A Calculation Model for Estimating Effect of COVID-19 Contact-Confirming Application (COCOA) on Decreasing Infectors
Yuto Omae, Jun Toyotani, Kazuyuki Hara, Yasuhiro Gon, Hirotaka Takahashi
IIEICE TRANS. FUNDAMENTALS, VOL.Exx–??, NO.xx XXXX 2020 LETTER
A Calculation Model for Estimating Effect of COVID-19Contact-Confirming Application (COCOA) on Decreasing Infectors
Yuto OMAE † , Member , Jun TOYOTANI † , Nonmember , Kazuyuki HARA † , Member ,Yasuhiro GON †† , Nonmember , and Hirotaka TAKAHASHI ††† , Member
SUMMARY
As of 2020, COVID-19 is spreading in the world. In Japan,the Ministry of Health, Labor and Welfare developed COVID-19 Contact-Confirming Application (COCOA). The researches to examine the effectof COCOA are still not sufficient. We develop a mathematical model toexamine the effect of COCOA and show examined result. key words:
COVID-19, Contact-Confirming Application, virus spreading,mathematical modeling
1. Introduction
On June 19, 2020, the Japanese government developed andreleased COVID-19 Contact-Confirming Application (CO-COA) [1], which was the smartphone app to decrease thenumber of COVID-19 infectors. By using COCOA, theusers can know whether or not they are contact with the in-fectors (refer to Fig. 1). If the close contact persons whoreceive the contact information from the app are staying athome, there is a possibility of decreasing the total infections(because they may be infections).We consider that the reduction effect of new infectorsincreases as the usage rate of the app increases. However,the usage rate of COCOA looks insufficient in Japan. As ofOct. 2020, the number of install is about 18 million [1]. Inother words, the usage rate is about 15%. To increase theusage rate, it is important to report the reduction effect bythe research (e.g. mathematical model and simulation etc.).There are many researches on the model for estimatingthe number of COVID-19 infectors. For example, Hou etal. [2] showed that a measure of decreasing the contact withthe persons could effectively decrease the total infectors.Chatterjee et al. [3] also conducted a simulation experimentas a case study in India. The other simulation models ofCOVID-19 were also reported in [4] [5] [6]. However, theseresearches did not show the reduction effect of the app suchas COCOA.The simulation model for estimating the effect of theapp on decreasing infectors were developed. For example,
Manuscript received October x, 2020.Manuscript revised October x, 2020. † The authors are with College of Industrial Technology, NihonUniversity, 1-2-1, Izumi, Narashino, Chiba 275-8575, Japan †† The author is with Nihon University School of Medicine, 30-1,Kami, Ooyaguchi, Itabashi, Tokyo 173-8610, Japan †††
The author is with Research Center for Space Science, Ad-vanced Research Laboratories, Tokyo City University, 8-15-1Todoroki, Setagaya, Tokyo 158-8557, JapanDOI: 10.1587/transfun.E0.A.1
Hinch et al. [7] used the individual-based network model andOmae et al. [8] used the multi-agent simulation. However,in the case of the infection disease estimation, the reliabilityverification of the simulation results is difficult because theresearcher cannot experiment in the real world (i.e. we can-not calculate the differences between actual and estimatedresults). As the alternative method to verify the reliability,it is necessary to examine the effects by the various simula-tion methods and verify that the obtained results are similar.However, the research to survey the effect of the app suchas COCOA to the number of total infectors is still insuffi-cient. Therefore, we develop a calculation model to knowthe app’s effect in this letter. We also examine whether ornot the results are similar between previous researches andour model.
