**Temporal transmission probability of undetected cases**

The undetected group includes asymptomatic cases, cases with mild symptoms but untested, and cases with mild symptoms and positive test results but failed to isolate themselves. Zou28 found that the viral load detected in asymptomatic patients was similar to that of symptomatic patients. We assume they have the same generation time profile as symptomatic cases and used Weibull distribution estimated from Ferretti et al.29 They used maximum likelihood estimates to infer generation time distribution and fit a Weibull distribution with a mean of 5.5 days and a standard deviation of 1.8 days. This translates to a Weibull distribution with a shape parameter of 1.6749 and a scale parameter of 6.1577. We apply this distribution to *R*0 to assign the probability that one infected person can spread the virus to other persons each day, as shown in Fig 1.

**Temporal transmission probability of detected cases**

For detected cases, to estimate the probability that an infected person can spread the virus each day, we combine incubation period distribution[3] estimated from Lauer et al.30 with the infectious profile[4] estimated from He et al.31 and updated by Ashcroft et al.32 to deduce generation time, as shown in Table 1. The rows in the table represent possible cases with different incubation periods, that can range from 1 to 21 days, with associated probabilities from Lauer et al.30 shown in the first column. Each row illustrates the probability that an infected person with a given incubation period will transmit the virus each day during the course of the infection. The infectiousness profile data are from He et al.31 and updated by Ashcroft et al.32. The “weighted average” row shows the incubation probability-weighted average of the infectiousness probabilities. The normalized numbers are shown in the “normalized” row to make the probabilities sum to one. Note that we cut the tail of the infectiousness profile after 21 days because, when multiplying with incubation period probability, the results are negligible. Hence, we assume that the maximum number of days that an infected person can transmit the virus,* I*, is 21. If an infected person does not recover in 21 days, he/she should be hospitalized and should no longer be able to transmit the virus.

**Table 1: Estimation of temporal transmission probability. Column day numbers represent the incubation period and the row day numbers represent infectiousness probability distribution.**

In many cases, we can improve the control of pandemics by imposing isolation after detection. This is usually done by quarantine, hospitalization, or self-isolation. We define the lag between the time an infected person develops symptoms and the time that he/she is isolated as *testing delay* or *k*g, which can vary from 0 to 7 days34 while the infectiousness probabilities after *k*g day from the onset will be set to zero. The example truncated probabilities with *k*g = 3 are shown in Table 2. The truncated probability numbers are then normalized with the factor from no isolation cases because isolation after detection should have no impact on the transmission probability before isolation.

**Table 2: Estimation of temporal transmission probability assuming that infected persons are isolated 3 days after symptom onset. Column day numbers represent the incubation period and the row day numbers represent infectiousness probability distribution.**

The results of the cases with 1–3 days of isolation after symptom onset are shown in Fig. 1.

**Contract tracing modeling**

We define incubation period distribution, which is estimated from Lauer et al.30 as *P*, where *p(j)* represents the probability that the incubation period is *j* days. Suppose a detected infected individual has an incubation period of *j* days, the time that he/she will be isolated is *j*+*k*g days, and the time that proportion *k*c of his/her contacts will be isolated is *j*+*k*g*+k*l days. On a given day *t*, the expected numbers of patients with *j* days incubation period that will be detected and isolated on that day are for the cooperative group and for the uncooperative group. The number of contacts that they may have transmitted the virus to, after having the virus for *i* days, can be computed as . It will take *k*l days until their contacts are isolated. As a result, on day *t*, contact tracing will reduce the number of contagious people in each state of the Markov chain as follows.

for all *j*=1, …, 21-*k*g and for all *i*= max(1,* k*l +* j *+ *k*g – *I *), …, min(*I*, *j+kg *-1).

