Deterministic, stochastic, and mean-field PDE models in neuroscience.

Experimental results have shown that anti-cancer therapies, such as radiotherapy and chemotherapy, can modulate the cell cycle and generate cell cycle phase-dependent responses. As a result, obtaining a detailed understanding of the cell cycle is one possible path towards improving the efficacy of many of these therapies. Here, we consider a basic structured partial differential equation (PDE) model for cell progression through the cell cycle, and derive expressions for key quantities, such as the population growth rate and cell phase proportions. These quantities are shown to be periodic and, as such, we compare the PDE model to a corresponding ordinary differential equation (ODE) model in which the parameters are linked by ensuring that the long-term ODE behaviour agrees with the average PDE behaviour. By design, we find that the ODE model does an excellent job of representing the mean dynamics of the PDE model within just a few cell cycles. However, by probing the parameter space we find cases in which this mean behaviour is not a good measure of the PDE population growth. Our analytical comparison of two caricature models (one PDE and one ODE system) provides insight into cases in which the simple ODE model is an appropriate approximation to the PDE model.

In this work, we propose novel epidemic models (named, susceptible–infected–recovered–susceptible-bacteria) for cholera dynamics by incorporating a general formulation of bacteria growth and spatial variation. In the first part, a generalized ordinary differential equation (ODE) model is presented and it is found that bacterial growth contributes to the increase in the basic reproduction number, . With the derived basic reproduction number, we analyse the local and global dynamics of the model. Particularly, we give a rigorous proof on the endemic global stability by employing the geometric approach. In the second part, we extend the ODE model to a partial differential equation (PDE) model with the inclusion of diffusion to capture the movement of human hosts and bacteria in a heterogeneous environment. The disease threshold of this PDE model is studied again by using the basic reproduction number. The results on the threshold dynamics of the ODE and PDE models are compared, and verified through numerical simulation. Additionally, our analysis shows that incorporating diffusive spatial spread does not produce a Turing instability when associated with the ODE model is less than the unity.

In this work, we construct a 2-dimensional (2D) stochastic partial differential equation (PDE) model for a thin film deposition process and design a multivariable predictive controller based on the constructed model to control thin film thickness and surface roughness. We focus on a thin film deposition process governed by three microscopic processes including molecule adsorption, migration and desorption. A 2D linear stochastic PDE model is initially constructed following the methodology proposed in our previous work [Ni, D. and Christofides, P. D., 2005]. Then, a stochastic PDE model-based multivariable controller is designed using the constructed stochastic PDE model. The control problem is formulated as a predictive control problem, in which the constructed stochastic PDE model is used to predict both the thin film thickness and the surface roughness. Moreover, the controller design is performed based on a finite stochastic ordinary differential equation (ODE) approximation of the stochastic PDE model to achieve high computational efficiency. The model-based predictive controller is applied to the kinetic Monte-Carlo (kMC) simulation of the deposition process to simultaneously control the thin film thickness and surface roughness. Closed-loop system simulation results demonstrate that the model is adequately accurate and that the controller is effective.

This work focuses on the global switching temperature control for microwave heating system with the input and output constraints. The traditional partial differential equation (PDE) model contains a spatial differential operator whose eigenspectrum can be divided into an infinite dimensional complement and finite dimensional one. This feature indicates that the existence of finite dimensional ordinary differential equation (ODE) model can approximately captures the dominant dynamics of the initial PDE model. The global switching controller with input constraints is proposed based on the ODE model to regulate the dynamic characteristics of traditional PDE model. Moreover, the exponential stability of closed-loop temperature system is also discussed and proved using the semigroup characteristics. The synthetic controller is implemented on a one-dimensional cavity model and the extensive numerical simulations demonstrate the controller performance and stability.

Mathematical models to study quorum sensing (QS) have become an important tool to explore all aspects of this type of bacterial communication. A wide spectrum of mathematical tools and methods such as dynamical systems, stochastics, and spatial models can be employed. In this chapter, we focus on giving an overview of models consisting of differential equations (DE), which can be used to describe changing quantities, for example, the dynamics of one or more signaling molecule in time and space, often in conjunction with bacterial growth dynamics. The chapter is divided into two sections: ordinary differential equations (ODE) and partial differential equations (PDE) models of QS. Rates of change are represented mathematically by derivatives, i.e., in terms of DE. ODE models allow describing changes in one independent variable, for example, time. PDE models can be used to follow changes in more than one independent variable, for example, time and space. Both types of models often consist of systems (i.e., more than one equation) of equations, such as equations for bacterial growth and autoinducer concentration dynamics. Almost from the onset, mathematical modeling of QS using differential equations has been an interdisciplinary endeavor and many of the works we revised here will be placed into their biological context.

