# Kermack-McKendrick SIR model (Brown & Rothery, 1993) In GillespieSSA: Gillespie's Stochastic Simulation Algorithm (SSA)

```set.seed(1)
knitr::opts_chunk\$set(fig.width = 8, fig.height = 6)
```

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
```

```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)
```

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GillespieSSA documentation built on March 18, 2022, 7:55 p.m.