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# Dongni Zhang: Epidemic models with contact tracing: manual and digital tracing

Time: Wed 2022-08-31 15.15 - 16.00

Location: Albano, House 1, Floor 3, Room 12

Doctoral student: Dongni Zhang

Opponent: Peter Olofsson (Jönköping University)

Supervisor: Tom Britton

Examiner: Daniel Ahlberg

### Abstract

Contact tracing is an effective measure for controlling the epidemic spreading. In this thesis, we consider a Markovian SIR epidemic model in a homogeneous community with a constant rate of diagnosis (testing) and investigate the preventive effect of contact tracing. The first paper concerns the traditional manual contact tracing, meaning that once an infectious individual tests positive, it is immediately isolated. Each of its contacts are traced and tested independently with some fixed probability. The second paper focuses on the more recent digital contact tracing via a contact tracing app (only app-users can trigger and be traced by digital tracing) and the combined effect of two types of tracing. We assume that manual or digital contact tracing occurs instantaneously and recursively for mathematical tractability.

In the first paper, 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. 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). Based on these approximations, the analytic expressions for the reproduction numbers (for the components and individuals), the probability of a minor outbreak, and the final fraction getting infected are derived and numerically evaluated. Our numerical results suggest that the manual tracing probability seems more effective in reducing the reproduction number than the testing fraction.   In the second paper, we add the digital tracing to the similar epi-demic model in the first paper. This model is then analysed using a two-type branching process relying on a large community, where one type of ”individuals” are ”app-using components” and another is non- app-users. Further, we investigate the combined preventive effect of two tracing methods. This combined model is analysed 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 are derived. We conclude that it is more essential to control the epidemic to have a large fraction of app-users compared to the manual tracing probability. Another important conclusion is that the combined preventive effect is bigger than the product of the two preventive effects.