The contribution of modelling to our understanding of infectious diseases

The modelling of vector dynamics in disease research

The impact of modelling on animal disease control

*G.F. Medley* and R.M. Anderson*^{+}

* Department of Biological Sciences

Warwick University

Coventry, CV4 7AL, UK

^{+}Department of Zoology

University of Oxford

South Parks Road

Oxford, OX1 3PS, UK

Abstract

Introduction

Population versus individual perspective

Quantitative approach

Static evaluation of helminth disease

Dynamic impact on helminth populations of chemotherapy application

Qualitative versus quantitative prediction

Vaccination against childhood viral diseases

Demographic impact of aids

Conclusions

References

Study of infectious diseases requires an ecological perspective: understanding of the biology of the host and pathogen (and intermediate hosts) is essential to comprehension of the transmission of infection and consequently patterns of disease. The veterinary and medical perspective of infectious disease is directed largely at the consequences of infection to the individual host, and tends to overlook the importance of population-level transmission processes. This paper will emphasize the value of modelling in development of understanding of population level issues in infectious disease control by reference to a variety of cases where this approach has led, and may lead in the future, to insights not as easily available to other approaches.

The quantitative approach is likely to become increasingly important in infectious disease epidemiological research and infection control. This is due to the increasing role of economic considerations in infectious disease control. By way of illustration, we can consider the industrialized and nonindustrialized regions of the world, between which the epidemiology and control of infectious disease differs markedly.

In developed countries, infectious disease is typified by either low incidence of disease or largely self-contained outbreaks. This is mainly due to resources available to vaccinate large proportions of the populations, the infra-structure available for continuous monitoring of infection, and the generally improved nutritional and health status of potential hosts. In these regions, the aim is to produce control strategies that cost less to implement than the productivity gained by the intervention. In contrast, the developing countries continue to bear an enormous infectious disease burden. In these countries, the aim is more comparative - given limited resources, which infections should be targeted by which method.

Quantitative frameworks are essential for rational consideration of policy in either instance. However, this approach requires that the population and transmission dynamics of the infection are well understood and characterized before economic and logistic variables can be combined into a single framework to consider the cost and benefits of different control strategies. Use of mathematical models within economic frameworks has been hampered by the lack of understanding of detail of transmission dynamics. However, as understanding is increased, so the models become more realistic and reliable, and they can be used as a basis for design of control strategies. This point is illustrated by consideration of several different infections under different states of development. Models of human helminths and human childhood viral infections have developed to the point where they have predictive value and can be combined with economic frameworks. In contrast, quantitative description of the transmission dynamics of HIV (and the consequent patterns of AIDS) is beset by inadequately described processes. In the latter case, models are useful for more qualitative predictions.

This paper is intended as a brief review of the usefulness and pitfalls associated with mathematical modelling of infectious disease agents. Readers are referred to Anderson and May (1991) for a fuller and more detailed account of methods and applications to particular diseases. Infectious diseases require special consideration for two reasons. First, they involve the interaction between two separate organisms: the host and the infectious agent. Second, in contrast with non-transmissible diseases, the infection (and disease) of one host provides the source of infection (and disease) to other hosts. This latter effect is inherently non-linear, generating epidemics that show complex patterns with time. Consequently, dynamic models (e.g. differential equations and Monte Carlo simulations) are the tools most commonly used in the study of infectious diseases, in contrast to the statistical models used in non-transmissible disease dynamics.

Study of infectious diseases requires an ecological perspective: understanding the biology of the host and pathogen (and intermediate hosts) is essential to a comprehension of the transmission of infection and consequently of the patterns of disease (Anderson and Thresh, 1988; Anderson, 1990). The veterinary and medical perspectives of infectious disease are directed largely at the consequences of infection to the individual host, and tends to undervalue the importance of population level transmission processes. This paper will emphasize the value of modelling in development of understanding of population level issues in infectious disease control by reference to a variety of cases where this approach has led, and may lead in the future, to insights not as easily available to other approaches.

The strategy by which infectious disease is controlled is increasingly determined by economic considerations, and consequently the quantitative approach is likely to become increasingly important in infectious disease epidemiological research. By way of illustration, we can consider the industrialized and nonindustrialized regions of the world, between which the epidemiology and control of infectious disease differs markedly.

In the developed countries, infectious disease is typified by either low incidence of disease or largely self-contained outbreaks. This is mainly due to resources available to vaccinate large proportions of the populations, the infra-structure available for continuous monitoring of prevalence of infections and the generally improved nutritional and health status of potential hosts. In these regions, the aim is to produce control strategies that cost less to implement than the productivity gained by the intervention. In contrast, the developing countries continue to bear an enormous infectious disease burden. In these countries, the aim is more comparative - given limited resources, which infections should be targeted by which method.

