## Abstract

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The Chinese comedy Never Say Die has brought in an impressive $326 million worldwide to date. "Perhaps it is no big deal whether or not you are in the group discussion, but if you were absent, 670,000 people would be absent, if you were silent, 670,000 people would be silent." ['hedwei] RSPCA inspectors found 13 dead cats and an emaciated survivor when they raided the property in Adelaide, South Australia, in September 2015. NBS senior statistician Sheng Guoqing attributed the slowdown mainly due to a 1.4-percent decline in food prices, which were down for the first time in 15 years. Golden State equaled the 1957-58 Celtics as the only defending champions to win their initial 14 games. The Warriors are one of five teams in NBA history to begin 14-0, and will travel to Denver looking to keep it going Sunday. 10. Run the Jewels “Run the Jewels 2” (Mass Appeal) Speaking of urgent and stinging, Killer Mike and El-P, veteran underground rappers from different scenes, found firm traction on their second round as Run the Jewels. Their flow is strong and their focus is furious on an album that calls out power structures but lets nobody off the hook. 4.菲亚特 品牌喜爱度：-7%/排名：85 Lori Steele It's time to make New Year resolutions for traveling! Backpacker bible Lonely Planet has published its annual list of best value destinations in 2017 for people looking for wallet-friendly sightseeing experiences. Take a look. adj. 社会的，社交的 同时，2016年应届毕业生选择就业的比例有所上升，比例由2015年的71.2%提高到 75.6%。 *喜剧类最佳客串女演员：蒂娜?菲(Tina Fey)和艾米?波勒(Amy Poehler)，《周六夜现场》(Saturday Night Live) Here are five things consumers and investors can count on (probably) in 2015: We learned about the causes and consequences of rising obesity around the world. A heartfelt coming-of-age story that perfectly captures the bittersweet transition from adolescence to dawning adulthood... 虽然她五个月之前才刚学会走路，但是这个被人们称作“宝贝碧昂斯”的小宝宝已经开始在各种盛会的舞台上展露风采了。 ## I. Introduction 1896年，美国首任总统的夫人玛莎·华盛顿的画像印在一美元银圆券上，此后美元纸币上再没有出现过女性画像。 无论你是否认为Snapchat配得上Facebook开出的30亿美元报价，有一点确定无疑：所谓的“阅后即焚”（内容被接收数秒后自动消失）社交网络拥有广阔的市场应用空间。此外，与流行的看法相反，Snapchat传播的内容不仅仅是色情短信和限制级的自拍照片，尽管它也的确是传播这类信息的最佳平台。随着主流社交平台的内容变得越来越商品化，以Snapchat为代表的“阅后即焚”平台在某种程度上重新恢复了社交媒体本应具备的趣味性和自发性。就像现实生活中的互动交流一样——创意自由流动，一般不用担心一切事情都被记录下来留给子孙后代观瞻，传遍整个世界——SnapChat提供了一个真实的、未经过滤的交流渠道。孩子们真的很喜欢它。Facebook公司自己的首席财务官上个月已正式承认，青少年使用Facebook的次数正在下降.而据坊间传言，驻足于Snapchat的青少年数量正在呈爆炸式增长。 petroleum 单词fare 联想记忆： 5. The 2012 Ig Nobel Chemistry Prize 自2005年以来，年度最佳商业图书奖的评奖标准一直是“对现代商业问题提供了最令人叹服且最有趣味的深刻见解”。2014年的获奖者是托马斯皮凯蒂(Thomas Piketty)的《21世纪的资本》(Capital in the Twenty-First Century)。 The 62 universities account for 12.4% of the list. The only country with a larger number of universities listed is the US, which accounts for 27.8%. Before that, she starred in popular sitcom, and has also made a name for herself in films. n. 购买，购买的物品 Today's and tomorrow's technology sits on top of multiple layers, every one of which is changing and has to inter-operate with others. This makes our gadgets, the internet of things, phones and laptops unstable. And it makes consumers irritated. How many of your apps actually work--and actually make life easier, faster or more fun? I'd expect to see consumer cynicism grow, as delight is overtaken by disappointment. This will put pressure on hardware and software developers to deliver that most boring of qualities: reliability. Kunis was No. 9 on FHM's list last year. The scientists’ analysis comes only a month after nearly 200 governments struck a new climate agreement in Paris that aims to stop global temperatures from rising more than 2C from pre-industrial levels, and ideally limit warming to 1.5C. 威廉凯特夫妇多次让加拿大人等候多时，例如，抵达加拿大时，两人花了20分钟时间才从飞机上下来接受加方欢迎团队的问候。 美国上市交易相比2016年（多年来最疲弱的年份之一）出现起色。根据Dealogic的统计，今年迄今已有112笔IPO得到定价，而去年这个时候只有63笔。发行金额翻了一番多，达到316亿美元。 5Japan The education sector has remained largely unchanged by online service delivery — but could be transformed dramatically in 2018. The measures required to inhibit disease transmission can be very costly in economic and social terms, including depression and other ‘diseases of despair’ among the millions who lose their jobs. These costs must be weighed against the medical benefits of intervention. The decision when to intervene and on what scale is a classic optimal control problem. This paper explores the choices facing the government using a simple mathematical model that is inspired by optimal control theory.1 For clarity we omit details of the full optimal control model which are to be found in Rowthorn (2020). The paper complements the theoretical analysis with some illustrative simulations. These simulations should not be taken literally but they indicate some of the issues and orders of magnitude involved. The economic literature on the optimal control of disease is sparse and its models mostly deal with individual behaviour and the externalities of individual decision-making with regard to treatment, vaccination, or social distancing.2 These are not our concern here. Our interest is in the cost–benefit analysis of large-scale interventions such as lockdowns. This involves an approach that is unusual in the existing