Spatial Modeling Using R: A case study with three landscape types and a disturbance agent

NEMO 2010

Joseph PekolDr. David Heibeler

Dr. Aaron Weiskittel

Outline

• Introduction• SIRS model basics• Model framework• Framework to R code• Sample output/results• Conclusion

Using R as a spatial modeling tool

• Event driven models

• Accessible matrix manipulation

• Quickly write, debug, and test code

• Easy to produce graphical output

• Built in statistical theory

• No existing R packages available

SIRS basics• Susceptible-Infected-Recovered-Susceptible

• Simulates a birth-death model of continuous-time spatial populations

• Long-distance and local interactions between pathogen and target

• Assumptions based on Poisson distribution and random occurrence on exponential dist.

• Poisson used for simplicity – memoryloss property

Linking Poisson processes and simulations

At time t, assume N(t) infected sites. Each infected site:• Produces offspring at rate φ• Dies off at rate μ

E.g. each site acts as Poisson with rate φ+μ, so the total population acts as a Poisson process with rate λT = (φ+μ)N

Simulating an SIRS model in 3 steps

1. Choose when the next event will occur from an exponential distribution with mean 1/ λT.

2. Choose the origin of the event from all occupied sites with an equal probability.

3. Simulate an attempted infection of another site (pbirth) or else the death of the current site. If a birth occurs, choose a target site (specified as a percentage of local/long-distance infection)

Case study outlineLandscape generator• Three landscape types

– Hydric, Mesic, Xeric

SIRS model• Input 3 different sets of

infection parameters

• Output visual landscape, residual proportions of stand types

Land 1 Land 2 Land 3Hydric 0.50 0.35 0.15

Mesic 0.15 0.35 0.35Xeric 0.33 0.33 0.50

Trial ColSuccess DistSiteSuc Recov

1, 4, 7 (0.9, 0.7, 0.3) (0.3, 0.2, 0.1) (0.2,0.4,0.45)

2, 5, 8 (0.3, 0.7, 0.9) (0.1, 0.2, 0.3) (0.45,0.4,0.2)

3, 6, 9 (0.5, 0.5, 0.5) (0.2, 0.2, 0.2) (0.5, 0.5, 0.5)

However, we first need a landscape to work with….

• Three site types on a 100x100 lattice

• Input site type densities and clustering level

• Output includes matrix of stand types and visual representation

• Total lines of code: 273

Cell type Code Color

Hydric 0Mesic 1Xeric 2

Using a modified landscape from R script by David Hiebeler:

What R needs to know…

[,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10] [,11] [,12] [,13] [,14] [,15] [1,] 1 1 1 0 0 0 2 2 1 1 1 0 0 0 2 [2,] 1 1 1 0 0 0 1 1 1 1 1 1 1 0 0 [3,] 1 1 0 2 2 2 2 1 1 1 1 1 1 1 2 [4,] 1 1 1 1 2 2 2 1 1 1 1 1 1 1 2 [5,] 1 1 1 1 2 2 2 2 0 0 0 1 2 2 2 [6,] 1 1 1 1 1 0 0 0 0 0 2 2 2 2 2 [7,] 1 1 1 1 1 0 0 0 0 0 2 2 2 2 2 [8,] 1 1 1 0 0 1 1 1 1 1 2 2 2 2 2 [9,] 2 1 1 2 0 1 1 1 1 1 2 2 2 2 2[10,] 2 1 1 2 1 1 2 1 1 1 2 2 2 2 2[11,] 0 2 1 1 1 1 2 2 0 0 2 2 0 2 1[12,] 0 0 0 0 1 1 2 2 0 0 0 0 0 2 1[13,] 1 2 0 1 1 0 2 2 1 1 1 0 2 2 2[14,] 1 1 1 1 0 0 0 1 1 1 1 1 1 2 2[15,] 2 1 1 1 1 2 2 1 1 1 1 1 1 2 2[16,] 2 1 1 1 1 2 2 1 1 1 1 1 1 2 2[17,] 2 2 2 2 2 2 0 1 1 1 1 1 1 2 2[18,] 1 1 2 2 1 1 2 0 2 0 0 1 1 0 2[19,] 0 2 2 2 1 1 1 0 0 0 0 1 1 1 1[20,] 0 0 2 2 1 1 1 0 0 0 0 1 1 1 1

=

SIRS model• Accepts landscape matrix

• Input infection, dist. Site infection, recovery, and death rates

• Randomly infects initial individuals

• Outputs visual landscape; data frame of residual stand types proportions

• Total lines of code: 335Cell type Code ColorHydric 0Mesic 1Xeric 2

Infected 3Damaged 4

Dead 5

SIRS model framework

Infections remain?

