Nonstandard Discretization and Stability Analysis of a novel type Malaria-Ross Model

Öz

Malaria is still a deadly disease in most developing countries. In order to prevent this and many other diseases in all countries, it is necessary to understand the dynamics of the disease well. For this reason, in this study, a new type of Malaria-Ross equation, Distributed order, is discussed. In this new type, the dynamics of the disease can be understood better and quicker in different situations with the density function included in such equations. Numerical discretization of this model is done with the help of a nonstandard finite difference scheme. Afterward, stability analyses of the equilibrium points obtained from the model that were performed. At the same time, comparisons were made with other numerical methods. Finally, the findings are expressed with graphs and tables.

Anahtar Kelimeler

• Distributed order differential equations
• Malaria-Ross model
• numerical Analysis
• discretization
• stability Analysis

Destekleyen Kurum

Scientific and Technological Research Council of Turkey (TUBITAK)

Proje Numarası

2211-E Program

Teşekkür

One of the authors, Mehmet KOCABIYIK, thanks the Scientific and Technological Research Council of Turkey (TUBITAK) for providing financial and moral support with the 2211-E Program.

Kaynakça

1. Aminikhah H, Refahi A, Rezazadeh H, 2013. Stability analysis of distributed order fractional Chen system. The Scientific World Journal, 2013.
2. Anderson RM, May RM, 1991. Infectious diseases of humans: dynamics and control London: Oxford University Press.
3. Aron JL, 1988. Mathematical modeling of immunity to malaria. Math Bioscience, 90: 385-396.
4. Aron JL, May RM, 1982. The population dynamics of malaria. In Population Dynamics of Infectious Disease. Edited by: Anderson RM. London: Chapmanand Hall, pp. 139-179.
5. Bagley RL, Torvik PJ, 2000 a. 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.
6. Bagley RL, Torvik PJ, 2000b. 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.
7. Caputo M, 1969. Elasticita e dissipazione. Zanichelli.
8. Caputo M, 1995. Mean fractional-order-derivatives differential equations and filters. Annali dell’Universita di Ferrara, 41(1): 73-84.

9. Caputo M, 2001. Distributed order differential equations modelling dielectric induction and diffusion. Fractional Calculus and Applied Analysis, 4(4): 421-442.
10. Caputo M, 2003. Diffusion with space memory modelled with distributed order space fractional differential equations. Annals of Geophysics.
11. Chitnis N, 2005. Using mathematical models in controlling the spread of malaria. PhD thesis University of Arizona, Program in Applied Mathematics.
12. Diethelm K, Ford NJ, 2009. Numerical analysis for distributed-order differential equations. Journal of Computational and Applied Mathematics, 225(1): 96-104.
13. Dietz K, 1988. Mathematical models for transmission and control of malaria. In Principles and Practice of Malariology. Edited by: Wernsdorfer W, McGregor Y. Edinburgh: Churchill Livingston, pp. 1091-1133.
14. Dimitrov DT, Kojouharov HV, 2007. Nonstandard numerical methods for a class of predator-prey models with predator interference. Electronic Journal of Differential Equations (EJDE) pp. 67-75.
15. Dimitrov DT, Kojouharov HV, 2008. Nonstandard finite-difference methods for predator–prey models with general functional response. Mathematics and Computers in Simulation, 78(1): 1-11.
16. Dorciak L, 1994. Numerical models for simulation the fractional-order control systems, UEF-04-94, The Academy of Sciences, Institute of Experimental Physic, Kosiice, Slovak Republic.
17. Elsheikh S, Ouifki R, Patidar KC, 2014. A non-standard finite difference method to solve a model of HIV Malaria co-infection. Journal of Difference Equations and Applications, 20(3): 354–378. doi: 10.1080/10236198.2013.821116.
18. Hartley TT, Lorenzo CF, 2003. Fractional-order system identification based on continuous order-distributions. Signal processing, 83(11): 2287-2300.
19. Katsikadelis JT, 2014. Numerical solution of distributed order fractional differential equations. Journal of Computational Physics, 259: 11-22.
20. Kocabıyık M, Özdoğan N, Ongun MY, 2020. Nonstandard Finite Difference Scheme for a Computer Virus Model. Journal of Innovative Science and Engineering (JISE), 4(2): 96-108.
21. Li XY, Wu BY, 2016. A numerical method for solving distributed order diffusion equations. Applied Mathematics Letters, 53: 92-99.
22. Luchko Y, 2009. Boundary value problems for the generalized time-fractional difusion equation of distributed order. Fractional Calculus and Applied Analysis, 12 (4): 409-422.
23. Macdonald G, 1957. The epidemiology and control of malaria London: Oxford University Press.
24. Mandal S, Sarkar RR, Sinha S, 2011. Mathematical models of malaria-a review. Malaria journal, 10(1): 1-19.
25. Meerschaert MM, Tadjeran C, 2004. Finite difference approximations for fractional advection–dispersion flow equations. Journal of computational and applied mathematics, 172(1): 65-77.
26. Mickens RE, 1989. 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.
27. Mickens RE, 1994. Nonstandard finite difference models of differential equations. World scientific.
28. Mickens RE, 2002. Nonstandard finite difference schemes for differential equations. Journal of Difference Equations and Applications, 8(9): 823-847.
29. Ngwa GA, 2004. Modelling the dynamics of endemic malaria in growing populations. Discrete Contin Dyn System- Ser B, 4: 1173-1202.
30. Ngwa GA, Shu WS, 2000. A mathematical model for endemic malaria with variable human and mosquito populations. Math Comput Model, 32: 747-763.
31. Nyang'inja R, Lawi G, Okongo M, Orwa A, 2019. Stability analysis of Rotavirus-malaria co-epidemic model with vaccination. Dyn. Syst. Appl, 28: 371-407.
32. Ongun MY, Arslan D, 2018. Explicit and Implicit Schemes for Fractional orders Hantavirus Model. Iranian Journal of Numerical Analysis and Optimization, 8(2): 75–93.
33. Ongun MY, Arslan D. Garrappa R, 2013. Nonstandard finite difference schemes for a fractional-order Brusselator system. Advances in Difference equations, 2013(1), 1-13.
34. Ongun MY, Turhan I, 2012. A numerical comparison for a discrete HIV infection of CD4+ T-Cell model derived from nonstandard numerical scheme. Journal of Applied Mathematics, 2013.4.
35. Ross R, 1911. The prevention of malaria London: John Murray.
36. Ross R, 1915. Some a priori pathometric equations. Br Med J, 1: 546-447.
37. Ross R, 1916. An application of the theory of probabilities to the study of a priori pathometry- I. Proc R Soc, A92: 204-230.
38. Ross R, 1916. An application of the theory of probabilities to the study of a priori pathometry- II. Proc R Soc, A93: 212-225.
39. Ross R, Hudson HP, 1916. An application of the theory of probabilities to the study of a priori pathometry- III. Proc R Soc, A93: 225-240.
40. WHO, 2017. Diarrhoeal disease fact sheet. World Health Organization.