Effects of control measures and their impacts on COVID-19 transmission dynamics

Governments around the world have grappled with the COVID-19 pandemic for more than a year. Control measures such as social distancing, use of face masks in public places, business and school closures, city or transportation lockdowns, mass gathering bans, population education and engagement, contact tracing, and improved mass testing protocols are being used to contain the pandemic. Currently, there are no studies to date that rank the effectiveness of these measures, resulting in government responses that may be uncoordinated and inecient. In this study, we developed a Discrete Time Markov Chain model that captures the above control measures and ranks them. We found that the importance of the measures changes over time and depends on the stage of transmission dynamics, as well as the ecological environment. For example, contact tracing is a powerful measure to effectively control the pandemic, however, our results show that while it is indeed helpful during the early stages of the pandemic, it is much less important after a vaccination program takes effect. Besides, our model improved the standard SEIR compartmental model by taking into account the dynamic temporal transmission and recovery probabilities along with considering a portion of the population that will not comply with government-mandated control measures. If implemented, our novel and unique model may assist many countries in pandemic control decisions.


Introduction
Different countries have used different control strategies to combat the COVID-19 pandemic. China and New Zealand implemented a seal and eradicate strategy whereby they quickly applied strict measures very early to stop local transmission, and then sealed their borders to protect the epidemic from reoccurring. The United States and UK chose to strike a balance between controlling the pandemic and its economic impacts by implementing various measures to keep the effective reproduction rate, R t , low while implementing advanced vaccination programs to achieve population level immunity. Widelyaccepted control measures for the pandemic control are social distancing, face masks, business and school closures, city or transportation lockdowns, mass gathering bans, population education and engagement, contact tracing, vaccination programs, viral testing advancements, and targeted mass testing. To formulate an effective control strategy, decision-makers must understand the impact of each control measure on transmission dynamics and identify the control measures that should put more focus on and allocate more resources into.
Huang et al. 1 presented an SEIR compartmental model and showed that quarantine and isolation measures, as well as testing delays, could cause the largest difference in numbers of daily con rmed cases. Zhang et al. 2 found that limiting the mobility of the general population was more important than fast detection and isolation during the early stages of the outbreak in mainland China. Liu et al. 3 found that earlier and stricter lockdown implementation yielded better control than later or more relaxed measures. Indeed, many researchers focus on speci c control measures and show their positive bene t, however, they have not aggregated all of these measures into a single model to see their interaction and compare their relative bene ts. In this paper, we develop a Discrete Time Markov Chain model that captures the most, if not all, widely accepted control measures and ranks them so the policymakers can appropriately allocate resources to the most bene cial measures.

