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Edited by: Matjaž Perc, University of Maribor, Slovenia

Reviewed by: Valerio Capraro, Middlesex University, United Kingdom; Jose A. Cuesta, Universidad Carlos III de Madrid, Spain

This article was submitted to Social Physics, a section of the journal Frontiers in Physics

This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

How long should we self-isolate at home to reduce the chances of a second wave of COVID-19? This is a question that billions of people are wondering early 2020 due to the outbreak of the novel coronavirus SARS-CoV-2. This virus can produce a severe pneumonia that has killed over 230,000 people so far, was detected for the first time late 2019 in Wuhan (China), and has spread all over the world due, in part, to the difficulty of detecting and isolating asymptomatic or mild-symptomatic cases. In this paper, we explore how long suppression strategies (i.e., home confinement and social distancing) must be put into practice in highly populated cities to reduce the chances that a quick rebound of COVID-19 infections occur again over the next months. This is explored, using New York City (USA), San Francisco (USA), and Madrid (Spain) as case studies, through a simple but realistic Monte Carlo stochastic model that takes into account that part of the undetected infected individuals remain in circulation propagating the virus. Our simulations reflect that, if suppression strategies are not properly applied, they can be counterproductive because there are high chances that the confinement time has to be lengthened without reducing the total number of infections. We also estimate that, in the most conservative scenario and under the model assumptions, home confinement is effective if applied at least ~110 days in New York City, ~80 days in San Francisco, and ~70 days in Madrid, i.e., until mid-July 2020, early June 2020, and late May 2020, respectively.

COVID-19 pneumonia, produced by the novel coronavirus SARS-CoV-2, has become a global public threat a few months after several cases were reported late 2019 in Wuhan, China [

The preferred strategy to reduce the impact of COVID-19 is suppression [

We have developed a Monte Carlo stochastic framework to model local viral transmission (

(a) A synthetic population of ^{2} domain, two of which are assumed to be initially infected with SARS-CoV-2.

(b) Healthy individuals become infected if they are within the radius of influence (_{inf}) of virus carriers and if _{1} < _{inf}, where _{1} is a uniform-distribution-generated random number and _{inf} is the probability of infection, which decreases when reducing social interactions (e.g., less social gatherings, hand shaking, hugging, or kissing). Note that the larger the radius of influence and the probability of infection, the faster the disease can spread in the population.

(c) Infected individuals are removed from the domain if _{2} > _{und} (_{2} is a uniform-distribution-generated random number and _{und} is the probability that a person infected with the virus is not detected and thus not removed from circulation (detected virus carriers are assumed to be quarantined and not infect other individuals). Note that _{und} decreases with more severe symptoms and with the availability of accurate tests for early case detection.

(d) If a virus carrier is infected during τ_{shed} days without being detected, it becomes immune and does not have the ability to continue infecting. Note that τ_{shed} represents the period of viral shedding of an infected individual.

(e) The healthy, immune, and undetected infected individuals remaining after applying the previous rules are distributed randomly in the domain to start a new time step (we use 1 day time step).

(f) Rules b-e are repeated until time step τ_{free}, when a certain amount of individuals (chosen randomly among healthy, immune, and infected) are removed from the domain (_{free} represents the duration of free spread of the outbreak, i.e., when no suppression policies are applied.

(g) The system keeps evolving with time, repeating rules b-e, as long as there are virus carriers undetected in the domain (

We use this model to explore three different scenarios, mimicking the evolution of the COVID-19 epidemic in New York City (USA), San Francisco (USA), and Madrid (Spain). Each simulation is repeated 1,000 times, and we export the mean effective confinement time and the mean ratio of total infections (detected and undetected) in terms of the mobility reduction (i.e., the ratio of individuals confined) and the reduction of social interactions. The values used for the different parameters of the model are provided in

Examples of different simulations performed with our Monte Carlo stochastic model. _{und} = 100%), and (ii) no part of the population is confined (duration of free spread of the outbreak, τ_{free} → ∞). _{und} = 85% chance that virus carriers are not detected, (ii) infected individuals detected are removed from the domain, and (iii) no confinement of part of the population is imposed (τ_{free} → ∞). _{und} = 85% chance that virus carriers are not detected, (ii) infected individuals detected are removed from the domain, and (iii) confinement of 60% of the population is imposed after τ_{free} = 50 days of free spread of the outbreak. _{shed} = 20 days; radius of influence, _{inf} = 4 _{inf} = 50%. We use 1 day time step. A Matlab script with the model can be found in

Summary of the values of the model parameters and results.

