A Framework for Epidemic Models

Abstract

A framework is developed that enables the modeling of the various mechanisms of epidemic processes. A model within the framework is completely characterized by a set of transmission functions. These functions support the modeling of the infectivity of a new infective as a function of its age-of-infection. They also support the presence of a, possible time-dependent, outside source of infection and allow for the introduction of an inhibitor function that accounts for the decreasing number of available susceptibles in the course of the epidemic. In addition this inhibitor function could describe the effects of an inoculation program, or a governmental information campaign, or improvements in general health care on the course of the epidemic. The proposed models deliver a complete stochastic description of the infectious-age structure of the population at any moment of time. In particular the size of the epidemic, that is the probability distribution of the number of new infectives, is worked out. By letting time tend to infinity an analytical expression for the final size of the epidemic is found. This leads to new recursive procedures for the determination of this final size. Ludwig's recursive scheme for this purpose is considerably extended. Finally, using a specific type of model from within the framework as a building block, a compound model is introduced that describes an epidemic with a structured population of infectives. Although such a compound model goes beyond the boundaries of the framework, still important characteristics present in the framework are preserved in such a model. The size of the epidemic at a certain moment of time, according to a compound model, as well as its final size are easily found. Moreover, for the infinite case simple expressions for the final size distribution and its first moment are found. This leads to a precise quantitative threshold theorem, that discriminates between a minor outbreak or a major build-up for such an epidemic.