= Solution
The deterministic <SIR model> follows the <ordinary differential equations>
$$
\boxed{\dot S=-\beta SI,\qquad\dot I=\beta SI-\gamma I,\qquad\dot R=\gamma I,\qquad(S(0),I(0),R(0))=(N,1,0).}
$$
Use $\beta\ge0$ and $\gamma>0$. The state space is $S,I,R\ge0$, $S+I+R=N+1$; 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.
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