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Mathematical Biosciences and Engineering

2019, Volume 16, Issue 5: 3488-3511. doi: 10.3934/mbe.2019175
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The differential equation model of pathogenesis of Kawasaki disease with theoretical analysis

  • 1.
    School of Mathematics and Physics, University of Science and Technology Beijing, Beijing100083, P.R. China
  • 2.
    Beijing Key Laboratory for Magneto-Photoelectrical Composite and Interface Science, Universityof Science and Technology Beijing, Beijing 100083, P.R. China
  • 3.
    School of Chemistry and Bioengineering, University of Science and Technology Beijing, Beijing100083, P.R. China
  • Received: 29 January 2019 Accepted: 08 April 2019 Published: 19 April 2019

  • Fever is a extremely common symptom in infants and young children. Due to the lowresistance of infants and young, long-term fever may cause damage to the child’s body. Clinically,some children with long-term fever was eventually diagnosed with Kawasaki disease (KD). KD, anautoimmune disease, is a systemic vasculitis mainly affecting children younger than 5 years old. Dueto the delayed therapy and diagnosis, coronary artery abnormalities (CAAs) develop in children with KD, and leads to a high risk of acquired heart disease. Later, patients may have myocardial infarctionor even die a sudden death. Unfortunately, at present, the pathogenesis of KD remains unknownand KD lacks of specific and sensitive biomarkers, thus bringing difficulties to diagnosis and therapy.Therefore it is a highly focused topic to research on the mechanism of KD. Some scholars believethat KD is caused by the cross reaction of external infection and organ tissue composition, herebytriggering disorder of the immune system and producing a variety of cytokines. On the basis ofconsidering the cytokines such as vascular endothelial cells, inflammatory factors, adhesion factorsand chemokines, endothelial cell growth factors, put forward a kind of dynamic model of pathogenesisof KD by the theory of ordinary differential equation. It is found that the dynamic model can showcomplex dynamic behavior, such as the forward and backward bifurcation of the equilibria. This articlereveals the possible complexity of KD infection, and provides a theoretical references for the researchof pathogenic mechanism and clinical treatment of KD.

    Citation: Rong Qiang, Wanbiao Ma, Ke Guo, Hongwu Du. The differential equation model of pathogenesis of Kawasaki disease with theoretical analysis[J]. Mathematical Biosciences and Engineering, 2019, 16(5): 3488-3511. doi: 10.3934/mbe.2019175

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  • Fever is a extremely common symptom in infants and young children. Due to the lowresistance of infants and young, long-term fever may cause damage to the child’s body. Clinically,some children with long-term fever was eventually diagnosed with Kawasaki disease (KD). KD, anautoimmune disease, is a systemic vasculitis mainly affecting children younger than 5 years old. Dueto the delayed therapy and diagnosis, coronary artery abnormalities (CAAs) develop in children with KD, and leads to a high risk of acquired heart disease. Later, patients may have myocardial infarctionor even die a sudden death. Unfortunately, at present, the pathogenesis of KD remains unknownand KD lacks of specific and sensitive biomarkers, thus bringing difficulties to diagnosis and therapy.Therefore it is a highly focused topic to research on the mechanism of KD. Some scholars believethat KD is caused by the cross reaction of external infection and organ tissue composition, herebytriggering disorder of the immune system and producing a variety of cytokines. On the basis ofconsidering the cytokines such as vascular endothelial cells, inflammatory factors, adhesion factorsand chemokines, endothelial cell growth factors, put forward a kind of dynamic model of pathogenesisof KD by the theory of ordinary differential equation. It is found that the dynamic model can showcomplex dynamic behavior, such as the forward and backward bifurcation of the equilibria. This articlereveals the possible complexity of KD infection, and provides a theoretical references for the researchof pathogenic mechanism and clinical treatment of KD.




