A general model of non-communicable diseases and its qualitative analysis without finding the eigenvalues

Volume 24, Issue 1, pp 73--81 http://dx.doi.org/10.22436/jmcs.024.01.07

Publication Date: January 13, 2021 Submission Date: April 16, 2020 Revision Date: September 16, 2020 Accteptance Date: November 20, 2020

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Authors

Auni Aslah Mat Daud - Faculty of Ocean Engineering Technology and Informatics, University Malaysia Terengganu, Kuala Nerus, 21030, Malaysia. Toh Cher Qing - Faculty of Ocean Engineering Technology and Informatics, University Malaysia Terengganu, Kuala Nerus, 21030, Malaysia.

Abstract

Obtaining the analytical solutions of a linear ordinary differential equations is impossible without finding the eigenvalues. In this study, a general linear model of non-communicable disease (NCD) is formulated using the compartmental analysis and its qualitative properties are analyzed without finding the eigenvalues. NCDs are diseases which are not passed from person to person. The proof of the qualitative properties of the general model including the existence and uniqueness of its solution and equilibrium, and the positivity and boundedness of its solutions are provided. The global stability of the general model is analyzed using the theorem of compartmental matrix and Lyapunov function. It is found that the model has one unique non-negative equilibrium which is globally exponentially stable. As a real-world example, the general model and its qualitative analysis are implemented to a NCD, namely venous thromboembolism (VTE) among pregnant and postpartum women. VTE is selected in this study as it is a major global health burden due to its association with disability and lower quality of life and death.

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Keywords

• Mathematical modelling
• qualitative analysis
• non-communicable disease
• venous thromboembolism
• global stability
• compartmental analysis

MSC

•  97M10
•  92D25
•  37N15

References

• [1] D. H. Anderson, Compartmental modeling and tracer kinetics, Springer-Verlag, Berlin (1983)
  • View Article
  • Google Scholar


• [2] A. Boutayeb, E. H. Twizell, K. Achouayb, A. Chetouani, A mathematical model for the burden of diabetes and its complications, Biomed. Eng. Online, 3 (2004), 8 pages
  • View Article
  • Google Scholar


• [3] F. Brauer, J. A. Nohel, The qualitative theory of ordinary differential equations: An introduction, Dover Publications, New York (1969)
  • View Article
  • Google Scholar


• [4] J. Drife, Thromboembolism: reducing maternal death and disability during pregnancy, Br. Med. Bull., 67 (2003), 177--190
  • View Article
  • Google Scholar


• [5] K.-J. Engel, R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, Springer-Verlag, New York (2000)
  • View Article
  • Google Scholar


• [6] G. P. Garnett, An introduction to mathematical models in sexually transmitted disease epidemiology, Sex. Transm. Infect., 78 (2002), 7--12
  • View Article
  • Google Scholar


• [7] H. W. Hethcote, The mathematics of infectious diseases, SIAM Rev., 42 (2000), 599--653
  • View Article
  • Google Scholar


• [8] J. A. Jacquez, C. P. Simon, Qualitative theory of compartmental systems, SIAM Rev., 35 (1993), 43--79
  • View Article
  • Google Scholar


• [9] A. H. James, Venous thromboembolism in pregnancy, Arterioscler. Thromb. Vasc. Biol., 29 (2009), 326--331
  • View Article
  • Google Scholar


• [10] S. Liu, J. Rouleau, K. S. Joseph, R. Sauve, R. M. Liston, D. Young, M. S. Kramer, Epidemiology of pregnancy-associated venous thromboembolism: a population-based study in Canada, J. Obstet. Gynaecol. Can., 31 (2009), 611--620
  • View Article
  • Google Scholar


• [11] A. A. Mat Daud, Mathematical modeling and stability analysis of population dynamics, In Dynamical Systems, Bifurcation Analysis and Applications, Springer, Singapore, 2018 (295), 3--13
  • View Article
  • Google Scholar


• [12] A. A. Mat Daud, Some issues on the mathematical modeling of population dynamics using differential equations, Int. J. Math. Comput. Sci., 15 (2020), 501--513
  • View Article
  • Google Scholar


• [13] A. A. Mat Daud, C. Q. Toh, S. Saidun, A mathematical model to study the population dynamics of hypertensive disorders during pregnancy, J. Interdiscip. Math., 22 (2019), 433--450
  • View Article
  • Google Scholar


• [14] A. A. Mat Daud, C. Q. Toh, S. Saidun, Development and analysis of a mathematical model for the population dynamics of Diabetes Mellitus during pregnancy, Math. Models Comput. Simul., 12 (2020), 620--630
  • View Article
  • Google Scholar


• [15] P. H. Reitsma, H. H. Versteeg, S. Middeldorp, Mechanistic view of risk factors for venous thromboembolism, Arterioscler. Thromb. Vasc. Biol., 32 (2012), 563--568
  • View Article
  • Google Scholar


• [16] C. I. Siettos, L. Russo, Mathematical modeling of infectious disease dynamics, Virulence, 4 (2013), 295--306
  • View Article
  • Google Scholar


• [17] A. A. Sultan, J. West, L. J. Tata, K. M. Fleming, C. Nelson-Piercy, M. J. Grainge, Risk of first venous thromboembolism in and around pregnancy: A population-based cohort study, Br. J. Haematol., 156 (2012), 366--373
  • View Article
  • Google Scholar


• [18] R. A. Virkus, E. C. L. Løkkegaard, T. Bergholt, U. Mogensen, J. Langhoff-Roos, O. Lidegaard, Venous thromboembolism in pregnant and puerperal women in Denmark 1995--2005: A national cohort study, Thromb. Haemost., 106 (2011), 304--309
  • View Article
  • Google Scholar


• [19] S. A. Wirkus, R. J. Swift, A course in ordinary differential equations (2nd ed.), CRC Press, New York (2004)
  • View Article
  • Google Scholar