We consider an SIR (susceptible $\to$ infective $\to$ recovered) epidemic in a closed population of size n, in which infection spreads via mixing events, comprising individuals chosen uniformly at random from the population, which occur at the points of a Poisson process. This contrasts sharply with most epidemic models, in which infection is spread purely by pairwise interaction. A sequence of epidemic processes, indexed by n, and an approximating branching process are constructed on a common probability space via embedded random walks. We show that under suitable conditions the process of infectives in the epidemic process converges almost surely to the branching process. This leads to a threshold theorem for the epidemic process, where a major outbreak is defined as one that infects at least $\log n$ individuals. We show further that there exists $\delta \gt 0$, depending on the model parameters, such that the probability that a major outbreak has size at least $\delta n$ tends to one as $n \to \infty$.

During an epidemic outbreak, typically only partial information about the outbreak is known. A common scenario is that the infection times of individuals are unknown, but individuals, on displaying symptoms, are identified as infectious and removed from the population. We study the distribution of the number of infectives given only the times of removals in a Markovian susceptible–infectious–removed (SIR) epidemic. Primary interest is in the initial stages of the epidemic process, where a branching (birth–death) process approximation is applicable. We show that the number of individuals alive in a time-inhomogeneous birth–death process at time $t \geq 0$, given only death times up to and including time t, is a mixture of negative binomial distributions, with the number of mixing components depending on the total number of deaths, and the mixing weights depending upon the inter-arrival times of the deaths. We further consider the extension to the case where some deaths are unobserved. We also discuss the application of the results to control measures and statistical inference.

We consider a class of processes describing a population consisting of k types of individuals. The process is almost surely absorbed at the origin within finite time, and we study the expected time taken for such extinction to occur. We derive simple and precise asymptotic estimates for this expected persistence time, starting either from a single individual or from a quasi-equilibrium state, in the limit as a system size parameter N tends to infinity. Our process need not be a Markov process on $ {\mathbb Z}_+^k$; we allow the possibility that individuals’ lifetimes may follow more general distributions than the exponential distribution.

Ferri-fluoro-katophorite is the second species characterised involving the rootname katophorite in the sodium–calcium subgroup of the amphibole supergroup. The mineral and its name were approved by the International Mineralogical Association Commission on New Minerals, Nomenclature and Classification, IMA2015-096. It was found in the Bear Lake diggings, Bancroft area, Ontario, Canada, where coarse euhedral crystals of amphibole, phlogopite, sanidine solid-solution (now coarsely exsolved to microcline perthite), titanite, augite, zircon and fluorapatite crystallised from a low-viscosity silicocarbonatitic magma of crustal origin. Greenish grey prismatic crystals of ferri-fluoro-katophorite generally protrude from the walls into a body of coarsely crystalline calcite, but they also occur away from the walls, completely enclosed by calcite. The empirical formula derived from electron microprobe analysis and single-crystal structure refinement is: A(Na0.55K0.32)Σ0.87B(Na0.79Ca1.18Mn2+0.03)Σ2.00C(Mg3.29Mn2+0.02Fe2+1.19Fe3+0.31Al0.09Ti4+0.08Li0.02)Σ5.00T(Si7.39Al0.61)Σ8.00O22W[F1.23 (OH)0.77]Σ2.00. Ferri-fluoro-katophorite is biaxial (–), with α = 1.640(2), β = 1.652(2), γ = 1.658(2), 2Vmeas. = 68.9(2)° and 2Vcalc.. = 70.1°. The unit-cell parameters are a = 9.887(3), b = 18.023(9), c = 5.292(2) Å, β = 104.66(3)°, V = 912.3(6) Å3, Z = 2 and space group C2/m. The strongest ten lines in the powder X-ray pattern [d values (in Å) I (hkl)] are: 2.708, 100, (151); 2.388, 74, (131); 3.139, 72, (310); 8.449, 69, (110); 2.540, 65, ($\bar{2}$02); 2.591, 53, (061); 2.739, 47, ($\bar{3}$31); 2.165, 45, (261); 3.279, 44, ($\bar{2}$40); 2.341, 43, ($\bar{3}$51).