2. Calculation model 𝐼 𝑡 at the day 𝑡 as 𝐼 𝑡 = 𝐼 𝑡 − + 𝑅𝑚 − ( − 𝑝 ) 𝐼 𝑡 − − 𝑚 − 𝐼 𝑡 − = 𝐼 𝑡 − ( + 𝑅𝑚 − ( − 𝑝 ) − 𝑚 − ) , (1)where 𝑅 is a reproduction number. 𝑅 means the number ofnew infectors that the single infector infects other personsduring total infection period. 𝑚 is the number of averagedays for recovering from infection. Therefore, 𝑚 − in thethird term is recovery rate of single day. And 𝑅𝑚 − in thesecond term is the number of new infectors of single day that Fig. 1
Schematic view of the COCOA [1]
Copyright © a r X i v : . [ q - b i o . O T ] O c t IEICE TRANS. FUNDAMENTALS, VOL.Exx–??, NO.xx XXXX 2020 reproduced by one infector. 𝑝 is the usage rate of the appin the total population. The first term 𝐼 𝑡 − shows the totalinfectors at previous day. The second term 𝑅𝑚 − ( − 𝑝 ) 𝐼 𝑡 − shows the amount of new increased infectors. COCOA cannotice the contact information to a person who is contact withan infector, only if both use the app. If only one uses theapp, COCOA cannot notice the information to a close contactperson. Moreover, the app’s usage probability of infectors is 𝑝 . And the app’s usage probability of close contact personsis also 𝑝 . Because a joint probability of them is 𝑝 , only thispercentage of newly infectors receive contact information.We assume that close contact persons who receive the contactinformation from the app do not go outside (i.e stay home).In this case, they infect nobody. Therefore, we can definethe amount of new increased infectors is 𝑅𝑚 − ( − 𝑝 ) 𝐼 𝑡 − .The third term 𝑚 − 𝐼 𝑡 − is the amount of decreased infectorsby the recover or death. Eq.(1) was developed by partiallyreferring to the equation for solving the number of infectorin SIR model [9].In the case of using the number of initial infectors 𝐼 ,we can express 𝐼 𝑡 from Eq.(1) as: 𝐼 𝑡 = 𝐼 ( + 𝑅𝑚 − ( − 𝑝 ) − 𝑚 − ) 𝑡 . (2)The proof of Eq.(2) is given in the appendix.Then, the relative rate of the total number of infectorswhether to use the app is 𝐼 𝑡 ( 𝑝 )/ 𝐼 𝑡 ( 𝑝 = ) . Therefore, theeffect of the usage rate of the app 𝑝 on decreasing infectorscan be defined as: 𝐸 𝑡 ( 𝑝 ) = − 𝐼 𝑡 ( 𝑝 ) 𝐼 𝑡 ( 𝑝 = ) = − ( + 𝑅𝑚 − ( − 𝑝 ) − 𝑚 − ) 𝑡 ( + 𝑅𝑚 − − 𝑚 − ) 𝑡 , (3)2.2 Turning point by the usage rate of the app 𝑝 betweeninfection spread and convergenceIn this section, we describe the turning point between in-fection spread and convergence by using the app. Based onEq.(1), the coefficient of the amount increasing infectors is 𝑅𝑚 − ( − 𝑝 ) , and the coefficient of the amount decreasinginfectors is 𝑚 − . Therefore, if 𝑅𝑚 − ( − 𝑝 ) = 𝑚 − , the totalinfectors 𝐼 𝑡 does not increase. In other words, 𝑅𝑚 − ( − 𝑝 ) = 𝑚 − , i . e . 𝑅 ( − 𝑝 ) = , (4)is the condition of convergence of spreading infection dis-ease. We solve for 𝑝 , then, 𝑝 = ( − 𝑅 − ) / . (5)Eq.(5) means that, to lead the spreading infection to con-vergence, the required usage rate 𝑝 of app depends on thereproduction number of virus 𝑅 . The relationship between 𝑝 and 𝑅 by Eq.(5) are shown in Table.1. As 𝑅 increases, therequired 𝑝 also increases.The reproduction number 𝑅 of COVID-19 is between1.4 and 2.5 [10]. 𝑅 of SARS which is prevailed in 2003 Table 1
Turning point between infection spread and convergence: therelationship between 𝑅 and 𝑝 in Eq.(5) 𝑅 𝑝 in Hong-Kong is about 2.7 [11]. 𝑅 of seasonal influenza isabout 1.3 [12]. Therefore, we consider that the app such asCOCOA is effective to various infection disease.Moreover, we consider the limitation of 𝑝 for 𝑅 :lim 𝑅 →∞ 𝑝 = lim 𝑅 →∞ ( − 𝑅 − ) / = . (6)This means that even if the epidemic of infection diseaseof very high 𝑅 occuers, when everyone use the app i.e. 𝑝 =
3. Simulation 𝐼 is 50 persons.The maximum of simulation days is 50 days. We considerthe usage rates of the app: 𝑝 = , , · · · , , 𝑅 , which is the number of new infec-tors that the single infector infects other persons. Accordingto WHO, the reproduction number of COVID-19 is from 1.4to 2.5 [10]. Therefore, we use 𝑅 = . 𝑚 =
14. In otherwords, we use the coefficient in the second term of Eq.(1) 𝑅𝑚 − = / (cid:39) .
143 and the coefficient in third term ofEq.(1) 𝑚 − = / (cid:39) . 𝐼 𝑡 by Eq.(2)is shown in Fig. 2. In the case of the usage rate 𝑝 = 𝑝 increases, the total infectors decreases. The mostnotable point is 𝑝 = 𝑝 =
80 and 100%,the total infectors decrease. In other words, we can interpretthat the spread of COVID-19 is the end. This result hassimilar tendency to the result obtained by Hinch et al. [7]based on the individual-based network model and Omae etal. [8] based on multi-agent simulation. They reported thespreading COVID-19 is convergence if the usage rate of theapp over 60%. Therefore, we emphasize that our developed
ETTER Fig. 2
Total infectors 𝐼 𝑡 mathematical model supports the results of Hinch et al. [7]and Omae et al. [8]. The point of appearing similar result byvarious methods is important.The effect of the usage rate of the app 𝑝 in 𝐸 𝑡 (Eq.(3)) isshown in Fig. 3. As the basic trends, the reduction effect ofthe number of infectors increases over the time. Moreover,as 𝑝 increases, the effect increases. However, even if theusage rate of the app is low (e.g. 𝑝 =
4. Conclusion
In this letter, we reported the effect of the app such as CO-COA on decreasing infectors based on the simple calcula-tion model. As the result, we could understand the fea-tures/dynamics of the total infectors because we incorporatedthe usage rate of the app into the model. However, other im-portant parameters did not be incorporated and consideredin this letter. One of them is the registration rate of infection.If infectors that use the app reject the registration of infec-tion information, COCOA will not work. Our model assumethat all infectors who use the app register the infection in-formation. Moreover, we assume that close contact personswho received the contact information from the app do notgo outside (i.e stay home). We consider that some personsgo outside without worrying about the contact notifications.Thus, our developed model can be interpreted as the upperlimit of the effect. Therefore, we incorporate their pointsinto the model as future works. Finally, we will developmore desirable model to estimate the effect of COCOA.
