An agent-based model for interrelation between COVID-19 outbreak and economic activities
Takeshi Kano, Kotaro Yasui, Taishi Mikami, Munehiro Asally, Akio Ishiguro
aa r X i v : . [ phy s i c s . s o c - ph ] J u l An agent-based model for interrelation betweenCOVID-19 outbreak and economic activities
July 24, 2020
Takeshi Kano ∗ , Kotaro Yasui , , Taishi Mikami , Munehiro Asally , , , andAkio Ishiguro Research Institute of Electrical Communication, Tohoku University, 2-1-1 Katahira,Aobaku, Sendai 980-8577, Japan Frontier Research Institute for Interdisciplinary Sciences, Tohoku University,Aramaki aza Aoba 6-3, Aoba-ku, Sendai 980-8578, Japan School of Life Sciences, University of Warwick, Coventry, CV4 7AL, UK Warwick Integrative Synthetic Biology Centre, University of Warwick, Coven-try, CV4 7AL, UK Bio-Electrical Engineering Innovation Hub, University of Warwick, Coventry,CV4 7AL, UK ∗ Contact author. Tel: +81-22-217-5465. Email: [email protected]
Abstract
As of July, 2020, acute respiratory syndrome caused by coronavirusCOVID-19 is spreading over the world and causing severe economic dam-ages. While minimizing human contact is effective in managing the out-break, it causes severe economic losses. Strategies solving this dilemmaby considering interrelation between the spread of the virus and economicactivities are in urgent needs for mitigating the health and economic dam-age. Here we propose an abstract agent-based model for the outbreak ofCOVID-19 in which economic activities are taken into account. The com-putational simulation of the model recapitulated the trade-off betweenhealth and economic damage associated with lockdown measures. Basedon the simulation results, we discuss how macroscopic dynamics of infec-tion and economy emerge from the individuals behaviours. We believeour model can serve as a platform for discussing solutions to the above-mentioned dilemma.
Keywords
COVID-19, economy, infection, agent-based model
The acute respiratory syndrome caused by a coronavirus, COVID-19, first re-ported in Wuhan in December 2019 [1–4], has spread over the world causingsevere health and economic damages. To date (July 21th, 2020), more than 141illion people have been tested positive for the disease and more than 610 thou-sand people have died due to COVID-19 [5]. From the economic aspect, manypeople have suffered from economic losses and lost their jobs [6]. It should benoted here that there is a dilemma between mitigation of the spread of COVID19and reduction of economic losses: namely, the strategies for decreasing contactsbetween humans such as draconic lockdown and social distancing hinder normaleconomic activities. Hence, it is an urgent issue to solve this dilemma and tofind strategies to mitigate outbreak minimizing economic losses.To date, many mathematical models for the spread of infectious diseaseshave been proposed using differential equations [7–10] and agent-based models[11–13]. Recently, mathematical models for the spread of COVID-19 have beenproposed from various contexts, e.g. [14–23]. These studies solely modelled thespread of infectious diseases, yet they did not describe economic activities math-emtaically. Other studies focused on economic impacts of COVID-19 [24–28],some of which considered details of economic processes and predicted economicimpact under realistic assumptions [24–26]; thus, they are complex and difficultto capture the essence. Meanwhile, the others estimated the economic impactusing a simple model based on differential equations [27,28]. While these modelsdescribe the economic effects at a population level, how individuals behaviouraffects the macroscopic dynamics of infection and economy remains unclear. Adefinitive mathematical model that captures the essential mechanism of interre-lation between the spread of the virus and economic activities at an individuallevel could fill this gap.In this study, we propose a simple mathematical model that considers bothinfections of COVID-19 and economic activities. Our aim is to extract theessence of the relation between the spread of COVID-19 and the economic ac-tivities, rather than making a quantitative and accurate prediction of infectionand economy. Moreover, we are interested in how macroscopic dynamics ofinfection and economy emerge from individuals behaviours, rather than macro-scopic description through coarse graining. Hence, we propose a highly abstractand simple agent-based model without employing detailed and realistic assump-tions. More specifically, we propose a cellular automaton model in which mobileagents with the internal states regarding infection and economy interact withothers and update their internal states. We demonstrate through simulationshow the health and economic state evolves depending on parameters. Based onthe results, we discuss the effect of lockdown.
