The pluses and minuses of R0

Abstract

The concept of the basic reproduction number (R0) occupies a central place in epidemic theory. The value of R0 determines the proportion of the population that becomes infected over the course of a (modelled) epidemic. In many models, (i) an endemic infection can persist only if R0>1, (ii) the value of R0 provides a direct measure of the control effort required to eliminate the infection, and (iii) pathogens evolve to maximize their value of R0. These three statements are not universally true. In this paper, some exceptions to them are discussed, based on the extensions of the SIR model.

Figure 1

(a) Bifurcation diagram for the model with a susceptible R-class. Curves are top s^* and bottom i^* as functions of R₀. Broken lines signify unstable steady states and unbroken lines stable steady states. Curves are plotted for $\mathcal{P}$=0 (forward bifurcation), $\mathcal{P}$=$\mathcal{P}$_crit (shown in grey) and $\mathcal{P}$=2$\mathcal{P}$_crit (backward bifurcation), together with the trivial steady state s=1, i=0. Other parameters are μ=0.02 and γ=0.05. (b) An enlargement of the ($\mathcal{R}$₀, i^*) curve for $\mathcal{P}$=2$\mathcal{P}$_crit, showing the possibility of a hysteresis effect. For explanation of the lettering see the text.

Figure 2

Bifurcation diagram for the carriage model. The curves are 1−s^* as a function of $\mathcal{R}$₀, broken lines signify unstable steady states and unbroken lines stable steady states. Curves are plotted for $\mathcal{P}$=0 (lower curve), $\mathcal{P}$=$\mathcal{P}$_crit (shown in grey) and $\mathcal{P}$=2$\mathcal{P}$_crit (upper curve), together with the trivial steady state s=1. Parameter values are γ=0.05, μ=1/70, δ=0.025 and k=1. The function q(x)=exp(−x⁻¹), where x=β(i^*+$\mathcal{P}$c^*)/μ (appendix A.6) and $\mathcal{P}$_crit=0.8. For explanation of the lettering see the text.