Avian influenza persists as an infectious threat across wild and domestic bird populations. This paper develops deterministic SEIR models to study avian influenza transmission in a bird population. We first formulate a non-delayed model with a general saturated incidence function and derive the basic reproduction number ($ \mathcal{R}_0 $). The analysis shows that the disease-free equilibrium is locally and globally asymptotically stable when $ \mathcal{R}_0\leq 1 $. When $ \mathcal{R}_0 > 1 $, a unique endemic equilibrium exists and is locally and globally asymptotically stable. The model is then extended by including three distributed-time delays that describe the delayed transition into the exposed, infectious, and recovered classes. This delayed formulation reflects variability in latency, infectious progression, and recovery among infected birds. The delayed model is shown to be mathematically well posed through the non-negativity and boundedness of its solutions. The delayed reproduction number ($ \mathcal{R}_d $) is derived, and suitable Lyapunov functionals are constructed to establish the global threshold dynamics. In particular, the disease-free equilibrium is globally asymptotically stable when $ \mathcal{R}_d\leq 1 $, whereas the endemic equilibrium is globally asymptotically stable when $ \mathcal{R}_d > 1 $. Numerical simulations support the theoretical results and show how distributed delays affect the persistence of avian influenza in the bird population. Sensitivity analysis identifies the parameters that have the strongest influence on $ \mathcal{R}_d $. In addition, the control simulations indicate that sufficiently strong transmission-blocking measures can reduce $ \mathcal{R}_d $ and shift the system from endemic persistence to disease elimination.
Citation: Mohammed H. Alharbi. Threshold dynamics of a distributed-delay SEIR model for avian influenza transmission in bird populations[J]. AIMS Mathematics, 2026, 11(10): 35631-35654. doi: 10.3934/math.20261382
Avian influenza persists as an infectious threat across wild and domestic bird populations. This paper develops deterministic SEIR models to study avian influenza transmission in a bird population. We first formulate a non-delayed model with a general saturated incidence function and derive the basic reproduction number ($ \mathcal{R}_0 $). The analysis shows that the disease-free equilibrium is locally and globally asymptotically stable when $ \mathcal{R}_0\leq 1 $. When $ \mathcal{R}_0 > 1 $, a unique endemic equilibrium exists and is locally and globally asymptotically stable. The model is then extended by including three distributed-time delays that describe the delayed transition into the exposed, infectious, and recovered classes. This delayed formulation reflects variability in latency, infectious progression, and recovery among infected birds. The delayed model is shown to be mathematically well posed through the non-negativity and boundedness of its solutions. The delayed reproduction number ($ \mathcal{R}_d $) is derived, and suitable Lyapunov functionals are constructed to establish the global threshold dynamics. In particular, the disease-free equilibrium is globally asymptotically stable when $ \mathcal{R}_d\leq 1 $, whereas the endemic equilibrium is globally asymptotically stable when $ \mathcal{R}_d > 1 $. Numerical simulations support the theoretical results and show how distributed delays affect the persistence of avian influenza in the bird population. Sensitivity analysis identifies the parameters that have the strongest influence on $ \mathcal{R}_d $. In addition, the control simulations indicate that sufficiently strong transmission-blocking measures can reduce $ \mathcal{R}_d $ and shift the system from endemic persistence to disease elimination.
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Figures(9) / Tables(5)
Mohammed H. Alharbi. Threshold dynamics of a distributed-delay SEIR model for avian influenza transmission in bird populations[J]. AIMS Mathematics, 2026, 11(10): 35631-35654. doi: 10.3934/math.20261382
Flow diagram of the avian influenza transmission model in a bird population
Numerical solutions of the non-delayed avian influenza model (3.1) for different initial bird populations when $ \mathcal{R}_0 = 0.859 < 1 $
Numerical solutions of the non-delayed avian influenza model (3.1) for different initial bird populations when $ \mathcal{R}_0 = 11.4583 > 1 $
Numerical solutions of the delayed avian influenza model (5.1) for different initial conditions when $ \mathcal{R}_d = 0.704 < 1 $
Numerical solutions of the delayed avian influenza model (5.1) for different initial conditions when $ \mathcal{R}_d = 9.381 > 1 $
Local sensitivity indices and PRCC values of $ \mathcal{R}_d $ with respect to the main parameters of the delayed avian influenza model
Effect of infection-blocking control efficacy $ p $ on the four bird classes in the delayed avian influenza model
Bifurcation diagram of the infectious class $ I^* $ at equilibrium as a function of the total delay $ \tau = \tau_1 + \tau_2 $, with $ \tau_1 = \tau_2 = \tau/2 $
Variation of the delayed basic reproduction number $ \mathcal{R}_d $ with respect to time delays. (a) $ \mathcal{R}_d $ as a function of $ \tau_1 $ for different values of $ n_1 $, with $ \tau_2 = 0.1 $ and $ n_2 = 1 $. (b) $ \mathcal{R}_d $ as a function of $ \tau_2 $ for different values of $ n_2 $, with $ \tau_1 = 0.1 $ and $ n_1 = 1 $. (c) Contour plot of $ \mathcal{R}_d $ as a function of both $ \tau_1 $ and $ \tau_2 $ with $ n_1 = n_2 = 1 $
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