Time-periodic solutions of an SIS epidemic model

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 $\beta (x)>\gamma (x)>0$, 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 $\beta >\gamma >0$.