Epidemic modeling with heterogeneity and social diffusion.

We propose and analyze a family of epidemiological models that extend the classic Susceptible-Infectious-Recovered/Removed (SIR)-like framework to account for dynamic heterogeneity in infection risk. The family of models takes the form of a system of reaction-diffusion equations given populations structured by heterogeneous susceptibility to infection. These models describe the evolution of population-level macroscopic quantities S, I, R as in the classical case coupled with a microscopic variable f, giving the distribution of individual behavior in terms of exposure to contagion in the population of susceptibles. The reaction terms represent the impact of sculpting the distribution of susceptibles by the infection process. The diffusion and drift terms that appear in a Fokker-Planck type equation represent the impact of behavior change both during and in the absence of an epidemic. We first study the mathematical foundations of this system of reaction-diffusion equations and prove a number of its properties. In particular, we show that the system will converge back to the unique equilibrium distribution after an epidemic outbreak. We then derive a simpler system by seeking self-similar solutions to the reaction-diffusion equations in the case of Gaussian profiles. Notably, these self-similar solutions lead to a system of ordinary differential equations including classic SIR-like compartments and a new feature: the average risk level in the remaining susceptible population. We show that the simplified system exhibits a rich dynamical structure during epidemics, including plateaus, shoulders, rebounds and oscillations. Finally, we offer perspectives and caveats on ways that this family of models can help interpret the non-canonical dynamics of emerging infectious diseases, including COVID-19.

Plateaus, rebounds and the effects of individual behaviours in epidemics.

Plateaus and rebounds of various epidemiological indicators are widely reported in Covid-19 pandemics studies but have not been explained so far. Here, we address this problem and explain the appearance of these patterns. We start with an empirical study of an original dataset obtained from highly precise measurements of SARS-CoV-2 concentration in wastewater over nine months in several treatment plants around the Thau lagoon in France. Among various features, we observe that the concentration displays plateaus at different dates in various locations but at the same level. In order to understand these facts, we introduce a new mathematical model that takes into account the heterogeneity and the natural variability of individual behaviours. Our model shows that the distribution of risky behaviours appears as the key ingredient for understanding the observed temporal patterns of epidemics.

Propagation of Epidemics Along Lines with Fast Diffusion.

It has long been known that epidemics can travel along communication lines, such as roads. In the current COVID-19 epidemic, it has been observed that major roads have enhanced its propagation in Italy. We propose a new simple model of propagation of epidemics which exhibits this effect and allows for a quantitative analysis. The model consists of a classical SIR model with diffusion, to which an additional compartment is added, formed by the infected individuals travelling on a line of fast diffusion. The line and the domain interact by constant exchanges of populations. A classical transformation allows us to reduce the proposed model to a system analogous to one we had previously introduced Berestycki et al. (J Math Biol 66:743-766, 2013) to describe the enhancement of biological invasions by lines of fast diffusion. We establish the existence of a minimal spreading speed, and we show that it may be quite large, even when the basic reproduction number [Formula: see text] is close to 1. We also prove here further qualitative features of the final state, showing the influence of the line.