The World Health Organization (WHO) declared the new coronavirus, identified as COVID-19, as a pandemic due to its fast rate of spread around the globe. As of May 15 2020, the outbreak of the virus has already generated 4,338,658 cases and 297,119 deaths globally 1 . The numbers of infections and deaths, as well as the economic disruptions are much more severe than those caused by SARS-CoV in 2002-2003 (8,906 infections and 774 deaths) [2] [3] [4] [5] [6] . A short-term spike of the number of cases can temporarily overwhelm the healthcare capacities in well-resourced countries 7 . Even worse, results from Ref. 8 indicate that the COVID-19 may exhibit spiral and seasonal patterns of the outbreak. Ending the global COVID-19 pandemic requires not only the effective pharmaceutical interventions, which are not expected to be available for months given the current medical knowledge (e.g., the hosts and transmission potential still remain unclear), but also the implementation of effective non-pharmaceutical interventions (NPIs). It has been confirmed that humanto-human transmission primarily occurs through respiratory droplets. It can also occur through contact with contaminated surfaces such as stainless steels and plastics 9, 10 .
Existing evidences have shown that NPIs including physical distancing and travel restrictions etc. would significantly change the social mixing patterns and reduce effective interactions between infected and susceptible individuals, hence interrupting the transmission from susceptible to exposed [11] [12] [13] [14] . Recent reports 8, 11, 12 suggest that NPIs can also delay the peak of the infection curve and ease the abrupt burden on healthcare systems.
So far various non-pharmaceutical interventions have been implemented to contain the spread of the virus and mitigate the cross-infection, such as inter-city travel restrictions, self-isolations, quarantine, social distancing, wearing face masks, travel-and contacttracing, mass gatherings bans, etc. These NPIs measures aim to reduce the transmission rate, flatten out the epidemic curve, reduce the peak demand on healthcare services, and buffer more time for vaccine development.
One of the governmental decision difficulties in battling the COVID-19 pandemic is to determine which NPIs are optimal and how stringent those NPIs should be implemented.
Since there are no targeted therapeutics nor effective vaccines 8 , the long-term NPIs can render a high economic loss and social disruptions. Minimization of deaths from COVID-19 virus and the economic impact of virus spreading cannot be achieved at the same time, as human and human-interaction are the pillars of economy health and growth. Once imposing the stringent NPIs such as city lockdowns and travel bans, the economy can be heavily affected. As the WHO Director-General noted, at the mission briefing on COVID-19 on March 12 2020, that "all countries must strike a fine balance between protecting health, preventing economic and social disruption, and respecting human rights" 15 .
Predictive mathematical model for epidemics can be a key to understand the COVID-19 spreading dynamics, providing a quantitative insight for the estimation of the medical requirements and capacities. It can also help to prevent, detect, treat or cure COVID-19 16 .
The susceptible-infected-recovered (SIR) model and its variants (reduced or enriched) are the most commonly-used ones for human-to-human transmission modeling [17] [18] [19] [20] . Regarding the predictive modeling of COVID-19, Koo et al. used an agent-based influenza epidemic simulation model (FluTE) to estimate the likelihood of human-to-human transmission in Singapore, and found that the estimated median number of infections was greatly reduced (99.3%) by NPIs in a relatively mild outbreak 12 Towards resolving the existing difficulties in making reliable projections, we took a further step to propose a time-varying SEIR model based on the two fundamental facts observed so far, (1) the effective basic reproduction number is highly correlated with the NPI measures and its stringency, and (2) the effective basic reproduction number varies with time. The former fact is clear, the past and ongoing experiences of China, South Korea, and Italy on reducing the total number of cases and containing the spreading are encouraging, comparing with other countries with more relaxed implementations of NPIs. 
In general, we can divide the NPI measures into two categories, individual actions and governmental actions. The governmental actions require a systematic policy-making process and a top-down supervised implementation strategy; therefore, the governmental actions can be executed with different degrees of stringency, and the individual actions are usually suggested or encouraged as public advices. We summarized the typical government actions implemented and individual actions advised by epidemic experts in Table 1 .
Governmental actions implemented by many countries in dealing with COVID-19 pandemic include, but may not be limited to, nationwide large-scale testing, virus-contact tracing, population flow tracing, town or city lockdown, border control, closure of schools and workplaces, and mass-gathering bans.
Quarantine plays an exigent role in the process of mitigating the spread of the virus, especially for countries with high population densities in large cities 23 . Scientific evidence shows that the incubation periods for other types of coronaviruses are between 1 and 14 days. Recent reports indicate that the median observed incubation period for COVID-19 is 5 days with the 95% confidence interval of 4.5-5.8 days, and 97.5% of those who get infected show symptoms within 11.5 days with a 95% confidence interval of 8.2-15.6 days 24 . The observation and statistical estimation form the basis of making the 14-day selfisolation policy for those who travel from virus-impacted regions with a potential viruscontact history.
