Total Positivity Research Articles

Total Positivity and Accurate Computations Related to q-Abel Polynomials

The attainment of accurate numerical solutions of ill-conditioned linear algebraic problems involving totally positive matrices has been gathering considerable attention among researchers over the last years. In parallel, the interest of q-calculus has been steadily growing in the literature. In this work the q-analogue of the Abel polynomial basis is studied. The total positivity of the matrix of change of basis between monomial and q-Abel bases is characterized, providing its bidiagonal factorization. Moreover, well-known high relative accuracy results of Vandermonde matrices corresponding to increasing positive nodes are extended to the decreasing negative case. This further allows to solve with high relative accuracy several algebraic problems concerning collocation, Wronskian and Gramian matrices of q-Abel polynomials. Finally, a series of numerical tests support the presented theoretical results and illustrate the goodness of the method where standard approaches fail to deliver accurate solutions.

Evaluation of Enterobius Vermicularis Infection in Primary School and Relationship with Nocturnal Enuresis

Enterobiasis or pinworms infection is one of the most prevalent worms found in children worldwide . There was a possible association between certain infection and the propensity to develop nocturnal enuresis including intestinal helminthes infection. Prospective study included (100) randomly selected children aged range (6-12) years from both sex, have been acquired from primary schools in the center of Al-Kut city. The children were examined for E.vermicularis infection by microscopic examination .Total positivity rate of Enterobiasis among children from 6 -12 years was 14% & the number of positive sample were 14 out of 100 child examined .37 (16.2%) were males and 63 (12.6%) were females..Males had a slightly higher prevalence than females However, difference was non-Significance (NS) P>0.05. The results showed the highest prevalence (28.57 %)in the group age 6 years , while the lower infection rate was found in the group age 10-12 years (0%), The statistical analysis showed significant differences at the level of 0.05 between the age groups. E. vermicularis is one of the most common intestinal helminthes with nocturnal enuresis among children in Al-Kut city., there are no significant differences (NS) P>0.05 between males and females for Enterobius vermicularis infection and nocturnal enuresis.

On the positivity of B-spline Wronskians

A proof that Wronskians of non-zero B-splines are positive is given, using only recursive formulas for B-splines and their derivatives. This could be used to generalize the de Boor–DeVore geometric proof of the Schoenberg–Whitney conditions and total positivity of B-splines to Hermite interpolation. For Wronskians of maximal order with respect to a given degree, positivity follows from a simple formula.

Five-year longitudinal surveillance reveals the continual circulation of both alpha- and beta-coronaviruses in Plateau and Gansu pikas (Ochotona spp.) at Qinghai Lake, China1

ABSTRACT The discovery of alphacoronaviruses and betacoronaviruses in plateau pikas (Ochotona curzoniae) expanded the host range of mammalian coronavirus (CoV) to a new order – Lagomorpha. However, the diversity and evolutionary relationships of CoVs in these plateau-region-specific animal population remains uncertain. We conducted a five-year longitudinal surveillance of CoVs harboured by pikas around Qinghai Lake, China. CoVs were identified in 33 of 236 plateau pikas and 2 of 6 Gansu pikas (Ochotona cansus), with a total positivity rate of 14.5%, and exhibiting a wide spatiotemporal distribution across seven sampling sites and six time points. Through meta-transcriptomic sequencing and RT–PCR, we recovered 16 near-complete viral genome sequences. Phylogenetic analyses classified the viruses as variants of either pika alphacoronaviruses or betacoronaviruses endemic to plateau pikas from the Qinghai-Tibet Plateau region. Of particular note, the pika-associated betacoronaviruses may represent a novel subgenus within the genus Betacoronavirus. Tissue tropism, evaluated using quantitative real-time PCR, revealed the presence of CoV in the rectal and/or lung tissues, with the highest viral loads at 103.55 or 102.80 RNA copies/μL. Surface plasmon resonance (SPR) assays indicated that the newly identified betacoronavirus did not bind to human or pika Angiotensin-converting enzyme 2 (ACE2) or Dipeptidyl peptidase 4 (DPP4). The findings highlight the ongoing circulation and broadening host spectrum of CoVs among pikas, emphasizing the necessity for further investigation to evaluate their potential public health risks.

Correlation of PD-L1 expression with different clinico-pathological and immunohistochemical features of ovarian surface epithelial tumors.