optimal control literature on disease. Costs and benefits in existing optimal control models are typically functions of the health status of individuals, computed by assigning values or weights to individuals according to their health status. This is a procedure followed here. However, unlike these models we also make an explicit allowance for the more general costs of comprehensive interventions such as lockdowns. Such costs depend on the scale and type of intervention but they are not linked in a direct way to the health status of individuals. These costs are given a central role in this paper. Since the outbreak of the epidemic there has been a spate of thought-provoking articles on economic aspects of COVID-19. Two, in particular, deserve special mention. Acemoglu et al. (2020) examine targeted lockdowns in a multi-group SIR model where infection, hospitalization, and fatality rates vary between groups—in particular between the ‘young’, ‘the middle-aged’, and the ‘old’. They also allow for the fact that lockdown damages the economy and reduces the productivity of non-infected members of the workforce. Their paper, incidentally, contains a good review of the recent literature. Giordano et al. (2020) draw on the experience of the Italian epidemic. Their model distinguishes between detected and undetected infection cases, and between cases with different severity of illness. They argue that social-distancing measures are necessary and effective, and should be promptly enforced at the earliest stage. They also argue that lockdown measures can only be relieved safely when an effective system of testing and contact tracing is in place. These are both excellent articles, and nothing in the present article contradicts their findings. A system of testing and tracing is most effective when the number of people to be tested or contacted is relatively small. It may be feasible to test small subgroups of the population on a frequent basis and trace their contacts if they test positive (Cleevely et al., 2020). Care home workers are an example. However, a policy of targeted testing is of limited use as a means of infection control if the disease is widespread, since most of the infected population will not be in the groups selected for testing. The alternative is universal and frequent random testing, but this is likely to be prohibitively expensive, as Cleevely et al. point out. If the scale of infection is too large for the system of testing and tracing to handle unaided, and if there is currently no treatment or vaccine, some form of social distancing will be required. This is the case in the present article. Indeed, our basic model goes further. It assumes that a perfect vaccine will become available on a known date in the future and that prior to this date there exists no testing and tracing regime at all. There is also no currently available treatment for the disease. Hence social distancing is the only feasible means of disease control. However, in one simulation we consider a scenario in which a test and trace regime is established in advance of vaccination. The analysis assumes that the scale of social distancing is determined by government fiat alone. In reality, as the disease spreads and people become aware of the risks involved, there will be a degree of voluntary social distancing. As a result, the more apocalyptic predictions of what would happen without draconian intervention may be wide of the mark. The implications of endogenous behaviour are not explored here, but are the subject of another paper (Ormerod et al., 2020). The theoretical section of this paper was written the day after Prime Minister Johnson announced a full-scale lockdown. The first batch of simulations was completed shortly thereafter with the aim of influencing the ongoing policy debate. The paper including simulations was published in mid-April in the CEPR real-time online journal Covid Economics (Rowthorn, 2020). These simulations were comprehensively revised in May and June for this issue of the Oxford Review of Economic Policy. By the time the journal appears, the die will have been cast and the actual policy choices of the government will be there for all to see. However, we hope that this paper will continue to provide a useful framework for thinking about the cost–benefit analysis of disease control. Our study is based on a standard but simple epidemiological model, and should therefore be regarded as presenting a methodological framework rather than giving policy prescriptions. ## II. The model The analysis in this paper uses a modified version of the standard SIR model of disease propagation. Ignoring births and deaths from non-COVID-19 causes, the initial population will divide in the future into three groups of people: susceptible, infected, and removed—denoted, respectively, by S(t), I(t), and R(t). The removed group includes people who have died from the disease. They are denoted by D(t). The population at the start of the epidemic is normalized to 1, so these various quantities can be interpreted as shares. Individuals who are infected remain infectious until they recover or die. Infected individuals who