Influenza and influenza-like illnesses pose significant challenges to healthcare systems globally. Mathematical models play a crucial role in understanding their dynamics, calibrating them to specific scenarios and making projections about their evolution over time. This study proposes a calibration process for three different but well-known compartmental models-SIR, SEIR/SLIR and SLAIR-using influenza data from the 2016-2017 season in the Valencian Community, Spain. The calibration process involves indirect calibration for the SIR and SLIR models, requiring post-processing to compare model output with data, while the SLAIR model is directly calibrated through direct comparison. Our calibration results demonstrate remarkable consistency between the SIR and SLIR models, with slight variations observed in the SLAIR model due to its unique design and calibration strategy. Importantly, all models align with existing evidence and intuitions found in the medical literature. Our findings suggest that at the onset of the epidemiological season, a significant proportion of the population (ranging from 29.08% to 43.75% of the total population) may have already entered the recovered state, likely due to immunization from the previous season. Additionally, we estimate that the percentage of infected individuals seeking healthcare services ranges from 5.7% to 12.2%. Through a well-founded and calibrated modeling approach, our study contributes to supporting, settling and quantifying current medical issues despite the inherent uncertainties involved in influenza dynamics. The full Mathematica code can be downloaded from https://munqu.webs.upv.es/software.html.

The intestinal microbiota play a critical role in human health and disease, maintaining metabolic and immune/inflammatory health, synthesizing essential vitamins and amino acids and maintaining intestinal barrier integrity. The aim of this paper is to develop a mathematical model to describe the complex interactions between the microbiota, vitamin D/vitamin D receptor (VDR) pathway, epithelial barrier and immune response in order to understand better the effects of supplementation with probiotics and vitamin D. This is motivated by emerging data indicating the beneficial effects of vitamin D and probiotics individually and when combined. We propose a system of ordinary differential equations determining the time evolution of intestinal bacterial populations, concentration of the VDR:1,25(OH)$_{2}$D complex in epithelial and immune cells, the epithelial barrier and the immune response. The model shows that administration of probiotics and/or vitamin D upregulates the VDR complex, which enhances barrier function and protects against intestinal inflammation. The model also suggests co-supplementation to be superior to individual supplements. We explore the effects of inflammation on the populations of commensal and pathogenic bacteria and the vitamin D/VDR pathway and discuss the value of gathering additional experimental data motivated by the modelling insights.

Alzheimer's disease (AD) presents a perplexing question: why does its development span decades, even though individual amyloid beta (Aβ) deposits (senile plaques) can form rapidly in as little as 24 hours, as recent publications suggest? This study investigated whether the formation of senile plaques can be limited by factors other than polymerization kinetics alone. Instead, their formation may be limited by the diffusion-driven supply of Aβ monomers, along with the rate at which the monomers are produced from amyloid precursor protein and the rate at which Aβ monomers undergo degradation. A mathematical model incorporating the nucleation and autocatalytic process (via the Finke-Watzky model), as well as Aβ monomer diffusion, was proposed. The obtained system of partial differential equations was solved numerically, and a simplified version was investigated analytically. The computational results predicted that it takes approximately 7 years for Aβ aggregates to reach a neurotoxic concentration of 50 μM. Additionally, a sensitivity analysis was performed to examine how the diffusivity of Aβ monomers and their production rate impact the concentration of Aβ aggregates.

This paper investigates intimal growth in arteries, induced by hemodynamical shear stress, through finite element simulation using the FEniCS computational environment. In our model, the growth of the intima depends on cross-section geometry and shear stress. In this work, the arterial wall is modeled as three distinct layers: the intima, the media and the adventitia, each with different mechanical properties. We assume that the cross-section of the vessel does not change in the axial direction. We further assume that the blood flow is steady, non-turbulent and unidirectional. Blood flow induces shear stress on the endothelium and stimulates the release of platelet derived growth factor (PDGF) which drives the growth. We simulate intimal growth for three distinct arterial cross section geometries. We show that the qualitative nature of intimal thickening varies depending on arterial geometry. For cross section geometries that are annular, the growth of the intima is uniform in the angular direction, and the endothelium stays circular as the intima grows. For non-annular cross section geometries, the intima grows more quickly where it is thicker, and shear stress and intimal thickening are negatively correlated with the distance from the flow center, where the flow velocity is maximal. Over time, the maxima and minima of the curvature increase and decrease, respectively, the PDGF concentration increases and the lumen becomes more polygonal. The model provides a framework for coupling hemodynamics simulations to mathematical descriptions of atherosclerosis, both of which have been modeled separately in great detail.

