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Further models for the distribution on pasture of infective larvae of the strongyloid nematode parasites of sheep
MATHEMATICAL
179
BIOSCIENCES
Further Models for the Distribution on Pasture of Infective Larvae of the Strongyloid Nematode Parasites of Sheep G. M. TALLIS Division of Mathematical C.S.I.R.O.
Newtown,
Statistics,
N.S. W., Australia
A. D. DONALD Division C.S.I.R.O.
of Animal Health, McMaster
Laboratory
Glebe, N.S. W., Australia
Communicated
by K. E. F. Watt
ABSTRACT This paper discusses a general approach to the problem of constructing useful models for the distribution on pasture of infective larvae of sheep nematode parasites. The work is related to earlier results obtained by the authors, and modifications are introduced as a result of practical experience with the original models. An explicit form for the function f(t), the probability that an egg reaches the infective larval stage in time t, is derived. The new version off(t) has components with direct biological interpretation.
INTRODUCTION
In earlier papers [5, 61, stochastic models were developed to describe the distribution on pasture of the infective larvae of some nematode parasites of sheep. These models were based on current knowledge of the distribution of fecal deposits in a paddock and on the assumption that the distribution of eggs between fecal deposits is Poisson. Subsequent experience [2, 41 has led to modifications in the original postulates and it seems appropriate at this stage to redevelop some of the previous results in the light of the present ideas. Studies of fecal distributions on paddocks of different sizes carrying varying numbers of sheep have emphasised that deposits are definitely Mathematical
Biosciences
7 (1970), 179-190
Copyright @ 1970 by American Elsevier Publishing Company, Inc.
180
G.
M. TALLIS
AND
A. D. DONALD
nonuniformly distributed over the total area. However, it seems possible to divide the paddock into subareas within which the distribution is, to a good approximation, uniform, although there may be considerable between area differences in concentration due to the grazing and resting habits of the flock. With this information in mind, it is the primary purpose of this paper to develop completely general distributions for the numbers of larvae in the various subareas of a paddock carrying S sheep. From these expressions certain reasonable simplifications will be introduced to make the formulae useful in practical investigations. Most of the argument is carried through using the general forms for the means and variances of the various processes. In the sequel we pay particular attention to the functionf(t), which is the probability that an egg dropped onto the pasture at time zero is in the infective larval stage on the herbage at time f. Explicit expressions for f(r) are derived from underlying biological hypotheses to give the formulae further interpretational value.
RESULTS
Distribution of the Total Number of Infective Larvae on Pasture. Let the paddock under consideration be of area A and suppose S sheep are introduced at zero time, T = 0. In accordance with the Introduction, the paddock is divided into k plots of area Aj, I:=, Aj = A. Within each plot the defecation pattern of the flock is such that fecal deposits are approximately uniformly distributed over the area. Initially we focus attention on plot j and sheep i. Let the number of eggs per fecal deposit for the ith sheep at T = x be a random variable with probability generating function, p.g.f., gi(s, x) with mean A(x) = A(l, x) and variance (T:(x) = g:l(l, x) + Ai
-
A;(X).
Now define the step function N,,(t) as the total number of fecal deposits associated with the ith sheep on plot j by T = t, and letf(t) be as defined in the Introduction. Then the p.g.f. for the number of live larvae at T = t emanating from a deposit dropped at T = x is Us,
x, t) = g,{[l
and the p.g.f. for the total Mathematical
Biosciences
7 (1970).
+f(t
number 179-190
- x)(s of live larvae
l)l, xl, on plot j due to the
DISTRIBUTION
OF INFECI-IVE
LARVAE
181
ith sheep at T = t is &(sj, t) = exp( /log
hi(sj, x, t) &V,,(X)].
(I)
0
Formula (1) assumes stochastic independence between the numbers of eggs in the deposits. Although this assumption can be made plausible by some mathematical argument, the details will not be presented since they are somewhat involved. The restriction of independence is, of course, easily removed at the cost of increasing the complexity of the expressions and the estimation procedures. Similarly, the p.g.f. for the number of larvae on plot j due to the S sheep is
+Csj,t>= eV( it Slog[Usj, x9 l)l dNij(xI)*
(2)
0
and the joint p.g.f. for all k plots is
If the prime interest is the total number of larvae on pasture, then the appropriate p.g.f., $(s, t) is obtained from (3) by setting sj = s for allj. Equations (I), (2), and (3) give the required expressions in the most general form, but they are of limited practical value since, for instance, the step functions Nij(t) must be known. Below, some reasonable simplifications are introduced to illustrate the use of the various p.g.f.‘s. However, before proceeding, general expressions for the means and variances of the various processes are obtained. Let L,(t) be the number of live larvae on plotj at T = t, then
E{Lj(t)} =
iilJk(x)f(t - X>dNij(x), 0
(4)
t
where Vi(x, t) = $(x)f’(t - x)‘+ &(x)f(t - x)[l -f
Ut) =
iLjCr),
j=l
E{L(t)Il
=~~lEILj(t)~9 .E
Muthenzaticol Biosciences 7 (1970), 179-190
182
G.
