Narrow Results

In this work, a mathematical model of six compartments is formulated, showing the dynamic flow of pneumonia disease in the human population with treatment and vaccination interventions. Equilibria points and stability analyses were carried out using the Lyapunov function approach. Analytically, it is found that at the disease-free equilibrium state, local and global asymptotic stability behaviors are achieved when $R_0<1,$ with instability if $R_0>1$. However, at the endemic equilibrium point, asymptotic stability is attainable if $R_0>1$ and instability otherwise. The study indicates that pneumonia disease infection is successfully reduced when treatment and vaccination interventions are administered to the patients. The work also proposes an adaptive sliding mode control approach with a closed-loop control system to manage pneumonia epidemic model uncertainties. This approach intends to reduce disease transmission and infection through successful tracking of defined trajectories and managing uncertainties. For the control rates $(u_1,u_2\ne 0)$, the technique managed to track the disease carriers and infectious agents accurately even in the presence of parameter uncertainties. In conclusion, an increase in the control rates $(u_1,u_2)$ in the existence of parameter uncertainty control systems significantly reduces the number of disease transmitters and infectious agents quicker than in their absence. Hence, this study signifies the pivotal role of treatment and vaccination in the control of pneumonia infection as well as the control of parameter uncertainties by the proposed method.