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

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.