M. TALLIS AND A. D. DONALD
and V{L(f)} = i V{&(r)). j=l
In order to simplify
(4) to a manageable
approximation
set
2Ai(
SX(x)=
i=l
$ a:(x),
SS2(x)=
i=l
and Ni j(x) = n jx, where nj is the daily fecal output sheep; then
per sheep on plot j averaged
over all
t
E{Lj(t)}
N
Snj
s s
X(x)f(t
-
X)
dx,
0
(5)
t
V{Lj(t)}
N
Snj
7(x, t) dx,
0
where F(x, t) = ?(x)J’“(t
-
x) + i(x)f(r
- x)[l
-f(t
- x)].
By suitable sampling, estimates of the functions nj, x(x), and C?“(X) must be obtained for values of x in the range of interest. The integrals can then be approximated by numerical quadrature provided f(t) is known. The amount of work required to carry out the preceding estimation is considerable and, at best, tedious. Some further assumptions will now be made which should facilitate the application of (5). Let Z,(x) be the random variable associated with gi(s, x), and suppose that the weight of each fecal deposit, W, is a random variable. Then for fixed W = w let gi(s, x) be Poisson with parameter
A,(x) 1w = WY,(X). Then E{Zi(X) I w> = Vi(X) and hence A,(x) = E{Zi(X)} =
ECWYiW
and, similarly, c?(x) = I’{&(x)> Mathematical
= Yi(x)E(W
Biosciences7 (1970), 179-190
+ Y:(x)wo.
DISTRIBUTION
183
OF INFECTIVE LARVAE
In order to use (5) under the assumptions above it is necessary to estimate E(W) and V(W), B(W), and F(W) say, for the flock of sheep. Then, since y,(x) can be estimated for each sheep by standard methods, average values of A(x) and a”(x) are easily obtained at different time points. The integrals in (5) can then be approximated by numerical quadrature.
A Modelforf(t). own right. Consider
The functionf(t) is of some biological interest the elementary flow diagram in Fig. 1.
PASTURE HERBAGE
FECAL DEPOSIT
FIG. 1. Elementary
in its
flow diagram.
(i) q is the probability that an egg does not get to infective larva and p = 1 - q is the probability that it (ii) Given that an egg completes development to the the resulting larva migrates to the pasture, X, is the time ment and migration from fecal deposit to herbage. (iii) X2 is the length of life as an infective larva on
Suppose that X1 and X2 have distribution independently distributed, then
the herbage as an will. infective stage and taken for developthe herbage.
functions
a1 and Q2 and are
[l - @(t - x)] &&(x).
(6)
t f(t)
= p
s
The assumption of independence is not necessary, but a more general approach leads to complexities of estimation which are not pursued here. In order to establish (6) let A be the event that an egg never develops into a larva that completes the migration process to the grass. Thus Pr{A} = q and Pr{K} = p, where K is the complementary event to A. Define the set s = {x,,
x2;
0
Q
x1
G
2,
t
-
Mathematical
x1
Q
x21,
Biosciences
7 (1970), 179-190
G. M. TALLIS
184
AND
A. D. DONALD
then f(t)
= Pr{egg d eve 10 p s into an infective larva alive at time t 1A}q + Pr{egg develops into an infective larva alive at time I 1 A)p
t = p
s
[l - @z(t - x)] d@,(x).
0
Iff*(S)
is the Laplace
transform, f*(s)
L.T., off(t),
= PS-WWI
then
Ms>l,
-
where $:(s) is the L.T. of +i(x), the derivative to exist. From (7) it is easily verified that
(7)
of Qi(x) which is assumed
. J a0
pr =
t’f(t) dt
0
= pr ! i
p~‘,,u~~~_,/n ! (r +
1 -
n)!
