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Global dynamics analysis of a time-delayed dynamic model of Kawasaki disease pathogenesis

  • * Corresponding author: Wanbiao Ma

    * Corresponding author: Wanbiao Ma 

The authors were supported by Beijing Natural Science Foundation (No.1202019) and National Natural Science Foundation of China (No.11971055)

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  • Kawasaki disease (KD) is an acute febrile vasculitis that occurs predominantly in infants and young children. With coronary artery abnormalities (CAAs) as its most serious complications, KD has become the leading cause of acquired heart disease in developed countries. Based on some new biological findings, we propose a time-delayed dynamic model of KD pathogenesis. This model exhibits forward$ / $backward bifurcation. By analyzing the characteristic equations, we completely investigate the local stability of the inflammatory factors-free equilibrium and the inflammatory factors-existent equilibria. Our results show that the time delay does not affect the local stability of the inflammatory factors-free equilibrium. However, the time delay as the bifurcation parameter may change the local stability of the inflammatory factors-existent equilibrium, and stability switches as well as Hopf bifurcation may occur within certain parameter ranges. Further, by skillfully constructing Lyapunov functionals and combining Barbalat's lemma and Lyapunov-LaSalle invariance principle, we establish some sufficient conditions for the global stability of the inflammatory factors-free equilibrium and the inflammatory factors-existent equilibrium. Moreover, it is shown that the model is uniformly persistent if the basic reproduction number is greater than one, and some explicit analytic expressions of eventual lower bounds of the solutions of the model are given by analyzing the properties of the solutions and the range of time delay very precisely. Finally, some numerical simulations are carried out to illustrate the theoretical results.

    Mathematics Subject Classification: Primary: 34K20, 34K18; Secondary: 92B05.

    Citation:

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  • Figure 1.  The phase trajectory and solution curves of model (2) with the initial value (4.74, 0.732, 0.79, 1.236) and $ \tau = 2.8\in[0, \tau_{1}^{(0)}) $. Here the inflammatory factors-existent equilibrium $ Q^{*} $ is locally asymptotically stable

    Figure 2.  The phase trajectory and solution curves of model (2) with the initial value (4.74, 0.732, 0.79, 1.236) and $ \tau = 7.1\in(\tau_{1}^{(0)}, \tau_{2}^{(0)}) $. Here the inflammatory factors-existent equilibrium $ Q^{*} $ is unstable and periodic oscillations occur

    Figure 3.  The phase trajectory and solution curves of model (2) with the initial value (4.74, 0.732, 0.79, 1.236) and $ \tau = 13.6\in(\tau_{2}^{(0)}, \tau_{1}^{(1)}) $. Here the inflammatory factors-existent equilibrium $ Q^{*} $ is locally asymptotically stable

    Figure 4.  The phase trajectory and solution curves of model (2) with the initial value (4.74, 0.732, 0.79, 1.236) and $ \tau = 16\in(\tau_{1}^{(1)}, +\infty) $. Here the inflammatory factors-existent equilibrium $ Q^{*} $ is unstable and periodic oscillations occur. Figure 4 (e) is a partial enlarged view of Figure 4 (a) near the inflammatory factors-existent equilibrium $ Q^* $

    Table 1.  Biological meanings of the parameters in model (1) [28]

    Parameters Biological meanings
    $ \; r $ Proliferation rate of normal endothelial cells (pg$ ^{-1} $ml$ ^{-1} $day$ ^{-1} $)
    $ \; d_{1}\; $ Apoptosis rate of normal endothelial cells (day$ ^{-1} $)
    $ \; d_{2}\; $ Hydrolytic rate of endothelial growth factors (day$ ^{-1} $)
    $ \; d_{3}\; $ Hydrolytic rate of activated adhesion factors$ / $chemokines (day$ ^{-1} $)
    $ \; d_{4}\; $ Hydrolytic rate of inflammatory factors (day$ ^{-1} $)
    $ \; k_{1}\; $ The rate of injury of endothelial cells caused by inflammatory factors(pg$ ^{-1} $ml$ ^{-1} $day$ ^{-1} $)
    $ \; k_{2}\; $ Production rate of endothelial growth factors caused by inflammatory factors (pg$ ^{-1} $ml$ ^{-1} $day$ ^{-1} $)
    $ \; k_{3}\; $ Production rate of activated adhesion factors$ / $chemokines caused by inflammatory factors (pg$ ^{-1} $ml$ ^{-1} $day$ ^{-1} $)
    $ \; k_{4}\; $ Production rate of activated adhesion factors$ / $chemokines caused by endothelial growth factors (day$ ^{-1} $)
    $ \; k_{5}\; $ Production rate of inflammatory factors by increasing of abnormally activated immune cells (day$ ^{-1} $)
    $ \; k_{6}\; $ Proliferation rate of endothelial cells promoted by endothelial growth factors (day$ ^{-1} $)
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