Abstract
In simple infection models, the susceptible proportion s^{*} in endemic equilibrium is related to the basic reproduction number R_{0} by s^{*} = 1/R_{0}. 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 superspreaders. For a homogeneous population, we find that: (i) the susceptible proportion is s* = 1/R0e, where R0e is a modified reproduction number, equal to R_{0} 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/R_{0} for many human infections by noting conditions under which R0e is approximately equal to R_{0}, while pointing out other circumstances under which this approximation fails. For heterogeneous populations, the formula s^{*} = 1/R_{0} does not hold in general, but we are able to exhibit symmetry conditions under which it is valid.
Original language  English 

Pages (fromto)  3036 
Number of pages  7 
Journal  Mathematical Biosciences 
Volume  269 
DOIs  
Publication status  Published  Nov 2015 
Keywords
 Basic reproduction number
 Endemic equilibrium
 Spatial heterogeneity
 Superspreaders
ASJC Scopus subject areas
 Applied Mathematics
 Statistics and Probability
 Modelling and Simulation
 General Agricultural and Biological Sciences
 General Biochemistry,Genetics and Molecular Biology
 General Immunology and Microbiology
 General Medicine
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Damian Clancy
 School of Mathematical & Computer Sciences  Professor
 School of Mathematical & Computer Sciences, Actuarial Mathematics & Statistics  Professor
Person: Academic (Research & Teaching)