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

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## 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.

## References

- Anderson RM, May RM (1991) Infectious diseases of humans: dynamics and control. Oxford University Press, OxfordGoogle Scholar
- Anderson RM, Medley GF, May RM, Johnson AM (1986) A preliminary study of the transmission dynamics of the human immunodeficiency virus (HIV), the causative agent of AIDS. IMA J Math Appl Med Biol 3:229–263MATHMathSciNetCrossRefGoogle Scholar
- Andersson H, Britton T (1998) Heterogeneity in epidemic models and its effect on the spread of infection. J Appl Probab 35:651–661MATHMathSciNetCrossRefGoogle Scholar
- Ball F (1985) Deterministic and stochastic epidemics with several kinds of susceptibles. Adv Appl Probab 17:1–22MATHCrossRefGoogle Scholar
- Coutinho FAB, Massad E, Lopez LF, Burattini MN, Struchiner CJ, Azevedo-Neto RS (1999) Modelling heterogeneities in individual frailties in epidemic models. Math Comput Model 30:97–115MATHMathSciNetCrossRefGoogle Scholar
- Diekmann O, Heesterbeek (1999) Mathematical epidemiology of infectious diseases., Wiley Series in Mathematical and Computational BiologyWiley, ChichesterMATHGoogle Scholar
- Dwyer G, Dushoff J, Elkinton JS, Levin SA (2000) Pathogen-driven outbreaks in forest defoliators revisited: building models from experimental data. Am Nat 156:105–120CrossRefGoogle Scholar
- Gomes MGM, Aguas R, Lopes JS, Nunes MC, Rebelo C, Rodrigues P, Struchiner CJ (2012) How host heterogeneity governs tuberculosis reinfection. Proc R Soc B 279:2473–2478CrossRefGoogle Scholar
- Gomes MGM, Lipsitch M, Wargo AR, Kurath G, Rebelo C, Medley GF, Coutinho A (2014) A missing dimension in measures of vaccination impacts. PLoS Pathog 10(3):e1003849. doi: 10.1371/journal.ppat.1003849 CrossRefGoogle Scholar
- Hickson RI, Roberts MG (2014) How population heterogeneity in susceptibility and infectivity influences epidemic dynamics. J Theor Biol 350(7):70–80MathSciNetCrossRefGoogle Scholar
- Hyman JM, Stanley EA (1988) Using mathematical models to understand the AIDS epidemic. Math Biosci 90:415–473MATHMathSciNetCrossRefGoogle Scholar
- Katriel G (2012) The size of epidemics in populations with heterogeneous susceptibility. J Math Biol 65:237–262MATHMathSciNetCrossRefGoogle Scholar
- Lloyd-Smith JO, Schreiber JS, Kopp PE, Getz W (2005) Superspreading and the effect of individual variation on disease emergence. Nature 438:355–359CrossRefGoogle Scholar
- Miller JC (2007) Epidemic size and probability in populations with heterogeneous infectivity and susceptibility. Phys Rev E 76:010101CrossRefGoogle Scholar
- Novozhilov A (2008) On the spread of epidemics in a closed heterogeneous population. Math Biosci 215:177–185MATHMathSciNetCrossRefGoogle Scholar
- Pastor-Satorras R, Vespignani V (2001) Epidemic dynamics and endemic states in complex networks. Phys Rev E 63:066117CrossRefGoogle Scholar
- Pessoa D, Souto-Maior C, Lopes JS, Gjini E, Ceña B, Codeço CT, Gomes MGM (2014) Unveiling time in dose-response models to infer host susceptibility to pathogens. PLoS Comput Biol 10(8):e1003773Google Scholar
- Rodrigues P, Margheri A, Rebelo C, Gomes MGM (2009) Heterogeneity in susceptibility to infection can explain high reinfection rates. J Theor Biol 259:280–290MathSciNetCrossRefGoogle Scholar
- Smith DL, Dushoff J, Snow RW, Hay SI (2005) The entomological inoculation rate and
*Plasmodium falciparum*infection in African children. Nature 438:492–495CrossRefGoogle Scholar