🤖 AI Summary
This work presents a formal verification in Isabelle/HOL of the global qualitative properties of the SIR epidemic model’s differential equations, including existence and uniqueness of solutions, non-negativity invariance, conservation of total population, monotonicity, and the threshold condition for infection growth. Building upon the Picard–Lindelöf flow framework from the Archive of Formal Proofs, the authors establish—for the first time—a complete formal bridge from local flows to global forward solutions and develop a reusable library of scalar compartment lemmas. By integrating the existence–uniqueness theorem, compactness arguments, conserved quantity analysis, and sign analysis of derivatives, all key properties are rigorously proved with no unproven assumptions, thereby providing a scalable foundation for the formal verification of epidemiological models.
📝 Abstract
We present a mechanically checked Isabelle/HOL bridge from the Picard-Lindelof flow infrastructure in the Archive of Formal Proofs (AFP) to selected qualitative facts for the mass-action, closed-population SIR epidemic ODE. The epidemiological facts are classical; the contribution is reusable theorem infrastructure connecting the AFP local-flow construction to global forward existence, uniqueness, forward invariance of the nonnegative orthant, conservation, monotonicity, the Kermack-McKendrick conserved phase-plane relation, compartment bounds, and threshold-ratio conditions for infectious growth and monotonicity.
The proof first establishes sign and conservation facts for local AFP flow segments, then uses the conserved nonnegative simplex as the compactness witness for extending the flow to all forward times. The finite-interval qualitative facts are then transferred to the unique AFP forward flow on arbitrary intervals [0,b] with b>0, so the results apply to the constructed Isabelle/AFP SIR solution rather than to an assumed trajectory.
The reusable layer provides homogeneous-linear scalar compartment lemmas for equations X'(t)=f(t)X(t), derivative-sign monotonicity, three-compartment conservation, and an SIR transfer bridge to the AFP flow infrastructure. We do not formalize stability, final-size, or asymptotic theory. The accompanying Isabelle artifact builds with Isabelle 2024 and AFP 2024 and contains no sorry or oops proof placeholders.