COVID-19 spreading under containment actions.

We propose an epidemiological model that explores the effect of human mobility on the spatio-temporal dynamics of the COVID-19 outbreak, in the spirit to those considered in Refs. Barmak et al. (2011, 2016) and Medus and Dorso (2011) [1]. We assume that people move around in a city of 120 × 120 blocks with 300 inhabitants in each block. The mobility pattern is associated to a complex network in which nodes represent blocks while the links represent the traveling path of the individuals (see below). We implemented three confinement strategies in order to mitigate the disease spreading: (1) global confinement, (2) partial restriction to mobility, and (3) localized confinement. In the first case, it was observed that a global isolation policy prevents the massive outbreak of the disease. In the second case, a partial restriction to mobility could lead to a massive contagion if this was not complemented with sanitary measures such as the use of masks and social distancing. Finally, a local isolation policy was proposed, conditioned to the health status of each block. It was observed that this mitigation strategy was able to contain and even reduce the outbreak of the disease by intervening in specific regions of the city according to their level of contagion. It was also observed that this strategy is capable of controlling the epidemic in the case that a certain proportion of those infected are asymptomatic.

Introduction

In the absence of a vaccine, strategies based on non-pharmaceutical interventions were proposed to contain the COVID-19 pandemic. Social distancing policies, specific mobility restrictions and lockdowns, among others, were the more common ones. Such policies should be implemented for long periods (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 changes in the mobility patterns and the disease propagation. Ref. [2] reports a correlation between the mobility pattern and the reduction of new infections. However, a delay of 2–3 weeks exist between both magnitudes due to the incubation period of the disease. Ref. [3] also finds that traveling restrictions and social distancing were effective actions (among others) in the containment of the COVID-19 disease in China. Additionally, Ref. [4] and Ref. [5] analyze the European and US cases, respectively. The later arrives to the conclusion that a small minority of “super-spreader” places are the responsible for the wide propagation of the infection.

The changes in the mobility patterns are a consequence of the kind of quarantine. Ref. [6], for instance, proposes a special kind of quarantine that reduces the disease infection while allowing the socio-economic activity. In a similar way, Ref. [7] proposes a cyclic schedule of 4-days work and 10-days lockdown. An improved version of this strategy can also be found in Ref. [8]. Moreover, Ref. [9] concludes that an alternated strategy of lockdown and release will gradually reduce the “cost” of an epidemic, as suggested by the Parrondo’s games.

Ref. [10] carried out a detailed comparison between the strategies applied in different countries for the disease control. The investigation concluded that either the mobility control, the coordination with neighboring countries and the testing procedures are the key issues for containing and reducing the number of new cases. However, it was also observed in Ref. [11] that other social factors, like globalization or extreme centralization, can run the food (or goods) supply to a critical stage.

Many other complex scenarios can be found in the literature. For instance, Ref. [12] analyzes the influence of human behavior on the infectious disease transmission. Refs. [13], [14] focus on the effects of vaccination during the pandemic and the vaccination repurposing, respectively. Ref. [15] studies the potential effectiveness of targeted mass prophylactic use of antiviral drugs as a containment strategy. The role of “super-spreader” events can also be found in Refs. [16], [17].

Our concern is actually placed on the effectiveness of mobility restrictions. Ref. [18] shows real evidence of the consequences of restrictions on the daily records of deaths. These authors state that deaths are expected to diminish after 3–4 weeks of establishing travel restrictions. However, this point has been questioned in the context of limited healthcare resources. Ref. [19] considers a scenario of limitations in the availability of mechanical ventilators, among other resources. They notice that early mobility restrictions are useful while upgrading the healthcare capacity (short-term efficiency), but do not necessarily ensure a long term reduction of deaths.

The mobility restrictions may also apply to quite different scales. In a quite enlightening work, in Ref. [20] the authors developed a dual network, attaining for short-scale and long scale mobilities (say, the land commuting system and the airline traffic, respectively). Both exhibit different dynamics, and thus, the mobility restrictions are expected to differ according to the global or local context. Further considerations on topological issues were explored in Refs. [1], [21], [22].

In this work we consider the spread of a disease mimicking the COVID-19. We propose a simple model of a city consisting of a square lattice of NN nodes. Each node represents a block populated, at the start of the simulations, by M agents. Some of the agents will remain in the block while others will move around in the city (see Section 2 for full description). This agents will be subjected to a SEIR kind of epidemic dynamics. We simulated and compared different confinement strategies in order to mitigate the disease propagation. We limit our investigation to the mobility within the city area. We emphasize, however, that our analysis focuses on the containment strategies at the level of a city like the one we live in, and in which this kind of containment has been (partially) applied.

