Nonintegrability of the SEIR epidemic model

TL;DR

This paper proves that the SEIR epidemic model is nonintegrable in the Bogoyavlenskij sense, extending existing theories and systems to demonstrate its complex mathematical structure and the nonexistence of certain integrals.

Contribution

It extends Morales-Ramis theory and applies it to the SEIR model, proving its nonintegrability and differentiating it from the simpler SIR model.

Findings

SEIR model is Bogoyavlenskij-nonintegrable.

Extended the system to a six-dimensional framework.

Proved the non-elementary nature of the incomplete gamma function.

Abstract

We prove the nonintegrability of the susceptible-exposed-infected-removed (SEIR) epidemic model in the Bogoyavlenskij sense. This property of the SEIR model is different from the more fundamental susceptible-infected-removed (SIR) model, which is Bogoyavlenskij-nonintegrable. Our basic tool for the proof is an extension of the Morales-Ramis theory due to Ayoul and Zung. Moreover, we extend the system to a six-dimensional system to treat transcendental first integrals and commutative vector fields. We also use the fact that the incomplete gamma function $\Gamma(\alpha,x)$ is not elementary for $\alpha\not\in\mathbb{N}$, of which a proof is included.