Contributions to the Mathematical Theory of Epidemics

William Ogilvy Kermack and Anderson Gray McKendrick created what is now called the Kermack-McKendrick theory of infectious diseases. This highly mathematical approach was developed by the authors in a series of five papers. Below we give the beginning of the Introduction and the summary of each of the five papers. More than 75 years after they began working on the mathematical theory of infectious diseases, a paper discusses how their work is seen in the 21st century. We give a brief extract from that paper.

Click on a link below to go to the information about that paper.

A Contribution to the Mathematical Theory of Epidemics

Contributions to the Mathematical Theory of Epidemics. II. The Problem of Endemicity

Contributions to the Mathematical Theory of Epidemics. III. Further Studies of the Problem of Endemicity

Contributions to the Mathematical Theory of Epidemics. IV. Analysis of Experimental Epidemics of the Virus Disease Mouse Ectromelia

Contributions to the Mathematical Theory of Epidemics. V. Analysis of Experimental Epidemics of Mouse-Typhoid; A Bacterial Disease Conferring Incomplete Immunity

A Bright Future for Biologists and Mathematicians? by A Hastings and M A Palmer.

1. A Contribution to the Mathematical Theory of Epidemics, by W O Kermack and A G McKendrick.
1.1. Reference.

Proceedings of the Royal Society of London. Series A 115 (772) (1927), 700-721.

1.2. From the Introduction.

One of the most striking features in the study of epidemics is the difficulty of finding a causal factor which appears to be adequate to account for the magnitude of the frequent epidemics of disease which visit almost every population. It was with a view to obtaining more insight regarding the effects of the various factors which govern the spread of contagious epidemics that the present investigation was undertaken. Reference may here be made to the work of Ross and Hudson (1915-17) in which the same problem is attacked. The problem is here carried to a further stage, and it is considered from a point of view which is in one sense more general. The problem may be summarised as follows: One (or more) infected person is introduced into a community of individuals, more or less susceptible to the disease in question. The disease spreads from the affected to the unaffected by contact infection. Each infected person runs through the course of his sickness, and finally is removed from the number of those who are sick, by recovery or by death. The chances of recovery or death vary from day to day during the course of his illness. The chances that the affected may convey infection to the unaffected are likewise dependent upon the stage of the sickness. As the epidemic spreads, the number of unaffected members of the community becomes reduced. Since the course of an epidemic is short compared with the life of an individual, the population may be considered as remaining constant, except in as far as it is modified by deaths due to the epidemic disease itself. In the course of time the epidemic may come to an end. One of the most important problems in epidemiology is to ascertain whether this termination occurs only when no susceptible individuals are left, or whether the interplay of the various factors of infectivity, recovery and mortality, may result in termination, whilst many susceptible individuals are still present in the unaffected population.

It is difficult to treat this problem in its most general aspect. In the present communication discussion will be limited to the case in which all members of the community are initially equally susceptible to the disease, and it will be further assumed that complete immunity is conferred by a single infection. It will be shown in the sequel that with these reservations, the course of an epidemic is not necessarily terminated by the exhaustion of the susceptible members of the community. It will appear that for each particular set of infectivity, recovery and death rates, there exists a critical or threshold density of population. If the actual population density be equal to (or below) this threshold value the introduction of one (or more) infected person does not give rise to an epidemic, whereas if the population be only slightly more dense a small epidemic occurs. It will appear also that the size of the epidemic increases rapidly as the threshold density is exceeded, and in such a manner that the greater the population density at the beginning of the epidemic, the smaller will it be at the end of the epidemic. In such a case the epidemic continues to increase so long as the density of the unaffected population is greater than the threshold density, but when this critical point is approximately reached the epidemic begins to wane, and ultimately to die out. This point may be reached when only a small proportion of the susceptible members of the community have been affected.

Two of the reasons commonly put forward as accounting for the termination of an epidemic, are (1) that the susceptible individuals have all been removed, and (2) that during the course of the epidemic the virulence of the causative organism has gradually decreased. It would appear from the above results that neither of these inferences can be drawn, but that the termination of an epidemic may result from a particular relation between the population density, and the infectivity, recovery, and death rates.

Further, if one considers two populations identical in respect of their densities, their recovery and death rates, but differing in respect of their infectivity rates, it will appear that epidemics in the population with the higher infectivity rate may be great as compared with those in the population with the lower infectivity rate, especially if the density of the former population is in the neighbourhood of the threshold value. If, then, the density of a particular population is normally very close to its threshold density it will be comparatively free from epidemic, but if this state is upset, either by a slight increase in population density, or by a slight increase in the infectivity rate, a large epidemic may break out. Such great sensitiveness of the magnitude of the epidemic with respect to these two factors, may help to account for the apparently sporadic occurrence of large epidemics, from very little apparent cause. Further, it will appear that a similar state of affairs holds with respect to diseases which are transmitted through an intermediate host. In this case the product of the two population densities is the determining, factor, and no epidemic can occur when the product falls below a certain threshold value.