The closure of workplaces, universities, schools, daycare centers, non-essential businesses and stores are alternative governmental means to reduce person-to-person contact. Although the detailed contributions from each individual measures are not yet clear given current COVID-19 data 25 , these measures must be implemented in conjunction with effective personal actions, for example, staying at home, avoiding unnecessary trips, eliminating group activities and mass gathering; otherwise, the governmental actions will not achieve the maximum mitigation strength.
Another governmental action is the implementation of nationwide virus-contact tracing and travel history tracing for those who develop symptoms 26 . With the help of personally shared map and navigation data as well as the telecommunication location service data 27 , the travel and contact history of a confirmed case can be used for backward tracing. The possibly affected people can be identified for further test, isolation, and treatment. In China, temperature testing at the entrance of public buildings and residential communities are also mandatory, providing a massive coverage in detecting the potential symptoms.
Individual actions such as wearing face masks, keeping social distance, reducing nonessential trips, staying at home, working from home, washing hands frequently, and so on, are equally important in mitigating the spread of virus during the pandemic. Without individual actions, the effectiveness of governmental actions will be largely reduced.
The different measures are just one dimension of the NPIs. The stringency in terms of mandatory actions is another dimension worth mentioning. The stringency can loosely be defined as whether the actions are mandatory or suggested.
Based on the stringency of the implementation of NPIs, countries with the COVID-19 outbreaks can be divided into three categories, namely, strong non-pharmaceutical interventions (S-NPI); moderate non-pharmaceutical interventions (M-NPI) and light nonpharmaceutical Interventions (L-NPI). We grouped the top 15 economies of the world into the three categories accordingly in Table 1 . The intervention timing refers to the time imposing the strictest NPI measures such as city lockdowns.
For the countries in the category of S-NPI, China implemented the stringent NPI measures to mitigate the spread of COVID-19 virus. Since January 23, Wuhan had been locked down and more than 30,000 medical staff from all over the country were mobilized to Hubei battling the virus as of Feb. 17 28 . The lockdown was also imposed on many cities in Hubei Province with inter-city travel bans. In dealing with the potential diffusion of the virus through the mass flow, China also rolled out the contact-tracing using mobile apps.
It allows to analyze the travel history and provide up-to-date status report on smartphones, and ensures that almost end-to-end (departure-arrival locations) and targeted isolation can be made on demand. The nationwide NPIs in China also include the closure of the daycare centers, schools, universities, the postpone of back to work for nonessential workers after the Chinese New Year holiday. Everyone travelling to and from epidemic-stricken cities are subject to 14-day at home self-isolation before participating other activities. The entries to residential communities, grocery stores, workplaces and other public places are all subject to temperature screening and the requirement of wearing face masks 29 . South Korea was the first countries to provide a drive-through testing system, and it also has the virus- In the category of M-NPI, French throughout the country were required to stay at home except those who purchase necessities and seek medical and healthcare services from March 16. Germany also implemented the statewide policy on restricting people from going out for unnecessary activities. Schools and daycare centers were closed and gatherings were banned. In Spain, people throughout the country were not allowed to leave their homes except under certain circumstances from March 14.
For L-NPI countries, the US has the first confirmed case of COVID-19 reported from Washington State on January 31, 2020 31 . Soon after, California and Washington reported outbreaks. At this moment the cases in the US exceed cases reported in China and Italy combined. Different states and major cities have imposed different policies. In states and cities that are greatly impacted by the virus, e.g., NYC 32 and New Jersey, staying at home was suggested and most companies asked their employees to work from home. Schools and daycare centers were also closed. Recently New York State doubled testing capacity to reach 40,000 diagnostic tests per day with more than 700 testing sites. Residents of New
York were encouraged to get tested at nearby testing sites. The state also implemented the Contact Tracing Program to help slow the spread of COVID-19 and ease the social isolation without triggering renewed virus spreading 33 . In UK, isolating towns or cities was not part of the British government plan at the beginning of the COVID-19 outbreak, in hope of a degree of herd immunity can help to reduce and broaden the peak 14 . Given the rapid development of the epidemic and the quickly growing number of infected people, the British government imposed the lockdown strategy in mid-March, with only essential trips to medical centers and grocery stores, and exercise, allowed. The government asked the citizen to stay at home as much as possible, work from home if that is possible, limit the contact with other people, keep 2-meter distance apart when going out. The government also published a guidance on April 15 for COVID-19 test by-appointment for their essential workers 34 . Australia declared on March 19 that non-Australian citizens were not allowed to enter the country. Some states closed their non-essential business places from March 22.
Japan imposed travel restrictions in many areas starting from March 26.
