Modeling hepatitis B transmission dynamics with spatial diffusion and disability potential in the chronic stage

Kamel Guedri; Rahat Zarin; Ashfaq Khan; Amir Khan; Basim M. Makhdoum; Hatoon A. Niyazi; Kamel Guedri; Rahat Zarin; Ashfaq Khan; Amir Khan; Basim M. Makhdoum; Hatoon A. Niyazi

doi:10.3934/math.2025061

AIMS Mathematics

2025, Volume 10, Issue 1: 1322-1349. doi: 10.3934/math.2025061

Previous Article Next Article

Research article Special Issues

Modeling hepatitis B transmission dynamics with spatial diffusion and disability potential in the chronic stage

1.
Mechanical Engineering Department, College of Engineering and Architecture, Umm Al-Qura University, P.O. Box 5555, Makkah 21955, Saudi Arabia
2.
Department of Mathematics, Faculty of Science, King Mongkut's University of Technology, Thonburi (KMUTT), Bangkok 10140, Thailand
3.
Department of Mathematics and Statistics, University of Swat, Khyber Pakhtunkhwa, Pakistan
4.
King Salman Center for Disability Research, Riyadh 11614, Saudi Arabia
5.
Department of Clinical Microbiology and Immunology, Faculty of Medicine, King Abdulaziz University, Jeddah 21589, Saudi Arabia

Received: 29 November 2024 Revised: 28 December 2024 Accepted: 13 January 2025 Published: 21 January 2025
MSC : 35Q92, 65M06, 92D30

In this study, we introduce a novel reaction-diffusion epidemic model to analyze the transmission dynamics of the hepatitis B virus (HBV). The model captured the interactions between five population groups: Susceptible individuals, those in the latent stage, acutely infected individuals, chronically infected individuals, and those who have recovered, while considering the spatial movement of these groups. Chronic HBV infection contributes to severe liver diseases such as cirrhosis and hepatocellular carcinoma. It is also a major cause of long-term disability due to complications that impair daily functioning. The stability conditions for the model were derived, and the basic reproductive number, $ R_0 $, was calculated using the next-generation matrix approach. Numerical simulations were performed using the Crank-Nicolson operator splitting method and the Unconditionally Positivity Preserving technique to solve the model under scenarios with and without diffusion. The stability of the endemic equilibrium point was analyzed comprehensively. Detailed simulation results are presented, highlighting a comparative analysis of the numerical findings in cases where exact solutions were unavailable. The reliability of the numerical results was validated by their alignment with theoretical expectations.
- hepatitis B virus,
- long-term disability,
- epidemic modeling,
- reaction-diffusion
Citation: Kamel Guedri, Rahat Zarin, Ashfaq Khan, Amir Khan, Basim M. Makhdoum, Hatoon A. Niyazi. Modeling hepatitis B transmission dynamics with spatial diffusion and disability potential in the chronic stage[J]. AIMS Mathematics, 2025, 10(1): 1322-1349. doi: 10.3934/math.2025061

Related Papers:

Abstract

In this study, we introduce a novel reaction-diffusion epidemic model to analyze the transmission dynamics of the hepatitis B virus (HBV). The model captured the interactions between five population groups: Susceptible individuals, those in the latent stage, acutely infected individuals, chronically infected individuals, and those who have recovered, while considering the spatial movement of these groups. Chronic HBV infection contributes to severe liver diseases such as cirrhosis and hepatocellular carcinoma. It is also a major cause of long-term disability due to complications that impair daily functioning. The stability conditions for the model were derived, and the basic reproductive number, $ R_0 $, was calculated using the next-generation matrix approach. Numerical simulations were performed using the Crank-Nicolson operator splitting method and the Unconditionally Positivity Preserving technique to solve the model under scenarios with and without diffusion. The stability of the endemic equilibrium point was analyzed comprehensively. Detailed simulation results are presented, highlighting a comparative analysis of the numerical findings in cases where exact solutions were unavailable. The reliability of the numerical results was validated by their alignment with theoretical expectations.

