doi: 10.3934/dcdsb.2021136
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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

Received  May 2020 Revised  January 2021 Early access May 2021

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

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.

Citation: Ke Guo, Wanbiao Ma, Rong Qiang. Global dynamics analysis of a time-delayed dynamic model of Kawasaki disease pathogenesis. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021136
References:
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M. AyusawaT. SonobeS. UemuraS. OgawaY. NakamuraN. KiyosawaM. Ishii and K. Harada, Revision of diagnostic guidelines for Kawasaki disease (the 5th revised edition), Pediatr. Int., 47 (2005), 232-234.  doi: 10.1111/j.1442-200x.2005.02033.x.  Google Scholar

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S. ChenC. Cheng and Y. Takeuchi, Stability analysis in delayed within-host viral dynamics with both viral and cellular infections, J. Math. Anal. Appl., 442 (2016), 642-672.  doi: 10.1016/j.jmaa.2016.05.003.  Google Scholar

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H. DahariA. LoR. M. Ribeiro and A. S. Perelson, Modeling hepatitis C virus dynamics: Liver regeneration and critical drug efficacy, J. Theoret. Biol., 247 (2007), 371-381.  doi: 10.1016/j.jtbi.2007.03.006.  Google Scholar

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Y. EnatsuY. Nakata and Y. Muroya, Lyapunov functional techniques for the global stability analysis of a delayed SIRS epidemic model, Nonlinear. Anal. Real World Appl., 13 (2012), 2120-2133.  doi: 10.1016/j.nonrwa.2012.01.007.  Google Scholar

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C. GaleottiS. V. KaveriR. CimazI. Koné-Paut and J. Bayry, Predisposing factors, pathogenesis and therapeutic intervention of Kawasaki disease, Drug. Discov. Today., 21 (2016), 1850-1857.  doi: 10.1016/j.drudis.2016.08.004.  Google Scholar

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K. Guo, W. Ma and R. Qiang, On global stability of the equilibria of an ordinary differential equation model of Kawasaki disease pathogenesis, Appl. Math. Lett., 106 (2020), 106319, 10pp. doi: 10.1016/j.aml.2020.106319.  Google Scholar

[10]

S. Guo and W. Ma, Global behavior of delay differential equations model of HIV infection with apoptosis, Discrete Contin. Dyn. Syst. Ser. B, 21 (2016), 103-119.  doi: 10.3934/dcdsb.2016.21.103.  Google Scholar

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F. Jiao, A. K. Jindal, V. Pandiarajan, R. Khubchandani, N. Kamath, T. Sabui, R. Mondal, P. Pal and S. Singh, The emergence of Kawasaki disease in India and China, Glob. Cardiol. Sci. Pract., 2017 (2017), e201721. doi: 10.21542/gcsp.2017.21.  Google Scholar

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T. Kawasaki, Acute febrile mucocutaneous syndrome with lymphoid involvement with specific desquamation of the fingers and toes in children, Arerugi, 16 (1967), 178–222 (in Japanese). Google Scholar

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A. KentsisA. ShulmanS. Ahmed and et al., Urine proteomics for discovery of improved diagnostic markers of Kawasaki disease, EMBO Mol. Med., 5 (2013), 210-220.  doi: 10.1002/emmm.201201494.  Google Scholar

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A. J. Kucharski, S. Funk, R. M. Eggo, H. P. Mallet, W. J. Edmunds and E. J. Nilles, Transmission dynamics of Zika virus in island populations: a modelling analysis of the 2013-14 French Polynesia outbreak, PLoS. Negl. Trop. Dis., 10 (2016), e0004726. doi: 10.1371/journal.pntd.0004726.  Google Scholar

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K. KudoS. HasegawaY. SuzukiR. HiranoH. WakiguchiS. Kittaka and T. Ichiyama, $1\alpha$, $25$-Dihydroxyvitamin $D_3$ inhibits vascular cellular adhesion molecule-1 expression and interleukin-8 production in human coronary arterial endothelial cells, J. Steroid Biochem. Mol. Biol., 132 (2012), 290-294.  doi: 10.1016/j.jsbmb.2012.07.003.  Google Scholar

