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which cause epidemics; on the conditions of living which favour the spread of infectious disease.

With the discovery of the organisms which cause disease and with the careful observation in the field as to the manner in which disease spreads from person to person, many new points of view have emerged. It is no longer sufficient to talk vaguely of fomites. Most diseases have their special forms of spreading which account for practically all the cases. Thus measles and smallpox are exceedingly infectious from person to person. Enteric fever is nearly always carried by contaminated water or contaminated food. Cholera is spread by water and flies. Other diseases have been found to be practically non-infectious from person to person unless by means of an intermediate parasite. Thus typhus and trench fever are carried by lice, while yellow fever and malaria require the intervention of the mosquito. The mode of spread of some diseases, however, is still obscure. Among these scarlet fever must be placed. While direct infection undoubtedly takes place a satisfactory elucidation of J.he prob- lems of its dissemination has not yet been arrived at.

For accurate thinking on infectious diseases it must be noted that disease-producing organisms possess two qualities: one, the power of causing the disease, and the second the power of producing a severe attack of disease. The first may be termed infectivity and the second virulence. These qualities must not be confused. In point of fact they are not associated in any constant degree. Sometimes an epidemic begins with a large number of severe cases and sometimes the reverse. In certain diseases the height of the epidemic seems to be associated with severe disease, in others with that of milder type. The former at least holds for a certain number of large epidemics of measles of which the statistics have been investigated. The latter is the case both in Glasgow and London in regard to the autumnal prevalence of scarlet fever.

That an epidemic might possess a definite form capable of calculation seems to have been advanced first by Dr. Farr. In 1840 he graduated the decline of the great smallpox epidemic in England to the normal curve of error, and obtained a very close representation of the facts. He promised further discussion but seems to have given none till 1867. In this year he returned to the subject in connexion with the cattle plague, writing a letter to the Daily News in which it was stated that though in the popular conception plague was advancing with such rapidity that all the cattle of the country might be destroyed, in reality the force of the epidemic was spent, and that if the form of the epidemic curve up to that point were taken as a basis of calculation the future course could be foretold. The prediction proved to be very near the truth.

The theory of the course of the epidemic, however, as a guide to the solution of the problem has unfortunately not proved so fertile as might have been hoped. Some facts are quite definite. The curve of the epidemic is generally found to be symmetrical, the fall cor- responding closely to the rise, though in some diseases the ascent is more rapid than the descent, and in some the reverse. The equation of the curve which describes the majority of epidemics, as found by trial apart from theory, is

(-4)"

where y is the number of cases at time t, t being measured from the centre of the epidemic. Curves closely resembling that given by the above equation arise on a number of hypotheses of which two are discussed. First, the organism may be assumed to possess at the beginning of the disease a high degree of infectivity which decreases as the epidemic goes on. If the loss of infectivity is according to geometric law, the normal curve of error already used by Dr. Farr is the result. It is sufficient to state that on various probable hypotheses regarding exposure to infection, etc., the normal curve may be so modified as to take the form found by observation. Secondly, a similar type of curve arises if we consider an epidemic dies out from lack of susceptible persons. It is not possible to distinguish statistically these hypotheses from the consideration of the epidemic form alone. In one case, however, the second hypothesis can be tested. If the form of the epidemic be calculated by assuming different degrees of infectivity on the part of the organism, an in- fectivity which remains constant during the epidemic, it is found that this curve becomes flatter and flatter the smaller the degree of

infectivity. Now with regard to plague in India among brown and black rats living more or less in the same circumstances, it is ob- served that many more brown rats are infected than 'black. In such circumstances the form of the epizootic should be different in the two species if the decline is due to lack of susceptible individuals. As a matter of fact it is nearly identical : a fact which tells strongly in favour of the hypothesis that the epidemic ends because of loss of infectivity on the part of the organisms. This example would be crucial but for the fact that the flea on which the spread of the epi- zootic depends has a law of seasonal prevalence of its own to which both the epizootics must conform. In many cases, however, the only feasible explanation of the course of an epidemic is that the organism loses the power of infecting as the epidemic proceeds. It is impossible to suppose, for instance, with regard to the great epi- demic of smallpox in London in 1901-2 that there were only 8,000 people susceptible, out of a population of 6,000,000. As the course of this epidemic was typical, rising and falling in the manner found to be characteristic, it cannot be argued that the decline was due to the action of the health authorities ; all they can have done is to limit the extent of the epidemic, leaving its course unchanged. It is clear, therefore, that in circumstances like this there is some biological factor at work as distinct from a statistical factor. It may then be taken that epidemics in general have a particular form which is identical in many different diseases: plague, influenza, scarlet fever, etc. Even great differences of time dp not bring about much change, the form of the epidemic of plague in Sydney in 1900 being nearly identical with that in London in 1665.

The next point requiring consideration is the periodicity in the epidemics of infectious diseases. Taking measles as an example, the common explanation is that each epidemic ends from the exhaustion of the number of susceptible persons, and that it is only when a new population of susceptible children has accumulated that a further outbreak occurs. This explanation fails to account for many of the facts. Even after the very large epidemic of measles in Glasgow in 1906, it was found that nearly half of the children admitted to the fever hospitals immediately thereafter suffering from other diseases had not suffered from measles so that there must have been, with the high infectivity of the epidemic, plenty of sus- ceptible material. The disease subject to the most extensive in- quiry hitherto has been measles. Using the method of the periodo- gram the statistics of London and all the chief towns of the British Isles have been analyzed. It is found that in almost no case is there only one period to be discovered. In London there are several, the chief of which is 97 weeks. This periodicity is found over the whole city. If the application of this mathematical method of analysis be admitted, this coexistence of epidemics of different periods, each appearing at its own time, seems to prove that the termination of an outbreak of the disease is due to loss of infectivity on the part of the organism. Periodicity in other diseases is well known. Thus in the city of Liverpool the epidemics of scarlet fever occurred at regular intervals of four years from 1850-78. On one occasion alone was there an exception when the interval between two epidemics was three years in place of four. A similar periodicity of five years has been observed in Glasgow. There is one specially interesting example, namely the occurrence of plague in Bombay. In many places, such as Hong-Kong, the period between each epi- demic is rigidly a year. In such a case the influence of the season of the year seems a sufficient explanation. But the case of Bombay is different. The first epidemic in 1897 had its maximum about the 4Oth day of the year. From this point until the last year for which statistics are available (1918), the date of the maximum of the epidemic has steadily advanced into the year, advancing about 80 days in 20 years or an average four days a year. It is difficult to account for a phenomenon like this except as being due to some property of the organism. The conclusion must be arrived at that while some periodicities of disease are strictly seasonal, others are not so, and require some further explanation.

A further important application of mathematics to epidemiology has been made by Sir Ronald Ross in his studies on malaria. Here the factors influencing the spread of the disease are numerous. Rain- fall and temperature, the number of persons carrying the organism in their blood, and the number of mosquitoes and the proximity of the breeding-places of the mosquito to the abodes of men are all capable of quantitative measurement, and of furnishing guidance in the adoption of suitable administrative measures.

Climate and Weather. The relationship of epidemics to cli- mate has received much attention in recent years, though in many cases the cause of seasonal prevalence is elusive. Thus why scarlet fever should be so regularly an autumnal disease is not at all clear. On, many cases, however, much light has been thrown. The discovery, for instance, that malaria was carried by the mosquito elucidates the seasonal distribution of that disease. A temperature of a certain height with associated pools of water is necessary for the rapid development of the mosquito and also a certain degree of temperature for the development of the parasite in the mosquito. In the same way the zone to which