The Volterra lattice, Abel's equation of the first kind, and the SIR epidemic models

The Volterra lattice, when imposing non-zero constant boundary values, admits the structure of a completely integrable Hamiltonian system if the system size is sufficiently small. Such a Volterra lattice can be regarded as an epidemic model known as the SIR model with vaccination, which extends the celebrated SIR model to account for vaccination. Upon the introduction of an appropriate variable transformation, the SIR model with vaccination reduces to an Abel equation of the first kind, which corresponds to an exact differential equation. The equipotential curve of the exact differential equation is the Lambert curve. Thus, the general solution to the initial value problem of the SIR model with vaccination, or the Volterra lattice with constant boundary values, is implicitly provided by using the Lambert W function.