Impact of supervise neural network on a stochastic epidemic model with Levy noise

Rukhsar Ikram; Amir Khan; Aeshah A. Raezah; Rukhsar Ikram; Amir Khan; Aeshah A. Raezah

doi:10.3934/math.20241033

AIMS Mathematics

2024, Volume 9, Issue 8: 21273-21293. doi: 10.3934/math.20241033

Previous Article Next Article

Research article

Impact of supervise neural network on a stochastic epidemic model with Levy noise

1.
Department of Mathematics & Statistics, University of Swat, KPK, Pakistan
2.
Department of Mathematics, Faculty of Science, King Khalid University, Abha 62529, Saudi Arabia

Received: 20 March 2024 Revised: 03 June 2024 Accepted: 12 June 2024 Published: 02 July 2024
MSC : 92B20, 93E03

This paper primarily focused on analyzing a stochastic $ \mathcal{SVIR} $ epidemic model that incorporates Levy noises. The population may be divided into four distinct compartments: vulnerable class ($ \mathcal{S} $), vaccinated individuals ($ \mathcal{V} $), infected individuals ($ \mathcal{I} $), and recovered individuals ($ \mathcal{R} $). To achieve this, we chose existing and unique techniques as the most feasible solution. In the nexus, the stochastic model was theoretically analyzed using a suitable Lyapunov function. This analysis broadly covered the existence and uniqueness of the non-negative solution, as well as the dynamic properties related to both the disease-free equilibrium and the endemic equilibrium. In order to eradicate diseases, a stochastic threshold value denoted as "$\textbf{R}_0$" was used to determine if they may be eradicated. If $ \textbf{R}_0 < 1, $ it means that the illnesses have the potential to become extinct. Moreover, we provided numerical performance results of the proposed model using the artificial neural networks technique combined with the Bayesian regularization method. We firmly believe that this study will establish a solid theoretical foundation for comprehending the spread of an epidemic, the implementation of effective control strategies, and addressing real-world issues across various academic disciplines.
- white noise,
- Levy noise,
- epidemic model,
- artificial neural networks,
- numerical solutions
Citation: Rukhsar Ikram, Amir Khan, Aeshah A. Raezah. Impact of supervise neural network on a stochastic epidemic model with Levy noise[J]. AIMS Mathematics, 2024, 9(8): 21273-21293. doi: 10.3934/math.20241033

Related Papers:

Abstract

This paper primarily focused on analyzing a stochastic $ \mathcal{SVIR} $ epidemic model that incorporates Levy noises. The population may be divided into four distinct compartments: vulnerable class ($ \mathcal{S} $), vaccinated individuals ($ \mathcal{V} $), infected individuals ($ \mathcal{I} $), and recovered individuals ($ \mathcal{R} $). To achieve this, we chose existing and unique techniques as the most feasible solution. In the nexus, the stochastic model was theoretically analyzed using a suitable Lyapunov function. This analysis broadly covered the existence and uniqueness of the non-negative solution, as well as the dynamic properties related to both the disease-free equilibrium and the endemic equilibrium. In order to eradicate diseases, a stochastic threshold value denoted as "$\textbf{R}_0$" was used to determine if they may be eradicated. If $ \textbf{R}_0 < 1, $ it means that the illnesses have the potential to become extinct. Moreover, we provided numerical performance results of the proposed model using the artificial neural networks technique combined with the Bayesian regularization method. We firmly believe that this study will establish a solid theoretical foundation for comprehending the spread of an epidemic, the implementation of effective control strategies, and addressing real-world issues across various academic disciplines.