Brandãoite, [BeAl2(PO4)2(OH)2(H2O)4](H2O), is a new Be–Al phosphate mineral from the João Firmino mine, Pomarolli farm region, Divino das Laranjeiras County, Minas Gerais State, Brazil, where it occurs in an albite pocket with other secondary phosphates, including beryllonite, atencioite and zanazziite, in a granitic pegmatite. It occurs as colourless acicular crystals <10 µm wide and <100 µm long that form compact radiating spherical aggregates up to 1.0–1.5 mm across. It is colourless and transparent in single crystals and white in aggregates, has a white streak and a vitreous lustre, is brittle and has conchoidal fracture. Mohs hardness is 6, and the calculated density is 2.353 g/cm3. Brandãoite is biaxial (+), α = 1.544, β = 1.552 and γ = 1.568, all ± 0.002; 2Vobs = 69.7(10)° and 2Vcalc = 71.2°. No pleochroism was observed. Brandãoite is triclinic, space group P$\bar{1}$, a = 6.100(4), b = 8.616(4), c = 10.261(5) Å, α = 93.191(11), β = 95.120(11), γ = 96.863(11)°, V = 532.1(8) Å3 and Z = 2. Chemical analysis of a 4 µm wide needle-shaped crystal by electron microprobe and secondary-ion mass spectrometry gave P2O5 = 28.42, Al2O3 = 20.15, BeO = 4.85, H2O = 21.47 and sum = 74.89 wt.%. The empirical formula, normalised on the basis of 15 anions pfu with (OH) = 2 and (H2O) = 5 apfu (from the crystal structure) is Be0.98Al1.99P2.02H12O15. The crystal structure was solved by direct methods and refined to an R1 index of 7.0%. There are two P sites occupied by P5+, two Al sites occupied by octahedrally coordinated Al3+, and one Be site occupied by tetrahedrally coordinated Be2+. There are fifteen anions, two of which are (OH) groups and five of which are (H2O) groups. The simplified ideal formula is thus [BeAl2(PO4)2(OH)2(H2O)4](H2O) with Z = 2. Beryllium and P tetrahedra share corners to form a four-membered ring. Aluminium octahedra share a common vertex to form an [Al2φ11] dimer, and these dimers are cross-linked by P tetrahedra to form a complex slab of polyhedra parallel to (001). These slabs are cross-linked by BeO2(OH)(H2O) tetrahedra, with interstitial (H2O) groups in channels that extend along [100].

Beusite-(Ca), ideally Ca${\rm Mn}_{\rm 2}^{2 +} $(PO4)2, is a new graftonite-group mineral from the Yellowknife pegmatite field, Northwest Territories, Canada. It occurs in a beryl–columbite–phosphate rare-element pegmatite where it is commonly intergrown with triphylite–lithiophilite or sarcopside, and may form by exsolution from a high-temperature (Li,Ca)-rich graftonite-like parent phase. It occurs as pale-brown lamellae 0.1–1.5 mm wide in triphylite, and is pale brown with a vitreous lustre and a very pale-brown streak. It is brittle, has a Mohs hardness of 5, and the calculated density is 3.610 g/cm3. Beusite-(Ca) is colourless in plane-polarized light, and is biaxial (+) with α = 1.685(2), β = 1.688(2), γ = 1.700(5), and the optic axial angle is 46.0(5)°. It is non-pleochroic with X || b; Y ˄ a = 40.3° in β obtuse; Z ˄ a = 49.7° in β acute. Beusite-(Ca) is monoclinic, has space group P21/c, a = 8.799(2), b = 11.724(2), c = 6.170(1) Å, β = 99.23(3)°, V = 628.3(1) Å3 and Z = 4. Chemical analysis by electron microprobe gave P2O5 41.63, FeO 19.43, MnO 23.63, CaO 15.45, sum 100.14 wt.%. The empirical formula was normalized on the basis of 8 anions pfu: (Ca0.94Fe0.92Mn1.13)Σ2.99(PO4)2.00. The crystal structure was refined to an R1 index of 1.55%. Beusite-(Ca) is a member of the graftonite group with Ca completely ordered at the [8]-coordinated M(1) site.