In Section 2 we detail the characteristics of our epidemiological model and the different mitigation strategies. Section 3 describes the simulation procedure. Section 4 displays the results of our investigation, while Section 5 is dedicated to the discussion of the main results. The conclusions are drawn in Section 6.

The epidemiological model

There are three main ingredients in the description of the COVID-19 contagion and spatial spreading: the scenario where the process takes place (Section 2.1), the mobility patterns of the individuals (Section 2.2) and the epidemiological dynamics of the individuals (Section 2.3).

The stage

City

The simulation of the evolution of COVID-19 is performed in a schematic city in which the basic unit is the block. The city is represented by an homogeneous urban area of blocks placed in a square grid. The size of each block is  m  m. The simulated grid corresponds to a big city, like Buenos Aires, Argentina. Each block hosts 300 people but this quantity may vary during the day (see next Section). The total population of the simulated 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 building a weighted and directed network. The nodes represent the blocks, while the links represent the human mobility from one block to another one. We consider two different links: short and long links.

Each node is linked with its first neighbors by a short link, as we illustrate in Fig. 1(a). They represent the human mobility pattern between all neighboring blocks. Notice that these links generate a connected graph. As we will see later, this characteristic of the network will allow the disease to reach all parts of the city (as long as it is not locked during the epidemic).

Fig. 1

Schematic representation of a city network. The city is composed by 120 × 120 blocks placed on a square grid. They are linked by (a) four short links connecting neighboring nodes and (b) long links connecting far away nodes (see text for more details). The city corresponds to the union of these two sub-networks.

On the other side, recent investigation on human mobility shows that the traveling lengths of the individuals follows a Levy distribution given by [23] where stands for the probability that an individual reaches a distance , and  m and correspond to empirical parameters.

We built a network of long links following the Levy mobility pattern. This network is illustrated in Fig. 1(b). The procedure was as follows

We randomly chose a node.

We then select randomly (according to the Levy distribution) the length of the next link.

We link the node from step 1 to any (random) node located at the distance in any direction.

Notice that each node may have more than one long link. Recall that these links are complementary to the short links connecting neighboring blocks (say, four neighbors per block).

The total number of long links will depend on how many times we repeat the above steps. Thus, we looped these steps until the geodesic path of the network resembled the one expected for a “small world” network [24]. As a general rule, we found that this condition was fulfilled after 25% of the nodes had at least one long link. Table 1 summarizes the final set of parameter.

Table 1

Most relevant parameters used in the simulation of COVID-19 epidemic along the mobility network.

Number of nodes (blocks)	120 × 120
Population per block	300
Short links	Between first neighbors
Long links	Over 25% of the nodes
Total number of links	31825
r_0 (Levy distribution)	100 m
β (Levy distribution)	1.75

Finally, we stress the fact that the links between different blocks are assigned at the beginning of the simulation. Thereafter, the links remain fixed until the end of the simulation.

Mobility pattern

We follow a similar scheme as in Ref. [24], [25] for the individuals’ mobility. This means that we divide the population of each block into two groups: stationary and non-stationary individuals. The former is composed by infected individuals while the latter is made up of susceptible, exposed and removed individuals.

We will assume that stationary individuals stay in their original block. We will further assume, in the spirit of Ref. [25], that 50% of the non-stationary stay in their original block, while the other 50% is considered to be mobile during the simulation. In this sense, we assume that 60% of the non-stationary individuals travel (via short links) from their original block to the neighboring block. And, the remaining 40% travel (via long links) from their original block to far blocks as indicated in Fig. 2.

Fig. 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 the 50% of the susceptible, exposed and removed individuals move to another blocks (see text for more details).

Daily displacements of the individuals

Each moving individual is assumed to stay 1/2 of the day in its original block while the other 1/2 of the day he (she) moves to another block. They go to his (her) destination (work) everyday, and at the end of the day return back to their original block (home). This mechanism is performed reversing the direction of the links between blocks.

Fig. 3 shows a schematic representation of the human mobility pattern during each day (we illustrate this by a small city of 4 blocks for a better visualization). Recall from the last section that the links represent the mobility pattern between different blocks. Thus, moving humans travel through the links from one block to another one.

Fig. 3

Schematic illustration of the epidemic model incorporating individual human mobility during each day. The node colors represent the population of each 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 to the pre-assigned mobility pattern. The arrows represent the movement direction. (c) The population of each block after mobile humans traveled to the working block. (d) Then, at 24 o’clock those humans that traveled from home to work return back to home. At the end of the day, the population of each block is the same as that in the beginning of the day. The sequence (a–d) repeats every day until the end of the simulation.

