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The COVID-19 outbreak of 2020 has required many governments to develop and adopt mathematical-statistical models of the pandemic for policy and planning purposes. To this end, this work provides a tutorial on building a compartmental model using Susceptible, Exposed, Infected, Recovered, Deaths and Vaccinated (SEIRDV) status through time. The proposed model uses interventions to quantify the impact of various government attempts made to slow the spread of the virus. Furthermore, a vaccination parameter is also incorporated in the model, which is inactive until the time the vaccine is deployed. A Bayesian framework is utilized to perform both parameter estimation and prediction. Predictions are made to determine when the peak Active Infections occur. We provide inferential frameworks for assessing the effects of government interventions on the dynamic progression of the pandemic, including the impact of vaccination. The proposed model also allows for quantification of number of excess deaths averted over the study period due to vaccination. [ABSTRACT FROM AUTHOR]

For each of the six-dimensional indecomposable nilpotent Lie algebras, the geodesic equations of the associated canonical Lie group connection are given. In each case, a basis for the associated Lie algebra of symmetries is constructed and analyzed. [ABSTRACT FROM AUTHOR]

We obtain minimal dimension matrix representations for each of the Lie algebras of dimensions five, six, seven and eight obtained by Turkowski that have a non-trivial Levi decomposition. The key technique involves using the invariant subspaces associated to a particular representation of a semi-simple Lie algebra to help in the construction of the radical in the putative Levi decomposition. [ABSTRACT FROM AUTHOR]

A diffusion problem involving a time derivative acting on two time scales represented by two fractional derivatives is investigated. The orders of the fractional derivatives are both between 0 and 1 and therefore the problem corresponds to the subdiffusion case. It is considered on a semi-infinite axis and the forcing term and the initial data are assumed compactly supported. To reduce the problem to that support there is a risk of being lead to an 'infected' problem due to the reflected waves on the new settled boundary. To avoid this undesirable effect of reflected waves on the standard boundaries, we establish artificial boundaries and find the appropriate artificial boundary conditions. Then, using the properties of fractional derivatives, a generalized version of the Mittag-Leffler function and some adequate manipulations of inverse Laplace transforms we find the explicit solution of the reduced problem. [ABSTRACT FROM AUTHOR]

Predator-prey models are often used in ecology to represent realistic encounters between species. Lotka-Volterra differential equations are famously used by ecologists to model the relationship between a predator and prey. Given ecological data collection is known to be time consuming and difficult, the purpose of this paper is to optimize the process and improve sampling efficiency. The method of sequential optimality is applied to a more complex population model with multiple basins of attraction, the Lotka-Volterra differential equations. In this application, various theoretical scenarios are simulated using varying sample sizes and timeframes to determine optimal designs. The first scenario uses a standard sample size and design window to represent a realistic budget for ecological data collection. The second scenario decreases the budget and extends the design window as a representation of a limited budget. The third scenario represents ample resources available for sampling. The three scenarios are used to then compare the optimal designs and help ecologists evaluate the method of sequential optimality. I, A and D optimal designs are compared in the simulation results. The I-optimality criterion tends to outperform the A and D criteria and is recommended as a primary selection when using the sequential optimality algorithm. This simulation study has further developed sequential optimality by predicting optimal sampling regimes according to the dynamics associated with the Lotka-Volterra differential equations. [ABSTRACT FROM AUTHOR]

We construct new examples of four-dimensional Einstein metrics with neutral signature and two-dimensional holonomy Lie algebra. [ABSTRACT FROM AUTHOR]

In this investigation, we present symmetry algebras of the canonical geodesic equations of the indecomposable solvable Lie groups of dimension five, confined to algebras A 5 , 7 a b c to A 18 a . For each algebra, the related system of geodesics is provided. Moreover, a basis for the associated Lie algebra of the symmetry vector fields, as well as the corresponding nonzero brackets, are constructed and categorized. [ABSTRACT FROM AUTHOR]

During the current COVID-19 pandemic, decision-makers are tasked with implementing and evaluating strategies for both treatment and disease prevention. In order to make effective decisions, they need to simultaneously monitor various attributes of the pandemic such as transmission rate and infection rate for disease prevention, recovery rate which indicates treatment effectiveness as well as the mortality rate and others. This work presents a technique for monitoring the pandemic by employing an Susceptible, Exposed, Infected, Recovered, Death model regularly estimated by an augmented particle Markov chain Monte Carlo scheme in which the posterior distribution samples are monitored via Multivariate Exponentially Weighted Average process monitoring. This is illustrated on the COVID-19 data for the State of Qatar. [ABSTRACT FROM AUTHOR]

The Lie algebra isomorphism between su(1,1)xsu(1,1) and o(2,2) is used to obtain a list of subalgebras of the latter. The resulting list of 32 subalgebras is then examined on a case by case basis to see if each can be the Lie algebra of the holonomy group of a neutral metric in four dimensions. The conclusions, taken in conjunction with previously known results, furnish a classification of such Lie subalgebras of o(2,2), with only one case remaining unresolved. © 2001 American Institute of Physics. [ABSTRACT FROM AUTHOR]

Algorithms for embedding certain types of nilpotent subalgebras in maximal subalgebras of the same type are developed, using methods of real algebraic groups. These algorithms are applied to determine non-conjugate subalgebras of the symmetry algebra of the wave equation, which in turn are used to determine a large class of invariant solutions of the wave equation. The algorithms are also illustrated for the symmetry algebra of a classical system of differential equations considered by Cartan in the context of contact geometry. [ABSTRACT FROM AUTHOR]