key: cord-1042320-1gmc5c8a
authors: Chao, Sung-Po
title: Simplified model on the timing of easing the lockdown
date: 2021-01-28
journal: Chin J Phys
DOI: 10.1016/j.cjph.2021.01.007
sha: 72c4a118e4bb79b16ba01995fde1351692457b9a
doc_id: 1042320
cord_uid: 1gmc5c8a

Lockdown procedures have been proven successful in mitigating the spread of the viruses in this COVID-19 pandemic, but they also have devastating impact on the economy. We use a modified Susceptible-Infectious-Recovered-Deceased model with time dependent infection rate to simulate how the infection is spread under lockdown. The economic cost due to the loss of workforce and incurred medical expenses is evaluated with a simple model. We find the best strategy, meaning the smallest economic cost for the entire course of the pandemic, is to keep the strict lockdown as long as possible.

The COVID-19 pandemic is an ongoing global pandemic of corona virus disease (COVID) first identified in Wuhan, China, in December 2019 [1] . As of 20 July 2020, more than 14.6 million confirmed cases of COVID-19 have been reported globally [2] . The virus is mainly spread between healthy and infected 5 people during close contact, most often through small droplets produced by coughing, sneezing, and talking [1] . Thus maintaining proper physical distance from other people, keeping good indoor ventilation, and wearing a face mask [3] in public places should be effective [4] in reducing the spread of the virus. People may also get infected by touching a contaminated surface and then touching 10 their face. Regularly disinfecting surfaces in public places and keeping the habit of hand washing for everyone should also curb COVID-19 infection.

3 Lockdown refers to the adoption of strict rules imposed by the local government to decrease individual mobility and increase physical distancing, some-15 times accompanied by mandatory use of masks or other personal protective equipment in public places. Based on the studies of mobile phone tracking done in China [6] and Italy [7] , the effectiveness of lockdown in controlling the spread of the virus depends on how strict the lockdown procedures are implemented. With a tighter lockdown, sufficient low mobility does bring down 20 transmission below the level needed to sustain the epidemic [7] . As vaccine against COVID-19 is currently not available [8] , lockdown remains to be one of the proven effective approach [18] for curbing the spread of infection, especially if contact tracing[9, 10] were not likely to be done efficiently. However these lockdown measures also come with many serious economic impacts, especially 25 for the low income households [11] , along with some social and psychological tensions within the populations [12, 13] . How to strike the balance between controlling the spread of the virus and maintain social economic stability becomes the key issue for navigating out the crisis brought by this pandemic. Some countries, such as Sweden and South Korea, have not fully locked down [14] , and its 30 success[15] depends on efficient contact tracing and massive tests to identify and quarantine those who are infected. Frequent random testing [16] may also help to avoid complete lockdowns. In the event of major outbreak, the large amount of infected people would make contact tracing ineffective against the spread of infection [17] , and some forms of locked down seems to be the most 35 effective approach to curb the spread of infection.

The goal of this paper is to find the optimal timing, in the sense of lowest economic cost for the entire course of the pandemic, for easing the strict lockdown. Less stringent controls are required following the end of the strict lock- 40 down to temporarily end the pandemic. We model the pandemic using the sim- down. The economic cost during the pandemic is vastly simplified to include only two parts: the economic cost due to the loss of available workforce and 45 lockdown regulations, and the medical expenses for the infected. Based on this model study, in a three-stages scenario starting with natural spread, followed by strict lockdown, and finally less stringent lockdown, the best strategy is to keep the strict lockdown as long as possible. Here the best strategy refers to least economic cost during the entire period of pandemic, which is shortened 50 with longer strict lockdown. This is not a trivial result as prolonged lockdown damaging the economy and causing livelihood issues even for healthy people.

Thus some sort of government subsidies are required to assist those who are in need to live through the strict lockdown. Three countries, U.S.A., Italy, and China, are selected to see if indeed the longest strict lockdown, which is indeed 55 good for temporarily stopping the pandemic, is also good for the economy. The changes in their quarterly GDP [21] seems to suggest that is correct, but no clear indication suggests this from their unemployment rate data[20]. More detailed comparisons in different economic indexes should be made. 60 There are several existing literatures discussing economic trade-offs and optimal policy analysis within the SIR framework [22, 26, 27, 28, 29, 23, 25, 24] , or its extension such as SEIR [30, 31] (E stands for "exposed"), or other more sophisticated compartment models [32, 33] . Other than these so called equations based approach, agent based model of COVID-19 epidemic [34, 35, 36, 37] , which 65 takes more computational efforts, is also applied on these issues. A nice relevant literature review can be found in [32] . Most of the studies support strict and long lockdown as advocated by this paper. The main difference between this work and others is that the end of pandemic defined here (for evaluation of economic cost during the pandemic) is not the real ending of the pandemic. That 70 is, the herd immunity is not discussed in this model study. The reason is that, without sufficient study on the duration of immunity and reliable vaccines for this virus, it would inevitably cost many more lives, as briefly discussed in section 3, to achieve the herd immunity. New lifestyles of epidemic prevention [38] 2 MODEL DISCUSSION 5 should be followed during the post pandemic time.