1.3. Another quote from the paper.

These results account in some measure for the frequency of occurrence of epidemics in populations whose density has been increased by the importation of unaffected individuals. They also emphasise the role played by contagious epidemics in the regulation of population densities. It is quite possible that in many regions of the world the actual density of a population may not be widely different from the threshold density with regard to some dominant contagious disease. Any increase above this threshold value would lead to a state of risk, and of instability. The longer the epidemic is withheld the greater will be the catastrophe, provided that the population continues to increase, and the threshold density remains unchanged. Such a prolonged delay may lead to almost complete extinction of the population. Similar results, though of a somewhat more complicated form, hold for epidemics transmitted through an intermediate host. In this case, in place of the threshold density we have to consider the threshold product.

1.4. Summary.

  1. A mathematical investigation has been made of the progress of an epidemic in a homogeneous population. It has been assumed that complete immunity is conferred by a single attack, and that an individual is not infective at the moment at which he receives infection. With these reservations the problem has been investigated in its most general aspects, and the following conclusions have been arrived at.

  2. In general a threshold density of population is found to exist, which depends upon the infectivity, recovery and death rates peculiar to the epidemic. No epidemic can occur if the population density is below this threshold value.

  3. Small increases of the infectivity rate may lead to large epidemics; also, if the population density slightly exceeds its threshold value the effect of an epidemic will be to reduce the density as far below the threshold value as initially it was above it.

  4. An epidemic, in general, comes to an end, before the susceptible population has been exhausted.

  5. Similar results are indicated for the case in which transmission is through an intermediate host.
2. Contributions to the Mathematical Theory of Epidemics. II. The Problem of Endemicity, by W O Kermack and A G McKendrick.
2.1. Reference.

Proceedings of the Royal Society of London. Series A 138 (834) (1932), 55-83.

2.2. From the Introduction.

In a previous communication an attempt was made to investigate mathematically the course of an epidemic in a closed population of susceptible individuals. In order to simplify the problem certain definite assumptions were made, namely, that all individuals were equally susceptible, and that death resulted, or complete immunity was conferred, as the result of an attack. The infectivity of the individual and his chances of death or recovery were represented by arbitrary functions, and the chance of a new infection occurring was assumed to be proportional to the product of the infected and susceptible members of the population. In spite of the introduction of the arbitrary functions, it was shown that in general a critical density of population existed, such that if the actual density was less than this, no epidemic could occur, but if it exceeded this by n an epidemic would appear on the introduction of a focus of infection, and further that if n was small relative to the population density, the size of the epidemic would be 2n per unit area. It was shown that these conclusions could be readily extended to the case of a metaxenous disease, that is, one in which transmission takes place through an intermediate host.

It is the purpose of the present paper to consider the effect of the continuous introduction of fresh susceptible individuals into the population. It appeared desirable to investigate this point, since it might make it possible to interpret certain aspects of the incidence of disease not only in human communities where there is usually an influx of fresh susceptible individuals either by immigration or by birth, but also in the animal experiments carried out by Topley and others - where fresh animals were introduced at a constant rate into the cages in which cases of disease were already present-from which certain definite results were obtained.

In order to make the present enquiry more general an attempt has also been made to include the effect of the partial immunity which may follow an attack of disease. The susceptibility of the individual who has recovered from an attack is assumed to be a function of the time which has elapsed since his complete recovery. In general this function will either remain constant or will increase with the time. This point is rather important as it will be shown that it enables us to reach certain conclusions even when the function characterising the susceptibility is otherwise quite arbitrary.

2.3. Summary.

  1. A mathematical investigation has been made of the prevalence of a disease in a population from which certain individuals are being removed as the result of the disease, whilst fresh individuals are being introduced as the result of birth or immigration. Allowance is made for the effects of the immunity produced as the result of an attack of the disease, but the effect of deaths from, other causes is not taken into account, and the action of the disease is supposed to be independent of the age of the individual.

  2. As a special case of the above, results have been obtained for a closed population in which no deaths occur and to which no fresh individuals are added, but in which the individuals after being infected acquire immunity, and then may be again infected. A threshold density of population exists analogous to that described in the previous paper, which is such that no disease can exist in a population, the density of which is below the threshold.