?2=0
and also lim sf*(s)
= 0, implying
limf(r)
S-*0
(8)
= 0. In (8),
t+m co
Pi’
=
xn
&D,(x),
I 0
and it is interesting to notice that the area underf(r) equations specified by (8) are useful for fittingf(t) below.
isp@. to data,
The moment as illustrated
Fitting the f (t) Model to Empirical Data. Empirical estimates of the function f (t) have been obtained in the course of field ecological studies, which will be reported in detail elsewhere, on the free-living stages of Trichostrongylus colubriformis and Haemonchus contortus, two important nematode parasites of sheep. Donald [4] has given a preliminary account of some of this work, including the methods used. Briefiy, a known amount of sheep faeces containing parasite eggs is scattered on a small plot of pasture. Estimates are made initially of the total number of eggs of each species placed on the plot, and at weekly intervals the numbers of infective larvae present on the herbage are estimated from samples. Point estimates of the function f (t) at weekly intervals are obtained by dividing the Muthermtical
Biosciences 7 (1970),
179-190
DISTRIBUTION
OF INFECTIVE
185
LARVAE
numbers of infective larvae recovered at time t by the number of eggs exposed on the plot at t = 0. Beginning in June 1967, a fresh plot has been set up every four weeks and herbage sampling of each plot has continued until at Ieast three consecutive zero recoveries have been obtained For the present purpose of illustrating the fitting of the model, data for T. colubriformis from each of four plots have been selected as representative.
FIG. 2. Plots of the observed and fitted values of,f(r) for the four groups. -I,
.O& -03 c -02 .ol
0
2
12
lr,
16
18
20
22
24
FIG. 2b Mathematical
Biosciences
7 (1970),
179-190
186
G. M. TALLIS AND A. D. DONALD
-04*03*02a01-
0 FIG. 2c
2
4
6
8
10
12
14
16
18
20
22
2
4
6
8
10
12
14
16
18
20
22
In order to fit (6) to these data assume
that
-03 -
0
24
FIG. 2d
d@,Jx) = ayi lW
XY+-le-a= dx;
then f(t)
= P&t,
where Z(x, y) is the incomplete are
71) -
gamma
Z(G
function.
Yl
rz>l,
+
The first three moments
Lcl = PY2bI+ (Yz+ 1)/21 3 a2
ru = PY&
2
Trapezoidal
+ lb’,
+ Y&‘Z + 1) + (yz + l)(yz + 2)/3]
a3 approximation
to ,LQcan be calculated
A = 4 F If(4+&:++, +_7(4)W+l Mathematical Biosciences 7 (1970), 179-190
-
using the formula 4)
DISTRIBUTION
OF INFECTIVE
LARVAE
187
for j = 0, 1, 2. The values of f(t) must be estimated time points as described above. Set &I&,= 1 and define i(j) = ,Gj//ii+l, then pZ = [l + 12[i(2) PI
=
(92
i(1) - 2
+
at equally
spaced
i(1)2 + i(l)]]““, 1)
,
and
Other methods of selecting 2 can be used, but the one chosen above is very convenient and seems to produce satisfactory results in practice. There does not appear to be a simple way of using another equation, TABLE REQUIRED
I STEPS
TO
FIT f(t)
1
BY
THE
METHOD
2
OF
MOMENTS
3
4
PO
0.866920
0.148050
0.321000
llil
5.311610
1.380660
3.066130
2.523180
43.240090
13.818520
35.586030
30.365180
1
1
Pc”s
i(l)
i(2) II
Ya
0.242340
7.067538
62.989755
29.756408
42.963339
66.366593
4258.300410
1075.881509
2133.538826
10.638966
52.270613
43.922382
54.200773
1.248055
36.354449
7.295217
15.362953
0.093994
0.019131
0.022767
0.018500
1.081964
5.382276
2.341765
3.723058
9.223132
7.738664
14.099085
13.135015
1
Yl
,u~, to obtain a better fit. Notice that E{X,} = yi/a is an important parameter in determining the relative influences of each stage of the pasture cycle onf(t). The observed and fittedf(t) function for the four plots are shown in Fig. 2 and the appropriate steps in the calculations of $, gl, gl and E{_?i} are given in Table I.
