Numerical analysis of distributed order SVIR model by nonstandard finite difference method

Abstract

Differential equations are used in mathematical modelling in many disciplines of science. In this study, distributed order differential equations which are more comprehensive are considered instead of the ordinary, partial and fractional order differential equations. SVIR (Susceptible, Vaccinated, Infectious, Recovered) model which is an epidemic model is defined with the help of distributed order differential equation and numerical solution is investigated by (NSFD). The model used in examination of contagious disease exhibits the effect and improvement of vaccination with the term V. The basic idea of the usage of distributed order differential equations is that these kinds of differential equations are the general case of ordinary and partial differential equations in some way and these equations leads to make interpretation with an only equation about different cases with the help of the intensity function included. Numerical solution and analysis of SVIR model are done in the study and later on, stability analysis of the discrete system is presented. In consequence of these studies, it is seen that the usage of the distributed order modelling is possible for this kind of epidemic modelling.

Keywords

• Distributed order differential equations
• SVIR epidemic model
• vaccination

Project Number

2211-E

Standart olmayan sonlu fark metodu ile dağılımlı mertebeden SVIR modelinin nümerik analizi

Abstract

Çoğu bilim dalındaki matematiksel modellemelerde diferansiyel denklemler kullanılmaktadır. Ancak genelde kullanılan adi, kısmi ve kesirli mertebeden diferansiyel denklemlerin kullanımı yerine bu çalışmada daha kapsamlı bir diferansiyel denklem olan dağılımlı (distributed) mertebeden diferansiyel denklem ele alınmıştır. Bu çalışmada dağılımlı mertebeden diferansiyel denklem yardımı ile epidemik model olan SVIR (Susceptible, Vaccinated, Infectious, Recovered) modeli tanımlanmış ve nümerik çözümü standart olmayan sonlu fark metodu (NSFD) ile araştırılmıştır. Bulaşıcı hastalıkların incelenmesinde kullanılan bu model aynı zamanda içinde barındırdığı V terimi ile hastalık evresinde aşılamanın etkisini ve gelişimini ortaya koymaktadır. Dağılımlı mertebeden diferansiyel denklemlerin kullanılmasında ki temel düşünce hem bu tip denklemlerin bir nevi adi ve kesirli diferansiyel denklemlerin genel hali olması hem de içinde tanımlanan yoğunluk fonksiyonu ile farklı durumlar hakkında tek bir denklem ile yorum yapılabilmesindendir. SVIR modelinin nümerik çözümü ve analizi çalışma içerisinde yapılmış ve sonrasında ayrıklaştırılmış sisteme ait kararlılık analizi ifade edilmiştir. Bu çalışmalar neticesinde dağılımlı mertebeden modellemenin bu tip epidemik modellemelerde kullanımının mümkün olduğu görülmüştür.

Keywords

• Dağılımlı mertebeden diferansiyel denklemler
• SVIR epidemik model
• aşılama

Supporting Institution

Türkiye Bilimsel ve Teknolojik Araştırma Kurumuna (TÜBİTAK)

Project Number

2211-E

Thanks

Yazarlardan Mehmet KOCABIYIK, 2211-E Programı ile maddi ve manevi destek veren Türkiye Bilimsel ve Teknolojik Araştırma Kurumuna (TÜBİTAK) teşekkür eder.

References

1. Kermark, M. ve Mckendrick, A. Contributions to the mathematical theory of epidemics. Part I. Proceedings of the royal society A, 115(5), 700-721, (1927).
2. Liu, X., Takeuchi, Y. ve Iwami, S. SVIR epidemic models with vaccination strategies. Journal of Theoretical Biology, 253(1), 1-11, (2008).
3. Kribs-Zaleta, C. M. ve Velasco-Hernández, J. X. A simple vaccination model with multiple endemic states. Mathematical biosciences, 164(2), 183-201, (2000).
4. Alexander, M. E., Bowman, C., Moghadas, S. M., Summers, R., Gumel, A. B. ve Sahai, B. M. A vaccination model for transmission dynamics of influenza. SIAM Journal on Applied Dynamical Systems, 3(4), 503-524, (2004).
5. Li, J., Yang, Y. ve Zhou, Y. Global stability of an epidemic model with latent stage and vaccination. Nonlinear Analysis: Real World Applications, 12(4), 2163-2173, (2011).
6. Caputo, M. Elasticita e dissipazione. Zanichelli, (1969).
7. Caputo, M. Mean fractional-order-derivatives differential equations and filters. Annali dell’Universita di Ferrara, 41(1), 73-84, (1995).
8. Caputo, M. Distributed order differential equations modelling dielectric induction and diffusion. Fractional Calculus and Applied Analysis, 4(4), 421-442, (2001).

