Stability analysis of pneumonia mathematic model with prevention using vaccines and treatment

Pneumonia is an infectious disease caused by living microorganisms such as bacteria, viruses, and fungi. Pneumonia transmission is the leading cause of death in children worldwide. Mathematics provides an important role in seeing the development of pneumonia, and by using mathematical modeling, pneumonia is modeled with five subpopulations, susceptible without vaccine (Su), susceptible with vaccines (Sv), carriers(C), infection (I), and treatment (T). From the results of the analysis of the mathematical model, were obtained two equilibrium points, the endemic equilibrium point and the disease-free equilibrium point, and obtained the type of stability from the mathematical model is asymptotically stable. From the characteristics of the mathematical model it can be seen that, in the initial 20 months, the population is free from disease. Where the number of individuals in the subpopulation of carriers, infection, and treatment has reached zero, along with the increasing use of vaccines to prevent the spread of pneumonia.