Two new minerals of the graftonite group, graftonite-(Mn), ideally M(1)MnM(2),M(3)Fe2(PO4)2, and graftonite-(Ca), ideally M(1)CaM(2),M(3)Fe2(PO4)2, were discovered in phosphate nodules of two beryl–columbite–phosphate pegmatites at Lutomia and Michałkowa, respectively, in the Góry Sowie Block, Lower Silesia, southwest Poland. Graftonite-(Mn) is pinkish brown, whereas graftonite-(Ca) shows more brownish colouration. Both minerals have a vitreous lustre, a good cleavage observed along (010) and irregular fracture; both are transparent and neither of them is fluorescent. They are brittle and have a Mohs hardness of ~5. The minerals are non-pleochroic, colourless in all orientations, biaxial (+), with mean refractive indices α = 1.710(2) and 1.690(2), β = 1.713(2) and 1.692(2), and γ = 1.725(2) and 1.710(5), respectively. With complete order of Ca at the M(1) site, the formulae of the holotype crystals are M(1)(Mn0.70Ca0.30)M(2),M(3)(Fe1.34Mn0.60Mg0.06Zn0.01)Σ3(PO4)2 for graftonite-(Mn) and M(1)(Ca0.98Mn0.02)M(2),M(3)(Fe1.38Mn0.56Mg0.05)Σ3(PO4)2 for graftonite-(Ca). Both crystal chemistry and crystal-structure refinement (R1 = 2.34 and 1.63%, respectively) indicate that the M(1) site is occupied dominantly by Mn in graftonite-(Mn) and by Ca in graftonite-(Ca), and the M(2) and M(3) sites are occupied by Fe2+ and Mn2+, with Fe2+ dominant over Mn2+ at the aggregate M(2) + M(3) sites. Graftonite-(Mn) and graftonite-(Ca) are isostructural with graftonite, M(1)FeM(2),M(3)Fe2(PO4)2 (monoclinic system; space-group symmetry P21/c), with the unit-cell parameters a = 8.811(2) Å, b = 11.494(2) Å, c = 6.138(1) Å, β = 99.23(3)° and V = 613.5(4) Å3, and a = 8.792(2) Å, b = 11.743(2) Å, c = 6.169(1) Å, β = 99.35(3)° and V = 628.5(1) Å3, respectively. The densities calculated on the basis of molar weights and unit-cell volumes are 3.793 g/cm3 for graftonite-(Mn) and 3.592 g/cm3 for graftonite-(Ca). The eight strongest lines in powder X-ray diffraction patterns on the basis of single-crystal data are, respectively [d, Å, I (hkl)]: 2.874, 100, (230 + 040); 2.858, 79, (221); 3.506, 73, (130); 2.717, 79, ($\bar{3}$11); 2.952, 55, (131); 2.916, 53, ($\bar{1}$12); 2.899, 44, (300); 3.016, 35, ($\bar{1}$02); and 3.654, 100, (130); 2.979, 85, (221); 3.014, 77, (230); 3.042, 76, (040 + $\bar{1}$12); 2.834, 68, ($\bar{3}$11); 3.097, 57, (131); 3.133, 56, ($\bar{1}$02); 2.542, 30, (311). Both minerals are common primary phosphates in phosphate nodules, occurring as lamellar intergrowths with sarcopside ± triphylite/lithiophilite, products of exsolution from a (Li,Ca)-rich graftonite-like parent phase crystallized at high temperature from P-bearing hydrosaline melts.

