Tuberculosis (TB) remains a major global health concern due to its infectious nature and complex treatment process. In this study, we developed a mathematical model incorporating TB progression, vaccination, latency delays, and disability outcomes. The compartmental model includes seven stages: Susceptible, vaccinated, latent, infectious, quarantined, recovered, and disabled, with time-delay terms capturing disease progression dynamics. The stability analysis of the equilibria was performed, and the sensitivity analysis was conducted using the direct differentiation method. The basic reproduction number $R_0$ was derived to assess TB spread under different interventions. Model parameters were estimated using Ordinary Least Squares (OLS) based on Saudi Arabia's TB data (2000–2023). Numerical simulations, solved via the Adams-Bashforth-Moulton method, highlight the impact of delayed latency and quarantine on TB control, emphasizing the need for timely interventions.

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.