Notice that the population inside each block is always 300 during the night hours. This remains fixed during the day hours only in the “Home” scenario (see Fig. 3(a)). The disease evolution in each node is therefore determined by the health care conditions within the block. However, in the “Work” scenario (see Fig. 3(c)), people arrive or depart from each block during the day according to the in–out links (network heterogeneity). The disease evolution, thus, mixes between visitors and inhabitants of each block, yielding to a much complex dynamic than in the “Home” scenario.

The “Work” scenario exhibits similarity to the type-II metapopulation models (see Ref. [21] for details). These are quite suitable models for describing the healthy to infected transition in a complex network. However, we are not interested in the stationary situation, but in the time evolution since a few people become infected. Therefore, unlike the usual metapopulation analysis, we do not consider mean population values at each node (i.e. block). The daily fluctuations of the amount of people at each node is a relevant issue in our model.

We finally call the attention on the fact that we do not include specific features in our model, such as geographical population distribution, detailed medical and demographical factor, etc. Instead, we will investigate those common characteristics that affect the disease dynamic during a epidemic. The vaccination and the use of antiviral drugs as a containments strategies or “super-spreader” events, will not be part of the investigation.

Epidemiological dynamics of the individuals

In order to describe the time evolution of a given population when a fraction gets infected, we resort to the SEIR compartmental models. These models consider that the individuals can be in four successive states: susceptible (S), exposed (E), infected (I) and removed (R). Details on each state can be found in Refs. [8], [24], [25], [26], [27]. The “exposed” state appears whenever the disease undergoes an “incubation” period, as occurs in the context of the COVID-19. The “removed” state includes either recovered and dead people.

The equations that describe the evolution of the infection read as follows [26], [27]

where , , etc. correspond to the fraction of people in each state. For the purpose of simplicity, we will consider the coefficients , as fixed parameters. The parameter (infection rate) depends on intrinsic ingredients like the infectivity of the virus under consideration and extrinsic ones like the contact frequency. Besides, the parameters and depend exclusively on the illness under consideration.

The basic reproduction number is defined at the beginning of the propagation process as [28]

This quantity represents the number of individuals that are infected by the first infected individuals (say, at the beginning of the disease). It is straight forward that the infection will blossom if is larger than 1. On the contrary, if this quantity is smaller than 1 the infection vanishes.

Fig. 4 shows the time evolution of either the SEIR model for a single block and for the whole mobility model (see caption for details). Notice that the latter shifts the date for the maximum infection to approximately 150. We will analyze this behavior in more detail in Section 4.

Fig. 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 scenario corresponds to a city composed by 120 × 120 blocks placed on a square grid (see Section 2.1). 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 any kind of intervention strategy. The infection rate equals all along the propagation process. 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 to be in the susceptible state.

Lockdown types

In this section we define the proposed containment strategies which we have found useful in order to mitigate the disease. We divide the different types of quarantine according to their level of real life implementation difficulty

Global lockdown

Imperfect lockdown

Local lockdown

Global lockdown scenario (GLs)

This type of lockdown consists in the isolation of each block. This means that people remain in their “home” blocks. We stress the fact that, within this strategy, society as a whole enters in lockdown. We illustrate this containment action in Fig. 5(b).

Fig. 5

(a–d) Schematic representation of different types of mitigation strategies (exemplified by a 4 × 4 grid). The lines represent the human mobility pattern between blocks. (a) People are allowed to evolve freely, equivalent to a “continuous activity” scenario. That is, there is no intervention during the epidemic. (b–d) Before a given day, individuals move freely from one block to another according to the mobility pattern. After that, (b) all blocks are isolated, (c) the flow between blocks (represented by red arrows) is reduced but the mobility pattern is unaffected, (d) those blocks with a number of infected individuals greater than a certain threshold are isolated (labeled with a red flag). The other blocks remain linked to each other (labeled with a green flag). See text for more details.

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 between different blocks. Recall from Section 2.2 that 50% of the non-stationary individuals move from their original block to another one. In this strategy, we reduce the level of mobility between blocks. Notice that the global quarantine corresponds to a full reduction of the mobility. We scheme this containment action in Fig. 5(c).

Local lockdown scenario (LLs)

Unlike the other lockdown types, this applies to certain blocks and not to the whole society. Only those blocks that have a certain number of infected people are isolated. In this sense, the population of the isolated block are prohibited to travel around the city. This is achieved in practice by “cutting” the links to/from the infected blocks. We illustrate this containment action in Fig. 5(d).

