We present a nonlinear fractional order epidemic model to investigate the spreading dynamical behavior of the avian influenza. The population of the model contains susceptible individuals, asymptomatic but infective latent individuals, and infective individuals. We first establish the existence, uniqueness, nonnegativity, and positive invariance of the solution, then we study the reproduction number of the model and the stability of the disease‐free equilibrium. We observe that the reproduction number varies with the order of the fractional derivative ν. In terms of epidemics, this suggests that varying ν induces a change in the avian's epidemic status. Furthermore, we derive the sufficient conditions for the existence and the stability of the endemic equilibrium. Finally, we carry out some numerical simulations to validate the analytical results. We find from simulations that the solution of the fractional order model tends to a stationary state over a longer period of time with decreasing the value of the fractional derivative, and the size of epidemic decreases with decreasing ν.

In this paper, we study the Galerkin spectral approximation to an unconstrained convex distributed optimal control problem governed by the time fractional diffusion equation. We construct a suitable weak formulation, study its well-posedness, and design a Galerkin spectral method for its numerical solution. The contribution of the paper is twofold: a priori error estimate for the spectral approximation is derived; a conjugate gradient optimization algorithm is designed to efficiently solve the discrete optimization problem. In addition, some numerical experiments are carried out to confirm the efficiency of the proposed method. The obtained numerical results show that the convergence is exponential for smooth exact solutions.

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In this paper, we study the fractional optimal control problem and its spectral approximation. The problem under investigation consists in finding the optimal solution governed by the time fractional diffusion equation with constraints on the control variable. We construct a suitable weak formulation, study its wellposedness, and design a Galerkin spectral method for its numerical solution. The main contribution of the paper includes: (1) a priori error estimates for the spacetime spectral approximation is derived; (2) a projection gradient algorithm is designed to efficiently solve the discrete minimization problem; (3) some numerical experiments are carried out to confirm the efficiency of the proposed method. The obtained numerical results show that the convergence is exponential for smooth exact solutions.

In this paper, a fractional order avian-human influenza epidemic model with logistic growth for avian population is investigated. The dynamical behavior of this model is discussed. We first establish the existence, uniqueness, non-negativity and positive invariance of the solution. Then we analyze the existence of various equilibrium points, and some sufficient conditions are derived to ensure the global asymptotic stability of the disease free equilibrium point and endemic equilibrium point. Finally, we take some numerical simulations to validate the analytical results.