General finite-volume framework for saddle-point problems of various physics

This article is dedicated to the general finite-volume framework used to discretize and solve saddle-point problems of various physics. The framework applies the Ostrogradsky–Gauss theorem to transform a divergent part of the partial differential equation into a surface integral, approximated by the summation of vector fluxes over interfaces. The interface vector fluxes are reconstructed using the harmonic averaging point concept resulting in the unique vector flux even in a heterogeneous anisotropic medium. The vector flux is modified with the consideration of eigenvalues in matrix coefficients at vector unknowns to address both the hyperbolic and saddle-point problems, causing nonphysical oscillations and an inf-sup stability issue. We apply the framework to several problems of various physics, namely incompressible elasticity problem, incompressible Navier–Stokes, Brinkman–Hazen–Dupuit–Darcy, Biot, and Maxwell equations and explain several nuances of the application. Finally, we test the framework on simple analytical solutions.

An equilibrated a posteriori error estimator for an Interior Penalty Discontinuous Galerkin approximation of the p-Laplace problem

We are concerned with an Interior Penalty Discontinuous Galerkin (IPDG) approximation of the p-Laplace equation and an equilibrated a posteriori error estimator. The IPDG method can be derived from a discretization of the associated minimization problem involving appropriately defined reconstruction operators. The equilibrated a posteriori error estimator provides an upper bound for the discretization error in the broken W 1,p norm and relies on the construction of an equilibrated flux in terms of a numerical flux function associated with the mixed formulation of the IPDG approximation. The relationship with a residual-type a posteriori error estimator is established as well. Numerical results illustrate the performance of both estimators.

Stability analysis of functionals in variational data assimilation with respect to uncertainties of input data for a sea thermodynamics model

The problem of stability and sensitivity of functionals of the optimal solution of the variational data assimilation of sea surface temperature for the model of sea thermodynamics is considered. The variational data assimilation problem is formulated as an optimal control problem to find the initial state and the boundary heat flux. The sensitivity of the response functions as functionals of the optimal solution with respect to the observation data is studied. Computing the gradient of the response function reduces to the solution of a non-standard problem being a coupled system of direct and adjoint equations with mutually dependent initial and boundary values. The algorithm to compute the gradient of the response function is presented, based on the Hessian of the original cost functional. Stability analysis of the response function with respect to uncertainties of input data is given. Numerical examples are presented for the Black and Azov seas thermodynamics model.

Numerical-statistical study of the prognostic efficiency of the SEIR model

A comparative analysis of the differential and the corresponding stochastic Poisson SEIR-models is performed for the test problem of COVID-19 epidemic in Novosibirsk modelling the period from March 23, 2020 to June 21, 2020 with the initial population N = 2 798 170. Varying the initial population in the form N = n m with m ⩾ 2, we show that the average numbers of identified sick patients is less (beginning from April 7, 2020) than the corresponding differential values by the quantity that does not differ statistically from C(t)/m, with C ≈ 27.3 on June 21, 2020. This relationship allows us to use the stochastic model for big population N. The practically useful ‘two sigma’ confidential interval for the time interval from June 1, 2020 to June 21, 2020 is about 108% (as to the statistical average) and involves the corresponding real statistical estimates. The influence of the introduction of delay on the prognosis, i.e., the incubation period corresponding to Poisson model is also studied.

Numerical modelling of the transition of infected cells and virions between two lymph nodes in a stochastic model of HIV-1 infection

The paper is focused on stochastic modelling of the process of transition of infected cells and virions of HIV-1 infection between two lymph nodes. The model is based on the following assumptions: (1) the duration of transition of infected cells and virions between two lymph nodes is set using a time-dependent function, (2) infected cells produce virions in the process of transition between two lymph nodes, (3) infected cells and virions may die when moving between two lymph nodes. The methods of the theory of branching random processes are used to study analytically the model variables. An algorithm for statistical modelling of the number of infected cells and virions in the second lymph node is presented. The results of computational experiments studying the distribution law of the number of virions produced by one infected cell depending on the duration of movement between two lymph nodes are presented.

Herd immunity levels and multi-strain influenza epidemics in Russia: a modelling study

In the present paper, we consider a compartmental epidemic model which simulates the co-circulation of three influenza strains, A(H1N1)pdm09, A(H3N2), and B, in a population with the history of exposure to these virus strains. A strain-specific incidence data for the model input was generated using long-term weekly ARI incidence and virologic testing data. The algorithm for model calibration was developed as a combination of simulated annealing and BFGS optimization methods. Two simulations were carried out, assuming the absence and the presence of protected individuals in the population, with 2017– 2018 and 2018–2019 epidemic seasons in Moscow as a case study. It was shown that strain-specific immune levels defined by virologic studies might be used in the model to obtain plausible incidence curves. However, different output parameter values, such as fractions of individuals exposed to particular virus strain in the previous epidemic season, can correspond to similar incidence trajectories, which complicates the assessment of herd immunity levels based on the model calibration. The results of the study will be used in the research of the interplay between the immunity formation dynamics and the circulation of influenza strains in Russian cities.

Frontiers in mathematical modelling of the lipid metabolism under normal conditions and its alterations in heart diseases

Pathophysiology of ischemic heart disease is a complex phenomenon determined by the interaction of multiple processes including the inflammatory, immunological, infectious, mechanical, biochemical and epigenetic ones. A predictive clinically relevant modelling of the entire trajectory of the human organism, from the initial alterations in lipid metabolism through to atherosclerotic plaque formation and finally to the pathologic state of the ischemic heart disease, is an open insufficiently explored problem. In the present review, we consider the existing mathematical frameworks which are used to describe, analyze and predict the dynamics of various processes related to cardiovascular diseases at the molecular, cellular, tissue, and holistic human organism level. The mechanistic, statistical and machine learning models are discussed in detail with special focus on the underlying assumptions and their clinical relevance. All together, they provide a solid computational platform for further expansion and tailoring for practical applications.

Two-scale haemodynamic modelling for patients with Fontan circulation

Palliation of congenital single ventricle heart defects suggests multi-stage surgical interventions that divert blood flow from the inferior and superior vena cava directly to the right and left pulmonary arteries, skipping the right ventricle. Such system with cavopulmonary anastomoses and single left ventricle is called Fontan circulation, and the region of reconnection is called the total cavopulmonary connection (TCPC). Computational blood flow models allow clinicians to predict the results of the Fontan operation, to choose an optimal configuration of TCPC and thus to reduce negative postoperative consequences. We propose a two-scale (1D3D) haemodynamic model of systemic circulation for a patient who has underwent Fontan surgical operation. We use CT and 4D flow MRI data to personalize the model. The model is tuned to patient’s data and is able to represent measured time-averaged flow rates at the inlets and outlets of TCPC, as well as pressure in TCPC for the patient in horizontal position.We demonstrate that changing to quiescent standing position leads to other patterns of blood flow in regional (TCPC) and global haemodynamics. This confirms clinical data on exercise intolerance of Fontan patients.