3 - Parameterising landscape and dispersal

To simulate epidemics in a heterogeneous landscape, landsepi needs (among others) these three elements which are related one each other:
- the spatial coordinates of fields composing the landscape (represented as polygons),
- the allocation of croptypes in the different fields,
- a dispersal matrix for between-field pathogen migration.

landsepi includes built-in landscapes (and associated dispersal matrices for rust pathogens) and an algorithm to allocate croptypes, but is it possible to use your own landscape, dispersal matrix and croptype allocation.

Using your own landscape

Any landscape can be used to simulate epidemics in landsepi, provided that it is in sp or sf format and contains, at least, polygon coordinates.

mylandscape <- st_read(dsn = "myshapefile.shp")
simul_params <- createSimulParams(outputDir = getwd())
simul_params <- setLandscape(simul_params, mylandscape)

Then you can simply call the method allocateLandscapeCroptypes to allocate croptypes to the fields of the landscape with controlled proportions and spatio-temporal aggregation (see tutorial on how to run a simple simulation). Otherwise, you can use your own allocation (see below).

Using your own croptype allocation

You must define for each year of simulation the index of the croptype (“croptypeID”) cultivated in each feature (polygons). Each feature has a field identified by “year_XX” (XX <- seq(1:Nyears+1)) and containing the croptype ID. Note that the allocation must contain one more year than the real number of simulated years (this is only for simulation purpose, the content of the allocation in year Nyears+1 does not affect the result).

Features/fields year_1 year_2 … year_Nyears+1
polygons1 13 10 13
polygonsX 2 1 2

An example for sf landscape:

mylandscape$year_1 <- c(13,2,4,1,1) # croptypes ID allocated to the different polygons
mylandscape$year_2 <- c(2,2,13,1,1)

Then simply add your landscape to the simulation parameters:

simul_params <- setLandscape(simul_params, mylandscape)

Computing the dispersal matrix

To simulate pathogen dispersal, landsepi needs a vectorized matrix giving the probability of propagule dispersal from any field of the landscape to any other field. This matrix must be computed before running any simulation with landsepi. It is a square matrix whose size is the number of fields in the landscape and whose elements are, for each line \(i\) and each column \(i'\) the probability \(\mu_{ii'}\) that propagules migrate from field \(i\) (whose area is \(A_i\)) to field \(i'\) (whose area is \(A_{i'}\)). This probability is computed from:
\[\mu_{ii'} = \frac { \int_{A_i} \int_{A_{i'}} g(\mid\mid z'-z \mid\mid).dz.dz' } { A_i }\]
with \(\mid\mid z'-z \mid\mid\) the Euclidian distance between locations \(z\) and \(z'\) in fields \(i\) and \(i'\), respectively, and \(g(.)\) the two-dimensional dispersal kernel of the propagules. Note that \(\sum_i \mu_{ii'} = 1\).

landsepi includes built-in dispersal matrices to represent rust dispersal in the five built-in landscapes. These have been computed from a power-law dispersal kernel: \[g(\mid\mid z'-z \mid\mid) = \frac {(b-2).(b-1)} {2.\pi.a^2} . (1+ \frac {\mid\mid z'-z \mid\mid} {a})^{-b}\] with \(a\) the scale parameter and \(b\) a parameter related to the width of the dispersal kernel.

A new dispersal matrix must be computed to run simulations with a different landscape or a different dispersal kernel.

The computation of \(\mu_{ii'}\) is performed using the CaliFloPP algorithm from the R package RCALI. The RCALI package has a limited number of built-in dispersal kernels. However, users can code for their own dispersal kernel. See section “Details” in the documentation of the function califlopp to learn how to implement your own kernel.

An example

Here is an example illustrating the computation of the dispersal matrix on the first landscape supplied in landsepi.

landscape <- landscapeTEST1
Npoly <- length(landscape)

For compatibility with the function califlopp, the landscape can be modified with specific functions of package sf relative to geographic projection (st_transform), polygon simplification (st_simplify).

The function califlopp needs a specific format for the coordinates of each polygon (i.e. fields) composing the landscape.

file_land <- "land_rcali.txt"  ## input for califlopp
file_disp <- "disp_rcali.txt"  ## output for califlopp (DO NOT WRITE A PATH)

## Formatting the polygons-file for califlopp
cat(Npoly, file=file_land)
for (k in 1:Npoly) {
  ## extract coordinates of polygon vertices
  coords <- landscape@polygons[[k]]@Polygons[[1]]@coords 
  ## alternatively:
  # coords <- as.data.frame(landscape$geometry[[k]][[1]])
  n <- nrow(coords)
  cat(NULL, file=file_land, append=T, sep="\n")
  cat(c(k,k,n), file=file_land, append=T, sep="\t")
  cat(NULL, file=file_land, append=T, sep="\n")
  cat(coords[1:n,1], file=file_land, append=T, sep="\t")
  cat(coords[1:n,2], file=file_land, append=T, sep="\t")
cat(NULL, file=file_land, append=T, sep="\n")

