Threshold dynamics in a within-host infection model with Crowley–Martin functional response considering periodic effects

Ibrahim Nali; Attila Dénes; Ibrahim Nali; Attila Dénes

doi:10.3934/math.2026197

AIMS Mathematics

2026, Volume 11, Issue 2: 4818-4836. doi: 10.3934/math.2026197

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Threshold dynamics in a within-host infection model with Crowley–Martin functional response considering periodic effects

Ibrahim Nali ^{1
,
,},
Attila Dénes ^1,2

1.
Bolyai Institute, University of Szeged, Aradi vértanúk tere 1., H-6720 Szeged, Hungary
2.
National Laboratory for Health Security, Hungary

Received: 07 November 2025 Revised: 07 January 2026 Accepted: 09 January 2026 Published: 27 February 2026
MSC : 34D05, 92D30

In this work, we investigate the threshold dynamics of a within host viral infection model that incorporates a Crowley–Martin functional response together with periodically varying parameters. For this nonautonomous system, we establish the essential analytic properties, namely existence and uniqueness of solutions, positivity, and uniform boundedness of the periodic trajectories. A key component of the analysis is the basic reproduction number $ \mathcal{R}_{0} $, formulated as the spectral radius of an associated integral operator. This quantity acts as the decisive threshold governing the global behavior of the system: if $ \mathcal{R}_{0} < 1 $, all solutions converge to the virus free $ \omega $ periodic orbit, whereas $ \mathcal{R}_{0} > 1 $ guarantees uniform persistence and the existence of at least one strictly positive periodic solution. Numerical experiments are provided to illustrate these threshold driven transitions and to complement the theoretical findings.
- within-host virus model,
- periodic effects,
- stability analysis,
- Crowley–Martin functional response
Citation: Ibrahim Nali, Attila Dénes. Threshold dynamics in a within-host infection model with Crowley–Martin functional response considering periodic effects[J]. AIMS Mathematics, 2026, 11(2): 4818-4836. doi: 10.3934/math.2026197

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Abstract

In this work, we investigate the threshold dynamics of a within host viral infection model that incorporates a Crowley–Martin functional response together with periodically varying parameters. For this nonautonomous system, we establish the essential analytic properties, namely existence and uniqueness of solutions, positivity, and uniform boundedness of the periodic trajectories. A key component of the analysis is the basic reproduction number $ \mathcal{R}_{0} $, formulated as the spectral radius of an associated integral operator. This quantity acts as the decisive threshold governing the global behavior of the system: if $ \mathcal{R}_{0} < 1 $, all solutions converge to the virus free $ \omega $ periodic orbit, whereas $ \mathcal{R}_{0} > 1 $ guarantees uniform persistence and the existence of at least one strictly positive periodic solution. Numerical experiments are provided to illustrate these threshold driven transitions and to complement the theoretical findings.

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