Another "flatten the COVID-19 curve" simulation... in R

 Hi there,

This is the best meme I've found during these days...

Well, here it is my "BUT" contribution. Some weeks ago The Washington Post published this simulations about how "social distancing" could help to "flat the curve" of COVID-19 infections. I fell in love with these simulations because their simplicity and explanatory power. Indeed, you can use the pretty similar principles to simulate predator hunt behavior and other movement patterns... I wrote some R code to have a tiny version of them...

 library(raster)  
 library(dismo)  
   
 r <- raster(nrows = 100, ncols = 100, xmn = 0, xmx = 100, ymn = 0, ymx = 100)   
 r[] <- 0  
   
 steps <- 500  
 n <- 100  
   
 locations <- data.frame(matrix(ncol = n, nrow = steps))  
   
 pp <- randomPoints(r,n)  
 cell <- cellFromXY(r, pp)  
 infected <- 1:10  
 pob <- 1:n  
 ratio <- data.frame(1:steps, NA)  
 pob_inf <- list()  
   
 for (j in 1:steps){  
  print(j)  
  locations[j,] <- cell  
    
  for(i in 1:n){  
     
   cell[i] <- adjacent(r, cell[i], 8)[round(runif(1,0.51,nrow(adjacent(r, cell[i], 8))+0.49),0),2]  
     
  }  
    
  new_inf <- cell %in% cell[infected]  
  infected <- pob[new_inf]  
  ratio[j,2] <- length(infected)  
  pob_inf[[j]] <- infected  
   
 }  
   
 locations2 <- data.frame(matrix(ncol = n, nrow = steps))  
   
 cell2 <- cellFromXY(r, pp)  
 infected2 <- 1:10  
 pob2 <- 1:n  
 ratio2 <- data.frame(1:steps, NA)  
 pob_inf2 <- list()  
   
 for (j in 1:steps){  
  print(j)  
  locations2[j,] <- cell2  
    
  for(i in 1:25){  
     
   cell2[i] <- adjacent(r, cell2[i], 8)[round(runif(1,0.51,nrow(adjacent(r, cell2[i], 8))+0.49),0),2]  
     
  }  
    
  new_inf2 <- cell2 %in% cell2[infected2]  
  infected2 <- pob2[new_inf2]  
  ratio2[j,2] <- length(infected2)  
  pob_inf2[[j]] <- infected2  
   
 }  

Let's make some plots to put them together in a GIF and better visualize the results...

 num <- seq(1,500,4)  
   
 for (p in 1:125){  
  id <- sprintf("%03d", num[p])  
  png(paste("corona_",id,".png", sep=""), width=780, height=800, units="px", pointsize = 15)  
  layout(matrix(c(1,1,2,3),2,2,byrow = TRUE))  
  plot(ratio[1:num[p],],pch=19, xlim = c(0,500), ylim = c(0,100), ylab = "nº of infected",   
     xlab = "time", col = "red", cex.axis = 1.4, cex.lab = 1.4, main = "Infected curves", cex.main = 2)  
  points(ratio2[1:num[p],],pch=19,col="blue")  
  legend("topright", legend = c("free movement", "restricted movement"),lwd = c(4,4), col = c("red","blue"), cex = 1.5 )  
  plot(r, col = "white", legend = FALSE, axes = FALSE, main = "free movement", cex.main = 1.8)  
  points(xyFromCell(r,as.numeric(locations[num[p],])), cex = 1.2, col = "grey40", pch = 18)  
  points(xyFromCell(r,as.numeric(locations[num[p],]))[pob_inf[[num[p]]],], pch = 18, col = "red", cex = 1.4)  
  plot(r, col = "white", legend = FALSE, axes = FALSE, main = "restricted movement", cex.main = 1.8)  
  points(xyFromCell(r,as.numeric(locations2[num[p],])), cex = 1.2, col = "grey40", pch = 18)  
  points(xyFromCell(r,as.numeric(locations2[num[p],]))[pob_inf2[[num[p]]],], pch = 18, col = "blue", cex = 1.4)  
    
  dev.off()   
 }  

Done!

 Stay safe!

eWalk, an R function to simulate two-dimensional evolutionary walks

Hey there! 

 During last months, I was thinking with my friend Javi Fajardo (see his work here!) a little bit more about Brownian / OU process and their applications in biology. Remember that an OU model describes a stochastic Markov process that consist of two parts: first, a Brownian motion (random walk) model that represents a completely random process of variable evolution, and a second term that represents the attraction of an optimum value of the variable. For this, OU process can be considered as a variation of Brownian motion model. In the last posts we were talking about these kinds of models used to drive trait character evolution, specifically an ecological trait, such as optimum temperature or precipitation.

