Computergestützte Epidemiologie
Mathematische Modellierung
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Critical population size for endemic transmission of measles and other infectious diseases

Infections which lead to temporal or life-long immunity can only persist in a population which is large enough. In small, isolated settlements, the pool of susceptible individuals soon is exhausted and the resulting bottle-neck in transmission causes a spontaneous extinction of the transmission. Using stochastic simulations, we examined how the critical population size depends on various demographic and epidemiologic parameters. 

Homogeneously mixing population

Using a single, homogeneously mixing population, we obtained the following results:

infectivity increases → critical population size decreases
duration of latency increases → critical population size decreases
duration of infectivity increases → critical population size decreases
vaccination coverage increases → critical population size increases
life expectancy increases → critical population size increases.

Increasing the variance of the duration of the infectious period (ie. using a gamma-distributed durations instead of exponentially distributed ones) also increases the critical population size, whereas an additional increase in the variance of the latent period or of the life expectancy of humans was of minor influence. When using a (positive or negative) population growth rate, it gets difficult to compare the results to simulations with a constant population size, but it was observed, that the infection could persist much better in growing populations, even if at all times the population size was smaller than in a corresponding simulation with constant size.

Fractured Populations

In further studies, we examined what happens if one subdivides the population in several subpopulations, assuming that most of the contacts of an individual remains within the population and that only a small fraction of contacts goes to inhabitants of other populations (without any spacial preferences). The result can be summarized as followed:

number of subpopulations increases → critical population size increases
isolation of subpopulations increases → critical population size increases

It can be shown that (unvaccinated) homogeneously mixing populations of several million inhabitants are needed to make infectious diseases like measles persist, if realistic parameter values are chosen. If one chooses to remove the unrealistic assumption of homogeneous mixing and uses fractured populations instead, the critical population size further increases. 

Spatially structured populations

Persistence in smaller populations could only be achieved after assuming a spatial structure such that individuals could only have contacts with the inhabitants of their own subpopulation and with those of a few selected "neighbouring" subpopulations. If, for example, one arranges all subpopulations in the form of a ring, and chooses an appropriate contact structure between neighbouring settlements, an infection like measles can persist in an unvaccinated population with a total size of less than 100,000 inhabitants. The spatial structure causes completely novel effects so that the dependency of the critical population size on parameter changes as shown in the above boxes holds no longer true.

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