Numerical analysis of a novel mathematical model of measles-pneumonia co-infection with treated-vaccinated compartment

Measles and pneumonia in Pakistan cause severe health deterioration and fatality due to weak immunity, medical accessibility challenges, and financial instability. This study proposes a new mathematical model integrating vaccine and treatment classes to investigate measles-pneumonia co-infection dynamics in Pakistan. The model's validity is assured by proving key characteristics such as positivity and invariant region. The basic reproduction number $R_0$ and equilibrium points are determined to evaluate whether the diseases endure or fade away. The Routh-Hurwitz criteria are used to evaluate the local stability, and global stability is established by using a Lyapunov function at the infection-free equilibrium. The model is visually shown to accurately fit actual cases after determining parameters using real data on measles cases in Pakistan. Numerical simulations are performed to investigate the effect of enhancing vaccination efficiency on infection rates and give insights into how both diseases behave when the reproduction number $R_0$ is less or more than one. A global sensitivity analysis is performed to identify the impact of crucial parameters on co-infection disease transmission and their impact on both disease reproduction numbers. Our results demonstrate the effectiveness of the implemented techniques in enhancing the accuracy of outbreak predictions.