 Solution

The deterministic SIR model follows the ordinary differential equationsUse $\beta \ge 0$ and $\gamma >0$. The state space is $S,I,R\ge 0$, $S+I+R=N+1$; summing the three equations proves conservation of this population. Infection is supported only where $S>0$ and $I>0$, and recovery only where $I>0$. The rates vanish at the corresponding boundaries, preventing negative compartment sizes.

This convention has an unnormalized population infection rate $\beta SI$, as required by the printed threshold. A convention with infection rate $\beta SI/(N+1)$ would use a differently scaled parameter.