SEIQR modelEvaluate the effectiveness of contact tracing

The world is seeing the global pandemic of COVID-19.

It's been over 2 weeks that London, where I live, is under lockdown.

Countries under lockdown are facing the dilemma of when to lift this restriction because once we go back to the life we used to lead, the resurgences of the outbreak wave is inevitable.

Here, I experimented on how contact tracing & active testing can be the longterm solution to keep the infection under control at the same time as keeping social & economic activities running.

1. Introduction: what is the value of identifying asymptomatic cases?

Photo by NeONBRAND on Unsplash

When we become sick, we can stay at home and prevent the virus to spread around. But what is the case if you don’t know that you are shedding the virus because you have no symptoms?

According to the report from the outbreak in Diamond Princess cruise ship in February 2020, 46% of those tested PCR positive remained asymptomatic.

https://www.mhlw.go.jp/stf/seisakunitsuite/bunya/0000164708_00001.html

We don’t know how reliable it is, but there is a growing consensus that some people remain asymptomatic or with very little symptoms for the whole course of their infection.

Those may be the ones who, unknowingly, transmit the virus and contribute to the spread of the disease.

To identify infected individuals who are asymptomatic, that’s where contact tracing plays a role.

Diagnosing and quarantining them shorten the period where they shed virus to others and results in decreasing reproductive number.

2. Method: SEIQR model

SEIR (Susceptible-Exposed-Infectious-Recovered) model is a model often used in viral infections such as measles and rubella.

Here, a compartment of “Q” (Quarantined) was added in-between “I” (Infectious) and “R” (Recovered).

The assumption is that those who are diagnosed are instantly quarantined and do not transmit the virus any further.

Here is the clinical course of those diagnosed and those undiagnosed. We did not distinguish individuals who are symptomatic or asymptomatic here.

β is the rate at which two specific individuals come into effective contact per day
p is the proportion who are diagnosed (=reported) of all infected cases.
Dq is the period between the onset of infectiousness and quarantine.

The duration of the period where an infected individual transmits the virus to others is shorter in quarantined(=reported) cases (Dq) than unreported cases (Dr).

Overall reproductive number (Rn) decreases by reducing β, increasing p and shortening Dq.

We examined the combination of the threshold of β, p, and Dq to achieve Rn<1.

3. Assumptions and parameters: example of the Japanese case
De = 4.2*
Dq = 9**
Dr = 15***
p = 0.4****
Japanese population = 120 million

It is known that infected individual becomes infectious one ~ a few days before the onset of symptom. We assume here that individual becomes infectious one day before the onset of symptom.

References
*Incubation period = 5.2 days (Qun Li, 2020 NEJM)
**Mean time-lag between onset to diagnosis = 8 days (data from Hokkaido)
http://www.pref.hokkaido.lg.jp/hf/kth/kak/hasseijoukyou.htm
***Young BE et al. 2020 JAMA. doi:10.1001/jama.2020.3204
****Nishiura et al. https://www.medrxiv.org/content/10.1101/2020.03.09.20033183v1.full.pdf?fbclid=IwAR1_S4He1ol7g6axOFBpWZYw0SzBliUM6kGCZJZaWPWWCe1-JLcfAkGM1Cc
Also see: http://statmodeling.hatenablog.com/entry/covid19-estimate-total-number-of-positives-in-japan?

4. Data fitting

We used published cumulative reported cases in Japan until 8 April that is accessible from here: https://signate.jp/competitions/261#misc

Berkeley Madonna was used

Fitting with the above parameters gives recent β to be
β = 2.437*10^(-9)

5. Simulation

Below is the set of thresholds of β, p, and Dq to achieve Rn<1

Considering that the current mean time-lag between symptom onset to diagnosis is 8 days for symptom-based diagnosis and 6 days for exposure-based diagnosis, shortening this time-lag less than 6 days require significant improvement in contact-tracing strategy.

Long-term contact reduction of more than 50% would require massive sacrifice on social and economic activity. A 20% reduction seems to be reasonable but it would require robust contact tracing to keep proportion reported above 80% and shorten the time-lag to be less than 5 days.

If methodologically it is possible to maintain robust contact tracing, that would be very cost-effective considering the implication to the economy.