The formulation of mathematical models using differential equations has become crucial in predicting the evolution of viral diseases in a population in order to take preventive and curative measures. In December 2019, a novel variety of Coronavirus (SARS-CoV-2) was identified in Wuhan, Hubei Province, China, which causes a severe and potentially fatal respiratory syndrome. Since then, it has been declared a pandemic by the World Health Organization and has spread around the globe. A reaction–diffusion system is a mathematical model that describes the evolution of a phenomenon subjected to two processes: a reaction process, in which different substances are transformed, and a diffusion process, which causes their distribution in space. This article provides a mathematical study of the Susceptible, Exposed, Infected, Recovered, and Vaccinated population model of the COVID-19 pandemic using the bias of reaction–diffusion equations. Both local and global asymptotic stability conditions for the equilibria were determined using a Lyapunov function, and the nature of the stability was determined using the Routh–Hurwitz criterion. Furthermore, we consider the conditions for the existence and uniqueness of the model solution and show the spatial distribution of the model compartments when the basic reproduction rate ${\mathcal{R}}_0<1$ and ${\mathcal{R}}_0>1$. Thereafter, we conducted a sensitivity analysis to determine the most sensitive parameters in the proposed model. We demonstrate the model’s effectiveness by performing numerical simulations and investigating the impact of vaccination, together with the significance of spatial distribution parameters in the spread of COVID-19. The findings indicate that reducing contact with an infected person and increasing the proportion of susceptible people who receive high-efficacy vaccination will lessen the burden of COVID-19 in the population. Therefore, we offer to the public health policymakers a better understanding of COVID-19 management.

In this paper we present a unified treatment of three-body atoms and molecules using numerical tensor methods. The Schrödinger equation in perimetric coordinates is recast in a canonical tensor format. It is shown that the Schrödinger equation can be solved in this full-tensor format but that by using a low-rank tensor decomposition, in particular the tensor-train and quantized-tensor-train formats, energies accurate to at least the nanohartree can be obtained for the He atom (in which the mass of the uniquely charged particle is much greater than the other two particles), the positronium negative ion (${\mathrm{Ps}}^{-}$, in which all the masses are equal), and the non-Born-Oppenheimer ${\mathrm{H}}_2{}^{+}$ molecule (in which the mass of the uniquely charged particle is much smaller than the other two particles).

In this paper we derive a numerical tensor method to solve the many-body Schrödinger equation for three-particle systems. The Schrödinger equation is first represented in a canonical tensor format using a triple orthogonal Laguerre-based wave function in scaled perimetric coordinates. We demonstrate that the antisymmetry of the wave function can be explicitly included in the tensor method formulation by coupling the indices associated with the coordinates representing the identical fermions, e.g., electrons or muons. It is shown that energies accurate to the picohartree can be achieved in full tensor format in less than 3300 s for atomic and muonic molecular systems. Furthermore, this CPU time is reduced to less than 300 s, without compromising the accuracy of the energy or quality of the wave function, using a low-rank tensor-train decomposition.

In this paper, a mathematical analysis of the global dynamics of a viral infection model in vivo is carried out. We study the dynamics of a hepatitis C virus (HCV) model, under therapy, that considers both extracellular and intracellular levels of infection. At present most mathematical modeling of viral kinetics after treatment only addresses the process of infection of a cell by the virus and the release of virions by the cell, while the processes taking place inside the cell are not included. We prove that the solutions of the new model with positive initial values are positive, exist globally in time and are bounded. The model has two virus-free steady states. They are distinguished by the fact that viral RNA is absent inside the cells in the first state and present inside the cells in the second. There are basic reproduction numbers associated to each of these steady states. If the basic reproduction number of the first steady state is less than one then that state is asymptotically stable. If the basic reproduction number of the first steady state is greater than one and that of the second less than one then the second steady state is asymptotically stable. If both basic reproduction numbers are greater than one then we obtain various conclusions which depend on different restrictions on the parameters of the model. Under increasingly strong assumptions we prove that there is at least one positive steady state (infected equilibrium), that there is a unique positive steady state and that the positive steady state is stable. We also give a condition under which every positive solution converges to a positive steady state. This is proved by methods of Li and Muldowney. Finally, we illustrate the theoretical results by numerical simulations.

In this paper, we solve the non-relativistic many-body Schrödinger equation (SE) for the excited P states (L = 1) of three-particle systems with natural and unnatural parity. The exact separation of the rotational degrees of freedom from the internal degrees of freedom presented by Sadhukhan et al. [arXiv:2506.23962 (2025)] is exploited. The angular matrix elements involve the product of two bipolar harmonics and are calculated using an explicit analytical expression containing the Wigner 3j- and 6j-symbols. Using a symmetry and parity adapted wavefunction for P states in perimetric coordinates, the variational form of the SE in canonical tensor format is derived. This consists of a sum of triple tensor products of factor matrices and is an exact representation. From this canonical format, the SE can be solved in full tensor format or used to derive an exact representation as a second-order tensor train. Two formulations are presented. In the first, scaled perimetric coordinates are used; it is assumed that the like-charged particles have the same asymptotic behavior so that the orthonormality of the Laguerre functions can be exploited and the matrices are sparse. In the second, the like-charged particles are not constrained to have the same asymptotic behavior and an analytical integral is used which results in dense matrices. It is shown that by optimizing either one or two non-linear variational parameters, it is possible to obtain excited P state energies that are accurate to the picohartree, for atomic and molecular systems in less than 25 min on a standard desktop computer.