Tracing the Spread: A Mathematical Modelof Manual and Digital Contact Tracing

During the early phase of the COVID- 19 pandemic, contact tracing was a key tool for controlling transmission by identifying and interrupting transmission chains. Manual tracing typically relies on interviews or self-reporting: diagnosed cases are asked to report recent contacts, who are then notified and tested. At the same time, digital tracing apps were introduced in several countries to accelerate the process. When an app-user was diagnosed, close contacts received immediate notifications recommending testing and isolation. But how do these two approaches interact, and under what conditions are they most effective?

In our study (Zhang and Britton, 2024), which received the Best Paper Award 2025 of the Swedish Society of Medical Statistics (FMS), we investigated these questions using an SEIR (Susceptible → Exposed → Infectious → Recovered) network epidemic model with additional random contacts. Infections can occur through local network contacts (e.g., friends or colleagues) or random global contacts (e.g., on public transport). When a case is diagnosed, manual tracing reaches network contacts after a random delay (with a certain reporting probability), whereas digital tracing – if both parts use the app – notifies contacts instantaneously, both local and global. For mathematical convenience, we assume one-step tracing only, meaning that traced contacts do not trigger further tracing.

We studied the early epidemic phases using multi-type branching process approximations and derived the effective reproduction numbers under three tracing scenarios: manual only, digital only and combined. The reproduction number R has a critical threshold property: if R>1, a major outbreak may occur, while if R≤1, there will be a minor outbreak with probability 1.

Our results show that:

• A larger fraction of app-users improves the effectiveness of digital tracing.
• Manual tracing is more effective when transmission is network-driven, or when the contact network is highly heterogeneous.
• Combining both manual and digital tracing can substantially reduce the reproduction number, though additional measures are needed to keep it below 1, thereby preventing large outbreaks.

Beyond COVID-19, this modelling framework provides a flexible mathematical tool for exploring how different tracing strategies interact with epidemic dynamics and may offer useful insights for understanding future outbreaks.