ALBERT

Notes: Summary This paper discusses a deterministic model of the spread of an infectious disease in a closed population that was proposed byKermack &McKendrick (1927). 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.

Notes: Summary 1. This paper discusses results of simulation studies with population models that were set up to illustrate the ideas about stabilization of population fluctuations and spreading of the risk of extinction expounded by den Boer (1968). In particular, the number of factors influencing net reproduction, the heterogeneity of the habitat and the possibility of a population's containing animals of different age classes were considered as possibly contributing to stabilization and to spreading of risk. 2. The model defined by equation (3.1.2), where r(t) denotes the net reproduction from t to t+1, f i (t) denotes the value of the i-th environmental factor in year t, and where the other symbols denote positive constants, was simulated by choosing for the f i(t) sequences of meteorological data from published tables. Such sequences may be serially correlated as well as correlated among themselves and using such real data was considered to be more realistic than working with sequences of independent random numbers, for example. Increasing the number k of factors turned out to stabilize fluctuations in the density. This fact could also be mathematically proved under not very restrictive assumptions. In a model where the logarithm of the net reproduction on the average is some-what greater than zero, and where “crashes” may occur at high densities, the population may persist for a very long time, even if the “size” of the crashes does not depend on density, and the times at which the crashes occur are chosen at random. 3. A model formulated in terms of matrices and vectors, in which a population was supposed to consist of 9 subpopulations and of several age classes was simulated. It was assumed that after a reproduction period the animals migrate between the subpopulations or emigrate from the whole population. It turned out that increasing the number of age classes may increase stability and that models where there is exchange of individuals between subpopulations by \ldmigration\rd are more stable than populations consisting of isolated subpopulations. Letting the exchange between subpopulations be \lddensity-dependent\rd had some stabilizing effect too, but not very conspicuously so.