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F. Miksch:
"Mathematische Modelle für neue Erkenntnisse über Epidemien mittels Herdenimmunität und Serotypenverschiebung";
Supervisor, Reviewer: M. Atanasijevic-Kunc, F. Breitenecker; Institut für Analysis und Scientific Computing, 2012; oral examination: 2012-11-22.

Motivation:
Models for simulation of epidemics have a long tradition in the field of mathematics and other sciences. In the past, simple laboratory experiments and models were mainly used. But today, simulations are getting more complex and detailed due to
increasing calculation power of computers and availability of more comprehensive data.
Hence it is possible to consider questions that could not be examined before.
Goals and methods:
The aim of this work is to examine, de ne and interpret dynamic
e ects on the spread of epidemics through the use of di erent models. This approach is heavily based on the deliberate application of agent based models, which simulate the spread of a disease as a consequence of transmission from person to person, and Markov models, which are based on stochastic processes. Thanks to its structure, the former
approach is able to create real e ects, while the latter approach uses given transition probabilities, and hence is not able to produce feedback e ects.
Results:
Simple epidemics consist of a single pathogen that spreads among a population. In more complex cases, several types of the pathogen, which are called serotypes, occur. Often they are in competition and interfere with each other in their propagation.
Interventions against such epidemics, for example by vaccinations, generally cause nonlinear e ects. This nonlinear system behavior has been observed and documented several times in the real world and also occurs in agent based models. One important e ect in this context is herd immunity which describes the bene t of unvaccinated people by a vaccination strategy. In case of competitive serotypes, vaccinations might additionally change the balance of powers of serotypes, which is called serotype shift. Suitable, exible de nitions allow meaningful interpretations of these phenomena that provide additional information on the spread of epidemics. Markov models do not allow generating such e ects due to given transition probabilities, hence their results di er from the ones of the agent based model. By taking into consideration the herd immunity and, in case of competitive sero-
types, also the serotype shift, the results of the Markov model can be corrected so that they match the agent based model. This relationship is demonstrated in representative test cases and is also proved analytically.