Rama Cont

Artur Kotlicki

Renyuan Xu

Modelling COVID-19 Contagion: Risk Assessment and Targeted Mitigation Policies

Keywords: COVID-19, SARS-n-COV, metapopulation epidemic models, network model, compartmental models, SEIAR model, nowcasting

We use a spatial epidemic model with demographic and geographic heterogeneity to study the regional dynamics of COVID-19 across 133 regions in England.

Our model emphasises the role of variability of regional outcomes and heterogeneity across age groups and geographic locations, and provides a framework for assessing the impact of policies targeted towards sub-populations or regions. We define a concept of efficiency for comparative analysis of epidemic control policies and show targeted mitigation policies based on local monitoring to be more efficient than country-level or non-targeted measures. In particular, our results emphasise the importance of shielding vulnerable sub-populations and show that targeted policies based on local monitoring can considerably lower fatality forecasts and, in many cases, prevent the emergence of second waves which may occur under centralised policies.

Overview

The novel coronavirus pandemic of 2019-2020 has led to disruption on a global scale, leading to more than 1.4 million deaths worldwide at the time of writing, and prompted the implementation of government policies involving a variety of 'non-pharmaceutical interventions' [START_REF] Ferguson | ) to reduce COVID-19 mortality and healthcare demand[END_REF] including school closures, workplace restrictions, restrictions on social gatherings, social distancing and, in some cases, general lockdowns for extended periods. This has led to a range of different public health policies across the world, and the efficiency of specific policy choices has been subject to much debate.

While the nature of these restrictions has been justified by the severe threat to public health posed by the virus, their design and implementation necessarily involves a trade-off, often implicit in the decision-making process, between health outcomes and the socioeconomic impact of such social restrictions.

An important feature of the COVID-19 pandemic has been the heterogeneity of epidemic dynamics and the resulting mortality across different regions, age classes and population categories. The importance of these heterogeneities suggests that homogeneous models -often invoked in discussions on reproduction number and herd immunity -may provide misleading insights, and points to the need for more granular modeling to take into account geographic, demographic and social factors which may influence epidemic dynamics.

We propose a flexible modelling framework which can serve as a decision aid to policy makers and public health experts by quantifying this trade-off between health outcomes and social cost. Using a structured population model for epidemic dynamics which accounts for geographic and demographic heterogeneity, we formulate this trade-off as a control problem for a partially observed distributed system and provide a quantitative framework for comparative analysis of various mitigation policies. We illustrate the usefulness of the framework by applying it to the study of COVID-19 dynamics across regions in England and showing how it may be used to reconstruct the latent progression of the epidemic and perform a comparative analysis of various mitigation policies through scenario projections.

Several recent studies have used homogeneous compartmental models [START_REF] Aguilar | [END_REF][START_REF] Lourenco | Fundamental principles of epidemic spread highlight the immediate need for large-scale serological surveys to assess the stage of the SARS-CoV-2 epidemic[END_REF][START_REF] Donnat | Modeling the heterogeneity in COVID-19 reproductive number and its impact on predictive scenarios[END_REF][START_REF] Adam | Early dynamics of transmission and control of COVID-19: a mathematical modelling study[END_REF][START_REF] Robert S Pindyck | Covid-19 and the welfare effects of reducing contagion[END_REF][START_REF] Roques | Impact of lockdown on the epidemic dynamics of COVID-19 in France[END_REF][START_REF] Rawson | How and when to end the covid-19 lockdown: An optimization approach[END_REF][START_REF] Rowthorn | A cost-benefit analysis of the Covid-19 disease[END_REF] or age-stratified versions of such models [START_REF] Acemoglu | A multi-risk SIR model with optimally targeted lockdown[END_REF][START_REF] Chikina | Modeling strict age-targeted mitigation strategies for covid-19[END_REF][START_REF] Nicholas G Davies | Age-dependent effects in the transmission and control of COVID-19 epidemics[END_REF][START_REF] Lipton | Mitigation strategies for COVID-19: Lessons from the K-SEIR model[END_REF][START_REF] Prem | The effect of control strategies to reduce social mixing on outcomes of the COVID-19 epidemic in wuhan, china: a modelling study[END_REF][START_REF] Singh | Age-structured impact of social distancing on the COVID-19 epidemic in india[END_REF] to analyse the dynamics and impact of the COVID-19 epidemic in various countries. Our framework, while compatible with such homogeneous models at aggregate level, accounts for demographic and spatial heterogeneity in a more detailed manner, leading to regional outcomes which may substantially deviate from homogeneous models Similar, though somewhat less detailed, heterogeneous models have been recently used to study COVID-19 outbreaks by Danon et al. [START_REF] Danon | A spatial model of covid-19 transmission in England and Wales: early spread and peak timing[END_REF] for the UK, Birge et al. [START_REF] John R Birge | Controlling epidemic spread: Reducing economic losses with targeted closures[END_REF] for New York City and Roques et al. [START_REF] Roques | A parsimonious model for spatial transmission and heterogeneity in the covid-19 propagation[END_REF] for France.

We first present below an overview of the main features of our approach and the key findings, before going into more detail on the methodology and results.

Methodology

We formulate a stochastic compartmental (SEIAR) epidemic model with spatial and demographic heterogeneity (age stratification) for modeling the dynamics of the COVID-19 epidemic and apply this model to the study of COVID-19 dynamics across regions in England.

The model takes into account:

• epidemiological features estimated by previous studies on COVID-19;

• the lack of direct observability of the total number of infectious cases and the presence of a non-negligible fraction of asymptomatic cases;

• the demographic structure of UK regions (age distribution, density);

• social contact rates across age groups derived from survey data;

• data on inter-regional mobility; and

• the presence of other random factors, not determined by the above.

We first demonstrate that this model is capable of accurately reproducing the early regional dynamics of the disease, both pre-lockdown and a month into lockdown, using a detailed calibration procedure that accounts for demographic heterogeneity across regions, low testing rates, and existence of asymptomatic carriers. The calibration reveals interesting regional patterns in social contact rates before and during lockdown. Underlying any public health policy is a trade-off between a health outcome -which may relate to mortality or hospitalisations -and the socio-economic impact of measures taken to mitigate the magnitude of the impact on public health. We present an explicit formulation of this trade-off and use it to perform a comparative analysis of various 'social distancing' policies, based on two criteria:

• the benefit, in terms of reduction in projected mortality; and

• the cost, in terms of restrictions on social contacts.

The goal of our analysis is to make explicit the policy outcomes for decision-makers, without resorting to (questionable) concepts such as the 'economic value of human life' used in some actuarial and economic models [START_REF] Acemoglu | A multi-risk SIR model with optimally targeted lockdown[END_REF][START_REF] Robert S Pindyck | Covid-19 and the welfare effects of reducing contagion[END_REF][START_REF] Rowthorn | A cost-benefit analysis of the Covid-19 disease[END_REF].

In our comparative analysis, we consider a broad range of policies and pay particular attention to population-wide versus targeted mitigation policies, feedback control based on the number of observed cases and the benefits of broader testing. We introduce a concept of efficient policy, and show how this concept allows to identify decision parameters which lead to the most efficient outcomes for each type of mitigation policy. The granular nature of our model, together with validation based on epidemiological data, provide a more detailed picture of the relative merits of various public health policies.

Summary of findings

Our first set of results concerns the reconstruction of the progression of the pandemic in England, in particular its latent spread through asymptomatic carriers.

• Using a baseline epidemic model consistent with epidemiological data and observations on fatalities and cases reported in England up to June 2020, we estimate more than 17.8 million persons in England (31.7% of the population) to have been exposed to COVID-19 by August 1, 2020. These estimates are much higher than numbers discussed in media reports, based on the number of reported cases.

• Based on a comparison of fatality counts and reported cases, we infer that less than 5% of cases in England had been detected prior to June 2020. This low detection probability implies in particular that the number of reported cases may severely underestimate the latent progression of the epidemic.

• We observe significant differences in epidemic dynamics across regions in England, with higher fatality and contagion levels in northern regions compared to southern regions, both before and during the lockdown period, pointing to the importance of demographic and geographic heterogeneity for modeling the impact of COVID-19.

After calibrating the model to replicate the regional progression of COVID-19 in England for the period March 1 to May 31, 2020, we use it for scenario projections under various mitigation policies. Comparative analysis of mitigation policies reveals that measures targeting sub-populations -such as regions with outbreaks -are more efficient than population-wide measures in terms of the tradeoff between health outcomes and social cost. More specifically:

• Shielding of elderly populations is by far the most effective single measure for reducing the number of fatalities. • By contrast, school closures and workplace restrictions are seen to be less effective than social distancing measures outside of school and work environments.

• Adaptive policies ('feedback control') which trigger measures when the number of daily observed cases exceed a threshold, are shown to be more effective than preplanned policies, leading to a substantial improvement in health outcomes. As such policies are based on monitoring of new cases, broader testing significantly improves their outcome.

• A decentralised policy which triggers regional confinement measures based on regional daily reported cases is found to be more efficient than centralised policies based on national indicators, resulting on average in an overall reduction of 20,000 in fatalities and, in many cases, significant damping of a 'second wave'.

• Comparative analysis of policies (Table 10) shows a wide range of health outcomes. The most effective policy in terms of reducing fatalities involves triggering of regional confinement measures decentralised based on monitoring of new cases, coupled with shielding of elderly populations.

The present work should be seen as an illustration of what may be done using our methodology, rather than an exhaustive analysis of different policy options and scenarios. We have made available an online implementation of the model, which may be used to explore other scenarios and policies than those presented below: http://covid19.kotlicki.pl

Outline

The modeling framework is described in Section 2. Data sources and parameter estimations are detailed in Section 3. Section 4 highlights the implications of partial observability of state variables and the associated model uncertainty.

The outcomes of various epidemic control policies are then discussed in Sections 5 and 6. Pre-planned policies are discussed in Sections 5.1 and 5.2, while Section 6 discusses adaptive ('feedback') control policies, in which measures are triggered when the daily number of new reported cases exceeds a threshold, and concludes with a comparative analysis of health outcomes and social cost of various types of mitigation policies.