    [1] C. Shao and S. Zhu, Clinical disease and immunity, Science Press, Beijing, 2002.
    [2] T. Kawasaki, Acute febrile mucocutaneous syndrome with lymphoid involvement with specificdesquamation of the fingers and toes in children, Arerugi, 16 (1967), 178–222.
    [3] S. Bayers, S. T. Shulman and A. S. Paller, Kawasaki disease: Part I. Diagnosis, clinical features,and pathogenesis, J. Am. Acad. Dermatol., 69 (2013), 501e1–501e11.
    [4] R. Uehara and E. D. Belay, Epidemiology of Kawasaki Disease in Asia, Europe, and the UnitedStates, J. Epidemiol., 22 (2012), 79–85.
    [5] H. Lue, L. Chen, M. Lin, et al., Epidemiological features of Kawasaki disease in Taiwan, 1976–2007: results of five nationwide questionnaire hospital surveys, Pediatr. Neonatol., 55 (2014),92–96.
    [6] C. Wei, J. Tsai, C. Lin, et al., Increased risk of idiopathic nephrotic syndrome in children withatopic dermatitis, Pediatr. Nephrol., 29 (2014), 2157–2163.
    [7] A. Kentsis, A. Shulman, S. Ahmed, et al., Urine proteomics for discovery of improved diagnosticmarkers of Kawasaki disease, EMBO Mol. Med., 22 (2012), 210–220.
    [8] M. Ayusawa, T. Sonobe, S. Uemura, et al., Revision of diagnostic guidelines for Kawasaki disease(the 5th revised edition), Pediatr. Int, 47 (2005), 232–234.
    [9] C. Galeotti, S. V. Kaveri, R. Cimaz, et al., Predisposing factors, pathogenesis and therapeuticintervention of Kawasaki disease, Drug. Discov. Today, 21 (2016), 1850–1857.
    [10] M. Terai and S. T. Shulman, Prevalence of coronary artery abnormalities in Kawasaki diseaseis highly dependent on gamma globulin dose but independent of salicylate dose, J. Pediatr., 131(1997), 888–893.
    [11] G. B. Kim, J. J. Yu, K. L. Yoon, et al., Medium- or higher-dose acetylsalicylic acid for acuteKawasaki disease and patient outcomes, J. Pediatr., 184 (2016), 125–129.e1.
    [12] J. C. Burns, B. M. Best, A. Mejias, et al., Infliximab treatment of intravenous immunoglobulin-resistant Kawasaki disease, J. Pediatr., 153 (2008), 833–838.
    [13] T. Yang, M. Lin, C. Lu, et al., The prevention of coronary arterial abnormalities in Kawasakidisease: A meta-analysis of the corticosteroid effectiveness, J. Microbiol. Immunol. Infect., 51(2018), 321–331.
    [14] J. Shen, L. Liang and C. Wang, Perifosine inhibits lipopolysaccharide (LPS)-induced tumornecrosis factor (TNF)- production via regulation multiple signaling pathways: new implicationfor Kawasaki disease (KD) treatment, Biochem. Biophys. Res. Commun., 437 (2013), 250–255.
    [15] F. Bajolle, J. F. Meritet, F. Rozenberg, et al., Markers of a recent bocavirus infection in childrenwith Kawasaki disease: a year prospective study, Pathol. Biol., 62 (2014), 365–368.
    [16] L. Chang, C. Lu, P. Shao, et al., Viral infections associated with Kawasaki disease, J. Formos.Med. Assoc., 113 (2014), 148–154.
    [17] N. Principi, D. Rigante and S. Esposito, The role of infection in Kawasaki syndrome, J. Infect.,67 (2013), 1–10.
    [18] A. Harnden, B. Alves and A. Sheikh, Rising incidence of Kawasaki disease in England: analysisof hospital admission data, Bmj, 324 (2002), 1424–1425.
    [19] H. Murata, Experimental candida-induced arteritis in mice. Relation to arteritis in themucocutaneous lymph node syndrome, Microbiol. Immunol., 23 (1979), 825–831.
    [20] A.H.Rowley, S.M.Wolinsky, D.A.Relman, etal., Searchforhighlyconservedviralandbacterialnucleic acid sequences corresponding to an etiologic agent of Kawasaki disease, Pediatr. Res., 36(1994), 567–571.
    [21] R. S. M. Yeung, The etiology of Kawasaki disease: a superantigen-mediated process, Progress Pediatr. Cardiol., 19 (2004), 115–122.
