Dynamics of acute bacterial diseases. Epidemiological models and their application in public health. Part I. Theory and practice of epidemiological models.

PART I. THEORY AND PRACTICE OF EPIDEMIOLOGICAL MODELS INTRODUCTION The object of this publication is to help solve the practical problems posed by the need for effective control of major bacterial diseases in developing countries where resources are scarce and must therefore be put to the best use. This is why attention is focused on major infections, such as typhoid, cholera, tetanus, diphtheria, whooping cough, and cerebro- spinal meningitis, which are prevalent in these countries. The models of these diseases are conceived to permit a study of their dynamics and to simulate the effect of preventive and curative measures, such as sanitation, vaccination, and chemoprophylaxis. By means of models, attempts have been made to evaluate the cost-effectiveness and cost-benefit of alternative preventive measures, thus enabling optimal control strategies to be developed. The models are aimed at assisting public health workers in decision-making in planning disease control programmes and also in their evaluation. Such applications do not preclude their use for other purposes, such as the analysis of past and future trends of diseases or the elaboration and testing of some hypotheses on the epidemiology of these diseases and their control. An epidemiological model represents a system that operates in a way resembling the natural course of a disease and its epidemic spread. It incorporates and interrelates major epidemiological factors that determine the dynamics of infection. A model necessarily represents a simplification of natural processes, but nevertheless, if properly constructed, it can simulate the natural evolution ofan epidemic or an endemic situation, thus permitting the study of the disease dynamics and the effect of deliberate interventions on the natural course of transmission of the infection and, hence, on the incidence of the disease. Observations of epidemics of infectious diseases led epidemiologists to the conclusion that they present some regular features and that there must be some definite principles that determine the evolution of infectious processes. Efforts have therefore been made to express in precise quantitative terms time-related changes in the dynamics of infections and to formulate a mathematical theory of epidemics. One of the general principles of the mechan- ism of epidemics was established in 1927 by Kermack & McKendrick (1) in mathematical terms when they formulated their theory ofthe critical " threshold ofdensity " of susceptible populations as the determining factor in epidemics. Later they extended their theory of epidemics to the study of endemicity (2, 3). The deterministic theory of the epidemic process put forward by Kermack & McKendrick is paralleled by stochastic analogues developed by Bailey (4) and Kendall (5). Other general theories of epidemics, such as those of Muench (6) on catalytic models in epidemiology, permitted the formulation of a variety of general deterministic and stochastic mathematical models; these have been reviewed in detail by Bailey (7) and will not be discussed here. On the basis ofthese general theories, models have been proposed for individual diseases, such as malaria (8), schistosomiasis and other parasitic diseases with intermediate hosts (9), measles (6, 10), other viral diseases (11), and tuberculosis (12). As far as is known, mathe- - 11- DYNAMCS OF ACUTE BACTERLAL DISEASES. PART I matical models for acute bacterial diseases have been attempted only in the study of the immune response to antigens in diphtheria (6) and tetanus (13). In these and similar studies a variety of approaches have been taken, mostly aimed at contributing to the mathematical theory of epidemics and for possible use in predicting epidemic trends, but not for planning practical control measures and formulating health strategies through simulation and goal-seeking techniques. There is at present a variety of concepts and tech- niques of modelling for various purposes. In modelling acute bacterial diseases we selected among various methods and approaches those that we thought were most suitable for meet- ing our primary aim, namely to contribute through epidemiological models towards more effective control of these diseases. Our models are called epidemiological rather than mathematical because they are not based on strictly mathematical solutions of epidemic processes but rather on simulations of the natural course and the dynamics of the diseases expressed as time-related numerical flows of populations to and from various states of infection. Such simulations became possible with the advent of modern electronic computing facilities. Our models were thus computerized and their use depends therefore on the availability of these facilities. Generally, modest electronic data processing (EDP) equip- ment will meet the requirements. The theoretical principles and methods of construction of our models will be described in Chapter 1 and the practical uses of the models in Chapter 2. However, since diseases differ in many ways, these general principles on which the models are constructed and their use are further developed, elaborated, and applied in the particular disease models described in Part II of this publication. 