Modeling framework

To take into account the role of geographic and demographic heterogeneity, we use a stochastic compartmental (SEIAR) model with age stratification, mobility across sites, social contact across age stratification, and the impact of asymptomatic infected individuals. For general concepts on deterministic and stochastic compartmental models we refer to Anderson and May [START_REF] Anderson | Infectious Diseases of Humans: Dynamics and Control[END_REF], Brauer and Castillo-Chavez [START_REF] Brauer | Mathematical Models for Communicable Diseases[END_REF], Britton et al. [START_REF] Britton | Stochastic epidemic models with inference[END_REF], Lloyd and Jansen [START_REF] Alun | Spatiotemporal dynamics of epidemics: synchrony in metapopulation models[END_REF].

State variables

We consider a regional meta-population model with K regions labeled r = 1, . . . , K. Each region r has a population N (r) which is further subdivided into M age classes labeled a ∈ {1, 2, 3, 4, . . . , M }. We denote N (r, a) the population in region r in age category a, with M a=1 N (r, a) = N (r). Individuals in each region and age group are categorized into six compartments:

• Susceptible (S) individuals who have not yet been exposed to the virus;

• Exposed (E) individuals who have contracted the virus but are not yet infectious.

Exposed individuals may then become infectious after a certain incubation period;

• Infectious (I) individuals who manifest symptoms;

• Asymptomatic (A) infectious individuals;

• Recovered (R) individuals. In line with current experimental and clinical observations on COVID-19, we shall assume that individuals who have recovered have temporary immunity, at least for the horizon of the scenarios considered, and cannot be reinfected [START_REF] Bao | Reinfection could not occur in SARS-CoV-2 infected rhesus macaques[END_REF]; and

• Deceased (D) individuals.

The progression of the disease in the population is monitored by keeping track of the respective number

S t (r, a), E t (r, a), I t (r, a), A t (r, a), R t (r, a), D t (r, a)
of individuals in each compartment. As the model focuses on the dynamics of the epidemic over a short period (1000 days), we neglect demographic changes over this period and assume that the population size N (r, a) in each location and age group is approximately constant, that is

S t (r, a) + E t (r, a) + I t (r, a) + A t (r, a) + R t (r, a) + D t (r, a) = N (r, a)
is constant.

A metapopulation SEIAR model

S E I D A R λ t p β E (1 -f )γI f γI (1 -p) β E γA Figure 1: Epidemic dynamics.
When each subpopulation (r, a) is large and homogeneous, the dynamics of state variables may be described through the following system of equations, represented in Figure 1:

                         Ṡt (r, a) = -λ t (r, a) S t (r, a), Ėt (r, a) = λ t (r, a) S t (r, a) -βE t (r, a), İt (r, a) = p a βE t (r, a) -γI t (r, a), Ȧt (r, a) = (1 -p a )βE t (r, a) -γA t (r, a), Ḋt (r, a) = γf a I t (r, a), Ṙt (r, a) = γ(1 -f a )I t (r, a) + γA t (r, a) N (r, a) = S t (r, a) + A t (r, a) + E t (r, a) + I t (r, a) + R t (r, a) + D t (r, a).
(2.1) where • 0 < α < 1 is the infection rate per contact, that is the probability of infection conditional on contact;

• β is the incubation rate, and 1/β is the average incubation period;

• γ is the rate at which infectious individuals recover;

• 0 < p a < 1 is the probability for an infected individual in age group a to develop symptoms;

• f a is the infection fatality rate for age group a, representing the probability that an infected individual in age group a dies from the disease; and

• The force of infection λ t (r, a), which measures the rate of exposure at location r for age group a, is given by

λ t (r, a) = α a / ∈W σ r a,a (t) κI t (r, a ) + A t (r, a ) N (r, a ) + α a ∈W σ r a,a (t) K r =1 M r,r (t) κI t (r , a ) + A t (r , a ) N (r , a ) . (2.2)
The force of infection in each subpopulation (r, a) depends on the rate of contact with (infected) individuals in other subpopulations, which differentiates this model from a homogeneous model. These interactions occur through:

• Contacts across age groups in the same region: the term σ r a,a (t) represents the average number of persons from age class a encountered per day by a person from age class a in region r on a day t. For infectious individuals with symptoms, we assume a lower contact rate κσ < σ due to (partial) self-isolation. This leads to the first term in (2.2).

• Inter-regional mobility: the second term in (2.2) corresponds to contacts between individuals in region r and age class a and those in the working population (age classes a ∈ W) commuting from other regions r = r. M r,r (t) represents the proportion of individuals from region r among the population of adults at a location r at date t.

Stochastic dynamics

The deterministic dynamics (2.1) ignores the variability of outcomes [START_REF] Isham | Assessing the variability of stochastic epidemics[END_REF] due to random factors not taken into account in the model. To account for this variability of outcomes we model the variables (S(t), E(t), I(t), A(t)) as a continuous-time Markov point process [START_REF] Linda | A primer on stochastic epidemic models: Formulation, numerical simulation, and analysis[END_REF][START_REF] Britton | Stochastic epidemic models with inference[END_REF] defined through its transition rates conditional on the history H t up to date t:

                             P(∆S t (r, a) = -1|H t ) = -λ t (
(2.
3)

The stochastic dynamics (2.3) are consistent with the deterministic dynamics of (2.1) for large populations, in the sense that the population fractions represented by each compartment converge to those represented by the solution of (2.1) as min r N (r) increases. However, even when the overall population is large, the stochastic dynamics (2.3) can substantially deviate from the deterministic model (2.1), especially in small subpopulations and in the early phases of the epidemic when the number of infected individuals in each region may be small, leading to random flare-ups and breakouts not present in the deterministic model. In the sequel we use the stochastic model (2.3) for the dynamics of the state variables.

Policies for epidemic control

Social distancing policies (and lockdowns) affect epidemic dynamics by influencing (lowering) the social contact rates σ r ij and the inter-regional mobility M r,r . To discuss targeted policies which may influence differently social contact rates at different locations, we decompose the baseline social contact matrix σ r as

σ r (0) = σ r,H + σ r,W + σ r,S + σ r,O , (2.4)
where the components correspond respectively to contacts at home (σ r,H ), at work (σ r,W ), school (σ r,S ) and other locations (σ r,O ). Social distancing policies are then parameterised in terms of their impact on various components of the social contact matrix:

σ r ij (t) = u r,H ij (t)σ r,H ij + u r,S ij (t)σ r,S ij + u r,W ij (t)σ r,W ij + u r,O ij (t)σ r,O ij ≤ σ r ij (0), (2.5)
where 0 ≤ u r,X ij (t) ≤ 1 are modulating factors which measure the impact of the policy on social contacts between age groups i and j at a location X in region r. In absence of social distancing or confinement measures, we have u r,X ij (t) = 1; the value of u r,X ij (t) reflects the fraction of social contacts when between age groups i and j at location X in region r when the policy is applied. This parameterisation allows us to consider policies targeted towards sub-population or specific regions. For example, school closure in region r during time period [t 1 , t 2 ] corresponds to setting u r,S ij (t) = 0 for t ∈ [t 1 , t 2 ], while 0 < u r,S ij < 1 corresponds to social distancing in schools, with lower values of u r,S ij corresponding to stricter enforcement of measures.

In most cases u r,X ij (t) does not explicitly depend on the age groups i, j as it is infeasible to discriminate between age groups when implementing social distancing requirements. Dependence on age groups arises when certain types of contacts are primarily related to certain age groups:

• Shielding of elderly populations: such policies affect the contact rates between elderly populations and other age groups.

• Work restrictions, which affect contacts between age groups of the working population

(denoted W): u r,W ij (t) = u r,W 1 i∈W 1 j∈W .
Regarding the inter-regional mobility matrix M , following the interpretation discussed in Section 3.2, we modulate its value according to the fraction u r,W of the population who continue to commute, that is

M r,r (t) = u r,W (t)M r,r + (1 -u r,W (t))I.
Here M r,r is the fraction of population in region r whose habitual residence is in region r .

The modulating factors u r,X ij may be chosen in advance or expressed as a function of the state of the system. We distinguish:

• Pre-planned (also called 'open-loop') policies, in which target values of modulating factors u r,X ij (t) are decided in advance; and • Adaptive policies (also called 'closed loop' or feedback control), in which actions are decided and updated as a function of observed quantities such as number of daily reported cases or number of daily fatalities.

Comparative analysis of mitigation policies

To perform comparative analysis across different policies, we need to evaluate policy outcomes across two dimensions: health outcome and socio-economic impact. We quantify the health outcome of each policy by the total number of fatalities during a reference period, taken to be t max = 1000 days after the reference date of March 1, 2020. The length of this reference period is chosen such that it takes into account an eventual 'second wave' of fatalities. We denote this outcome by D tmax (u), which represent the total fatalities at date t max associated with policy u.

To quantify the socio-economic impact of a policy, we use as metric the reduction in social contact resulting from the policy over the horizon [0, t max ], that is

J(u) = tmax t=1 K r=1 M i,j=1 σ r ij (0) -σ r ij (t) N (r, i), (2.6)
defined in terms of person×day units. The range of policies examined below lead to different outcomes in terms of fatalities D tmax (u) and social cost J(u). A policy v dominates (or improves upon) a policy u if it leads to a similar or better health outcome at an equal or lower cost:

J(v) ≤ J(u) and D tmax (v) ≤ D tmax (u),
with at least one inequality being strict. A policy u is efficient among a class of policies U if it cannot be improved upon by any policy in this class. Given a set of policies U , the subset of efficient policies forms the efficient frontier of U . Some recent economic models [START_REF] Acemoglu | A multi-risk SIR model with optimally targeted lockdown[END_REF][START_REF] Robert S Pindyck | Covid-19 and the welfare effects of reducing contagion[END_REF][START_REF] Rowthorn | A cost-benefit analysis of the Covid-19 disease[END_REF] formulate the trade-off in different terms, by introducing a concept of monetary value of human life in order to build a (monetary) welfare function combining both terms. Aside from ethical issues linked to the very concept of monetisation of human life, there is no consensus on its actual value, which is a key determinant of the trade-off in this approach. Our approach avoids specifying such a value and aims at identifying the range of efficient policies, leaving the final choice of the trade-off to policymakers.

In what follows, the goal is to determine the set of efficient policies and describe the characteristics and outcomes of such policies. Pre-planned policies are discussed in Sections 5.1 and 5.2, while adaptive policies are discussed in Section 6.

Data sources and parameter estimation

We now describes the model inputs as well as the methodology used in the parameter estimation. Table 4 contains a summary of model parameters.