Suppose that a contact of an index case was isolated on day *t*. This means that the index case was detected and isolated on day *t-k*l. On day *t-k*l, the index case has had the virus for *j*+*k*g days, where *j* is his/her incubation period. The first day that he/she got the virus would be day *t* -* k*l - (* j*+*k*g -1). Since then he/she has been potentially spreading the virus for *i* days, *i* = 1, …., *j+k*g-1, corresponding to day *t* -* k*l - (*j*+*k*g -*i*), from day *t* -* k*l - (*j*+*k*g -1) to day *t* -* k*l - (*j*+*k*g – (*j+k*g-1)) = *t* -* k*l –1. Hence, on day *t*, the contact who contracted the virus from the index case on the index case’s *i*th day of infection, which was day *t* -* k*l - (*j*+*k*g -*i*), would be infected for *t *– (*t* -* k*l - (*j*+*k*g -*i*)) = *k*l +* j *+ *k*g -*i* days. This explains the superscript of *UC* in Equation (1).

We only considered the cases in which the contact has been infected for less than *I* days (i.e., *k*l +* j *+ *k*g -*i* ≤ *I*) because after *I* days, the virus can no longer be transmitted through contact. Thus, *i* ≥ *k*l +* j *+ *k*g - *I* and *i* ≥ max(1,* k*l +* j *+ *k*g – *I*). Moreover, the index cases can spread the virus at most *I* days. Hence, *i* ≤ min(*I*, *j+kg *-1). This explains the range of *i* in Equation (1).

The reduction in the numbers in states *UU*t follows the same form as *UC*t in Equation (1), except that *SC* in (1) is replaced by *SU*. The reductions in numbers in states *TC*t and* TU*t follow the same forms as states *UC*t and *UU*t, respectively, except the term (35% - *k*d) is replaced by term (65% + *k*d) to appropriately represent the detected group. Moreover, the sum of reductions from states *UC*t and *UU*t are added to state *M*t because these isolated contagious contacts will no longer be able to transmit the virus. However, only 97.29% of the sum of the reductions from state *TC*t and *TU*t are added to state *M*t because 2.71% of them will die, which in this case, are added to state *D*t.

**Vaccination modeling**

Due to the possibility of reinfection, countries such as the UK35 and United States36 and currently provide vaccination to their populations regardless of whether they were previouslly infected. We define *m* to be the proportion of COVID-19 recovered individuals that can get a vaccination. Based on the current vaccination policy, we assume that *m* = 1. Let *k*e be the number of days for the vaccine to give full protection. CDC37 currently suggests that *k*e be 14 days. As a result, the number of people receiving vaccination at time* t* is *k*v(t)(N-Dt) - *k*v(t-1)(N-Dt-1). The population that can receive a vaccination at time *t* is *N-D*t-Vt-(1-*m**)M*t. Therefore, the number of cooperative susceptible individuals *SC*t to get the vaccine at time *t* is *(**k*v(t)(N-Dt) - *k*v(t-1)(N-Dt-1))∙ *SC*t */ (N-D*t-Vt-(1-*m**)M*t). These vaccinated individuals would still be susceptible for the next *k*e days as they may become infected before the vaccine reaches its maximum protection. We assume that, during the *k*e days before the vaccine takes effect, the chance that susceptible individuals become infected is the same regardless of whether they were vaccinated or not. Hence, at time *t*, the number of cooperative susceptible *SC*t will be reduced by *(**k*v(t-ke)(N-Dt-ke) - *k*v(t-ke-1)(N-Dt-ke-1))∙ *SC*t */ (N-D*t-ke-Vt-ke-(1-*m**)M*t-ke) and moved to* V*t. The number of infected individuals that were vaccinated at time *t-k*e is *(**k*v(t-ke)(N-Dt-ke) - *k*v(t-ke-1)(N-Dt-ke-1))∙ (*SC*t-ke *-* *SC*t*) / (N-D*t-ke-Vt-ke-(1-*m**)M*t-ke), which is subtracted from* M*t and moved to* V*t. The formula for state *SU*t follows the same fashion as *SC*t but with *SU* replacing all *SC*.

