One Dimensional Reduction of a Renewal Equation for a Measure-Valued Function of Time Describing Population Dynamics

Despite their relevance in mathematical biology, there are, as yet, few general results about the asymptotic behaviour of measure valued solutions of renewal equations on the basis of assumptions concerning the kernel. We characterise, via their kernels, a class of renewal equations whose measure-valued solution can be expressed in terms of the solution of a scalar renewal equation. The asymptotic behaviour of the solution of the scalar renewal equation, is studied via Feller’s classical renewal theorem and, from it, the large time behaviour of the solution of the original renewal equation is derived.

Regularization algorithm and its implementation in general systems theory and its implications in physics

In recent years a new physics known as complexity has emerged, whose fundamental structure is based on the general theory of systems or the so-called mathematical biology. There, problems arise with a large number of variables on which decisions must be made, and it is important to have selection techniques that are robust and efficient, In this research work, a couple of techniques are presented that will serve the aforementioned purpose; these are: the Lasso regularized linear regression technique and the Ridge regression technique, the latter with the slight difference in the penalty, since it makes use of a different rule than the Euclidean one, which implies very different consequences. One of these techniques was applied in a clinical problem and corresponds to the study of patients with diabetes. With this objective, the algorithm of descent of coordinates was implemented for the type of regularization loop and the analytical solution of regression of ridges. In addition, Kernels was used for the implementation of the regularized vector machine. The above algorithms were compared with the Ridge regularization and the Euclidean regularized vector machine. These types of techniques are common in physics, and it is vitally important to be able to implement other types of rules for a problem in which the choice of them can be simplified.

Existence, Uniqueness, and Stability Analysis of the Probabilistic Functional Equation Emerging in Mathematical Biology and the Theory of Learning

Probabilistic functional equations have been used to analyze various models in computational biology and learning theory. It is worth noting that they are linked to the symmetry of a system of functional equations’ transformation. Our objective is to propose a generic probabilistic functional equation that can cover most of the mathematical models addressed in the existing literature. The notable fixed-point tools are utilized to examine the existence, uniqueness, and stability of the suggested equation’s solution. Two examples are also given to emphasize the significance of our findings.

History and Future Perspectives on the Discipline of Quantitative Systems Pharmacology Modeling and Its Applications

Mathematical biology and pharmacology models have a long and rich history in the fields of medicine and physiology, impacting our understanding of disease mechanisms and the development of novel therapeutics. With an increased focus on the pharmacology application of system models and the advances in data science spanning mechanistic and empirical approaches, there is a significant opportunity and promise to leverage these advancements to enhance the development and application of the systems pharmacology field. In this paper, we will review milestones in the evolution of mathematical biology and pharmacology models, highlight some of the gaps and challenges in developing and applying systems pharmacology models, and provide a vision for an integrated strategy that leverages advances in adjacent fields to overcome these challenges.