[20]

D. Li and W. Ma, Asymptotic properties of a HIV-1 infection model with time delay, J. Math. Anal. Appl., 335 (2007), 683-691.  doi: 10.1016/j.jmaa.2007.02.006.  Google Scholar

[21]

C. Y. LinC. C. LinB. Hwang and B. Chiang, Serial changes of serum interleukin-6, interleukin-8, and tumor necrosis factor alpha among patients with Kawasaki disease, J. Pediatr., 121 (1992), 924-926.  doi: 10.1016/S0022-3476(05)80343-9.  Google Scholar

[22]

N. MakinoY. NakamuraM. YashiroR. AeS. TsuboiY. AoyamaT. KojoR. UeharaK. Kotani and H. Yanagawa, Descriptive epidemiology of Kawasaki disease in Japan, 2011-2012: from the results of the 22nd nationwide survey, J. Epidemiol., 25 (2015), 239-245.  doi: 10.2188/jea.JE20140089.  Google Scholar

[23]

B. W. McCrindle, A. H. Rowley, J. W. Newburger, et al., Diagnosis, treatment, and long-term management of Kawasaki disease: a scientific statement for health professionals from the American Heart Association, Circulation, 135 (2017), e927–e999. doi: 10.1161/CIR.0000000000000484.  Google Scholar

[24]

A. U. NeumannN. P. LamH. DahariD. R. GretchT. E. WileyT. J. Layden and A. S. Perelson, Hepatitis C viral dynamics in vivo and the antiviral efficacy of interferon-$\alpha$ therapy, Science, 282 (1998), 103-107.  doi: 10.1126/science.282.5386.103.  Google Scholar

[25]

J. W. NewburgerM. TakahashiM. A. Gerber and et al., Diagnosis, treatment, and long-term management of Kawasaki disease: A statement for health professionals from the Committee on Rheumatic Fever, Endocarditis and Kawasaki Disease, Council on Cardiovascular Disease in the Young, American Heart Association, Circulation, 110 (2004), 2747-2771.  doi: 10.1161/01.CIR.0000145143.19711.78.  Google Scholar

[26]

M. A. Nowak and C. R. M. Bangham, Population dynamics of immune responses to persistent virus, Science, 272 (5258), 74-79.  doi: 10.1126/science.272.5258.74.  Google Scholar

[27]

A. S. Perelson and P. W. Nelson, Mathematical analysis of HIV-1 dynamics in vivo, SIAM Rev., 41 (1999), 3-44.  doi: 10.1137/S0036144598335107.  Google Scholar

[28]

R. QiangW. MaK. Guo and H. Du, The differential equation model of pathogenesis of Kawasaki disease with theoretical analysis, Math. Biosci. Eng., 16 (2019), 3488-3511.  doi: 10.3934/mbe.2019175.  Google Scholar

[29]

C. M. Saad-RoyJ. Ma and P. van den Driessche, The effect of sexual transmission on Zika virus dynamics, J. Math. Biol., 77 (2018), 1917-1941.  doi: 10.1007/s00285-018-1230-1.  Google Scholar

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J. J. E. Slotine and W. Li, Applied Nonlinear Control, Prentice hall, Englewood Cliffs, 1991. Google Scholar

[31]

J. Tam, Delay effect in a model for virus replication, IMA J. Math. Appl. Med. Biol., 16 (1999), 29-37.  doi: 10.1093/imammb/16.1.29.  Google Scholar

[32]

M. TeraiK. YasukawaS. NarumotoS. TatenoS. Oana and Y. Kohno, Vascular endothelial growth factor in acute Kawasaki disease, Am. J. Pediatr., 83 (1999), 337-339.  doi: 10.1016/S0002-9149(98)00864-9.  Google Scholar

[33]