Quantitative frameworks are essential for rational consideration of policy in either instance. However, this approach requires that the population and transmission dynamics of the infection are well understood and characterized before economic and logistic variables can be combined into a single framework to consider total cost and benefits of different control strategies. Use of mathematical models within economic frameworks has been hampered by the lack of understanding of detail of transmission dynamics. However, as understanding is increased, the models become more realistic and reliable and they can be used as a basis for the design of control strategies. This point is illustrated by consideration of several different infections under different states of development. Models of human helminths and human childhood viral infections have developed to the point where they have predictive value and can be combined with economic frameworks. In contrast quantitative description of the transmission dynamics of human immunodeficiency virus (HIV), and the consequent patterns of AIDS, is beset by inadequately described processes. In the latter case, models are useful for more qualitative predictions.

Infectious diseases occur as a consequence of parasitism by one organism of another. Parasites (including all members of the *Protista*) and hosts are ecological entities which are subject to evolutionary pressures to maximize reproduction. Parasitism is one life-style that organisms have adopted. Consequently, the study of disease transmission and propagation is an essentially ecological subject and in order to understand the parasite it is necessary to consider its ecology in terms of the parasite's reproduction and survival. In this respect, the parasite's effect on the host (disease) is only relevant in terms of the repercussions that these have on the survival and reproduction of the parasite. This concept of considering the parasite's point of view is contrary to the education and training that the veterinary profession currently receives. In this subject, the emphasis is strictly on the host, the pathological effects of parasites and the cure of disease within single hosts. This is not intended as a criticism of the veterinary profession, which should be concerned with the wellbeing of individuals, but is intended to demonstrate that understanding of infectious disease transmission dynamics is not best served by the veterinary perspective alone.

As an example, consider helminth parasites. The usual pattern is for parasites to have a highly clumped distribution within the host population: most hosts have few or no parasites, while a few have many parasites (Anderson and May, 1991). A simple assumption is that the more helminth parasites that a host harbours, the more likely that host suffers disease (clinical signs of infection) and the greater the loss of production of that host. Figure la shows those two assumptions. These can then be combined to consider the herd (community) loss due to parasites, shown in Figure 1b, in terms of the production loss by degree of parasitism. The greatest individual loss occurs in those hosts with intermediate burdens (the curve is peaked). Individuals with intermediate burdens are individually less affected than those with the heaviest burdens, but this is outweighed by greater representation of intermediate burdens within the herd. Consequently, if the aim of control is to reduce disease within the herd, targeting those individuals with the highest burdens will not be as effective as targeting those with intermediate burdens as well, or even instead.

A mathematical model of infectious disease agent transmission is a framework of ideas including individual level processes expressed in a mathematical formulation. Consequently, an understanding of the processes at the individual level is required, for example: mode of transmission, rates of reproduction, mortality and density dependence in reproduction, survival and establishment. When these are combined into population level models, their amalgamation can produce counter-intuitive results, as a result of the non-linear nature of the individual processes. The comparison of individual and herd production losses illustrated above (Figure 1) serves as an example of this. Once such understanding of the population dynamics exists and a framework has been developed, it can be used in several ways.

**Figure 1. (b) shows the combination of the two relationships in (a) as a function of parasite burden. The line represents the product of the frequency with which a burden occurs and the production loss associated with that burden, and is therefore the production loss associated with each parasite burden in a herd.**

The simplest use is to examine the population effects of disease. M.S. Chan, D.A.P. Bundy, G.F. Medley and D. Jamison (manuscript under preparation) used the understanding of the population dynamics of human helminth infections to assess the health impact of these parasites at a global level. They considered six levels of heterogeneity within their framework. First was the distribution of helminths within communities, which shows surprising consistency across infected communities (Guyatt *et al.,* 1990). Second was the distribution of community level prevalences within geographical locations, which was estimated by collation of many observations of prevalences (proportion of individuals infected) within different communities. For example, the overall prevalence of infection within a country may be 50%, but there will be a distribution of prevalences such that some communities will have much higher prevalences, some lower and some will be uninfected. Third was the observed geographical heterogeneity on a country basis, such that some countries have higher overall prevalences than others. Fourth was the effect of age on the prevalence of infection. Fifth was the effect of age on disease. The health impact of infection was measured simply as the proportion of people with helminth burdens greater than some threshold value, and this phase value changes with age, such that younger children have a lower threshold. Sixth was the demographic heterogeneity in terms of the age structure of populations within different countries, which was used to convert the proportions into the numbers of individuals suffering disease. The results indicate that even with conservative estimates of the threshold and prevalence of infection, the number of people with deleterious health consequences of helminth infection is up to two orders of magnitude greater than previously estimated from case reports.

