  We introduce a new type of point process model to describe the incidence of
contagious diseases. The model is a variant of the Hawkes self-exciting process
and exhibits similar clustering but without the restriction that the component
describing the contagion must remain static over time. Instead, our proposed
model prescribes that the degree of contagion (or productivity) changes as a
function of the conditional intensity; of particular interest is the special
case where the productivity is inversely proportional to the conditional
intensity. The model incorporates the premise that when the disease occurs at
very low frequency in the population, such as in the primary stages of an
outbreak, then anyone with the disease is likely to have a high rate of
transmission to others, whereas when the disease is prevalent in the
population, then the transmission rate is lower due to human mitigation actions
and prevention measures and a relatively high percentage of previous exposure
in the total population. The model is said to be recursive, in the sense that
the conditional intensity at any particular time depends on the productivity
associated with previous points, and this productivity in turn depends on the
conditional intensity at those points. Some basic properties of the model are
derived, estimation and simulation are discussed, and the recursive model is
shown to fit well to historic data on measles in Los Angeles, California, a
relevant example given the 2017 outbreak of this disease in the same region.
