Modelling and Foot and Mouth Disease (FMD):

An Evaluation of the Techniques used in 2001 for Determining Control Policy

 

 

Susan Haywood PhD, BVSc, MRCVS and George Haywood BSc, MSc (Structural Eng), MSc (Knowledge Based Systems)

 

The modelling of biological systems is increasingly being used to predict their behaviour and to assess the relative merits of strategies that can be used to modify this behaviour; the control of infectious diseases being one application. Modelling methods have been used to determine control policy in the 2001 FMD outbreak in the UK. The criticism this has evoked, together with a lack of understanding of the issues involved, has created a need to assess how the respective modelling techniques were used in the 2001 outbreak and to evaluate their relative performance.

 

The aim of this paper is to present the arguments from the modelling perspective integrated within a veterinary and social framework.

 

The Modelling Paradigm

Modelling tries to represent real world events by utilising just those features that contribute to real world behaviour. Modelling, although seemingly objective, should be seen as being a subjective activity in which the world-view of the modeller is an integral part of the process. Thus, models can either be qualitative, in which non-dimensional properties of the world, human desires, needs and intentions are explored, or quantitative, in which numerical values for the quantifiable features such as cost/benefit can be obtained.

 

There are two main types of quantitative model: deterministic and non-deterministic. Deterministic models are those in which a set of mathematical formulae is used to calculate the required values. Non-deterministic models allow variations, usually on a probabilistic basis, in the values of the variables modelling the features of the problem, and whose outcome varies from run to run. Both types have been used in modelling the FMD epidemic. The adjective, ‘parametric’ is used in the reports of both types of model, but is different in sense. For consistency, we have used the following terms: ‘parameter’, a mathematical quantity which represents the features in the world and which will take different values according to the needs of the model (distance between farms, for example); ‘coefficient’, a modifier to the parameter which remains constant during the model’s behaviour, (susceptibility of an animal to infection, for example).

   

There are a number of criteria that should be taken into account when devising a quantitative model:

 

a)       type of model that should be used. If the medium is assumed to be homogeneous then this can be realistically modelled using differential equations that can be solved numerically. If however, the domain happens to be a set of discrete entities that are related in such a way that cannot be expressed in a continuous function, then a non-mathematical model will be required.

b)       choice of factors with which to represent the domain. These will become the parameters of the model and will have values attached to them. The choosing of the parameters is guided by the way the modeller sees the situation, by the requirements of the problem to be solved and the type of model chosen.  The level of detail will also determine the parameters. If the domain is modelled in too great a detail, then some of the parameters will be extraneous, and including them might only confuse the output from the model. Some parameters might not even contribute to the problem solving, although this can be tested by a sensitivity analysis in which a particular parameter is varied, all others being kept constant. If there should be an appreciable effect in the output, then that parameter would be deemed to be sufficiently important and relevant to be included in the model.

c)       values of the coefficients that are required to modify the parameters.

d)       correct data for populating the model. This must be accurate and consistent with the domain being modelled. The verification of the model is concerned with the model performing the calculations correctly (the syntax of the model); the validation is concerned with the model modelling the right thing (the semantics of the model).

e)       proper methodological use of the model. The model itself, even if verified, must still be operated correctly to give valid results.

 

Description of the Models

Four modelling teams were used to inform policy in the FMD outbreak:

a)       Imperial College model developed by Professor Anderson and colleagues from Imperial College. This is a deterministic mathematical model in which the disease is assumed to flow through the population, rather like heat flow through a medium.

b)       Cambridge/Edinburgh combined model developed by Professor Grenfell from Cambridge and Professor Woolhouse from Edinburgh and their colleagues. This is a non-deterministic model that simulates the progress of the disease from farm to farm on a daily basis, using the Monte Carlo technique to determine the likelihood that a farm will be infected.

c)       Veterinary Laboratories Agency (VLA) model developed by Professor Wilesmith from the state Veterinary Labs Agency backed up by colleagues from Massey University, New Zealand. This is a non-deterministic model based on InterSpread, a computer program which can be used to model an FMD epidemic and also uses the Monte Carlo method.

