Agent-based perspectives on epidemiological models: analysis of interviews with upper high-school students

In disease spread modeling, a prevalent approach employs differential equations to depict the dynamics of susceptible, infectious, and recovered populations over time. Nonetheless, alternative avenues exist through agent-based epidemiological models, drawing inspiration from interaction models from the physics of complex systems. This study delves into the formulation of such models by upper high-school students who attended a teaching-learning module on computational simulations. The paper focuses on their development of agent-based virus spread models, exploring their ability to forge analogies with previously encountered models of complex systems - namely, predator-prey, opinion dynamics, and cooperative behaviour models. Through the qualitative analysis of individual interviews, our findings reveal that effective strategies of analogy’s construction embed a comprehensive exploration of the underlying interaction mechanisms governing the evolution of the system under study. Conversely, in instances where the mechanistic dimension remains unexplored or vague, the depth and quality of the model elaborated is lower and the potential of comparing models to construct a more robust analogy remains unexploited.


Introduction
Computational modelling and simulations are nowadays a prominent part of scientific research and practice, and they have applications in most disciplines, from physics to social sciences [1,2].With the climate change emergency and more recently with the COVID-19 pandemic, also citizens have been called to enter social debates and decision-making processes also based on these tools.For these reasons, the educational research community is investigating new ways not only to embed simulations in disciplinary teaching but also to foster learners' awareness of these tools and their role as the "third pillar" of scientific inquiry [3].When addressing complex systems like social ones, agent-based models and simulations play an important role both from a theoretical and an educational point of view.In the wide panorama of educational research about agent-based models and simulation, one topic remains largely unexplored [4]: what happens when students engage in modelling processes by trying to model a phenomenon by analogy with other agent-based models and simulations?
This paper aims at addressing this literature gap by presenting the analysis of interviews conducted with upper high-school students at the end of a teaching-learning module on simulations of complex systems.More specifically, the goal is to characterize how students construct analogies between known agent-based models (a predator-prey model, an opinion dynamics model, and a model for cooperative behaviour) and a new issue proposed to them: the spread of a virus in a population.
The paper is structured as follows.In section 2, we present the disciplinary basis of the paper, the basics of equation-based and agent-based approaches to simulation and their application to 2 epidemiological modelling.In section 3, we introduce the research educational framework, with the main research results achieved on equation-based and agent-based models and simulations and a summary of the research literature on the topic of analogies' development and their relationship with agent-based simulations.Then, we move to the description of the study which is the object of this paper, providing an overview of the teaching-learning module (section 4), the data collection tool (section 5), and the research questions and the methodology of data analysis (section 6).Section 7 is dedicated to the presentation of the findings that are finally discussed in Section 8.

Equation-based and agent-based approaches to simulation
Most simulations of complex systems can be categorized as equation-based or agent-based [5].According to the equation-based approach, a system is described through differential equations.When analytical solutions are not available or highly costlyfor example for non-linear differential equations numerical methods are used.These methods are implemented from the simulation and allow us to solve the model step by step, obtaining the evolution of a system depending on the initial conditions.The evolution of the system is deterministically derived: from the present state, the numerical methods used by the simulation lead to the future state of the target system in a univocal and pre-determined way.This means that multiple runs of the same simulation, starting from the same initial conditions, produce the same results.Another important feature of equation-based simulations is that, in the computational model, only macroscopic, global variables appear.There is no reference to the single components of the system because it is modelled as an undifferentiated whole, essentially continuous.The properties of the system that the developers of the model consider relevant for its description are summarized in the equations in the form of "rate" variables which describe simultaneously the whole system.An example of this modelling approach is the Lotka-Volterra model presented in section 4 of this paper.
The agent-based approach is very different.The initial step of the simulation process consists in defining and creating the minimal components of the system (they can be for example the individuals in a population) which are named the agents of the model.Each agent has associated a set of local rules that determine its behaviour in the interaction with the other agents and with the simulated environment.Agents are heterogenous, since they can have different attributes and different rules associated, and autonomous since their interactions are determined exclusively by the rules assigned to them.From these interactions among the individual components and the consequent "evolution" of the agents, the evolution of the system emerges and can be observed in patterns and regularities at the macroscopic level.This feature of agent-based models and simulations is referred to as their bottom-up character or, following Keller [6], "modelling from above", since there is no mathematical theory governing the behaviour of the agents.The macroscopic way in which the aggregate system evolves is not apriori known nor coded but is the result of the relatively simple actions of many single entities that interact at the microscopic level.An example of this approach is the voter model presented in section 4.

