COVID-19 spreading under containment actions
CCOVID-19 spreading under containment actions
F.E. Cornes a , G.A. Frank b , C.O. Dorso a,c, a Departamento de F´ısica, Facultad de Ciencias Exactas y Naturales,Universidad de Buenos Aires,Pabell´on I, Ciudad Universitaria, 1428 Buenos Aires, Argentina. b Unidad de Investigaci´on y Desarrollo de las Ingenier´ıas, Universidad Tecnol´ogicaNacional, Facultad Regional Buenos Aires, Av. Medrano 951, 1179 Buenos Aires,Argentina. c Instituto de F´ısica de Buenos Aires,Pabell´on I, Ciudad Universitaria, 1428 Buenos Aires, Argentina.
Abstract
We propose an epidemiological model that includes the mobility patternsof the individuals, in the spirit to those considered in Refs. [1–3]. We assumethat people move around in a city of 120 ×
120 blocks with 300 inhabitantsin each block. The mobility pattern is associated to a complex network inwhich nodes represent blocks while the links represent the traveling path ofthe individuals (see below). We implemented three confinement strategiesin order to mitigate the disease spreading: 1) global confinement, 2) partialrestriction to mobility, and 3) localized confinement. In the first case, it wasobserved that a global isolation policy prevents the massive outbreak of thedisease. In the second case, a partial restriction to mobility could lead toa massive contagion if this was not complemented with sanitary measuressuch as the use of masks and social distancing. Finally, a local isolationpolicy was proposed, conditioned to the health status of each block. It wasobserved that this mitigation strategy was able to contain and even reducethe outbreak of the disease by intervening in specific regions of the cityaccording to their level of contagion. It was also observed that this strategyis capable of controlling the epidemic in the case that a certain proportionof those infected are asymptomatic.
Keywords:
COVID-19, Pandemic, Human mobility
PACS:
Preprint submitted to Elsevier February 5, 2021 a r X i v : . [ phy s i c s . s o c - ph ] F e b . Introduction In the absence of a vaccine, strategies based on non-pharmaceutical inter-ventions were proposed to contain the COVID-19 pandemic. Social distanc-ing policies, specifically mobility restrictions and lockdowns, among otherswere the more common ones. Such policies should be implemented for longperiods (typically months) to avoid re-emergence of the epidemic once lifted.Therefore, quantitative research is still needed to assess the efficacy of non-pharmaceutical interventions and their timings.Many works analyze real-time mobility data in order to relate the changesin the mobility patterns and the disease propagation. Ref. [4] reports a cor-relation between the mobility pattern and the reduction of new infections.Besides, they found that it takes two to three weeks to see results due to theincubation time of the disease. Also, Ref. [5] carries out a detailed study ofthe effects of containment measures during the first 50 days of the COVID-19epidemic in China. These researchers found that traveling restrictions andsocial distancing measures (among others) were effective in the containmentof the disease. Ref. [6] analyzes real mobility datasets in many US metropoli-tan areas. They found that a small minority of “super-spreaders” places arethe responsible for the wide propagation of the infection.The changes in the mobility patterns are a consequence of the implemen-tation of different quarantines. Ref. [7] analyzes different quarantine typesconsidering a complex SEIR scheme. They suggested an alternative type ofquarantine to reduce the infection disease while allowing a socio-economicactivity. In a similar way, Ref. [8] proposes a cyclic schedule of 4-days workand 10-days lockdown. Also, an improved version of this strategy can befound in Ref. [9]. The influence of human behaviors on infectious diseasetransmission [10], the effects of vaccination during a pandemic [11] and therole of the “super-spreaders” [12, 13] are many other complex scenarios ana-lyzed in the literature. However, a compressive and quantitative comparisonof the effectivenesses of different lockdown and their timing appears to belacking.In this work we consider the spread of a disease mimicking the COVID-19, assuming a spatio-temporal SEIR model of mobile agents. We simulatedand compared different confinement strategies in order to mitigate the disease2ropagation. In Section 2 we describe the characteristics of the epidemio-logical model and the different mitigation strategies. Section 3 details thesimulation procedure. Section 4 displays the results of our investigation,while Section 5 is dedicated to the discussion of the main results. The con-clusions are drawn in Section 6.
2. The epidemiological model
There are three main ingredients in the description of the COVID-19contagion and spatial spreading: the scenario where the process takes place(Section 2.1), the mobility patterns of the individuals (Section 2.2) and theepidemiological dynamics of the individuals (Section 2.3).
The simulation of the evolution of COVID-19 is performed in a schematiccity in which the basic unit is the block. The city is represented by an ho-mogeneous urban area of 120 ×
120 blocks placed in a square grid. The sizeof each block is 100 m ×
100 m. The simulated grid corresponds to a big city,like Buenos Aires, Argentina. Each block hosts 300 people but this quantitymay vary during the day (see next Section). The total population of thesimulated city is 4 .
32 M similar to the Ciudad Autonoma de Buenos Aires’population.
Network construction
The human mobility pattern between the blocks is accomplished by build-ing a weighted and directed network. The nodes represent the blocks, whilethe links represent the human mobility from one block to another one. Weconsider two different links: short and long links.Each node is linked with its first neighbors by a short link, as we illus-trate in Fig. 1a. They represent the human mobility pattern between allneighboring blocks. Notice that these links generate a connected graph. Aswe will see later, this characteristic of the network will allow the disease to3each all parts of the city (as long as it is not locked during the epidemic).On the other side, recent investigation on human mobility shows that thetraveling lengths of the individuals follows a Levy distribution given by [14] P ( r ) ∝ ( r + r ) − β (1)where P ( r ) stands for the probability that an individual reaches a distance r , and r = 100 m and β = 1 .