This is an example of the use of understanding of population level effects. It is not a dynamic model (there is no time component), but a static examination of the distribution of helminth parasites to assess their health impact. Realistically, this will never be estimable by any other means than a modelling approach, emphasizing the value of mathematical modelling. Further, it is necessary to measure the impact of infection to enable health policy-makers to prioritize resource allocation, and, as it must be done, is best done by those with some knowledge of the population biology of helminths.

In order to extend this analysis of the effect of parasite burdens to the benefits that can be accrued from chemotherapeutic interventions, the models must contain some dynamic component, i.e. the parasite population must change with time. Chemotherapy application reduces the helminth population (across all hosts) to a proportion of its pre-treatment level dependent on the treatment regime implemented. Following the application, the helminth population increases to recover its precontrol level. A framework incorporating this dynamic aspect plus the distribution of parasites required to estimate their health impact has been developed by Medley *et al. * (1993).

One of the problems with constructing the framework is the lack of knowledge concerning the mechanisms that generate the observed distribution of helminth parasites (Bundy and Medley, 1992). Chemotherapy will alter the parasite distribution, but during the period of helminth population expansion following chemotherapy, the original pretreatment distribution of parasites is regained, and some mechanism must be postulated for this to occur. The simplest, most general mechanism that is consistent with empirical observation was used, but this illustrates that the biological and ecological processes must be understood to generate a population level model.

The most striking result obtained was the non-linear relationship between disease prevention and treatment effort measured as the proportion of people treated (coverage) and drug efficacy. This arises from the fact that increased treatment reduces the rate of establishment of worms across the whole host population (untreated as well as treated), so that those hosts that went untreated gained more benefit as the helminth population was reduced. The aim of this research is to incorporate economic and logistic costs of treatment in interventions. Without consideration of costs of treatment the most effective control program is to treat everybody all the time. Incorporation of costs allows different programs to be compared in terms of both the benefits gained (disease prevented) and the cost of that benefit.

Mathematical models of infectious agent transmission provide quantitative results, i.e. numbers. However, the reliability of the results is not dependent on the numerical precision to which they are quoted, but on the biological and ecological understanding that underpins the model. It is possible to generate mathematically-complex models that produce virtually worthless results because the biological assumptions are unfounded. Even if the biological assumptions and estimates of the parameters of the biological processes are perfect, there remain many problems in interpreting numerical results of mathematical models in terms of predictions. It is unlikely that models of infectious diseases will have the predictive power of, say, models of the solar system, given the degree of complexity and heterogeneity within biological systems. Consequently, it is very unlikely that models may be used to predict the parasite burden of individual hosts.

Returning to the example of the helminth control program model described above, it is again unlikely that such a framework will be able to predict accurately the quantitative effects of chemotherapeutic interventions. However, what is required of the model is that it provides the correct rank ordering of different programs, i.e. that it correctly predicts that program A will cost more and be less effective than program B. This type of quantitative prediction is perfectly feasible, and more robust than qualitative prediction.

Viral infections of childhood, especially measles, remain significant causes of morbidity and mortality throughout the nonindustrialized countries. Models of the transmission of these disease agents have developed significantly over the past decade to the point where they are used for quantitative examination of different vaccination policies (Anderson and Grenfell, 1986; Nokes *et al.,* 1990; Anderson and May, 1991). In developing countries there is difficulty in vaccinating children before they become infected. Live virus vaccines fail to provide protection if the child has significant levels of maternally-derived antibodies so that ideally it would be better to wait until this passive protection had waned (about one year) before vaccination. However, transmission may be high in these areas, so that if a significant proportion of children have lost maternal protection, they will become infected and suffer disease before one year of age. Consideration of different vaccine formulations that overcome the maternal protection to some extent and can be administered to younger infants has been usefully done within a model framework to assess the impact of such a change in vaccination policy (McLean *et al.,* 1991).

In developed countries, the problems are different. Vaccination coverage is generally at very high levels, and transmission is on the verge of being halted. Here, the problems are associated with epidemic outbreaks in groups of unimmunized people, and with vaccine safety. As transmission is reduced, so individuals are at considerably diminished risk of acquiring infection and suffering disease. Consequently, the number of people who suffer adverse effects due to the vaccine can outweigh those suffering the adverse effects of disease. This poses a dilemma in regard to the benefits of vaccination to communities and individuals. A quantitative analysis of the situation, in particular with regard to the use of more immunogenic vaccines with higher reactogenicity versus less immunogenic and reactogenic vaccines, can only be performed with the use of a transmission model (Nokes and Anderson, 1991). Models of this sort can only be used quantitatively when the epidemiological and biological details of transmission and infection are well understood.