In order to achieve prescribed drug release kinetics some authors have been investigating bi-phasic and possibly multi-phasic releases from blends of biodegradable polymers. Recently, experimental data for the release of paclitaxel have been published by Lao et al. (Lao and Venkatraman in J. Control. Release 130:9---14, 2008; Lao et al. in Eur. J. Pharm. Biopharm. 70:796---803, 2008). In Blanchet et al. (SIAM J. Appl. Math. 71(6):2269---2286, 2011) we validated a two-parameter quadratic ordinary differential equation (ODE) model against their experimental data from three representative neat polymers. In this paper we provide a gradient flow interpretation of the ODE model. A three-dimensional partial differential equation (PDE) model for the drug release in their experimental set up is introduced and its parameters are related to the ones of the ODE model. The gradient flow interpretation is extended to the study of the asymptotic concentrations that are solutions of the PDE model to determine the range of parameters that are suitable to simulate complete or partial drug release.

In the traditional microwave heating partial differential equation (PDE) model, one of the main characteristics is the infinite-dimensional nature, which does not allow to readily design and implement a controller. Motivated by this obstruction, this paper proposes a microwave heating finite-dimensional ordinary differential equation (ODE) model, which can not only describe the thermodynamics field with nonhomogeneous boundary conditions but also be coupled with the variation of electromagnetic field in temperature-dependent dielectric media. Initially, the equivalent PDE model with a homogeneous boundary condition is derived by constructing an auxiliary function in order to directly derive the eigenspectrum of the spatial differential operator. With the help of model-reduction techniques, the dominant dynamics of temperature distribution are subsequently captured with a reasonable Galerkin truncation. The simulation results on microwave heating a water prototype show that the temporal and the spatial evolution of the temperature profile can be described by solving the temperature-dependent electromagnetic field and the finite-dimensional ODE model. Moreover, the effectiveness of the model is verified by comparing with the numerical results from the traditional COMSOL model. A further development of this ODE model may provide a useful numerical tool for the design and synthesis of microwave heaters to avoid thermal runaway phenomena.

We find concrete evidence for a recently discovered form of intermittency, referred to as in–out intermittency, in both partial differential equation (PDE) and ordinary differential equation (ODE) models of mean field dynamos. This type of intermittency [introduced in P. Ashwin, E. Covas, and R. Tavakol, Nonlinearity 9, 563 (1999)] occurs in systems with invariant submanifolds and, as opposed to on–off intermittency which can also occur in skew product systems, it requires an absence of skew product structure. By this we mean that the dynamics on the attractor intermittent to the invariant manifold cannot be expressed simply as the dynamics on the invariant subspace forcing the transverse dynamics; the transverse dynamics will alter that tangential to the invariant subspace when one is far enough away from the invariant manifold. Since general systems with invariant submanifolds are not likely to have skew product structure, this type of behavior may be of physical relevance in a variety of dynamical settings. The models employed here to demonstrate in–out intermittency are axisymmetric mean-field dynamo models which are often used to study the observed large-scale magnetic variability in the Sun and solar-type stars. The occurrence of this type of intermittency in such models may be of interest in understanding some aspects of such variabilities.

ABSTRACTThis paper introduces a novel partial differential equation immuno-eco-epidemiological model of competition in which one species is affected by a disease while another can compete with it directly and by lowering the first species' immune response to the infection, a mode of competition termed stress-induced competition. When the disease is chronic, and the within-host dynamics are rapid, we reduce the partial differential equation model (PDE) to a three-dimensional ordinary differential equation (ODE) model. The ODE model exhibits backward bifurcation and sustained oscillations caused by the stress-induced competition. Furthermore, the ODE model, although not a special case of the PDE model, is useful for detecting backward bifurcation and oscillations in the PDE model. Backward bifurcation related to stress-induced competition allows the second species to persist for values of its invasion number below one. Furthermore, stress-induced competition leads to destabilization of the coexistence equilibrium and sustained oscillations in the PDE model. We suggest that complex systems such as this one may be studied by appropriately designed simple ODE models.

To model biochemical systems in which both noise in the chemical reaction process and spatial movement of molecules is important, both the reaction-diffusion master equation (RDME) and Smoluchowski diffusion-limited reaction (SDLR) partial differential equation (PDE) models have been used. In previous work we showed that the solution to the RDME may be interpreted as an asymptotic approximation in the reaction radius to the solution of the SDLR PDE [S. A. Isaacson, SIAM J. Appl. Math. 70, 77 (2009)]. The approximation was shown to be divergent in the limit that the lattice spacing in the RDME approached zero. In this work we expand upon these results for the special case of the two-molecule annihilation reaction, A+B-->Ø. We first introduce a third stochastic reaction-diffusion PDE model that incorporates a pseudopotential based bimolecular reaction mechanism. The solution to the pseudopotential model is then shown to be an asymptotic approximation to the solution of the SDLR PDE for small reaction radii. We next illustrate how the RDME may be obtained by a formal discretization of the pseudopotential model, motivating why the RDME is itself an asymptotic approximation of the SDLR PDE. Finally, we give a more detailed numerical analysis of the difference between solutions to the RDME and SDLR PDE models as a function of both the reaction-radius and the lattice spacing (in the RDME).