75 correspond to empirical parameters. (a)
Network with only short links (b)
Network with only long links
Figure 1: Schematic representation of a city network. The city is composed by120 ×
120 blocks placed on a square grid. They are linked by (a) four short linksconnecting neighboring nodes and (b) long links connecting far away nodes (see text formore details). The city corresponds to the union of these two sub-networks.
We built a network of long links following the Levy mobility pattern. Thisnetwork is illustrated in Fig. 1b. The procedure was as follows1. We randomly chose a node.2. We then select randomly (according to the Levy distribution) the length r of the next link.3. We link the node from step 1 to any (random) node located at thedistance r in any direction.Notice that each node may have more than one long link . Recall that theselinks are complementary to the short links connecting neighboring blocks(say, four neighbors per block). 4 able 1: Most relevant parameters used in the simulation of COVID-19 epidemic alongthe mobility network. Number of nodes (blocks) 120 × r (Levy distribution) 100 m β (Levy distribution) 1.75The total number of long links will depend on how many times we repeatthe above steps. Thus, we looped these steps until the geodesic path of thenetwork resembled the one expected for a “small world” network [1]. As ageneral rule, we found that this condition was fulfilled after 25% of the nodeshad at least one long link. Table 1 summarizes the final set of parameter.Finally, we stress the fact that the links between different blocks are as-signed at the beginning of the simulation. Thereafter, the links remain fixeduntil the end of the simulation. We follow a similar scheme as in Ref. [1, 2] for the individuals’ mobility.This means that we divide the population of each block into two groups: sta-tionary and non-stationary individuals. The former is composed by infectedindividuals while the latter is made up of susceptible, exposed and removedindividuals.We will assume that stationary individuals stay in their original block. Wewill further assume, in the spirit of Ref. [2], that 50% of the non-stationary stay in their original block, while the other 50% is considered to be mo-bile during the simulation. In this sense, we assume that 60% of the non-stationary individuals travel (via short links) from their original block to theneighboring block. And, the remaining 40% travel (via long links) from theiroriginal block to far blocks as indicated in Fig. 2.5 igure 2: Schematic representation of the behavior of the population for a single block.The population is composed by susceptible (S), exposed (E), infected (I) and removed (R)individuals. Infected individuals stay at home during the simulation process while the50% of the susceptible, exposed and removed individuals move to another blocks (see textfor more details).
Daily displacements of the individuals
Each moving individual is assumed to stay 1/2 of the day in its originalblock while the other 1/2 of the day he (she) moves to another block. Theygo to his (her) destination (work) everyday, and at the end of the day returnback to their original block (home). This mechanism is performed reversingthe direction of the links between blocks.Fig. 3 shows a schematic representation of the human mobility patternduring each day (we illustrate this by a small city of 4 × In order to describe the time evolution of a given population when afraction gets infected, we resort to the SEIR compartmental models. Thesemodels consider that the individuals can be in four successive states: suscep-tible (S), exposed (E), infected (I) and removed (R). Details on each statecan be found in Refs. [1, 2, 9, 15, 16]. The “exposed” state appears wheneverthe disease undergoes an “incubation” period, as occurs in the context of theCOVID-19. The “removed” state includes either recovered and dead people.6 (a)
Home (0-12 h) (b)
Traveling to work (c)
Work (12-24 h) (d)
Returning to home
Figure 3: (Color on-line only) Schematic illustration of the epidemic model incorporatingindividual human mobility during each day. The node colors represent the population ofeach block (see scale bar on the right). (a) The day starts with all individuals at home.(b) At 12 o’clock, 50% of the population of each block moves to other cells, according tothe pre-assigned mobility pattern. The arrows represent the movement direction. (c) Thepopulation of each block after mobile humans traveled to the working block. (d) Then, at24 o’clock those humans that traveled from home to work return back to home. At theend of the day, the population of each block is the same as that in the beginning of theday. The sequence (a-d) repeats every day until the end of the simulation.
The equations that describe the evolution of the infection read as follows[15, 16] ˙ s ( t ) = − βi ( t ) s ( t )˙ e ( t ) = βi ( t ) s ( t ) − σe ( t )˙ i ( t ) = σe ( t ) − γi ( t )˙ r ( t ) = γi ( t ) s ( t ) + e ( t ) + i ( t ) + r ( t ) = 1 (2)where s ( t ) = S ( t ) /N , e ( t ) = E ( t ) /N , etc. correspond to the fraction ofpeople in each state. For the purpose of simplicity, we will consider the co-efficients β , σ y γ as fixed parameters. The parameter β (infection rate)depends on intrinsic ingredients like the infectivity of the virus under consid-eration and extrinsic ones like the contact frequency. Besides, the parameters σ and γ depend exclusively on the illness under consideration.The basic reproduction number R is defined at the beginning of thepropagation process as [17] R = βγ (3)7his quantity represents the number of individuals that are infected bythe first infected individuals (say, at the beginning of the disease). It isstraight forward that the infection will blossom if R is larger than 1. Onthe contrary, if this quantity is smaller than 1 the infection vanishes.Fig. 4 shows the time evolution of either the SEIR model for a singleblock and for the whole mobility model (see caption for details). Notice thatthe latter shifts the date for the maximum infection to approximately 150.We will analyze this behavior in more detail in Section 4. time (days) . . . . . n o r m a li z e d nu m b e r (a) basic SEIR model time (days) . . . . . n o r m a li z e d nu m b e r (b) spatio-temporal SEIR model Figure 4: Normalized number of susceptibles ( ), exposed ( ), infected ( ) and removed( ) as a function of time. In (a) we simulated a single block, while in (b) the scenariocorresponds to a city composed by 120 ×
120 blocks placed on a square grid (see Section2.1). The plots are normalized with respect to the total population in the city: (a) 300 and(b) 4 .
