Hidden Oscillations: How Minimal Immunological Interactions Destabilize Two-Strain Epidemic Models

Abstract

Abstract. The foundational two-strain susceptible infectious recovered (SIR) model with lifelong partial cross-immunity has been analytically proven to be universally stable under weak, nearly symmetric cross-immunity. However, numerical observations in closely related models exhibit complex, oscillatory behavior in this exact parameter regime. This raises a fundamental question regarding the robustness of the theoretical consensus: does this stability hold beyond the minimal model? Or is there perhaps an intrinsic oscillatory mechanism driving the dynamics across these extended models, only to mathematically degenerate, and thus cease to exist, in the strict limit of the minimal model? We demonstrate a third reality. The minimal model does give rise to oscillations under weak, nearly symmetric cross-immunity: the accepted stability paradigm is actually an artifact of a critical breakdown in standard asymptotic methods. We show that this breakdown is driven by a previously overlooked mathematical degeneracy: the leading-order coalescence of Jacobian eigenvalues at the coexistence equilibrium. By developing a rigorous singular perturbation approach that resolves this degeneracy, we analytically derive the precise parameter boundary where stability is lost. This result overturns previous analytical guarantees, confirming that the classical minimal framework supports sustained oscillations and demonstrating that subtle immunological shifts, such as those characteristic of antigenic drift, are sufficient to trigger self-sustained outbreaks. Relevance to Life Sciences. This work identifies a novel, intrinsic mechanism for generating complex dynamics in disease systems such as seasonal influenza, where cross-protection is often weak. Previous theoretical frameworks posited that strong cross-immunity or significant asymmetry between strains was necessary to drive oscillations in these systems. Our findings correct this view by demonstrating that self-sustained outbreaks can arise in those systems even when strains interact weakly. Rather than minimal immunological interactions being inherently stabilizing, we reveal that an underlying oscillatory mechanism exists within the minimal framework but was mathematically masked by standard approximations. The discovery of this hidden oscillatory regime provides a rigorous new theoretical basis for understanding the emergence of complex dynamics in multi-strain pathogens, without requiring external seasonal forcing, strong competition, or additional structural model complexity. Mathematical Content. We identify the driver of the instability as a previously overlooked mathematical degeneracy: the leading-order coalescence of Jacobian eigenvalues at the coexistence equilibrium. The prior mathematical guarantee of stability stems directly from the failure of standard regular perturbation theory to resolve this critical transition. Specifically, this algebraic degeneracy causes the high-order correction terms of the standard asymptotic expansion to diverge. We resolve this breakdown by developing a tailored singular perturbation approach that incorporates a fractional scaling to correctly capture the system dynamics near the coalescing region. This analytical refinement not only overturns the asymptotic stability guarantees found in prior literature but also provides a uniform asymptotic approximation for the eigenvalues that is valid across the entire parameter space, effectively bridging the gap between the standard and singular scaling regimes.