As a final remark, we stress the fact that full and localized isolation of blocks solely occurs in GLs and LLs, respectively. But, in IGLs there is no isolation of the blocks at all. Actually, the number of people moving between the blocks is partially reduced.

Numerical simulations

We integrate the SEIR equations by means of the Runge Kutta 4th-order method. The chosen time step was 0.1 (days). The SEIR equations were updated twice a day (after the people left their homes and after they returned back (see Fig. 3, Fig. 3).

As mentioned in Section 2.3, the parameters and represent the incubation rate and the recuperation rate, respectively. Therefore, and correspond to the mean incubation time and the mean recovery time, respectively. According to preliminary estimations for COVID-19, we consider the following parameter values for the SEIR model:  days and  days [3], [29], [30], [31], [32].

Infected individuals remain at their “Home” until evolving into the removed state. Susceptible, exposed and removed individuals are able to move from one block to another. The simulation started with 20 infected individuals located at the central block of the city, while the rest of the individuals were assumed to be in the susceptible state.

According to preliminary estimations for COVID-19, the basic reproduction number is (approximately) 3 [3], [33], [34]. This means that the infection rate is 0.75 (considering  days). The implementation of complementary health policies (use of mask, social distancing policies, among others) tends to reduce the contact frequency, and, therefore, the infection rate (). Thus, we will also examine situations accomplishing infection rates of a fraction of .

Results

We will examine three major scenarios affecting the human mobility:

The (global) lockdown scenario assumes that people remain confined at home until the epidemic is almost over. See details in Section 4.1.

The scenario where the confinement recommendation is followed by a fraction of the city inhabitants. We assume that the traveling individuals move around according to the Levy pattern explained in Section 2. See further details on this scenario in Section 4.2.

Mobility is suppressed only for the inhabitants of “infected blocks”. That is, common life mobility is sustained between blocks where no symptoms 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 set of “non-symptomatic” individuals in this scenario, in order to understand possible 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 city in the context of our model. We assume, however, that people may still get in contact within their own block.

In this case, we consider the mobility suppression as the only heath-care policy. 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. 6(a) shows the number of new infected people along time for three different lockdown periods (see caption for details). It is also shown the evolution for the case of no lockdown at all. The mobility cutoff prevents the infection curves in Fig. 6(a) from growing almost immediately after the beginning of the lockdown. The disease, however, disappears (approximately) 50 days (or 7 weeks) after. This is the time it take the susceptible or exposed individuals in each block to surpass the disease.

Fig. 6

(a) Number of new infected and (b) normalize number of removed as a function of 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,  10k and  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 . That is, there is no complementary health-care policies during the lockdown. The different lockdown implementation days are indicated by vertical lines.

Fig. 6(b) exhibits the number of removed individuals as a function of time (see caption for details). These correspond to those individuals that previously appear as infected in Fig. 6(a). It can be seen that the number of removed individuals increases since the outbreak of the disease, but it reaches a plateau soon after the lockdown is established. The plateau level, however, depends strongly on the starting date of the lockdown. Recall that the disease evolves only within the infected blocks after the lockdown implementation. Thus, the earlier the lockdown, the less number of infected (and removed) blocks at the end of the disease.

We now turn to the city map, in order to get a more accurate picture of the 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 pixel color is associated to the corresponding scale on the left, which states the number of infected and the normalized number of removed per block. The snapshots capture the disease propagation from a single block located at the center of the map. A complete lockdown occurs at day 100.

Fig. 7

(a) Total number of new infected and (b) normalized number of removed as a function of time. The plot in (b) is normalized with respect to the total population in the city (4.32 M). The GLs started at the day 100 after the first infected was detected. This is indicated by a vertical line in (a) and (b). The red continuous line corresponds to the scenario 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 different dates (indicated in (a) and (b) by blue squares). The scale bar on the right corresponds to the number of infected and the normalized number of removed per block, respectively. The normalization was computed with respect to the population of each block. The infection rate remains constant along the simulation process and equals . The city was composed by blocks placed on a square grid with 300 individuals per block.

The successive snapshots display a seemingly symmetrical propagation pattern, shortly after the outbreak. However, “secondary” focuses appear around the main focus due to those long traveling individuals. Recall that we assumed that human mobility follows a Levy-flight distribution (see Section 2.1).

Notice that the lockdown implementation (from day 100 onwards) somehow “freezes” the picture until the disease disappears (say, 50 days after). People continue to get infected within each block during the “quarantine” period. Complementary health-care recommendations may be required for the disease control within each block.

