= Discrete susceptible-infective model with recruitment
{title2=$S_{t+1}=S_t+B-\alpha S_tI_t,\quad I_{t+1}=\alpha S_tI_t$}
This generation-based <compartmental model> replenishes the <susceptible compartment> by $B>0$ per step and transfers $\alpha S_tI_t$ individuals into the next infectious generation. Members of the current <infectious compartment> leave it after one step. In scaled variables $x=\alpha S$, $y=\alpha I$, $r=\alpha B$, the update is $x'=x+r-xy$, $y'=xy$.
Its only fixed point is $(S_*,I_*)=(1/\alpha,B)$. There is no finite disease-free fixed point, since susceptible recruitment continues when $I=0$. The <Jacobian matrix> at the fixed point is $M=\begin{pmatrix}1-r&-1\\r&1\end{pmatrix}$, with determinant one and trace $2-r$. For $0<r<4$, its two distinct eigenvalues have modulus one, so the linearized iterates are bounded and oscillatory. At $r=4$, the repeated eigenvalue $-1$ has a nontrivial <Jordan block>, making generic linearized iterates grow linearly. For $r>4$, one eigenvalue has modulus greater than one.
Thus $0<r<4$ gives neutral linear stability, not asymptotic stability. <Unit-modulus multipliers do not determine nonlinear stability>. Recruitment can restore susceptibility above $1/\alpha$ after a wave, allowing repeated infection growth; a closed <SIR model> without replenishment has a different susceptible balance.
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