This paper is organized in the following way. In section 2, we introduce a slightly modified SIRD model and a simple model for economic cost due to the pandemic and lockdown. In section 3, economic costs for different duration of lockdown are evaluated with simplified time dependent infection rate. Results 80 are summarized in Table 1 . More realist time dependent infection rate for three countries (U.S.A., Italy, and China; data collected up to July 9th, 2020) and some of the associated economic cost are shown in section 4. In section 5 we summarize our results, and comment on the limitations of the studies done in this paper.

To model the pandemic for some number of population N , we use a slightly modified SIRD model [39] , incorporating the loss of lives when the infection rate goes above some "medical threshold". The equations describing the ratio 90 of susceptible S(t), currently infected I(t), the recovered ones R(t), and the deceased ones D(t) are given by:

In this SIRD model, we assume the disease is spread between those who are infected and those we are susceptible in a well-mixed system. The spatial inhomogeneity, say the population density variations between the city and country-95 side, is averaged over in the model parameters β(t), γ(t), and η(t). Here β(t) 6 is a time dependent infection rate, which decreases from some maximal value set by the nature of the virus and the average population density in the region we discuss to a positive number close to zero if very strict lockdowns were imposed. γ(t) is the recovery rate given by the average time it takes for the 100 infected one to recover from this disease. η(t) is a time dependent death rate, and Θ(x) is the Heaviside step function being 1 when x ≥ 0 and 0 otherwise.

Parameter I m can be evaluated by the fatality rate in the region we discuss, and is related to the medical resources available in that region during this pandemic. 105 We use a naive model to describe the economic impact brought by the lock- The economic cost at time t due to the lockdown and the additional cost on the medical treatment is then given by:

(2) we note that the crucial factor is M E/GDP as the value of GDP and N are assumed to be fixed (time independent) in this model. If pandemic. In between we may have two special points in this time frame: the time to start the lockdown, and the time for easing or lifting of the lockdown, denoted as s l T and e l T respectively. It follows that 0 < s l < e l ≤ 1. 

In the Table 1 , we list the ending day of the pandemic T , the economic cost from day 1 to T (labeled as E T otal cost ), and the economic cost from day 1 to t = 90 (labeled as E 90days cost with three different α values mentioned above. The mortality rate, shown at the bottom of the table, in all three cases are kept at 0.01%. The fatality rate, defined as number of deaths divided by those who 220 have been infected, at the end of the pandemic, is given by

The fatality rate is listed in the 5th row of the Table 1 . From the row atop the bottom of Table 1 , we see that longer lockdown makes the fatality rate slightly higher (around 18% for 60 days lockdown compared with 15% for 30 days). This is because the total number of people infected is larger for shorter Mortality rate 0.01% 0.01% 0.01% Table 1 : Table for In the simplified scenarios, the company can survive in the short term (i.e. the strict lockdown as early as possible, without knowing that shorter lockdown actually brings even more cost in the long run. That is, E max cost < E T cost for some 90 < T < T even for (e l − s l )T = 30 case. The best strategy for the company then is to keep the strict lockdown as long as it can, and how long the

14 company may survive under the lockdown crucially depends on E max cost . To have longer survival time, the company needs to increase E max cost with the help from banks (lowering interests rate) and timely government subsides, and to modify the business patterns to lower its α value (changing the in person meetings to online ones, for example). Some passenger airplanes having been modified to 275 serve as cargo planes in some airlines is one good example of lowering the α value in the airline business.

In short, "the more haste, the less speed" or "haste makes waste" accurately depicts the shortcomings of the hasty lifting of the lockdown. During this pan-280 demic we shall "overcome impetuosity and exercise patience, and go steady so we can go far". Let us use the modified SIRD model described in the Eq.(1) to study more realistic cases, and compare the lockdown procedures and timings between these cases in the following section. The methods used to obtain the parameters β(t), γ(t), η(t), and I m in this modified SIRD model are introduced 285 in the beginning of the next section.

