Epidemiological significance of the 3-dimensional Volterra lattice with constant boundary values

Determine the epidemiological interpretation and significance of the three-dimensional Volterra lattice with constant boundary condition (a0 = α, aM+1 = β) as defined by condition (III) with system size M = 3, which is completely integrable but lacks a clear role as an epidemic model, in contrast to the M = 2 case that corresponds to the SIR model with vaccination (SIRv).

Background

The paper shows that imposing constant boundary values on the Volterra lattice yields completely integrable dynamics for small system sizes. For M = 2, the resulting two-dimensional system coincides with the SIR model with vaccination (SIRv), and the authors provide an explicit integrability framework via Abel’s equation of the first kind and the Lambert W function.

For M = 3, the authors also obtain complete integrability (e.g., under α = β in the three-dimensional case) and present conserved quantities. They further show a reduction to a two-dimensional integrable subsystem using a conserved quantity, which approaches the classical SIR model as a parameter tends to zero. However, despite these structural results, the authors explicitly state that the epidemiological meaning of this higher-dimensional Volterra lattice model has not been clarified.

This gap contrasts with the clear epidemiological interpretation of the M = 2 case (SIRv), motivating the explicit open question about identifying or establishing the model’s significance in epidemic modeling for M = 3.