DISCUSSION
In their original models, Tallis and Donald [6] proposed a negative binomial model for the distribution of fecal deposits on whole paddocks grazed by sheep. From investigations on the distribution of fecal deposits, Mathematical Biosciences 7 (1970), 179-190 13
G. M. TALLIS
188
AND
A. D. DONALD
Donald and Leslie [4] concluded that, although the negative binomial provided a reasonable empirical description of the distributions, there was some evidence of departure from the model at a low rate of stocking. More importantly, however, the hypothesis that the distribution of feces is additive and independent with respect to time was found to be unacceptable. This finding arose from the tendency shown by flocks of sheep to deposit heavy concentrations of feces in the same circumscribed area of a paddock during consecutive resting periods. Donald and Leslie [4] concluded that, in the presence of heterogeneities of pasture and topography, the known tendency for subflock formation in large flocks and the different behavior patterns of different age classes of sheep, it seemed doubtful whether any simple two-parameter probability distribution could adequately describe the distribution of fecal deposits in all situations. The present model overcomes these difliculties by using the general forms for the means and variances of the different processes and is therefore much more flexible. It has the added advantage that particular subareas of paddocks are easily considered separately. For example, there is some evidence that sheep do not graze on resting areas (or “camps”) while these areas carry heavy concentrations of freshly deposited feces but may do so later when such areas have ceased to be used for resting [l, 41. Thus, when potential rates of infective larval intake by grazing sheep are being considered, it may be necessary to derive separate estimates of infective larval abundance for grazing and resting areas, respectively. Turning to the model forf(t), this is a considerable advance over the original models of Tallis and Donald [6] in which this component was The parameters p, E(X,), and E(X,) left to be estimated empirically. have simple biological meaning, and it would not be difficult to design experiments to estimate them independently. The fit of the model to the four sets of data, shown in Fig. 2, appears reasonable, particularly if the error variance of the estimates, which is unavoidably rather large, is taken into account. Each empirical point estimate off(t) is the mean of four samples and an average standard error of .Ol for each sample mean has been calculated. During the course of the field studies from which the data used to illustrate the fitting of the f(t) model were derived, some independent evidence was also obtained relating to E(X,), namely, the average time taken for development to the infective stage and migration to the herbage. When herbage samples were collected each week, the associated fecal material was also collected, and estimates were made of the numbers of surviving eggs and preinfective larvae which had not yet completed development to the infective stage and of surviving infective larvae which had not yet migrated from the feces. These data are presented in Table II in the Mathematical
Biosciences
7 (1970),
179-190
DISTRIBUTION
189
OF INFECTIVE LARVAE
TABLE II ESTIMATES THE?
OF THE AVERAGE
INFECTIVE
DERIVED
STAGE
FROM
FITTING
ESTIMATES
OF MAXIMUM
OBTAINED
FROM
TIME TAKEN
AND THE
f(f)
MODEL
DEVELOPMENT
EXAMINATTON
FGR DEVELOPMENT
MIGRATION
AND
TO
THE
TO
HERBAGE
COMPARED
WITH
MIGRATION
TIMES
OF FECES
Plot
E{$l,) (weeks)
“Development completed,“’ by week
“Migration completed,“” by week
1 3 4 2
1.08 2.34 3.72 5.38
2 2 2 6
3 6 13 10
d For explanation, see text. form of times recorded for the completion of development and of migration for the majority of eggs originally present in the sample. The former estimate represents the first weekly sampling at which no viable preinfective stages were recovered, and the latter is the first sampling at which no infective larvae were found in the feces. Because the estimate of E(X,) derived from the fitting of f(f) is an average value whereas the estimates of development and migration times obtained from fecal samples represent maximum values, they are not directly comparable. However, inspection of Table II reveals at least that there are no gross anomalies. The f(t) model may be particularly useful in two situations. First, estimates of the parameters derived from fitting the model might constitute suitable dependent variables for the application of such techniques as Second, multiple regression analysis against components of environment. simulation studies with the whole model will be greatly facilitated if various forms of the functionf(t) can be generated by choosing parameter values appropriate to particular sets of climatic and other environmental conditions.
REFERENCES 1 H. D. Crofton, Nematode parasite populations in sheep on lowland farms. VI. Sheep behaviour and nematode infections, Parasitology 48(1958), 251-260. 2 A. D. Donald, Population studies on the infective stage of some nematode parasites of sheep. III. The distribution of strongyloid egg output in flocks ofsheep, Parasitology 58(1968),
951-960. Mathematical
Biosciences
7 (1970), 179-190
190
G. M. TALLIS AND A. D. DONALD
3 A. D. Donald, Ecology of the free living stages of nematode parasites of sheep, Aus~raliun I’er. J. 44(1968), 139-144. 4 A. D. Donald and R. T. Leslie, Population studies on the infective stage of some nematode parasites of sheep. II. The distribution of faecal deposits on fields grazed by sheep, Purasifology 59(1969), 141-157. 5 G. M. Tallis, A note on the estimation of larval concentrations on pasture, Australian J. biol. Sci. 17(1964), 1016-1019. 6 G. M, Tallis and A. D. Donald, Models for the distribution on pasture of infective larvae of the gastrointestinal nematode parasites of sheep. Australian J. biol. Sci. 17(1964),
Mathematical
504-513.
