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Compartmental transmission models have become an invaluable tool to study the dynamics of infectious diseases. The Susceptible-Infectious-Recovered (SIR) model is known to have an exact semi-analytical solution. In the current study, the approach of Harko et al. (Appl. Math. Comput. 236:184–194, 2014) is generalised to obtain an approximate semi-analytical solution of the Susceptible-Exposed-Infectious-Recovered (SEIR) model. The SEIR model curves have nearly the same shapes as the SIR ones, but with a stretch factor applied to them across time that is related to the ratio of the incubation to infectious periods. This finding implies an approximate characteristic timescale, scaled by this stretch factor, that is universal to all SEIR models, which only depends on the basic reproduction number and initial fraction of the population that is infectious.

To constrain the formation history of an exoplanet, we need to know its chemical composition 1 – 3 . With an equilibrium temperature of about 4,050 kelvin 4 , the exoplanet KELT-9b (also known as HD 195689b) is an archetype of the class of ultrahot Jupiters that straddle the transition between stars and gas-giant exoplanets and are therefore useful for studying atmospheric chemistry. At these high temperatures, iron and several other transition metals are not sequestered in molecules or cloud particles and exist solely in their atomic forms 5 . However, despite being the most abundant transition metal in nature, iron has not hitherto been detected directly in an exoplanet because it is highly refractory. The high temperatures of KELT-9b imply that its atmosphere is a tightly constrained chemical system that is expected to be nearly in chemical equilibrium 5 and cloud-free 6 , 7 , and it has been predicted that spectral lines of iron should be detectable in the visible range of wavelengths 5 . Here we report observations of neutral and singly ionized atomic iron (Fe and Fe + ) and singly ionized atomic titanium (Ti + ) in the atmosphere of KELT-9b. We identify these species using cross-correlation analysis 8 of high-resolution spectra obtained as the exoplanet passed in front of its host star. Similar detections of metals in other ultrahot Jupiters will provide constraints for planetary formation theories. Cross-correlation analysis of high-resolution spectra obtained as the exoplanet KELT-9b transited its host star reveals neutral and singly ionized atomic iron and singly ionized atomic titanium in the exoplanet’s atmosphere.

A longitudinal thermal brightness map of the super-Earth exoplanet 55 Cancri e reveals strong day–night temperature contrast, indicating inefficient heat redistribution consistent with 55 Cancri e either being devoid of atmosphere or having an optically thick atmosphere with heat recirculation confined to the planetary dayside. Night and day on a super-Earth Super-Earth 55 Cancri e is a nearby exoplanet with a diameter less than around twice that of Earth, but a mass that is about eight times Earth's mass. 55 Cancri e is among the best candidates for the study of the nature of a class of exoplanets that is unknown to our Solar System: previous observations of super-Earths have yielded only featureless spectra. Here Brice-Olivier Demory et al . report a longitudinal thermal brightness map of 55 Cancri e in transit across its host star. The map reveals strong day–night temperature contrast, suggesting inefficient heat redistribution consistent with the planet either lacking an atmosphere, or having an optically thick atmosphere with heat recirculation confined to the planetary dayside. Over the past decade, observations of giant exoplanets (Jupiter-size) have provided key insights into their atmospheres 1 , 2 , but the properties of lower-mass exoplanets (sub-Neptune) remain largely unconstrained because of the challenges of observing small planets. Numerous efforts to observe the spectra of super-Earths—exoplanets with masses of one to ten times that of Earth—have so far revealed only featureless spectra 3 . Here we report a longitudinal thermal brightness map of the nearby transiting super-Earth 55 Cancri e (refs 4 , 5 ) revealing highly asymmetric dayside thermal emission and a strong day–night temperature contrast. Dedicated space-based monitoring of the planet in the infrared revealed a modulation of the thermal flux as 55 Cancri e revolves around its star in a tidally locked configuration. These observations reveal a hot spot that is located 41 ± 12 degrees east of the substellar point (the point at which incident light from the star is perpendicular to the surface of the planet). From the orbital phase curve, we also constrain the nightside brightness temperature of the planet to 1,380 ± 400 kelvin and the temperature of the warmest hemisphere (centred on the hot spot) to be about 1,300 kelvin hotter (2,700 ± 270 kelvin) at a wavelength of 4.5 micrometres, which indicates inefficient heat redistribution from the dayside to the nightside. Our observations are consistent with either an optically thick atmosphere with heat recirculation confined to the planetary dayside, or a planet devoid of atmosphere with low-viscosity magma flows at the surface 6 .