In contrast to the situation with childhood viral infections is the situation with regard to the human immunodeficiency virus (HIV) and the acquired immunodeficiency syndrome (AIDS). It was only in 1982 that this syndrome was recognized. The incubation time to AIDS (the length of time between infection with HIV and clinical diagnosis of AIDS) is very long (median about 10 years) and variable (Hendriks *et al.,* 1993). Consequently, although knowledge of the biochemical and immunological effects of the virus is growing rapidly, epidemiological knowledge is growing at a much slower rate. The longest cohort study has been following individuals since 1979, and the whole range of incubation periods has still not been observed. It is also not known what proportion of those infected will develop AIDS and over what time scale, nor the fate of those that do not develop AIDS.

Of particular interest with regard to the implications of the AIDS pandemic is the effect that it will have on the demography of developing countries, which are currently suffering the greatest prevalence of HIV infection (Anderson *et al.,* 1988; Anderson and May, 1991). Because of the lack of epidemiological and biological knowledge, mathematical models of this situation cannot be considered predictive. However, they do serve to provide qualitative results. They indicate that if demographic impact is to be marked (for example to produce a reduction in population size), then this will be seen over a time scale of decades. They also highlight those epidemiological and biological variables that are most influential in determining the epidemic pattern, and therefore those variables that must be better understood if quantitative prediction is to be possible. The most important of these variables are the full incubation period distribution, the infectivity of individuals throughout the incubation period, and the patterns of sexual behaviour (the number of different sexual partners and how they are chosen with respect to their sexual behaviour).

Mathematical modelling is becoming increasingly recognized amongst those working on infectious diseases as a useful tool for researching the quantification of infectious processes. In the research arena, models provide a framework for the amalgamation of field and laboratory data and can be used to highlight those areas where more data and investigation are required. This process is likely to continue in the future, but perhaps the greatest growth will be in the area of designing control policy. As modelling science develops and the understanding of transmission dynamics of specific disease agents increases, epidemiological models will become increasingly useful within the context of health planning and disease prevention. Health policymakers and resource managers already make use of quantitative techniques to assess costs of disease control programs, but all too frequently the epidemiological detail is missing. The coalition of these two areas is likely to be an important aspect of mathematical modelling of infectious disease epidemiology in the future.

Another area of growth is likely to be the development of models of the dynamic processes within individual hosts in contrast to the transmission dynamics between hosts. This is likely to provide insights into immunological, biochemical and genetic processes and hopefully to guide research to produce increased understanding of the mechanisms that govern host/parasite interactions at an individual level. At present, models in this area are largely speculative as the requisite biological knowledge is lacking, but as researchers in this area become more aware of the potential usefulness of mathematical modelling, so models will become more established in empirical observation (Schweitzer and Anderson, 1992).

ANDERSON, R.M. 1990. Populations and infectious diseases: ecology or epidemiology? *Journal of Animal Ecology *60: 1-50.

ANDERSON, R.M. and GRENFELL, B.T. 1986. Quantitative investigation of different vaccination policies for the control of congenital rubella syndrome (CRS) in the UK. *Journal of Hygiene *96: 305-333.

ANDERSON, R.M. and MAY, R.M. 1991. *Infectious Diseases of Humans: Dynamics and Control*. Oxford: Oxford University Press, 757 pp.

ANDERSON, R.M. and THRESH, J.M., eds. 1988. *The Epidemiology and Ecology of Infectious Disease Agents*. London: The Royal Society.

ANDERSON, R.M., MAY, R.M. and McLEAN, A.R. 1988. Possible demographic consequences of AIDS in developing countries. *Nature *332: 228-234.

BUNDY, D.A.P. and MEDLEY, G.F. 1992. Immuno-epidemiology of human geohelminthiasis: ecological and immunological determinants of worm burden. *Parasitology* 104: S105-S119.

GUYATT, H.L., BUNDY, D.A.P., MEDLEY, G.F. and GRENFELL, B.T. 1990. The relationship between the frequency distribution of *Ascaris lumbricoides* and the prevalence and intensity of infection in human communities. *Parasitology* 101, 139-143.

