A STATE-TRANSITION MODEL OF EPIDEMIC FOOT-AND-MOUTH DISEASE

William M. MillerEpidemiology Unit, Central Veterinary Laboratory,

New Haw, Weybridge, Surrey.

INTRODUCTION

In the course of a study of the likely economic effects of an outbreak ofFoot-and-Mouth Disease (FMD) in the USA, it was decided to investigatethe feasibility of mathematical modelling in the prediction of such anoutbreak.

Such a technique is inevitably required in the case of a cost-benefitanalysis of a disease control strategy, as it is necessary to hypothesizewhat would be expected to happen if the strategy were not applied.Ellis (1972) first used this approach in the evaluation of Swine Fevereradication in Great Britain.

Little has been published of a specifically veterinary nature in modellingor simulation, although the first attempts were made as early as 1866 byWilliam Farr, concerning the prediction of the cattle plague outbreak ofthat time. He had previously (1844) analysed the English small poxepidemic of 1937-39 and calculated that the second ratios of incidences (orratio of ratios) were approximately constant (which was analagous to fittinga symmetrical curve to the data).

In the case of the cattle plague, he employed the third ratio (for anunexplained reason) and made a remarkably precise prediction (Brownlee, 1915).

It was 40 years later when Brownlee began to publish work of a similarnature, in which he further developed the curve-fitting approach and notedthe tendency to positive skewness in epidemic curves (which he attributedto the diminishing efectivity of the organism). He also illustrated therelationship between "infectivity" length of epidemic and proportionremaining uninfected at termination (Brownlee, 1918).

From 1915 onwards, Sir Ronald Ross began to challenge the empirical approachand using malaria data from a particularly well-described situation) heproposed a more mechanistic technique based upon a set of differentialequations (Ross and Hudson, 1917). Epidemic models have taken two ratherseparate paths of development since Roes's time. One approach has beenthat of the pure mathematicians who have developed a number of theoreticalconcepts e.g., Kermack and McKendrick (1927) - 'Threshold Theory' andBailey (1957) - 'Generalized Stochastic Epidemics'. (An example ofveterinary interest is the work of Taylor (1968) on Optimal Control inBovine Viral Diarrhoea).

On the other hand, a number of models of specific diseases have beendeveloped recently, with the emphasis on realism rather than mathematicalrigour. In medical epidemiology, examples are found in:

Tuberculosis - Waaler & Piot (1969)- Waaler, et al (1974)

Leprosy - Lechat (1971)Common Cold - Hammond & Tyrrell (1971)Trachoma - Sundaresan & Assaad (1973)Malaria - Dietz, et al (1974)

New Techniques in Veterinary Epidemiology and Economics, 1976 (ISVEE 1)Available at www.sciquest.org.nz

57

Models have also been used to explore the economic aspects of controlprogrammes in diseases such as:

Tuberculosis - Revelle, et al (1969)Typhoid - Cvjetanovic, et al (1971)Tetanus - Cvjetanovic, et al (1972)

In the veterinary field, examples are:

Haemonchiasis - Ratcliffe, et al (1969)Mastitis - Morris (1972)Foot Mouth Disease Tinline (1972).

- Hugh-Jones (1973)Rubinstein (1975)

Brucellosis • Roe (1972)- Hugh-Jones, et al (1975)

Swine Fever Ellis (1972)

For an excellent introduction to epidemic simulation, the reader shouldconsult Bailey's text (1975), and possibly the historical review ofSerfling (1952).

What is plain from a review of the literature is that there is no singleapproach applicable to all infectious diseases and that the problems facedby . the veterinary epidemiologist are quite different to those in humandisease. Indeed, farm animal epidemics have much in common with thoseof plants and for that reason the conference reported by Kranz (1974)is of considerable interest.

AIMS

The aim of this work was to construct a model of FMD which wouldsimulate the spread of the disease across the USA starting at the stagewhere the disease was well-established and traditional control measureswere no longer effective.