Primary carcinoma of the ovary (OCs) are responsible for a significant number of deaths related to cancer, and have the highest rate of death related to cancers of the female reproductive organs. Programmed cell death 1 (PD1) protein, acts as an immune checkpoint, and has an important role in the down-regulation of the immune system by preventing the activation of T-cells, which will weaken the autoimmunity and increases self-tolerance. This study aimed at the evaluation of the immunohistochemical (IHC) expression of PD-L1 in various primary surface ovarian epithelial tumours and to test its correlation with different clinicopathological parameters together with the expression of a panel of P53, ER and PR. A set of 102 cases of primary ovarian surface epithelial neoplasms (benign, borderline and malignant) were collected to construct Tissue Microarray (TMA) using 3 tissue cores from each case. IHC for PD-L1, p53, PR and ER was performed. The expression of PD-L1 was evaluated in relation to some clinicopathological parameters and to the expression patterns of other markers. Expression of PD-L1 was detected in about 51% (n = 36) of malignant tumours. The malignant group significantly showed PD-L1 positivity compared to borderline and benign groups. The malignant tumours significantly showed PD-L1 and total p53 positivity in comparison to borderline group. Also, malignant tumours significantly showed higher combined positivity of PD-L1 and either PR or ER compared to borderline and benign lesions. No significant correlation was appreciated between PD-L1 expression and with any of the studied clinicopathological parameters. This study showed a significant PD-L1 expression in malignant primary surface epithelial tumours. Construction of a panel of IHC markers, including PD-L1, could have a potential value to define patients those would benefit from the addition of immunotherapy to the treatment plan.

On Lattice Boltzmann Methods based on vector-kinetic models for hyperbolic partial differential equations

In this paper, we are concerned about the lattice Boltzmann methods (LBMs) based on vector-kinetic models for hyperbolic partial differential equations. In addition to usual lattice Boltzmann equation (LBE) derived by explicit discretisation of vector-kinetic equation (VKE), we also consider LBE derived by semi-implicit discretisation of VKE and compare the relaxation factors of both. We study the properties such as H-inequality, total variation boundedness and positivity of both the LBEs, and infer that the LBE due to semi-implicit discretisation naturally satisfies all the properties while the LBE due to explicit discretisation requires more restrictive condition on relaxation factor compared to the usual condition obtained from Chapman-Enskog expansion. We also derive the macroscopic finite difference form of the LBEs, and utilise it to establish the consistency of LBEs with the hyperbolic system. Further, we extend this LBM framework to hyperbolic conservation laws with source terms, such that there is no spurious numerical convection due to imbalance between convection and source terms. We also present a D2Q9 model that allows upwinding even along diagonal directions in addition to the usual upwinding along coordinate directions. The different aspects of the results are validated numerically on standard benchmark problems.

On the total positivity of q-Bernstein mass matrices and their accurate computations

In this paper the total positivity of the Gramian (mass) matrices of q-Bernstein bases is analyzed. Furthermore, we provide an efficient method to obtain a bidiagonal decomposition of these mass matrices allowing us to calculate their singular values and inverses to high relative accuracy. Numerical examples are provided to illustrate the high accuracy of the performed computations using the proposed decompositions.

Assessment of cytotoxic T-lymphocyte antigen-4 (CD152) Levels Associated with HBV in Patients with Diabetes Mellitus

Background: Diabetes mellitus is a chronic disease characterized by an imbalance in glucose homeostasis. The hepatitis B virus is a liver-attacking virus that can cause viral hepatitis, cirrhosis, and liver cancer. A cytotoxic T-lymphocyte antigen-4 called CTLA-4 is an immune checkpoint protein that is necessary for T cells to control immune responses and stop the HBV infection from spreading. It does this by acting as an inhibitory receptor, limiting liver damage during an acute infection, and making it easier for the infection to stay in the body during a chronic case. Objective: The study aims to assess CTLA-4 levels in relation to HBV infection in diabetes mellitus patients. Subjects and methods: A cross-sectional study was conducted from July to October 2023. The serum was obtained from 200 diabetic patients who were Iraqis. All of the patients were tested using an ELISA technique for CTLA-4 and HBc IgG. They were tested using a five-panel kit for HBsAb, HBsAg, HBcAb, HBeAg, and HBeAb and measured blood sugar levels. The statistical analysis approach was conducted using SPSS version 26. Results: Serum CTLA-4 levels were correlated with serum HBc IgG positivity (P = 0.000), total HBc Ab positivity (P = 0.000), and HBs Ab positivity (P = 0.000), and CTLA-4 level was correlated with diabetes mellitus (P = 0.034). Conclusions: The study concluded that elevated serum CTLA-4 levels in diabetic patients with HBV infection and type I diabetes compared to type II diabetes. This suggests that diabetes affects the immune system's response to HBV.

