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Models of the spread of infectious diseases commonly have to deal with the problem of multiple timescales which naturally occur in the epidemic models. In the most cases, this problem is implicitly avoided with the use of the so-called constant population size assumption. However, applicability of this assumption can require a justification (which is typically omitted). In this paper we consider some multiscale phenomena that arise in a reasonably simple Susceptible- Infected-Removed (SIR) model with variable population size. In particular, we discuss examples of the canard cascades and a black swan that arise in this model. © 2020 the Author(s).

The comprehensive list of various eye disorders includes congenital malformations, eye infections, and other acquired pathologies. If inherited and infectious diseases are excluded from the list, the most common causes of blindness are glaucoma, age-related macular degeneration, cataract, and diabetic retinopathy. Among these, cataract is the one most easily diagnosed and treatable by surgery. Diabetic retinopathy is the late complication of diabetes mellitus. Early detection and the timely treatment of diabetes mellitus help prevent diabetic retinopathy. The following chapter focuses on glaucoma, cataract, and age-related macular degeneration as the most common eye disorders with no effective treatment yet.

Despite the efforts of researchers all over the world, the mechanisms underlying the pathology of neurological diseases remain unclear. In this chapter, we review the molecular pathogenesis of Alzheimer’s disease and amyotrophic lateral sclerosis because these disorders are among the most challenging. Familial hemiplegic migraine is also reviewed in this chapter as an example of a disease with known mutations in genes regulating neuronal function.

In most papers focused on the system order reduction models, describing processes of heating, evaporation and ignition in fuel sprays, it is assumed that all functions in corresponding differential equations are sufficiently smooth and consequently Lipschitzian. In many cases, however, these functions are non-Lipschitzian. This means that the conventional approach to system order reduction, based on the theory of integral manifolds, cannot be applied. It is pointed out that the order reduction of systems with non-Lipschitzian non-linearities can be performed, using a concept of positively invariant manifolds. This concept is discussed and applied to the analysis of spray ignition based on five ODEs (for gas temperature, fuel vapour and oxygen concentrations, and droplet temperatures and radii). This system is reduced to single ordinary differential equations for the gas temperature or fuel concentrations. It is shown that the equation for gas temperature predicts an increase in gas temperature up to its limiting value during finite time. The reaching of this temperature is accompanied by the complete depletion of either fuel vapour or oxygen depending on their initial concentrations, as follows from the analysis of the equations for gas temperature and fuel concentration.

In women, the flow of psoriasis is influenced by each phase of a woman’s life cycle. According to previous findings, significant changes in the levels of sex hormones affect the severity of the disease. Aim: The aim of this study was to identify the estrogen-responsive genes that could be responsible for the exacerbation of psoriasis in menopausal women. Methods: Skin samples of lesional skin donated by psoriasis patients (n = 5) were compared with skin samples of healthy volunteers (n = 5) using liquid chromatography–tandem mass spectrometry (LC–MS/MS). The set of differentially expressed proteins was subjected to protein ontology analysis to identify differentially expressed estrogen-responsive proteins. The expression of discovered proteins was validated by qPCR and ELISA on four groups of female participants. The first group included ten psoriasis patients without menopause, the second included eleven postmenopausal patients, the third included five healthy volunteers without menopause, and the fourth included six postmenopausal volunteers. Moreover, the participants’ blood samples were used to assess the levels of estradiol, progesterone, and testosterone. Results: We found that the levels of estradiol and progesterone were significantly lower and the levels of testosterone were significantly higher in the blood of patients compared to the control. The protein ontology analysis of LC–MS/MS data identified six proteins, namely HMOX1, KRT19, LDHA, HSPD1, MAPK1, and CA2, differentially expressed in the lesional skin of female patients compared to male patients. ELISA and qPCR experiments confirmed differential expression of the named proteins and their mRNA. The genes encoding the named proteins were differentially expressed in patients compared to volunteers. However, KRT19 and LDHA were not differentially expressed when we compared patients with and without menopause. All genes, except MAPK1, were differentially expressed in patients with menopause compared to the volunteers with menopause. HMOX1, KRT19, HSPD1, and LDHA were differentially expressed in patients without menopause compared to the volunteers without menopause. However, no significant changes were found when we compared healthy volunteers with and without menopause. Conclusion: Our experiments discovered a differential expression of six estrogen-controlled genes in the skin of female patients. Identification of these genes and assessment of the changes in their expression provide insight into the biological effects of estrogen in lesional skin. The results of proteomic analysis are available via ProteomeXchange with identifier PXD021673.

