Qualitative properties of mathematical model of English language education

This paper presents a mathematical model that examines the impacts of traditional and modern educational programs. We calculate two reproduction numbers. By using the Chavez and Song theorem, we show that backward bifurcation occurs. In addition, we investigate the existence and local and global stability of boundary equilibria and coexistence equilibrium point and the global stability of the coexistence equilibrium point using compound matrices.

A Mathematical Study of a Generalized SEIR Model of COVID-19

In this work (Part I), we reinvestigate the study of the stability of the Covid-19 mathematical model constructed by Shah et al. (2020) [1]. In their paper, the transmission of the virus under different control strategies is modeled thanks to a generalized SEIR model. This model is characterized by a five dimensional nonlinear dynamical system, where the basic reproduction number can be established by using the next generation matrix method. In this work (Part I), it is established that the disease free equilibrium point is locally as well as globally asymptotically stable when . When , the local and global asymptotic stability of the equilibrium are determined employing the second additive compound matrix approach and the Li-Wang’s (1998) stability criterion for real matrices [2]. In the second paper (Part II), some control parameters with uncertainties will be added to stabilize the five-dimensional Covid-19 system studied here, in order to force the trajectories to go to the equilibria. The stability of the Covid-19 system with these new parameters will also be assessed in Intissar (2020) [3] applying the Li-Wang criterion and compound matrices theory. All sophisticated technical calculations including those in part I will be provided in appendices of the part II.

A Mathematical Model for the Vector Transmission and Control of Banana Xanthomonas Wilt

Banana Xanthomonas wilt is currently wrecking havoc in East and Central Africa. In this paper, a novel theoretical model for the transmission of banana Xanthomonas wilt by insect vectors is formulated and analyzed. The model incorporates roguing of infected plants and replanting using healthy suckers. The model is analyzed for the existence and stability of the equilibrium points. The global stability of the disease-free equilibrium point was determined by using a Lyapunov function and LaSalle's invariance principle. For the global stability of the endemic equilibrium point, the theory of competitive systems, compound matrices and stability of periodic orbits were used. It was established that if the basic reproduction number satisfies $R_0 \leq 1$, the disease-free equilibrium point is globally stable and the disease will be wiped out and if $R_0 > 1,$ the endemic equilibrium is stable and the disease persists. A numerical simulation of the model was also carried out. It was found out that at appropriate roguing and replanting, the disease can be contained.

Global stability of an SEIR epidemic model with vaccination

In this paper, an SEIR epidemic model with vaccination is formulated. The results of our mathematical analysis indicate that the basic reproduction number plays an important role in studying the dynamics of the system. If the basic reproduction number is less than unity, it is shown that the disease-free equilibrium is globally asymptotically stable by comparison arguments. If it is greater than unity, the system is permanent and there is a unique endemic equilibrium. In this case, sufficient conditions are established to guarantee the global stability of the endemic equilibrium by the theory of the compound matrices. Numerical simulations are presented to illustrate the main results.

Generalizations of the Cauchy and Fujiwara Bounds for Products of Zeros of a Polynomial

The Cauchy bound is one of the best known upper bounds for the modulus of the zeros of a polynomial. The Fujiwara bound is another useful upper bound for the modulus of the zeros of a polynomial. In this paper, compound matrices are used to derive a generalization of both the Cauchy bound and the Fujiwara bound. This generalization yields upper bounds for the modulus of the product of $m$ zeros of the polynomial.

Robust Synchronization of the Unified Chaotic System

<p>The paper investigates the synchronization problem of the unified chaotic system. The case of identical, but unknown master and slave unified chaotic systems is considered. Based on compound matrices formalism, a unified synchronization control scheme is proposed independently of the unknown system parameter. Simulation results are provided to show the effectiveness of the presented scheme.</p>

The Stability Criteria with Compound Matrices

The bifurcation problem is one of the most important subjects in dynamical systems. Motivated by M. Li et al. who used compound matrices to judge the stability of matrices and the existence of Hopf bifurcations in continuous dynamical systems, we obtained some effective methods to judge the Schur stability of matrices on the base of the spectral property of compound matrices, which can be used to judge the asymptotical stability and the existence of Hopf bifurcations of discrete dynamical systems.