The ability to predict clinically relevant exposure to potentially hazardous compounds that can leach from polymeric components can help reduce testing needed to evaluate the biocompatibility of medical devices. In this manuscript, we compare two physics-based exposure models: 1) a simple, one-component model that assumes the only barrier to leaching is the migration of the compound through the polymer matrix and 2) a more clinically relevant, two-component model that also considers partitioning across the polymer-tissue interface and migration in the tissue away from the interface. Using data from the literature, the variation of the model parameters with key material properties were established, enabling the models to be applied to a wide range of combinations of leachable compound, polymer matrix and tissue type. Exposure predictions based on the models suggest that the models are indistinguishable over much of the range of clinically relevant scenarios. However, for systems with low partitioning and/or slow tissue diffusion, the two-component model predicted up to three orders of magnitude less mass release over the same time period. Thus, despite the added complexity, in some scenarios it can be beneficial to use the two-component model to provide more clinically relevant estimates of exposure to leachable substances from implanted devices.

The linear functional analysis, historically founded by Fourier and Legendre (Fourier's supervisor), has provided an original vision of the mathematical transformations between functional vector spaces. Fourier, and later Laplace and Wavelet transforms, respectively defined using the simple and damped pendulum, have been successfully applied in numerous applications in Physics and engineering problems. However the classical pendulum basis may not be the most appropriate in several problems, such as biological ones, where the modelling approach is not linked to the pendulum. Efficient functional transforms can be proposed by analysing the links between the physical or biological problem and the orthogonal (or not) basis used to express a linear combination of elementary functions approximating the observed signals. In this study, an extension of the Fourier point of view called Dynalet transform, is described. The approach provides robust approximated results in the case of relaxation signals of periodic biphasic organs in human physiology.

Prion-like proteins play crucial parts in biological processes in organisms ranging from yeast to humans. For instance, many neurodegenerative diseases are believed to be caused by the production of prion-like proteins in neural tissue. As such, understanding the dynamics of prion-like protein production is a vital step toward treating neurodegenerative disease. Mathematical models of prion-like protein dynamics show great promise as a tool for predicting disease trajectories and devising better treatment strategies for prion-related diseases. Herein, we investigate a generic model for prion-like dynamics consisting of a class of non-linear ordinary differential equations (ODEs), establishing constraints through a linear stability analysis that enforce the expected properties of mammalian prion-like toxicity. Furthermore, we identify that prion toxicity evolves through three distinct phases for which we provide analytical descriptions using perturbation analyses. Specifically, prion-toxicity is initially characterised by the healthy phase, where the dynamics are dominated by the healthy form of prions, thereafter the system enters the mixed phase, where both healthy and toxic prions interact, and lastly, the system enters the toxic phase, where toxic prions dominate, and we refer to these phases as HeMiTo-dynamics. These findings hold the potential to aid researchers in developing precise mathematical models for prion-like dynamics, enabling them to better understand underlying mechanisms and devise effective treatments for prion-related diseases.

This paper investigates some properties of the large time behaviour of the solutions of a spatially distributed system of equations modelling the evolutionary epidemiology of a plant-pathogen system. The model takes into account the phenotypic trait and the mutation of the pathogen, which is described by a non-local operator. We roughly speaking prove that the solutions separate the phenotype trait from the spatio-temporal evolution in the large time asymptotic. This feature is obtained by investigating the positive and bounded entire solutions of the problem, that are shown to exhibit such a separation of the variables property, by reformulating them as the positive solutions of suitable integral equations in some ordered Banach space. In addition, some numerical simulations are performed to support our theoretical results.

Experimental and theoretical properties of amino acids as building blocks of peptides and proteins have been extensively researched. Each such method assigns a number to each amino acid, and one such assignment is called amino-acid scale. Their usage in bioinformatics to explain and predict behaviour of peptides and proteins is of essential value. The number of such scales is very large. There are more than a hundred scales related just to hydrophobicity. A large number of scales can be a computational burden for algorithms that try to define peptide descriptors combining several of these scales. Hence, it is of interest to construct a smaller, but still representative set of scales. Here, we present software that does this. We test it on the set of scales using a database constructed by Kawashima and collaborators and show that it is possible to significantly reduce the number of scales observed without losing much of the information. An algorithm is implemented in C#. As a result, we provide a smaller database that might be a very useful tool for the analyses and construction of new peptides. Another interesting application of this database would be to compare the artificial intelligence construction of peptides having as an input the complete Kawashima database and this reduced one. Obtaining in both cases similar results would give much credibility to the constructs of such AI algorithms.

