Andrzej F. Nowakowski

We propose an epidemic compartment model which includes mortality caused by the disease, but excludes demographic birth and death processes. Individuals are represented by random walkers which are in one of the states (compartments) S (susceptible for infection), E (exposed: infected but not infectious corresponding to the latency period), I (infected and infectious), R (recovered, immune), D (dead). The disease is transmitted with a certain probability at contacts of I to S walkers. The compartmental sojourn times are independent random variables drawn from specific (here Gamma-) distributions. We implement this model into random walk simulations. Each walker performs an independent simple Markovian random walk on a graph, where we consider a Watts-Strogatz (WS) network. Only I walkers may die. For zero mortality we prove the existence of an endemic equilibrium for basic reproduction number ${\cal R}_0>1$ and for which the disease free (globally healthy) state is unstable. We explore the effects of long-range-journeys (stochastic resetting) and mortality. Our model allows for various interpretations, such as certain chemical reactions, the propagation of wildfires, and in population dynamics.

We introduce a compartment model with memory for the dynamics of epidemic spreading in a constant population of individuals. Each individual is in one of the states $\mathbf{S}=\text{susceptible}$, $\mathbf{I}=\text{infected}$, or $\mathbf{R}=\text{recovered}$ (SIR model). In state $\mathbf{R}$ an individual is assumed to stay immune within a finite-time interval. In the first part, we introduce a random lifetime or duration of immunity which is drawn from a certain probability density function. Once the time of immunity is elapsed an individual makes an instantaneous transition to the susceptible state. By introducing a random duration of immunity a memory effect is introduced into the process which crucially determines the epidemic dynamics. In the second part, we investigate the influence of the memory effect on the space-time dynamics of the epidemic spreading by implementing this approach into computer simulations and employ a multiple random walker's model. If a susceptible walker meets an infectious one on the same site, then the susceptible one gets infected with a certain probability. The computer experiments allow us to identify relevant parameters for spread or extinction of an epidemic. In both parts, the finite duration of immunity causes persistent oscillations in the number of infected individuals with ongoing epidemic activity preventing the system from relaxation to a steady state solution. Such oscillatory behavior is supported by real-life observations and not captured by the classical standard SIR model.

The paper investigates shock-induced vortical flows within inhomogeneous media of nonuniform thermodynamic properties. Numerical simulations are performed using an Eularian type mathematical model for compressible multi-component flow problems. The model, which accounts for pressure non-equilibrium and applies different equations of state for individual flow components, shows excellent capabilities for the resolution of interfaces separating compressible fluids as well as for capturing the baroclinic source of vorticity generation. The developed finite volume Godunov type computational approach is equipped with an approximate Riemann solver for calculating fluxes and handles numerically diffused zones at flow component interfaces. The computations are performed for various initial conditions and are compared with available experimental data. The initial conditions promoting a shock-bubble interaction process include: weak to high planar shock waves with a Mach number ranging from $1.2$ to $3$ and isolated cylindrical bubble inhomogeneities of helium, argon, nitrogen, krypton and sulphur hexafluoride. The numerical results reveal the characteristic features of the evolving flow topology. The impulsively generated flow perturbations are dominated by the reflection and refraction of the shock, the compression and acceleration as well as the vorticity generation within the medium. The study is further extended to investigate the influence of the ratio of the heat capacities on the interface deformation.

We study the clustering of inertial particles using a periodic kinematic simulation. Particle clustering is observed for different pairs of Stokes number and Froude number and different spectral power laws $\left(1.4\le p\le 2.1\right)$. The main focus is to identify and then quantify the effect of $p$ on the clustering attractor—by attractor we mean the set of points in the physical space where the particles settle when time tends to infinity. It is observed that spectral power laws can have a dramatic effect on the attractor shape. In particular, we observed an attractor type which was not present in previous studies for Kolmogorov spectra ($p=5/3$).

In this paper, we study the clustering of inertial particles using a periodic kinematic simulation. The systematic Lagrangian tracking of particles makes it possible to identify the particles' clustering patterns for different values of particle inertia and drift velocity. The different cases are characterized by different pairs of Stokes number (St) and Froude number (Fr). For the present study, $0\le \mathrm{St}\le 1$ and $0.4\le \mathrm{Fr}\le 1.4$. The main focus is to identify and then quantify the clustering attractor—when it exists—that is the set of points in the physical space where the particles settle when time goes to infinity. Depending on the gravity effect and inertia values, the Lagrangian attractor can have different dimensions, varying from the initial three-dimensional space to two-dimensional layers and one-dimensional attractors that can be shifted from a horizontal to a vertical position.