A numerical approach for a dynamical system of fractional infectious disease problem

Abstract

In this investigation, we study for a dynamical system aimed at elucidating a disease model under the influence of environmental stress from a broad perspective. The model is articulated through both standard differential equations and their Caputo fractional form. Our methodology involves a numerical approach using the Adams-Bashforth-Moulton technique to solve the system of differential equations, including the initial conditions. The existence, uniqueness and convergence of the technique are also briefly discussed. This study aims not only to improve the current technique, but also to introduce a novel design for obtaining numerical solutions to issues discussed in the existing literature, thus paving the way for further research. We also perform a stability analysis focusing on the coexistence equilibrium. In addition, we present visualisations of the results to elucidate the behaviour of the system, time evolution and phase plane plots with respect to specific parameters.

Keywords

• disease modelling
• Adams-Bashforth-Moulton method
• Caputo fractional derivative
• stability analysis

References

1. [1] T. Akman Yıldız, Optimal control problem of a non-integer order waterborne pathogen model in case of environmental stressors, Front. Phys. 7, 95, 2019.
2. [2] M.E. Alexander, S.M. Moghadas, Bifurcation analysis of an SIRS epidemic model with generalized incidence, SIAM J. Appl. Math. 65(5), 1794-1816, 2005.
3. [3] R. Almeida, A.M.C.B. da Cruz, N. Martins, M.T.T. Monteiro, An epidemiological MSEIR model described by the Caputo fractional derivative, Int. J. Dyn. Control. 7(2), 776-784, 2019.
4. [4] J.K.K. Asamoah, Fractalfractional model and numerical scheme based on Newton polynomial for Q fever disease under AtanganaBaleanu derivative, Results Phys. 34, 105189, 2022.
5. [5] J.K.K. Asamoah, G.Q. Sun, Fractional Caputo and sensitivity heat map for a gonorrhea transmission model in a sex structured population, Chaos Solitons Fractals. 175, 114026, 2023.
6. [6] J.K.K. Asamoah, A fractional mathematical model of heartwater transmission dynamics considering nymph and adult amblyomma ticks, Chaos Solitons Fractals. 174, 113905, 2023.
7. [7] L.C.D. Barros, M.M. Lopes, F.S. Pedro, E. Esmi, J.P.C.D. Santos, D.E. Sánchez, The memory effect on fractional calculus: an application in the spread of COVID-19, Comput. Appl. Math. 40, 1-21, 2021.
8. [8] H.M. Baskonus, H. Bulut, On the numerical solutions of some fractional ordinary differential equations by fractional Adams-Bashforth-Moulton method, Open Math. 13(1), 000010151520150052, 2015.