WHY USE A MODEL?

Whenever we summarize a system verbally, physically or mathematically, weconstruct a 'model'. It is a symbolic description of that part of thesystem in which we are particularly interested. In epidemiology, math-ematical models (commonly evaluated by a computer) are of increasingimportance because they facilitate the study of the complex, interlockingsystems which influence disease spread. They can be used to simulatelarge outbreaks of disease which would be too costly or dangerous toexperiment with and allow numerous repetitions at minimal cost. Theyalso serve to organize knowledge and may indicate mechanisms of particular(and unexpected) importance. There may be many models of the same disease,all with a contribution to make. The choice of an appropriate one is oftendifficult and flexibility in this matter is vital. What is necessary is tostrike "... an appropriate balance between realism and abstraction for thepurpose in hand" (Patten, 1971).

HOW DOES ONE JUDGE THE WORTH OF A MODEL?

1. Its mechanisms should be intuitively acceptable.

2. It should behave in a biologically and mathematically reasonable way(e.g. it should be 'sensitive' to appropriate variables).

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3. It should mimic real life situations.

4. It should be simple enough for rigorous testing and yet complexenough to adequately represent the aspect of the system understudy.

5. It it is to be used as a tool in decision-making, its scope andlimitations should be fully comprehended by the decision-maker.

WHY THE STATE-TRANSITION MODEL WAS USED

There are, as has already been stated, a wide range of possible approachesto the simulation of disease spread. The present model was developed froma Markov Chain model, suggested by Drs. Reimann and Franti from Universityof California, Davis. (The Markov Chain is a special case of a State-Transition model.) However the basic assumptions (no history and constanttransition matrix - which will be discussed later) are not appropriate tothe epidemic situation. Rubinstein (1975) is currently developing a Markovapproach to endemic FMD in Colombia.

The State-Transition approach is more general and permits the building of amodel in which a variety of control strategies can be examined. It isalso easy to program and cheap to run. As the application involves large-scale epidemics, the fact that it is deterministic (not having an elementof randomness) is of no consequence. It can be used on populations ofalmost any size (e.g. that of the US FMD-susceptible population).

THE STATE TRANSITION MODEL

In this model, the basic unit is the herd (or premises). Herds areconsidered to be in one of four mutually exclusive "states" - 'Susceptible','Infectious', 'Immune' and 'Removed' or Dead. In the current version ofthe model (FMC?), only the following pathways (or 'transitions') shown inFig. I are considered.

Each week, the probability of every transition is calculated and theproportion of herds in each state during the next week are thus derived.This is termed a discrete-time process (the calculations are done at theend of each week).

It is convenient to represent the pathways in a transition matrix (or table):

TO (States of destination)FROM SUSCEPTIBLE INFECTIOUS IMMUNE REMOVED(States of origin)

SUSCEPTIBLE Remaining Infection Effective 'Contact'susceptible vaccination slaughter

INFECTIOUS •••••n• Convalescentimmunity

Slaughterof affected

IMMUNE Waningimmunity

n••••• Remainingimmune

REMOVED Restocking ••• 4WD Remainingdepopulated

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EaLn cell contains the probabilities of transition from one state to anotherduring a particular week. They are re-set each week, if necessary.

Not all transitions areconsidered in FMD7. Some are obviously impossible (e.g. Infectious toSusceptible) and others are unimportant (e.g. Removed to Infectious or'Recrudescncel).

By scanning each row horizontally, one can see all possible outcomes foreach state, in one week. The sum of these probabilities must be unity.That is to say, an 'Infectious' herd must become 'Immune' (through con-valescence) or 'Removed' (by slaughter) after one week. Conversely, eachcolumn shows the source of the immediate predecessors to each state (i.e.herds currently in the 'Immune' state must be derived from those that wereeither effectively vaccinated, acquired convalescent immunity or remainedimmune, in the previous week).

To run the model, it is necessary to calculate a list (or vector) ofproportions of herds expected to be in each state at the starting point.

