Controlling the Outbreak of COVID-19: A Noncooperative Game Perspective
Anupam Kumar Bairagi, Mehedi Masud, Do Hyeon Kim, Md. Shirajum Munir, Abdullah Al Nahid, Sarder Fakhrul Abedin, Kazi Masudul Alam, Sujit Biswas, Sultan S Alshamrani, Zhu Han, Choong Seon Hong
JJOURNAL OF XXX, VOL. XX, NO. XX, JULY 2020 1
Controlling the Outbreak of COVID-19: ANoncooperative Game Perspective
Anupam Kumar Bairagi,
Member, IEEE,
Mehedi Masud,
Senior Member, IEEE,
Do Hyeon Kim, Md. ShirajumMunir,
Student Member, IEEE,
Abdullah Al Nahid, Sarder Fakhrul Abedin,
Student Member, IEEE,
KaziMasudul Alam, Sujit Biswas,
Member, IEEE,
Sultan S Alshamrani, Zhu Han,
Fellow, IEEE, and Choong Seon Hong,
Senior Member, IEEE
Abstract — COVID-19 is a global epidemic. Till now, there is noremedy for this epidemic. However, isolation and social distancingare seemed to be effective preventive measures to control thispandemic. Therefore, in this paper, an optimization problem isformulated that accommodates both isolation and social distanc-ing features of the individuals. To promote social distancing,we solve the formulated problem by applying a noncooperativegame that can provide an incentive for maintaining socialdistancing to prevent the spread of COVID-19. Furthermore, thesustainability of the lockdown policy is interpreted with the helpof our proposed game-theoretic incentive model for maintainingsocial distancing where there exists a Nash equilibrium. Finally,we perform an extensive numerical analysis that shows theeffectiveness of the proposed approach in terms of achieving thedesired social-distancing to prevent the outbreak of the COVID-19 in a noncooperative environment. Numerical results showthat the individual incentive increases more than with anincreasing percentage of home isolation from to forall considered scenarios. The numerical results also demonstratethat in a particular percentage of home isolation, the individualincentive decreases with an increasing number of individuals. Index Terms —COVID-19, health economics, isolation, socialdistancing, noncooperative game, nash equilibrium.
I. I
NTRODUCTION
The novel Coronavirus (2019-nCoV or COVID-19) is con-sidered to be one of the most dangerous pandemics of thiscentury. COVID-19 has already affected every aspect of in-dividual’s life i.e. politics, sovereignty, economy, education,religion, entertainment, sports, tourism, transportation, and
Anupam Kumar Bairagi, Do Hyeon Kim, Md. Shirajum Munir, SarderFakhrul Abedin, and Choong Seon Hong are with the Department of ComputerScience and Engineering, Kyung Hee University, Yongin-si 17104, Republicof Korea (e-mail: [email protected], [email protected], [email protected],[email protected], [email protected]).Mehedi Masud is with the Department of Computer Science, Taif Univer-sity, Taif, KSA (e-mail: [email protected]).Mehedi Masud and Sultan S Alshamrani is with the Department ofComputer Science, Taif University, Taif, KSA (e-mail: [email protected]).Abdullah Al Nahid is with the Electronics and Communication EngineeringDiscipline, Khulna University, Bangladesh (e-mail: [email protected]).Kazi Masudul Alam is with the Computer Science and Engineering Disci-pline, Khulna University, Bangladesh (e-mail:[email protected]).Sujit Biswasis is with the Department of Computer Science and Engineer-ing, Faridpur Engineering College, Bangladesh (e-mail: [email protected]).Sultan S Alshamrani is with theDepartment of Information Technology, TaifUniversity, Taif, KSA (e-mail: [email protected]).Zhu Han is with the Electrical and Computer Engineering Department,University of Houston, Houston, TX 77004, USA, and also with the Depart-ment of Computer Science and Engineering, Kyung Hee University, Yongin-si17104, Republic of Korea (e-mail: [email protected]).Corresponding author: Choong Seon Hong (e-mail: [email protected]) manufacturing. It was first identified in Wuhan City, China onDecember 29, 2019, and within a short span of time, it spreadout worldwide [1], [2]. The World Health Organization (WHO)has announced the COVID-19 outbreak as a Public HealthEmergency of International Concern (PHEIC) and identifiedit as an epidemic on January 30, 2020 [3]. Till July 23,2020, COVID-19 has affected countries and territoriesthroughout the globe and 2 international conveyances [4].The recent statistics on COVID-19 also indicate that morethan , , persons have been affected in different ways[4], [5]. Currently, the ten most infected countries are USA,Brazil, India, Russia, South Africa, Peru, Mexico, Chile,Spain, and UK, where affected people of these countries morethan of the worldwide cases. Since the outbreak, thetotal number of