 

 My colleague Javi and I were talking about the applications of this models in two dimensions, representing for example space coordinates (latitude/longitude). They have been used, for example, to simulate predator/prey movements, home range movements, etc. We developed an R function similar to the previous eMotion function in order implement this kind of movement models: eWalk (from evolutiveWalk...). eWalk is similar to other movement-simulation functions in adehabitatLT package (Calenge, 2006). However, you can use eWalk to simulate a variety of processes as Brownian motion, OU processes or Lèvy flights (see below).

Here you have the (commented!) code to load the function into R console:

   
   
eWalk <- function(sigma,lon,lat, generations,alpha_x,theta_x,alpha_y,theta_y, color,levy=FALSE, plot=TRUE) {
  route=data.frame(1:generations,1:generations)                             # a data.frame to store the route
  names(route)<-c("lon","lat")
  generation= generations                                               # generation counter
  
  while (generation > 0){                                                   # stops the simulation when generation counter is 0
    if (levy == TRUE){                                                      # only if levy=TRUE: Levy Flight!
      rndm=abs(rcauchy(1))                                                  # process to select if I should jump or not
      if (rndm > 15){                                                       # Levy jump probability threshold
        l=15                                                                # length of the jump
        x= lon + (l * (sigma * rnorm(1))) + (alpha_x * ( theta_x - lon))    # Brownian or OU process for longitude
        y= lat + (l * (sigma * rnorm(1))) + (alpha_y * ( theta_y - lat))    # Brownian or OU process for latitude
        loc=cbind(x,y)
        route[generation,]<- loc                                            # store new location
        generation= generation - 1                                        # advance to next generation
        lon<- x                                                        # go to the new position!
        lat<- y
        
      }
      else{                                                                 # if Levy jump is not selected, l=1 i.e. proceed with  
        l=1                                                                 #normal Brownian motion or OU process
        x= lon + (l * (sigma * rnorm(1))) + (alpha_x * ( theta_x - lon))
        y= lat + (l * (sigma * rnorm(1))) + (alpha_y * ( theta_y - lat))
        loc=cbind(x,y)
        route[generation,]<- loc
        generation= generation - 1  
        lon<- x      
        lat<- y
      }
    }
    else{                                                                   # if levy=FALSE, l=1, only Brownian motion or OU
      l=1
      x= lon + (l * (sigma * rnorm(1))) + (alpha_x * ( theta_x - lon))
      y= lat + (l * (sigma * rnorm(1))) + (alpha_y * ( theta_y - lat))
      loc=cbind(x,y)
      route[generation,]<- loc
      generation= generation - 1  
      lon<- x      
      lat<- y
    }
  }
  
  if (plot == TRUE) {                                                       # if plot=TRUE, plto the route!
    lines(route, col= color, lwd=3)  
  }
  return(route)                                                             # print the data.frame with each location during the walk!
  
}
>

 The needed parameters to run the function are similar to those for eMotion, but taking into account that now you have two variables (latitude/longitude, for example) at the same time.

eWalk(sigma,initial_lon,initial_lat,number of generations, lon_alpha, lon_optimum, y_alpha, y_optimum, color, levy=FALSE, plot=TRUE)

In addition, I have added a complement to simulate Lèvy flights, a non-gaussian process similar to Brownian motion but allowing for jumps during the motion. Lévy flights pattern has been found in searching movements of predators (Bartumeus et al., 2005), or have been used to simulate punctuated equilibrium in traits evolution (Gould and Eldredge, 1977; Landis et al., 2013). To perform Lèvy flight, eWalk selects a random number from Cauchy distribution and if it is higher than a threshold (15 by default), Brownian motion term is multiplied by a constant representing a jump (also 15 by default). Although this is not the formal expression of a Lèvy process, this is a simple way to simulate a negative exponential rate in for the length of each step.

Here you have some examples using eWalk and their code to reproduce them:

   
 par(mfrow=c(2,2))

a=eWalk(0.15,0,0,1000, 0, 0, 0, 0, color="blue", levy=FALSE, plot=FALSE)
plot(a,type="n", main="Brownian motion")
lines(a,lwd=3, col="blue")
points(a, cex=.3, col="black", pch=21)

a=eWalk(0.15,0,0,1000, 0, 0, 0, 0, color="red", levy=TRUE, plot=FALSE)
plot(a,type="n", main="Lêvy flight")
lines(a,lwd=3, col="red")
points(a, cex=.3, col="black", pch=21)

a=eWalk(0.15,0,0,1000, 0.01, 20, 0, 0, color="orange", levy=FALSE, plot=FALSE)
plot(a,type="n", main="OU one optimum")
lines(a,lwd=3, col="orange")
abline(v=20, col="red", lwd="2")
points(a, cex=.3, col="black", pch=21)

a=eWalk(0.15,0,0,1000, 0.01, 20, 0.01, 20, color="green", levy=FALSE, plot=FALSE)
plot(a,type="n", main="OU two optimum")
lines(a,lwd=3, col="green")
points(a, cex=.3, col="black", pch=21)
abline(v=20, col="red", lwd="2")
abline(h=20, col="red", lwd="2")

 References

-Bartumeus, F., da Luz, M. E., Viswanathan, G. M., & Catalan, J. (2005). Animal search strategies: a quantitative random‐walk analysis. Ecology, 86(11), 3078-3087.

