key: cord-0481629-l0u86vjz authors: Bach, Philipp; Chernozhukov, Victor; Spindler, Martin title: Insights from Optimal Pandemic Shielding in a Multi-Group SEIR Framework date: 2020-11-02 journal: nan DOI: nan sha: efe57f12ffdefae9779c05b4c571301116f07d1c doc_id: 481629 cord_uid: l0u86vjz The COVID-19 pandemic constitutes one of the largest threats in recent decades to the health and economic welfare of populations globally. In this paper, we analyze different types of policy measures designed to fight the spread of the virus and minimize economic losses. Our analysis builds on a multi-group SEIR model, which extends the multi-group SIR model introduced by Acemoglu et al.~(2020). We adjust the underlying social interaction patterns and consider an extended set of policy measures. The model is calibrated for Germany. Despite the trade-off between COVID-19 prevention and economic activity that is inherent to shielding policies, our results show that efficiency gains can be achieved by targeting such policies towards different age groups. Alternative policies such as physical distancing can be employed to reduce the degree of targeting and the intensity and duration of shielding. Our results show that a comprehensive approach that combines multiple policy measures simultaneously can effectively mitigate population mortality and economic harm. In a recent contribution, [1] extend the classical SIR model, which is well-known from the epidemiological literature, by explicitly incorporating the trade-off that policy makers must consider in times of the pandemic. The authors derive the efficient frontier for different policies and show that efficiency gains can be achieved by targeting lockdown policies at different age groups, each of which is, in turn, characterized by different productivity and mortality risks. 1 In a setting calibrated to the U.S. population and economy, they show that protecting the most vulnerable group (i.e., those aged 65 and older) with stricter shielding rules (i.e., targeted shielding) is associated with fewer losses than a blanket shielding policy (also referred to as a uniform shielding, i.e., a policy that applies equally to all groups). [1] briefly mention and discuss a potential extension of the multi-group SIR model to the SEIR case. Here, we continue their analysis and analyze a variety of policy measures within the SEIR model. We explicitly state the key equations of this model and calibrate it to social interaction patterns as estimated in [15] . In this paper, we consider a model that is calibrated to Germany -that is, we adjust it to the country's demographic and economic characteristics, as well as its system of health care provision. Germany and the U.S. differ in many regards, such as the demographic structure of the population, age-specific employment and income patterns, and the capacities of the health system. We present the results of the model and discuss various policy measures, such as group distancing, test strategies, contact tracing, and combinations of these. We also discuss in detail how a targeted policy, protecting vulnerable groups like old people, might be implemented in practice and discuss some policy examples. Mortality from COVID-19 is particularly high among older people, [9] , whose productivity is relatively low. Hence, a targeted shielding policy that limits face-to-face contacts with persons aged 65 or older might lead to lower mortality in this population group and less damage to the economy. Additionally, a set of potentially voluntary policies that reduce transmission rates and social contacts could, in principle, be considered as an alternative to age-targeted shielding. Indeed, in our analysis, we find that testing, contact tracing, group distancing and improved conditions for working from home help to reduce the economic costs of the pandemic and the intensity and duration of age-targeted shielding. Moreover, if these measures are combined in a comprehensive approach as described in the initial quote by Tedros Adhamom Ghebreyesus, population mortality and economic outcomes improve substantially. Throughout our analysis, the efficiency gains associated with age-targeting remain relatively stable and sizable, and we recommend exploiting these gains by improving conditions for individuals at high risk, for example by providing services such as special shopping or consultation hours for older people, as well as testing capacities for those who have contact with high-risk groups to decrease the probability of infections. The rest of this paper is structured as follows: In Section 2 we briefly introduce the multi-group SEIR model. In Section 3 we describe our specification of the parameters for the SEIR model for Germany. Section 4 presents the results and describes the optimal policies comprising measures such as group distancing, testing, contact tracing and improved medical treatment. Finally, a conclusion summarizes the results and makes a range of policy recommendations. Because there is still so much that we do not know about SARS-CoV-2, including the transmission rate, mortality rates and aspects related to immunity, all of the results reported throughout the paper must be interpreted with caution. As in the study by [1] , we do not focus on presenting absolute