Pan Weiqiu, Li Tianzeng, Ali Safdar
School of Mathematics and Statistics, Sichuan University of Science and Engineering, Zigong, 643000 China.
South Sichuan Center for Applied Mathematics, Yibin, 644000 China.
Adv Differ Equ. 2021;2021(1):161. doi: 10.1186/s13662-021-03272-5. Epub 2021 Mar 10.
The Ebola outbreak in 2014 caused many infections and deaths. Some literature works have proposed some models to study Ebola virus, such as SIR, SIS, SEIR, etc. It is proved that the fractional order model can describe epidemic dynamics better than the integer order model. In this paper, we propose a fractional order Ebola system and analyze the nonnegative solution, the basic reproduction number , and the stabilities of equilibrium points for the system firstly. In many studies, the numerical solutions of some models cannot fit very well with the real data. Thus, to show the dynamics of the Ebola epidemic, the Gorenflo-Mainardi-Moretti-Paradisi scheme (GMMP) is taken to get the numerical solution of the SEIR fractional order Ebola system and the modified grid approximation method (MGAM) is used to acquire the parameters of the SEIR fractional order Ebola system. We consider that the GMMP method may lead to absurd numerical solutions, so its stability and convergence are given. Then, the new fractional orders, parameters, and the root-mean-square relative error are obtained. With the new fractional orders and parameters, the numerical solution of the SEIR fractional order Ebola system is closer to the real data than those models in other literature works. Meanwhile, we find that most of the fractional order Ebola systems have the same order. Hence, the fractional order Ebola system with different orders using the Caputo derivatives is also studied. We also adopt the MGAM algorithm to obtain the new orders, parameters, and the root-mean-square relative error which is . With the new parameters and orders, the fractional order Ebola systems with different orders fit very well with the real data.
2014年的埃博拉疫情导致了许多感染和死亡病例。一些文献提出了一些模型来研究埃博拉病毒，如SIR、SIS、SEIR等。事实证明，分数阶模型比整数阶模型能更好地描述疫情动态。在本文中，我们首先提出了一个分数阶埃博拉系统，并分析了该系统的非负解、基本再生数以及平衡点的稳定性。在许多研究中，一些模型的数值解与实际数据拟合得不是很好。因此，为了展示埃博拉疫情的动态，采用了戈伦弗洛-马伊纳尔迪-莫雷蒂-帕拉迪西格式（GMMP）来获得SEIR分数阶埃博拉系统的数值解，并使用改进的网格近似方法（MGAM）来获取SEIR分数阶埃博拉系统的参数。我们认为GMMP方法可能会导致荒谬的数值解，因此给出了它的稳定性和收敛性。然后，得到了新的分数阶、参数以及均方根相对误差。利用新的分数阶和参数，SEIR分数阶埃博拉系统的数值解比其他文献中的模型更接近实际数据。同时，我们发现大多数分数阶埃博拉系统具有相同的阶数。因此，还研究了使用卡普托导数的不同阶数的分数阶埃博拉系统。我们还采用MGAM算法来获得新的阶数、参数以及均方根相对误差 。利用新的参数和阶数，不同阶数的分数阶埃博拉系统与实际数据拟合得非常好。
