An Algebraic Approach to Analyze an MSEIR Epidemic Model

This study introduces a fully algorithmic framework for analyzing the dynamics of an MSEIR (Maternal, Susceptible, Exposed, Infectious, Recovered) epidemic model. Unlike traditional approaches that rely on heuristic, model-specific Lyapunov functions for global stability or manual algebraic manipulations for finding equilibria, we employ computational algebraic geometry. By computing Gröbner bases, the coupled nonlinear ordinary differential equations are systematically reduced to a triangular form, guaranteeing the exhaustive determination of all equilibria. Local stability is rigorously established via the Liénard–Chipart criterion applied to the Jacobian's characteristic polynomial, bypassing explicit eigenvalue extraction. Furthermore, we demonstrate a transcritical bifurcation where the disease-free and endemic equilibrium exchange stability. Finally, the global asymptotic stability of the disease-free equilibrium for R0