Equation-based and agent-based formulations of the SIR model
The equation-and agent-based approaches introduced above can be used to simulate the SIR model, developed in 1927 by the mathematicians Kermack and McKendrick to study the spread of a disease in a population [7].This model is based on the assumption that the population could be divided into compartments, i.e. mutually exclusive classes to which individuals of the population belong.Three compartments are considered: Susceptible (healthy individuals that can contract the disease), Infectious (individuals that have the disease and can infect others), Removed (individuals that have recovered from the disease and have been immunized, or that have died due to the disease).The temporal evolution of the three compartments is described by a set of three ordinary non-linear differential equations:  is the infection rate,  the recovery-or-death rate and  the total population size, while (), () and () represent the sizes of the susceptible, infectious and removed compartments over time.
The SIR model is usually treated as one of the main examples of equation-based models since it is already expressed as a system of differential equations.Indeed, the equation-based approach to simulate the SIR model consists of the numerical integration of the system of equations (1)(2)(3).The SIR model can be addressed also with an agent-based approach.In this case, the model itself and its equations (1-3) can be reformulated from an agent-based perspective.For example, equation ( 1) can be re-written as: (4) Following Sterman [8] and Macal [9], we interpret the rate of infection β as the product of a number β c that quantifies the mean number of contacts that each individual has (per unit of time) and of a number β i that quantifies the probability for a susceptible individual of getting infected because of contact with an infectious one.β c is also called the "contact rate" and it is measured in people contacted per person per time, while β i is called "infectivity".In this way, the probability that a susceptible becomes infectious depends on the probability that a susceptible meets an infectious and on the probability that a susceptible becomes infectious.Similar considerations can be done for the equations (2-3) for the infectious and removed compartments, where the recovery-or-death rate  can be modelled as the reciprocal of the average duration of infectivity  [8]:   [9]).The novelty of the agent-based approach, in a nutshell, is that in the new formulation, we refer to the individual components of the population: the agents.To perform an agent-based simulation, we must create as many agents as the size of the population previously considered.Then to each agent of the system are associated variables and rules.The variables contain information on the epidemiological status of the agent (susceptible, infectious, or removed).The rules describe how the agents interact with each other according to their epidemiological status: an example of the rules set is reported in Figure 1.

Educational research on equation-based and agent-based models and simulations
In science education, equation-based and agent-based simulations are mainly introduced when complex systems are addressed.In particular, they are used when the science of complex systems is emphasized as the discipline that studies how the behaviour of phenomena at different scales is related to the interdependent components at lower scales [10].Equation-based and agent-based models and simulations have also become a way within educational research to address teaching different forms of reasoning about dynamic systems, with special regard to the formulation of explanations [11,12].The first form of reasoning is positioned on a "macro" level: the focus is on the system conceived as a population (sometimes as composed of different groups) with its macroscopic properties that evolve according to rates of change, for example of transitions between groups.On the contrary, the second form of reasoning acts at a "micro" level: the attention is on the minimum elements of the system, the agents, which interact according to local rules.Here, we can easily recognize the parallelism between these two levels of reasoning and the features of the equation-and agent-based simulations described in section 2. Traditional mathematical and science education mainly encourage aggregate reasoning, also through the introduction of differential equations as descriptive tools of dynamic systems.More recently, since the 90s, the importance of agent-based reasoning has been emphasized within education as a way to foster an understanding of the systems and to enter the mechanistic dimension of local interactions [13].Nowadays, the two forms of reasoning are considered both essential to reach a profound understanding of complex systems and in particular to comprehend the emergence of global patterns and behaviours from the local interactions among agents.This concept has been expressed as the "embedded complementarity" of aggregate and agent-based reasoning [14].Even if they have their own very different features, they are not incompatible, nor necessarily working against one another.On the opposite, they are complementary for reaching mature reasoning about emergent phenomena in complex systems.Moreover, this complementarity is "embedded" because it requires not to consider the two forms of reasoning as juxtaposed but to move from one to the other, in a dynamic mutual relationship where elements of connection can be pointed out.Several strategies can be found to connect aggregate and agent-based reasoning: Levy and Wilensky [15] have identified relevant on this account the construction of mid-level groups, that are in-between the level of the agents and that of the emergent property.Barth-Cohen [16] has instead focused on the role of transitional explanations between the microscopic and macroscopic levels of the system.As anticipated in the introduction to this paper, what remains largely unexplored in the literature is the issue of how agent-based simulations can be developed by analogy construction.In the next paragraph, we will provide some elements in this regard.

