The COVID-19 sanitary crisis has shed an unprecedented light on the role of modellers, mathematicians, computer scientists and data scientists in the monitoring and in the understanding of the dynamics of an epidemic. The general audience became rapidly acquainted with the concepts of R0, of quasi-exponential growth of the number of cases, as well as with the burden that such an explosive dynamics puts on medical structures. But mathematical and computational approaches have even more to offer. From the early stages of the pandemic, many organisational issues arose to cope with the wave of patients, which overwhelmed many hospitals and care structures. Emergency call centres faced a sharp increase in the number of calls, which had to be efficiently sorted according to the urgency of the patient’s condition. The saturation of critical care services led to the need to re-orient incoming patients to other hospitals with available beds, or to organise the transfer of patients from a heavily burdened region to a region under a weaker capacity strain. Hospital services had to find humanly manageable ways to deal with the work surplus, in conditions where health workers were not only at risk of physical and psychological exhaustion, but also at risk of being infected with SARS-CoV-2. In this context, already existing collaborations between mathematicians or computer scientists and medical doctors or emergency call centres gave rise to the extraordinarily quick emergence of tools to help operators and medical staff with logistic decisions. Below, we present four such initiatives, which arose during the first COVID-19 wave in France in 2020. They illustrate what close collaborations between medical staff and researchers from the fields of operations research, modelling of complex dynamics or data sciences can bring to the management of critical situations in health services. Beyond the current pandemic, they also pose the question of how to improve our preparedness to future crises.