Model
Some studies 4,5 use a Markov Chain model with four states (Healthy, Active, Recovered, and Death) to model COVID-19 evolution. Although having few states makes the model easy to manage, these models fail to accurately capture other important dynamics of the epidemic, such as the chance of viral transmission and the likelihood of recovery, both of which uctuate depending on the length of time an individual patient has been symptomatic. Our proposed model has numerous states to capture this temporal information, and therefore appropriately model dynamic transmission and recovery probabilities.
Our model also accounts for the portion of the population that will not comply with governmentmandated control measures by classifying populations as either "cooperative" or "uncooperative." With effective communication and promotional campaigns, a government might be able to improve the proportion of cooperative people, k o .
There is a signi cant portion of asymptomatic cases. Most asymptomatic cases will never be detected but can still spread the virus similar to symptomatic patients. Moreover, those with mild symptoms that have recovered by themselves may not get tested, also contributing to the undetected rate in the population. These two types are classi ed as the "undetected infected" class. There may also be some infected person, possibly with mild symptoms, who did not comply with self-isolation. Those who did not self-isolate were also assigned to the undetected infected class because they may go out and spread the virus like undetected infected individuals. Thus, the undetected infected class includes asymptomatic and mild symptom cases who do not get tested and those that tested positive but failed to isolate themselves. Recent studies have estimated asymptomatic cases to anywhere from 4% to 52% [6,7,8,9,10,11]. Without loss of generality, we set the asymptomatic proportion at 35% and implement a control parameter, k d , as a percentage adjustment to the asymptomatic proportion to allow manipulation of asymptomatic and symptomatic detection and isolation. Therefore, the proportion of undetected infected over total infected is 35% -k d . For example, if we can detect 30% of asymptomatic cases and isolate 80% of symptomatic case, we have k d = (30% × 35%) -(20% × 65%) = -2.5%. We estimate that k d can range from -10% to 30%, depending on the effectiveness of the control measures being implemented.
Our Discrete Time Markov Chain model at time t has the following states: SC t = Susceptible and cooperative with control measure on day t SU t = Susceptible and uncooperative with control measure on day t UC i t = Undetected infected for i days and cooperative with control measure on day t UU i t = Undetected infected for i days and uncooperative with control measure on day t TC i t = Detected or will be detected infected for i days and cooperative with control measure on day t TU i t = Detected or will be detected infected for i days and uncooperative with control measure on day t M t = Cumulative immune on day t V t = Cumulative vaccinated on day t D t = Cumulative dead on day t, where i = 1, …, I , and I is the maximum number of days [1] that an infected person can transmit the virus.
In other words, f u (i) and f d (i) should be negligible when i > I.
The probability for an infected individual to spread the virus after becoming infected for i days follows the temporal transmission probability mass functions, f u (i) and f d (i), for undetected and detected infected individuals, respectively. Let R 0 be the basic reproduction number, i.e. the expected number of cases directly generated by one infected individual. Let us de ne an aggregated epidemic control measure k r (t) as the measures implemented on day t that directly affect the reproduction number, such as mask wearing, social distancing, and lockdown. For example, Oraby 5 found that lockdown reduces R 0 by 64-85% across 155 countries. By design, only the cooperative populations will implement this k r (t) measure while the uncooperative populations will not. Hence, the expected number of new cases generated from the contagious population on day t can be calculated as For a country with a population size N, the number of newly-infected individuals on day t+1 can be computed as UC 1 t+1 = E t × SC t /(N-D t ) × (35% -k d ), UU 1 t+1 = E t × SU t /(N-D t ) × (35% -k d ), TC 1 t+1 = E t × SC t /(N-D t ) × (65% + k d ), and TU 1 t+1 = E t × SU t /(N-D t ) × (65% + k d ). The number of susceptible individuals on day t+1, SC t+1 andSU t+1 , are then to be reduced by the newly-infected cases accordingly. Infected states will transition into the next day infected states where , , , and , for all t and i =1,…, I-1. Undetected cases are asymptomatic or mildly symptomatic, therefore we assume they will all recover and become immune. For detected cases, some of them will die. Recent global case fatality rate (CFR) 12 is 2.71%. Hence, the probability that patients in the detected group will die is assumed to be 2.71% and the remaining 97.29% will recover. The estimated CFR of 2.71% is also in a reasonable range with other studies. [2] As a result, the immune state M t+1 equals M t +(UC I t + UU I t )+ 97.29% (TC I t + TU I t ) and the death state D t+1 equals D t + 2.71% (TC I t + TU I t ).
