Scheduling mechanisms to control spread of Covid-19 without lockdowns

Gopal Pandurangan, University of Houston

Summary

We present and study mechanisms for reopening economic activities
that explore the trade off between containing the spread of COVID-19 and maximizing economic (or human activity) impact. This is of current importance, as many organizations (including universities, schools, and large businesses), cities, and states are formulating strategies to keep the virus in check while still keeping the economy running without lockdowns.

Our mechanisms, referred to as group scheduling, are based on partitioning the population randomly into groups and scheduling each group on appropriate days with possible gaps (when no one is working and all are quarantined). Each group interacts with no other group and, importantly, any person who is symptomatic in a group is quarantined. Specifically, our mechanisms are characterized by three parameters (g,d,t), where g is the number of groups, d is the number of days a group is continuously scheduled, and t is the gap between cycles. For example, the (2,5,0) mechanism partitions the population into two groups that alternatively work for five days each, and the (2,3,2) mechanism partitions the population into two groups that work alternatively for 3 days each with a gap of two days between the cycles (i.e., every 6 days).

The main findings of our work are:

1. We show that our mechanisms effectively trade off economic activity for more effective control of the COVID-19 virus. This interpolates between full lockdowns where there is very little economic activity and stronger control of the virus and to almost full opening with lot of economic activity but with very little control of the virus.

2 . We show that the (2,5,0) mechanism flatlines the number of COVID-19 cases quite effectively, while still maintaining economic activity at 70% of pre-COVID-19 level. Mechanisms such as (2,3,2) and (3,3,0) achieve even more aggressive control of the virus at the cost of a somewhat lower economic output (about 50%); these could be applicable in situations when the disease spread is more rampant in the population. Depending on the disease spread, one can use an appropriate mechanism that achieves a desired control of the virus at a certain level of economic activity.

3. Our mechanisms can provide a basis for safely reopening large organizations such as universities and schools where such mechanisms can be effectively implemented. We demonstrate the efficacy of our mechanisms by theoretical analysis and extensive experimental simulations on various epidemiological models. Our results show that both the peak number of cases per day as well as the total number of infections can be significantly controlled by following appropriate mechanisms.

4. Our mechanisms prove beneficial just by regulating human interaction.
Moreover, our results show that if the disease transmission rate (or reproductive rate) is made lower by following social distancing, mask wearing, and other public health guidelines, it can further increase the efficacy of our mechanisms.

This article is a summary of a paper posted at arXiv written with co-authors Khalid Hourani and Adi Pasic (University of Houston), John Augustine (IIT, Madras) and Anisur Rahaman Molla (ISI, Kolkata).

Introduction

The COVID-19 pandemic that is sweeping the world currently has already spread to a large number of people. In many parts of the world it has already infected a significant fraction of the population. For example, in New York City, antibody testing suggests that as many as quarter of the population could have been infected by June 2020. However, this is still nowhere near the fraction required for herd immunity. Many states have already (at least partially) reopened their businesses. Large organizations such as universities and schools are also considering reopening (or already partially opening) in the Fall of 2020 and beyond. In many instances, there have been closures of schools and universities and businesses after reopening due to spike in cases. In many places, after a fall in cases for a while, there is a surge, leading again to lockdowns and closures and the cycle keeps repeating. Thus one needs effective strategies to reopen and to keep opened organizations functioning safely for a longer time, by keeping the virus under check. Such strategies may significantly help in containing, controlling, and slowing the spread of COVID-19, even though it may not fully eliminate it. This will be a “new normal’’ wherein we learn to live with the virus, while simultaneously keeping it under control.

The goal of our work is to study intervention mechanisms that can help in safely reopening society. Our main contribution is the study of a class of simple intervention mechanisms called group scheduling. The inspiration for group scheduling comes from COVID-19 characteristics whereby individuals remain asymptomatic and (possibly) less infectious for around 3–5 days from contraction. Subsequently, they either turn symptomatic (and can therefore be quarantined) or remain asymptomatic (and still spread the disease). Group scheduling randomly partitions the population (say, the students in a university or school, or the work force in a company or organization) and schedules them (i.e., allows them to work) on different days with possible gaps (i.e., when no group is scheduled). A group is considered quarantined (at home, say) when it is not scheduled to work. Any individual who is symptomatic is quarantined as soon as he/she exhibits symptoms.