12 CHAPTER 1 Construction of epidemiological models An epidemiological model represents a dynamic system of strictly interrelated epidemiological factors, able to " mirror " epidemic processes. A model is constructed by identifying categories of individuals that play a well-defined and important role in the dynamics of the disease. In order to make the model relatively simple and manageable it is desirable to eliminate unimportant factors and retain only those that significantly influence and determine the mechanism of the epidemiological processes. The aim of the model construction is to arrive at a system that is able to " mimic " the natural processes, such as the past outbreaks and trends of diseases, and thus to simulate various real or hypothetical situations. It is obvious that the models must be based on the natural history of the disease and should be an expression of that history. Before embarking on the construction of a model of any particular disease, it is essential to consider the purpose of the model and the practical aims it is required to serve. While the motivation for model construction may also be to satisfy scientific curiosity or to learn more about mastering the art of modelling, the only socially relevant aim of a model is to facilitate improvement in the control and/or treatment of infectious diseases by a more rational application of existing preventive and curative measures and available resources. Once the purpose that the model should serve has been clarified, the requirements for construction of the model become explicit. Thus, the epidemio- logical factors relevant to the purpose of the model gain importance and prominence, while others can be neglected. It may therefore happen that for one and the same disease several different models could be developed, each to suit a specific purpose. Ihis does not preclude the same model serving several purposes. We considered it more appropriate to construct simple purposeful models that can be used in public health practice for the planning and evaluation of control programmes than to develop large comprehensive systems, however perfect, that have only limited practical use. This is not to deny the scientific importance of modelling, which can greatly add to epidemiological theory and increase understanding of epidemic processes, and thus ultimately lead to improvements in public health practice. Even in " pure " modelling research the practical uses of the products of such studies need to be kept in mind. NATURAL HISTORY OF THE DISEASES An epidemiological model must reflect faithfully the natural history of the disease and the relevant epidemic processes. These processes have been described with more or less accuracy in published observations and studies, and these descriptions can serve as a basis for model construction. To construct the model, accurate data are needed on quantitative measurements of various factors and parameters, such as incubation period, duration of infectivity and illness, duration of carrier states, and degree and duration of immunity. Most of the present knowledge on the natural history ofinfectious diseases has been inherited from the past when epidemiologists were less statistically and quantita- tively m-inded. Often the absence of precise informa- tion and great variations in some epidemiological parameters make the construction of epidemiological models difficult, if not unrealistic or impossible. The natural history of the disease must be well known and the infection well defined to permit construction of the model. The starting point is perhaps information on incidence and prevalence. One of the difficulties in evaluating the disease incidence in the population is the fact that most infectious diseases are underreported in routine - 13 DYNAMICS OF ACUTE BACTERIAL DISEASES. PART I health statistics. Results of special surveys are much more meaningful. Further, usually only a part of the infected population is recognized clinically as ill in most infections, making it difficult to deter- mine the actual extent of the infection. These aspects must of course be taken into account in a sound epidemiological model. In defining the natural history of the disease and the parameters on which to build the model, it is necessary to screen critically the available data and sometimes to correct commonly accepted but erroneous ideas about the mode of tansmission, frequency of infection, and mechanism of trnsmission. This points to the necessity for making a thorough and critical review of the common knowledge of the biology and pathogenesis of the disease before embarking on the construction of the model. Sometimes special studies and surveys must be undertaken to obtain reliable data on which the model can be built. When sound information on the natural history of a disease is available and the purpose of the model is well defined, it is essential to proceed with the identification of each epidemiological factor and its relative importance. If a certain factor is considered to play an important role in the dynamics of infection, then an accurate quantitative determina- tion of this factor must be made. Selection of important factors among many is sometimes difficult, and in some instances preliminary model simulations may be needed to reveal their real importance, if any. On the other hand, it would be erroneous to include all possible factors, making the model unnecessarily complex without necessarily contribut- ing to its usefulness. Besides, it is unlikely that in public health practice all the relevant data on a multitude of factors would be available or easy to collect. STRUCIURE OF THE MODEL A disease model can be defined as a logical and quantitative expression of the relationships existing between all the epidemiological and other variables that are involved in the natural history of the disease. The naturl history of infectious diseases shows that there are numerous epidemiological states, such as susceptibility, resistance, incubation, illness, and infectivity. Accordingly, multistate models are constructed. For the purpose of the model, the population is divided