Global Dynamics of a Within‐Host Model for Immune Response With a Generic Distributed Delay

• Published in: Mathematical methods in the applied sciences Vol. 48; no. 12; pp. 12186 - 12206
• Main Author: Al‐Darabsah, Isam
• Format: Journal Article
• Language: English
• Published: Freiburg Wiley Subscription Services, Inc 01.08.2025
• Subjects: COVID-19; distributed delay; Equilibrium; global dynamics; Health services; Immune response; Immune system; Infections; Lymphocytes; Qualitative analysis; Replication; Sensitivity analysis; Statistical methods; Time lag; Viruses; within‐host modeling
• ISSN: 0170-4214; 1099-1476
• DOI: 10.1002/mma.11021

ABSTRACT Epidemics caused by infectious agents, including viruses, are a major threat to humanity, and understanding the underlying dynamics is essential for developing effective strategies to mitigate their impact. Elucidating viral behavior, like the absence of detectable infectious viruses during the eclipse phase at the early stages of infection, highlights a key aspect of viral replication that is crucial for preventing and managing infectious diseases. This paper introduces a mathematical model of the immune response incorporating a delay distribution that accounts for the eclipse phase, the period between viral entry into a host cell, and the activation of immune cells for viral production. We perform a qualitative analysis of the model dynamics under a general delay distribution kernel with respect to the basic reproduction number. Specifically, we investigate the existence of a positive infection equilibrium, the global stability of the virus‐free equilibrium, the global attractivity of the infection equilibrium, and the persistence of infection. To understand the influence of time delay, we calibrate the model to CD8 +$$ {}^{+} $$ T cell (immune cells) clinical data on coronavirus disease 2019 (COVID‐19) patients from the literature using a variety of delay distributions: Dirac delta, uniform, and gamma. The uniform distribution seems to be the most appropriate model based on different statistical methods, such as the Akaike information criterion (AIC). Then, we conduct a global sensitivity analysis considering the three distributions to identify potential strategies for reducing the viral load when the infection cannot be eradicated. We find that the considered parameters affect the steady state of the viral load to varying extents, depending on the delay distribution. Finally, we modify the model into a treatment model that integrates two treatment strategies: antiviral and immune‐modulating treatments, to target viral replication and modulate the immune response. Using computational analysis, we study the impact of these interventions on viral load and immune cell density.