Data sources

The basic inputs of the model are panel data on number of cases and fatalities reported at the level of Upper Tier Local Authorities (UTLA) level in England, provided by the Public Health England and NHSX [START_REF] Gov | UK Coronavirus (COVID-19) in the UK[END_REF]. This defines the geographic granularity of the model: we partition the population of England into 133 regions as defined by the Nomenclature of Territorial Units for Statistics at level 3 (NUTS-3) [START_REF]NUTS -Nomenclature Of Territorial Units For Statistics[END_REF].

For the purpose of our study we distinguish M = 16 age groups, as shown in Table 1, which is the maximum granularity allowed by the available estimates of age-dependent social contact rates and fatality rates. The size N (r, a) of age group a in region r is retrieved using the population dataset provided by Eurostat [START_REF]Population on 1 January by age group, sex and NUTS 3 region[END_REF]. Appendix A provides the list of UK regions used in this study and outlines the performed mapping procedure from UTLA to NUTS-3 regions to ensure consistency across data sources.

Modeling of inter-regional mobility

For our baseline estimate of inter-regional mobility we use the 2011 Census data on location of usual residence and place of work in the United Kingdom, provided by the Office for National Statistics [START_REF]Census: Special Workplace Statistics (United Kingdom)[END_REF]. The dataset classifies people aged 16 and over in employment during March 2011 and shows the movement between their area of residence and workplace, defined in Local Administrative Units at level 1 (LAU-1) terms. We then map this data onto NUTS-3 regions using the lookup table between LAU-1 and NUTS-3 areas provided by the Office for National Statistics [START_REF]LAU2 to LAU1 to NUTS3 to NUTS2 to NUTS1[END_REF].

The data is then represented in the model through the inter-regional mobility matrix M , whose elements M r,j represent the fraction of population in region r whose habitual residence is in region j. Denote by Π(r, j) the population with residence registered in region j and workplace registered in region r for r = j. In addition, we denote by Π(r, r) = a∈W N (a, r), where W = {5, 6, 7, 8, 9, 10, 11, 12}, the total population at location r in the age category [START_REF]NUTS -Nomenclature Of Territorial Units For Statistics[END_REF]60) years. Then, we estimate the coefficients of M r,j by M r,j = Π(r, j)

K i=1 Π(r, i) . (3.1)

Epidemiological parameters

Epidemiological parameters were either estimated from publicly available sources [START_REF] Adhikari | Inference, prediction and optimization of non-pharmaceutical interventions using compartment models: the PyRoss library[END_REF][START_REF] Mossong | Polymod social contact data[END_REF] or set to values consistent with recent clinical and epidemiological studies in COVID-19 [START_REF] Dorigatti | Severity of 2019-novel coronavirus (ncov)[END_REF][START_REF] Khalili | Epidemiological characteristics of COVID-19: A systemic review and meta-analysis[END_REF][START_REF] Verity | Estimates of the severity of coronavirus disease 2019: a model-based analysis[END_REF].

Social contact rates Contact rates across age classes have been estimated in studies by Mossong et al. [START_REF] Mossong | Social contacts and mixing patterns relevant to the spread of infectious diseases[END_REF][START_REF] Mossong | Polymod social contact data[END_REF] and Béraud et al. [START_REF] Béraud | The French connection: the first large population-based contact survey in france relevant for the spread of infectious diseases[END_REF]. We use the estimates of social contact rates provided by Mossong et al. [START_REF] Mossong | Social contacts and mixing patterns relevant to the spread of infectious diseases[END_REF] for the 16 age groups defined in Table 1. Using the PyRoss methodology [START_REF] Adhikari | Inference, prediction and optimization of non-pharmaceutical interventions using compartment models: the PyRoss library[END_REF], we further decompose the contact matrix, as in (2.4), into four components representing contacts at home (σ H ), work (σ W ), school (σ S ) and other locations (σ O ). Estimation methods and parameter values for these matrices are discussed in Appendix B. Contact rates may vary across different regions due to the heterogeneity in socioeconomic composition structure and specific regional characteristics, such as population density, level of urbanisation and the level of use of public transport. To account for this heterogeneity, we parameterise the (pre-lockdown) contact matrix in region r as σ r (0) = d r σ where the regional adjustment factors {d r : r = 1, . . . , 133} are estimated to reproduce the regional growth rate of reported cases before the lockdown period. The results are displayed in Figure 2. Table 2 provides a summary of selected characteristics of five regions with the highest values of the regional adjustment factors d r . As seen in Figure 2, our findings imply heterogeneity of social contact rates across regions. As we will observe below, these differences have a considerable impact on regional epidemic dynamics. Incubation rate Following the study of Ferguson et al. [START_REF] Ferguson | ) to reduce COVID-19 mortality and healthcare demand[END_REF], we use an incubation rate β = 0.2, which corresponds to an incubation period of approximately 5 days. This is further supported by several empirical studies on diagnosed cases in China outside Hubei province. An early study of Backer et al. [START_REF] Jantien | Incubation period of 2019 novel coronavirus (2019-ncov) infections among travellers from Wuhan, China[END_REF] based on 88 confirmed cases, which uses data on known travel to and from Wuhan to estimate the exposure interval, indicates a mean incubation period of 6.4 days with a 95% confidence interval (CI) of 5.6-7.7 days. Linton et al. [START_REF] Natalie M Linton | Incubation period and other epidemiological characteristics of 2019 novel coronavirus infections with right truncation: a statistical analysis of publicly available case data[END_REF], based on 158 confirmed cases, estimate a median incubation period of 5.0 days with 95% CI of 4.4-5.6 days and estimate the incubation period to have a mean of around 5 days with 95% CI of 4.2-6.0 days. Lauer et al. [START_REF] Stephen A Lauer | The incubation period of coronavirus disease 2019 (COVID-19) from publicly reported confirmed cases: estimation and application[END_REF] estimates a median of incubation period to be 5.1 days with 95% CI of 4.5-5.8 days, based on 181 cases over the period of January 4 to February 24, 2020.

Region

Proportion of symptomatic and asymptomatic infections

The probability p that an infected individual develops symptoms is an important parameter for epidemic dynamics, yet subject to a high degree of uncertainty: studies on various data sets [START_REF] Buitrago-Garcia | The role of asymptomatic SARS-CoV-2 infections: A rapid systematic review[END_REF][START_REF] Nicholas G Davies | Age-dependent effects in the transmission and control of COVID-19 epidemics[END_REF][START_REF] Mizumoto | Estimating the asymptomatic proportion of Coronavirus disease 2019 (COVID-19) cases on board the Diamond Princess cruise ship[END_REF][START_REF] Flaxman | Estimating the number of infections and the impact of non-pharmaceutical interventions on COVID-19 in 11 European countries[END_REF][START_REF]Coronavirus (COVID-19) infections in the community in England[END_REF] are based on small samples and yield a wide range of estimates. In particular, an early estimates from the Diamond Princess cruise ship [START_REF] Mizumoto | Estimating the asymptomatic proportion of Coronavirus disease 2019 (COVID-19) cases on board the Diamond Princess cruise ship[END_REF] and Japanese evacuation flights from Wuhan yielded estimates as high as p 0.7 -0.8 [START_REF] Nishiura | Estimation of the asymptomatic ratio of novel coronavirus infections (covid-19)[END_REF], while a July 2020 study by the Office for National Statistics [START_REF]Coronavirus (COVID-19) infections in the community in England[END_REF], based on a much larger sample, showed that p can be as low as 0.23. However, clinical studies [START_REF] Nicholas G Davies | Age-dependent effects in the transmission and control of COVID-19 epidemics[END_REF] indicate that this probability may strongly depend on the age group considered.

We use a range of values for the age-dependent probability p a whose upper bound is consistent with Davies et al. [START_REF] Nicholas G Davies | Age-dependent effects in the transmission and control of COVID-19 epidemics[END_REF] and whose lower bound is consistent with the estimates provided by the Office for National Statistics [START_REF]Coronavirus (COVID-19) infections in the community in England[END_REF]. These values are displayed in Table 3. Given the much larger sample size used in the study of Office for National Statistics [START_REF]Coronavirus (COVID-19) infections in the community in England[END_REF], we use the corresponding estimates ('low values', denoted as p low in Table 3) as benchmark unless stated otherwise.

Age group [0,5) [START_REF] Anderson | Infectious Diseases of Humans: Dynamics and Control[END_REF][START_REF] Brauer | Mathematical Models for Communicable Diseases[END_REF] [START_REF]Coronavirus (COVID-19) infections in the community in England[END_REF] and Davies et al. [START_REF] Nicholas G Davies | Age-dependent effects in the transmission and control of COVID-19 epidemics[END_REF].

Recovery rate γ In line with Cao et al. [START_REF] Cao | Estimating the effective reproduction number of the 2019ncov in china[END_REF], Li et al. [START_REF] Li | Early transmission dynamics in Wuhan, China, of novel coronavirus-infected pneumonia[END_REF] and Rocklöv et al. [START_REF] Rocklöv | COVID-19 outbreak on the diamond princess cruise ship: estimating the epidemic potential and effectiveness of public health countermeasures[END_REF], we use a recovery rate γ = 0.1, which corresponds to an average infectious period of 10 days.

Infection fatality rates We denote by f a the (infection) fatality rate for age group a.

In practice, these parameters are difficult to estimate during outbreaks and estimates may be subject to various biases [START_REF] Lipsitch | Potential biases in estimating absolute and relative case-fatality risks during outbreaks[END_REF]. Note that the infection fatality rate (IFR) is different from (and generally much smaller than) the case fatality rate.

Fatality rates for COVID-19 have been observed to be highly variable across age groups [START_REF] Khalili | Epidemiological characteristics of COVID-19: A systemic review and meta-analysis[END_REF][START_REF] Salje | Estimating the burden of SARS-CoV-2 in France[END_REF][START_REF] Verity | Estimates of the severity of coronavirus disease 2019: a model-based analysis[END_REF]. Based on the infection fatality rates provided in Verity et al. [START_REF] Verity | Estimates of the severity of coronavirus disease 2019: a model-based analysis[END_REF] for different age groups and the UK population distribution, we derive the aggregated IFR for the respective 16 age groups of interest as summarised in Table 5. These estimates are consistent with data obtained from other countries; for example, see Salje et al. [START_REF] Salje | Estimating the burden of SARS-CoV-2 in France[END_REF].

