Agriculture Reference
In-Depth Information
(a) Host growth
Host growth becomes important if there is large change in the size of susceptible
host tissues relative to the duration of active epidemic development. Disease
incidence or intensity may apparently decline over time even though the absolute
disease level increases. For example, the proportion of leaves diseased may decline
if the rate of new leaves produced is greater than the rate of leaf infection. Ignoring
host growth may lead to underestimation or even negative values of the rate of
disease increase (Kushalappa and Ludwig, 1982). Several methods have been
proposed to incorporate the host growth into growth models.
Van der Plank (1963) proposed a correction for the rate of disease increase based
on exponential change in mass of susceptible host tissues such that
⎡
⎤
1
⎛
y
⎞
⎛
y
⎞
(8.2)
(
)
(
)
r
L
=
ln
⎜
⎜
⎝
2
⎟
⎟
⎠
−
ln
⎜
⎜
⎝
1
⎟
⎟
⎠
+
ln
H
−
ln
H
⎢
⎣
⎥
⎦
2
1
t
−
t
1
−
y
1
−
y
2
1
2
1
where
H
1
and
H
2
represent the host size at time 1 and 2, respectively. This empirical
method was judged to be satisfactory in a large comparative study (Kranz, 1975).
However, even this corrected rate may have a negative value as demonstrated for
coffee rust by Kushalappa and Ludwig (1982). To overcome this, these authors
proposed a new method to correct host growth in calculating the apparent infection
rate. Assuming a logistic disease increase, the corrected rate is
⎡
⎤
1
⎛
y
h
⎞
⎛
y
h
⎞
(8.3)
r
L
=
ln
⎜
⎜
⎝
2
2
⎟
⎟
⎠
−
ln
⎜
⎜
⎝
1
1
⎟
⎟
⎠
⎢
⎣
⎥
⎦
t
−
t
1
−
y
h
1
−
y
h
2
1
2
2
1
1
where
h
=
H/H
max
. In this model, no particular models were assumed for host
growth.
A more general approach to accommodate host growth is to use linked
differential/difference equations (LDE) (Jeger, 2000). This approach has been
widely adopted and is regarded as a standard approach in medical epidemiology and
modelling. In this approach, host growth and disease dynamics are described by
separate equations but linked together via the fact that the change in disease at time
t
is a function of the total susceptible host area at that time. For example, if
we assume that a logistic model can describe both host growth and disease
development, then
⎧
dY
H
−
Y
⎛
⎞
=
r
Y
⎜
⎝
⎟
⎠
⎪
⎪
⎨
D
dt
H
(8.4)
⎛
⎞
dH
K
−
H
⎪
⎩
⎜
⎜
⎝
⎟
⎟
⎠
=
r
H
H
⎪
H
dt
K
H