 

Imperial College Model

The findings from the development of this model, published in May and October 2001, show that there was a development of the model during the progress of the epidemic. Because government policy decisions were made during March 2001, and were informed by the early model, it is this earlier version of the Imperial College model that is described and assessed in this critique. 

 

The model is a deterministic, mathematical model that was derived from experience with human sexually transmitted disease and which originally assumed a homogeneous animal population, with no distinction being made between species, and with a constant rate of disease spread from an infected source. The progress of the disease is estimated from the value of Ro, the transmission potential of the infectious agent. Ro measures the average number of secondarily infected farms generated from a primary source of infection in a susceptible group of farms. If Ro is < 1, then there will be no further spread of the disease, and is the test condition at which the disease is assumed to be under control. It is estimated from the contact data by multiplying the average number of infectious days by the average number of farms infected per infectious day. Ro is used to predict the behaviour of the set of farms in the region, or in the country as a whole; it is not used to predict where the disease might occur next.

 

Factors such as contact spread, variation in farm size, differences in species and numbers were only incorporated into the model at a later date.

 

Cambridge/Edinburgh Model

This model uses the individual farm as the primary unit. An uninfected farm is susceptible to infection according to the number and types of livestock it holds. Each animal on the farm is accorded a susceptibility to infection and transmissibility value on a species basis. These coefficients, in conjunction with the inter-farm contacts (the dispersal kernel), are used to determine the probability of a particular farm being infected on a particular day. In each run of the simulation, the event that a susceptible farm becomes infected is determined by whether the random number generated for that farm is at least equal to the calculated probability. The progress of the disease each day, is then simulated by performing the set of simulations 100 times, and observing which of the farms are most likely to become infected.

 

The values of the coefficients for infectiousness, susceptibility and transmissibility are obtained by relating the observed incidences of the disease from the figures collected from MAFF/DEFRA against the predicted incidences derived from using the model.

 

 VLA Model

 The base population for this model is the spatial locations of all farms and animal markets. The underlying operation for InterSpread is that of the Cambridge/Edinburgh model, but the number of factors that are modelled in InterSpread are much greater. A review of the parameters used in the program show it to be particularly biased towards dairy farming; these include the proportion of farms with lactating cattle, the probability of a farm being selected for a particular tanker route etc. Thus, whilst the Cambridge/Edinburgh model contents itself with just three parameters, InterSpread has over fifty. It has coefficients for susceptibility of all types of animal, but considers only vaccination of cattle and pigs – revealing its subjective bias!

The disadvantage of having so many parameters is that each simulation run of the model can be intensive in computational power, and so the number of runs to determine the daily spread of the disease was limited to only five.

 

Using the models

 

Because the models were intended to help formulate policy, different control strategies were assessed by the models. The two types of control examined were culling in which the area of cull and the speed of cull were the two key parameters, and vaccination.

 

The Imperial College model showed that Ro was sensitive to delays in identifying, reporting and confirming the presence of the disease; and the speed by which the infected farm could be culled. Ro could be reduced if the animals on an IP could be slaughtered within 24 hours, without waiting for laboratory confirmation, although Ro persisted at >1. However, Ro could become < 1 if contiguous premises were culled within 48 hours of the infected premise being culled. Analyses showed that both ring culling or ring vaccination to be both equally highly effective strategies if implemented rigorously. However, the team deemed ring vaccination would have to be more extensive than ring culling because of potential infected non-responders in the vaccinated population. Given this, together with the EU ruling on eventual culling of all vaccinates to resume trade in the short term, they concluded the superiority of ring culling.