HENDRIKS, J.C.M., MEDLEY, G.F., Van GRIENSVEN, J.P.G., COUTINHO, R.A., HEISTERKAMP, S.H. and Van DRUTEN, H.A.M. 1993. The treatment-free incubation period of AIDS in a cohort of homosexual men. *AIDS*, in press.

McLEAN, A.R., NOKES, D.J. and ANDERSON, R.M. 1991. Model-based comparisons of measles immunisation strategies using high dose Edmonston-Zagreb type vaccines. *International Journal of Epidemiology *20: 1107-1117.

MEDLEY, G.F., GUYATT, H.L. and BUNDY, D.A.P. 1993. A quantitative framework for evaluating the effect of community treatment on the morbidity due to ascariasis. *Parasitology *106: 201-221.

NOKES, D.J. and ANDERSON, R.M. 1991. Vaccine safety versus vaccine efficacy in mass immunisation programmes. *Lancet* II: 1309-1313.

NOKES, D.J., McLEAN, A.R., ANDERSON, R.M. and GRABOWSKY, M. 1990. Measles immunisation strategies for countries with high transmission rates: interim guidelines predicted using a mathematical model. *International* *Journal of Epidemiology *of 19: 703-710.

SCHWEITZER, A.N. and ANDERSON, R.M. 1992. Dynamic interactions between CD4+ T cell subsets and parasitic helminths: mathematical models of heterogeneity in outcome. *Parasitology* 105: 513-522.

*D.J. Rogers*

Department of Zoology

University of Oxford

South Parks Road

Oxford OX 1 3PS

England

The relationship between disease risk and vector challenge may be used for quite different purposes; to determine the amount of intervention (i.e. vector control) required to reduce disease by a certain amount, or to test the output of a full epidemiological model of disease transmission. The relationship is necessary and sufficient for the first exercise, but not for the second. If understanding the dynamics of disease transmission is our aim, we require precise information on the determinants of risk to vertebrates, the challenge by vectors and the relationship between the two.

Attempts are made to reconcile the rather distinct objectives of those who wish to control vector-borne tropical diseases and of those who wish to understand the transmission of such diseases through detailed epidemiological studies.

I begin with an overview of direct and indirectly transmitted diseases and conclude that the basic reproductive number of the latter is often considerably higher than that of the former. The implications for the spread and control of indirectly transmitted diseases are outlined. The equation for the basic reproductive number for vector-borne diseases shows that most of the components of importance are related to the vectors rather than the hosts. The study of vector populations dynamics therefore forms a vital part of vector-borne disease epidemiology.

A brief discussion of risk-challenge relationships shows that the vector modelling requirements will vary from one disease to another. For some diseases, such as malaria, the vector models need to concentrate on what happens at low vector densities, whilst for others, such as onchocerciasis, the higher vector densities are relatively more important. African animal trypanosomiasis appears to fall between these two extremes. Whilst we already know the major ingredients of models for vector (and other animal) species, the density-independent and density-dependent components have rarely been quantified. Models allow us to guess the relative importance of density dependence *vis-a-vis* density independence, and they even allow us to guess that certain stages of the vector's life cycle are more vulnerable to density-independent mortality than others. Here, therefore, it is not the models which are lacking, but the field data to test them.

Vector models eventually need to be integrated into testable models for disease transmission. Here we believe we understand the broad generalities of transmission, but not its details in particular situations. I suggest that we may borrow from insect ecologists, who faced an analogous problem many years ago, an analytical technique to be (re) named 'transmission factor analysis' (tfa) which will identify the causes of changes in the effective reproductive number of vector-borne (and other) diseases. This will both highlight areas where data are still lacking and act as an interface between those charged with controlling disease and those who believe that better control methods will only arise from a more thorough understanding of disease transmission.

*R.S. Morris* and W.E. Marsh*^{+}

* Department of Veterinary Clinical Sciences

Massey University

Palmerston North, New Zealand

^{+}Department of Clinical and Population Sciences

College of Veterinary Medicine

University of Minnesota

St Paul, Minnesota, USA

Abstract

Introduction

Types of models

Applications of the various modelling approaches

Examples of models

References

Over the last 20 years computer modelling has gone from being an esoteric interest of a few people (with little understanding by others of its value and application) to being a recognized tool both for research and policy formulation in animal disease and its control. As with most innovations in research method, acceptance and application of the technique has been patchy and slower than might have been hoped, but both the number of people using the technique and the range of practical application has grown rapidly over the last ten years. This paper reviews the techniques which have been used in computer modelling and the degree to which the method has received practical application. Examples will be provided of different types of models and their application.