    [22] C. Lin, C. Lin, B. Hwang, et al., Serial changes of serum interleukin-6, interleukin-8, and tumornecrosis factor alpha among patients with Kawasaki disease,J. Pediatr, 121 (1992), 924–926.
    [23] M. Xiao, L. Men, M. Xu, et al., Berberine protects endothelial progenitor cell from damage ofTNF- via the PI3K/AKT/eNOS signaling pathway, J. Pediatr., 743 (2014), 11–16.
    [24] J. S. Hui-Yuen, T. T. Duong and R. S. Yeung, TNF- is necessary for induction of coronary arteryinflammation and aneurysm formation in an animal model of Kawasaki disease, J. Immunol., 176(2006), 6294–6301.
    [25] R. Fukazawa, Y. Uchikoba, Y. Kuramochi, et al., Leukocyte adhesion factor Mac-1 and migrationinhibitory factor-related protein (MRP) on granulocyte plays the essential role for causingvasculitis in kawasaki disease and the gamma globulin therapy inhibit leukocyte-endothelial celladhesion, JACC, 39 (2002), 409.
    [26] M. Terai, K. Yasukawa, S. Narumoto, et al., Vascular endothelial growth factor in acute Kawasakidisease, J. Pediatr., 83 (1999), 337–339.
    [27] S. T. Shulman and A. H. Rowley, Etiology and pathogenesis of Kawasaki disease, Progress Pediatr. Cardiol., 6 (1997), 187–192.
    [28] Y. Kuang, Delay Differential Equations with Applications in Population Dynamics, AcademicPress, Boston, 1993.
    [29] L. Chen, X. Meng and J. Jiao, Biodynamics, Science Press, Beijing, 2009.30. Z. Ma, Y. Zhou and W. Wang, et al., Mathematical Modeling and Research of Epidemic Dynamics,Science Press, Beijing, 2004.
    [30] 31. M. A. Nowak and C. R. M. Bangham, Population dynamics of immune responses to persistentvirus, Science, 272 (1996), 74–79.
    [31] 32. A. S. Perelson, A. U. Neumann, M. Markowitz, et al., HIV-1 dynamics in vivo: virion clearancerate, infected cell life-span, and viral generation time, Science, 271 (1996), 1582–1586.
    [32] 33. D. E. Kirschner and G. F. Webb, A model for the treatment strategy in the chemotherapy of AIDS,Bull. Math. Biol., 58 (1996), 367–390.
    [33] 34. A. S. Perelson and P. W. Nelson, Mathematical analysis of HIV-1 dynamics in vivo, SIAM Rev.,41 (1999), 3–44.
    [34] 35. R. Culshaw and S. Ruan, A delay-differential equation model of HIV infection of CD4 + T-cells,Math. Biosci., 165 (2000), 27–39.
    [35] 36. M. A. Nowak and R. M. May, Virus Dynamics: Mathematical Principles of Immunology and Virology, Oxford University Press, Oxford, 2000.
    [36] 37. Y. Iwasa, F. Michor and M. A. Nowak, Some basic properties of immune selection, J. Theor. Biol.,229 (2004), 179–188.
    [37] 38. R. Kumar, G. Clermont, Y. Vodovotz, et al., The dynamics of acute inflammation, J. Theor. Biol.,230 (2004), 145–155.
    [38] 39. Y. Xiao, S. Tang and J. Wu, Media impact switching surface during an infectious disease outbreak,Sci. Rep., 5 (2015), 78
    [39] 40. S. K. Sasmal, Y. Dong and Y. Takeuchi, Mathematical modeling on T-cell me diate d adaptiveimmunity in primary dengue infections, J. Theor. Biol., 429 (2017), 229–240.
    [40] 41. J. K. Hale and S. M. V. Lunel, Introduction to Functional Differential Equations, Springer Verlag,New York, 1993.
    [41] 42. O. Diekmann, J. A. P. Heesterbeek and J. A. J. Metz, On the definition and the computation ofthe basic reproduction ratio R 0 in models for infectious diseases in heterogeneous populations, J. Math. Biol., 28 (1990), 365–382.
    [42] 43. P. van den Driessche and J. Watmough, Reproduction numbers and sub-threshold endemicequilibria for compartmental models of disease transmission, Math. Biosci., 180 (2002), 29–48.

