On the stability analysis of a fractional order epidemic model including the general forms of nonlinear incidence and treatment function

Year 2024, Volume: 73 Issue: 1, 285 - 305, 16.03.2024

Esra Karaoğlu

https://doi.org/10.31801/cfsuasmas.1258454

Abstract

In this paper, we propose to study a SEIR model of fractional order with an incidence and a treatment function. The incidence and treatment functions included in the model are general nonlinear functions that satisfy some meaningful biological hypotheses. Under these hypotheses, it is shown that the disease free equilibrium point of the proposed model is locally and globally asymptotically stable when the reproduction number $R_{0} $ is smaller than 1. When $ R_{0}>1 $, it is established that the endemic equilibrium of the studied system is uniformly asymptotically stable. Finally, some numerical simulations are provided to illustrate the theory.

Keywords

Fractional order SEIR model, uniform asymptotic stability, nonlinear incidence function, treatment function

References

• Ahmed, E., El-Sayed, A. M. A., El-Saka, H. A. A., Equilibrium points, stability and numerical solutions of fractional-order predator-prey and rabies models, Journal of Mathematical Analysis and Applications, 325 (1) (2007), 542–553. https://doi: 10.1016/j.jmaa.2006.01.087
• Almeida, R., Analysis of a fractional SEIR model with treatment, Applied Mathematics Letters, 84 (2018), 56–62. https://doi: 10.1016/j.aml.2018.04.015
• Almeida, R., Brito da Cruz, A. M. C., Martins, N., Monteiro, M. T. T., An epidemiological MSEIR model described by the Caputo fractional derivative, International Journal of Dynamics and Control, 7 (2)(2019), 776–784. https://doi: 10.1007/s40435-018-0492-1
• Anderson, R. M., May, R. M., Infectious Diseases of Humans, Oxford: Oxford University Press, 1991.
• Atede, A. O., Omame, A., Inyama, S. C., A fractional order vaccination model for COVID-19 incorporating environmental transmission: a case study using Nigerian data, Bulletin of Biomathematics, 1 (1) (2023), 78—110. https://doi.org/10.59292/bulletinbiomath.2023005
• Bhattacharya, P., Paul, S., Biswas, P., Mathematical modeling of treatment SIR model with respect to variable contact rate, International Proceedings of Economics Development and Research, 83 (2015), 34–41.
• Brauer, F., Compartmental Models in Epidemiology. In: Brauer F., van den Driessche P., Wu J. (eds) Mathematical Epidemiology. Lecture Notes in Mathematics, vol 1945, Springer, Berlin, Heidelberg. https://doi:10.1007/978-3-540-78911-6-2
• Brauer, F., Mathematical epidemiology: past, present, and future, Infectious Disease Modelling, 2 (2) (2017), 113–127. https://doi: 10.1016/j.idm.2017.02.001
• Castillo-Chavez, C., Feng, Z., To treat or not to treat: the case of tuberculosis transmission, Journal of Mathematical Biology, 35 (6) (1997), 629–656.
• Delavari, H., Baleanu, D., Sadati, J., Stability analysis of Caputo fractional-order nonlinear systems revisited, Nonlinear Dynamics, 67 (1) (2012), 2433–2439. https://doi:10.1007/s11071-011-0157-5
• Elkhaiar, S., Kaddar, A., Stability analysis of an SEIR model with treatment, Research in Applied Mathematics, 1 (2017) article id 101266. https://doi: 10.11131/2017/101266
• Garrapa, R., Predictor-corrector PECE method for fractional differential equations, MATLAB Central File Exchange, 2012, File ID:32918.
• Gonzalez-Parra, G., Arenas, A. J., Chen-Charpentier, B. M., A fractional order epidemic model for the simulation of outbreaks of influenza A(H1N1), Mathematical Methods in Applied Sciences, 37 (15) (2014), 2218-2226. https://doi: 10.1002/mma.2968
• Hethcote, H. W., Qualitative analyses of communicable disease model, Mathematical Biosciences, 28 (1976), 335–356.
