Stochastic Compartment Model with Mortality and Its Application to Epidemic Spreading in Complex Networks.

We study epidemic spreading in complex networks by a multiple random walker approach. Each walker performs an independent simple Markovian random walk on a complex undirected (ergodic) random graph where we focus on the Barabási-Albert (BA), Erdös-Rényi (ER), and Watts-Strogatz (WS) types. Both walkers and nodes can be either susceptible (S) or infected and infectious (I), representing their state of health. Susceptible nodes may be infected by visits of infected walkers, and susceptible walkers may be infected by visiting infected nodes. No direct transmission of the disease among walkers (or among nodes) is possible. This model mimics a large class of diseases such as Dengue and Malaria with the transmission of the disease via vectors (mosquitoes). Infected walkers may die during the time span of their infection, introducing an additional compartment D of dead walkers. Contrary to the walkers, there is no mortality of infected nodes. Infected nodes always recover from their infection after a random finite time span. This assumption is based on the observation that infectious vectors (mosquitoes) are not ill and do not die from the infection. The infectious time spans of nodes and walkers, and the survival times of infected walkers, are represented by independent random variables. We derive stochastic evolution equations for the mean-field compartmental populations with the mortality of walkers and delayed transitions among the compartments. From linear stability analysis, we derive the basic reproduction numbers RM,R0 with and without mortality, respectively, and prove that RM1, the healthy state is unstable, whereas for zero mortality, a stable endemic equilibrium exists (independent of the initial conditions), which we obtained explicitly. We observed that the solutions of the random walk simulations in the considered networks agree well with the mean-field solutions for strongly connected graph topologies, whereas less well for weakly connected structures and for diseases with high mortality. Our model has applications beyond epidemic dynamics, for instance in the kinetics of chemical reactions, the propagation of contaminants, wood fires, and others.

A Markovian random walk model of epidemic spreading.

We analyze the dynamics of a population of independent random walkers on a graph and develop a simple model of epidemic spreading. We assume that each walker visits independently the nodes of a finite ergodic graph in a discrete-time Markovian walk governed by his specific transition matrix. With this assumption, we first derive an upper bound for the reproduction numbers. Then, we assume that a walker is in one of the states: susceptible, infectious, or recovered. An infectious walker remains infectious during a certain characteristic time. If an infectious walker meets a susceptible one on the same node, there is a certain probability for the susceptible walker to get infected. By implementing this hypothesis in computer simulations, we study the space-time evolution of the emerging infection patterns. Generally, random walk approaches seem to have a large potential to study epidemic spreading and to identify the pertinent parameters in epidemic dynamics.

Four-compartment epidemic model with retarded transition rates.

We study an epidemic model for a constant population by taking into account four compartments of the individuals characterizing their states of health. Each individual is in one of the following compartments: susceptible S; incubated, i.e., infected yet not infectious, C; infected and infectious I; and recovered, i.e., immune, R. An infection is visible only when an individual is in state I. Upon infection, an individual performs the transition pathway S→C→I→R→S, remaining in compartments C, I, and R for a certain random waiting time t_{C}, t_{I}, and t_{R}, respectively. The waiting times for each compartment are independent and drawn from specific probability density functions (PDFs) introducing memory into the model. The first part of the paper is devoted to the macroscopic S-C-I-R-S model. We derive memory evolution equations involving convolutions (time derivatives of general fractional type). We consider several cases. The memoryless case is represented by exponentially distributed waiting times. Cases of long waiting times with fat-tailed waiting-time distributions are considered as well where the S-C-I-R-S evolution equations take the form of time-fractional ordinary differential equations. We obtain formulas for the endemic equilibrium and a condition of its existence for cases when the waiting-time PDFs have existing means. We analyze the stability of healthy and endemic equilibria and derive conditions for which the endemic state becomes oscillatory (Hopf) unstable. In the second part, we implement a simple multiple-random-walker approach (microscopic model of Brownian motion of Z independent walkers) with random S-C-I-R-S waiting times in computer simulations. Infections occur with a certain probability by collisions of walkers in compartments I and S. We compare the endemic states predicted in the macroscopic model with the numerical results of the simulations and find accordance of high accuracy. We conclude that a simple random-walker approach offers an appropriate microscopic description for the macroscopic model. The S-C-I-R-S-type models open a wide field of applications allowing the identification of pertinent parameters governing the phenomenology of epidemic dynamics such as extinction, convergence to a stable endemic equilibrium, or persistent oscillatory behavior.

Simple model of epidemic dynamics with memory effects.

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 S=susceptible, I=infected, or R=recovered (SIR model). In state 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.

Piezoacoustic wave spectra using improved surface impedance matrix: application to high impedance-contrast layered plates.

Starting from the general modal solutions for a homogeneous layer of arbitrary material and crystalline symmetry, a matrix formalism is developed to establish the semianalytical expressions of the surface impedance matrices (SIM) for a single piezoelectric layer. By applying the electrical boundary conditions, the layer impedance matrix is reduced to a unified elastic form whether the material is piezoelectric or not. The characteristic equation for the dispersion curves is derived in both forms of a three-dimensional acoustic SIM and of an electrical scalar function. The same approach is extended to multilayered structures such as a piezoelectric layer sandwiched in between two metallic electrodes, a Bragg coupler, and a semi-infinite substrate as well. The effectiveness of the approach is numerically demonstrated by its ability to determine the full spectra of guided modes, even at extremely high frequencies, in layered plates comprising up to four layers and three materials. Negative slope in f-k curve for some modes, asymptotic behavior at short wavelength regime, as well as wave confinement phenomena made evident by the numerical results are analyzed and interpreted in terms of the surface acoustic waves and of the interfacial waves in connection with the bulk waves in massive materials.

Piezoelectric Love waves on rotated Y-cut mm2 substrates.

Consider a layer consisting of a m3m dielectric crystal, with faces cut parallel to a symmetry plane. Then bond it onto a semi-infinite mm2 piezoelectric substrate. For a X- or Y-cut of the substrate, a Love wave can propagate in the resulting structure and the corresponding dispersion equation is derived analytically. It turns out that when the upper (free) face of the layer is metalized, a fully explicit treatment can also be conducted in the case of a Y-cut rotated about Z. In the case of a germanium layer over a potassium niobate substrate, the wave exists at any wavelength for X-and Y-cuts but this ceases to be the case for rotated cuts, with the appearance of forbidden ranges. By playing on the cut angle, the Love wave can be made to travel faster than, or slower than, or at the same speed as, the shear bulk wave of the layer. A by-product of the analysis is the derivation of the explicit secular equation for the Bleustein-Gulyaev wave in the substrate alone, which corresponds to an asymptotic behavior of the Love wave. The results are valid for other choices for the layer and for the substrate, provided they have the same, or more, symmetries.