into categories of individuals who belong to the specific epidemiological states and are considered as epidemiological classes. In order to make the model operative, the number of classes is reduced to a minimum, keeping in mind that all important classes must be represented in order to meet the requirements of the model. In the multistate models, the population at any time is conceived as composed of various epidemiological classes. In the course of infectious processes indi- viduals change their respective epidemiological states and classes in well-defined sequence. A susceptible individual, after being infected, first enters the class of incubating persons; after the incubation period is over, he moves into the class of sick persons (infectious or non-infectious) and later into the carrier class (when a carrier state exists) or to recovery or death. There are additional classes in some diseases, such as the class of resistant, artificially or naturally iunized individuals. Classes and possible subclasses are clearly distin- guished as well as the direction of flow of individuals from one class to another. It is convenient to draw a flow chart to visualize the various epidemiological classes and the movements between them. Such a system is not entirely closed as it is subject to population dynamics, namely the entry of the new- born in it, the removal of the deceased, and the effect of possible migrations. The reader is referred to the flow charts given in Part II for each of the 6 acute bacterial diseases, which illustrate the above description (see pp. 30, 48, 66, 84, 104, and 117). RATES OF TRANSMON AND FORCE OF CTION The rates of transition between various classes are estimated from the available quantitative evidence on the history of the disease. The rate of transition is considered simply as the probability that a person belonging to one class will be transfer- red to another class per unit of time, e.g., an hour, a day, a week, or an even longer period. In models of some acute bacterial diseases one day was con- sidered as a suitable time unit, but in other models longer periods have been used. When the rates of transition have been defined and quantified and all relationships between the epidemiological classes or categories determined, these can be formulated and written in mathematical terms. Special consideration is needed concerning the rate of transition of the susceptibles to the infected class. This rate, which is most critical in model construction, depends on the variety of factors that determine the bansmission of infection to suscep- tibles. 14 CONSTRUCnION OF EPIDEMIOLOGICAL MODELS The number of new cases per unit of time is con- sidered to be the result of an interaction between the number of susceptibles, the number of infectious persons, and a " force of infection ". The force of infection represents the totality of a variety of factors that determine bansmission of infection to susceptible individuals. It depends on the number of effective contacts between the sus- ceptible person and the source of infection (in- fectious person, animal, or containated environ- ment) per unit of time. It is expressbd as the average number of persons with whom each infectious individual (or other source of infection) has sufficient contact, directly or indirectly, per unit of time to infect them, assuming them all to be susceptible to the disease. Thus, the number infected during a unit of time (e.g., per day) in the community is estimated to be: number of force proportion infectious x of of susceptible pons in .infectin thepopulanpersons in the population n th population The force of infection depends on numerous environmental, biological, social, and economic factors. Theoretically, and also sometmes in practice, it is possible to determine all those factors closely in a quantitative way as well as to define their relationships. We have, for example, elaborated practically all the components of the force of infection in tetanus (see chapter 3). It goes without saying that when the force of infection can be broken down into its component factors the epide- miological model i much. In such a case, the force of infection itself represents a kind of model of its own, which is subject to quantitative changes through alteration of the factors involved or by interference, e.g., by changing environmental saitation, personal hygiene, or food preparation. The immunity status of the host, which is an ex- tremely important factor, is taken into account in the third term of the above formula and does not figure among the factors that determine the force of infection. MATEMATICAL FORMULATION OF HETMODEL When the structure of the model has been defined and the epidemiological classes and rates of ransi- tion decided, the dynamics of the disease in the population can be expessed mathematically by a system of differential equations. Usually, however, the system is not linear and is too complex to be solved analytically with mathematical rigour. But today the large capacity and very high calculation speeds of modern electronic computers bring an elegant -solution to the problem; the parameter and time differentials are replaced by finite differen- ces, choosing a time unit that is relatively short compared with the time needed for measurable changes in the structure of the model. The system is then solved by the method of successive iterations; the inantous rates of entry into and exit out of the various epidemiological classes are, of course, replaced by rate exps in terms of the selected unit of time. If the unit of time is short enough (for instance one day) the modification of the system, although discontinuous, will be sfficently precise to study trends over periods ofseveral years. For