Theoretical Population Biology 78 (2010) 71–76 Contents lists available at ScienceDirect Theoretical Population Biology journal homepage: www.elsevier.com/locate/tpb Fitting parameters of stochastic birth–death models to metapopulation data Heinrich zu Dohna a , Mario Pineda-Krch b,∗ a Center for Animal Disease Modelling, Department of Veterinary Medicine, University of California Davis, One Shields Avenue, Davis, CA 95618, USA b Centre for Mathematical Biology, Department of Mathematical & Statistical Sciences, 632 Central Academic Building, University of Alberta, Edmonton, Alberta T6G 2G1, Canada article info Article history: Received 11 August 2009 Available online 17 June 2010 Keywords: Metapopulation Birth–death processes SIS model Logistic growth Maximum likelihood abstract Populations that are structured into small local patches are a common feature of ecological and epidemiological systems. Models describing this structure are often referred to as metapopulation models in ecology or household models in epidemiology. Small local populations are subject to demographic stochasticity. Theoretical studies of household disease models without resistant stages (SIS models) have shown that local stochasticity can be ignored for between patch disease transmission if the number of connected patches is large. In that case the distribution of the number of infected individuals per household reaches a stationary distribution described by a birth–death process with a constant immigration term. Here we show how this result, in conjunction with the balancing condition for birth–death processes, provides a framework to estimate demographic parameters from a frequency distribution of local population sizes. The parameter estimation framework is applicable to estimate parameters of disease transmission models as well as metapopulation models. © 2010 Elsevier Inc. All rights reserved. 1. Introduction Metapopulation models in ecology and household models in epidemiology describe populations whose individuals are distributed over many loosely connected small local populations. An important approach in metapopulation biology is the incidence function model which estimates extinction and colonization probabilities of local populations from one or multiple snapshots of patch occupancy (Hanski, 1994; Moilanen, 1999). The incidence function model and similar approaches distinguish between empty and occupied patches but do not take data on local population sizes into account. Household models in epidemiology have been used to estimate parameters of within and between household disease transmission from a frequency distribution of infected individuals per household for models that contain susceptible, infected and resistant individuals (SIR models) (Longini and Koopman, 1982; Longini et al., 1982). Parameter estimation approaches have been lacking so far for household models of diseases without a resistant stage (SIS models). Theoretical studies of SIS household models have shown that at equilibrium within household dynamics can be treated as stochastic birth–death process with constant immigration term when the number of households is large (Ball, 1999; Ghoshal et al., 2004). ∗ Corresponding author. E-mail addresses: hzudohna@ucdavis.edu (H. zu Dohna), mpineda@math.ualberta.ca (M. Pineda-Krch). Here we show how approximation results from household SIS models can be combined with analytical results for stochastic birth–death processes to estimate parameters for within and be- tween household disease transmission from a frequency distribu- tion of infected individuals per household. The same approach can be used to estimate immigration, birth and death rates from a fre- quency distribution of local population sizes in metapopulation data. 2. Methods 2.1. Problem description Metapopulations with small local populations undergo demo- graphic stochasticity at the local scale. If the immigration rate to any local population depends on a large number of other local populations, the local demographic stochasticity does not translate into stochastic fluctuations of the immigration rate to each patch (however, demographic stochasticity in the number of immigrants per time still occurs). In that case the distribution of local popu- lation sizes at equilibrium can be well approximated by a station- ary distribution of a stochastic immigration–birth–death process whose immigration term depends on the mean local population size (Ghoshal et al., 2004). Such a stationary distribution whose mean equals the mean population size in its immigration term has been called a ‘self-consistent field’ and has been shown to approx- imate simulations of a household SIS model well for a surprisingly wide range of conditions (Ghoshal et al., 2004). 0040-5809/$ – see front matter © 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.tpb.2010.06.004