Folvikite, Sb5+Mn3+(Mg,Mn2+)10O8(BO3)4, is a new oxyborate mineral from the Kitteln mine, Värmland, Sweden, where it occurs as a primary skarn mineral embedded in calcite. It forms striated prismatic crystals up to 0.3 mm, and is black to dark reddish-brown with submetallic lustre and a reddish-brown streak. It is brittle, has a Mohs hardness of 6, and the calculated density is 4.14 g/cm3. Folvikite is biaxial with indeterminate optic sign due to pervasive twinning. The optic axial angle is 68.9(4)°. Refractive indices were not measured; the calculated mean refractive index is 1.85. Strong pleochroism was observed in plane-polarized light: AB = brown (intermediate), OB = dark brown (maximum) and ON = honey brown (minimum). Folvikite is monoclinic, space group P2, a = 5.3767(10), b = 6.2108(10), c = 10.9389(18) Å, β = 94.399(9)°, V = 364.22(16) Å3 and Z = 1. Chemical analysis by electron microprobe gave Sb2O5 18.15, MgO 24.11, MnO 29.73, Mn2O3 11.62, Al2O3 0.27, Fe2O3 0.45, B2O3 15.27, sum 99.60 wt.%. The B2O3 content was assigned as B = 4 apfu and the Mn2O3 / (MnO + Mn2O3) ratio was determined from the crystal structure. The empirical formula was normalized on the basis of 20 anions pfu: (Sb5+1.02Mn3+1.34Al0.05Fe3+0.05Mg5.46Mn2+3.82□0.26)Σ12O8(BO3)4. A simplified formula may be written as Sb5+Mn3+(Mg,Mn2+)10O8(BO3)4 with Z = 1. The crystal structure was solved by direct methods and refined to an R1 index of 4.1%. Folvikite is a member of the (3 Å) zigzag wallpaper-borate structures in which chains of edge-sharing octahedra extend along the c axis and are cross-linked by BO3 groups. There are five X sites partly occupied by Mn2+ > Mg, one octahedrally coordinated M-site occupied by Sb5+ > Mg, two M sites occupied by Mg ≥ Mn > Sb5+, two M sites occupied by Mn3+ > Mn2+, two M sites occupied by Mg > Mn2+, and one M-site occupied by Mg > □; plus two [3]-coordinated B sites occupied by B. As with the other zigzag borates, the polyhedra are arranged in F-walls, C-walls and S-columns.

Pink-orange crystals of a composition within the ferri-obertiite compositional space were found in vesicles in a pale beige silicate vein found from a basalt quarry at Mount Rothenberg, Eifel district, Germany. Associated minerals are potassic feldspar, alpha quartz paramorphic afterbeta quartz, eifelite (the second occurrence after the Caspar quarry at Bellerberg volcano, Eifel region), tridymite, rutile, roedderite and other amphiboles. The ideal formula of ferri-obertiite is ANaBNa2C(Mg3Fe3+Ti)TSi8O22WO2; the empirical formula derived for the holotype specimen from Mount Rothenberg from the results of electron-microprobe analysis and single-crystal structure refinement is A(Na0.76K0.22)∑0.98B(Na1.61Ca0.35Mn0.042+)∑2.00C(Mg3.58Mn0.112+Fe0.623+Ti0.664+Cr0.013+Zn0.01Ni0.01)∑5.00T(Si7.82Ti0.124+Al0.06)∑8.00O22W[O1.26F0.55(OH)0.19]∑2.00. The unit-cell dimensions are a = 9.7901(7), b = 17.9354(13), c = 5.2892(4)Å, β= 104.142(2)°, V = 900.58 (11) Å3. The space group is C2/m, Z = 2. Ferri-obertiite is biaxial (+), with α = 1.664, β = 1.680, γ = 1.722, all ±0.002 and 2V (meas.) = 66.4(3)o, 2V (calc.) = 64.7o.The strongest eight reflections in the powder X-ray pattern [d values (in Å), I, (hkl)] are: 2.704, 100, (151); 3.116, 76, (310); 3.388, 72, (131); 8.931, 72, (110); 2.529, 67, (202); 2.583, 39, (061); 2.160, 38, (261); 3.260, 37, (240). Both the mineral and thename have been approved by the Commission on New Minerals, Nomenclature and Classification of the International Mineralogical Association (IMA 2015-079); the rock specimen has been deposited at the Museo di Mineralogia, Dipartimento di Scienze della Terra e dell'Ambiente, Universitàdegli Studi di Pavia, under the code 2015-02.