recover acquire complete immunity, so the journey from S(t) via I(t) to R(t) is in one direction only. Writing for a Comedy Series: Aziz Ansari and Alan Yang, “Master of None” (“Parents”) $dS(t)dt=−β(t)S(t)I(t)$ (1) $dI(t)dt=β(t)S(t)I(t)−γI(t)$ (2) $dR(t)dt=γI(t)$ (3) $dD(t)dt=δγI(t)$ (4) $S(0)=S0≥0$ (5) $I(0)=I0≥0$ (6) $R(0)=R0≥0$ (7) $D(0)=D0≥0$ (8) $S(t)+I(t)+R(t)=1$ (9) where $γ$ and $δ$ are constant. These constants indicate, respectively, the rate at which infected individuals cease to be infectious, and the probability that an infected individual will die. Note that there are only two genuinely independent state variables in this model. For example, if the trajectories of I(t) and R(t) are known, the trajectories of S(t) and D(t) are uniquely determined by equations (1) and (4).. Equation (1) indicates how the pool of susceptible individuals is depleted by the outflow of newly infected individuals. Assuming that social encounters are random, the probability that a susceptible individual will be infected in a given unit of time is proportional to the prevalence of infection in the population. Equation (2) indicates how the pool of currently infected individuals is augmented by the inflow of newly infected individuals and depleted by the outflow of infected individuals who recover or die. The rate of outflow is $γI(t)$ of whom a fraction $δ$ are dead. Equation (3) indicates how the removed category is augmented by the inflow of newly recovered or dead individuals. The coefficient $β(t)$ in equation (1) is a variable which depends on the current intensity of social interaction. The intensity of social interaction depends, in turn, on the measures that the government puts in place to inhibit the spread of the disease. Specifically, it is assumed that: $β(t)=(1−q(t))β0$ (10) where $q(t)$ is an index of policy severity. The effective reproduction rate of the disease is $r(t)=(1−q(t))S(t)r0$ (11) where $r0=β0γ$ The number $r0$ indicates how many people the average infected person would infect in a situation where everyone was susceptible to the disease and there was no government intervention to control its spread. The number $r(t)$ indicates how many people are infected if there is government intervention and some people are immune. If $r(t)<1$, the prevalence of the disease will be diminishing through time. Government intervention comes at a cost $C(q(t))$ in the form of damage to the economy. This cost is independent of the number of people currently infected and is the result of society-wide measures to control the disease. It is in addition to the various costs arising directly from infection. The function $C(.)$ is assumed to be twice differentiable and such that (12) where $qmax<1$ is an upper limit beyond which it is not feasible to increase $q$. Thus, $C(q)$ is strictly convex over the relevant range. Examples are shown in Figure 1 which plots the function $C(q)=Cmax(qqmax)1+ϕ$ for various values of $ϕ>0$. When $q$ is close to zero, the marginal cost of intervention is low but rises steeply at higher values of $q$. These are realistic assumptions. Think of hand-washing at one end of the scale and the closure of shops, pubs, cafés, and restaurants at the other. Figure 1: Weekly cost C(q): £’000 per capita Figure 1: Weekly cost C(q): £’000 per capita The government is assumed to have perfect foresight. Thus, the entire control trajectory is decided at the very outset. The system is therefore open loop, whereas in a closed loop system the control is modified in the light of new information. We assume that an effective vaccine will become available at time T at negligible cost.3 For simplicity we also assume that a cure will become available at the same time as the vaccine at zero cost. The government chooses the trajectory q(t) so as to minimize the following quantity subject to the foregoing equations: $J=∫0T[πAI(t)+C(q(t))]dt+πD[D(T)−D(0)]$ (13) where $πA$ is the monetary value that planners assign to each person who is currently alive and infected and $πD$ the additional value they assign to those who die. “The steady and now record-breaking rise in average global temperatures is not an issue for another day,” Michael R. Bloomberg, the former New York mayor who is spending tens of millions of dollars of his personal fortune to battle climate change, said in a statement. “It’s a clear and present danger that poses major economic, health, environmental and geopolitical risks.” 说到底，我预期这些新的领导人将开始从更广泛的候选人中选拔人员，并把不同背景的人任命为自己的直接下属，从而驳斥那些根据当前的失衡来外推还需数十年才能实现领导层男女平衡的悲观者。 After all, during the last round of collections, the most striking pieces — from Louis Vuitton, Dior, Proenza Schouler, Narciso Rodriguez — had a streamlined momentum that wasn’t dragged down by any decade-related reference, or identity. They were clothes that went striding into the future, freed from the weight of the past. Justin Bieber剃发募捐 集得4万美金 1.It wasn’t me! – Because some things just aren’t worth taking credit for。 $E=∫0T[πAI(t)+C(q(t))]dt$ (14) Thus, $J=E+πD[D(T)−D(0)]$ (15) The monetary allowance for death $πD$ is not included in economic cost since most of the people who die from the disease are not economically active, so their death does not have a significant effect on output. Their cost of treatment prior to death is included in the $πA$ term which is an average for all infected individuals, including those who die and those who are asymptomatic or require no treatment. ## III. Simulation The optimization problem defined above has no explicit solution. In the absence of such a solution, the obvious procedure is to explore the properties of the system by means of numerical simulation. We solved the optimization problem by posing it as a nonlinear programming problem.4 ### Assumptions One of the things that makes Guardians such a great superhero franchise is its sense of humor—which is full of self-deprecation and sarcasm. It's not like you're going to watch this movie and laugh your way through it, but you'll at least have some moments of "ha ha, Groot," and "lolololol Chris Pratt." $C(q)=Cmax(qqmax)1+ϕ$ (16) where $Cmax$ is the cost of the maximum feasible lockdown and $ϕ>0$. The larger is the value of $ϕ$ the lower is the cost of the other interventions relative to lockdown and the greater is the economic benefit of moving to less draconian forms of intervention (see Figure 1). Our simulations use parameter values that we hope are realistic, although given the paucity of reliable data, a fair amount of guesswork is involved. The simulations take 1 April 2020 as their notional starting point for optimization, although the epidemic is assumed to have started some weeks earlier.. The lockdown was officially announced on 23 March, but it was not until 1 April that it had a clear effect on the number of people infected (King’s College, 2020). The unit of time is a week and the time horizon is T=52. The monetary unit of account is thousands of UK pounds. There are initially 2m people currently infected and therefore infectious. In addition a further 1.4m have had the disease and recovered or died. The initial conditions are thus I0=0.030, R0=0.021. The death rate is $δ=0.7$ per cent. The UK population is assumed to be 66.8m. The parameters in the baseline scenario have the following values: $β0=4.8$, $γ=1.6$, $Cmax=0.20$,$πA=2$,$πD=2,000$. Infected individuals cease to be infectious at an exponential rate of –1.6 per week, which implies that after 2 weeks 96 per cent are no longer infectious. They have either recovered or died. In the absence of intervention the net reproduction rate $r0=3$. The per capita weekly cost of full lockdown is £200 which is approximately 35 per cent of per capita GDP at factor cost, in line with the Office for Budget Responsibility’s prediction of what the lockdown might do to the UK economy (OBR, 2020). The values $πA=2$ and $πD=2,000$ assume that planners assign a monetary value of £2,000 per week to the average currently infected person, plus a further £2m to each fatality. The latter figure is what the UK Treasury assumes in project evaluation as the value of a prevented fatality (Dolan and Jenkins, 2020). To derive the path before 1 April, we assume that 4.7 weeks previously the state of the system was $S−4.7=1−10−8,I−4.7=10−8,R−4.7=0,D−4.7=0.$ From this starting point the system is assumed to grow freely with parameters $β=4.8,γ=1.6,δ=0.007$ until 1 April, when government intervention in our simulations begins. We ignore the limited interventions of the government before 1 April. ## IV. Results Tables 1 and 2 provide information about the optimum path under various scenarios. The numbers for deaths and total economic cost in these tables have been adjusted to include the pre-intervention weeks. This is a small adjustment which does not materially affect the results. It makes it easier to compare scenarios with different starting dates for intervention. Table 1: Optimal paths compared $ϕ$Value of life (£m)Duration of lockdown (weeks)Peak infection (m)Total deaths (thousands)Total economic cost (£ per capita) Do nothing 2.0 20.1 439.8 14,342 Baseline 2.0 5.3 2.0 59.9 6,589 Low relative cost 2.0 1.8 2.0 67.1 4,811 High relative cost 2.0 7.9 2.0 57.6 7,660 Long time horizon: unconstrained 2.0 7.0 270.5 1,916 constrained 2.0 3.3 268.1 2,093 Early start 2.0 0.9 0.3 8.3 7,360 Test & trace 2.0 6.0 2.0 60.1 3.551 $ϕ$Value of life (£m)Duration of lockdown (weeks)Peak infection (m)Total deaths (thousands)Total economic cost (£ per capita) Do nothing 2.0 20.1 439.8 14,342 Baseline 2.0 5.3 2.0 59.9 6,589 Low relative cost 2.0 1.8 2.0 67.1 4,811 High relative cost 2.0 7.9 2.0 57.6 7,660 Long time horizon: unconstrained 2.0 7.0 270.5 1,916 constrained 2.0 3.3 268.1 2,093 Early start 2.0 0.9 0.3 8.3 7,360 Test & trace 2.0 6.0 2.0 60.1 3.551 Table 1: Optimal paths compared $ϕ$Value of life (£m)Duration of lockdown (weeks)Peak infection (m)Total deaths (thousands)Total economic cost (£ per capita) Do nothing 2.0 20.1 439.8 14,342 Baseline 2.0 5.3 2.0 59.9 6,589 Low relative cost 2.0 1.8 2.0 67.1 4,811 High relative cost 2.0 7.9 2.0 57.6 7,660 Long time horizon: unconstrained 2.0 7.0 270.5 1,916 constrained 2.0 3.3 268.1 2,093 Early start 2.0 0.9 0.3 8.3 7,360 Test & trace 2.0 6.0 2.0 60.1 3.551 $ϕ$Value of life (£m)Duration of lockdown (weeks)Peak infection (m)Total deaths (thousands)Total