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Substantial advancements have been made in applying fractional calculus to understanding the epidemiology of viral diseases. However, it remains troubling that over 2,000,000 children, particularly those under the age of 5 and elderly individuals over 65, succumb to pneumonia annually in developing nations. This paper uses a deterministic SVEIR (susceptible, vaccinated, exposed, infected, and recovered) model to investigate pneumonia disease dynamics from a mathematical perspective. The governing model has been generalized from Caputo fractional derivatives, and the generalized Euler's method has been used to calculate the approximate solution of the governing model. The dynamical systems theory examines the model's stability analysis, basic reproduction number, and equilibrium points. The linearization method for spatial equilibrium and the Lyapunov functional method demonstrate that the model is locally asymptotic stable for disease-free equilibrium. Graphical results are showcased through MATLAB21 to illustrate the resolution of the problem. The research findings suggest that the disease will eventually disappear from the population if vaccination coverage rises over the necessary vaccination threshold. According to sensitivity analysis, the most sensitive factors are the transmission rate and the pace at which exposed individuals become contagious. According to the study, we must strengthen treatment effectiveness to prevent disease spreading.
Kumawat et al. (Mon,) studied this question.