Invasion reproductive numbers for periodic epidemic models

Abstract

There are many cases within epidemiology where infections compete to persist within a population. In studying models for such cases, one of the goals is to determine which infections can invade a population and persist when other infections are already resident within the population. Invasion reproductive numbers (IRN), which are tied to the stability of boundary endemic equilibria, can address this question. By reinterpreting resident infections epidemiologically, this study extends methods for finding IRNs to periodic systems, and presents some examples which illustrate the often complex computations required. Results identify conditions under which a simple time-average can be used to derive IRNs, and apply the methods to examine how seasonal fluctuations in influenza incidence facilitate the year-round persistence of bacterial respiratory infections.

Keywords: Basic reproductive number; Mathematical epidemiology; Periodic models.

Fig. 1

Graphs of system (12) that show ${\tilde{R}}_{1T}$ (dashed) and ${\tilde{R}}_1$ (solid) for varying values of τ. At the cutoff point, ${\tilde{R}}_{1T}=R_{1T}=R_1={\tilde{R}}_1$. The solid gray line is $R_1[=R_{1T}]$.

Fig. 2

Graphs of system (12) that show ${\tilde{R}}_{2T}$ (dashed) and ${\tilde{R}}_2$ (solid) for varying values of τ. At the cutoff point, ${\tilde{R}}_{2T}=R_{2T}=R_2={\tilde{R}}_2$. The solid gray line is $R_2[=R_{2T}]$.

Fig. 3

Simulation of system (12) for a set of parameters in which ${\tilde{R}}_1,{\tilde{R}}_2>1$ ($\tau =0.5$ and see Table 1).

Fig. 4

Graph showing 4 regions in $\left(R_1,R_2\right)$ parameter space representing different behaviors of system (12). $E_0$ is disease-free, $E_1$ is only infection 1 prevalent, $E_2$ is only when infection 2 is prevalent, and $E_3$ is coinfection. The dashed curves indicate where an IRN (derived via the linear operator method) equals 1, while the solid curves represent thresholds generated by the time-average method.

Fig. 5

$I_2\left(t\right)$ (bacterial infection, solid curve), $I_{12}\left(t\right)$ (influenza/bacterial coinfection, dashed curve), and their sum (dotted curve) vs. time, over a period of one year.