Studying the albedos of the planets and moons of the Solar System dates back at least a century 1 – 4 . Of particular interest is the relationship between the albedo measured at superior conjunction, known as the ‘geometric albedo’, and the albedo considered over all orbital phase angles, known as the ‘spherical albedo’ 2 , 5 , 6 . Determining the relationship between the geometric and spherical albedos usually involves complex numerical calculations 7 – 11 , and closed-form solutions are restricted to simple reflection laws 12 , 13 . Here we report the discovery of closed-form solutions for the geometric albedo and integral phase function, which apply to any law of reflection that only depends on the scattering angle. The shape of a reflected light phase curve, quantified by the integral phase function, and the secondary eclipse depth, quantified by the geometric albedo, may now be self-consistently inverted to retrieve globally averaged physical parameters. Fully Bayesian phase-curve inversions for reflectance maps and simultaneous light-curve detrending may now be performed due to the efficiency of computation. Demonstrating these innovations for the hot Jupiter Kepler-7b, we infer a geometric albedo of 0.2 5 − 0.02 + 0.01 , a phase integral of 1.77 ± 0.07, a spherical albedo of 0.4 4 − 0.03 + 0.02 and a scattering asymmetry factor of 0.0 7 − 0.11 + 0.12 . An ab initio closed-form analytical solution for the geometric albedo and the integral phase function, which is valid for any law of reflection provided that it depends only on the scattering angle, is proposed. Such solutions can be applied to determine the scattering properties of planetary atmospheres or surfaces.

Spectral retrieval has long been a powerful tool for interpreting planetary remote sensing observations. Flexible, parameterised, agnostic models are coupled with inversion algorithms in order to infer atmospheric properties directly from observations, with minimal reliance on physical assumptions. This approach, originally developed for application to Earth satellite data and subsequently observations of other Solar System planets, has been recently and successfully applied to transit, eclipse and phase curve spectra of transiting exoplanets. In this review, we present the current state-of-the-art in terms of our ability to accurately retrieve information about atmospheric chemistry, temperature, clouds and spatial variability; we discuss the limitations of this, both in the available data and modelling strategies used; and we recommend approaches for future improvement.

What does it mean to understand the natural world? To a classically trained physicist, it means that one is able to construct a model that not only accounts for currently measured phenomena but is also capable of predicting future phenomena. The modern challenge posed by artificial intelligence is that if one has a sufficiently large dataset of a system, in principle one may train a machine learning model to predict future phenomena without having any understanding of the underlying physical mechanisms. It reignites the debate of what\"understanding\" really means. In a mechanistic view of the world, any phenomenon may be understood as a large ensemble of interacting billiard balls. The reductionist asserts that as long as enough basic units (\"billiard balls\") are present in such a system, one may model or simulate it exhaustively.

Supernova remnants are among the most spectacular examples of astrophysical pistons in our cosmic neighborhood. The gas expelled by the supernova explosion is launched with velocities ~1000 kilometers per second into the ambient, tenuous interstellar medium, producing shocks that excite hydrogen lines. We have used an optical integral-field spectrograph to obtain high-resolution spatial-spectral maps that allow us to study in detail the shocks in the northwestern rim of supernova 1006. The two-component Hα line is detected at 133 sky locations. Variations in the broad line widths and the broad-to-narrow line intensity ratios across tens of atomic mean free paths suggest the presence of suprathermal protons, the potential seed particles for generating high-energy cosmic rays.