Perkins Embedding for General Starting Laws

The Skorokhod embedding problem (SEP) is to represent a given probability measure as a Brownian motion $B$ at a particular stopping time. In recent years particular attention has gone to solutions which exhibit additional optimality properties due to applications to martingale inequalities and robust pricing in mathematical finance. Among these solutions, the Perkins embedding sticks out through its distinct geometric properties. Moreover is the only optimal solution to the SEP which so far has been limited to the case of Brownian motion started in a dirac distribution. In this paper we provide for the first time an optimal solution to the Skorokhod embedding problem for the general SEP which leads to the Perkins solution when applied to Brownian motion with start in a dirac. This solution to the SEP also suggests a new geometric interpretation of the Perkins solution which better clarifies the relation to other optimal solutions of the SEP

Land‐use change differentially affects endemic, forest and open‐land butterflies in Madagascar

The conversion of tropical forests into agriculture reduces biodiversity dramatically. However, species might differ in their responses, depending on their habitat specialisation and geographic origin. In this study, we assess how butterfly assemblages differ between old-growth forests, forest fragments, forest-derived vanilla agroforests, fallow-derived vanilla agroforests, woody fallows, herbaceous fallows, and rice paddies in Madagascar. We recorded 88 butterfly species, of which 65 species are endemic to Madagascar. Land-use types with woody vegetation sustained many endemic (mean: 6.8 species) and forest butterfly species (mean: 4.8 species). Rice paddies and herbaceous fallows were richer in open-land species (mean: 7.6 species) and poorer in forest species (mean: 1.7 species) compared to other land-use types. Compared to herbaceous fallows, fallow-derived vanilla agroforests hosted more endemic (+164%) and forest (+239%) species. Richness of open-land species in forest-derived vanilla agroforests was six times higher than in forest fragments. Overall, 27% of species occurred exclusively in one land-use type and 19% of all species when old-growth forests were excluded. We found the highest number of exclusive species in forest-derived vanilla agroforests. We conclude that all studied land-use types contribute to butterfly conservation in our study region. Especially the woody vegetation in forest fragments, vanilla agroforests, and woody fallows supports a high butterfly diversity and is pivotal for maintaining a broad diversity of forest butterflies in the agricultural matrix. Our study highlights the importance of preserving the diversity of small-scale land-use types, including agroforestry, forests and fallow land in this tropical biodiversity hotspot

Disease momentum: Estimating the reproduction number inthe presence of superspreading

A primary quantity of interest in the study of infectious diseases is the average number ofnew infections that an infected person produces. This so-called reproduction number has significant implications for the disease progression. There has been increasing literature suggesting that superspreading, the significant variability in number of new infections caused by individuals, plays an important role in the spread of SARS-CoV-2. In this paper, we consider the effect that such superspreading has on the estimation of the reproduction number and subsequent estimates of future cases. Accordingly, we employ a simpleextension to models currently used in the literature to estimate the reproduction number and present a case-study of the progression of COVID-19 in Austria. Our models demonstrate that the estimation uncertainty of the reproduction number increases with superspreading and that this improves the performance of prediction intervals. Of independent interest is the derivation of a transparent formula that connects the extent of superspreading to the width of credible intervals for the reproduction number. This serves as a valuable heuristic for understanding the uncertainty surrounding diseases with superspreading

Win-win opportunities combining high yields with high multi-taxa biodiversity in tropical agroforestry

Resolving ecological-economic trade-offs between biodiversity and yields is a key challenge when addressing the biodiversity crisis in tropical agricultural landscapes. Here, we focused on the relation between seven different taxa (trees, herbaceous plants, birds, amphibians, reptiles, butterflies, and ants) and yields in vanilla agroforests in Madagascar. Agroforests established in forests supported overall 23% fewer species and 47% fewer endemic species than old-growth forests, and 14% fewer endemic species than forest fragments. In contrast, agroforests established on fallows had overall 12% more species and 38% more endemic species than fallows. While yields increased with vanilla vine density and length, non-yield related variables largely determined biodiversity. Nonetheless, trade-offs existed between yields and butterflies as well as reptiles. Vanilla yields were generally unrelated to richness of trees, herbaceous plants, birds, amphibians, reptiles, and ants, opening up possibilities for conservation outside of protected areas and restoring degraded land to benefit farmers and biodiversity alike

Robust models of disease heterogeneity and control, with application to the SARS-CoV-2 epidemic.

