Models of animal health problems

MODELS

OF ANIMAL

HEALTH

PROBLEMS*

ANDREW D. JAMES

Dept of Agriculture & Horticulture, University of Reading, Earley Gate, Reading, RG6 2A T, Great Britain

SUMMAR Y

This paper discusses the usefulness of modelling techniques in solving two problems of estimation arising in the economic assessment oJ'animal health programmes. The first problem is that of predicting the effects that control measures will have upon the incidence of a disease. The development of a model of joot-and-mouth disease is described to illustrate this application of models. The second problem lies in assessing the effect of the disease upon the productivity of the animal. A possible approach to this type of problem is illustrated by a teehnique jor improving estimates of the milk loss resulting from infertility in dairy herds.

In making economic appraisals of animal health programmes, two problems of estimation arise, which modelling techniques can play a part in solving. The first problem is that of predicting the effects that control measures will have on the incidence of a disease. Usually the relationships are very complex. Vaccination, for example, besides protecting the animals which have been properly vaccinated against a certain degree of exposure to infective agent, will also protect to some degree animals which have not been vaccinated. This is because less infective agent will be released, as the animals protected by vaccination will not become infected and disseminate more infection. The second problem of estimation lies in assessing the effect of the disease on the productivity of the animal. This is particularly important in milk production, as many diseases cause infertility, as well as a direct depression of milk yield in dairy animals. Thus both the quality and the quantity of lactations are reduced. Seasonal * Paper presented at a meeting of the Society for General Systems Research held at Reading University on Nov. 10, 1976. (Sole sponsors: UK Regional Division of the SGSR.)

183 Agricultural Systems (2) (1977)--©Applied science Publishers Ltd, England, 1977 Printed in Great Britain

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ANDREW D. JAMES

differences in the quality of lactations, and many other factors also affect the situation, making a modelling approach potentially very valuable, as it enables the effects of many such factors to be taken into account. In this paper I shall discuss the applications of one model to each of these problems. Until recently most of the use of models in the animal health field has been directed at the understanding of the epidemiology of the major infectious diseases, particularly in man, but also in farm animals. At present these diseases are of much less economic significance in developed countries than they have been. However, their significance in developing countries, rather than diminishing, is increasing, and they are seriously hampering livestock development programmes. There is also the continued threat of reintroduction of a disease to countries which have eradicated it. A major epidemic of foot-and-mouth disease in a c o u n t r y where the livestock population has no natural immunity would have disastrous consequences on the livestock producing industries, as witness the 1967-68 outbreak of foot-and-mouth disease in England and Wales. It was with this situation in mind that W. Miller, of the Central Veterinary Laboratory, Weybridge, who was at that time working on the development of an action programme in the event of a major outbreak of foot-andmouth disease in the USA, produced a simple model of the disease, which we have modified to some extent in order to widen its usefulness. The model works on the Markov-chain principle. It considers that a herd of cattle can be in one of four states; susceptible, affected, immune, or removed. Because the disease is so contagious, the assumption that the whole of a herd will be in the same state at any time is not unrealistic. By defining a matrix of probabilities of a herd transferring between any two states in a given time period, it is possible to postmultiply a row vector of the number of herds in each state by the transition probability matrix, and thereby derive the new row vector of the number of herds in each state at the beginning of the following time ~eriod:

(Us, Ua, U~, U~)

S~ Psa

Psi

Psr "~

P:~ P"~ Pis P i a

Pai Pii

Par Pi~

Pri

Prr

Iip rs

STATE V E C T O R

P~a

x TRANSITION PROBABILITY MATRIX

= (N'~, N'~, U I, U'r)

= NEXTSTATE VECTOR

This procedure provides an efficient 'accounting' framework. Most of the transition probabilities are easy to define; there is no possibility of a herd transferring from removed to affected, unless it was restocked with affected animals; it is certain that an affected herd will become immune if it is not slaughtered. The key transition probability is the probability of a susceptible herd becoming

