Anja Slim

Edge-based compartmental modelling for infectious disease spread

The primary tool for predicting infectious disease spread and intervention effectiveness is the mass action susceptible–infected–recovered model of Kermack & McKendrick. Its usefulness derives largely from its conceptual and mathematical simplicity; however, it incorrectly assumes that all individuals have the same contact rate and partnerships are fleeting. In this study, we introduce edge-based compartmental modelling, a technique eliminating these assumptions. We derive simple ordinary differential equation models capturing social heterogeneity (heterogeneous contact rates) while explicitly considering the impact of partnership duration. We introduce a graphical interpretation allowing for easy derivation and communication of the model and focus on applying the technique under different assumptions about how contact rates are distributed and how long partnerships last.

Onset and cessation of time-dependent, dissolution-driven convection in porous media

Motivated by convection in the context of geological carbon dioxide sequestration, we present the conditions for free, dissolution-driven convection in a horizontal, ideal porous layer from a time-dependent, pure-diffusion base state. We assume that solute as a separate phase is instantaneously placed in the pores above a given horizontal level at time zero, and gradually diffuses into the underlying liquid. As the concentration of dissolved solute in the liquid increases, its density increases and the system may eventually become gravitationally unstable and convection may begin. We define the amplitude of a perturbation as the mean square of the difference of the concentration profile and the pure-diffusion profile. To identify instability, we calculate the maximum possible instantaneous growth rate of the amplitude over all possible infinitesimal and finite perturbations. Instability exists where this growth rate is positive. We consider two scenarios. In the first scenario, the underlying liquid canno...

Dissolution-driven convection in a Hele-Shaw cell

Motivated by convection in the context of geological carbon-dioxide (CO2) storage, we present an experimental study of dissolution-driven convection in a Hele–Shaw cell for Rayleigh numbers R in the range 100 < R < 1700. We use potassium permanganate (KMnO4) in water as an analog for CO2 in brine and infer concentration profiles at high spatial and temporal resolution and accuracy from transmitted light intensity. We describe behavior from first contact up to 65% average saturation and measure several global quantities including dissolution flux, average concentration, amplitude of perturbations away from pure one-dimensional diffusion, and horizontally averaged concentration profiles. We show that the flow evolves successively through distinct regimes starting with a simple one-dimensional diffusional profile. This is followed by linear growth in which fingers are initiated and grow quasiexponentially, independently of one-another. Once the fingers are well-established, a flux-growth regime begins as fresh fluid is brought to the interface and contaminated fluid removed, with the flux growing to a local maximum. During this regime, fingers still propagate independently. However, beyond the flux maximum, fingers begin to interact and zip together from the root down in a merging regime. Several generations of merging occur before only persistent primary fingers remain. Beyond this, the reinitiation regime begins with new fingers created between primary existing ones before merging into them. Through appropriate scaling, we show that the regimes are universal and independent of layer thickness (equivalently R) until the fingers hit the bottom. At this time, progression through these regimes is interrupted and the flow transitions to a saturating regime. In this final regime, the flux gradually decays in a manner well described by a Howard-style phenomenological model.

Surface wrinkling of a channelized flow

We consider an inclined rectangular duct of constant cross-section conveying viscous fluid and covered by an elastic plate. The fluid is described by the Stokes equations and the plate by the Föppl–von Kármán equations. The equations admit an equilibrium solution in which the plate is flat and fluid flows underneath due to gravity. This base flow induces a varying traction across the plate, which can lead to out-of-plane buckling due to the associated in-plane shear. Linear stability analysis demonstrates that buckling occurs for sufficiently thin plates on steep slopes and deep channels. The most unstable modes take the form of either symmetric or antisymmetric downslope-directed chevron patterns that travel downstream at a fraction of the average speed of the base flow. An analogous analysis shows that similar buckling also occurs if the elastic plate is replaced by a thin skin of very viscous fluid. Our description provides a simple model for the formation of ropy pahoehoe lava.

