This paper concerns with the existence and uniqueness of the Cauchy problem for a system of fuzzy fractional differential equation with Caputo derivative of order $q\in (1,2]$, ${}_0^c{D}_{0^{+}}^{q}u(t)=\lambda u(t)\oplus f(t,u(t))\oplus B(t)C(t),t\in [0,T]$ with initial conditions $u(0)=u_0,u^{\prime }(0)=u_1.$ Moreover, by using direct analytic methods, the $E_q$–Ulam-type results are also presented. In addition, several examples are given which show the applicability of fuzzy fractional differential equations.

A novel coronavirus infection system is established for the analytical and computational aspects of this study, using a fuzzy fractional evolution equation (FFEE) stated in Caputo’s sense for order (1,2). It is constructed using the FFEE formulated in Caputo’s meaning. The model consist of six components illustrating the coronavirus outbreak, involving the susceptible people ${\mathbf{K}}_{\ell }(\omega )$, the exposed population ${\mathbf{L}}_{\ell }(\omega )$, total infected strength ${\mathbf{C}}_{\ell }(\omega )$, asymptotically infected population ${\mathbf{M}}_{\ell }(\omega )$, total number of humans recovered ${\mathbf{E}}_{\ell }(\omega )$, and reservoir ${\mathbf{Q}}_{\ell }(\omega )$. Numerical results using the fuzzy Laplace approach in combination with the Adomian decomposition transform are developed to better understand the dynamical structures of the physical behavior of COVID-19. For the controlling model, such behavior on the generic characteristics of RNA in COVID-19 is also examined. The findings show that the proposed technique of addressing the uncertainty issue in a pandemic situation is effective.

This paper concerns with the existence and uniqueness of fuzzy fractional evolution equation with uncertainty involves function of form ${}^c{D}^{\alpha }x\left(t\right)=f(t,x\left(t\right),D^{\beta }x\left(t\right)),I^{\alpha }x\left(0\right)=x_0,x^{\prime }\left(0\right)=x_1,$ where $1<\alpha <2,0<\beta <1$. After determining the equivalent integral form of solution we establish existence and uniqueness by using Rogers conditions, Kooi type conditions and Krasnoselskii-Krein type conditions. In addition, various numerical solutions have been presented to ensure that the main result is true and effective. Finally, a few examples which express fuzzy fractional evolution equations are shown.