Biosciences
7 (1970), 179-190
179
BIOSCIENCES
Further Models for the Distribution on Pasture of Infective Larvae of the Strongyloid Nematode Parasites of Sheep G. M. TALLIS Division of Mathematical C.S.I.R.O.
Newtown,
Statistics,
N.S. W., Australia
A. D. DONALD Division C.S.I.R.O.
of Animal Health, McMaster
Laboratory
Glebe, N.S. W., Australia
Communicated
by K. E. F. Watt
ABSTRACT This paper discusses a general approach to the problem of constructing useful models for the distribution on pasture of infective larvae of sheep nematode parasites. The work is related to earlier results obtained by the authors, and modifications are introduced as a result of practical experience with the original models. An explicit form for the function f(t), the probability that an egg reaches the infective larval stage in time t, is derived. The new version off(t) has components with direct biological interpretation.
INTRODUCTION
In earlier papers [5, 61, stochastic models were developed to describe the distribution on pasture of the infective larvae of some nematode parasites of sheep. These models were based on current knowledge of the distribution of fecal deposits in a paddock and on the assumption that the distribution of eggs between fecal deposits is Poisson. Subsequent experience [2, 41 has led to modifications in the original postulates and it seems appropriate at this stage to redevelop some of the previous results in the light of the present ideas. Studies of fecal distributions on paddocks of different sizes carrying varying numbers of sheep have emphasised that deposits are definitely Mathematical
Biosciences
7 (1970), 179-190
Copyright @ 1970 by American Elsevier Publishing Company, Inc.
180
G.
M. TALLIS
AND
A. D. DONALD
nonuniformly distributed over the total area. However, it seems possible to divide the paddock into subareas within which the distribution is, to a good approximation, uniform, although there may be considerable between area differences in concentration due to the grazing and resting habits of the flock. With this information in mind, it is the primary purpose of this paper to develop completely general distributions for the numbers of larvae in the various subareas of a paddock carrying S sheep. From these expressions certain reasonable simplifications will be introduced to make the formulae useful in practical investigations. Most of the argument is carried through using the general forms for the means and variances of the various processes. In the sequel we pay particular attention to the functionf(t), which is the probability that an egg dropped onto the pasture at time zero is in the infective larval stage on the herbage at time f. Explicit expressions for f(r) are derived from underlying biological hypotheses to give the formulae further interpretational value.
RESULTS
Distribution of the Total Number of Infective Larvae on Pasture. Let the paddock under consideration be of area A and suppose S sheep are introduced at zero time, T = 0. In accordance with the Introduction, the paddock is divided into k plots of area Aj, I:=, Aj = A. Within each plot the defecation pattern of the flock is such that fecal deposits are approximately uniformly distributed over the area. Initially we focus attention on plot j and sheep i. Let the number of eggs per fecal deposit for the ith sheep at T = x be a random variable with probability generating function, p.g.f., gi(s, x) with mean A(x) = A(l, x) and variance (T:(x) = g:l(l, x) + Ai
-
A;(X).
Now define the step function N,,(t) as the total number of fecal deposits associated with the ith sheep on plot j by T = t, and letf(t) be as defined in the Introduction. Then the p.g.f. for the number of live larvae at T = t emanating from a deposit dropped at T = x is Us,
x, t) = g,{[l
and the p.g.f. for the total Mathematical
Biosciences
7 (1970).
+f(t
number 179-190
- x)(s of live larvae
l)l, xl, on plot j due to the
DISTRIBUTION
OF INFECI-IVE
LARVAE
181
ith sheep at T = t is &(sj, t) = exp( /log
hi(sj, x, t) &V,,(X)].
(I)
0
Formula (1) assumes stochastic independence between the numbers of eggs in the deposits. Although this assumption can be made plausible by some mathematical argument, the details will not be presented since they are somewhat involved. The restriction of independence is, of course, easily removed at the cost of increasing the complexity of the expressions and the estimation procedures. Similarly, the p.g.f. for the number of larvae on plot j due to the S sheep is
+Csj,t>= eV( it Slog[Usj, x9 l)l dNij(xI)*
(2)
0
and the joint p.g.f. for all k plots is
If the prime interest is the total number of larvae on pasture, then the appropriate p.g.f., $(s, t) is obtained from (3) by setting sj = s for allj. Equations (I), (2), and (3) give the required expressions in the most general form, but they are of limited practical value since, for instance, the step functions Nij(t) must be known. Below, some reasonable simplifications are introduced to illustrate the use of the various p.g.f.‘s. However, before proceeding, general expressions for the means and variances of the various processes are obtained. Let L,(t) be the number of live larvae on plotj at T = t, then
E{Lj(t)} =
iilJk(x)f(t - X>dNij(x), 0
(4)
t
where Vi(x, t) = $(x)f’(t - x)‘+ &(x)f(t - x)[l -f
Ut) =
iLjCr),
j=l
E{L(t)Il
=~~lEILj(t)~9 .E
Muthenzaticol Biosciences 7 (1970), 179-190
182
G.