9. [9] D. Baleanu, M.H. Abadi, A. Jajarmi, K.Z. Vahid, J.J. Nieto, A new comparative study on the general fractional model of COVID-19 with isolation and quarantine effects, Alex. Eng. J. 61(6), 4779-4791, 2022.
10. [10] J.C. Butcher, Numerical methods for ordinary differential equations in the 20th century, J. Comput. Appl. Math. 125(1-2), 1-29, 2000.
11. [11] A.H. Bukhari, M. Sulaiman, M.A.Z. Raja, S. Islam, M. Shoaib, P. Kumam, Design of a hybrid NAR-RBFs neural network for nonlinear dusty plasma system, Alex. Eng. J. 59(5), 3325-3345, 2020.
12. [12] V. Daftardar-Gejji, Y. Sukale, S. Bhalekar, A new predictorcorrector method for fractional differential equations, Appl. Math. Comput. 244, 158-182, 2014.
13. [13] K. Diethelm, A.D. Freed, The FracPECE subroutine for the numerical solution of differential equations of fractional order, Forschung und wissenschaftliches Rechnen. 1999, 57-71, 1998.
14. [14] K. Diethelm, N.J. Ford, A.D. Freed, Detailed error analysis for a fractional Adams method, Numer. Algorithms. 36, 31-52, 2004.
15. [15] K. Diethelm, Efficient solution of multi-term fractional differential equations using P(EC)mE methods, Computing. 71, 305-319, 2003.
16. [16] K. Diethelm, N.J. Ford, A.D. Freed, A predictor-corrector approach for the numerical solution of fractional differential equations, Nonlinear Dyn. 29, 3-22, 2002.
17. [17] K. Diethelm, N.J. Ford, Analysis of fractional differential equations, J. Math. Anal. 265(2), 229-248, 2002.
18. [18] A. Erdélyi, Higher transcendental functions, in: Higher transcendental functions, 59, McGraw-Hill, New York, 1953.
19. [19] K. Diethelm, The analysis of fractional differential equations: An application-oriented exposition using differential operators of Caputo type, in: Lecture Notes in Mathematics 393, 1-247, Springer-Verlag, Berlin, 2010.
20. [20] A. Dobson, Climate variability, global change, immunity, and the dynamics of infectious diseases, Ecol. 90(4), 920-927, 2009.
21. [21] R. Douaifia, S. Bendoukha, S. Abdelmalek, A Newton interpolation based predictorcorrector numerical method for fractional differential equations with an activatorinhibitor case study, Math. Comput. Simulation. 187, 391-413, 2021.
22. [22] R. Garrappa, Numerical solution of fractional differential equations: A survey and a software tutorial, Mathematics. 6(2), 16, 2018.
23. [23] R. Garrappa, On linear stability of predictorcorrector algorithms for fractional differential equations, Int. J. Comput. Math. 87(10), 2281-2290, 2010.
24. [24] R. Garrappa, Predictor-corrector PECE method for fractional differential equations, The MathWorks Inc. MATLAB Central File Exchange. Retrieved May 6, 2022.
25. [25] B. Ghanbari, A new model for investigating the transmission of infectious diseases in a preypredator system using a nonsingular fractional derivative, Math. Methods Appl. Sci. 46(7), 8106-8125, 2023.
26. [26] A.G.O. Goulart, M.J. Lazo, J.M.S Suarez, D.M. Moreira, Fractional derivative models for atmospheric dispersion of pollutants, Phys. A: Stat. Mech. Appl. 477, 9-19, 2017.
27. [27] A.G. Goulart, M.J. Lazo, J.M.S Suarez, A new parameterization for the concentration flux using the fractional calculus to model the dispersion of contaminants in the Planetary Boundary Layer, Phys. A: Stat. Mech. Appl. 518, 38-49, 2019.
28. [28] A. Gökçe, A dynamic interplay between Allee effect and time delay in a mathematical model with weakening memory, Appl. Math. Comput. 430, 127306, 2022.
29. [29] S. Gupta, S.K. Bhatia, N. Arya, Effect of incubation delay and pollution on the transmission dynamics of infectious disease, Ann. Univ. Ferrara. 69(1), 23-47, 2023.
30. [30] B. Gürbüz, A.D. Rendall, Analysis of a model of the Calvin cycle with diffusion of ATP, Discrete Continuous Dyn. Syst. Ser. B. 27(9), 5161-5177, 2022.
31. [31] E. Hairer, C. Lubich, M. Schlichte, Fast numerical solution of nonlinear Volterra convolution equations, SIAM J. Sci. Comput. 6(3), 532-541, 1985.
32. [32] E. Hairer, S.P. Nørsett, G. Wanner, Solving Ordinary Differential Equations I: Nonstiff Problems, in: Springer Series in Comput. Mathematics, 8, 1-247, Springer-Verlag, Berlin-Heidelberg, 1993.
33. [33] V.F. Hatipoglu, S. Alkan, A. Secer, An efficient scheme for solving a system of fractional differential equations with boundary conditions, Adv. Differ. Equ. 2017(1), 1-13, 2017.
34. [34] M.B. Hoshen, A.P. Morse, A weather-driven model of malaria transmission, Malar. J. 3(1), 114, 2004.
35. [35] A. Jhinga, V. Daftardar-Gejji, A new numerical method for solving fractional delay differential equations, Comput. Appl. Math. 38, 118, 2019.