quantitative results, such as GDP forecasts, but rather qualitative insights into potential policy measures that are considered in variation-of-parameters analyses. Literature review. The classical SIR and SEIR models are used widely in epidemiology and described in many standard textbooks. Driven by the COVID-19 crisis, various extensions of the standard epidemiological models have been developed and modified to consider economic factors. For example, [4] include individual choices about the amount of time spent on activities outside the house, such as work or consumption, to the standard SIR epidemiological model. These activities are associated with externalities, i.e., higher risk of transmission to and from others. The model also incorporates heterogeneity in terms of age and different policy measures, such as testing or quarantines. [3] provide an extended SEIR model focusing on testing and quarantine measures and thereby explicitly address the imperfect information that arises due to the fact that cases can be symptomatic or asymptomatic. A recent study by [13] extends a classical SEIR model by introducing a high and low risk group that differ, for example, in hospitalization and mortality rates. Their study focuses on the evolution of infected, recovered and deceased, i.e., the epidemiological aspects of the SEIR model in a parametrization calibrated to Germany. While a blanket shielding policy (i.e., for the entire population) is, of course, the optimal way to protect everyone from infection, the associated economic losses might become substantial. The multi-group SEIR model incorporates economic costs that arise due to sick leave, productivity losses when individuals work from home and discounted lifetime income losses from deaths due to COVID-19. Moreover, important indirect health consequences are associated with strict shielding measures, such as missed appointments for other conditions, less exercise, mental health issues, increased alcohol consumption, social isolation and increased levels of domestic abuse. While these indirect, non-pecuniary costs are not incorporated in our study, it might be useful to model them in future work. We build on the work of [1] , who study targeted shielding policies in a multi-group SIR model, and thereby address the trade-off between mortality and economic losses. They consider two possible targeting strategies: finding separate, optimal shielding policies for the young, middle-aged and senior groups (the so-called "fully targeted" policy) or imposing two separate shielding policies, one for the senior group and the other for the young and the middle-aged (so-called "semi-targeted" shielding). In their baseline results, semi-targeted policies are associated with substantial efficiency gains that cannot be improved substantially by fully targeted policies. While [1] analyze the optimal policy for the U.S., we extend their framework and calibrate it to Germany. Our baseline model is a SEIR model that incorporates contact patterns as estimated by [15] , who evaluate data from the BBC pandemic project in 2017 and 2018. Moreover, we consider a broader set of policy measures, such as testing and contact tracing, as well as various forms of group distancing. 2. Multi-Group SEIR Model. In this section, we briefly describe a SEIR model based on [1] , who focus in their analysis on the SIR model and state that their conclusions also hold for the SEIR version. For an in-depth discussion with additional information on the theoretical set up of the original SIR model, we refer to [1] . One of the major features of the framework is that it allows the population to be partitioned into subgroups that are heterogeneous in terms of their productivity and mortality rates. In particular, we consider the following three subgroups: young (20-49 years), middle-aged (50-64 years) and senior citizens (65+ years). Accordingly, there are age-group specific compartments for susceptible (S j ), infectious (I j ), recovered (R j ) and deceased (D j ) persons, with j = y, m, s referring to the young, middle-aged and senior groups. The epidemiological SEIR model extends the SIR model by the compartment of exposed individuals − that is, those who have been infected by the virus but whose infection is not yet sufficiently severe that they have symptoms or are infectious. Hence, the model considered in the following incorporates a compartment E j for each age group in addition to compartments S j , I j , and R j at each point in time t ∈ [0, ∞). N j is the number of initial members in each group, j = y, m, s. The compartment structure of a two-group SEIR model is illustrated in Figure 1 with the red arrows indicating the paths of transmissions through contacts of infectious and susceptible. Without any policy intervention that enforces shielding of the population or isolation of those who are infected, the (gross) number of new infections in the segment of exposed (E j ) and infectious (I j ) is governed by the following equations New exposed in group j = M j (S, E, I, R; α) · β · S j · k ρ jk I k (2.1) where {ρ jk } are parameters for the contact rate between group j and k and M j (·) refers to a matching technology, with M j (·) = 1 if α = 2 which is our baseline case. The parameter β denotes the transmission rate from contacts between individuals in I j and S j and γ E j is the exit rate from the latent state to the infectious state. 