Guy Katriel

Mountain pass theorems and global homeomorphism theorems

Abstract We show that mountain-pass theorems can be used to derive global homeomorphism theorems. Two new mountain-pass theorems are proved, generalizing the “smooth” mountain-pass theorem, one applying in locally compact topological spaces, using Hofer’s concept of mountain-pass point, and another applying in complete metric spaces, using a generalized notion of critical point similar to the one introduced by Ioffe and Schwartzman. These are used to prove global homeomorphism theorems for certain topological and metric spaces, generalizing known global homeomorphism theorems for mappings between Banach spaces.

Eigenvalues of Non-selfadjoint Operators: A Comparison of Two Approaches

The central problem we consider is the distribution of eigenvalues of closed linear operators which are not selfadjoint, with a focus on those operators which are obtained as perturbations of selfadjoint linear operators. Two methods are explained and elaborated. One approach uses complex analysis to study a holomorphic function whose zeros can be identified with the eigenvalues of the linear operator. The second method is an operator theoretic approach involvingthe numerical range. General results obtained by the two methods are derived and compared. Applications to non-selfadjoint Jacobi and Schrodinger operators are considered. Some possible directions for future research are discussed.

Synchronization of oscillators coupled through an environment

Abstract We study synchronization of oscillators that are indirectly coupled through their interaction with an environment. We give criteria for the stability or instability of a synchronized oscillation. Using these criteria we investigate synchronization of systems of oscillators which are weakly coupled, in the sense that the influence of the oscillators on the environment is weak. We prove that arbitrarily weak coupling will synchronize the oscillators, provided that this coupling is of the ‘right’ sign. We illustrate our general results by applications to a model of coupled GnRH neuron oscillators proposed by Khadra and Li [A. Khadra, Y.X. Li, A model for the pulsatile secretion of gonadotropin-releasing hormone from synchronized hypothalamic neurons, Biophys. J. 91 (2006) 74–83.], and to indirectly weakly-coupled λ – ω oscillators.

Modelling the initial phase of an epidemic using incidence and infection network data: 2009 H1N1 pandemic in Israel as a case study

This paper presents new computational and modelling tools for studying the dynamics of an epidemic in its initial stages that use both available incidence time series and data describing the populations infection network structure. The work is motivated by data collected at the beginning of the H1N1 pandemic outbreak in Israel in the summer of 2009. We formulated a new discrete-time stochastic epidemic SIR (susceptible-infected-recovered) model that explicitly takes into account the diseases specific generation-time distribution and the intrinsic demographic stochasticity inherent to the infection process. Moreover, in contrast with many other modelling approaches, the model allows direct analytical derivation of estimates for the effective reproductive number (Re) and of their credible intervals, by maximum likelihood and Bayesian methods. The basic model can be extended to include age–class structure, and a maximum likelihood methodology allows us to estimate the models next-generation matrix by combining two types of data: (i) the incidence series of each age group, and (ii) infection network data that provide partial information of ‘who-infected-who’. Unlike other approaches for estimating the next-generation matrix, the method developed here does not require making a priori assumptions about the structure of the next-generation matrix. We show, using a simulation study, that even a relatively small amount of information about the infection network greatly improves the accuracy of estimation of the next-generation matrix. The method is applied in practice to estimate the next-generation matrix from the Israeli H1N1 pandemic data. The tools developed here should be of practical importance for future investigations of epidemics during their initial stages. However, they require the availability of data which represent a random sample of the real epidemic process. We discuss the conditions under which reporting rates may or may not influence our estimated quantities and the effects of bias.

Modeling and statistical analysis of the spatio-temporal patterns of seasonal influenza in Israel

Background Seasonal influenza outbreaks are a serious burden for public health worldwide and cause morbidity to millions of people each year. In the temperate zone influenza is predominantly seasonal, with epidemics occurring every winter, but the severity of the outbreaks vary substantially between years. In this study we used a highly detailed database, which gave us both temporal and spatial information of influenza dynamics in Israel in the years 1998–2009. We use a discrete-time stochastic epidemic SIR model to find estimates and credible confidence intervals of key epidemiological parameters. Findings Despite the biological complexity of the disease we found that a simple SIR-type model can be fitted successfully to the seasonal influenza data. This was true at both the national levels and at the scale of single cities.The effective reproductive number Re varies between the different years both nationally and among Israeli cities. However, we did not find differences in Re between different Israeli cities within a year. R e was positively correlated to the strength of the spatial synchronization in Israel. For those years in which the disease was more “infectious”, then outbreaks in different cities tended to occur with smaller time lags. Our spatial analysis demonstrates that both the timing and the strength of the outbreak within a year are highly synchronized between the Israeli cities. We extend the spatial analysis to demonstrate the existence of high synchrony between Israeli and French influenza outbreaks. Conclusions The data analysis combined with mathematical modeling provided a better understanding of the spatio-temporal and synchronization dynamics of influenza in Israel and between Israel and France. Altogether, we show that despite major differences in demography and weather conditions intra-annual influenza epidemics are tightly synchronized in both their timing and magnitude, while they may vary greatly between years. The predominance of a similar main strain of influenza, combined with population mixing serve to enhance local and global influenza synchronization within an influenza season.

