Motivation: We consider continuous-time Markov chains that describe the stochastic evolution of a dynamical system by a transition-rate matrix Q which depends on a parameter θ. Computing the probability distribution over states at time t requires the matrix exponential exp(tQ), and inferring θ from data requires its derivative ∂ exp(tQ)/∂θ. Both are challenging to compute when the state space and hence the size of Q is huge. This can happen when the state space consists of all combinations of the values of several interacting discrete variables. Often it is even impossible to store Q. However, when Q can be written as a sum of tensor products, computing exp(tQ) becomes feasible by the uniformization method, which does not require explicit storage of Q.Results: Here we provide an analogous algorithm for computing ∂ exp(tQ)/∂θ, the differentiated uniformization method. We demonstrate our algorithm for the stochastic SIR model of epidemic spread, for which we show that Q can be written as a sum of tensor products. We estimate monthly infection and recovery rates during the first wave of the COVID-19 pandemic in Austria and quantify their uncertainty in a full Bayesian analysis. Availability: Implementation and data are available at https://github.com/spang-lab/TenSIR.

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New exploration studies have revealed a near-vertical, plug-like gabbroic intrusive body emplaced along the feeder conduit of the funnel-shaped Eagle East ultramafic intrusion in the Upper Peninsula of Michigan. U-Pb baddeleyite analysis on this gabbro yielded an age of 1103.4 ± 1.2 Ma. Although a sulfide-deficient gabbro is the principal rock unit, peripheral zones of the intrusion contain feldspathic pyroxenite and melagabbro with minor blebs of chalcopyrite and pyrrhotite. A small pocket-like body of peridotite with significant quantities of sulfide minerals occurs along the edge of the main gabbroic unit. The modes of occurrence of sulfide minerals in the pocket peridotite are similar to the larger Eagle East and Eagle intrusions nearby. The composition of olivine in the main gabbroic rock is Fo24–30, but olivine in pocket peridotite shows a composition of ~Fo74. The Ni-concentration in the olivine of the main gabbro unit is below 100 ppm, but in the Mg-rich olivine of the pocket peridotite, it is between 2,100 and 2,800 ppm. Although δ34S values of sulfide minerals in the pyroxenite rim and in the pocket peridotite range from –0.6 to 7.0‰, the average value is higher than those reported from sulfides in the Eagle East and Eagle intrusions. The origin of the gabbro unit could be explained by a closed-system crystallization of a fractionated late derivative of an original high-MgO magma, while the peridotite pocket is a xenolithic representative of the old peridotitic intrusion dislodged and entrained in a late magmatic pulse.