human death and recovery to/from COVID-19 are , and , , , respectively [4], [5] (till July23, 2020). The fatality of human life due to COVID-19 isfrightening in numerous countries. For instance, among thehighest mortality rates countries, of the mortality belongsto the top countries due to COVID-19. Furthermore, thepercentages of affected cases for male and female are around . and . , whereas these values are about and , respectively in death cases globally [6]. Differentcountries are undertaking different initiatives to reduce theimpact of the COVID-19 epidemic, but there is no clear-cutsolution to date.One of the most crucial tasks that countries need to dofor understanding and controlling the spread of COVID-19 istesting. Testing allows infected bodies to acknowledge thatthey are already affected. This can be helpful for taking careof them, and also to decrease the possibility of contaminatingothers. In addition, testing is also essential for a properresponse to the pandemic. It allows carrying evidence-basedsteps to slow down the spread of COVID-19. However, to date,the testing capability for COVID-19 is quite inadequate inmost countries around the world. South Korea was the secondCOVID-19 infectious country after China during February2020. However, mass testing may be one of the reasons why itsucceeded to diminish the number of new infections in the firstwave of the outbreak since it facilitates a rapid identificationof potential outbreaks [7]. For detecting COVID-19, two kindsof tests are clinically carried out: (i) detection of virus particlesin swabs collected from the mouth or nose, and (ii) estimatingthe antibody response to the virus in blood serum.This COVID-19 epidemic is still uncontrolled in most a r X i v : . [ c s . C Y ] J u l OURNAL OF XXX, VOL. XX, NO. XX, JULY 2020 2 countries. As a result, day by day, the infected cases anddeath graph are rising exponentially. However, researchers arealso focusing on the learning-based mechanism for detectingCOVID-19 infections [8]–[14]. This approach can be cost-effective and also possibly will take less time to perform thetest. Meanwhile, other studies [15]–[20] focus on analyzing theepidemiological and/or clinical characteristics of COVID-19.However, the infected cases of the COVID-19 can be reducedby maintaining a certain social distance among the people.In particular, to maintain such social distancing, self-isolation,and community lockdown can be possible approaches. Thus, itis imperative to develop a model so that the social communitycan take a certain decision for self-isolation/lockdown toprevent the spread of COVID-19.To the best of our knowledge, there is no study that focuseson the mathematical model for monitoring and controllingindividual in a community setting to prevent this COVID-19 epidemic. Thus, the main contribution of this paper isto develop an effective mathematical model with the help ofglobal positioning system (GPS) information to fight againstCOVID-19 epidemic by monitoring and controlling individual.To this end, we make the following key contributions: • First, we formulate an optimization problem for maxi-mizing the social utility of individual considering bothisolation and social distancing. Here, the optimizationparameters are the positions of individual. • Second, we reformulate the objective function whichis incorporated with the social distancing feature of anindividual as a noncooperative game. Here, we show thathome isolation is the dominant strategy for all the playersof the game. We also prove that the game has a NashEquilibrium (NE). • Third, we interpret the sustainability of lockdown policywith the help of our model. • Finally, we evaluate the effectiveness of the proposedapproach with the help of extensive numerical analysis.The remainder of the paper is organized as follows. In Sec-tion II, we present the literature review. We explain the systemmodel and present the problem formulation in Section III. Theproposed solution approach of the above-mentioned problemis addressed in Section IV. We interpret the sustainability oflockdown policy with our model in Section V. In SectionVI, we provide numerical analysis for the proposed approach.Finally, we draw some conclusions in Section VII.II. L
ITERATURE R EVIEW
