Dynamics of a novel fractional order Encephalitis model incorporating immune disorders persons with optimal vaccination and treatment control strategies

This study investigates the heightened susceptibility of immune-disordered individuals to encephalitis through a novel fractional-order compartmental model. Using the Liouville-Caputo fractional order derivative framework, we formulate a seven-dimensional epidemic Susceptible-Vaccinated-Infected-Hospitalized-Recovered (SVIHR) model of Encephalitis dynamics that explicitly distinguishes between susceptible and vaccinated populations through dedicated compartments. We rigorously elucidate the uniqueness, existence, positiveness, and boundedness of solutions, ensuring the models epidemiological validity. We evaluate the basic reproduction number ($\mathcal{R}_{0}$) to determine epidemic potential, establishing threshold conditions for disease persistence. Through comprehensive stability analysis, we characterize both equilibrium points: the Encephalitis infection-free and the Encephalitis infection endemic equilibrium points, identifying their local and global stability properties. A systematic sensitivity analysis further elucidates the influence of key epidemiological parameters on spreading dynamics. Utilizing rigorous bifurcation analysis, we derive the exact parameter thresholds that demarcate phase transitions in encephalitis dynamics, characterizing the precise conditions for epidemic emergence. In addition, we develop a fractional optimal control problem using Pontryagin’s maximum principle to derive time-dependent vaccination and treatment strategies that minimize encephalitis transmission in immunocompromised populations. Our framework provides policymakers with cost-effective intervention scenarios, aligning with sustainable health goals while rigorously optimizing outbreak response. Numerical experiments, conducted using the Adams-Bashforth-Moulton method and complemented by graphical illustrations, provide quantitative insights into the impact of optimized control strategies—highlighting the practical significance of the theoretical developments. These findings provide actionable insights for public health policies targeting encephalitis mitigation in high-risk populations.