Abstract
In this work, the concept of the fractional derivative is used to improve a mathematical model for the transmission dynamics of pneumonia in the Al-Baha region of the Kingdom of Saudi Arabia. We establish a dynamics model to predict the transmission of pneumonia in some local sheep and goat herds. The proposed model is a generalization of a system of five ordinary differential equations of the first order, regarding five unknowns, which are the numbers of certain groups of animals (susceptible, vaccinated, carrier, infected, and recovered). This consists of investigating the equilibrium, basic reproduction number, stability analysis, and bifurcation analysis. It is observed that the free equilibrium point is local and global asymptotic stable if the basic reproduction number is less than one, and the endemic equilibrium is local and global asymptotic stable if the basic reproduction number is greater than one. The optimal control problem is formulated using Pontryagin's maximum principle, with three control strategies: Disease prevention through education, treatment, and screening. The most cost-effective intervention strategy to combat the pneumonia pandemic is a combination of prevention and treatment, according to the cost-effectiveness analysis of the adopted control techniques. A numerical simulation is performed, and the significant data are graphically displayed. The results predicted by the model show a good agreement with the actual reported data.