key: cord-280470-dtb2nza1
authors: Bairagi, A. K.; Masud, M.; Kim, D. H.; Munir, M. S.; Nahid, A. A.; Abedin, S. F.; Alam, K. M.; Biswas, S.; Alshamrani, S. S.; Han, Z.; Hong, C. S.
title: A Noncooperative Game Analysis for Controlling COVID-19 Outbreak
date: 2020-05-26
journal: nan
DOI: 10.1101/2020.05.22.20110783
sha: 
doc_id: 280470
cord_uid: dtb2nza1

COVID-19 is a global epidemic. Till now, there is no remedy for this epidemic. However, isolation and social distancing are seemed to be effective to control this pandemic. In this paper, we provide an analytical model on the effectiveness of the sustainable lockdown policy that accommodates both isolation and social distancing features of the individuals. To promote social distancing, we analyze a noncooperative game environment that provides an incentive for maintaining social distancing. Furthermore, the sustainability of the lockdown policy is also interpreted with the help of a game-theoretic incentive model for maintaining social distancing. Finally, an extensive numerical analysis is provided to study the impact of maintaining a social-distancing measure to prevent the Covid-19 outbreak. Numerical results show that the individual incentive increases more than 85% with an increasing percentage of home isolation from 25% to 100% for all considered scenarios. The numerical results also demonstrate that in a particular percentage of home isolation, the individual incentive decreases with an increasing number of individuals.

1 Introduction 1 acknowledge that they are already affected. This can be helpful for taking care of them, 22 and also to decrease the possibility of contaminating others. Testing is also essential for 23 a proper reply to the pandemic. It allows carrying evidence-based steps to slow down 24 the spread of COVID-19. However, to date, the testing capability for COVID-19 is quite 25 inadequate in most countries around the world. South Korea was the second COVID-19 26 infectious country after China during February 2020. However, mass testing may be one 27 of the reasons why it succeeded to diminish the number of new infections since it 28 facilitates a rapid identification of potential outbreaks [8] . For detecting COVID-19, two 29 kinds of tests are clinically carried out: (i) detection of virus particles in swabs collected 30 from the mouth or nose and (ii) estimates the antibody response to the virus in blood 31 serum. 32 This COVID-19 epidemic is still uncontrolled in most countries. Infected cases and 33 death graph are rising every day. However, researchers are also focusing on the 34 learning-based mechanism for detecting COVID-19 infections [16] [17] [18] [19] [20] [21] [22] . This approach 35 can be cost-effective and also possibly will take less time to perform the test. Other 36 studies [9] [10] [11] [12] [13] [14] focus on analyzing the epidemiological and/or clinical characteristics of 37 COVID-19. As per our knowledge, there is no study that focuses on mathematical 38 model is for monitoring and controlling individual to prevent this COVID-19 epidemic. 39 Thus, the main contribution of this paper is to develop an effective mathematical model 40 with the help of global positioning system (GPS) information to fight against 41 COVID-19 epidemic by monitoring and controlling individual. To this end, we make the 42 following key contributions: 43 First, we study the real-world dataset to realize the worldwide severity of 44 COVID-19 epidemic and also show the predicted results for infected and active 45 cases of COVID-19. 46 Second, we consider an optimization problem that maximizes the social utility of 47 individuals by taking into account isolation and social distancing policies. Here, 48 the optimization parameters are the positions of an individual. 49 Third, we analyze the objective function by incorporating the social distancing 50 feature of an individual in a noncooperative game environment. Here, we observe 51 that home isolation is the dominant strategy for all the players of the game. 52 Finally, we interpret the sustainability of lockdown policy for controlling the 53 impact of COVID-19 outbreak through extensive numerical analysis. 54 The remainder of the paper is organized as follows. The global phenomenon of consider solution approach of the above-mentioned problem is addressed in Section 5. 58 We interpret the sustainability of lockdown policy with our model in Section 6. In 59 Section 7, we provide an extensive numerical analysis to study the impact of 60 maintaining a social-distancing measure. Finally, we draw the conclusions in Section 8. In this subsection, we present the analysis of available COVID-19 data [3] [4] [5] [6] . Fig. 1 64 shows that the daily cumulative infected, death, recovered, and active cases of all over 65 May 19, 2020 3/22