Magnesio-arfvedsonite, the CFe3+-dominant analogue of eckermannite, has been found in a sample of “szechenyite” in the mineral collection of the American Museum of Natural History (AMNH H35024). It comes from the northern part of the Jade Mine Tract near Hpakan, Kachin State, Myanmar. Associated minerals are kosmochlor–jadeite solid-solution pyroxene and clinochlore. The ideal formula of magnesio-arfvedsonite is ANaBNa2C(Mg4Fe3+)TSi8O22W(OH)2, and the empirical formula derived from electron microprobe analysis and single-crystal structure refinement for the sample of this work is A(Na0.96K0.04)Σ=1.00B(Na1.57Ca0.40Fe0.022+Mn0.01)Σ=2.00C(Mg4.26Fe0.192+Fe0.413+Al0.11Ti0.034+)Σ=5.00T(Si7.99Al0.01)Σ=8.00O22W[F0.02(OH)1.98]Σ=2.00. The unit-cell dimensions are a = 9.867(1), b = 17.928(2), c = 5.284(1) Å, β = 103.80(2)°, V = 907.7 (2) Å3, Z = 2. Magnesio-arfvedsonite is biaxial (–), with α = 1.624, β = 1.636, γ = 1.637, all ± 0.002 and 2Vobs = 36(1)°, 2Vcalc = 32°. The ten strongest reflections in the X-ray powder pattern [d values (in Å), I, (hkl)] are: 2.708, 100, (151); 3.399, 68, (131); 3.144, 63, (310); 2.526, 60, (202); 8.451, 46, (110); 3.273, 39, (240); 2.167, 37, (261); 2.582, 34, (061); 2.970, 34, (221); 2.326, 33, [(251) (421)].

Katophorite has the ideal formula ANaB(NaCa)C(Mg4Al)T(Si7Al)O22W(OH)2 (Hawthorne et al., 2012). No published analyses of amphiboles fall in the katophorite compositional field, except that of Harlow and Olds (1987) for an amphibole from near Hpakan in the Jade Mine Tract, Myanmar. This amphibole was approved by the International Mineralogical Association Commission on New Minerals, Nomenclature and Classification (vote 2013-140) as katophorite, and is reported here. Holotype katophorite is monoclinic, space group C2/m, a = 9.8573(8), b = 17.9617(15), c = 5.2833(4) Å, β = 104.707(2)°, V = 904.78(13) Å3, Z = 2. The calculated density is 3.091 g cm–3. In plane-polarized light, katophorite is pleochroic, X = pale blue (medium), Y = light blue-green (strongest), Z = colourless; X ∧ a = 30.6° (β obtuse), Y || b, Z ∧ c = 15.8 (β acute). It is biaxial negative, α = 1.638, β = 1.642, γ = 1.644, all ± 0.002; 2Vobs = 73(1)°, 2Vcalc = 70°. The eight strongest lines in the powder X-ray diffraction pattern are [d in Å (I)(hkl)]: 2.700 (100)(151), 3.129 (69)(310), 2.536 (65)(202), 3.378 (61)(131), 8.421 (55)(110), 2.583 (46)(061), 2.942 (43)(221) and 2.334 (41)(351). Electron-microprobe analysis of the refined crystal gave SiO251.74, Al2O37.38, TiO2 0.14, FeO 1.55, Fe2O3 2.82, MgO 18.09, CaO 8.17, Na2O 6.02, K2O 0.24, F 0.06, H2Ocalc. 1.80, Li2Ocalc. 0.09, sum 100.55 wt.% (Li2O and H2O based on the results of single-crystal structure refinement). The formula unit, calculated on the basis of 24 (O,OH,F) with (OH + F + O) = 2 is: A(Na0.85K0.04)Σ=0.89B(Ca1.22Na0.78)Σ=2.00C(Mg3.76Al0.43Fe0.303+Cr0.273+Fe0.182+Li0.05Ti0.014+)Σ=5.00T(Si7.21Al0.79)Σ=8.00O22W[(OH)1.67O0.30F0.03)]Σ=2.00.