Educational research on analogies and their relationship with agent-based simulations
The role and potential of analogies to support conceptual learning in science education have been debated for decades [17].According to the constructivist view of learning, analogies reflect a fundamental need to understand the unknown (the target of the analogy) based on what is already known (the source).They are also used as a bridge to facilitate an "intuitive shift" from the realm known to the learner to other situations that are initially more problematic and not perceived as analogous to the former [18].Moreover, cognitive research pointed out significant differences between the analogies built on similarities of surface elements and those based on similarities of processes [19].The first type of analogy does not involve a causal characterization of the system or reference object, is easy to map, and favours analogy identification, whereas the latter leads to deeper thinking and conceptual understanding of both source and target systems [20].
In the attempt to connect agent-based modelling and analogy's development, previous research has shown that, when students try to model real-life issues constructing analogies with agent-based models of reference, they alternate reasonings that are typical either of the macroscopic or the microscopic level of the simulation [4].In the following, we list the categories by providing examples of students' sentences.Macroscopic levels of analogy's development are recognized when students reason in terms of aggregate, collective features of the system, i.e. when they reason on the: Macro 1) Similarity between source and target contexts of application ("We are talking about different ways to see the school, so we are talking about opinions as in the voting simulation") Macro 2) Similarity between the imagined scenario of time evolution for the target system and the simulated evolution of the source system ("I was thinking that voting, in the end, remains always constant because it does not describe an evolution… if we want to analyse a teaching method that evolves, voting is not correct because in a while we reach an equilibrium without analysing how the change occurred… so, with predation is more adequate because we can see how one method disappears and the other grows") Macro 3) Similarity between the emergent interactions among groups of agents ("As in the Lotka-Volterra model there is a fight between groups of people with different opinions") On the opposite, microscopic levels of analogy are detected when students focus on the: Micro 1) Characterization of the types of agents of the system ("With the voting simulation we could do that the blue colour is the traditional teaching method while the green colour is the innovative method that uses new technologies") Micro 2) Meaning of parameters and other elements of the simulation ("We could do that grass, in this case, is the level of patience of the students") Micro 3) Interaction among individual agents embedded in the simulation ("Greedy cows are the old educational traditions, and the cooperative cows are the innovative methods.Both are fed by the consensus of students but there are differences.The cooperative tend not to impose themselves to avoid consuming the grass, they spare the food and let it grow, as the students have time to get acquainted with the new methods.The others, the greedy cows, eat and force the students to their will and do not leave them to grow") We stress that in both categories related to interaction (Macro 3 and Micro 3), there is a reference to the mechanistic dimension and causal processes.The difference between the two is that, in a microscopic perspective, the interaction occurs between individual agents (as coded in the simulation), while, in a macroscopic perspective, the students reason about an interaction between groups of agents that can only be observed as a result of the simulation's run.

