Journal of Applied Mathematics

In this paper, we develop the theoretical provisions of symmetric cryptographic algorithms based on the polynomial residue number system for the first time. The main feature of the proposed approach is that when reconstructing the polynomial based on the method of undetermined coefficients, multiplication is performed not on the found base numbers but on arbitrarily selected polynomials. The latter, together with pairwise coprime residues of the residue class system, serve as the keys of the cryptographic algorithm. Schemes and examples of the implementation of the developed polynomial symmetric encryption algorithm are presented. The analytical expressions of the cryptographic strength estimation are constructed, and their graphical dependence on the number of modules and polynomial powers is presented. Our studies show that the cryptanalysis of the proposed algorithm requires combinatorial complexity, which leads to an NP-complete problem.

Gumboro disease is a viral poultry disease that causes immune suppression on the infected birds leading to poor production, mortality, and exposure to secondary infections, hence a major threat in the poultry industry worldwide. A mathematical model of the transmission dynamics of Gumboro disease is developed in this paper having four compartments of chicken population and one compartment of Gumboro pathogen population. The basic reproduction number is derived, and the dynamical behaviors of both the disease-free equilibrium (DFE) and endemic equilibrium are analyzed using the ordinary differential equation theory. From the analysis, we found that the system exhibits an asymptotic stable DFE whenever and an asymptotic stable EE whenever . The numerical simulation to verify the theoretical results was carried out using MATLAB ode45 solver, and the results were found to be consistent with the theoretical findings.

In this paper, we study asymptotic model change detection in multivariate linear regression by using the Kolmogorov–Smirnov function of the partial sum process of recursive residuals. We approximate the rejection region and also the power function of the test by establishing a functional central limit theorem for the sequence of the partial sum processes of the recursive residuals of the observations. When the assumed model is true, the limit process is given by the standard multivariate Brownian motion which does not depend on the regression functions. However, when the assumed model is not true (some models change), the limit process is represented by a vector of deterministic trend plus the standard multivariate Brownian motion. The finite sample size rejection region and the power of the test are investigated by means of Monte Carlo simulation. The simulation study shows evidence that the proposed test is consistent in the sense that it attains the power larger than the size of the test when the hypothesis is not true. We also demonstrate the application of the proposed test method to Indonesian economic growth data in which we test the adequacy of three-variate low-order polynomial model. The test result shows that the growth of the Indonesian economy is neither simultaneously constant nor linear. The test has successfully detect the appearance of a change in the model which is mainly caused by the COVID-19 pandemic in 2020.

Lymphatic filariasis is a leading cause of chronic and irreversible damage to human immunity. This paper presents deterministic and continuous-time Markov chain (CTMC) stochastic models regarding lymphatic filariasis dynamics. To account for randomness and uncertainties in dynamics, the CTMC model was formulated based on deterministic model possible events. A deterministic model’s outputs suggest that disease extinction is feasible when the secondary threshold infection number is below one, while persistence becomes likely when the opposite holds true. Furthermore, the significant contribution of asymptomatic carriers was identified. Results indicate that persistence is more likely to occur when the infection results from asymptomatic, acutely infected, or infectious mosquitoes. Consequently, the CTMC stochastic model is essential in capturing variabilities, randomness, associated probabilities, and validity across different scales, whereas oversimplification and unpredictability of inherent may not be featured in a deterministic model.

This work presents a stochastic model of malaria spread. We first calculated the basic reproduction number of the models and in order to show that the malaria-free equilibrium is asymptotically stable; then, we used a finite Markov chain model to describe the interactions between the different compartments of the model . We carried out numerical simulations of our results for two types of transmission zones: a zone with low malaria transmission and an endemic zone. Through these simulations, we first determined the invariant stationary distribution of the model, and then, we found that the use of the indoor residual spraying (IRS) method by regular application of insecticides is more effective for the elimination of malaria than the use of long-acting impregnated mosquito nets (LLINs).

This paper introduces a novel approach for solving first-order stiff initial value problems through the development of a one-step family of three optimized second-derivative hybrid block methods. The optimization process was integrated into the derivation of the methods to achieve maximal accuracy. Through a rigorous analysis, it was determined that the methods exhibit properties of consistency, zero-stability, convergence, and A-stability. The proposed methods were implemented using the waveform relaxation technique, and the computed results demonstrated the superiority of these schemes over certain existing methods investigated in the study.