Modeling COVID-19 Nonpharmaceutical Interventions: Exploring periodic NPI strategies.

We developed a COVID-19 transmission model used as part of RAND's web-based COVID-19 decision support tool that compares the effects of nonpharmaceutical public health interventions (NPIs) on health and economic outcomes. An interdisciplinary approach informed the selection and use of multiple NPIs, combining quantitative modeling of the health/economic impacts of interventions with qualitative assessments of other important considerations (e.g., cost, ease of implementation, equity). This paper provides further details of our model, describes extensions, presents sensitivity analyses, and analyzes strategies that periodically switch between a base NPI level and a higher NPI level. We find that a periodic strategy, if implemented with perfect compliance, could have produced similar health outcomes as static strategies but might have produced better outcomes when considering other measures of social welfare. Our findings suggest that there are opportunities to shape the tradeoffs between economic and health outcomes by carefully evaluating a more comprehensive range of reopening policies.

Introduction

Coronavirus disease 2019 (COVID-19) is unprecedented in terms of scale and speed, affecting millions worldwide. Until recently, vaccines and effective treatments for COVID-19 were unavailable. National leaders have had to take extraordinary measures to mitigate the virus’s spread and prevent health care systems from being overwhelmed. Policymakers have implemented a range of nonpharmaceutical public health interventions (NPIs). These interventions include partial closings (e.g., schools and non-essential businesses, prohibiting large gatherings, quarantining the most vulnerable) and complete lockdown (e.g., placing all residents under stay-at-home orders). The goal of NPIs is to delay and reduce the peak number of cases per day, reduce pressure on health services, and allow time for vaccines to be distributions [1]. If NPIs are relaxed too soon a new wave of infections may occcur. However, NPIs have wide-ranging effects on the health, economy, and social well-being of populations, which has led to growing pandemic fatigue and a decline in adherence to NPIs since they were first initiated [2, 3]. Decision-makers are faced with tough decisions, such as how to sequence, relax, and possibly reinstate mitigation measures. Exacerbating these decisions are significant uncertainties, including new variants and behavioral responses to extended interventions.

Mathematical and simulation models of COVID-19 transmission dynamics are invaluable tools to help decision-makers forecast and compare intervention outcomes, predict the timing of peaks in cases and deaths, medical supply needs, and if and when we should expect additional waves. They enable projection and comparison of population-level outcomes over hypothetical scenarios. Model outcomes include the incidence and prevalence of the infection over time and for different population groups. The hypothetical scenarios can consist of the impact of varying pharmaceutical and nonpharmaceutical public health interventions, distributing vaccines, and the emergence of new strains.

We developed a COVID-19 transmission Population-Based Model (PBM) used as part of a web-based COVID-19 decision support tool that compares the effects of different nonpharmaceutical public health interventions (NPIs) on health and economic outcomes. An interdisciplinary approach informed the selection NPI portfolios, combining quantitative modeling of the health/economic impacts with qualitative assessments of cost, ease of implementation and equity. An in-depth description of our approach was previously published as a RAND report describing how the PBM, the economic model, and a systematic assessment of NPIs informed the web-tool [4].

We expanded our original model [4] to account additional uncertainties and consider an expanded set of NPI strategies. In this paper, we consider periodic NPI strategies. Recent research has demonstrated that high-frequency periodic NPIs [5] have the potential to mitigate COVID-19 resurgences while providing more predictability and alleviating the damaging effects on economic activity and social well-being. We use our updated model to explore if a periodic strategy could have provided benefits compared to fixed strategies that would keep R close to one. We find that a periodic strategy can dominated fixed strategies, improving both health and days spent under restrictions.

The paper is structured as follows. First, we provide an overview of our model structure. Then, we briefly analyze a set of illustrative scenarios, including periodic and fixed strategies, identifying if the periodic strategies used by other modelers modelers [5] produce similar results in our model. Finally, we provide detailed information on our mathematical model and present sensitivity analyses.

