A computational approach to identifiability analysis for a model of the propagation and control of COVID-19 in Chile.

A computational approach is adapted to analyze the parameter identifiability of a compartmental model. The model is intended to describe the progression of the COVID-19 pandemic in Chile during the initial phase in early 2020 when government declared quarantine measures. The computational approach to analyze the structural and practical identifiability is applied in two parts, one for synthetic data and another for some Chilean regional data. The first part defines the identifiable parameter sets when these recover the true parameters used to create the synthetic data. The second part compares the results derived from synthetic data, estimating the identifiable parameter sets from regional Chilean epidemic data. Experiments provide evidence of the loss of identifiability if some initial conditions are estimated, the period of time used to fit is before the peak, and if a significant proportion of the population is involved in quarantine periods.

An investigation of spatial-temporal patterns and predictions of the coronavirus 2019 pandemic in Colombia, 2020-2021.

Colombia announced the first case of severe acute respiratory syndrome coronavirus 2 on March 6, 2020. Since then, the country has reported a total of 5,002,387 cases and 127,258 deaths as of October 31, 2021. The aggressive transmission dynamics of SARS-CoV-2 motivate an investigation of COVID-19 at the national and regional levels in Colombia. We utilize the case incidence and mortality data to estimate the transmission potential and generate short-term forecasts of the COVID-19 pandemic to inform the public health policies using previously validated mathematical models. The analysis is augmented by the examination of geographic heterogeneity of COVID-19 at the departmental level along with the investigation of mobility and social media trends. Overall, the national and regional reproduction numbers show sustained disease transmission during the early phase of the pandemic, exhibiting sub-exponential growth dynamics. Whereas the most recent estimates of reproduction number indicate disease containment, with Rt<1.0 as of October 31, 2021. On the forecasting front, the sub-epidemic model performs best at capturing the 30-day ahead COVID-19 trajectory compared to the Richards and generalized logistic growth model. Nevertheless, the spatial variability in the incidence rate patterns across different departments can be grouped into four distinct clusters. As the case incidence surged in July 2020, an increase in mobility patterns was also observed. On the contrary, a spike in the number of tweets indicating the stay-at-home orders was observed in November 2020 when the case incidence had already plateaued, indicating the pandemic fatigue in the country.

Measuring differences between phenomenological growth models applied to epidemiology.

Phenomenological growth models (PGMs) provide a framework for characterizing epidemic trajectories, estimating key transmission parameters, gaining insight into the contribution of various transmission pathways, and providing long-term and short-term forecasts. Such models only require a small number of parameters to describe epidemic growth patterns. They can be expressed by an ordinary differential equation (ODE) of the type C'(t)=f(t,C;Θ) for t>0, C(0)=C0, where t is time, C(t) is the total size of the epidemic (the cumulative number of cases) at time t, C0 is the initial number of cases, f is a model-specific incidence function, and Θ is a vector of parameters. The current COVID-19 pandemic is a scenario for which such models are of obvious importance. In Bürger et al. (2019) it is demonstrated that some PGMs are better at fitting data of specific epidemic outbreaks than others even when the models have the same number of parameters. This situation motivates the need to measure differences in the dynamics that two different models are capable of generating. The present work contributes to a systematic study of differences between PGMs and how these may explain the ability of certain models to provide a better fit to data than others. To this end a so-called empirical directed distance (EDD) is defined to describe the differences in the dynamics between different dynamic models. The EDD of one PGM from another one quantifies how well the former fits data generated by the latter. The concept of EDD is, however, not symmetric in the usual sense of metric spaces. The procedure of calculating EDDs is applied to synthetic data and real data from influenza, Ebola, and COVID-19 outbreaks.

A non-local pedestrian flow model accounting for anisotropic interactions and domain boundaries.

This study revises the non-local macroscopic pedestrian flow model proposed in [R. M. Colombo, M. Garavello, and M. Lécureux-Mercier. A class of nonlocal models for pedestrian traffic. Math. Models Methods Appl. Sci., 22(4):1150023, 2012] to account for anisotropic interactions and the presence of walls or other obstacles in the walking domain. We prove the well-posedness of this extended model and we apply high-resolution numerical schemes to illustrate the model characteristics. In particular, numerical simulations highlight the role of different model parameters in the observed pattern formation.

Simulation and control of dissolved air flotation and column froth flotation with simultaneous sedimentation.

Flotation is a separation process where particles or droplets are removed from a suspension with the aid of floating gas bubbles. Applications include dissolved air flotation (DAF) in industrial wastewater treatment and column froth flotation (CFF) in wastewater treatment and mineral processing. One-dimensional models of flotation have been limited to steady-state situations for half a century by means of the drift-flux theory. A newly developed dynamic one-dimensional model formulated in terms of partial differential equations can be used to predict the process of simultaneous flotation of bubbles and sedimentation of particles that are not attached to bubbles. The governing model is a pair of first-order conservation laws for the aggregate and solids volume fractions as functions of height and time. An analysis of nonlinear ingredients of the governing equations helps to identify desired steady-state operating conditions. These can be chosen by means of operating charts, which are diagrams that visualize regions of admissible values of the volumetric flows of the feed input and underflow outlet. This is detailed for the DAF thickening process. Dynamic simulations are obtained with a recently developed numerical method. Responses to control actions are demonstrated with scenarios in CFF and DAF.