^{*}^{1} |
^{*}^{6}(%) |
||||||
---|---|---|---|---|---|---|---|

New York (USA) | 11,000 | 8–37 | 21 | 50–100 | 4 | 93 | ~110 |

San Francisco (USA) | 7,300 | 8–37 | 12 | 50–100 | 4 | 92 | ~80 |

Madrid (Spain) | 5,300 | 8–37 | 19 | 50–100 | 4 | 95 | ~70 |

^{1}Number of individuals randomly distributed on a 1 km^{2}. These values reproduce the average population density of the cities explored (we do not use neither actual population numbers nor actual size domains to reduce computational cost).

^{*}^{2}Duration of viral shedding by infected individuals. For each infected individual, we use a random number between 8 and 37 days (this is the range reported in a sample of 191 patients [

^{*}^{3}Time elapsed between the detection of the first case of COVID-19 and shelter-in-place was enforced/encouraged. Shelter-in-place was enforced/encouraged from 23 March 2020 in New York City, 17 March 2020 in San Francisco, and 14 March 2020 in Madrid [

^{*}^{4}Probability of infection (P_{inf}) and probability of not detecting virus carriers (P_{und}). Probabilities are assigned randomly between 50 and 100%, although only values providing exponential growth rates (at the beginning of the outbreak) in the range 0.2–0.4 day^{−1} (consistent with data [

^{*}^{5}Radius of influence of a virus carrier. Values used are consistent with maximum droplet dispersion distances obtained during coughing experiments [

^{6}Mobility reduction is estimated by averaging the Citymapper Mobility Index reported since shelter-in-place was enforced/encouraged and up to April 1, 2020. This index is the ratio of city moving compared to usual, as calculated using trips planned in the Citymapper application [

The main predictions of our Monte Carlo computational experiments are described below, using New York City as example (

Effective days of confinement and total (detected and undetected) people infected as a function of the mobility reduction (i.e., ratio of individuals confined). _{inf}) reduce by 50% once confinement is decreed. _{inf} = 4 _{free} = 21 days. The duration of viral shedding is chosen randomly in the range τ_{shed} = 8−37 days; and initial probability of infection, _{inf}, and probability of not detecting infected individuals, _{und}, are chosen randomly between 50 and 100%. Only combinations of the parameters providing exponential growth rates at the beginning of the outbreak in the range 0.2–0.4 day^{−1} are accepted. These simulations are for the case of New York City.

It is worth highlighting that the results described in the previous paragraph correspond to the end-member scenario in which only confinement is imposed/encouraged, i.e., with no other suppression policies. The other major suppression strategy consists of reducing social interaction through social distancing (e.g., by keeping distance with others, no hand-shaking, no kissing, no hugging, etc.), which this model can account for by assuming that the probability of infection _{inf} decreases once confinement begins at time τ_{free}. In such a case, the model yields three major predictions. First, the maximum of effective isolation time moves toward lower confinement ratios (

Finally, the confinement time that should be applied in the cities of interest (New York, San Francisco, and Madrid) to minimize the chances of a quick second wave of COVID-19 can be inferred if the reduction of mobility (i.e., the ratio of individuals confined) can be constrained. Mobility reduction can be estimated using the Citymapper Mobility Index, as calculated from the trips planned with the Citymapper application [_{inf} (

Effective confinement days for the three cities explored as a function of social interaction (if lockdown conditions are not relaxed). The lines represent the 2-sigma upper limit, i.e., there is, at least, a 95% chance that the confinement time required to reduce the density of infected individuals below one per square kilometer is below the lines depicted. Social interaction is expressed in terms of the percentage of the probability of infection at the beginning of the outbreak (i.e., before suppression strategies apply; note that more social interaction implies higher probability of infection). The values of the parameters of the model are provided in

Our model highlights the importance of applying in combination both home confinement and social distancing to reduce the duration that these strategies need to be applied to minimize the chances of a quick second wave of COVID-19. An important prediction is that, if suppression strategies are not properly applied, they not only are ineffective but they can be indeed counterproductive. In other words, a mild application of the suppression strategies can be worse than no applying suppression strategies at all because there are high chances that they lengthen the effective confinement time without reducing the total number of infections (see, for example, _{shed}); in such a case, the virus can propagate very quickly among the population, thus producing a large number of infections in a very short time period. A mild application of the suppression strategies implies that T and the shedding time are on the same order magnitude; in such a case, it is highly likely that many individuals become infected but when virus carriers are close to the end of their contagious period, thus producing a large number of infections in a long time. Mild suppression strategies can therefore diminish healthcare stress but without necessarily decreasing the number of infections and fatalities. However, a strict application of the suppression strategies implies that T is much larger than the shedding time; in such a case, the chances for asymptomatic virus carriers to infect other individuals reduce drastically, thus producing a low number of infections and the prompt elimination of contagious agents.

