# Kermack-McKendrick SIR model (Brown & Rothery, 1993)

The Kermack-McKendrick SIR model is defined as

    dS/dt = -beta*N*S
    dI/dt = beta*N*S - gamma*I
    dR/dt = gamma*I

This model consists of two reactions with the following per capita rates,

    transmission: beta
    recovery:     gamma

Load package

    library(GillespieSSA)

Define parameters

    parms <- c(beta=.001, gamma=.100)
    tf <- 100                                      # Final time
    simName <- "Kermack-McKendrick SIR"            # Name

Define initial state vector

    x0 <- c(S=500, I=1, R=0)

Define state-change matrix

    nu <- matrix(c(-1,0,1,-1,0,1),nrow=3,byrow=TRUE)

Define propensity functions

    a  <- c("beta*S*I", "gamma*I")

Run simulations with the Direct method

    set.seed(1)
    out <- ssa(
      x0 = x0,
      a = a,
      nu = nu,
      parms = parms,
      tf = tf,
      method = ssa.d(),
      simName = simName,
      verbose = FALSE,
      consoleInterval = 1
    )
    ssa.plot(out, show.title = TRUE, show.legend = FALSE)

Run simulations with the Explict tau-leap method

    set.seed(1)
    out <- ssa(
      x0 = x0,
      a = a,
      nu = nu,
      parms = parms,
      tf = tf,
      method = ssa.etl(),
      simName = simName,
      verbose = FALSE,
      consoleInterval = 1
    )
    ssa.plot(out, show.title = TRUE, show.legend = FALSE)

Run simulations with the Binomial tau-leap method

    set.seed(2) # for some reason, this does not work with seed = 1
    out <- ssa(
      x0 = x0,
      a = a,
      nu = nu,
      parms = parms,
      tf = tf,
      method = ssa.btl(),
      simName = simName,
      verbose = FALSE,
      consoleInterval = 1
    )
    ssa.plot(out, show.title = TRUE, show.legend = FALSE)

Run simulations with the Optimized tau-leap method

    set.seed(1)
    out <- ssa(
      x0 = x0,
      a = a,
      nu = nu,
      parms = parms,
      tf = tf,
      method = ssa.otl(),
      simName = simName,
      verbose = FALSE,
      consoleInterval = 1
    )
    ssa.plot(out, show.title = TRUE, show.legend = FALSE)