9. Caputo, M. Diffusion with space memory modelled with distributed order space fractional differential equations. Annals of Geophysics, 223-234, (2003).
10. Hartley, T. T. ve Lorenzo, C. F. Fractional-order system identification based on continuous order distributions. Signal processing, 83(11), 2287-2300, (2003).
11. Bagley, R. L. ve Torvik, P. J. On the existence of the order domain and the solution of distributed order equations-Part I. International Journal of Applied Mathematics, 2(7), 865-882, (2000).
12. Bagley, R. L. ve Torvik, P. J. On the existence of the order domain and the solution of distributed order equations-Part II. International Journal of Applied Mathematics, 2(8), 965-988, (2000).
13. Ford, N. J. ve Morgado, M. L. Distributed order equations as boundary value problems. Computers and Mathematics with Applications, 64(10), 2973-2981, (2012).
14. Diethelm, K. ve Ford, N. J. Numerical analysis for distributed order differential equations. Journal of Computational and Applied Mathematics, 225(1), 96-104, (2009).
15. Katsikadelis, J. T. Numerical solution of distributed order fractional differential equations. Journal of Computational Physics, 259, 11-22, (2014).
16. Li, X. Y., ve Wu, B. Y. A numerical method for solving distributed order diffusion equations. Applied Mathematics Letters, 53, 92-99, (2016).
17. Aminikhah, H., Refahi Sheikhani, A. ve Rezazadeh, H. Stability analysis of distributed order fractional Chen system. The Scientific World Journal, 1-13, (2013).
18. Najafi, H. S., Sheikhani, A. R. ve Ansari, A. Stability analysis of distributed order fractional differential equations. In Abstract and Applied Analysis, Hindawi, (2011).
19. Refahi, A., Ansari, A., Najafi, H. S. ve Merhdoust, F. Analytic study on linear systems of distributed order fractional differential equations. Le Matematiche, 67(2), 3-13, (2012).
20. Dorciak L. Numerical models for simulation the fractional order control systems, UEF-04-94, The Academy of Sciences, Institute of Experimental Physic,Kosice, Slovak Republic, (1994).
21. Meerschaert, M. M. ve Tadjeran, C. Finite difference approximations for fractional advection–dispersion flow equations. Journal of computational and applied mathematics, 172(1), 65-77, (2004).
22. Mickens, R. E. Exact solutions to a finite‐difference model of a nonlinear reaction‐advection equation: Implications for numerical analysis. Numerical Methods for Partial Differential Equations, 5(4), 313-325, (1989).
23. Mickens, R. E. Nonstandard finite difference models of differential equations. World scientific, (1994).
24. Mickens, R. E. Applications of nonstandard finite difference schemes. World Scientific, (2000).
25. Mickens, R. E. Nonstandard finite difference schemes for differential equations. Journal of Difference Equations and Applications, 8(9), 823-847, (2002).
26. Oldham, K. ve Spanier, J. The fractional calculus theory and applications of differentiation and integration to arbitrary order. Elsevier, (1974).
27. Podlubny, I. Fractional differential equations, vol. 198 of Mathematics in Science and Engineering, (1999).
28. Mickens, R. E. Calculation of denominator functions for nonstandard finite difference schemes for differential equations satisfying a positivity condition. Numerical Methods for Partial Differential Equations: An International Journal, 23(3), 672-691, (2007).
29. Ongun, M. Y. ve Turhan, I. A numerical comparison for a discrete HIV infection of CD4+ T-Cell model derived from nonstandard numerical scheme. Journal of Applied Mathematics, 2013, 1-9, (2013).
30. Modanlı, M. Kesirli telegraf kısmi diferansiyel denklemlerin fark şeması metodu ile nümerik çözümü. Balıkesir Üniversitesi Fen Bilimleri Enstitüsü Dergisi, 20(1), 440-449, (2018).
31. Ongun, M. Y., Arslan, D. ve Garrappa, R. Nonstandard finite difference schemes for a fractional order Brusselator system. Advances in Difference equations, 2013(1), 102, (2013).
32. Hiçdurmaz, B. Numerical analysis for coupled systems of two-dimensional time-space fractional Schrödinger equations with trapping potentials. Balıkesir Üniversitesi Fen Bilimleri Enstitüsü Dergisi, 22(1), 1-12, (2020).
33. Dimitrov, D. T. ve Kojouharov, H. V. Nonstandard finite-difference methods for predator prey models with general functional response. Mathematics and Computers in Simulation, 78(1), 1-11, (2008).
34. Elaydi S.N. An introduction to difference equations, Second edition, Undergraduate Texts in Mathematics, Springer-Verlag, New York, 2.3, 3, (1999).
35. Dimitrov, D. T. ve Kojouharov, H. V. Nonstandard numerical methods for a class of predator-prey models with predator interference. Electronic Journal of Differential Equations (EJDE), 67-75, (2007).
36. Ogata, K. Discrete time control systems. Englewood Cliffs, NJ: Prentice Hall, (1995).
37. Ahmad, A., Javeed, N., Farman, M., Ahmad, M. O., Hafeez, A. ve Raza, A. Dynamical Behavior of Fractional Order SVIR Epidemic Model. International Journal of Analysis and Applications, 17(2), 260-274, (2019).