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 parameterR 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 ofR 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 andR 0 . Preliminary analysis shows a similar behavior for the effect of stochastic advection.

Drying in porous media with gravity-stabilized fronts: experimental results

In a recent paper [Yiotis et al., Phys. Rev. E 85, 046308 (2012)] we developed a model for the drying of porous media in the presence of gravity. It incorporated effects of corner film flow, internal and external mass transfer, and the effect of gravity. Analytical results were derived when gravity opposes drying and hence leads to a stable percolation drying front. In this paper, we test the theory using laboratory experiments. A series of isothermal drying experiments in glass bead packings saturated with volatile hydrocarbons is conducted. The transparent glass cells containing the packing allow for the visual monitoring of the phase distribution patterns below the surface, including the formation of liquid films, as the gaseous phase invades the pore space, and for the control of the thickness of the diffusive mass boundary layer over the packing. The experimental results agree very well with theory, provided that the latter is generalized to account for the effects of corner roundness in the film region (which was neglected in the theoretical part). We demonstrate the existence of an early constant rate period (CRP), which lasts as long as the films saturate the surface of the packing, and of a subsequent falling rate period (FRP), which begins practically after the detachment of the film tips from the external surface. During the CRP, the process is controlled by diffusion within the stagnant gaseous phase in the upper part of the cells, yielding a Stefan tube problem solution. During the FRP, the process is controlled by diffusion within the packing, with a drying rate inversely proportional to the observed position of the film tips in the cell. Theoretical and experimental results compare favorably for a specific value of the roundness of the films, which is found to be constant and equal to 0.2 for various conditions, and verify the theoretical dependence on the capillary Ca(f), Bond Bo, and Sherwood Sh numbers.

Analytical solutions of drying in porous media for gravity-stabilized fronts

We develop a mathematical model for the drying of porous media in the presence of gravity. The model incorporates effects of corner flow through macroscopic liquid films that form in the cavities of pore walls, mass transfer by diffusion in the dry regions of the medium, external mass transfer over the surface, and the effect of gravity. We consider two different cases: when gravity opposes liquid flow in the corner films and leads to a stable percolation drying front, and when it acts in the opposite direction. In this part, we develop analytical results when the problem can be cast as an equivalent continuum and described as a one-dimensional (1D) problem. This is always the case when gravity acts against drying by opposing corner flow, or when it enhances drying by increasing corner film flow but it is sufficiently small. We obtain results for all relevant variables, including drying rates, extent of the macroscopic film region, and the demarkation of the two different regimes of constant rate period and falling rate period, respectively. The effects of dimensionless variables, such as the bond number, the capillary number, and the Sherwood number for external mass transfer are investigated. When gravity acts to enhance drying, a 1D solution is still possible if an appropriately defined Rayleigh number is above a critical threshold. We derive a linear stability analysis of a model problem under this condition that verifies front stability. Further analysis of this problem, when the Rayleigh number is below critical, requires a pore-network simulator which will be the focus of future work.

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.

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.

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.

Crossing the elliptic region in a hyperbolic system with change-of-type behavior arisingin flow between two parallel plates

Change-of-type behavior from hyperbolic to elliptic is common to quasilinear hyperbolic systems. This issue is addressed here for the particular case of miscible flow of three fluids between two parallel plates. Change of type occurs at the leading edge of the displacement front and reflects the failing of the equilibrium assumption, necessary for the quasilinear hyperbolic formalism, at the front. To cross the elliptic region requires the solution of the full, higher-dimensionality problem, obtained here using lattice gas simulations. For the specific example, it is found that the system self-selects a front structure independent of injection conditions.

Mixing and reaction fronts in laminar flows

Autocatalytic reaction fronts between unreacted and reacted mixtures in the absence of fluid flow propagate as solitary waves. In the presence of imposed flow, the interplay between diffusion and advection enhances the mixing, leading to Taylor hydrodynamic dispersion. We present asymptotic theories in the two limits of small and large Thiele modulus (slow and fast reaction kinetics, respectively) that incorporate flow, diffusion, and reaction. For the first case, we show that the problem can be handled to leading order by the introduction of the Taylor dispersion replacing the molecular diffusion coefficient by its Taylor counterpart. In the second case, the leading-order behavior satisfies the eikonal equation. Numerical simulations using a lattice gas model show good agreement with the theory. The Taylor model is relevant to microfluidics applications, whereas the eikonal model applies at larger length scales.

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).

Effect of liquid films on the isothermal drying of porous media

We study the effects of liquid films on the isothermal drying of porous media. They are important for the transport of liquid to an evaporation interface, far from the receding liquid clusters. Through a transformation, the drying problem is mapped to the Laplace equation around the percolation liquid clusters. From its solution, the properties of drying are obtained in terms of the capillary number. Consistent with experimental evidence, film flow is shown to accelerate drying significantly.

Nonlocal Kardar-Parisi-Zhang equation to model interface growth

The dynamics of the growth of interfaces in the presence of noise and when the normal velocity is constant, in the weakly nonlinear limit, are described by the Kardar-Parisi-Zhang (KPZ) equation. In many applications, however, the growth is controlled by nonlocal transport, which is not contained in the original KPZ equation. For these problems we are proposing an extension of the KPZ model, where the nonlocal contribution is expressed through a Hilbert transform and can act to either stabilize or destabilize the interface. The model is illustrated with a specific example from reactive infiltration. The properties of the solution of the resulting equation are studied in one spatial dimension in the linear and the nonlinear limits, for both stable and unstable growth. We find that the early-time behavior has a power-law scaling similar to that of the KPZ equation. However, in the case of stable growth, the scaling of the saturation width is logarithmic, which differs from the power law in the KPZ equation. This dependence reflects the stabilizing effect of nonlocal transport. In the unstable case, we obtain results similar to those of Olami et al. [Phys. Rev. E 55, 2649 (1997)].

Identification of the permeability field of a porous medium from the injection of a passive tracer

We propose a method for the direct inversion of the permeability field of a porous medium from the analysis of the displacement of a passive tracer. By monitoring the displacement front at successive time intervals (for example, using a tomographic method), the permeability can be directly obtained from the solution of a nonlinear boundary-value problem. Well posedness requires knowledge of the pressure profile or the permeability at no-flow boundaries. The method is tested using synthetic data in two dimensions (2D) (and some 3D) geometries for a variety of heterogeneous fields and found to work well when the permeability contrast is not too large. However, it is sensitive to sharp variations in permeability. In the latter case, a modified approach based on the successive injection in both directions and the use of an optimization technique leads to improved estimates. The sensitivity to measurement errors is analyzed. An important feature of the direct method is that it also applies to anisotropic porous media. When the principal axes of anisotropy are known, a suitable procedure is proposed and demonstrated using synthetic data.