Model parameter name

Estimation of the infection rate

We use a simulation-based indirect inference method [START_REF] Gourieroux | Indirect inference[END_REF] for estimating the infection rate α. We simulate the stochastic model (2.3) for a range of values 0.03 ≤ α ≤ 0.15. The value of α is estimated by matching the logarithmic growth rates of the simulated reported cases with that of reported cases C t in England.

For the simulation, we use parameters specified in Table 4 and the following initial conditions for t 0 = March 10, 2020:

E t 0 (r, a) = N (r, a) a ∈W N (r, a ) C t 0 +5 (r) p 2 π 1 a∈W (3.2)
where W = {5, 6, 7, 8, 9, 10, 11, 12} corresponds to age groups in the working population and A t 0 (r, a) = 0, D t 0 (r, a) = 0, I t 0 (r, a) = 0 for all a. These initial conditions ensure that the simulations agree on average with regional case numbers on March 15, 2020, for all values of α. This procedure yields an estimated value of α = 0.055 and a confidence interval [0.051, 0.062]. This value of α, together with the model parameters in Table 4, yields a good fit of the pre-lockdown evolution of case numbers.

These results are consistent with estimates obtained in Donnat and Holmes [START_REF] Donnat | Modeling the heterogeneity in COVID-19 reproductive number and its impact on predictive scenarios[END_REF] and Dorigatti et al. [START_REF] Dorigatti | Severity of 2019-novel coronavirus (ncov)[END_REF] using data from other countries.

Inter-regional mobility and social contact during confinement

Confinement measures were implemented across the United Kingdom starting March 23, 2020 via the Coronavirus Act2 . During this 'lockdown' period schools and workplaces were closed and social contact was reduced, as evidenced by mobility data. 3 However mobility data also reveal regional differences in the impact of the lockdown.

We model the reduction in inter-regional mobility through an adjusted mobility matrix M r,r (t) = q M r,r + (1 -q)I, where 0 < q < 1,

and M r,r is the inter-regional mobility matrix defined in (3.1). According to the Labor Force Survey data from 2018/19 [START_REF] Farquharson | Key workers: key facts and questions[END_REF], 7.1 million adults across the UK are considered as 'key workers'. We set q = 20% to take into account the fact that these key workers continued to access their workplace during the lockdown period. This is also consistent with the methodology in Rawson et al. [START_REF] Rawson | How and when to end the covid-19 lockdown: An optimization approach[END_REF] and empirical studies of Santana et al. [START_REF] Santana | Analysis of human mobility in the UK during the COVID-19 pandemic[END_REF] on mobility changes before and after lockdown in the UK. Figure 3 shows the submatrix corresponding to daily mobility across London boroughs, and illustrates the observed dramatic drop in commute patterns. We model the impact of confinement on the social contact matrix through a regional multiplier l r , σ r (t) = l r × σ(0), (3.4) where l r ≤ d r represents the reduction in social contacts during the lockdown period; l r = d r corresponds to the pre-lockdown level of social contact. The value of l r is estimated from panel data on regional epidemic dynamics during the period from March 23 to June 1, 2020, using a least-squares logarithmic regression on the number of observed regional cases.

The average value of this reduction factor is found to be

133 r=1 N (r)l r 133 r=1 N (r) = 0.12,
which is an average reduction of 88% in social contacts, an order of magnitude corroborated by mobility data [START_REF] Santana | Analysis of human mobility in the UK during the COVID-19 pandemic[END_REF], showing that the lockdown was very effective in reducing social contacts. 6: Estimated values for regional adjustments d r and l r in NUTS-1 regions.

NUTS

Goodness-of-fit

Having estimated the model parameters using data on reported cases between March 10 and May 20, 2020 we assess the goodness-of-fit and out-of-sample performance using reported cases and fatalities between May 21 and June 22, 2020. Figures 4 and5 show that the model is able to reproduce the in-sample and out-of-sample evolution of case numbers and fatalities, at national level as well as regional level.

Observable quantities and uncertainty

When applying such models to epidemic data, a key point is to realize that the state variables S, E, I, A, R are not directly observed (and certainly not in real time) but need to be inferred from other observable quantities.

In absence of widespread testing, public health authorities are faced with the problem of controlling a system under partial observation. This lack of direct observability has some implications for the estimation and interpretation of the model, which we briefly discuss here.

Observable quantities

The two main observables in COVID-19 data are

• the cumulative number of reported cases; and

• the cumulative number of COVID-19 fatalities D t .

Of the two, fatalities are generally considered more reliable, as deaths are nearly always reported, while identification of cases requires testing or self-reporting. We thus identify the observed number of fatalities with the state variable D t .

In absence of widespread testing, only a fraction π of cases are reported. This fraction may change with time due to testing campaigns4 . We therefore cannot assume the number of infectious cases to be directly observed: rather, we estimate it from the fatality count D t (see also Jombart et al. [START_REF] Jombart | Inferring the number of covid-19 cases from recently reported deaths[END_REF]).

Let C t be the cumulative number of (symptomatic) infectious cases. Assuming that

• the daily number r(t) of reported cases is a fraction π(t) of new cases, that is

r(t) = π(t) (C t+1 -C t ) ; (4.1)
• deaths occur on average T days after detection;

we obtain that the daily fatality count is proportional to the lagged number of new cases,

D t+T +1 -D t f (C t+1 -C t ) = f π(t) r(t), (4.2)
where f is the (average) infection fatality rate. We use these relations to obtain an estimate for the cumulative number C t of symptomatic infections and the reporting ratio π(t). Using Eq. (4.2) we estimate the average delay T between case reporting and death by identifying the lag T which maximizes the correlation between the D t+T +1 -D t and r(t). Using an average fatality rate of f = 0.9% for the UK as in Ferguson et al. [START_REF] Ferguson | ) to reduce COVID-19 mortality and healthcare demand[END_REF] (see discussion in Section 3.3), we estimate the reporting probability to be

π(t) = f r(t) D t+T +1 -D t (4.3)
which implies that the total number of cases in England is more than 20 times the reported number. As shown in Figure 6, prior to June 2020 this reporting ratio was around π(t) = 4.5%; with the subsequent increase in testing, the estimated reporting ratio has steadily increased to more than 20% in November 2020.

Implications of partial observability

A key issue in epidemic control is the availability of reliable indicators for the intensity of an ongoing epidemic. Public health authorities have communicated the daily number of reported cases and fatalities, and these have served as inputs for policy planning. An important corollary of the above discussion is that, given the combination of random factors affecting dynamics and the considerable uncertainty on the actual number of new infections, it is perfectly possible to observe a run of many consecutive days without new reported cases while in fact the actual number of infections is on the rise. Reporting probability is π = 4.5%. Figure 7 shows an example of scenario in our model where, for 60 consecutive days, although a small number of (symptomatic and asymptomatic) cases appear, due to the low detection probability (π = 4.5%), none of them is reported. Nevertheless, after a run of 60 days without any reported cases (blue shaded area in Figure 7), which may prompt public health authorities to lower their guard, the epidemic takes off again. Figure 7 displays in fact two sample paths with the same initial conditions, which differ only through the stochasticity of the dynamics. The fact that the break-out occurs only in one of the two scenarios (in blue) but no in the other illustrates how random flare-ups may originate from a small group of undetected cases.

Figure 8a shows the probability of observing a second peak in infections when social distancing measures are lifted after no reported cases for L consecutive days. This probability is estimated using 500 simulated paths from (2.3). It is striking to observe that, even after 60 days with no reported cases, the probability of observing a resurgence of the epidemic is around 40%. Figure 8a (blue dashed line) shows the same probability conditional on observing no fatalities for L consecutive days.

These observations point to the importance of broader testing: as shown in Figure 8b, an increase in the probability π of detecting new cases leads to a strong decrease in the probability of misdiagnosing the end of the epidemic, as in the scenario described above.

(a) Probability of having a second peak in infections after no reported cases (solid line) and no fatalities (dashed line) for L consecutive days (low symptomatic ratios).

(b) Probability of having a second peak in infections following 60 consecutive days with no reported cases, as a function of reporting probability π.

Comparative analysis of epidemic control policies

Confinement followed by social distancing

We first consider the impact of a national 'lockdown' followed by social distancing, which reflects the situation in the UK between March 2020 and August 2020. We examine in particular the impact of a lockdown duration T and the level of social distancing after lockdown on the number of fatalities and the associated social cost. To do so, we parameterise the contact matrix as σ r (t) = l r σ for t 0 ≤ t ≤ t 0 + T (lockdown), ((1 -m)l r + md r ) σ for t > t 0 + T (after lockdown),

where l r measures the level of social distancing under lockdown, as estimated from observations for the period from Mar 23 to May 31, and the parameter m ∈ [0, 1] measures the level of compliance with social distancing measures. A value of m close to zero indicates a level of social contact similar to lockdown, while m = 1 corresponds to normal levels of social contact. The origin date t = 0 corresponds to March 1, 2020. All scenario simulations include a lockdown starting at t 0 = March 23, 2020. We consider a range 105 ≤ T ≤ 335 for the lockdown duration and 0.2 ≤ m ≤ 1 for post-lockdown social distancing levels. Note that the actual duration of the first lockdown in England corresponded to T = 105.

As shown in Figure 9a, the level of social distancing after the confinement period is observed to be more important (Figure 9b) than the length of the confinement period (Figure 9a). This is consistent with the findings in Lipton and Lopez de Prado [START_REF] Lipton | Mitigation strategies for COVID-19: Lessons from the K-SEIR model[END_REF]. Smaller values of m, associated with stricter social distancing, lead to a lower of fatalities but for at an increased social cost (Figure 9b). On the other hand, the lengthening of the lockdown duration T , while significantly increasing the associated social cost, does not result in a significant reduction in the number of fatalities, especially if social distancing is not respected after lockdown. Figure 9 also shows that some of these policies are inefficient, in the sense that we can reduce fatalities and the social cost simultaneously by shortening the lockdown period or by relaxing social distancing constraints, as shown in Figure 10. 10.

By comparing the orange and blue plots, which represent the same post-lockdown compliance level (m = 0.5), we observe that extending the lockdown duration increases social cost without reducing the total number of fatalities. On the other hand, comparing the orange and green plots, which correspond to the same lockdown duration of T = 105 days, shows that moving the compliance level from m = 0.5 to m = 0.4 reduces the second peak amplitude by 35% and fatalities by 13.9%.

