Existence and stability analysis of fractional-order SVEIR epidemic models with generalized incidence

Yassine  Babrhou; Prof. Ali  El Mfadel; Brahim   El Boukari

doi:10.5269/bspm.75184

Yassine Babrhou Laboratory of Applied Mathematics and Scientific Computing, Sultan Moulay Slimane University, Beni Mellal, Morocco
Prof. Ali El Mfadel
Brahim El Boukari Laboratory of Applied Mathematics and Scientific Computing, Sultan Moulay Slimane University, Beni Mellal, Morocco

Abstract

This paper develops a fractional-order epidemic model ($SVEIR$) incorporating a generalized incidence rate, analyzed within the framework of the Caputo fractional derivative. The foundational properties of the model, including existence, uniqueness, non-negativity, and boundedness of solutions, are rigorously established to ensure its well-posedness. The basic reproduction number $R_0$ is computed using the next-generation matrix method, providing threshold criteria for disease elimination and persistence. The model exhibits two equilibria: the disease-free equilibrium and the endemic equilibrium. Through stability analysis, we prove the global stability of these equilibria by constructing suitable Lyapunov functions and applying LaSalle's invariance principle. To validate the theoretical results and examine the effects of various epidemiological parameters, numerical simulations are performed using MATLAB. These simulations provide deeper insights into the dynamic behavior of the proposed fractional model, highlighting its potential applicability in understanding and controlling disease spread.

Downloads

Download data is not yet available.

References

Jia, J. and Li, Q., 2007. Qualitative analysis of an SIR epidemic model with stage structure. Applied Mathematics and Computation, 193(1), pp.106-115.

Busenberg, S., Cooke, K. and Hsieh, Y. H., 1995. A model for HIV in Asia. Mathematical biosciences, 128(1-2), pp.185-210.

Hou, Q., Sun, X., Zhang, J., Liu, Y., Wang, Y. and Jin, Z., 2013. Modeling the transmission dynamics of sheep brucellosis in Inner Mongolia Autonomous Region, China. Mathematical biosciences, 242(1), pp.51-58.

Kermack, William Ogilvy, and Anderson G. McKendrick. A contribution to the mathematical theory of epidemics. Proceedings of the royal society of london. Series A, Containing papers of a mathematical and physical character 115, no. 772 (1927): 700-721.

Ahmed, E., and A. S. Elgazzar. On fractional order differential equations model for nonlocal epidemics. Physica A: Statistical Mechanics and its Applications 379, no. 2 (2007): 607-614.

Yu, Jimin, Hua Hu, Shangbo Zhou, and Xiaoran Lin. Generalized Mittag-Leffler stability of multi-variables fractional order nonlinear systems. Automatica 49, no. 6 (2013): 1798-1803.

Diethelm, K. and Ford, N.J., 2002. Analysis of fractional differential equations. Journal of Mathematical Analysis and Applications, 265(2), pp.229-248.

van den Driesche P. and Watmough J. Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission. Mathematical Biosciences. 2002;180:29-48.

Settati, A., Lahrouz, A., El Jarroudi, M. and El Jarroudi, M., 2016. Dynamics of hybrid switching diffusions SIRS model. Journal of Applied Mathematics and Computing, 52, pp.101-123.

Wei, C. and Chen, L., 2008. A delayed epidemic model with pulse vaccination. Discrete Dynamics in Nature and Society, 2008.

Kaddar, A., 2009. On the dynamics of a delayed SIR epidemic model with a modified saturated incidence rate. Electronic Journal of Differential Equations (EJDE)[electronic only], 2009, pp.Paper-No.

Li, B., Yuan, S. and Zhang, W., 2011. Analysis on an epidemic model with a ratio-dependent nonlinear incidence rate. International Journal of Biomathematics, 4(02), pp.227-239.