COVID-19 is the seventh coronavirus identified to contami-nate humans. Individuals were first affected by the 2019-nCoVvirus from bats and other animals that were sold at the seafoodmarket in Wuhan [21], [22]. Afterward, it began to spread fromhuman to human mainly through respiratory droplets producedwhile people sneeze cough or exhaling [3]. Epidemiologicaland/or clinical characteristics of COVID-19 are analyzed inthe studies [15]–[20].In [15], the authors investigate the epidemiologic and clini-cal characteristics based on 91 cases of COVID-19 patientsof Zhejiang, China. Among these samples, . were laboratory-confirmed COVID-19 tested positive for SARS-Cov-2 while . were clinical-diagnosed COVID-19 cases.The average age of the patients was 50 while femalesaccounted for . . The typical indications were fever( . ), cough ( . ) and fatigue ( . ). . othese patients were affected from local cases, . went toor were in Wuhan/Hubei, . came in contact with peoplesfrom Wuhan, and . were from aircraft transmission.The authors represent a detailed statistical analysis of , individuals collected from January 21 to February 14, 2020,and covering 18 regions of the Henan province, China [16].Among these cases, were male and ages of these patientswere from to years. Among these patients, . had Wuhan’s travel history. In [17], the authors investigateepidemiological, demographic, clinical, and radiological fea-tures and laboratory data for cases of 2019-nCoV collectedfrom Jinyintan Hospital, Wuhan, China. They found that of these patients traveled to the Huanan seafood market. Theaverage age of the victims was . years, and most of them( . ) were men. The main clinical manifestations werefever ( ), cough ( ), shortness of breath ( ). Amongthe sufferers, exhibited bilateral pneumonia also. Thework in [18] analyzes the clinical characteristics of 1,099patients with laboratory-confirmed 2019-nCoV ARD from 552hospitals in 31 provinces/provincial municipalities of Wuhan,China. This work concluded that the median age of thesepatients was 47 years where . of them were female.The most common symptoms of these patients were fever( . ) and cough ( . ). Most of these cases had a Wuhanconnection ( . had been to Wuhan, and . hadcontacted people from Wuhan). Epidemiological investigationswere conducted in [19] among all close contacts of COVID-19patients (or suspected patients) in Nanjing, Jiangsu Province,China. Among them, . recently traveled Hubei and theaverage age of these cases was . years including . male. . of these patients showed fever, cough, fatiguesymptoms during hospitalization whereas . cases showedtypical CT images of the ground-glass chest and . presented stripe shadowing in the lungs. The study in [20]estimates the clinical features of COVID-19 in pregnancy andthe intrauterine vertical transmission potential of COVID-19infection. The age range of the subjects was 26–40 yearsand everybody of them had laboratory-confirmed COVID-19 pneumonia. They showed a similar pattern of clinicalcharacteristics to non-pregnant adult patients. The authorsmainly found that no intrauterine fetal infections occurred as aresult of COVID-19 infection during a late stage of pregnancy.Machine learning can play an important role to detectCOVID-19 infected people based on the observatory data. Thework in [8] proposes an algorithm to investigate the readingsfrom the smartphone’s sensors to find the COVID 19 symp-toms of a patient. Some commons symptoms of COVID-19victims like fever, fatigue, headache, nausea, dry cough, lungCT imaging features, and shortness of breath can be capturedby using the smartphone. This detection approach for COVID-19 is faster than the clinical diagnosis methods. The authorsin [9] propose an artificial intelligence (AI) framework for ob-taining the travel history of people using a phone-based survey OURNAL OF XXX, VOL. XX, NO. XX, JULY 2020 3 to classify them as no-risk, minimal-risk, moderate-risk, andhigh-risk of being affected with COVID-19. The model needsto be trained with the COVID-19 infected information of theareas where s/he visited to accurately predict the risk level ofCOVID-19. In [10], the authors develop a deep learning-basedmethod (COVNet) to identify COVID -19 from the volumetricchest CT image. For measuring the accuracy of their system,they utilize community-acquired pneumonia (CAP) and othernon-pneumonia CT images. The authors in [11] also use deeplearning techniques for distinguishing COVID-19 pneumoniafrom Influenza-A viral pneumonia and healthy cases basedon the pulmonary CT images. They use a location-attentionclassification model to categorize the images into the abovethree groups. Depth cameras and deep learning are applied torecognize unusual respiratory pattern of personnel remotelyand accurately in [12]. They propose a novel and effectiverespiratory simulation model based on the characteristics oforiginal respiratory signals. This model intends to fill thegap between large training datasets and infrequent real-worlddata. Multiple retrospective experiments