In the present work we model multi-lane traffic flow in presence of two population of vehicles: cars and trucks. We first develop a finite-dimensional hybrid system which rely on continuous Bando-Follow-the-Leader dynamics coupled with discrete events motivated by the lane-change maneuvers. Then we rigorously prove that the mean-field limit is given by a system of Vlasov-type PDE with source terms generated by the lane-change maneuvers of the human-driven vehicles.

Genetic variations in the COVID-19 virus are one of the main causes of the COVID-19 pandemic outbreak in 2020 and 2021. In this article, we aim to introduce a new type of model, a system coupled with ordinary differential equations (ODEs) and measure differential equation (MDE), stemming from the classical SIR model for the variants distribution. Specifically, we model the evolution of susceptible $S$ and removed $R$ populations by ODEs and the infected $I$ population by a MDE comprised of a probability vector field (PVF) and a source term. In addition, the ODEs for $S$ and $R$ contains terms that are related to the measure $I$. We establish analytically the well-posedness of the coupled ODE-MDE system by using generalized Wasserstein distance. We give two examples to show that the proposed ODE-MDE model coincides with the classical SIR model in case of constant or time-dependent parameters as special cases.

During the Covid-19 pandemic a key role is played by vaccination to combat the virus. There are many possible policies for prioritizing vaccines, and different criteria for optimization: minimize death, time to herd immunity, functioning of the health system. Using an age-structured population compartmental finite-dimensional optimal control model, our results suggest that the eldest to youngest vaccination policy is optimal to minimize deaths. Our model includes the possible infection of vaccinated populations. We apply our model to real-life data from the US Census for New Jersey and Florida, which have a significantly different population structure. We also provide various estimates of the number of lives saved by optimizing the vaccine schedule and compared to no vaccination.

The COVID-19 pandemic highlighted the need to quickly respond, via public policy, to the onset of an infectious disease breakout. Deciding the type and level of interventions a population must consider to mitigate risk and keep the disease under control could mean saving thousands of lives. Many models were quickly introduced highlighting lockdowns, testing, contact tracing, travel policies, later on vaccination, and other intervention strategies along with costs of implementation. Here, we provided a framework for capturing population heterogeneity whose consideration may be crucial when developing a mitigation strategy based on non-pharmaceutical interventions. Precisely, we used age-stratified data to segment our population into groups with unique interactions that policy can affect such as school children or the oldest of the population, and formulated a corresponding optimal control problem considering the economic cost of lockdowns and deaths. We applied our model and numerical methods to census data for the state of New Jersey and determined the most important factors contributing to the cost and the optimal strategies to contained the pandemic impact.

Current research directions indicate that vehicles with autonomous capabilities will increase in traffic contexts. Starting from data analyzed in R. E. Stern et al. (2018), this paper shows the benefits due to the traffic control exerted by a unique autonomous vehicle circulating on a ring track with more than 20 human-driven vehicles. Considering different traffic experiments with high stop-and-go waves and using a general microscopic model for emissions, it was first proved that emissions reduces by about 25%. Then, concentrations for pollutants at street level were found by solving numerically a system of differential equations with source terms derived from the emission model. The results outline that ozone and nitrogen oxides can decrease, depending on the analyzed experiment, by about 10% and 30%, respectively. Such findings suggest possible management strategies for traffic control, with emphasis on the environmental impact for vehicular flows.