Heterogeneous population dynamics and scaling laws near epidemic outbreaks

Andreas  Widder; Christian  Kuehn

doi:10.3934/mbe.2016032

Article Contents

2016, Volume 13, Issue 5: 1093-1118. Doi: 10.3934/mbe.2016032

This issue Previous Article A two-sex matrix population model to represent harem structure Next Article

Heterogeneous population dynamics and scaling laws near epidemic outbreaks

Andreas Widder^1, and
Christian Kuehn^2,

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

Received: August 2015

Revised: April 2016

Published: June 2016

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Abstract

In this paper, we focus on the influence of heterogeneity and stochasticity of the population on the dynamical structure of a basic susceptible-infected-susceptible (SIS) model. First we prove that, upon a suitable mathematical reformulation of the basic reproduction number, the homogeneous system and the heterogeneous system exhibit a completely analogous global behaviour. Then we consider noise terms to incorporate the fluctuation effects and the random import of the disease into the population and analyse the influence of heterogeneity on warning signs for critical transitions (or tipping points). This theory shows that one may be able to anticipate whether a bifurcation point is close before it happens. We use numerical simulations of a stochastic fast-slow heterogeneous population SIS model and show various aspects of heterogeneity have crucial influences on the scaling laws that are used as early-warning signs for the homogeneous system. Thus, although the basic structural qualitative dynamical properties are the same for both systems, the quantitative features for epidemic prediction are expected to change and care has to be taken to interpret potential warning signs for disease outbreaks correctly.

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Mathematics Subject Classification: 34C60, 34D23, 37N25, 45J05, 91B69, 92B05.

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