Global stability of the coexistence endemic equilibrium in the two-strain model

Establish global stability of the endemic coexistence equilibrium φ* in the two-strain compartmental epidemic model (system (1): S, I1, I2, J1, J2, R1, R2, R3) under the parameter regime guaranteeing its existence and local stability (σ12>0 and R2 > (γ1+δ1+μ)/(γ1η2σ12) with sufficiently large β1>γ1). Specifically, determine whether, for all admissible initial conditions in the feasible region, trajectories converge to φ* rather than to single-strain equilibria or other long-term behaviors.

Background

The paper introduces and analyzes a two-strain epidemic model with compartments S, I1, I2, J1, J2, R1, R2, and R3, allowing for partial cross-immunity, waning immunity, and modified infectivity in secondary infections. The authors prove Theorem 2.2, which guarantees the existence, uniqueness, and local stability of a coexistence endemic equilibrium φ* when one strain is highly transmissible (large β1, hence large R1), partial cross-immunity from strain 1 to strain 2 holds (σ12>0), and strain 2’s reproduction number satisfies R2 > (γ1+δ1+μ)/(γ1η2σ12).

While numerical simulations suggest convergence to φ* from initial conditions far from equilibrium, the paper explicitly states that only local stability has been proved. The global stability of φ*—i.e., whether all admissible initial conditions converge to the coexistence equilibrium—remains an open question and is highlighted as a key direction for future mathematical analysis.