One important control measure is contact tracing. We de ne control parameters k l as tracing delay, the time between the isolation of a known case and isolation of its contacts, and k c as contact tracing coverage, the proportion of contacts detected and isolated. k c then equals the percentage of cases that conduct contact tracing multiplied by the proportion of contacts that can be isolated over all traced contacts. Lastly, one of the most effective control measures is vaccination. Vaccination coverage being administered up to time t is k v (t), which is equal to the percentage of the population that has received vaccination up to time t multiplied by vaccine e cacy (i.e., 95% for Moderna 13 and P zer-BioNTech 14 and 72% for Johnson & Johnson 15 ). Additional details about contact tracing and vaccination modeling can be found in the methods section.
Footnotes: [1] In the methods section, we show that I = 21.
[2] Wu et al. 24  of people are willing to take a COVID-19 vaccination if it was available to them. Hence, we should not expect k v (t) to go near 1.0 easily. Therefore, we assume for a base case that the vaccination program will continue at a 0.5% growth rate until it reaches 60% of the population. In the worst-case scenario, the program will continue with a 0.5% growth rate until it reaches 45% of the population. In the best-case scenario, the program will continue with a 0.55% growth rate until it reaches 75% of the population.
Because more than 98% of vaccines administered 18 in the United States are Moderna or P zer/BioNTech, which are 95% effective, we will assume the United States has a vaccine e cacy of 95%.
Gramlich's 2018 survey 19 revealed that 75% of Americans would cooperate with each other in a crisis.
This number is also consistent with the results of a survey 20 about mask use in March 2021. Hence, 75% will be the baseline for k o , with 50% and 95% for worst and best case scenarios, respectively.
Based on current CDC recommendations, 21 for the purpose of advancing public health planning, the best estimate of R 0 to use in our model is 2.5. We then t the parameters with actual data to obtain the baseline of a control measure k r (t). By minimizing the sum of square error in tting the seven day moving average of new case numbers from actual historical data and predictive model for the past three weeks, we found that a stationary k r (t) at 0.4354 gives the best t. As a result, our baseline for k r (t) is 0.4354. Table 1 shows a summary of control parameters and their assumptions. The United States is quite advanced in its vaccination program because as of March 21, 2021, nearly 25% of its population has received at least one dose of the vaccine. Figure 1 shows the number of daily new cases, under the baseline, worst case, and best case for control measure k r . In the baseline, when k r = 0.4354, it will take 136 days to get zero new cases. If a lockdown is imposed, and k r becomes 0.2177, then it will shorten this duration to only 95 days. When k r is at 0.8708 (e.g. people meeting twice as frequently) the number of new cases transitions to an upward trend but later descends in mid-May when the percentage of the population vaccinated increases, increasing the duration to 275 days.
Extend Data Fig. 1 shows the number of daily new cases, under different vaccination program scenarios. All three scenarios would not yield any signi cant difference concerning the daily new case numbers. With a 0.55% daily growth rate and a limit of 75%, it would take 124 days until there are zero new cases, while with a 0.5% daily growth rate and a limit of 45%, the duration will increase to 160 days. If the vaccination program stops on March 22, 2021, then the number of daily new cases will still go down if the control measure level is maintained as previously. This is a result of the effective reproduction rate R e being less than 1.0 due to the 25% vaccination coverage, but the duration until zero new case increases to 289 days. There is a herd immunity threshold 22 required to control the transmission of the virus. However, we show that the bene t of vaccination can be attained immeidately without having to wait to reach the herd immunity threshold, as long as the control measures to keep R t down continue to be enforced. Fig. 2 is a tornado diagram displaying the number of days required to achieve no new cases from both the best and worst case scenarios for each control measure. For example, reducing k r (t) to 0.2177 (best case) results in 95 days to zero new cases, while increasing k r (t) to 0.8708 (worst case) results in 275 days to zero new cases. The diagram shows that, in the United States, where the vaccination program is quite advanced, in order to further improve the speed of pandemic control, policymakers should focus on (1) lowering the effective reproduction even more or at least keep it at the base level by effectively inform the people to maintain social distancing and facemask wearing habits, (2) improving population cooperation through education and engagement, and (3) reducing testing delays. On the contrary, the contact tracing program and asymptomatic detection now have minimal impact on the control speed and becomes less of a priority (e.g. only an 11 day decrease, by improving these parameters).