32 M ( i.e.
300 individuals per block). In both cases, the disease spreads without anykind of intervention strategy. The infection rate equals β = 0 .
75 all along the propagationprocess. The simulation started with 20 infected individuals located in the single block(a) and at the central block of the city (b). The rest of the individuals were assumed tobe in the susceptible state.
In this section we define the proposed containment strategies which wehave found useful in order to mitigate the disease. We divide the differ-ent types of quarantine according to their level of real life implementationdifficulty • Global lockdown 8
Imperfect lockdown • Local lockdown
Global lockdown scenario (GLs)
This type of lockdown consists in the isolation of each block. This meansthat people remain in their “home” blocks. We stress the fact that, withinthis strategy, society as a whole enters in lockdown. We illustrate this con-tainment action in Fig. 5b.
Imperfect lockdown scenario (IGLs)
As in the previous case, the whole society adopts the same behavior.But, in this mitigation strategy, we partially reduce the movement betweendifferent blocks. Recall from Section 2.2 that 50% of the non-stationary in-dividuals move from their original block to another one. In this strategy, wereduce the level of mobility between blocks. Notice that the global quarantinecorresponds to a full reduction of the mobility. We scheme this containmentaction in Fig. 5c.
Local lockdown scenario (LLs)
Unlike the other lockdown types, this applies to certain blocks and not tothe whole society. Only those blocks that have a certain number of infectedpeople are isolated. In this sense, the population of the isolated block areprohibited to travel around the city. This is achieved in practice by “cutting”the links to/from the infected blocks. We illustrate this containment actionin Fig. 5d.
3. Numerical simulations
We integrate the SEIR equations by means of the Runge Kutta 4th-ordermethod. The chosen time step was 0.1 (days). The SEIR equations were up-dated twice a day (after the people left their homes and after they returnedback (see Figs. 3a and 3c). 9 a) Without lockdown (b)
Global lockdown (c)
Imperfect lockdown (d)
Local lockdown
Figure 5: (a-d) Schematic representation of different types of mitigation strategies (exem-plified by a 4 × As mentioned in Section 2.3, the parameters σ and γ represent the in-cubation rate and the recuperation rate, respectively. Therefore, σ − and γ − correspond to the mean incubation time and the mean recovery time,respectively. According to preliminary estimations for COVID-19, we con-sider the following parameter values for the SEIR model: σ − = 3 days and γ − = 4 days [5, 18–21].Infected individuals remain at their “Home” until evolving into the re-moved state. Susceptible, exposed and removed individuals are able to movefrom one block to another. The simulation started with 20 infected individ-uals located at the central block of the city, while the rest of the individualswere assumed to be in the susceptible state.According to preliminary estimations for COVID-19, the basic reproduc-tion number R is (approximately) 3 [5, 22, 23]. This means that the infectionrate β is 0.75 (considering γ − = 4 days). The implementation of comple-mentary health policies (use of mask, social distancing policies, among oth-ers) tends to reduce the contact frequency, and, therefore, the infection rate( β ). Thus, we will also examine situations accomplishing infection rates of afraction of β . 10 . Results We will examine three major scenarios affecting the human mobility:1. The (global) lockdown scenario assumes that people remain confinedat home until the epidemic is almost over. See details in Section 4.1.2. The scenario where the confinement recommendation is followed by afraction of the city inhabitants. We assume that the traveling individu-als move around according to the Levy pattern explained in Section 2.See further details on this scenario in Section 4.2.3. Mobility is suppressed only for the inhabitants of “infected blocks”.That is, common life mobility is sustained between blocks where nosymptoms of the disease appeared. See Section 4.3 for details.This last scenario is the most cumbersome one since “non-symptomatic”does not actually mean “non-infected”. We will explicitly introduce a setof “non-symptomatic” individuals in this scenario, in order to understandpossible flaws to confinement.
The GLs means that people remain confined within the block where she(he) lives. It corresponds to a sudden break of the mobility around the cityin the context of our model. We assume, however, that people may still getin contact within their own block.In this case, we consider the mobility suppression as the only heath-carepolicy. Additional health-care recommendations for the every day living (say,masks, common rooms disinfections, etc.) are considered in Appendix A.We will come back to this issue at the end of this Section.Fig. 6a shows the number of new infected people along time for threedifferent lockdown periods (see caption for details). It is also shown the evo-lution for the case of no lockdown at all. The mobility cutoff prevents theinfection curves in Fig. 6a from growing almost immediately after the begin-ning of the lockdown. The disease, however, disappears (approximately) 5011
50 100 150 200 time (days) n e w i n f e c t e d (a) Infected time (days) . . . . . r e m o v e d / p o p u l a t i o n (b) Removed
Figure 6: (a) Number of new infected and (b) normalize number of removed as a functionof time. The plot in (b) is normalized with respect to the total population in the city(4 .
