Local Stability and Threshold Analysis of an SEITR Model with Standard Incidence for Hepatitis B Transmission

Nawara Elfakhakhre; Hind Masoud; Noor Habil; Malak Alawami; Albatool Binzabiah; Maryam Elfakhakhre

doi:10.54361/ajmas.261129

Authors

Nawara Elfakhakhre Department of Mathematics, Faculty of Science, University of Derna, Derna, Libya https://orcid.org/0000-0002-2369-8435
Hind Masoud Department of Mathematics, Faculty of Science, University of Derna, Derna, Libya https://orcid.org/0009-0001-6695-745X
Noor Habil Department of Mathematics, Faculty of Education, University of Derna, Derna, Libya https://orcid.org/0009-0006-2283-6054
Malak Alawami Department of Mathematics, Faculty of Education, University of Derna, Derna, Libya https://orcid.org/0000-0002-7922-9523
Albatool Binzabiah Department of Mathematics, Faculty of Education, University of Derna, Derna, Libya https://orcid.org/0000-0002-1437-4455
Maryam Elfakhakhre Department of Mathematics, Faculty of Education, University of Derna, Derna, Libya https://orcid.org/0000-0003-1939-978X

DOI:

https://doi.org/10.54361/ajmas.261129

Keywords:

Hepatitis B Transmission, SEITR Model, Standard Incidence Rate, Basic Reproduction Number, Routh Hurwitz Criterion, Sensitivity Analysis

Abstract

This study analyses the SEITR model of hepatitis B virus (HBV) transmission with a standard incidence rate of βSI/N in a population of variable size. The model is reduced to an autonomous system expressed in population proportions. The disease-free and endemic equilibria and their Jacobian matrices are derived, and the basic reproduction number is obtained using the next generation matrix method. The Routh-Hurwitz criterion establishes that the disease-free equilibrium is locally asymptotically stable if R_0<1 and unstable if R_0>1. For disease-induced mortality μ_2>0, shows that the endemic equilibrium is unique and is locally asymptotically stable. For μ_2=0, the endemic equilibrium exists for R_0>1 and is locally asymptotically stable, with a forward bifurcation occurring at R_0=1. Numerical simulations calibrated to 2022 surveillance data from Ambon City, Indonesia, confirm the analytical results: the critical transmission rate is β^*≈0.432, and the critical vaccination coverage is σ^*≈0.622. The baseline value of R_0≈0.926 lies just below the threshold, resulting in very slow extinction of the infection over hundreds of years. Normalised sensitivity analysis indicates that R_0 is most sensitive to vaccination coverage, followed by the transmission rate and the treatment rate.