M. TALLIS AND A. D. DONALD
and V{L(f)} = i V{&(r)). j=l
In order to simplify
(4) to a manageable
approximation
set
2Ai(
SX(x)=
i=l
$ a:(x),
SS2(x)=
i=l
and Ni j(x) = n jx, where nj is the daily fecal output sheep; then
per sheep on plot j averaged
over all
t
E{Lj(t)}
N
Snj
s s
X(x)f(t
-
X)
dx,
0
(5)
t
V{Lj(t)}
N
Snj
7(x, t) dx,
0
where F(x, t) = ?(x)J’“(t
-
x) + i(x)f(r
- x)[l
-f(t
- x)].
By suitable sampling, estimates of the functions nj, x(x), and C?“(X) must be obtained for values of x in the range of interest. The integrals can then be approximated by numerical quadrature provided f(t) is known. The amount of work required to carry out the preceding estimation is considerable and, at best, tedious. Some further assumptions will now be made which should facilitate the application of (5). Let Z,(x) be the random variable associated with gi(s, x), and suppose that the weight of each fecal deposit, W, is a random variable. Then for fixed W = w let gi(s, x) be Poisson with parameter
A,(x) 1w = WY,(X). Then E{Zi(X) I w> = Vi(X) and hence A,(x) = E{Zi(X)} =
ECWYiW
and, similarly, c?(x) = I’{&(x)> Mathematical
= Yi(x)E(W
Biosciences7 (1970), 179-190
+ Y:(x)wo.
DISTRIBUTION
183
OF INFECTIVE LARVAE
In order to use (5) under the assumptions above it is necessary to estimate E(W) and V(W), B(W), and F(W) say, for the flock of sheep. Then, since y,(x) can be estimated for each sheep by standard methods, average values of A(x) and a”(x) are easily obtained at different time points. The integrals in (5) can then be approximated by numerical quadrature.
A Modelforf(t). own right. Consider
The functionf(t) is of some biological interest the elementary flow diagram in Fig. 1.
PASTURE HERBAGE
FECAL DEPOSIT
FIG. 1. Elementary
in its
flow diagram.
(i) q is the probability that an egg does not get to infective larva and p = 1 - q is the probability that it (ii) Given that an egg completes development to the the resulting larva migrates to the pasture, X, is the time ment and migration from fecal deposit to herbage. (iii) X2 is the length of life as an infective larva on
Suppose that X1 and X2 have distribution independently distributed, then
the herbage as an will. infective stage and taken for developthe herbage.
functions
a1 and Q2 and are
[l - @(t - x)] &&(x).
(6)
t f(t)
= p
s
The assumption of independence is not necessary, but a more general approach leads to complexities of estimation which are not pursued here. In order to establish (6) let A be the event that an egg never develops into a larva that completes the migration process to the grass. Thus Pr{A} = q and Pr{K} = p, where K is the complementary event to A. Define the set s = {x,,
x2;
0
Q
x1
G
2,
t
-
Mathematical
x1
Q
x21,
Biosciences
7 (1970), 179-190
G. M. TALLIS
184
AND
A. D. DONALD
then f(t)
= Pr{egg d eve 10 p s into an infective larva alive at time t 1A}q + Pr{egg develops into an infective larva alive at time I 1 A)p
t = p
s
[l - @z(t - x)] d@,(x).
0
Iff*(S)
is the Laplace
transform, f*(s)
L.T., off(t),
= PS-WWI
then
Ms>l,
-
where $:(s) is the L.T. of +i(x), the derivative to exist. From (7) it is easily verified that
(7)
of Qi(x) which is assumed
. J a0
pr =
t’f(t) dt
0
= pr ! i
p~‘,,u~~~_,/n ! (r +
1 -
n)!