 

Two points emerge from this analysis: the nature of the model meant that ring vaccination could only be modelled as the vaccination of all animals within a given radius from the infected farm. The more usual interpretation of ring vaccination, that in which all animals within an annulus several kilometres deep and several kilometres from the infected farm could not be considered. Neither could the effect of protecting rare pedigree stock or threatened genotypes be incorporated into the model due to its insufficient flexibility. Secondly, the team went outside its remit in promoting the cull option in  so much that it failed to consider the EU ruling which allows vaccinates to live after a longer period (12 months) of ‘quarantine’.

 

The Cambridge/Edinburgh model showed similar results in that speed of identification and cull was important – as would be expected. The high spatio-temporal resolution of the model allowed it to successfully predict and explain the long ‘tail-off’ of this particular epidemic. The vaccination of cattle as an alternative to neighbourhood culling was explored. This showed that prompt vaccination from the start of the epidemic resulted in a reduction in disease outbreaks, although the same result could be achieved by culling alone if not delayed. Again they promoted the cull strategy above vaccination on account of the supposed logistical difficulties of a mass vaccination programme, but without a similar examination of the difficulties inherent in the killing and disposal of millions of animals in an equally  limited time scale.

 

A major shortcoming in the use of this model was the failure to examine a more flexible approach to vaccination. The EU ruling allowed rare breeds to be vaccinated without compromising the return to disease-free status. The model was perfectly capable of representing such farms as isolated areas of vaccinated animals, and it would have been instructive to see the effects of having such protected farms capable of acting as buffers against the spread of disease.

 

The VLA model showed that pre-emptive slaughtering of an average of 1.4 farms for every infected farm meant that the disease could be eradicated by the end of October at the latest, and was in accordance with the other models. Vaccination as a control policy was modelled and summarily dismissed by the team. However  the strategy employed – the dividing of the country into five separate areas, separated by bands of vaccinated cattle has already been discussed by Paul Sutmoller and others in letters to the Veterinary Record and dismissed as "lacking disease-control logic."

 

 

Assessment of the Models against the Criteria

The Imperial College and the Cambridge/Edinburgh models, although using deterministic mathematic modelling and stochastic simulations respectively, were based on the use of similar factors: infectiousness and susceptibility of the animals and transmissibility of the disease. The VLA model had similar coefficients, but was larger and included many more parameters, all biased towards dairy farming. In all the models, the values of the coefficients had to be established from the progress of the disease: thus, susceptibilities, infectiousness of the different types of animals could only be determined after the infections had occurred. In the early stages of the disease, the models could only track its progress, and were not capable of making accurate predictions.

 

Some data, such the geographical locations of each farm, its size, the types and number of livestock were available from several databases held by MAFF, the VLA and county-parish-holdings (CPH). This data was often inconsistent because the databases were unreconciled. The quality and integrity of the data used by all the modelling teams has been severely criticised by Valerie Lusmore, a professional data analyst, who concluded that it was below the standard acceptable in modern practice. Furthermore she stated that until data validity was improved only simplistic models should be used as an adjunct to, rather than a means of, determining policy.

 

A model might have the relevant parameters and values of the coefficients consistent with the properties of the disease, but might still be used incorrectly. The Monte Carlo method usually requires multiple runs of 1000 - 2000 for each simulation to give valid results. The Cambridge/Edinburgh model reported using 100 runs; this should be just acceptable to comply with a stochastic approach to modelling. The VLA model however, only used five runs per simulation, which seriously calls into doubt the validity of the results.

 

Conclusions

The benefit of having a valid model is that different control strategies can be evaluated. The strategies examined in the papers were for cull and vaccination. All models showed that culling just the infected premises meant that the disease spread rapidly, so a combination of cull of infected and contiguous premises were also examined. The Imperial College and the Cambridge/Edinburgh models examined the use of ring or neighbourhood vaccination. Both models showed that vaccination controlled the disease as effectively as rapid culling, but vaccination was dismissed on logistical grounds that were outside the bounds of the model. The reasons given related both to EU ruling and to the difficulties in implementing a vaccination policy – but these were often based on spurious assumptions of vaccine availability, efficacy and personnel required.

 

At the beginning we commented on the subjectivity inherent in modelling. In the virtual world of the modeller only the selected criteria – in this case disease control – are considered. However in the real world the outcome of this approach immediately runs into difficulties of an ethical and legal nature. For example, once the decision has been taken to cull on suspicion rather than awaiting confirmation, veterinary professional ethics are breached as has been cogently argued by Alan Richardson MRCVS. Also the implementation of the 3km cull went beyond the 1981 Animal Health Act (Order 1983) – tacitly acknowledged by MAFF/DEFRA in the need to get the “voluntary” co-operation of farmers and the more recently proposed Animal Health Bill. After such guidelines were breached, the tactics employed to implement the cull policy created such dismay and loss of confidence in the government's handling of the FMD epidemic that the co-operation between the farming community and central government will probably never be regained.

 

Finally, it should be recognised that quantitative modelling is seductive in that it delivers a result that appears to be objective. In all the models used, it is the spread of the disease that is modelled. Crucially important factors such as preservation of  biological diversity in the form of pedigree stock and rare breeds together with the preservation of the rural economy were never part of the models, nor fell within their remit. And yet, it is this aspect that should be regarded as being of primary importance. Only when the complexity inherent within natural systems is fully recognised, and the hopes, fears, aspirations and expectations of all the stakeholders are taken into account, will the quantitative models have their place.

 

Sources

Ferguson, N M, Donnelly, C A and Anderson R M (2001a) The Foot-and –Mouth Epidemic in Great Britain:Pattern of Spread and impact Interventions. Science,vol 292, 11May 2001, 1155-1160.

 

Ferguson, N M, Donnelly, C A and Anderson R M (2001b) Transmission intensity and impact of control policies on the foot and mouth epidemic in Great Britain. Nature, vol. 413, 4 October 2001, 542-8.

 

Ferguson, N M, Donnelly, C A and Anderson R M.  The determinants of transmission intensity and the impact of control policies on the foot-and-mouth disease (FMD) epidemic in Great Britain. Supplementary Information

 

Keeling, M J, Woolhouse, M E J, Shaw, D J, Matthews,L, Chase-Topping, M, Haydon, D T, Cornell S J, Kappey, J, Wilesmith, J and Grenfell, B T. (2001) Dynamics of the 2001 UK Foot and Mouth epidemic: Stochastic Dispersal in a Heterogeneous Landscape. Science Express 04 October 2001, pages 1-7,/10.1126/science.1065973

 

Keeling, M J, Woolhouse, M E J, Shaw, D J, Matthews,L, Chase-Topping, M, Haydon, D T, Cornell S J, Kappey, J, Wilesmith, J and Grenfell, B T. Supplementary Material for Dynamics of the 2001 UK Foot and Mouth epidemic-dispersal in a heterogeneous landscape.

Science magazine www.sciencemag.org/cgi/content/full/1065973/DC1/1

 

Morris, R S, Wilesmith, J W, Stern, M W, Sanson, R L, and Stevenson, M A (2001) Predictive spatial modelling of alternative control strategies for the foot and mouth epidemic in Great Britain, 2001. VeterinaryRecord, Aug 4 2001, 137-14

 

Letters: Keith Sumption, Suzanne I Boardman et al, Paul Sutmoller and Kate Wood.  Modelling control strategies for foot-and-mouth disease. Veterinary Record, Aug 25, 2001.

 

Valerie Lusmore, Submission to the Royal Society of Edinburgh Inquiry into Foot and Mouth www.warmwell.com/mathsubmar11.htm

 

Alan Richardson. FMD: did MAFF/DEFRA bring the profession into disrepute? Veterinary Times Vol 32, No 1, page 1.

 

 

 

Authors affiliations

Susan Haywood is a Senior Fellow in the Department of Veterinary Pathology, Liverpool University.

George Haywood is a consultant in complex systems and a tutor with the Open University.