Infection Risk Assessment via Agent Simulation with Seat-Choice Behavior

. The human behavior has much impact on the infectious desease spreading. The pathogen generation is not uniform for the residential area since the human have some preference on where to be. The purpose of the present study is to evaluate the impact of the human preference on the seat choice behavior on the pathogen concentration. The preference of the seat selections in a train was modelled and the agent simulation was performed in order to evaluate the non-uniformity of pathogen. The concentration was calculated with Wells-Riley model. The results of the agent simulation showed the preference of seat choice based on the experimental results in a train environment. The seats at the end were more selected than other seats. The concentration at the seats at the end kept increasing because the agent sat there by turns and that is why there was no recovery time. The double effect of the more seat selection and the less recovery time has much impact on the non-uniformity of the concentration.


Introduction
Taking countermeasures against infectious diseases including COVID-19 is essential for our healthy life. Ones of the effective countermeasures are avoidance of the three Cs, which are closed space, crowded places, and close-contact settings. Although closed space can be resolved with ventilation, crowded places and closecontact setting are determined by human behavior. Therefore, issues of indoor air quality can not be discussed only based on the physical phenomena such as airflow movement and particle spreading, but also the human behavior. Infected person is a sort of contaminant source for the indoor environment. If he/she exhale, cough, and sneeze, pathogen is generated there. The location of the generation source depends on the location where the infected person is. The decision of where to sit is not random but is based on some preference of human. Thus, the human preference needs to be modelled in order to determine the boundary condition of pathogen generation for airflow simulation. The purpose of the present study is to clarify the impact of human preference on the pathogen concentration. In-train environment is employed and seat-choice behavior is modelled based on the results of experiments. The distribution and the time change of pathogen concentration is analyzed via agent simulation.

Agent simulation
An agent simulation method was used to model agent's behavior. It simulates the behavior of agents by setting up their actions based on modelling formula and placing an arbitrary number of agents in an arbitrary space defined by the simulation method. Evacuation simulation is one of the agent simulations. An agent means an entity that acts according to the surrounding situation. In the present study, it simply refers to a person. The greatest feature of agent simulation is that it is possible to see the effects of each agent's behavior on the spatial unit. In this respect, the agent simulation method is more advantageous than the mathematical epidemiological model, which is too large on a macrosocial scale, or the Computational Fluid Dynamics (CFD) simulation, which does not consider the human behavior.

Multinomial logit model
In modeling seat selection behavior, it is necessary to know what kind of factors people think about when they select a seat. In the case of choosing a seat, we considered that persons do not just choose a seat at random, but that they choose a seat in consideration of the benefits they can gain by sitting down, such as avoiding sitting next to other people or sitting in a position where they can hear others easily. In econometrics, this "benefit gained by choice" is called "utility". This "utility" can be calculated as in the following equation (1).
: Utility : Utility coefficient : Factor The consideration factor xi is the item to be taken into account when selecting a choice, for example, "Is there someone next to me?". The utility coefficient βi is a value that indicates how important the factor xi is in the selection. The linear model is a model in which the value of utility obtained by equation (1) is used as the selection probability as it is. The logit model is a model that uses the logistic function obtained by integrating the logistic distribution that takes a distribution close to the standard normal distribution, and the value of the probability is obtained by formula (2).
where, : probability of choosing However, this equation can only calculate the probability of choosing a certain option, and in a general choice situation, there are multiple options and the probabilities of choosing each option are interrelated. Therefore, we use the multinomial logit model [1] shown in equation (3) below.
where, : Utility of option : Probability of choosing option The probability of selecting an option j out of n options can be obtained by taking the sum of the exponential of the utilities of all options as the denominator. In this study, seat selection behavior was modelled by introducing this multinomial logit model into an agent simulation.

Wells-Riley model
For the analysis of virus concentrations, the formula used in the Wells-Riley model (4) (4) into the virus concentration at a certain time t, the following equation (5) is obtained by considering the virus concentration −1 before a small time elapsed.
:Virus concentration at time t [quanta/m 3 ] ：Probability of infected person The probability P of a passenger being an infected passenger is a value indicating the percentage of passengers who are infected.