To compare with the actual cases, we evaluate theβ(t) by fitting the actual data nicely organized by Wade Fagen-Ulmschneider [40] .β(t) is a time dependent infection rate in the modified susceptible-infected (SI) model with 290 equations given by: 

The actual data points used to obtainβ(t) is shown as red circular dots in the Fig.3 , in which the data from the U.S.A. is used as an example of how we 300 use the actual data. Those points represent the geometric growth number β 0 (t) of a given day obtained from the data of total confirmed cases shown in [40] . We fit β 0 (t) with some mathematical function β f (t), and part of the fitting function 

Time dependent infection rateβ(t) in the SI model is related to the fitted geometric growth function β f (t) viaβ(t) = ln(β f (t)). Using thisβ(t) we solve SI model numerically to obtain the daily new cases dĨ(t) dt . The result is shown as the thick blue line in the inset of Fig.3 , which is roughly consistent with the new cases shown in [40] . Fig.7 to Fig.10 , we use the same color codes as those used in the Fig.4. of Fig. 3 , is roughly consistent with that obtained from the SI model (blue thin line in the same figure) . The parameters γ, η, and I m are chosen by requiring the total confirmed case, the active (or currently infected) cases, the recovered, and the deceased ratio to be roughly consistent[41] with the actual data at the 320 last day (t = 122 in this U.S.A. case) of the data set.

Following aforementioned methods, we find that β(t) −β(t) = 0.0053, γ = 0.006, η = 0.0022, and I m = 1 × 10 −5 giving rise to the total confirmed case, the active cases, the recovered, and the deaths ratio for the U.S.A. up to July 9th. 

Italy case

Following the same methods as in the U.S.A. case, we find the fitted geometric growth function β f (t) for Italy is given by:

We choose β(t) −β(t) = 0.028, γ = 0.036, η = 0.01, and I m = 1.5 × 10 −6 to match the total confirmed case, the active cases, the recovered, and the deaths ratio for Italy up to July 9th. These computed results are plotted in the Fig.7 and Fig.8 . The logarithmic scale plot Fig.8 shows more clearly the onset of recovered and death ratio, which is around 7 and 10 days after the onset of the 380 active cases respectively. This is very close to the actual data set, with recovered and death ratio around 8 and 10 days after the onset. The starting and ending day corresponds to t 20 and t 86 in Fig.7 and has dropped down consistently from its peak value (at around t = 53 in Fig.7) .

Italy's current emergency rules remain in place until at least the end of July.

The current measures include the obligation to wear face masks on public transport and in shops, restaurants, public offices, hospitals, and workplaces where it is not possible for people to keep at least one meter apart at all times [48] . With the current trend, the pandemic is expected to stop (I(t) 10 −6 ), without the help of vaccination, for around one year and three months from now.

Following the same methods as in the U.S.A. case, the fitted geometric growth function β f (t) for China is given by:

We choose β(t) −β(t) = 0.021, γ = 0.056, η = 0.006, and I m = 8 × 10 −6 to match the total confirmed case, the active cases, the recovered, and the deaths 400 ratio for China up to July 9th. These computed results are plotted in the Fig.9 and Fig.10 . The logarithmic scale plot Fig.10 shows more clearly the onset of recovered and death ratio, which is around 6 and 19 days after the onset of the active cases respectively. This is roughly consistent with the actual data [40] with the onset of recovered and death ratio around 11 and 20 days after the The lockdown in China started from January 23, with partial lifting on 19

March [49] , and ends on April 8th for Wuhan [50] . The starting and ending day corresponds to t 1 and t 75 in Fig.9 and Fig.10 . It started in Wuhan city 410 in Hubei, China, but later expanded to many parts of China later. From Fig.10 we see that the pandemic stops at around t = 110. The actual data [40] actually shows the pandemic stops at around 86 days from the onset of the pandemic. This amount of population ratio is similar to the ones in the U.S.A. and Italy (c.f. Fig.5 and Fig.8 ).

Aforementioned three cases happen to roughly illustrating three cases in the [54] .

The next question would be that if this longer, stricter lockdown does do good for the economy, as is suggested in the simplified scenarios. The answer to 435 that is not clear, as we need more transparent data for making a fair comparison.

The second quarterly (Q2) GDP [21] in China rebounded to 3.2% growth from a record 6.8% contraction in the previous (Q1) quarterly GDP (announced on April 17 for Q1 and July 16 for Q2). Q1 GDP in the U.S.A. [21] for last year's Q4). The Trading Economics forecast [21] for Q2 of U.S.A. and Italy are −27% and −13.2%, respectively. If the forecast were correct, then this trend in the quarterly GDP suggests longer, stricter lockdown might indeed be 445 beneficial for the overall economy.

