A Probabilistic Model for Termite Attack
Robert H. LEICESTER
Honorary Research Fellow
CSIRO Australia
PO Box 56, Highett Vic 3190
[email protected]
Dr. Leicester has more
than forty years research
experience related to the
timber and construction
industries. He has been
involved in the drafting of
several Australian and
International Standards on
timber and structural
loading.
Chi-Hsiang Wang
Research Scientist
CSIRO Australia
Dr. Wang’s research
interests include timber
durability issues, risk and
reliability analysis,
stochastic structural
analysis, and inelastic
structural performance
under dynamic loads
L. J. Cookson
Research Scientist
CSIRO Australia
Summary
This paper describes a model to predict the risk of termite attack on a house in Australia. It is based
on a survey of expert opinion and data from 5000 houses. The model gives a quantitative estimate
of risk, and as such is useful for the development of risk management systems. An example of the
application of such a system is given.
Keywords: Termite, probability, risk, timber, expert opinion, calibration
1.
Introduction
For engineering purposes, it is useful for termite attack to be considered a probabilistic event. This
paper describes the development of a model to predict the risk of attack on a house in Australia
during a specified duration such as 50 years. Such a model is useful for assessing (in a quantified
manner) the value of various protection strategy proposals.
A preliminary model was first developed on the basis of opinions elicited from experts on termite
behaviour such as entomologists and pest controllers. The model was then calibrated through the
use of data on 5000 houses, denoted as “Termite Tally”, which was reported in detail by Cookson
[1]. It was based on an Australia wide survey undertaken by school children.
The model takes into consideration a great number of factors, such as for example the age of a
house and its surrounding suburbs, the inspection program, the use of physical and chemical termite
barriers, the type of house construction, details of the local environment and climate. Further details
of many aspects of this model can be found in previous papers [2, 3, 4].
2.
Probability Model
The probability density function of the time for a house to be attacked by termites is assumed to be
of the type shown in Fig. 1. The form of this function was chosen to fit the data found in the
Termite Tally. The equation for the density function is assumed to be
p
a bt
(1)
where a and b are the distribution parameters, and t is the time since time zero, the time at which
the house was constructed. The notation tmax will be used to denote the upper limit of the density
function, evaluated from the assumption that the area under the density graph must be unity, and ta
denotes the lower end of the density distribution. The mean time of attack is given by
a 2
b 3
tmax ta2 tmax
ta3
(2)
2
3
An interesting aspect of the risk model is to distinguish between a “true” risk and an “apparent”
risk. The apparent risk is the risk estimate based on the historical memory of the house occupant.
This is obviously less than the true risk as the householder is not aware of all past termite attacks. In
the Termite Tally the average time of occupancy of the householders interviewed was found to be
about 11 years. In this risk model the historical memory of the average occupant, denoted by tmem,
has been taken to be 20 years, hence the apparent risk is the probability that the houses have been
attacked during the previous 20 years.
mean(t )
p
p
p = a + bt
ta = 0
p = a + bt
a
tmax
t
tmax
ta
age of house (yrs)
t
age of house (yrs)
(a) for positive a
(b) for negative a
Fig. 1 Probability density functions of the time of a termite attack
Figure 2 shows plots of the true and apparent risk that a house of age t years will have been
attacked. For the case ta < t < tmax, the true probability Ptrue that a house has been attacked is
b 2 2
t ta
(3)
2
For the case (ta + tmem) < t < tmax, the apparent probability that a house has been attacked, Papparent, is
a t ta Probability that attack has occurred
Papparent
P
§b· 2
atmem ¨ ¸ tmem
btmem t
© 2¹
(4)
true risk of attack
1.0
apparent risk of attack
0
tmem
tmax
t
Probability that attack has occurred
Ptrue
P
1.0
true risk of attack
apparent risk of attack
0
ta
ta + tmem
tmax
age of house (yrs)
age of house (yrs)
(a) for positive a
(b) for negative a
Fig. 2 Schematic illustration of the cumulative distributions of the attack time
t
3.
Model based on Expert Opinion
The model for estimating the time to attack has been derived on the basis of expert opinion. It
applies to a house surrounded by 50 m of termite-free land. The distance of 50 m was chosen
because this is about the limit of the foraging distance of most termite species. The model used
endeavours to estimate the time taken for four sequential stages of attack (Fig. 3): (1) t1, time for
establishment of a mature colony; (2) t2, time for termite foraging galleries to progress to a house;
(3) t3, time for penetration or bypass of termite barrier; and (4) t4, time for ruin of a timber element.