We consider a cellular automaton model in which hexagonal cells are alignedregularly on a two-dimensional plane (Fig. 1). N agents are on the plane.Each agent has a home cell, and can stay at home or move to its adjacent cellevery time step. Each agent has a health state State i and money M i , which areupdated through interaction with other agents. Agents die when they do notrecover after infection or when their money becomes zero.Agents have their own business. We assume that agent i sells goods toagent j when agent j visits the home cell of agent i . Prime costs of goods2igure 1: Outline of the proposed model. When ρ i,k ( t ) exceeds integral multipleof ρ th , job type k is added to the bottom of the target list of agent i . Agent i visits the home of the nearest agent among agents whose job type correspondsto the top of the target list. When it reached the home, it pays a unit amountmoney to buy the goods. Then, since the demand of agent i is satisfied, ρ i,k decreases by ρ th . The top item in the target list is removed, and the other itemsin the target list move up. Then, agent i visits the next target. When there isno item in the target list, agent i goes back and stay its home.are not considered. There are several types of businesses, which range fromselling commodities to selling luxury goods. Each agent chooses one of them,and it does not change temporarily. Each agent has demands for goods. Whenthe demand exceeds a certain threshold, the agent goes out to buy them; itpays money when it reached its home of an agent who sells the needed goods(Fig. 1). When experiencing lockdown under outbreak, agents do not tendto demand luxury goods, while they demand commodities as usual. In thefollowing subsections, we describe the details of the proposed model. We model the spread of COVID-19 by drawing inspiration from a spatial susceptible-exposed-infectious-recovered (SEIR) model [13]. Each agent has a health state
State i , which can take a susceptible state (S), an exposed state (E), an infec-tious state (I), a quarantine state (Q), a recovered state (R), and a death state(D).The state changes according to the rule shown in Fig. 2. When an agentwith state S is in a cell occupied by an agent with state E or I , State i changesfrom S to E with probability p E every time step. Here, although the previ-ously proposed SEIR model [13] assumed that only agents with state I haveinfectability, we assumed that agents with state E in addition to state I haveinfectability since asymptomatic patients of COVID-19 have infectability [29].When τ E time steps have passed after the transition from state S to E , thestate changes to state I . Agents with state I changes to state Q and D with3igure 2: Rule for the transition of the health state.Figure 3: Definition of D i and e l .probability p D and p Q , respectively, every time step. Agents with state Q ,which represents hospitalized patients, do not have infectability and stop anyeconomic activity described in the next subsection. Agents with state Q alsodie with probability p D every time step. Regardless of whether quarantined ornot, the state changes to R when τ R time steps have passed after the transitionfrom state E to I . Each agent has money M i . The money M i decreases and increases when agent i buys goods and when other agents buy agent i ’s goods, respectively. In additionto buying and selling, high-income persons pay high taxes while low-incomepersons receive public assistance in real societies. Thus, we describe the timeevolution of M i as M i ( t + 1) = M i ( t ) + q in,i ( t ) − q out,i ( t ) + K ( ˆ M − M i ( t )) , (1)where K and ˆ M are positive constants, and q in,i ( t ) and q out,i (t) denote theamount of money increased by selling and that decreased by buying, respec-4ively. The fourth term on the right-hand side represents the effect of incomeredistribution: high-income persons pay high taxes and low-income persons re-ceive public assistance. Agent i dies when M i ( t ) becomes zero.Each agent chooses one of m job types, which does not change temporarily.Agent i ’s demand for goods produced by job type k ( i = 1 , , · · · , N and k =1 , , · · · , m ) is denoted by ρ i,k . The time evolution of ρ i,k is described as ρ i,k ( t + 1) = ρ i,k ( t ) + ǫ k + max { σ k ( M i − λU i ) , } , (2)where ǫ k , σ k , and λ are positive constants, and U i represents the degree of anoutbreak, which formulation is described below. Parameter ǫ k represents theincrease rate of the demand which does not depend on the money agent i ownsor the degree of the outbreak. In contrast, σ k represents the increase rate ofthe demand which is affected by the money agent i owns or the degree of theoutbreak. ǫ k and σ k are set to be large and small, respectively, when job type k treats commodities, while vice versa when job type k treats luxury goods. Thethird