The key intuition that inspires group scheduling is that individuals can be grouped — hence reducing the number of contacts (on average) than without grouping — and scheduled to work predominantly with other less infectious individuals. Effective group schedules work in such a way that most infected individuals turn symptomatic (though a significant percentage, about 40%, may remain asymptomatic) and contagious during their break — thereby facilitating quarantining before spreading the infection. On the flip side, instead of a full lockdown, our scheduling allows for significant and sustained economic activity. In fact, we showcase specific group schedules that operate at 70% of the economic activity compared to a typical five-day work week that can simultaneously dampen the spread of COVID-19 quite effectively.

Our results indicate three specific categories of mechanisms — corresponding to high, medium, and low economic activity — and their performance in controlling COVID-19. The lower the economic activity of the mechanism, the higher the control of disease. A main takeaway from our results is that group scheduling mechanisms help in significantly controlling the spread of COVID-19 — reducing the peak number of cases per day as well as the total number of infections. Depending on the rate of infection in a population, different mechanisms are applicable that control the disease spread while maintaining an appropriate level of economic activity.

Group Scheduling Mechanisms

Group scheduling partitions the population into different (randomly chosen) groups and schedules each group on different days with possible gaps between the schedules. A gap is when no group is scheduled. More precisely, a group scheduling mechanism is characterized by three parameters g,d, and t, where g is the number of groups, d is number of days a group is continuously scheduled, and t is the gap between the schedules. We call this a (g,d,t) schedule. The normal five day work week schedule is (1,5,2), where the entire workforce is just one group scheduled continuously for five days with two days off. A quintessential group schedule example that we highlight is the (2,5,0) schedule ((2,4,0) also has quite similar performance with the same economic output). It partitions individuals into two groups scheduled alternatively for five days each without any break between cycles. Thus, individuals cycle between five days of work and five days quarantined at home.

We evaluate our mechanisms in two main aspects:

1. How disease spreads under the mechanism and comparing it with simple baseline mechanisms. In particular, we posit the so-called flattening ratio which is the ratio of the peak number of cases (during the course of the epidemic simulation) under our mechanism compared to the peak number of cases under a baseline mechanism, e.g., the normal 5-day work week.

2. The economic output of the mechanism characterized by what we call as the work output or economic ratio defined as the ratio of the average number of working hours of an individual under our mechanism to the average number of working hours under the standard five days, 8 hours per day, working week (i.e., the (1,5,2) schedule). For a mechanism with parameters (g,d,t), the economic ratio is computed by the formula: 7d/(5(gd+t)). The economic ratio for a normal work week, i.e., the (1,5,2) schedule, is 100%, whereas for the (2,4,0) (and (2,5,0)) schedule it is 70%.

There are two advantages in group scheduling:

(1) The average number of contacts per group is reduced by a factor of 1/g (compared to a single group) which means that less individuals are infected by an infected person on average per day;

(2) Since infected symptomatic individuals in a group are quarantined when they are not scheduled, this means that the number of days a person is infectious is reduced. Thus even when the number of groups is relatively small (say 2, 3, or 4) and even for small d and t values, the spread of disease is significantly reduced, while still keeping the economic ratio reasonable.

Summary of Results

We demonstrate the efficacy of our mechanisms both theoretically and experimentally. For our theoretical analysis, we consider a branching process model and analyze how the disease spreads as a function of the mechanism parameters (g,d, and t) and the COVID-19 disease parameters. Our analysis is fruitful in determining what mechanisms are likely to work well. We also conduct extensive simulations of our mechanisms under various epidemiological models and compare their performance with baseline mechanisms. (We refer to full paper for detailed results.)

We show that our mechanisms gives qualitatively similar benefits regardless of the specific models used in our simulations as well as the specific choice of parameters of the respective models. For example, given a particular mechanism, say (2,4,0), we simulate the disease spread under this mechanism under various epidemiological models. Under each model, we vary the following parameters to study their effects. We vary transmission probability Tp (which captures the rate of infection spread in the population), the number of index patients (which captures the percentage of individuals currently infected among the population), and the percentage of asymptomatic carriers (we assume this to be 40% according to current estimates, but we also try other values). We then compare the disease spread, under the same set of respective parameters, to three baseline mechanisms —the basic model (where the disease spreads without any intervention), symptomatic quarantine or the (1,1,0) schedule (where infected individuals are quarantined after exhibiting symptoms), and the (1,5,2) schedule, which is the normal 5-day work week with symptomatic quarantine (note that in the latter two mechanisms there is only one group). The key metric of comparison is the flattening ratio — the ratio of the peak number of cases of the mechanism under consideration (say (2,4,0)) to that of the baseline mechanisms. (We also compare the total number of infections as well.)

