Mathematical Modeling of Epidemic Spread with Vaccination and Awareness: A Reaction–Diffusion Approach.

This study presents a comprehensive mathematical model designed to investigate the dynamics of infectious disease spread by integrating two critical
intervention strategies: vaccination and individual awareness. The population is structured into distinct compartments: susceptible, infected, vaccinated, fully aware, and partially aware individuals. In this framework, vaccination serves to reduce the direct risk of infection, while awareness campaigns influence social behavior, thereby slowing down the overall transmission rate. Furthermore, the model accounts for the loss of immunity over time, allowing vaccinated individuals to return to the susceptible class eventually. To capture the spatial dimension of an outbreak, the model is formulated as a system of nonlinear partial differential equations (PDEs) with reaction–diffusion terms. Methodologically, the positivity and global existence of solutions are rigorously established to ensure the model’s mathematical and biological consistency. We derive the equilibrium
points, including the disease-free and endemic states, and calculate the basic reproduction number using the next-generation matrix method. Local stability analysis is conducted to evaluate the system’s long-term behavior near these equilibria.

Numerical simulations are performed to illustrate the theoretical findings and quantify the impact of spatial diffusion on disease persistence.
The results demonstrate that while awareness significantly mitigates the peak of an epidemic, it is not sufficient to eradicate the disease on its own. In contrast, effective and sustained vaccination programs are shown to be the primary driver in reducing infection levels. This work emphasizes the necessity of combining behavioral interventions with clinical strategies to understand and manage real-world epidemic dynamics in spatial environments.