32 M). The GLs is applied when the number of new infected equals to: 5k, 10kand 30k new infected individuals. corresponds to the scenario of no lockdown at all.The infection rate remains constant along the simulation process and equals to β = 0 . days (or 7 weeks) after. This is the time it take the susceptible or exposedindividuals in each block to surpass the disease.Fig. 6b exhibits the number of removed individuals as a function of time(see caption for details). These correspond to those individuals that previ-ously appear as infected in Fig. 6a. It can be seen that the number of removedindividuals increases since the outbreak of the disease, but it reaches a plateausoon after the lockdown is established. The plateau level, however, dependsstrongly on the starting date of the lockdown. Recall that the disease evolvesonly within the infected blocks after the lockdown implementation. Thus,the earlier the lockdown, the less number of infected (and removed) blocksat the end of the disease.We now turn to the city map, in order to get a more accurate picture ofthe results so far. Fig. 7 displays the city as a square arrangement of blocks(see caption for details). Each “pixel” corresponds to a block, and the pixelcolor is associated to the corresponding scale on the left, which states thenumber of infected and the normalized number of removed per block. Thesnapshots capture the disease propagation from a single block located at the12enter of the map. A complete lockdown occurs at day 100.The successive snapshots display a seemingly symmetrical propagationpattern, shortly after the outbreak. However, “secondary” focuses appeararound the main focus due to those long traveling individuals. Recall thatwe assumed that human mobility follows a Levy-flight distribution (see Sec-tion 2.1).Notice that the lockdown implementation (from day 100 onwards) some-how “freezes” the picture until the disease disappears (say, 50 days after).People continue to get infected within each block during the “quaratine” pe-riod. Complementary health-care recommendations may be required for thedisease control within each block.Fig. 8 shows the number of “infected blocks”, regardless of the numberof infected people in the block (see caption for details). These curves quan-tify the infection map displayed in Fig. 7, and resumes the effects of the fulllockdown.We may conclude that the GLs appears as a reliable strategy for avoid-ing the disease propagation. But the main drawback is that “non-infected”blocks will enter the “quarantine”. Appendix A further shows the effects ofcomplementary health-care policies. In this case we model a situation in which the GLs cannot be imple-mented. We distinguish, however, two groups which cannot be kept confined:workers from essential activities (say, health care, food supply or public orderservices, etc.) or those who decide not to accept the confinement recommen-dation. The former are expected to follow complementary health-care rec-ommendations, while the latter might not. For this reason, we will examinerelaxed confinement conditions and infection rates reduction.13
50 100 150 200 time (days) n e w i n f e c t e d (a) Infected time (days) . . . . . r e m o v e d / p o p u l a t i o n (b) Removed InfectedRemovedDay 50 Day 80 Day 110 Day 130 Day 150
Figure 7: (a) Total number of new infected and (b) normalized number of removed as afunction of time. The plot in (b) is normalized with respect to the total population in thecity (4 .
32 M). The GLs started at the day 100 after the first infected was detected. Thisis indicated by a vertical line in (a) and (b). The red continuous line corresponds to thescenario of no lockdown at all, while the lockdown case is indicated by green triangles.(Lower) Spatial distribution of the number of infected individuals for the five differentdates (indicated in (a) and (b) by blue squares). The scale bar on the right corresponds tothe number of infected and the normalized number of removed per block, respectively. Thenormalization was computed with respect to the population of each block. The infectionrate remains constant along the simulation process and equals β = 0 .
75. The city wascomposed by 120 ×
120 blocks placed on a square grid with 300 individuals per block.
50 100 150 200 time (days) . . . . . i n f e c t e d b l o c k s / t o t a l b l o c k s Figure 8: Normalized number of infected blocks as a function of time. The plot isnormalized with respect to the total number of blocks (14 . β = 0 .
75. That is, there is no complementaryhealth-care policies during the lockdown. The different lockdown implementation days areindicated by vertical lines.
Fig. 9a shows how the infection curves change as the number of agentswhich do not accept the movement restriction increases (see caption for de-tails). The confinement starts at the vertical line. Notice that the propa-gation stops dramatically for the complete confinement situation. But thepossibility of stopping the outbreak vanishes if a fraction of people (as smallas 20%) still move around. A quick inspection of Fig. 9b confirms this point.We also notice from Fig. 9b that the total number of removed people atthe end of the lockdown is the same for any IGLs. This appears to be indisagreement with the stochastic point of view, where the mobility reductionyields to the reduction in the probability of meeting people. We should re-mark that the meeting probability is somehow included in the infection rate β within the SEIR model. Thus, for a complete picture of the IGLs, it isnecessary to explore different values of β , as described below.15
100 200 300 time (days) n e w i n f e c t e d (a) Infected time (days) . . . . . r e m o v e d / p o p u l a t i o n (b) Removed
Figure 9: (a) Number of new infected people and (b) normalized number of removedindividuals as a function of time. The plot in (b) is normalized with respect to the totalpopulation in the city (4 .
32 M). The IGLs is applied when the number of new infectedequals to 5k (say, at day 66). From the lockdown implementation day, the percentageof individuals that move around is: 0%, 20% and 50%. corresponds to thescenario of no lockdown at all (100% of moving people). We consider a partial break ofthe mobility as the only heath-care policy. Thus, the infection rate equals β = 0 .
75 allalong the propagation process.
