On the global stability of the discrete-time epidemic models: A new approach

Omaima Slimani; Bouchra Aylaj; Saber Elaydi; Omaima Slimani; Bouchra Aylaj; Saber Elaydi

doi:10.3934/mbe.2026027

Mathematical Biosciences and Engineering

2026, Volume 23, Issue 3: 702-721. doi: 10.3934/mbe.2026027

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On the global stability of the discrete-time epidemic models: A new approach

1.
Department of Mathematics and Computer Science, Faculty of Sciences Ain Chock, Hassan Ⅱ University of Casablanca, Casablanca, Morocco
2.
Department of Mathematics, Trinity University, San Antonio, USA

Received: 05 September 2025 Revised: 20 October 2025 Accepted: 03 November 2025 Published: 29 January 2026

We developed a unified analytical framework for the global dynamics of discrete-time susceptible infectious susceptible (SIS) epidemic models with nonlinear recruitment. Emphasis was placed on demographic feedback through Beverton-Holt and Ricker-type recruitment, which regulates host population size and thereby shapes transmission and long-term persistence (Persistence allows population densities to approach zero asymptotically, wheras uniform persistence requires them to remain bounded away from zero). Under minimal assumptions, we reduced non-autonomous systems to appropriately defined autonomous limiting systems and used this reduction to obtain a complete global threshold characterization: When the basic reproduction number $ R_{0} > 1 $, the endemic equilibrium existed and was globally asymptotically stable; when $ R_{0}\le 1 $, solutions converged to the disease-free state. The approach extended to periodically forced SIS models, which showed that the threshold and stability conclusions persisted in the periodic non-autonomous setting. The results unified and strengthened prior work and clarify how recruitment dynamics govern persistence in discrete-time epidemic systems.
Citation: Omaima Slimani, Bouchra Aylaj, Saber Elaydi. On the global stability of the discrete-time epidemic models: A new approach[J]. Mathematical Biosciences and Engineering, 2026, 23(3): 702-721. doi: 10.3934/mbe.2026027

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Abstract

We developed a unified analytical framework for the global dynamics of discrete-time susceptible infectious susceptible (SIS) epidemic models with nonlinear recruitment. Emphasis was placed on demographic feedback through Beverton-Holt and Ricker-type recruitment, which regulates host population size and thereby shapes transmission and long-term persistence (Persistence allows population densities to approach zero asymptotically, wheras uniform persistence requires them to remain bounded away from zero). Under minimal assumptions, we reduced non-autonomous systems to appropriately defined autonomous limiting systems and used this reduction to obtain a complete global threshold characterization: When the basic reproduction number $ R_{0} > 1 $, the endemic equilibrium existed and was globally asymptotically stable; when $ R_{0}\le 1 $, solutions converged to the disease-free state. The approach extended to periodically forced SIS models, which showed that the threshold and stability conclusions persisted in the periodic non-autonomous setting. The results unified and strengthened prior work and clarify how recruitment dynamics govern persistence in discrete-time epidemic systems.

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