Stability and sensitivity analysis of a pneumonia model with awareness component using differential equations

Pneumonia presents a persistent and significant global health challenge, with its complex transmission dynamics exacerbated by the prevalence of asymptomatic carriers and diverse clinical presentations. This study addresses the limitation of traditional epidemiological models by incorporating the impact of public awareness on disease dynamics. We propose and analyze a novel six-compartment mathematical model using a system of ordinary differential equations. The model stratifies the population into susceptible (S), Exposed E(t), unaware infected ( $\mathrm{I}_{\mathrm{u}}$ ), aware infected ( $\mathrm{I}_{\mathrm{a}}$ ), treated (T), and recovered (R) compartments. We rigorously establish the positivity and boundedness of solutions and determine the stability of both the disease-free and endemic equilibriums. A comprehensive sensitivity analysis of the basic reproduction number ( $\mathrm{R}_{0}$ ) is performed to identify the most influential parameters affecting disease spread. Our analysis confirms the stability of the disease-free equilibrium when $\mathrm{R}_{0} < 1 $ and the existence and stability of the endemic equilibrium when $\mathrm{R}_{0} > 1$ . The sensitivity analysis reveals that increased public awareness and enhanced treatment rates are the most critical parameters for reducing disease transmission. Numerical simulations validate these findings, demonstrating a significant decrease in disease prevalence with higher awareness and treatment rates. The findings underscore the vital role of public health campaigns in mitigating pneumonia outbreaks. We show that integrating behavioral factors like public awareness into epidemiological models provides a more accurate representation of disease dynamics and highlights the necessity of proactive, data-driven interventions for effective and long-term disease control.