2.1. Model Assumptions. In this section we describe and discuss the model assumptions. Infection, ICU, Fatality and Recovery. In the SEIR model described above, a transmission of SARS-CoV-2 arises through contact of susceptible individuals with infectious individuals. After an average latent period 1 γ E j , they become infectious themselves. Individuals in compartment I j may require ICU care. We assume for simplicity that a need for ICU is apparent immediately after entering state I j . ICU patients either recover with Poisson rate δ r j or die at Poisson rate δ d j . Non-ICU patients will always recover at Poisson rate γ I j . The death rate can vary with total ICU needs relative to capacity. We assume that This means that the proportions of ICU and non-ICU patients among the infected do not change over time in group j. H j (t) denotes the number of individuals needing ICU care at time t in group j, so that H j (t) = ι j I j (t). H(t) = j H j (t) is the total need for ICU. The probability of death is a non-decreasing function of the number of patients, such that the probability of death will rise if the capacity is exceeded: , for a given non-decreasing function ψ j . Testing, Contact Tracing and Isolation. Detection and isolation of infected individuals is not perfect. In the SIR model, [1] denote the probability that an individual in compartment I j is not detected and put in isolation by η j . In their analysis, comparative statics are performed to illustrate the consequences of variation of η j , for example due to intensified testing. Incorporating the group of exposed (E j ) in a SEIR model allows tests to be performed for those who have had contact with an infected person. This setting could be considered a simplified form of contact tracing, for instance enabled by a smartphone application that records physical contacts. Hence, quarantining those who have been in contact with infected individuals might enable policy makers to exclude these infected but not yet infectious individuals from social interactions. Accordingly, we denote the probability that a person in compartment E j or I j is not detected and isolated by η E j and η I j , respectively, and thereby avoid including additional state variables. In this manner, we can model the fact that only those infected who have not been detected and isolated in stage E j or I j contribute to the spread of the disease via their contacts. Shielding and Physical Distancing. Shielding policies describe all measures that reduce the rate of transmission of infections in social and business life and physical distancing. The productivity of members of j is w j without shielding and ξ j w j with shielding, with ξ j ∈ [0, 1]. L j (t) = 1 refers to a full shielding policy and L j (t) = 0 to a situation without any restrictions to social interactions. L j (t) ∈ (0, 1) would be partial shielding, for example by shielding a (potentially randomly and independently drawn) fraction of the population. It is assumed that shielding cannot be perfectly enforced and that, with shielding, the effective reduction in social interaction is only 1 − θ j L j (t) with θ j < 1. Contact Rates. We implement a version of the SEIR model that incorporates social interaction patterns to capture the major findings in [15] -that is, high rates of interaction within and by the group of young and decreasing intensity of interactions with age. The study evaluates large-scale data on the frequency and intensity of social interactions that were collected in the BBC Pandemic project in the UK in 2017 and 2018 and make it possible to derive age-specific contact rates. To model the group interaction within and between groups, let denote ρ 0 jk the elements of the contact matrix with the first row and column referring to the young group, the middle row and column referring to the middle-aged group and the third row and column referring to the senior citizen group. 2 The contact estimates of [15] refer to a pre-pandemic setting and, hence, constitute the benchmark scenario for comparison to social distancing policies. To incorporate voluntary reductions of physical contacts, we base our baseline results in Section 4 on a rescaled contact matrix that presumes a 25% reduction in physical contacts. Incorporating more realistic contact patterns in the SEIR model with multiple groups is important for evaluating policy measures that are targeted at different age groups. For example, lower rates of contact between the vulnerable group (i.e., senior citizens) and younger people might allow for less intense shielding patterns. Physical Distancing, Face Masks and Additional Hygiene Measures. Various mandatory or voluntary policies can be employed to reduce the transmission rate of SARS-CoV-2. These measures range from a general reduction in face-to-face or physical contacts (for example, by imposing strict physical distancing measures that apply equally to all age groups) or specific interventions that aim to protect especially those who are most vulnerable. The latter include, for example, a reduction in face-to-face contacts with senior citizens -for instance by placing restrictions on visits