Comparative analysis of phenomenological growth models applied to epidemic outbreaks.

Phenomenological models are particularly useful for characterizing epidemic trajectories because they often offer a simple mathematical form defined through ordinary differential equations (ODEs) that in many cases can be solved explicitly. Such models avoid the description of biological mechanisms that may be difficult to identify, are based on a small number of model parameters that can be calibrated easily, and can be utilized for efficient and rapid forecasts with quantified uncertainty. These advantages motivate an in-depth examination of 37 data sets of epidemic outbreaks, with the aim to identify for each case the best suited model to describe epidemiological growth. Four parametric ODE-based models are chosen for study, namely the logistic and Gompertz model with their respective generalizations that in each case consists in elevating the cumulative incidence function to a power p ∈ [0,1]. This parameter within the generalized models provides a criterion on the early growth behavior of the epidemic between constant incidence for p = 0, sub-exponential growth for 0 < p < 1 and exponential growth for p = 1. Our systematic comparison of a number of epidemic outbreaks using phenomenological growth models indicates that the GLM model outperformed the other models in describing the great majority of the epidemic trajectories. In contrast, the errors of the GoM and GGoM models stay fairly close to each other and the contribution of the adjustment of p remains subtle in some cases. More generally, we also discuss how this methodology could be extended to assess the "distance" between models irrespective of their complexity.

Numerical solution of a spatio-temporal gender-structured model for hantavirus infection in rodents.

In this article we describe the transmission dynamics of hantavirus in rodents using a spatio-temporal susceptible-exposed-infective-recovered (SEIR) compartmental model that distinguishes between male and female subpopulations [L.J.S. Allen, R.K. McCormack and C.B. Jonsson, Bull. Math. Biol. 68 (2006), 511--524]. Both subpopulations are assumed to differ in their movement with respect to local variations in the densities of their own and the opposite gender group. Three alternative models for the movement of the male individuals are examined. In some cases the movement is not only directed by the gradient of a density (as in the standard diffusive case), but also by a non-local convolution of density values as proposed, in another context, in [R.M. Colombo and E. Rossi, Commun. Math. Sci., 13 (2015), 369--400]. An efficient numerical method for the resulting convection-diffusion-reaction system of partial differential equations is proposed. This method involves techniques of weighted essentially non-oscillatory (WENO) reconstructions in combination with implicit-explicit Runge-Kutta (IMEX-RK) methods for time stepping. The numerical results demonstrate significant differences in the spatio-temporal behavior predicted by the different models, which suggest future research directions.

Numerical solution of a spatio-temporal predator-prey model with infected prey.

A spatio-temporal eco-epidemiological model is formulated by combining an available non-spatial model for predator-prey dynamics with infected prey [D. Greenhalgh and M. Haque, Math. Meth. Appl. Sci., 30 (2007), 911-929] with a spatio-temporal susceptible-infective (SI)-type epidemic model of pattern formation due to diffusion [G.-Q. Sun, Nonlinear Dynamics, 69 (2012), 1097-1104]. It is assumed that predators exclusively eat infected prey, in agreement with the hypothesis that the infection weakens the prey, making it available for predation otherwise we assume that the predator has essentially no access to healthy prey of the same species. Furthermore, the movement of predators is described by a non-local convolution of the density of infected prey as proposed in [R.M. Colombo and E. Rossi, Commun. Math. Sci., 13 (2015), 369-400]. The resulting convection-diffusion-reaction system of three partial differential equations for the densities of susceptible and infected prey and predators is solved by an efficient method that combines weighted essentially non-oscillatory (WENO) reconstructions and an implicit-explicit Runge-Kutta (IMEX-RK) method for time stepping. Numerical examples illustrate the formation of spatial patterns involving all three species.

Concentration-driven models revisited: towards a unified framework to model settling tanks in water resource recovery facilities.

A new perspective on the modelling of settling behaviour in water resource recovery facilities is introduced. The ultimate goal is to describe in a unified way the processes taking place both in primary settling tanks (PSTs) and secondary settling tanks (SSTs) for a more detailed operation and control. First, experimental evidence is provided, pointing out distributed particle properties (such as size, shape, density, porosity, and flocculation state) as an important common source of distributed settling behaviour in different settling unit processes and throughout different settling regimes (discrete, hindered and compression settling). Subsequently, a unified model framework that considers several particle classes is proposed in order to describe distributions in settling behaviour as well as the effect of variations in particle properties on the settling process. The result is a set of partial differential equations (PDEs) that are valid from dilute concentrations, where they correspond to discrete settling, to concentrated suspensions, where they correspond to compression settling. Consequently, these PDEs model both PSTs and SSTs.

On constitutive functions for hindered settling velocity in 1-D settler models: Selection of appropriate model structure.