Context of the study
In January-February 2021 a teaching-learning module on computational simulations was implemented for the first time online with 35 upper high-school students within an optional program of university orientation of an Italian physics department.The module, described in detail in [21][22][23], comprised conferences of experts, roundtables with researchers and interactive lectures.In particular, the students were progressively guided to analyse three NetLogo agent-based simulations of complex systems that are inspired by interaction models of statistical physics [24].
The first is the model for predation mechanism, based on the equation-based model by Lotka and Volterra, elaborated to describe the evolutionary dynamics of an ecosystem in which two species interact, one as a predator and the other as a prey [25].The mathematical model consists of a pair of ordinary differential equations which describe how the sizes of prey and predator population change over time depending on some parameters.The simulations of the model show the periodic evolution of the two populations.Adopting an agent-based approach, the model is built by defining two kinds of agents (wolves and sheep) that move randomly around a grid.When a predator and prey meet, the wolf eats the sheep.Each movement costs the wolves energy, and they must eat sheep to replenish their energy, otherwise, they die.At each time step, each wolf or sheep has a fixed probability of reproducing.For what concerns the behaviour of the sheep, the simulation has two variations.In the case the grass is modelled as infinite, sheep always have enough food and cannot die from starvation.In the second variation, the sheep must eat grass to maintain their energy and when they reach zero-energy die; moreover, once the grass is eaten it only regrows after a fixed amount of time.The two variations lead to different evolutions of the system: while with infinite grass unstable dynamics are produced, when a limit is set on sheep's resources, we observe stable dynamics.
The second model addressed in the module is the voter model which implements a model of opinion dynamics.Developed in analogy with the Ising model of magnetization [24], it is one of the simplest possible examples of an agent-based model.There are two kinds of agents identified with a binary variable -namely the "opinion".In the beginning, the agents are placed randomly on a grid.Each agent, at each time step, counts the number of neighbours of each kind and keeps or changes its own opinion following the majority.Letting the simulation run, the system reaches a configuration with clusters of agents with the same opinion.
The third module is an evolutionary biology model for cooperative behaviour.There are two kinds of agents that differ in their behaviour in consuming food: greedy cows eat the grass regardless of its quantity, while cooperative cows do not eat the grass if it is below a certain height (below a certain height, the grass grows at a slower rate).When the cows eat, their energy increases, and they can eventually reproduce.When the simulation runs, we observe that, if the cows can frequently move around, the greedy behaviour wins, otherwise the cooperative behaviour is evolutionarily successful.

Final interviews
At the end of the module, semi-structured interviews were conducted with 8 volunteer students.The protocol guided the participants to construct their agent-based model for the phenomenon of virus spreading.Even if this example was not addressed during the course, it was chosen both for its tragic connection with the pandemic and because it is often taken as an example to reflect on the interaction mechanisms in an agent-based, decentralized perspective.We recall for example the work by Barth-Cohen [26] who proposed to her students different problem contexts that exhibit complex systems behaviours that can be construed through decentralized causality.One of these problems is the spread of a virus, in which the phenomena at a macroscopic level (the time evolution of the number of susceptible, infected, and recovered individuals) are determined by the interactions of the subparts, individuals (the interactions between agents of different epidemiological statuses) at the microscopic level.The first questions we asked the students were exploratory (Q1: What can it mean to model a virus spreading phenomenon?What comes to your mind?): we did not want to implicitly suggest the idea of referring to the agent-based modelling strategy and that is why we did not mention terms like "rule", "population", "individual" or "agents", nor we did refer to the idea of "simulation" but only to "model".Then, the questions become more focused when we asked the students to design a simulation to model the spread of a virus.Here, we made the students focus on the agent-based approach to modelling and on the rules which characterize it (Q2: What would you do?What rules would you choose to model this phenomenon?).The last questions required the students to move even further, trying to recognize analogies between the three simulations explored in the course and this problem of the virus' spreading (Q3a: Do you see any possible links between these models and the issue of the virus spreading?If so, which ones?If not, why?).To help them focus on the level of the analogy, we asked at which level(s) they recognized the similarity between the virus' spreading and the three simulations (Q3b: If you identified any similarity, at what level were they?Does it concern a similarity of global, overall phenomena, or mechanisms of interaction between agents, or other?).With these final questions, we wanted to check whether the interaction models we introduced had become for the students a way to reinterpret the modelling they had reached on the spreading phenomenon.

This study aims at answering the following research questions: How do upper high-school students develop their agent-based model to describe the spread of a virus in a population? In particular, how do they use known agent-based models to construct their new model by analogy?
To answer them, we analysed the pseudonymized transcripts of the 8 video-recorded interviews.The analytical process used qualitative methods inspired by grounded theory with explicit back-and-forth dynamics, from bottomup data exploration to their theory-oriented interpretation [27][28].More specifically, the data analysis process was articulated in three main methodological phases.The first step consisted in performing the transcription and pseudonymisation of the video recordings of the interviews.The second step consisted in selecting the parts of the transcript in which the students referred to the three simulations to construct their analogies with the problem of virus spreading; these were coded using the macro-micro categories

Data analysis and results
In this section, we present the results of the analysis by commenting on excerpts of the transcripts of the interviews with Federica, Michele, and Karl (gender-indicative pseudonyms).The analysis will allow us to show the variety of ways in which the analogical dimension is switched on vary across the students, the different degrees of disciplinary robustness of the analogies developed, and the roles of the analogy in the development of the agent-based model for the spread of the virus.

