Journal of Signal Processing, Vol.26, No.6, pp.189-193, November 2022

RESEARCH NOTERESEARCH NOTE

Eﬀect of Social Distancing on COVID-19 Infection
Determined by a Multi-agent Simulation
Masaki Saito, Yoko Uwate and Yoshifumi Nishio
Graduate School of Sciences and Technology for Innovation, Tokushima University
2-1 Minami-Josanjima-cho, Tokushima 770-8506, Japan
E-mail: {saito, uwate, nishio}@ee.tokushima-u.ac.jp

Abstract

COVID-19 has spread all over the world, and the cumulative number of infected people isstill increasing dairy. Therefore, interventions to limit the spread of COVID-19 should be considered foreach social situation. Eﬀective interventions to minimize COVID-19 transmission vary for each situation inaccordance with a quantitative framework called event R. Of those various events, the example of a school
has the highest possibility of infection, but distancing as an intervention cuts the number of new infectionsin half. Therefore, perform we multi-agent simulation of COVID-19 transmission without measures andwith social distancing. To perform simulation under the same conditions as those in the event R calculation,
our simulation is performed under the following conditions: a single infected person enters the classroom
with the ﬁrst set of multiple uninfected people and a certain amount of time passes. After that, theinfected person enters another classroom with the same number of new uninfected people, and the process
is repeated eight times. The simulation results show that there is a relationship between social distancing
and the spread of infection, but the rate of decrease is not constant.

Keywords: COVID-19, multi-agent simulation, social distancing

1.

Introduction

Recently, there has been much discussion about coexisting with COVID-19, and some states are not requiring the wearing of masks. Therefore, the eﬀectiveness of interventions for infection control other thanmask wearing must be emphasized inlight of these discussions. An eﬀective intervention that can be considered is social distancing.
The interventions to limit the spread of COVID-19
should be considered for each social situation such as
public transport and high schools because the spreadhas not yet been suﬃciency controlled around theworld. Here, there is a quantitative framework to determine which intervention is likely to have the mosteﬀect in which situation. The concept of event R
refers to the expected number of new infected people
owing to the presence of a single infected person in asituation [1].
Then, in workplaces and businesses, it is necessary to determine whether the conditions are linear orsaturating, and whether people are strongly mixed orbubbling. Figure 1, adapted from Tuppera et al. [1],Journal of Signal Processing, Vol. 26, No. 6, November 2022

shows the values of event R for no measures and eachintervention. In Fig. 1, the settings of high schoolsare saturating and mixing, and the results show that
diﬀerent events have diﬀerent eﬀective interventions.
High schools, which are most likely to be susceptible to infection among these four settings, are focusedon. Then, event R is halved by halving the number ofcontacts in event R considering social distancing. Inaddition, the calculations are made under the following conditions in high schools: a single infected person
comes in contacts with the ﬁrst set of multiple uninfected people for a certain length of time. After that,the infected person comes in contact with the same
number of new uninfected people. This is repeatedeight times for a total time of one week.
We perform multi-agent simulation (MAS) of
COVID-19 transmission in a high school without measures and with social distancing. Then, we examine
whether the number of new infections is reduced byhalf by practicing social distancing, as in the event
R calculation. In addition, we examine the eﬀect ofinfection reduction on changes in the proportion ofpeople practicing social distancing.
189

4.
4.1

Simulation and Conditions

Conditions of all simulations

To perform simulation under the same conditions
as those in the event R calculation, the following conditions are adopted.
• The simulation is repeated eight times with one
initially infected person, 20 uninfected people,and a 400 m2 (20 m × 20 m) classroom. Because
the people are replaced seven times in the event
R calculations, the simulations in this study areadapted to these conditions as well.

Fig. 1
Four diﬀerent kinds of events: (left) linear (low transmission probability), (right) saturating(high transmission probability), (upper) static (samecontacts for whole event), and (lower) mixing (highturnover of contacts).

• In this one-week simulation, the initially infected
person is always in the classroom and infectious.
• The agent’s initial position and direction ofmovement are determined randomly.