No – Record final site types, calculate proportion site type left, output data Yes – Calculate rates based on infected population.

Choose where event originates and susceptible target.

Long distance interaction?- Choose random S coord.

Local interaction?- Choose neighbor using offset vector

If target Susc. (=0,1,2)Infect target based on P(susceptible site)

If target recovered. (=4)Infect target based on P(disturbed site)

Infected indiv. loses infectionwith rate γ

Site diesif runif(1) < P(Site recov.)

Site recoversif runif(1) > P(Site recov.)

Update pop. vectors &Landscape matrix

The key is good code for efficient performance

• Keeping track of population totals• Choosing sites• Keeping track of infected locations• Updating population indices

Keeping track of everythingchucksize = 500# keeps track of how long are state vectors currently arevectorlengths = chunksize# Create vectorsS = rep(0,chunksize)I = rep(0,chunksize)R = rep(0,chunksize)D = rep(0,chunksize)et = rep(0,chunksize)……..if (i == vectorlengths) { vectorlengths = vectorlengths + chunksize length(S) = vectorlengths length(I) = vectorlengths…..}

Initialize population vectors

Sets vector length, improves efficiancy so Rdoesn’t have to ‘grow as it goes’.

Increase vector lengths as needed…

coordInd = floor(runif(1)*currentI)+1 x = Ixv[coordInd] y = Iyv[coordInd]

X and Y vectors hold coordinates for infected individuals. Random choice of index determines event locations

Choosing event locationsChoose ‘origin’ site

Choose ‘interaction’ site as local or long distance

if (runif(1) < alpha) { # long-distance contact otherx = floor(runif(1)*L)+1 othery = floor(runif(1)*L)+1 } else { # local contact randInd = floor(runif(1)*4) + 1 otherx = x + xoffsets[randInd] othery = y + yoffsets[randInd] otherx = ((otherx - 1 + L) %% L) + 1 othery = ((othery - 1 + L) %% L) + 1 }

Wrap around code allows movement off one edge of matrix and onto the opposite edge.

Vector containing neighborhood offsets:nhoodOffsets=matrix(c(0,-1,0,1,-1,0,1,0),nrow=4,byrow=TRUE)xoffsets = nhoodOffsets[,1]yoffsets = nhoodOffsets[,2] Or just…

xoffsets = c(0,-1,0,1)yoffsets = c(-1,0,1,0)

if (stateArray[otherx,othery] == 0 || stateArray[otherx,othery] == 1 || stateArray[otherx,othery] == 2 && (runif(1)< pColSuccess[hab[x,y]+1])) { # target susceptible currentS = currentS - 1 currentI = currentI + 1

stateArray[otherx,othery] = 3

Ixv[currentI] = otherx Iyv[currentI] = othery siteTypeInf[currentI] = hab[otherx,othery] }

Tracking event occurrencesCheck whether an infection attempt is successful

Update current population totals

Update matrix with new stand classification

Update location of infection for indexing in next iteration

Results – Landscape maps

Results – SIRS maps

Results – Proportions by trial

Here, only one attempt per trial…

However, a simple script can run the model multiple times, producing means, standard error, etc.

Results – Output of populations over time

• deSolve package

• Provides functions to solve first order, ordinary differential equations (ODE) among others

• Modeled populations vs estimated ODE population curves

Conclusions – Pros and Cons of event driven models in RPros• Quick and easy data manipulation

• No need to compile: writing, testing, and debugging code is simplified

• Opportunity for very robust results

• Excellent packages make life easier

• Graphical and statistical output very accessible

Cons• Much slower than standard

programming languages

• Increased model complexity leads to trickier programming

• Efficient coding a must for quick run-times

But…the biggest Pro of all…

R is free!!!

Thank you!

Questions/Comments?