The simplest models incorporating disease information have been at the level of individual farms. Most have been used to improve understanding of particular biological systems, to evaluate research directions and priorities and to provide improved 'rules of thumb' for advisers in their work with farmers. They have also been used quite extensively as teaching tools in order to allow students to gain experience of making complex decisions without influencing the real-life outcome.

Current farm-scale models are reaching the point where they are quite practical for individual 'innovator' end 'early adopter' farmers to use as decision support tools, and that is occurring to a growing extent. The key requirement in moving to this stage is that the models be integrated with health and production recording software and other programs already used by farmers. Most models have so far been strategic in nature, concerned with medium-term policy issues. A growth area is the use of specialized models in short-term tactical decision-making, to answer limited questions such as whether or not to apply a particular treatment to animals or plants, given the specific circumstances at that time rather than general expectations as to the future.

Regional and national models have received far greater practical use. A wide range of these 'policy models' have been developed to answer questions about individual diseases and countries, or to investigate other situations where modelling has specific application. The range of diseases and situations covered has been quite wide. Current developments include the integration of models with other precisely targeted software to form a complete decision-support system which can assist in the management of a farm or of a national disease control program.

Over the last 20 years computer modelling has changed from being an esoteric interest of a few people (with little understanding by others of its value and application) to being a recognized tool both for research and for policy formulation in animal disease and its control. As with most innovations in research methods, acceptance and application of the technique has been patchy and slower than might have been hoped, but both the number of people using the technique and the range of practical applications has grown rapidly over the last ten years. This paper will review the techniques which have been used in computer modelling and the degree to which the method has received practical application.

The objective of modelling is to build a simplified representation of a complex system within the real world, in order to test procedures which would be too costly or impractical for various reasons to test on the real world system. Such models can include physical replicas, mental or conceptual models, mathematical representations which are solved by analytical methods, and representations within a computer which can be investigated by mathematical or purely computational methods.

The bulk of current modelling effort involves computer processing, although varied methods are used to carry out the modelling and produce results. Early computer models of livestock production and of animal disease (Morris, 1972) built between about 1965 and 1975 were all designed to run on mainframe computers, and could only be used by people with detailed technical knowledge of computer operation. A variety of modelling methods were used with varying degrees of success. As computing has moved to minicomputers and thence to personal computers, models have followed. They have become far more comprehensive and realistic, and in most cases far more accessible to non-modellers through the design of easily used interfaces for setting parameters and the development of more visual methods of presenting results. These now include spatial as well as temporal trends in disease occurrence, plus other relevant items such as economic consequences. Examples will be provided of different types of models and their applications.

There are two fundamentally different ways of representing disease processes in computer models. The first is deterministic, in which the processes built into the model are fixed by the coefficients set for each variable, and no biological variability is allowed for. Such models will always produce the same outcome for any given set of parameters and initial conditions. The second is stochastic or probabilistic, in which outcomes of at least some of the processes are obtained by drawing samples randomly from standard statistical distributions (binomial, normal, etc.), or empirical distributions based on field data. Such models produce different outcomes for each run, and it is necessary to run the model a number of times (commonly five, and in some cases as many as ten) in order to represent the range of likely outcomes and provide a reasonable estimate of the mean outcome. Deterministic models are faster to run, but it is more difficult to make them realistically represent the disease control issues of interest at a practical rather than a theoretical level. In some cases they are incapable of realism - for example in estimating the proportion of cases in which a disease would be successfully eradicated by a particular control strategy. They will always predict either success or failure under such circumstances - not a probability of success.

Both of the approaches have their uses, and the one chosen should depend on the nature of the problem and the kinds of answers required; deterministic models are valuable for deriving general principles, while stochastic models are applicable to analysing specific practical problems.

Differential and Difference Equation Models

The classical deterministic mathematical approach to analysing time-varying processes such as disease occurrence is through the formulation of differential equations (for continuous processes) or difference equations (for step processes), and using integration to predict the future values of the variable of interest. Over the years this has been a very fruitful approach for systems with relatively few important variables, provided that the equations describing the system were mathematically tractable (Anderson and May, 1981; Anderson, 1982). The approach is very valuable for deriving general principles about the behaviour of simplified 'representative' systems. As the model system being described is allowed to approach reality and hence increases in complexity, at some point for any system the set of differential equations becomes insoluble by standard mathematical methods.

Computer simulation is then normally used to solve the equations, in which case the approach becomes closer to other forms of simulation modelling, although the solutions derived are still deterministic. In some differential equation models certain of the coefficients have biological interpretations which are helpful in understanding fundamental systems dynamics. However again the insights offered through such coefficients tend to lose their clarity as model systems become more complex and approach closer to field reality. Some have argued (Onstad, 1988) that this general approach is inadequate to handle ecological systems modelling, because the coefficients are highly aggregated and not representative of true relationships and because the equilibrium solutions sought may be largely imaginary as far as real world systems are concerned.