Impact of parameter uncertainty

The above results are highly sensitive to the value of the symptomatic ratios which, as noted in Section 3, are highly uncertain (see Table 3). Figure 11 shows the policy outcomes for low versus high symptomatic ratios across different compliance levels and lockdown duration. As observed in this figure, while the overall pattern of the efficiency diagram is similar, the projected fatality levels shift considerably depending on the assumption on the symptomatic ratio: from 50,000-200,000 for low symptomatic ratios to 126,000-430,000 for high symptomatic ratios. Regional heterogeneity While the policies discussed here are applied uniformly across all regions, we observe a significant heterogeneity in mortality levels across regions, as well in terms of the timing and amplitude of a second peak tin infections. As shown in 12, some regions exhibit mortality levels up to 4 times higher than others. This huge disparity in mortality rates cannot be explained by demographic differences alone, which are much less pronounced: more important seem to be the differences in social contact patterns, as illustrated in Figure 2. Indeed, as shown in Figure 13a, there is a positive correlation (above 40%) between regional COVID-19 mortality and the intensity of social contact as measured by the parameter d r , defined in Section 3.3. Figure 13b shows that this heterogeneity is also reflected in the timing and amplitude of second peaks.

Targeted policies

We now consider the impact of social distancing measures targeting particular age groups or environments (school, work, etc.) following a lockdown of duration T , by setting

σ r ij (t) = l r σ for t 0 ≤ t ≤ t 0 + T (lockdown) σ r,H ij + u S ij σ r,S ij + u W ij σ r,W ij + u O ij σ r,O ij for t > t 0 + T. (5.2)
We consider different targeted measures after a lockdown period of T = 105 days (the actual duration of the lockdown in England): school closure, shielding of elderly populations and workplace restrictions, restrictions on social gatherings and combinations thereof. Note that there is no control over the social contacts at home.

School closures

Although most of the infected population in age group 1 is asymptomatic, they may in turn infect the population in age groups 3 and 4 who are more likely to develop symptoms. School closure corresponds to u S = 0, school reopening with social distancing correspond to u S = 0.5, and school reopening without social distancing correspond to u S = 1.

Shielding

The high infection fatality rates among elderly populations (age groups 3 and 4) have naturally lead to consider shielding policies for these populations. We model this as a reduction in social contacts of age groups 3 and 4 to the level observed under lockdown:

σ r i,j (t) = l r σ H i,j + σ S i,j + σ W i,j + σ O i,j if i ∈ {3, 4} or j ∈ {3, 4}.
Workplace restrictions We model the impact of a restricted return to work after confinement by assuming different proportion of workforce return after the lockdown period by choosing

0.2 < u W < 1 for t > t 0 + T, (5.3)
the lower bound u W = 0.2 corresponding to restricting workplace return to 'essential workers', as discussed in Section 3.5. Since workplace restrictions have an effect on commuting, such measures also have an impact on the inter-regional mobility matrix

C t = u W (t)C 0 + (1 -u W (t))I, (5.4)
where C 0 (r, r ) is the baseline mobility matrix defined in (3.1).

Restrictions on social gatherings Although social activities, such as gatherings at pubs or sports events, may aggravate the contagion of COVID-19, keeping certain levels of social activities is important to the economic recovery and the well-being of individuals. The parameter u O measures the fraction of social gatherings: during the lockdown this fraction was estimated to be as low as 20% (see Section 3.5). In what follows, we consider u O ∈ [0.3, 1.0] after the period of lockdown.

Pubs and schools

School Social distancing Normal school closure at school regime u S = 0 u S = 0.5 u S = 1.0 Social cost (10 11 ) 2.2 1.9 1.5 Projected fatalities 153,900 157,000 159,300

= u W = 1, u O = 0.5.
Table 8 show the impact of school closures and social distancing at schools on projected fatalities and social contacts. Reopening of schools, while reducing significantly the social cost, does not seem to lead to a significant increase in fatalities. We compare two post-confinement policies, one (labeled as 'schools') consisting in leaving schools open while social gatherings are restricted (u S = 1, u O = 0.2), and the other (labeled as 'pubs') consisting in closing schools while not restricting social gatherings (u S = 0, u O = 1). The social cost for the 'Pubs' policy is 2.3, while the cost for the 'Schools' policy is 3.0. However, as shown in Figure 15, the 'open school' policy leads to 35% fewer fatalities compared to the 'open pubs' policy.

Shielding of senior citizens

We have examined the impact of shielding in isolation and also in combination with other measures such as school closure and social distancing.

As shown in Figure 14a, whether applied in isolation or in combination with other measures, shielding of elderly populations is by far the most effective measure for reducing the number of fatalities. As clearly shown in Figure 14a, regardless of the trade-off between social cost and health outcome, a policy which neglects shielding of the elderly is not efficient and its outcomes can always be improved through shielding measures. However, as observed in Figure 17b, these policies are not efficient when shielding measures are put in place for the elderly. Under shielding, the spectrum of efficient policies is parameterised by the fraction u W of the workforce returning to work. As shown in Figure 17c, we can distinguish two classes of efficient policies under shielding:

• 'School and Pubs', consisting of policies without restrictions on schools or social gatherings (u S = 1, u O = 1) and different levels u W of restrictions on workplace gatherings.

• 'Restricted Work' policies, under which only 'essential' workers are allowed on-site work (u W = 0.2), with either (i) no school restrictions (u S = 1) and different levels of restrictions on social gatherings (0.2 ≤ u O ≤ 1) or (ii) restrictions on social gatherings (u O = 0.3, that is 'no pubs') and different levels of social distancing in school (0 ≤ u S ≤ 1).

As Figure 17d illustrates, 'School and Pubs' and 'Restricted Work' policies are not efficient without shielding.

In absence of shielding, social gatherings seem to be the main vector for contagion. When shielding measures are put in place, the social contacts associated with the elderly are reduced to the same level as under lockdown; in this case, contacts at work become the main vector of contagion.

Adaptive mitigation policies

We now consider adaptive mitigation policies, in which the daily number of (national or regional) reported cases is used as a trigger for social distancing measures. Such policies have been recently implemented, in the UK and elsewhere, at a local or national level with various degrees of success. We distinguish centralised policies, based on monitoring of national case numbers, from decentralised policies where monitoring and implementation of measures are done at the level of (NUTS-3) regions.

Country-wide restrictions

We first consider centralised policies which monitor the number of daily reported cases at country level. Whenever the number of daily reported cases (per 100,000 inhabitants) exceeds a threshold B on , confinement measures are imposed for a minimum of L days, until the number of daily reported cases falls below the threshold B off < B on . Outside these lockdown periods, we assume social distancing is in place with a compliance level m; we use a default value of m = 0.5.

This policy is implemented after the initial lockdown (that is, after July 4, 2020). In terms of the social contact matrix, we have, for t > t 0 + T ,

     σ r (t) = ((1 -m)l r + md r ) σ, and i s = 0, if (C t ≤ N
100,000 B off and Π t-1 s=t-L i s = 1); or (C t ≤ N 100,000 B on and i t-1 = 0); σ r (t) = l r × σ, and

i t = 1, if C t > N 100,000 B on or Π t-1 s=t-L i s = 1. (6.1)
Here T = 105, i t is the indicator of whether lockdown is applied on day t and C t is the daily reported cases in England on day t. Π t-1 s=t-L i s = 1 if lockdown has been applied for L consecutive days during the period [t -L, t -1].

We simulate the dynamics with various choices of B off and B on :

• B on ∈ {2, 4, 6, 8, 10} (daily reported cases per 100,000 inhabitants); and

• B off = 0.2 B on , B off = 0.4 B on or B off = 0.8 B on .
We assume that once a lockdown is triggered it lasts a minimum of L = 7 days and that, once lockdown is removed, individuals continue to observe social distancing as measured by the parameter m ∈ [0, 1]. Data on real-time mobility monitoring in the UK 5 , indicate mobility to be at 50% of normal level during the post-lockdown period, and thus we use m = 0.5 as a default value. Example Figure 19 shows an example of such an adaptive policy, where lockdown is triggered when daily cases exceeds 2240 nationally, and maintained until the count of new daily cases drops to 896. In the scenario shown in Figure 19a, this results in two short lockdowns, totaling 19 days in all, which bring under control the national progression of the epidemic and avoid a 'second peak' at national level. However, as shown in Figure 19b, this policy is less successful at regional level, resulting in a regional outbreak in Leicester. We observe in our simulations a second peak in I t for England when B on = 10, while we observe no second peak when B on = 2. When B on = 2, I t remains at level 2 × 10 5 with frequent interventions for 200 days and then decreases to zero. The social cost for policy B on = 10 and policy B on = 2 are 2.9 and 3.1, respectively. Policy B on = 2 have 18% fewer fatalities compared to policy B on = 10. Oxfordshire exhibits the same profile as England when B on = 10. However, the shape of I t is different for B on = 2 where Oxfordshire experiences a small outbreak around day 350.

In summary, smaller B on values correspond to more frequent lockdowns and result in damping or elimination of the 'second peak'.

Increasing testing capacity To study the effect of an increased testing capacity, we assume wide testing is adopted such that the reporting probability is increased from 4.5% to a significantly higher level (20%, 50%) on July 4, 2020 (see Figure 21). Table 9: Average social cost and fatalities for a given policy with different testing capacities (50 paths). Policy: B on = 6, B off = 0.4B on , and m = 0.5.

By increasing the testing capacity, the observable quantity of daily reported cases becomes more consistent with the underlying dynamics of I t . Compared to the policy with a reporting probability π = 4.5% throughout the reference period, we see that the dynamics of I t when π = 50% decrease to a small value rapidly. Increasing the testing capacity also implies a more efficient control and as a result leads to fewer fatalities. Impact of demographic granularity Several studies on the impact of public health policies on COVID-19 dynamics have used less granular models with fewer age groups [START_REF] Acemoglu | A multi-risk SIR model with optimally targeted lockdown[END_REF].

To assess whether such coarse-graining may result in a loss of accuracy for the model projections, we have compared our present model, which has 16 age groups, with coarsegrained versions of the model in which all individuals in the 20-59 age range are grouped into 2 age groups (leading to a total of 5 age groups) or a single group (leading to 4 age groups). 6 Parameters for the coarse-grained models are obtained as population-weighted averaged of the granular model.

