey = 0.7, col = pokepal('Charizard')[c(4, 14)])
axis(2, col = 'grey', labels = FALSE)
axis(1, col = 'grey', labels = FALSE)
#title(main = bquote(italic(bar(k)) ~ '='~ .(k4)), cex.main = titleSz, line = -1.18)
mtext(side = 3,  line = -0.3, text = 'B)', cex = 1, at = -22)

plotColonyNet(p3, area = side[5], cex = points, mar = c(bigx, bigy, 0, 2), alpha = 1, lowgrey = 0.2, highgrey = 0.7, col = pokepal('Charizard')[c(4, 14)])
axis(2, col = 'grey', cex.axis = axSz, col.axis = '#a4a4a4', at = c(0, 50, 100, 150, 200), labels = c('0', '', '100', '', '200'))
axis(1, col = 'grey', labels = FALSE)
axis(1, col = 'grey', cex.axis = axSz, col.axis = '#a4a4a4', at = c(0, 100, 200), labels = c('0', '100', '200'),  line = 0.5, lwd = 0)
#title(main = bquote(italic(bar(k)) ~ '='~ .(k3)), cex.main = titleSz, line = -1.18)
mtext(side = 2,  line = 4, text = 'Latitude', cex = labSz, col  =  "#8B8B8B")
mtext(side = 1,  line = 4.4, text = 'Longitude', cex = labSz, col  =  "#8B8B8B")
lines(x = c(0, 100, 100), y = c(100, 100, 0), lty = 2, col = 'grey')
mtext(side = 3,  line = -0.3, text = 'C)', cex = 1, at = -62)

plotColonyNet(p2, area = side[5], cex = points, mar = c(bigx, 6, 0, 2), alpha = 1, lowgrey = 0.2, highgrey = 0.7, col = pokepal('Charizard')[c(4, 14)])
axis(1, col = 'grey', labels = FALSE)
axis(1, col = 'grey', cex.axis = axSz, col.axis = '#a4a4a4', at = c(0, 100, 200), labels = c('0', '100', '200'),  line = 0.5, lwd = 0)
axis(2, col = 'grey', labels = FALSE)
#title(main = bquote(italic(bar(k)) ~ '='~ .(k2)), cex.main = titleSz, line = -1.18)
mtext(side = 1,  line = 4.4, text = 'Longitude', cex = labSz, col  =  "#8B8B8B")
mtext(side = 3,  line = -0.3, text = 'D)', cex = 1, at = -22)




%%end.rcode


%%begin.rcode plotKcapt

plotKcapt <- '
Change in average metapopulation network degree ($\\bar{k}$) with increasing geographic range size. 
Bars show the median, boxes show the interquartile range, vertical lines show the range and grey dots indicate outlier values.
Notches indicate the 95\\% confidence interval of the median.
All simulations had 20 groups, meaning 19 is the maximum value of $\\bar{k}$.
'

plotKtitle <- 'Change in average network degree with increasing geographic range size'

%%end.rcode

%%begin.rcode plotK, fig.cap = plotKcapt, fig.scap = plotKtitle, cache = FALSE, fig.height = 4, fig.show = TRUE, out.width = '0.7\\textwidth'

ggplot(dens1, aes(x = factor(area), y = meanK, colour = 'a', fill = 'a')) +
  geom_boxplot(outlier.size = 1, size = 0.3, outlier.colour = grey(0.3), 
    notch = TRUE, width = 0.4, notchwidth = 0.6) +
  scale_colour_manual(values = pokepal('vileplume')[4]) +
  scale_fill_manual(values = pokepal('vileplume')[7]) +
  stat_summary(geom = "crossbar", width = 0.2, fatten = 2, 
    fun.data = function(x){ return(c(y = median(x), ymin = median(x), ymax = median(x))) }) +
  theme(legend.position = 'none', panel.grid.major.x = element_blank()) + 
  scale_x_discrete(labels = c('2.5', '5', '10', '20', '40')) +
  xlab(expression(paste('Area ', (1000 %*% km^2)))) +
  ylab(expression(paste('Mean degree, ', italic(bar(k))))) +
  ylim(0, 20)

%%end.rcode



%%begin.rcode calcMeans
# Constant density, altered colony size
# Need to remove rows with errors to auto find classes of columns
dens1text <- readLines('Data/DensSims.csv')
rmerrors <- dens1text[!grepl('Error in', dens1text)]
rmerrors <- rmerrors[nchar(rmerrors) > 3]
dens1 <- read.csv(text = rmerrors) %>% 
           dplyr::select(-X) 

dens1Means <- dens1 %>%
                #dplyr::filter(nExtantDis > 0) %>%
                group_by(transmission, colonySize, colonyNumber) %>%
                summarise(success = sum(nExtantDis == 2), sampleSize = n(), pop = mean(pop)) %>%
                ungroup %>% 
                cbind(., vary = 'colonySize', binom.confint(.$success, .$sampleSize, conf.level = 0.95, methods = "exact"))