the computer to calculate the successive, extemely small, daily changes affcting the system-for instance over several ten-year periods-is generally a matter of minutes only. By this approach, and on the trial and error basis, it is possible to solve the system of equations and to fix the force of infection in order to arrive at the changes that would occur in certain populations with reard to the disease concerned, assuming a particular level of endemicity or particular types of epidemic cycles. For a study of endemic situations, once it has been verified that the computer output remains unchanged over time, the state of stable endemicity is reached and the model system can be considered suitable for application. The balance reached at the state of stable endemicity can be taken as the starting point for simulation of the effect of specific curative and/or preventive measures; this is done by introducing appropriate modifications of the rates of transfer or force of infection, which can also be mathematically formulated and inserted in the computer programme. The validaion of the model is essential before it can be considered realistic and used for meaningful simulation. The validation is performed by attempt- ing to simulate the natural course of infection and its modification through health interventions. A valid model must be able to reproduce actual data from real life. SIMULATION OF NATURAL COURS OF RNFCTON While properly constructed epidemiological mo- dels should be able to simulate endemic or epidemic 15 DYNAMCS OF ACUTE BACTERIAL DIS. PART I situations that occured in the past, they need not reproduce past epidemic or endemic situations in all their details. They must, however, at least ulate their main features; otherwise they are not realistic and cannot be considered as valid. A model, once constructed, should be put to the test of simulating several well known and well described situations. Should it fail, the causes of its failure can usually be found and appropriate corrections and adjust- ments made. The final model is developed through simulations by trial and error. Simulation of an endmic situation is relatively easy as all that is needed is to arrive at a stable endemicity level which is maintained by a stable force of infection. If the force of infection is modified slightly, it is likely that the model, after a certain time, will again establish its balance at another stable level of endemicity. However, should the force of infection be set at a considerably different level, in the model or in real life, an outbreak can be generated or a continuous decline of incidence may result. In most diseases the level of endemicity is likely to be stable through long periods of time- namely, many years-but seasonal variations can still take place each year. In such a case, the force of infection is likely to oscillate around its mean value. If such oscillations are relatively large, typical seasonal outbreaks may result. In our models for cholera and cerebrospinal meningitis, -we have used two different methods that brought about such seasonal changes of incidence resenbling closely actual seasonal patterns of these diseases (see chapters 5 and 6). Simulation of epidemic situations can be made by introducing seasonal and gradual changes in the force of infection or simply by sudden increases in the force of infection for shorter or longer periods of time. Such simulations would be appropriate for droplet or water-borne diseases where sudden crowding, in the first case, or failure of water supply and/or sanitation, in the second case, would bring ;an outbreak of varying intensity and duration. In fact we have produced such simulations with the cholera model (see chapter 5). Epidemics could also be simulated by leaving constant the level of force of infection but introducing a number of infectious individuals in a susceptible population. Simulations oflong-term trends ofdisease are made to study the historical evolution of infections. Their practical use is to determine the pace and degree of the change in the force of infection; we did this, for example, by studying the past history of typhoid in the United Kingdom and the United States of America (see chapter 4) and of whooping cough in the United Kingdom (see chapter 7). SIMULATION OF INTVENTIONS After successful simulation of the natural course of epidemics and/or of the evolution of endemic situations, which represents the basic test for the value of any model, the next step is simulation of the effect of planned interventions and other possible interferences with the natural course of epidemic pressures. The natural course of stable endemicity or of regular seasonal outbreaks can be modified by a change in the force of infection, or by the introduc- tion of infectious individuals or of susceptible ones into the population. The effect of various measures intentionally introduced by health authorities in order to interfere with the natural course of infection and to control the spread of the disease may be simulated by the model by introducing in the computerized system modifications able to reflect the mechanism specific to each envisaged interven- tion. Treatment, chemoprophylaxis, and isolation of infected persons is simulated by removing thiem from infectious classes into the appropriate non- infectious classes. The simulations will show to what extent this prevents further spread of infection. Treatment and chemoprophylaxis can be carried out in a variety of ways by various drugs and in different groups of the population. This gives an opportunity to examine through simulations the effect of various schemes of drug treatment or prophylaxis, various dosages, coverages of popula- tion, etc. Likewise, the impact of