Since 2019, a new strain of coronavirus has challenged global health systems. Due its fragile healthcare systems, Africa was predicted to be the most affected continent. However, past experiences of African countries with epidemics and other factors, including actions taken by governments, have contributed to reducing the spread of SARS-CoV-2. This study aims to assess the marginal impact of non-pharmaceutical interventions in fifteen African countries during the pre-vaccination period. To describe the transmission dynamics and control of SARS-CoV-2 spread, an extended time-dependent SEIR model was used. The transmission rate of each infectious stage was obtained using a logistic model with NPI intensity as a covariate. The results revealed that the effects of NPIs varied between countries. Overall, restrictive measures related to assembly had, in most countries, the largest reducing effects on the pre-symptomatic and mild transmission, while the transmission by severe individuals is influenced by privacy measures (more than $10\%$). Countries should develop efficient alternatives to assembly restrictions to preserve the economic sector. This involves e.g. training in digital tools and strengthening digital infrastructures.

Glioblastoma multiforme is a highly aggressive form of brain cancer, with a median survival time for diagnosed patients of 15 months. Treatment of this cancer is typically a combination of radiation, chemotherapy and surgical removal of the tumour. However, the highly invasive and diffuse nature of glioblastoma makes surgical intrusions difficult, and the diffusive properties of glioblastoma are poorly understood. In this paper, we introduce a stochastic interacting particle system as a model of in vitro glioblastoma migration, along with a maximum likelihood-algorithm designed for inference using microscopy imaging data. The inference method is evaluated on in silico simulation of cancer cell migration, and then applied to a real data set. We find that the inference method performs with a high degree of accuracy on the in silico data, and achieve promising results given the in vitro data set.

Epidemic models of susceptibles, exposed, infected, recovered and deceased (SΕIRD) presume homogeneity, constant rates and fixed, bilinear structure. They produce short-range, single-peak responses, hardly attained under restrictive measures. Tuned via uncertain I,R,D data, they cannot faithfully represent long-range evolution. A robust epidemic model is presented that relates infected with the entry rate to health care units (HCUs) via population averages. Model uncertainty is circumvented by not presuming any specific model structure, or constant rates. The model is tuned via data of low uncertainty, by direct monitoring: (a) of entries to HCUs (accurately known, in contrast to delayed and non-reliable I,R,D data) and (b) of scaled model parameters, representing population averages. The model encompasses random propagation of infections, delayed, randomly distributed entries to HCUs and varying exodus of non-hospitalized, as disease severity subdues. It closely follows multi-pattern growth of epidemics with possible recurrency, viral strains and mutations, varying environmental conditions, immunity levels, control measures and efficacy thereof, including vaccination. The results enable real-time identification of infected and infection rate. They allow design of resilient, cost-effective policy in real time, targeting directly the key variable to be controlled (entries to HCUs) below current HCU capacity. As demonstrated in ex post case studies, the policy can lead to lower overall cost of epidemics, by balancing the trade-off between the social cost of infected and the economic contraction associated with social distancing and mobility restriction measures.

In this paper we consider a tumor-immune system interaction model with immune response delay, in which a nonmonotonic function is used to describe immune response to the tumor burden and a time delay is used to represent the time for the immune system to respond and take effect. It is shown that the model may have one, two or three tumor equilibria, respectively, under different conditions. Time delay can only affect the stability of the low tumor equilibrium and local Hopf bifurcation occurs when the time delay passes through a critical value. The direction and stability of the bifurcating periodic solutions are also determined. Moreover, the global existence of periodic solutions is established by using a global Hopf bifurcation theorem. We also observe the existence of relaxation oscillations and complex oscillating patterns driven by the time delay. Numerical simulations are presented to illustrate the theoretical results.

Viral infection develops in the organism due to virus replication inside infected cells and its transmission from infected to uninfected cells through the extracellular matrix or cell junctions. In this work, we model infection spreading in tissue with a delay reaction-diffusion system of equations for the concentrations of uninfected cells, infected cells and virus. We prove the wave existence, determine its speed of propagation and introduce a simplified one-equation model obtained from the complete model using a quasi-stationary approximation.