Model Overview

Theory-based epidemiological models use a theoretical understanding of biological and social processes to represent a disease’s clinical and epidemiological course. The most typical model considers the population in four different disease states: susceptible, exposed, infected, and Removed (SEIR). Our PBM incorporates several extensions to the SEIR model of disease transmission. It is formulated deterministically by coupled ODEs and integrated numerically by solvers for stiff problems [6-9]. We extend the SEIR framework to better describe COVID-19 transmission by adding additional disease states and considering population strata based on age and chronic conditions. The PBM models the effects of different NPIs on health outcomes, from partial closings to complete lockdown. Unlike many other COVID-19 models, we simulate the impact of NPIs on different mixing modes (such as home, school, and work) separately, allowing us to model various interventions flexibly. Our PBM also includes population strata and specify mixing pattern heterogeneities across the population strata and for each mode. These heterogeneities included in our model allow us to set the NPI more specifically, with mixing rates reduced deferentially by mixing mode. Our model is designed to help policymakers understand the effects of NPIs, weigh tradeoffs among them, make decisions about which NPIs to implement, and estimate how long-term interventions should be enforced to control the virus.

Figure 1 shows a simplified illustration of the disease states included in the first basic version of our PBM. The model includes a pre-symptomatic highly infectious state, which leads to either an asymptomatic state or a state with mild symptoms, a fraction of which continue to severe disease. Most of those who develop severe symptoms are hospitalized. Non-hospitalized severely-symptomatic either recover or die. The hospitalized state includes compartments for both the main hospital and the ICU, where individuals are admitted if they develop critical symptoms. Capacities can be set for the hospital and ICU beyond which no additional patients can be added. At each of the infectious states, individuals can be tested for COVID-19. Each compartment is composed of ten population strata, five age groups, and two health states (those with and without at least one chronic condition). These strata allow the model to simulate how the disease impacts different population groups, including differences in population size, mixing mortality rate, and the proportion who are asymptomatic. Disease progression rates are based on figures given in the literature and are sampled from distributions if uncertain. The force of infection (the rate that susceptible individuals become exposed) is characterized by how many infectious people are in each state, the transmissibility of each state, and levels of mixing. We estimate transmissibilities for each state based on biological and social factors. For instance, viral loads are highest early in the disease [10], so these states have higher biological transmissibility. We assume that those who receive positive tests or exhibit symptoms have lower social transmissibility because they take measures to limit the exposure of others. The total transmissibility of a state is the product of the biological and social transmissibilities, and the population-weighted sum of transmission is proportional to the number of new infections.

Figure 1:

COVID-19 PBM disease states

In our model, NPIs are portfolios of restrictions mandated at the state level, as described in table 1. The set of NPIs levels used by each state is characterized by a discrete set of intervention levels ranging from 1 (no intervention) to 6 (close schools, bars, restaurants, and nonessential businesses; and issue a shelter-in-place order for everyone but essential workers). Each intervention level is associated with mixing matrices that describe how strata interact with each other in six different settings: household, work, school, commercial, recreation, and other. Interventions are modeled as changing the level of mixing which occurs in each of these settings. For instance, closing schools reduces school and work mixing but increases home mixing. Given the specified model structure, the NPI time-series, and the mixing matrices, we calibrate our model for each state using deaths time series. Appendix A provides a detailed description of the mathematical formulation of our model.

Table 1:

Nonpharmaceutical intervention levels.

NPI Level	Description
Level 1: No Intervention	No Intervention
Level 2: Close schools	All schools are closed.
Level 3: Close schools, bars, and restaurants; and ban large events	In addition to school closures, all bars’ and restaurants’ dine-in services are closed, only allowing for take-out options. Also, large gatherings are banned.
Level 4: Close schools, bars, and restaurants; ban large events; and close nonessential businesses	In addition to school, bar, and restaurant closures, all nonessential businesses are closed.
Level 5: Close schools, bars, and restaurants; ban large events; close nonessential businesses; and shelter in place for the most vulnerable	In addition to the closure of all nonessential businesses, a shelter in place recommended for the vulnerable population, including the elderly, children, and other at-risk populations.
Level 6: Close schools, bars, and restaurants; ban large events; close nonessential businesses; and shelter in place for everyone but essential workers	In addition to the interventions above, shelter in place order is issued for the everyone but essential workers.

Exploring Periodic NPI Strategies

In this section, we present an illustrative retrospective analysis of policies that can be tested with the model, using the state of California as an example. These analyses and findings do not constitute predictions. Yet, they illustrate that alternative plausible NPI strategies could have produced improved outcomes during 2020, in the absence of vaccines. The purpose of this analysis is two-fold. First, it demonstrates how our model can be used to trace many-objective trade-off curves to support the analysis of reopening strategies. Second, this analysis demonstrates that a periodic switching of NPIs might have placed society in a better position in these trade-off curves - that is, it could have represented a pareto-improvement.

