  We numerically address the stability analysis of linear age-structured
population models with nonlocal diffusion, which arise naturally in describing
dynamics of infectious diseases. Compared to Laplace diffusion, models with
nonlocal diffusion are more challenging since the associated semigroups have no
regularizing properties in the spatial variable. Nevertheless, the asymptotic
stability of the null equilibrium is determined by the spectrum of the
infinitesimal generator associated to the semigroup. We propose a numerical
method to approximate the leading part of this spectrum by first reformulating
the problem via integration of the age-state and then by discretizing the
generator combining a spectral projection in space with a pseudospectral
collocation in age. A rigorous convergence analysis proving spectral accuracy
is provided in the case of separable model coefficients. Results are confirmed
experimentally and numerical tests are presented also for the more general
instance.