immunization on the dynamics of the disease can be simulated by proper expression in the system of the vaccine characteristics and operational scheme of application. For instance, immunization with vaccines conferring high protec- tion against contracting the infection will result in a large transfer of susceptible persons to the resistant class for some time; it is, however, possible for such vaccines not to be highly effective against the development of the severe clinical forn of the disease among vaccinated persons. On the contrary, some vaccnes may reduce effectively the case letha- lity among infected susceptibles but not modify substantially the incidence of the infection; in such situations protection against infection will mainly be obtained through naturally acquired immunity. Simulation of the effect of vaccines with various degrees of potency, applying different immunization 16 CONSTRUCTION OF EPMIEMOLOGICAL MODELS schemes, schedules and coverages, can show the advantages and disadvantages of various immuniza- tion programmes. Environmental changes, such as the introduction of sanitation, can be simulated by change in the force of infection. The effect of sanitation measures, such as the provision of water-supplies or sewerage, unlike that of treatment, isolation, chemoprophylaxis, and uniation, is lasting and its effect is actually cumulative. Effects of socioeconomic chaes, changes in education, in personal hygiene, and in the level and efficiency of health services can be similarly simulated by appropriate modification of the level of the force of infection (see chapter 3). Finally, one can simulate and study the effects of various combinations of the health interventions mentioned, as well as some natural interferencsp, such as population movements or calamities leading to the interruption of sanitation or hygienic practices and thus to an increase in the force of infection. Some deliberate alterations in the parameters and rates of transfer in the model would cause a chain of changes. Their effect on the final results of simulations may be great or possibly make little difference, accordi to whether important factors have been altered to a significant degree or minor ones changed to a small extent. As new knowledge becomes available on quanti- tative aspects of various epidemiological factors, the models should be amended accordingly. Such amendments would not usually require basic substantial modifications in the structure of the system, but simply a change in one or more para- meters. The effect of new immunizing agents and remedies can be simulated with a simple change in the parameters when the model has been constructed to take treatment and immunoprophylaxis into account, but to simulate the effect of entirely new control methods, considerable adaptation of the model may be required. The effect of multiple infections and multiple interventions could be simulated by simultaneous use of several epidemiological models, one for each of the diseases concerned. 17 CHAPTER 2 Uses of epidemiological models The uses of epidemiological models are multiple. Some models have been oriented toward solving specific problems and may therefore be more useful for some purposes than for others. The models of great complexity have limited use in public health practice, because they usually require extensive information to be provided on actual epidemiological situations and such information may not be available, in particular in developing countries. More practical are relatively simple models that require only essential information, e.g., on population size, incidence, and, if necessary, the extent and cost of planned health interventions. We shall limit the discussion on the uses of the models to those developed by us for acute bacterial infections. PLANNING OF CONTROL PROGRAMMES The models can be used in the planning of control programmes. The starting point and the base line in the planning is the simulation of the natural course ofan infectious disease in certain populations, using data petinent to that population. The next step is to select a few well thought out, feasible control programmes and simulate their potential effects on the disease for longer or shorter periods of time. On the grounds ofthe first set of simulations, further exploration of the above programmes (modified as may be indicated) can be made. Finally, the cost-effectiveness and cost-benefit of selected programmes can be analysed using the model. In this way, the most effective programme can be identified. Often, various constraints are imposed on planning in view of limited resources, and in search of solutions unthought of possibilities may be found. The fallacies and inadequacies of various programmes may also be revealed, thus allowing such inadequate programmes to be abandoned or corrected. The absence of computer facilities should not discourage public health workers from using the models in the planning of health programmes. An analysis of the intended control programmes can be made elsewhere provided the pertinent data are sent to the institution where computers and relevant computer programs are available. Most health ad- ministrations, nowadays, have access to computer facilities either in their own country or in a neigh- bouring state, or at an international institution. It is rather the absence of skilled personnel than the absence of computer facilities that hampers wider use of epidemiological models in health planning. The problem that the computer is expected to deal with must be well and clearly formulated and necessary basic information made available in order to obtain meaningful results. Besides this there is a need for epidemiological knowledge and skill to interpret correctly the results