An SIS model

Let $S$ be the number of susceptible individuals, and let $I$ be the number of infected individuals. For an SIS model, infected individuals return to the susceptible class on recovery because the disease confers no immunity against reinfection. (For the SIR model covered in lecture, recovered individuals instead pass to the class R upon recovery.) The simplest SIS model is given by

$$\frac{dS}{dt} = -\beta SI + \alpha I,$$ (1)

$$\frac{dI}{dt} = \beta SI - \alpha I.$$ (2)

Let's briefly explore the meaning of these terms.

• The $\beta SI$ term is understood as follows: An average infected individual makes contact sufficient to infect $\beta N$ others per unit time. Also, the probability that a given individual that each infected individual comes in contact with is susceptible is $S/N$. Thus, each infected individual causes $(\beta N)(S/N) = \beta S$ infections per unit time. Therefore, $I$ infected individuals cause a total number of infections per unit time of $\beta SI$.
• The $\alpha I$ term is even simpler to understand: $\alpha$ is the fraction of infected individuals who recover (and re-enter the susceptible class) per unit time.

We see that

$$\frac{d}{dt} (S+I) = 0,$$

so

$$S+I = N = {\rm constant}.$$

Here $N$ is the total population. Substituting $I=N-S$ into (2), we obtain

$$\frac{dI}{dt} = \beta I (N-I) - \alpha I$$

$$= (\beta N - \alpha) I - \beta I^2.$$ (3)

Solving $dI/dt=0$, we see that there are two possible equilibria for this SIS model, one with $I=0$ and the other with $I = N - \alpha/\beta$. Defining the basic reproductive number as

$$R_0 \equiv \frac{\beta N}{\alpha},$$

it can be shown that

• $R_0 < 1 \; \Rightarrow$ the equilibrium with $I=0$ is stable,
• $R_0 > 1 \; \Rightarrow$ the equilibrium with $I = N - \alpha/\beta$ is stable.