# Kermack-McKendrick SIR model

The Kermack-McKendrick SIR model (Brown, Rothery, et al. 1993) is defined as

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

Note that simulations of this model can generate in all zero propensity, if the first reaction is a recovery of the single ‘Infected’ individual.

Define parameters

    library(GillespieSSA2)
    sim_name <- "Kermack-McKendrick SIR model"
    params <- c(beta = .001, gamma = .1)
    final_time <- 100
    initial_state <- c(S = 500, I = 1, R = 0)

Define reactions

    reactions <- list(
      reaction("beta * S * I", c(S = -1, I = +1), name = "transmission"),
      reaction("gamma * I", c(I = -1, R = +1), name = "recovery")
    )

Run simulations with the Exact method

    set.seed(1)
    out <- ssa(
      initial_state = initial_state,
      reactions = reactions,
      params = params,
      final_time = final_time,
      method = ssa_exact(),
      sim_name = sim_name
    )
    plot_ssa(out)

Run simulations with the Explict tau-leap method

    set.seed(1)
    out <- ssa(
      initial_state = initial_state,
      reactions = reactions,
      params = params,
      final_time = final_time,
      method = ssa_etl(),
      sim_name = sim_name
    )
    plot_ssa(out)

Run simulations with the Binomial tau-leap method

    set.seed(2)
    out <- ssa(
      initial_state = initial_state,
      reactions = reactions,
      params = params,
      final_time = final_time,
      method = ssa_btl(),
      sim_name = sim_name
    )
    plot_ssa(out)

## References

Brown, David, Peter Rothery, et al. 1993. Models in Biology: Mathematics, Statistics and Computing. John Wiley & Sons Ltd.