Kinetic modelling of epidemic dynamics: social contacts, control with uncertain data, and multiscale spatial dynamics

Abstract

In this survey we report some recent results in the mathematical modeling of epidemic phenomena through the use of kinetic equations. We initially consider models of interaction between agents in which social characteristics play a key role in the spread of an epidemic, such as the age of individuals, the number of social contacts, and their economic wealth. Subsequently, for such models, we discuss the possibility of containing the epidemic through an appropriate optimal control formulation based on the policy maker's perception of the progress of the epidemic. The role of uncertainty in the data is also discussed and addressed. Finally, the kinetic modeling is extended to spatially dependent settings using multiscale transport models that can characterize the impact of movement dynamics on epidemic advancement on both one-dimensional networks and realistic two-dimensional geographic settings.

Identification of Time Delays in COVID-19 Data

“…A full review of the relevant literature is beyond the current work. However, some general classes of models include data-driven and machine learning approaches [1,2,3,4,5,6], models based on partial differential equation (PDE) systems [7,8,9,10,11,12,13,14], agent-based models [15], and models based on ordinary differential equation (ODE) systems. This last category is by far the most common such model, with such articles numbering in the thousands.…”

“…In particular, the exposed compartment is not necessarily easily known from data, requiring its estimation. This may be done, for example, through rule-of-thumb estimates based on the infected population, which result in high uncertainty [13,12,9]. Compounding this problem further are the well-known presence of asymptomatic patients, who may spread the disease while never showing appreciable symptoms, and, at least in the earlier stages of the pandemic, were almost certainly under-counted [9,11].…”