Kayo–Kengne–Akgül fractional multi-delay model for avian influenza: Hopf bifurcation and optimal control analysis

Saeed M. Alamry; Saeed M. Alamry

doi:10.3934/math.20261342

AIMS Mathematics

2026, Volume 11, Issue 10: 34369-34416. doi: 10.3934/math.20261342

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Research article

Kayo–Kengne–Akgül fractional multi-delay model for avian influenza: Hopf bifurcation and optimal control analysis

Saeed M. Alamry ^,

Department of Mathematics and Statistics, College of Science, Taif University, P. O. Box 11099, Taif 21944, Saudi Arabia

Received: 09 July 2026 Revised: 30 September 2026 Accepted: 08 October 2026 Published: 22 October 2026
MSC : 34D20, 34K20, 34K60, 92C60, 92D45

This work develops a Kayo–Kengne–Akgül (KKA) fractional multi-delay framework for modeling avian-influenza transmission. The proposed formulation considers the interaction between bird and human populations while accounting for two biologically meaningful delays: an avian infectious-pressure delay and a human transmission-response delay. By incorporating the KKA fractional derivative, the model is able to represent memory-dependent effects that influence the spread of infection. The well-posedness of the system is verified by proving positivity and boundedness of solutions. The disease-free and endemic steady states are obtained, and the threshold quantity governing disease persistence is computed through the next-generation matrix method. Because the equilibria do not explicitly depend on time, their algebraic forms remain consistent with those of the corresponding classical delayed model. Nevertheless, the fractional operator alters the stability criteria and the Hopf bifurcation analysis through the associated KKA characteristic equation. In addition, an optimal-control formulation is presented by introducing avian biosecurity and contact-reduction measures, infected-bird culling, and awareness-based intervention as control measures. The theoretical findings are supported using a KKA-adapted Adams–Bashforth numerical method. Finally, numerical experiments demonstrate how the fractional order, delay parameters, and control interventions affect the evolution of avian influenza.
- avian influenza,
- Kayo–Kengne–Akgül derivative,
- fractional-order model,
- time delay,
- Hopf bifurcation,
- optimal control,
- Adams–Bashforth scheme
Citation: Saeed M. Alamry. Kayo–Kengne–Akgül fractional multi-delay model for avian influenza: Hopf bifurcation and optimal control analysis[J]. AIMS Mathematics, 2026, 11(10): 34369-34416. doi: 10.3934/math.20261342

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Abstract

This work develops a Kayo–Kengne–Akgül (KKA) fractional multi-delay framework for modeling avian-influenza transmission. The proposed formulation considers the interaction between bird and human populations while accounting for two biologically meaningful delays: an avian infectious-pressure delay and a human transmission-response delay. By incorporating the KKA fractional derivative, the model is able to represent memory-dependent effects that influence the spread of infection. The well-posedness of the system is verified by proving positivity and boundedness of solutions. The disease-free and endemic steady states are obtained, and the threshold quantity governing disease persistence is computed through the next-generation matrix method. Because the equilibria do not explicitly depend on time, their algebraic forms remain consistent with those of the corresponding classical delayed model. Nevertheless, the fractional operator alters the stability criteria and the Hopf bifurcation analysis through the associated KKA characteristic equation. In addition, an optimal-control formulation is presented by introducing avian biosecurity and contact-reduction measures, infected-bird culling, and awareness-based intervention as control measures. The theoretical findings are supported using a KKA-adapted Adams–Bashforth numerical method. Finally, numerical experiments demonstrate how the fractional order, delay parameters, and control interventions affect the evolution of avian influenza.

References

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