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

Ke Guo; Wanbiao Ma; Rong Qiang

doi:10.3934/dcdsb.2021136

Article Contents

2022, Volume 27, Issue 4: 2367-2400. Doi: 10.3934/dcdsb.2021136

This issue Previous Article On a generalized diffusion problem: A complex network approach Next Article Homogenization for stochastic reaction-diffusion equations with singular perturbation term

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

School of Mathematics and Physics, University of Science and Technology Beijing, Beijing100083, China

^* Corresponding author: Wanbiao Ma
^* Corresponding author: Wanbiao Ma

Received: April 30, 2020

Revised: December 31, 2020

Early access: May 2021

Published: April 2022

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

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Abstract

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.

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

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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

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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

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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

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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^* $

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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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