The effect of population heterogeneities upon spread of infection Article First Online: 02 September 2012 Received: 16 January 2012 Revised: 25 July 2012 DOI :
10.1007/s00285-012-0578-x

Cite this article as: Clancy, D. & Pearce, C.J. J. Math. Biol. (2013) 67: 963. doi:10.1007/s00285-012-0578-x
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Abstract It has often been observed that population heterogeneities can lead to outbreaks of infection being less frequent and less severe than homogeneous population models would suggest. We address this issue by comparing a model incorporating various forms of heterogeneity with a homogenised model matched according to the value of the basic reproduction number \(R_0\) . We mainly focus upon heterogeneity in individuals’ infectivity and susceptibility, though with some allowance also for heterogeneous patterns of mixing. The measures of infectious spread we consider are (i) the probability of a major outbreak; (ii) the mean outbreak size; (iii) the mean endemic prevalence level; and (iv) the persistence time. For each measure, we establish conditions under which heterogeneity leads to a reduction in infectious spread. We also demonstrate that if such conditions are not satisfied, the reverse may occur. As well as comparison with a homogeneous population, we investigate comparisons between two heterogeneous populations of differing degrees of heterogeneity. All of our results are derived under the assumption that the susceptible population is sufficiently large.

Keywords Basic reproduction number SIR epidemic SIS epidemic Outbreak size Endemic prevalence Fade-out of infection

References Adler FR (1992) The effects of averaging on the basic reproduction ratio. Math Biosci 111:89–98

CrossRef MATH Google Scholar Andersson H, Britton T (1998) Heterogeneity in epidemic models and its effect on the spread of infection. J Appl Probab 35:651–661

MathSciNet CrossRef MATH Google Scholar Andersson H, Britton T (2000) Stochastic epidemics in dynamic populations: quasi-stationarity and extinction. J Math Biol 41:559–580

MathSciNet CrossRef MATH Google Scholar Andreasen V (2003) Dynamics of annual influenza A epidemics with immuno-selection. J Math Biol 46:504–536

MathSciNet CrossRef MATH Google Scholar Andreasen V (2011) The final size of an epidemic and its relation to the basic reproduction number. Bull Math Biol 73:2305–2321

MathSciNet CrossRef Google Scholar Ball FG (1983) The threshold behaviour of epidemic models. J Appl Probab 20:227–241

MathSciNet CrossRef MATH Google Scholar Ball FG (1985) Deterministic and stochastic epidemics with several kinds of susceptibles. Adv Appl Probab 17:1–22

CrossRef MATH Google Scholar Ball FG (1986) A unified approach to the distribution of total size and total area under the trajectory of infectives in epidemic models. Adv Appl Probab 18:289–310

CrossRef MATH Google Scholar Ball FG (1999) Stochastic and deterministic models for SIS epidemics among a population partitioned into households. Math Biosci 156:41–68

MathSciNet CrossRef MATH Google Scholar Ball FG, Britton T, Lyne OD (2004) Stochastic multitype epidemics in a community of households: estimation of threshold parameter and secure vaccination coverage. Biometrika 91:345–362

MathSciNet CrossRef MATH Google Scholar Ball FG, Clancy D (1993) The final size and severity of a generalised stochastic multitype epidemic model. Adv Appl Probab 25:721–736

MathSciNet CrossRef MATH Google Scholar Barbour AD (1972) The principle of the diffusion of arbitrary constants. J Appl Probab 9:519–541

MathSciNet CrossRef MATH Google Scholar Barbour AD (1976) Quasi-stationary distributions in Markov population processes. Adv Appl Probab 8: 296–314

MathSciNet CrossRef MATH Google Scholar Becker N, Marschner I (1990) The effect of heterogeneity on the spread of disease. Lect Notes Biomath 86:90–103

Google Scholar Blackwell D (1951) Comparisons of experiments. In: Proceedings of the second Berkeley symposium on mathematical statistics and probability. University of California Press, Berkeley, pp 93–102

Blackwell D (1953) Equivalent comparisons of experiments. Ann Math Stat 24:265–272

MathSciNet CrossRef MATH Google Scholar Borwein JM, Lewis AS, Nussbaum RD (1994) Entropy minimization, DAD problems, and doubly stochastic kernels. J Funct Anal 123:264–307

MathSciNet CrossRef MATH Google Scholar Britton T, Lindenstrand D (2009) Epidemic modelling: aspects where stochasticity matters. Math Biosci 222:109–116

MathSciNet CrossRef MATH Google Scholar Clancy D, Mendy ST (2011) The effect of waning immunity on long-term behaviour of stochastic models for the spread of infection. J Math Biol 61:527–544

MathSciNet CrossRef Google Scholar Darroch J, Seneta E (1967) On quasi-stationary distributions in absorbing continuous-time finite Markov chains. J Appl Probab 4:192–196

MathSciNet CrossRef MATH Google Scholar Ethier SN, Kurtz TG (1986) Markov processes: characterization and convergence. Wiley, New Jersey

