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This is the SIR model with waning immunity. The mass-action interaction term $\beta IS$ transfers people from the susceptible compartment to the infected compartment, $\nu I$ transfers infectives to the recovered compartment, and $fR$ returns recovered people to susceptibility as immunity wanes. Adding the equations givesso the total population $N$ is conserved.

When $f=0$ and infection is rare, $S\simeq N$ andThus $I$ initially decays if $\beta N<\nu$. Equivalently, the basic reproduction number is ${\mathcal{R}}_0=\beta N/\nu <1$, so the epidemic invasion threshold is not crossed and no epidemic occurs.

The susceptible compartment retains $S_t$ individuals, gains $B$ recruits per generation and loses $\alpha S_t{I}_t$ through infection. The mass-action interaction term represents new infections generated by encounters between susceptible and infectious individuals. Those new cases form the next infectious compartment, $I_{t+1}$; the current infectious generation leaves that compartment after one time step.

The initial conditions $S_0,I_0$ specify the starting susceptible and infectious counts. The parameter $\alpha$ is the transmission coefficient in this deterministic approximation, while $B$ supplies new susceptible individuals. This is the discrete susceptible-infective model with recruitment.