Some New Results on Stochastic Comparisons of Spacings of Generalized Order Statistics from One and Two Samples

Generalized order statistics (GOSs) are often adopted as a tool for providing a unified approach to several stochastic models dealing with ordered random variables. In this contribution, we first recall various useful results based on the notion of total positivity. Then, some stochastic comparisons between spacings of GOSs from one sample, as well as two samples, are developed under the more general assumptions on the parameters of the model. Specifically, the given results deal with the likelihood ratio order, the hazard rate order and the mean residual life order. Finally, an application is demonstrated for sequential systems.

Coefficientwise Hankel-total positivity of the row-generating polynomials for the output matrices of certain production matrices

Total positivity of matrices is deeply studied and plays an important role in various branches of mathematics. The aim of this paper is to study the criteria for coefficientwise Hankel-total positivity of the row-generating polynomials of generalized m-Jacobi-Rogers triangles and their applications.Using the theory of production matrices, we present the criteria for coefficientwise Hankel-total positivity of the row-generating polynomials of the output matrices of certain production matrices. In particular, we gain a criterion for coefficientwise Hankel-total positivity of the row-generating polynomial sequence of the generalized m-Jacobi-Rogers triangle. This immediately implies that the corresponding generalized m-Jacobi-Rogers triangular convolution preserves the Stieltjes moment property of sequences and its zeroth column sequence is coefficientwise Hankel-totally positive and log-convex of higher order in all the indeterminates. In consequence, for m=1, we immediately obtain some results on Hankel-total positivity for the Catalan-Stieltjes matrices. In particular, we in a unified manner apply our results to some combinatorial triangles or polynomials including the generalized Jacobi Stirling triangle, a generalized elliptic polynomial, a refined Stirling cycle polynomial and a refined Eulerian polynomial. For the general m, combining our criterion and a function satisfying an autonomous differential equation, we present different criteria for coefficientwise Hankel-total positivity of the row-generating polynomial sequence of exponential Rirodan arrays. In addition, we also derive some results for coefficientwise Hankel-total positivity in terms of compositional functions and m-branched Stieltjes-type continued fractions. Finally, we apply our criteria to: (1) rook polynomials and signless Laguerre polynomials (confirming a conjecture of Sokal on coefficientwise Hankel-total positivity of rook polynomials), (2) labeled trees and forests (proving some conjectures of Sokal on total positivity and Hankel-total positivity), (3) rth-order Eulerian polynomials (giving a new proof for the coefficientwise Hankel-total positivity of rth-order Eulerian polynomials, which in particular implies the conjecture of Sokal on the coefficientwise Hankel-total positivity of reversed 2th-order Eulerian polynomials), (4) multivariate Ward polynomials, labeled series-parallel networks and nondegenerate fanout-free functions, (5) an array from the Lambert function and a generalization of Lah numbers and associated triangles, and so on.

The Application of the Bidiagonal Factorization of Totally Positive Matrices in Numerical Linear Algebra

The approach to solving linear systems with structured matrices by means of the bidiagonal factorization of the inverse of the coefficient matrix is first considered in this review article, the starting point being the classical Björck–Pereyra algorithms for Vandermonde systems, published in 1970 and carefully analyzed by Higham in 1987. The work of Higham briefly considered the role of total positivity in obtaining accurate results, which led to the generalization of this approach to totally positive Cauchy, Cauchy–Vandermonde and generalized Vandermonde matrices. Then, the solution of other linear algebra problems (eigenvalue and singular value computation, least squares problems) is addressed, a fundamental tool being the bidiagonal decomposition of the corresponding matrices. This bidiagonal decomposition is related to the theory of Neville elimination, although for achieving high relative accuracy the algorithm of Neville elimination is not used. Numerical experiments showing the good behavior of these algorithms when compared with algorithms that ignore the matrix structure are also included.