economic cost (£ per capita) Do nothing 2.0 20.1 439.8 14,342 Baseline 2.0 5.3 2.0 59.9 6,589 Low relative cost 2.0 1.8 2.0 67.1 4,811 High relative cost 2.0 7.9 2.0 57.6 7,660 Long time horizon: unconstrained 2.0 7.0 270.5 1,916 constrained 2.0 3.3 268.1 2,093 Early start 2.0 0.9 0.3 8.3 7,360 Test & trace 2.0 6.0 2.0 60.1 3.551 Table 2: 毁约涨价重现北京房地产市场 中介深夜排队取号 $ϕ$Value of life (£m)Duration of lockdown (weeks)Peak infection (m)Total deaths (thousands)Total economic cost (£ per capita) Baseline 2.0 5.3 2.0 59.9 6,589 High value of life 5.0 8.4 2.0 55.9 6,768 Low value of life: unconstrained 1.0 7.1 275.3 1,582 constrained 1.0 3.3 269.5 1,776 Nil value of life: unconstrained 8.8 333.0 1,139 constrained 3.3 335.1 1,327 $ϕ$Value of life (£m)Duration of lockdown (weeks)Peak infection (m)Total deaths (thousands)Total economic cost (£ per capita) Baseline 2.0 5.3 2.0 59.9 6,589 High value of life 5.0 8.4 2.0 55.9 6,768 Low value of life: unconstrained 1.0 7.1 275.3 1,582 constrained 1.0 3.3 269.5 1,776 Nil value of life: unconstrained 8.8 333.0 1,139 constrained 3.3 335.1 1,327 Table 2: 家居行业上演展会大战：血拼创意设计 $ϕ$Value of life (£m)Duration of lockdown (weeks)Peak infection (m)Total deaths (thousands)Total economic cost (£ per capita) Baseline 2.0 5.3 2.0 59.9 6,589 High value of life 5.0 8.4 2.0 55.9 6,768 Low value of life: unconstrained 1.0 7.1 275.3 1,582 constrained 1.0 3.3 269.5 1,776 Nil value of life: unconstrained 8.8 333.0 1,139 constrained 3.3 335.1 1,327 $ϕ$Value of life (£m)Duration of lockdown (weeks)Peak infection (m)Total deaths (thousands)Total economic cost (£ per capita) Baseline 2.0 5.3 2.0 59.9 6,589 High value of life 5.0 8.4 2.0 55.9 6,768 Low value of life: unconstrained 1.0 7.1 275.3 1,582 constrained 1.0 3.3 269.5 1,776 Nil value of life: unconstrained 8.8 333.0 1,139 constrained 3.3 335.1 1,327 Figure 2 shows what happens if the government does nothing to control the disease and restricts itself to the medical treatment of those infected. Within a few weeks, 90 per cent of the population has been infected and the cumulative death toll by the end of the year is 440,000 (Table 1). At the peak of the epidemic 20m people are currently infected and hence infectious. Figure 2: 靠农民工购房去库存 月收入至少得够5000元 Figure 2: 河北雄安新区政务服务中心投入运行 Under the Baseline scenario, the optimum response of the government is to impose a tight lockdown at the very beginning of the planning year. The lockdown lasts 5.3 weeks and brings the disease under control quite soon, although not before millions of people have been infected and many thousands have died (Figure 3). The eventual death toll is 60,000. The death toll is so high because the lockdown is not complete. Lockdown reduces the transmission of the disease but does not entirely prevent it. As a result there is inertia in the system. If the level of infection is already high when the lockdown is imposed, this will continue to be the case for some time thereafter. This is a good reason for acting swiftly before the disease has really taken hold. Once the lockdown is relaxed there is a prolonged period when it is optimal to maintain restrictions close to the minimal level required to contain the disease (Figure 4). During this period the effective reproduction rate r, although rising, is close to 1 (Figure 5). As the vaccination date draws near, restrictions are lifted at an accelerating pace until eventually they are largely abandoned. The result is a brief resurgence of infection which is halted by vaccination or treatment. Figure 3: “一盘散沙各自为营” 国内智能家居的真实写照 Figure 3: 卖场难觅婴儿家具 市民购买多依靠网络 Figure 4: Optimal path for q: baseline scenario Figure 4: Optimal path for q: baseline scenario Figure 5: Optimal path for r: baseline scenario Figure 5: Optimal path for r: baseline scenario Figure 6 compares the course of infection under various scenarios. Under the Early Start scenario, the lockdown is imposed a week earlier, with the result that infection and deaths are much lower. The eventual death toll is around 8,000 as compared to 60,000 under the Baseline scenario. The lockdown is also much shorter: 0.9 weeks as compared to 5.3 weeks. This comparison illustrates clearly the harm that may arise from even a short delay. Figure 6: 新型环保建材真的环保吗？ Figure 6: 不动产登记与房地产市场走向关系几何 In Rowthorn (2020), it was argued that extending the planning horizon does not greatly affect the results. This conclusion is not supported by the more sophisticated simulations reported here. Suppose the vaccine comes on stream after 2 years instead of one. The effect on the optimal path is dramatic. There is no lockdown and the final death toll is 271,000. Peak infection is 7m and eventually over 40 per cent of the population catches the disease. Infection on the peak scale would impose an intolerable burden on the health system. To avoid such an eventuality, we repeat the simulation with a ceiling of 3.3m on the permitted level of infection. This is just over 50 per cent more than the initial level of infection (2m). The existence of this constraint has little impact on the eventual death toll, although it does reduce the peak load on the health system. ### Relative costs The parameter $ϕ$ conveys information