key: cord-0764950-ag4faw4m
authors: Lampert, A.
title: Decentralized governance may lead to higher infection levels and sub-optimal releases of quarantines amid the COVID-19 pandemic
date: 2020-05-25
journal: nan
DOI: 10.1101/2020.05.20.20108167
sha: 38dbdfe3298b4d8b1ead7f1c3820e54324e59a6a
doc_id: 764950
cord_uid: ag4faw4m

The outbreak of the novel Coronavirus (COVID-19) has lead countries worldwide to administer quarantine policies. However, each country or state decides independently what mobility restrictions to administer within its borders, while aiming to maximize its own citizens' welfare. In turn, since individuals travel between countries and states, at least during periods when quarantines are less restrictive, the policy in one country may ultimately affect the infection level in other countries. Therefore, major questions are whether the policy dictated by a decentralized government is efficient, and if not, how the governments can coordinate a better policy. Here, we focus on the decision regarding the timing of releasing the quarantines. We consider a game theory model in which each of two governments decides when to switch from a restrictive to a non-restrictive quarantine and vice versa. We used parameter values driven by the literature and publically available data. We show that, if travel is sufficiently frequent during the non-restrictive quarantine periods, then the strategies are sub-optimal: Each governor tends to release the quarantine sooner, which ultimately leads to longer periods of restrictive quarantines and a higher prevalence of the disease. In turn, if the governments restrict international and interstate travel to a low level even when the quarantines are non-restrictive, the policy dictated by the decentralized governance may become optimal.

The outbreak of the novel Coronavirus (COVID-19) necessitates quarantine policies [1] [2] [3] [4] . While by multiple countries [14] [15] [16] , or independently by multiple states in countries like the U.S., or 23 even independently by multiple municipal authorities. Each governor is expected to dictate the 24 strategy that best serves her/his own citizens; however, in periods when the quarantines are less 25 restrictive, travelers can transmit the disease between countries, sates, and cities. Consequently, 26 the quarantine policy in one country/state may ultimately affect the outcome in other 27 countries/states. 28 Decentralized governance has a benefit: Each country or state may have a better knowledge of 29 its own citizens' lifestyle and needs and may dictate a policy that better suits its own citizens. 30 However, the decentralized policy also comes with a cost: Each government might ignore the 31 cost borne to citizens of other countries due to international and interstate travel. Accordingly, 32 various previous game theory studies have suggested that agents (individuals and countries) 33 under-invest in the prevention and control of diseases [17] [18] [19] [20] [21] [22] . 34 In this paper, we examine the case where each governor decides independently about the timing 35 of releasing the quarantine, and we ask what the inefficiency is due to such decentralized 36 governance. Namely, we examine how the strategy of different governments differ from the 37 socially optimal strategy of a hypothetical centralized government that aims to maximize the 38 welfare of all the citizens in all the countries (price of anarchy). Specifically, we consider two 39 countries/states, and we analyze the following three cases (Fig. 1 ) [14] [15] [16] 23] : (1) 

Our model assumptions and parameterization are explicitly motivated by the COVID-19 outbreak. 46 We consider two states or countries, 1 and 2, and in line with concurrent data about the COPVID- 47 19 outbreak [4, 15, 16, [23] [24] [25] [26] , we assume that the number of infected people in each country is 48 very small compared to the country's total population size. Accordingly, we consider only the 49 dynamics of the infection level in each country, , defined as the portion of individuals that are 50 infected in country . (Namely, in contrast to traditional SIR models [27], here we consider shorter 51 timescales during which the number of susceptible individuals is constant, which is in line with 52 the COVID-19 data as of May 5, 2020 [4, 15, 16, 23-26] ). In turn, we assume for simplicity that 53 each government can administer one of two types of quarantine at any given time: restrictive 54 and non-restrictive. Each government can choose when to switch between these two 55 quarantines. Specifically, we assume that under a restrictive quarantine in country , decreases 56 . CC-BY-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review)

The copyright holder for this preprint this version posted May 25, 2020. . https://doi.org/10.1101/2020.05.20.20108167 doi: medRxiv preprint exponentially after the first two weeks at a rate − 0 as long as > min . (Even a restrictive 57 quarantine is not expected to eliminate the disease entirely, and min characterizes some minimal 58 infection level that persists in the population.) In turn, according to evidence showing that the 59 infection level may still increase under non-restrictive quarantine conditions [4, 15, 16, 25, 26] , 60 we assume that increases at a rate under a non-restrictive quarantine, where is country-61 specific and can be higher in those countries or states where interactions among individuals are 62 more frequent. Also, we assume that if a country is under a restrictive quarantine, there is no 63 travel from or to that country, whereas if two countries are both under a non-restrictive 64 quarantine, some individuals travel between these countries (we denote as the number of 65 individuals that reside in country but are in travel to country ). We describe in more detail the 66 dynamics of 1 and 2 in Methods: Dynamics of the infection levels. 67 In turn, the government in each country dictates the timing at which it switches from the non-68 restrictive quarantine to the restrictive one and vice versa. We assume that a government will 69 not allow the health system in its country to collapse [1] , and therefore, it will always switch to a 70 restrictive quarantine if approaches some maximum capacity, = max . . CC-BY-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.

