Contributions to the mathematical theory of epidemics--III. Further studies of the problem of endemicity. 1933.
(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, namely, in Cases (4) and (2'd).(ABSTRACT TRUNCATED AT 400 WORDS)
Kermack
W O
WO
Laboratory of the Royal College of Physicians, Edinburgh, U.K.
McKendrick
A G
AG
D004813:Epidemiology / Q000266:history*; D049673:History, 20th Century; D006801:Humans; D008433:Mathematics / Q000266:history; D008962:Models, Theoretical