Modelling seasonal influenza: the role of weather and punctuated antigenic drift

Seasonal influenza appears as annual oscillations in temperate regions of the world, yet little is known as to what drives these annual outbreaks and what factors are responsible for their inter-annual variability. Recent studies suggest that weather variables, such as absolute humidity, are the key drivers of annual influenza outbreaks. The rapid, punctuated, antigenic evolution of the influenza virus is another major factor. We present a new framework for modelling seasonal influenza based on a discrete-time, age-of-infection, epidemic model, which allows the calculation of the models likelihood function in closed form. This framework may be used to perform model inference and parameter estimation rigorously. The modelling approach allows us to fit 11 years of Israeli influenza data, with the best models fitting the data with unusually high correlations in which r > 0.9. We show that using actual weather to modulate influenza transmission rate gives better results than using the inter-annual means of the weather variables, providing strong support for the role of weather in shaping the dynamics of influenza. This conclusion remains valid even when incorporating a more realistic depiction of the decay of immunity at the population level, which allows for discrete changes in immunity from year to year.

Mathematical modelling and prediction in infectious disease epidemiology

We discuss to what extent disease transmission models provide reliable predictions. The concept of prediction is delineated as it is understood by modellers, and illustrated by some classic and recent examples. A precondition for a model to provide valid predictions is that the assumptions underlying it correspond to the reality, but such correspondence is always limited—all models are simplifications of reality. A central tenet of the modelling enterprise is what we may call the ‘robustness thesis’: a model whose assumptions approximately correspond to reality will make predictions that are approximately valid. To examine which of the predictions made by a model are trustworthy, it is essential to examine the outcomes of different models. Thus, if a highly simplified model makes a prediction, and if the same or a very similar prediction is made by a more elaborate model that includes some mechanisms or details that the first model did not, then we gain some confidence that the prediction is robust. An important benefit derived from mathematical modelling activity is that it demands transparency and accuracy regarding our assumptions, thus enabling us to test our understanding of the disease epidemiology by comparing model results and observed patterns. Models can also assist in decision-making by making projections regarding important issues such as intervention-induced changes in the spread of disease.

Onset of a pandemic: characterizing the initial phase of the swine flu (H1N1) epidemic in Israel

BackgroundThe swine influenza H1N1 first identified in Mexico, spread rapidly across the globe and is considered the fastest moving pandemic in history. The early phase of an outbreak, in which data is relatively scarce, presents scientific challenges on key issues such as: scale, severity and immunity which are fundamental for establishing sound and rapid policy schemes. Our analysis of an Israeli dataset aims at understanding the spatio-temporal dynamics of H1N1 in its initial phase.MethodsWe constructed and analyzed a unique dataset from Israel on all confirmed cases (between April 26 to July 7, 2009), representing most swine flu cases in this period. We estimated and characterized fundamental epidemiological features of the pandemic in Israel (e.g. effective reproductive number, age-class distribution, at-risk social groups, infections between sexes, and spatial dynamics). Contact data collected during this stage was used to estimate the generation time distribution of the pandemic.ResultsWe found a low effective reproductive number (Re= 1.06), an age-class distribution of infected individuals (skewed towards ages 18-25), at-risk social groups (soldiers and ultra Orthodox Jews), and significant differences in infections between sexes (skewed towards males). In terms of spatial dynamics, the pandemic spread from the central coastal plain of Israel to other regions, with higher infection rates in more densely populated sub-districts with higher income households.ConclusionsAnalysis of high quality data holds much promise in reducing uncertainty regarding fundamental aspects of the initial phase of an outbreak (e.g. the effective reproductive number Re, age-class distribution, at-risk social groups). The formulation for determining the effective reproductive number Reused here has many advantages for studying the initial phase of the outbreak since it neither assumes exponential growth of infectives and is independent of the reporting rate. The finding of a low Re(close to unity threshold), combined with identification of social groups with high transmission rates would have enabled the containment of swine flu during the summer in Israel. Our unique use of contact data provided new insights into the differential dynamics of influenza in different ages and sexes, and should be promoted in future epidemiological studies. Thus our work highlights the importance of conducting a comprehensive study of the initial stage of a pandemic in real time.

Pandemic Dynamics and the Breakdown of Herd Immunity

In this note we discuss the issues involved in attempting to model pandemic dynamics. More specifically, we show how it may be possible to make projections for the ongoing H1N1 pandemic as extrapolated from knowledge of seasonal influenza. We derive first-approximation parameter estimates for the SIR model to describe seasonal influenza, and then explore the implications of the existing classical epidemiological theory for the case of a pandemic virus. In particular, we note the dramatic nonlinear increase in attack rate as a function of the percentage of susceptibles initially present in the population. This has severe consequences for the pandemic, given the general lack of immunity in the global population.

EXISTENCE OF TRAVELLING WAVES IN DISCRETE SINE-GORDON RINGS

We prove existence results for travelling waves in discrete, damped, dc-driven sine-Gordon equations with periodic boundary conditions. Methods of nonlinear functional analysis are employed. Some unresolved questions are raised.