The paper is devoted to the application of methods for constructing invariant integrals of fast-slow systems to the problems of "nonholonomic mechanics".

The review is devoted to genetic research in cancer of the vulva. In genetic changes, the mutation irreversibly changes the nucleotide sequence of DNA, or the number of copies of chromosomes changes per cell. In epigenetics, the nucleotide sequence remains unchanged, but gene activity is regulated by methylation of DNA or modification of histones. Most of the studies analyzed are devoted to the study of mutations in the TP53 gene. Many studies indicate that somatic mutations are more common in HPV-negative than in HPV-positive patients. Epigenetic studies in the main devoted to hypermethylation. The gene CDKN2A is most often studied in epigenetic terms. For most of the studied genes, hypermethylation occurs more often in squamous cell carcinoma of the vulva than in the precursors.

Peutz-Jeghers syndrome is a rare autosomal dominant disorder. Approximately 1:25,000 to 1:280,000 cases are registered annually. The pathogenesis of the disease is based on the mutation of the STK 11 gene on chromosome 19. Peutz-Jeghers syndrome is characterized by several symptoms: the formation of multiple hamartomatous polyps primarily in the gastrointestinal tract and hyperpigmentation of the mucous membranes and skin. Patients with Peutz-Jeghers syndrome often develop various malignant neoplasms, mainly localized in the pancreas and colon. We describe Peutz-Jeghers syndrome in a girl 4 years 7 months old. Initially, the child was diagnosed with vitiligo due to complaints of depigmentation of the skin of the face and hands. During re-examination after half a year, foci of hyperpigmentation on the lip and mucous membranes of the oral cavity were noted. Esophagogastroduodenoscopy showed the presence of a polypous lump in the stomach. Genetic consultation confirmed the diagnosis of Peutz-Jeghers syndrome. The absence of family history indicates a sporadic case characterized by diseases with an autosomal-dominant mode of inheritance. This clinical case demonstrates the need for gastroenterological and genetic examinations in the presence of lesions on the oral mucosa and the vermillion border of the lips to confirm or exclude Peutz-Jeghers syndrome.

The method of integral manifolds is used to study singularly perturbed systems of differential equations. The algorithms for the construction of the slow invariant manifolds in the case with different dimensions of the fast and slow variables was derived.

A specific class of cheap control problems is considered. The micro-drone quadrocopter model is used as an illustration.

The method of decomposition of singularly perturbed differential systems is developed and application of the method to the problems of non-holonomic mechanics is considered.

This article describes premises for the development of psychodermatology. An analysis of research literature and data is presented based on the example of psoriasis and anxiety disorder. Protein molecules with altered concentrations in patients with psoriasis and anxiety disorder compared to controls are identified (chemokine [C-C motif] ligand 2, corticotropin-release hormone, growth hormone 1, leptin, and tumor necrosis factor with increased concentration and brain-derived neurotrophic factor with decreased concentration). All molecules are secretory peptides. In the future, the information obtained may make it possible to pursue an in-depth study of the molecular mechanisms underlying psychodermatology.

We study an initial–boundary value problem for a singularly perturbed system of partial integro-differential equations. We prove a theorem on the passage to the limit. The result is used to decrease the dimension of a virus evolution model. We construct an asymptotic solution by the Tikhonov–Vasil’eva boundary function method. The analytic results obtained are compared with a numerical study of the system.

Annual and perennial plants represent two different evolutionary strategies based on differential synchronization of their reproductive development. The mobile signal protein FLOWERING LOCUS T (FT) plays a central role in mediating the onset of reproduction in both plant types. Two novel FT-like genes from pear (Pyrus communis)-PcFT1 and PcFT2-were isolated, and their expression profiles were determined for one annual cycle. The effects of PcFT2 on flowering were investigated in annual (tobacco) and perennial (apple) plants by means of grafting and generating transgenic plants. Long-distance graft transmission of PcFT2 in both annual and perennial plants was confirmed using a 35S::PcFT2-YFP construct. Ectopic overexpression of PcFT2 caused early flowering in tobacco but not in apple. Transgenic apples were less sensitive to short-day-induced dormancy, and this phenotype was also observed in wild-type apples grafted onto the transgenic plants. Comparison of PcFT2 protein structure to the paralogous FT proteins from apple and pear showed alterations that could influence protein structure and thus the florigen-activation complex. PcFT2 protein seems to function by promoting flowering as all other FT proteins in the annual plant tobacco while in the perennial plant apple PcFT2 does not promote flowering but delays senescence. This observation may hint to a modified function of FT2 in perennial plants.