Fig. 10 examines the number of removed individuals ( i.e. agents thathave undergone the complete cycle of the illness, and have either reached ahealthy state or have died) at the end of the epidemic in terms of the comple-mentary health-care policies. Two startup days for the lockdown are shown(see caption for details). We can see that the number of removed individualsexperiences a dramatic change at β/β ≈ . . . . . . . β/β . . . . . r e m o v e d / p o p u l a t i o n (a) . . . . . . β/β . . . . . r e m o v e d / p o p u l a t i o n (b)Figure 10: Normalized number of removed people as a function of the infection rate atthe end of the epidemic. The plot is normalized with respect to the total population in thecity (4 .
32 M). The IGLs is applied when the number of new infected equals to: (a) 5k and(b) 30k. Since the lockdown implementation day, the percentage of individuals that movearound is: 0%, 20% and 50%. corresponds to the scenario of no lockdown atall (100% of moving people). Also, from the lockdown implementation day, the infectionrate change from β = 0 .
75 to β . The city was composed by 120 ×
120 blocks placed on asquare grid with 300 individuals per block.
50% of the people still move around), then complementary health-care poli-cies should be heavily implemented, in order to avoid a massive contagion.This appears as an essential issue for late lockdowns.We close this section with the following conclusion: a complete mobil-ity suppression appears as the most effective way of reducing the number ofcasualties. Essential workers following strict health-care recommendations(say, masks, behavioral protocols, etc.) that move around, however, will notspread the disease to uncontrolled levels. But, a small fraction of peoplemoving around out of protocol can spoil the mitigation efforts.
The local lockdown means that people remain confined within the blockwhere he (she) lives, depending on the infection level of their block. That is,those blocks surpassing a certain “threshold” of infected people are immedi-ately isolated, while the others remain “open”. As in the case of the GLs,we assume that people from an isolated block may still get in contact within17 a) 0% of mobility (b) 50% of mobilityFigure 11: Normalized number of removed individuals (see scale on the right) as a functionof the lockdown implementation day and the infection rate ( β/β ). The normalization wasdone taking into account the city population. The infection rate changes from β = 0 . β since the lockdown implementation day. this block. We consider the mobility break as the only heath policy. time (days) n e w i n f e c t e d (a) Infected time (days) . . . . . r e m o v e d / p o p u l a t i o n (b) Removed
Figure 12: (a) Number of new infected people and (b) normalized number of removedindividuals as a function of time. The plot in (b) is normalized with respect to the totalpopulation in the city (4 .
32 M). The LLs is applied when the number of new infectedequals to 30k (indicate by the vertical line, at day 100). The locked blocks are those thatexceed the following thresholds: 0, 5 and 10 individuals (see text for details).corresponds to the scenario of no lockdown at all. We consider the isolation of the infectedblocks as the only heath-care policy. Thus, the infection rate equals β = 0 .
75 all alongthe propagation process. (a)
Infected threshold 0 (b)
Infected threshold 5 (c)
Infected threshold 20
Figure 13: Spatial distribution of the blocks according to their infected state at thelockdown implementation day (100). Those blocks without infected individuals are rep-resented in black, while those blocks with a number of infected people lower (greater)than the threshold (see legend) are represent in orange (blue). Blue blocks were isolatedfrom the rest, while black and orange blocks were “opened”. The infection rate remainsconstant along the simulation process and equals β = 0 .
75. The city was composed by120 ×
120 blocks placed on a square grid with 300 individuals per block.
The “threshold” of non-detected people is responsible for the time lapse19etween the lockdown day and the end of the disease. This can be confirmedthrough Fig. 13, that shows the contour maps for the infected blocks (seecaption for details). Notice that the non-detected blocks (say, the orangeones) become more relevant as the “threshold” level increases.Fig. 14 exhibits the fraction of isolated blocks with respect to those blocksattaining at least an infected individual. We can observe that half of the in-fected blocks are actually not detectable for a threshold as low as 5 individ-uals per block. This is a strong warning on the effectivity of the LLs. Publichealth officers will lock down as many blocks as detected, but the undetectedwill actually continue the propagation. time (days) . . . . . i s o l a t e b l o c k s / i n f e c t e d b l o c k s Figure 14: Normalized number of isolate blocks with respect to the total number ofinfected blocks as a function of time. The LLs is applied when the number of new infectedequals to 30k (indicated by the vertical line, at day 100). The locked blocks are thosethat exceed the following thresholds: 0, 5, 10, 20 and 40 individuals (seetext for details). The dashed lines corresponds to the behavior of both magnitudes beforethe lockdown implementation. The infection rate remains constant along the simulationprocess and equals β = 0 . Whatever the efforts to detect infected, it seems that 40 −
45% of theinfected individuals do not experience noticeable symptoms [24]. We in-troduced this phenomenon into our simulations. Fig. 15 shows the overallremoved people for an increasing number of “non-symptomatic” individuals.This confirms once more the lack of effectivity of the local lockdown if noother policy is established. 20 . . . . . . asymptomatic/infected . . . r e m o v e d / p o p u l a t i o n Figure 15: Normalized number of removed individuals (at the end of the epidemic) asa function of the percentage of asymptomatic individuals. The normalization was donetaking into account the city population. The LLs started at the day 100 after the firstinfected was detected. The infection threshold is five individuals. From the lockdownimplementation day, the infection rate switches from β = 0 .
75 to β = 0 .