  3. In other special cases investigated when either immigration or birth is operative in the supply of fresh individuals, as well as in the general case, only one steady state of disease is possible. To reach this state the population must be of a certain density which will be determined by the functions characterising the infectivity, morbidity, etc., of the disease.

  4. Increase of the immigration rate or of the birth-rate results in an increase in the rate of infection of the healthy individuals and also in the percentage rate of infection, the percentage of sick, and in the percentage mortality from the disease. This result is, of course, a necessary consequence of our assumption that the disease is the only cause of death.

  5. More particular results have been obtained by substituting constants in the place of the undetermined functions assumed in the general theory.
Further, under these conditions the nature of the steady states has been more fully investigated and it has been shown that in all cases, except one, the steady states are stable ones. In the exception, a disturbance would result in purely periodic oscillations about the steady state.
3. Contributions to the Mathematical Theory of Epidemics. III. Further Studies of the Problem of Endemicity, by W O Kermack and A G McKendrick.
3.1. Reference.

Proceedings of the Royal Society of London. Series A 141 (843) (1933), 94-122.

3.2. From the Introduction.

In a previous paper (Part II of this series) an attempt was made to treat from a general point of view the problem of a single disease in a population which consisted of three categories of people - namely, never infected, sick and recovered - and in which the infectivity of the disease was a function of the period of illness, whilst the susceptibility of a recovered person was a function of the period which had elapsed since the time of his recovery. New individuals entering the population either by birth or by immigration naturally entered the category of the never infected which for convenience we called " virgins." It was pointed out that the results obtained were subject to two important limitations: (1) that the disease under consideration was the only cause of death, and (2) that the age of the individuals did not affect their in susceptibility or reproductiveness.

It is the purpose of the present paper to remove the first of these limitations by the introduction of constant non-specific death rates, which for the sake of generality are assumed to be different for virgins, sick, and recovered. It may be stated at once that the introduction of this additional factor produces surprisingly little change in the general nature of the results previously obtained, and that the conclusions of the previous paper hold with very little modification.

In the previous paper the results were first of all worked out for constant infectivity, recovery and death rates, and the more general problem, in which these rates were variable, was thereafter considered. The algebra for constant coefficients was relatively very simple. It is now found, however, that with the introduction of non-specific death rates, the simple case increases in complexity relatively much more than does the general case, so that the advantage of treating it separately largely disappears. In particular, whereas the various expressions for steady state levels previously came out explicitly as fairly simple functions of the constant coefficients, they are now dependent on a somewhat complicated quadratic equation. On the other hand the equations which refer to the case with variable rates, although they now contain a few extra terms, remain qualitatively similar in type to those previously obtained, and the same method of treatment leads at once to identical or closely similar results. We shall not therefore, in the present communication, treat in detail the case of constant rates but only give some of the main results. It may be mentioned, however, that the general equations have been checked at the various points by the introduction of constant rates, and comparison has been made between the formulae so obtained and those found when constant rates were used throughout.

As in the previous paper the equations which describe the progress of small variations about the steady state are formulated, but their fuller discussion has at present been reserved.

It will be recalled that in Part II a number of special cases were discussed either because they had some special practical importance, or because they exhibited peculiarities from the mathematical point of view. With the introduction of non-specific death rates, a number of new special cases requiring detailed consideration came to light. To appreciate the relationship between all these cases it became necessary to adopt a scheme of classification, which although multiplying the total number of special cases considerably, made their treatment much simpler, and except in two instances only very brief discussion was required. The five special cases of Part II are readily accommodated in the new scheme.

A question of some practical importance is the effect upon the size of the population, the number of sick, and the relative incidence of the disease, of changes in the various parameters which characterise either the population or the disease. These points were investigated to some extent in the previous paper, but are more fully considered in the present communication both in connection with the general case, and with a number of the special cases referred to above. The results obtained are not always in accordance with expectation.

3.3. Summary.

  1. The mathematical investigation of the progress of an infectious disease in a community of susceptible individuals has been extended to include the case where members of the community are removed as the result of some general cause of death acting according to constant non-specific death rates, as well as by death from the disease itself. Under the more general conditions here dealt with the main conclusions arrived at in the previous paper remain qualitatively unaltered. The limitations which remain are that the susceptibility and the infective power of the individual are supposed to be independent of his age, and further that specific individual immunity does not exist in the sense that the part of the population which escapes infection is assumed to be just as susceptible as the whole population would have been if it had not been infected.

  2. In the general case a unique steady state is found to exist provided that certain relatively simple conditions are satisfied. In the special cases considered a unique steady state in general exists when these conditions continue to be satisfied; but in particular instances, when these conditions are not satisfied, unique steady states will exist provided that certain other requirements are fulfilled.