term on the right-hand side of Eq. (2) means that the demand increasesrapidly when agent i has much money, yet the increase stops under the outbreak.Parameter λ represents the degree of a lockdown under the outbreak.The degree of outbreak U i is formulated as U i ( t + 1) = (1 − κ ) U i ( t ) + κn i ( t ) , (3)where n i is the number of agents with state Q within the radius r from agent i ,which represents the number of patients monitored in the residential area. Here,only the number of quarantined agents is counted, based on the assumptionthat people cannot notice that people who are exposed to the virus but nothospitalized (state E and I ) are actually exposed to the virus. Eq. (3) meansthat U i ( t ) increases/decreases followed by the increase/decrease of n i ( t ). U i ( t )is updated rapidly when κ is large. Thus, κ characterizes how fast lockdown isperformed in response to the spread of the virus.Each agent has a target list, which represents the priority of purchasinggoods (Fig. 1). When the demand ρ i,k exceeds the integral multiple of ρ th , jobtype k is added to the bottom of the target list of agent i . Agent i visits thehome of the nearest agent among agents whose job type corresponds to the topof the target list. When it reached its home, it pays a unit amount of moneyto buy the goods. Here we simply assumed that this deal holds even when theselling agent is not at its home. Then, since the demand of agent i is satisfied, ρ i,k decreases by ρ th . The top item in the target list is removed, and the otheritems in the target list move up. Then, agent i visits the next target. Whenthere is no item in the target list, agent i goes back and stay its home.The rule for the movement of each agent is described as follows (Fig. 3).First, the target position of agent i is defined as the home of the nearest agentamong agents whose job type corresponds to the top of the target list. Whenthere is no item in the target list, the target position is defined as agent i ’shome. A unit vector which points the target position from the current positionof agent i is denoted by D i . When the target position is identical to the currentposition, D i = . Agent i moves to its adjacent cells or stays at the currentposition with the following probability p i ( l ): p i ( l ) = 16 (1 + e l · D i ) , ( l = 1 , , , , , i (0) = 0 , (4)in the case of D i = , and p i ( l ) = 0 , ( l = 1 , , , , , p i (0) = 1 , (5)in the case of D i = , where the definition of the subscript l in e l is as shownin Fig. 3. Thus, owing to the bias term e l · D i in Eq. (4), agent i tends toapproach the target position. We performed simulations of the proposed model. The number of cells alongthe horizontal and vertical direction was 43 and 50, respectively, and periodicboundary condition was adopted. Except for the experiment shown in Fig. 9,the total number of agents N was set to 1000. Initial amount of money was setto ˜ M for all agents. The homes of the agents were randomly placed without anyoverlap. Each agent was initially located at its home. Each trial was performedfor 30000 time steps.There were four job types in the simulation. The number of people for eachjob type was set to be identical. Parameters ǫ k and σ k ( k = 1 , , ,
4) weredetermined as ǫ k = 0 . k − σ k = 0 . − ǫ k ˜ M . (6)Thus, persons whose job type number was large sold commodities while thosewhose job type number was small sold luxury goods. From Eqs. (2) and (6),the increase rate of the demand ρ i,k does not depend on the job type when M i = ˜ M .Non-dimensional parameters were used in the simulation. Since the proposedmodel is highly abstract, it is difficult to determine the parameters based onrealistic data. However, the duration of latent period τ E and that of infection τ R were chosen to roughly mimic the property of COVID-19. Specifically, τ E and τ R were set to be 600 and 1200; these non-dimensional values correspond to 7and 14 days, respectively, if each time step is rescaled to 16.8 minutes, and theseTable 1: Parameter values employed in the simulations. parameter value parameter value N R p E p D p Q κ τ E τ R ρ th K M
60 ˆ M V [ M ] and (b) D eco are shown for several values of K . In (b), lines for K = 0 . p D was set so that the death rate becomes about 10% if there was no lockdown, i.e. , λ = 0. Economic parameters such as ˜ M , ˆ M , ρ th , and K were determinedso that death caused by economic loss becomes almost zero in the absence ofoutbreak but some people die due to economic loss under lockdown during anoutbreak. The parameter values thus determined are shown in Table 1. Thesevalues were used when unspecified hereafter. To capture the basic property of the proposed model, simulations were per-formed under the condition where no patient exists, i.e. , State i = S for all i . By removing the factor of infection from the model, we can