A sharp threshold for disease persistence in host metapopulations

Abstract

A sharp threshold is established that separates disease persistence from the extinction of small disease outbreaks in an S→E→I→R→S type metapopulation model. The travel rates between patches depend on disease prevalence. The threshold is formulated in terms of a basic replacement ratio (disease reproduction number), ℛ₀, and, equivalently, in terms of the spectral bound of a transmission and travel matrix. Since frequency-dependent (standard) incidence is assumed, the threshold results do not require knowledge of a disease-free equilibrium. As a trade-off, for ℛ₀>1, only uniform weak disease persistence is shown in general, while uniform strong persistence is proved for the special case of constant recruitment of susceptibles into the patch populations. For ℛ₀<1, Lyapunov's direct stability method shows that small disease outbreaks do not spread much and eventually die out.

Keywords:

• Persistence
• Extinction
• Basic reproduction number

Acknowledgements

The authors thank two anonymous referees for helpful comments. This study was partially supported by NSF grant DMS 0314529 (T.D. and H.R.T.) and by NSERC and MITACS (P.v.d.D).