CrossRef MATH Google Scholar Gardiner CG (2009) Stochastic methods: a handbook for the natural and social sciences. Springer, Berlin

Google Scholar Hagenaars TJ, Donnelly CA, Ferguson NM (2004) Spatial heterogeneity and the persistence of infectious diseases. J Theor Biol 229:349–359

MathSciNet CrossRef Google Scholar Hethcote HW (1996) Modeling heterogeneous mixing in infectious disease dynamics. In: Isham V, Medley GFH (eds) Models for infectious human diseases. Cambridge University Press, Cambridge, pp 215–238

Hethcote HW, Yorke JA (1984) Gonorrhea transmission dynamics and control. Springer, Berlin

MATH Google Scholar Jagers P (1975) Branching processes with biological applications. Wiley, London

MATH Google Scholar Keeling MJ, Rohani P (2007) Modeling infectious diseases in humans and animals. Princeton University Press, Princeton

Google Scholar Laub AJ (2005) Matrix analysis for scientists and engineers. SIAM publications, Philadelphia

CrossRef MATH Google Scholar Lajmanovich A, Yorke JA (1976) A deterministic model for gonorrhea in a nonhomogeneous population. Math Biosci 28:221–236

MathSciNet CrossRef MATH Google Scholar Lefèvre C, Malice M-P (1988) Comparisons for carrier-borne epidemics in heterogeneous and homogeneous populations. J Appl Probab 25:663–674

MathSciNet CrossRef MATH Google Scholar Lindholm M (2008) On the time to extinction for a two-type version of Bartlett’s epidemic model. Math Biosci 212:99–108

MathSciNet CrossRef MATH Google Scholar Lloyd-Smith JO, Schreiber SJ, Kopp PE, Getz WM (2005) Superspreading and the effect of individual variation on disease emergence. Nature 438:355–359

CrossRef Google Scholar Ma J, Earn DJD (2006) Generality of the final size formula for an epidemic of a newly invading infectious disease. Bull Math Biol 68:679–702

MathSciNet CrossRef Google Scholar Marschner IC (1992) The effect of preferential mixing on the growth of an epidemic. Math Biosci 109:39–67

CrossRef MATH Google Scholar Marshall AW, Olkin I, Arnold BC (2010) Inequalities: theory of majorization and its applications. Springer, Berlin

Google Scholar Meester R, Trapman P (2011) Bounding basic characteristics of spatial epidemics with a new percolation model. Adv Appl Probab 43:335–347

MathSciNet CrossRef MATH Google Scholar Metz JAJ (1978) The epidemic in a closed population with all susceptibles equally vulnerable; some results for large susceptible populations and small initial infections. Acta Biother 27:75–123

CrossRef Google Scholar Miller JC (2008) Bounding the size and probability of epidemics on networks. J Appl Probab 45:498–512

MathSciNet CrossRef MATH Google Scholar Mode CJ (1971) Multitype branching processes. Elsevier, New York

MATH Google Scholar Nåsell I (1999) On the time to extinction in recurrent epidemics. J Roy Stat Soc B 61:309–330

CrossRef MATH Google Scholar Nåsell I (2002) Stochastic models of some endemic infections. Math Biosci 179:1–19

MathSciNet CrossRef MATH Google Scholar Nåsell I (2005) A new look at the critical community size for childhood infections. Theor Popul Biol 67:203–216

CrossRef MATH Google Scholar Neal PJ (2006) Stochastic and deterministic analysis of SIS household epidemics. Adv Appl Probab 38: 943–968

MathSciNet CrossRef MATH Google Scholar Nishiura H, Cook AR, Cowling BJ (2011) Assortativity and the probability of epidemic extinction: a case study of pandemic Influenza A (H1N1-2009). Interdiscip Perspect Infect Dis 2011, Article ID 194507

Nold A (1980) Heterogeneity in disease-transmission modelling. Math Biosci 52:227–240

MathSciNet CrossRef MATH Google Scholar Scalia-Tomba (1986) Asymptotic final size distribution of the multitype Reed–Frost process. J Appl Probab 23:563–584

MathSciNet CrossRef MATH Google Scholar Seneta E (1986) Non-negative matrices and Markov chains. Springer, New York

Google Scholar Vergu E, Busson H, Ezanno P (2010) Impact of the infection period distribution on the epidemic spread in a metapopulation model. PLoS ONE 5:e9371

CrossRef Google Scholar Xiao Y, Clancy D, French NP, Bowers RG (2006) A semi-stochastic model for Salmonella infection in a multi-group herd. Math Biosci 200:214–233

MathSciNet CrossRef MATH Google Scholar Yates A, Antia R, Regoes RR (2006) How do pathogen evolution and host heterogeneity interact in disease emergence? Proc Roy Soc B 273:3075–3083

CrossRef Google Scholar Authors and Affiliations 1. Department of Mathematical Sciences University of Liverpool Liverpool UK