075 in , whileremains constant ( β ) in . A thoughtful policy should include either strategic testings and backwardtracing of the infected, from our point of view. We simulated this policyby tracing back any infected people to where she (he) belonged before beingdetected. The procedure in a nutshell is as follows: • Test a random block. If infected, trace back all the individuals to theblock they visited before. • Test all the blocks recognized as visited immediately before. • Lock down any of the above if infected.The results from these simulations are shown in Appendix C. The test-ing procedure exhibits a noticeable efficiency (say, a noticeable decay in thenumber of removed people) if at least 80% of the blocks can be tested. Theback-tracing procedure further reduces this fraction significantly. Other com-plementary health-care policies (like masks, distancing, etc.) can also im-prove significantly the number of removed individuals (see Appendix C for21etails). In summary, the simulation results confirm our intuition on theeffectivity of the back-tracing methodology.We conclude from this Section that the effectivity of any LLs will stronglydepend on breaking off the mobility of infected people. If this fails (becauseof asymptomatic or unreachable people), the disease will spread dramaticallythroughout the city. The strategic testing and back-tracing of the infectedshould be considered as an essential tool for the disease mitigation.
5. Analysis of the effects of the different strategies
In this Section we discuss the performance of the GLs, IGLs and LLs.We limit our analysis to the following points(a) The performance of the lockdown is actually associated to the mitiga-tion of the disease. Smooth infection curves are preferred in order toavoid stressing the medical care system.(b) Lockdowns seriously damage the economy. The less disturbing andshorter lasting actions on non-infected people are therefore preferred.We propose a merit function in order to rate the performance of the dif-ferent lockdown strategies above mentioned, with respect to conditions (a)and (b). We will consider the fraction of the new infected people at any time i ( t ) (or i n at step n of the simulation) and the mobility µ ( t ) as the mostrelevant quantities for building the merit function (see below for the precisedefinitions). Thus, the merit function will be expressed as C = C ( i, µ ).Notice from Section 4 that the maximum number of new infected peopleis quite different for the examined scenarios (see, for example, Figs. 9 and12). Our merit function will consider the new infected people ( i ) normalizedwith respect to the maximum number of new infected when no lockdown iscarried out. Accordingly, we will consider the mobility fraction ( µ ) as theamount of traveling people normalized with respect to the traveling peoplebefore the lockdown. 22he topic (a) concerns the infected people. The successful lockdown willmitigate the overall number of infected people. Our first proposal would beto rate the performance as the cumulative value of i along the lockdown pe-riod. This approach, however, does not consider the stressing of the medicalcare system. For instance, it does not make any difference between sharpinfection curves and smooth ones, provided that the total number of infectedpeople are the same. We can therefore improve the proposed function by cu-mulating the fractions i α , for α >
1. The coefficient α introduces a penaltyto the sharp maximum (see below).The topic (b) concerns the non-infected people. The lockdown breaks theroutine of the traveling fraction of people µ (1 − i ), and consequently, theeconomic activity. We propose rating the performance of the non-infectedmotion as the aggregate of a linear function of µ (1 − i ).We express our merit function as follows C = N (cid:88) n =1 (cid:20) (a) (cid:122)(cid:125)(cid:124)(cid:123) i αn + (b) (cid:122) (cid:125)(cid:124) (cid:123) A + B µ n · (1 − i n ) (cid:21) (4)where n = 1 ...N stands for the day of the lockdown. The term (a) refers tothe medical care cost, and the term (b) refers to the economical cost.Notice that in regular working days i = 0 and µ = 1. This yields a dailyeconomical cost equal to A + B , according to (4). We rate this cost as thenull cost ( C = 0) for practical reasons. Thus, we set A + B = 0 to hold thiscondition. The cost function then reads C = N (cid:88) n =1 (cid:20) (a) (cid:122)(cid:125)(cid:124)(cid:123) i αn + (b) (cid:122) (cid:125)(cid:124) (cid:123) A [1 − µ n · (1 − i n )] (cid:21) (5)This expression shows that an increase in the number of new infected peo-ple (although keeping µ = 1) yields an increase in the medical cost (a) andthe economical cost (b). The implementation of a strict lockdown ( µ → A is the decisive parameter in the balance between the medicalcare, and economical cost.We stress that the proposed function (5) is limited to items (a) and (b),while other presumably important arguments could have been left aside forsimplicity. We also fixed α = 2 for practical reasons, but we checked that C behaves qualitatively the same for other values α > A is actually the only free parameter in our cost function.It stands as a weighting factor for the economical cost. We will discuss thebehavior of C for the lockdown strategies appearing in Section 4 while vary-ing the values of A .Let us first examine the medical care cost (a) and economical cost (b) sep-arately. Fig. 16 plots these costs for the situations shown in Figs. 9 and 12,respectively (see caption for details). The varying parameter is different oneach plot, that is, the horizontal axis corresponds to the mobility in Fig. 16aand to the threshold level in Fig. 16b. The horizontal scale in Fig. 16a runsfrom the most strict situation at the origin ( µ = 0) to the normal movingsituation ( µ = 1). Analogously, the horizontal scale in Fig. 16b runs fromthe most early detection at the origin to a detection level of 10 individuals.We present, however, both plots together in order to visualize the behaviorof the GLs, IGLs and LLs as the lockdown becomes more and more relaxed.Fig. 16 resumes, indeed, the costs as a function of the lockdown strictness.We notice from Fig. 16 that the medical cost, although different, shows aquite similar behavior in the IGLs and the LLs (see green squares in there).The economical costs, however, differ from each other when the lockdownsare very strict. The mutual differences vanish for the relaxed situations, nomatter if the lockdown is global or local.The dramatic increase of the economical cost (for the strict global lock-downs) is a matter of concern. Fig. 16a reports this phenomenon for smallmobility values (IGLs), although not for the null mobility situation (GLs).The null mobility situation means that the disease remains confined to eachblock. But if a few agents avoid the lockdown, they can additionally spreadthe disease to other non-infected blocks. Thus, the perfect lockdown situa-24
20 40 60 80 100 mobility ( % ) C o s t mobility ( % ) T o t a l c o s t (a) infected threshold C o s t infected threshold T o t a l c o s t (b)Figure 16: Medical care cost ( ) and economical cost ( ) for the scenarios analyzed inSection 4. The inset shows the total cost C assuming A = 0 .