The power of formulating agent-based models by activating analogies: the case of Federica
The case of Federica is emblematic to see how the activation of the analogical dimension changes the type of discourse of the student.Before activating any kind of explicit reference to the previously encountered models, while answering Q2, the student says: To

. I can put a person who follows the rules or does not follow the rules. I would have to decide how many people follow the rules, how many do not follow the rules and then maybe even which rules to follow more precisely, maybe one puts a mask on someone who doesn't hold it, a person who often sanitizes their hands one doesn't often sanitize his hands, I should add as many factors as possible.
Then also lockdowns, schools, workplaces... Federica is engaging in a modelling procedure by trying her best to describe with as many details as possible the features of the system at stake ("the elements that I need", "as many factors as possible").She names the agents she wants ("the possible agents") and their most fundamental characteristics (the epidemiological state of being "infected or uninfected").Then, she adds other features to these agents, calling them "variables", e.g. the willingness of an agent to obey the laws ("a person who follows the rules or does not follow the rules").Moreover, she specifies that she would need the number of agents with each specific set of features ("how many people follow the rules, how many do not follow the rules") and the laws at stake ("maybe one puts a mask", "a person […] sanitizes their hands").Finally, she cites the possibility to add other things of which she does not specify the role in the simulation, such as "lockdowns, schools, workplaces".So far, Federica is not referring to any specific simulations encountered previously in the module.Still, she is only using the basic lexicon of agent-based simulations: "agents", "variables", and "rules".After five seconds of silence, Federica says: So… okay… let's make it simpler.Well, for example, an agent could be individuals, another agent just be the virus (Micro 1) and I don't know for example now in the simulations we have seen, I don't know for example when you have a contact when the virus comes into contact with an individual, I don't know, even the individual becomes a virus (Micro 3) and here with a series of... you can see how the pandemic spreads (Macro 2) a bit like, more or less... it's not really what we have seen in Voting, but more or less it is like that.
The activation of analogical reasoning spontaneously happens for Federica when she realizes that what she mentioned in the previous piece of the transcript was not framed within a coherent picture: all the elements were scattered, and it was difficult for her to cognitively manage them.This becomes evident as she initiates her explanation with the phrase "let's make it simpler", indicating her intention to streamline her discourse.By building an analogy with the voter simulation, Federica reorganizes three distinct layers of the simulations that intersect the macroscopic and the microscopic levels.The first layer operates from a microscopic standpoint.It encompasses the aspect of agents within the system, where an agent could pertain to an individual and another could be the virus ("an agent could be individuals, another agent just the virus").Continuing with her analogical reasoning, the second layer that Federica introduces maintains its focus on the microscopic level.This layer revolves around the mechanism of local interactions between two agents via physical contact ("when the virus comes into contact with an individual, […] the individual becomes a virus").The third layer, a significant pivot, transitions from the microscopic to the macroscopic realm.This shift in perspective enables Federica to delve into the broader, temporal evolution of the system.Her use of the analogy helps her construct a comprehensive narrative describing how the pandemic disseminates over time through a sequence of events ("with a series of... you can see how the pandemic spreads").By encapsulating this process within the analogy, she facilitates a more intuitive grasp of the pandemic's progression.
From a disciplinary point of view, we need to clarify both the potential and the limits of the analogy established by Federica between the spread of the virus and the voter agent-based model.For what concerns the agents, the analogy is fully valid: in the voter simulation, we have two types of agents that vary for a binary feature (e.g.individuals who vote right and individuals who vote left), while for her virus simulation, Federica imagines the two types of agents one as a human and the other as a virus.Regarding the mechanism of local interaction between agents, in the voter simulation as well as in that imagined by Federica, the local result of the behavioural rules is the transformation of one agent into another.However, Federica's analogy has an important limitation that the student does not recognize: in the voter simulation it is allowed that both types of agents transform themselves into the other, while in the virus spread model, only a person can become a virus while the opposite does not happen.It could be imagined as a mechanism of recovery in which, after a certain time, an ill person (the virus) recovers from the disease and changes into a person, but Federica does not mention such a mechanism.Finally, about the overall evolution of the system, Federica mentions for the virus simulation a spread of the virus (with more virus-agents than person-agents), while in the voter model, we usually observe a configuration in which neighbourhoods with agents of the same type are formed, without any sharp prevalence of a type over the other.

