Journal of Environmental Accounting and Management 13(2) (2025) 125--141 | DOI:10.5890/JEAM.2025.06.002

Mudassar Rafique$^{1}$, Muhammad Aziz Ur Rehamn$^{1}$, Muhammad Rafiq$^{2,3}$, Zafar Iqbal$^{4}$,\\ Nauman Ahmed$^{3,4}$, Ali Akg\"{u}l$^{3,5}$

$^{1}$ Department of Mathematics, University of Management and Technology, Lahore, Pakistan

$^{2}$ Department of Mathematics, Faculty of Science and Technology, University of Central Punjab, Lahore, 54500, Pakistan

$^{3}$ Department of Computer Science and Mathematics, Lebanese American University, Beirut, 1102-2801, Lebanon

$^{4}$ Department of Mathematics and Statistics, The University of Lahore, Lahore, Pakistan

$^{5}$ Siirt University, Art and Science Faculty, Department of Mathematics, 56100 Siirt, Turkey

Abstract

This study aims to investigate the solution of fractional order delayed lumpy skin infection model with Caputo operator numerically as well as analytically. This delay factor helps to control and slow down the spread of infection in individuals. In this study, existence and uniqueness of the underlying model is discussed. Equilibria of the lumpy skin model are computed along with reproductive number ($R_0$), if $R_0>1$ refers spread of illness and if $R_0<1$${}_{\ }$means control of disease. Local and global stability of fraction delayed model is also presented. Moreover, positive and bounded solution of proposed model are investigated. For the numerical solution of this model, we use Grunwald Letnikov non-standard finite difference scheme. The key properties of the numerical scheme are also investigated like positivity and boundedness. Numerical example is given to present the graphical solution of the fractional order delay epidemic model.

References

1. [1]

   Kononov, A., Prutnikov, P., Shumilova, I., Kononova, S., Nesterov, A., Byadovskaya, O., Pestova, Y., Diev, V., and Sprygin, A. (2019), Determination of lumpy skin disease virus in bovine meat and offal products following experimental infection, Transboundary and Emerging Diseases, 66(3), pp.1332-1340.

2. [2]	Abdulqa, H.Y., Rahman, H.S., Dyary, H.O., and Othman, H.H. (2016), Lumpy skin disease, Reproductive Immunology: Open Access, 1(25), 1-6, https://doi.org/10.21767/2476-1974.100025.

3. [3]

   Roche, X., Rozstalnyy, A., TagoPacheco, D., Pittiglio, C., Kamata, A., Beltran Alcrudo, D., Bisht, K., Karki, S., Kayamori, J., and Larfaoui, F. (2021), Introduction and spread of lumpy skin disease in South, East and Southeast Asia: Qualitative Risk Assessment and Management, 1-183. https://doi.org/10.4060/cb1892en.

4. [4]	Abera, Z., Degefu, H., Gari, G., and Ayana, Z. (2015), Review on epidemiology and economic importance of lumpy skin disease, International Journal of Basic and Applied Virology, 4(8), 8-21.

5. [5]	Weiss, K.E. (1968), Lumpy Skin Disease Virus, Springer-Verlag Wien, 111--131.

6. [6]	Hailu, B. and Alemayehu, G. (2015), Epidemiology, economic importance and control techniques of lumpy skin disease: a review, InJAR, 3, 197-205.

7. [7]	European Food Safety Authority (EFSA) (2017), Lumpy skin disease: I, Data collection and analysis. EFSA J. 15(4).

8. [8]	Carn, V.M. and Kitching, R.P. (1995), An investigation of possible routes of transmission of lumpy skin disease virus (Neethling), Epidemiology $\&$ Infection, 114(1), 219-226.

9. [9]	Kreyszig, E. (1993), Introductory Functional Analysis with Applications, John wiley and sons, New York.

10. [10]	Butt, A.I.K., Imran, M., Chamaleen, D.B.D., and Batool, S. (2023), Optimal control strategies for the reliable and competitive mathematical analysis of Covid‐19 pandemic model, Mathematical Methods in the Applied Sciences, 46(2), 1528-1555.

11. [11]	Baleanu, D., Ghassabzade, F.A., Nieto, J.J., and Jajarmi, A. (2022), On a new and generalized fractional model for a real cholera outbreak, Alexandria Engineering Journal, 61(11), 9175-9186. doi: 10.1016/j.aej.2022.02.054.

12. [12]	Ahmad, W. and Abbas, M. (2021), Effect of quarantine on transmission dynamics of Ebola virus epidemic: a mathematical analysis, The European Physical Journal Plus, 136(4), 1-33.

13. [13]	Ahmad, M.D., Usman, M., Khan, A., and Imran, M. (2016), Optimal control analysis of Ebola disease with control strategies of quarantine and vaccination, Infectious Diseases of Poverty, 5, pp.1-12.

14. [14]	Agusto, F.B. (2009), Optimal chemoprophylaxis and treatment control strategies of a tuberculosis transmission model, World Journal of Modelling and Simulation, 5(3), 163-173.

15. [15]	Butt, A.I.K., Imran, M., Batool, S., and Nuwairan, M.A. (2023), Theoretical analysis of a COVID-19 CF-fractional model to optimally control the spread of pandemic, Symmetry, 15(2), p.380. https:// doi.org/10.3390/sym15020380

16. [16]	Butt, A.I.K., Imran, M., Batool, S., and Nuwairan, M.A. (2023), Theoretical analysis of a COVID-19 CF-fractional model to optimally control the spread of pandemic, Symmetry, 15(2), p.380.

17. [17]	Butt, A.I.K., Aftab, H., Imran, M., and Ismaeel, T. (2023), Mathematical study of lumpy skin disease with optimal control analysis through vaccination, Alexandria Engineering Journal, 72, 247-259. Elsevier BV. https://doi.org/10.1016/j.aej.2023.03.073

18. [18]	Arenas, A.J., Gonzalez-Parra, G., and Chen-Charpentier, B.M. (2016), Construction of nonstandard finite difference schemes for the SI and SIR epidemic models of fractional order, Mathematics and Computers in Simulation, 121, 48-63.

19. [19]	Podlubny, I. (1998), Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications, Elsevier.

20. [20]	Ilhan, Z. and Sahin, G. (2023), A numerical approach for an epidemic SIR model via Morgan-Voyce series, International Journal of Mathematics and Computer in Engineering, 2(1), 125-140. https://doi.org/10.2478/ijmce-2024-0010.