• Hu, Z., Ma, W., Ruan, S., Analysis of SIR epidemic models with nonlinear incidence rate and treatment, Mathematical Biosciences, 238 (1) (2012), 1–20. https://doi:10.1016/j.mbs.2012.03.010
• Joshi, H., Yavuz, M., Townley, S., Jha, B. K., Stability analysis of a non-singular fractional-order COVID-19 model with nonlinear incidence and treatment rate, Physica Scripta, 98 (4) (2023), 045216. https://doi:10.1088/1402-4896/acbe7a
• Joshi, H., Jha, B. K., Chaos of calcium diffusion in Parkinson’s infectious disease model and treatment mechanism via Hilfer fractional derivative, Mathematical Modelling and Numerical Simulation with Applications, 1 (2) (2021), 84–94. https://doi:110.53391/mmnsa.2021.01.008
• Joshi, H., Jha, B. K., Yavuz, M., Modelling and analysis of fractional-order vaccination model for control of COVID-19 outbreak using real data, Mathematical Biosciences and Engineering, 20 (1) (2023), 213–240. https://doi: 10.3934/mbe.2023010
• Kaddar, A., Stability analysis in a delayed SIR epidemic model with a saturated incidence rate, Nonlinear Analysis: Modelling and Control, 15 (3) (2010), 299–306.
• Karaji, P. T., Nyamoradi, N., Analysis of a fractional SIR model with general incidence function, Applied Mathematics Letters, 108 (2020) 106499. https://doi:10.1016/j.aml.2020.106499
• Kermack, W. O., McKendrick, A. G., A contribution to the mathematical theory of epidemics, Proceedings of the Royal Society A, 115 (772) (1927), 700–721. https://doi:10.1098/rspa.1927.0118
• Korobeinikov, A., Lyapunov functions and global stability for SIR and SIRS epidemiological models with non-linear transmission, Bulletin of Mathematical Biology, 30 (2006), 615–626. https://doi:10.1007/s11538-005-9037-9
• Korobeinikov, A., Global properties of infectious disease models with nonlinear incidence, Bulletin of Mathematical Biology, 69 (2007), 1871–1886. https://doi: 10.1007/s11538-007-9196-y
• Korobeinikov, A., Maini, P. K., Nonlinear incidence and stability of infectious disease models, Mathematical Medicine and Biology: A Journal of the IMA, 22 (2) (2005), 113–128.
• Korobeinikov, A., Wake, G. C., Lyapunov functions and global stability for SIR, SIRS and SIS epidemiological models, Applied Mathematics Letters, 15 (8) (2002), 955–961. https://doi:10.1016/S0893-9659(02)00069-1
• Kuddus, M. A., Rahman, A., Analysis of COVID-19 using a modified SLIR model with nonlinear incidence, Results in Physics, 27 (2021), 104478. https://doi: 10.1016/j.rinp.2021.104478
• Li, M., Liu, X., An SIR epidemic model with time delay and general nonlinear incidence rate, Abstract and Applied Analysis, 2014 (2014), 131257. https://doi: 10.1155/2014/131257
• Li, X. Z., Li, W. S., Ghosh, M., Stability and bifurcation of an SIR epidemic model with nonlinear incidence and treatment, Applied Mathematics and Computation, 210 (1) (2009), 141–150. https://doi:10.1016/j.amc.2008.12.085
• Lin, W., Global existence theory and chaos control of fractional differential equations, Journal of Mathematical Analysis and Applications, 332 (1) (2007), 709–726. https://doi:10.1016/j.jmaa.2006.10.040
• Matignon, D., Stability results for fractional differential equations with applications to control processing, In: Computational Engineering in Systems Applications, 2 (1996), 963–968.
• McCluskey, C. C., Global stability of an SIR epidemic model with delay and general nonlinear incidence, Mathematical Biosciences and Engineering, 7 (4) (2010), 837–850. https://doi:10.3934/mbe.2010.7.837
• Naik, P.A., Yavuz, M., Qureshi, S., Zu, J., Townley, S., Modeling and analysis of COVID-19 epidemics with treatment in fractional derivatives using real data from Pakistan, The European Physical Journal Plus, 135(795) (2020). https://doi:10.1140/epjp/s13360-020-00819-5
