فصلنامه مدل سازی پیشرفته ریاضی
سال چهاردهم شماره 1 (بهار 1403)

In general, the process of the spread of any virus in the human body consists of five stages, which include attachment of the virus to the host cell, penetration, preparation for reproduction, reproduction and propagation. However, different viruses have different life cycles. In this article, we will model the behavior of the corona virus in the body of each affected person and analyze the behavior of this virus in the body using dynamic systems. For this purpose, we study the dynamics of the evolutionary competition between the strategy of the corona virus and the body’s immune cells, especially lymphocytes T and B.

‎A sequence of continuous linear mappings $\{\Phi_n\}_{n=0}^\infty$ form a Hilbert C* -‎module‎ M into M is called a local higher derivation if for each $a\in\mathfrak{M}$ there is a continuous higher derivation $\{\varphi_{a,n}\}_{n=0}^\infty$ ‎on‎ M such that $\Phi_n(a)=\varphi_{a,n}(a)$ for each non-negative integer n‎. ‎In this paper w‎e show that if M is a Hilbert C* -‎module‎ such that every local derivation on M is a derivation, then each local higher derivation on M‎ is a higher derivation‎. Also, we prove that each local higher derivation on a unital C*-algebra is automatically continuous‎.

Crime control and security are among the essential needs of development in any society. One of the frequently used theories of criminology that its birth was in the 1990s is postmodern or eclectic criminology. Postmodern criminology through combining different scientific theories, including mathematics tries to do a comprehensive analysis of crime. Nowadays, mathematical models are used to study the dynamic behavior of many phenomena and processes in engineering sciences, basic sciences and humanities. Mathematical modeling is one of the important tools of scientists in controlling and predicting the future of various dynamic phenomena. In the field of law and especially in criminology, these models are very useful for evaluating crime control strategies. In this article, using a new mathematical model, we deal with mass dynamic modeling and its analysis.

In this paper, a simple and highly accurate approximate method for solving optimal control problems (OCPs) is presented. In this method, initially, by using the necessary optimality conditions based on the Pontryagin's maximum principle, the given OCP is transformed into a two-point boundary value problem (BVP). Then, by applying a second order finite difference formula, the resulting BVP is discretized, and a system of algebraic equations is formulated. Convergence analysis of the proposed method is also discussed, and matrix formulations are provided for ease of implementation. The numerical results obtained in this study are compared with some other methods, demonstrating the high accuracy, speed, and efficiency of the proposed method for solving both linear and nonlinear OCPs.

In this note, considering the main properties of closed ideals of C*-algebras, we will determine the structure of closed ideals of the C*-algebra C(X, A), the space of all continuous functions from compact Hausdorff space X to C*_algebra A. Indeed, we will show that for every closed ideal of C(X, A), there is some closed subset F of topological space X × prim(A), such that {f ∈ C(X, A) : ∀ (x, P )∈F , f(x) ∈ P}=I.

In this paper, we first calculate the measure theoretic Duggal transform of Lambert conditional operators. Next, by using the polar decomposition of operators, we obtain Moore-Penrose inverse $(\widehat{T}^p)$ and Drazin inverse $(\widehat{T }^d)$ of these types of operators, and then we will check the relationships between these types of inverses for the Duggal transformation. Finally, by using various examples including matrix representation, we will show the correctness of the obtained results.

This article introduces an iterative method for solving discrete optimal control problems involving interconnected nonlinear systems. Using this approach, the discrete and coupled nonlinear boundary value problem (BVP) obtained from the necessary optimality conditions transforms into a sequence of linear time invariant BVPs. Furthermore, the linear BVP at each iteration of the proposed method consists of several decoupled sub-problems, which can be solved in parallel and are unrelated to each other. The solution of these problems, employing common techniques for solving linear difference equations, leads to an optimal control law in a converging series form with uniform convergence. Moreover, a practical approach is presented to extend the designed optimal control to a feedback form. Subsequently, the implementation of the proposed method involves the design of a highly accurate iterative algorithm with low computational complexity, ensuring that the suboptimal control law is obtained with a minimal number of iterations. Finally, the efficacy of this technique is demonstrated through simulation and the solution of various numerical examples.

‎In this paper‎, ‎the time-space distributed-order fractional diffusion equations with weakly singular solutions are considered‎. We provide the difference scheme using a nonuniform mesh to solve ‎equations‎‎. ‎The stability and convergence of the difference scheme are discussed‎, ‎We prove that the difference scheme is unconditionally stable‎. ‎We find that the difference scheme is convergent‎. ‎We also show that the temporal convergence order on the nonuniform mesh is higher than on the uniform mesh‎. Finally, some numerical examples are given to verify the theoretical analysis‏.

‎‎‎In this article‎, ‎using the technique of the measure of non-compactness and the Petryshyn's fixed point theorem in Banach algebra an existence theorem for some Hadamard proportional fractional integral equations is provided.‎ The study of these integral equations are important because they contain lots of particular cases of integral equations that arise in many branches of nonlinear analysis and its applications. ‎Comparing Petryshyn's fixed point theorem to Schauder and‎ ‎Darbo's fixed point theorems‎, ‎that is, it enables us to skip demonstrating closed‎, ‎convex, and compactness‎ ‎properties on the investigated operators‎. ‎‎ Finally, some examples are provided for the accuracy and efficiency of the obtained results.

Using mathematical models to describe infectious diseases and then how to control and eliminate them by vaccines or other treatments, is a great help to public health organizations. Eradication of this category of diseases is possible when treatments are prescribed at the right time and with the right amount and process; In this regard, optimal control theory has been applied as a successful tool. The main purpose of this article is to review the existing literature considering such strategies in dealing with infectious diseases in the form of the famous basic model SIR. For this purpose, this study, deals with the way to use the control functions and how to explain the provided solutions, indeed the aims are to investigate susceptible, infected, and recovered populations in terms of the required goals, among the performed activities by evaluation and analyzing. Based on the number of used control variables in the treatment model, which indicate different methods of simultaneous prevention, including vaccination, treatment of infection, quarantine, and like or, the type of model, this study has been categorized and the results are presented. Also, in this review, the methods of implementing models from a numerical computation point of view and reality have also been discussed and time delay, stochastic, and discrete-time models in the case of SIR are also investigated. This review would help the researchers in order to have knowledge about the subject and activities carried out to continue research in this area are very helpful.