Equations with Infinite Delay: Pseudospectral Discretization for Numerical Stability and Bifurcation in an Abstract Framework

We consider nonlinear delay differential and renewal equations with infinite delay. We extend the work of Gyllenberg et al. [Appl. Math. Comput., 333 (2018), pp. 490–505] by introducing a unifying abstract framework, and we derive a finite-dimensional approximating system via pseudospectral discretization. For renewal equations, we consider a reformulation in the space of absolutely continuous functions via integration. We prove the one-to-one correspondence of equilibria between the original equation and its approximation, and that linearization and discretization commute. Our most important result is the proof of convergence of the characteristic roots of the pseudospectral approximation of the linear(ized) equations when the collocation nodes are chosen as the family of scaled zeros or extrema of Laguerre polynomials. This ensures that the finite-dimensional system correctly reproduces the stability properties of the original linear equation if the dimension of the approximation is large enough. The result is illustrated with several numerical tests, which also demonstrate the effectiveness of the approach for the bifurcation analysis of equilibria of nonlinear equations. The new approach used to prove convergence also provides the exact location of the spectrum of the differentiation matrices for the Laguerre zeros and extrema, adding new insights into properties that are important in the numerical solution of differential equations by pseudospectral methods

Pseudospectral discretization of nonlinear delay equations: new prospects for numerical bifurcation analysis

We apply the pseudospectral discretization approach to nonlinear delay models described by delay differential equations, renewal equations or systems of coupled renewal equations and delay differential equations. The aim is to derive ordinary differential equations and to investigate the stability and bifurcation of equilibria of the original model by available software packages for continuation and bifurcation for ordinary differential equations. Theoretical and numerical results confirm the effectiveness and the versatility of the approach, opening a new perspective for the bifurcation analysis of delay equations, in particular coupled renewal and delay differential equations

Map of the sarcoglycan sub-complex in rat brain

The sarcoglycan-sub-complex is made up of six glycoproteins which play mechanosignaling functions, connecting the extracellular matrix to cytoskeleton. This protein complex has been identified in different kind of tissues; in central nervous system, instead, only the ζ-sarcoglycans and the ε-sarcoglycans are considered to be present, where they seem to play a different role from the role played in muscle. Although that, previous our study have shown the expression of the entire sarcoglycan sub-complex in some region of the rat brain and the colocalization of this complex with post-synaptic receptors as GABA and DOPA receptors. Since we found that each sarcoglycan changes in staining pattern level among the brain regions, in the present study we performed, for the first time, a map of sarcoglycans expression in whole brain of rat and we examined which kind of post-synaptic receptor colocalizes with sarcoglycans in each part of brain. Results have shown that in rat brain the staining pattern level for each sarcoglycan and the different kind of colocalization between sarcoglycans and post-synaptic receptors, sarcoglycan/GABA or sarcoglycan/ DOPA, are characteristic of each brain region. These results support a role of the sarcoglycan sub-complex in post-synaptic neurotransmission, maybe modulating post-synaptic receptor assembly and stabilization

Prevalence of thyroid nodules in an occupationally radiation exposed group: a cross sectional study in an area with mild iodine deficiency

BACKGROUND: Thyroid nodules and thyroid cancer occur more frequently in people exposed to radiation for therapeutic purposes, and to nuclear fallout. Furthermore, it is known that a moderate degree of iodine deficiency may be responsible for an increased prevalence of thyroid nodules, while it is suspected that radiation exposure could induce changes in thyroid autoimmunity. The iodine intake of people resident in Bari, S. Italy, is mildly deficient, which could be presumed to cause a higher prevalence of thyroid pathology. This study was conducted to evaluate the prevalence of thyroid nodules in a population occupationally exposed to radiation, in an area of mild iodine deficiency. METHODS: A cross-sectional study was designed to evaluate the prevalence of thyroid nodules in radiation exposed workers, compared with a stratified sample of non exposed workers. After giving written consent to participate in the study, all the recruited subjects (304 exposed and 419 non exposed volunteers) were interviewed to fill in an anamnestic questionnaire, and underwent a physical examination, ultrasound thyroid scan, serum determinations of fT3, fT4 and TSH, fine needle aspiration biopsy. The sample was subdivided into one group exposed to a determined quantity of radiation (detected by counter), one group exposed to an undetectable quantity of radiation, and the non exposed control group. RESULTS: The prevalence of thyroid nodules <1 cm in diameter, defined as incidentalomas, in the exposed group with detected doses, was 11.28% in males and 9.68% in females, while in the exposed group with undetectable dose the prevalence was 10.39% in males and 16.67% in females. In the non exposed group the prevalence of incidentalomas was 9.34% in males and 13.20% in females. These prevalences were not statistically different when analysed by a multiple test comparison with the bootstrap method and stratification for sex. Instead, the prevalence of thyroid nodules > 1 cm in diameter resulted statistically different in exposed and non exposed health staff: 18.68% in non exposed males vs exposed: 3.76% (determined dose) and 9.09% (undetectable dose) in males, and 20.30% in non exposed females versus 3.23% (detected dose) and 9.52% (undetectable dose) in exposed females. There was a higher proportion of healthy staff in the exposed group than in the non exposed: (80.45% vs 68.68% in males; 80.65% vs 57.87% in females). CONCLUSION: In our study, occupational exposure to radiation combined with mild iodine deficiency did not increase the risk of developing thyroid nodules. The statistically significant higher prevalence of thyroid nodules in the non exposed group could be explained by the high percentage (22%) of people with a familial history of, and hence a greater predisposition to, thyroid disease. The endemic condition of mild iodine deficiency, demonstrated in other studies, played a major role in determining the thyroid pathology in our study groups

Microsatellite instability in thyroid tumours and tumour-like lesions

Fifty-one thyroid tumours and tumour-like lesions were analysed for instability at ten dinucleotide microsatellite loci and at two coding mononucleotide repeats within the transforming growth factor β (TGF-β) type II receptor (TβRII) and insulin-like growth factor II (IGF-II) receptor (IGFIIR) genes respectively. Microsatellite instability (MI) was detected in 11 out of 51 cases (21.5%), including six (11.7%) with MI at one or two loci and five (9.8%) with Ml at three or more loci (RER+ phenotype). No mutations in the TβRII and IGFIIR repeats were observed. The overall frequency of MI did not significantly vary in relation to age, gender, benign versus malignant status and tumour size. However, widespread MI was significantly more frequent in follicular adenomas and carcinomas than in papillary and Hürthle cell tumours: three out of nine tumours of follicular type (33.3%) resulted in replication error positive (RER+), versus 1 out of 29 papillary carcinomas (3.4%, P = 0.01), and zero out of eight Hürthle cell neoplasms. Regional lymph node metastases were present in five MI-negative primary cancers and resulted in MI-positive in two cases. © 1999 Cancer Research Campaig

Approximating reproduction numbers: a general numerical method for age-structured models

In this paper, we introduce a general numerical method to approximate the reproduction numbers of a large class of multi-group, age-structured, population models with a finite age span. To provide complete flexibility in the definition of the birth and transition processes, we propose an equivalent formulation for the age-integrated state within the extended space framework. Then, we discretize the birth and transition operators via pseudospectral collocation. We discuss applications to epidemic models with continuous and piecewise continuous rates, with different interpretations of the age variable (e.g., demographic age, infection age and disease age) and the transmission terms (e.g., horizontal and vertical transmission). The tests illustrate that the method can compute different reproduction numbers, including the basic and type reproduction numbers as special cases