Numerical study of a two-dimensional eco-epidemiological model with diffusion and convex incidence rate: Unconditionally positivity preserving method

Document Type : Research Article

Authors

1 Department of Mathematical Sciences-Yazd University-Yazd-Iran

2 Department of Mathematics-Yazd University-Yazd-Iran.

3 Department of Mathematical Sciences, Yazd University, Yazd, Iran.

4 Department of Mathematical Sciences-Yazd University-Yazd- Iran.

Abstract

In this paper,  a two-dimensional eco-epidemiological model with diffusion and convex incidence rate is studied, that is, the density of population depends on time and two spatial variables.  The main challenge in investigation of population models is finding a numerical method to obtain non-negative solutions. Some numerical methods, for instance Euler's method, based on the standard finite difference formulas are inefficient for solving such models because they are not always able to produce non-negative approximate solutions. On the other hand, the non-standard finite difference schemes can provide  non-negative approximations conditionally.  In { the} current work,  first, the stability of the dynamic proposed eco-epidemiological model is examined.  Then, a numerical method that provides unconditional acceptable solutions is introduced.  In what follows, the consistency and stability of the numerical method are discussed. Finally, using numerical simulation,  the efficiency of this method is compared with the Euler and non-standard methods. Furthermore, we examined the role of initial functions in interpreting species-environment interactions and deliberated on predator-prey behaviors in various scenarios.

