Study of eigenvalue problems associated with matrix polynomials with single or multiple parameters appears in many practical applications. Polynomial two-parameter eigenvalue problem (PTEP) is a special type of such problems, which consists of a system of two bivariate matrix polynomials with two spectral parameters. The numerical study of polynomial eigenvalue problem is an emerging area of research in recent times and sustainable progress has been made in the areas of numerical algorithms, sensitivity and backward error analysis. To know the error in a computed solution of eigenpairs, it is very much essential to analyse the backward errors and condition numbers of the problem. However, for PTEP of grade k, this analysis becomes more complex, than in the single-parameter case due to the interaction between the two parameters. In this paper, we consider the PTEP of grade k and provided new estimates of the normwise backward error and condition numbers.

Human immunodeficiency virus (HIV) remains a major global health concern. This study presents a mathematical model using ordinary differential equations with a saturated incidence rate to analyse HIV transmission dynamics. The basic reproduction number R0 is derived, indicating disease-free equilibrium stability when R0 < 1, and endemic stability when R0 >1. Unlike conventional models, we employ structure-preserving numerical schemes to ensure biologically consistent solutions and unconditional stability. Two numerical approaches are developed, the standard finite difference (SFD) and non-standard finite difference (NSFD) methods. While SFD shows conditional convergence, the NSFD scheme provides unconditionally stable and positive solutions, even with large time steps. Numerical experiments confirm the accuracy and practicality of the proposed NSFD approach. This model and method can be extended to other infectious diseases, such as hepatitis and whooping cough, offering a robust tool for long-term epidemiological analysis and control strategy assessment.

In this paper, several shrunken estimators for the ratio of scale parameters of the exponential two-sample problem using two types of shrinkage functions are proposed. The first shrinkage function is based on the maximum likelihood estimator, and the second is based on the uniformly minimum variance unbiased estimator. The shrunken estimators expressions are obtained, computed and compared. Moreover, the performances of the shrunken estimators are evaluated. The proposed estimators dominate the classical estimators in the surprisingly large neighbourhood of the prior estimate of the ratio of scale parameters. Ultimately, using the data of a real experiment, the usefulness of these estimators are also illustrated.