This article describes premises for the development of psychodermatology. An analysis of research literature and data is presented based on the example of psoriasis and anxiety disorder. Protein molecules with altered concentrations in patients with psoriasis and anxiety disorder compared to controls are identified (chemokine [C-C motif] ligand 2, corticotropin-release hormone, growth hormone 1, leptin, and tumor necrosis factor with increased concentration and brain-derived neurotrophic factor with decreased concentration). All molecules are secretory peptides. In the future, the information obtained may make it possible to pursue an in-depth study of the molecular mechanisms underlying psychodermatology.

We consider a simple model of a passive dynamic biped robot with point feet and legs without knee. The model is a switched system, which includes an inverted double pendulum. We present an asymptotic solution of the model. The first correction to the zero order approximation is shown to agree with the numerical solution with high degree of accuracy for a limited parameter range.

We explore the possibility of applying the method of order reduction of the optimal estimation problem with low measurement noise for singularly perturbed systems. It is shown that matrix differential Riccati equation for the Kalman–Bucy filter has a periodic solution which may be used instead the solution of the original initial value problem for the matrix Riccati equation.

Photosystem II (PSII) reaction centre D1 protein of oxygenic phototrophs is pivotal for sustaining photosynthesis. Also, it is targeted by herbicides and herbicide-resistant weeds harbour single amino acid substitutions in D1. Conservation of D1 primary structure is seminal in the photosynthetic performance in many diverse species. In this study, we analysed built-in and environmentally-induced (high temperature and high photon fluency – HT/HL) phenotypes of two D1 mutants of Chlamydomonas reinhardtii with Ala250Arg (A250R) and Ser264Lys (S264K) substitutions. Both mutations differentially affected efficiency of electron transport and oxygen production. In addition, targeted metabolomics revealed that the mutants undergo specific differences in primary and secondary metabolism, namely, amino acids, organic acids, pigments, NAD, xanthophylls and carotenes. Levels of lutein, β-carotene and zeaxanthin were in sync with their corresponding gene transcripts in response to HT/HL stress treatment in the parental (IL) and A250R strains. D1 structure analysis indicated that, among other effects, remodelling of H-bond network at the QB site might underpin the observed phenotypes. Thus, the D1 protein, in addition to being pivotal for efficient photosynthesis, may have a moonlighting role in rewiring of specific metabolic pathways, possibly involving retrograde signalling.

We present a clinical case reflecting the difficulties of differential diagnosis of pemphigus erythematosus (PE) in a 40-yearold male presented with erythematous topic lesions, pustules, and crusts on the skin of the scalp and face, which had existed for the previous 20 years of his life. In our case, clinical manifestations of dermatosis were atypical due to the presence of pustules and crusts impregnated with seropurulent exudates, resistant to antibiotic therapy. The patient had areas of atrophy on the scalp that could be a sign of discoid lupus erythematosus. The negative results of tests for LE cells and double-strand DNA antibodies, as well as ineffective treatment with plaquenil, antibiotics and itraconazole, helped to verify the diagnosis of PE.

Accurate prediction of protein function in humans is important for understanding biological processes at the molecular level in biomedicine and drug design. Over a third of proteins are commonly held to bind metal, and ∼10% of human proteins, to bind zinc. Therefore, an initial step in protein function prediction frequently involves predicting metal ion binding. In recent years, methods have been developed to predict a set of residues in 3D space forming the metal-ion binding site, often with a high degree of accuracy. Here, using extensions of these methods, we provide an extensive list of human proteins and their putative metal ion binding site residues, using translated gene sequences derived from the complete, resolved human genome. Under conditions of ∼90% selectivity, over 900 new human putative metal ion binding proteins are identified. A statistical analysis of resolved metal ion binding sites in the human metalloproteome is furnished and the importance of remote homology analysis is demonstrated. As an example, a novel metal-ion binding site involving a complex of a botulinum substrate with its inhibitor is presented. On the basis of the location of the predicted site and the interactions of the contacting residues at the complex interface, we postulate that metal ion binding in this region could influence complex formation and, consequently, the functioning of the protein. Thus, this work provides testable hypotheses about novel functions of known proteins. Proteins 2015; 83:931–939. © 2015 Wiley Periodicals, Inc.