about the relative cost of various interventions. When $ϕ$ is small the economic benefit from a partial relaxation of the lockdown is also small. This creates an incentive to extend the duration of lockdown. Why relax an effective policy for so little economic gain? Conversely, if $ϕ$ is large, the economic gain from a partial relaxation is large. The duration of lockdown is therefore short. Under the Baseline scenario $ϕ=2$ and the lockdown lasts for 5.3 weeks. If $ϕ=1$, the lockdown lasts for 7.9 weeks. If $ϕ=4$, it lasts for 1.8 weeks. ### Test and trace A test and trace system is designed to isolate infectious individuals and their contacts, so that they cannot infect the general population. Within the framework of the present model it is equivalent to either a reduction in the transmission coefficient $β0$ or else an increase in $γ$, the rate at which infected people cease to be infectious. To explore the implications of the system introduced by the UK government, we assume that it becomes fully operational in week 20. This is later than the government’s initial target, but we allow for teething problems. The system has a capacity of 200,000 tests per day. We assume it has negligible cost. The effectiveness of a test and trace system depends on the following factors: (i) the number of tests carried out, (ii) the share of infected individuals in the tested population, (ii) the fraction of infected individuals who are available for testing, and (iv) the number of infected contacts who self-isolate following a positive diagnosis. The roles of these various factors are discussed in the Appendix. The parameters we use for our simulation are somewhat arbitrary, but the results illustrate clearly the impact of test and trace on the optimum path. Figure 7 plots the optimum paths with and without test and trace. The effect of test and trace is to lower the trajectory of the control variable q. The reason for this is as follows. The existence of a test and trace system reduces the impact of present interventions on the future course of infection. Planners therefore have less need to be concerned about the future. They can afford to relax since test and trace will help deal with the outcome. This is true both before and after test and trace comes into operation. The test and trace system in our simulation is not perfect, so some degree of social distancing is still required after this system becomes operational. Figure 7: Optimal paths for q: scenarios compared Figure 7: Optimal paths for q: scenarios compared ### The value of life Any cost–benefit analysis of optimal policy towards COVID-19 requires some assumption about the value of human life (Social Value UK, 2016; Dolan and Jenkins, 2020). This assumption may be explicit or it may be implicit. Governments may reject the whole idea of valuing life in the context of disease control, but to the extent that their actions are consistent, they imply some tacit valuation of life. In other policy areas, such as transport and drug evaluation, it is normal for government agencies to put a value on life. In our simulations a reduction in the value of life implies a shorter lockdown or maybe no lockdown at all (Table 2). This is true even if we impose a ceiling on the permitted scale of infection. Under the Baseline scenario, the value of life is £2m and the optimal lockdown lasts for 5.3 weeks. Holding other parameters constant, it becomes optimal to dispense with the lockdown altogether once the value of life drops below £1.68m. If we impose the condition that peak infection must not exceed what the health service can handle, it is optimal to dispense with lockdown when the value of life is below £1.56m. At the other end of the spectrum, the optimal duration of lockdown becomes rather insensitive to further increases in the value of life. The optimal lockdown is not much different if the value of life is £10m or £20m (Figure 8). Figure 8: More than 27,000 vacancies are offered by over 120 central departments and their affiliated public institutions in the 2017 civil servant recruitment drive. The number of vacancies is about the same as 2016. Figure 8: 但是他们摆烂的原因还是各不相同的。比如热火队，他们试图通过输球来锻炼队中的年轻人并且确立一个核心；比如魔术队，他们试图摆脱停滞不前的过程。 Figure 9 plots the relationship between total deaths and total economic cost. Through its impact on optimal policy, the value of life affects both the economic cost of the disease and the number of people who die from it. Each point on the curve corresponds to a certain value of life, and the variables shown are calculated on the assumption that the government behaves optimally given this value of life.5 Figure 9: 这是5年来非合办EMBA项目首次跻身前5名，全球EMBA项目前5名的名次非常稳定。 Figure 9: 许多人对此不屑一顾，《纽约每日新闻》采访到的一位营养师声称，黄金披萨中最昂贵的成分：厄瓜多尔进口黄金薄片缺乏营养价值。 如果Lloyd Grossman有机会进入奥斯卡得主们的家中，他将会在很多不同的地方看到他们各自的小金人，从高级房产的壁炉台到积尘良久的壁橱深处，都可能看到它的身影。 顾名思义，这个真人秀节目还专门拍摄学生们的母亲，展现母亲如何教育孩子与他人竞争。 注册人数：511人 Wonder Woman Ashley Graham, 30, is the first ever curve model to make the highest-paid list, coming in at 10th place after banking$5.5 million from her lingerie and swimsuit lines contracts.
protein
“疲软的国内需求和大宗商品价格下跌继续拖累中国的进口增长，”澳新银行(ANZ bank)经济学家刘利刚表示。“展望未来，中国的出口行业仍将面临一些重大不利因素。”