This section explores two types of strategies that could have been followed to manage NPIs in the year 2020. The first set of strategies are “fixed” NPI levels. This type of strategy holds the NPI level constant over time. Although this strategy has not been followed in California explicitly, the NPI mandates imposed in California are best approximated in our model by setting the NPI level to three. This NPI level was stable between July of 2020 through the end of the year, and this represents our baseline scenario and is the scenario used to calibrate our model.

Fixed NPIs are not, however, the only way to control the pandemic. Alternatively, one could use periodic strategies to curb transmission [5]. A periodic NPI can represent a strategy wherein society goes into more severe periods of NPIs then relaxes to lower levels of stringency. This strategy’s rationale is that those newly infected during the relaxed periods would take a few days before becoming infectious themselves. The enforcement of stringent NPIs would then limit the time and possibilities for the virus to spread further from these infectious individuals during the time before they either recover or are hospitalized. This periodical switching would systematically reduce transmissions and force the dynamics of the epidemics to be controlled by the strategy’s frequency.

In essence, a periodic strategy uses the natural timescales of disease progression and infectivity to induce a synchronization phase that helps align the times when people are more likely to be infected together, allowing for the social distancing NPIs to be more effective. An example of a similar strategy includes schools adopting a hybrid learning model wherein students go to school every other week. Similarly, restaurants could open for indoor dining periodically. In absence of vaccines, such policies may be desirable as they would provide stability, regularity, and increased predictability for businesses to plan against. The policies could, in principle, help suppress the transmission of the virus and simultaneously reduce uncertainties in economic activity.

Figure 2 illustrates the dynamics of periodic and fixed NPI strategies. The fixed NPI strategies represented in the figure suggest that under the NPI level three, R closely followed one and increased towards the end of December, driven by our model’s seasonal effect. Because R was close to one in the model, a departure from the current NPI level would be expected to produce a significant departure from the R = 1. Therefore, a policy that reopens the state (F-1) would be expected to produce a spike in prevalence and subsequently in the number of deaths, and a more stringent, constant policy (F-5) would be expected to reduce the number of deaths.

Figure 2:

Model Dynamics with Fixed and Periodic NPIs.

Fixed NPI strategies are coded with an “F” followed by the intervention level included in the NPI. Periodic NPI strategies are coded with a “P” followed by the maximum NPI level in the periodic strategy and the period in days. In that case, P-3–14 means that the state will switch between the NPI level 3 and 1 every 14 days.

As figure 2 shows, the periodic strategy P-5–14 switches between NPI levels 1 and 5 every two weeks. This switching causes R to oscillate such that prevalence does not increase unchecked. As a consequence, the number of deaths is controlled. The choice of two weeks is based on the typical timescale describing the disease progression for the majority of infected people. However, other choices for the periodicity could be explored.

When judging alternative strategies, policymakers often have to weigh multiple criteria to make decisions, so one needs to translate model outcomes to meaningful criteria. One criterion could be the number of days of school closures, which has been an important concern during the COVID-19 pandemic. However, the number of days of school closures does not distinguish scenarios in which non-essential businesses are closed for long periods, so other proxies for welfare loss are needed. One approach could be to use weights for each NPI level, such that those weights are proportional to the marginal daily welfare loss induced by each NPI level. If one defines those weights such that one day under lockdown is equivalent to one, and one day under no restrictions is set to zero, one can compute a proxy to social welfare that can be used to judge alternative strategies. Our weighted lockdown days metric corresponds to this criterion.

There are other plausible ways to compute NPI costs. NPIs arguably induce income loss. Our third metric uses the income loss under each NPI level estimated by a general equilibrium model [11] to account for the economic consequences of NPI restrictions. Although all these proxies are imperfect measures of social welfare loss induced by NPIs, our conclusions do not rely on their precision, but on the assumption that NPI costs are increasing in the level of restriction. This structural assumption allows us to illuminate trade-offs and reveal pareto-dominated strategies that rely on the structure of the epidemiological model[1].

Figure 3 demonstrates that using a small set of alternative measures can support those decisions and reveal pareto-dominated strategies. Strategy F-3, our baseline strategy, is pareto-dominated by a wide range of strategies that oscillate between the NPI level of 5 and 1, using many periods, which is in line with prior research [5].

Figure 3:

Tradeoff surface implied by alternative policies.

The vertical axis presents the number of Deaths / 100k at the end of the simulation run (Feb. 2021) in California, and the horizontal axis contains several proxies that represent alternative criteria to evaluate the costs of NPIs. Across all these criteria, we find that periodic NPIs tend to dominate fixed NPIs. The plot demonstrates that strategies with fixed NPIs generally are dominated by periodic NPI strategies.

