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Spatial Modeling Using R: A case study with three landscape types and a disturbance agent NEMO 2010 Joseph Pekol Dr. David Heibeler Dr. Aaron Weiskittel

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

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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

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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

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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 λ T = (φ+μ)N

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Simulating an SIRS model in 3 steps 1.Choose when the next event will occur from an exponential distribution with mean 1/ λ T. 2.Choose the origin of the event from all occupied sites with an equal probability. 3.Simulate an attempted infection of another site (p birth ) 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)

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Case study outline Landscape 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 1Land 2Land 3 Hydric Mesic Xeric TrialColSuccessDistSiteSucRecov 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)

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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 typeCodeColor Hydric0 Mesic1 Xeric2 Using a modified landscape from R script by David Hiebeler:

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What R needs to know… [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10] [,11] [,12] [,13] [,14] [,15] [1,] [2,] [3,] [4,] [5,] [6,] [7,] [8,] [9,] [10,] [11,] [12,] [13,] [14,] [15,] [16,] [17,] [18,] [19,] [20,] =

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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: 335 Cell typeCodeColor Hydric0 Mesic1 Xeric2 Infected3 Damaged4 Dead5

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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 infection with rate γ Site dies if runif(1) < P(Site recov.) Site recovers if runif(1) > P(Site recov.) Update pop. vectors & Landscape matrix

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The key is good code for efficient performance Keeping track of population totals Choosing sites Keeping track of infected locations Updating population indices

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Keeping track of everything chucksize = 500 # keeps track of how long are state vectors currently are vectorlengths = chunksize # Create vectors S = 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 R doesn’t have to ‘grow as it goes’. Increase vector lengths as needed…

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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 locations Choose ‘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 L) % L) + 1 othery = ((othery 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)

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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 occurrences Check 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

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Results – Landscape maps

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Results – SIRS maps

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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.

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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

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Conclusions – Pros and Cons of event driven models in R Pros 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!!!

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Thank you! Questions/Comments?

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