Capturing individual heterogeneity

In the original Wallinga and Teunis method, the only characteristic that differs between cases is the symptom onset date. Cases with the same onset date have exactly the same probability of having infected a specific secondary case regardless of their physical proximity to the infectee, and have the same individual reproductive number. This partly contributes to the R(t) estimates from the Wallinga and Teunis method being smoothed, even during the periods of super-spreading events.

Incorporating contact tracing data into the method can improve the accuracy of the reproduction number estimates by incorporating prior data on who infected who. However, in reality, contact tracing data are seldom obtainable from all cases, and in large scale outbreaks, resources required for contact tracing of cases become unaffordable as the epidemic grows (Ghani, Baguelin et al. 2009). On the other hand, only contact with people who are known by the case can be collected in contact tracing, and it is also subject to recall bias. In the Hong Kong SARS outbreak, a high proportion of cases had data on possible epidemiological source of transmission collected, and some of them had specific contacts traced (Severe Acute Respiratory Syndrome Expert Committee 2003). However, still around one-tenth of the cases could not identify any place or person from which infection might have been acquired (source of infection data described in chapter 2).

Spatial information for cases, for instance residential location, is more likely to remain available for identified cases even at the later stages of the epidemic. The prerequisite for infectious disease transmission is that infected and susceptible individuals are in the same place at the same time. Capturing this spatial condition allows more accurate and realistic reconstruction of the infection tree. A study on the UK Foot and Mouth disease epidemic in 2001 (Ferguson, Donnelly et al. 2001) had used a kernel of spatial distance from the infected farm as a multiplicative factor to the farm- specific susceptibility and infectiousness to estimate the relative risk of transmission.

Human movements are less traceable and more complicated that than livestock animals, but gravity models had proved successful in modelling human travel patterns and thus representing the spatial heterogeneity in the probability of contact between individuals based on their spatial location in epidemics.

Past work has explored both infector driven (Xia, Bjornstad et al. 2004; Eggo, Cauchemez et al. 2010) and susceptible driven (Merler and Ajelli 2010) assumptions in linking gravity models to the probability of contact in epidemic models, but the effects of these assumptions were not compared. The essence of the Wallinga and Teunis method is to identify the potential infectors of cases, and then assign the relative likelihood of infection to all the potential infection pairs. Spatial context has a role to play in both processes. The original method identifies a case as potential infector regardless whether the potential infection pairs would have been in the same location or not. The use of spatial data can refine the list of potential infectors to those who were present at the infection location at the date of infection. The probability of infection between cases can be considered to be composed of two components – probability of contact based on the spatial location of the cases, and the probability of transmission given the timeline of infection. This chapter uses these principles to

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derive an extended Wallinga and Teunis method incorporating both spatial and temporal components.

5.2 Study objectives

I consider two levels of spatial extension of the Wallinga and Teunis method. On a coarser spatial resolution, spatial data were used only to estimate the probability of contact between cases, referred as the “partially extended method”. On a more refined spatial level that used the source of infection (e.g. hospital or community) and hospitalisation data of the cases, eligible potential infectors were those present at the infection location at the date of infection. The method with this additional spatial extension is referred as the “fully extended method”.

Because the occurrence of disease transmission is not observable, the date of infection was inferred from the probability distribution of the incubation period and the symptom onset date to identify potential infectors who fulfilled the spatial criteria. Accordingly the temporal component of the probability of infection were estimated by a convoluted incubation and infectious (or infectivity) distribution instead of the serial interval density. Therefore, the first objective of this chapter is to derive such a deconvoluted distribution for the serial interval density of the SARS outbreak to be applied in both the partially and fully extended methods.

The second objective is to derive the partially extended method and evaluate different spatial mixing assumptions under which gravity models are used to derive the probability of contact. Gravity models predict the flows from a home location to the destination, and therefore give the probability of travel from home to a destination for certain trip purpose. This unilateral direction of movement, regardless of whether one considers the susceptible or infector to be travelling, risks oversimplifying the infection risks associated with travel. Therefore I also considered bilateral (both susceptible and infector could travel to each other’s home location) and multilateral (susceptible and infector could travel to any location and meet there) movement, and under different trip types (work, work-study, adjusted overall) and trip data (census and survey), to find the most appropriate proxy for the probability of contact leading to the observed spatiotemporal spread of the SARS epidemic in Hong Kong.

Building on the result of this chapter, the fully extended method is derived in the next chapter to study the hospital and community transmission and their dynamics in the SARS outbreak in Hong Kong. The following section describes the method and discusses the results of deconvoluting the serial interval distribution – the first objective of this chapter.