On a Pandemic Threshold Theorem of the Early Kermack-McKendrick Model with Individual Heterogeneity

A pandemic threshold theorem of the Kermack-McKendrick epidemic system with individual heterogeneity is proved from the definition of R_{<italic>0</italic>} by Diekmann, Heesterbeek, and Metz. The early Kermack-McKendrick epidemic model is extended to recognize individual heterogeneity, where the state variable indicates an epidemiological state or genetic, physiological, or behavioral characteristics such as risk of infection. With the basic reproduction number R_{<italic>0</italic>} for the heterogeneous population, the final size equation of the limit epidemic starting from a completely susceptible steady state at t = &minus∞ has a unique positive solution if and only if R_{<italic>0</italic>} > 1. The main result is that the positive solution of the final size equation gives the lower bound of the intensity of any epidemic starting from a host population composed of susceptible and a few infected individuals who spread on a noncompact domain of the trait variable.