 Solution

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.