A multiscale framework integrating within-host infection kinetics with airborne transmission dynamics

TL;DR

The paper develops a multiscale framework that links within-host viral replication to indoor airborne transmission by representing hosts as patches within a diffusion-dominated air space. Through matched asymptotics, it reduces a coupled ODE-PDE system to a nonlinear ODE framework that preserves spatial structure via a Neumann Green's function, and proves well-posedness. In the well-mixed limit, it recovers the classical TCL dynamics and provides insight into how diffusion and spatial arrangement influence within-host kinetics and transmission timing. The approach enables tractable analysis of transient infection dynamics in indoor settings and offers a platform for evaluating ventilation- and space-based interventions.

Abstract

Coupling within-host infection dynamics with population-level transmission remains a major challenge in infectious disease modeling, especially for airborne pathogens with potential to spread indoor. The frequent emergence of such diseases highlight the need for integrated frameworks that capture both individual-level infection kinetics and between-host transmission. While analytical models for each scale exist, tractable approaches that link them remain limited. In this study, we present a novel multiscale mathematical framework that integrates within-host infection kinetics with airborne transmission dynamics. The model represents each host as a patch and couples a system of ordinary differential equations (ODEs) describing in-host infection kinetics with a diffusion-based partial differential equation (PDE) for airborne pathogen movement in enclosed spaces. These scales are linked through boundary conditions on each patch boundary, representing viral shedding and inhalation. Using matched asymptotic analysis in the regime of intermediate diffusivity, we derived a nonlinear ODE model from the coupled ODE-PDE system that retains spatial heterogeneity through Neumann Green's functions. We established the existence, uniqueness, and boundedness of solutions to the reduced model and analyzed within-host infection kinetics as functions of the airborne pathogen diffusion rate and host spatial configuration. In the well-mixed limit, the model recovers the classical target cell limited viral dynamics framework. Overall, the proposed multiscale modeling approach enables the simultaneous study of transient within-host infection dynamics and population-level disease spread, providing a tractable yet biologically grounded framework for investigating airborne disease transmission in indoor environments.

Figures (6)

• Figure 1: Schematic illustration of our modeling framework. (A) Illustration of individuals in a region showing virus exhalation, diffusion, and inhalation. (B) Mathematical description of the problem, where individuals are represented by circular patches ($\Omega_j$, for $j=1,\dots,5$), located in a bounded domain ($\Omega$). The red dots represent diffusing viral particles in both panels. Diagram is created by BioRender.
• Figure 2: Sensitivity analysis for the multiscale ODE model. Partial Rank Correlation Coefficients (PRCC) for the basic reproduction number $(\mathcal{R}_0)$ in Eq. \ref{['Eq:R0_SingVIralParticle']} and viral load peak size for the multiscale model in Eq. \ref{['eqn:SingleInd_Model']} with respect to selected model parameters.
• Figure 3: Sensitivity analysis for the TCL model. Partial Rank Correlation Coefficients (PRCC) for the basic reproduction number $(\mathcal{R}_0)$ in Eq. \ref{['Eq:R0_SingInd_VDModel']} and viral load peak size for the TCL model in Eq. \ref{['eqn:TCL_Model']} with respect to selected model parameters.
• Figure 4: Effect of diffusion on within-host infection and the basic reproduction number. Contour plots of the leading-order approximation of the basic reproduction number ($\mathcal{R}_1$) given in Eq. \ref{['Eq:R0_SingInd_NoLoc']} (top row) and the two-term expansion of the basic reproduction number ($\mathcal{R}_0$) in Eq. \ref{['Eq:R0_SingVIralParticle']} (bottom row). Results are shown as functions of the transmission rates $\beta_1$ and $\beta_2$ for different diffusion rates, ranging from $D_0 = 0.002$ to $D_0 = 2$. Parameters are given in Table \ref{['Table:TableParameter']}.
• Figure 5: Effect of diffusion on within-host infection dynamics. Time series of airborne viral concentration and within-host infection kinetics for a single host located at the origin $(0,0)$, computed using the multiscale model \ref{['eqn:SingleInd_Model']} (solid lines) and the TCL model \ref{['eqn:TCL_Model']} (dashed lines). The figure illustrates how diffusion influences within-host infection dynamics. Model parameters are provided in Table \ref{['Table:TableParameter']}.

Theorems & Definitions (6)

• Lemma 4.1: Non-negativity
• proof
• Lemma 4.2: Boundedness
• proof
• Theorem 4.1: Existence and uniqueness
• proof