Fig. 8 shows the number of  “infected blocks”, regardless of the number of infected people in the block (see caption for details). These curves quantify the infection map displayed in Fig. 7, and resumes the effects of the full lockdown.

Fig. 8

Normalized number of infected blocks as a function of time. The plot is normalized with respect to the total number of blocks (14.4 k). Any block is classified as “infected” if at least one individual in the block is infected. We only consider people “living in the block” (say, those that sleep in there). The global lockdown is applied when the number of new infected equals to:  5k,  10k and 30k new infected individuals.  corresponds to the scenario of no lockdown at all. The infection rate remains constant along the simulation process and equals . That is, there is no complementary health-care policies during the lockdown. The different lockdown implementation days are indicated by vertical lines.

We may conclude that the GLs appears as a reliable strategy for avoiding the disease propagation. But the main drawback is that “non-infected” blocks will enter the “quarantine”. Appendix A further shows the effects of complementary health-care policies.

The imperfect global scenario (IGLs)

In this case we model a situation in which the GLs cannot be implemented. We distinguish, however, two groups which cannot be kept confined: workers from essential activities (say, health care, food supply or public order services, etc.) or those who decide not to accept the confinement recommendation. The former are expected to follow complementary health-care recommendations, while the latter might not. For this reason, we will examine relaxed confinement conditions and infection rates reduction.

Fig. 9(a) shows how the infection curves change as the number of agents which do not accept the movement restriction increases (see caption for details). The confinement starts at the vertical line. Notice that the propagation stops dramatically for the complete confinement situation. But the possibility of stopping the outbreak vanishes if a fraction of people (as small as 20%) still move around. A quick inspection of Fig. 9(b) confirms this point.

Fig. 9

(a) Number of new infected people and (b) normalized number of removed individuals as a function of time. The plot in (b) is normalized with respect to the total population in the city (4.32 M). The IGLs is applied when the number of new infected equals to 5k (say, at day 66). From the lockdown implementation day, the percentage of individuals that move around is:  0%, 20% and 50%.  corresponds to the scenario of no lockdown at all (100% of moving people). We consider a partial break of the mobility as the only heath-care policy. Thus, the infection rate equals all along the propagation process.

We also notice from Fig. 9(b) that the total number of removed people at the end of the lockdown is the same for any IGLs. This appears to be in disagreement with the stochastic point of view, where the mobility reduction yields to the reduction in the probability of meeting people. We should remark that the meeting probability is somehow included in the infection rate within the SEIR model. Thus, for a complete picture of the IGLs, it is necessary to explore different values of , as described below.

Fig. 10 examines the number of removed individuals (i.e. agents that have undergone the complete cycle of the illness, and have either reached a healthy state or have died) at the end of the epidemic in terms of the complementary health-care policies. Two startup days for the lockdown are shown (see caption for details). We can see that the number of removed individuals experiences a dramatic change at for the different moving people situations, regardless of the startup day of the lockdown. This confirms the necessity of proper policies when these situations are expected to occur.

Fig. 10

Normalized number of removed people as a function of the infection rate at the end of the epidemic. The plot is normalized with respect to the total population in the city (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 move around is:  0%, 20% and 50%.  corresponds to the scenario of no lockdown at all (100% of moving people). Also, from the lockdown implementation day, the infection rate change from to . The city was composed by blocks placed on a square grid with 300 individuals per block.

We improved on the above results by displaying in Fig. 11 the contour maps for different mobility situations, as a function of the lockdown date. Notice that Fig. 10 corresponds to date 66 in both contour maps (see caption in Fig. 10).

Fig. 11

Normalized number of removed individuals (see scale on the right) as a function of the lockdown implementation day and the infection rate (). The normalization was done taking into account the city population. The infection rate changes from to since the lockdown implementation day.

Fig. 11 expresses the fact that whenever the human mobility is suppressed, the number of removed individuals depends (almost exclusively) on the timing of the lockdown implementation. That is, the earlier the lockdown, the lower the number of removed. Besides, if mobility cannot be suppressed (say, 50% of the people still move around), then complementary health-care policies 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 mobility suppression appears as the most effective way of reducing the number of casualties. Essential workers following strict health-care recommendations (say, masks, behavioral protocols, etc.) that move around, however, will not spread the disease to uncontrolled levels. But, a small fraction of people moving around out of protocol can spoil the mitigation efforts.

The local lockdown means that people remain confined within the block where he (she) lives, depending on the infection level of their block. That is, those blocks surpassing a certain “threshold” of infected people are immediately 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 within this block. We consider the mobility break as the only heath policy.

