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.

We consider $n$ tubes exiting a junction and filled with a non
 viscous isentropic or isothermal fluid. In each tube a copy of the
 $p$-system in Euler coordinates is considered. The aim of the
 presentation is to compare three different notions of solutions at
 the junctions: p-solutions, Q-solutions and P-solutions.

Comparison among different notions of solution for the $p$-system at a junction

This paper considers a system of two conservation laws with a Lipschitz continuous flow and deals with solutions to the Cauchy problems at a node. The system considered is the two-phase traffic mod...

Global weak solutions to the Cauchy Problem for a two-phase model at a node

A compressible fluid flows through a junction between two different pipes. Its evolution is described by the 2D or 3D Euler equations, whose analytical theory is far from complete and whose numerical treatment may be rather costly. This note compares different 1D approaches to this phenomenon.

https://www.aimsciences.org/article/exportPdf?id=9afd01dc-0597-46fb-8e17-1a0fbc2c818f

On the 1D Modeling of Fluid Flowing through a Junction

Autonomous vehicles (AVs) allow new ways of regulating the traffic flow on road networks. Most of available results in this direction are based on microscopic approaches, where ODEs describe the evolution of regular cars and AVs. In this paper, we propose a multiscale approach, based on recently developed models for moving bottlenecks. Our main result is the proof of existence of solutions for open-loop controls with bounded variation.

Mauro Garavello

Papers

Comparison among different notions of solution for the $p$-system at a junction

Global weak solutions to the Cauchy Problem for a two-phase model at a node

On the 1D Modeling of Fluid Flowing through a Junction

Comparison among different notions of solution for the $p$-system at a junction

A multiscale model for traffic regulation via autonomous vehicles