Keywords

Main Subjects

References

[1] N. Ahmed, S.S. Tahira, M. Rafiq, M.A. Rehman, M. Ali, M.O. Ahmad, Positivity preserving oper-
ator splitting nonstandard finite difference methods for SEIR reaction diffusion model, Open Math.
17 (2019) 313–330.
[2] N. Ahmed, Z. Wei, D. Baleanu, M. Rafiq, M.A. Rehman, Spatio-temporal numerical modeling of
reaction-diffusion measles epidemic system, Chaos 29 (2019) 103101.
[3] F. Al-Showaikh, Numerical modelling of some systems in the biomedical sciences, Ph.D. thesis,
Brunel University, School of Information Systems, Computing and Mathematics, 1998.
[4] J.A. Aledo, C. Andreu-Vilarroig, J.-C. Cortes, J.C. Orengo, R.-J. Villanueva, On the epidemiologi-
cal evolution of colistin-resistant acinetobacter baumannii in the city of valencia: An agent-based
modelling approach, Math. Model. Nat. Phenom. 18 (2023) 33.
[5] S. Bagheri, M.H. Akrami, G.B. Loghmani, M. Heydari, Traveling wave in an eco-epidemiological
model with diffusion and convex incidence rate: Dynamics and numerical simulation, Math. Com-
put. Simul. 216 (2024) 347–366.
[6] A.M. Bate, F.M. Hilker, Preytaxis and travelling waves in an eco-epidemiological model, Bull.
Math. Biol. 81 (2019) 995–1030.
[7] B. Buonomo, D. Lacitignola, On the dynamics of an seir epidemic model with a convex incidence
rate, Ric. Mat. 57 (2008) 261–281.
[8] B. Buonomo, S. Rionero, On the lyapunov stability for SIRS epidemic models with general nonlin-
ear incidence rate, Appl. Math. Comput. 217 (2010) 4010–4016.
[9] Y. Cai, Y. Kang, W. Wang, A stochastic SIRS epidemic model with nonlinear incidence rate, Appl.
Math. Comput. 305 (2017) 221–240.
[10] B.M. Chen-Charpentier, H.V. Kojouharov, An unconditionally positivity preserving scheme for
advection-diffusion reaction equations, Math. Comput. Model 57 (2013) 2177–2185.
[11] C. Cosner, D.L. DeAngelis, J.S. Ault, D.B. Olson, Effects of spatial grouping on the functional
response of predators, Theor. Popul. Biol. 56 (1999) 65–75.
[12] K. Dariva, T. Lepoutre, Influence of the age structure on the stability in a tumor-immune model for
chronic myeloid leukemia, Math. Model. Nat. Phenom. 19 (2024) 1.
[13] R.U. Din, E.A. Algehyne, Mathematical analysis of COVID-19 by using SIR model with convex
incidence rate, Results Phys. 23 (2021) 103970.
[14] P. Dutta, D. Sahoo, S. Mondal, G. Samanta, Dynamical complexity of a delay-induced eco-epidemic
model with Beddington–DeAngelis incidence rate, Math. Comput. Simul. 197 (2022) 45–90.
[15] D. Greenhalgh, Q.J. Khan, F.A. Al-Kharousi, Eco-epidemiological model with fatal disease in the
prey, Nonlinear Anal. Real World Appl. 53 (2020) 103072.
[16] E.E. Holmes, M.A. Lewis, J.E. Banks, R.R. Veit, Partial differential equations in ecology: spatial
interactions and population dynamics, Ecology 75 (1994) 17–29.
[17] F. Izadi, Using nonstandard finite difference methods for solving converted schrodinger equation to
an ODE, Iran. J. Optim. 13 (2021) 13–27.
[18] Y. Jin, W. Wang, S. Xiao, An SIRS model with a nonlinear incidence rate, Chaos Solit. Fract. 34
(2007) 1482–1497.
[19] A. Khan, R. Zarin, G. Hussain, N.A. Ahmad, M.H. Mohd, A. Yusuf, Stability analysis and optimal
control of covid-19 with convex incidence rate in khyber pakhtunkhawa (pakistan), Results Phys.
20 (2021) 103703.
[20] A. Khan, R. Zarin, G. Hussain, A.H. Usman, U.W. Humphries, J.G. Aguilar, Modeling and sensi-
tivity analysis of HBV epidemic model with convex incidence rate, Results Phys. 22 (2021) 103836.
[21] A. Korobeinikov, P.K. Maini. Non-linear incidence and stability of infectious disease models, Math.
Med. Biol. 22 (2005) 113–128.
[22] G.-H. Li, Y.-X. Zhang, Dynamic behaviors of a modified SIR model in epidemic diseases using
nonlinear incidence and recovery rates, Plos One. 12 (2017) 0175789.
[23] L. Liu, D.P. Clemence, R.E. Mickens, A nonstandard finite difference scheme for contaminant
transport with kinetic langmuir sorption, Numer. Methods Partial Differ. Equ. 27 (2011) 767–785.
[24] R.E. Mickens, A best finite-difference scheme for the fisher equation, Numer. Methods Partial Dif-
fer. Equ. 10 (1994) 581–585.
[25] R.E. Mickens, Exact solutions to a finite-difference model of a nonlinear reaction-advection equa-
tion: Implications for numerical analysis, Numer. Methods Partial Differ. Equ. 5 (1989) 313–325.
[26] A. Mondal, A.K. Pal, G.P. Samanta, Analysis of a delayed eco-epidemiological pest–plant model
with infected pest, Biophys Rev. Lett. 14 (2019) 141–170.
[27] A. Mondal, A. Pal, G. Samanta, On the dynamics of evolutionary leslie-gower predator-prey eco-
epidemiological model with disease in predator, Ecol. Genet. Genom. 10 (2019) 100034.
[28] J.D. Murray, Mathematical Biology: II: Spatial Models and Biomedical Applications, volume 3.
Springer, 2003.
[29] A. Nauman, M. Rafiq, M.A. Rehman, M. Ali, M.O. Ahmad, Numerical modeling of seir measles
dynamics with diffusion, Commun. Math. Appl. 9 (2018) 315.
[30] A. Pal, A. Bhattacharyya, A. Mondal, Qualitative analysis and control of predator switching on
an eco-epidemiological model with prey refuge and harvesting, Results Control Optim. 7 (2022)
100099.
[31] A.K. Pal, A. Bhattacharyya, A. Mondal, S. Pal, Qualitative analysis of an eco-epidemiological
model with a role of prey and predator harvesting, Z. Naturforsch. A. 77 (2022) 629–645.
[32] A. Pal, G. Samanta, Stability analysis of an eco-epidemiological model incorporating a prey refuge,
Nonlinear Anal. Model. Control 15 (2010) 473–491.
[33] M.S. Rahman, S. Chakravarty, A predator-prey model with disease in prey, Nonlinear Anal. Model.
Control 18 (2013) 191–209.
[34] G.U. Rahman, K. Shah, F. Haq, N. Ahmad, Host vector dynamics of pine wilt disease model with
convex incidence rate, Chaos Solit. Fract. 113 (2018) 31–39.
[35] L.-I.W. Roeger, Dynamically consistent discrete-time SI and SIS epidemic models, Conf. Publ. 2013
(2013) 653–662.
[36] S. Saha, A. Maiti, G.P. Samanta, A Michaelis–Menten predator–prey model with strong Allee effect
and disease in prey incorporating prey refuge, Int. J. Bifurc. Chaos 28 (2018) 1850073.
[37] S. Saha, G.P. Samanta, A prey-predator system with disease in prey and cooperative hunting strat-
egy in predator, J. Phys. A: Math. Theor. 53 (2020) 485601.
[38] G. Samanta, Deterministic, Stochastic and Thermodynamic Modelling of some Interacting Species,
Forum for Interdisciplinary Mathematics. Springer Singapore, 2021.
[39] X.-F. San, Z.-C. Wang, Z. Feng, Spreading speed and traveling waves for an epidemic model in a
periodic patchy environment, Commun. Nonlinear Sci. Numer. Simul. 90 (2020) 10538.
[40] N. Santra, S. Saha, G. Samanta, Exploring cooperative hunting dynamics and prcc analysis: in-
sights J. Phys. A: Math. Theor. 57 (2024) 305601.
[41] N. Sapoukhina, Y. Tyutyunov, R. Arditi, The role of prey taxis in biological control: a spatial
theoretical model, Am. Nat. 162 (2003) 61–76.
[42] M. Sieber, H. Malchow, F.M. Hilker, Disease-induced modification of prey competition in eco-
epidemiological models, Ecol. Complex. 18 (2014) 74–82.
[43] G.D. Smith, Numerical solution of partial differential equations: finite difference methods, Oxford
University Press, 1985.
[44] H. Yang, Y. Wang, S. Kundu, Z. Song, Z. Zhang, Dynamics of an SIR epidemic model incorporating
time delay and convex incidence rate, Results Phys. 32 (2022) 105025.
Journal of Mathematical Modeling
Volume 13, Issue 4
December 2025
Pages 983-1005

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