Discussion
Case study II: Thailand as of January 6, 2021 To understand the effects of the control measures in a different environment, we applied the model with parameters estimated from Thailand data as of January 6, 2021. Table 2 shows the worst-case, baseline, and best-case scenarios used in the analysis. By tting with the actual data and k r (t) set at 1.0, we found that R 0 for the 2 nd wave in Thailand was 2.026. Thus, the worst case scenario for k r (t) is 1.0 which is the same level as the no new case period immediately preceeding the start of the second wave. Rotejanaprasert et al. 23

estimated R t in Bangkok
and showed that it can drop to as low as 0.5 on May 1, 2020. This implies k r (t) = 0.2, which is the same level as when Thailand was in lockdown during April/May 2020. For the base case scenario, we assume an intermediate k r (t) of 0.6. Fig. 3 shows the daily new case numbers when similar lockdown measures implemented in April 2020 (k r (t) = 0.2) were re-implemented after the second wave was detected for 2, 9, and 16 days delay. The graph shows that the longer the control measure was delayed, the longer the duration required to achieve zero new daily cases. For example, if the lockdown was implemented in 2, 9, 16, and 25 days, then it would take 30, 43, 55, and 67 days, respectively, to bring the number of new cases to zero.
As seen from Fig. 4, even though Thailand's vaccination program lags behind many other nations, if it implements a medium control level k r at 0.6, it will begin to see a positive effect of the vaccination program when k v reaches 5%, corresponding to 4.6 million people having received the vaccine. At this point, the number of daily new cases will start to descend. This is because the effective reproduction rate is near 1, meaning that a minor reduction in the susceptible population would be satisfactory to decrease the effective reproduction rate to less than 1.

Discussion
The tornado diagram of Thailand in Fig. 5 indicates that, even though Thailand currently has a low number of new cases, it will take more than a year to get zero new cases if all control measures are kept at baseline. Some serious effort is required in order to improve some, if not all, of these control measures.
Most bene ts would be obtained from implementing measures that reduce reproduction rate and/or testing delay, such as (1) mass gathering bans, (2) increasing the number of mobile testing units particularly for those who live/work in highly-populated areas, and (3) quickly, practically, and effectively isolating those with positive test results so that they can no longer spread the virus. In addition, contact tracing and asymptomatic detection ability still need to be implemented and improved. Moving any of these parameters from worst-case to best-case levels will decrease the amount of time to zero cases by more than 100 days. Most importantly, there are three measures, reducing the reproduction rate, increasing cooperation levels, and implementing vaccination programs, that require special attention to ensure that they do not fall to their worst-case scenario otherwise the pandemic can get out of control.

Conclusion
In conclusion, we presented a Discrete Time Markov Chain model that enhances the standard SEIR compartmental model by taking into account various control measures, the dynamic temporal transmission and recovery probabilities, and different cooperation levels of the population. We implemented the model in two cases, the United States and Thailand, to nd the relative effectiveness of various control measures. The model identi ed that the United States should put more focus on lowering the effective reproduction, improving population cooperation, and reducing testing delays and less focus on contact tracing program and asymptomatic detection. The model also identi ed that Thailand should focus on all measures with special attention to reducing the reproduction rate, increasing population cooperation, and implementing vaccination programs.

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. Zou 28 found that the viral load detected in asymptomatic patients was similar to that of symptomatic patients. We assume they have the same generation time pro le as symptomatic cases and used Weibull distribution estimated from Ferretti et al. 29 They used maximum likelihood estimates to infer generation time distribution and t 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 pro le [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 rst 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 pro le 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 pro le 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.
usually done by quarantine, hospitalization, or self-isolation. We de ne 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 days 34 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 de ne 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.
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 rst day that he/she got the virus would be day tk 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 tk l -(j+k g -i), from day tk l -(j+k g -1) to day tk l -(j+k g -(j+k g -1)) = tk 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 tk 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 UK 35 and United States 36 and currently provide vaccination to their populations regardless of whether they were previouslly infected. We de ne 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. CDC 37 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-D t ) -k v (t-1)(N-D t-1 ). The population that can receive a vaccination at time t is N-D t -V t -(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-D t ) -k v (t-1)(N-D t-1 ))• SC t / (N-D t -V t -(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-k e )(N-D t-ke ) -k v (t-k e -1) (N-D t-ke-1 ))• SC t / (N-D t-ke -V t-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-k e )(N-D t-ke ) -k v (t-k e -1)(N-D t-ke-1 ))• (SC t-ke -SC t ) / (N-D t-ke -V t-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 days 26 from exposure to symptoms onset. Lauer et al. 26 t 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 t 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 pro le were updated in Ashcroft et al. 32 Figure 1 The number of daily new cases, under the base, worst, and best cases of control measure kr.