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    69. Jinliang Wang, Ran Zhang, Toshikazu Kuniya, The dynamics of an SVIR epidemiological model with infection age: Table 1., 2016, 81, 0272-4960, 321, 10.1093/imamat/hxv039
    70. A note for the global stability of a delay differential equation of hepatitis B virus infection, 2011, 8, 1551-0018, 689, 10.3934/mbe.2011.8.689
    71. Jinliang Wang, Ran Zhang, Toshikazu Kuniya, The stability analysis of an SVEIR model with continuous age-structure in the exposed and infectious classes, 2015, 9, 1751-3758, 73, 10.1080/17513758.2015.1006696
    72. Michael Y. Li, Hongying Shu, Global dynamics of a mathematical model for HTLV-I infection of CD4+ T cells with delayed CTL response, 2012, 13, 14681218, 1080, 10.1016/j.nonrwa.2011.02.026
    73. Salih Djilali, Tarik Mohammed Touaoula, Sofiane El-Hadi Miri, A Heroin Epidemic Model: Very General Non Linear Incidence, Treat-Age, and Global Stability, 2017, 152, 0167-8019, 171, 10.1007/s10440-017-0117-2
    74. Shengqiang Liu, Shaokai Wang, Lin Wang, Global dynamics of delay epidemic models with nonlinear incidence rate and relapse, 2011, 12, 14681218, 119, 10.1016/j.nonrwa.2010.06.001
    75. Liming Cai, Bin Fang, Xuezhi Li, A note of a staged progression HIV model with imperfect vaccine, 2014, 234, 00963003, 412, 10.1016/j.amc.2014.01.179
    76. Yasin Ucakan, Seda Gulen, Kevser Koklu, Analysing of Tuberculosis in Turkey through SIR, SEIR and BSEIR Mathematical Models, 2021, 27, 1387-3954, 179, 10.1080/13873954.2021.1881560
    77. Soufiane Bentout, Salih Djilali, Sunil Kumar, Tarik Mohammed Touaoula, Threshold dynamics of difference equations for SEIR model with nonlinear incidence function and infinite delay, 2021, 136, 2190-5444, 10.1140/epjp/s13360-021-01466-0
    78. Chunyue Wang, Jinliang Wang, Ran Zhang, Global analysis on an age‐space structured vaccination model with Neumann boundary condition, 2022, 45, 0170-4214, 1640, 10.1002/mma.7879
    79. Vijay Pal Bajiya, Jai Prakash Tripathi, Vipul Kakkar, Jinshan Wang, Guiquan Sun, Global Dynamics of a Multi-group SEIR Epidemic Model with Infection Age, 2021, 42, 0252-9599, 833, 10.1007/s11401-021-0294-1
    80. Qianqian Cui, Jiabo Xu, Qiang Zhang, Kai Wang, An NSFD scheme for SIR epidemic models of childhood diseases with constant vaccination strategy, 2014, 2014, 1687-1847, 10.1186/1687-1847-2014-172
    81. S.Y. Tchoumi, H. Rwezaura, J.M. Tchuenche, A mathematical model with numerical simulations for malaria transmission dynamics with differential susceptibility and partial immunity, 2023, 3, 27724425, 100165, 10.1016/j.health.2023.100165
    82. Jiaxin Nan, Wanbiao Ma, Stability and persistence analysis of a microorganism flocculation model with infinite delay, 2023, 20, 1551-0018, 10815, 10.3934/mbe.2023480
    83. Aboudramane Guiro, Dramane Ouedraogo, Harouna Ouedraogo, 2023, Chapter 10, 978-3-031-27660-6, 259, 10.1007/978-3-031-27661-3_10
    84. Ying He, Junlong Tao, Bo Bi, Stationary distribution for a three-dimensional stochastic viral infection model with general distributed delay, 2023, 20, 1551-0018, 18018, 10.3934/mbe.2023800
    85. F. Najm, R. Yafia, M. A. Aziz Alaoui, A. Aghriche, A. Moussaoui, A survey on constructing Lyapunov functions for reaction-diffusion systems with delay and their application in biology, 2023, 10, 23129794, 965, 10.23939/mmc2023.03.965
    86. Abderrazak NABTi, Dynamical analysis of an age-structured SEIR model with relapse, 2024, 75, 0044-2275, 10.1007/s00033-024-02227-6
    87. Jinxiang Zhan, Yongchang Wei, Dynamical behavior of a stochastic non-autonomous distributed delay heroin epidemic model with regime-switching, 2024, 184, 09600779, 115024, 10.1016/j.chaos.2024.115024
    88. Narges M. Shahtori, S. Farokh Atashzar, Temporal Dynamics and Interplay of Transmission Rate, Vaccination, and Mutation in Epidemic Modeling: A Poisson Point Process Approach, 2024, 11, 2327-4697, 5023, 10.1109/TNSE.2024.3421308
    89. Isam Al‐Darabsah, Global Dynamics of a Within‐Host Model for Immune Response With a Generic Distributed Delay, 2025, 0170-4214, 10.1002/mma.11021
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