A Model for Reinfections and the Transition of Epidemics.

Reinfections of infected individuals during a viral epidemic contribute to the continuation of the infection for longer periods of time. In an epidemic, contagion starts with an infection wave, initially growing exponentially fast until it reaches a maximum number of infections, following which it wanes towards an equilibrium state of zero infections, assuming that no new variants have emerged. If reinfections are allowed, multiple such infection waves might occur, and the asymptotic equilibrium state is one in which infection rates are not negligible. This paper analyzes such situations by expanding the traditional SIR model to include two new dimensionless parameters, ε and θ, characterizing, respectively, the kinetics of reinfection and a delay time, after which reinfection commences. We find that depending on these parameter values, three different asymptotic regimes develop. For relatively small θ, two of the regimes are asymptotically stable steady states, approached either monotonically, at larger ε (corresponding to a stable node), or as waves of exponentially decaying amplitude and constant frequency, at smaller ε (corresponding to a spiral). For θ values larger than a critical, the asymptotic state is a periodic pattern of constant frequency. However, when ε is sufficiently small, the asymptotic state is a wave. We delineate these regimes and analyze the dependence of the corresponding population fractions (susceptible, infected and recovered) on the two parameters ε and θ and on the reproduction number R0. The results provide insights into the evolution of contagion when reinfection and the waning of immunity are taken into consideration. A related byproduct is the finding that the conventional SIR model is singular at large times, hence the specific quantitative estimate for herd immunity it predicts will likely not materialize.

A comprehensive spatial-temporal infection model.

Motivated by analogies between the spread of infections and of chemical processes, we develop a model that accounts for infection and transport where infected populations correspond to chemical species. Areal densities emerge as the key variables, thus capturing the effect of spatial density. We derive expressions for the kinetics of the infection rates, and for the important parameter R 0 , that include areal density and its spatial distribution. We present results for a batch reactor, the chemical process equivalent of the SIR model, where we examine how the dependence of R 0 on process extent, the initial density of infected individuals, and fluctuations in population densities effect the progression of the disease. We then consider spatially distributed systems. Diffusion generates traveling waves that propagate at a constant speed, proportional to the square root of the diffusivity and R 0 . Preliminary analysis shows a similar behavior for the effect of stochastic advection.

Pore-network study of the mechanisms of foam generation in porous media.

Understanding the role of pore-level mechanisms is essential to the mechanistic modeling and simulation of foam processes in porous media. Three different pore-level events can lead to foam formation: snapoff, leave behind, and lamella division. The initial state of the porous medium (fully saturated with liquid or already partially drained), as surfactant is introduced, also affects the different foam-generation mechanisms. Bubbles created by any of these mechanisms cause the formation of new bubbles by snapoff and leave behind as gas drains liquid-saturated pores. Lamellae are stranded unless the pressure gradient is sufficient to mobilize those that have been created. To appreciate the roles of these mechanisms, their interaction at the pore-network level was studied. We report an extensive pore-network study that incorporates the above pore-level mechanisms, as foam is created by drainage or by the continuous injection of gas and liquid in porous media. Pore networks with up to 10 000 pores are considered. The study explores the roles of the pore-level events, and by implication, the appropriate form of the foam-generation function for mechanistic foam simulation. Results are compared with previous studies. In particular, the network simulations reconcile an apparent contradiction in the foam-generation model of Rossen and Gauglitz [AIChE J. 36, 1176 (1990)], and identify how foam is created near the inlet of the porous medium when lamella division controls foam generation. In the process, we also identify a new mechanism of snap-off and foam generation near the inlet of the medium.

Pore-network study of the characteristic periods in the drying of porous materials.

We study the periods that develop in the drying of capillary porous media, particularly the constant rate (CRP) and the falling rate (FRP) periods. Drying is simulated with a 3-D pore-network model that accounts for the effect of capillarity and buoyancy at the liquid-gas interface and for diffusion through the porous material and through a boundary layer over the external surface of the material. We focus on the stabilizing or destabilizing effects of gravity on the shape of the drying curve and the relative extent of the various drying periods. The extents of CRP and FRP are directly associated with various transition points of the percolation theory, such as the breakthrough point and the main liquid cluster disconnection point. Our study demonstrates that when an external diffusive layer is present, the constant rate period is longer.

Pattern formation in reverse filtration combustion.

Using a pore-network simulator we study pattern formation in reverse filtration combustion in porous media. The two-dimensional pore network includes all relevant pore-level mechanisms, including heat transfer through the pore space and the solid matrix, fluid and mass transfer through the pore space, and reaction kinetics of a solid fuel embedded in the pores. Both adiabatic and nonadiabatic cases are considered, the latter modeled with the inclusion of heat losses from the pore network to the ambient. The simulation results show the development of unstable, fingered patterns of the burned fuel, similar to previously reported in the literature in the related problem of reverse combustion in a Hele-Shaw cell. We study the sensitivity of the patterns obtained on a number of parameters, including the Peclet number. The results on finger spacing and finger width are analyzed in terms of a selection principle, similar to that used in the theory for unstable Laplacian growth.

The critical gas saturation in a porous medium in the presence of gravity.

The critical gas saturation, S(gc), denotes the volume fraction of the gas phase at the onset of bulk gas flow during the depressurization of a supersaturated liquid in a porous medium. In the absence of gradients due to viscous or gravity forces, S(gc) is controlled by nucleation, capillary forces, and the rate of decline of the supersaturation. In this paper we address one important additional effect, that of buoyancy. We use 2-D pore-network simulations, based on invasion percolation in a gradient (IPG), and corresponding scaling relations to obtain the dependence of S(gc) on the gravity Bond number, B, under conditions of slow growth, namely when mass transfer is sufficiently fast. The critical gas saturation approaches two plateau values at low and high Bond numbers. In the in-between region it scales as a power law of B, which for a 2-D lattice is S(gc) approximately B(-0.91).