  3. Increase of birth rates, in general, increases both the absolute and the relative prevalence of the disease in its steady state. The effect of increase in the non-specific death rates is less simple, but has been worked out at some length.

    Decrease in the infectivity of the disease or in the susceptibility of the uninfected results in an increase in the whole population density as well as in an increase in the number of infected. The effect upon the relative incidence of the disease cannot be simply expressed, but it has been worked out in detail in the text. In the absence of immigration, and with the birth rates and also the non-specific death rates equal for virgins and recovered, variation in infectivity or susceptibility will not alter the relative incidence of the disease.

    The total population, however, will increase with decrease of either of these two factors, whilst the number of diseased will also increase proportionately.

  4. Two types of threshold values have been encountered. In the first type the quantity in question must initially exceed the threshold value if the event or process is to occur in the population. Two examples of this type have been found. In the second type the quantity in question need not initially exceed the threshold value, but may gradually change as the system develops. The event or process to which the threshold refers can only take place when the threshold value has been exceeded. The total population density has a threshold value of this second kind with reference to the existence of steady states.
4. Contributions to the Mathematical Theory of Epidemics. IV. Analysis of Experimental Epidemics of the Virus Disease Mouse Ectromelia, by W O Kermack and A G McKendrick.
4.1. Reference.

The Journal of Hygiene 37 (2) (1937), 172-187.

4.2. From the Introduction.

During recent years we have been engaged in the study of the mathematical theory of the spread of an infectious disease in a community of susceptible individuals, and some of the earlier results have been published in a series of papers (Kermack & McKendrick, 1927, 1932, 1933, 1936). We have assumed that the recovery rate, death-rate, infection rate, etc., are represented by functions of the period of the disease, and these in particular cases may be constants. The results obtained even in the absence of definite a priori knowledge of the values of these functions or constants allow us to reach certain qualitative conclusions and to interpret qualitative observations. So far, however, it has been difficult to apply the theory in a quantitative sense, chiefly because of the absence of satisfactory data referring to actual diseases. Such data might refer either to human or animal communities under ordinary non-experimental conditions, or to the results of animal experiments under strict control. The data referring to animal and human communities are usually meagre and far from homogeneous. On the other hand the experimental study of epidemics is still in its infancy and it is only recently that the pioneer work of Greenwood et al. (1936) has made available a certain amount of experimental data which though limited in range is of the highest value. It is obviously desirable to find out whether the theory as developed in our previous communications can be applied to these data, and if so to work out the results in a quantitative sense.

The mathematical theory which we have developed accommodates the experimental fact that an individual who has recovered from a disease is frequently found to be relatively immune. In the special case where the immunity is complete so that no second infection ever occurs the theory proves to be comparatively simple. This is especially so when we can assume that the community does not reproduce itself by birth, but is recruited solely by immigration. It happens that one of the cases investigated experimentally by Greenwood et al. (1936) is of this simple type. This is the epidemic of the virus disease ectromelia in mice which is fully reported in a memoir by the above authors, to which the reader is referred for details. Two separate epidemics are actually reported upon, and referred to as ectromelia 1 and ectromelia 2. The purpose of the present paper is to discuss the data derived from these two experiments in the light of our general mathematical theory.

4.3. Summary.

  1. The experimental data obtained by Greenwood et al. (1936) relating to epidemics of the virus disease ectromelia in mice have been examined in the light of mathematical theory. Attention has been directed in particular to the life tables calculated from the observed data, which give the chance of survival and death after various periods of cage life.

  2. The life table relating to ectromelia 1 during the steady state phase shows very close agreement over the range 0-550 days with that predicted by the theoretical equation which involves only four arbitrary constants. A slight discrepancy over the first few days is evidently due to the fact that representation of the death-rate and the recovery rate by constant coefficients does not accommodate an incubation period. The values of the constants obtained from the data give a measure of the essential characteristics of the epidemic.

  3. General agreement, though not so complete, is also found when the theory is applied to ectromelia 2 During this phase however, although conditions were otherwise apparently uniform, a steady state had not actually been attained. On the other hand the equations do not apply so satisfactorily to the obviously inhomogeneous period.

  4. In the present analysis the assumption of constant rates gives a satisfactory account of the progress of an infection in a susceptible community. This result suggests that for many purposes the assumption of constant rates may be adequate.

