Transmission

# Discrete-time SIR (Susceptible-Infectious-Recovered) difference equation model
S <- 999; I <- 1; R <- 0; N <- S + I + R
beta <- 0.3; gamma <- 0.1; days <- 100
out <- matrix(NA, nrow=days, ncol=3, dimnames=list(NULL, c("S", "I", "R")))

for (t in 1:days) {
  out[t, ] <- c(S, I, R)
  incidence <- (beta * S * I) / N
  recovery <- gamma * I
  S <- S - incidence
  I <- I + incidence - recovery
  R <- R + recovery
}

matplot(out, type="l", lty=1, lwd=2, col=c("blue", "red", "darkgreen"),
        xlab="Time (Days)", ylab="Prevalence", main="SIR Epidemic Trajectory")
legend("right", legend=colnames(out), col=c("blue", "red", "darkgreen"), lty=1, lwd=2)

The basic reproduction number ($R_0$) for this deterministic model is defined by $R_0 = \frac{\beta}{\gamma}$. Given the parameters $\beta = 0.3$ (effective contact rate) and $\gamma = 0.1$ (removal rate), $R_0 = 3.0$. Because $R_0 > 1$, the infectious disease will propagate through the susceptible population, creating a classic epidemic curve.This discrete-time recursive approach approximates the continuous-time ordinary differential equations (ODEs) of the Kermack-McKendrick formulation natively, bypassing the need for external numerical integration packages such as deSolve.