Ferro-ferri-hornblende is a new member of the amphibole supergroup (IMA-CNMNC 2015-054). It has been found in a rock specimen from the historical collection of Leandro De Magistris, which was collected at the Traversella mine (Val Chiusella, Ivrea, Piemonte, Italy). The specimen was catalogued as ‘speziaite', and contains a wide range of amphibole compositions from tremolite/actinolite to magnesio-hastingsite. The end-member formula of ferro-ferri-hornblende is A□BCa2c(Fe+Fe3+)T(Si7Al) O22W(OH)2 , which requires SiO2 43.41, Al2O3 5.26, FeO 29.66, Fe2O3 8.24 CaO 11.57, H2O 1.86, total 100.00 wt.%. The empirical formula derived from electron microprobe analysis and single-crystal structure refinement for the holotype crystal is A(Na0.10K0.13) Σ=0.23B(Ca 1.93Na0.07)Σ=2.00C(Mg1.16Fe2+3.21Mn0.O6Fe3+0.45 Al0.12Ti 0.01)Σ=5.01T(Si7.26Al0. 74)Σ=8.00 O22W(OH1.89F0.01C10.10)Σ=2.00- Ferro-ferri-hornblende is biaxial (-), with α = 1.697(2), P = 1 .722(5), γ = 1.726(5) and 2V (meas.) = 35.7(1.4)°, 2V (calc.) = 43.1°. The unit-cell parameters are a = 9.9307(5), b = 18.2232(10), c = 5.3190(3) Å, β = 104.857(1)°, V= 930.40 (9) Å3, Z= 2, space group C2/m. The a:b:c ratio is 0.545:1:0.292. The strongest eight reflections in the powder X-ray pattern [d values (in Å), I, (hkl)] are: 8.493, 100, (110); 2.728, 69, (151); 3.151, 47, (310); 2.555, 37, (); 2.615, 32, (061); 2.359, 28, (); 3.406, 26, (131); 2.180, 25, (261). Type material is deposited in the collections of the Museo di Mineralogia, Dipartimento di Scienze della Terra e dell'Ambiente, Università di Pavia, under the catalogue number 2015-01. Sample M/U15285 from the historical collection of Luigi Colomba, presently at the Museo Regionale di Scienze Naturali di Torino, was also checked, and the presence of ferro-ferri-hornblende was confirmed.

Magnesio-ferri-fluoro-hornblende has the ideal formula A□B Ca2C(Mg4Fe3+)T(Si7Al)O22WF2 (Hawthorne et al., 2012). The holotype sample described in this work occurs as prismatic crystals in vugs of volcanic rocks (Seruci ignimbrites), found along the coast road ∼5.5 km northeast of Portoscuso, Cagliari, Sardinia; associated minerals are tridymite, todorokite, magnetite, and hematite. The name and the mineral were approved by the IMA CNMNC (2014-091). Holotype magnesio-ferri-fluoro-hornblende is monoclinic, space group C2/m, a = 9.839(5), b = 18.078(9), c = 5.319(3) Å, β = 104.99(3)°, V = 913.9(9) Å3, Z = 2. The density calculated from the empirical formula is 3.315 g cm–3. In plane-polarized light, magnesio-ferri-fluoro-hornblende is pleochroic, X = pale grey (least), Y = dark grey (most), Z = pale brownish grey (intermediate); X^ a = 47.6° (β obtuse), Y // b, Z^ c = 33.4° (β acute). It is biaxial negative, α = 1.669, β = 1.676, γ = 1.678, all ±0.002; 2Vobs = 74(1)°, 2Vcalc = 56°. The strongest eight lines in the powder X-ray diffraction pattern are [d in Å (I)(hkl)]: 2.711 (100)(151), 8.412 (89)(110), 3.121 (64)(310), 2.553 (61)(2̄02), 3.389 (55)(131), 2.599 (45)(061), 2.164 (36)(261), and 2.738 (34)(3̄31). Electron-microprobe analysis of the refined crystal gave SiO2 45.34, Al2O3 6.18, TiO2 1.22, FeO 15.24, Fe2O3 6.27, MgO 9.71, MnO 0.78, ZnO 0.06, CaO 10.18, Na2O 1.35, K2O 1.15, F 3.22, Cl 0.30, H2Ocalc 0.37, sum 99.95 wt.%. The empirical formula unit, calculated on the basis of 24 (O, OH, F, Cl) apfu with (OH + F + Cl) = 2 apfu is: (Na0.15K0.22)∑0.37(Na0.25Ca1.66Mn0.09)∑2.00(Mg2.20Fe2+1.94Mn0.01Zn0.01Fe3+0.72Ti0.13)∑5.01 (Al1.11Si6.89)∑8.00O22[F1.55(OH)0.37Cl0.08)∑2.00.

