Biology Reference
In-Depth Information
0
10
20
30
40
50
60
70
Age (years)
FIGURE 7.6
An immigrationedeath model fitted by trajectory matching to baseline
Ascaris lumbricoides
worm burdens from a community in India.
19
Small gray circles
represent individual worm counts and the large squares represent the mean worm burden
in the age categories: 1
e
e
e
e
e
e
e
e
e
þ
3.5; 4
5; 6
7; 7.5
10.5; 11
13; 14
20; 21
26; 27
35; 36
50; 51
e
years. The thick solid line is the fitted immigration
death model with per capita worm
1 year
1
, and age, a, -dependent force of infection (FOI),
mortality rate,
m ¼
L
(a), given by an
exponentially damped linear model
240
of the form
L
(a)
¼
(
b
1
a
e b
0
) exp(
eb
2
a)
þ b
0
. The
(a) are such that the FOI at birth is 0 year
1
,
properties of
L
L
(0)
¼
0; the initial (and
b
0
b
2
þ b
1
year
2
, and the long-term
maximum) instantaneous rate of increase in the FOI is
b
0
year
1
. The thin dashed lines denote 95% confidence intervals
around the fitted model. The model was fitted by maximum likelihood to the individual
data, assuming a negative binomial distribution with constant overdispersion parameter k.
Maximum likelihood estimates to 2 significant figures are as follows:
residual FOI as a
is
/N
4.0 year
1
;
b
0
¼
0.17 year
2
;
5.0 year
1
; k
b
1
¼
b
2
¼
¼
0.98.
baseline age-worm burden data one could calculate the mean worm
burden in different age groups and fit the model trajectory by least
squares, weighted by the sample size of each age group.
To fit a model to individual data, other distributional properties of the
data must be either assumed or derived from the dynamic model. The
former approach remains firmly within the domain of (traditional)
empirical inference and is, in essence, analogous to trajectory matching
using individual data (
Figure 7.6
). Revisiting the previous example, rather
than grouping the worm burden data by age groups and calculating the
mean, one could fit directly to the individual data assuming a negative
