The following is an excerpt of\"A Simple Planning Problem for COVE19 Lock-down, Testing, and Tracthg\" by Alvarez, Fernando, David W and Francesco Lippi from American Economic Review: Insights 2921, 3(3): SFSSE- Use the scrim-pt to answer questions I. A Model of Lockdown, Testing, and Tracing We start with a modied version of the SIR model by W and MW (192?) as described in m (REED)- Agents are divided between those susceptible to be infected, 3,; those infected, It; and those recovered, RE; that is, (11st, =s,+r,+s, forallt :_:- e- The \"recovered" include those that have been infected, survived the disease, and are assumed to be (forever) immune. Einee we only include those that are alive, Mr is changing through time, and normalize the initial population to Na = 1- The planner can lockdown a'action L, E [GI] ofthe population, where E 5 1 allows us to consider that even in a crisis scenario some economic activity, such as energy and basic food production, will contimie. We assume that the lock-down is only partially elfective in eliminating the transmission of the virus- 1|when L agents are in lock-down, then (1 3L) agents can transmit the virus, where E E [ELI] is a measure of the lock-down effectiveness. [f3 = 1, the policy is fully effective in curbing the diffusion, but since some contacts will still happen in the population even under a full economic lockdown, we allow 3 s: 1. In addition to the lock-down policy, we assume the planner can \"test and trace" infected agents and place them into quarantine. We refer to this policy as a 'I'I'Q. Let Q, E I, denote the stock of quarantined agents at time t. The quarantined agents are removed from the pool of the active infected and thus do not contribute to the propagatiqp. of the new infections. The law of motion of the susceptible agents then is :12) s} = vireo some: rs] (1 ea]. In the case where no control is exercised, L = t] and Q = D, the uncontrolled evolution of the system obeys the wellknown 3 = ,Ei'5f equation, where ,8 2:: D is the number of susceptible agents per unit of time to whom an infected agent can transmit the virus. It is evident that locking down a part of the population can be powerful in reducing the rate at which susceptible agents become infected. This is because it is the product of the infected and susceptible that determines the new infections per unit of time- Hence, the new infections are reduced by the square of the lockdown rate- Likewise, if a action of the infected is quarantined, that is, if Q 2:"- II], the remoduction rate of new infections is controlled by reducing the number of infected agents who have contacts with others, namelyr Q). A fraction y (per unit of time) of the infected recovers, thus: (3] II: = _5-r _ TI:- The stock of quarantined agents follows the law of motion (4) Qt = Tt - VQt: where 7, $ 7 denotes the flow per unit of time of agents that are traced, tested (positive), and placed into quarantine, and 7 is a capacity constraint on the number of agents that can be traced per unit of time. A rate 0 > 0, and we assume that with probability v per unit of time both a vaccine and a cure appear, so that the planner discount rate is r + v. The problem consists of minimizing the following present value: (7) V(Solo. Qo) = min S e-(7+) [wer+ why[t(S+ + he - Q ) + (1 - D) (1 - Q.)] + vslo (It)It + c(Tt: 5,, It. Q )]dt subject to the laws of motion equation (2), equation (3), and equation (4) and an initial condition (Solo. Qo) with lo > 0 and So + Io 2 No. Note that, as the vaccine and cure arrive, there is no more cost, and the continuation value is zero. Question 1. According to equation (2), which of the following best explains the option or options that a government has to stop the transmission of an infectious disease? A. Lock-down a part of the population B. Quarantine the infected agents C. Both A and B D. No option can stop the transmission completelyQuestion 2. Which of the following best explains the role of [S.(1 - OL,)] in equation (2)? A. To show that new infections are reduced by the square of the lock-down rate B. To show that the behaviour of susceptible agents is described by the number of susceptible agents, proportion of population under lock-down and effectiveness of lock-down C. To show that the effectiveness of lock-down depends on the proportion of population under lock-down D. To show that increasing the proportion of population under lock-down increases the effectiveness of lock-down Question 3. Which of the following best explains the role of [(It - Q.)