R. Uehara and E. D. Belay, Epidemiology of Kawasaki Disease in Asia, Europe, and the United States, J. Epidemiol., 22 (2012), 79-85.  doi: 10.2188/jea.JE20110131.  Google Scholar

[34]

W. Wang, Global behavior of a SEIRS epidemic model with time delays, Appl. Math. Lett., 15 (2002), 423-428.  doi: 10.1016/S0893-9659(01)00153-7.  Google Scholar

[35]

X. WangS. Liu and L. Rong, Permanence and extinction of a non-autonomous HIV-1 model with time delays, Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), 1783-1800.  doi: 10.3934/dcdsb.2014.19.1783.  Google Scholar

[36]

M. XiaoL. MenM. XuG. WangH. Lv and C. Liu, Berberine protects endothelial progenitor cell from damage of TNF-$\alpha$ via the PI3K/AKT/eNOS signaling pathway, Eur. J. Pharmacol., 743 (2014), 11-16.  doi: 10.1016/j.ejphar.2014.09.024.  Google Scholar

[37]

R. S. M. Yeung, The etiology of Kawasaki disease: A superantigen-mediated process, Prog. Pediatr. Cardiol., 19 (2004), 115-122.  doi: 10.1016/j.ppedcard.2004.08.004.  Google Scholar

show all references

References:
[1]

M. AyusawaT. SonobeS. UemuraS. OgawaY. NakamuraN. KiyosawaM. Ishii and K. Harada, Revision of diagnostic guidelines for Kawasaki disease (the 5th revised edition), Pediatr. Int., 47 (2005), 232-234.  doi: 10.1111/j.1442-200x.2005.02033.x.  Google Scholar

[2]

E. j. Benjamin, S. S. Virani, C. W. Callaway, et al., Heart disease and stroke statistics-2018 update: A report from the American Heart Association, Circulation, 137 (2018), e67–e492. doi: 10.1161/CIR.0000000000000558.  Google Scholar

[3]

M. Cartwright and M. Husain, A model for the control of testosterone secretion, J.Theoret. Biol., 123 (1986), 239-250.  doi: 10.1016/S0022-5193(86)80158-8.  Google Scholar

[4]

R. V. Culshaw and S. Ruan, A delay-differential equation model of HIV infection of CD$4^{+}$ T-cells, Math. Biosci., 165 (2000), 27-39.  doi: 10.1016/S0025-5564(00)00006-7.  Google Scholar

[5]

S. ChenC. Cheng and Y. Takeuchi, Stability analysis in delayed within-host viral dynamics with both viral and cellular infections, J. Math. Anal. Appl., 442 (2016), 642-672.  doi: 10.1016/j.jmaa.2016.05.003.  Google Scholar

[6]

H. DahariA. LoR. M. Ribeiro and A. S. Perelson, Modeling hepatitis C virus dynamics: Liver regeneration and critical drug efficacy, J. Theoret. Biol., 247 (2007), 371-381.  doi: 10.1016/j.jtbi.2007.03.006.  Google Scholar

[7]

Y. EnatsuY. Nakata and Y. Muroya, Lyapunov functional techniques for the global stability analysis of a delayed SIRS epidemic model, Nonlinear. Anal. Real World Appl., 13 (2012), 2120-2133.  doi: 10.1016/j.nonrwa.2012.01.007.  Google Scholar

[8]

C. GaleottiS. V. KaveriR. CimazI. Koné-Paut and J. Bayry, Predisposing factors, pathogenesis and therapeutic intervention of Kawasaki disease, Drug. Discov. Today., 21 (2016), 1850-1857.  doi: 10.1016/j.drudis.2016.08.004.  Google Scholar

[9]

K. Guo, W. Ma and R. Qiang, On global stability of the equilibria of an ordinary differential equation model of Kawasaki disease pathogenesis, Appl. Math. Lett., 106 (2020), 106319, 10pp. doi: 10.1016/j.aml.2020.106319.  Google Scholar

[10]