  5. The short period fluctuations observed in the ectromelia epidemic were probably random in character and have no relationship to periodic fluctuation such, for example, as those detected by Brownlee in the case of measles.
5. Contributions to the Mathematical Theory of Epidemics. V. Analysis of Experimental Epidemics of Mouse-Typhoid; A Bacterial Disease Conferring Incomplete Immunity, by W O Kermack and A G McKendrick.
5.1. Reference.

The Journal of Hygiene 39 (3) (1939), 271-288.

5.2. From the Introduction.

In a recently published paper (Kermack & McKendrick, 1937) the observational data relating to epidemics of ectromelia in populations of mice maintained under experimental conditions (Greenwood et al. 1936) has been analysed in the light of a mathematical theory of epidemics developed by us during recent years (Kermack & McKendrick, 1927, 1932, 1933, 1936). It was shown that the life table giving the chance of mice surviving for various lengths of time in infected communities is very closely represented by a formula calculated on the assumption that the various rates - infection rate, recovery rate, death rate, etc. - are constants. It is, of course, realised that this simplifying assumption can only be regarded as approximately true. It renders the application of the general theory practicable, and the result of the investigation justifies its use, in so far as the theory so simplified does actually conform to the experimental results.

The ectromelia epidemics, however, are of rather special type in that they refer to a disease in which the immunity conferred by an attack is almost, if not quite, complete. The disease is also peculiar in having a very short incubation period. In the simplified theory with constant coefficients no allowance at all was made for an incubation period, and it was to be expected that in the case of a disease with a longer incubation period, some special method would have to be devised to accommodate it.

It was therefore especially desirable that an attempt should be made to extend the work already done in relation to the virus disease ectromelia to the case of a typical bacterial disease, in which immunity was not so complete and the incubation period was somewhat longer.

5.3. Conclusions.

  1. An analysis has been made of the progress of an epidemic of mouse typhoid (previously described by Greenwood et al.) in a relatively large herd of mice. This epidemic differs from that of ectromelia analysed in our previous paper, in that it is caused by a bacterium (Bact. aertrycke), and not by a virus. In this case the immunity resulting from an attack of the disease is only partial, and the incubation period is somewhat longer, being about 13 days as compared with 3 or 4 in the case of ectromelia.

  2. Although the existence of partial instead of complete immunity leads to more complicated mathematical expressions involving an extra constant, it does not introduce any essential difficulty. It is found however that the longer incubation period requires special treatment.

  3. The life tables which have been calculated by Greenwood et al. for the epidemic B_6, have been fitted on the basis of the present theory. Reasonably good agreement between theoretical and observed values is obtained, provided that the theory is suitably modified so as to accommodate the incubation period. As a result of the symmetry of the equations involved, two alternative interpretations of the parameters, in terms of the constants which characterise the disease, are admissible. From the available data we have not been able to determine which of the two interpretations is the correct one.

  4. When the immigration rate is three mice or one mouse per day, the steady state levels, according to the published data, appear to be approximately one-half, or one-sixth respectively of that found in the case where the immigration rate is six mice per day. This result is in agreement with the present theory.

  5. The above results indicate that the present theory using constant coefficients is adequate to explain the main features of the mouse typhoid epidemics, provided that allowance be made for the somewhat prolonged incubation period.
6. A Bright Future for Biologists and Mathematicians?, by A Hastings and M A Palmer.
6.1. Reference.

Science, New Series 299 (5615) (2003), 2003-2004.

6.2. Extract from the paper.

Kermack and McKendrick developed the threshold theorem to determine the conditions under which infectious disease epidemics occur. This theorem has proved crucial for calculating the level of vaccination (less than complete coverage) required to eradicate diseases like polio and smallpox, and for preventing outbreaks of diseases such as pertussis. This theorem relates the occurrence of an epidemic to the number of susceptible individuals, the duration of the infectious period, and the infectivity of the disease. The threshold theorem was initially developed to answer two fundamental biological questions: Why do infectious disease epidemics occur, and why do they typically die out before all susceptible individuals contract the disease? These questions were answered by using the threshold theorem to develop the SIR (susceptible, infective, removed) model (1), which consists of three differential equations. The SIR model assumes that over the time scale of an epidemic, births and deaths in the host population can be ignored. The model includes the rate of removal (through death or recovery) of infected persons from the group passing on the infection, instead of specifying the more correct but harder to analyse assumption that there is a fixed time period during which an individual can infect others. The threshold theorem was originally illustrated using methods that relied on the graphic display of the number of infective and susceptible individuals during an infectious disease outbreak. The graphic representation of the threshold theorem reveals that the density of susceptible individuals must exceed a certain critical value for an epidemic to occur. This theorem has unquestionable relevance, given heightened concerns about the deliberate introduction of new infectious bioterrorist agents.

Last Updated November 2020