Exact closed-form solution of a modified SIR model

The paper states and proves a closed-form solution for the modified SIR system Ṡ = −βSI/(S+I), İ = βSI/(S+I) − αI, Ṙ = αI, giving S(t), I(t), R(t) in terms of τ=(β−α)−1 (its Eq. (3)) and derives it via a Hamiltonian/Poisson–Casimir reduction leading to a logistic ODE for a transformed variable B(t) (Eqs. (11)–(20)) . The candidate solution independently reduces the system by setting E=I/(S+I) and Z=S+I to obtain E′=(β−α)E(1−E) (logistic) and Z′/Z=−αE, then reconstructs S and I, recovering exactly the same formulas for S(t), I(t), R(t). Notably, the paper’s B(t) equals 1−E(t), so both arguments hinge on the same logistic reduction but follow different routes (Casimir transformation vs. direct ratio), and they agree on initial data, dynamics, and normalization S+I+R≡1 .