Product structure and regularity theorem for totally nonnegative flag varieties

The totally nonnegative flag variety was introduced by Lusztig. It has enriched combinatorial, geometric, and Lie-theoretic structures. In this paper, we introduce a (new) J\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$J$\\end{document}-total positivity on the full flag variety of an arbitrary Kac-Moody group, generalizing the (ordinary) total positivity.We show that the J\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$J$\\end{document}-totally nonnegative flag variety has a cellular decomposition into totally positive J\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$J$\\end{document}-Richardson varieties. Moreover, each totally positive J\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$J$\\end{document}-Richardson variety admits a favorable decomposition, called a product structure. Combined with the generalized Poincare conjecture, we prove that the closure of each totally positive J\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$J$\\end{document}-Richardson variety is a regular CW complex homeomorphic to a closed ball. Moreover, the J\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$J$\\end{document}-total positivity on the full flag provides a model for the (ordinary) totally nonnegative partial flag variety. As a consequence, we prove that the closure of each (ordinary) totally positive Richardson variety is a regular CW complex homeomorphic to a closed ball, confirming conjectures of Galashin, Karp and Lam in (Adv. Math. 351:614–620, 2019). We also show that the link of the totally nonnegative part of U−\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$U^{-}$\\end{document} for any Kac-Moody group forms a regular CW complex. This generalizes the result of Hersh (Invent. Math. 197(1):57–114, 2014) for reductive groups.

Estimating the incidence of COVID-19, influenza and respiratory syncytial virus infection in three regions of Queensland, Australia, winter 2022: findings from a novel longitudinal testing-based sentinel surveillance programme

ObjectiveThe 2022 Australian winter was the first time that COVID-19, influenza and respiratory syncytial virus (RSV) were circulating in the population together, after two winters of physical distancing, quarantine and...

Total positivity and least squares problems in the Lagrange basis

SummaryThe problem of polynomial least squares fitting in the standard Lagrange basis is addressed in this work. Although the matrices involved in the corresponding overdetermined linear systems are not totally positive, rectangular totally positive Lagrange‐Vandermonde matrices are used to take advantage of total positivity in the construction of accurate algorithms to solve the considered problem. In particular, a fast and accurate algorithm to compute the bidiagonal decomposition of such rectangular totally positive matrices is crucial to solve the problem. This algorithm also allows the accurate computation of the Moore‐Penrose inverse and the projection matrix of the collocation matrices involved in these problems. Numerical experiments showing the good behaviour of the proposed algorithms are included.

Concurrent infection of dengue virus with malaria parasites among outpatients attending healthcare facilities in Benin city, Nigeria.

Dengue virus (DENV) and malaria parasites (MP) are among the common febrile diseases affecting the tropics and subtropics of the world. Both are mosquito-borne pathogens affecting humans and other animals. Blood samples were collected from 280 consented out-patients attending the selected hospitals and were analyzed. Malaria parasites were detected using microscopy and Malaria Ag Pf/Pan Rapid Test Device. Dengue virus was detected by serology and heminested reverse transcriptase PCR (hnRT-PCR) to target the flavivirus polymerase (NS5) gene. Malaria parasites recorded a total positivity of 151 patients (53.9%) using microscopy, while DENV antibodies (DENV IgM and DENV IgG) were positive in 16 (5.7%) and 39 (13.9%) patients, respectively. There was a concurrent infection between MP/DENV IgM in 13 (4.6%) patients and MP/DENV IgG in 27 (9.6%) patients. Molecular identification revealed DENV serotype 2 in circulation. This study documents molecular evidence of dengue virus coexisting with malaria parasites in the study population, hence the need for efficient surveillance and control system.

The geometry of the modular bootstrap

We explore the geometry behind the modular bootstrap and its image in the space of Taylor coefficients of the torus partition function. In the first part, we identify the geometry as an intersection of planes with the convex hull of moment curves on R+⊗ℤ, with boundaries characterized by the total positivity of generalized Hankel matrices. We phrase the Hankel constraints as a semi-definite program, which has several advantages, such as the validity of bounds irrespective of spin truncation. We derive bounds on the gap, twist-gap, and the space of Taylor coefficients themselves. We find that if the gap is above ∆gap∗\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {\\Delta }_{\ extrm{gap}}^{\\ast } $$\\end{document}, where c−112<Δgap∗<c12\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\frac{c-1}{12}<{\\Delta}_{\ extrm{gap}}^{\\ast }<\\frac{c}{12} $$\\end{document}, all coefficients become bounded on both sides and kinks develop in the space. In the second part, we propose an analytic method of imposing the integrality condition for the degeneracy number in the spinless bootstrap, which leads to a non-convex geometry. We find that even at very low derivative order this condition rules out regions otherwise allowed by bootstraps at high derivative order.

Total positivity and high relative accuracy for several classes of Hankel matrices

SummaryGramian matrices with respect to inner products defined for Hilbert spaces supported on bounded and unbounded intervals are represented through a bidiagonal factorization. It is proved that the considered matrices are strictly totally positive Hankel matrices and their catalecticant determinants are also calculated. Using the proposed representation, the numerical resolution of linear algebra problems with these matrices can be achieved to high relative accuracy. Numerical experiments are provided, and they illustrate the excellent results obtained when applying the theoretical results.