The local lockdown relies on the idea that the early detection of the infected can prevent the disease from spreading around the city. This idea presumes insignificant or null detection flaws. But in practice, a few infected individuals may not be detected due to wrong testings, or do not present symptoms at all. Our concern is on these situations. We will assume that a small fraction of infected people per block cannot actually be detected. This means that the lockdown occurs after surpassing a “threshold” of infected people (although not detected).

Fig. 12 represent the usual infection and passed over curves as a function of time, respectively (see caption for details). A quick inspection of these curves show that the disease decays almost immediately if the lockdown process occurs just after the first infected appeared in the block. Otherwise, the number of infected people continues increasing until the day 150 (approximately).

Fig. 12

(a) Number of new infected people and (b) normalized number of removed individuals as a function of time. The plot in (b) is normalized with respect to the total population in the city (4.32 M). The LLs is applied when the number of new infected equals to 30k (indicate by the vertical line, at day 100). The locked blocks are those that exceed 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 infected blocks as the only heath-care policy. Thus, the infection rate equals all along the propagation process.

Fig. 12 can be compared to its counterpart in the context of the IGLs, that is, Fig. 9(a). The time scales in both situations appear quite similar, although the curves in Fig. 12(a) do not extend further than 200 days.

The “threshold” of non-detected people is responsible for the time lapse between the lockdown day and the end of the disease. This can be confirmed through Fig. 13, that shows the contour maps for the infected blocks (see caption for details). Notice that the non-detected blocks (say, the orange ones) become more relevant as the “threshold” level increases.

Fig. 13

Spatial distribution of the blocks according to their infected state at the lockdown implementation day (100). Those blocks without infected individuals are represented 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 isolated from the rest, while black and orange blocks were “opened”. The infection rate remains constant along the simulation process and equals . The city was composed by blocks placed on a square grid with 300 individuals per block.

Fig. 14 exhibits the fraction of isolated blocks with respect to those blocks attaining at least an infected individual. We can observe that half of the infected blocks are actually not detectable for a threshold as low as 5 individuals per block. This is a strong warning on the effectivity of the LLs. Public health officers will lock down as many blocks as detected, but the undetected will actually continue the propagation.

Fig. 14

Normalized number of isolate blocks with respect to the total number of infected blocks as a function of time. The LLs is applied when the number of new infected equals to 30k (indicated by the vertical line, at day 100). The locked blocks are those that exceed the following thresholds: 0, 5,  10, 20 and 40 individuals (see text for details). The dashed lines corresponds to the behavior of both magnitudes before the lockdown implementation. The infection rate remains constant along the simulation process and equals .

Whatever the efforts to detect infected, it seems that of the infected individuals do not experience noticeable symptoms [35]. We introduced this phenomenon into our simulations. Fig. 15 shows the overall removed people for an increasing number of “non-symptomatic” individuals. This confirms once more the lack of effectivity of the local lockdown if no other policy is established.

Fig. 15

Normalized number of removed individuals (at the end of the epidemic) as a function of the percentage of asymptomatic individuals. The normalization was done taking into account the city population. The LLs started at the day 100 after the first infected was detected. The infection threshold is five individuals. From the lockdown implementation day, the infection rate switches from to in , while remains constant () in .

A thoughtful policy should include either strategic testings and backward tracing of the infected, from our point of view. We simulated this policy by tracing back any infected people to where she (he) belonged before being detected. The procedure in a nutshell is as follows:

Test a random block. If infected, trace back all the individuals to the block 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 testing procedure exhibits a noticeable efficiency (say, a noticeable decay in the number of removed people) if at least 80% of the blocks can be tested. The back-tracing procedure further reduces this fraction significantly. Other complementary health-care policies (like masks, distancing, etc.) can also improve significantly the number of removed individuals (see Appendix C for details). In summary, the simulation results confirm our intuition on the effectivity of the back-tracing methodology.

We conclude from this Section that the effectivity of any LLs will strongly depend on breaking off the mobility of infected people. If this fails (because of asymptomatic or unreachable people), the disease will spread dramatically throughout the city. The strategic testing and back-tracing of the infected should be considered as an essential tool for the disease mitigation.

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

The performance of the lockdown is actually associated to the mitigation of the disease. Smooth infection curves are preferred in order to avoid stressing the medical care system.

Lockdowns seriously damage the economy. The less disturbing and shorter lasting actions on non-infected people are therefore preferred.

We propose a merit function in order to rate the performance of the different lockdown strategies above mentioned, with respect to conditions (a) and (b). We will consider the fraction of the new infected people at any time (or at step of the simulation) and the mobility as the most relevant quantities for building the merit function (see below for the precise definitions). Thus, the merit function will be expressed as .