References

[1]	WHO, Hepatitis B, The World Health Organisation. Available from: http://www.who.int/mediacentre/factsheets/fs204/en/
[2]	Canadian Centre for Occupational Health and Safety, Hepatitis B. Available from: http://www.ccohs.ca/oshanswers/diseases/hepatitis_b.html
[3]	A. Schweitzer, J. Horn, R. T. Mikolajczyk, G. Krause, J. J. Ott, Estimations of worldwide prevalence of chronic hepatitis B virus infection: A systematic review of data published between 1965 and 2013, The Lancet, 386 (2015), 1546–1555. https://doi.org/10.1016/S0140-6736(15)61412-X doi: 10.1016/S0140-6736(15)61412-X
[4]	G. F. Medley, N. A. Lindop, W. J. Edmunds, D. J. Nokes, Hepatitis-B virus endemicity: Heterogeneity, catastrophic dynamics and control, Nat. Med., 7 (2001), 619–624. https://doi.org/10.1038/87953 doi: 10.1038/87953
[5]	A. Khan, R. Zarin, G. Hussain, A. H. Usman, U. W. Humphries, J. F. Gomez-Aguilar, Modeling and sensitivity analysis of HBV epidemic model with convex incidence rate, Results Phys., 22 (2021), 103836. https://doi.org/10.1016/j.rinp.2021.103836 doi: 10.1016/j.rinp.2021.103836
[6]	R. Zarin, Modeling and numerical analysis of fractional order hepatitis B virus model with harmonic mean type incidence rate, Comput. Method Biomech Biomed, 26 (2023), 1018–1033. https://doi.org/10.1080/10255842.2022.2103371 doi: 10.1080/10255842.2022.2103371
[7]	J. Mann, M. Roberts, Modelling the epidemiology of hepatitis B in New Zealand, J. Theor. Biol., 269 (2011), 266–272. https://doi.org/10.1016/j.jtbi.2010.10.028 doi: 10.1016/j.jtbi.2010.10.028
[8]	MMWR Recommendations and Reports, Hepatitis B virus: A comprehensive strategy for eliminating transmission in the United States through universal childhood vaccination. Recommendations of the Immunization Practices Advisory Committee (ACIP), 1991, 1–25.
[9]	M. K. Libbus, L. M. Phillips, Public health management of perinatal hepatitis B virus, Public Health Nurs., 26 (2009), 353–361. https://doi.org/10.1111/j.1525-1446.2009.00790.x doi: 10.1111/j.1525-1446.2009.00790.x
[10]	F. B. Hollinger, D. T. Lau, Hepatitis B: The pathway to recovery through treatment, Gastroenterol. Clin. N., 35 (2006), 425–461. https://doi.org/10.1016/j.gtc.2006.03.002 doi: 10.1016/j.gtc.2006.03.002
[11]	C. L. Lai, M. F. Yuen, The natural history and treatment of chronic hepatitis B: A critical evaluation of standard treatment criteria and end points, Ann. Intern. Med., 147 (2007), 58–61. https://doi.org/10.7326/0003-4819-147-1-200707030-00010 doi: 10.7326/0003-4819-147-1-200707030-00010
[12]	R. Zarin, A. Raouf, A. Khan, U. W. Humphries, Modeling hepatitis B infection dynamics with a novel mathematical model incorporating convex incidence rate and real data, Eur. Phys. J. Plus, 138 (2023), 1056. https://doi.org/10.1140/epjp/s13360-023-04642-6 doi: 10.1140/epjp/s13360-023-04642-6
[13]	R. Zarin, U. W. Humphries, T. Saleewong, Advanced mathematical modeling of hepatitis B transmission dynamics with and without diffusion effect using real data from Thailand, Eur. Phys. J. Plus, 139 (2024), 385. https://doi.org/10.1140/epjp/s13360-024-05154-7 doi: 10.1140/epjp/s13360-024-05154-7
[14]	R. M. Anderson, R. M. May, Infectious disease of humans: Dynamics and control, Oxford: Oxford University Press, 1991. https://doi.org/10.1093/oso/9780198545996.001.0001
[15]	S. Thornley, C. Bullen, M. Roberts, Hepatitis B in a high prevalence New Zealand population: A mathematical model applied to infection control policy, J. Theor. Biol., 254 (2008), 599–603. https://doi.org/10.1016/j.jtbi.2008.06.022 doi: 10.1016/j.jtbi.2008.06.022
[16]	S. Zhao, Z. Xu, Y. Lu, A mathematical model of hepatitis B virus transmission and its application for vaccination strategy in China, Int. J. Epidemiol., 29 (2000), 744–752. https://doi.org/10.1093/ije/29.4.744 doi: 10.1093/ije/29.4.744
[17]	K. Wang, W. Wang, S. Song, Dynamics of an HBV model with diffusion and delay, J. Theor. Biol., 253 (2008), 36–44. https://doi.org/10.1016/j.jtbi.2007.11.007 doi: 10.1016/j.jtbi.2007.11.007
