Leslie and Gardner classified H. pertussis cultures into four types and established an experimental basis for the development of an effective vaccine.

"... the Leslie matrix is a discrete, age-structured model of population growth that is very popular in population ecology.... The Leslie matrix (also called the Leslie model) is one of the most well known ways to describe the growth of populations (and their projected age distribution), in which a population is closed to migration, growing in an unlimited environment, and where only one sex, usually the female, is considered.

"The Leslie matrix is used in ecology to model the changes in a population of organisms over a period of time. In a Leslie model, the population is divided into groups based on age classes. A similar model which replaces age classes with ontogenetic stages is called a Lefkovitch matrix,^[1] whereby individuals can both remain in the same stage class or move on to the next one. At each time step, the population is represented by a vector with an element for each age class where each element indicates the number of individuals currently in that class" (Wikipedia article on Leslie Matrix, accessed 03-2018).