S. Guo and W. Ma, Global behavior of delay differential equations model of HIV infection with apoptosis, Discrete Contin. Dyn. Syst. Ser. B, 21 (2016), 103-119.  doi: 10.3934/dcdsb.2016.21.103.  Google Scholar

[11]

J. K. Hale, Theory of Functional Differential Equations, 2$^{nd}$ edition, Springer-Verlag, New York, 1977. doi: 10.1007/978-1-4612-9892-2.  Google Scholar

[12]

J. S. Hui-YuenT. T. Duong and R. S. Yeung, TNF-$\alpha$ is necessary for induction of coronary artery inflammation and aneurysm formation in an animal model of Kawasaki disease, J. Immunol., 176 (2006), 6294-6301.  doi: 10.4049/jimmunol.176.10.6294.  Google Scholar

[13]

F. Jiao, A. K. Jindal, V. Pandiarajan, R. Khubchandani, N. Kamath, T. Sabui, R. Mondal, P. Pal and S. Singh, The emergence of Kawasaki disease in India and China, Glob. Cardiol. Sci. Pract., 2017 (2017), e201721. doi: 10.21542/gcsp.2017.21.  Google Scholar

[14]

T. Kawasaki, Acute febrile mucocutaneous syndrome with lymphoid involvement with specific desquamation of the fingers and toes in children, Arerugi, 16 (1967), 178–222 (in Japanese). Google Scholar

[15]

A. KentsisA. ShulmanS. Ahmed and et al., Urine proteomics for discovery of improved diagnostic markers of Kawasaki disease, EMBO Mol. Med., 5 (2013), 210-220.  doi: 10.1002/emmm.201201494.  Google Scholar

[16]

D. Kirschner and J. C. Panetta, Modelling immunotherapy of the tumor-immune interaction, J. Math. Biol., 37 (1998), 235-252.  doi: 10.1007/s002850050127.  Google Scholar

[17] Y. Kuang, Delay Differential Equations with Applications in Population Dynamics, Academic Press, Boston, 1993.   Google Scholar
[18]

A. J. Kucharski, S. Funk, R. M. Eggo, H. P. Mallet, W. J. Edmunds and E. J. Nilles, Transmission dynamics of Zika virus in island populations: a modelling analysis of the 2013-14 French Polynesia outbreak, PLoS. Negl. Trop. Dis., 10 (2016), e0004726. doi: 10.1371/journal.pntd.0004726.  Google Scholar

[19]

K. KudoS. HasegawaY. SuzukiR. HiranoH. WakiguchiS. Kittaka and T. Ichiyama, $1\alpha$, $25$-Dihydroxyvitamin $D_3$ inhibits vascular cellular adhesion molecule-1 expression and interleukin-8 production in human coronary arterial endothelial cells, J. Steroid Biochem. Mol. Biol., 132 (2012), 290-294.  doi: 10.1016/j.jsbmb.2012.07.003.  Google Scholar

[20]

D. Li and W. Ma, Asymptotic properties of a HIV-1 infection model with time delay, J. Math. Anal. Appl., 335 (2007), 683-691.  doi: 10.1016/j.jmaa.2007.02.006.  Google Scholar

[21]

C. Y. LinC. C. LinB. Hwang and B. Chiang, Serial changes of serum interleukin-6, interleukin-8, and tumor necrosis factor alpha among patients with Kawasaki disease, J. Pediatr., 121 (1992), 924-926.  doi: 10.1016/S0022-3476(05)80343-9.  Google Scholar

[22]

N. MakinoY. NakamuraM. YashiroR. AeS. TsuboiY. AoyamaT. KojoR. UeharaK. Kotani and H. Yanagawa, Descriptive epidemiology of Kawasaki disease in Japan, 2011-2012: from the results of the 22nd nationwide survey, J. Epidemiol., 25 (2015), 239-245.  doi: 10.2188/jea.JE20140089.  Google Scholar

[23]

B. W. McCrindle, A. H. Rowley, J. W. Newburger, et al., Diagnosis, treatment, and long-term management of Kawasaki disease: a scientific statement for health professionals from the American Heart Association, Circulation, 135 (2017), e927–e999. doi: 10.1161/CIR.0000000000000484.  Google Scholar

