Mathematical Models for Infectious Disease Statistics

Numerous mathematical models have been developed to gain better insight into the transmission and control of infectious diseases. Yet there are many unsolved problems, partly because the models are still too simple, partly because detailed epidemiologic records are notoriously lacking. The present survey concentrates on virus infections in humans. It is shown that available data do not allow a discrimination between various plausible models for the spread of common cold in households. Similar problems of model identification arise in the analysis of age-specific sero-prevalence-data of antibodies with so-called catalytic models. From such data alone one cannot derive contact rates between different age groups, although knowledge of these rates is needed in order to evaluate the effects of mass immunization and to describe the fluctuating infection incidence patterns. A new deterministic model is presented which takes into account increased infection transmission inside schools. This provides an explanation for one- and two-year periods of recurrent measles epidemics. The paper provides an outlook to future developments in this field.