### Mathematical Biosciences and Engineering

2016, Issue 5: 1093-1118. doi: 10.3934/mbe.2016032

# Heterogeneous population dynamics and scaling laws near epidemic outbreaks

• Received: 01 August 2015 Accepted: 29 June 2018 Published: 01 July 2016
• MSC : 34C60, 34D23, 37N25, 45J05, 91B69, 92B05.

• 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.

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

### Related Papers:

• 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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