Same Resident Strains, Different Attractors: Opposite Local Growth Signs for a Rare Third Strain

Abstract

Most models assess whether a newly introduced pathogen strain can grow when rare by testing it against a steady endemic background. Resident strains, however, can have more than one long-term behavior under the same parameters. They may settle to a steady state or continue through recurring outbreaks, leaving different fractions of hosts with different infection histories. We study this possibility in a stylized immune-history model. For two resident strains, we identify a parameter point where a stable endemic equilibrium and a stable outbreak cycle are both locally attracting. Numerical continuation maps a Hopf branch and a branch of folds of periodic orbits that bound a candidate bistability region, organized locally by a generalized Hopf (Bautin) point. We then embed the resident model exactly in a 20-state, three-strain system. When strain 3 infectiousness and total removal are common across four host-history classes, rare growth is determined by a weighted sum of the corresponding uninfected host fractions. The equilibrium and cycle yield different weighted sums, so one fixed strain 3 parameter set grows near one resident attractor but declines near the other. A second parameter set gives the reverse ordering, and both patterns persist over open regions of invader parameter space. Sixty full-model simulations with progressively smaller inocula reproduce these linear predictions across cycle phases and numerical solvers. Thus, knowing which resident strains are present may not suffice to predict initial invasion; the realized resident attractor can also matter. These results concern local growth from rarity, not eventual establishment, long-term coexistence, or post-invasion dynamics.