MODELS OF ANIMAL HEALTH PROBLEMS

185

affected, and our modifications to the model mostly concern this probability. The important factor is the contact rate, which is the expected number of herds that will come into contact with each infected herd during a given time period. Simply using such a contact rate has an obvious defect. Suppose, for example, that 500 of a population of 1000 herds were infected, and the contact rate was three, 1500 new cases would be predicted for the following week. This problem was overcome in the original model by the use of an exponential relationship to ensure t h a t the new infection rate asymptotes to zero as the number of cases rises. There were two objections to this relationship. The first was that it had no theoretical basis, and t h e second was that for contact rates of less than one, it overestimated the new infection rate. The problem can be alleviated t o some extent in two stages. In the first stage it is assumed that the herds in the population are all equally likely to be infected. Then, if the probability of any herd being infected in any time period is low, which in practice it always is, there is a Poisson situation. In this case the probability of any herd not being infected is exp ( - m ) m ° 0! where m is the expected number of times that a herd will be infected. Therefore, the probability of it being infected at least once is 1 - exp(-m) This makes the model much more acceptable, but because all herds have the same probability of being infected, the model has no spatial dimension. In the second stage this condition m a y be relaxed by assuming that most of the infections will occur within a sub-set of the population, around the centre of infection. For example, it might be possible to assume that in the next week 95 ~ of the new outbreaks will occur among 35 ~ of the population. Under these conditions, the assumption that all herds within the two sub-sets have the same probability of infection is much more realistic. It would of course be possible to extend the number of sub-sets, but the difference that this makes to the behaviour of the model appears to be negligible. An additional advantage of this form of model is that it represents the effect of vaccination much more accurately. I f a herd is immune by vaccination or recovery, it is likely to be near the centre of infection, and so the model can consider most of the immune herds to be a m o n g the sub-set of herds in the infected area, instead of being randomly distributed through the whole population. Thus the effects of such policies as ring vaccinations can be examined. The example of a model designed to demonstrate the effect of disease upon production is one which investigates the relationship between infertility and production of a dairy herd. This type of disease loss is of much more significance in developed countries than the diseases causing major epizootics. In most dairy herds a cow is expected to calve and commence a new lactation before the natural end of

186

ANDREW D. JAMES

the old lactation, so she will be dried off artificially some time before calving. If, because of infertility, the cow calves later, then the onset of the new lactation will be delayed, and the present lactation prolonged. Cows give far more milk in the early stages of a lactation than they d o at the end, so the delay in conception and calving will entail a loss of milk. Data giving the average yields of milk from each stage of the lactation are available, but this makes no allowance for cows which have dried off. The difficulty in estimating the magnitude of this loss of milk lies in finding what proportion of cows would have dried off naturally before they calved. There are no reliable data on the average length of a lactation under different conditions and an indirect approach is therefore necessary. It is necessary to hypothesise some form of relationship for the effect of the number of days from the beginning of the lactation upon the proportion of cows which have dried off naturally (see Fig. 1). I I

Proportion of cowe still lactating at drying off (%L) I

i

ol

220 Days from start of lactation

Fig. 1.

i----~ 0

Calving to conception interval

Possible relationships between calving to conception interval and the proportion of cows still lactating at drying off.

It is assumed that the shortest normal lactation length (LL) is 220 days--which is the gestation period minus the standard 60-day dry period. Thus a 220-day lactation implies a calving to conception interval (CC) of zero. As the calving to conception interval lengthens, the lactation would be extended, until the cow dries off naturally. If it is assumed that the relationship between calving to conception interval and the proportion of cows still lactating when they have to be dried off is linear, then it can be stated that: %L= 1 +flCC

(1)

where fl is a constant. The average lactation length will increase by one day for each additional day on the average calving to conception interval, while the cows are all lactating at drying off. If only 80 ~ of the cows were still lactating at drying off, then for each additional day on the average calving to conception interval, the average lactation length would

MODELS OF ANIMAL HEALTH PROBLEMS

187

Lactation

length (days) (LL)

220

0 Calving to conception interval ( CC ) Fig. 2.

The relationship between calving to conception interval and lactation length.

increase by only 0.8 of a day. Thus it can be seen that the gradient of the line in Fig. 2 at any value of calving to conception interval will be the proportion of cows still lactating at drying off. Therefore: LL = 220 + ( ~ L ) C C

(2)

Substituting from (1): LL = 220 + (1 + flCC)CC LL = 220 + CC + flCC 2 The value o f fl can be estimated from this relationship by least-square regression, and substituted back into eqn. (1) to derive the proportion of cows still lactating when they have to be dried off, which is the n u m b e r of extra days of lactation obtained for each additional day on the calving to conception interval. Thus the cost o f infertility in reduced milk production can be estimated. The linear relationship used in eqn. (1) is obviously unrealistic. It seems reasonable to hypothesise that lactation length is a normally distributed variable, in which case the proportion of cows still lactating when they have to be dried off would be the area under the normal distribution function. We have been unable, as yet, to reduce this to an estimatable relationship. A cubic form does give rise to an estimatable relationship and at the centre of the curve the error is probably small. The resulting equations are: %L = 1 + fl~CC + j~2CC 2 -[- j~3CC 3 LL = 220 + CC + fllCC / + fl2CC 3 + fl3CC 4