?2=0
and also lim sf*(s)
= 0, implying
limf(r)
S-*0
(8)
= 0. In (8),
t+m co
Pi’
=
xn
&D,(x),
I 0
and it is interesting to notice that the area underf(r) equations specified by (8) are useful for fittingf(t) below.
isp@. to data,
The moment as illustrated
Fitting the f (t) Model to Empirical Data. Empirical estimates of the function f (t) have been obtained in the course of field ecological studies, which will be reported in detail elsewhere, on the free-living stages of Trichostrongylus colubriformis and Haemonchus contortus, two important nematode parasites of sheep. Donald [4] has given a preliminary account of some of this work, including the methods used. Briefiy, a known amount of sheep faeces containing parasite eggs is scattered on a small plot of pasture. Estimates are made initially of the total number of eggs of each species placed on the plot, and at weekly intervals the numbers of infective larvae present on the herbage are estimated from samples. Point estimates of the function f (t) at weekly intervals are obtained by dividing the Muthermtical
Biosciences 7 (1970),
179-190
DISTRIBUTION
OF INFECTIVE
185
LARVAE
numbers of infective larvae recovered at time t by the number of eggs exposed on the plot at t = 0. Beginning in June 1967, a fresh plot has been set up every four weeks and herbage sampling of each plot has continued until at Ieast three consecutive zero recoveries have been obtained For the present purpose of illustrating the fitting of the model, data for T. colubriformis from each of four plots have been selected as representative.
FIG. 2. Plots of the observed and fitted values of,f(r) for the four groups. -I,
.O& -03 c -02 .ol
0
2
12
lr,
16
18
20
22
24
FIG. 2b Mathematical
Biosciences
7 (1970),
179-190
186
G. M. TALLIS AND A. D. DONALD
-04*03*02a01-
0 FIG. 2c
2
4
6
8
10
12
14
16
18
20
22
2
4
6
8
10
12
14
16
18
20
22
In order to fit (6) to these data assume
that
-03 -
0
24
FIG. 2d
d@,Jx) = ayi lW
XY+-le-a= dx;
then f(t)
= P&t,
where Z(x, y) is the incomplete are
71) -
gamma
Z(G
function.
Yl
rz>l,
+
The first three moments
Lcl = PY2bI+ (Yz+ 1)/21 3 a2
ru = PY&
2
Trapezoidal
+ lb’,
+ Y&‘Z + 1) + (yz + l)(yz + 2)/3]
a3 approximation
to ,LQcan be calculated
A = 4 F If(4+&:++, +_7(4)W+l Mathematical Biosciences 7 (1970), 179-190
-
using the formula 4)
DISTRIBUTION
OF INFECTIVE
LARVAE
187
for j = 0, 1, 2. The values of f(t) must be estimated time points as described above. Set &I&,= 1 and define i(j) = ,Gj//ii+l, then pZ = [l + 12[i(2) PI
=
(92
i(1) - 2
+
at equally
spaced
i(1)2 + i(l)]]““, 1)
,
and
Other methods of selecting 2 can be used, but the one chosen above is very convenient and seems to produce satisfactory results in practice. There does not appear to be a simple way of using another equation, TABLE REQUIRED
I STEPS
TO
FIT f(t)
1
BY
THE
METHOD
2
OF
MOMENTS
3
4
PO
0.866920
0.148050
0.321000
llil
5.311610
1.380660
3.066130
2.523180
43.240090
13.818520
35.586030
30.365180
1
1
Pc”s
i(l)
i(2) II
Ya
0.242340
7.067538
62.989755
29.756408
42.963339
66.366593
4258.300410
1075.881509
2133.538826
10.638966
52.270613
43.922382
54.200773
1.248055
36.354449
7.295217
15.362953
0.093994
0.019131
0.022767
0.018500
1.081964
5.382276
2.341765
3.723058
9.223132
7.738664
14.099085
13.135015
1
Yl
,u~, to obtain a better fit. Notice that E{X,} = yi/a is an important parameter in determining the relative influences of each stage of the pasture cycle onf(t). The observed and fittedf(t) function for the four plots are shown in Fig. 2 and the appropriate steps in the calculations of $, gl, gl and E{_?i} are given in Table I.
