  We introduce an extension to Kermack and McKendrick's classic
susceptible-infected-recovered (SIR) model in epidemiology, whose underlying
mechanism of infection consists of individuals attending randomly generated
social gatherings. This gives rise to a system of ODEs where the force of
infection term depends non-linearly on the proportion of infected individuals.
Some specific instances yield models already studied in the literature, to
which the present work provides a probabilistic foundation. The basic
reproduction number is seen to depend quadratically on the average size of the
gatherings, which may be helpful to understand how restrictions on social
gatherings affect the spread of the disease. We rigorously justify our model by
showing that the system of ODEs is the mean-field limit of the jump Markov
process corresponding to the evolution of the disease in a finite population.

Probability