Notice from Section 4 that the maximum number of new infected people is quite different for the examined scenarios (see, for example, Fig. 9, Fig. 12). Our merit function will consider the new infected people () normalized with respect to the maximum number of new infected when no lockdown is carried out. Accordingly, we will consider the mobility fraction () as the amount of traveling people normalized with respect to the traveling people before the lockdown.

The topic (a) concerns the infected people. The successful lockdown will mitigate the overall number of infected people. Our first proposal would be to rate the performance as the cumulative value of along the lockdown period. This approach, however, does not consider the stressing of the medical care system. For instance, it does not make any difference between sharp infection curves and smooth ones, provided that the total number of infected people are the same. We can therefore improve the proposed function by cumulating the fractions , for . The coefficient introduces a penalty to the sharp maximum (see below).

The topic (b) concerns the non-infected people. The lockdown breaks the routine of the traveling fraction of people , and consequently, the economic activity. We propose rating the performance of the non-infected motion as the aggregate of a linear function of .

We express our merit function as follows

where stands for the day of the lockdown. The term (a) refers to the medical care cost, and the term (b) refers to the economical cost.

Notice that in regular working days and . This yields a daily economical cost equal to , according to (4). We rate this cost as the null cost () for practical reasons. Thus, we set to hold this condition. The cost function then reads

This expression shows that an increase in the number of new infected people (although keeping ) yields an increase in the medical cost (a) and the economical cost (b). The implementation of a strict lockdown () further increases the economical cost (b) to its maximum value. The intermediate situations will be more or less costly according to the balance between the medical care cost (a) and the economical cost(b) in expression (5). The parameter is the decisive parameter in the balance between the medical care, 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 for simplicity. We also fixed for practical reasons, but we checked that behaves qualitatively the same for other values (not shown).

The parameter is actually the only free parameter in our cost function. It stands as a weighting factor for the economical cost. We will discuss the behavior of for the lockdown strategies appearing in Section 4 while varying the values of .

Let us first examine the medical care cost (a) and economical cost (b) separately. Fig. 16 plots these costs for the situations shown in Fig. 9, Fig. 12, respectively (see caption for details). The varying parameter is different on each plot, that is, the horizontal axis corresponds to the mobility in Fig. 16(a) and to the threshold level in Fig. 16(b). The horizontal scale in Fig. 16(a) runs from the most strict situation at the origin () to the normal moving situation (). Analogously, the horizontal scale in Fig. 16(b) runs from the most early detection at the origin to a detection level of 10 individuals. We present, however, both plots together in order to visualize the behavior of 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.

Fig. 16

Medical care cost ( ) and economical cost ( ) for the scenarios analyzed in Section 4. The inset shows the total cost assuming . (a) Implementation of the GLs and IGLs vs. . The dashed lines means that the economical cost for mobilities between 0% and 5% are greater than 300. (b) Implementation of the LLs vs. the infection threshold. In both cases, the lockdown is applied when the number of new infected equals to 5k. The infection rate remains constant along the simulation process and equals .

We notice from Fig. 16 that the medical cost, although different, shows a quite similar behavior in the IGLs and the LLs (see green squares in there). The economical costs, however, differ from each other when the lockdowns are very strict. The mutual differences vanish for the relaxed situations, no matter if the lockdown is global or local.

The dramatic increase of the economical cost (for the strict global lockdowns) is a matter of concern. Fig. 16(a) reports this phenomenon for small mobility values (IGLs), although not for the null mobility situation (GLs). The null mobility situation means that the disease remains confined to each block. But if a few agents avoid the lockdown, they can additionally spread the disease to other non-infected blocks. Thus, the perfect lockdown situation and the imperfect situation are quite different ones. This can be verified by observing Fig. 9. The infection curve smoothens and widens as the mobility switches from 0% to 20%. For a mobility fraction of 50% the curve narrows back again.

Either Fig. 9(a) and Fig. 16(a) point out that small mobility values induce a slow spreading dynamics (i.e. no massive propagation). This does not stress the medical care system, but yields long disruptions of the working routines. Notice from the cost expression (5) that the term (b) is linear to if . Thus, the long lasting lockdowns are responsible for the increase in the economical costs.

The LLs isolates the infected blocks only. The mobility is locally suppressed, and the whole scene undergoes a mixture of blocks with full mobility and blocks that are isolated. This is a dynamical process where the blocks become isolated and are opened back again. The net result is that the overall mobility is not significantly reduced along the lockdown, and therefore, the spreading dynamic does not “slow down” completely, as already noticed in Fig. 12(a). Recall that the curves therein do not extend further than 200 days, no matter the detection threshold. This prevents the dramatic increase in the economic costs, as shown in Fig. 16(b).

