Exploring Modeling by Programming: Insights from Numerical Experimentation

Abstract

This chapter aims to provide students the background necessary to use computational methods and numerical experimentation to pursue mathematical research. We provide a brief refresher on differential equations, model building, and programming, followed by in-depth discussions of how numerical experimentation was critical to three undergraduate research projects involving parasite transmission in feral cats, the role of protein regulation in establishing circadian rhythms, and treatment costs in an influenza outbreak. Code (written in R) is provided for most of the models and nearly all of the figures presented in the chapter, so that the reader may learn the programming as they go along. We also propose ideas for extending the research projects presented in the text. Both deterministic and stochastic models are presented and the critical role of computational methods in obtaining solutions is illustrated.

Electronic supplementary material The online version of this chapter (https://doi.org/10.1007/978-3-030-33645-5_4) contains supplementary material, which is available to authorized users.

Appendices

Appendix 1: Notes on Programming

Where possible we have attempted to simplify code in such a way to allow you to more simply reproduce the figures in this Chapter. We expect that the results you are able to produce from running code snippets will match those that we have included in the chapter, but please do not be alarmed by slight aesthetic differences. In particular, we prefer to use the mtext(...) command in R to generate axis labels which can be adjusted slightly more easily than can the defaults. For example, to produce many of our figures in the text, by plotting the model output ..., we recommend the code

plot(..., xlab=’Time’, ylab=’Population size’)

For our own use, we prefer the slightly more complex code below, which results in slightly better looking output, though this is a matter of preference.

par(mfrow=c(1, 1), mar=c(4.1, 4.1, 1.1, 1.1)) plot(..., xlab=’’, ylab=’’) # empty axis labels mtext(’Time’, 1, line=2.5, font=2, cex=1.25) mtext(’Population size’, 2, line=2.5, font=2, cex=1.25)

The first line of this code sets the layout of the plot, here one row by one column, and the second sets the plot margins in the order bottom-left-top-right. We add our plot (with blank labels) to a window of that shape and configuration and add text to the margins. The ambitious reader may be interested in exploring the graphics capabilities of the ggplot2 package.

Figures can be saved from menu selections or using the following approach

pdf(’filename.pdf’, height=5, width=5) ## insert arbitrary figure code dev.off()

Entering ?pdf into the R console will show how the pdf(...) command works and related commands like png(...} or jpeg(...).

A few examples of syntax and useful commands for basic calculations or plotting are given in Table 4. A brief overview of available differential equations solvers from the deSolve package is given in Table 5.

Table 4 List of useful commands and operations included in the base installation of R

Table 5 List of differential equations solvers and commands included in deSolve

Appendix 2: Circadian Rhythm Code

Start by defining the model. We have included some if statements that allow us to turn the alcohol parameter on or off at a given time (t.switch1 and t.switch2), to test how the system responds. Earlier we mentioned having to find and replace in order to replace the printed single quote used to specify many arguments in R if you copy code from an electronic source. Here we use double asterisks (∗∗) to enter exponents rather than the more familiar caret (ˆ), since the latter does not copy well from electronic text.

## the model CLOCKmod <- function(t, x, parms)  {   with(as.list(c(parms, x)), {     ## alcohol value switch     if(t > t.switch1){     a <- 1     }     if(t > t.switch2){     a <- 1     } #conc. of Per2/Cry mRNA     dY1 <- (a∗v1b∗(Y7+c))/(k1b∗(1+(Y3/k1i)∗∗p+Y7+c))        -k1d∗Y1 # conc. of PER2/CRY complex in the cytoplasm     dY2 <- k2b∗(Y1)∗∗q - k2d∗Y2 - k2t∗Y2 + k3t∗Y3 # conc. of PER2/CRY complex in the nucleus     dY3 <- k2t∗Y2 - k3t∗Y3 - k3d∗Y3 # conc. of Bmal1 mRNA     dY4 <- (v4b∗(Y3)∗∗r)/((k4b)∗∗r + (Y3)∗∗r) - k4d∗Y4 # conc. of BMAL1 protein in the cytoplasm     dY5 <- k5b∗Y4 - k5d∗Y5 - k5t∗Y5 + k6t∗Y6 # conc. of BMAL1 protein in the nucleus     dY6 <- k5t∗Y5 - k6t∗Y6 - k6d∗Y6 + k7a∗Y7 - k6a∗Y6 # conc. of transcriptionally active form BMAL1     dY7 <- k6a∗Y6 - k7a∗Y7 - k7d∗Y7 # the output     list(c(dY1, dY2, dY3, dY4, dY5, dY6, dY7)) })}

Again to verify our assignments, we print the parameters and initial conditions to the screen, but since times refers to a very long vector, we assign it but do not print it to the screen.