However, there are limitations to our analysis. First, we do not consider practicality: these periodic strategies might be regarded as unfeasible, impractical, or undesirable by policymakers and the public. This consideration is particularly important because the strength of the periodic NPIs relies on the ability to abruptly reduced mixing, which can only be achieved with a high level of compliance. Further, people are may to shift their mixing to the open periods reducing or even canceling the mitigation effects on transmission intended by the periodic NPI policy. Nevertheless, we use them in this paper as an example to illustrate that alternative NPIs strategies, if implemented with high levels of compliance, might pareto-dominate fixed strategies and might shift the trade-off surface among health and economic/social outcomes.

Conclusion

The scenarios presented in this paper serve the purpose to illustrate that alternative policies using periodic NPIs to manage the COVID-19 pandemic might have produced the same health outcomes while allowing essential activities, such as in-person education, to have happened in a controlled manner. While we do not advocate for any particular strategy, this brief analysis demonstrated that alternative NPI strategies have the potential to shift the trade-off curves among the relevant outcomes.

Including social welfare loss measures induced by NPIs in analyses seeking to inform COVID-19 reopening decisions is essential. Only including health outcomes in those analyses leaves the task of weighing other concerns to the policymaker, who may or may not be able to do so consistently. Metrics of welfare loss induced by NPIs can be either derived directly from the model outcomes (e.g., days of school closures), or use a weighted sum based on the NPIs stringency level, potentially using economic models in our prior work[4]. Even if the analysis estimates are not precise or could become less precise with time, they can still be useful. As figure 3 demonstrated, if one ignores all the horizontal axes under the argument that those estimates might be imprecise, policymakers might not be properly informed that alternative policies dominate some policies. This statement and the pattern seen in the trade-off curves do not rely on the precision of economic estimates but on the theory-based epidemiological model structure.

The trade-off curves we present also should not be seen as static. Many other factors that have been held constant in our analysis might also shift this curve. Widespread adoption of high-quality masks, for example, would shift every point inwards, making society systematically better off. The emergence of new, more transmissible variant strains can shift the curve outwards. A more stringent strategy to eliminate community spread and prevent re-seeding (such as New Zealand’s strategy) can remove the health-economic trade-offs curve completely if successfully implemented. Adaptation measures to prevent transmission within schools would also shift this curve, strengthening periodic strategies even more attractive to allow in-person education. Moreover, the introduction of vaccines also shapes this trade-off curve over time. As vaccination roll-out advances, the marginal benefit of an additional day under stringent NPIs will decrease. Accounting for the uncertainties mentioned before and the vaccination dynamics will be essential to guide society to a new normal through a robust reopening plan.

Table 2:

Disease states included in the model. The dependence on time t is implicitly assumed.

Disease State X	Description	Infectious XI	Diagnosed
S	Noninfected and susceptible.	No	No
E	Exposed and infected but not yet infectious.	No	No
P	Presymptomatic infectious.	Yes	No
I Sm	Nondiagnosed Infected with mild symptoms.	Yes	No
I Ss	Nondiagnosed infected with severe symptoms.	Yes	No
Y Sm	Diagnosed infected with mild symptoms.	Yes	Yes
Y Ss	Diagnosed infected with severe symptoms.	Yes	Yes
H	Hospitalized not in the intensive-care unit.	Yes	Yes
H ICU	Hospitalized in the intensive-care unit.	Yes	Yes
I A	Nondiagnosed infected asymptomatic.	Yes	No
Y A	Diagnosed infected asymptomatic.	Yes	Yes
R s	Recovered who were symptomatic.	No	Yes & No
R a	Recovered who were asymptomatic.	No	Yes & No
D	Those who have died.	No	Yes & No

Table 3:

Model strata and corresponding populations

Non-aggregated Strata	US Unique Population (millions)	Aggregated Strata
Young	73.4	Young
FLEW without high-risk conditions	36.0	FLEW
Employed, but not FLEW, without high-risk conditions	60.0	Working age
Not employed without high-risk conditions	33.2	Working age
Old without high-risk conditions	11.8	Old
FLEW with high-risk conditions	14.1	FLEW
Employed, but not FLEW, with high-risk conditions	37.6	Working age with high-risk conditions
Not employed with high-risk conditions	20.8	Working age with high-risk conditions
Old with high-risk conditions	37.6	Old

Table 4:

