Bovine tuberculosis

The Epidemiology of Mycobacterium bovis Infection in Cattle 49

transmission is a common (McCallum et al.,
2001), but controversial (Begon et al., 2002),
assumption for epidemic models in animal popu-
lations. The controversy comes from the funda-
mental assumption at the heart of most epidemic
models that the rate of transmission is mediated
by the direct contact, or interaction of suscepti-
ble and infectious individuals. Given this
assumption, at least for a well-mixed, homoge-
nous population, we would expect the rate of
encounters between individuals to scale with the
number of susceptible individuals and the pro-
portion of infective animals. In this situation, so-
called frequency-dependent transmission, the
value of R 0 scales independently of herd size:

R 0 = βL

Following Francis’s observation, epidemic
models for bovine tuberculosis typically assume
density-dependent transmission where the rate
of infection depends on both the numbers of
susceptible and infectives and R 0 increases lin-
early with herd size (H):

R 0 = βLH

While this assumption is more consistent with
the empirical patterns of prevalence for bovine
tuberculosis, it raises technical concerns not the

least of which is that R 0 potentially increases

without bound with the size of the population. It

is also unclear what a herd size dependence on

rate of transmission means mechanistically as

the empirical association could be confounded

by differences in husbandry, demography or

even the balance of direct and indirect transmis-

sion acting upon herds.

This distinction in how transmission is

modelled is not just a technical matter, but has

fundamental consequences for the likely efficacy

of controls. If transmission is density dependent,

then we would expect controls, in particular

vaccination, to become increasingly less effec-

tive with the size of the herd. Different authors

have chosen to use either density-dependent

transmission (Barlow et al., 1997; Kao et al.,

1997; O’Hare et al., 2014) or frequency-

dependent transmission (Fischer et al., 2005;

van Asseldonk et al., 2005) with potentially

profound implications for their conclusions

of the relative merits of alternative control

interventions.

Conlan and colleagues attempted to resolve

this controversy by introducing a non-linearly

density-dependent term where R 0 scales with

an additional parameter q, that measures the

strength of density dependence:

Fig. 4.2. (a) Average percentage of animals reacting to the tuberculin skin test with increasing herd
size. (b) Estimated reproduction ratio based upon the apparent average prevalence of tuberculin reactors
within these coarse herd size ranges. Adapted from Francis, J. (1947) Bovine Tuberculosis, Table XI.

A

20

10

1–20 21–40

Herd size (Range)

41–70

% Reactors

B

1.1

1.2

1.3

1–20 21–40

Herd size (Range)

41–70

Estimated R

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