Comparison of model projections, shown in Figure 22, indicate that the results are robust to changes in model granularity. Some quantitative differences may emerge when assessing the impact of targeted policies, but the overall dynamics of infections, cases and fatalities are rather insensitive to the demographic granularity.

Decentralised policies

We now consider a decentralised version of the above policies, based on monitoring of regional number of cases as triggers for regional confinement measures. In terms of the social contact matrices, we have, for t > t 0 + T ,

     σ r (t) = ((1 -m)l r + md r ) σ, i r t = 0, if (C t (r) ≤ N (r) 100,000 B off and Π t-1 s=t-L i r s = 1) or (C t (r) ≤ N (r)
100,000 B on and i r t-1 = 0); σ r (t) = l r × σ, and

i r t = 1, if C t (r) > N (r) 100000 B on or Π t-1 s=t-L i r s = 1.
Here i r t is the indicator of whether lockdown is applied in region r on day t and C t (r) is the daily number of cases reported in region r on day t. The term Π t-1 s=t-L i r s is used to track if lockdown has been applied in region r for L consecutive days during [t -L, t -1]. We use the same values of B on and B off as in Section 6.1. Figure 24 compares the outcomes of centralised and decentralised triggering policies. Decentralised policies are observed to always improve over centralised policies.

As an example, for B on = 4 and B off = 0.4B on fatalities in England are 133, 000 under the centralised policy and 122, 000 under the decentralised policy, that is 8% lower.

Figure 26 compares regional fatalities per 100,000 habitants for these policies. For more than 90% of the regions, decentralised measures lead to fewer fatalities. The most effective reductions are in Dorset, South West England (UKK22) with 23% fewer fatalities and in Cornwall and Isles of Scilly (UKK30) with 21% fewer fatalities. There are a few exceptions (see regions in light blue in Figure 26c). These regions are already under control before adaptive policies are applied. Therefore the improvement of moving from centralised policy to decentralised policy is limited.

Figure 27a compares the dynamics of symptomatic infections (I t ) for the same example. There is a reduction of 100,000 in the amplitude of the second peak value when moving from the centralised policy to decentralised one. Decentralised policy also damps the second-peak values in most of the regions. Similar effects are observed for York (Figure 27c) and Leicester (Figure 27b).

On June 29, 2020, Leicester became the first city in Britain to be placed in a local lockdown, after public health officials voiced concern at the city's alarming rise in COVID-19 cases. Earlier in June, the Government announced that parts of the city would be released from lockdown, while a 'targeted' approach will see pockets remain under tighter restrictions. Our simulations indicate a 60% reduction of the second-peak value in Leicester when a decentralised policy is implemented (Figure 27b).

Example Figure 25 shows an example of such a decentralised triggering policy, with the same triggering thresholds as in the centralised example in Figure 19. At regional level, we see in Figure 25a that this policy is more successful than the centralised policy in taming the local outbreaks in Leicester, substantially reducing the second peak through 4 one-week regional lockdowns. At the national level this results in a strong damping of 'second wave' infections, as shown in Figure 25b (compare with Figure 19a). We observe that

• adaptive policies, in which measures are triggered when the number of daily new cases exceeds a threshold, are more efficient than pre-planned policies; and

• as shown in Figures 28a and28b, a decentralised policy is more efficient than both centralised policy and pre-planned policy.

In Table 10, we provide a summary of outcomes for five different types of policies.

• Confinement of T = 105 days followed by social distancing (m = 0.3 or m = 0.5), no shielding.

• Pre-planned policy: social distancing at work and school (u H = 1, u S = 0.5, u W = 0.5), restrictions on social gatherings (u O = 0.3) and no shielding.

• Centralised and decentralised triggering policies (Section 6.1 and 6.2) with m = 0.5, B on = 4, B off = 0.4B on and no shielding;

• Decentralised triggering combined with shielding of elderly populations: m = 0.5, B on = 4, B off = 0.4B on ;

• 'Protect Lives' policy: in the range of efficient policies, the one which results in the fewest fatalities is a decentralised triggering policy with B on = 2, B off = 0.2B on (so more frequent triggering of confinement measures than the above), high degree of social distancing (m = 0.25) and shielding of elderly populations. This policy corresponds to the point in the lower right corner of Figure 28b. The social cost is 4.52, which is much higher than for the other considered policies. Outcomes are averaged across 50 scenarios, starting from the same initial conditions on July 4 (end of the UK lockdown).

I
Regional outcomes Comparing the regional outcomes of the centralised, decentralised and pre-planned policies displayed in Table 10 shows that the decentralised triggering policies are able in many cases to considerably damp the 'second wave' of infections. Figure 29 illustrates this in the case of Mid Lancashire, York, Leicester and Birmingham: the decentralised triggering policy reduces the second peak amplitude by around one half compared to the pre-planned policy.

A Demographic regions

B Baseline parameters for social contact rates

This appendix outlines the sources used for the baseline social contact rate parameters. In particular, two sources have been used: the POLYMOD study [START_REF] Mossong | Polymod social contact data[END_REF], processed using the methodology of PyRoss [START_REF] Singh | Age-structured impact of social distancing on the COVID-19 epidemic in india[END_REF], and the BBC Pandemic study [START_REF] Klepac | Contacts in context: large-scale setting-specific social mixing matrices from the bbc pandemic project[END_REF] is used as a robustness check. It should be noted that these parameters are used as a baseline, and a further detailed calibration is carried out region by region to account for heterogeneity of social contact patterns across UK regions. We use estimates for social contact rates across the 16 age groups (detailed in Table 1) given in Mossong et al. [START_REF] Mossong | Polymod social contact data[END_REF]:

σ =                       1
                     
We use the Bayesian hierarchical framework provided by the PyRoss library [START_REF] Adhikari | Inference, prediction and optimization of non-pharmaceutical interventions using compartment models: the PyRoss library[END_REF] to decompose contact rates into 'work', 'home', 'school', and 'other' [START_REF] Prem | Projecting social contact matrices in 152 countries using contact surveys and demographic data[END_REF]. The results are provided below, and also visualized in Figure 30.

σ H =                      
0.48 0.55 0.33 0.13 0.14 0.28 0.41 0.49 0.11 0.07 0.04 0.02 0.01 0.00 0.00 0.00 0.26 0.92 0.52 0.12 0.03 0.17 0.45 0.58 0.32 0.07 0.03 0.01 0.00 0.00 0.00 0.00 0.17 0.54 1.08 0.39 0.04 0.01 0.22 0.59 0.49 0.13 0.04 0.02 0.00 0.01 0.00 0.00 0.09 0.15 0.42 0.98 0.13 0.03 0.06 0.25 0.42 0.21 0.07 0.05 0.01 0.02 0.00 0.00 0.17 0.08 0.07 0.36 0.80 0.21 0.07 0.04 0.16 0.28 0.08 0.09 0.02 0.01 0.00 0.00 0.49 0.30 0.04 0.07 0.13 0.66 0.21 0.03 0.01 0.05 0.17 0.11 0.03 0.00 0.00 0.00 0.32 0.47 0.27 0.08 0.05 0.09 0.64 0.15 0.03 0.00 0.01 0.04 0.02 0.00 0.00 0.00 0.38 0.70 0.56 0.20 0.02 0.01 0.09 0.59 0.15 0.00 0.02 0.01 0.01 0.01 0.00 0.00 0.17 0.52 0.73 0.42 0.07 0.00 0.09 0.19 0.46 0.10 0.02 0.00 0.03 0.01 0.01 0.00 0.13 0.15 0.33 0.71 0.35 0.08 0.01 0.08 0.07 0.53 0.11 0.03 0.01 0.00 0.00 0.01 0.12 0.10 0.18 0.26 0.31 0.07 0.07 0.04 0.08 0.06 0.37 0.13 0.01 0.00 0.00 0.00 0.02 0.01 0.11 0.28 0.23 0.18 0.06 0.02 0.01 0.08 0.09 0.39 0.10 0.00 0.00 0.00 0.02 0.00 0.05 0.03 0.09 0.10 0.11 0.07 0.08 0.02 0.04 0.11 0.48 0.06 0.01 0.00 0.05 0.07 0.11 0.15 0.02 0.01 0.02 0.10 0.22 0.03 0.02 0.05 0.08 0.51 0.05 0.00 0.00 0.02 0.16 0.25 0.01 0.00 0.00 0.00 0.30 0.08 0.03 0.00 0.06 0.14 0.16 0.09 0.02 0.00 0.04 0.00 0.02 0.00 0.04 0.00 0.07 0.05 0.08 0.00 0.00 0.00 0.11 0.27

                      σ S =                      
0.97 0.15 0.01 0.03 0.01 0.09 0.13 0.09 0.04 0.03 0.00 0.00 0.01 0.00 0.00 0.00 0.24 2.35 0.06 0.00 0.02 0.04 0.06 0.06 0.06 0.05 0.04 0.00 0.00 0.00 0.00 0.00 0.00 1.12 2.56 0.12 0.01 0.05 0.04 0.12 0.09 0.07 0.03 0.02 0.01 0.00 0.00 0.00 0.04 0.08 1.17 4.14 0.06 0.12 0.08 0.08 0.07 0.05 0.04 0.03 0.00 0.00 0.00 0.00 0.00 0.13 0.00 0.27 0.23 0.01 0.02 0.03 0.01 0.01 0.01 0.00 0.00 0.00 0.00 0.00 0.11 0.07 0.01 0.00 0.00 0.06 0.04 0.06 0.04 0.03 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.10 0.01 0.01 0.01 0.00 0.04 0.00 0.02 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.04 0.17 0.07 0.03 0.00 0.02 0.05 0.07 0.07 0.05 0.00 0.01 0.00 0.01 0.00 0.00 0.07 0.10 0.03 0.00 0.04 0.06 0.03 0.06 0.02 0.01 0.01 0.01 0.00 0.00 0.00 0.00 0.02 0.00 0.02 0.21 0.00 0.00 0.05 0.01 0.07 0.05 0.06 0.04 0.01 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.02 0.00 0.00 0.04 0.00 0.00 0.00 0.00 0.05 0.14 0.05 0.00 0.00 0.00 0.03 0.00 0.15 0.03 0.00 0.08 0.00 0.00 0.00 0.00 0.00 0.13 0.00 0.00 0.02 0.01 0.05 0.11 0.02 0.03 0.01 0.01 0.00 0.00 0.00 0.00 0.00 0.06 0.04 0.00 0.00 0.00 0.04 0.03 0.00 0.00 0.00 0.03 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.06 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