easily under-stand the basic property of economic activity. In this experiment, the pa-rameter K was changed to investigate the effect of income redistribution bytaxes and public assistances. The results were evaluated by the number ofdeaths caused by economic loss D eco and the variance of the amount of money V [ M ] ≡ N − P Ni =1 ( M i − ¯ M ) , where ¯ M is the average of M i .Supplementary videos 1–3 provided in [30] show the results when K =0 , . V [ M ]and D eco for several values of K . It is found that V [ M ] increase as the timepasses. The increase rate is smaller as K is larger (Fig. 4(a)). The numberof deaths D eco increased as the time passes for small K , while it remains zerofor large K (Fig. 4(b)). This result is reasonable because frequent buying andselling events cause the distribution of M i to spread in a diffusive manner. As K increased, the difference between the rich and the poor decreased, and thus,the number of deaths caused by economic loss decreased.7 .2 Effects of lockdown We performed simulations under the condition that the health state changedfrom S to E with the probability of 0.005 when the time step is 5000. Parameter K was set to be 0.00007. Cases of λ = 0, 120, and 210 were examined toinvestigate the effect of lockdown.Supplementary videos 4–6 provided in [30] show the results when λ = 0, 120,and 210, respectively. Figs. 5 (a)–(c) show the time evolution of the numberof agents with each health state. When λ = 0, the infection spread rapidlyand more than 80 % of the agents were infected. The maximum number of thequarantined patients (state Q ) was 278, and the number of deaths caused byinfection D inf ( t ) finally reached 101 (Fig. 5(a)). As λ increased, the number ofpatients and the deaths caused by infection decreased considerably (Figs. 5(b)and (c)). When λ = 210, less than 20% of the agents were infected (Fig. 5(c)).Next, we observed the economic tendency under these cases. The resultswere evaluated by the total number of deaths caused by economic loss andthe total amount of money for each job type, denoted by D eco,k ( t ) and T k ( t ),respectively ( k = 1 , , , T k ( t ) is given by T k ( t ) = X i ∈ job type k M i ( t ) . (7)The time evolutions of D eco,k ( t ) and T k ( t ) for the cases of λ = 0, 120, and 210are shown in Figs. 5(d)–(f) and (g)–(h), respectively. For λ = 0, only a fewagents died because of economic loss (Fig. 5(d)). The total amount of money T k ( t ) was kept almost constant (Fig. 5(g)). When agents undergo lockdown, i.e. , λ >
0, the economic gap among job types is generated. Specifically, T k ( t )became small for small k and vice versa for large k followed by the outbreak.The economic gap is the highest around 15000 time steps, and it is mitigatedgradually until around 30000 time steps (Figs. 5(h) and (i)). The number ofdeaths caused by economic loss is larger for small k (Fig. 5(e)), and it is largeras λ is larger (Fig. 5(f)). This result suggests that people who sell luxury goodssuffer from economic loss caused by lockdown, while those who sell commoditiesdo not. However, the persons who are suffering from economic loss can comeback to their normal life after the outbreak is over.In summary, lockdown is effective for mitigating an outbreak, while it induceseconomic gap and increases the number of deaths caused by economic loss. For further understanding of the properties of the proposed model, we per-formed simulations by changing several parameters expected to affect resultantbehaviours. Specifically, we changed the infectious rate p E , the duration of in-fection τ E and τ R , the amount of money ˜ M and ˆ M , the total number of agents N , and the quickness for the response to outbreak κ , since we expected thatthese parameters would affect infection and economic dynamics. For each pa-rameter, we also changed λ and made color maps. Other simulation conditionswere the same as the previous subsection. Results were evaluated by the ratio ofthe number of deaths caused by infection D inf to the number of agents N andthat of the number of deaths caused by economic loss D eco to the number ofagents N at 30000 time step. Ten trials were performed for each parameter, and8 = 0 λ = 120 λ = 210 !" !" T k ( t ) D eco,k ( t ) D eco,k ( t ) D eco,k ( t ) T k ( t ) T k ( t ) S E I R Q D inf D eco ( ≡ ! k D eco,k ) D eco ( ≡ ! k D eco,k ) D eco, D eco, D eco, D eco, T T T T !"