# fit glms to each transmission value

colonySizeModel   <- dens1 %>% 
                       mutate(invasion = nExtantDis == 2 ) %>%
                       mutate(ValueChange = log2(colonySize / median(colonySize))) %>%
                       #filter(nExtantDis != 0) %>%
                       group_by(transmission) %>%
                       do(glm(invasion ~ ValueChange, data = ., family = 'binomial') %>% tidy(conf.int = TRUE)) 

colonySizePredictions <- dens1 %>% 
                           mutate(invasion = nExtantDis == 2 ) %>%
                           mutate(ValueChange = colonySize / median(colonySize)) %>%
                           #filter(nExtantDis != 0) %>%
                           group_by(transmission) %>%
                           do(augment(glm(invasion ~ log2(ValueChange), data = ., family = 'binomial'),
                                      newdata = data.frame(ValueChange = seq(min(.$ValueChange), max(.$ValueChange), 
                                                                                 length.out = 1000)),
                                      type.predict = 'response')) %>%
                           mutate(vary = 'colonySize')




# Constant density, altered colony number

dens2 <- read.csv('Data/Dens2Sims.csv') %>% dplyr::select(-X) %>% as.data.frame

dens2Means <- dens2 %>%
                #dplyr::filter(nExtantDis > 0) %>%
                group_by(transmission, colonySize, colonyNumber) %>%
                summarise(success = sum(nExtantDis == 2), sampleSize = n(), pop = mean(pop)) %>% 
                ungroup %>%
                cbind(., vary = 'colonyNumber',  binom.confint(.$success, .$sampleSize, conf.level = 0.95, methods = "exact"))



# Fit glms to each transmission value

colonyNumberModel <- dens2 %>% 
                       mutate(invasion = nExtantDis == 2 ) %>%
                       mutate(ValueChange = log2(colonyNumber / median(colonyNumber))) %>%
                       #filter(nExtantDis != 0) %>%
                       group_by(transmission) %>%
                       do(glm(invasion ~ ValueChange, data = ., family = 'binomial') %>% tidy(conf.int = TRUE))
                       
                       
colonyNumberPredictions <- dens2 %>% 
                             mutate(invasion = nExtantDis == 2 ) %>%
                             mutate(ValueChange = colonyNumber / median(colonyNumber)) %>%
                             #filter(nExtantDis != 0) %>%
                             group_by(transmission) %>%
                             do(augment(glm(invasion ~ log2(ValueChange), data = ., family = 'binomial'),
                                        newdata = data.frame(ValueChange = seq(min(.$ValueChange), max(.$ValueChange), 
                                                                                   length.out = 1000)),
                                        type.predict = 'response')) %>%
                             mutate(vary = 'colonyNumber')

                       
# Constant population, altered area

pop <- read.csv('Data/PopSims.csv') %>% dplyr::select(-X) %>% as.data.frame

popMeans <- pop %>%
                #dplyr::filter(nExtantDis > 0) %>%
                group_by(transmission, dens) %>%
                summarise(success = sum(nExtantDis == 2), sampleSize = n()) %>%
                ungroup %>%
                cbind(., binom.confint(.$success, .$sampleSize, conf.level = 0.95, methods = "exact"))


# fit glms to each transmission value

areaModel <- pop %>% 
               mutate(invasion = nExtantDis == 2 ) %>%
               mutate(ValueChange = log2(dens / median(dens))) %>%
               #filter(nExtantDis != 0) %>%
               group_by(transmission) %>%
               do(glm(invasion ~ ValueChange, data = ., family = 'binomial') %>% tidy(conf.int = TRUE))


areaPredictions <- pop %>% 
                     mutate(invasion = nExtantDis == 2 ) %>%
                     mutate(ValueChange = dens / median(dens)) %>%
                     #filter(nExtantDis != 0) %>%
                     group_by(transmission) %>%
                     do(augment(glm(invasion ~ log2(ValueChange), data = ., family = 'binomial'),
                                newdata = data.frame(ValueChange = seq(min(.$ValueChange), max(.$ValueChange), 
                                                                           length.out = 1000)),
                                type.predict = 'response')) %>%
                     mutate(vary = 'Density')