were demonstratedto examine the performance of the system in the detectionof speculated COVID-19 thoracic CT characteristics in [13].A 3D volume review, namely "Corona score" is employedto assess the evolution of the disease in each victim overtime. In [14], the authors use a pre-trained UNet to fragmentthe lung region for automatic detection of COVID-19 froma chest CT image. Afterward, they use a 3D deep neuralnetwork to estimate the probability of COVID-19 infectionsover the segmented 3D lung region. Their algorithm uses 499CT volumes as a training dataset and 131 CT volumes asa test dataset and achieves 0.959 ROC AUC and 0.976 PRAUC. The study in [23] presents evidence of the diversityof human coronavirus, the rapid evolution of COVID-19, andtheir clinical and Epidemiological characteristics. The authorsalso develop a deep learning model for identifying COVID-19. and trained the model using a small CT image datasets.They find an accuracy of around using a small CT imagedataset.In [24], the authors propose a stochastic transmission modelfor capturing the phenomenon of the COVID-19 outbreakby applying a new model to quantify the effectiveness ofassociation tracing and isolation of cases at controlling a severeacute respiratory syndrome coronavirus 2 (SARS-CoV-2)-likepathogen. In their model, they analyze synopses with a varyingnumber of initial cases, the basic reproduction number, thedelay from symptom onset to isolation, the probability thatcontacts were traced, the proportion of transmission that oc-curred before symptom start, and the proportion of subclinicalinfections. They find that contact tracing and case isolationare capable enough to restrain a new outbreak of COVID-19within 3 months. In [25], the authors present a risk-sensitivesocial distance recommendation system to ensure privatesafety from COVID-19. They formulate a social distance rec-ommendation problem by characterizing Conditional Value-at-Risk (CVaR) for a personal area network (PAN) via Bluetoothbeacon. They solve the formulated problem by proposing a twophases algorithm based on a linear normal model. In [26], theauthors mainly dissect the various technological interventions
MarketEducational Institution Train StationHome HomeHome
Figure 1: Exemplary System model. Isolation indicates stayingat home whereas social distancing measures the distance of aindividual from others.made in the direction of COVID-19 impact management.Primarily, they focus on the use of emerging technologies suchas Internet of Things (IoT), drones, artificial intelligence (AI),blockchain, and 5G in mitigating the impact of the COVID-19pandemic.The works [8]–[20], [23]–[26] focused on COVID-19 detec-tion and analyzed the characteristic of its respiratory pattern.Hence, the literature has achieved a significant result in termsof post responses. In fact, it is also imperative to controlthe epidemic of COVID-19 by maintaining social distance.Therefore, different from the existing literature, we focus onthe design of a model that can measure individual’s isolationand social distance to prevent the epidemic of COVID-19. Themodel considers both isolation and social distancing featuresof individuals to control the outbreak of COVID-19.III. S
YSTEM M ODEL AND P ROBLEM F ORMULATION
Consider an area in which a set N of N individuals areliving under COVID-19 threat and must decide whether tostay at home or go leave their homes to visit a market,shop, train station, or other locations, as shown in Figure 1.Everyone has a mobile phone with GPS. From analyzing theGPS information, we can know their home locations of eachindividuals, and longitude and latitude of these locations aredenoted by X h , and Y h , respectively. We consider one timeperiod (e.g., 15 or 30 minutes) for our scenario and this timeperiod is divided into T smaller time steps in a set T . Foreach of time step t ∈ T , we have the GPS coordinates X and Y of every individual.Now, the deviation from home for any individual i ∈ N inbetween two time steps can be measured by using Euclideandistance as follows: δ ti = (cid:113) ( X hi − X ti ) + ( Y hi − Y ti ) , if t = , (cid:113) ( X t − i − X ti ) + ( Y t − i − Y ti ) , otherwise . (1) OURNAL OF XXX, VOL. XX, NO. XX, JULY 2020 4