2.

MAS [2] is based on the idea that programs exhibit behaviors entirely described by the program instructions. It is possible to simulate an artiﬁcial world
inhabited by interacting processes by relating an individual to a program. We can simulate by transposing
the population of a real biosystem to its artiﬁcial counterpart in which particular hypotheses can be exploredby repeating experiments. Each organism of the population is represented as an agent whose behavior isprogrammed with all the required details.
The simulation of the number of people infectedwith infectious diseases is sometimes performed using
a mathematical model based on diﬀerential equations.
In addition, overall parameters such as infection rate
and mortality rate are deﬁned, in the simulation using
a mathematical model. However, in MAS, it is possible to consider more details than in a simulation using
a mathematical model because several parameters aregiven to each agent.
3.

• If an infected person enters within a radius of
1[m] around an uninfected person, the uninfected person becomes infected, and the time required for virus transmission is 15 minutes. Thisis based on the deﬁnition of intensive contact inJapan.
• For all agents, behavior and contact outside theclassroom are not considered.
• To obtain results under the same conditions as
those in the event R calculation, the time of oureight repeated simulations is set to one week.
Weekends are not taken into account, and thetime for one simulation is set at 8 hours.
4.2

Conditions of simulation with socialdistancing

Model

We use the SIR model in our one-week MAS simulation of the number of new infections. In the SIR
model, agents are given the state S (susceptible), I(infectious), or Re (recovered), which changes in turn.
The SIR model is also applied to the early spread ofSARS-CoV-2 and the ﬁtting of the reported COVID19 cases in Italy [3].
In this study, MAS is performed for the cases without measures and with social distancing. Recoveryand death are not considered in this one-week simulation because the infectious period is approximately
eight days [4].

190

• All of the agents are between the ages of 15 and18, and all of the infected people are infectious
and asymptomatic, and there is no quarantineof the infected people.

MASThe simulations are performed without measures
and with social distancing. In the simulation with social distancing, a repulsion force that is inversely proportional to the distance between the agents should
be applied if another agent enters within a radius of
2[m] around one agent. Furthermore, agents are programmed to predict where other agents are going, sowhen there is interference by many other agents, theycannot avoid contact completely.

Journal of Signal Processing, Vol. 26, No. 6, November 2022

5.

Proposed Method

The total number of initially uninfected people in
the eight simulations is 160 because the simulation of
20 uninfected people is performed eight times. The
proportion of the total number of new infected people
to the total number of these 160 people is calculated.
Then, this simulation is performed 10 times and themean and median of the data from these 10 times areobtained. Furthermore, the simulations are performed
without measures and with the number of agents practicing social distancing set to 10 diﬀerent values from10 to 100% of the number of uninfected people. However, in the simulation with social distancing, the initially infected person practices social distancing in allsituations. For example, the total number of agentspracticing social distancing is 11 if 50% of the agentspractice social distancing.
6.

Results

Figure 2 shows a scatter plot and graph of the result of one of the eight repeated simulations withoutmeasures. The scatter plot shows the agents, and themovement of the agents and the infection situation
can be seen in a video. The red points are the infectedagents and the blue points are the uninfected agents.
The graph below shows the number of infected anduninfected people at the same time as that indicatedin the scatter plot above. In this graph, the horizontal axis is time and the vertical axis is the number ofinfected people. The red area increases as the numberof infected people increases. In the case of the simulation shown in Fig. 2, one initially infected person
eventually generates 14 new infected people.
Likewise, Fig. 3 shows a scatter plot of the resultof one of the eight repeated simulations with socialdistancing. the orange points are the agents infected
despite practicing social distancing. In the case of
the simulation shown in Fig. 3, one initially infected
person eventually generates 11 new infected people.

Fig. 3 Scatter plot and graph of result of one of thesimulations with social distancing
The means and medians obtained from all simulations are shown in Figs. 4 and 5. In these ﬁgures, 0%
people practicing social distancing corresponds to theresult of the simulation without measures.