We turn now to the discussion of the different lockdown strategies performance. Fig. 17 shows the total cost as a function of the weighting factor (see caption for details). Notice that although 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 bounding value to , as shown in red in Fig. 17 (see caption for details).

Fig. 17

(a) Cost function as a function of the weighting parameter for the GLs and IGLs. From the lockdown implementation day, the percentage of individuals that move around is: 0% (GLs), 5%, 10%, 25% and  50%. (b) Cost function as a function of the weighting parameter for the LLs. The locked blocks are those that exceed 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 when the number of new infected equals to 5k. The infection rate remains constant along the simulation process and equals .

Fig. 17 reports the minimum cost situation in blue color ( ) for the GLs and LLs. These correspond to either the most strict global lockdown () 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 the explored values of .

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 daily cost for this situation is negligible. This is in agreement with an early detection, where the fraction of infected people is quite small () and most of the non-infected people are allowed to move around (). The reader can check that this yields to vanishing values for the term (b) in our cost model (5).

We further notice that the curves’ slope increases for increasing detection thresholds in Fig. 17(b). This occurs because of the increase in the daily economical cost, although the lockdown period remains close to  days. Interestingly, the lockdown curves in Fig. 17(b) meet the no-lockdown curve for thresholds surpassing 2–3 infected people. This makes the lockdown curves only valid for small values of (say, below 0.7).

Recall that the IGLs experiences a dramatic increment of the economical cost for small mobility fractions (see Fig. 16(a)). As a result, the lockdown curves always meet the no-lockdown curve at some point (except for ), as shown in Fig. 17(a). This is quite a difference with respect to the local strategy, since the lockdown curves are always limited to small values of (except for ).

Let us close the discussion with the following comments. We showed that the most strict implementation of either the global or local lockdown leads to the optimum performance (although the local one is preferred, as discussed above). But we noticed that there is some space left for partially effective lockdowns, if the most strict conditions are not attainable. The degree of effectiveness depends on the balance between the medical care costs and the economical costs, within this model.

Conclusions

This work concerns with the effect of human mobility in the context of a model for the spatio-temporal evolution of the COVID-19 outbreak. People move according to the Levy distribution and get into contact with each other during their daily routine. The lockdowns prevent these contacts from occurring, and thus, mitigate the propagation of the disease. Our investigation studies different lockdown scenarios and performs a careful evaluation of their effectiveness. We draw some recommendations for the better performance of the lockdown.

We assumed a SEIR compartmental model (with constant infection rates) for the inhabitants of a block. Each block was considered as a node within a square network of 120 × 120 nodes. People were allowed to travel twice a day between blocks. The initial conditions for the simulations considered 20 infected individuals located at the central block of the network, while the rest of the individuals were assumed to be in the susceptible state.

We focused on three scenarios: the full lockdown of all the blocks (perfect scenario), the partial lockdown of all the blocks (imperfect scenario), and the lockdown of only the infected blocks (local lockdown). We sustained the lockdown until the disease propagation was (almost) over.

We first noticed that the success of any control action depends strongly on canceling the mobility around the city. But a small number of individuals may spoil the effectiveness of these actions if they do not follow the confinement recommendations. This is, in our opinion, the major risk when implementing any lockdown strategy.

We further built a cost function to rate the three strategies. We arrived to the conclusion that full lockdowns, or, the (very) early detection and isolation strategy are the most effective ones. The local isolation strategy is preferred, though, since it appears as the less costly in the context of our model.

It is important to emphasize that strict lockdown policies also allow for short periods of isolation. More relaxed lockdowns are less costly daily, but cumulate large costs after an extended period of time. The full lockdown looses efficacy if the mobility is not completely canceled, as already mentioned. Our model shows that local lockdowns can still be quite effective even if a small number of infected people is not detected (and isolated).

Finally, we emphasize that the goal of our investigation is to show that each lockdown strategy yields different medical care and economical costs. And therefore, a quite different overall cost. The ultimate decision on whether to choose a global or local lockdown on a specific circumstance will depend on the chosen balance between the medical care cost and the economical cost due to routine disruptions. We consider that this work is a useful framework for the policy makers. We will leave this discussion open for future research.

CRediT authorship contribution statement

F.E. Cornes: Epidemiological dynamics, Investigation. G.A. Frank: Numerical simulations, Investigation, Supervision. C.O. Dorso: Epidemiological dynamics, Investigation, Supervision.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.