### Mathematical Biosciences and Engineering

2021, Issue 6: 8883-8904. doi: 10.3934/mbe.2021438
Research article Special Issues

# Crank-Nicholson difference scheme for the system of nonlinear parabolic equations observing epidemic models with general nonlinear incidence rate

• Received: 16 July 2021 Accepted: 09 September 2021 Published: 15 October 2021
• In this work, we study second order Crank-Nicholson difference scheme (DS) for the approximate solution of problem (1). The existence and uniqueness of the theorem on a bounded solution of Crank-Nicholson DS uniformly with respect to time step $\tau$ is proved. In practice, theoretical results are presented on four systems of nonlinear parabolic equations to explain how it works on one and multidimensional problems. Numerical results are provided.

Citation: Allaberen Ashyralyev, Evren Hincal, Bilgen Kaymakamzade. Crank-Nicholson difference scheme for the system of nonlinear parabolic equations observing epidemic models with general nonlinear incidence rate[J]. Mathematical Biosciences and Engineering, 2021, 18(6): 8883-8904. doi: 10.3934/mbe.2021438

### Related Papers:

• In this work, we study second order Crank-Nicholson difference scheme (DS) for the approximate solution of problem (1). The existence and uniqueness of the theorem on a bounded solution of Crank-Nicholson DS uniformly with respect to time step $\tau$ is proved. In practice, theoretical results are presented on four systems of nonlinear parabolic equations to explain how it works on one and multidimensional problems. Numerical results are provided.

 [1] M. Li, X. Liu, An SIR epidemic model with time delay and general nonlinear incidence rate, Abstr. Appl. Anal., (2014), Article ID 131257, http://dx.doi.org/10.1155/2014/131257. [2] B. Kaymakamzade, E. Hincal, Delay epidemic model with and without vaccine, Qual. Quant., 52 (2018), 695–709. doi: 10.1007/s11135-017-0647-8 [3] J. J. Wang, K. H. Reilly, H. Han, Z. H. Peng, N. Wang, Dynamic characteristic analysis of HIV mother to child transmission in China, Biol. Env. Sci., 23 (2010), 402–408. [4] H. M. Yang, A. R. R. Freitas, Biological view of vaccination described by mathematical modellings: From rubella to dengue vaccines, Math. Biosci. Eng., 16 (2019), 3195–-3214. doi: 10.3934/mbe.2019159 [5] E. Hincal, M. Sayan, B. Kaymakamzade, T. Sanlidag, F. T. Saad, I. A. Baba, Springer Proceedings in Mathematics & Statistics, Switzerland, 2020. [6] A. Ashyralyev, E. Hincal, B. Kaymakamzade, Numerical solutions of the system of PDEs for observing epidemic models, AIP Conference Proceedings, ICAAM 2018, 1997 (2018), 020050. [7] A. Ashyralyev, E. Hincal, B. Kaymakamzade, Bounded solution of the system of nonlinear parabolic equations observing epidemic models with general nonlinear incidence rate, Math. Model Nat. Phenom., in press. [8] M. Sayan, E. Hincal, T.Sanlidag, B.Kaymakamzade, F. T. Sa'ad, I. A. Baba, Dynamics of HIV/AIDS in Turkey from 1985 to 2016, Qual. Quant., 52 (2018), 711–723. doi: 10.1007/s11135-017-0648-7 [9] S. G. Krein, Linear Differential Equations in Banach Space, Nauka: Moscow, 1966. [10] A. A. Samarskii, The Theory of Difference Schemes, CRC Press; 1 edition, 2001. [11] A. Ashyralyev, Mathematical Methods in Engineering, Dordrecht, 2007. [12] A. Ashyralyev, A. Sarsenbi, Well-posedness of an elliptic equation with involution, Elect. J. Diff. Eqn., 284 (2015), 1–8. [13] P. E. Sobolevskii, Difference Methods for the Approximate Solution of Differential Equations, Voronezh, 1975.
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