The SIR model is an Euler–Lagrange equation

We establish a variational formulation for Bailey’s susceptible–infected–removed epidemic model using the Jacobi last multiplier method. An explicit first-order Lagrangian is derived whose Euler–Lagrange equations reproduce the susceptible–infected subsystem, while the removed compartment is recovered through conservation of the total population. The autonomous character of the Lagrangian yields a nontrivial first integral, which provides a description of the epidemic phase curves. We also show that the reduced dynamics admits a noncanonical Hamiltonian formulation. These results reveal an underlying variational and geometric structure in a classical epidemiological model usually regarded as nonconservative.

Published in Examples and Counterexamples. The abstract is shown as published and remains the work of its authors; read the full article at the source.

doi.org/10.1016/j.exco.2026.100232