                      σ W =                      
0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.02 0.00 0.00 0.00 0.10 0.00 0.04 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.01 1.08 0.87 0.28 0.11 0.28 0.15 0.48 0.32 0.06 0.02 0.00 0.00 0.00 0.00 0.00 0.02 0.44 0.64 0.71 0.56 0.71 0.35 0.37 0.17 0.16 0.02 0.00 0.00 0.00 0.00 0.00 0.01 0.49 0.61 0.84 0.77 0.62 0.95 0.54 0.55 0.09 0.02 0.00 0.00 0.00 0.00 0.00 0.05 0.10 0.59 0.63 0.65 1.05 0.58 0.69 0.27 0.21 0.02 0.00 0.00 0.00 0.00 0.00 0.02 0.33 0.37 0.59 0.54 0.70 0.62 0.66 0.52 0.19 0.01 0.00 0.00 0.00 0.00 0.00 0.01 0.23 0.39 0.53 0.55 0.83 0.88 0.77 0.35 0.25 0.02 0.00 0.00 0.00 0.00 0.00 0.26 0.28 0.39 0.76 0.76 0.68 0.85 1.04 0.42 0.19 0.03 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.12 0.43 0.47 0.48 0.88 0.52 0.42 0.16 0.05 0.00 0.00 0.00 0.00 0.00 0.01 0.09 0.33 0.36 0.31 0.25 0.29 0.33 0.30 0.14 0.01 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.03 0.07 0.03 0.09 0.07 0.05 0.06 0.04 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 Note that these baseline values are modulated to reflect regional differences, using the approach described in Section 3.

                      σ O =                      

Figure 2 :

 2 Figure 2: Regional multiplier d r for social contact matrix, implied by epidemic dynamics pre-lockdown (before March 23, 2020).

  (a) Pre-lockdown: before March 23, 2020. (b) During lockdown: 23/03-10/06/2020.

Figure 3 :

 3 Figure 3: Inter-regional mobility across London boroughs.

Figure 4 :

 4 Figure 4: Fatalities in England: comparison of model with data. Grey dashed line: separation between estimation sample and test data; orange line: model simulation; blue dot: in-sample data; green triangle: out-of-sample data.

Figure 5 :

 5 Figure 5: Cumulative reported cases in selected regions. Grey dashed line: separation between estimation sample and test data; orange line: average of 50 simulated scenarios; blue dot: in-sample data; green triangle: out-of-sample data.

Figure 6 :

 6 Figure 6: Estimate of case reporting ratio π(t) based on a comparison of fatalities and reported cases.

  (a) Number of infected I t . (b) Reported cases.

Figure 7 :

 7 Figure 7: Example of latent progression of the epidemic with zero reported case for 60 consecutive days (red shaded area for scenario 1 and blue shaded area for scenario 2). Reporting probability is π = 4.5%.

Figure 8 :

 8 Figure 8: Probability of observing a second peak after a period with no cases reported.

  (a) Impact of lockdown duration T . (b) Impact of compliance level m.

Figure 9 :

 9 Figure 9: Fatalities against social cost for different T and m values. (Results for low symptomatic ratios).

  (a) Cumulative fatalities in England. (b) Dynamics of I t in England.

Figure 10 :

 10 Figure 10: Comparison of three policies: blue dotted line: m = 0.5 and T = 335; orange dashed line: m = 0.5 and T = 105; green solid line: m = 0.4 and T 2 = 105. Average across 50 simulated scenarios.

Figure 11 :

 11 Figure 11: Trade-off between fatalities and social cost for a T -day lockdown followed by social distancing (0.2 ≤ m ≤ 1, 105 ≤ T ≤ 335): low symptomatic ratio (orange) and high symptomatic ratio (blue).

Figure 12 :

 12 Figure 12: Lockdown of 105 days followed by social distancing (m = 0.3): regional mortality per 100,000 inhabitants.

  (a) Level of social contact (d r ) against COVID-19 mortality (per 100,000 inhabitants).(b) Regional dynamics of symptomatic infections (I t ):North Northamptonshire (UKF25), Birmingham (UKG31), Berkshire (UKJ11) and East Cumbria (UKD12). Dotted lines denote the secondpeak times.

Figure 13 :

 13 Figure 13: Regional outcomes for lockdown of 105 days followed by social distancing (m = 0.3).

( a )

 a Impact of the shielding measure for senior citizens. (b) Social distancing outside work and school: impact of the parameter u O . (c) Social distancing at work: impact of the parameter u W . (d) Social distancing at school: impact of the parameter u S .

Figure 14 :

 14 Figure 14: Efficiency plot of social cost against projected fatalities for the shielding measure and various values of u S , u W , and u O (u H = 1 and T = 105).

Figure 15 :

 15 Figure 15: 'Open pubs' versus 'open schools' policy.

  (a) Symptomatic infections in England. (b) Fatalities in England.

Figure 16 :

 16 Figure 16: Comparison of policies with and without shielding in place, u = (1, 0.0, 1.0, 0.5). Blue: no shielding; orange: shielding in place.

  (a) Efficient policies without shielding: 'School and Work' and 'No Pubs'. (b) Policies 'School and Work' and 'No Pubs' are not efficient when shielding is applied. (c) Efficient policies with shielding: 'School and Work' and 'No work'. (d) Policies 'School and Pubs' and 'No work' are not efficient when shielding is removed.

Figure 17 :

 17 Figure 17: Impact of shielding on the efficiency frontier.

( a )

 a Influence of the threshold B on to resume lockdown. (b) Influence of the threshold B off to lift lockdown.

Figure 18 :

 18 Figure 18: Social cost against fatalities when m = 0.5.

( a )

 a Daily reported cases in England. (b) Daily reported cases in Leicester.

Figure 19 :

 19 Figure 19: Simulation of reported cases in England and Leicester under a centralised triggering policy with B on = 4, B off = 0.4 × R on , m = 0.5 and no shielding.

Figure 20 :

 20 Figure 20: Comparison between triggering thresholds B on = 10 and B on = 2.

  (a) Dynamics of I t in England. (b) Cumulative fatalities in England.

Figure 21 :

 21 Figure 21: Reporting probabilities π = 4.5% (blue line) versus π = 20% (orange line) and π = 50% (green line). Policy: B on = 6, B off = 0.2B on , and m = 0.5.

  (a) Symptomatic infections (I t ) in England. (b) Projected fatalities. (c) Projections for reported cases: England. (d) Projections for reported cases: Leicester.

Figure 22 :

 22 Figure 22: Impact of model granularity: projections for an adaptive policy with R on = 4, R off = 0.4 × R on , m = 0.5 and no shielding.

  (a) Influence of the threshold B on for triggering lockdown. (b) Influence of the threshold B off for lifting lockdown.

Figure 23 :

 23 Figure 23: Decentralised confinement triggered by regional daily case numbers: social cost versus fatalities (m = 0.5).

  (a) Low symptomatic ratios.(b) High symptomatic ratios.

Figure 24 :

 24 Figure 24: Efficiency analysis for centralised (blue) and decentralised (orange) adaptive mitigation policies. Outcomes are averaged across 100 simulated scenarios.

  (a) Daily reported cases in Leicester.(b) Daily reported cases in England.

Figure 25 :

 25 Figure 25: Reported cases in England and Leicester under a decentralised triggering policy: average of 50 simulated scenarios with B on = 4, B off = 0.4 × R on , m = 0.5, no shielding.

Figure 26 :

 26 Figure 26: Fatalities per 100,000 inhabitants for centralised (left) verus regional (right) adaptive mitigation policies. Same triggering thresholds are used in both cases: B on = 4 and B off = 0.4B on .

Figure 27 :

 27 Figure 27: Number of infected individuals under under centralised (blue dashed line) and decentralised (orange solid line) policies. Same triggering thresholds are used in both cases: B on = 4 and B off = 0.4B on .

Figure 28 :

 28 Figure 28: Efficiency plot: pre-planned versus adaptive mitigation policies.

Figure 28

 28 Figure 28 compares the health outcome and social cost of the efficient policies considered in Sections 5.2, 6.1 and 6.2. The efficient frontier of pre-planned policies are among policies with u S ∈ {0, 0.5, 1}, 0.2 ≤ u W ≤ 1.0 and 0.3 ≤ u O ≤ 1.0. For centralised and decentralised policies, m = 0.25, 0.5, 0.75, 1; B on = 2, 4, 6, 8, 10; and B off = p × B on with p = 0.2, 0.4, 0.8.We observe that

  (a) I t in Mid Lancashire. (b) I t in York. (c) I t in Leicester. (d) I t in Birmingham.

Figure 29 :

 29 Figure 29: Regional comparison of pre-planned and adaptive mitigation policies.

0. 26

 26 0.10 0.05 0.13 0.19 0.26 0.19 0.34 0.31 0.07 0.15 0.11 0.06 0.04 0.04 0.01 0.18 0.77 0.13 0.09 0.06 0.22 0.23 0.18 0.26 0.17 0.16 0.06 0.09 0.06 0.04 0.01 0.15 0.35 0.88 0.31 0.09 0.21 0.09 0.17 0.26 0.13 0.11 0.05 0.06 0.02 0.04 0.05 0.04 0.26 0.68 1.67 0.27 0.20 0.16 0.37 0.25 0.21 0.16 0.08 0.06 0.04 0.01 0.00 0.15 0.04 0.18 0.96 0.75 0.37 0.24 0.30 0.16 0.32 0.09 0.14 0.08 0.03 0.02 0.05 0.25 0.09 0.08 0.20 0.99 0.85 0.46 0.27 0.29 0.32 0.20 0.10 0.09 0.05 0.02 0.00 0.16 0.17 0.09 0.18 0.32 0.44 0.67 0.44 0.23 0.23 0.22 0.25 0.10 0.06 0.01 0.02 0.13 0.22 0.10 0.10 0.24 0.30 0.31 0.52 0.51 0.25 0.15 0.20 0.21 0.10 0.11 0.04 0.00 0.25 0.15 0.15 0.27 0.18 0.36 0.35 0.39 0.39 0.24 0.20 0.10 0.02 0.05 0.00 0.00 0.01 0.03 0.13 0.19 0.13 0.17 0.39 0.46 0.56 0.32 0.12 0.16 0.10 0.06 0.06 0.03 0.03 0.10 0.39 0.30 0.48 0.24 0.46 0.41 0.60 0.23 0.29 0.20 0.14 0.12 0.01 0.11 0.06 0.06 0.11 0.36 0.57 0.55 0.46 0.49 0.22 0.46 0.64 0.40 0.20 0.12 0.11 0.00 0.09 0.07 0.03 0.29 0.22 0.28 0.31 0.32 0.27 0.28 0.40 0.26 0.09 0.11 0.08 0.02 0.11 0.06 0.06 0.11 0.49 0.29 0.46 0.42 0.43 0.39 0.52 0.29 0.22 0.14 0.06 0.07 0.03 0.09 0.24 0.51 0.44 0.24 0.14 0.49 0.36 0.34 0.47 0.48 0.55 0.24 0.25 0.00 0.00 0.04 0.11 0.07 0.04 0.13 0.22 0.20 0.41 0.35 0.00 0.47 0.00 0.19 0.47

Figure 30 :

 30 Figure 30: Baseline social contact matrices with 16 age groups.

Table 1 :

 1 Age group distribution for England, 2019. Source: Eurostat[START_REF]Population on 1 January by age group, sex and NUTS 3 region[END_REF].

