# On the correlation between variance in individual susceptibilities and infection prevalence in populations

## Abstract

The hypothesis that infection prevalence in a population correlates negatively with variance in the susceptibility of its individuals has support from experimental, field, and theoretical studies. However, its generality has never been formally demonstrated. Here we formulate an endemic SIS model with individual susceptibility distributed according to a discrete or continuous probability function to assess the generality of such hypothesis. We introduce an ordering among susceptibility distributions with the same mean, analogous to that considered in Katriel (J Math Biol 65:237–262, 2012) to order the attack rates in an epidemic SIR model with heterogeneity. It turns out that if one distribution dominates another in this order then it has greater variance and corresponds to a lower infection prevalence for \(R_0\) varying in a suitable maximal interval of the form \(]1, R_0^*].\) We show that in both the discrete and continuous frameworks \(R_0^*\) can be finite, so that the expected correlation among variance and prevalence does not always hold. For discrete distributions this fact is demonstrated analytically, and the proof introduces a constructive procedure to find ordered pairs for which \(R_0^*\) is arbitrarily close to \(1.\) For continuous distributions our conclusion is based on numerical studies with the beta distribution. Finally, we present explicit partial orderings among discrete susceptibility distributions and among symmetric beta distributions which guarantee that \(R_0^*=+\infty \).

## Keywords

Epidemiological model Heterogeneous population Susceptibility distribution Variance## Mathematics Subject Classification

92D25 92D30## Notes

### Acknowledgments

We thank the referees for their useful comments, which helped to improve both the presentation and the mathematical contents of this work.

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