Initially it is calculated as follows:

P(I)= Proportion of herds susceptible at the onsetP(2)= Proportion of herds infectious at the onsetP(3)= Proportion of herds immune at the outsetP(4)= Proportion of herds removed at the outset

Each succeeding week the transition matrix is then up-dated and multipliedby the probability vector. The productindicates the probability of herds being in each state at the end of theweek. The probabilities of the herds being in a particular state areextrapolated to indicate the proportions of the herds expected to be ineach state and thus the expected numbers in each state. This processcontinues for a specified number of weeks.

The approach used in FMD7 differs from that of Tin line (1972) in which adetailed analysis of the 1967-1968 FMD outbreak in the U.K. led to aspecial model capable of examining the wind-bourne spread hypothesis.In the FMD7 application, where the unusual meteorological conditions ofthe U.K. epidemic could not be assumed to be present, it was not possibleto utilize such parameters, nor was it possible to set-up a truly spatialmodel, due to the size constraints on a computer model of the entire USA.

SIMULATION OF DISEASE SPREAD

The probability of a susceptible herd becoming infected in a particularweek is considered as a function of the number of infectious herds (whichindicates the number of point sources of agent) in the previous week andthe dissemination rate (its propensity to spread to other herds).

Because herds are stationary and generally have only indirect or 'passive'contact with other herds, the traditional concept of 'adequate contact'occurring at random amongst a homogeneously mixing population (Abbey,1952) is inappropriate. An alternative approach is therefore suggested.The Dissemination Rate (DR) represents the average number of herds (orpremises) to which agent is delivered by each infected herd, irrespectiveof that herd's status.

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Dissemination Rate depends upon a number of factors:

TopographyHerd densityWeather, etc.HusbandryFomite opportunities (e.g. Milk tankers)MarketingPasture seekingMovementsDisease securityQuarantineMovements standstills

Environmental

Type of farming -

Animal movements -

Farmer behaviour -

Disease control efforts -

In epidemics affecting farm animals it is most unusual for the outbreakto be halted by lack of susceptibles alone. Very often diminishing DRhas the stronger influence upon termination.

Diminishing DR may be due to:

Factors initially favourable to dissemination no longer acting.Disease control effects.Increased awareness amongst stock owners.Less favourable topography being encountered.l Easy t targets having been used up.

As can be seen from Fig. 2 (data from 1967-1968 FMD outbreak in GreatBritain), Estimated Dissemination Rate (EDR) (which is equivalent togrowth rate in the following week) decays exponentially towards a valueof about .75.

Transformation of the data reveals that log In EDR can be convenientlydecomposed into two linear regressions, the gecond of which is approximatelyhorizontal.

In other words, Log DR can be estimated by three parameters:

BO

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So: Current Log DR = Bo + (B 1 * Current Week Number).

Subject to a minimum value ay. The constant DR (homogeneous mixing)situation is a special case where B is zero. This model is admittedlysimplistic, and has no explicitly Oatial characteristics, but does behaveIn an analagous way. A truly spatial model on a national scale wouldrequire a far more complex (and possibly unworkable) formulation.

To calculate the 'force of disease' i.e. the probability of a susceptibleherd communicating with at least one infectious premises, it is consideredas a Poisson variable, which estimates the probability of a herd beingexposed at least once, i.e.

Transition Probability = 1 e-DR * Pa

Where Pa=Proportion of herds infectious in the previous week. (Assumingthat herds excrete virus in significant quantities only during the weekfollowing infection).

Its value at time zero (intercept)

Its rate of decay (slope)

Its minimum value ('plateau')

63

SIMULATION OF CONTROL STRATEGIES

Provided that it is possible to represent a control strategy (or diseasepathway) in the transition matrix, it is a relatively simple matter toincorporate it in the program because such strategies can be representedas changes in particular transition probabilities.