. CC-BY 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review) the world till April 17, 2020, and showing the significant increase of confirmed cases 66 over time. In fact, a sharp increase in infected cases from the third week of March 2020 67 is observed in Fig. 1 . This is due to the massive spread of COVID-19 incumbents in 68 Europe and USA. Fig. 2 shows the percentage of top 10 infected countries of COVID-19 69 till April 18, 2020, and they contributed 77% of inmates over all the infected cases. Moreover, within the short period of time, the infected cases in USA reached to around 71 one-third of the global confirmed cases. The global increase rate of COVID-19 infected 72 cases over the time (till April 17, 2020) is illustrated in Fig. 3 the loss of the globe is shown in Fig. 4 . However, only three countries namely USA,

Italy and Spain contribute ore than 50% of COVID-19 deaths as presented in Fig. 4 . . CC-BY 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review)

The copyright holder for this preprint this version posted May 26, 2020. pandemic. Fig. 6 also exhibits that the recovery cases from COVID-19 are satisfactory 94 in China (93.14%), Iran (69.36%) and Germany (61.10%) only, other countries are still 95 straggling to find the right ways. By the way, the average recovered rate is only 34% for 96 these countries till now. The demographic information of the infected and death cases 97 are presented in Fig. 7 , 8, and 9. Fig. 7 shows the percentage of COVID-19 infected 98 cases by sex based on [32] and infection among females are more. However, the death 99 percentage for COVID-19 is more for male than female as shown in Fig. 8 . Fig. 9 shows 100 the percentages of infected and death cases among different age groups based upon [32] . 101 The figure depicts that the infection cases are higher for ages from 21 to 70. The main 102 cause of the characteristics is that these groups are working class and come contact with 103 many peoples. However, the death cases are higher for ages above 60 years as shown in 104 Fig. 9 . The main cause is the weak immune system for older peoples.

May 19, 2020 6/22

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The copyright holder for this preprint this version posted May 26, 2020. . CC-BY 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.

The copyright holder for this preprint this version posted May 26, 2020. . were first affected by the 2019-nCoV virus from bats and other animals that were sold 120 at the seafood market in Wuhan [15, 24] . Afterward, it began to spread from human to 121 human mainly through respiratory droplets produced while people sneeze cough or 122 exhaling [3] . Epidemiological and/or clinical characteristics of COVID-19 are analyzed 123 in the studies [9] [10] [11] [12] [13] [14] .

In [9] , the authors investigate the epidemiologic and clinical characteristics based on 125 91 cases of COVID-19 patients of Zhejiang, China. Among these samples, 96.70% were 126 laboratory-confirmed COVID-19 tested positive for SARS-Cov-2 while 3.30% were 127 clinical-diagnosed COVID-19 cases. The average age of the patients was 50 while 128 females accounted for 59.34%. The typical indications were fever (71.43%), cough 129 (60.44%) and fatigue (43.96%). 43.96%o these patients were affected from local cases, 130 34.07% went to or were in Wuhan/Hubei, 8.79% came in contact with peoples from 131 Wuhan, and 12.09% were from aircraft transmission. The authors represent a detailed 132 statistical analysis of 1, 212 individuals collected from January 21 to February 14, 2020, 133 and covering 18 regions of the Henan province, China [10] . Among these cases, 55% 134 were male and ages of these patients were from 21 to 60 years. Among these patients, 135 20.63% had Wuhan's travel history. In [11] , the authors investigate epidemiological, 136 demographic, clinical, and radiological features and laboratory data for 99 cases of 137 2019-nCoV collected from Jinyintan Hospital, Wuhan, China. They found that 49% of 138 these patients traveled to the Huanan seafood market. The average age of the victims 139 was 55.5 years, and most of them (67.68%) were men. The main clinical manifestations 140 were fever (83%), cough (82%), shortness of breath (31%). Among the sufferers, 75% 141 exhibited bilateral pneumonia also. The work in [12] analyzes the clinical characteristics 142 of 1,099 patients with laboratory-confirmed 2019-nCoV ARD from 552 hospitals in 31 143

. CC-BY 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.

The copyright holder for this preprint this version posted May 26, 2020. . CC-BY 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.

The copyright holder for this preprint this version posted May 26, 2020. . . CC-BY 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.