Thus, the total deviation from home by each individual i ∈ N in a particular time period can be calculated as follows: δ i = (cid:213) t ∈T δ ti , ∀ i ∈ N (2)On the other hand, at the end of a particular time period, thedistance between an individual i ∈ N and any other individuals j ∈ N , j (cid:44) i is as follows: d ji = (cid:113) ( X Ti − X Tj ) + ( Y Ti − Y Tj ) . (3)Hence, the total distance of individual i ∈ N from otherindividuals N i ⊆ N , who are in close proximity with i ∈ N ,can be expressed as follows: d i = (cid:213) j ∈N i d ji , ∀ i ∈ N . (4)Our objective is to keep δ minimum for reducing thespread of COVID-19 from infected individuals, which is anisolation strategy. Meanwhile, we want to maximize socialdistancing which mathematically translates into maximizing d for reducing the chance of infection from others. However,we can use log term to bring fairness [27], [28] in theobjective function among all individuals. Hence, we can posethe following optimization problem: max X , Y ω (cid:213) i ∈N log ( Z − δ i ) + ( − ω ) (cid:213) i ∈N log d i (5)s.t. δ i ≤ δ max , (5a) d ji ≥ d min , ∀ i , j (5b) ω ∈ [ , ] . (5c)In (5), Z is a large number for changing the minimiza-tion problem to maximization one, and Z > δ i , ∀ i ∈ N .The optimization variables X and Y indicate longitude, andlatitude, respectively, of the individuals. Moreover, the firstterm in (5) encourages individual for isolation whereas thesecond term in (5) encourages individual to maintain fair social distancing . In this way, solving (5) can play a vitalrole in our understanding on how to control the spread ofCOVID-19 among vast population in the society. Constraint(5a) guarantees small deviation to maintain emergency needs,while Constraint (5b) assures a minimum fair distance amongall the individuals to reduce the spreading of COVID-19 fromone individual to another. Constraint (5c) shows that ω cantake any value between 0 and 1 which captures the importancebetween two key factors captured in the objective function of(5). For example, if COVID-19 is already spreading in a givensociety, then most of the weight would go to isolation termrather than social distancing. The objective of (5) is difficultto achieve as it requires the involvement and coordinationamong all the N individual. Moreover, if the individuals are notconvinced then it is also difficult for the government to attainthe objective forcefully. Thus, we need an alternative solutionapproach that encourage individual separately to achieve theobjective and game theory, which is successfully used in [29],[30], can be one potential solution, which will be elaboratedin the next section. IV. A N ONCOOPERATIVE G AME S OLUTION
To attain the objective for a vast population, governmentscan introduce incentives for isolation and also for socialdistancing. Then every individual wants to maximize theirutilities or payoffs. In this way, government can play its rolefor achieving social objective. Hence, the modified objectivefunction is given as follows: U ( δ , d ) = α (cid:213) i ∈N log ( Z − δ i ) + β (cid:213) i ∈N log d i , (6)where α = α (cid:48) ω and β = β (cid:48) ( − ω ) with α (cid:48) > and β (cid:48) > are incentives per unit of isolation and social distancing.In practice, α and β can be monetary values for per unitof isolation and social distancing, respectively. In (6), oneindividual’s position affects the social distancing of others, andhence, the individuals have partially conflicting interest on theoutcome of U . Therefore, the situation can be interpreted withthe noncooperative game [31], [32].A noncooperative game is a game that exhibit a competitivesituation where each player needs to make choices independentof the other players, given the possible policies of the otherplayers and their impact on the player’s payoffs or utilities.Now, a noncooperative game in strategic form or a strategicgame G is a triplet G = (N , (S i ) i ∈N , ( u i ) i ∈N ) [35] for any timeperiod where: • N is a finite set of players, i.e., N = { , , · · · , N } , • S i is the set of available strategies for player i ∈ N , • u i : S → R is the payoff function of player i ∈ N , with S = S × S × .. × S N .In our case S i = { s hi , s mi } where s hi and s mi indicate thestrategies of staying at home and moving outside for player i ∈ N , respectively. The payoff or incentive function of anyplayer i ∈ N in a time period can be defined as follows: u i ( . ) = (cid:40) α log Z + β log ˜ d i , if strategy is s hi ,α log ( Z − δ i ) + β log d i , if strategy is s mi . (7)where ˜ d i = (cid:205) j ∈N i (cid:113) ( X hi − X j ) + ( Y hi − Y j ) .The Nash equilibrium [33] is the most used solution conceptfor a noncooperative game. Formally, Nash equilibrium can bedefined as follows [34]: Definition 1. : A pure strategy Nash equilibrium for a non-cooperative game G = (N , (S i ) i ∈N , ( u i ) i ∈N ) is a strategyprofile s ∗ ∈ S where u i ( s ∗ i , s ∗− i ) ≥ u i ( s i , s ∗− i ) , ∀ s i ∈ S i , ∀ i ∈ N . However, to find the Nash equilibrium, the following twodefinitions can be helpful.