The copyright holder for this preprint this version posted May 25, 2020. . https://doi.org/10.1101/2020.05.20.20108167 doi: medRxiv preprint decentralized governance, the solution is sub-optimal, the total amount of time during which a 91 restrictive quarantine is administered is greater, and the average infection level is higher. This 92 result is consistent with previous results that suggested that agents under-invest in control of 93 diseases [17] [18] [19] [20] [21] [22] . In turn, notice that in both the optimal solution and the Nash equilibrium, the 94 governments tend to be coordinated in the sense that the second country to switch does so when 95 its infection level is similar to that of the other country (Fig. 2D ,F). 96 The difference between the optimal solution and the Nash equilibrium emerges if the number of 97 travelers between the authorities is above a certain threshold, in which case the price of anarchy 98 increases with the number of travelers (Fig. 3) . On the other hand, if the number of travelers is 99 below the threshold, the Nash equilibrium and the optimal solution are identical (Fig. 3) . This 100 suggests that one way to prevent the inefficiency due to the decentralized governance is to 101 declare in advance that travel will be restricted to a low level even when the quarantines are 102 non-restrictive. This may be applicable at the international scale, but much more difficult at the 103 municipal scale. 104 There are two mutually dependent reasons why decentralized governance results in releasing 105 the restrictive quarantine sooner. First, each governor ignores the damages that its own travelers 106 inflict on other countries, and thus, keeping a higher level of infection is perceived by the 107 governor as less costly. Second, as a consequence of the first reason, each country hosts more 108 infected travelers from the other country, and its own travelers are also hosted in countries with 109 higher infection levels. Consequently, if the infection level in a given country is low, it increases 110 rapidly due to travel, and therefore, it is not worthwhile for the country to reduce its infection 111 level beyond a certain value (where due to travel, this value may be lower than min ). 112 Finally, note that we have made numerous simplifying assumptions in our model, which suggests 113 various future directions for the examination of the consequences of relaxing these assumptions. 114 First, we considered only two types of quarantine, whereas in reality, more options are available. 115 In particular, governments can try to administer an intermediate level of quarantine that keeps 116 the infection at a constant level. Examining whether this policy is better than the ones that we 117 considered is beyond the scope of this paper (see [1] [2] [3] ); however, it is likely that a similar result 118 will hold: Decentralized governance might maintain a higher infection level than the optimum. 119 Second, we assumed that travel is allowed under non-restrictive quarantine. However, it would 120 be interesting to examine policies that also dictate how to best integrate the quarantine policy 121 with travel policy. Specifically, further restrictions on travel might mitigate the problem (Fig. 3) ; 122 however, it should be noted that restrictions on travel come with economic cost [13] . Third, we 123 considered only two countries, whereas considering more countries is generally expected to is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.

The copyright holder for this preprint this version posted May 25, 2020. . CC-BY-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.

The copyright holder for this preprint this version posted May 25, 2020. . https://doi.org/10.1101/2020.05.20.20108167 doi: medRxiv preprint Figure 2 : The equilibrium solution dictates that governments release the quarantines earlier 132 than optimal. We consider the three cases illustrated in Fig. 1: identical countries (A, B) , non- is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.

The copyright holder for this preprint this version posted May 25, 2020. . https://doi.org/10.1101/2020.05.20.20108167 doi: medRxiv preprint residents of country that get infected inside country by residents of country . Note that we 175 assume that travel is temporary (Lagrangian approach [29, 30] ); that is, a traveler from country 176 In turn, without the simplifying assumption that ≪ , Eq. (3) can be written in a more general 179 form:

In our simulations, we used values of that are much smaller than (see Parameterization   181 subsection), and therefore, the results obtained using Eq. (3) and those obtained using Eq. (4) 182 were almost indistinguishable.

The government in each country chooses the time periods during which each type of quarantine 184 is administered in that country. We assume that the objective of each government is to maximize 185 the relative time during which the quarantine is non-restrictive in its country, under the 186 constraint that the health system does not collapse, and therefore, a restrictive quarantine has 187 to be administered in country whenever approaches max [1] . 188 Specifically, denote tot as the time to complete a cycle during which increases to max and 189 returns back to its initial value, (0). In turn, denote non as the total time within such a cycle 

Since the growth rates of 1 and 2 are piecewise-linear, and since we considered a time delay 197 after the government switches to restrictive quarantine before the infection level starts to 198 decline, it follows from Pontryagin's maximum principle [28, 31] that the optimal strategy of each 199 . CC-BY-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.

The copyright holder for this preprint this version posted May 25, 2020. . https://doi.org/10.1101/2020.05.20.20108167 doi: medRxiv preprint government is to choose a time, ≥ 0, at which it switches to a restrictive quarantine and then 200 to wait until = max before switching to a restrictive quarantine again (Fig. 2) . 201 Then, non , is the period from until approaches max , and the time during which the 202 quarantine is restrictive, tot − non , is given by the sum of three distinct periods: In turn, we assume that max is given approximately by the infection level that was approached 230 in countries in Europe before the restrictive quarantine was administered or two weeks after it 231 was administered, where considering min ≈ 0.01% − 0.1% of the total population size is a 232 reasonable estimate. Next, parameters like min and the travel rates 12 and 21 are harder to 233 . CC-BY-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review)

The copyright holder for this preprint this version posted May 25, 2020. . https://doi.org/10.1101/2020.05.20.20108167 doi: medRxiv preprint estimate, as they depend on the specific location and scale of the countries and state considered. 234 Therefore, we performed sensitivity analyses and examined a variety of values (e.g., Fig. 3 ). 235 Finally, the ratio between 1 and 2 reflects the relative population sizes of the two countries: 

. CC-BY-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review)

The copyright holder for this preprint this version posted May 25, 2020. . https://doi.org/10.1101/2020.05.20.20108167 doi: medRxiv preprint

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