untitled

=−γI (11.2)

Here we assume γis a constant and that transmission is directly related to the

proportion of infected individuals in the population (I/N) times the size of the

susceptible population (S).

The rate of change of the recovered population is given by the rate of recovered

infected individuals. Thus:

=γI (11.3)

When does a disease become epidemic, that is, start to spread through a population?

The answer to this question depends on the net reproductive rate, R 0 , of the

pathogen. For microparasites R 0 is the average number of secondary infections pro-

duced by one infected individual, and for macroparasites it is the average number of

offspring per parasite that grow to maturity. If the parasite has two sexes it can also

be defined as the average number of daughters reaching maturity per adult female.

If R 0 is less than unity the initial inoculum of parasites will decay to extinction.

R 0 is not a constant for a parasitic species but is determined by the varying charac-

teristics of both the parasite and the host populations, particularly the density of the

host. The conditions leading to persistence of the infection are given by Anderson

and May (1986) and Anderson (1991) as the ratio of the rate at which new infec-

tives are generated (β) to the rate at which they are lost:

R 0 = (11.4)

An epidemic occurs if R 0 >1, meaning that more infectives are generated than are

lost. An epidemic stops when R 0 =1. We can stop an epidemic by vaccinating a pro-

portion, C, of the susceptible individuals. We can then reduce R 0 by

R 0 = (11.5)

The proportion to be vaccinated to prevent an epidemic (i.e. to keep R 0 =1) is:

C> 1 −= 1 − (11.6)

Thus, the proportion to be vaccinated is critically dependent on R 0. If R 0 =2 then 50%

of susceptibles must be vaccinated, if R 0 =10 then 90% must be vaccinated (Krebs 2001).

The relationship can be expressed also in terms of a threshold host density NTbelow

which the infection will die out:

NT=(α+b+γ)/β>N (11.7)

where b is the mortality rate of uninfected hosts.

1

R 0

γ

β

(1 −C)β

γ

β

γ

dK

dt

βSI

N

dI

dt

PARASITES AND PATHOGENS 181

11.3.2Thresholds of
infection and the
transmission
coefficient