A refractory density approach to a multi-scale SEIRS epidemic model

TL;DR

The paper introduces a multi-scale refractory-density framework that connects microscopic within-host viral-immune dynamics to mesoscopic and macroscopic population-level descriptions of SEIRS-like epidemics with infection-age structure. By implementing two micro-scale noise paradigms—Gaussian white noise and escape noise—it derives RD-based PDEs for the population and a stochastic RD formulation for finite populations, enabling consistent analysis of transient dynamics and finite-size fluctuations across scales. Simulations demonstrate epidemic waves, seasonality effects, and the impact of network size on stochasticity, while linking the micro-to-meso-to-macro descriptions through shared variables like the infection rate and hazard function. The approach offers a principled bridge from measured within-host trajectories to population-level patterns, with potential extensions to contact networks, age-structured and asymptomatic spread, and inverse problems for parameter estimation, providing a versatile tool for understanding and predicting real-world epidemic dynamics.

Abstract

We propose a novel multi-scale modeling framework for infectious disease spreading, borrowing ideas and modeling tools from the so-called Refractory Density (RD) approach. We introduce a microscopic model that describes the probability of infection for a single individual and the evolution of the disease within their body. From the individual-level description, we then present the corresponding population-level model of epidemic spreading on the mesoscopic and macroscopic scale. We conclude with numerical illustrations taking into account either a white Gaussian noise or an escape noise to showcase the potential of our approach in producing both transient and asymptotic complex dynamics as well as finite-size fluctuations consistently across multiple scales. A comparison with the epidemiology of coronaviruses is also given to corroborate the qualitative relevance of our new approach.

Figures (7)

• Figure 1: Schematic of the micro- and macroscopic models described in sections \ref{['sec:individual']} and \ref{['sec:population']}. A, Microscopic model. The state variable $V_i$ of an individual $i$ evolves in time $t$ (bottom trace); its crossings of the threshold $V^{T}$ determine the onsets of disease, the infection time moments, when the drive $D(t)$ appear, which describes virus production and immune response (top). The time since the last infection $t^{*}$ is zeroed at the onset of the disease. B, Macroscopic model. At time $t$, individuals with different $t^{*}$ are distributed with the density $\rho$ in the $t^{*}$-space (bottom). With time, the density transports rightwards (as $t^{*}$ increases with $t$) and undergoes a return flux to the infection state $t^{*}=0$ according to the hazard function $H$, so the dashed arrows represent subsequent infections. The hazard depends on the mean state variable $U$, which is similar to $V_i$ in A but is extrapolated over the threshold.
• Figure 2: Escape noise in the form of \ref{['Eq_escape']}. Simulation of the response to the rapid change of the rate of viral load $k$ from 0.6 to 2.4 and the short infecting pulse $I_0(t)$, at $t=100$. A, top to bottom: $I_0(t)$, $k(t)$, the fraction of infected individuals $I(t)$, the state variable $V_i(t)$ of a representative individual, and the rate of new cases $\nu(t)$. For $I(t)$ and $\nu(t)$, the solutions obtained with both macro- (black lines) and microscopic equations for $N=20000$ (blue lines) are shown. B, the distribution in the $t^{*}$-space of the mean state variable $U$, the source term $\rho H$, and the density $\rho$, in 4 time moments. Parameters: $m=1$, $c=0.015$ days$^{-1}$, $\tau=50$, $\tau_I=10$, $\tau_r=1$, $\tau_R=20$, $a=100$, and $V^{T}=2$.
• Figure 3: Escape noise in the form of \ref{['Eq_escape']} with $m=$ 0.5, 1 and 2. The remaining parameter values are as in Fig. \ref{['fig:U_nu5']}.
• Figure 4: Epidemiology of Seasonal Coronaviruses. A, Data from Nickbakhsh2020; specifically, the monthly prevalence of seasonal coronaviruses (sCoVs) detected among patients with respiratory illness virologically tested in NHS Greater Glasgow and Clyde, Scotland, United Kingdom. B, The case of oscillating $k$. Escape noise in the form of \ref{['Eq_escape']}. The remaining parameter values are as in Fig. \ref{['fig:U_nu5']}.
• Figure 5: White noise. Simulation of the response to the rapid change of the rate of viral load $k$ from 3 to 1.2 and the noise amplitude $\sigma(t)$ from 0.5 to 2, at $t=t_0=0$. A, top to bottom: $k(t)$, $\sigma(t)$, the fraction of potentially infected population $I(t)$, the state variable $V_i(t)$ of a representative individual, and the rate of new cases $\nu(t)$. For $I(t)$ and $\nu(t)$, the solutions obtained with both macro (black lines) and microscopic equations for $N=20000$ (blue lines) are shown. B, the distribution in the $t^{*}$-space of the mean state variable $U$, the source term $\rho H$, and the density $\rho$, in 4 time moments. C, Response to periodically changing $k$ mimicking change of seasons ($V^{T}=4$). D, Response to periodically changing $k$ mimicking change of seasons ($V^{T}=3.7$). Parameters: $\tau=50$, $\tau_I=10$, $\tau_r=1$, $\tau_R=20$, $a=50$, $V^{T}=4$, and $\sigma=2\sqrt{\pi}$.