!! !" Figure 5: Simulation results when patients appear at the time step 5000:(a)–(c) Time evolutions of the number of agents in each health state ( S , E , I , R , Q , D inf , and D eco ). (d)–(g) Time evolutions of D eco,k ( t ) and D eco ( ≡ P k D eco,k ( t )). (h)–(i) Time evolutions of T k ( t ). λ = 0 for (a)(d)(g),120 for (b)(e)(h), and 210 for (c)(f)(i).the average values of D inf /N and D eco /N are shown hereafter. In the followingsubsections, we will show the results when the parameters are varied. p E dependence Figure 6 shows the result when the infectious rate p E was changed. As isexpected, the number of deaths is smaller as p E is smaller. When p E is large, D inf /N is large for small λ while D eco /N is large for large λ . Namely, whereasinfection spreads in the absence of lockdown, economic loss becomes large iflockdown is excessive. This trade-off becomes severer as the infectious rate islarger. τ E and τ R dependence Figure 7 shows the result when the duration of infection τ E and τ R were changedwith holding the relation τ R = 2 τ E . When τ E and τ R are large, D inf /N is largefor small λ while D eco /N is large for large λ . Thus, the above-mentioned tradeoff becomes severer as the duration of infection is longer. ˜ M and ˆ M dependence Figure 8 shows the result when the initial amount of money ˜ M is changed. ˆ M in Eq. (1) was also changed so that ˆ M = ˜ M . The number of death by infection D inf /N is not much affected by ˜ M . However, D eco /N is larger as ˜ M is smaller.9igure 6: Simulation results when p E and λ are changed. D inf /N and D eco /N are shown in (a) and (b), respectively.Figure 7: Simulation results when τ E , τ R , and λ are changed with satisfying τ R = 2 τ E . D inf /N and D eco /N are shown in (a) and (b), respectively.This means that people cannot tolerate lockdown if they do not have enoughmoney. N dependence Figure 9 shows the result when the total number of agents N is changed. Forsmall λ , D inf /N is larger as N is larger, which indicates that infection tends tospread when population density is large. Meanwhile, D eco /N tends to increaseas λ increases. κ dependence Figure 10 shows the result when the quickness for the response to outbreak κ is changed. When κ is small, the outbreak cannot be mitigated owing to theslow response. As a consequence, D inf /N tends to become large. Furthermore,since agents have to undergo lockdown for a long time, D eco /N tends to be alsolarge. Thus, responding quickly is important for both mitigating outbreak andmaintaining economic activities. 10igure 8: Simulation results when ˜ M and λ are changed. D inf /N and D eco /N are shown in (a) and (b), respectively.Figure 9: Simulation results when N and λ are changed. D inf /N and D eco /N are shown in (a) and (b), respectively. We proposed a simple and abstract mathematical model of COVID-19 outbreakwith economic activities. Simulation results showed that lockdown measuresenable mitigating outbreak, while it generates economic gap among job types.The reason why the economic gap generates can be explained as follows. Theincome of agents who sell luxury goods decreases during an outbreak becausedemands for luxury goods decrease owing to lockdown. Since they have tobuy commodities as usual, they become poorer. In contrast, agents who sellcommodities can get income as usual, yet they do not buy luxury goods owingto lockdown. As a consequence, they become richer. This suggests that inreal societies, governments should make efforts to reduce the economic gap;otherwise, lockdown measures cannot be continued, which makes the mitigationof an outbreak difficult.The result shown in Fig. 8 suggests that people cannot tolerate lockdownif their overall economic level is low. Thus, it is particularly harder for thecommunities with low economic power to mitigate an outbreak while maintain-ing economic activities. A possible solution to this is to respond to the spreadof the virus as early as possible. As suggested from Fig. 10, if the responseis done earlier (which corresponds to large κ in our model), it is possible tostop an outbreak without causing severe economic damages. If a governmenthas failed to respond quickly and the virus has spread, people should anticipate11igure 10: Simulation results when κ and λ are changed. D inf /N and D eco /N are shown in (a) and (b), respectively.other possibilities such as the development of antiviral drugs and weakening ofvirus, which correspond to decreasing p E in our model.Because we did not consider production activities and simply assumed thatprime costs of goods are zero, our model is not suitable for making preciseand quantitative predictions, especially on the economic impact. However, ourmodel captures the essence of the interrelation between the spread of virus andeconomic activities, and thus, we believe that our model can become a platformfor discussing strategies for mitigating outbreak while maintaining economicactivities. Indeed, our model potentially has many possibilities in the future. Forexample, it may be possible to take the effect of long-distance movement withtransportations like airplanes and trains into account. Defining the capacityof hospitals may enable us to discuss how to avoid overwhelming hospitals. 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