Fig. 4Means of all simulations

Fig. 5Medians of all simulations

Fig. 2 Scatter plot and graph of result of one of thesimulations without measures
Journal of Signal Processing, Vol. 26, No. 6, November 2022

191

Figures 4 and 5 show that the number of new infected people decreased as the proportion of people
practicing social distancing increased. However, thenumber of new infected people is not reduced by half
compared with the simulation without measures unlike in the event R calculation. Social distancing isrepresented by halving the number of people in contact in the event R calculation, whereas the agents ina MAS move randomly and always try to keep a distance of 2[m] from each other. Even if the frequency
of contact between the initially infected person anduninfected people is halved in MAS, the new infectedpeople come in contact with each other in one simulation. Therefore, the number of new infected people
decreases by only about 30% when people practice social distancing of 2[m] because of secondary infectionsin the simulation MAS.
Furthermore, the results in Fig. 4 for the case without measures and the case with 10% of the people
practicing social distancing show that there is a difference of about 10% between these two values. Thisdiﬀerence is larger than the diﬀerences in other parameters. Then, one initially infected person practicessocial distancing in all simulations with social distancing. Therefore, this conﬁrms that the eﬀect of social
distancing on the spread of infection by infected people is signiﬁcant.
In addition, Fig. 6 shows the proportion of new
infected people among the number of agents without
any measures and the proportion of new infected people among the number of agents practicing social distancing. The blue graph shows the number of agents
without any measures and the orange graph shows thenumber of agents practicing social distancing. The
proportion ranges from 10% to 90% of the number ofpeople practicing social distancing because the aim isto compare the eﬀects of no measures and social distancing in the same simulation. In Fig. 6, the agents
without measures consistently infected more than 60%of the population, while the agents practicing social
distancing consistently infected less than 50% of thenew infected people.

Fig. 6192

7.

Conclusion

The purposes of this study were as follows: to conﬁrm whether the number of new COVID-19 infectedpeople with social distancing was reduced by half compared with no measures and to analyze the eﬀect of
the number of people practicing social distancing on
the spread of infection.
MAS was performed without measures and with
the number of agents practicing social distancing set
to 11 diﬀerent values ranging from 0 to 100% of thetotal number of agents. As a result, in the simulation
with social distancing, the number of new infectedpeople was not reduced by half compared with thesimulation without measures. However, the numberof new infected people decreases as the proportion ofpeople practicing social distancing increased.
In a school, students do not always move. They sit
in their classes, and during breaks, some move whileothers do not. In the future, it is necessary to recreate
this situation in MAS to make the simulation more
realistic. In addition, although the number of new
infected people was not reduced by half even with social distancing, it is necessary to examine whether thenumber of new infected people can be reduced by halfwhen the distance between agents is greater than 2[m].

References[1] P. Tuppera, H. Bouryb, M. Yerlanova and C. Colijn: Eventspeciﬁc interventions to minimize COVID-19 transmission,
Proc. National Academy of Sciences of the United States ofAmerica, Vol. 117, No. 50, pp. 32038-32044, 2020.
[2] A. Drogoul and J. Ferber: Multi-agent simulation as a toolfor modeling societies: Application to social diﬀerentiationin ant colonies, MAAMAW 1992. Lecture Notes in Computer
Science (Lecture Notes in Artiﬁcial Intelligence), Vol. 830, p.
2, July 2005.
[3] M. D’Arienzo and A. Coniglio: Assessment of the SARSCoV-2 basic reproduction number, R0, based on the earlyphase of COVID-19 outbreak in Italy, Contents lists available
at ScienceDirect Biosafety and Health, Vol. 2, Issue 2, pp.
57-59, June 2020.
[4] J. M. Carcione, J. E. Santos, C. Bagaini and J. Ba: A
simulation of a COVID-19 epidemic based on a deterministic
SEIR model, Frontiers in Public Health, Vol. 8, Article 230,
pp. 4-12, May 2020.
[5] S. He, Y. Peng and K. Sun: SEIR modeling of the COVID19 and its dynamics, Nonlinear Dynamics An International
Journal of Nonlinear Dynamics and Chaos in Engineering
Systems, Vol. 101, pp. 1667-1680, June 2020.