%%end.rcode



%%begin.rcode plotCoeffsCapts
plotCoeffsCapt <- '
GLM regression coefficients for for three sets of simulations at three transmission, $\\beta$ values.
To examine whether host density or population size affects pathogen invasion more strongly we compare the orange and blue bars.
To examine whether group size or number of groups affects pathogen invasion more strongly we compare the second and third bar in each facet.
'

plotCoeffsTitle <- 'Regression coefficients'
%%end.rcode



%%begin.rcode plotCoeffs, fig.cap = plotCoeffsCapt, fig.scap = plotCoeffsTitle, out.width = '0.7\\textwidth', fig.show = TRUE, cache = FALSE, fig.height = 3.9, fig.width = 9

allModels <- rbind(areaModel %>% mutate(varied = 'Host density'),
                   colonyNumberModel %>% mutate(varied = 'Number of groups'),
                   colonySizeModel %>% mutate(varied = 'Group size')) %>%
               filter(term == 'ValueChange') %>%
               mutate(variedSort = factor(varied, levels = unique(varied)[c(2, 3, 1)])) %>%
               mutate(Density = ifelse(varied == 'Host density', 'Host density', 'Pop. size'))
allModels$transmission <- factor(allModels$transmission, labels = c('beta == 0.1', '0.2', '0.3'))


ggplot(allModels, aes(x = variedSort, y = estimate, colour = Density)) + 
  facet_grid(. ~ transmission, labeller = label_parsed) +
  geom_point() +
  geom_errorbar(aes(ymax = conf.high, ymin = conf.low), width = 0.3) +
  xlab('Population-level trait') +
  ylab('Regression estimate') +
  scale_colour_poke(pokemon = 'Charizard', spread = 2, guide_legend(title = '')) +
  coord_flip() +
  theme(legend.position = 'bottom')

%%end.rcode






%%begin.rcode transMeansCapt

transMeansCapt <- '
Comparison of the effect of host population size on probability of invasion when population size is altered by changing group size or number of groups.
Relationships are shown separately for each transmission value, $\\beta$.
It can be seen that changes in group size give a much greater increase in invasion probability than changes in the number of groups.
Note that this is the same data as Figure~3 but with the $x$-axis scaled by population size, rather than relative parameter change.
'

transMeansTitle <- 'Comparison of the probability of the affect of group size and number of groups on probability of invasion'

%%end.rcode


%%begin.rcode plotTransMeans, fig.cap = transMeansCapt, fig.scap = transMeansTitle, out.width = '0.7\\textwidth', fig.show = TRUE, cache = FALSE, fig.height = 5

# Convert to percentage change in density, colonysize or colony number.

d2 <- rbind(dens1Means %>% 
             mutate(ValueChange = colonySize/median(colonySize), vary = 'colonySize') %>%
             dplyr::select(pop, transmission, mean, lower, upper, vary),
           dens2Means %>% 
             mutate(ValueChange = colonyNumber/median(colonyNumber), vary = 'colonyNumber') %>%
             dplyr::select(pop, transmission, mean, lower, upper, vary)
     )

d2$vary <- factor(d2$vary, levels = unique(d2$vary)[c(2, 3, 1)])
d2$transmission <- factor(d2$transmission, labels = c('beta == 0.1', '0.2', '0.3'))

# Join predictions

colonySizePredictions2 <- dens1 %>% 
                             mutate(invasion = nExtantDis == 2 ) %>%
                             #filter(nExtantDis != 0) %>%
                             group_by(transmission) %>%
                             do(augment(glm(invasion ~ pop, data = ., family = 'binomial'),
                                        newdata = data.frame(pop = seq(min(.$pop), max(.$pop), 
                                                                                   length.out = 1000)),
                                        type.predict = 'response')) %>%
                             mutate(vary = 'colonySize')
                       
colonyNumberPredictions2 <- dens2 %>% 
                             mutate(invasion = nExtantDis == 2 ) %>%
                             #filter(nExtantDis != 0) %>%
                             group_by(transmission) %>%
                             do(augment(glm(invasion ~ pop, data = ., family = 'binomial'),
                                        newdata = data.frame(pop = seq(min(.$pop), max(.$pop), 
                                                                                   length.out = 1000)),
                                        type.predict = 'response')) %>%
                             mutate(vary = 'colonyNumber')

percChangePreds2 <- rbind(colonySizePredictions2, colonyNumberPredictions2)

percChangePreds2$transmission <- factor(percChangePreds2$transmission, labels = c('beta == 0.1', '0.2', '0.3'))

ggplot(d2, aes(x = pop, y = mean, colour = vary)) +
  geom_line(data = percChangePreds2, aes(x = pop, y = .fitted, colour = vary)) +
  geom_point() +
  geom_errorbar(aes(ymin = lower, ymax = upper),
              width=.2) +
  ylab('Prop. Invasions') +
  xlab('Population size') +
  ylim(0, 1) +
  scale_colour_manual(name = "Population-level trait", 
                      labels = c('Number of groups', 'Group Size'), 
                      values = pokepal('Swampert', spread = 3)[2:1]) +
  scale_y_continuous(breaks = c(0, 0.5, 1), labels = c('0.0', '0.5', '1.0')) +
  scale_x_continuous(limits = c(0, 32000), 
                     breaks = c(0, 1e4, 2e4, 3e4), 
                     label = c('0', '10000', '20000', '30000')) +
  facet_grid(transmission ~ ., labeller = label_parsed) +
	theme(legend.position = 'bottom', legend.direction = 'vertical')

%%end.rcode