obtained. EVALUATION OF CONTROL PROG]RAMMES Once a control programme, adopted on the grounds of model simulations, has been executed, its results should be evaluated with the aid of the model. Any significant discrepancy between the results predicted by the model and those obtained in practice should be studied and the reasons for the differences found. These discrepancies may reveal important causes of failure such as: leaving high risk groups inadequately protected, low quality of vaccine or drug, or untimely use of the control measures. They may also point to inadequacies in the original model and to the need for its improve- ment. Established routine control procedures should also be evaluated from time to time as not only the - 19 DYNAMICS OF ACUTE BACTERIAL DISEASES. PART I epidemiological situation but also the effectiveness of the control procedures and their respective cost- benefit assessments may change with time. Evaluation of new control measures deserves particular attention as some activities that would seem effective in simulations during the planning phase may prove ineffective in reality for a variety of reasons. It should be kept in mind that evaluation must be planned in advance at the very onset of control operations so that all necessary records can be kept ready for this purpose (14). COST-EFFECTIVENESS AND COSTrBENEFlT ANALYMS The actual value of different preventive measures used individually or in combination in various control programmes can be fully appreciated only when their effect in terms of the decreases in morbi- dity and mortality is compared with the cost of these measures. The total cost of the control pro- gramme and of the expenditure on disease should be compared with the costs that would have been incurred if no control measures had been undertaken (cost-benefit analysis). In public health practice, owing to the limited funds and resources that are available for health services, economic factors are often decisive in the selection of the strategy for disease control. There is therefore a need for analysis of the cost-effective- ness of various control operations. Cost-effectiveness and cost-benefit analyses are based on information on actual costs, which is not always easy to obtain. Assessment of the cost of disease comprises assessment of direct costs of illness such as those of treatment (drugs, doctor's and nurse's time, etc.), costs of wages, invalidity and death, and indirect costs, such as disturbances in trade and traffic (e.g., in the case of cholera). In cost-effectiveness analysis ofa preventive measure, the total cost of the control programme and disease are compared with the number of cases of an illness prevented, the result being often expressed as the cost per prevented case. In cost-benefit assessment, economic benefit is usually expressed as savngs accrued mainly on treatment but also on other direct and indirect disease costs by the prevention of cases of illness through prophylactic measures (15). The benefit can also be measured by saved suffering and human lives but in that case certain money values need to be ascribed to human lives and wellbeing. Although some health economists have attempted to do so (16) it is questionable whether human lives and suffering can ever be expressed in terms of money. Some measures, such as the provision of water supplies and sanitation, have multiple economic effects of which the health benefits may not, in fact, be the greatest or the most important. Furthermore, the benefit can be regarded either from a general public point of view or from the standpoint of the govern- ment's budget. Benefits will be calculated differently according to the socioeconomic structure of the community and the degrees of responsibility assumed by the government and the individual in the field of health. When the cost-effectiveness and cost-benefit aspects of various measures have been studied and determined, a comparison can be made. Thus, one can select the optimal disease-control programme that produces the greatest health and/or financial benefit per monetary unit and resources invested. Our cost-effectiveness and/or cost-benefit analyses in the fields of typhoid, cholera, and tetanus have been based on the preventive measures used in practice. Cost-effectiveness and cost-benefit were considered in view of the long-term effect of respec- tive programmes, since short-term analyses may be misleading. The examples of cost-effectiveness and cost-benefit analysis and their interpretations are given with descriptions of specific models of bacterial diseases in the second part of this publication. FORMULATION OF HEALTH STRATEGIES Decision on control strategy is based on selecting individual or combined curative and/or preventive measures and applying them to all or to specific population groups at high degrees of risk. Through simulations the models permit the study and selection of the most cost-effective control strategy. It would be erroneous to believe that for the formulation of health strategy nothing else besides epidemiological models should be used. First of all, there are as yet no epidemiological and mathematical models for a number of infectious diseases. Further, for the proper formulation of health strategy much wider views on health and economics must be taken and more complete analyses made than can be offered by epidemiological models alone today. Epidemiological models are used in the study of a variety of public health activities (17) and in the medical industry (18). There are mathematical 20 USES OF EPIDEMIOLOGICAL MODELS models for determination of the resource allocations for control of diseases such as tuberculosis (19). Nu- merous other studies on the rational uses ofresources in health