36. [36] N.K. Kamboj, S. Sharma, S. Sharma, Modelling and sensitivity analysis of COVID- 19 under the influence of environmental pollution, in: Mathematical Analysis for Transmission of COVID-19, 309-323, Springer, Singapore, 2021.
37. [37] W.O. Kermack, A.G. McKendrick, A contribution to the mathematical theory of epidemics, Proc. R. Soc. Lond. A. 115(772), 700-721, 1927.
38. [38] A.A. Khan, R. Amin, S. Ullah, W. Sumelka, M. Altanji, Numerical simulation of a Caputo fractional epidemic model for the novel coronavirus with the impact of environmental transmission, Alex. Eng. J. 61(7), 5083-5095, 2022.
39. [39] N.A. Khan, O.A. Razzaq, S.P. Mondal, Q. Rubbab, Fractional order ecological system for complexities of interacting species with harvesting threshold in imprecise environment, Adv. Difference Equ. 2019(1), 1-34, 2019.
40. [40] A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and applications of fractional differential equations, in: Theory and Applications of Fractional Differential Equations, 204, 1-525, Elsevier, Netherlands, UK, USA, 2006.
41. [41] S. Kumar, S. Sharma, A. Kashyap, R.P. Agarwal, Modelling the effect of environmental pollution on Zika outbreak: A case study of Brazil, Discrete Continuous Dyn. Syst. Ser. S., 2023.
42. [42] N. Kumari, S. Sharma, Modeling the dynamics of infectious disease under the influence of environmental pollution, Int. J. Comput. Math. 4, 1-24, 2018.
43. [43] N. Kumari, S. Kumar, S. Sharma, F. Singh, R. Parshad, Basic reproduction number estimation and forecasting of COVID-19: A case study of India, Brazil and Peru, Commun. Pure Appl. Anal. 22(2), 417-440, 2023.
44. [44] M. Lipsitch, T. Cohen, B. Cooper, J.M. Robins,..., M. Murray, Transmission dynamics and control of severe acute respiratory syndrome, Science. 300(5627), 1966-1970, 2003.
45. [45] S. Liu, S. Ruan, X. Zhang, On avian influenza epidemic models with time delay, Theory Biosci. 134, 75-82, 2015.
46. [46] K.D. Lafferty, The ecology of climate change and infectious diseases, Ecol. 90(4), 888-900, 2009.
47. [47] Z. Lin, W. Xu, X. Yue, Q. Han, Study on the effect of environmental pollution based on a fractional derivative resource depletion model, Chaos Solitons Fractals. 104, 705-715, 2017.
48. [48] A. Marciniak, M.A. Jankowska, Interval methods of Adams-Bashforth type with variable step sizes, Numer. Algorithms 84, 651-678, 2020.
49. [49] S. Mashayekhi, M. Razzaghi, Numerical solution of nonlinear fractional integrodifferential equations by hybrid functions, Eng. Anal. Bound. Elem. 56, 81-89, 2015.
50. [50] D. Moreira, P. Xavier, E. Nascimento, New approach to solving the atmospheric pollutant dispersion equation using fractional derivatives, Int. J. Heat Mass Transf. 144, 118667, 2019.
51. [51] M. Nasir, S. Jabeen, F. Afzal, A. Zafar, Solving the generalized equal width wave equation via sextic-spline collocation technique, Int J. Math. Comput. Sci. 1(2), 229242, 2023
52. [52] I. Podlubny, Fractional differential equations: an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications, in: Fractional Differential Equations: An Introduction to Fractional Derivatives, 198, 1-340, Elsevier, USA, 1998.
53. [53] S. Qureshi, A. Yusuf, Modeling chickenpox disease with fractional derivatives: From caputo to Atangana-Baleanu, Chaos Solitons Fractals. 122, 111-118, 2019.
54. [54] F.A. Rihan, Q.M. Al-Mdallal, H.J. AlSakaji, A. Hashish, A fractional-order epidemic model with time-delay and nonlinear incidence rate, Chaos Solitons Fractals. 126, 97-105, 2019.
55. [55] S.G. Samko, Fractional integrals and derivatives. Theory and applications, in: Fractional Differential Equations: An Introduction to Fractional Derivatives, 1993.
56. [56] S.V. Sharif, P.H. Moshfegh, H. Kashani, Simulation modeling of operation and coordination of agencies involved in post-disaster response and recovery, Reliab. Eng. Syst. Saf. 235, 109219, 2023.
57. [57] A. Sirijampa, S. Chinviriyasit, W. Chinviriyasit, Hopf bifurcation analysis of a delayed SEIR epidemic model with infectious force in latent and infected period, Adv. Difference Equ. 2018(1), 1-24, 2018.
58. [58] J. Weissinger, Zur theorie und anwendung des iterationsverfahrens, Math. Nachrichten. 8(1), 193-212, 1952.
59. [59] L. Wen, X. Yang, Global stability of a delayed SIRS model with temporary immunity, Chaos Solitons Fractals. 38(1), 221-226, 2008.
60. [60] P. Yan, Z. Feng, Variability order of the latent and the infectious periods in a deterministic SEIR epidemic model and evaluation of control effectiveness, Math. Biosci. 224(1), 43-52, 2010.
61. [61] H.M. Yang, M.L.G. Macoris, K.C. Galvani, M.T.M. Andrighetti, D.M.V. Wanderley, Assessing the effects of temperature on dengue transmission, Epidemiol Infect. 137(8), 1179-1187, 2009.