Beyond the Threshold: How a Distributed Eclipse-Phase Delay Reshapes Dynamics and Bifurcations in a Within-Host Virus–Interferon Model
- 1Department of Mathematics and Statistics, Faculty of Science and Arts, Jordan University of Science and Technology, P.O. Box 3030, Irbid 22110, Jordan
- 2Mechanical and Industrial Engineering Department, Al-Ahliyya Amman University, Amman 19328, Jordan
Abstract
In this paper, we propose a within-host virus–interferon model with a distributed eclipse phase. We assume that exposed target cells pass through an eclipse phase with a distributed duration and are subject to survival during this period before becoming either productively infected cells or interferon-protected refractory cells. The proposed model is formulated as a functional differential equation with infinite delay, and we first establish its well-posedness in an exponentially weighted fading-memory space. The model admits a continuum of infection-free equilibria. We associate a reproduction number R0(c) with each infection-free equilibrium and identify a critical susceptible fraction ccri that divides the continuum into transversely stable and unstable segments. The continuum, and with it the point-dependent invasion thresholds, exists because refractory cells do not revert to susceptibility. A positive waning rate removes this degeneracy, and the results based on the continuum then describe the dynamics on a timescale inversely proportional to that rate. We show that R0(1)>1 does not necessarily guarantee infection invasion from all infection-positive initial states. In particular, uniform persistence fails on the whole infection-positive state space for all parameter values, and clearance and persistence basins coexist whenever a locally asymptotically stable endemic equilibrium exists. We prove that, under a nondegeneracy condition, such an equilibrium exists near the invasion threshold when interferon protection is inactive at baseline. We also show that the eclipse distribution affects the equilibria and invasion thresholds only through its mean survival S, while the stability depends on the full Laplace transform of the distribution. Thus, concentrating the eclipse phase while keeping S fixed can destabilize the endemic equilibrium and lead to oscillatory dynamics. Moreover, we investigate the effect of the curvature of the incidence function on the local direction of the endemic bifurcation. We show that a concave incidence function leads to a forward branch, while a sufficiently convex incidence function can generate a backward branch and bistability below the threshold. Finally, numerical computations using three complementary schemes are provided to cross-validate the simulations and stability calculations and confirm the theoretical results.
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BibTeX
@article{aldarabsah2026beyond,
title = {{Beyond the Threshold: How a Distributed Eclipse-Phase Delay Reshapes Dynamics and Bifurcations in a Within-Host Virus–Interferon Model}},
author = {Isam Al-Darabsah and Hussam Alothman},
journal = {Mathematics},
year = {2026},
month = oct,
doi = {10.3390/math14203756},
url = {https://www.mdpi.com/2227-7390/14/20/3756},
note = {CC BY 4.0},
}
RIS
TY - JOUR
TI - Beyond the Threshold: How a Distributed Eclipse-Phase Delay Reshapes Dynamics and Bifurcations in a Within-Host Virus–Interferon Model
AU - Isam Al-Darabsah
AU - Hussam Alothman
PY - 2026
DA - 2026/10/16
JO - Mathematics
DO - 10.3390/math14203756
UR - https://www.mdpi.com/2227-7390/14/20/3756
AB - In this paper, we propose a within-host virus–interferon model with a distributed eclipse phase. We assume that exposed target cells pass through an eclipse phase with a distributed duration and are subject to survival during this period before becoming either productively infected cells or interferon-protected refractory cells. The proposed model is formulated as a functional differential equation with infinite delay, and we first establish its well-posedness in an exponentially weighted fading-memory space. The model admits a continuum of infection-free equilibria. We associate a reproduction number R0(c) with each infection-free equilibrium and identify a critical susceptible fraction ccri that divides the continuum into transversely stable and unstable segments. The continuum, and with it the point-dependent invasion thresholds, exists because refractory cells do not revert to susceptibility. A positive waning rate removes this degeneracy, and the results based on the continuum then describe the dynamics on a timescale inversely proportional to that rate. We show that R0(1)>1 does not necessarily guarantee infection invasion from all infection-positive initial states. In particular, uniform persistence fails on the whole infection-positive state space for all parameter values, and clearance and persistence basins coexist whenever a locally asymptotically stable endemic equilibrium exists. We prove that, under a nondegeneracy condition, such an equilibrium exists near the invasion threshold when interferon protection is inactive at baseline. We also show that the eclipse distribution affects the equilibria and invasion thresholds only through its mean survival S, while the stability depends on the full Laplace transform of the distribution. Thus, concentrating the eclipse phase while keeping S fixed can destabilize the endemic equilibrium and lead to oscillatory dynamics. Moreover, we investigate the effect of the curvature of the incidence function on the local direction of the endemic bifurcation. We show that a concave incidence function leads to a forward branch, while a sufficiently convex incidence function can generate a backward branch and bistability below the threshold. Finally, numerical computations using three complementary schemes are provided to cross-validate the simulations and stability calculations and confirm the theoretical results.
SN - 2227-7390
ER -