An initial boundary value problem for a singularly perturbed system of partial integro-differential equations involving two small parameters multiplying the derivatives is studied. The problem arises in a virus evolution model. An asymptotic solution of the problem is constructed by the Tikhonov-Vasil’eva method of boundary functions. The analytical results are compared with numerical ones.

The problem of stabilizing a spin satellite by means of passive dampers is considered. The application of the method of integral manifolds allows us to find conditions for the loss of stability in the analytical form.

Mathematical model of a single-link manipulator is considered. It describes the motion of the manipulator in the case of non-smooth path. Interpolation of the trajectory of motion is used, which makes it possible to reduce the amount of calculations and allows you to take into account the restrictions on the movement of the manipulator. Integral manifold method is used for the system order reduction. As a result, the reduced system of the investigated object is obtained, and the control function for the manipulation robot model in the case of a non-smooth periodic trajectory is constructed.

Proinflammatory cytokines TNF, IFNG, and IL17 play an important role in eruption of psoriasis. The activation of epidermal keratinocytes with the named cytokines alters their terminal differentiation program and causes their hyperproliferation in the diseased skin. HaCaT cells, which are immortalized human keratinocytes, are often used as a cellular model of psoriasis. The aim of this study was to evaluate changes in gene expression and the proliferation rates in cultured HaCaT cells treated with TNF, IFNG, and IL17. We found that HaCaT cells decrease their proliferation rate in response to either IL17 or a combination TNF and IFNG. The analysis of microarray data discovered a group of 12 genes, which were downregulated in HaCaT after treatments with the named cytokines and upregulated in psoriatic lesional skin. Eight genes were important for DNA replication and they also contributed to two larger networks that regulated cell progression through the cell cycle. We conclude that HaCaT cells have a sufficient limitation as a cellular model of psoriasis due to their treatment with proinflammatory cytokines, namely TNF, IFNG, and IL17 does not increase their proliferation rate. Thus, the studies of psoriasis based on HaCaT cells as an experimental model shall take in account this important phenomenon.

The possibility of applying the method of integral manifolds to the reduction of optimal filtering problems for systems with low energy dissipation is explored. For such systems, it is shown that the slow subsystem of matrix Riccati differential equations turns out to have a higher dimension than expected, which leads to an increase in the dimension of the reduced problems. An optimal filter is constructed for the Langevin equation and for a dynamic model of a single-link flexible manipulator.

Some new approaches to the solution of complex problems, focused on spray modelling, using relatively simple mathematical tools are briefly summarised. The following problems are considered: modelling of spray drying with pharmaceutical applications, modelling of heating and evaporation of suspended droplets with applications to water sprays for fire suppression, modelling of micro-explosions with automotive applications, and parameterisations of slow invariant manifolds with applications to spray ignition modelling. The modelling of spray drying is based on the assumption that the array of small suspended solid particles inside droplets can be treated as a non-evaporating liquid component. The modelling of the effect of a support rod on droplet heating and evaporation is based on the assumption that this heat is homogeneously distributed inside the whole volume of a droplet. In the modelling of a puffing/micro-explosion it is assumed that a water sub-droplet is located exactly at the centre of a fuel droplet and the temperature at the surface of the fuel droplet is fixed. In the presentation of slow invariant manifolds in parametric form these manifolds are found using asymptotic expansions.

The problem of the splitting of singularly perturbed differential systems is invesigated and the application to the problem of suicide substrate kinetics is given.

In the paper, we consider a new type of travelling waves, a profile of which is a pure slow heteroclinic canard. To construct the profile we use the so-called canard cascade. Such travelling waves are critical because they are a watershed between travelling waves of different types and simulate critical phenomena in various applied problems.

