Agriculture Reference
In-Depth Information
Table 8.1. Summary of differential and integrated equations and their linearised forms for
common growth curve models used in plant disease epidemiology
Model
dy/dt
y
Linearising
transformation
⎛
⎞
K
[
]
(
)
(
)
r
M
K
−
y
K
1
−
B
exp
−
r
M
t
ln
Monomolecular
⎜
⎟
K
−
y
⎝
⎠
⎛
⎞
⎛
⎝
⎞
⎠
K
K
−
y
y
r
L
y
ln
⎜
⎟
Logistic
⎜
⎟
()
1
+
B
exp
−
r
L
t
K
K
−
y
⎝
⎠
⎡
⎤
⎛
⎞
y
K
[
]
[
]
()
− ln
y
()
()
r
G
y
ln
K
K
exp
−
B
exp
−
r
G
t
−
ln
−
ln
⎜
⎟
Gompertz
⎢
⎥
⎝
⎠
⎣
⎦
K
= maximum level of disease or asymptote of disease progress curve,
y
= disease at time
of observation,
B
= a parameter related to the level of initial disease or point of inflection,
r
= rate of disease increase and
t
= time.
The biological basis of these common growth models results from the fact that
these growth models share a common form of the differential equation:
dy
rfyg y
()
(
)
dt
=
•
1
−
(8.1)
•
where
dy/dt
is the temporal change in disease and
r
is a rate parameter. This
equation states that the rate of disease (
y
) change at time (
t
) over time is related to
the amount of diseased (
y
) and healthy susceptible area (
1-y
). In the above equation,
the maximum disease is assumed to be 1 (i.e. all healthy tissues will be eventually
diseased); otherwise we can use
K - y
to represent healthy susceptible tissues where
K
is a variable representing the maximum disease. The precise form of two functions
(
f
,
g
) may depend on biological characteristics of the pathosystem concerned. For
example, in the case of monocyclic diseases, since newly diseased tissue in the
current season will not produce new inoculum that may lead to new infections in the
same season,
f
(
y
) is omitted from the differential equation. Furthermore, if we
assume that
g
(
1-y
) =
1-y
, the resulting model is the monomolecular model. On the
other hand, for polycyclic diseases, as the rate of disease change is expected to be
dependent on both diseased and healthy susceptible tissues, the simplest model is a
logistic model where f(
y
) =
y
and g(
1-y
) =
1-y
. Similarly, in the Gompertz model
g(
1-y
) = ln(
1
) - ln(
y
) = -ln(
y
). Van der Plank used the exponential, monomolecular
and logistic models primarily as biological models in his analysis. However, it
should be pointed out that the biological nature of the pathosystem could not be
ascertained purely on the basis that one particular model was statistically more
appropriate than the other. This is particularly so since statistical criteria used to
