  When modelling metapopulation dynamics, the influence of a single patch on
the metapopulation depends on the number of individuals in the patch. Since
there is usually no obvious natural upper limit on the number of individuals in
a patch, this leads to systems in which there are countably infinitely many
possible types of entity. Analogous considerations apply in the transmission of
parasitic diseases. In this paper, we prove central limit theorems for quite
general systems of this kind, together with bounds on the rate of convergence
in an appropriately chosen weighted $\ell_1$ norm.