-Calenge, C. (2006). The package “adehabitat” for the R software: a tool for the analysis of space and habitat use by animals. Ecological modelling, 197(3), 516-519.

-Gould, S. J., & Eldredge, N. (1977). Punctuated equilibria: the tempo and mode of evolution reconsidered. Paleobiology, 3(02), 115-151.

-Landis, M. J., Schraiber, J. G., & Liang, M. (2013). Phylogenetic analysis using Lévy processes: finding jumps in the evolution of continuous traits. Systematic biology, 62(2), 193-204.

eMotion, an R function for evolutionary motion of character values

Hi all!

 Some time ago, I was studying an article from Tamara Münkemüller et al. 2015: "Phylogenetic niche conservatism – common pitfalls and ways forward". It´s a nice study where the ability of current tools in order to identify the evolutionary model that drives the changes in a trait (in this case, an ecological suitability) is discussed.
 They simulate the evolution of a trait under several evolutionary processes and then, try to apply the tools that are commonly used to test what kind of process is behind the evolution of the trait, allowing us to infer several patterns or processes as phylogenetic niche conservatism. Their results are really interesting, and I recommend you to take a look to the whole research!

However, I would like to focus in one of the main pictures of the article:

 Looks really nice! As a beginner, it was very difficult to me to understand this awesome figure. From my biological background, to study all this mathematical stuff requires a lot of time that I should invert in laboratory or field work... but it´s also very important to know the underlying statistical background of all the analysis that I would like to apply to my data in the future! So I decided to try to replicate some of those boxes by myself using R language. Many times, it´s the best way to learn!

 The figure shows trait evolution over time under Ornstein–Uhlenbeck 1 parameter in the left column, Ornstein–Uhlenbeck evolutionary at middle column, and Ornstein–Uhlenbeck extended (multiple optimum) at right column, with different selection strength in each row. This means that the first row is actually a Brownian Motion process, since alpha (selection strength) is 0.

 The graphic is the result of applying the OU process equation:

x(t+dt)= x(t)+ alpha(theta(t)-x(t))dt + (sigma dt E)

where , x(t) is the trait value at time=t, alpha is the selection strength with which a specific trait value (optimum) is wanted, theta is the optimal value of the trait in time=t, sigma is Brownian motion rate (the amount of random trait variability per time) and E is a normal distribution(0,1).

 Following this equation, I developed a simple and intuitive R function that simulate an OU evolutionary process for a hypothetical character in two species: eMotion (from "evolutionary motion")

 You can provide each parameter of the OU equation for the two species, and then you should define the number of generations (time) and replicates (number of simulations per species).

 Here you have the code to load the function into R console:

 eMotion <- function (sigma, theta, alpha, sigma2, theta2, alpha2, generations, n) {  
   plot(seq(1:generations),seq(1:generations), ylim=c(-200,200), type="n",   
       xlab="generations", ylab="trait value", main=paste("Sp1- s=", sigma,  
       " th=", theta, " a=", alpha,"  Sp2- s=", sigma2, " th=", theta2,   
       " a=", alpha2, sep=""))  
   trait_a=numeric()  
   simulations= list(1:n)  
   for (j in 1:n){  
     a=0  
     for (i in 1:generations) {   
       a= a + (sigma * rnorm(1)) + (alpha * ( theta - a))  
       trait_a[i]=a  
     }  
     aa=data.frame(trait_a)  
     simulations[j]=aa  
   }  
   for(k in 1:n){  
     bb=data.frame(simulations[k])  
     lines(bb, col="red")  
   }  
   #######  
   trait_a=numeric()  
   simulations= list(1:n)  
   for (j in 1:n){  
     a=0  
     for (i in 1:generations) {   
       a= a + (sigma2 * rnorm(1)) + (alpha2 * ( theta2 - a))  
       trait_a[i]=a  
     }  
     aa=data.frame(trait_a)  
     simulations[j]=aa  
   }  
   for(k in 1:n){  
     bb=data.frame(simulations[k])  
     lines(bb, col="blue")  
   }  
 }  

 Once that you have loaded the function, you can use it to create simple plots to observe the effect of each parameter of OU equation in resultant graphic. For instance, here we describe an OU process of a trait in two species. The species 1 (red) follows an OU process of character evolution, with a sigma value (Brownian motion rate) of 2 and an optimum in 100, while species 2 (blue) follows an OU model with a sigma value of 0.5 and an optimum in -100. In both cases, alpha is 0.002.

eMotion(sigma, theta, alpha, sigma2, theta2, alpha2, generations, n)

 eMotion(2, 100, 0.002, 0.5,-100, 0.002, 10000, 20)  

The function automatically generates this simple plot, where you can observe that species 1 (red), is looking for a trait value around 100 (optimum), while species 2 (blue) is trying to get an optimum near to -100. Although both species have the same alpha (selection strength), the red one has a Brownian rate (sigma) much higher than the blue one. That explains why species 1 (red) has a higher range of variation than species 2 (blue), where all values are always near to the optimum (-100).

