Dynamics of Fractional-Order Epidemic Models with General Nonlinear Incidence Rate and Time-Delay

Nonlinear Reaction–Diffusion Equations with Delay: Partial Survey, Exact Solutions, Test Problems, and Numerical Integration

The paper describes essential reaction–diffusion models with delay arising in population theory, medicine, epidemiology, biology, chemistry, control theory, and the mathematical theory of artificial neural networks. A review of publications on the exact solutions and methods for their construction is carried out. Basic numerical methods for integrating nonlinear reaction–diffusion equations with delay are considered. The focus is on the method of lines. This method is based on the approximation of spatial derivatives by the corresponding finite differences, as a result of which the original delay PDE is replaced by an approximate system of delay ODEs. The resulting system is then solved by the implicit Runge–Kutta and BDF methods, built into Mathematica. Numerical solutions are compared with the exact solutions of the test problems.

Asymptotic behavior of a stochastic SIR model with general incidence rate and nonlinear Lévy jumps

In this paper, we consider a stochastic SIR epidemic model with general disease incidence rate and perturbation caused by nonlinear white noise and L e ´ vy jumps. First of all, we study the existence and uniqueness of the global positive solution of the model. Then, we establish a threshold λ by investigating the one-dimensional model to determine the extinction and persistence of the disease. To verify the model has an ergodic stationary distribution, we adopt a new method which can obtain the sufficient and almost necessary conditions for the extinction and persistence of the disease. Finally, some numerical simulations are carried out to illustrate our theoretical results.