A NOVEL STOCHASTIC HEPATITIS B VIRUS EPIDEMIC MODEL WITH SECOND-ORDER MULTIPLICATIVE &#x03B1;-STABLE NOISE AND REAL DATA

Anwarud DIN; Yassine SABBAR; Peng WU

doi:10.1007/s10473-024-0220-1

Acta mathematica scientia, Series B >

2024 , Vol. 44 >Issue 2: 752 - 788

DOI: https://doi.org/10.1007/s10473-024-0220-1

A NOVEL STOCHASTIC HEPATITIS B VIRUS EPIDEMIC MODEL WITH SECOND-ORDER MULTIPLICATIVE α-STABLE NOISE AND REAL DATA

Anwarud DIN ,
Yassine SABBAR ,
Peng WU

Expand

1. Department of Mathematics, Sun Yat-Sen University, Guangzhou 510275, China;
2. MAIS Laboratory, MAMCS Group, FST Errachidia, Moulay Ismail University of Meknes, P.O. Box 509, Errachidia 52000, Morocco;
3. School of Science, Hangzhou Dianzi University, Hangzhou 310018, China; School of Data Sciences, Zhejiang University of Finance and Economics, Hangzhou 310018, China

Yassine SABBAR, E-mail: y.sabbar@umi.ac.ma; Peng WU, E-mail: hzpengwu@163.com

Received date: 2022-11-20

Revised date: 2023-01-08

Online published: 2024-04-16

Supported by

NSFC (12201557) and the Foundation of Zhejiang Provincial Education Department, China (Y202249921).

Fold

Abstract

This work presents an advanced and detailed analysis of the mechanisms of hepatitis B virus (HBV) propagation in an environment characterized by variability and stochasticity. Based on some biological features of the virus and the assumptions, the corresponding deterministic model is formulated, which takes into consideration the effect of vaccination. This deterministic model is extended to a stochastic framework by considering a new form of disturbance which makes it possible to simulate strong and significant fluctuations. The long-term behaviors of the virus are predicted by using stochastic differential equations with second-order multiplicative $\alpha$-stable jumps. By developing the assumptions and employing the novel theoretical tools, the threshold parameter responsible for ergodicity (persistence) and extinction is provided. The theoretical results of the current study are validated by numerical simulations and parameters estimation is also performed. Moreover, we obtain the following new interesting findings: (a) in each class, the average time depends on the value of $\alpha$; (b) the second-order noise has an inverse effect on the spread of the virus; (c) the shapes of population densities at stationary level quickly changes at certain values of $\alpha$. The last three conclusions can provide a solid research base for further investigation in the field of biological and ecological modeling.

Key words： HBV model; nonlinear perturbation; probabilistic bifurcation; long-run forecast; numerical simulation

Cite this article

Anwarud DIN , Yassine SABBAR , Peng WU . A NOVEL STOCHASTIC HEPATITIS B VIRUS EPIDEMIC MODEL WITH SECOND-ORDER MULTIPLICATIVE α-STABLE NOISE AND REAL DATA[J]. Acta mathematica scientia, Series B, 2024 , 44(2) : 752 -788 . DOI: 10.1007/s10473-024-0220-1