[18]	S. Zhang, Y. Zhou, The analysis and application of an HBV model, Appl. Math. Model., 36 (2012), 1302–1312. https://doi.org/10.1016/j.apm.2011.07.087 doi: 10.1016/j.apm.2011.07.087
[19]	T. Khan, G. Zaman, M. I. Chohan, The transmission dynamic and optimal control of acute and chronic hepatitis B, J. Biol. Dyn., 11 (2016), 172–189. https://doi.org/10.1080/17513758.2016.1256441 doi: 10.1080/17513758.2016.1256441
[20]	G. P. Samanta, S. Sharma, Analysis of a delayed Chlamydia epidemic model with pulse vaccination, Appl. Math. Comput., 230 (2014), 555–569. https://doi.org/10.1016/j.amc.2013.12.123 doi: 10.1016/j.amc.2013.12.123
[21]	T. Khan, G. Zaman, O. Algahtani, Transmission dynamic and vaccination of hepatitis B epidemic model, Wulfenia J., 22 (2015), 230–241.
[22]	J. Danane, K. Allali, Mathematical analysis and treatment for a delayed hepatitis B viral infection model with the adaptive immune response and DNA-containing capsids, High-Throughput, 7 (2018), 35. https://doi.org/10.3390/ht7040035 doi: 10.3390/ht7040035
[23]	A. A. Raezah, A. Raouf, R. Zarin, A. Khan, Exploring the effectiveness of control measures and long-term behavior in Hepatitis B: An analysis of an endemic model with horizontal and vertical transmission, Results Phys., 53 (2023), 106966. https://doi.org/10.1016/j.rinp.2023.106966 doi: 10.1016/j.rinp.2023.106966
[24]	R. Zarin, U. W. Humphries, Modeling the dynamics of COVID-19 Epidemic with a reaction-diffusion framework: A case study from Thailand, Eur. Phys. J. Plus, 139 (2024), 1076. https://doi.org/10.1140/epjp/s13360-024-05870-0 doi: 10.1140/epjp/s13360-024-05870-0
[25]	I. Zada, M. N. Jan, N. Ali, D. Alrowail, K. S. Nisar, G. Zaman, Mathematical analysis of hepatitis B epidemic model with optimal control, Adv. Differ. Equ., 2021 (2021), 451. https://doi.org/10.1186/s13662-021-03607-2 doi: 10.1186/s13662-021-03607-2
[26]	O. Diekmann, J. A. P. Heesterbeek, J. A. J. Metz, On the definition and the computation of the basic reproduction ratio $R_0$ in models for infectious diseases in heterogeneous populations, J. Math. Biol., 28 (1990), 365–382. https://doi.org/10.1007/BF00178324 doi: 10.1007/BF00178324
[27]	P. van den Driessche, Reproduction numbers of infectious disease models, Infect. Dis. Model., 2 (2017), 288–303. https://doi.org/10.1016/j.idm.2017.06.002 doi: 10.1016/j.idm.2017.06.002
[28]	A. Chakrabrty, M. Singh, B. Lucy, P. Ridland, Predator-prey model with prey-taxis and diffusion, Math. Comput. Model., 46 (2007), 482–498. https://doi.org/10.1016/j.mcm.2006.10.010 doi: 10.1016/j.mcm.2006.10.010
[29]	N. Ahmed, S. S. Tahira, M. Rafiq, M. A. Rehman, M. Ali, M. O. Ahmad, Positivity preserving operator splitting nonstandard finite difference methods for SEIR reaction diffusion model, Open Math., 17 (2019), 313–330. https://doi.org/10.1515/math-2019-0027 doi: 10.1515/math-2019-0027
[30]	R. E. Mickens, Nonstandard finite difference models of differential equations, World Scientific, 1993. https://doi.org/10.1142/2081
[31]	C. Basdevant, M. Devile, P. Haldenwang, J. M. Lacroix, J. Ouazzan, R. Peyret, et al., Spectral and finite difference solutions of the Burgers equation, Comput. Fluids, 14 (1986), 23–41. https://doi.org/10.1016/0045-7930(86)90036-8 doi: 10.1016/0045-7930(86)90036-8
[32]	D. Gottlieb, S. A. Orszag, Numerical analysis of spectral methods: Theory and applications, SIAM, 1977.
[33]	N. Ahmed, M. Rafiq, M. A. Rehman, M. S. Iqbal, M. Ali, Numerical modelling of three-dimensional Brusselator reaction diffusion system, AIP Adv., 9 (2019), 015205. https://doi.org/10.1063/1.5070093 doi: 10.1063/1.5070093
[34]	N. Ahmed, M. Rafiq, M. A. Rehman, M. Ali, M. O. Ahmad, Numerical modeling of SEIR measles dynamics with diffusion, Commun. Math. Appl., 9 (2018), 315–326.

Reader Comments

Your name:*

Email:*
© 2025 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)