During this COVID-19 pandemic, we are all required to make important moral decisions (1). Most has been written in these days concerning the complex choices that our leaders had to make when setting up lockdown measures, which involve a trade off between potential benefits (e.g., saving more lives and avoiding a collapse of health care services) and costs (e.g., economic costs).
Yet not just our leaders, but also we as individuals make important moral decisions, such as whether or not to stick to lockdown or quarantine rules (2) (3) (4) . Indeed, lockdowns and quarantines have been described from the beginning as altruistic acts, i.e., we should stay at home not (or not just) for our own sake, but especially to help others, and especially elder and fragile persons.
Crucially, the moral decisions we make in this period have unprecedented effects at the collective level. In standard conditions, a minority of "cheaters" who break the rules (e.g., when voting, paying taxes or breaking traffic rules) can be tolerated and does not significantly affect collective dynamics (5) . Rather, in this pandemic situation, even the choice of a few individuals may matter at the collective level -for the good or the bad.
To illustrate the unprecedented effect of the choices of few individuals at the collective level, we model a paradigmatic case of moral decision: the "escape from Milan" that occurred in the early days of the COVID-19 spread in Italy. On March 7th, there were rumors of an imminent lockdown in Lombardy, which had a large number of COVID-19 cases. During the weekend of March 7th-8th, there was a massive escape from Milan (the biggest city of Lombardy): thousands of persons literally assaulted trains to reach southern regions of Italy, such as Campania and Lazio, where few COVID-19 had been reported. This situation was extensively covered and stigmatized in the Italian public debate, with public authorities and media accused to disclose critical information too early and travelers accused to spread the virus in southern regions of Italy, whose health care services were unprepared.
We used the SEIR model (6) to simulate this situation, and to quantify the potential impact of "travelers" who putatively reached Campania -some of which may have been active COVID-19 cases.
COVID-19 total cases, active cases and deaths in Campania start to increase more steeply during the period of time comprised between March 18th and March 23rd, i.e., around 10 to 15 days after the escape from Milan (Figure 1 ; circles are real data and lines are fits from three SEIR models). To show that this abrupt increase of COVID-19 cases constitutes a novel trend, and not the simple continuation of the previous trend, we trained a SEIR model with data from Feb 24th to March 23rd, and used it to simulate the next days. This model (called without-travelers, see Figure 1A -B, dashed lines) severely underestimates total cases, active cases and deaths in the successive days, from March 24th to April 7th (RMSE error 8.4054), suggesting that the trend changed around March 23rd. 23th to April 5th) and the corresponding real data of deaths, active and total cases. All results are obtained using the same generalized SEIR model (6) , which extends the original SEIR model (7) to cover deaths and recovered; and has been validated with public data of National Health Commission of China.
To understand what could have caused the change, we trained another instance of the same SEIR model (called without-travelers, see Figure 1A -B, dotted lines) using the same data as before, i.e., from Feb 24th to March 23rd. However, crucially, we added a varying number (1 to 100) of new "active cases" to the model. Our results show that a very small number (28 to 43) of new "active cases" permits predicting very well total cases, active cases and deaths in the successive days, from March 24th to April 7th (the best fit is with 41 novel active cases, see Figure 1C ; RMSE error 0.6087).
To further validate the SEIR model, we also trained a third instance of the same SEIR model (called ground truth, see Figure 1 , solid lines) using all the available data, from Feb 24th to April 7th; which unsurprisingly achieves an excellent fit (RMSE error 0.0718).
These simulations indicate that very few novel active cases (namely, 41 cases) are sufficient to explain the changed COVID-19 trend in Campania, which occurred around March 18th -23th, i.e., 10 -15 days after the massive escape from Milan. This novel trend had dramatic effects: up until April 5th, there are 341 additional total cases and 156 additional deaths compared to what could have happened without 41 novel active states (compare solid and dashed lines).
We used the same approach to simulate the case of Lazio: another favorite destination of the escape from Milan. Analogous to Campania, our simulation shows a changed trend of virus expansion around March 23rd, which can be explained by few novel active cases (Figure 2 ; RMSE of without-travelers SEIR model: 3.5261; RMSE of with-travelers SEIR model: 0.0480; best fit with 52 active cases). Rather, a simulation of Lombardy (whose capital is Milan) does not show any change of trend (RMSE of without-travelers SEIR model: 1.2523). Note that we selected Campania, Lazio and Lombardy for our simulations, as they are the most populated regions of Italy and those for which more data are available (https://github.com/pcmdpc/COVID-19).