In light of the continuing emergence of new SARS-CoV-2 variants and vaccines, we create a robust simulation framework for exploring possible infection trajectories under various scenarios. The situations of primary interest involve the interaction between three components: vaccination campaigns, non-pharmaceutical interventions (NPIs), and the emergence of new SARS-CoV-2 variants. Additionally, immunity waning and vaccine boosters are modeled to account for their growing importance. New infections are generated according to a hierarchical model in which people have a random, individual infectiousness. The model thus includes super-spreading observed in the COVID-19 pandemic which is important for accurate uncertainty prediction. Our simulation functions as a dynamic compartment model in which an individual's history of infection, vaccination, and possible reinfection all play a role in their resistance to further infections. We present a risk measure for each SARS-CoV-2 variant, [Formula: see text], that accounts for the amount of resistance within a population and show how this risk changes as the vaccination rate increases. [Formula: see text] highlights that different variants may become dominant in different countries-and in different times-depending on the population compositions in terms of previous infections and vaccinations. We compare the efficacy of control strategies which act to both suppress COVID-19 outbreaks and relax restrictions when possible. We demonstrate that a controller that responds to the effective reproduction number in addition to case numbers is more efficient and effective in controlling new waves than monitoring case numbers alone. This not only reduces the median total infections and peak quarantine cases, but also controls outbreaks much more reliably: such a controller entirely prevents rare but large outbreaks. This is important as the majority of public discussions about efficient control of the epidemic have so far focused primarily on thresholds for case numbers

Land‐use intensification increases richness of native and exotic herbaceous plants, but not endemics, in Malagasy vanilla landscapes

Abstract Aim North‐eastern Madagascar is a hotspot of plant diversity, but vanilla and rice farming are driving land‐use change, including slash‐and‐burn management. It still remains unknown how land‐use change and land‐use history affect richness and composition of endemic, native and exotic herbaceous plant species. Location North‐eastern Madagascar. Methods We assessed herbaceous plants along a land‐use intensification gradient ranging from unburned land‐use types (i.e. old‐growth forest, forest fragment and forest‐derived vanilla agroforest) to burned land‐use types (i.e. fallow‐derived vanilla agroforest, woody fallow and herbaceous fallow) and rice paddy. We compared land‐use types and analysed the effects of land‐use history, canopy closure and landscape forest cover on species richness. Additionally, we analysed species compositional changes across land‐use types. Results Across 80 plots, we found 355 plant species (180 native non‐endemics, 57 exotics, 60 endemics and 58 species of unknown origin). Native and exotic species richness increased with increasing land‐use intensity, whereas endemics decreased. Unburned land‐use types had higher endemic species richness (4.28 ± 0.37 [mean ± SE]) than burned ones (2.4 ± 0.21). Exotic and native species richness, but not endemics, decreased with increasing canopy closure. Increasing landscape forest cover reduced exotic, but not native or endemic richness. Species composition of old‐growth forests was unique compared to all other land uses and forest‐derived, not fallow‐derived vanilla agroforests, had a similar endemic species composition to forest fragments. Main conclusions Our results indicate that old‐growth forests and forest fragments are indispensable for maintaining endemic herbaceous plants. We further show that the land‐use history of agroforests should be considered in conservation policy. In forest‐derived vanilla agroforests, management incentives are needed to halt loss of canopy closure, thereby maintaining or even enhancing endemics. In conclusion, considering species origin (endemic, native and exotic) and composition is essential for the identification of suitable management practices to avoid irreversible species loss