(parms <- c(v1b = 9, k1b = 1, k1i = 0.56, c = 0.01,    p = 8, k1d = 0.12, k2b = 0.3, q = 2, k2d = 0.05,    k2t = 0.24, k3t = 0.02, k3d = 0.12, v4b = 3.6,    k4b = 2.16, r = 3, k4d = 0.75, k5b = 0.24,    k5d = 0.06, k5t = 0.45, k6t = 0.06, k6d = 0.12,    k6a = 0.09, k7a = 0.003, k7d = 0.09,    a = 1, t.switch1 = 5000, t.switch2 = 5000)) times <- seq(0, 5000, by=0.1) (xstart <- c(Y1 = 0.25, Y2 = 0.3, Y3 = 1.15,    Y4 = 0.85, Y5 = 0.72, Y6 = 1.35, Y7 = 1.075))

To assess the role of the alcohol sensitivity parameter on dynamics, we first numerically solve the model for the original parameter value a = 1, store, and plot the solution over the interval of interest.

## calculate solution    out  <- lsoda(xstart, times, CLOCKmod, parms)    out  <- as.data.frame(out) ## plots    plot(out$time, out$Y3, type=’l’, col=’gray’,       lty=1, lwd=2, ylim=c(0, 5), xlim=c(0, 250),       xlab=’Time (hours)’,       ylab=’Protein amounts (nM)’, las=1)    lines(out$time, out$Y5 + out$Y6 + out$Y7,       type=’l’, col=’black’, lty=2, lwd=2)

Then, we change the value of a in the list of parameters by manually resetting it. Since the initial conditions, solution times, and the structure of the model itself are unchanged, we can simply calculate a new solution with the updated parameters and add the desired results to the existing plot. This realization allows us to greatly simplify our computer code by leaving out large, redundant chunks that would otherwise be tempting to repeat.

## fix parameter value, repeat as necessary    parms[’a’] <- 0.1    out2  <- lsoda(xstart, times, CLOCKmod, parms)    out2  <- as.data.frame(out2)

After computing the solution for the parameter of interest, we can add these solutions to our existing plot to better understand the role of the parameter.

## add corresponding solutions    lines(out2$time, out2$Y3, type=’l’, col=’red’,       lty=1, lwd=2)    lines(out2$time, out2$Y5 + out2$Y6 + out2$Y7,       type=’l’, col=’red4’, lty=2, lwd=2)

Finally, since the plot is now quite complex, we add a nice legend.

## add legend    legend(’topleft’, c(’PER2/CRY Protein’,       ’BMAL1 Protein’, ’PER2/CRY Protein (w/ alcohol)’,       ’BMAL1 Protein (w/ alcohol)’), col=c(’gray’,       ’black’, ’red’, ’red4’), lty=c(1, 2, 1, 2), lwd=2,       bg=’white’)

For space purposes, we have not included the code in which peak concentrations were calculated and stored. This was done using the pracma package [6], and can be found in the project’s GitHub page.

Appendix 3: Influenza Code

First we define the model, since the equation for the change in cost is long, the + at the end of the first two parts of the equation ensures that each subsequent line is counted as continuation of the previous line.

