Persistence time of SIS infections in heterogeneous populations and networks

Abstract

For a susceptible–infectious–susceptible infection model in a heterogeneous population, we present simple formulae giving the leading-order asymptotic (large population) behaviour of the mean persistence time, from an endemic state to extinction of infection. Our model may be interpreted as describing an infection spreading through either (1) a population with heterogeneity in individuals’ susceptibility and/or infectiousness; or (2) a heterogeneous directed network. Using our asymptotic formulae, we show that such heterogeneity can only reduce (to leading order) the mean persistence time compared to a corresponding homogeneous population, and that the greater the degree of heterogeneity, the more quickly infection will die out.

Precise Estimates of Persistence Time for SIS Infections in Heterogeneous Populations

“…As Clancy (2018) points out, see also Wilkinson and Sharkey (2013), it follows from the network duality results of Harris (1976) and Holley and Liggett (1975) that provided infectious periods are exponentially distributed, the value of τ is unchanged if we interchange the roles of λ, μ. Theorem 1(ii) then follows immediately from Theorem 1(i).…”

“…Equation ( 18) has been previously studied in numerous specific applications (e.g. Assaf and Meerson 2010;Clancy 2018;Dykman et al 1994;Elgart and Kamenev 2004;Hindes and Schwartz 2016;Kamenev and Meerson 2008;Lindley et al 2014). Equation ( 19) has been analysed in some detail in the k = 1dimensional case (Assaf and Meerson 2017), but does not seem to have been much studied in the context of multidimensional problems.…”

“…For the case λ = 1, the solution to Eq. ( 18) with boundary condition V ( y * ) = 0 is known, by a slight generalisation of formula (20) of Clancy (2018), to be…”

“…A different extension of the result of Andersson and Djehiche (1998) has recently been established in Clancy (2018). For an SIS model incorporating heterogeneity in susceptible individuals' levels of susceptibility or infected individuals' levels of infectiousness, an explicit formula was found for the leading-order constant A in the relationship lim N →∞ (ln τ )/N = A.…”

“…Infectious periods were allowed to follow an Erlang distribution, and the value of A shown to depend only upon the mean of the distribution, under this assumption. In the current paper, we build on the work of Clancy (2018) to establish results of the much more precise form (1), including simple explicit formulae for the prefactor constant C. At the same time, we extend the model of Clancy (2018) in two ways: firstly, we allow for heterogeneity in individuals' infectious period distributions in addition to the heterogeneities in susceptibility and infectiousness of Clancy (2018); secondly, following the approach of Ball et al (2016), we allow for infectious periods following quite general (not necessarily Erlangian) distributions. Clancy (2018) showed that for a sufficiently large population, greater heterogeneity (in the sense of majorization ordering, see Marshall et al 2011), whether in susceptibility or infectiousness, leads to a reduction in mean persistence time of infection in the population.…”