Time-periodic solutions of an SIS epidemic model
Abstract
We prove the existence of strictly positive, nonstationary time-periodic solutions for the spatially heterogeneous SIS epidemic model with standard incidence. On a bounded interval with Neumann boundary conditions, we construct smooth, time-independent transmission and recovery rates satisfying β(x)>γ(x)>0, together with diffusion coefficients dI>dS>0, such that periodic solutions coexist with the unique endemic equilibrium for every prescribed positive total population. Thus, pointwise dominance of transmission over recovery does not ensure convergence to equilibrium. Our approach combines an inverse construction of an unstable endemic equilibrium with a coefficient path that preserves both the equilibrium and the pointwise inequalities. A change in the number of unstable eigenvalues along this path yields small-amplitude periodic solutions through a center-manifold reduction and a generalized Hopf bifurcation argument. The proof uses an AI-assisted construction of an unstable endemic equilibrium under the pointwise condition β>γ>0.
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BibTeX
@misc{li2026timeperiodic,
title = {{Time-periodic solutions of an SIS epidemic model}},
author = {Fang Li and Pengfei Song and Kexin Zhang and Maolin Zhou},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.29062},
eprint = {2610.29062},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.29062},
}
RIS
TY - UNPB
TI - Time-periodic solutions of an SIS epidemic model
AU - Fang Li
AU - Pengfei Song
AU - Kexin Zhang
AU - Maolin Zhou
PY - 2026
DA - 2026/10/21
PB - arXiv
DO - 10.48550/arxiv.2610.29062
UR - https://arxiv.org/abs/2610.29062
AB - We prove the existence of strictly positive, nonstationary time-periodic solutions for the spatially heterogeneous SIS epidemic model with standard incidence. On a bounded interval with Neumann boundary conditions, we construct smooth, time-independent transmission and recovery rates satisfying β(x)>γ(x)>0, together with diffusion coefficients d_I>d_S>0, such that periodic solutions coexist with the unique endemic equilibrium for every prescribed positive total population. Thus, pointwise dominance of transmission over recovery does not ensure convergence to equilibrium. Our approach combines an inverse construction of an unstable endemic equilibrium with a coefficient path that preserves both the equilibrium and the pointwise inequalities. A change in the number of unstable eigenvalues along this path yields small-amplitude periodic solutions through a center-manifold reduction and a generalized Hopf bifurcation argument. The proof uses an AI-assisted construction of an unstable endemic equilibrium under the pointwise condition β>γ>0.
ER -