References
Fig. 3
Effect of the app on reduction of infectors 𝐸 𝑡 Model,” Medical Journal Armed Forces India, vol.76, no.2, pp.147-155, 2020.[4] Z. Yang et al., “Modified SEIR and AI Prediction of the EpidemicsTrend of COVID-19 in China Under Public Health Interventions,”Journal of Thoracic Disease, vol.12, no.3, pp.165-174, 2020.[5] S. He, Y. Peng and K. Sun, “SEIR Modeling of the COVID-19 andIts Dynamics,” Nonlinear Dynamics, pp.1-14, 2020.[6] S. Annas, M. I. Pratama, M. Rifandi, W. Sanusi, and S. Side, “Sta-bility Analysis and Numerical Simulation of SEIR model for Pan-demic COVID-19 Spread in Indonesia,” Chaos, Solitons & Fractals,vol.139, 2020.[7] R. Hinch, W. Probert, A. Nurtay, M. Kendall, C. Wymant,M. Hall and C. Fraser, “Effective Configurations of aDigital Contact Tracing App: A Report to NHSX,”https://cdn.theconversation.com/static_files/files/1009/Report_-_Effective_App_Configurations.pdf, accessed Oct. 14. 2020.[8] Y. Omae, J. Toyotani, K. Hara, Y. Gon and H. Takahashi,“Effectiveness of the COVID-19 Contact-Confirming Application(COCOA) based on a Multi Agent Simulation,” arXiv preprintarXiv:2008.13166, 2020.[9] J. Satsuma, R. Willox, A. Ramani, B. Grammaticos, A. S. Carstea,“Extending the SIR Epidemic Model,” Physica A: Statistical Me-chanics and its Applications, vol.336, no.3-4, pp.369-375, 2004.[10] Y. Liu, A. A. Gayle, A. Wilder-Smith and J. Rocklov, “The Repro-ductive Number of COVID-19 is Higher Compared to SARS Coro-navirus,” Journal of Travel Medicine, vol.27, no.2, pp.1-4, 2020.[11] S. Riley, C. Fraser, C. A. Donnelly, A. C. Ghani, “TransmissionDynamics of the Etiological Agent of SARS in Hong Kong: Impactof Public Health Interventions,” Science, vol.300, no.5627, pp.1961-1966, 2003.[12] G. Chowell, M. A. Miller, C. Viboud, “Seasonal Influenza in theUnited States, France, and Australia: Transmission and Prospectsfor Control,” Epidemiology and Infection, vol.136, no.6, pp.852-864, 2008.
Appendix: Proof of Eq.(2)
We explain the proof of Eq.(2) from Eq.(1) of all naturalnumber 𝑡 by using mathematical induction.When 𝑡 =
1, Eq.(1) is 𝐼 = 𝐼 ( + 𝑅𝑚 − ( − 𝑝 ) − 𝑚 − ) , (7)and Eq.(2) is 𝐼 = 𝐼 ( + 𝑅𝑚 − ( − 𝑝 ) − 𝑚 − ) . (8) IEICE TRANS. FUNDAMENTALS, VOL.Exx–??, NO.xx XXXX 2020
Then, Eq.(7) equals Eq.(8) i.e. when 𝑡 =
1, Eq.(1) equalsEq.(2).After that, we assume Eq.(1) equals Eq.(2) when 𝑡 = 𝑘 ( 𝑘 is a natural number). In other words, 𝐼 𝑘 = 𝐼 ( + 𝑅𝑚 − ( − 𝑝 ) − 𝑚 − ) 𝑘 , (9)is true by Eq.(2).When 𝑡 = 𝑘 +
1, Eq.(1) is 𝐼 𝑘 + = 𝐼 𝑘 ( + 𝑅𝑚 − ( − 𝑝 ) − 𝑚 − ) = 𝐼 ( + 𝑅𝑚 − ( − 𝑝 ) − 𝑚 − ) 𝑘 + . (10)Note that we use Eq.(9).When 𝑡 = 𝑘 +
1, Eq.(2) is 𝐼 𝑘 + = 𝐼 ( + 𝑅𝑚 − ( − 𝑝 ) − 𝑚 − ) 𝑘 + . (11)Thus, Eq.(10) equals Eq.(11) when 𝑡 = 𝑘 + 𝑡𝑡