Oxo-mangano-leakeite, a newly approved end-member of the amphibole supergroup (IMA-CNMNC 20150-35), has been found in a rock containing manganese silicate and oxide at the Hoskins Mine, a Mn deposit 3 km west of Grenfell, New South Wales. The end-member formula of oxo-mangani-leakeite is ANaBNa2C(Mn3+4Li)TSi8 O22WO2, which would require SiO2 53.15, Mn2O3 34.91, Li2O 1.66, Na2O 10.28, total 100.00 wt.%. The empirical formula derived for the sample of this work from electron and ion microprobe analysis using constraints resulting from single-crystal structure refinement is A(Na0.65K0.36)∑ = 1.01B(Na1.94Ca0.06)∑ = 2.00C(Mg1.60Zn0.01 Li0.58)∑ = 5.01T(Si7.98Al0.02)∑ = 8.00O22W(O1.34OH0.66)∑ = 2.00. Oxo-mangano-leakeite is biaxial (–), with α = 1.681, β = 1.712, γ = 1.738, all ± 0.002, and 2V (meas.) = 81.0(4)°, 2V (calc.) = 83.5°. The unit-cell dimensions are a = 9.875(5), b = 17.873(9), c = 5.295(2) Å, β = 104.74(3)°, V = 903.8 (7) Å3; the space group is C2/m, with Z = 2. The strongest ten reflections in the powder X-ray pattern [d values (in Å), I, (hkl)] are: 8.423, 100, (110); 3.377, 46, (131); 4.461, 40, (040); 4.451, 40, (021); 3.134, 37, (310); 2.694, 37, (151); 2.282, 27, (); 2.734, 25, (31); 2.575, 24, (061); 2.331, 24, [() ()]. The holotype material is deposited in the Canadian Museum of Nature, Ottawa, under the catalogue number CMNMC 86895.

Jointed goatgrass (Aegilops cylindrica) is a serious problem for winter wheat producers throughout the western U.S. Interference from this weed can severely reduce grain yield and contaminate harvested grain, resulting in dockage losses. There are currently no selective herbicides registered for controlling jointed goatgrass in wheat. Imazamox, an imidazolinone herbicide, was applied to an imidazolinone herbicide-resistant (IMI) wheat mutant of the winter wheat cultivar ‘Fidel.’ Jointed goatgrass control from spring postemergence application of imazamox ranged from 61 to 97% when applied at 36 g/ha. Negligible crop injury from imazamox treatment was observed at 36 g/ha at several locations under dryland environments in the U.S. Pacific Northwest. Wheat yield was increased 19 to 41% by imazamox treatment in three of four experiments. Percent dockage resulting from jointed goatgrass spikelet contamination of harvested wheat grain was eliminated by imazamox treatment. Introduction of the IMI trait into commercial wheat cultivars could provide an effective method for selective control of jointed goatgrass in winter wheat.

Jointed goatgrass is an invasive winter annual grass weed that is a particular problem in the low to intermediate rainfall zones of the Pacific Northwest (PNW). For the most part, single-component research has been the focus of previous jointed goatgrass studies. In 1996, an integrated cropping systems study for the management of jointed goatgrass was initiated in Washington, Idaho, and Oregon in the traditional winter wheat (WW)–fallow (F) region of the PNW. The study evaluated eight integrated weed management (IWM) systems that included combinations of either a one-time stubble burn (B) or a no-burn (NB) treatment, a rotation of either WW–F–WW or spring wheat (SW)–F–WW, and either a standard (S) or an integrated (I) practice of planting winter wheat. This study is the first, to our knowledge, to evaluate and identify complete IWM systems for jointed goatgrass control in winter wheat. At the Idaho location, in a very low weed density, no IWM system was identified that consistently had the highest yield, reduced grain dockage, and reduced weed densities. However, successful IWM systems for jointed goatgrass management were identified as weed populations increased. At the Washington location, in a moderate population of jointed goatgrass, the best IWM system based on the above responses was the B:SW–F–WW:S system. At the Washington site, this system was better than the integrated planting system because the competitive winter wheat variety did not perform well in drought conditions during the second year of winter wheat. At the Oregon site, a location with a high weed density, the system B:SW–F–WW:I produced consistently higher grain yields, reduced grain dockage, and reduced jointed goatgrass densities. These integrated systems, if adopted by PNW growers in the wheat–fallow area, would increase farm profits by decreasing dockage, decreasing farm inputs, and reducing herbicide resistance in jointed goatgrass.