The goal of this work is to understand and quantify how a line with nonlocal diffusion given by an integral enhances a reaction-diffusion process occurring in the surrounding plane. This is part of a long term programme where we aim at modelling, in a mathematically rigorous way, the effect of transportation networks on the speed of biological invasions or propagation of epidemics. We prove the existence of a global propagation speed and characterise in terms of the parameters of the system the situations where such a speed is boosted by the presence of the line. In the course of the study we also uncover unexpected regularity properties of the model. On the quantitative side, the two main parameters are the intensity of the diffusion kernel and the characteristic size of its support. One outcome of this work is that the propagation speed will significantly be enhanced even if only one of the two is large, thus broadening the picture that we have already drawn in our previous works on the subject, with local diffusion modelled by a standard Laplacian. We further investigate the role of the other parameters, enlightening some subtle effects due to the interplay between the diffusion in the half plane and that on the line. Lastly, in the context of propagation of epidemics, we also discuss the model where, instead of a diffusion, displacement on the line comes from a pure transport term.

Predicting the endemic/epidemic transition during the temporal evolution of a contagious disease.

This paper is concerned with the existence of transition fronts for a one-dimensional two patch model with KPP reaction terms. Density and flux conditions are imposed at the interface between the two patches. We first construct a pair of suitable super- and sub solutions by making full use of information of the leading edges of two KPP fronts and gluing them through the interface conditions. Then, an entire solution obtained thanks to a limiting argument is shown to be a transition front moving from one patch to the other one. This propagating solution admits asymptotic past and future speeds, and it connects two different fronts, each associated with one of the two patches. The paper thus provides the first example of a transition front for a KPP-type two-patch model with interface conditions.

Macrophages play a wide range of roles in resolving the inflammatory damage that underlies many medical conditions and have the ability to adopt different phenotypes in response to different environmental stimuli. Categorising macrophage phenotypes exactly is a difficult task, and there is disparity in the literature around the optimal nomenclature to describe these phenotypes; however, what is clear is that macrophages can exhibit both pro- and anti-inflammatory behaviours dependent upon their phenotype, rendering mathematical models of the inflammatory response potentially sensitive to their description of the macrophage populations that they incorporate. Many previous models of inflammation include a single macrophage population with both pro- and anti-inflammatory functions. Here, we build upon these existing models to include explicit descriptions of distinct macrophage phenotypes and examine the extent to which this influences the inflammatory dynamics that the models emit. We analyse our models via numerical simulation in MATLAB and dynamical systems analysis in XPPAUT, and show that models that account for distinct macrophage phenotypes separately can offer more realistic steady state solutions than precursor models do (better capturing the anti-inflammatory activity of tissue resident macrophages), as well as oscillatory dynamics not previously observed. Finally, we reflect on the conclusions of our analysis in the context of the ongoing hunt for potential new therapies for inflammatory conditions, highlighting manipulation of macrophage polarisation states as a potential therapeutic target.

Myeloproliferative neoplasms (MPN) are blood cancers that appear after acquiring a driver mutation in a hematopoietic stem cell. These hematological malignancies result in the overproduction of mature blood cells and, if not treated, induce a risk of cardiovascular events and thrombosis. Pegylated IFN$\alpha $ is commonly used to treat MPN, but no clear guidelines exist concerning the dose prescribed to patients. We applied a model selection procedure and ran a hierarchical Bayesian inference method to decipher how dose variations impact the response to the therapy. We inferred that IFN$\alpha $ acts on mutated stem cells by inducing their differentiation into progenitor cells; the higher the dose, the higher the effect. We found that the treatment can induce long-term remission when a sufficient (patient-dependent) dose is reached. We determined this minimal dose for individuals in a cohort of patients and estimated the most suitable starting dose to give to a new patient to increase the chances of being cured.

We discuss the mathematical modelling of two of the main mechanisms that pushed forward the emergence of multicellularity: phenotype divergence in cell differentiation and between-cell cooperation. In line with the atavistic theory of cancer, this disease being specific of multicellular animals, we set special emphasis on how both mechanisms appear to be reversed, however not totally impaired, rather hijacked, in tumour cell populations. Two settings are considered: the completely innovating, tinkering, situation of the emergence of multicellularity in the evolution of species, which we assume to be constrained by external pressure on the cell populations, and the completely planned-in the body plan-situation of the physiological construction of a developing multicellular animal from the zygote, or of bet hedging in tumours, assumed to be of clonal formation, although the body plan is largely-but not completely-lost in its constituting cells. We show how cancer impacts these two settings and we sketch mathematical models for them. We present here our contribution to the question at stake with a background from biology, from mathematics and from philosophy of science.