Notions of synergy for combinations of interventions against infectious diseases in heterogeneously mixing populations

In public health programmes interventions are frequently combined with hoped for 'synergies' [22]. However, there is not yet a precise definition for synergy between interventions that captures the idea that there is added benefit at the population-level in using them together. To explore the synergy between interventions in the context of endemic disease, we consider a general model of infection spread in a heterogeneously mixing population. We consider interventions which may alter individuals' infectiousness, susceptibility, profile of infectiousness through time and survival while infected. Allowing general patterns of overlap and targeting in those receiving the interventions, we show how to compute changes to epidemiological indices such as R0, and introduce a simple technique for calculating equilibrium prevalences and incidences via an iterated map. We argue for a particular definition of synergy and investigate its behaviour, both analytically and numerically, concluding that it is easiest to achieve synergy between interventions which perform poorly in isolation; implementation strategies that minimize the overlap of different interventions in the population tend to achieve more synergy; and that in populations with heterogeneous risk, interventions that are redundant when universally targeted can regain substantial synergy when applied in a targeted manner.