Fractional model for malaria transmission under control strategies

We study a fractional model for malaria transmission under control strategies.Weconsider the integer order model proposed by Chiyaka et al. (2008) in [15] and modify it to become a fractional order model. We study numerically the model for variation of the values of the fractional derivative and of the parameter that models personal protection, b. From observation of the figures we conclude that as b is increased from 0 to 1 there is a corresponding decrease in the number of infectious humans and infectious mosquitoes, for all values of α. This means that this result is invariant for variation of fractional derivative, in the values tested. These results are in agreement with those obtained in Chiyaka et al.(2008) [15] for α = 1.0 and suggest that our fractional model is epidemiologically wellposed

New findings on the dynamics of HIV and TB coinfection models

In this paper we study a model for HIV and TB coinfection. We consider the integer order and the fractional order versions of the model. Let α∈[0.78,1.0] be the order of the fractional derivative, then the integer order model is obtained for α=1.0. The model includes vertical transmission for HIV and treatment for both diseases. We compute the reproduction number of the integer order model and HIV and TB submodels, and the stability of the disease free equilibrium. We sketch the bifurcation diagrams of the integer order model, for variation of the average number of sexual partners per person and per unit time, and the tuberculosis transmission rate. We analyze numerical results of the fractional order model for different values of α, including α=1. The results show distinct types of transients, for variation of α. Moreover, we speculate, from observation of the numerical results, that the order of the fractional derivative may behave as a bifurcation parameter for the model. We conclude that the dynamics of the integer and the fractional order versions of the model are very rich and that together these versions may provide a better understanding of the dynamics of HIV and TB coinfection

Exploration of Various Fractional Order Derivatives in Parkinson's Disease Dysgraphia Analysis

Parkinson's disease (PD) is a common neurodegenerative disorder with a prevalence rate estimated to 2.0% for people aged over 65 years. Cardinal motor symptoms of PD such as rigidity and bradykinesia affect the muscles involved in the handwriting process resulting in handwriting abnormalities called PD dysgraphia. Nowadays, online handwritten signal (signal with temporal information) acquired by the digitizing tablets is the most advanced approach of graphomotor difficulties analysis. Although the basic kinematic features were proved to effectively quantify the symptoms of PD dysgraphia, a recent research identified that the theory of fractional calculus can be used to improve the graphomotor difficulties analysis. Therefore, in this study, we follow up on our previous research, and we aim to explore the utilization of various approaches of fractional order derivative (FD) in the analysis of PD dysgraphia. For this purpose, we used the repetitive loops task from the Parkinson's disease handwriting database (PaHaW). Handwritten signals were parametrized by the kinematic features employing three FD approximations: Gr\"unwald-Letnikov's, Riemann-Liouville's, and Caputo's. Results of the correlation analysis revealed a significant relationship between the clinical state and the handwriting features based on the velocity. The extracted features by Caputo's FD approximation outperformed the rest of the analyzed FD approaches. This was also confirmed by the results of the classification analysis, where the best model trained by Caputo's handwriting features resulted in a balanced accuracy of 79.73% with a sensitivity of 83.78% and a specificity of 75.68%.Comment: Print ISBN 978-3-031-19744-

COVID-19 Mathematical Study with Environmental Reservoir and Three General Functions for Transmissions

In this paper, the ongoing new coronavirus (COVID-19) epidemic is being investigated using a mathematical model. The model depicts the dynamics of infection with several transmission pathways and general infection functions, plus it highlights the significance of the environment as a reservoir for the disease’s propagation and dissemination. We have studied the qualitative behavior of the proposed model representing a system of fractional differential equations. Under a set of conditions on the general functions and the parameters, we have proven the global asymptotic stability of all steady states by using the Lyapunov method and LaSalle’s invariance principle. We also carried some numerical results to confirm the analytical results we obtained

Uniform asymptotic stability of a fractional tuberculosis model

We propose a Caputo type fractional-order mathematical model for the transmission dynamics of tuberculosis (TB). Uniform asymptotic stability of the unique endemic equilibrium of the fractional-order TB model is proved, for any α (0, 1). Numerical simulations for the stability of the endemic equilibrium are provided.publishe