Definition 2. [35]: A strategy s i ∈ S i is the dominant strategyfor player i ∈ N if u i ( s i , s − i ) ≥ u i ( s (cid:48) i , s − i ) , ∀ s (cid:48) i ∈ S and ∀ s − i ∈S − i , where S − i = (cid:206) j ∈N , j (cid:44) i S j is the set of all strategy profilesfor all players except i. Definition 3. [35]: A strategy profile s ∗ ∈ S is the dominantstrategy equilibrium if every elements s ∗ i of s ∗ is the dominantstrategy of player i ∈ N . Thus, if we can show that every player of our game G hasa strategy that gives better utility irrespective of other players OURNAL OF XXX, VOL. XX, NO. XX, JULY 2020 5
Table I: Game matrix for 2-players P s h s m P s h ( u ( s h , s h ) , u ( s h , s h )) ( u ( s h , s m ) , u ( s h , s m )) s m ( u ( s m , s h ) , u ( s m , s h )) ( u ( s m , s m ) , u ( s m , s m )) strategies, then with the help of Definition 2 and 3, we cansay that Proposition 1 is true. Proposition 1. G has a pure strategy Nash equilibrium when α > β .Proof. Let us consider a 2-player simple matrix game asshown in Table I with the mentioned strategies. For simplicity,we consider a distance Laplacian distance ∆ that each playercan pass in any timestamp.Thus, the utilities of P : u ( s h , s h ) = α log Z + β log d , u ( s h , s m ) = α log Z + β log ( d ± ∆ ) , u ( s m , s h ) = α log ( Z − ∆ ) + β log ( d ± ∆ ) , u ( s m , s m ) = α log ( Z − ∆ ) + β log ( d ± ∆ ) , (8)where ± indicates the movement of player to other player andopposite direction, respectively. Now, u ( s h , s h ) − u ( s m , s h ) = α log (cid:18) ZZ − ∆ (cid:19) + β log (cid:18) d d ± ∆ (cid:19) , u ( s h , s m ) − u ( s m , s m ) = α log (cid:18) ZZ − ∆ (cid:19) + β log (cid:18) d ± ∆ d ± ∆ (cid:19) , (9)As α > β , so the following conditions hold from (9): u ( s h , s h ) − u ( s m , s h ) ≥ , u ( s h , s m ) − u ( s m , s m ) ≥ , (10)Hence, rewriting (10), we get the followings: u ( s h , s h ) ≥ u ( s m , s h ) , u ( s h , s m ) ≥ u ( s m , s m ) . (11)Hence, s h is the dominant strategy of P . Moreover, for theplayer P , the utilities are as follows: u ( s h , s h ) = α log Z + β log d , u ( s m , s h ) = α log Z + β log ( d ± ∆ ) , u ( s h , s m ) = α log ( Z − ∆ ) + β log ( d ± ∆ ) , u ( s m , s m ) = α log ( Z − ∆ ) + β log ( d ± ∆ ) , (12)Now, u ( s h , s h ) − u ( s h , s m ) = α log (cid:18) ZZ − ∆ (cid:19) + β log (cid:18) d d ± ∆ (cid:19) , u ( s m , s h ) − u ( s m , s m ) = α log (cid:18) ZZ − ∆ (cid:19) + β log (cid:18) d ± ∆ d ± ∆ (cid:19) , (13) As α > β , so the following conditions hold from (13): u ( s h , s h ) − u ( s m , s m ) ≥ , u ( s m , s h ) − u ( s m , s m ) ≥ , (14)Hence, rewriting (14), we get the followings: u ( s h , s h ) ≥ u ( s h , s m ) , u ( s m , s h ) ≥ u ( s m , s m ) . (15)Hence, s h is the dominant strategy of player P .When there are N -players ( N > ) in the game, incentiveof player i ∈ N (takes strategy s hi , without considering othersstrategy), is given as follows: u i ( s hi , . . . ) = α log Z + β log ˜ d i . (16)However, if the player i ∈ N takes the strategy s mi , i.e.,the player visits some crowded place like market, shop, trainstation, school, or other location, then a person may come inclose contact with many others. Thus, the incentive of player i ∈ N with this strategy is given as follows: u i ( s mi , . . . ) = α log ( Z − δ i ) + β log d i , (17)where δ i is calculated from (2) and d i is measured from(4) for that particular location. Moreover, d i < ˜ d i as theseplaces are crowded and individuals are in short distance withone another. Hence, u i ( s hi , . . . ) > u i ( s mi , . . . ) as Z > Z − δ i and ˜ d i > d i for any player i ∈ N . That means, s hi isthe dominant strategy for player i ∈ N irrespective of thestrategies of other players in the game G . Thus, there is astrategy profile s ∗ = { s h , s h , · · · , s hN } ∈ S where each element s ∗ i is a dominant strategy. Hence, by Definition 3, s ∗ is adominant strategy equilibrium. Moreover, a dominant strategyequilibrium is always a Nash equilibrium [35]. Hence, thegame G has always a pure strategy Nash equilibrium. (cid:4) Thus, Nash equilibrium is the solution of the noncooperativegame G . In this equilibrium, no player of N has the benefit ofchanging their strategy while others remain in their strategies.That means, the utility of each player i ∈ S is maximizedin this strategy, and hence ultimately maximize the utilityof (6). In fact, incentivizing the social distancing mechanismis promoting social distancing to each individual. To thisend, maximizing U of (6) ultimately maximize the originalobjective function of (5).Moreover, the Nash equilibrium point has a greater implica-tion on controlling the spread of COVID-19 in the society. Atthe NE point, every individual stays at home. So, if someonegets affected by COVID-19, the individual will not go incontact with others. Similarly, an unaffected individual has noprobability to come in contact with an affected individuals.Unfortunately, the family members have the chance to beaffected if they don’t follow fair distance and health norms.V. S USTAINABILITY OF L OCKDOWN P OLICY WITH THE S YSTEM M ODEL
The sustainability of the lockdown policy can be interpretedby using the outcome of the Nash equilibrium point that isachieved in the noncooperative game in Section IV.