The copyright holder for this preprint this version posted May 26, 2020. . https://doi.org/10.1101/2020.05.22.20110783 doi: medRxiv preprint by X h , and Y h , respectively. We consider one time period (e.g., 15 or 30 minutes) for 221 our scenario and this time period is divided into T smaller time steps in a set T . For 222 each of time step t ∈ T , we have the GPS coordinates X and Y of every individual. 223 Now, the deviation from home for any individual i ∈ N in between two time steps 224 can be measured by using Euclidean distance as follows:

Thus, the total deviation from home by each individual i ∈ N in a particular time 226 period can be calculated as follows:

Hence, the grand deviation of all the individuals of N can be summarized as follows:

On the other hand, at the end of a particular time period, the distance between an 229 individual i ∈ N and any other individuals j ∈ N , j = i is as follows:

Hence, the total distance of individual i ∈ N from other individuals N i ⊆ N , who 231 are in close proximity with i ∈ N , can be expressed as follows:

In the similar fashion of (3), the grand distance of all N individual can be summarized 233 as follows:

Our objective is to keep δ minimum for reducing the spread of COVID-19 from infected individuals, which is an isolation strategy. Meanwhile, we want to maximize social distancing which mathematically translates into maximizing d for reducing the chance of infection from others. However, we can use log term to bring fairness [26, 29] in the objective function among all individuals. Hence, we can pose the following optimization problem:

In (7), Z is a large number and Z > δ i , ∀i ∈ N . The optimization variables X and Y 235 indicate longitude, and latitude, respectively, of the individuals. Moreover, the first 236 term in (7) encourages individual for isolation whereas the second term in (7) 237 encourages individual to maintain fair social distancing. In this way, solving (7) . CC-BY 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.

The copyright holder for this preprint this version posted May 26, 2020. . https://doi.org/10.1101/2020.05.22.20110783 doi: medRxiv preprint vast population in the society. Constraint (7a) guarantees small deviation to maintain 240 emergency needs, while Constraint (7b) assures a minimum fair distance among all the 241 individuals to reduce the spreading of COVID-19 from one individual to another.

242 Constraint (7c) shows that ω can take any value between 0 and 1 which captures the 243 importance between two key factors captured in the objective function of (7). For 244 example, if COVID-19 is already spreading in a given society, then most of the weight 245 would go to isolation term rather than social distancing. The objective of (7) is difficult 246 to achieve as it requires the involvement and coordination among all the N individual. 247 Moreover, if the individuals are not convinced then it is also difficult for the government 248 to attain the objective forcefully. Thus, we need an alternative solution approach that 249 encourage individual separately to achieve the objective and game theory, which is 250 successfully used in [27, 28] , can be one potential solution, which will be elaborated in 251 the next section. To attain the objective for a vast population, governments can introduce incentives for 255 isolation and also for social distancing. Then every individual wants to maximize their 256 utilities or payoffs. In this way, government can play its role for achieving social 257 objective. Hence, the modified objective function is given as follows: 

The Nash equilibrium [31] is the most used solution concept for a noncooperative 276 game. Formally, Nash equilibrium can be defined as follows [34] :

May 19, 2020 12/22

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The copyright holder for this preprint this version posted May 26, 2020. . Definition 1. : A pure strategy Nash equilibrium for a non-cooperative game

However, to find the Nash equilibrium, the following two definitions can be helpful. 281 remain in their strategies. That means, the utility of each player i ∈ S is maximized in 293 this strategy, and hence ultimately maximize the utility of (8) . To this end, maximizing 294 U of (8) ultimately maximize the original objective function of (7). The total amount of incentive a particular time period is presented in (8) . In a 298 particular day, we have T s = 24×60 T0 time period where T 0 is the length of a time period 299 in minutes. Thus, we can denote the incentive of a time stamp t s in a particular day p 300 as follows:

Hence, the incentive in a particular day, p can be given as follows:

Now, if we are interested to find the sustainability of lockdown policy for a 303 particular country till a particular number of days, denoted by P , we have to satisfy the 304 following inequality:

where R 0 is the initial resource of a particular country starting of the lockdown policy 306 and r p is the collected resources in a particular day, p, of the lockdown period. Here, r p 307 includes governmental revenue, donation from different individuals, organizations and 308 even countries. Moreover, the unit of α, β, R 0 and r p are same.

If we assume for simplicity that U p and r p are same for every day and they are 310 denoted byŨ andr, respectively, then we can rewrite (12) as follows:

May 19, 2020 13/22

. CC-BY 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.

The copyright holder for this preprint this version posted May 26, 2020. . Hence, if we are interested to find the upper limit of sustainable days for a particular 312 country using lockdown policy, then we have the following equality:

Thus, by simplifying (14), we have the following:

Here, the sustainable days P depends on R 0 ,Ũ , andr. However, we cannot change 315 R 0 but government can predictr. Moreover, depending on R 0 andr, government can 316 formulate its policy to set α and β so that individuals are encouraged to follow the 317 lockdown policy. Alongside, we cannot continue lockdown policy infinitely based upon 318 the limited total resources. Hence, the governments should formulate and update its 319 lockdown policy based on the predicted sustainable capability to handle COVID-19, 320 otherwise resource crisis will be a further bigger pandemic in the world. . CC-BY 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.

