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

Abstract

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.

The Dynamics of Murid Gammaherpesvirus 4 Within Wild, Sympatric Populations of Bank Voles and Wood Mice

“…Thus, the dynamics of MuHV-4 and cowpox differ considerably within this multihost population. Cowpox dynamics are consistent with a seasonal flush of susceptible animals during the breeding season and some density dependence in transmission rates (Begon et al, 1998). In contrast, MuHV-4 dynamics show no strong seasonal or density-dependent patterns and infections appear to be more concentrated within one section of the population (old males).…”

“…Consequently, MuHV-4 is not strongly influenced by either the influx of young animals during the breeding season or overall population density. If transmission in such infections is density-dependent, infection risk is likely to be related to the density of the high-risk group, rather than overall population density (Begon et al, 2002).…”

Effect of strain and inoculation dose of classical swine fever virus on within-pen transmission

“…The pigs inoculated with the Zoelen strain did not transmit the virus to the contact pigs at all. This was not expected since this strain was isolated in 1975 from foetuses of a ''carrier sow'' during a field outbreak 2 . In several experimental studies 3 [28], virus titres in blood and secretions and excretions were always low, which could explain the inability of this strain to transmit to contact pigs in the present experiment.…”

“…1 Pigs from the low inoculation dose trial (10 2 TCID 50 ) were not included in the table since none of the pigs were infected. 2 Start and end of the infectious period coincided with virus excretion in faeces. 3 Start and end of the infectious period coincided with virus excretion in faeces and/or oropharyngeal fluid.…”

“…In a SEIR model, the transmission dynamics of infectious diseases between individuals are described by the change in the number of susceptible (S), exposed (E), infectious (I), recovered or removed (dead) (R) and total number (N) of animals. Assuming that the number of contacts per animal is independent of the herd size (frequency dependent mixing) [2], susceptible pigs become infected with a rate dS t /dt = ÀbAES t AEI t /N t . In this formula, b is the transmission rate and can be interpreted as the average number of new infections for a typical infectious animal in a susceptible population per unit of time.…”