A striking feature of Figure 9 is the discontinuity indicated by the broken line. This was unexpected, but appears genuine. We checked it using two different programs. This break in the curve marks the transition between two radically different types of policy. To the right of the break, the optimal policy is lockdown with a low death rate. To the left, the optimal policy is no lockdown and a high death rate. This transition occurs abruptly when the value of life is around £1.68m. It is clearly visible in Figure 8.

What light does this discussion throw on the actual policy of the UK government? The period of maximum lockdown lasted approximately 10 weeks. With the baseline cost structure ($ϕ=2,πA=2$), a lockdown of this length is only optimal when the value of life exceeds £10m. If $ϕ=1$,$πA=3$ the figure is £4m. These numbers are much larger than the value of life implied by the official guidelines for drug evaluation (£200,000 to £300,000).6 To the extent that the government is behaving optimally, these comparisons imply that it values the lives of potential COVID-19 victims a lot more highly than those of other types of victim.

## V. Concluding remarks

Soon after the implications of lockdown became evident people began to ask the obvious question: ‘Is the cure worse than the disease?’ (Miles et al., 2020). Governments began to seek cost-effective policies that would enable them to exit the lockdown without setting off a renewed surge of infection. Although they are speculative in nature and limited in their methodology, the simulations presented here and their underlying theory may throw some light on government policy.

"China is really in a tough position," Dr. Peters said. "Emissions have grown so much in the last 10 years or so that no matter how you look at China, it has an immense task."
007电影中，反派人物往往没有自己的主题曲，但是《金枪人》中克里斯多弗·李扮演的暗杀者出场时有一段轻快动人的绝妙曲调，衬托出暗杀者如地狱使者般帅气冷酷。LuLu的演唱完全演绎出这个反派斯卡拉孟加的过人之处。即使这首歌在今天来说欢快得有点可笑，但这也增添了它的魅力。
The report found that students majoring in art, agriculture and engineering were more willing to start businesses, while those majoring in history and science showed relatively low interest.

One of the main strengths of the LBS programmes is the wide range of students from different countries. More than 90 per cent in its 2015 MBA cohort were from overseas, coming from about 60 different countries.
Several other parties have support that is only slightly lower, including the centrist liberal D66, the Christian Democratic Appeal and leftwing GreenLeft.
It is revealed that Hadid's visa application was turned down after she offended many by squinting her eyes in an attempt to impersonate the Buddha in February.

Interest in wearable technology isn’t limited to technology companies. Mercedes-Benz is porting its mobile experience to a wearable device, while Virgin Atlantic is exploring the customer service aspect of Google Glass on a trial basis. Kenneth Cole is also using Glass as part of a marketing campaign.
The regulator said that although this year's growth will be slightly lower than 35 percent due to the Chinese currency's depreciation, the film market will still see robust growth.

In his Covid Economics paper, Rowthorn (2020) argued that, if a relatively inexpensive way can be found to maintain an r value close to 1, this is the policy to aim for in the medium term. A lockdown may (or may not) be necessary to halt the explosive spread of the disease, but once this aim has been achieved it would be a costly mistake to stick with expensive social distancing policies that aim to keep r well below 1. This conclusion is reinforced by our example of test and trace. If there is an effective test and trace system in the offing, it may even be optimal to let r exceed 1 during the weeks before this system becomes operational. This will cause infection to increase somewhat, but the potential explosion will be prevented when test and trace comes on stream. The same is true during the run-up to mass vaccination.