Deciphering the dynamics of lamellipodium in a fish keratocytes model

The present work aims to extend the climate change energy balance models using a heat source. An ordinary differential equations (ODEs) model is extended to a partial differential equations (PDEs) model using the effects of diffusion over the spatial variable. In addition, numerical schemes are presented using the Taylor series expansions. For the climate change model in the form of ODEs, a comparison of the presented scheme is made with the existing Trapezoidal method. It is found that the presented scheme converges faster than the existing scheme. Also, the proposed scheme provides fewer errors than the existing scheme. The PDEs model is also solved with the presented scheme, and the results are displayed in the form of different graphs. The impact of the climate feedback parameter, the heat uptake parameter of the deep ocean, and the heat source parameter on global mean surface temperature and deep ocean temperature is also portrayed. In addition, these recently developed techniques exhibit a high level of predictability. Doi: 10.28991/CEJ-2022-08-07-04 Full Text: PDF

The role of memory and cognition in the movement of individuals (e.g. animals) within a population, is thought to play an important role in population dispersal. In response, there has been increasing interest in incorporating spatial memory effects into classical partial differential equation (PDE) models of animal dispersal. However, the specific detail of the transport terms, such as diffusion and advection terms, that ought to be incorporated into PDE models to accurately reflect the memory effect remains unclear. To bridge this gap, we propose a straightforward lattice-based model where the movement of individuals depends on both crowding effects and the historic distribution within the simulation. The advantage of working with the individual-based model is that it is straightforward to propose and implement memory effects within the simulation in a way that is more biologically intuitive than simply proposing heuristic extensions of classical PDE models. Through deriving the continuum limit description of our stochastic model, we obtain a novel nonlinear diffusion equation which encompasses memory-based diffusion terms. For the first time we reveal the relationship between memory-based diffusion and the individual-based movement mechanisms that depend upon memory effects. Through repeated stochastic simulation and numerical explorations of the mean-field PDE model, we show that the new PDE model accurately describes the expected behaviour of the stochastic model, and we also explore how memory effects impact population dispersal.

Environmental stochasticity refers to unpredictable spatiotemporal fluctuation in environmental conditions. The term is often used in the literature on ecology and evolution. Unpredictability is defined as an inability to predict the future state precisely such that only its distribution can be known. The environment is typically defined as any set of abiotic (e.g. temperature and nutrient availability) and biotic (e.g. predator, competitor and food) conditions that organisms experience. Environmental stochasticity influences how population abundance fluctuates and affects the fate (e.g. persistence or extinction) of populations. In an evolutionary timescale, environmental stochasticity also affects the life history strategy of organisms. Environmental stochasticity is included in population models using univariate difference equations, stochastic matrix population models, stochastic differential equations and partial differential equations. Ecological data are analysed to determine the effect of environmental stochasticity using methods such as spectral analysis, capture–recapture analysis, state‐space analysis, generalised linear models and multivariate statistical analyses.Key ConceptsEnvironmental stochasticity is unpredictable spatiotemporal fluctuations in environmental conditions.Observed population dynamics consist of fluctuation due to environmental stochasticity, but it is often confounded with other factors such as observational errors, deterministic fluctuation and demographic stochasticity.Environmental stochasticity is reflected in the fluctuations in ecological processes and affects their fate (e.g. extinction or persistence of populations).Environmental stochasticity plays an important role in the evolution of life history strategies of organisms by affecting their fitness.Stochastic discrete‐time models, stochastic matrix population models, stochastic differential equation models and partial differential equation models are the four basic population models that include environmental stochasticity.Spectral analysis, state‐space model analysis, capture–recapture analysis, generalised linear models and multivariate statistical analysis are commonly used for separating the effects of environmental stochasticity in data.

The outbreak of COVID-19 in 2020 has led to a surge in interest in the mathematical modeling of infectious diseases. Such models are usually defined as compartmental models, in which the population under study is divided into compartments based on qualitative characteristics, with different assumptions about the nature and rate of transfer across compartments. Though most commonly formulated as ordinary differential equation models, in which the compartments depend only on time, recent works have also focused on partial differential equation (PDE) models, incorporating the variation of an epidemic in space. Such research on PDE models within a Susceptible, Infected, Exposed, Recovered, and Deceased framework has led to promising results in reproducing COVID-19 contagion dynamics. In this paper, we assess the robustness of this modeling framework by considering different geometries over more extended periods than in other similar studies. We first validate our code by reproducing previously shown results for Lombardy, Italy. We then focus on the U.S. state of Georgia and on the Brazilian state of Rio de Janeiro, one of the most impacted areas in the world. Our results show good agreement with real-world epidemiological data in both time and space for all regions across major areas and across three different continents, suggesting that the modeling approach is both valid and robust.