A clarification of transmission terms in host-microparasite models: numbers, densities and areas

Transmission is the driving force in the dynamics of any infectious disease. A crucial element in understanding disease dynamics, therefore, is the ‘transmission term’ describing the rate at which susceptible hosts are ‘converted’ into infected hosts by their contact with infectious material. Recently, the conventional form of this term has been increasingly questioned, and new terminologies and conventions have been proposed. Here, therefore, we review the derivation of transmission terms, explain the basis of confusion, and provide clarification. The root of the problem has been a failure to include explicit consideration of the area occupied by a host population, alongside both the number of infectious hosts and their density within the population. We argue that the terms ‘density-dependent transmission’ and ‘frequency-dependent transmission’ remain valid and useful (though a ‘fuller’ transmission term for the former is identified), but that the terms ‘mass action’, ‘true mass action’ and ‘pseudo mass action’ are all unhelpful and should be dropped. Also, contrary to what has often been assumed, the distinction between homogeneous and heterogeneous mixing in a host population is orthogonal to the distinction between density- and frequency-dependent transmission modes.