Constructing the effect of alternative intervention strategies on historic epidemics

Abstract

Data from historical epidemics provide a vital and sometimes under-used resource from which to devise strategies for future control of disease. Previous methods for retrospective analysis of epidemics, in which alternative interventions are compared, do not make full use of the information; by using only partial information on the historical trajectory, augmentation of control may lead to predictions of a paradoxical increase in disease. Here we introduce a novel statistical approach that takes full account of the available information in constructing the effect of alternative intervention strategies in historic epidemics. The key to the method lies in identifying a suitable mapping between the historic and notional outbreaks, under alternative control strategies. We do this by using the Sellke construction as a latent process linking epidemics. We illustrate the application of the method with two examples. First, using temporal data for the common human cold, we show the improvement under the new method in the precision of predictions for different control strategies. Second, we show the generality of the method for retrospective analysis of epidemics by applying it to a spatially extended arboreal epidemic in which we demonstrate the relative effectiveness of host culling strategies that differ in frequency and spatial extent. Some of the inferential and philosophical issues that arise are discussed along with the scope of potential application of the new method.

Figure 1

Diagram representing the different approaches described in the paper. In this simple scenario, the actual intervention (control) ρ is implemented at time t_actual, lasts until the end of the outbreak and results in epidemic trajectory X. The choice of ρ makes use of parametric (θ) and trajectory-based (X) information until time t_actual. After the outbreak is over, an alternative mooted intervention ρ′ is considered which would have been implemented at time t_mooted. This would have resulted in the notional trajectory X′, which has a distribution reflecting our uncertainty in the parameters and trajectory. In the semi-retrospective approach, all available parametric information is used, but the only trajectory information used is that occurring before the intervention. In the (fully) retrospective approach described in this paper, all available information (on X and θ) is used.

Figure 2

The difference between the Poisson and Sellke constructions in terms of the conditional distribution of t_B and $t_{\text{B}}^{\prime }$. In the observed epidemic, background sources infect host A at time t_A; B is then infected at time t_B either by background sources or by A. In the notional epidemic, A is removed immediately after infection and so it is unable to infect B. Under the Sellke construction, B is infected later (green dot). Under the Poisson construction, B may be infected at the same time as before (if it was infected by background sources in the observed epidemic, orange dot); alternatively its infection time follows a shifted exponential distribution (orange line) if it was infected by A in the observed epidemic.

Figure 3

(a–d) The difference between the Poisson and Sellke constructions in terms of the retrospective joint distribution of the number of infective and removed hosts at time 7 under the two interventions. (a) Retrospective: Poisson and (b) retrospective: Sellke. Also shown is the joint distribution taking realizations under the two interventions to be independent after times t=0 and 3, corresponding to what we termed the semi-retrospective approach. (c) Semi-retrospective: fully independent and (d) semi-retrospective: independent post intervention. Marginal distributions are shown in the margins of the plots. These have the same distribution regardless of the method used to generate them.

Figure 4

Effect of two notional intervention strategies on an historic epidemic of the common cold on the island of Tristan da Cunha. These take the form of scaling the infection rate by ξ after t=3; time has been translated so that the first removal occurs on day 1. (a,c) The semi-retrospective distributions are what we would obtain using the posterior for the model parameters based on the whole of the epidemic but assuming that the trajectory of the epidemic after t=3 is independent of the actual epidemic. (b,d) The fully retrospective distributions show the distribution of the effect of the interventions on the epidemic conditional on the actual epidemic. Predictions take the form of credible regions.

Figure 5

Retrospective effect of three removal strategies on lost citrus trees in an area of Florida. In (a), solid lines are posterior means and dashed lines 95% credible intervals. Points mark the observed number of infections (there were no removals in reality). Colours distinguish three notional control strategies: ${\rho }_1^{\prime }$: sample the population every 60 days and remove all infected trees (corresponding to (b)); ${\rho }_2^{\prime }$: sample every 120 days and remove all infected trees (c); and ${\rho }_3^{\prime }$: sample every 120 days but only detect and remove infection with probability 40% (d). The three lower panels show maps of the areas, with coloured symbols indicating the posterior mean probability that the corresponding tree would have been infected by time 360 had the intervention taken place (legend in top panel).