Impact of vaccination and immune response in reducing Hepatitis B-related liver disease model

Thangavel Megala; Thangaraj Nandha Gopal; Mehmet Yavuz; Muthurathinam Sivabalan; Manickasundaram Siva Pradeep; Thangavel Megala; Thangaraj Nandha Gopal; Mehmet Yavuz; Muthurathinam Sivabalan; Manickasundaram Siva Pradeep

doi:10.3934/nhm.2026075

Networks and Heterogeneous Media

2026, Volume 21, Issue 5: 1987-2006. doi: 10.3934/nhm.2026075

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Impact of vaccination and immune response in reducing Hepatitis B-related liver disease model

1.
Department of Mathematics, Sri Ramakrishna Mission Vidyalaya College of Arts and Science, Periyanaickenpalayam, Coimbatore, Tamil Nadu 641 020, India
2.
Department of Mathematics and Computer Sciences, Necmettin Erbakan University, 42090 Konya, Türkiye
3.
Department of Applied Mathematics and Informatics, Kyrgyz-Turkish Manas University, Bishkek 720038, Kyrgyzstan
4.
Centre for Environmental Mathematics, Faculty of Environment, Science and Economy, University of Exeter, Cornwall, TR10 9FE, United Kingdom
5.
Department of Mathematics, United Institute of Technology, Periyanaickenpalayam, Coimbatore, Tamil Nadu 641 020, India

Received: 17 December 2025 Revised: 16 September 2026 Accepted: 24 September 2026 Published: 09 October 2026

Hepatitis B virus (HBV) infection can progress from acute infection to chronic liver disease, posing a major public health concern. In this work, a mathematical model incorporating immune response and vaccination is developed using susceptible, infected, chronic, and recovered populations. Positivity and boundedness of solutions are established, and disease-free and endemic equilibria are analyzed. The basic reproduction numbers and stability conditions are derived using the next-generation matrix and Jacobian methods. Critical thresholds are obtained as $ \Upsilon^* = 0.2237 $ for immune response and $ \Gamma^* = 0.7466 $ for vaccination. Numerical simulations using the fourth-order Runge–Kutta method demonstrate that enhanced immune response and vaccination can reduce HBV infection and chronic cases. The findings provide quantitative evidence on the role of immune response and vaccination in controlling HBV transmission and disease progression.
- vaccination,
- disease progression,
- liver disease,
- chronic infection
Citation: Thangavel Megala, Thangaraj Nandha Gopal, Mehmet Yavuz, Muthurathinam Sivabalan, Manickasundaram Siva Pradeep. Impact of vaccination and immune response in reducing Hepatitis B-related liver disease model[J]. Networks and Heterogeneous Media, 2026, 21(5): 1987-2006. doi: 10.3934/nhm.2026075

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Abstract

Hepatitis B virus (HBV) infection can progress from acute infection to chronic liver disease, posing a major public health concern. In this work, a mathematical model incorporating immune response and vaccination is developed using susceptible, infected, chronic, and recovered populations. Positivity and boundedness of solutions are established, and disease-free and endemic equilibria are analyzed. The basic reproduction numbers and stability conditions are derived using the next-generation matrix and Jacobian methods. Critical thresholds are obtained as $ \Upsilon^* = 0.2237 $ for immune response and $ \Gamma^* = 0.7466 $ for vaccination. Numerical simulations using the fourth-order Runge–Kutta method demonstrate that enhanced immune response and vaccination can reduce HBV infection and chronic cases. The findings provide quantitative evidence on the role of immune response and vaccination in controlling HBV transmission and disease progression.

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