Disease-economy trade-offs under alternative epidemic control strategies

Here we provide an overview of the key elements of our framework including describing the contact function that links economic activities to contacts, the SIRD (Susceptible-Infectious-Recovered-Dead) model, the dynamic economic model governing choices, and calibration. The core of our approach is a dynamic optimization model of individual behavior coupled with an SIRD model of infectious disease spread. Additional details are found in the SI.

Contact function

We model daily contacts as a function of economic activities (labor supply, measured in hours, and consumption demand, measured in dollars) creating a detailed mapping between contacts and economic activities. For example, all else equal, if a susceptible individual reduces their labor supply from 8 to 4 h, they reduce their daily contacts at work from 7.5 to 3.75. Epidemiological data is central to calibrating this mapping between epidemiology and economic behavior. Intuitively, the calibration involves calculating the mean number of disease-transmitting contacts occurring at the start of the epidemic and linking it to the number of dollars spent on consumption and hours of labor supplied before the recession begins.

We use an SIRD transmission framework to simulate SARS-CoV-2 transmission for a population of 331 million interacting agents. This is supported by several studies (e.g.,^77,78) that identify infectiousness prior to symptom onset. We consider three health types m ∈ {S, I, R} for individuals, corresponding to epidemiological compartments of susceptible (S), infectious (I), and recovered (R). Individuals of health type m engage in various economic activities ({A}_{i}^{m}), with i denoting the activities modeled. One of the ({A}_{i}^{m}) is assumed to represent unavoidable other non-economic activities, such as sleeping and commuting, which occur during the hours of the day not used for economic activities (see SI 2.3.1). Disease dynamics are driven by contacts between susceptible and infectious types, where the number of susceptible-infectious contacts per person is given by the following linear equation:

while similar in several respects to prior epi-econ models^15,16,74, a methodological contribution is that ρ_i converts hours worked and dollars spent into contacts. For example, ρ_c has units of contacts per squared dollar spent at consumption activities, while ρ_l has units of contacts per squared hour worked.

We also consider robustness to different functional forms in Fig. 6F, G as a reduced-form way to consider multiple consumption and labor activities with heterogeneous contact rates. Formally:

where α > 1 (convex) corresponds to a contact function where higher-contact activities are easiest to reduce or individuals with more contacts are easier to isolate. α < 1 (concave) corresponds to a contact function where higher-contact activities are hardest to reduce or individuals with fewer contacts are easier to isolate. The baseline case (α = 1) implies all consumption or labor activities and individuals have identical contact rates (See SI 2.3.2 for further discussion and intuition).

Calibrating contacts

To calibrate the contact function, we use US-specific age and location contact matrices generated in ref. ⁶, which provide projected age-specific contact rates at different locations in 2017 (shown in SI section 2.3.1). We group these location-specific contact matrices into matrices for contacts during consumption, labor, and unavoidable other activities. The transmission rate was calibrated to give a value of ({{{{{{{{mathcal{R}}}}}}}}}_{0}) = 2.6, reflective of estimates⁷⁹. For this, we use the next-generation matrix⁴⁰. The next-generation matrix describes the “next generation” of infections caused by a single infected individual; the ({{{{{{{{mathcal{R}}}}}}}}}_{0}) is the dominant eigenvalue of the next-generation matrix (see SI 2.5.3). This calculation is done at the disease-free steady state of the epidemiological dynamical system, when all the population is susceptible. Specifically, we calculate the benchmark number of contacts from each activity in the pre-epidemic equilibrium (e.g., ρ_cc^Sc^I for consumption from equation (1)), under pre-epidemic consumption and labor supply levels. We then calculate the coefficients ρ_c, ρ_l, ρ_o (for consumption, labor, unavoidable other) using (1) such that pre-epidemic consumption and labor supply levels equal the benchmark number of contacts. To account for contacts that are not related to economic activities, the “unavoidable other” contact category is normalized to 1, so that the coefficient ρ_o is simply the number of contacts associated with unavoidable other activities. While pre-pandemic contact structures are necessary to calibrate ({{{{{{{{mathcal{R}}}}}}}}}_{0}), our model allows contacts to evolve over time as a function of individual choices, which respond to disease dynamics.

The contact matrices in⁶ measure only contacts between individuals in different age groups by activity, without noting which individuals are consuming and which are working. Given the lack of precise data on contacts between individuals engaging in different activities, we simplify by assuming individuals who are consuming only contact others who are consuming, and individuals who are working only contact others who are working. However, in reality individuals who are consuming also interact with individuals who are working (e.g., a bar or restaurant). Future work could collect more detailed contact data describing contacts between individuals engaging in different activities.