1. (a) Implementation ofthe GLs and IGLs vs. µ . The dashed lines means that the economical cost for mobilitiesbetween 0% and 5% are greater than 300. (b) Implementation of the LLs vs. the infectionthreshold. In both cases, the lockdown is applied when the number of new infected equalsto 5k. The infection rate remains constant along the simulation process and equals β =0 . tion and the imperfect situation are quite different ones. This can be verifiedby observing Fig. 9. The infection curve smoothens and widens as the mo-bility switches from 0% to 20%. For a mobility fraction of 50% the curvenarrows back again.Either Fig. 9a and Fig. 16a point out that small mobility values induce aslow spreading dynamics ( i.e. no massive propagation). This does not stressthe medical care system, but yields long disruptions of the working routines.Notice from the cost expression (5) that the term (b) is linear to N if µ ≈ C as a function of the weighting factor A (see caption for details). Notice that although A can be chosen freely,it seems unrealistic to allow values yielding to costs beyond the cost of no-lockdown at all. We will consider the unlocked situation as the boundingvalue to C , as shown in red in Fig. 17 (see caption for details). . . . . . . A C o s t . . . (a) . . . . . . A C o s t . . . (b)Figure 17: (a) Cost function C as a function of the weighting parameter A for the GLsand IGLs. From the lockdown implementation day, the percentage of individuals thatmove around is: 0% (GLs), 5%, 10%, 25% and 50%. (b) Cost function C as a function of the weighting parameter A for the LLs. The locked blocks are those thatexceed the following thresholds: 0, 2, 3, 5 and 10 individuals. In both cases,corresponds to the scenario of no lockdown at all. Also, the lockdown is applied whenthe number of new infected equals to 5k. The infection rate remains constant along thesimulation process and equals β = 0 . Fig. 17 reports the minimum cost situation in blue color ( ) for the GLsand LLs. These correspond to either the most strict global lockdown ( µ = 0)or the most early detection for the local lockdown (say, the null threshold).The latter attains a better performance with respect to the former for all theexplored values of A .The economical cost (b) is responsible for the slope of the curves in Fig. 17.The almost flat slope observed for the null threshold means that the daily26ost for this situation is negligible. This is in agreement with an early detec-tion, where the fraction of infected people is quite small ( i ≈
0) and most ofthe non-infected people are allowed to move around ( µ ≈ N ≈
200 days.Interestingly, the lockdown curves in Fig. 17b meet the no-lockdown curve forthresholds surpassing 2-3 infected people. This makes the lockdown curvesonly valid for small values of A (say, below 0.7).Recall that the IGLs experiences a dramatic increment of the economicalcost for small mobility fractions (see Fig. 16a). As a result, the lockdowncurves always meet the no-lockdown curve at some point (except for µ = 0),as shown in Fig. 17a. This is quite a difference with respect to the localstrategy, since the lockdown curves are always limited to small values of A (except for µ = 0).Let us close the discussion with the following comments. We showed thatthe most strict implementation of either the global or local lockdown leads tothe optimum performance (although the local one is preferred, as discussedabove). But we noticed that there is some space left for partially effectivelockdowns, if the most strict conditions are not attainable. The degree ofeffectiveness depends on the balance between the medical care costs and theeconomical costs, within this model.