Detailed equations for expert opinion on these estimates of t1, t2, t3, and t4 are given in [4].
Destruction
Stage 4
Nest
Stage 3
Stage 1
Stage 2
Fig. 3 Illustration of termite progress
The model also needs to take into account the possibility that the target house is closer than 50 m to
the adjoining suburbs and mature nests exist nearby at time zero, the year in which the house is
constructed. An approximation to the time for termite attack given by expert, denoted by texpert, is
d 10
º
ª
Pgarden t2 1 Pgarden « Psuburb
t2 1 Psuburb t1 t2 » t3 t4
(5)
20
¬
¼
where Pgarden is the probability that a mature nest exists in the garden at time zero, and Psuburb, is the
probability that a mature nest exists in the suburb at time zero. A suitable equation for estimating
Pgarden and Psuburb is
texpert
tsuburb d 100 years;
­tsuburb 100,
(6)
®
tsuburb ! 100 years.
¯1
where tsuburb denotes the age of the suburb at time zero, the year in which the target house was built.
Pgarden
4.
Psuburb
Model Calibration
Apparent probability that
house has been attacked
As discussed in a previous paper [4], data from the Termite Tally indicates for the distribution
function of attack times, suitable calibration choices are for a parameter b = 0.0002, and the mean
time to attack mean(t) = 1.5 mean(texpert), where mean(texpert) denotes the meant time estimate based
on expert opinion. Figure 4 shows how predictions on apparent risk, based on these assumptions,
match the Termite Tally data for average hazard zone conditions.
0.6
Model
mean(t) = 44 yrs
0.4
Termite Tally:
Temperature
zonation
(zone 2)
Temperature (zone
2)
0.2
Agro-ecological (zone
2
Agro-ecological
zonation
& 3)
0
(zone 2 and 3)
0
20 40 60 80 100
Age of house (years)
Fig. 4 Comparison of apparent risks derived from the model and the Termite Tally
1003
5.
Concluding Comments
Fig. 5 shows the risk for the cases of hazards for extreme and mean conditions. The envelope
represents the extremes of risk predictions by the model. For houses of a given age, there is such a
wide range of possible risk that it is obviously not economical to treat all houses similarly. Use of a
risk prediction model represents a practical tool for cost optimised design against termite attack.
1.5
mean(tmodel) indicated 15 yrs
1
45 yrs
0.5
180 yrs
0
0
50
100
150
age of house (years)
(a) Apparent risk
200
250
true probability that a house
has been attacked
Apparent probability that a
house has been attacked
The model described assesses the risk of termite attack in quantitative terms. As such it can be used
as a risk management tool.
There are two major improvements that can be made to the model. The first is to obtain information
in order to effectively interpolate the hazard zones between the major city clusters covered by the
data of the Termite Tally. The second is to obtain more accurate quantitative information on the
performance of termite barriers.
The model is not perfect. However it does provide a framework into which knowledge can be
placed as this becomes available. The model provides a tool for making a quantitative assessment of
the monetary value of obtaining new knowledge, new barrier systems and new attack mitigation
strategies.
1.5
mean(t model) indicated 15 yrs
1
4 5 yrs
0.5
18 0 yrs
0
0
50
100
150
200
250
age of house (yrs)
(b) True risk
Fig. 5 Possible ranges of apparent and true risks of termite attack
6.
References
[1]
Cookson, L.J., “Termite Survey and Hazard Mapping,” Client Report No. 664, CSIRO
Forestry and Forest Products, Melbourne, Australia, August, 1999.
Leicester, R.H. and Wang, C-H., “A probabilistic model of termite attack,” Proceedings of
International Conference on Structural Safety and Reliability, Newport Beach, California,
17–21 June, 2001.
Leicester, R.H., Wang, C-H., Cookson, L.J., and Creffield, J., “A Model for Termite Hazard in
Australia,” Proceedings of 9th International Conference on Durability of Building Materials
and Components, Brisbane, Australia, March, 2002.
Leicester, R.H., Wang, C-H., and Cookson, L.J., “A Risk Model for Termite Attack in
Australia,” Proceedings of 34th IRGWP Annual Meeting, Brisbane, Australia, 19–23 May,
2003, Paper No. IRG/WP/03.
[2]
[3]
[4]