to be done in this area. Face recognition is a problem that scarcely clamoured for attention before the computer age but, having surfaced, has involved a wide range of techniques and has attracted the attention of some fine minds (David Mumford was a Fields Medallist in 1974). This singular application of mathematical modelling to a messy applied problem of obvious utility and importance but with no unique solution is a pretty one to share with students: perhaps, returning to the source of our opening quotation, we may invert Duncan's earlier observation, 'There is an art to find the mind's construction in the face!'.

Linear and nonlinear waves, by G. B. Whitham. Pp.636. £50. 1999. ISBN 0 471 35942 4 (Wiley).

Autonomous vehicle (AV) is regarded as the ultimate solution to future automotive engineering; however, safety still remains the key challenge for the development and commercialization of the AVs. Therefore, a comprehensive understanding of the development status of AVs and reported accidents is becoming urgent. In this article, the levels of automation are reviewed according to the role of the automated system in the autonomous driving process, which will affect the frequency of the disengagements and accidents when driving in autonomous modes. Additionally, the public on-road AV accident reports are statistically analyzed. The results show that over 3.7 million miles have been tested for AVs by various manufacturers from 2014 to 2018. The AVs are frequently taken over by drivers if they deem necessary, and the disengagement frequency varies significantly from 2 × 10−4 to 3 disengagements per mile for different manufacturers. In addition, 128 accidents in 2014–2018 are studied, and about 63% of the total accidents are caused in autonomous mode. A small fraction of the total accidents (∼6%) is directly related to the AVs, while 94% of the accidents are passively initiated by the other parties, including pedestrians, cyclists, motorcycles, and conventional vehicles. These safety risks identified during on-road testing, represented by disengagements and actual accidents, indicate that the passive accidents which are caused by other road users are the majority. The capability of AVs to alert and avoid safety risks caused by the other parties and to make safe decisions to prevent possible fatal accidents would significantly improve the safety of AVs. Practical applications. This literature review summarizes the safety-related issues for AVs by theoretical analysis of the AV systems and statistical investigation of the disengagement and accident reports for on-road testing, and the findings will help inform future research efforts for AV developments.

/pdf/safety-of-autonomous-vehicles-17ojztzbq2.pdf

Safety of Autonomous Vehicles

We present a new fluid-dynamical model of traffic flow. This model generalizes the model of Aw and Rascle [SIAM J. Appl. Math. 60 916-938] and Greenberg [SIAM J. Appl. Math 62 729-745] by prescribing a more general source term to the velocity equation. This source term can be physically motivated by experimental data, when taking into account relaxation and reaction time. In particular, the new model has a (linearly) unstable regime as observed in traffic dynamics. We develop a numerical code, which solves the corresponding system of balance laws. Applying our code to a wide variety of initial data, we find the observed inverse-$\lambda$ shape of the fundamental diagram of traffic flow.

On the fundamental diagram of traffic flow

We present an epidemic model capable of describing key features of the Covid-19 pandemic. While capturing several qualitative properties of the virus spreading, it allows to compute the basic reproduction number, the number of deaths due to the virus and various other statistics. Numerical integrations are used to illustrate the adherence of the evolutions described by the model to specific well known real features of the present pandemic. In particular, this model is consistent with the well known relevance of quarantine, shows the dramatic role of care houses and accounts for the increase in the death toll when spatial movements are not constrained.

/pdf/an-age-and-space-structured-sir-model-describing-the-covid-3qh2qxpjgw.pdf

An age and space structured SIR model describing the Covid-19 pandemic

The adoption of containment measures to reduce the amplitude of the epidemic peak is a key aspect in tackling the rapid spread of an epidemic. Classical compartmental models must be modified and studied to correctly describe the effects of forced external actions to reduce the impact of the disease. The importance of social structure, such as the age dependence that proved essential in the recent COVID-19 pandemic, must be considered, and in addition, the available data are often incomplete and heterogeneous, so a high degree of uncertainty must be incorporated into the model from the beginning. In this work we address these aspects, through an optimal control formulation of a socially structured epidemic model in presence of uncertain data. After the introduction of the optimal control problem, we formulate an instantaneous approximation of the control that allows us to derive new feedback controlled compartmental models capable of describing the epidemic peak reduction. The need for long-term interventions shows that alternative actions based on the social structure of the system can be as effective as the more expensive global strategy. The timing and intensity of interventions, however, is particularly relevant in the case of uncertain parameters on the actual number of infected people. Simulations related to data from the first wave of the recent COVID-19 outbreak in Italy are presented and discussed.

/pdf/control-with-uncertain-data-of-socially-structured-26foioisyy.pdf

Control with uncertain data of socially structured compartmental epidemic models.

https://hal.archives-ouvertes.fr/hal-02308788/document

A multiscale model for traffic regulation via autonomous vehicles

We present a modeling framework based on a structured SIR model where different vaccination strategies can be tested and compared. Vaccinations can be dosed at prescribed ages or at prescribed times to prescribed portions of the susceptible population. Different choices of these prescriptions lead to entirely different evolutions of the disease. Once suitable "costs" are introduced, it is natural to seek, correspondingly, the "best" vaccination strategies. Rigorous results ensure the Lipschitz continuous dependence of various reasonable costs on the control parameters, thus ensuring the existence of optimal controls and suggesting their search, for instance, by means of the steepest descent method.

Optimizing vaccination strategies in an age structured SIR model.

SIR models, also with age structure, can be used to describe the evolution of an infectious disease. A vaccination campaign influences this dynamics immunizing part of the susceptible individuals, essentially turning them into recovered individuals. We assume that vaccinations are dosed at prescribed times or ages which introduce discontinuities in the evolution of the S and R populations. It is then natural to seek the “best” vaccination strategies in terms of costs and/or effectiveness. This paper provides the basic well posedness and stability results on the SIR model with vaccination campaigns, thus ensuring the existence of optimal dosing strategies.

Well Posedness and Control in a NonLocal SIR Model

SIR models, also with age structure, can be used to describe the evolution of an infective disease. A vaccination campaign influences this dynamics immunizing part of the susceptible individuals, essentially turning them into recovered individuals. We assume that vaccinations are dosed at prescribed times or ages which introduce discontinuities in the evolutions of the S and R populations. It is then natural to seek the 'best' vaccination strategies in terms of costs and/or effectiveness. This paper provides the basic well posedness and stability results on the SIR model with vaccination campaigns, thus ensuring the existence of optimal dosing strategies.

Mauro Garavello

Papers

An age and space structured SIR model describing the Covid-19 pandemic

A multiscale model for traffic regulation via autonomous vehicles

Optimizing vaccination strategies in an age structured SIR model.

Well Posedness and Control in a NonLocal SIR Model

Well Posedness and Control in a NonLocal SIR Model