There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

抗原变异可使得多种致病微生物易于逃避宿主免疫应答。表达在感染红细胞表面的恶性疟原虫红细胞表面蛋白1（PfPMP1）与感染红细胞、内皮细胞、树突状细胞以及胎盘的单个或多个受体作用，在黏附及免疫逃避中起关键的作用。每个单倍体基因组var基因家族编码约60种成员，通过启动转录不同的var基因变异体为抗原变异提供了分子基础。

疟原虫var基因转换速率变化导致抗原变异[英]／Paul H, Robert P, Christodoulou Z, et al//Proc Natl Acad Sci U S A

http://www.maths.qmul.ac.uk/%7Elatora/report_06.pdf

Complex networks: Structure and dynamics

We will review some of the major results in random graphs and some of the more challenging open problems. We will cover algorithmic and structural questions. We will touch on newer models, including those related to the WWW.

Random graphs

We review in detail a number of approaches that have been adopted to try and explain the remarkable observation of our accelerating universe. In particular we discuss the arguments for and recent progress made towards understanding the nature of dark energy. We review the observational evidence for the current accelerated expansion of the universe and present a number of dark energy models in addition to the conventional cosmological constant, paying particular attention to scalar field models such as quintessence, K-essence, tachyon, phantom and dilatonic models. The importance of cosmological scaling solutions is emphasized when studying the dynamical system of scalar fields including coupled dark energy. We study the evolution of cosmological perturbations allowing us to confront them with the observation of the Cosmic Microwave Background and Large Scale Structure and demonstrate how it is possible in principle to reconstruct the equation of state of dark energy by also using Supernovae Ia observational data. We also discuss in detail the nature of tracking solutions in cosmology, particle physics and braneworld models of dark energy, the nature of possible future singularities, the effect of higher order curvature terms to avoid a Big Rip singularity, and approaches to modifying gravity which leads to a late-time accelerated expansion without recourse to a new form of dark energy.

Dynamics of dark energy

Recurrent epidemics in small world networks.

We analyze the dynamics of models of warm inflation with general dissipative effects. We consider phenomenological terms both for the inflaton decay rate and for viscous effects within matter. We provide a classification of the asymptotic behavior of these models and show that the existence of a late-time scaling regime depends not only on an asymptotic behavior of the scalar field potential, but also on an appropriate asymptotic behavior of the inflaton decay rate. There are scaling solutions whenever the latter evolves to become proportional to the Hubble rate of expansion regardless of the steepness of the scalar field exponential potential. We show from thermodynamic arguments that the scaling regime is associated to a power-law dependence of the matter-radiation temperature on the scale factor, which allows a mild variation of the temperature of the matter/radiation fluid. We also show that the late time contribution of the dissipative terms alleviates the depletion of matter, and increases the duration of inflation.

Asymptotic behavior of the warm inflation scenario with viscous pressure

There is a consensus that mass vaccination against SARS-CoV-2 will ultimately end the COVID-19 pandemic. However, it is not clear when and which control measures can be relaxed during the rollout of vaccination programmes. We investigate relaxation scenarios using an age-structured transmission model that has been fitted to age-specific seroprevalence data, hospital admissions, and projected vaccination coverage for Portugal. Our analyses suggest that the pressing need to restart socioeconomic activities could lead to new pandemic waves, and that substantial control efforts prove necessary throughout 2021. Using knowledge on control measures introduced in 2020, we anticipate that relaxing measures completely or to the extent as in autumn 2020 could launch a wave starting in April 2021. Additional waves could be prevented altogether if measures are relaxed as in summer 2020 or in a step-wise manner throughout 2021. We discuss at which point the control of COVID-19 would be achieved for each scenario. Despite the consensus that mass vaccination against SARS-CoV-2 will ultimately end the pandemic, it is not clear when and which control measures can be relaxed during the rollout of vaccination programmes. Here, the authors investigate relaxation scenarios using an age-structured transmission model that has been fitted to data for Portugal.

/pdf/controlling-the-pandemic-during-the-sars-cov-2-vaccination-39lreurhpk.pdf

Controlling the pandemic during the SARS-CoV-2 vaccination rollout.

The effects of demographic stochasticity on the long-term behaviour of endemic infectious diseases have been considered for long as a necessary addition to an underlying deterministic theory. The latter would explain the regular behaviour of recurrent epidemics and the former the superimposed noise of observed incidence patterns. Recently, a stochastic theory based on a mechanism of resonance with internal noise has shifted the role of stochasticity closer to the centre stage, by showing that the major dynamic patterns found in the incidence data can be explained as resonant fluctuations, whose behaviour is largely independent of the amplitude of seasonal forcing, and by contrast very sensitive to the basic epidemiological parameters. Here we elaborate on that approach, by adding an ingredient which is missing in standard epidemic models, the ‘mixing network’ through which infection may propagate. We find that spatial correlations have a major effect on the enhancement of the amplitude and the coherence of the resonant stochastic fluctuations, providing the ordered patterns of recurrent epidemics, whose period may differ significantly from that of the small oscillations around the deterministic equilibrium. We also show that the inclusion of a more realistic, time-correlated recovery profile instead of exponentially distributed infectious periods may, even in the random-mixing limit, contribute to the same effect.

Ana Nunes

Papers

Recurrent epidemics in small world networks.

Asymptotic behavior of the warm inflation scenario with viscous pressure

Controlling the pandemic during the SARS-CoV-2 vaccination rollout.

Stochastic fluctuations in epidemics on networks

On the potentials yielding cosmological scaling solutions