  Some basic aspects of the recently discovered phenomenon of local
immunodeficiency \cite{pnas} generated by antigenic cooperation in
cross-immunoreactivity (CR) networks are investigated. We prove that local
immunodeficiency (LI) that's stable under perturbations already occurs in very
small networks and under general conditions on their parameters. Therefore our
results are applicable not only to Hepatitis C where CR networks are known to
be large \cite{pnas}, but also to other diseases with CR. A major necessary
feature of such networks is the non-homogeneity of their topology. It is also
shown that one can construct larger CR networks with stable LI by using small
networks with stable LI as their building blocks. Our results imply that stable
LI occurs in networks with quite general topology. In particular, the
scale-free property of a CR network, assumed in \cite{pnas}, is not required.
