A fractional order dynamical system and simulation with an application on infectious diseases

Fractional calculus provides a powerful mathematical framework that extends classical calculus to capture memory and hereditary properties in dynamic systems. Its application to epidemiology offers deeper insights into the complex dynamics of infectious disease transmission. This study focuses on compartmental epidemic models, beginning with the classical SIR model, which divides the population into Susceptible (S), Infected (I), and Recovered (R) groups. Extensions of this framework include the SEIR model, which accounts for an Exposed (E) class, the SIS model, which allows recovered individuals to return to susceptibility, and the Carrier model, which introduces asymptomatic or persistent carriers. A more generalized model, the SEIQRV, divides the population into six compartments: Susceptible (S), Exposed (E), Infected (I), Quarantined (Q), Recovered (R), and Vaccinated (V). The basic reproduction number R_0 serves as a fundamental threshold metric for assessing disease spread, with stability analysis conducted through equilibrium points and the Jacobian matrix. The system's stability depends on the eigenvalues of the Jacobian, which determine whether the disease-free or endemic equilibrium is attained. Numerical simulations, performed in MATLAB, examine the SEIQRV model under fractional-order derivatives with different orders (α=1,0.9,0.8). The results demonstrate that incorporating quarantine and vaccination compartments effectively lowers the reproduction number, thereby controlling the spread of infection. Overall, the study highlights the importance of fractional-order epidemic models in evaluating intervention strategies and emphasizes their role in public health planning and disease management.