This paper discusses a deterministic model of the spread of an infectious disease in a closed population that was proposed byKermack &McKendrick . The mathematical assumptions on which the model is based are listed and criticized. The ‘threshold theorem’ according to which an epidemic develops if, and only if, the initial population density exceeds a certain value determined by the parameters of the model, is discussed. It is shown that the theorem is not true. A weaker result is stated and (...) proved.A new simplified version of the model is discussed. In this model it is assumed that an infected individual becomes infectious after a constant time and that it will removed after a constant time. Numerical simulations of this model show that the form of the fluctuations in the number of diseased individuals depends on the values of the parameters. The number of infectious individuals may oscillate up and down under certain circumstances. Necessary conditions for the number of infectious individuals to rise above previous levels are stated and proved. (shrink)

Deterministic models in population dynamics often are really approximations to stochastic models, justified by an appeal to the law of large numbers. It is proposed to call such models pseudodeterministic. Four questions are discussed in this article: (1) What errors may be made by equating deterministically predicted values to expectations? (2) When, and in what sense, may numbers be assumed to be large? (3) How large are the variances, coefficients of variations, etc., as assigned to the variables in the stochastic (...) versions of the models? (4) What role may pseudodeterministic models play in empirical research, where problems of statistical reliability arise?As an example, a modified Nicholson-Bailey model of the interaction between insect parasitoids and their hosts is discussed; the modification consists of assigning a random (density-independent) mortality to the parasitoid population. A stochastic version of this model is discussed. The expectation of the final host density is compared with the value computed from the deterministic model. The latter value is systematically lower than the former. The magnitude of the difference depends on parameter values. The variability to be expected with the stochastic model is characterized by the coefficient of variation of the final host density; its dependence on parameter values and initial conditions is discussed. It is concluded that it is worthwhile in practical applications to estimate parasitoid mortality, and that the coefficient of variation in real situations may be far from negligible. (shrink)