The most common strategy recommended for management of jointed goatgrass infestations is to rotate from winter wheat to a spring crop for several years. A field study was conducted at three locations in 1998 and 1999 to determine the effects of spring seeding date on the ability of jointed goatgrass to flower and produce viable seed in the presence or absence of spring wheat and to determine the effect of jointed goatgrass competition and crop seeding date on spring wheat grain yield. Spring wheat was seeded on four dates at each location in both hand-sown and natural jointed goatgrass infestations. Jointed goatgrass plants from hand-sown spikelets flowered and developed spikelets on all seeding dates except the last; viable seed was produced on the two earliest seeding dates. Jointed goatgrass plant densities from natural infestations were from 1 to 12 plants m−2, and spikelet production ranged from 0 to 480 spikelets m−2. Natural jointed goatgrass infestations produced spikelets containing viable seed on all seeding dates at one location in 1998, the driest location. Spring wheat yield was not affected by jointed goatgrass competition; however, jointed goatgrass spikelet production was reduced by spring wheat competition compared with that of monoculture jointed goatgrass. The last seeding date of spring wheat was associated with 51% less crop yield compared with the recommended seeding date. The decision to manage jointed goatgrass infestations with a spring crop rotation should consider delayed seeding dates to minimize viable spikelet production by spring-germinating jointed goatgrass; however, the cost of this decision may include grain yield reduction.

Jointed goatgrass is most commonly described as a winter annual species. However, it has been observed to produce spikes in spring crops, apparently without being exposed to vernalizing conditions. A controlled environment study was conducted to determine the reproductive response of jointed goatgrass plants grown from seeds of fall- and spring-emerging parent plants to various vernalization durations. Winter wheat was included as a control. Winter wheat spikelet production was dependent on vernalization, and the number of spikes per plant was 10-fold greater if the plants were exposed to 4 C for 10 wk. In contrast, jointed goatgrass spike production without vernalization remained as high as 50% of that produced by plants exposed to 10 wk of vernalization conditions. Jointed goatgrass is thus not as dependent on vernalization for reproduction as the comparative winter wheat standard. Apparently, jointed goatgrass is more a facultative rather than an obligate winter annual. Rotating to a spring-seeded crop should not be expected to completely prevent jointed goatgrass seed production. Fields rotated to spring wheat to eliminate jointed goatgrass seed production should be monitored, and jointed goatgrass should be hand pulled or otherwise controlled to ensure zero seed production.

We prove that, for Poisson transmission and recovery processes, the classic susceptible→infected→recovered (SIR) epidemic model of Kermack and McKendrick provides, for any given time t>0, a strict lower bound on the expected number of susceptibles and a strict upper bound on the expected number of recoveries in the general stochastic SIR epidemic. The proof is based on the recent message passing representation of SIR epidemics applied to a complete graph.

We study continuous-time birth–death type processes, where individuals have independent and identically distributed lifetimes, according to a random variable Q, with E[Q] = 1, and where the birth rate if the population is currently in state (has size) n is α(n). We focus on two important examples, namely α(n) = λ n being a branching process, and α(n) = λn(N - n) / N which corresponds to an SIS (susceptible → infective → susceptible) epidemic model in a homogeneously mixing community of fixed size N. The processes are assumed to start with a single individual, i.e. in state 1. Let T, An, C, and S denote the (random) time to extinction, the total time spent in state n, the total number of individuals ever alive, and the sum of the lifetimes of all individuals in the birth–death process, respectively. We give expressions for the expectation of all these quantities and show that these expectations are insensitive to the distribution of Q. We also derive an asymptotic expression for the expected time to extinction of the SIS epidemic, but now starting at the endemic state, which is not independent of the distribution of Q. The results are also applied to the household SIS epidemic, showing that, in contrast to the household SIR (susceptible → infective → recovered) epidemic, its threshold parameter R* is insensitive to the distribution of Q.