 You can use the function to explore other options in parameter values, which could be useful in order to teach about evolutionary processes or to understand the Ornstein–Uhlenbeck equation. Here you have one more example, enjoy!

 eMotion(1, 0, 0, 1,0, 0.01, 10000, 20)  

nichEvolve documentation and code

Here you have some documentation about the nichEvolve model and a link to download the NetLogo code. I will write soon some practical experiments to test with the model... enjoy!
Link:  nichEvolve.nlogo

*(Don't you have NetLogo installed in your computer? NetLogo is a nice tool to implement simple agent-based models, and very useful to introduce yourself in the world of dynamic models!) Download from here: NetLogo

 WHAT IS IT?

Here is a simple approach to explain how different models of trait evolution can work over one or two different species. Two available models of trait evolution are available in the general framework of Ornstein-Uhlenbeck process, with the possibility of adapting it into a simpler Brownian Motion process by changing the settings.
The model could be usefull as a virtual laboratory to teach about this concepts.

author: Javier Fernandez-Lopez  jflopez.bio@gmail.com

HOW IT WORKS

The model represent the evolution of a single ecological trait that could be defined as "ecological suitability", and could be understood for instance, as a specific requirement of temperature or humidity. The program allows an investigator to observe the effect of changes in each parameter of the Ornstein-Uhlenbeck equation

x(t+dt)= x(t)+ alpha(theta(t)-x(t))dt + (sigma dt E)

which define the evolution through time of a trait, where x(t) is the trait value at time=t, sigma is Brownian motion rate (the amount of random variability of the trait per time), theta is the optimal value of the trait in time=t, and alpha is the selection strengh with which a specific trait value is wanted.

 By changing equation parameters, the researcher can check the evolution of the trait value over time in the graphic box at the right side of the model. The "optimum" boxes show as well the theoretical value of the trait value. If speciation switch is "off", the model represents just one species. When speciation is turned "on", the species is split (red and blue) and the trait evolution occur independently. Now, theta parameter (OU-optimum) could differ for each species.

 Left side of the model shows a espatialy-splicit representation of the model. 200 mobile agents of each colour are shown over a grid that simules different ecological conditions. Remember that if speciation is "off", both colours represent the same species with identical ecological requirements. When the simulation starts, each agent try to find a patch around him that best match with his ecological requirements (theorical optimun niche values). The "realized" boxes show the mean of the values of the environmental conditions that the mobile agents have found by tracking the surronding environment. You can also compare the theorical optimum with the realized niche of mobile agents.

 Following KISS philosophy (Keep It Simple, Stupid!) we have not included some processes as extinction when theoretical and realized niches doesn't match, or reproduction of mobile agents.

 Finally, we have included as an experiment the option of "climate-change". When this switch is "on", it only affects to the spatially explicit representation of the model. The environment value of each patch increases by time, changing the niche availability for the mobile agents and affecting (or not) their distribution. The background process of trait evolution is not affected by "climate-change".

HOW TO USE IT

Click "Setup" to initialize the model. Then, you can use "Step" to see how the model runs step by step, or use "Go" button to run the model in a continuous way.

1) Start with a simple Brownian Motion model for a single species (speciation "off"). For this, alpha slide should be 0, theta's values are not important, since alpha (optimum selection strengh) is zero. You could test for changes in graphical box when sigma values are too large or too small.

2) Once that you know the effect of the Brownian motion rate (sigma), you could turn on speciation switch, to split the species. Now you can observe how both species are evolving independently, and, if you are lucky, you could see how their distributions differs from each other in the 2D display.

3) Now you can explore OU models increasing the value of alpha. You can try first using the same theta value for both species and changing it latter. You can also observe how interact Brownian motion rate (sigma) with optimum slection strengh (alpha)

4) Finally you could check the behaivour of mobile agents when climate-change is "on"


THINGS TO NOTICE

 By observing the patterns drawn in graphical box by different combinations of each parameter values, you could think about how difficult it is to guess the actual background process that drives a specific trait evolution if we only know the final state of the values.  


HOW TO CITE

Fernandez-Lopez, J. 2016. nichEvolve: a niche evolution model