The size of epidemics in populations with heterogeneous susceptibility.

We formulate and study a general epidemic model allowing for an arbitrary distribution of susceptibility in the population. We derive the final-size equation which determines the attack rate of the epidemic, somewhat generalizing previous work. Our main aim is to use this equation to investigate how properties of the susceptibility distribution affect the attack rate. Defining an ordering among susceptibility distributions in terms of their Laplace transforms, we show that a susceptibility distribution dominates another in this ordering if and only if the corresponding attack rates are ordered for every value of the reproductive number R0. This result is used to prove a sharp universal upper bound for the attack rate valid for any susceptibility distribution, in terms of R0 alone, and a sharp lower bound in terms of R0 and the coefficient of variation of the susceptibility distribution. We apply some of these results to study two issues of epidemiological interest in a population with heterogeneous susceptibility: (1) the effect of vaccination of a fraction of the population with a partially effective vaccine, (2) the effect of an epidemic of a pathogen inducing partial immunity on the possibility and size of a future epidemic. In the latter case, we prove a surprising '50% law': if infection by a pathogen induces a partial immunity reducing susceptibility by less than 50%, then, whatever the value of R0>1 before the first epidemic, a second epidemic will occur, while if susceptibility is reduced by more than 50%, then a second epidemic will only occur if R0 is larger than a certain critical value greater than 1.