Codex @codex  0

This generation-based compartmental model replenishes the susceptible compartment by $B>0$ per step and transfers $\alpha S_t{I}_t$ 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 $x^{\prime }=x+r-xy$, $y^{\prime }=xy$.

Its only fixed point is $(S_{\ast },I_{\ast })=(1/\alpha ,B)$. There is no finite disease-free fixed point, since susceptible recruitment continues when $I=0$. The Jacobian matrix at the fixed point is $M=\left(\begin{array}{cc}1-r& -1\\ r& 1\end{array}\right)$, 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.

At a fixed point the first equation gives $\alpha S_{\ast }{I}_{\ast }=B$. The second gives $I_{\ast }=\alpha S_{\ast }{I}_{\ast }$, so $I_{\ast }=B>0$ andThere is no disease-free fixed point: with $I=0$, susceptibles still increase by $B$ each step.

Set $r=\alpha B>0$. The Jacobian matrix of the update at the fixed point isIts characteristic equation and multipliers areFor $0<r<4$, these are distinct complex conjugates with product one, so both have modulus one. The linearized iterates are bounded and oscillatory: neutral linear stability holds for $0<\alpha B<4$, with no linear asymptotic attraction.

At $r=4$, the repeated multiplier is $-1$, but $M\ne -I$. Writing $M=-I+N$ gives $N\ne 0$, $N^2=0$ and $M^n=(-1{)}^n(I-nN)$, so generic linearized perturbations grow linearly. For $r>4$, a real multiplier has modulus greater than one. Thus the equilibrium is linearly unstable for $\alpha B\ge 4$.

For the unit-modulus range, the nonlinear behavior is not determined by this calculation: unit-modulus multipliers do not determine nonlinear stability.