In this paper, a fractional-order model for African swine fever with limited medical resources is proposed and analyzed. First, the existence and uniqueness of a positive solution is proven. Second, the basic reproduction number and the conditions sufficient for the existence of two equilibriums are obtained. Third, the local stability of the two equilibriums is studied. Next, some numerical simulations are performed to verify the theoretical results. The mathematical and simulation results show that the values of some parameters, such as fractional order and medical resources, are critical for the stability of the equilibriums.

The stability of the nabla discrete distributed-order nonlinear dynamic systems is investigated in this paper. Firstly, a sufficient condition for the asymptotic stability of the nabla discrete distributed-order nonlinear systems is proposed based on Lyapunov direct method. In addition, some properties of the nabla distributed-order operators are derived. Based on these properties, a simpler criterion is provided to determine the stability of such systems. Finally, two examples are given to illustrate the validity of these results.

African swine fever is a highly contagious virus that causes pig disease. Its onset process is short, but the mortality rate is as high as 100%. There are still no effective drugs that have been developed to treat African swine fever, and prevention and control measures are currently the best means to avoid infection in pig herds. In this paper, two fractional order mathematical models with media coverage are constructed to describe the transmission of African swine fever. The first model is a basic model with media coverage, and no control measures are considered. For this model, the reproduction number is obtained by using the next generation matrix method. Then, the sufficient conditions for the existence and stability of two equilibriums are obtained. Based on the first model, the second model is established incorporating two control measures. By using Pontryagin’s maximal principle, the optimal control solution is derived. After that, some numerical simulations are performed for the two models to verify the theoretical results. Both the qualitative analysis and numerical results indicate that timely media coverage combined with disinfection control measures is crucial to preventing the spread of disease.