Contact tracing strategies in heterogeneous populations

Abstract

Contact tracing is a well-established disease control measure that seeks to uncover cases by following chains of infection. This paper examines mathematical models of both single-step and iterative contact tracing schemes and analyses the ability of these procedures to trace core groups and the sensitivity of the intervention to the timescale of tracing. An iterative tracing process is shown to be particularly effective at uncovering high-risk individuals, and thus it provides a powerful public health tool. Further targeting of tracing effort is considered. When the population exhibits like-with-like (assortative) mixing the required effort for eradication can be significantly reduced by preferentially tracing the contacts of high-risk individuals; in populations where individuals have reliable information about their contacts, further gains in efficiency can be realized. Contact tracing is, therefore, potentially an even more potent tool than its present usage suggests.

Fig. 1

An example of a simple mixing network. Individuals are represented by circles, partnerships by lines. The infection status of three individuals is included for illustration. The two highlighted individuals contribute 1 towards [SI], the number of susceptible–infected partnerships; along with the individual top right they comprise a SII triple.

Fig. 2

Equilibrium prevalence plotted against the infection parameter, r, for a range of tracing fractions, f, in a single-step tracing model, g=1 throughout, with r varied by altering τ. A heterogeneous network was used to parameterize the model, with individuals of neighbourhood size ranging from 1 to 13.

Fig. 3

The region in parameter space for which single-step tracing can eradicate infection in a homogeneous network with uniform neighbourhood size k, shown in grey. Above the shaded region infection persists whatever the tracing fraction; below the shaded region persistence is impossible even when there is no tracing.

Fig. 4

Equilibrium prevalence plotted against the infection parameter, r, for a range of tracing fractions in an iterative tracing model, g=1 throughout, with r varied by altering τ. The tracing fraction is given by c/(c+a). The same network was used as in Figure 2.

Fig. 5

Mean neighbourhood size of infected individuals plotted against equilibrium prevalence for a range of tracing fractions, using the simulation results shown in Figures 2 and 4. (a) Single-step tracing; (b) iterative tracing. Also shown (dashed line) is the mean neighbourhood size of the population.

Fig. 6

Critical tracing fraction necessary to eradicate infection for the iterative tracing model plotted against R₀ for a range of timescale separations. The timescale separation, Λ, is defined by the ratio of the tracing and infection timescales: g=1 and a=Λ×g throughout; R₀ and tracing fraction are varied by changing τ and c respectively.

Fig. 7

The effect of mixing pattern on the required effort to eradicate infection, using the iterative tracing model in a simplified population consisting of two types of individual. Relative targeted effort is defined to be the effort needed when tracing is optimally targeted divided by the effort required when tracing is applied uniformly. Within-group mixing is defined to be the proportion of contacts that are between individuals in the same group. In this case, effort is targeted according to the properties of the index case.

Fig. 8

The effect of assortativity on the required effort to eradicate infection. As in Figure 7, but here effort is targeted according to the properties of the contact.