Global Conference on Mathematics and Applications (GCMA)

The incorporation of memory effects into epidemic models has become
increasingly important for accurately describing disease transmission.
In this work, we investigate a fractional-order SEIR epidemic model with
asymptomatic infectious individuals and a general nonlinear incidence
rate. The model is formulated using the Caputo fractional derivative of
order $\gamma \in (0,1)$ and is governed by the system
\begin{align}
{}^{C}!D_t^{\gamma} S &= \Lambda - f(S)\big(g_1(I_1) + g_2(I_2)\big) - \mu S,\
{}^{C}!D_t^{\gamma} E &= f(S)\big(g_1(I_1) + g_2(I_2)\big) - (\epsilon + \mu)E,
\end{align}
together with the corresponding equations for the symptomatic,
asymptomatic and recovered populations.

We first establish the fundamental properties of the model, including
positivity, boundedness and existence of solutions. The basic
reproduction number $\mathcal{R}_0$ is then derived and used as a
threshold parameter for disease persistence. By constructing suitable
Lyapunov functionals and using tools from fractional dynamical systems
theory, we obtain global stability conditions for both equilibria. In
particular, the disease-free equilibrium is globally asymptotically
stable whenever $\mathcal{R}_0 \le 1$, whereas the endemic equilibrium is
globally asymptotically stable when $\mathcal{R}_0 > 1$.

Numerical simulations validate the theoretical analysis and illustrate
the influence of the fractional order on epidemic evolution. The results
highlight the significant role of memory effects in the transmission and
control of infectious diseases.