	Age group	[0,5)	[5,10) [10,15) [15,20) [20, 25) [25, 30) [30, 35) [35, 40)
	Size (millions) 3.3	3.5	3.3	3.1	3.5	3.8	3.8	3.7
	Fraction	5.9%	6.3%	5.9%	5.5%	6.2%	6.8%	6.8%	6.6%
	Age group	[40,45) [45,50) [50,55) [55,60) [60, 65) [65, 70) [70, 75) [75, 100)
	Size (millions) 3.4	3.8	3.9	3.6	3.1	2.8	2.8	4.7
	Fraction	6.0%	6.7%	7.0%	6.5%	5.5%	5.0%	4.9%	8.4%

Table 2 :

 2 Summary of regions with the highest regional multiplier d

		d r	Density	Inward mobility Outward mobility Population
	UKC12 1.80 925.9 (#59) 17.6% (#82)	19.1% (#105)	276988 (#102)
	UKI62	1.68 4518.4 (#67) 16.6% (#87)	43.2% (#12)	389473 (#59)
	UKG32 1.64 1205.5 (#59) 14.5% (#96)	46.2% (#6)	215055 (#120)
	UKI53	1.62 6161.9 (#11) 23.4% (#50)	41.6% (#16)	587575 (#25)
	UKC23 1.52 2026.9 (#49) 13.7% (#103)	24.0% (#78)	277733 (#99)

r for social contact matrix. Number in brackets signify the respective rank of the measured quantity.

Table 3 :

 3 Age-dependent symptomatic ratios, p. Source: Office for National Statistics

				[10,15) [15,20) [20, 25) [25, 30) [30, 35) [35, 40)
	p low	0.075	0.075	0•05	0•05	0.15	0.15	0.21	0.21
	p high	0.15	0.15	0.1	0.1	0.3	0.3	0.42	0.42
	Age group [40,45) [45,50) [50,55) [55,60) [60, 65) [65, 70) [70, 75) [75, 100)
	p low	0.23	0.23	0.28	0.28	0.41	0.41	0.375	0.375
	p high	0.45	0.45	0.56	0.56	0.82	0.82	0.75	0.75

Table 4 :

 4 Summary of parameters for the COVID-19 model.

	Symbol Value	Source

Table 5 :

 5

Age-dependent infection fatality rates. Source: Verity et al.

[START_REF] Verity | Estimates of the severity of coronavirus disease 2019: a model-based analysis[END_REF]

.

Table 7 :

 7 Outcomes for policies represented in Figure

Table 8 :

 8 Impact of school closures and social distancing at schools: outcomes averaged across 50 simulated scenarios, u H

Table 10 :

 10 Summary of outcomes for different policies, starting from the same initial conditions on July 4, 2020.

		t	A t	Fatalities max I t	Social Projected
	Policy				(2 nd peak) cost	fatalities
		(Aug 1) (Aug 1) (Aug 1)		(10 11 ) (1000 days)
	Confinement followed	47,400	188,700 39,400	255,700	3.8	96,600
	by strict social dis-
	tancing (m=0.3)
	Confinement followed	98,400	392,400 40,700	766,800	2.9	146,100
	by moderate social
	distancing (m=0.5)
	Pre-planned	84,700	360,100 45,500	613,300	2.9	122,900
	Centralised triggering 80,300	321,200 40,500	423,200	3.0	133,500
	Decentralised trigger-	80,100	320,200 40,400	292,200	3.0	122,100
	ing
	Decentralised trigger-	55,000	266,100 39,700	267,700	3.4	65,900
	ing and shielding
	'Protect Lives'	25,900	118,400 39,600	63,900	4.3	51,700

Table 11

 11 details the used mapping between Upper Tier Local Authority (UTLA) region codes and the Nomenclature of Territorial Units for Statistics at level 3 (NUTS-3) codes7 . If more than one UTLA region falls within the boundary of a single NUTS-3 region, the data is then aggregated. On the other hand, if a single UTLA region lies within more than one NUTS-3 region, the data is distributed among NUTS-3 regions in proportion to the total number of people living in each region.

	UTLA Code UTLA Region Name	NUTS-3 Code Mapping
	E06000001	Hartlepool	UKC11
	E06000002	Middlesbrough	UKC12
	E06000003	Redcar and Cleveland	UKC12
	E06000004	Stockton-on-Tees	UKC11
	E06000005	Darlington	UKC13
	E06000006	Halton	UKD71
	E06000007	Warrington	UKD61
	E06000008	Blackburn with Darwen	UKD41
	E06000009	Blackpool	UKD42
	E06000010	Kingston upon Hull, City of	UKE11
	E06000011	East Riding of Yorkshire	UKE12
	E06000012	North East Lincolnshire	UKE13
	E06000013	North Lincolnshire	UKE13
	E06000014	York	UKE21
	E06000015	Derby	UKF11
	E06000016	Leicester	UKF21
	E06000017	Rutland	UKF22
	E06000018	Nottingham	UKF14
	E06000019	Herefordshire, County of	UKG11
	E06000020	Telford and Wrekin	UKG21
	E06000021	Stoke-on-Trent	UKG23
	E06000022	Bath and North East Somerset UKK12
	E06000023	Bristol, City of	UKK11
	E06000024	North Somerset	UKK12
	E06000025	South Gloucestershire	UKK12
	E06000026	Plymouth	UKK41
	E06000027	Torbay	UKK42
	E06000030	Swindon	UKK14
	E06000031	Peterborough	UKH11
	E06000032	Luton	UKH21
	E06000033	Southend-on-Sea	UKH31
	E06000034	Thurrock	UKH32
	E06000035	Medway	UKJ41
	E06000036	Bracknell Forest	UKJ11
	E06000037	West Berkshire	UKJ11
	E06000038	Reading	UKJ11
	E06000039	Slough	UKJ11
	E06000040	Windsor and Maidenhead	UKJ11

Table 11 :

 11 Mapping between the Upper Tier Local Authority (UTLA) regions and the Nomenclature of Territorial Units for Statistics at level 3 codes (NUTS-3).

  .96 0.91 0.87 0.75 1.02 1.41 1.87 0.75 0.64 0.51 0.26 0.32 0.33 0.34 0.34 0.37 0.57 0.68 0.88 0.81 0.82 0.90 0.82 0.74 0.98 0.46 0.31 0.28 0.76 0.24 0.20 0.35 0.32 0.50 0.69 0.67 0.66 0.66 0.66 0.91 1.17 0.73 0.43 0.20 0.46 0.24 0.40 0.25 0.25 0.38 0.54 0.65 0.80 0.82 0.65 0.53 0.91 0.65 0.55 0.30 0.66 0.19 0.31 0.29 0.39 0.32 0.52 0.42 0.74 0.60 0.39 0.42 0.63 0.64 0.70 0.52 0.20 0.15 0.11 0.28 0.40 0.26 0.29 0.17 0.28 0.66 0.55 0.44 0.34 0.40 0.60 0.59 0.57 0.01 0.18 0.12 0.24 0.14 0.10 0.24 0.32 0.43 0.28 0.60 0.39 0.45 0.12 0.29 0.86

	.92 0.81 0.47 0.30 0.49 0.79 0.89 1.07 0.44 0.27 0.35 0.27 0.22 0.15 0.10 0.02
	0.78 6.64 1.24 0.58 0.49 0.72 1.09 1.40 1.10 0.36 0.35 0.23 0.35 0.24 0.07 0.23
	0.42 1.16 6.85 1.30 0.25 0.37 0.57 1.10 1.18 0.64 0.35 0.35 0.2 0.2 0.17 0.14
	0.26 0.52 1.26 6.71 1.24 0.72 0.47 0.87 0.97 0.97 0.52 0.31 0.2 0.26 0.24 0.28
	0.43 0.44 0.24 1.26 2.59 1.36 0.84 0.76 0.83 0.93 0.63 0.5 0.31 0.22 0.16 0.17
	0.73 0.68 0.38 0.76 1.42 1.83 1.13 0.92 0.9 0.92 0.85 0.72 0.45 0.38 0.18 0.12
	0.73 0.93 0.53 0.44 0.79 1.02 1.67 1.27 0.98 0.72 0.7 0.63 0.48 0.27 0.09 0.27
	0.79 1.06 0.89 0.74 0.63 0.74 1.12 1.5 1.27 0.86 0.63 0.55 0.53 0.43 0.14 0.31
	0.32 0.83 0.97 0.83 0.69 0.73 0.87 1.28 1.35 1.21 0.7 0.55 0.55 0.35 0.33 0.43
	0.24 0.32 0.62 0

We do recognise that the implementation of such shielding measures may be extremely challenging in practice.

See https://www.legislation.gov.uk/ukpga/2020/7/contents/enacted.

See https://www.oxford-covid-19.com/

See https://ourworldindata.org/coronavirus-testing.

See https://www.oxford-covid-19.com/.

This model was implemented in a previous version of this paper: https://www.medrxiv.org/ content/10.1101/2020.08.26.20182477v2.

See https://geoportal.statistics.gov.uk/datasets/c893dfece45f465f857ac34641041863_0 for a lookup table used in the process of mapping.