Dongni Zhang: Stochastic epidemic models with contact tracing

Abstract.

The COVID-19 pandemic has significantly heightened research interest in infectious disease modelling. Specifically, the pandemic has underscored the critical role of epidemic models in understanding and predicting epidemic dynamics as well as evaluating the impact of public health interventions. This thesis delves into stochastic epidemic modelling, concentrating especially on the effectiveness of contact tracing in preventing large outbreaks.

In Paper I, we consider a Markovian SIR epidemic model in a homogeneous community with a rate of diagnosis (testing). This model is then incorporated with traditional manual contact tracing: once an infectious individual tests positive, it is immediately isolated. Each of its contacts is traced and tested independently with some fixed probability. Using large population approximations, we analyzed the early stage of the epidemic when the process of “{to-be-traced components}” behaves like a branching process. Based on these approximations, the analytic expressions for the reproduction numbers (for the components and individuals), the probability of a major outbreak, and the final fraction getting infected are derived and numerically evaluated. For the main stage of the epidemic, the process of to-be-traced components converges to a deterministic process (defined by a system of differential equations). Our numerical results suggest that the manual tracing probability is more effective in reducing the reproduction number than the testing fraction.

Paper II concerns a similar epidemic model as in Paper I but focuses on digital contact tracing (based on a contact tracing app) and a hybrid approach combining both manual and digital tracing. We assume manual as well as digital contact tracing occurs instantaneously and recursively for mathematical tractability. This model is then analyzed using a two-type branching process relying on a large community, where one type of “individuals” are “{app-using components}” and the other is non-app-users. Further, we investigate the combined preventive effect of two tracing methods. This combined model is analyzed by a different two-type branching process, with both types being the “to-be-traced components” but starting with different {“roots”}. The corresponding reproduction numbers \(R\) are derived. It is proven that the critical fraction app-users for which digital contact tracing to \(R=1\) is larger than the critical manual reporting probability to reach \(R=1\).

Paper III presents an SEIR epidemic model allowing for network and random contacts, incorporating manual and digital (app-based) contact tracing. Manual tracing is only allowed to happen on the network and has random tracing delays. In contrast, instantaneous digital tracing occurs among network and random contacts (both need to be app-users). Both manual and digital tracing are assumed to be forward only and non-iterative. We show that the initial phase of the epidemic with both manual and/or digital tracing can be approximated by different multi-type branching processes, leading to the derivation of three respective reproduction numbers. This paper sets a lower bound on the effectiveness of contact tracing through its conservative assumptions, while Paper II offers an upper bound by presenting an optimistic scenario. The actual effectiveness of contact tracing in real-world situations is expected to fall somewhere in between.

Paper IV explores an epidemic model with sideward contact tracing. Individuals are involved in mixing events that occur at some rate. Infection is driven by interaction with at least one infective at the mixing event rather than pairwise interaction outlined in Paper I, II and III. Once an infective is diagnosed, each individual who got infected at the same event as the diagnosed individual is traced with a certain probability. Assuming few initial infectives within a large population, the initial stage of the epidemic is approximated by a limiting process where individuals can be traced by their siblings (individuals who become infected at the same event), thus creating dependencies. Treating sibling groups as macro-individuals allows for a macro branching process interpretation, from which we derive the reproduction number. Finally, we present some numerical results showing how the reproduction number varies with the size of the mixing event, the fraction of diagnosis and the tracing probability.