to nursing homes or prescribing mandatory (reusable or disposable) face masks during for contacts with senior citizens. For example, [8] undertook a systematic review and meta-analysis of studies that examined the effectiveness of face masks and physical distancing for COVID-19 and related diseases (e.g., MERS and SARS). Accordingly lower transmission rates are associated with greater physical distance and the use of N95 face masks and comparable respirators rather than disposable surgical masks. There are a huge number of potential policy measures that aim to reduce the transmission of SARS-CoV-2, all of which can be employed in combination. We list a few examples of such measures in Section 4.3. Something that all of these measures have in common is that they effectively change or rescale the elements in the contact matrix ρ. In our analysis, we focus mainly on two variants of group distancing, namely (i) so-called uniform group distancing, which effectively reduces the contact rates in ρ for all groups (corresponding to a multiplication of the matrix (corresponding to a multiplication of the matrix ρ with a scalar ν), and (ii) group distancing policies with a focus on the vulnerable that refer only to interactions with the group of seniors and the elements ρ sj with j = y, m, s, and ρ js , respectively. Moreover, it is possible to simulate settings in which the level of interactions within the senior group might be left unchanged, thus reducing the impact on daily interactions with others at the same age. Improved Conditions for Working from Home. Working from home can be an effective way to reduce the costs of the pandemic and of shielding policies. To host a scenario with improved conditions for working from home, we (i) implement a parameter constellation with respect to the contact rates within and between the young and middle-aged group and between these groups and the senior group and (ii) decrease the productivity loss associated with working from home, ξ j .We believe that this captures some aspects of working from home in that those who are most likely to be employed can reduce their social interactions with lower economic losses. Changes in terms of (i) are imposed by scaling the entries of the contact matrix ρ yy , ρ mm , ρ ym by a factor π 1 and a scaling the contact rates ρ ys and ρ ms by π 2 with π 1 < π 2 . Vaccine and Cure. [1] assume that a vaccine and an effective drug for all infected individuals becomes available at some date T and that full immunity is achieved and maintained after an infection. 3 In our analysis, we will evaluate changes in T resulting from a faster development of a vaccine -for example after one year or six months. Currently, there are various treatments for COVID-19 that have been approved or are being evaluated in clinical trials. We assess the implications of a medical treatment with respect to the optimal shielding policy. Put simply, a new treatment could have any of the following three effects: (i) reduce the length of hospitalization, (ii) reduce the probability of dying from COVID-19, (iii) reduce the probability that an infection with SARS-CoV-2 becomes severe. We will focus on the availability of a treatment that leads to a reduction in mortality from COVID-19 for the group of senior citizens because most deaths and severe cases have been observed in this age group (e.g., as reported for Germany in [16] ). Dynamics in the MG-SEIR Model. If vaccine and cure are unavailable, the number of individuals in the exposed compartment for group j evolves according to the differential equations for all t ∈ (0, T ) for nonnegative β and contact coefficients ρ jk and where In the quadratic case M j (S, E, I, R, L) = 1. The parameter κ j refers to the share of recovered individuals that can return to work and social life while being exempted from shielding policies due to immunity. 4 Setting η E j = 1 for all j refers to a setting where it is not possible to test and isolate exposed individuals. However, a value η E j < 1 means that the effective number of individuals who contribute to further spread of the disease can be reduced by contact tracing and isolating those who have been exposed. The rest of the laws of motion for t ∈ (0, T ) arė where H j = ι j I j denotes the number of ICU patients in group j. After discovery of a vaccine and cure at T , every individual alive is in the recovered category. The government can control the degree of shielding L j (t) for each group j at any point in time t ∈ [0, T ). In particular we will compare uniform policies (i.e., blanket policies with L j (t) = L(t)) and group-specific (i.e., targeted) policies. The goal of the social planner is to minimize the overall costs of the pandemic, which consist of two parts: 1. Lives Lost = j D j (T ). The economic losses for group j are given by where the second term refers to the income loss of exposed individuals under shielding. The third term in the economic cost function is now adjusted to the case with the testing and isolation of exposed individuals, as well. ∆ j captures the present discounted value of a group j member's remaining employment time until retirement, which is lost due to death. The objective function is a weighted sum of both losses with weight factor χ and the task is to choose a