[24]

A. U. NeumannN. P. LamH. DahariD. R. GretchT. E. WileyT. J. Layden and A. S. Perelson, Hepatitis C viral dynamics in vivo and the antiviral efficacy of interferon-$\alpha$ therapy, Science, 282 (1998), 103-107.  doi: 10.1126/science.282.5386.103.  Google Scholar

[25]

J. W. NewburgerM. TakahashiM. A. Gerber and et al., Diagnosis, treatment, and long-term management of Kawasaki disease: A statement for health professionals from the Committee on Rheumatic Fever, Endocarditis and Kawasaki Disease, Council on Cardiovascular Disease in the Young, American Heart Association, Circulation, 110 (2004), 2747-2771.  doi: 10.1161/01.CIR.0000145143.19711.78.  Google Scholar

[26]

M. A. Nowak and C. R. M. Bangham, Population dynamics of immune responses to persistent virus, Science, 272 (5258), 74-79.  doi: 10.1126/science.272.5258.74.  Google Scholar

[27]

A. S. Perelson and P. W. Nelson, Mathematical analysis of HIV-1 dynamics in vivo, SIAM Rev., 41 (1999), 3-44.  doi: 10.1137/S0036144598335107.  Google Scholar

[28]

R. QiangW. MaK. Guo and H. Du, The differential equation model of pathogenesis of Kawasaki disease with theoretical analysis, Math. Biosci. Eng., 16 (2019), 3488-3511.  doi: 10.3934/mbe.2019175.  Google Scholar

[29]

C. M. Saad-RoyJ. Ma and P. van den Driessche, The effect of sexual transmission on Zika virus dynamics, J. Math. Biol., 77 (2018), 1917-1941.  doi: 10.1007/s00285-018-1230-1.  Google Scholar

[30]

J. J. E. Slotine and W. Li, Applied Nonlinear Control, Prentice hall, Englewood Cliffs, 1991. Google Scholar

[31]

J. Tam, Delay effect in a model for virus replication, IMA J. Math. Appl. Med. Biol., 16 (1999), 29-37.  doi: 10.1093/imammb/16.1.29.  Google Scholar

[32]

M. TeraiK. YasukawaS. NarumotoS. TatenoS. Oana and Y. Kohno, Vascular endothelial growth factor in acute Kawasaki disease, Am. J. Pediatr., 83 (1999), 337-339.  doi: 10.1016/S0002-9149(98)00864-9.  Google Scholar

[33]

R. Uehara and E. D. Belay, Epidemiology of Kawasaki Disease in Asia, Europe, and the United States, J. Epidemiol., 22 (2012), 79-85.  doi: 10.2188/jea.JE20110131.  Google Scholar

[34]

W. Wang, Global behavior of a SEIRS epidemic model with time delays, Appl. Math. Lett., 15 (2002), 423-428.  doi: 10.1016/S0893-9659(01)00153-7.  Google Scholar

[35]

X. WangS. Liu and L. Rong, Permanence and extinction of a non-autonomous HIV-1 model with time delays, Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), 1783-1800.  doi: 10.3934/dcdsb.2014.19.1783.  Google Scholar

[36]

M. XiaoL. MenM. XuG. WangH. Lv and C. Liu, Berberine protects endothelial progenitor cell from damage of TNF-$\alpha$ via the PI3K/AKT/eNOS signaling pathway, Eur. J. Pharmacol., 743 (2014), 11-16.  doi: 10.1016/j.ejphar.2014.09.024.  Google Scholar

[37]

R. S. M. Yeung, The etiology of Kawasaki disease: A superantigen-mediated process, Prog. Pediatr. Cardiol., 19 (2004), 115-122.  doi: 10.1016/j.ppedcard.2004.08.004.  Google Scholar

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 (e) is a partial enlarged view of Figure 4 (a) near the inflammatory factors-existent equilibrium $ Q^* $">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} $)
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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