Footnotes:

[3] The incubation period has a median time of 4–5 days26 from exposure to symptoms onset. Lauer et al.26 fit the incubation period into log-normal distribution with mean and SD of the natural logarithm of the distribution of 1.621 and 0.418, respectively. The incubation period for COVID-19 is thought to extend to 14 days, with a median time of 4–5 days from exposure to symptoms onset. Lauer et al.26 reported that 97.5% of people with COVID-19 who have symptoms will do so within 11.5 days of infection.

[4] He et al.27 fit serial interval distribution with gamma distribution. Together with a log-normal incubation period distribution estimated from Li et al.33, they inferred that infectiousness started from 12.3 days before symptom onset, peaked at symptom onset, and then declined quickly within 7 days. This aligns with recent advisories by the EU that 10 days after symptom onset would be safe to not be in isolation. Some parameters of the infectious profile were updated in Ashcroft et al.32

**References**

28. Zou, Lirong, Feng Ruan, Mingxing Huang, Lijun Liang, Huitao Huang, Zhongsi Hong, Jianxiang Yu, et al. 2020. “SARS-CoV-2 Viral Load in Upper Respiratory Specimens of Infected Patients.” The New England Journal of Medicine 382 (12): 1177–79

29. Ferretti, L., et al, The timing of COVID-19 transmission, *medRxiv*09.04.20188516; doi: https://doi.org/10.1101/2020.09.04.20188516

30. Lauer SA, Grantz KH, Bi Q, et al. The Incubation Period of Coronavirus Disease 2019 (COVID-19) From Publicly Reported Confirmed Cases: Estimation and Application. *Ann Intern Med*. 2020 May 5;172(9):577–82. doi:10.7326/M20-0504external icon.

31. He, X., Lau, E.H.Y., Wu, P. *et al.* Temporal dynamics in viral shedding and transmissibility of COVID-19. *Nat Med* **26, **672–675 (2020). https://doi.org/10.1038/s41591-020-0869-5

32. Ashcroft P, Huisman JS, Lehtinen S, Bouman JA, Althaus CL, Regoes RR, Bonhoeffer S. COVID-19 infectivity profile correction. Swiss Med Wkly. 2020 Aug 5;150:w20336. doi: 10.4414/smw.2020.20336. PMID: 32757177.

33. Li Q, Guan X, Wu P, Wang X, Zhou L, Tong Y, Ren R, Leung KS, Lau EH, Wong JY, Xing X, Xiang N, Wu Y, Li C, Chen Q, Li D, Liu T, Zhao J, Liu M, Tu W, Chen C, Jin L, Yang R, Wang Q, Zhou S, Wang R, Liu H, Luo Y, Liu Y, Shao G, Li H, Tao Z, Yang Y, Deng Z, Liu B, Ma Z, Zhang Y, Shi G, Lam TT, Wu JT, Gao GF, Cowling BJ, Yang B, Leung GM, Feng Z. Early transmission dynamics in Wuhan, China, of novel coronavirus–infected pneumonia. *N* *Engl J Med*. 2020;382(13):1199–207.

34. Kretzschmar, M., G. Rozhnova, M. C J Bootsma, M. van Boven, J. H H M van de Wijgert, M. J M Bonte, Impact of delays on effectiveness of contact tracing strategies for COVID-19: a modelling study, *Lancet Public Health* 2020; 5: e452–59 Published Online July 16, 2020, https://doi.org/10.1016/ S2468-2667(20)30157-2

35. Matt Webster, Should people who have had COVID be at the back of the vaccine queue?, January 30, 2021, https://theconversation.com/should-people-who-have-had-covid-be-at-the-back-of-the-vaccine-queue-153894

36. CDC, Frequently Asked Questions about COVID-19 Vaccination, Updated Mar. 12, 2021, https://www.cdc.gov/coronavirus/2019-ncov/vaccines/faq.html

37. CDC, When You’ve Been Fully Vaccinated: How to Protect Yourself and Others, Updated Mar. 21, 2021, https://www.cdc.gov/coronavirus/2019-ncov/vaccines/fully-vaccinated.html