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. 2019 Jul 8;374(1776):20180279.
doi: 10.1098/rstb.2018.0279.

Perfect counterfactuals for epidemic simulations

Joshua Kaminsky  1 , Lindsay T Keegan  1 , C Jessica E Metcalf  1   2 , Justin Lessler  1
Affiliations

Perfect counterfactuals for epidemic simulations

Joshua Kaminsky et al. Philos Trans R Soc Lond B Biol Sci. .

Abstract

Simulation studies are often used to predict the expected impact of control measures in infectious disease outbreaks. Typically, two independent sets of simulations are conducted, one with the intervention, and one without, and epidemic sizes (or some related metric) are compared to estimate the effect of the intervention. Since it is possible that controlled epidemics are larger than uncontrolled ones if there is substantial stochastic variation between epidemics, uncertainty intervals from this approach can include a negative effect even for an effective intervention. To more precisely estimate the number of cases an intervention will prevent within a single epidemic, here we develop a 'single-world' approach to matching simulations of controlled epidemics to their exact uncontrolled counterfactual. Our method borrows concepts from percolation approaches, prunes out possible epidemic histories and creates potential epidemic graphs (i.e. a mathematical representation of all consistent epidemics) that can be 'realized' to create perfectly matched controlled and uncontrolled epidemics. We present an implementation of this method for a common class of compartmental models (e.g. SIR models), and its application in a simple SIR model. Results illustrate how, at the cost of some computation time, this method substantially narrows confidence intervals and avoids nonsensical inferences. This article is part of the theme issue 'Modelling infectious disease outbreaks in humans, animals and plants: epidemic forecasting and control'. This theme issue is linked with the earlier issue 'Modelling infectious disease outbreaks in humans, animals and plants: approaches and important themes'.

Keywords: counterfactuals; infectious disease dynamics; infectious disease epidemiology; network models.

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Conflict of interest statement

We declare we have no competing interests.

Figures

Figure 1.
Figure 1.
Anatomy of an event. Each event has preconditions based on the state of one or more individuals p in the population at time ti, a probability of occurrence, and an outcome that sets the state of an individual at time ti+1. (a) In its full representation, an event is represented as a node on the graph (diamonds) with incoming edges representing the preconditions individuals (circle nodes) must meet for that event to occur. In this case, both events require individual pm to be infectious (orange) and the bottom event additionally requires individual pn to be susceptible (purple). The outgoing edge captures the outcome of the event, in this case removed (green) and infectious (orange). (b) We use a reduced representation where the event nodes are implicit. Edges are coloured by their outcome, and infectious edges (orange) carry an implicit precondition that the target is susceptible. Here, the reduced representation in panel (b) is equivalent to the full graph in panel (a).
Figure 2.
Figure 2.
‘Single-world’ simulation process. To simulate an initial epidemic and intervention, we start with a (implicit) complete graph. We represent potential states using coloured wedges; an individual can be susceptible (purple pie slice), infectious (orange) or removed (green). We start by using our initial conditions to eliminate all but one potential state for each individual in the population at time t0 (b); here individual p1 starts infectious, and other individuals start susceptible. We next prune all events with at least one precondition that we know is not satisfied by the initial state (c). We then prune those events selected not to occur in this particular simulation according to our underlying infection model (d). We set the possible states of each individual at t1 based on the remaining events in the graph, so each individual’s set of potential states encompasses both the outcomes of any remaining events and (for events potentially prevented by the intervention) the outcomes of their absence (e). We now repeat steps (c–e) for events connecting t1 to t2 (f–h), t2 to t3 (i–k) and so on. Note that when we prune the events with unattainable preconditions (i), we still keep the events for p1 and p2 infecting p3, even though p3 was potentially infectious at t1, because susceptible is still a potential state for p3 at t2 in an intervention scenario (i.e. removing the transmission event at t1 could lead to a scenario where p3 is susceptible at t2). This final potential epidemic graph (PEG) can then be used to obtain simulated epidemics with and without interventions (figure 3).
Figure 3.
Figure 3.
‘Single-world’ simulation process (continued). In order to measure the impact of the intervention on the epidemic, we use the PEG (figure 2) to construct two epidemics: one uncontrolled (left), one with intervention (right). We start by setting the actual state (denoted by colouring the whole circle) of each individual at time t0 to their initial condition, and then changing their state according to the intervention, in this case setting p2 as vaccinated (dark green). We then prune events between t0 and t1: removing inconsistent events and allowing the intervention to stochastically remove events (b,c). We then use those events to determine the actual state of each individual at t1, allow the intervention to alter that state, then prune inconsistent events again (d,e). We repeat the process for times t2 (f,g), t3 (h,i) and so on. This final graph can be used to extract our outcome of interest. Any graphs made from the same PEG represent the results of different interventions in a ‘single world’.
Figure 4.
Figure 4.
Two types of transitions commonly present in compartmental models: (a) independent transitions occur with probability αi,j independent of the state of the model and (b) contact transitions occur between two specific individuals with probability βi,j,k.
Figure 5.
Figure 5.
Time series showing cumulative number of cases averted at each time caused by the intervention calculated using our method (single-world) and a standard method. Shaded regions denote 90% confidence intervals. Note that there is more variation in the middle of the epidemic, so it may seem as though the number of cases averted is large during those times. (Online version in colour.)
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