Several stochastic modeling approaches have been proposed recently to simulate different aspects of the COVID-19 epidemic [

(a) Scaling. We assume that results obtained on a 1 km^{2} domain are realistic as long as the number of individuals used in the simulations reproduces the average population density of the cities under study. This approach also implies that confinement is considered to be effective when the average density of virus carriers reduces below one per square kilometer.

(b) Closed system. We assume that, in the cities or areas studied, there is no flow of individuals moving in or out from the domain (only those infected individuals that are detected and therefore quarantined and removed from circulation). In such a case, the ratio of infected/healthy individuals is only a function of the interactions in previous time steps. If new individuals (healthy or asymptomatic virus carriers) were imported with time, the effective confinement time would tend to increase.

(c) Person-to-person transmission. We consider that infection occurs predominantly through close contact with virus carriers, which is thought to be the main transmission method [

(d) Population distribution. We assume for simplicity that encounters between different individuals are controlled by a uniform random distribution. More complex random distributions could be incorporated in the model to account for non-uniform population density and for different confinement conditions in distinct neighborhoods; for example, confinement is probably stricter in richer areas because more people is expected to be able to work remotely. However, more complex random distributions would lead to new tuning parameters that are difficult to constrain because it is impossible to know the actual mobility of free individuals. An outcome from our simulations is therefore that suppression strategies may need to be applied longer in poorer, highly populated, neighborhoods with lower mobility restrictions.

(e) Constant mobility restrictions and social interactions. We assume that, once suppression strategies are put into practice, the degree of mobility reduction and social distancing does not change with time. Variations in the degree of applicability of the suppression strategies might lengthen or shorten the effective confinement time, although that effect may not be significant due to its intrinsic uncertainty.

(f) Interaction with confined individuals. We assume that confined individuals no longer interact with the rest of the population and therefore cannot infect nor be infected, i.e., they are considered a second-order factor in the spreading of the disease. This implies that they are assumed to apply extreme social distancing and cleaning habits (e.g., when they go to the supermarket).

(g) Tuning parameters. Our model contains six different parameters, most of which can be constrained based on previous studies (see _{shed}) is chosen randomly (assuming uniform distribution) for each individual in the range 8–37 days (range reported from a sample of 191 patients [_{inf}) and the probability that a virus carrier is not detected (_{und}) are randomly assigned between 50 and 100%. The actual values of these three parameters are not well-known, but we assume that realistic values are those that produce an exponential growth rate at the beginning of the outbreak in the range 0.2–0.4 day^{−1} [

The epidemic of COVID-19 spreads quickly due, in part, to the difficulty of detecting and isolating asymptomatic or mild-symptomatic cases, a factor that must be taken into account to forecast the evolution of the outbreak. This is accounted for in this work, focused on estimating how long suppression strategies (i.e., home confinement and social distancing) must be put into practice in highly populated cities in order to reduce the chances that a quick second wave of COVID-19 cases emerge over the next months. In particular, the questions addressed in this work are: How long should suppression strategies last to be effective, i.e., to avoid quick rebounds in the transmission once interventions are relaxed? How does the effective intervention time depend on the mobility restrictions imposed to the population and social interaction? These questions are addressed through a set of Monte Carlo stochastic simulations, using New York City (USA), San Francisco (USA), and Madrid (Spain) as case studies. Our main conclusions are: (1) If suppression strategies are not properly applied, they not only are ineffective but they can be indeed counterproductive because there are high chances that they lengthen the effective confinement time without reducing the total number of infections. This results from a non-linear interplay between degree of confinement, confinement time, and social distancing. (2) Confinement is effective, beyond the 95% confidence level and under the model assumptions, if it is applied ~110 days in New York City, ~80 days in San Francisco, and ~70 days in Madrid. As a general guide, we conclude that these cities should keep >90% of mobility reduction until, at least, mid-July 2020, early June 2020, and late May 2020, respectively; this would minimize the chances of an uncontrolled resurgence of the disease right after restrictions are alleviated.

The original contributions presented in the study are included in the article/

TG conceived the idea and led the modeling, analysis, and writing of the manuscript.

The author declares that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

California Institute of Technology Government sponsorship acknowledged. The research was carried out at the Jet Propulsion Laboratory, California Institute of Technology, under a contract with the National Aeronautics and Space Administration (80NM0018D0004). The author appreciate discussions on the matters of this paper with R. Zinke, A. Probst, and E. Havazli.

The Supplementary Material for this article can be found online at: