Stochastic SIR Epidemic Model

Definition

The Stochastic SIR (Susceptible-Infected-Removed) Epidemic Model is a bivariate continuous-time Markov chain where recovered individuals have permanent immunity.

• $N$: constant total population ($S(t)+I(t)+R(t)=N$)
• $S(t)$: number of susceptibles
• $I(t)$: number of infectives
• $R(t)$: number of removed (recovered/immune)
• $\beta$: contact (transmission) rate
• $\gamma$: recovery rate

State Space

The process $\{(S(t),I(t)):t\ge 0\}$ has state space $\{(i,j):i=0,\dots ,N;j=0,\dots ,N-i\}$. $R(t)=N-S(t)-I(t)$.

Transition Rates

$$\begin{array}{rl}{P}_{(i,j)\to (i-1,j+1)}(\Delta t)& =\frac{\beta ij}{N}\Delta t+o(\Delta t)\text{(infection)}\\ P_{(i,j)\to (i,j-1)}(\Delta t)& =\gamma j\Delta t+o(\Delta t)\text{(recovery)}\end{array}$$

Interpretation

Unlike the SIS model, the SIR model is not a simple birth-death process — it is bivariate. The state $(i,j)$ tracks both susceptibles and infectives. All states with $j=0$ (no infectives) are absorbing — the epidemic ends.

Key Properties

• Not a standard B/D process: The absorbing states are $\{(i,0):i=0,\dots ,N\}$ (epidemic always ends)
• Basic Reproduction Number: ${\mathcal{R}}_0=\beta /\gamma$, same as SIS
• Final Epidemic Size: $S(\infty )$ satisfies the implicit equation:

$S(\infty )=S(0)\exp \left[-\frac{{\mathcal{R}}_0(N-S(\infty )+I(0))}{N}\right]$

• Expected Duration: Computed from the submatrix $Q$ of the generator, solving $Q\mathbf{\tau }=-1$ over transient states

Deterministic Analog

$\frac{dS}{dt}=-\frac{\beta SI}{N},\frac{dI}{dt}=\frac{\beta SI}{N}-\gamma I,\frac{dR}{dt}=\gamma I$

If ${\mathcal{R}}_{0}S(0)/N\le 1$, $I(t)$ decreases monotonically. If ${\mathcal{R}}_{0}S(0)/N>1$, $I(t)$ first increases to a maximum, then decreases to zero (the epidemic “burns through” the susceptibles).

Related

• Stochastic SIS Epidemic Model
• Birth and Death Process
• Continuous-Time Markov Chain
• Probability of Population Extinction