Gravity currents from a line source in an ambient flow

We present a mainly theoretical study of high-Reynolds-number planar gravity currents in a uniformly flowing deep ambient. The gravity currents are generated by a constant line source of fluid, and may also be supplied with a source of horizontal momentum and a source of particles. We model the motion using a shallow-water approximation and represent the effects of the ambient flow by imposing a Froude-number condition in a moving frame. We present analytic and numerical expressions for the threshold ambient flow speed above which no upstream propagation can occur at long times. For homogeneous gravity currents in an ambient flow below threshold, we find similarity solutions in which the up- and downstream fronts spread at a constant rate and the current propagates indefinitely in both directions. For gravity currents consisting of both interstitial fluid of a different density to the ambient and a sedimenting particle load, we find long-time asymptotic solutions for ambient flow strengths below threshold. These consist of a steady particle-rich near-source region, in which settling and advection of particles balance, and an effectively particle-free frontal region. The homogeneous behaviour of the fronts ensures that they also spread at a constant rate and therefore can propagate upstream indefinitely. For gravity currents driven solely by a sedimenting particle load, we find numerically that a single regime exists for ambient flow strengths below threshold. In these solutions, settling balances advection near the source leading to a steady region, which joins on to a complex frontal boundary layer. The upstream front progressively decelerates. Our solutions for homogeneous and particle-driven gravity currents compare well with published experimental results.

Self-similar solutions of the axisymmetric shallow-water equations governing converging inviscid gravity currents

A phase-plane approach is used to determine similarity solutions of the axisymmetric shallow-water equations which represent inwardly propagating, inviscid gravity currents. A Froude number condition characterizes the movement of the front. The unique similarity exponent is found numerically as a function of the frontal Froude number and the height and velocity profiles are presented for three different Froude numbers. The fluid speed and height are seen to increase monotonically towards the front except very close to the front where the height decreases. The maxima in both height and speed increase as the Froude number increases, reflecting the change in ambient resistance. For the Froude number that has been obtained experimentally for lock-exchange Boussinesq flows ( Fr =1 .19) for which the similarity exponent is 0.859094, the similarity solution is compared to the numerical solution of the initial value problem, obtained recently by Hallworth, Huppert & Ungarish (2003). Our similarity solution compares reasonably well with their integration of the shallow-water equations in the neighbourhood of the front and at times close to collapse (when the front reaches the origin); however, near this point their numerics begin to fail. The solution at collapse and the similarity solution after collapse are also found for Fr =1 .19. This similarity solution describes the formation of a shock, as well as its initial propagation.

Saturation effects and the concurrency hypothesis: Insights from an analytic model

The existence of sexual partnerships that overlap in time (concurrent relationships) is believed by some to be a significant contributing factor to the spread of HIV, although this is controversial. We derive an analytic model which allows us to investigate and compare disease spread in populations with and without concurrency. We can identify regions of parameter space in which its impact is negligible, and other regions in which it plays a major role. We also see that the impact of concurrency on the initial growth phase can be much larger than its impact on the equilibrium size. We see that the effect of concurrency saturates, which leads to the perhaps surprising conclusion that interventions targeting concurrency may be most effective in populations with low to moderate levels of concurrency.Sexual partnerships that overlap in time (concurrent relationships) may play a significant role in the HIV epidemic, but the precise effect is unclear. We derive edge-based compartmental models of disease spread in idealized dynamic populations with and without concurrency to allow for an investigation of its effects. Our models assume that partnerships change in time and individuals enter and leave the at-risk population. Infected individuals transmit at a constant per-partnership rate to their susceptible partners. In our idealized populations we find regions of parameter space where the existence of concurrent partnerships leads to substantially faster growth and higher equilibrium levels, but also regions in which the existence of concurrent partnerships has very little impact on the growth or the equilibrium. Additionally we find mixed regimes in which concurrency significantly increases the early growth, but has little effect on the ultimate equilibrium level. Guided by model predictions, we discuss general conditions under which concurrent relationships would be expected to have large or small effects in real-world settings. Our observation that the impact of concurrency saturates suggests that concurrency-reducing interventions may be most effective in populations with low to moderate concurrency.