  We develop a model of infection spread that takes into account the existence
of a vulnerable group as well as the variability of the social relations of
individuals. We develop a compartmentalized power-law model, with power-law
connections between the vulnerable and the general population, considering
these connections as well as the connections among the vulnerable as parameters
that we vary in our tests. We use the model to study a number of vaccination
strategies under two hypotheses: first, we assume a limited availability of
vaccine but an infinite vaccination capacity, so that all the available doses
can be administered in a short time (negligible with respect to the evolution
of the epidemic). Then we assume a limited vaccination capacity, so that the
doses are administered in a time non-negligible with respect to the evolution
of the epidemic. We develop optimal strategies for the various social
parameters, where a strategy consists of (1) the fraction of vaccine that is
administered to the vulnerable population and (2) the criterion that is used to
administer it to the general population. In the case of a limited vaccination
capacity, the fraction (1) is a function of time, and we study how to optimize
it to obtain a maximal reduction in the number of victims.