Double and Triple Binomial Models

These models have been used by a few workers (Bear and McCallon, 1983) to describe animal disease processes, based on the fact that most disease transmission events are binomial in character, and it is possible to formulate epidemiological problems by means of these more complex binomial functions. However they have no clear advantage over other approaches and their use has not spread widely.

Markov Chain Models

These are well suited to the deterministic description of disease transmission processes, and can be formulated in terms that are mathematically manageable and realistically handle infectious disease epidemiology (Kristensen, 1987; Carpenter, 1988; Dijkhuizen, 1989). They are not as well suited to other types of disease, such as parasitic diseases, where the issue of interest is the severity of disease rather than its presence or absence.

Other Mathematical Approaches

A wide variety of other specific mathematical techniques have found application in various specific instances for investigating disease, but in general they have been used because they fitted a particular special case and they have not found wide applicability.

Electronic Spreadsheet Models

Although electronic spreadsheets were originally designed as accounting tools, they have evolved into very powerful methods for representing many different types of quantitative problems, using the capacity to embed mathematical equations within individual cells of the spreadsheet. It is not difficult to build various forms of simulations within spreadsheet software, and this can be a good structure within which to formulate a disease model. If appropriate it can subsequently be converted to a standard computer program, but the development process will always benefit from the design work done within the spreadsheet format.

Although simple spreadsheet models are deterministic in nature, it is also possible to build stochastic models within a spreadsheet, using the random number generator built into current spreadsheets. This can be done from scratch, but it is now possible to use add-on modules such as the program @Risk, which integrates with selected spreadsheets and adds to them the capacity to run full stochastic simulations involving sampling on any one of about 20 statistical distributions, with automated processing of sequences of runs using both standard Monte Carlo procedures and the faster variant termed Latin hypercube simulation. Although such models are slower to run than those which are written in a programming language, the flexibility and speed of development and adjustment make them an attractive option for some modelling activities.

Monte Carlo Models

These use random number generators to sample from statistical distributions and hence create the sequence of events and results which form the model outcome. Each single outcome is a chance event (hence the gambling association which produced the name of the technique), but in current Monte Carlo systems where millions of such random number selections take place in a single run, the behaviour of the system is stable yet reflects the variability seen in real-world systems. By repeating runs five or more times with different random number seeds to start the process, estimates of natural variability can be made. The technique has advantages over analytical approaches in that it is far easier for non-mathematicians to understand and use, it can approximate much more closely to the real nature of events and processes using a mix of analytical and empirical representations, and it can far better represent sequences over time and conditional probabilities. It does not however have the mathematical purity and same potential for providing conceptual insights which mathematical analyses can in some cases offer, nor does it automatically identify optimal parameter settings for disease control. However the optima it identifies through structured sensitivity analysis (Marsh *et al.,* 1987) are usually more realistic for practical applications than analytical solutions, so this is not a major disadvantage.

A Monte Carlo model is structured by defining the biological processes believed to be involved in the system of interest, and then creating a computer program which carries out a simulation of all the relevant processes in the time sequence believed to occur in reality. The structure of the model is based on the nature of the data resources which can be used to construct it and hence it is normally possible to get data for most aspects of the model from published research. Where an essential item is unavailable, a guesstimate is used and a decision made in the light of initial model runs whether field data collection will be necessary to refine the estimate.

Optimizing Models

For some disease control purposes, it may be useful to have a true optimizing procedure which mathematically finds the best combination of resource inputs to achieve the desired goal. Linear programming, parametric programming and dynamic programming have all been used in such applications. Dynamic programming is the most powerful of the techniques, but requires considerable mathematical understanding to apply and is not yet readily available as a package procedure on microcomputers. It is likely to find increasing use in the future as a goal-seeking tool in the definition of control policies in combination with more traditional simulation methods (Huirne *et al.,* 1992).

Each of the modelling approaches is gradually finding its niche in the spectrum of techniques, as modelling matures as a research tool. Analytical mathematical models are best suited to identifying central issues in relation to a particular disease and establishing broad principles concerning constraints to the effectiveness of alternative approaches to control. As the focus moves from principle to specific guidance on the detailed merits and risks of specific control measures, the mathematical methods reach a point where they cannot approach realism much closer than they have already done because the issues which determine differences between strategies are not capable of adequate representation within a tractable mathematical function.

