Distributed Intracellular Memory Stabilizes a Diffusive Virus-Immune Model

Standard within-host virus models often summarize the eclipse phase—the interval between infection of a cell and the start of viral production—by a single fixed delay. A new study in Boundary Value Problems argues that this simplification hides dynamically important structure, because intracellular production times in fact vary across infected cells.

The research team, comprising Mohamed Biomy, Luca Guerrini, A. M. A. Abo-Bakr and Stefania Ragni, replaced the fixed lag in a diffusive virus-immune model with two gamma-distributed representations of intracellular memory, one weak and one strong. Using the linear-chain technique, they converted these distributed delays into finite-dimensional reaction-diffusion systems that can be analyzed with standard tools.

The analysis establishes positivity and equilibrium preservation for the reformulated systems, derives mode-dependent characteristic polynomials, and formulates explicit Routh-Hurwitz stability criteria. These results give closed-form conditions under which the spatially uniform equilibrium of the model remains stable.

The most striking numerical finding concerns the reference delay value. At the mean delay T = 21.034, which corresponds to the Hopf threshold of the original discrete-delay model, the homogeneous mode remains strictly stable under both gamma kernels. The spectral abscissa is approximately −0.0358 for the weak kernel and −0.0243 for the strong kernel, while the discrete-delay formulation sits at a near-zero crossing of the real part, the signature of impending oscillatory instability.

Numerical continuation in the gamma order shows that the dominant real part moves slowly toward zero as the kernel becomes more concentrated. In other words, the more the distribution of intracellular times is squeezed toward a single fixed value, the closer the system comes to the Hopf boundary of the classical model.

Time-domain and phase-plane simulations support the spectral results: distributed memory damps the oscillatory feedback, and the broader weak kernel produces the strongest attenuation of the oscillations. The authors therefore identify the variance and shape of intracellular memory, independently of its mean, as dynamically relevant stability parameters.

From a modeling perspective, the work suggests that temporal dispersion of the eclipse phase can suppress delay-induced Hopf instability in spatially structured virus-immune systems. Because the effect is driven by the distribution’s shape rather than its average, two models with identical mean delays can behave qualitatively differently once diffusion and immune dynamics are included.

The article, published open access on 2 October 2026, is a theoretical and computational study of a mathematical model rather than an experimental measurement of viral dynamics, and its conclusions apply to the analyzed model class. The authors note that the version shared at publication is a peer-reviewed accepted manuscript subject to further edits before the final Version of Record.

Subject of Research: Mathematical modeling of how distributed intracellular production delays affect stability in a diffusive virus-immune model.

Article Title: Stability effects of distributed intracellular memory in a diffusive virus-immune model

Article References: Biomy, M., Guerrini, L., Abo-Bakr, A. M. A., & Ragni, S. (2026). Stability effects of distributed intracellular memory in a diffusive virus-immune model. Boundary Value Problems. https://doi.org/10.1186/s13661-026-02368-9

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DOI: 10.1186/s13661-026-02368-9

Keywords: virus-immune model, intracellular memory, gamma distribution, delay differential equations, reaction-diffusion, Hopf bifurcation, Routh-Hurwitz criteria, linear-chain technique, stability analysis, eclipse phase, spectral abscissa, mathematical biology