Stochastic SIS Epidemic Model

Definition

The Stochastic SIS (Susceptible-Infected-Susceptible) Epidemic Model is a birth and death process where individuals recover without immunity.

• $N$: constant total population ($S(t)+I(t)=N$)
• $X(t)$: number of infectives at time $t$, state space $\{0,1,\dots ,N\}$
• $\beta$: contact (transmission) rate
• $\gamma$: recovery rate

Birth and Death Rates

${\lambda }_i=\frac{\beta i(N-i)}{N},{\mu }_i=\gamma i$

with ${\lambda }_0={\mu }_0=0$ and ${\lambda }_N=0$ (state $0$ is absorbing — disease extinction).

Interpretation

A “birth” is a new infection (susceptible → infective), proportional to the number of contacts between infectives and susceptibles. A “death” is a recovery (infective → susceptible).

Basic Reproduction Number

${\mathcal{R}}_0=\frac{\beta }{\gamma }$

• If ${\mathcal{R}}_0<1$: disease dies out quickly
• If ${\mathcal{R}}_0>1$: an epidemic can occur, but extinction is still certain (${\lim }_{t\to \infty }{p}_0(t)=1$); expected extinction time grows rapidly with ${\mathcal{R}}_0$

Quasistationary Distribution

Since state $0$ is absorbing, the quasistationary distribution (conditional on non-extinction) is:

$q_i^{\ast }=\frac{{\mathcal{R}}_0^i\left(\genfrac{}{}{0ex}{}{N}{i}\right)}{{\sum }_{j=1}^N{\mathcal{R}}_0^j\left(\genfrac{}{}{0ex}{}{N}{j}\right)},i=1,\dots ,N$

Deterministic Analog

The deterministic SIS model is logistic: $\frac{dI}{dt}=\lambda I(1-I/K)$ with $\lambda =\beta -\gamma$ and $K=N(1-1/{\mathcal{R}}_0)$.

The equilibrium $I=0$ is stable if ${\mathcal{R}}_0<1$. The endemic equilibrium $I=N(1-1/{\mathcal{R}}_0)$ exists and is stable if ${\mathcal{R}}_0>1$.

NOTE

The stochastic mean of infectives is always less than the deterministic equilibrium, and ultimate extinction is certain in the stochastic model but not in the deterministic one.

Related

• Birth and Death Process
• Quasistationary Probability Distribution
• Stochastic SIR Epidemic Model
• Probability of Population Extinction