Epidemiology: SIR & SIS Models
Epidemiology: SIR & SIS Models
Compartmental models divide population $N$ into Susceptible ($S$), Infectious ($I$), and Recovered ($R$) individuals. The SIR model applies to diseases conferring lasting immunity (measles); the SIS model applies where recovery returns individuals to the susceptible pool (many bacterial infections).
The basic reproduction number $\mathcal{R}_0=\beta N/\gamma$ determines epidemic fate: $\mathcal{R}_0>1$ means growth; $\mathcal{R}_0<1$ means extinction. Herd immunity requires vaccination coverage $p_c=1-1/\mathcal{R}_0$.
SIR Equations
$$\dot{S}=-\beta SI/N,\quad \dot{I}=\beta SI/N-\gamma I,\quad \dot{R}=\gamma I.$$ $S+I+R=N$ is conserved at all times.
Epidemic Threshold
An epidemic grows iff $\mathcal{R}_0>1$. Peak infection at $S^*=N/\mathcal{R}_0$; final size satisfies $r_\infty=-\mathcal{R}_0^{-1}\ln(1-r_\infty)$.
Example 1
Measles $\mathcal{R}_0\approx15$. What vaccination coverage achieves herd immunity?
Solution: $p_c=1-1/15\approx93.3\%$.
Example 2
In the SIS model, find the endemic equilibrium $I^*$.
Solution: $I^*=N(1-1/\mathcal{R}_0)$ for $\mathcal{R}_0>1$; otherwise $I^*=0$.
Practice
- How does $\mathcal{R}_0$ differ from the effective reproduction number $\mathcal{R}_t$?
- Why does a SIR epidemic end before all susceptibles are infected?
- Write the SEIR equations adding a latent (exposed) class.
- Find the herd-immunity threshold for $\mathcal{R}_0=3$.
Show Answer Key
1. $\mathcal{R}_0$ is the expected number of secondary infections from one infected individual in a fully susceptible population. $\mathcal{R}_t = \mathcal{R}_0 \cdot S(t)/N$ accounts for the current fraction of susceptibles, decreasing as immunity builds up.
2. As infected individuals recover and become immune, $S(t)$ decreases. When $S < N/\mathcal{R}_0$, $\mathcal{R}_t < 1$ and new infections decline faster than recoveries. The epidemic ends with some susceptibles remaining because the effective reproduction number drops below 1.
3. $dS/dt = -\beta SI/N$, $dE/dt = \beta SI/N - \sigma E$, $dI/dt = \sigma E - \gamma I$, $dR/dt = \gamma I$. Here $1/\sigma$ is the mean latent period and $1/\gamma$ is the mean infectious period.
4. Herd immunity threshold $= 1 - 1/\mathcal{R}_0 = 1 - 1/3 = 2/3 \approx 67\%$. At least 67% of the population must be immune to prevent sustained transmission.