Delayed immune responses and heterogeneous exposure shape within-host viral dynamics

Abdelkarim Lamghari; Aissam Jebrane; Abdelkarim Lamghari; Aissam Jebrane

doi:10.3934/mbe.2026066

Mathematical Biosciences and Engineering

2026, Volume 23, Issue 6: 1799-1843. doi: 10.3934/mbe.2026066

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Research article

Delayed immune responses and heterogeneous exposure shape within-host viral dynamics

Abdelkarim Lamghari ^{1
,
,},
Aissam Jebrane ²

1.
LAMAI, Faculty of Sciences and Technics, Department of Mathematics, Cadi Ayyad University, Marrakesh 40140, Morocco
2.
École Centrale Casablanca, Complex Systems and Interactions Research Center (CSIRC), Ville Verte, Bouskoura 27182, Morocco

Received: 13 April 2026 Revised: 02 May 2026 Accepted: 12 May 2026 Published: 22 May 2026

In this paper, we present a delayed within-host viral dynamics model that integrates nonlinear immune responses and heterogeneous exposure. The model combines logistic viral replication, delayed immune-mediated clearance, and time-dependent external forcing. We introduce a $ 4 \times 4 $ factorial framework that systematically couples four canonical immune response architectures (linear, Michaelis–Menten, Hill-type, and switch-like) with four representative exposure profiles (constant, periodic, event-driven, and adaptive), yielding sixteen distinct dynamical scenarios. A rigorous theoretical analysis established positivity, boundedness, equilibrium structure, and delay-induced Hopf bifurcations. Extensive numerical simulations quantified transient amplification, timing, clearance efficiency, oscillatory behavior, and long-term persistence using a comprehensive set of nine quantitative metrics. The results reveal robust dynamical hierarchies governed by immune sensitivity and exposure intermittence rather than maximal clearance capacity. In particular, ultrasensitive immune responses amplified early overshoots and cumulative burden, whereas impulsive or adaptive exposure substantially reduced persistence and accelerated post-peak decay. These findings revealed fundamental trade-offs between early transient control and long-term viral persistence in delayed immune feedback systems.
- within-host viral dynamics,
- immune delay,
- nonlinear immune response,
- external forcing,
- delay differential equations,
- Hopf bifurcation
Citation: Abdelkarim Lamghari, Aissam Jebrane. Delayed immune responses and heterogeneous exposure shape within-host viral dynamics[J]. Mathematical Biosciences and Engineering, 2026, 23(6): 1799-1843. doi: 10.3934/mbe.2026066

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Abstract

In this paper, we present a delayed within-host viral dynamics model that integrates nonlinear immune responses and heterogeneous exposure. The model combines logistic viral replication, delayed immune-mediated clearance, and time-dependent external forcing. We introduce a $ 4 \times 4 $ factorial framework that systematically couples four canonical immune response architectures (linear, Michaelis–Menten, Hill-type, and switch-like) with four representative exposure profiles (constant, periodic, event-driven, and adaptive), yielding sixteen distinct dynamical scenarios. A rigorous theoretical analysis established positivity, boundedness, equilibrium structure, and delay-induced Hopf bifurcations. Extensive numerical simulations quantified transient amplification, timing, clearance efficiency, oscillatory behavior, and long-term persistence using a comprehensive set of nine quantitative metrics. The results reveal robust dynamical hierarchies governed by immune sensitivity and exposure intermittence rather than maximal clearance capacity. In particular, ultrasensitive immune responses amplified early overshoots and cumulative burden, whereas impulsive or adaptive exposure substantially reduced persistence and accelerated post-peak decay. These findings revealed fundamental trade-offs between early transient control and long-term viral persistence in delayed immune feedback systems.

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