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This paper is devoted to the multidisciplinary modelling of a pandemic initiated by an aggressive virus, specifically the so-called SARS–CoV–$2$ Severe Acute Respiratory Syndrome, corona virus n.$2$. The study is developed within a multiscale framework accounting for the interaction of different spatial scales, from the small scale of the virus itself and cells, to the large scale of individuals and further up to the collective behaviour of populations. An interdisciplinary vision is developed thanks to the contributions of epidemiologists, immunologists and economists as well as those of mathematical modellers. The first part of the contents is devoted to understanding the complex features of the system and to the design of a modelling rationale. The modelling approach is treated in the second part of the paper by showing both how the virus propagates into infected individuals, successfully and not successfully recovered, and also the spatial patterns, which are subsequently studied by kinetic and lattice models. The third part reports the contribution of research in the fields of virology, epidemiology, immune competition, and economy focussed also on social behaviours. Finally, a critical analysis is proposed looking ahead to research perspectives.

The outbreak of COVID-19 resulted in high death tolls all over the world. The aim of this paper is to show how a simple SEIR model was used to make quick predictions for New Jersey in early March 2020 and call for action based on data from China and Italy. A more refined model, which accounts for social distancing, testing, contact tracing and quarantining, is then proposed to identify containment measures to minimize the economic cost of the pandemic. The latter is obtained taking into account all the involved costs including reduced economic activities due to lockdown and quarantining as well as the cost for hospitalization and deaths. The proposed model allows one to find optimal strategies as combinations of implementing various non-pharmaceutical interventions and study different scenarios and likely initial conditions.

The COVID-19 pandemic lit a fire under researchers who have subsequently raced to build models which capture various physical aspects of both the biology of the virus and its mobility throughout the human population. These models could include characteristics such as different genders, ages, frequency of interactions, mutation of virus, etc. Here, we propose two mathematical formulations to include virus mutation dynamics. The first uses a compartmental epidemiological model coupled with a discrete-time finite-state Markov chain. If one includes a nonlinear dependence of the transition matrix on current infected, the model is able to reproduce pandemic waves due to different variants. The second approach expands such an idea to a continuous state-space leveraging a combination of ordinary differential equations with an evolution equation for measure. This approach allows to include reinfections with partial immunity with respect to variants genetically similar to that of first infection.