While not all the active cases in our simulations necessarily correspond to actual travelers from Milan, our results strongly indicate that the choices of few individuals can have significant effects at the collective level. As remarked above, this is very uncommon in the moral decisions we usually do. Are we aware that our moral decisions in these days can be so impactful?
The COVID-19 pandemic exposes the limits of our adaptive rationality When deciding whether or not to leave Milan, potential travelers were balancing their obvious individualistic benefits with potential dangers from the safety of themselves, their families and other residents of southern regions. While opportunistic reasons (8) may have prevailed, is it plausible that travelers also underappreciated the potential societal costs of their choices.
Indeed, the COVID-19 pandemic poses significant cognitive challenges to our capacity to correctly understand the situation and make adaptive decisions. The impending sense of danger, stress, urgency and isolation that we all face, together with an over-exposure to multiple and sometimes conflicting sources of information, all contribute to create a challenging context to deploy our cognitive skills.
These problems are exacerbated by some of our cognitive biases. We are unable to correctly understand and predict phenomena that grow exponentially -this is called an exponential growth bias, and we are not aware of it (9) (10) (11) . We also tend to discount steeply (positive or negative) events that occur after some temporal delay, or at some spatial distance from us (13); and we tend to disregard information that is not compatible with our prior belief, i.e., a confirmation bias (12) . There may be an adaptive rationality beyond these and other cognitive biases. Exponential phenomena (or more broadly, phenomena that start small and can become big very fast) are uncommon in in our daily experiences -they are not part of our "natural statistics". Phenomena that are far in time and space are less likely to affect us and in most cases discounting them is safe.
Yet the COVID-19 pandemic challenges the adaptive rationality beyond these and other cognitive biases. The virus seems to propagate, at least initially, at exponential (or similar) rate, hence effectively "compressing" the temporal horizon that we normally consider when discounting; and it propagates at a global scale, making physical distance less relevant.
This implies that we are all asked to make complex moral decisions in situations that are not just stressing, but also expose the limits of our adaptive rationality. We might have seen the effects of this problem in the moral decisions made by travelers from Milan. If travelers evaluated the costs of their decision in terms of a linear growth (instead of the more complex pattern shown by the SEIR model), they would have severely underestimated them. To give a measure of such underestimation, the total cases reported in Campania on April 5th are 2960, whereas a linear projection from the data available on March 8th (the day of the escape from Milan) would result in 295 total cases -the difference is a factor of ten.
A similar underestimation of risks may help explaining other cases that are nowadays popular in the media, from individuals deciding to leave quarantine for a walk, to public authorities deciding to postpone lockdowns of companies, towns (e.g., the case of Ischgl in Austria) or countries. As an example of the latter, when the problems with COVID-19 were already apparent in China, and there were already indications that the virus was spreading in the north of Italy, the dangers were initially neglected. This is testified by the widespread slogan "Milano does not stop" (in Italian, "Milano non si ferma") and the widespread invitation to keep going as usual by prominent politicians (14) .
Was the Italian case an exception, perhaps given the novelty of the situation (Italy was the first country to experience problems with COVID-19 outside China, where the virus plausibly originated)? It appears that other countries, like UK and USA, failed to learn from the cases of China and Italy and delayed important measures. The reasons for these decisions are of course multifarious and cannot be reduced just to cognitive biases. Yet, it is important not to disregard the unusual cognitive challenges that accompany the decisions we make in these days -as individuals and as policy makers -which go beyond usual considerations about costs-benefits and uncertainty.
The pattern of results we discussed, with a very limited number of novel active cases significantly changing COVID-19 dynamics in entire regions, defies our intuitions about the societal costs of our individual moral decisions -and calls for an unprecedented sense of responsibility. While there are repeated public appeals to responsibility in the media in these days (e.g., campaigns to "stay home") they should be accompanied by measures that increase our awareness and understanding of the potential costs of our individual decisions, and the cognitive challenges we face.
What we need are novel educational strategies aiming to make available for the large public the most relevant knowledge from cognitive and social sciences, as well as the statistical tools that may help us making more informed decisions. There seems to be an increased trust -and hope -in scientific thinking in these days; but to reach their full transformative potential, trust and hope should be accompanied by understanding.