fraction stoppped = (usage fraction in population)2 * (fraction of infections recorded as contacts) * (fraction of detected contacts who follow up correctly)
That's obviously a massive over simplification; for example real population dynamics don't give full random mixing between all members of the population. But every factor there is potentially depressed because of opt in and anomnymity; the first obviously so, the third because people are less likely to followup correctly if there's no consequence for not doing so, and the second because it will be impossible to tune the detection parameters without any data to work from. In that — again oversimplified — model if you get 50% o the population using the app (compared to 80% smartphone usage in western Europe), it detects 80% of transmissions and (say) 50% of people follow up correctly before themselves passing the disease on, you end up with 10% of infection chains traced. That doesn't seem like it's going to have a big impact.
So assuming that model's not wildly wrong — which it could be, again, I'm really looking for a link to an expert discussion of this, I can see a few possibilities: * Digitial contract tracing doesn't have much impact on our ability to control the virus * Governments seek to outsource the privacy-invasion needed to companies by e.g. requiring evidence that people are using the contact tracing software and have followed up on potential transmission events before allowing people to buy food etc. * Ineffective opt-in contact tracing and a second wave of pandemic deaths/lockdown is used to bounce people into accepting the need for mandatory, non-anonymous contact tracing with fewer privacy concessions than you'd get right now.
To be clear I'm very concerned with the idea of govenments and in particular the current UK government having access to non-anonymised contact data; it's already being run by someone who considers population-level manipulation using data science and social media to be his core skillset. But I'd also like to understand what tradeoffs are being made in terms of disease control.