The world's first flying bicycle flew on November 9, 1961, when Derek Pigott of the University of Southampton flew in a bicycle with an airplane-like body. It was called the Southampton University Man Powered Aircraft (sumpac). Derek furiously pedaled the air-bike to get it off the ground. It then flew 1.8 meters (about 6 ft) above the ground over a distance of 64 meters (210 ft). While the flight was short and slow, it still does not change the fact that it was the first bicycle to fly and at the same time, the first human-powered flight.
Insecure

“这是关于‘我们是谁’的基本问题，”伦敦大学伯克贝克学院的政治学教授埃里克?考夫曼(Eric Kaufmann)说，“作为这个国家的一员意味着什么？它是否已经不再是‘我们’的国家？——‘我们’是指占多数的民族。

That even at a lower profit margin (say, 40%) and a 1/3 cannibalization rate (i.e. customers buy one third fewer full-priced iPhones), the cheaper iPhone would increase Apple'srevenue and gross profits (see her spreadsheet above).

Corkin died this year but shortly after, journalist Luke Dittrich published a book claiming Corkin buried inconvenient findings, shredded files, and acted unethically in gaining HM's consent.

### Appendix: Test and trace

Throughout this appendix the symbol $I$ refers to infected individuals who are not isolated and can therefore infect the susceptible population. Isolated individuals who are infectious are classified as removed.

Suppose that a fraction $a$ of the infected population $I$ is currently available for testing. The rest are either asymptomatic or unwilling to undergo testing. For those available for testing; the probability of not being tested positive in a period of length $s$ is equal to $e−ps$where $p$ is constant. The probability that an infected individual will cease to be infectious in the small time interval $Δs$ is $[−dds(e−γs)]Δs=γe−γsΔs$. Thus, the probability of recovering or dying without testing positive is:

$∫0∞γe−(p+γ)sds=γp+γ$
(A1)

$1−γp+γ=pp+γ$
(A2)

The average length of time that an individual remains infected is $1γ$. Thus, the probability that he or she will test positive during a small time interval of length $Δt$. is equal to:

$pp+γ(Δt1γ)=(γpp+γ)Δt$
(A3)

The number of infected individuals who are available for testing is $aI$. The number of such individuals who test positive in the time interval $Δt$ is

$aI(γpp+γ)Δt$

$(γpp+γ)aI$

Suppose there is no constraint on testing. Then $p=∞$ and the rate of testing infected persons is:

$A=γaI$
(A4)

In the constrained case assume that $M$ is the maximum number of tests per week. Assume also that a constant fraction $b$ of these tests is directed at infected persons. Then access to testing will be capacity constrained if $bM<γaI$. In this case

$A=(γpp+γ)aI=bM$
(A5)

Thus,

$p=γaIγaI−bM$
(A6)

Assume that for each person who tests positive the number of infected persons who self-isolate (including the tested person) is $c$. Then infected persons are isolated at the rate $cA$. They are classified as removed.

Suppose the test and trace system comes on stream at time $T∗$. Define the following function:

(A7)

8月水泥行业实现利润80亿元 旺季涨价幅度将超过预期

$dIdt=(1−q)β0SI−γI−Q(t,I)dRdt=γI+Q(t,I)$
(A8)

Our simulation assumes a daily capacity of 200,000 for test and trace. This amounts to 1,400,000 per week, which is equal to a fraction 0.021 of the population. Thus, $M=0.021$. It is also assumed that the test and trace system becomes fully operational in week 20 and that $a=0.5,b=0.5,c=1.6.$

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## Footnotes

1

3

In a game theoretic paper on social distancing Reluga (2010) also assumes that vaccination will occur on a fixed date in the future. In their recent paper, Acemoglu et al. (2020) assume that a vaccine and a cure become simultaneously available.

4

Cnzz.com的报告还讨论了目前在中国网络游戏行业盛行的装备收费模式问题。举例来说，很多美国游戏公司都是按照时间收费的，但大多数中国网络游戏都采取了装备收费模式，玩家可以免费试玩。用户玩游戏的时间越长，就越有可能花钱购买游戏装备，以获得更高的游戏级别。但这也意味着花钱最多的玩家就在游戏中表现最好。
Management consultancies pride themselves on being expert advisers on “change” — from helping clients integrate after mergers, to keeping them at the cutting edge of new technology. Several of the world’s largest consultancies will have to make use of their own advice in 2018 as they prepare for leadership changes — the first in years — that will have significant ramifications for their workforce.

5

Technically speaking, the curve is parameterized by $πD$.

6

Best chances: Best film, and best original screenplay recognition looks certain, and Frances McDormand is currently the favorite for the best actress Oscar.