Navigating macroscopic and microscopic levels of analogy's development by remaining at the surface: the case of Michele
A different transition pattern between the macroscopic and microscopic levels is exhibited by Michele.Differently from Federica, he activates analogical reasoning only when Q3 is posed by the interviewer: The most obvious simulation that immediately comes to my mind is that of Voting.Right?For the concept of exchange (Macro 1), that is, for the interaction between the individual and the rest of the population that surrounds it (Micro 3).In my opinion, the voting model would be the closest to virus simulation.But another one that I think might be a good idea would be that of greedy cows.Because we could use... in the simulation there was the concept of altruism (Macro 1)… if you choose certain rules, we could say that responsible people use protection systems and obey social rules while irresponsible do not use them (Micro 1) and consequently make the situation evolve in a certain way (Macro 2).
At first, Michele links the virus situation with the voter model, then moves to the cooperation model.Unlike Federica, Michele initiates his analogy by connecting the virus situation not with the microscopic level description of the agents involved, but rather with a broader perspective that centers on the overarching characteristics shared by the systems under consideration.These shared characteristics are referred to as "concepts" (specifically, "the concept of exchange" and "the concept of altruism").When viewed through our analytical lens, these concepts appear to be founded on a macroscopic analogy, drawing parallels between the general nature of the two contexts.For what concerns the analogy with the voter model, his focus remains on the microscopic interactions transpiring between an individual and their neighbouring members of the population ("for the interaction between the individual and the rest of the population that surrounds it").When Michele transitions to the cooperation model, he shifts his attention to the microscopic level once more, this time delving into the categorization of agent types based on their behaviour.He elucidates the dichotomy between "responsible people", who employ protective measures and adhere to social regulations, and "irresponsible" individuals who disregard such measures ("responsible people use protection systems and obey social rules while irresponsible do not use them").Then, he transitions again to the macroscopic level of the overall evolution of the system ("and consequently make the situation evolve in a certain way").
It can be noticed that his analogy is much less accurate than that developed by Federica.The "concepts", the local interactions and the time evolutions shared by the spread of the virus and the voter and cooperation models are mentioned but not fully expounded upon.Unlike Federica's well-structured and layered analogy, Michele's comparison remains somewhat surface-level, leaving certain aspects of the analogy underexplored.For these reasons, it is not possible to comment from a disciplinary point of view on the analogy constructed by Michele.

The pivotal role of reflecting on the local mechanisms of interaction to discriminate between sources of the analogy: the case of Karl
A concise but accurate evaluation of the elements of the analogy is performed by Karl who says: If I think of cooperative cows, there could be selfish people who think only of themselves, so they do not respect the restrictions (Micro 1).But the virus is not like in the cooperation simulation... in this case the individuals, the agents influence each other, like in Wolf Sheep Predation (Micro 3).In Cooperation, the agents live, and if they die in the end it is an accident because it is not someone who kills them or infects them (Micro 3).They die because there's no more food.On the other hand, in sheep and wolves, there is direct action (Micro 3).The similarity with Wolf Sheep is more obvious because agents influence others through direct contact (Micro 3).
Karl embarks on the construction of his virus spread model by drawing inspiration from the cooperation model and focusing on the microscopic level.In his exploration, he identifies a potential type of agent ("there could be selfish people who think only to themselves").While Karl does not explicitly mention the connection, we can infer that behind his association of selfish individuals in the epidemic scenario with the greedy cows in an evolutionary biology simulation lies a shared macroscopic similarity rooted in the concept of "selfishness".This concept seemingly links both situations, although Karl doesn't overtly articulate this, leading us to abstain from labeling it as a "Macro 1".
As Karl delves deeper into his analogy, he uncovers a crucial inadequacy within the cooperation model.He contrasts the direct and immediate cause-and-effect relationship characterizing the spread of the virus (where contagion leads to infection) with the absence in the cooperation model of a direct link to the death of a particular cow.He underscores that, within the cooperation model, cow deaths emerge as a consequence of dwindling resources, rather than a clear, direct causal relationship between agents ("In Cooperation, the agents live, and if they die in the end it is an accident because it is not someone who kills them or infect them... they die because there is no more food").The ambiguity surrounding the causality of cow deaths within the cooperation model prompts Karl to seek an alternative framework that can encapsulate the essence of direct interaction vital for modeling the virus spread.
Finally, Karl identifies the predator-prey model as a fitting candidate.In this context, the action of wolves preying on sheep directly results in the demise of the sheep.This alignment between the predatory action and its effect satisfies Karl's need for a more direct and unambiguous interaction-a critical requirement for accurately modeling the transmission of the virus.He underscores this newfound alignment, expressing that in this case, individual agents within the analogy exert a tangible influence on one another, and their actions propagate through direct contact ("in this case the individuals, the agents influence each other", "agents influence others through direct contact").

