Differential Equation Models in Applied Mathematics: Theoretical and Numerical Challenges

The articles published in the Special Issue “Differential Equation Models in Applied Mathematics: Theoretical and Numerical Challenges” of the MDPI Mathematics journal are here collected [...]

Calculating university education model based on finite element fractional differential equations and macro-control analysis

Firstly, based on the charging theory of ‘education cost-sharing,’ under appropriate assumptions, two basic differential equation models are proposed to describe the problem of college education charges; secondly, through qualitative analysis of the basic model, it is concluded that colleges and universities maintain or impose several conditions for stabilising its education fees; finally, through the analysis of two basic models in three unique models under three situations, some new conclusions and suggestions on the macro-control of college education fees and enrolment scale are given. Also, three extended differential equation models are proposed.

College Students’ Mental Health Climbing Consumption Model Based on Nonlinear Differential Equations

College students continue to improve their consumption levels due to compassion or vanity, but it not only increases the financial burden of the family, but is also not conducive to their personal health. In order not to affect the mental health of college students, to solve the problem of mutual comparison consumption, this paper establishes differential equation models to describe this phenomenon, from qualitative and quantitative angles to interpret and predict the results of comparison consumption. The results show that the current consumption behaviour of college students also has a reasonable place for being unreasonable, mainly from college students’ own and living environment.

Differential equation model of financial market stability based on Internet big data

In the context of Internet big data, the market characteristics of the financial market can be used to feed back its stability with the help of differential equation models. China's financial market is roughly divided into three main markets: stocks, currency and foreign exchange. The interaction of the three has promoted the development of the financial market. With this as a background, the paper aims at these three financial markets and selects relevant indicators that can reflect the indications of the financial market to construct differential equations to analyse the relationship between the three. The paper uses the nonlinear characteristics of ordinary differential equations and related algorithms to solve the three types of market models. It uses an example to demonstrate that the differential equation model proposed in this paper can feed back the evolutionary characteristics of the three, and this model can help investors produce more correct investment decisions.

Bayesian uncertainty quantification for data-driven equation learning

Equation learning aims to infer differential equation models from data. While a number of studies have shown that differential equation models can be successfully identified when the data are sufficiently detailed and corrupted with relatively small amounts of noise, the relationship between observation noise and uncertainty in the learned differential equation models remains unexplored. We demonstrate that for noisy datasets there exists great variation in both the structure of the learned differential equation models and their parameter values. We explore how to exploit multiple datasets to quantify uncertainty in the learned models, and at the same time draw mechanistic conclusions about the target differential equations. We showcase our results using simulation data from a relatively straightforward agent-based model (ABM) which has a well-characterized partial differential equation description that provides highly accurate predictions of averaged ABM behaviours in relevant regions of parameter space. Our approach combines equation learning methods with Bayesian inference approaches so that a quantification of uncertainty can be given by the posterior parameter distribution of the learned model.

Population Dynamics and the Functional Response

In this chapter, I show how the functional response can drive predator–prey cycles (and dynamics more generally). I introduce predator–prey differential equation models and fit them to real dynamic data on classic predator–prey systems (lynx–hare and Daphnia–algae). This coupling achieves two things. First, it allows me to demonstrate that the models are capable of describing real predator–prey dynamics and that the functional response really does have a role in driving predator–prey cycles (even if it is not the driver of all cycles). Second, it allows me, from an empirically grounded starting point, to vary the parameters of the functional response to show how changes in the functional response parameters change the dynamics.

Stability and Bifurcations of Equilibria in Networks with Piecewise Linear Interactions

In this paper, we study equilibria of differential equation models for networks. When interactions between nodes are taken to be piecewise constant, an efficient combinatorial analysis can be used to characterize the equilibria. When the piecewise constant functions are replaced with piecewise linear functions, the equilibria are preserved as long as the piecewise linear functions are sufficiently steep. Therefore the combinatorial analysis can be leveraged to understand a broader class of interactions. To better understand how broad this class is, we explicitly characterize how steep the piecewise linear functions must be for the correspondence between equilibria to hold. To do so, we analyze the steady state and Hopf bifurcations which cause a change in the number or stability of equilibria as slopes are decreased. Additionally, we show how to choose a subset of parameters so that the correspondence between equilibria holds for the smallest possible slopes when the remaining parameters are fixed.

Rational design of complex phenotype via network models

We demonstrate a modeling and computational framework that allows for rapid screening of thousands of potential network designs for particular dynamic behavior. To illustrate this capability we consider the problem of hysteresis, a prerequisite for construction of robust bistable switches and hence a cornerstone for construction of more complex synthetic circuits. We evaluate and rank most three node networks according to their ability to robustly exhibit hysteresis where robustness is measured with respect to parameters over multiple dynamic phenotypes. Focusing on the highest ranked networks, we demonstrate how additional robustness and design constraints can be applied. We compare our results to more traditional methods based on specific parameterization of ordinary differential equation models and demonstrate a strong qualitative match at a small fraction of the computational cost.

Assessing the Spatio-temporal Spread of COVID-19 via Compartmental Models with Diffusion in Italy, USA, and Brazil

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.