Adnani, J., Hattaf, K. and Yousfi, N., 2013. Stability analysis of a stochastic SIR epidemic model with specific nonlinear incidence rate. International Journal of Stochastic Analysis, 2013, pp.1-4.

Du, N. H., Dieu, N. T. and Nhu, N. N., 2019. Conditions for permanence and ergodicity of certain SIR epidemic models. Acta Applicandae Mathematicae, 160, pp.81-99.

Zhou, X. and Cui, J., 2011. Global stability of the viral dynamics with Crowley-Martin functional response. Bulletin of the Korean Mathematical Society, 48(3), pp.555-574.

Tuong, T. D., Nguyen, D. H., Dieu, N.T. and Tran, K., 2019. Extinction and permanence in a stochastic SIRS model in regime-switching with general incidence rate. Nonlinear Analysis: Hybrid Systems, 34, pp.121-130.

Du, N. H. and Nhu, N.N., 2020. Permanence and extinction for the stochastic SIR epidemic model. Journal of Differential Equations, 269(11), pp.9619-9652.

Lin, W., 2007. Global existence theory and chaos control of fractional differential equations. Journal of Mathematical Analysis and Applications, 332(1), pp.709-726.

La Salle, J. P., 1966. An invariance principle in the theory of stability (No. NASA-CR-74165).

Yi, N., Zhang, Q., Mao, K., Yang, D. and Li, Q., 2009. Analysis and control of an SEIR epidemic system with nonlinear transmission rate. Mathematical and computer modelling, 50(9-10), pp.1498-1513.

Beretta, E. and Breda, D., 2011. An SEIR epidemic model with constant latency time and infectious period. Mathematical Biosciences and Engineering, 8(4), pp.931-952.

Wu, L. I. and Feng, Z., 2000. Homoclinic bifurcation in an SIQR model for childhood diseases. Journal of differential equations, 168(1), pp.150-167.

Kilbas, Anatoli Aleksandrovich, Hari M. Srivastava, and Juan J. Trujillo. Theory and applications of fractional differential equations. Vol. 204. elsevier, 2006.

Jena, Rajarama Mohan, Snehashish Chakraverty, Hadi Rezazadeh, and Davood Domiri Ganji. On the solution of timefractional dynamical model of Brusselator reaction-diffusion system arising in chemical reactions. Mathematical Methods in the Applied Sciences 43, no. 7 (2020): 3903-3913.

Srinivasa, Kumbinarasaiah, and Hadi Rezazadeh. Numerical solution for the fractional-order one-dimensional telegraph equation via wavelet technique. International Journal of Nonlinear Sciences and Numerical Simulation 22, no. 6 (2021): 767-780.

Ahmed, Nauman, Muhammad Rafiq, Waleed Adel, Hadi Rezazadeh, Ilyas Khan, and Kottakkaran Sooppy Nisar. Structure preserving numerical analysis of HIV and CD4+ T-cells reaction diffusion model in two space dimensions. Chaos, Solitons and Fractals 139 (2020): 110307.

Korobeinikov, Andrei, and Philip K. Maini. Non-linear incidence and stability of infectious disease models. Mathematical medicine and biology: a journal of the IMA 22, no. 2 (2005): 113-128.

Korobeinikov, Andrei. Lyapunov functions and global stability for SIR and SIRS epidemiological models with non-linear transmission. Bulletin of Mathematical biology 68 (2006): 615-626.

Korobeinikov, Andrei. Global properties of infectious disease models with nonlinear incidence. Bulletin of Mathematical Biology 69 (2007): 1871-1886.

Korobeinikov, Andrei. Stability of ecosystem: global properties of a general predator-prey model. Mathematical Medicine and Biology: A Journal of the IMA 26, no. 4 (2009): 309-321.

Vargas-De-Leon, Cruz. Volterra-type Lyapunov functions for fractional-order epidemic systems. Communications in Nonlinear Science and Numerical Simulation 24, no. 1-3 (2015): 75-85.