References

[1] Akbari R, Kamyad A V, Heydari A A. Stability analysis of the transmission dynamics of an HBV model. Int J Indu Math, 2016, 8: 119-219
[2] Akdim A K, Ez-Zetouni, Zahid M. The influence of awareness campaigns on the spread of an infectious disease: a qualitative analysis of a fractional epidemic model. Modeling Earth Systems and Environment, 2021, 2(3): 1-9
[3] Alade T O, Shafeek A G, Saud M A. Global stability of a class of virus dynamics models with general incidence rate and multitarget cells. The European Physical Journal Plus, 2021, 136: 1-20
[4] Allen L J S, Van D P. Stochastic epidemic models with a backward bifurcation. Mathematical Biosciences & Engineering, 2006, 3: 445
[5] Bandekar R, Ghosh M. Mathematical modeling of COVID-19 in India and its states with optimal control. Modeling Earth Systems and Environment, 2021, 2: 1-16
[6] Costa A, Pires M, Resque R, Almeida S S. Mathematical modeling of the infectious diseases: key concepts and applications. Journal of Infectious Diseases and Epidemiology, 2021, 7: 209
[7] Davtyan M, Brandon B, Morenike O F. Addressing Ebola-related stigma: lessons learned from HIV/AIDS. Global Health Action, 2014, 7: 26058
[8] Din A, Yongjin L. Stationary distribution extinction and optimal control for the stochastic hepatitis B epidemic model with partial immunity. Physica Scripta, 2021, 96: 074005-074020
[9] Din A, Yongjin L, Abdullahi Y. Delayed hepatitis B epidemic model with stochastic analysis. Chaos, Solitons & Fractals, 2021, 146: 110839
[10] Din A, Yongjin L, Khan T, Zaman G. Stochastic dynamics of hepatitis B epidemics. Results in Physics, 2021, 20: 103730-102740
[11] Emerenini B O, Inyama S. Mathematical model and analysis of hepatitis B virus transmission dynamics. Research, 2017, 7: 1-12
[12] Ghita M, Copot M, Ionescu C M. Lung cancer dynamics using fractional order impedance modeling on a mimicked lung tumor setup. Journal of Advanced Research, 2021, 32: 61-71
[13] Hussain G, Khan A, Zahri M, Zaman G. Ergodic stationary distribution of stochastic epidemic model for HBV with double saturated incidence rates and vaccination. Chaos, Solitons &Fractals, 2022, 160: 112195-112205
[14] Ji C, Jiang D, Shi N. Multigroup SIR epidemic model with stochastic perturbation. Physica A: Statistical Mechanics and Its Applications, 2011, 390: 1747-62
[15] Kahn J O, Bruce D W. Acute human immunodeficiency virus type 1 infection. New England Journal of Medicine, 1998, 339: 33-39
[16] Khan T, Jung H I, Zaman G. A stochastic model for the transmission dynamics of hepatitis B virus. Journal of Biological Dynamics, 2018, 13: 328-344
[17] Khan T, Khan A, Zaman G. The extinction and persistence of the stochastic hepatitis B epidemic model. Chaos, Solitons & Fractals, 2018, 108: 123-128
[18] Kiouach D, Sabbar Y. The long-time behavior of a stochastic SIR epidemic model with distributed delay and multidimensional Lêvy jumps. International Journal of Biomathematics, 2022, 15: 2250004
[19] Lotfy W M. Plague in Egypt: Disease biology, history and contemporary analysis: A minireview. Journal of Advanced Research, 2015, 6(4): 549-554
[20] Mao X.Stochastic Differential Equations and Applications. Oxford: Horwood, 2007
[21] Mann J, Roberts M. Modelling the epidemiology of hepatitis B in New Zealand. J Theor Biol, 2011, 269: 266-272
[22] Marquardt D. An algorithm for least squares estimation of nonlinear parameters. SIAM Journal Applied Mathematics, 1963, 11: 431-441
[23] Michelsen L, Thomsen Per G, Morten K.Parameter Estimation in Nonlinear Dynamical Systems. Lyngby: Technical University of Denmark, 2004
[24] Mohammed K B. Modeling the effectiveness of social distancing interventions on the epidemic curve of coronavirus disease in Ethiopia. Modeling Earth Systems and Environment, 2021, 40: 1-11
[25] Nowak M, Robert M M.Virus Dynamics: Mathematical Principles of Immunology and Virology: Mathematical Principles of Immunology and Virology. Oxford: Oxford University Press, 2000
[26] Nowak M A, Sebastian B, Andrew M H, et al. Viral dynamics in hepatitis B virus infection. Proceedings of the National Academy of Sciences, 1996, 93: 4398-4402
[27] Pattyn J, Hendrickx G, Vorsters D V. Hepatitis B Vaccines. The Journal of Infectious Diseases, 2021, 224: 1-12
[28] Rosinski J. Tempering stable processes. Stochastic Processes and Their Applications, 2007, 117: 677-707
[29] Sabbar Y, Kiouach D, Rajasekar S P, El-idrissi S E. The influence of second-order Lêvy noise on the dynamic of an SIC contagious illness model: New framework, critical comparison and an application to COVID-19 (SARS-CoV-2) case. Chaos, Solitons & Fractals, 2022, 159: 112110-112120
[30] Shen Z, Zhang H, Du L, et al. Initiation and termination of epilepsy induced by Lévy noise: A view from the cortical neural mass model. Chaos, Solitons & Fractals, 2023, 167: 113038
[31] Stettner L. On the existence and uniqueness of invariant measure for continuous-time markov processes. Technical Report1986, LCDS, Brown University,Province: 18-86
[32] Tong J, Zhang Z, Bao J. The stationary distribution of the facultative population model with a degenerate noise. Statistics and Probability Letters, 2013, 83: 655-664
[33] Waikhom P, Jain R Tegar S. Sensitivity and stability analysis of a delayed stochastic epidemic model with temperature gradients. Modeling Earth Systems and Environment, 2016, 2(1): 1-18
[34] World Health Organization. Hepatitis B.[2023-10-11]. https://www.who.int/news-room/fact-sheets/detail/hepatitis-b
[35] Zhang S, Xu X. A mathematical model for hepatitis B with infection-age structure. Discrete Contin Dyn Syst-Ser B, 2016, 21: 1329-1346
[36] Zhang S, Zhou Y. Dynamic analysis of a hepatitis B model with three-age-classes. Commun Nonlinear Sci, 2014, 19: 2466-2478
[37] Zhang T, Li N, Xie W, Ding X. Mathematical analysis and simulation of a Hepatitis B model with time delay: A case study for Xinjiang, China. Mathematical Biosciences & Engineering, 2020, 17: 1757-1775
[38] Zhao Y, Jiang D. The threshold of a stochastic SIS epidemic model with vaccination. Applied Mathematics and Computation, 2014, 243: 718-727
[39] Zou L, Ruan S, Zhang W. On the sexual transmission dynamics of hepatitis B virus in China. J Theor Biol, 2015, 369: 1-12

Options

Abstract

Outlines

模态框（Modal）标题

Abstract

Cite this article

References