Then the function califlopp calculates the flow of particles between polygons using an integration method. Here we use the dispersal kernel of oilseed rape pollen (available in RCALI: use dispf=1 in the arguments of function califlopp, see ?califlopp for details).

param <- list(input=2, output=0, method="cub", dp=6000, dz=6000
              , warn.poly=FALSE, warn.conv=FALSE, verbose=FALSE)
califlopp(file=file_land, dispf=1, param=param, resfile=file_disp)

The RCALI package has a limited number of built-in dispersal kernels (dispf = 1 in our example). However, users can code for their own dispersal kernel (let say the name of your kernel is f) using dispf=f in the function califlopp:

my_df <-function(x, a=40, b=7) ((b-2)*(b-1)/(2*a^2*pi)*(1+(abs(x)/a))^(-b))

param <- list(input=2, output=0, method="cub", dp=6000, dz=6000, warn.poly=FALSE,
              warn.conv=FALSE, verbose=FALSE)
califlopp(file=file_land, dispf=my_df, param=param, resfile=file_disp)

However, if there are many polygons in the landscape, computations may be long. In this situation, we recommend to replace one of the built-in functions of RCALI by your own function in the source code, and to recompile RCALI. See paragraph “The individual dispersion functions” in the details of the documentation of the califlopp function (?califlopp).

The output of califlopp must then be reformatted to generate the dispersal matrix that will be further used in landsepi. The vector of field areas can also be generated.

## Import califlopp results
disp_df <- getRes(file_disp)
## Double the table because only half of the flows have been calculated
emitter <- c(disp_df$poly1, disp_df$poly2)
receiver <- c(disp_df$poly2, disp_df$poly1)

## Write a text file containing a vector of areas of all polygons
area_e <- c(disp_df$area1, disp_df$area2)
area_r <- c(disp_df$area2, disp_df$area1)
area <- as.vector(by(area_e, emitter, mean))
write(area, file="area.txt", sep=",")

## Generation of the dispersal matrix
name_f <- "mean.flow"
flow_mean <- c(disp_df[,name_f], disp_df[,name_f])
flow_f <- cbind(emitter, receiver, flow_mean, area_e, area_r)

## Remove the doublons (i.e. half the lines where emitter == receiver)
flow_f[1:nrow(disp_df),][(disp_df$poly2 - disp_df$poly1) == 0,] <- NA
flow_f <- flow_f[is.na(apply(flow_f, 1, sum)) == F,]
flow_f <- as.data.frame(flow_f)
colnames(flow_f) <- c("emitter", "receiver", "flow", "area_e", "area_r")
flow_f <- flow_f[order(flow_f$emitter),]

## lines: emitter
## columns: receiver
matrix_f <- NULL
for(k in 1:Npoly){
  ## flow divided by the emitter area
  matrix_f <- cbind(matrix_f, flow_f$flow[flow_f$receiver==k] / area)

## Normalisation of the matrix (reflecting boundaries)
## (do not normalise for absorbing boundaries)
flowtot_f <- apply(matrix_f,1,sum)
for(k in 1:Npoly){
  matrix_f[k,] <- (matrix_f[k,] / flowtot_f[k]) ## In order to have sum == 1

write(as.vector(matrix_f), file="dispersal.txt", sep=",")

Then, to read the file, use:

disp_patho <- scan("dispersal.txt", sep=",")

Plotting landscape and dispersal

Landscape structure can be plotted using the basic function plot(), or using the landsepi function plotland():

landscape <- landscapeTEST1

To highlight a specific field:

poly <- 10
colFields <- rep("white", length(landscape))
colFields[poly] <- "red"
plot(landscape, col = colFields)

To check the dispersal matrix and represent in a graphic the flow emitted by a specific polygon, use:

## convert dispersal in matrix
mat <- matrix(disp_patho, nrow=sqrt(length(disp_patho)))
poly <- 1
dispToPlot <- log10(mat[poly,] +1E-20)  ## 1E-20 to avoid log(0)

## Colour palette
nCol <- 11
whiteYellowRed <- colorRampPalette(c("white", "#FFFF99", "#990000"))
col_disp <- whiteYellowRed(nCol)
intvls <- seq(min(dispToPlot) - 1, max(dispToPlot) + 1, length.out=nCol)
intvls_disp <- findInterval(dispToPlot, intvls)

## Plot
plot(landscape, col = col_disp[intvls_disp], main=paste("Dispersal from polygon", poly))

With package ggplot2:

ggplot(landscape) + ggtitle(paste("Dispersal from polygon", poly)) +
    geom_sf(colour="black", aes(fill = dispToPlot)) + 
    scale_fill_gradientn(name="Prob. of\ndispersal", colours=rev(heat.colors(10)), breaks=-1:-10, labels=10^(-1:-10)) +
    # theme_classic() +