References

[1]	P. Agarwal, R. Singh, Modelling of transmission dynamics of Nipah virus (Niv): a fractional order approach, Physica A, 547 (2020), 124243. https://doi.org/10.1016/j.physa.2020.124243 doi: 10.1016/j.physa.2020.124243
[2]	J. Amador, D. Armesto, A. Gómez-Corral, Extreme values in SIR epidemic models with two strains and cross-immunity, Math. Biosci. Eng., 16 (2019), 1992–2022. https://doi.org/10.3934/mbe.2019098 doi: 10.3934/mbe.2019098
[3]	K. Okuwa, H. Inaba, T. Kuniya, Mathematical analysis for an age-structured SIRS epidemic model, Math. Biosci. Eng., 16 (2019), 6071–6102. https://doi.org/10.3934/mbe.2019304 doi: 10.3934/mbe.2019304
[4]	S. Kim, J. H. Byun, I. H. Jung, Global stability of an SEIR epidemic model where empirical distribution of incubation period is approximated by Coxian distribution, Adv. Differ. Equ., 2019 (2019), 469. https://doi.org/10.1186/s13662-019-2405-9 doi: 10.1186/s13662-019-2405-9
[5]	H. Qi, L. Liu, X. Meng, Dynamics of a nonautonomous stochastic SIS epidemic model with double epidemic hypothesis, Complexity, 2017 (2017), 4861391. https://doi.org/10.1155/2017/4861391 doi: 10.1155/2017/4861391
[6]	Q. Liu, D. Jiang, Dynamical behavior of a stochastic multigroup SIR epidemic model, Physica A, 526 (2019), 120975. https://doi.org/10.1016/j.physa.2019.04.211 doi: 10.1016/j.physa.2019.04.211
[7]	Q. Liu, D. Jiang, Stationary distribution and extinction of a stochastic SIR model with nonlinear perturbation, Appl. Math. Lett., 73 (2017), 8–15. https://doi.org/10.1016/j.aml.2017.04.021 doi: 10.1016/j.aml.2017.04.021
[8]	Z. Liu, C. Tian, A weighted networked SIRS epidemic model, J. Differ. Equations, 269 (2020), 10995–11019. https://doi.org/10.1016/j.jde.2020.07.038 doi: 10.1016/j.jde.2020.07.038
[9]	C. M. Kribs-Zaleta, J. X. Velasco-Hernández, A simple vaccination model with multiple endemic states, Math. Biosci., 164 (2000), 183–201. https://doi.org/10.1016/S0025-5564(00)00003-1 doi: 10.1016/S0025-5564(00)00003-1
[10]	X. Liu, Y. Takeuchi, S. Iwami, SVIR epidemic models with vaccination strategies, J. Theor. Biol., 253 (2008), 1–11. https://doi.org/10.1016/j.jtbi.2007.10.014 doi: 10.1016/j.jtbi.2007.10.014
[11]	S. M. A. Rahman, X. Zou, Modelling the impact of vaccination on infectious diseases dynamics, J. Biol. Dyn., 9 (2015), 307–320. https://doi.org/10.1080/17513758.2014.986545 doi: 10.1080/17513758.2014.986545
[12]	W. Halota, M. Muszyska, M. Pawowska, Hepatitis B virus serologic markers and anti-hepatitis B vaccination in patients with diabetes, Med. Sci. Monit., 8 (2002), 516–519.
[13]	X. Duan, S. Yuan, X. Li, Global stability of an SVR model with age of vaccination, Appl. Math. Comput., 226 (2014), 528–540. https://doi.org/10.1016/j.amc.2013.10.073 doi: 10.1016/j.amc.2013.10.073
[14]	P. Raúl, C. Vargas-De-León, P. Miramontes, Global stability results in a SVIR epidemic model with immunity loss rate depending on the vaccine-age, Abstr. Appl. Anal., 2015 (2015), 341854. http://doi.org/10.1155/2015/341854 doi: 10.1155/2015/341854
[15]	Y. Geng, J. Xu, Stability preserving NSFD scheme for a multi-group SVIR epidemic model, Math. Method. Appl. Sci., 40 (2017), 4917–4927. https://doi.org/10.1002/mma.4357 doi: 10.1002/mma.4357
[16]	W. Li, Y. Ding, Stability and branching analysis of a class of time-delay SVIR model with saturation incidence, Journal of Lanzhou University of Arts and Science (Natural Science Edition), 32 (2018), 1–6.
[17]	R. Zhang, S. Liu, Traveling waves for SVIR epidemic model with nonlocal dispersal, Math. Biosci. Eng., 16 (2019), 1654–1682. https://doi.org/10.3934/mbe.2019079 doi: 10.3934/mbe.2019079
[18]	S. Liao, W. Yang, A SVIR optimal control model with vaccination, (Chinese), Journal of Southwest University (Natural Science), 37 (2015), 72–78. https://doi.org/10.13718/j.cnki.xdzk.2015.01.011 doi: 10.13718/j.cnki.xdzk.2015.01.011