We projected the infection curve, the pandemic duration, the susceptible and recovered The model allows us to quantify the strength of NPIs imposed by different countries using the PIF (λ). The quantitative evaluation of NPIs can be leveraged to help decisionmakings at the early stage of the epidemic. This advantage has not been realized using the classical SEIR model. According to our results, the S-NPI corresponds to the range that λ > 0.05, M-NPI corresponds to 0.03 < λ ≤ 0.05, and L-NPI is correlated with the range of λ ≤ 0.03, as shown in Fig. 1.   Fig. 1 The Correlation between the stringency of NPIs and policy intensity factor .
In Fig. 4 Experiences with 1918-19H1N1 influenza pandemic have shown that NPIs have significant effects on reducing the amount of people been infected. Several cities in US adopted a variety of NPIs such as, the closure of school and work place in that pandemic.
However, transmission rebounded once controls were lifted 38 . This phenomenon indicates that the early termination of NPIs may cause a rebound of the pandemic even with stringent policies.
To investigate the potential impact of the NPI duration in the countries, the total number of affected people given three different NPIs durations of 90, 180, and 360 days are projected. The detailed results of the affected fraction of population were presented in Fig.   3 . The projection results indicated that for S-NPI countries, a short-term (90 days) implementation can be an effective and efficient means to cease the spreading of the virus and reduce the number of affected people. For L-NPI countries, a long-term implementation is necessary to reduce the total number of affected people. However, the economy can be greatly impacted due to the implementation of nationwide stringent NPIs. For example, the most effective governmental measures such as city lockdown, self-isolation, closure of work places and schools, inter-city travel bans, border control and so on, may cause serious social and economic burdens 39 . The International Labor Organization (ILO) estimates the job loss due to the COVID-19 epidemic may reach 25 million 40 . Achieving the delicate balance between the mitigation measures and the economy impact is highly nontrivial.
As noted in Ref. 14 , "personal, rather than government action, in western democracies might be the most important issue". It is no doubt that personal responses to the governmental actions will be crucial to control the spread of COVID-19 in an outbreak. infection chain is used. It is known that the infection chain can be decomposed into more intermediate or detailed states. For example, the infected can be divided into "symptomatic infected but undetected" and "diagnosed" sub-states 3 . The concept of PIF can also be adopted to a model with more state variables. Thus, when such detailed survey data become available, these states can directly be incorporated in the VR-SEIR model. 
Mandatory Suggested 
where the parameter is the decay constant, and is the time variable. The solution to Eq.
(1) is ,
where 0 is the initial value at = 0 . The parameter λ characterizes the dynamical behavior of under the influence of NPIs. Therefore, we call this parameter policy intensity factor (PIF).
By introducing the so-called policy intensity factor (PIF), we can incorporate the influence of the NPI measures on the basic reproduction number, and obtain the following variable-rate SEIR model, influence of NPIs , (5) noting that the orignial constant basic reproduction number 0 is defined as the ratio of the constant transition rate of → over that of → 43 .
With the proposed VR-SEIR model, we defined the pandemic duration as the time 
Another criterion can be used to define the duration of the pandemic is the double or single-sided α − quantile (e.g., 95%) interval of projected infection curve 12 . Using the quantile approach, we can define the end of pandemic as ,
where ̂ is the projected nubmer of infected people, and −1 (̂) is the α − quantile of the cumulative summation of the projected number of infected people.
Without the loss of generality, in this study, we define the projected pandemic duration as the minimal value between the two quantities. (8) where = 1% and α = 0.95 are used to project the pandemic duration throughout this study.
The COVID-19 data (as of May 15, 2020 † ) of the top 15 economies in the world are obtained. We use the data to estimate the model parameters ( 0 , λ, α, γ) by minizing the difference between the model projection and the actual numbers of the four states using the method of least squares. In both parameter estimation and projection, we used one-day interval as the time-step since it is the maximum time resolution in the data. In projection, we assumed that the natural birth and death are relatively stable; therefore, the total population does not reflect the natural birth and death during the projection. The initial values (I.V.) of ( 0 , 0 , 0 , 0 ) for the state variables , , , are set accordingly as follows, ,
where (1) is the reported number of confirmed cases on day 1 in the record. The initial value of 0 is set to the infected number of people on day 7, 
where is the total population of the country, and ̃ is the value when the VR-SEIR model converges, i.e., .
Projections of the NPIs durations of 90, 180, and 360 days were made for the top 15 economies. We assume that when the NPIs are not imposed, the virus will spread in a natural manner, characterized by the basic reproduction number at that moment. The basic reproduction number is assumed to remain constant after that time as no further NPI measures are imposed. The resulting model after the termination of NPIs, essentially reduces to the regular SEIR model, as shown in Eq. (11) .
, (11) where is the NPI duration, and ( ) is given in Eq. (4). Eq. (11) determines the transition rates among the four states considering the NPIs duration length . Another piece of crucial information is the initial value of the numbers of people of the four states. Here we set the initial numbers of and as the average numbers of the two states from = 0 to τ.