فصلنامه مدل سازی پیشرفته ریاضی
سال سیزدهم شماره 4 (Winter 2024)

A recent report indicated that one hundred and thirty thousand Nigerians were lately infected with HIV with the majority of infections resulting from unguarded vaginal sex. About two-thirds of new HIV infections in adults exist in women. Male circumcision and male condoms limit the danger of HIV infection, but the adoption of these procedures is beyond the control of Nigerian women due to gender inequality and gender-based violence against women in the country. In an attempt to provide a mathematical framework to examine a potential female-controlled strategy of HIV acquisition in Nigeria, a mathematical model is modified and analyzed for the transmission of HIV by incorporating pre-exposure prophylaxis (PrEP) in the form of a microbicide. The solutions of the model are proved to be positive. The critical points of the model and the epidemic threshold known as the reproduction number are also derived. The restricted case of total compliance to the use of the microbicide is analyzed by proving the global stability of the disease-free equilibrium (DFE) of the model. The general case which permits individuals to default the microbicide is also investigated by proving the global stability of the endemic equilibrium (EE) of the model. Numerical simulation is carried out to verify the analytical results and the results of the simulation show that strict compliance and consistent use of the microbicide may tend HIV acquisition among Nigerian women to zero.

In this paper, we prove a monotonicity result related to the principal eigenvalue for Dirichlet-Laplacian with a drift operator in a punctured ball.

In this note, we intend to explain and interpret the proofs of some of the classic mathematical results that are made easy by the beautiful mind of Karamzadeh, who has an incredible love for mathematics and has been doing well for years with regards to its popularization in our country.

The purpose of this paper is to present an efficient numerical method for finding numerical solutions Fokker-Planck-Kolmogorov time-fractional differential equations.The Legendre wavelet approach was employed for this objective. The Legendre wave was the first to be introduced. The Fokker-Planck-Kolmogorov time-fractional differential equation is converted to the linear equation using the Legendre wavelet operation matrix in this technique. This method has the advantage of being simple to solve. The simulation was carried out using MATLAB software. Finally, the proposed strategy was used to solve certain problems. The absolute value of the error between the precise and approximate answers provided by the numerical technique was then introduced, and the numerical method's error was analyzed.The results revealed that the suggested numerical method is highly accurate and effective when used to Fokker-Planck-Kolmogorov time fraction differential equations. The results for some numerical examples are documented in table and graph form to elaborate on the efficiency and precision of the suggested method. Moreover, for the convergence of the proposed technique, inequality is derived in the context of error analysis. In this paper and for the first time: the authors presented results on the numerical simulation for classes of time-fractional differential equations. The authors applied the reproducing Legendre wavelet method for the numerical solutions of nonlinear Fokker-Planck-Kolmogorov time-fractional differential equation.The method presented in the present study can be used by programmers, engineers and other researchers in this field.

The study of clinical observations in the family planning of intensity-modulated radiation therapy (IMRT), indicates that the target dose prescribed within the framework of trapezoidal fuzzy numbers, more closely matches the oncologist's goals. In this study, optimal treatment planning was described as a solution of an optimization problem using a quadratic objective function, where the prescribed target dose is a trapezoidal fuzzy number. First the problem was transformed into a non-fuzzy optimization problem, then the optimal solution was obtained based on the gradient method and projection operations. In this paper, we used Computational Entertainment for Radiotherapy Research (CERR) for treatment planning, importing the patient scans, and calculating the influence matrix. Numerical simulation was performed for a head and neck cancer case. Numerical results were presented in the form of Dose-Volume Histograms (DVH) and compared with the deterministic state. These results showed that the treatment planning that we provided based on the trapezoidal fuzzy target dose, is more consistent with the goals of oncologists.

In this article, we study modules that satisfy the double infinite chain condition on uncountably generated submodules, briefly called u.c.g.-DICC modules. We show that if a quotient finite dimensional module M satisfies the double infinite chain condition on uncountably generated submodules, then it has Krull dimension. We study submodules N of a module M such that whenever $\frac{M}{N}$ satisfies the double infinite chain condition so does M. Moreover, we observe that an α-atomic module, where α>\ω_{1}$ is an ordinal number, satisfies the previous chain condition if and only if it satisfies the descending chain condition on uncountably generated submodules.

A non-parallel submodules graph of M, denoted by G ∦ (M), is an undirected simple graph whose vertices are in one-to-one correspondence with all non-zero proper submodules of M and two distinct vertices are adjacent if and only if they are not parallel to each other. In this paper, we investigate the interplay between some of the module-theoretic properties of M and the graph-theoretic properties of G ∦ (M) . It is shown that if G ∦ (M) is connected, then diam(G ∦ (M)) ≤ 3 and if G ∦ (M) is not connected, then G ∦ (M) is a null graph. It is proved that G ∦ (M) is null if and only if M contains a unique simple submodule. In particular, M is a strongly semisimple R -module if and only if G ∦ (M) is a complete graph, and from this it follows that if G ∦ (M) is complete, then every R -module with finite Goldie dimension is Artinian and Noetherian. In addition, G ∦ (M) is a finite star graph if and only if M∼= Z pq, for some distinct prime numbers p and q.

‎In this paper, the Lie symmetry method and Lie brackets of vector fields are used in order to find some new solutions of the (3+1)-dimensional sourceless wave equation‎. ‎The obtained solutions are classified into two categories; polynomial and non-polynomial exact solutions‎. ‎Because of the properties of the Lie brackets and the symmetries‎, ‎a generalized method is implemented for constructing new solutions from old solutions‎. ‎We demonstrate the generation of such polynomial and non-polynomial solutions through the medium of the group theoretical properties of the equation‎. ‎It is noteworthy that this method could be used when the equations have two special kinds of symmetries which will be mentioned below‎.