In this paper we treat the effect of lockdown as changing the values of time dependent infection rate β(t) in the SIRD model described in the Eq.(1). Other parameters γ(t) and η(t) are kept to be time independent to simplify the cal-450 culation. Combining the obtained results with a naive economic cost function described in Eq.

(2), we compute the economic cost for the entire course of lockdown and economic cost after 90 days (counting from the beginning of the pandemic) in simplified scenarios, where β(t) changes discontinuously during different stages of lockdown. The results, summarized in Table. 1, suggest that 455 long strict lockdown is the most beneficial approach for economy in the long run, even with the same number of deaths due to the pandemic [55] . More realistic computations for β(t) are done in the section.4 for U.S.A., Italy, and China, and

their respective economic indicators during this pandemic are discussed. Quarterly GDP seems to suggest the conclusion reached in the simplified scenarios, 460 but the monthly unemployment rate does not support that. We emphasize that lockdown should be viewed as a last resort for controlling the spread of diseases, and to shorten the lockdown efficient, regular mass testings are required [15, 16] to differentiate and separate the susceptible and infected ones. 

COVID-19 Dashboard by the Center for Systems Science and Engineer-500 ing

Face masks during the COVID-19 pandemic Efficiency studies for COVID-19

Physical distancing, face masks, 505 and eye protection to prevent person-to-person transmission of SARS-CoV-2 and COVID-19: a systematic review and meta-analysis

COVID-19) advice for the public

The effect of human mobility and control measures on the COVID-19 epidemic in China

Lockdown 515 timing and efficacy in controlling COVID-19 using mobile phone tracking

Scenario analysis of non-pharmaceutical interventions on global COVID-19 transmissions

Socio-Economic Impacts of COVID-19 on Household Consumption and Poverty

Data-driven Simulation and Optimization for Covid-19 Exit Strategies

Twit-530 ter and Census Data Analytics to Explore Socioeconomic Factors for Post-COVID-19 Reopening Sentiment

Using random testing to manage a safe exit from the COVID-19 lockdown

The Macroeconomics of Epidemics

An analytical model of covid-19 lockdowns, mimeo LSE

Asaf Cohen, and April Nellis, A Macroeconomic SIR Model for COVID-19

Susceptible-Infected-Recovered (SIR) Dynamics of COVID-19 and Economic Impact

A Simple Plan-555 ning Problem for COVID-19 Lockdown, Working Paper 26981

Optimal Mitigation Policies in a Pandemic: Social Distancing and Working from Home, Working Paper 26984

Internal and External Effects of Social Distancing in a Pandemic

Optimal Targeted Lockdowns in a Multi-Group SIR Model. Working Paper 27102

Optimal quarantine strategies for COVID-19 control models

Sustaining the economy under partial lockdown: A pandemic centric approach

Mor-575 tality containment vs. economics opening: Optimal policies in a SEIARD model

COVID-ABS: An agent-based model of COVID-19 epidemic to simulate health and economic effects of social distancing interventions

An agent-based model for interrelation between COVID-19 outbreak 585 and economic activities

Analysing the combined health, social and economic impacts of the corovanvirus pandemic using agent-based social simulation, Minds and Machines 30

The propagation of economic impacts through supply chains: The case of a mega-city lockdown to prevent the spread of COVID-19

For I m = 0 and parameters β(t), γ(t), η(t) being time independent, the model is called Susceptible-Infectious-Recovered-Deceased (SIRD) model

This website is made by Prof. Wade Fagen-Ulmschneider (UIUC), with data from John Hopkins' Center for Systems Science and Engineering

We can also obtain all time dependent parameters empirically by setting I m = 0 (or some positive constant). β(t) is determined from 1st line of Eq.(1) with input of daily new infection −dS(t)/dt, susceptible ratio S(t), and active cases ratio I(t). η(t) is computed from 2nd line of Eq.(1) with input of daily deaths ratio and active cases ratio I(t)

) with input of daily recovered ratio dR(t)/dt and active cases ratio I(t)

Psychosocial impact of COVID-19, Diabetes and 630 metabolic syndrome

More realistically, shortening the strict lockdown prolongs the duration of the pandemic. This also leads to increase of number of deaths (as can be checked by requiring I m = 0), leading to even larger loss in the simple economic cost function described in the Eq

At the 640 same time I also convinced a group of undergraduate students to switch from solid state physics study club (Steve Simon's book) to the compartmental models in epidemiology

Longitudinal observation and decline of neutralizing antibody responses in the three months fol-645 lowing SARS-CoV-2 infection in humans

A decision-support 650 framework to optimize border control for global outbreak mitigation

Spatially explicit models for exploring COVID-19 lockdown strategies. Transactions in GIS

☒ The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work

This work is done with the financial support from MOST in Taiwan