We analyze a number of group scheduling mechanisms that showcase the trade off between economic ratio and the disease spread. We categorize them into three broad groups as:
1. High-ER mechanisms (70% economic ratio),
2. Mid-ER mechanisms (40–50%), and
3. Low-ER mechanisms (about 30%).

We summarize our results below. These particular results assume a population of 50,000 (typical population in a large university) and the number of index patients is 3% of the population (i.e., 3% of the population is initially infected as is currently estimated in Harris County, TX). We have also simulated higher populations (up to a million) and have varied the number of index patients. More importantly, we have analyzed a wide variety of Tp values — the higher the Tp value, the higher the reproductive rate of the virus.

The canonical example for the high-ER category is the (2, 5, 0) schedule, which achieves a (1,5,2) flattening ratio as low as 12% (i.e., the ratio of the peak number of cases is 12% compared to that of the standard work week schedule) even when Tp is 0.1 (corresponding to a high reproductive rate). When Tp is lower, say around 0.01 the flattening ratio becomes much lower. (In general, lower the Tp, the better the flattening, in general, for any given mechanism.) In fact, several mechanisms of the form (2,d,0) yield the same economic ratio and essentially the same flattening ratio for d ≥ 4. In the mid-ER category, we get even better flattening ratios. The canonical example for this case is (3,3,0) mechanism yielding an economic ratio of 46%, but with the flattening ratio down to 4% even under Tp = 0.1. For mechanisms in low-ER category, such as (4,4,0) with still a reasonable economic ratio of 35%, the flattening is down to about 1%.

The mid-ER and low-ER mechanisms are attractive when Tp values are high, as they not only lead to a low flattening ration, but also low number of peak cases (in absolute numbers) with respect to the total population. Moreover, the total number of infections is a small fraction of the total population and the infection dies off quickly in the population.

Discussion

Our results crucially imply the existence of mechanisms that can trade off between their effectiveness in damping the spread of COVID-19 and the economic ratio. Our mechanisms take a principled approach to disease control that interpolates between extreme measures — lockdowns on one hand that severely cripple the economy and a “herd immunity” approach that advocates normal behavior for most people (except the most vulnerable). The latter approach, though it helps economic activity, has the danger that even younger or middle-aged people who apparently are less vulnerable can still get the disease in severe form (and even die), and this can happen in large numbers, overwhelming the health care system.

An important point to note is that if the transmission probability (reproductive rate) is smaller, then the efficacy of group scheduling is increased even further. Thus reducing the transmission probability by following public health guidelines like wearing masks, social distancing, and hand washing will be very beneficial.

The mechanisms we study are elementary and can be easily coupled with other policies such as social distancing and wearing face masks that can further reduce the spread of COVID-19. For convenience, we have categorized our mechanisms based on their economic ratios. The low end of this spectrum provides the best flattening ratio. So naturally, when cases surge (characterized by a higher transmission probability or reproductive rate) or when easing out of complete lockdown, we may wish to opt for the low economic ratio options that have an improved flattening ratio. As case numbers decrease, we have a couple of strategies that we can choose between. On the one hand, low case numbers afford us the ability to contact trace more effectively and thereby stop the spread. On the other hand, we can also move to mechanisms with improved economic ratio. As demonstrated by our results, if the transmission probability is reduced, then the peak number of cases (as well as the total number of cases, in many mechanisms) goes down significantly. Following social distancing and mask wearing and other public health guidelines, reduces the transmission probability and thus increases the efficacy of the mechanisms, allowing deployment of high-ER mechanisms as well.

We are guided by two main principles. Firstly, our mechanisms — by partitioning the population into groups — reduce the number of contacts per person. Secondly, we schedule employees to with sufficient breaks so that a large number of infected people become symptomatic (and therefore quarantined) during their breaks.

We believe that policy makers can be guided by these principles, taking other local considerations into account. In general, a policy A that (with respect to another policy B) either decreases the number of contacts and/or improves the probability that people become symptomatic during their break will dominate over B and lead to better flattening. This means that adding common work-breaks — e.g., Sundays off — is likely to improve the flattening ratio and unlikely to worsen it. Policy makers can use the above two principles to address scenarios that we have not addressed directly. Shift workers, for example, may need to work on a more fine-grained schedule. Consider a shift schedule that requires two shifts per day. We could consider four groups with each alternating between four working days and four off days. The groups may be staggered so that — on any given day — two groups are working and two are off.