Slaughter

In the model, slaughter of 'contact' herds is interpreted as 'removing'those herds which would have become infectious in the current week. Thestrategy is very effective in halting an epidemic, however, in reality itis difficult to identify a high percentage of such herds in time, unlessa 'scorched earth' policy is adopted.

Slaughter of infectious herds in the epidemic situation is necessary ifthey are not to remain as possible long-term foci of infection, butbecause the vast majority of virus is excreted before lesions appear,this strategy is not considered to affect the force of disease. Althoughit Is a simplistic view, it seems adequate for its purposes of thisepidemic model. In the endemic situation, it might be inadequate.

Infectious and contact herds which have been slaughtered join the'removed' category until re-stocking is due. They then revert to the sus-ceptible state.

Vaccination

It is possible to simulate a programme of vaccination during a specifiedtime period. This can be done by storing the number of herds becomingimmune each week in a vector, which is updated to account for newvaccinations, re-vaccinations, and waning immunity.

Movement Restrictions

It is theoretically possible to represent market closures, quarantine, etc.as a , reduction in DR. However, as these control ars difficult to quantifyand their specific relationship to DR is not known, It has not beenpossible to simulate them, beyond stating that a reduction of DR leads toa marked reduction in the epidemic size.

APPLICATION OF THE MODEL TO THE USA SITUATION

The objective was to simulate the most serious possible FMD epidemic inthe USA. Of course, epidemics of FMD are, by definition, rare events(only 8 epidemics in the U.S.A. since 1870) and the most severe are rarerstill (even in the UK where there were 632 'primary outbreaks" from1942 to 1967, 58% were confined to one premises). However, if the diseaseentered the marketing system, became established in a number of states andtraditional controls were abandoned, a national pandemic would beinevitable (Leading to an endemic situation).

How soon could this point be reached? Unfortunately, the only data fromwhich one can discover the possible rate of spread of FMD in whollysusceptible stock is that from the UK. One of the reasons for the term-ination of that epidemic was the topography - by the time of the peak,most of the Cheshire plains were penetrated. Assuming that no suchlimitation occurred in the USA (e.g. a Mid-West introduction), the DRwould not be expected to diminish as it did in the UK.

*"A primary outbreak is one that cannot be linked with any known sourcesof infection and is therefore attributed to the virus having been intro-duced from abroad".

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Fig. 3 demonstrates how, with DR's diminishing at a rate only slightly more slowly than UK figures, that a situation demanding the abandonmentof traditional controls could be reached in 4 or 5 weeks after intorduction.Of course, it could take much longer, but it is epidemiologically feasible,even in 4 weeks.

After abandoning traditional control measures, what then? For thepurposes of this simulation, the USA was divided into three regions (seeFig. 4. . The Range States (Region II)were treated separately, on the grounds that:

1. Rates of spread would be much slower under range conditions.

2. The disease would reach Region II, only when it was well-establishedin Region III (as the movement of cattle is generally out of Region II).Regions I and III were considered together, because of ma..etingconnections and the likelihood of spread into California feedlots anddairies.

The choice of DR was obviously crucial, but given that the objective wasto simulate the worst feasible situation, it was reasonable to be guidedby data (where available) and intuition.

A pandemic does not necessarily imply no control whatever and it is likelythat self-imposed quarantine and inter-state embargoes would at leasttemporarily halt the spread of the epidemic. Such factors cannot bequantified so the simulation was made on the basis that disease waswidespread (say 25 herds in each of the 40 states in Regions I and Ill)and that it would merely diffuse across each state, limited largely bylack of susceptibles (as the convalescent herds became immune).

The results are shown in Fig. 5. Inreality, because of the factors such as quarantine, the peak wouldprobably be smaller and have a more pronounced 'tail', due to the varioussub-epidemics (each with their own DR's) which tend to 'blurr' the peak.The predicted attack rates (56% in Regions I and III and 66% in Region Iare probably under-estimates for the same reason).