(1 - 0L,)] in equation (2)? A. To show that completely quarantining the infected is not possible B. To show that lock-down is not as effective as intended C. Both A and B D. Neither A nor B Question 4. Which of the following best explains the role of f in equation (2)? A. To show that lock-down and TTQ have negative impacts on citizens B. To allow for a range of disease transmission rates C. Both A and B D. Neither A nor B Question 5. Which of the following best describes $, in equation (2)? A. New infections per unit of time B. Square of the lock-down rate C. A fraction of the infected that is quarantined D. The number of infected agents who have contacts with others at time t Question 6. Which of the following best describes the proportion of the population that is infected but not quarantined at a given time t? A. St B. [S.(1 - OL.)]/5, C. (It - Qt) D. (It - Q:)/5, Question 7. Which of the following best describes L = 1? A. A government tests and traces the infected with complete success B. A government imposes complete lock-down C. The infected are quarantined without fail D. Quarantine is for an indefinite period of time -Question S. In the case where government control is exercised such that L = 1, is disease transmission still possible? A. Yes, because lock-down may not be completely effective B. Yes, because testing and tracing all of the infected are not possible C. No, because the infected are unable to contact the susceptible D. No, because government control is assumed to be complete Question 9. Which of the following best describes Q = 1? A. A government tests and traces the infected with complete success B. A government imposes complete lock-down C. The infected are quarantined with complete success D. Quarantine is for an indefinite period of time Question 10. Which of the following best describes 0 = 0? A. Citizens do not respond to government lock-down B. A government refuses to impose lock-down C. Both A and B D. Neither A nor B Question 11. Which of the following best describes yl, in equation (3)? A. The portion of the infected that recover at time t B. The portion of the infected that are quarantined at time t C. The portion of the infected that are not quarantined at time t D. None of the above Question 12. Which of the following best describes the reason why I, depends on $, in equation (3)? A. There is no reason as the authors made an error B. The susceptible are more likely to be infected The infected become less likely to be susceptible D. As the susceptible become infected, the decline in the susceptible represents a corresponding increase in the infected Question 13. Which of the following best describes a mechanism by which , the upper bound of , in equation (4), may be increased? A. Superior technology to identify the infected B. Increased personnel to enforce the quarantine C. Both A and B D. Neither A nor B can increase TQuestion 14. Which of the following best describes the reason why yQ, is negative in equation (4)? A. The stock of quarantined agents decreases as the disease evolves B. The stock of quarantined agents decreases as they die C. The stock of quarantined agents decreases as they recover D. The stock of quarantined agents decreases as the social behaviour changes Question 15. Do the authors allow for the infected recovery rate to change? A. Yes, the authors allow for the infected recovery rate to change over time B. Yes, the authors allow for the infected recovery rate to change with I C. Yes, the authors allow for the infected recovery rate to change with D. No, the authors do not allow for the infected recovery rate to change Question 16. Suppose vel is the value of a statistical life and c is the cost of TTQ. Which of the following best describes equation (7)? A. The cost of lock-down, TTQ, and death B. The time it takes for the vaccine and cure to arrive C. Both A and B D. Neither A nor B Question 17. Which of the following best describes w@, in equation (7)? A. Cost to enforce quarantine at time t Output of TTQ at time t C. Output lost by the quarantined at time t D. Output lost due to the lock-down Question 18. Suppose r = 1. Which of the following best describes we: + why[(S, + , - Qt) + (1 - T)(1 - Q-)] in equation (7)? A. Output lost by the quarantined, susceptible and infected who are unable to work at time t B. Output of TTQ and lock-down at time t C. Cost to enforce quarantine and lock-down at time t D. Output lost due to quarantine Question 19. Suppose usl is the value of a statistical life. Which of the following best describes vslo (I )It in equation (7)? A. Output lost by death B. Benefit of reduced congestion in the health care system C. Benefit of reduced disease propagation due to fewer infected D. Both B and CQuestion 20. Which of the following best describes the role of @ (7+1) in equation (7)? A. To account for the probability that a vaccine and a cure may appear B. To account for the time that a vaccine and a cure may appear C. Both A and B D. To represent the cost in terms of present value Question 21. Suppose . = 1 represents a state where the recovered can be identified and cannot be re- infected. Suppose + = 0 represents a state where the recovered cannot be identified. Explain the practical difference between the two states as represented in equation (7)