Global stability of the multi-strain Kermack-McKendrick (renewal) epidemic model

TL;DR

This paper proves the global stability of multi-strain epidemic models based on the Kermack-McKendrick framework, showing how the basic reproduction number determines whether strains die out or the fittest strain persists.

Contribution

It extends previous work to multi-strain models, establishing global stability criteria based on the maximum basic reproduction number.

Findings

If all R0j ≤ 1, the infection dies out and the population is infection-free.

If the maximum R0j > 1, the fittest strain persists and the endemic equilibrium is globally stable.

The basic reproduction number acts as a sharp threshold for strain persistence or extinction.

Abstract

We extend a recent investigation by Meehan et al. (2019) regarding the global stability properties of the general Kermack-McKendrick (renewal) model to the multi-strain case. We demonstrate that the basic reproduction number of each strain $R_{0j}$ represents a sharp threshold parameter such that when $R_{0j} \leq 1$ for all $j$ each strain dies out and the infection-free equilibrium is globally asymptotically stable; whereas for $R_{01} \equiv \mathrm{max}_j\, R_{0j} > 1$ the endemic equilibrium point $\bar{P}^1$, at which only the fittest strain (i.e. strain 1) remains in circulation, becomes globally asymptotically stable.