Mathematical Biosciences and Engineering, 2016, 13(5): 1093-1118. doi: 10.3934/mbe.2016032.

34C60, 34D23, 37N25, 45J05, 91B69, 92B05.

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Heterogeneous population dynamics and scaling laws near epidemic outbreaks

1. ORCOS, Institute of Statistics and Mathematical Methods in Economics, Vienna University of Technology, Wiedner Hauptstrasse 8, A-1040 Vienna
2. Faculty of Mathematics, Technical University of Munich, Boltzmannstrasse 3, 85748 Garching

In this paper, we focus on the influence of heterogeneity and stochasticity of the population on thedynamical structure of a basic susceptible-infected-susceptible (SIS) model. Firstwe prove that, upon a suitable mathematical reformulation of the basic reproduction number, thehomogeneous system and the heterogeneous system exhibit a completely analogous globalbehaviour. Then we consider noise terms to incorporate the fluctuation effects and therandom import of the disease into the population and analyse the influence of heterogeneityon warning signs for critical transitions (or tipping points). This theory shows that one maybe able to anticipate whether a bifurcation point is close before it happens. We use numericalsimulations of a stochastic fast-slow heterogeneous population SIS model and show various aspectsof heterogeneity have crucial influences on the scaling laws that are used as early-warningsigns for the homogeneous system. Thus, although the basic structural qualitative dynamicalproperties are the same for both systems, the quantitative features for epidemic predictionare expected to change and care has to be taken to interpret potential warning signs for diseaseoutbreaks correctly.
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Keywords warning signs; critical transition; transcritical bifurcation; heterogeneous population; stochastic perturbation; SIS-model; reproduction number.; Epidemics; tipping point

Citation: Andreas Widder, Christian Kuehn. Heterogeneous population dynamics and scaling laws near epidemic outbreaks. Mathematical Biosciences and Engineering, 2016, 13(5): 1093-1118. doi: 10.3934/mbe.2016032

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Copyright Info: 2016, Andreas Widder, et al., licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution Licese (http://creativecommons.org/licenses/by/4.0)

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