An epidemic with individual infectivities and susceptibilities

This paper derives approximations and bounds for the mean duration time of a simple epidemic in a population of individuals i = 1,…,N, each of whom has a specific susceptibility αi to infection, and a specific infectivity βi once infected, starting from an initial infective with infectivity βo. Some algebraic results are obtained for the mean duration E(T) of this epidemic. An inequality is proved and a conjecture formulated for E(T) compared with the mean duration of an epidemic with homogeneous susceptibility and infectivity rates equal to the averages of the αi and βi, respectively. Bounds for E(T) are found and illustrated in four different cases of {αi} and {βi}, for which simulations are provided. It is concluded that variable susceptibility is of greater effect than variable infectivity in changing the mean and variability of the duration T relative to its characteristics under homogeneous assumptions.