Some properties of combinatorial triangles related to Horadam polynomials

The Horadam polynomials unify many well-known polynomials, such as the Fibonacci polynomials, the Lucas polynomials, the Pell polynomials, the Jacobsthal polynomials and the Chebyshev polynomials. We investigate some properties of various combinatorial triangles related to Horadam polynomials, including their properties as almost-Riordan arrays and Riordan arrays, their total positivity, the real-rootedness of the generating functions of their rows, and the asymptotic normality (by central and local limit theorems).

SARS-CoV-2 Antibody Prevalence in Unvaccinated Children and Change in Antibody Levels Between 2020-2022

After the declaration of the COVID-19 pandemic by the World Health Organization (WHO), various restrictions were implemented, and face-to-face education was suspended in schools. In some countries, COVID-19 vaccines emergency use authorization has been granted for the pediatric age group. This study aimed to investigate COVID-19 antibody level in children who have not been vaccinated against COVID-19. A total of 1,000 pediatric patients aged 0-12 years, unvaccinated against COVID-19, admitted to Necmettin Erbakan University Medical Faculty Hospital Pediatrics clinics with different symptoms between October 2020 and April 2022 were included in this study. SARS-CoV-2 S total antibody levels were tested by ELISA method from the sera samples of these children. Of the children included in the study, 535 (53.5%) were male and 465 (46.5%) were female. Anti-SARS-CoV-2 S total Ig positivity rates were 75.9% in girls and 78.5% in boys and the difference between them was not statistically significant (p&gt;0.05). When the anti-SARS-CoV-2 S total Ig positivity rates of children were evaluated according to years, 23 (43.4%) in 2020, 183 (60%) in 2021, and 567 (88.3%) in 2022 was found positive. The difference between antibody results by years was found to be statistically significant (p&lt;0.001). Median value of SARS-CoV-2 antibody levels of patients with positive antibody test in 2020 46 AU/ml (min 5, max: 214 AU/ml), in 2021 130 AU/ml (min:1, max: 250 AU/ml), and in 2022 it was found to be 250 AU/ml (min:1, max: 250 AU/ml). When the antibody levels of patients with anti-SARS-CoV-2 S Ig total positive were evaluated, the difference between years was found to be statistically significant (p&lt;0.05). In our study, it was determined that the rate of SARS-CoV-2 S total antibody positivity reached 90% in unvaccinated children aged 12 and younger. Determining the SARS-CoV-2 antibody status of children aged 12 and under who unvaccinated against COVID-19 in our country will both give information about immunization by natural immunity and will be effective in deciding and planning the need for COVID-19 vaccination for children.

The frequency of hepatitis B virus reactivation in patients receiving anti-TNF treatment: a single center, retrospective study

Objective : The equilibrium between the host immune response counter the hepatitis B virus (HBV) and the amount of viral replication is a crucial factor in the pathogenesis of HBV-associated liver disease. Tumor necrosis factor-alpha (TNF-α) is an considerable proinflammatory and immune regulatory cytokine in the pathogenesis of various inflammatory and autoimmune conditions. There is no consensus on using antiviral prophylaxis treatments in cases who have been exposed to hepatitis B but have not become chronically ill, and are thus planned to receive anti-TNF-α treatment.The aim of this study is to determine the frequency of reactivation after anti-TNF treatment in cases with isolated anti-HBc total positivity who have been exposed to hepatitis B virus. Matarail and Methods: Serological HBV infection markers (HBsAg, anti-HBc IgG and anti-HBs) of 1467 adult cases who received anti-TNF therapy for the indications of various rheumatological diseases in the rheumatology and physical therapy clinics between the years 2010-2021 were retrospectively screened using the hospital’s electronic information system. Results: 140 rheumatologic disease cases who took a TNF-α inhibitor (infliximab, adalimumab, etanercept, golimumab, certolizumab) treatment were included in this study. Before the cases were started on TNF-α treatment, all cases were anti-HBc total positive, 110 were anti-HBs positive, 30 were anti-HBs negative, and 4 were HbsAg positive and HBV-DNA negative. The median pre-treatment anti-HBc total and anti-HBs values of the cases were 5.6 IU/L and 79.29 IU/L, respectively. No HBV reactivation was observed in any of the 140 cases after a median follow-up duration of 71.5 (min. 8, max. 185) months. Conclusion: In conclusion, HBV reactivation was not detected in any of the anti-HBc positive cases included in this study, which suggest that anti-HBc positive cases can be followed up with close follow-up without starting them on anti-TNF therapies and antiviral prophylaxis.