[19]	Z. Wang, R. Xu, Global dynamics of an SVIR epidemiological model with infection age and nonlinear incidence, J. Biol. Syst., 25 (2017), 419–440. https://doi.org/10.1142/S0218339017500206 doi: 10.1142/S0218339017500206
[20]	T. Khan, A. Khan, G. Zaman, The extinction and persistence of the stochastic hepatitis B epidemic model, Chaos Soliton. Fract., 108 (2018), 123–128. https://doi.org/10.1016/j.chaos.2018.01.036 doi: 10.1016/j.chaos.2018.01.036
[21]	M. Song, W. Zuo, D. Jiang, T. Hayat, Stationary distribution and ergodicity of a stochastic cholera model with multiple pathways of transmission, J. Frank. Inst., 357 (2020), 10773–10798. https://doi.org/10.1016/j.jfranklin.2020.04.061 doi: 10.1016/j.jfranklin.2020.04.061
[22]	Q. Liu, D. Jiang, T. Hayat, A. Alsaedi, B. Ahmad, Dynamical behavior of a higher order stochastically perturbed SIRI epidemic model with relapse and media coverage, Chaos Soliton. Fract., 139 (2020), 110013. https://doi.org/10.1016/j.chaos.2020.110013 doi: 10.1016/j.chaos.2020.110013
[23]	A. Lahrouz, A. Settati, A. Akharif, Effects of stochastic perturbation on the SIS epidemic system, J. Math. Biol., 74 (2017), 469–498. https://doi.org/10.1007/s00285-016-1033-1 doi: 10.1007/s00285-016-1033-1
[24]	Z. Cao, W. Feng, X. Wen, L. Zu, M. Cheng, Dynamics of a stochastic SIQR epidemic model with standard incidence, Physica A, 527 (2019), 121180. https://doi.org/10.1016/j.physa.2019.121180 doi: 10.1016/j.physa.2019.121180
[25]	Y. Cai, Y. Kang, W. Wang, A stochastic SIRS epidemic model with nonlinear incidence rate, Appl. Math. Comput., 305 (2017), 221–240. https://doi.org/10.1016/j.amc.2017.02.003 doi: 10.1016/j.amc.2017.02.003
[26]	X.-B. Zhang, X.-H. Zhang, The threshold of a deterministic and a stochastic SIQS epidemic model with varying total population size, Appl. Math. Model., 91 (2021), 749–767. https://doi.org/10.1016/j.apm.2020.09.050 doi: 10.1016/j.apm.2020.09.050
[27]	S. Wang, G. Hu, T. Wei, L. Wang, Permanence of hybrid competitive Lotka-Volterra system with Lévy noise, Physica A, 540 (2020), 123116. https://doi.org/10.1016/j.physa.2019.123116 doi: 10.1016/j.physa.2019.123116
[28]	A. El Koufi, A. Bennar, N. Yousfi, Dynamics behaviors of a hybrid switching epidemic model with levy noise, Appl. Math. Inform. Sci., 15 (2021), 131–142. http://dx.doi.org/10.18576/amis/150204 doi: 10.18576/amis/150204
[29]	Y. Zhou, S. Yuan, D. Zhao, Threshold behavior of a stochastic SIS model with Levy jumps, Appl. Math. Comput., 275 (2016), 255–267. https://doi.org/10.1016/j.amc.2015.11.077 doi: 10.1016/j.amc.2015.11.077
[30]	Y. Liu, Y. Zhang, Q. Wang, A stochastic SIR epidemic model with Lévy jump and media coverage, Adv. Differ. Equ., 2020 (2020), 70. https://doi.org/10.1186/s13662-020-2521-6 doi: 10.1186/s13662-020-2521-6
[31]	J. Wu, Dynamics of a two-predator one-prey stochastic delay model with Lévy noise, Physica A, 539 (2020), 122910. https://doi.org/10.1016/j.physa.2019.122910 doi: 10.1016/j.physa.2019.122910
[32]	M. El Fatini, I. Sekkak, Lévy noise impact on a stochastic delayed epidemic model with Crowly-Martin incidence and crowding effect, Physica A, 541 (2020), 123315. https://doi.org/10.1016/j.physa.2019.123315 doi: 10.1016/j.physa.2019.123315
[33]	A. Din, Y. Li, Stationary distribution extinction and optimal control for the stochastic hepatitis B epidemic model with partial immunity, Phys. Scr., 96 (2021), 074005. https://doi.org/10.1088/1402-4896/abfacc doi: 10.1088/1402-4896/abfacc
[34]	Y. Zhao, D. Jiang, The threshold of a stochastic SIRS epidemic model with saturated incidence, Appl. Math. Lett., 34 (2014), 90–93. https://doi.org/10.1016/j.aml.2013.11.002 doi: 10.1016/j.aml.2013.11.002

Reader Comments

Your name:*

Email:*
© 2024 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)