Generality of endemic prevalence formulae

In simple infection models, the susceptible proportion s^* in endemic equilibrium is related to the basic reproduction number R₀ by s^* = 1/R₀. We investigate the extent to which this relationship remains valid under more realistic modelling assumptions. In particular, we relax the biologically implausible assumptions that individuals' lifetimes and infectious periods follow exponential distributions; allow a general recruitment process; allow for multiple stages of infection; and consider extension to a multigroup model in which the groups may represent, for instance, spatial heterogeneity, or the existence of super-spreaders. For a homogeneous population, we find that: (i) the susceptible proportion is s* = 1/R0e, where R0e is a modified reproduction number, equal to R₀ only in certain circumstances; (ii) the proportions of the population in each stage of infection are proportional to the expected time spent by an infected individual in that stage before recovery or death. We demonstrate robustness of the formula s^* = 1/R₀ for many human infections by noting conditions under which R0e is approximately equal to R₀, while pointing out other circumstances under which this approximation fails. For heterogeneous populations, the formula s^* = 1/R₀ does not hold in general, but we are able to exhibit symmetry conditions under which it is valid.