The size of epidemics in populations with heterogeneous susceptibility

Abstract

We formulate and study a general epidemic model allowing for an arbitrary distribution of susceptibility in the population. We derive the final-size equation which determines the attack rate of the epidemic, somewhat generalizing previous work. Our main aim is to use this equation to investigate how properties of the susceptibility distribution affect the attack rate. Defining an ordering among susceptibility distributions in terms of their Laplace transforms, we show that a susceptibility distribution dominates another in this ordering if and only if the corresponding attack rates are ordered for every value of the reproductive number R0. This result is used to prove a sharp universal upper bound for the attack rate valid for any susceptibility distribution, in terms of R0 alone, and a sharp lower bound in terms of R0 and the coefficient of variation of the susceptibility distribution. We apply some of these results to study two issues of epidemiological interest in a population with heterogeneous susceptibility: (1) the effect of vaccination of a fraction of the population with a partially effective vaccine, (2) the effect of an epidemic of a pathogen inducing partial immunity on the possibility and size of a future epidemic. In the latter case, we prove a surprising '50% law': if infection by a pathogen induces a partial immunity reducing susceptibility by less than 50%, then, whatever the value of R0>1 before the first epidemic, a second epidemic will occur, while if susceptibility is reduced by more than 50%, then a second epidemic will only occur if R0 is larger than a certain critical value greater than 1.

Epidemics spread in heterogeneous populations

“…Resistant individuals are capable of building firewalls [14], while irresistant individuals determine paths along which epidemics can easily spread. Figure 1 indicates that increasing σ I , in line with earlier investigations [10,23], might decrease the epidemic size, see the top, bottom and middle panels of Figure 1.…”

“…If only the susceptibility is varied, less homogeneous populations are less likely to be invaded and consequently epidemic size is reduced [10,15,23]. Increasing heterogeneity reduces the chances of large outbreaks due to the possibility of creating impenetrable firewalls [14].…”

“…e.g. susceptibility or infectivity, follow continuous distributions [8][9][10][11][12][13]. Some of such heterogeneous models, by means of nonlinear transformations [9], can be mapped into homogeneous models.…”

Population-level effects of suppressing fever

“…Because the final size Z is a decelerating function of the reproduction number R (figure 1a), antipyresis always enhances transmission more for less transmissible diseases (which have smaller R 0 : figure 1b). The precise quantitative predictions in figure 1b depend on our use of the standard final size relation; however, the qualitative conclusions are very general because the expected final size always increases (typically in a decelerating fashion) as R increases [27][28][29][30][31].…”

Set-membership estimations for the evolution of infectious diseases in heterogeneous populations

“…Theoretical results about the influence of heterogeneity on the basic reproduction number in such models are available (e.g. Katriel 2012; Margheri et al. 2015), as well as results about intervention strategies, e.g.…”