SIRD epidemiological model

The SIRD model is given by:

Where S, I, R, D represent the fractions of the population in those compartments. Because the contact function ({{{{{{{{mathscr{C}}}}}}}}}^{SI}({{{{{{{bf{A}}}}}}}})) returns the number of contacts per person as a function of activities A, then τ is a property of the pathogen that determines the infections per contact. This decomposes the classic “β” in epidemiological modeling into a biological component that is a function of the pathogen (τ) and a behavioral component linked to economic activity (({{{{{{{mathscr{C}}}}}}}}({{{{{{{bf{A}}}}}}}}))), such that (beta ={{{{{{{mathscr{C}}}}}}}}({{{{{{{bf{A}}}}}}}})tau) (e.g.,¹⁶).

A key input into individual decision making is the probability of infection for a susceptible individual, which per the SIRD model above depends on the properties of the pathogen, contacts generated through economic activities, and the share of infectious individuals in the population:

If a susceptible individual reduces their activities (and thus contacts) today, they reduce the probability they will get infected, which in turn reduces the growth of the infection. However, if they keep their economic behavior the same, they enjoy those benefits today, but take the risk of becoming infected in the future. Finally, P^R is the rate at which infectious individuals recover, and P^D is the rate at which they die. Both are assumed to be constant over time and independent of economic activities and contacts.

Our framework can be generalized to other structured compartmental models beyond mean-field (homogeneous) SIRD models. The key feature to translate is the contact function. For example, in an age-structured model the contact function would need to reflect age-specific consumption and labor supply patterns.

Choices

In order to analyze the three control strategies (voluntary isolation, blanket lockdown, targeted isolation), we solve two types of constrained optimization problems: a decentralized problem and a social planner problem. The decentralized problem reflects atomistic behavior by individuals—they aim to maximize their personal utility and make choices regarding economic activity. The decentralized problem is used to analyze the voluntary isolation and blanket lockdown strategies. Conversely, in the social planner problem, a social planner considers the utility of the population as a whole and coordinates economic activity to jointly maximize the utilities of all individuals in the population. Importantly, the social planner internalizes the full economic costs to the population associated with disease transmission. The social planner problem is used to analyze the targeted lockdown strategy.

In the decentralized problem, individuals observe the disease dynamics, know their own health state, and make consumption and labor choices in each period accounting for the risks incurred by contacts with potentially infectious individuals. Individuals’ knowledge of their own daily health state is consistent with a testing system where individuals use a daily test which reveals their health state. Let ({{{{{{{bf{A}}}}}}}}={{c}_{t}^{m},{l}_{t}^{m}}) represent the economic activities of consumption and labor chosen in period t by individuals of health type m. Individuals maximize their lifetime utility by choosing their economic activities, ({c}_{t}^{m}) and ({l}_{t}^{m}), accounting for the effects of infection and recovery on their own welfare:

Per-period utility (u({c}_{t}^{m},{l}_{t}^{m})) captures the contemporaneous net benefits from consumption and labor choices. In particular, susceptible individuals in period t recognize their personal risk of infection ({P}_{t}^{I}) is related to their choices regarding economic activity ({c}_{t}^{S},{l}_{t}^{S}), and if they do become infected in period t + 1, they have some risk of death in period t + 2. Death imposes a constant utility of Ω, calibrated to reflect the value of a statistical life (see SI 2.1.3). The daily discount factor δ reflects individuals’ willingness to trade consumption today for consumption tomorrow.

Finally, individuals exchange labor (which they dislike), for consumption (which they do like) such that their budget balances in each period:

The wage rate w_t is paid to all individuals, per effective unit of labor ({phi }_{t}^{m}{l}_{t}^{m}), and is calculated from per-capita GDP. We represent the degree to which individuals are able to be productive at work by ϕ^m (labor productivity). We assume that symptomatic individuals are less productive, such that ϕ^S = ϕ^R = 1 and ϕ^I < 1, reflecting the average decrease in productivity of infectious individuals (accounting for the share of asymptomatic and pre-symptomatic individuals, similar to²⁴—see SI 2.1.1). Following standard practice, the price of consumption p is normalized to 1. Finally, market equations that state how individuals are embedded in a broader economy are described in the SI.

