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

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.

💡 Research Summary

The paper tackles the long‑standing problem of linking within‑host viral dynamics to population‑level airborne transmission in indoor environments. The authors construct a multiscale mathematical framework in which each individual is represented as a small circular “patch” embedded in a bounded two‑dimensional domain that mimics a room, classroom, or hospital ward. Viral particles released by infected hosts diffuse through the air, are removed by natural decay, and can be inhaled by susceptible individuals.

Mathematically, the airborne virus concentration r(X,τ) obeys a linear diffusion‑decay partial differential equation (PDE) with diffusion coefficient Dr and decay rate kr. On the boundary of each host patch Ωj the authors impose a Robin (mixed) condition Dr ∂n r = −γj uj, where uj is the intracellular viral load of host j and γj is a shedding coefficient. This condition couples the PDE to a system of ordinary differential equations (ODEs) that describe the within‑host dynamics of target cells (Tj), eclipse‑phase infected cells (Ej), productively infected cells (Ij), and free intracellular virus (uj). The ODEs follow the classic target‑cell limited model, with infection occurring both from inhaled airborne virus (rate b1j) and from intracellular virus (rate b2j).

The model is nondimensionalized using the domain size L and the viral decay rate kr as characteristic scales, introducing a small geometric parameter ε = R/L (the ratio of host radius to room size). The dimensionless diffusion coefficient becomes D = Dr/(kr L²). The authors focus on the intermediate‑diffusivity regime where D = O(μ⁻¹) with μ = −1/ln ε → 0 as ε → 0. In this regime, singular perturbation (matched asymptotic) analysis is applied.

First, the authors define the spatial average of the airborne virus, V(t), over the bulk region (the room minus the host patches). Integrating the PDE and using the divergence theorem yields an evolution equation for V:

 ∂t V + V = (1/|Ω\Ωh|) ∑ ξj vj,

where ξj = γj L/kr and vj = uj/rc are dimensionless shedding rates. The term on the right represents the total contribution of all infected hosts to the airborne pool.

Next, the authors solve the diffusion problem in the limit ε → 0 by expanding the solution in powers of μ. The leading‑order term is spatially uniform (the well‑mixed assumption). The first‑order correction involves the Neumann Green’s function G(x, x′) for the domain with no‑flux boundaries, which captures how the location of each host perturbs the otherwise uniform concentration. This correction appears as an O(μ) term in the reduced ODE system, preserving the influence of spatial heterogeneity without solving the full PDE.

The final reduced model consists of a set of nonlinear ODEs for each host:

- dTj/dt = −β1j Tj ∮∂Ωεj V dS − β2j Tj vj
- dEj/dt = β1j Tj ∮∂Ωεj V dS + β2j Tj vj − kj Ej
- dIj/dt = kj Ej − δj Ij
- dvj/dt = pj Ij − cj vj

together with the bulk equation for V that includes the Green’s‑function correction. The authors prove existence, uniqueness, and boundedness of solutions using standard ODE theory and maximum‑principle arguments for the underlying elliptic operator.

Parameter analysis shows that decreasing the diffusion coefficient D (or equivalently, reducing ventilation) leads to higher local airborne concentrations and larger infection peaks, especially when hosts are clustered. In the opposite limit of very large D, the Green’s‑function term vanishes, the system collapses to a spatially homogeneous “well‑mixed” model, and the reduced equations recover the classic target‑cell limited viral dynamics (dV/dt = p I − c V).

The paper concludes that the proposed framework bridges a critical gap between detailed within‑host immunology and realistic indoor transmission physics. It provides a tractable yet mechanistically grounded tool for exploring how room geometry, ventilation rates, and occupant placement affect epidemic outcomes. The authors suggest extensions to incorporate non‑linear airflow, stochastic shedding, and immune feedback, which would further enhance the model’s applicability to real‑world outbreak mitigation and building‑design guidelines.