Mixing in Meta-Population Models

Among the means by which heterogeneity can be modeled, Levins' (Bull Entomol Soc Am 15:237–240, 1969) meta-population approach preserves the most analytical tractability. When model populations are stratified, contacts among their respective sub-populations must be described. Using a simple meta-population model, Feng et al. (J Theor Biol 386:177–187, 2015) showed that mixing among sub-populations, as well as heterogeneity in characteristics affecting sub-population reproduction numbers, must be considered when evaluating public health interventions to prevent or control infectious disease outbreaks. We employed the convex combination of preferential within- and proportional among-group contacts devised by Nold (Math Biosci 52:227–240, 1980) and generalized by Jacquez et al. (Math Biosci 92:119–199, 1988). As the utility of meta-population modeling in support of public policymaking depends on more realistic mixing functions, Glasser et al. (Math Biosci 235:1–7, 2012) included preferential contacts between parents and children and among co-workers as well as contemporaries. Feng et al. (Math Biosci 287:93–104, 2017) omitted workplace contacts, but added those between grandparents and grandchildren. We also devised a general scheme for multi-level mixing that meets the conditions for mixing functions specified by Busenberg and Castillo-Chavez (IMA J Math Appl Med Biol 8:1–29, 1991) and provided several two-level examples.