Mathematical Model of an Infectious Respiratory Disease with a Double-Dose Vaccine, Isolation and Use of Face-Mask

Mathematical modeling plays a crucial role in epidemiology by helping us understand how an epidemic unfolds under different conditions. Respiratory infectious diseases have emerged in our history, the virus has significantly impacted all aspects of life. In the absence of a definitive treatment, vaccination and Non-Pharmaceutical Interventions (NPIs) such as social distancing, handwashing, wearing face masks, quarantine, isolation, and contact tracing have been essential in controlling its spread. This study develops a deterministic mathematical model to explore the dynamics of respiratory infectious diseases under key mitigation measures, including vaccination, face mask usage, quarantine, and isolation. The system of Ordinary Differential Equations (ODEs) is solved using Wolfram Mathematica, while the Next Generation Matrix (NGM) method is employed to determine the basic reproduction number. Stability analysis is conducted using the Jacobian matrix, and numerical simulations are carried out in Python using Jupyter Notebook. The analysis indicates that the model has a disease-free equilibrium (DFE), which is locally asymptotically stable when the basic reproduction number is less than one. This suggests that respiratory infectious diseases can be effectively controlled if vaccination and NPIs are implemented together. Sensitivity analysis highlights that the most critical factors for eradicating respiratory infectious diseases are the vaccine coverage rate (the proportion of susceptible individuals vaccinated) and vaccine efficacy.