Discussion and conclusions
The data analysis pointed out that how the analogical dimension is switched on varies a lot across the students.Some, like Federica, spontaneously refer to the three simulations of predator-prey, voter, or cooperation to ground on them their formulation of the agent-based model.Others, like Michele, consider the given models only when explicitly requested and limit themselves to mentioning the existence of shallow similarities, without specifying which or why.
Even when the analogy is activated, we can still observe a diversity of forms of students' reasoning while modelling a new target phenomenon based on previously encountered sources.This diversity is well represented by the six macroscopic and microscopic categories of analogies development illustrated in section 3.2.We remark that the two sets of categories do not correspond to the surface-likeness and process-likeness identified by Carey [19].If a student maps agents or parameters of the simulations with the analogous elements in the problem of virus spread, we interpret it as a microscopic level of the simulation's analysis (Micro 1 for the agents or Micro 2 for the parameters): nevertheless, what the student is pointing to is an "issue of surface", since it lacks explicit reference to the processes represented by the relationships between the agents.On the opposite, the process-likeness can be traced both at the microscopic and at the macroscopic levels, since in the analogy's development both interactions between agents (Micro 3) and between groups (Macro 3) can be mapped.
The comparison of the cases of Michele and Karl showed that reflecting on the source simulations to model the target phenomenon set the basis for reflecting on the different characterizing mechanisms of interactions between individuals, allowing the students to validate their analogy based on the mechanisms they point out.Michele remains on the surface of the problem, recalling the similarities of what he calls "concepts" ("exchange" and "altruism") but without going really into the analogy: the result is that he is unable to discriminate which model, between the voter and the cooperation, is the more appropriate to study the phenomenon of the spread of the virus.Karl, instead, centres his reasoning on the microscopic level of the interaction among agents and focuses on understanding which simulation embeds the mechanism that better suits the phenomenon he needs to address.Doing this, he starts with the cooperation simulation as the source for its analogy but, after having recognized its limits (i.e., the absence of direct interaction among the agents, since it is mediated only by the growth of grass), he arrives at the predator-prey simulation that satisfies his needs concerning the epidemiological modelling.
In conclusion, we can argue that productive strategies of analogy's construction merge surface and process, macroscopic and microscopic levels, and the pivotal role is played by the identification and discussion of the mechanisms of interaction underlying the evolution of systems at stake.On the opposite, when the mechanistic dimension remains unexplored or vague, the different interpretations proposed by the students cannot be distinguished and the potential of their comparison to construct a more robust analogy remains unexploited.
= − (  ) = −  (  ) (  ) = −  [      * (ℎ   )] *   = − where the recovery-or-death rate  can be modelled as the reciprocal of the average duration of infectivity [8].The mathematical model in (4-6) is formally equivalent to the SIR classical model of equations(1)(2)(3).The novel element is in the interpretation of the variables in the equations.While in (1-3) we had rates of infection and recovery-or-death, in (4-6) we are referring to contacts that individuals of different compartments have, to the probability that a single susceptible individual is infected after contact with an infectious individual, to the average duration of infectivity.

Figure 1 .
Figure 1.Diagram representation of the rules associated with each agent of the SIR agent-based model (adapted from [9]).The novelty of the agent-based approach, in a nutshell, is that in the new formulation, we refer to the individual components of the population: the agents.To perform an agent-based simulation, we must create as many agents as the size of the population previously considered.Then to each agent of the of analogies' development illustrated in section 3.2.The final step consisted in comparing the succession of codes obtained for the different interviews to recognize possible recurrent patterns. 7