Field experiment in reference
Based on these methods and theories, the seat selection behavior in train was modelled based on the seat selection of actual human (Konishi et al. 2018) [2]. In that study, 84 persons were employed and the six cases of seating congestion in a space simulating a train consisting of seven seats and one standing room. They were asked to choose which of the available seats they would choose, and the probability of choosing all seats in each case was calculated. In the present study, same condition with seven seats and seven standing rooms in front of them in the simulation space were modelled (see Figure 1). Note that the standing rooms are selected based on utility value because they are a measure to prevent passengers from overflowing as they board, virus concentration on standing room were not discussed.
Seat  For these items xi, the utility coefficient βi was identified to reproduce the selection probabilities of the experiment in the previous study. As for standing seats, the sum of the selection probabilities for all standing seats is identified to be close to the value for standing seats in the previous study. The identified values are shown in Table 2. Substituting the values in Tables 1  and 2 into Equation (1) gives the utility of the seat, and further substituting that value into Equation (3) gives the selection probability of that seat. Table 2. Identified values of utility coefficient For the discussion of virus concentration, the diffusion of exhaled virus was not considered in the simulation, since the virus concentration at a particular seat was considered to act only on that seat. Therefore, the room volume was calculated based on the size of the seats, which are 460 mm in both width and depth according to the JIS standard, and the height of the seats was considered to be one meter because passengers sit on the seats. The present study focuses not on the diffusion characteristics of virus but the characteristics of the diffusion source. As mentioned above, many studies have analysed the airflow, the diffusion and the droplet generated by infected person treated as the stable source. In the actual environment, however, human can move and then the source of the virus is not uniform nor constant. The agent simulation can clarify these characteristics of the location of the virus source and its time change.
The values of the virus generation rate E and the virus attenuation rate λ are those of the COVID-19, which is familiar and easy to visualize in the literature [3] [4]. Table 3 below shows a summary of the various values necessary for the discussion of virus concentrations.

Result and discussion
3.1 Choice frequency of seat Figure 2 shows the simulated results of the choice frequency of seat. The error bar indicates the standard deviation of five trials of simulation. The seats at the end of the row, which were seat A and G, were selected more often. The seats in between, which were seat B to F, were selected almost the same number of times. The results indicates that the human preference has much impact on the seat choice and then the generation probability of virus is not uniform entire the space.

Time change of concentration
Time change of virus concentration at each seat is shown in Figure 3. This result is just based on one trial of simulation. Although the results also do not account for viral spread, which is an extreme case, it does validate the trend in the relationship between seat selection and viral concentration. Concentration kept increasing during the seat was occupied and then the agent exhaled virus there. Only in the period when the seat was vacant, the concentration decayed. The seat A and G, which were the seats at the end, showed the highest concentration because the agents sat there by turns and then there was no recovery time.
In comparison between seat A and E, the concentration at seat A was four times larger than that of the seat E at the end of the simulation. As for the choice frequency, shown in Figure 2, the seat A showed only three times larger than that of the seat E. The difference is thought to be caused by the non-linear nature of concentration decay. It indicates that the preferred seat has higher risk of infection because the more persons select the seat and also the recovery time becomes short. This double effect needs to be considered when the CFD simulation is performed since the source intensity of pathogen is not uniform for the residential area.

Conclusion
The preference of the seat selections in a train was modelled and the agent simulation was performed in order to evaluate the non-uniformity of pathogen. The concentration was calculated with Wells-Riley model.
The results of the agent simulation showed the preference of seat choice based on the experimental results in a train environment. The seats at the end were more selected than other seats. The concentration at the seats at the end kept increasing because the agent sat there by turns and that is why there was no recovery time. The double effect of the more seat selection and the less recovery time has much impact on the non-uniformity of the concentration.