Disease duration parameter estimates

Parameter	Mode	Sample Range	Sample Dist	Formula	Sources
Duration in days of incubation phase	5	4 – 6	PERT	1ν+1γ_A+γ_S	[32, 33, 35, 39]
Proportion of incubation phase which is non-infectious	60%	50 – 70%	PERT	γ_A+γ_Sγ_A+γ_S+ν	[32, 40]
Infectious duration in days of asymptomatic and mild disease	5	4 – 7	PERT	1ξ_A	[32–36]
Expected days spent in hospital (including ICU) at hospitalization	8	6 – 10	PERT	1ξ_H+μ_H+A_ICUχ×(1+χξ_ICU+μ_ICU)	[35, 37]
Expected days spent in the ICU at ICU admission as a proportion of expected days spent in the hospital at hospitalization	90%	80 – 100%	PERT	ξ_H+μ_H+A_ICUχξ_ICU+μ_ICU+χ	[35, 37]
Months before loss of natural immunity	20	10 – 40%	PERT	ρ	[41]

Table 5:

Disease prognosis parameter estimates

Parameter	Mode	Sample Range	Sample Dist	Formula	Sources
Proportion of infections which are asymptomatic	25%	15 – 50%	PERT	γ_Aγ_A+γ_S	[35, 42–44]
Proportion of symptomatic infections which are severe (require hospitalization)	5%	3 – 7%	Uniform	υυ+ξ_m+μ_S	See Section B.2.2
Proportion of severe cases which are critical (require ICU admission)	32%	26 – 38%	PERT	χχ+ξ_H+μ_H	[35, 38]
Initial proportion of critical cases which result in death	75%	70 – 80%	PERT	μ_ICUI_ICU+μ_ICU	[38, 47, 48]
Reduction in ICU death proportion as treatment improves	50%	40 – 60%	PERT	N/A	[47]
Time period over which treatment improves (months)	6	4.8 – 7.2	PERT	N/A	[47]

Table 6:

Relative infectivity parameter estimates

Infectivity of phase relative to asymptomatic phase	Mode	Sample Range	Sample Dist	Sources
Pre-symptomatic infectious	2.07	1.65 – 2.48	PERT	[10, 31, 35, 50]
Mild symptomatic	0.83	0.66 – 1.00	PERT	[10, 31, 35, 50]
Severe symptomatic	0.14	0.11 – 0.17	PERT	[10, 31, 35, 50]
Hospitalized	0.07	0.06 – 0.08	PERT	[10, 31, 35, 50]
Tested mild symptomatic	0.28	0.22 – 0.33	PERT	[10, 31, 35, 50]
Tested severe symptomatic	0.14	0.11 – 0.17	PERT	[10, 31, 35, 50]
Tested asymptomatic	0.20	0.16 – 0.24	PERT	[10, 31, 35, 50]

Table 7:

Other parameter estimates

Parameter	Mode	Sample Range	Sample Dist	Formula	Sources
Per person daily detection rate for those not yet exposed	0.002	0.001 – 0.004	PERT	ζ	See Section A.3
Per person daily detection rate for those who are asymptomatic	0.01	0.005 – 0.04	PERT	ζ A	See Section A.3
Per person daily detection rate for those who are symptomatic	0.2	0.1 – 0.8	PERT	ζ S	See Section A.3
Impact of seasonality	0.2	0.15 – 0.3	PERT	s	See Section A.11
Increased progression rate for tested individuals	2	1.33 – 4	PERT	Κ ξ	See Section A.1
Effectiveness of NPIs	2	0.5 – 3.5	PERT	θ	See Section A.9

Table 8:

Shortened parameter labels of the most significant parameters found by the sensitivity analyses.

Parameter	Denoted
Proportion of incubation phase which is non-infectious	Proportion Non- infectious Incubation
Proportion of symptomatic infections which are severe (require hospitalization)	Proportion Severe
Proportion of severe cases which are critical (require ICU admission)	Proportion Critical
Proportion of infections which are asymptomatic	Proportion Asymptomatic
Initial proportion of critical cases which result in death	Proportion die in the ICU
Duration of incubation phase	Incubation Days
Expected time spent in hospital (including ICU) at hospitalization	Hospitalized Days
Infectious duration of asymptomatic and mild disease	Symptomatic Mild Days
Per person daily detection rate for those who are asymptomatic	zetaA
Per person daily detection rate for those who are symptomatic	zetaS
Infectivity of phase of the mild symptomatic relative to asymptomatic phase	m.Sm
Infectivity of phase of the severe symptomatic relative to asymptomatic phase	m.Ss