## the model FLUmod <- function(t, x, parms)  {   with(as.list(c(parms, x)), { #for all, m is parameter for natural mortality #Susceptible    dS <- p - m∗S - beta∗S∗I - zeta∗S + omega∗(V + N + R) #Vaccination    dV <- zeta∗S - m∗V - omega∗V #Infectious    dI <- beta∗S∗I - gamma∗I - m∗I #Recovered Naturally    dN <- (1 - delta)∗(1 - rho)∗gamma∗I - m∗N - omega∗N #Treated, coded as R to avoid confusion with T for TRUE    dR <- (1 - delta)∗(rho)∗gamma∗I - m∗R - omega∗R #Mortality    dD <- delta∗gamma∗I #Added cost equation    dC <- z∗zeta∗S + b∗beta∗S∗I + d∗delta∗gamma∗I +      + n∗(1 - delta)∗(1 - rho)∗gamma∗I  +      + r∗(1 - delta)∗(rho)∗gamma∗I     list(c(dS, dV, dI, dN, dR, dD, dC)) # the output })}

Then we define parameters, the list of times for which to calculate the solution, and initial conditions. The factor of 1∕52 included in many of the parameter values was used to scale all values to be measured in weeks. To provide example dynamics in Fig. 12, we solve the model over a period of 2 years.

## parameters    parms <- c(p = (13)∗(1/52), m = (0.013)∗(1/52),       beta = (0.08)∗(1/52), zeta = (1.2)∗(1/52),       gamma = (12)∗(1/52), omega = (0.8)∗(1/52),       delta = 0.0005, rho = (1/3), z = 52.92,       b = 3.00, d = 542430.70, n = 110.28, r = 5613.65) ## time points    times <- seq(0, 52∗2, by=0.1) ## initial values    xstart <- parms[’p’]/parms[’m’]∗c(S = 0.98,V = 0.01,       I = 0.01, N = 0, R = 0, D = 0, C = 0)

Finally, we use the numerical solver lsoda to find the model solution.

out <- lsoda(xstart, times, FLUmod, parms) out <- as.data.frame(out)

The matplot(...) command lets us plot multiple columns simultaneously (even allowing for different line styles or widths in addition to colors). We can access multiple columns of output, each of which corresponds to a model variable, by referring to their numbers (e.g., out[, c(2, 4, 6)]), a range of numbers (e.g., out[, 2:6], which is all of the entries 2 through 6), or by name (as shown in the sample code below). Since the vector of custom colors is rather long, we define the cols at the start, which can be referenced anytime the variables are plotted.

## define colors cols <- c("red", "gray", "green", "blue", "magenta") ## panel 1 matplot(out$time, out[, c(’S’, ’V’, ’I’, ’R’,’N’)],    type=’l’, lty=1, lwd=2, col=cols,    xlab=’Time (weeks)’, ylab=’Population    densities (number/area)’, las=1) legend(’topright’, c(’Susceptible’, ’Vaccinated’,    ’Infectious’, ’Treated’, ’Natural’),    col=cols, lty=1, lwd=2, bg=’white’) ## panel 2 (cost scaled to millions) plot(out$time, out$C/1e6, type=’l’, lty=1,    xlab=’Time (weeks)’, ylab=’Cost (millions USD)’,       las=1)

To solve the model for a range of ζ parameter values, then plot the long-term behaviors as a function of ζ as it varies, we use the code below.

## time vector    times <- seq(0, 52∗200, by=0.1) ## range of zeta values    zetas <- c(seq(0.0, 0.2, by=0.01)) ## object to store model output    vals <- NULL for(i in 1:length(zetas)){    parms[’zeta’] <- zetas[i]    out  <-  lsoda(xstart, times, FLUmod, parms)    out  <- as.data.frame(out)    if(i/10==floor(i/10))print(i) ## progress bar    vals <- rbind(vals, out[nrow(out), ]) }

The above code assigns the parameter value of interest, calculates the corresponding solution, and stores the final output of the solution in an object called vals. Here we take advantage of the power of the computer to generate solutions over a very long span of time, just to ensure that solutions have equilibrated. Inspecting plots of solutions against time (as in Fig. 12) offers some explanation, as the dynamics of the two recovered groups are especially slow.

matplot(zetas, vals[, c(’S’, ’V’, ’I’, ’R’, ’N’)],    type=’l’, lty=1, lwd=2, col=cols,    xlim=c(0, 0.2), ylim=c(0, 1000), xlab=’Vaccination    rate’, ylab=’Population steady-state values’, las=1) legend(’topleft’, c(’Susceptible’, ’Vaccinated’,    ’Infectious’, ’Treated’, ’Natural’),    col=cols, lty=1, lwd=2, bg=’white’)