Numerical periodic normalization for codim 2 bifurcations of limit cycles : computational formulas, numerical implementation, and examples

Explicit computational formulas for the coefficients of the periodic normal forms for codimension 2 (codim 2) bifurcations of limit cycles in generic autonomous ODEs are derived. All cases (except the weak resonances) with no more than three Floquet multipliers on the unit circle are covered. The resulting formulas are independent of the dimension of the phase space and involve solutions of certain boundary-value problems on the interval [0, T], where T is the period of the critical cycle, as well as multilinear functions from the Taylor expansion of the ODE right-hand side near the cycle. The formulas allow one to distinguish between various bifurcation scenarios near codim 2 bifurcations of limit cycles. Our formulation makes it possible to use robust numerical boundary-value algorithms based on orthogonal collocation, rather than shooting techniques, which greatly expands its applicability. The implementation is described in detail with numerical examples, where numerous codim 2 bifurcations of limit cycles are analyzed for the first time

Symmetries and Cluster Synchronization in Multilayer Networks

Real-world systems in epidemiology, social sciences, power transportation, economics and engineering are often described as multilayer networks. Here we first define and compute the symmetries of multilayer networks, and then study the emergence of cluster synchronization in these networks. We distinguish between independent layer symmetries which occur in one layer and are independent of the other layers and dependent layer symmetries which involve nodes in different layers. We study stability of the cluster synchronous solution by decoupling the problem into a number of independent blocks and assessing stability of each block through a Master Stability Function. We see that blocks associated with dependent layer symmetries have a different structure than the other blocks, which affects the stability of clusters associated with these symmetries. Finally, we validate the theory in a fully analog experiment in which seven electronic oscillators of three kinds are connected with two kinds of coupling

A theoretical analysis of complex armed conflicts

The introduction and analysis of a simple idealized model enables basic insights into how military characteristics and recruitment strategies affect the dynamics of armed conflicts, even in the complex case of three or more fighting groups. In particular, the model shows when never ending wars (stalemates) are possible and how initial conditions and interventions influence a conflict’s fate. The analysis points out that defensive recruitment policies aimed at compensating for suffered losses lead to conflicts with simple dynamics, while attack groups sensitive to the damages they inflict onto their enemies can give rise to conflicts with turbulent behaviours. Since non-governmental groups often follow attack strategies, the conclusion is that the evolution of conflicts involving groups of that kind can be expected to be difficult to forecast

Churg-Strauss syndrome: outcome and long-term follow-up of 38 patients from a single Italian centre

Objective: This study was aimed at verifying any potential correlation between anti-myeloperoxidase antineutrophil cytoplasmic antibodies (ANCA-MPO) and clinical features and outcome indices in Churg-Strauss Syndrome (CSS). Methods: Thirhty-eight Churg-Strauss syndrome patients were selected from the medical records of all vasculitis patients attending the Rheumatology and Immunology Unit at the Department of Internal Medicine of the University of Pisa in the decades between 1989 and 2008. Data were analysed retrospectively. Statistical analyses of the results were carried out using the Mann-Whitney test to determine the correlations between the clinical and serological parameters. Qualitative variables were compared using contingency table analysis and Fisher's exact test. Results: ANCA-MPO were detected in15/38 (39%) patients. Positive ANCA status was associated with peripheral neuropathy (p=0.0006), whereas negative ANCA status was associated with lung involvement (p=0.002). Relapses were strongly associated with positive ANCA status (p=0.01) and with an increase in- or a reappearence of ANCA-MPO levels (p=0.006). Finally, ANCA-MPO were significantly associated with neurological damage (p=0.003). Conclusions: The presence or absence of ANCA-MPO identify different clinical subsets in CSS. Overall, ANCA-MPO appears as a useful tool in the monitoring of CSS and in particular a good predictor of CSS relapse

Occurrence of hashimoto thyroiditis among the first- and second-degree relatives of systemic lupus erythematosus patients with Hashimoto thyroiditis

Occurrence of Hashimoto thyroiditis among the first- and second-degree relatives of systemic lupus erythematosus patients with Hashimoto thyroiditis

Long-term cyclic intravenous iloprost in systemic sclerosis: clinical experience from a single center