shielding policy which minimizes Varying the values for χ makes it possible to identify the efficient frontier -in other words, to find the policy that minimizes the objective function for a given χ. Hence, the policy recommendations that can be obtained from an analysis of the efficient frontiers do not depend heavily on a specific choice of χ but rather reflect the difficult trade-off that policy makers face in the pandemic [1] . Before we discuss optimal shielding policies in the multigroup SEIR model, we will first comment on how we set and calibrated the parameters for Germany. We will present adaptations of country-specific parameters that would also apply to a calibration of the initial multi-group SIR model in [1] . These parameters refer to demographic and economic conditions, as well as to characteristics of health care provision in Germany. Second, we will discuss the adaptations of the SIR model parameters to a SEIR version based on information from the Robert Koch Institute (RKI) as of July 2020. Finally, we will comment on the ation of the basic reproduction number R 0 . Calibration of Socio-Demographic and Economic Parameters. Germany has a demographic composition that is substantially different from that of the U.S. In particular, the share of the group aged 65 and older is larger and that of the young group is smaller than in the U.S. For example, the median age in the U.S. is around 38 [7] years whereas it is around 45 years in Germany [12] . Using data from German micro census from 2018 as provided by the German Federal Statistical Office [5] and [6] , we calculated the remaining lifetime earnings as displayed in Table 1 assuming retirement at age 67. An interesting difference that we observed in the comparison of Germany and the U.S. is the distinct employment patterns in the group aged 65 and above. Whereas approximately 20% of individuals in this group are still employed in the U.S., the corresponding share for Germany amounts only to around 7%, leading to the re-weighted per-capita earnings in Table 1 . In both countries, the median earnings are relatively similar for those who are employed in the middle-aged group and the senior groups. Table 2 Hospital beds and ICU beds. Source: [10] , [2] , [11] . [2] refers to adult ICU beds and population only. The demographic distribution of the population in Germany implies that the share of persons who have a higher risk of dying from COVID-19 is relatively large. Thus, uniform shielding policies that aim to keep mortality in the entire population at a low level are expected to be more costly in terms of economic damage. At the same time, the group of senior citizens accounts for a relatively low share of GDP, implying that targeted policies are more favorable. Shielding targeted only towards the elderly therefore makes it possible to reduce overall mortality while allowing the younger and economically more productive groups to continue working. Variables. Calibrating the model in terms of parameters that are related to health care provision is challenging -for example due to limited comparability of hospital capacities and their dynamic expansion in reaction to the pandemic [10] . We performed various variations to parameters of the original SIR model of [1] and its SEIR version which are provided, in part, in the Appendix and chose one of these parameter configurations as a baseline setting in our analysis as described in the following. Health care provision in Germany is considerably different from that in the U.S. In a recent report, the Organisation for Economic Cooperation and Development (OECD) [10] compares health care provision across different countries. We list the numbers of hospital and ICU beds for the U.S. and Germany in Table 2 . Due to the dynamic expansion of hospital capacities during the COVID-19 pandemic in both countries, we add more recent, constantly updated data from the American Hospital Association (AHA) [2] and the German Interdisciplinary Association for Intensive and Emergency Medicine (DIVI) [11] . Compared to the OECD data, the number of ICU beds reported by AHA and DIVI has increased by around 21% in the U.S. and around 15% in Germany. The number of ICU beds is frequently reported to be one of the crucial measures of whether countries are able to keep mortality from COVID-19 low. According to the report by [10] , Germany is the country with the highest ICU capacities among all OECD members. Germany not only has more ICU beds per capita than the U.S.; other measures, such as the number of hospital beds or coverage with public health insurance [10] , suggest that the health care system in Germany has comparably greater capacities (per capita) than that of the U.S. To take account of these differences, we adjust the parameter λ, which enters the relationship of the daily mortality rate, δ d j , and hospital capacities at time t, H(t), to a default value λ = 0.6, which is smaller than λ = 1 as chosen in the analysis of [1] . whereδ d j is the baseline mortality rate for group j with δ y = 0.001, δ m = 0.01 and δ s = 0.06. 5 We refer to Figure 18 in the Appendix for illustrations of the variation of health-provision-related parameters. An alternative to specifying the parameter λ would be to impose a hard ICU constraint by enforcing H(t)