Another factor is that the people who must make use of the information at the practical level must be able to understand and believe what has been done, and in most cases they have difficulty with mathematical approaches in which typically some of the variables and many of the coefficients do not translate into measurable items in the field situation.

Decision-makers are also very interested in the probability of failure, as well as the expected outcome, since a control policy which succeeds 'on average' may fail a substantial proportion of the time. A model which fails to make estimates of variability around expected outcomes is not very helpful in practice.

Thus as the investigation of a major disease problem moves from establishment of principles to field implementation of control policies, modelling support needs to move from deterministic to stochastic, and from a mathematically solved solution to a solution by repeated simulation with sensitivity analysis. The extent to which model parameters match items which are measurable in the field also becomes increasingly important. In selected cases optimizing rather than evaluative models can offer useful insights.

The point along this continuum where modelling stops is usually determined by money, as far as investment in research and evaluation models is concerned. If the problem is an easy one or not of great importance then a simple mathematical or spreadsheet approach may be adequate. If the problem is a biologically difficult one or is seen to be very important, then a larger investment in more detailed modelling may be justified, as a continuing aid to decision-making and to justification for the chosen research and control strategy. The selection of approach is always a compromise between ease and speed of development on one hand, versus realism and power to investigate approaches in detail on the other hand. The background of the modeller will also be very influential, with those from a mathematical background espousing the mathematical approach and those with a biological training seeking closer approximation to the reality they see in the field. Unfortunately there have been very few examples where the two approaches have been compared effectively in dealing with the same problem, so intuition rather than evidence largely determines the approach adopted.

We choose to use Monte Carlo modelling as our principal, but not our only modelling method. One of us (RSM) has a training both in veterinary science and mathematics, but exploration of various alternative approaches over a period of years led to dissatisfaction with the mathematical approaches as veterinary tools, and increasing reliance on Monte Carlo modelling. It is a very flexible technique which can readily be adapted to deal with quite diverse animal disease issues, and models can be formulated around existing knowledge and available data, rather than having to process the data to fit it to the requirements of a mathematically determined set of coefficients. With current computing techniques, it is possible to achieve high speed of model operation and prompt output of results, without sacrificing biological validity. Interactions between factors in complex biological systems are usually crucial in determining actual system behaviour, and Monte Carlo models can represent such interactions in a much more biologically realistic manner than alternative approaches. Because the model is formulated around current understanding of the particular problem, it is relatively easy to incorporate new knowledge as it becomes available. It is also much easier to represent biological heterogeneity, including spatial variability, through various procedures such as the use of multiple sub-models and linking models to geographical information systems. It is far easier to provide a model interface which is easy to use and requests information in a familiar form, and to explain to potential users what the variables are and how the model operates. As modellers at the applied end of the spectrum, we find that these advantages justify the additional development effort, and allow the models to be useful over a longer time period and a wider geographical area.

The simplest models incorporating disease information have been at the level of individual farms. A few such models have been applied for specific decision-making in the past, but most have been used to improve understanding of particular biological systems, to evaluate research directions and priorities and to provide improved 'rules of thumb' for advisers in their work with farmers. They have also been used quite extensively as teaching tools in order to allow students to gain experience of making complex decisions without influencing the real-life outcome. Most of the earlier models were used almost exclusively by their developers and close associates, because until recently the portability of models between computers and between different field environments has been low. With standardization of computer hardware this problem (which was previously a major issue) has virtually disappeared, all except a few models now being available on MS-DOS personal computers.

Farm models deal either with a specific disease within the farm, but without including a representation of the production system. Current farm-scale models are reaching the point where they are quite practical for individual 'innovator' and 'early adopter' farmers to use as decision support tools, and that is occurring to a growing extent. The key requirement in moving to this stage is that the models be integrated with health and production recording software and other programs already used by farmers. If they do not have to re-enter data they are more likely to use the program. Most models have so far been strategic in nature, concerned with medium-term policy issues. A growth area is the use of specialized models in short-term tactical decision-making (Jalvingh, 1993) and to answer limited questions such as whether or not to apply a particular treatment to animals or plants, given the specific circumstances at that time rather than general expectations as to the future.

Regional and national models have received far greater practical use than farm models, largely because typically only one or two people are needed to work directly with the model, and if important decisions were to be made they could devote sufficient time and effort to developing and then applying the models. A wide range of these 'policy models' have been developed to answer questions about individual diseases and countries, or to investigate other situations where modelling has specific application. The range of diseases and situations covered has been quite wide. Current developments include the integration of models with other precisely targeted software to form a complete decision support system which can assist in the management of a national disease control program (Morris *et al.,* 1992).

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