The social planner problem coordinates the economic activities of the individuals described above. Instead of economic activities being individually chosen to maximize personal utility, the social planner coordinates consumption and labor choices of each type (({{{{{{{{bf{c}}}}}}}}}_{{{{{{{{bf{t}}}}}}}}}={c}_{t}^{S},{c}_{t}^{I},{c}_{t}^{R}), ({{{{{{{{bf{l}}}}}}}}}_{{{{{{{{bf{t}}}}}}}}}={l}_{t}^{S},{l}_{t}^{I},{l}_{t}^{R})) to maximize the utility of the population over the planning horizon, subject to the disease dynamics (3) and budget constraints (9):

Additional structure (e.g., age compartments, job types, geography) can be incorporated here either by creating additional utility functions or by introducing type-specific constraints. For example, with age compartments, each age type would have a set of utility functions like equations (5)–(8). These would then be calibrated to reflect age-specific economic activity levels, structural parameters, and observed risk-averting behaviors.

Both the decentralized problem and the social planner problem are solved for optimal daily consumption and labor supply choices in response to daily state variable updates, and we normalize the total initial population size to 1 for computational convenience. The assumption that individuals use a daily test that reveals their health state is maintained across both the decentralized and the social planner problems. We abstract from the cost of the testing system. Since the cost is common to both problems, it does not affect the relative comparison between the two.

Utility calibration

Details of the utility function calibration and data sources are found in the SI. Briefly, economic activity levels and structural economic parameters are calibrated to match observed pre-epidemic variables for the US economy. We calibrate risk aversion and the utility cost of death to match the value of a statistical life. This approach ensures both the levels of economic choice variables and their responses to changes in the probability of infection are consistent with observed behaviors in other settings.

Model applications

We add information frictions and individual non-compliance to our baseline model to study how plausible magnitudes of such distortions may affect our policy conclusions. These are modeled by altering the inputs into agents’ optimal choice rules (known as “policy functions” in dynamic optimization problems, not to be confused with pandemic control policies) that specify their (c^*, l^*) choice given the (S, I, R) information they have. The choice rules take the form shown in the equation below, where the only addition to the usual sub/superscripts is [P] denoting the policy type {V, T, L} for voluntary isolation, targeted isolation and blanket lockdown policies respectively:

all three types of agent choose consumption and labor consistent with these choice rules depending on what they know of the state of the world (i.e. (S_t, I_t, R_t)). These choice rules are the main output of the value function iteration process described in SI 3.1. By feeding different information into the choice rules or taking weighted averages under different policies, we can model the frictions described below as different scenarios.

Test reporting lags

Test reporting lags force agents to react to population-level infection information from x days ago. This is modeled as feeding (S_t−x, I_t−x, R_t−x) into the choice rules above when finding (({c}_{t}^{* },{l}_{t}^{* })). We select x to be roughly consistent with observed lags during the COVID-19 pandemic: initially 8 days at the outset of the pandemic, before falling to 5 days on day 60 and 3 days at day 75. The choice rules become:

Test quality

Tests for individual health status differ in quality throughout the course of a novel pandemic, starting from very low quality before becoming progressively more accurate. We assume that due to test quality q (for the specific foundation of this single-metric quality notion related to specificity and sensitivity see SI 3.3.2), individuals take a weighted average of the choice-rule-prescribed action for their true health type and a “no information” action which is averaged uniformly across the actions for each type. This is equivalent to either of the following behavioral microfoundations:

• individuals realize they do not know their type with certainty, so can do no better than using q to mix between the choice-rule-prescribed action for their test-reported type and an average across actions for each of the three types; or

• a fraction q of agents of a given type trust their test result and follow the associated choice-rule-prescribed action, while the remaining 1 − q fraction either do not get tested or do not trust their test and uniformly mix across actions for all health types.

We consider two types of test quality scenarios: first a “limited testing” scenario where test quality is low throughout the whole pandemic, and second a more-realistic “improving test quality” scenario where test quality linearly improves over the course of the pandemic, becoming perfect at day 75. The choice rules become:

we examine the robustness of our conclusions to an equilibrium model of behavior under low-quality information or limited cognitive capacity in SI 3.3.4, finding that the qualitative results regarding policy effectiveness are unchanged.

Compliance

Some individuals may not comply with policy mandates. We model this as a share of agents (bar{c}) of any type that choose the decentralized (i.e. voluntary isolation) action rather than complying with the targeted isolation or blanket lockdown mandates. We consider two types of scenarios, “low compliance” with 10% compliance and “partial compliance” with 75% compliance. The choice rules become:

Reporting summary

Further information on research design is available in the Nature Research Reporting Summary linked to this article.

Source: Ecology - nature.com