Qualitative behavior of a highly non-linear Cutaneous Leishmania epidemic model under convex incidence rate with real data.

In the context of this investigation, we introduce an innovative mathematical model designed to elucidate the intricate dynamics underlying the transmission of Anthroponotic Cutaneous Leishmania. This model offers a comprehensive exploration of the qualitative characteristics associated with the transmission process. Employing the next-generation method, we deduce the threshold value $ R_0 $ for this model. We rigorously explore both local and global stability conditions in the disease-free scenario, contingent upon $ R_0 $ being less than unity. Furthermore, we elucidate the global stability at the disease-free equilibrium point by leveraging the Castillo-Chavez method. In contrast, at the endemic equilibrium point, we establish conditions for local and global stability, when $ R_0 $ exceeds unity. To achieve global stability at the endemic equilibrium, we employ a geometric approach, a Lyapunov theory extension, incorporating a secondary additive compound matrix. Additionally, we conduct sensitivity analysis to assess the impact of various parameters on the threshold number. Employing center manifold theory, we delve into bifurcation analysis. Estimation of parameter values is carried out using least squares curve fitting techniques. Finally, we present a comprehensive discussion with graphical representation of key parameters in the concluding section of the paper.

Chaos in Cancer Tumor Growth Model with Commensurate and Incommensurate Fractional-Order Derivatives.

Analyzing the dynamics of tumor-immune systems can play an important role in the fight against cancer, since it can foster the development of more effective medical treatments. This paper was aimed at making a contribution to the study of tumor-immune dynamics by presenting a new model of cancer growth based on fractional-order differential equations. By investigating the system dynamics, the manuscript highlights the chaotic behaviors of the proposed cancer model for both the commensurate and the incommensurate cases. Bifurcation diagrams, the Lyapunov exponents, and phase plots confirm the effectiveness of the conceived approach. Finally, some considerations regarding the biological meaning of the obtained results are reported through the manuscript.

A fuzzy fractional model of coronavirus (COVID-19) and its study with Legendre spectral method.

The virus which belongs to the family of the coronavirus was seen first in Wuhan city of China. As it spreads so quickly and fastly, now all over countries in the world are suffering from this. The world health organization has considered and declared it a pandemic. In this presented research, we have picked up the existing mathematical model of corona virus which has six ordinary differential equations involving fractional derivative with non-singular kernel and Mittag-Leffler law. Another new thing is that we study this model in a fuzzy environment. We will discuss why we need a fuzzy environment for this model. First of all, we find out the approximate value of ABC fractional derivative of simple polynomial function ( t - a ) n . By using this approximation we will derive and developed the Legendre operational matrix of fractional differentiation for the Mittag-Leffler kernel fractional derivative on a larger interval [ 0 , b ] , b ⩾ 1 , b ∈ N . For the numerical investigation of the fuzzy mathematical model, we use the collocation method with the addition of this newly developed operational matrix. For the feasibility and validity of our method we pick up a particular case of our model and plot the graph between the exact and numerical solutions. We see that our results have good accuracy and our method is valid for the fuzzy system of fractional ODEs. We depict the dynamics of infected, susceptible, exposed, and asymptotically infected people for the different integer and fractional orders in a fuzzy environment. We show the effect of fractional order on the suspected, exposed, infected, and asymptotic carrier by plotting graphs.

Using Probabilistic Approach to Evaluate the Total Population Density on Coarse Grids.

Evaluation of the population density in many ecological and biological problems requires a satisfactory degree of accuracy. Insufficient information about the population density, obtained from sampling procedures negatively, impacts on the accuracy of the estimate. When dealing with sparse ecological data, the asymptotic error estimate fails to achieve a reliable degree of accuracy. It is essential to investigate which factors affect the degree of accuracy of numerical integration methods. When the number of traps is less than the recommended threshold, the degree of accuracy will be negatively affected. Therefore, available numerical integration methods cannot guarantee a satisfactory degree of accuracy, and in this sense the error will be probabilistic rather than deterministic. In other words, the probabilistic approach is used instead of the deterministic approach in this instance; by considering the error as a random variable, the chance of obtaining an accurate estimation can be quantified. In the probabilistic approach, we determine a threshold number of grid nodes required to guarantee a desirable level of accuracy with the probability equal to one.

Some new and modified fractional analysis of the time-fractional Drinfeld-Sokolov-Wilson system.

In this paper, we present a presumably new approach in order to solve the time-fractional Drinfeld-Sokolov-Wilson system, which is based upon the Liouville-Caputo fractional integral (LCFI), the Caputo-Fabrizio fractional integral, and the Atangana-Baleanu fractional integral in the sense of the LCFI by using the Adomian decomposition method. We compare the approximate solutions with the exact solution (if available), and we find an excellent agreement between them. In the case of a non-integer order, we evaluate the residual error function, thereby showing that the order of the error is very small. In all of our calculations, we apply the software package, Mathematica (Version 9).

Some new mathematical models of the fractional-order system of human immune against IAV infection.

Fractional derivative operators of non-integer order can be utilized as a powerful tool to model nonlinear fractional differential equations. In this paper, we propose numerical solutions for simulating fractional-order derivative operators with the power-law and exponential-law kernels. We construct the numerical schemes with the help the fundamental theorem of fractional calculus and the Lagrange polynomial interpolation. These schemes are applied to simulate the dynamical fractional-order model of the immune response (FMIR) to the uncomplicated influenza A virus (IAV) infection, which focuses on the control of the infection by the innate and adaptive immunity. Numerical results are then presented to show the applicability and efficiency on the FMIR.

An efficient spectral collocation method for the dynamic simulation of the fractional epidemiological model of the Ebola virus.

This article investigates a family of approximate solutions for the fractional model (in the Liouville-Caputo sense) of the Ebola virus via an accurate numerical procedure (Chebyshev spectral collocation method). We reduce the proposed epidemiological model to a system of algebraic equations with the help of the properties of the Chebyshev polynomials of the third kind. Some theorems about the convergence analysis and the existence-uniqueness solution are stated. Finally, some numerical simulations are presented for different values of the fractional-order and the other parameters involved in the coefficients. We also note that we can apply the proposed method to solve other models.

New fractional derivatives with non-singular kernel applied to the Burgers equation.

In this paper, we extend the model of the Burgers (B) to the new model of time fractional Burgers (TFB) based on Liouville-Caputo (LC), Caputo-Fabrizio (CF), and Mittag-Leffler (ML) fractional time derivatives, respectively. We utilize the Homotopy Analysis Transform Method (HATM) to compute the approximate solutions of TFB using LC, CF, and ML in the Liouville-Caputo sense. We study the convergence analysis of HATM by computing the interval of the convergence, the residual error function (REF), and the average residual error (ARE), respectively. The results are very effective and accurate.