programmes have been published. The epidemiological models that we have devel- oped have been used in cost-effectiveness and cost- benefit analyses and in the formulation of optimal health strategy. Models permit the formulation of alternative strategies and their effectiveness and costs, and thus provide necary information not only on optimal programmes but also on other alternatives that, in view of constraints, may be preferred. They provide the basis for sound decision- making on the strategy to be adopted. Thus the advantages ofa control programme by immunization can be compared with a programme ofchemoprophy- laxis and/or sanitation. Similarly, various immuniza- tion programmes can be compared and their merits evaluated. The examples of such uses are given in Part II of this publication. EPIDEMIOLOGICAL INVESIIGATIONS Epidemiological models represent the most convenient tool for investigation of the dynamics of infections, namely their quantitative time changes. When an accurate and validated model is available it can be used to reconstruct the actual distribution of epidemiological categories and classes of popula- tion from fragmentary information. If, for example, the incidence of typhoid fever is known but a search for carriers has never been done, the model would provide information on the number of carriers in the community (not, of course, on who they are). The models can be used to verify possible changes in patterns of the outbreaks, thus indicating that some changes of a biological or socioeconomic nature have taken place. Various patterns of infec- tions in different countries can be compared and the factors that are responsible for these differences determined. In the analysis of past and the simulation of future trends of diseases, certain regularities can be found and some of them may be of both practical and scientific value. For example, in the study of trends of typhoid in developing countries it was observed that typhoid has a tendency to self-limitation and to decline continually once it has reached a certain lower level of incidence (see chapter 4). This finding is obviously of great interest as it indicates the mechanism that apparently exists in the epidemic processes of typhoid and that could be exploited in practice for eventual eradication of this infection in the distant future. Such studies of long-term trends can reveal certain regularities, which can be used for predictions of future trends and the fate of an infection in a country. This type of study is of practical and theoretical interest in view of the disputes over the feasibility of eradication of some infectious diseases, a problem that could be better tackled with the assistance of a proper disease model than without it. TRAINING AND EDUCATION Last but not least, the models represent an excellent tool for teaching epidemiology, and epide- mic processes in particular. The role of various factors and the mechanism and dynamics of infec- tions can be vividly explained and demonstrated by models. When a computer terminal is available the student himself can see the meaning of changing some parameter, e.g., the incubation period, duration of immunity, or force of infection in an epidemic. The epidemiological models have shown that the dynamic processes of the acute bacterial diseases that we have studied differ greatly one from another. The models pointed to the intricacies and the com- plexity of the epidemic processes in each specific disease, but they also permitted better understanding of these processes. Simulations are useful in enabling students to gain a better understanding of the natural course of infections and the impact of control measures. Simulation of the epidemics has become in recent years an integral part of the training of medical students at advanced universities. Professor R. A. Deininger (20) of the School of Public Health at the University of Michigan has used our typhoid model for this purpose since 1971, and so has the School of Public Health " Andrija Stampar " of the Univer- sity of Zagreb and some other institutions. It is perhaps right to say that without simulations it is difficult for a student to understand fully the dynamics of infections and appreciate the relative importance and role of various epidemiological factors. Thus, one of the important uses of the models appears to be in the teaching of modern quantitative epidemiology and public health. The models open the way for epidemiologists and health administrators to replace intuitive approxim- tions by strictly quantitative and logical scientific analysis. With the ever-increasing availability and use of modern electronic data processing systems, 21 DYNAMCS OF ACUTE BACIERIAL DBEASES. PART I models are likely to become commonly used tools for health planning and evaluation of control programmes; teaching the construction and use of models seems, therefore, to be aining iQmportance. More practical experience and theoretical studies in modelling are still needed, however, before it will be possible to assess the contribution that models could make in furthering epidemiological theory and public health practice and in leading to an improvement in health. 22 USES OF EPIDEMIOLOGICAL MODELS 23 REFERENCES 1. KERmAcK, W. 0. & MCKENDRICK, A. G. A con- tribution to the mathematical theory of epidemics. [Part 1.] Proc. r. Soc. A, 111: 700.721 (1927). 2. KERMACK, W. 0. & McKENDRicK, A. G. A con- tribution to the mathematical theory of epidemics. II. The problem of endemicity. Proc. r. Soc. A, 138: 55-83 (1932). 3. KERMACK, W. 0. & McKEmnnIcK, A. G. A con- tribution to the mathematical theory of epidemics. III. Further studies of the problem of endemicity. Proc. r. Soc. 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