Means of all simulations

Journal of Signal Processing, Vol. 26, No. 6, November 2022

Masaki Saito received his
B.E. degree in electrical andelectronic engineering from
Tokushima University, Tokushima, Japan, in 2022. His research interests include simulation and neural networks. His
main research is on a multiagent simulation of COVID-19infection.

Yoko Uwate received her
B.E., M.E., and Ph.D. degrees
in electrical and electronic engineering from Tokushima University, Tokushima, Japan, in2003, 2005, and 2006, respectively. From October 2006 toMarch 2008, she was a postdoctoral resarch fellow (PD) ofthe Japan Society for the Promotion Science (JSPS) at Tokushima University. Shewas a visiting postdoctoral research fellow at theInstitute of Neuroinformatics (INI), University andETH Zurich, Switzerland. From April 2008 to March
2010, she worked as a postdoctoral fellow for research abroad of the Japan Society for the Promotion
Science (JSPS) at the Institute of Neuroinformatics(INI), University and ETH Zurich, Switzerland. FromApril 2010, she has been working at Tokushima University, Tokushima, Japan, and is currently an associate professor. Her research interests include complex phenomena in chaotic circuits and neural networks. She was the Chair of the IEEE CAS Society
Technical Committee on Neural Systems and Applications (NSA) (2019-2021), and is a member of the IEEE
CAS Society Board of Governors (2013-2015 and 2021present). Currently, she is the WiCAS-YP Chair ofthe IEEE CAS Society (since 2021). She was an associate editor of IEEE Transactions on Circuits and SystemsI: Regular Papers (2012-2013) and IEEE Transactions on Circuits and SystemsII: Express Briefs (20172019), and is currently an associate editor for IEEE
CAS Society Newsletter (since 2012).

Journal of Signal Processing, Vol. 26, No. 6, November 2022

Yoshifumi Nishio received
his B.E., M.E., and Ph.D. degrees in electrical engineeringfrom Keio University, Yokohama Japan, in 1988, 1990,and 1993, respectively.
In1993, he joined the Department of Electrical and Electronic Engineering at Tokushima University, Tokushima, Japan, where he is currently a professor. From May 2000, he spent one year
in the Laboratory of Nonlinear Systems (LANOS) atthe Swiss Federal Institute of Technology Lausanne(EPFL) as a visiting professor. He has been an adjunct professor of Xi’an Jiaotong University since 2012
and University of Electronic Science and Technologyof China (UESTC) since 2019. His research interests
are in the areas of nonlinear circuits engineering, including the analysis and application of chaos in electrical circuits, analysis of synchronization in coupledoscillatory circuits, and complex networks. He was
the Chair of the IEEE CAS Society Technical Committee on Nonlinear Circuits and Systems (NCAS)(2004–2005), the Steering Committee Secretary of theIEICE Research Society of Nonlinear Theory and its
Applications (NOLTA) (2004–2007), the Chair of theIEEE CAS Society Shikoku Chapter (2011–2014), amember of the IEEE CAS Society Board of Governors (2012–2017), and the vice president on Regional Activities and Membership of the IEEE CASSociety (2018–2021). He was an associate editor ofIEEE Transactions on Circuits and Systems–I: Regular Papers (2004–2005), IEEE Transactions on Circuits and Systems–II: Express Briefs (2012–2013 and2016–2019), and IEEE CAS Magazine (2008–2009).
He is currently serving as Secretary for NOLTA, IEICE (since 2009), and an associate editor for IEEE
CAS Society Newsletter (since 2007) and International
Journal of Bifurcation and Chaos (since 2015). He isalso a member of the Editorial Board of International
Journal of Circuit Theory and Applications.

(Received May 16, 2022; revised July 25, 2022)

193