In this work we provide the novel hysteretic approach to control the chaotic dynamics of the van der Pol oscillator under periodic excitation. Using the phenomenological Bouc-Wen model, as well as in the frame of the small parameter approach we investigate the influence of the hysteretic control to the dynamical characteristics of this system. Based on the analysis of numerical results in the form of bifurcation diagrams and Lyapunov exponents, the efficiency of the stabilizing role of the hysteretic control element is established. In addition, we investigate the synchronization of oscillations in the system of coupled van der Pol oscillators (with “simple” linear coupling, as well as under hysteretic control). The stabilizing role of the hysteretic control in this case is also demonstrated.

The problem of the decomposition of singularly perturbed differential systems by the method of integral manifolds is studied and the application of the method to the problems of enzyme kinetics is considered.

Three-dimensional models of skin and epidermis imitate the structure of real tissues and provide accurate information about certain skin conditions, such as psoriasis. A three-dimensional model of mouse epidermis was generated from the epidermal keratinocytes of newborn mice and treated with cytokines. The aim of this study was to evaluate this model as an experimental model of psoriasis and to assess the changes occurring in its structure and gene expression after the exposure to proinflammatory cytokines. Treatment of the three-dimensional model with either interleukin 17 or a combination of tumor necrosis factor and interferon was shown to produce morphological changes, which were similar to acanthosis in psoriatic skin. The observed changes in gene expression of metalloproteinases and certain psoriasis biomarkers, such as mki67, krt16 and fosl1, were similar to the changes in patients skin. Notably, changes caused by interleukin 17 were less evident than those caused by the combination of interferon and tumor necrosis factor. On the contrary, HaCaT cells exhibited no significant changes in the expression of fosl1 and had decreased levels of mki67 after being treated with a combination of TNF and IFNG. Moreover, treatment with IL17 had no significant effect on krt16 and mki67 expression and even reduced the fosl1 levels. The findings suggest that artificially generated three-dimensional models of murine skin can be used to study psoriasis.

This review summarizes the existing knowledge regarding the role of receptor for advanced glycation end products in pathogenesis of psoriasis. This receptor plays a crucial role in the inflammatory response. By interacting with multiple ligands and activating several signaling mechanisms, receptor for advanced glycation end products regulates gene expression via a group of well-characterized transcription factors, such as NFkB and AP1. The expression of receptor for advanced glycation end products in both immune cells and their targets, a high stability of this receptor in complexes with ligands as well as a positive feedback loop, upregulating the expression of its certain ligands, suggest receptor for advanced glycation end products as a possible principal factor that promotes the development of psoriasis. Considering receptor for advanced glycation end products as a potential master regulator of several processes that play a crucial role in development of psoriatic plaques, we believe that further experimental studies are needed to elucidate how exactly this receptor converts a transient inflammatory reaction to a sustainable inflammatory response. These studies are also needed for the development of novel medications that target receptor for advanced glycation end products and signaling mechanisms that this receptor activates.

We use the method of integral manifolds to reduce optimal control and filtering problems in application to mechanical systems with low friction.

Approximately one third of proteins bind metal ions for stability and/or enzymatic function. However, on a structural level, only a small fraction of binding sites have been resolved. Metal binding site predictions can serve as a first step in putative function assignment for many unannotated proteins. Sequence based and structure based methods for metal binding site predictions are reviewed here. The CHED and SeqCHED methods of prediction from apo protein structures and translated gene sequences, respectively, are described in detail, including their web server applications. The relevance of SeqCHED to the analysis of single nucleotide polymorphisms (SNPs) associated with disease related metal binding sites is illustrated.

The mathematical model of the differential cascade with a symmetrical division of stem cells is considered in the paper. The using of the RouthHurwitz’s criterion and the Andronov-Hopf theorem [1] allows to find the conditions of bifurcation cycle in the model under consideration, and the phenomenon of "soft loss of stability" was stated.

The aim of the paper is to describe the main critical cases in the theory of singularly perturbed optimal control problems and to give examples which are typical for slow/fast systems. The theory has traditionally dealt only with perturbation problems near normally hyperbolic manifold of singularities and this manifold is supposed to isolated. We reduce the original singularly perturbed problem to a regularized one such that the existence of slow integral manifolds can be established by means of the standard theory. We illustrate our approach by several examples of control problems.