• Naik, P. A., Zu J., Owolabi K. M., Global dynamics of a fractional order model for the transmission of HIV epidemic with optimal control, Chaos Solitons Fractals 138, (2020), 109826.
• Naim, M., Sabbar, Y., Zeb, A., Stability characterization of a fractional-order viral system with the non-cytolytic immune assumption, Mathematical Modelling and Numerical Simulation with Applications, 2(3) (2022), 164–176. https://doi:10.53391/mmnsa.2022.013
• Odibat, Z. M., Momani, S., An algorithm for the numerical solution of differential equations of fractional order, Journal of Applied Mathematics & Informatics, 26(1-2) (2008), 15-27.
• Özköse, F., Yavuz, M., Investigation of interactions between COVID-19 and diabetes with hereditary traits using real data: A case study in Turkey, Computers in Biology and Medicine, 141 (2022), 105044, https://doi.org/10.1016/j.compbiomed.2021.105044
• Pinto, C. M. A., Carvalho, A. R. M., A latency fractional order model for HIV dynamics, Journal of Computational and Applied Mathematics, ICMCMST 2015, 312(2017) (2015), 240–256. https://doi:10.1016/j.cam.2016.05.019
• Pinto, C. M. A., Tenreiro, Machado J. A., Fractional model for malaria transmission under control strategies, Computers & Mathematics with Applications, Special issue: Fractional Differentiation and its Applications, 66(5) (2013), 908–916. https://doi:10.1016/j.camwa.2012.11.017
• Podlubny, I., Fractional Differential Equations, New York, Academic Press, 1999.
• Silva, C. J., Torres, D. F., Stability of a fractional HIV/AIDS model, Mathematics and Computers in Simulation, 164 (2019), 180–190. https://doi: 10.1016/j.matcom.2019.03.016
• Uçar, S., Existence and uniqueness results for a smoking model with determination and education in the frame of non-singular derivatives, Discrete and Continuous Dynamical Systems-S, 14(7) (2021), 2571–2589. https://doi: 10.3934/dcdss.2020178
• Ullah, S., Khan, M. A., Farooq, M., A fractional model for the dynamics of TB virus, Chaos, Solitons & Fractals, 116 (2018), 63–71. https://doi: 10.1016/j.chaos.2018.09.001
• van den Driessche, P., Watmough, J., Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Mathematical Biosciences, 180(1) (2002), 29–48. https://doi: 0.1016/S0025-5564(02)00108-6
• Vargas-De-Leon, C., Volterra-type Lyapunov functions for fractional-order epidemic systems, Communications in Nonlinear Science and Numerical Simulation, 24(1) (2015), 75–85. https://doi: 10.1016/j.cnsns.2014.12.013
• Wang, J., Zhang, J,. Jin, Z., Analysis of an SIR model with bilinear incidence rate, Nonlinear Analysis, 11(4) (2010), 2390–2402. https://doi: 10.1016/j.nonrwa.2009.07.012
• Xu, C., Yu, Y., Chen, Y., Lu, Z., Forecast analysis of the epidemics trend of COVID-19 in the USA by a generalized fractional-order SEIR model, Nonlinear Dynamics, 101(3) (2020), 1621–1634. https://doi: 10.1007/s11071-020-05946-3
• Yang, Y., Li, J., Ma, Z., Liu, L., Global stability of two models with incomplete treatment for tuberculosis, Chaos, Solitons & Fractals, 43(1) (2010), 79–85. https://doi:10.1016/j.chaos.2010.09.002
• Yang, Y., Xu, L., Stability of a fractional order SEIR model with general incidence, Applied Mathematics Letters, 105 (2020), 106303. https://doi: 10.1016/j.aml.2020.106303
• Yavuz, M., Sene, N., Stability analysis and numerical computation of the fractional predator–prey model with the harvesting rate. Fractal and Fractional, 4(3), (2020), 35. https://doi.org/10.3390/fractalfract4030035
• Zhang, F., Li, Z., Zhang, F., Global stability of an SIR epidemic model with constant infectious period, Applied Mathematics and Computation, 199(1) (2008), 285–291. https://doi:10.1016/j.amc.2007.09.053