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The total amount of incentive a particular time period ispresented in (6). In a particular day, we have T s = × T timeperiod where T is the length of a time period in minutes.Thus, we can denote the incentive of a time stamp t s in aparticular day p as follows: U t s p ( δ , d ) = α (cid:213) i ∈N log ( Z − δ i ) + β (cid:213) i ∈N log d i . (18)Hence, the amount of resources/money that is necessary toincentivize peoples in a particular day, p can be expressed asfollows: U p = T s (cid:213) t s = U t s p ( δ , d ) . (19)Now, if we are interested to find the sustainability oflockdown policy for a particular country till a certain numberof days, denoted by P , we have to satisfy the followinginequality: P (cid:213) p = U p ≤ R + P (cid:213) p = r p , (20)where R is the amount of resource/money of a particularcountry at the starting of lockdown policy that can be usedas incentive and r p is the collected resources in a particularday, p , of the lockdown period. Here, r p includes governmentalrevenue and donation from different individuals, organizationsand even countries. Moreover, the unit of α , β , R and r p aresame.If we assume for simplicity that U p and r p are same forevery day and they are denoted by ˜ U and ˜ r , respectively, thenwe can rewrite (20) as follows: P × ˜ U ≤ R + P × ˜ r . (21)Hence, if we are interested to find the upper limit of sustain-able days for a particular country using lockdown policy, thenwe have the following equality: P × ˜ U = R + P × ˜ r . (22)Thus, by simplifying (22), we have the following: P = R ˜ U − ˜ r . (23)Here, the sustainable days P depends on R , ˜ U , and ˜ r .However, we cannot change R but government can predict ˜ r .Moreover, depending on R and ˜ r , government can formulateits policy to set α and β so that individuals are encour-aged to follow the lockdown policy. Alongside, we cannotcontinue lockdown policy infinitely based upon the limitedtotal resources. Hence, the governments should formulate andupdate its lockdown policy based on the predicted sustainablecapability to handle COVID-19, otherwise resource crisis willbe a further bigger worldwide pandemic.VI. N UMERICAL A NALYSIS
In this section, we assess the proposed approach usingnumerical analyses. We consider an area of , m × , m for our analysis where individuals’ position are randomly Table II: Value of the principal simulation parameters Symbol Value N { , , , } α . β . ω [ , ] Z R { E + , . E + , E + , . E + , E + } ˜ r { ˜ U ∗ . , ˜ U ∗ . , ˜ U ∗ . , ˜ U ∗ . , ˜ U ∗ . } Value of T o t a l i n c en t i v e N =500(100% home quarantine) N =500(100% Random location) N =1000(100% home quarantine) N =1000(100% Random location) Figure 2: Comparison of incentive (in log scale) for varyingvalue of ω .distributed. The value of the principal simulation parametersare shown in the Table II.Figure 2 illustrates a comparison between home isolation(stay at home) and random location in the considered area fora varying value of ω . In this figure, we consider two cases of N = and N = , . In both the cases, home isolation(quarantine) is beneficial over staying in random location andthe differences between two approaches are increased withthe increasing value of ω . Moreover, the difference of payoffsbetween two approaches are increased with the increasingvalue of ω as the more importance are given in home isolation.Figure 3 shows the empirical cumulative distribution func-tion (ecdf) of incentives for different numbers of individuals.The figure revels that the incentive values increase with theincreasing number of home quarantine individuals in all thefour cases. Figure 3a exhibits that the incentives are below , , and , for , and sure, respectively, for and home quarantine cases whereas the incentivesare sure in between , and , for homeisolation case. Further, the same values are at least , for sure in case of full home isolation. Figure 3b depictsthat the incentive of being below , is sure for home isolation case, however, the same values of being OURNAL OF XXX, VOL. XX, NO. XX, JULY 2020 7
Total incentive e c d f Empirical CDF
25% Home Quarantine50% Home Quarantine75% Home Quarantine100% Home Quarantine (a) N = Total incentive e c d f Empirical CDF
25% Home Quarantine50% Home Quarantine75% Home Quarantine100% Home Quarantine (b) N = , Total incentive e c d f Empirical CDF
25% Home Quarantine50% Home Quarantine75% Home Quarantine100% Home Quarantine (c) N = , Total incentive e c d f Empirical CDF
25% Home Quarantine50% Home Quarantine75% Home Quarantine100% Home Quarantine (d) N = , Figure 3: Ecdf of incentives (in log scale) for different valueof N with α = . and β = . using runs. T o t a l i n ce n ti v e ( a n ti l og s ca l e ) Home isolation
N=500 N=1,000 N=1,500 N=2,000
Figure 4: Total incentive (average of 50 runs) for varyingpercentage of home isolation individuals when α = . and β = . .