Advanced 1-D models for Secondary Settling Tanks (SSTs) explicitly account for several phenomena that influence the settling process (such as hindered settling and compression settling). For each of these phenomena a valid mathematical expression needs to be selected and its parameters calibrated to obtain a model that can be used for operation and control. This is, however, a challenging task as these phenomena may occur simultaneously. Therefore, the presented work evaluates several available expressions for hindered settling based on long-term batch settling data. Specific attention is paid to the behaviour of these hindered settling functions in the compression region in order to evaluate how the modelling of sludge compression is influenced by the choice of a certain hindered settling function. The analysis shows that the exponential hindered settling forms, which are most commonly used in traditional SST models, not only account for hindered settling but partly lump other phenomena (compression) as well. This makes them unsuitable for advanced 1-D models that explicitly include each phenomenon in a modular way. A power-law function is shown to be more appropriate to describe the hindered settling velocity in advanced 1-D SST models.

On time discretizations for the simulation of the batch settling-compression process in one dimension.

The main purpose of the recently introduced Bürger-Diehl simulation model for secondary settling tanks was to resolve spatial discretization problems when both hindered settling and the phenomena of compression and dispersion are included. Straightforward time integration unfortunately means long computational times. The next step in the development is to introduce and investigate time-integration methods for more efficient simulations, but where other aspects such as implementation complexity and robustness are equally considered. This is done for batch settling simulations. The key findings are partly a new time-discretization method and partly its comparison with other specially tailored and standard methods. Several advantages and disadvantages for each method are given. One conclusion is that the new linearly implicit method is easier to implement than another one (semi-implicit method), but less efficient based on two types of batch sedimentation tests.

Modelling the spatial-temporal progression of the 2009 A/H1N1 influenza pandemic in Chile.

A spatial-temporal transmission model of 2009 A/H1N1 pandemic influenza across Chile, a country that spans a large latitudinal range, is developed to characterize the spatial variation in peak timing of that pandemic as a function of local transmission rates, spatial connectivity assumptions for Chilean regions, and the putative location of introduction of the novel virus into the country. Specifically, a metapopulation SEIR (susceptible-exposed-infected-removed) compartmental model that tracks the transmission dynamics of influenza in 15 Chilean regions is calibrated. The model incorporates population mobility among neighboring regions and indirect mobility to and from other regions via the metropolitan central region ('hub region'). The stability of the disease-free equilibrium of this model is analyzed and compared with the corresponding stability in each region, concluding that stability may occur even with some regions having basic reproduction numbers above 1. The transmission model is used along with epidemiological data to explore potential factors that could have driven the spatial-temporal progression of the pandemic. Simulations and sensitivity analyses indicate that this relatively simple model is sufficient to characterize the south-north gradient in peak timing observed during the pandemic, and suggest that south Chile observed the initial spread of the pandemic virus, which is in line with a retrospective epidemiological study. The 'hub region' in our model significantly enhanced population mixing in a short time scale.

A consistent modelling methodology for secondary settling tanks: a reliable numerical method.

The consistent modelling methodology for secondary settling tanks (SSTs) leads to a partial differential equation (PDE) of nonlinear convection-diffusion type as a one-dimensional model for the solids concentration as a function of depth and time. This PDE includes a flux that depends discontinuously on spatial position modelling hindered settling and bulk flows, a singular source term describing the feed mechanism, a degenerating term accounting for sediment compressibility, and a dispersion term for turbulence. In addition, the solution itself is discontinuous. A consistent, reliable and robust numerical method that properly handles these difficulties is presented. Many constitutive relations for hindered settling, compression and dispersion can be used within the model, allowing the user to switch on and off effects of interest depending on the modelling goal as well as investigate the suitability of certain constitutive expressions. Simulations show the effect of the dispersion term on effluent suspended solids and total sludge mass in the SST. The focus is on correct implementation whereas calibration and validation are not pursued.

A consistent modelling methodology for secondary settling tanks in wastewater treatment.

The aim of this contribution is partly to build consensus on a consistent modelling methodology (CMM) of complex real processes in wastewater treatment by combining classical concepts with results from applied mathematics, and partly to apply it to the clarification-thickening process in the secondary settling tank. In the CMM, the real process should be approximated by a mathematical model (process model; ordinary or partial differential equation (ODE or PDE)), which in turn is approximated by a simulation model (numerical method) implemented on a computer. These steps have often not been carried out in a correct way. The secondary settling tank was chosen as a case since this is one of the most complex processes in a wastewater treatment plant and simulation models developed decades ago have no guarantee of satisfying fundamental mathematical and physical properties. Nevertheless, such methods are still used in commercial tools to date. This particularly becomes of interest as the state-of-the-art practice is moving towards plant-wide modelling. Then all submodels interact and errors propagate through the model and severely hamper any calibration effort and, hence, the predictive purpose of the model. The CMM is described by applying it first to a simple conversion process in the biological reactor yielding an ODE solver, and then to the solid-liquid separation in the secondary settling tank, yielding a PDE solver. Time has come to incorporate established mathematical techniques into environmental engineering, and wastewater treatment modelling in particular, and to use proven reliable and consistent simulation models.