The aim of the present study was to retrospectively evaluate response to therapy in 73 patients affected by systemic sclerosis (SSc) who underwent long-term cyclic treatment with intravenous iloprost for peripheral vascular involvement (average duration of treatment 54.12±41.04 months). Seventy-three SSc patients were enrolled. Data were collected by reviewing clinical records and by phone or direct interview. Patients underwent a thorough physical examination at the end of follow up. The incidence of severe vascular manifestations was also assessed. Statistical analysis was performed by Wilcoxon's signed rank test and descriptive statistics using Statview software. In this study cohort, 55 of 73 (75.2%) patients had a history of ischemic digital ulcers (DUs); 28 patients (38.4%) had active DUs at the beginning of treatment. Skin ulcers healed completely in 25 of 28 patients (89.3%) at the end of the first treatment. However, 40 of 55 patients (72.6%) relapsed after an average of 24 months. There was a significant correlation between relapse rate and/or number of ulcers and clinical factors (diffuse subset, changes in results of Allen's test, NT-pro BNP levels). The annual incidence of pulmonary arterial hypertension (PAH) was 2.34 (95%CI: 0.94-4.83) per 100 person years, the rate of gangrene was 2.7%, and no cases of scleroderma renal crisis were recorded. The incidence of PAH and of digital gangrene was higher than that observed in unselected SSc case series. These data suggest that our patients treated with iloprost have a higher vascular involvement than large case series of unselected SSc patients. A number of clinical factors are correlated to the severity of vascular involvement and could have an impact on the response to therapy. The clinical significance of these findings requires clarification and further investigation is needed

A network model of Italy shows that intermittent regional strategies can alleviate the COVID-19 epidemic

The COVID-19 epidemic hit Italy particularly hard, yielding the implementation of strict national lockdown rules. Previous modelling studies at the national level overlooked the fact that Italy is divided into administrative regions which can independently oversee their own share of the Italian National Health Service. Here, we show that heterogeneity between regions is essential to understand the spread of the epidemic and to design effective strategies to control the disease. We model Italy as a network of regions and parameterize the model of each region on real data spanning over two months from the initial outbreak. We confirm the effectiveness at the regional level of the national lockdown strategy and propose coordinated regional interventions to prevent future national lockdowns, while avoiding saturation of the regional health systems and mitigating impact on costs. Our study and methodology can be easily extended to other levels of granularity to support policy- and decision-makers

Evoluzione verso connettiviti definite in una popolazione di pazienti affetti da “early undifferentiated connective tissue disease” [Analysis of the evolution to defined connective tissue diseases of patients with early undifferentiated connective tissue diseases (UCTD)]

The term undifferentiated connective tissue diseases (UCTD) is used to identify systemic autoimmune diseases not fulfilling classificative criteria for defined connective tissue diseases (CTD). Aim of the present study was to evaluate the evolution to defined CTD of an historical cohort of 91 UCTD patients followed at our Unit and to describe clinical and serological characteristics of stable UCTD patients with a disease duration of more than 5 years. Patients, previously described, were selected for having an undifferentiated profile after 1 year of follow up. These patients have been regularly followed at our Unit and their diagnosis has been reassessed annually based on the existing classificative criteria. Seven UCTD patients with a follow up of less than 5 years have been excluded from the study, therefore 84 patients (F: 81, M: 3) have been analysed. During the follow up 28 patients (33%) developed a defined CTD. In particular 22 patients developed systemic lupus erythematosus (SLE), while the remaining 6 patients developed other CTDs (2 primary Sjögren's syndrome, 2 overlap syndromes, 1 Systemic Sclerosis, 1 rheumatoid arthritis). The evolution to a defined CTD occurred after a mean disease duration of 80.6+/- 66.8 months (min 14, max 336, median 72); the evolution to SLE occurred after a mean disease duration of 66.8+/-43.3 months (min 17, max 216, median 57). Anti-cardiolipin antibodies were the only variable correlated with the evolution to SLE (p<0.05). Stable UCTD were characterized by a simplified clinical picture with no major organ involvement and by a simplified autoantibody profile (anti-Ro/SSA antibodies and anti-RNP antibodies were the single antibody specificities observed in 22% and 13% of patients respectively). These results confirm previous data showing that about 30% of UCTD patients will develop a defined CTD, the predictive role of anti-cardiolipin antibodies for the evolution to SLE, and the existence of stable UCTD, distinct clinical entities with a simplified clinico-serological profile. The early identification of stable UCTD is very important both from a clinical and a research point of view. Future research is needed to define a new set of classification criteria