25% 50% 75% 100%05E+171E+181.5E+182E+182.5E+183E+18
N=500 N=1,000 N=1,500 N=2,000 H o m e i s o l a ti on A v e r a g e i nd i v i du a l i n ce n ti v e ( a n til og s ca l e ) Number of individuals
Figure 5: Average individual incentive for varying percentageof home quarantine individuals when α = . and β = . .above , , and , are , and , sure, respective,for , and cases. Moreover, for home isolationcase, the values are in between , to , for sure. Theincentives for , , , and home isolation casesare above , , , , , , and , , respectively,with probability . , . , . , and . , respectively, asshown in Figure 3c. Additionally, the same values are at least , , , , , , and , with . , . , . , and . probabilities, respectively, which is presented in Figure3d.The total incentive (averaging of 50 runs) for varyingpercentage of home isolation individuals with different samplesize are shown in Figure 4. From this figure, we observe thatthe total payoff increases with increasing number of home iso-lation individuals for all considered cases. The incentives are , , , and better from home quarantineof to for N = , N = , , N = , , and N = , , respectively. Moreover, for a particular percentageof home isolation, the total incentive is related with the samplesize. In case of individuals in the home isolation, the OURNAL OF XXX, VOL. XX, NO. XX, JULY 2020 8
25 50 75 100
Percentage of home isolation D a ys N = 500 N = 1,000 N = 1,500 N = 2,000 Figure 6: Maximum possible lockdown period with varyingnumber of individuals when R = E + , ˜ r = ˜ U ∗ . , andusing total incentive shown in Figure .
25 50 75 100
Percentage of home isolation (
N=1,000) D a ys Figure 7: Maximum possible lockdown period with varying R and ˜ r with total payoff shown in Figure .incentive for N = , is . , . , and . morethan that of N = , N = , , and N = , , respectively.Figure 5 shows the average individual payoff for varyingparentage of home isolation individuals for different scenarios.The figure exhibits that the average individual incentive in-creases with an increasing percentage of home isolation as thedeviation δ decreases and hence, the value of home isolationincentive increases. For N = , the incentive of homeisolation is . more than that of home isolation.Moreover, in a particular percentage of home isolation, theincentive decreases with an increasing number of consideredindividuals as the social distancing decreases due to the morenumber of individuals. In case of home isolation, theindividual incentive for N = is . more than thatof N = , . Figure 6 shows the maximum possible lockdown periodfor a varying number of individuals within a fixed amountof resource R . The figure reveals that with the increasingpercentage of home isolation individuals, the maximum lock-down period significantly decreases for all considered cases.The reason behind this is that the more individuals are in homeisolation, the more it is necessary to pay the incentives. With afixed amount of resources, a country with less individuals cansurvive a longer lockdown period. With more percentages ofhome isolation individuals, the number of loackdown periodis less, and possible of spreading of COVID-19 is also less.Therefore, the governments can consider a trade-off betweenincreasing expenditure as a incentive and lockdown period.For , individuals, the maximum possible lockdown periodfor varying amount of R and ˜ r is presented in Figure 7.The figure also illustrates that with the increasing percentagesof home isolation individuals, the continuity of the lockdownperiod reduces for every scenarios. However, for a particularpercentage of home isolation individuals where total number ofindividuals are fixed, a country can continue higher lockdownperiod who has more am amount of resources, R . Addition-ally, ˜ r also play an important role to continue the lockdownperiod. VII. C ONCLUSIONS
In this paper, we have introduced a mathematical model forcontrolling the outbreak of COVID-19 by augmenting isolationand social distancing features of individuals. We have solvedthe utility maximization problem by using a noncooperativegame. Here, we have proved that staying home (home iso-lation) is the best strategy of every individual and there isa Nash equilibrium of the game. By applying the proposedmodel, we have also analyzed the sustainability period of acountry with a lockdown policy. Finally, we have performed adetailed numerical analysis of the proposed model to controlthe outbreak of the COVID-19. In future, we will furtherstudy and compare with extended cases such as centralizedand different game-theoretic models. In particular, an extensiveanalysis between the government-controlled spread or peoplecontrolled spread under more diverse epidemic models.R