The copyright holder for this preprint this version posted May 26, 2020. is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.

The copyright holder for this preprint this version posted May 26, 2020. simulation parameters are shown in the Table 1 .

326 Fig. 13 illustrates a comparison between home isolation (stay at home) and random 327 location in the considered area for a varying value of ω. In this figure, we consider two 328 cases of N = 500 and N = 1, 000. In both the cases, home isolation (quarantine) is 329 beneficial over staying in random location and the differences between two approaches 330 May 19, 2020 16/22

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The copyright holder for this preprint this version posted May 26, 2020. are increased with the increasing value of ω. Moreover, the difference of payoffs between 331 two approaches are increased with the increasing value of ω as the more importance are 332 given in home isolation. Figure 18 . and 20, 000 for 50%, and 48% sure, respectively, for 25% and 50% home quarantine 338 cases whereas the incentives are 90% sure in between 20, 500 and 21, 000 for 75% home 339 isolation case. Moreover, the same values are at least 21, 500 for 50% sure in case of full 340 home isolation. Fig. 15 depicts that the incentive of being below 38, 000 is 40% sure for 341 25% home isolation case, however, the same values of being above 40, 000, and 41, 000 342 are 40%, and 60%, sure, respective, for 50%, and 75% cases. Moreover, for 100% home 343 isolation case, the values are in between 42, 000 to 43, 000 for sure. The incentives for 344 25%, 50%, 75%, and 100% home isolation cases are above 57, 000, 59, 000, 61, 000, and 345 63, 000, respectively, with probability 0.60, 0.65, 0.65, and 0.80, respectively, as shown in 346 Fig. 16 . Additionally, the same values are at least 77, 000, 79, 000, 81, 000, and 83, 500 347 with 0.50, 0.50, 0.72, and 1.00 probabilities, respectively, which is presented in Fig. 17 . 348 The total incentive (averaging of 50 runs) for varying percentage of home isolation 349 individuals with different sample size are shown in Fig. 18 . From this figure, we observe 350 that the total payoff increases with increasing number of home isolation individuals for 351 all considered cases. The incentives are 578%, 571%, 571%, and 571% better from home 352 quarantine of 25% to 100% for N = 500, N = 1, 000, N = 1, 500, and N = 2, 000, 353 respectively. Moreover, for a particular percentage of home isolation, the total incentive 354 is related with the sample size. In case of 50% individuals in the home isolation, the 355 incentive for N = 2, 000 is 97.08%, 42.50%, and 15.96% more than that of N = 500, 356 N = 1, 000, and N = 1, 500, respectively. Fig. 19 shows the average individual payoff 357 for varying parentage of home isolation individuals for different scenarios. The figure   358 exhibits that the average individual incentive increases with an increasing percentage of 359 home isolation as the deviation δ decreases and hence, the value of home isolation 360 incentive increases. For N = 500, the incentive of 100% home isolation is 85.25% more 361 than that of 25% home isolation. Moreover, in a particular percentage of home isolation, 362

May 19, 2020 18/22

. CC-BY 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. significantly decreases for all considered cases. The reason behind this is that the more 369 individuals are in home isolation, the more it is necessary to pay the incentives. With a 370 fixed amount of resources, a country with less individuals can survive a longer lockdown 371 period. With more percentages of home isolation individuals, the number of loackdown 372 period is less, and possible of spreading of COVID-19 is also less. Therefore, the 373 governments can consider a trade-off between increasing expenditure as a incentive and 374 lockdown period. For 1, 000 individuals, the maximum possible lockdown period for 375 varying amount of R 0 andr is presented in Fig. 21 . The figure also illustrates that with 376 the increasing percentages of home isolation individuals, the continuity of the lockdown 377 period reduces for every scenarios. However, for a particular percentage of home 378 isolation individuals where total number of individuals are fixed, a country can continue 379 higher lockdown period who has more am amount of resources, R 0 . Additionally,r also 380 play an important role to continue the lockdown period. In this paper, we have studied an analytical model for controlling the outbreak of . CC-BY 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review)

The copyright holder for this preprint this version posted May 26, 2020. . performed a detailed numerical analysis of the proposed model to control the outbreak 388 of the COVID-19. In future, we will further study and compare with extended cases 389 such as centralized and different game-theoretic models. In particular, an extensive 390 analysis between the government-controlled spread or people controlled spread under 391 more diverse epidemic models.

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