DISCUSSION
In their original models, Tallis and Donald [6] proposed a negative binomial model for the distribution of fecal deposits on whole paddocks grazed by sheep. From investigations on the distribution of fecal deposits, Mathematical Biosciences 7 (1970), 179-190 13
G. M. TALLIS
188
AND
A. D. DONALD
Donald and Leslie [4] concluded that, although the negative binomial provided a reasonable empirical description of the distributions, there was some evidence of departure from the model at a low rate of stocking. More importantly, however, the hypothesis that the distribution of feces is additive and independent with respect to time was found to be unacceptable. This finding arose from the tendency shown by flocks of sheep to deposit heavy concentrations of feces in the same circumscribed area of a paddock during consecutive resting periods. Donald and Leslie [4] concluded that, in the presence of heterogeneities of pasture and topography, the known tendency for subflock formation in large flocks and the different behavior patterns of different age classes of sheep, it seemed doubtful whether any simple two-parameter probability distribution could adequately describe the distribution of fecal deposits in all situations. The present model overcomes these difliculties by using the general forms for the means and variances of the different processes and is therefore much more flexible. It has the added advantage that particular subareas of paddocks are easily considered separately. For example, there is some evidence that sheep do not graze on resting areas (or “camps”) while these areas carry heavy concentrations of freshly deposited feces but may do so later when such areas have ceased to be used for resting [l, 41. Thus, when potential rates of infective larval intake by grazing sheep are being considered, it may be necessary to derive separate estimates of infective larval abundance for grazing and resting areas, respectively. Turning to the model forf(t), this is a considerable advance over the original models of Tallis and Donald [6] in which this component was The parameters p, E(X,), and E(X,) left to be estimated empirically. have simple biological meaning, and it would not be difficult to design experiments to estimate them independently. The fit of the model to the four sets of data, shown in Fig. 2, appears reasonable, particularly if the error variance of the estimates, which is unavoidably rather large, is taken into account. Each empirical point estimate off(t) is the mean of four samples and an average standard error of .Ol for each sample mean has been calculated. During the course of the field studies from which the data used to illustrate the fitting of the f(t) model were derived, some independent evidence was also obtained relating to E(X,), namely, the average time taken for development to the infective stage and migration to the herbage. When herbage samples were collected each week, the associated fecal material was also collected, and estimates were made of the numbers of surviving eggs and preinfective larvae which had not yet completed development to the infective stage and of surviving infective larvae which had not yet migrated from the feces. These data are presented in Table II in the Mathematical
Biosciences
7 (1970),
179-190
DISTRIBUTION
189
OF INFECTIVE LARVAE
TABLE II ESTIMATES THE?
OF THE AVERAGE
INFECTIVE
DERIVED
STAGE
FROM
FITTING
ESTIMATES
OF MAXIMUM
OBTAINED
FROM
TIME TAKEN
AND THE
f(f)
MODEL
DEVELOPMENT
EXAMINATTON
FGR DEVELOPMENT
MIGRATION
AND
TO
THE
TO
HERBAGE
COMPARED
WITH
MIGRATION
TIMES
OF FECES
Plot
E{$l,) (weeks)
“Development completed,“’ by week
“Migration completed,“” by week
1 3 4 2
1.08 2.34 3.72 5.38
2 2 2 6
3 6 13 10
d For explanation, see text. form of times recorded for the completion of development and of migration for the majority of eggs originally present in the sample. The former estimate represents the first weekly sampling at which no viable preinfective stages were recovered, and the latter is the first sampling at which no infective larvae were found in the feces. Because the estimate of E(X,) derived from the fitting of f(f) is an average value whereas the estimates of development and migration times obtained from fecal samples represent maximum values, they are not directly comparable. However, inspection of Table II reveals at least that there are no gross anomalies. The f(t) model may be particularly useful in two situations. First, estimates of the parameters derived from fitting the model might constitute suitable dependent variables for the application of such techniques as Second, multiple regression analysis against components of environment. simulation studies with the whole model will be greatly facilitated if various forms of the functionf(t) can be generated by choosing parameter values appropriate to particular sets of climatic and other environmental conditions.
REFERENCES 1 H. D. Crofton, Nematode parasite populations in sheep on lowland farms. VI. Sheep behaviour and nematode infections, Parasitology 48(1958), 251-260. 2 A. D. Donald, Population studies on the infective stage of some nematode parasites of sheep. III. The distribution of strongyloid egg output in flocks ofsheep, Parasitology 58(1968),
951-960. Mathematical
Biosciences
7 (1970), 179-190
190
G. M. TALLIS AND A. D. DONALD
3 A. D. Donald, Ecology of the free living stages of nematode parasites of sheep, Aus~raliun I’er. J. 44(1968), 139-144. 4 A. D. Donald and R. T. Leslie, Population studies on the infective stage of some nematode parasites of sheep. II. The distribution of faecal deposits on fields grazed by sheep, Purasifology 59(1969), 141-157. 5 G. M. Tallis, A note on the estimation of larval concentrations on pasture, Australian J. biol. Sci. 17(1964), 1016-1019. 6 G. M, Tallis and A. D. Donald, Models for the distribution on pasture of infective larvae of the gastrointestinal nematode parasites of sheep. Australian J. biol. Sci. 17(1964),
Mathematical
504-513.
Biosciences
7 (1970), 179-190