6. Conclusions
This work concerns with the effect of human mobility in the context of amodel for the spatio-temporal evolution of the COVID-19 outbreak. Peoplemove according to the Levy distribution and get into contact with each otherduring their daily routine. The lockdowns prevent these contacts from oc-curring, and thus, mitigate the propagation of the disease. Our investigationstudies different lockdown scenarios and performs a careful evaluation of theireffectiveness. We draw some recommendations for the better performance of27he lockdown.We assumed a SEIR compartmental model (with constant infection rates)for the inhabitants of a block. Each block was considered as a node withina square network of 120 ×
120 nodes. People were allowed to travel twicea day between blocks. The initial conditions for the simulations considered20 infected individuals located at the central block of the network, while therest of the individuals were assumed to be in the susceptible state.We focused on three scenarios: the full lockdown of all the blocks (per-fect scenario), the partial lockdown of all the blocks (imperfect scenario), andthe lockdown of only the infected blocks (local lockdown). We sustained thelockdown until the disease propagation was (almost) over.We first noticed that the success of any control action depends stronglyon canceling the mobility around the city. But a small number of individ-uals may spoil the effectiveness of these actions if they do not follow theconfinement recommendations. This is, in our opinion, the major risk whenimplementing any lockdown strategy.We further built a cost function to rate the three strategies. We arrivedto the conclusion that full lockdowns, or, the (very) early detection and iso-lation strategy are the most effective ones. The local isolation strategy ispreferred, though, since it appears as the less costly in the context of ourmodel.It is important to emphasize that strict lockdown policies also allow forshort periods of isolation. More relaxed lockdowns are less costly daily, butcumulate large costs after an extended period of time.The full lockdown looses effectivenesses if the mobility is not completelycanceled, as already mentioned. Our model shows that local lockdowns canstill be quite effective even if a small number of infected people is not detected(and isolated). But the ultimate decision on whether to choose a global orlocal lockdown on a specific circumstance will depend on the right balancebetween the medical care cost and the economical cost due to routine dis-ruptions. We will leave this discussion open to future research.28 cknowledgments
This work was supported by the National Scientific and Technical Re-search Council (spanish: Consejo Nacional de Investigaciones Cient´ıficas yT´ecnicas - CONICET, Argentina) and grant Programaci´on Cient´ıfica 2018(UBACYT) Number 20020170100628BA. G. Frank thanks Universidad Tec-nol´ogica Nacional (UTN) for partial support through Grant PID NumberSIUTNBA0006595. 29 ppendix A. Global lockdown with complementary health poli-cies
We analyze here the effects of complementary health policies during theglobal lockdown. Fig. A.18a shows the number of new infected people alongtime for three different infection rate (see caption for details). It is also shownthe evolution for the case of no lockdown at all. The mobility cutoff preventsthe infection curves in Fig. A.18a from growing almost immediately after thebeginning of the lockdown. In this sense, the more additional health policies,the faster decrease of the new infected individuals. time (days) n e w i n f e c t e d (a) Infected time (days) . . . . . r e m o v e d / p o p u l a t i o n (b) RemovedFigure A.18: (a) Number of new infected people and (b) normalized number of removedindividuals as a function of time. The plot in (b) is normalized with respect to the totalpopulation of the city (4 .
32 M). The global lockdown started at the day 100 after the firstinfected was detected. This is indicated by a vertical line in (a) and (b). Before this day,the infection rate equals to β = 0 .
75. However, from this day, the infection rate equals:(1 / β , (1 / β and β . corresponds to the scenario of no lockdown at all. Inthe last case, the infection rate remains constant along the simulation process and equalsto β = 0 . Fig. A.18b exhibits the number of individuals that passed over the dis-ease as a function of time (see caption for details). These correspond to thoseindividuals that previously appear as infected in Fig. A.18a. First, it can beseen that the number of removed individuals increase since the outbreak ofthe disease. Second, it reaches a plateau soon after the lockdown is estab-lished. The plateau level, however, strongly depends on the complementaryhealth policies. The more “intense” health policies ( i.e. the lower infection30ate), the lower number of removed individuals.
Appendix B. Comparison between global and local lockdown
This appendix compare the global and local lockdown in terms of thenew infected individuals. Recall that the global lockdown “cutoff” the hu-man mobility around the city. In this sense, all blocks are isolated from therest regardless of whether they are “infected” or not. On the contrary, themobility is suppressed (depending the infected threshold) only for the inhab-itants of “infected blocks” in the local lockdown. time (days) n e w i n f e c t e d Figure B.19: Comparison between the number new infected in the global ( ) and local( ) lockdown scenario. The global and local lockdown started at the day 100 after thefirst infected was detected. This is indicated by a vertical line. From this day, all thoseinfected blocks in which the number of infected exceeds 0 are isolated (in the local lockdownscenario). corresponds to the scenario of no lockdown at all. From the lockdownimplementation day, the infection rate switches from β = 0 .
75 to (1 / β (in both cases). Fig. B.19 shows the number of new infected people along time implement-ing a global and local lockdown (see caption for details). It is also shownthe evolution for the case of no lockdown at all. It can be seen that bothcurves matches along time. This means that it is not necessary isolated allblocks to reduce the infection. In this sense, as can be expected, only isnecessary isolated those infected blocks. Therefore, we can conclude that thelocal lockdown is the ”efficient” case of the global lockdown.31 ppendix C. Implementation of the testing & back-tracing strat-egy
We represent in this Section a possible testing procedure by means of aprobability p as follows:1. We first chose (and test) a block at random with probability p .2. If the chosen block is infected, we lock down the block until the diseasedisappears. If not, the block remains “open”.3. We trace back and test all the individuals who visited the infectedblock.4. We lock down any of the above blocks if infected.5. We repeat this procedure every day.Fig. C.20 shows the number of new infected people along time for threedifferent testing probabilities (see caption for details). It is also shown thetime evolution for the case of no lockdown at all. Notice that the propagationdecreases dramatically for the ideal tested scenario. But the possibility ofstopping the outbreak vanishes in a bad (or poor) testing scenario. As canbe seen in Fig. C.20 the back-tracing of the infected improves the mitigationstrategy.Fig. C.21 shows the number of removed individuals at the end of the epi-demic in terms of the testing probability p for three different complementaryhealth policies (see caption for details). It can be seen that, in a scenariowithout complementary health policies (blue squares in Fig. C.21), a massivetesting (of more than 80%) is the only tool to avoid a wide spreading on thepopulation. Note that this percentage is significantly reduced to 60% if aback tracing policy is also implemented.Notice from Fig. C.21 that the implementation of strict complementaryhealth policies (such as the use of a mask and social distancing) improvesthe mitigation of the disease. ReferencesReferences [1] D. H. Barmak, C. O. Dorso, M. Otero, H. G. Solari, Dengue epidemicsand human mobility, Phys. Rev. E 84 (2011) 011901.32
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