The purpose of this talk is to explore how heterogeneity in the connectivity of an epidemiological network impacts the interaction between strains in a co-infection SIS framework.

In the first part, we will present an epidemiological model of co-infection with multiple strains under the assumption of homogeneous connectivity among hosts. We will show that, under the assumption of neutrality, this model satisfies a neutral null property, an ecologically important concept. Mathematically, this neutrality means that the $\omega$-limit set of any trajectory is a central manifold that can be parameterized by the proportion $(z_i)$ of each strain $i$.

This neutrality can be relaxed using a slow-fast argument, leading to an equation for $(z_i)$ that describes the slow dynamics on the central manifold. This equation is the well-known replicator equation, whose parameters are explicitly linked to macro-level ecological parameters. This connection allows us to understand pathogen ecology through the epidemiological dynamics at the host scale.

In the second part, we will explain how this approach can be extended when considering a heterogeneous network of host connectivity. After precisely describing this new model, we will show that the final replicator equation retains traces of this heterogeneity, providing a direct way to model how the complexity of host interactions affects the dynamics of a pathogen and its multiple variants.

The phenomenon of dispersion in a physical system occurs whenever the elementary building blocks of the system, whether they are particles or waves, overall move away from each other, because each evolves according to a distinct momentum. This physical process limits the superposition of particles or waves, and leads to remarkable mathematical properties of the densities or amplitudes, including local and global decay, Strichartz estimates, and smoothing.

In kinetic theory, the effects of dispersion in the whole space were notably well captured by the estimates developed by Castella and Perthame in 1996, which, for instance, are particularly useful in the analysis of the Boltzmann equation to construct global solutions. However, these estimates are based on the transfer of integrability of particle densities in mixed Lebesgue spaces, which fails to apply to general settings of kinetic dynamics.

Therefore, we are now interested in characterizing the kinetic dispersive effects in the whole space in cases where only natural principles of conservation of mass, momentum and energy, and decay of entropy seem to hold. Such general settings correspond to degenerate endpoint cases of the Castella–Perthame estimates where no dispersion is effectively measured. However, by introducing a suitable kinetic uncertainty principle, we will see how it is possible to extract some amount of entropic dispersion and, in essence, measure how particles tend to move away from each other, at least when they are not restricted by a spatial boundary.

A simple application of entropic dispersion will then show us how kinetic dynamics in the whole space inevitably leads, in infinite time, to an asymptotic thermodynamic equilibrium state with no particle interaction and no available heat to sustain thermodynamic processes, thereby providing a provocative interpretation of the heat death of the universe.

We present the theory of local and nonlocal minimal surfaces in relation to models of phase coexistence, with special attention to regularity and geometric properties.

We present the theory of local and nonlocal minimal surfaces in relation to models of phase coexistence, with special attention to regularity and geometric properties.