The DR in the pandemic was derived from that in the latter stages of theUK epidemic and was found to allow a reasonably protracted outbreak.

In summary, a 'run-away' FMD outbreak could peak in a little as 15 weeksand might affect 100,000 herds per week at that time. Such a pandemicwould be expected to significantly decline by the 30th week, havingaffected a minimum of 60% of the population. The epidemic would subsidedue to lack of susceptibles and would probably re-appear after the 60thweek, if immunity were to wane (e.g. due to herd turnover and natural lossof protection). It would then begin a series of endemic cycles (uponwhich seasonal patterns would probably be superimposed).

THE IMPACT OF 'CONTACT' SLAUGHTER

One of the tasks to which the model is well-suited is the demonstration ofthe (theoretical) relationship between levels of control and their outcome.Contact slaughter (CS) is a case in point.

The model was set to simulate the UK outbreak. Although a considerableamount of at the time 'dangerous contact' slaughter was undertaken, itis effectively included in the changing DR. So the CS rates are super-imposed on what was already happening.

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Effective CS requires that herds are removed before they become infectious.It is therefore extremely difficult to achieve rates of above 5)%,especially if the epidemic is of any size, but the results can be dramaticas Fig. 6 shows. Notice that 100% CS (the 'scorched earth' policy) is mosteffective. Fig. 7 shows the diminishing returns to effect beyond about60% CS and that if 19% of the infectious herds in the UK outbreak couldhave been anticipated by the CS, then the size of the epidemic could havebeen halved.

DISCUSSION

Harris (1960) states the... "even a bad model may be useful in providingsome kind of intellectual foothold In a new and difficult problem..."Although one can make no special claims regarding the reliability of theFMD7 simulation, its construction certainly brought a number of issues intosharper focus. The main problem has been to compromise between realism(which leads to impossible complexity and demands for data) and practicality(which may be too simplistic). The outcome has been a simple, flexiblemodel, which is economical to use and this has facilitated testing. Ithas provided a basis on which to estimate the scale of a 'runaway' epidemic.

Such a model can have other applications:

I. In the theoretical investigation of the relationship betweenepidemiological parameters e.g. Vaccination and stage of epidemic,which may provide some unexpected results, leading to further datacollection.

2. In the prediction of actual epidemics. Where the Dissemination Rateparameters can be measured from field data, and alternative strategyexamined. This version of the model is currently being developed.

3. In teaching (both students of epidemiology and decision-makers)the relative merits of the various strategies. This approachhas already been tried with some success and further developmentsare underway.

CONCLUSIONS

The State-Transition model provides a simple, versatile yet reasonablyrealistic representation of a large epidemic. Its non-spatial aspects canto some extent be compensated by a variable Dissemination Rate. The modelsuggests that in the event of a 'runaway' epidemic of FMD in the USA, that60% of the susceptible herds could be affected within 30 weeks and that inthe absence of vaccination, the epidemic would re-appear after 60 weeks andbegin a series of endemic cycles.

Even with dissemination rates only slightly greater than those seen in theUK epidemic, a 'run-away' situation (1000 herds per week) could be reachedin as little as five weeks.

The beneficial effects of contact slaughter are demonstrated and the modelindicates that if 18% of potentially infectious herds could be slaughteredbefore excreting virus, an epidemic (similar to the UK 67-68) could bereduced in size by 50%.

71

BIBLIOGRAPHY

1. Bailey, N. .J. (1957) The mathematical theory of epidemics. (1st Edn.)London: Griffin.

2. Bailey, N.T.J. (1975) The mathematical theory of infectious diseasesand its applications. London: Griffin.

3. Brownlee, J. (1915) Historical note on Farr's theory of the epidemic.Brit. Med. J. p.250-252

4. Brownlee, J. (19special relation

18) Certain aspects of the theory of epidemiology into Plague. Proc. Roy. Soc. Med. II: p. 58-132

5. Cvjetanovic, B.,of Typhoid Feverimmunization andp. 53-75

Grab, B., and Uemura, K. (1971) Epidemiological modeland its use in planning and evaluation of anti-typhoidsanitation programmes. Bull. Md. Hlth. Org. 45:

6. Cvjetanovic, B., Grab, B., Uemura, K. and Blytchenko, B. (1972)Epidemiological model of Tetanus and its use in the planning ofimmunization programmes. Int. J. Epid. 1: p. 125-137

7. Dietz, K., Molineaux, L. and Thomas, A. (1974) A malaria model testedin African Savannah. Bull. Med. Hlth. Org. 50: p. 347-357

8. Farr, W. (1840) Progress of Epidemics. Second Report of the RegistrarGeneral of England and Wales. p. 91-98.

9. Hammond, B.J. and Tyrrell, D.A.J. (1971) A mathematical model ofcommon-cold epidemics in Tristan da Cunha. J. Hyg. Camb. 69: p.423-433.

10. Harris, J.E. (1960) Models and analogs in biology (Ed., J.W.L. Beament).Cambridge: University Press. p. 250-255

II. Hugh-Jones, M.E., Ellis, P.R. and Felton, M.R. (1975) An Assessment ofthe eradication of Bovine Brucellosis in England and Wales. Universityof Reading, Dept. of Agriculture and Horticulture. Study No. 19,75 pages.

12. Kermack, W.O. and McKendrick, A.G. (1927) A contribution to the mathe-matical theory of epidemics. Proc. Roy. Soc. A. 115: p. 700-721.

13. Kranz, J. (1974) Epidemics of plant diseases. Mathematical analysisand modelling. London: Chapman and Hall.

14. Lechat, M.F. (1971) An epidemiometric approach for planning andevaluating Leprosy control activities. Int. J. Leprosy 39: p.603-607

15. Morris, R. (1972) Mastitis model.

16. Patten, B.C. (1971) Systems analysis and simulation in ecology.New York & London: Academic Press.

17. Ratcliffe, L.H., Taylor, H.M., Whitlock, J.H. and Lynn, W.R. (1969)Systems analysis of a host-parasite intraction. Parasitology 59:p. 649-661.

18. Revelle, C., Feldmann, F. and Lynn, W. (1969) An optimisation modelof Tuberculosis epidemiology. Management Sci. l&: B. p. 19)-211

19. Roe R, (1976) Personal communication.

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20. Ross, R. (1995) Some of a priori pathometric equations. Brit.Med. J. I: p. 546-5471

21. Ross, R. and Hudson, H.P. (1997) An application of the study of theoryof probabilities to the study of a priori pathometry. Parts II andIll. Proc. Roy. Soc. A., pgs. 212-225, 225-240.

22. Rubinstein, E.M. de (1975) Methodology proposed for the economicevaluation of Foot and Mouth Disease Control Unpublished UATdocument (December 1975).

23. Serfling, R.E. (1952) Historical review of epidemic theory.Hum. Biol. 24: p. 146-166

24. Sundaresan, T.K. and Assaad, F.A. (1973) The use of simple epidemio-logical models in the evaulation of disease control programmes: astudy of trachoma. Bull. Wld. Hlth. Org. 43: p. 709-714.

25. Taylor, H.M. (1968) Some models in epidemic control. Math.Biosci., 3: p. 383-398.

26. Tinline, R. (1972) A simulation study of the 1967-68 Foot-and-MouthEpizootic in Great Britain. Thesis: Bristol., 351 pages.

27. Waaler, H.T. Gothi, G.D., Bally, G.V.J. and Nair, S.S. (1947)Tuberculosis in rural South India. A study of possible trends andpotential impact of antituberculosis programmes. Bull. Wld.Rog. 51: p.263-271.

28. Waaler, H.T. and Piot, M.A. (1969) The use of an epidemiologicalmodel for estimating the effectiveness of Tuberculosis controlmeasures. Bull. Md. Hlth. Org. 41: p. 75-93.