Bayesian estimation of ordinary differential equation models when the likelihood has multiple local modes

2018 ◽  
Vol 24 (2) ◽  
pp. 117-127
Author(s):  
Baisen Liu ◽  
Liangliang Wang ◽  
Jiguo Cao
Keyword(s):  
Likelihood Function ◽  
Local Modes ◽  
Predator Prey ◽  
Complex Dynamic Systems ◽  
Ode Models ◽  
Differential Equation Models ◽  
Model Complex ◽  
Noisy Measurements ◽  
Optimizing Algorithm ◽  
Trapping Problem

Abstract Ordinary differential equations (ODEs) are popularly used to model complex dynamic systems by scientists; however, the parameters in ODE models are often unknown and have to be inferred from noisy measurements of the dynamic system. One conventional method is to maximize the likelihood function, but the likelihood function often has many local modes due to the complexity of ODEs, which makes the optimizing algorithm be vulnerable to trap in local modes. In this paper, we solve the global optimization issue of ODE parameters with the help of the Stochastic Approximation Monte Carlo (SAMC) algorithm which is shown to be self-adjusted and escape efficiently from the “local-trapping” problem. Our simulation studies indicate that the SAMC method is a powerful tool to estimate ODE parameters globally. The efficiency of SAMC method is demonstrated by estimating a predator-prey ODEs model from real experimental data.

Overview of Gene Regulatory Network Inference Based on Differential Equation Models

2020 ◽  
Vol 21 (11) ◽  
pp. 1054-1059
Author(s):  
Bin Yang ◽  
Yuehui Chen
Keyword(s):  
Differential Equation ◽  
Gene Regulatory Networks ◽  
Regulatory Networks ◽  
Network Inference ◽  
New Drugs ◽  
Gene Regulatory Network Inference ◽  
Ode Models ◽  
Differential Equation Models ◽  
Linear Ode ◽  
Gene Regulatory

: Reconstruction of gene regulatory networks (GRN) plays an important role in understanding the complexity, functionality and pathways of biological systems, which could support the design of new drugs for diseases. Because differential equation models are flexible androbust, these models have been utilized to identify biochemical reactions and gene regulatory networks. This paper investigates the differential equation models for reverse engineering gene regulatory networks. We introduce three kinds of differential equation models, including ordinary differential equation (ODE), time-delayed differential equation (TDDE) and stochastic differential equation (SDE). ODE models include linear ODE, nonlinear ODE and S-system model. We also discuss the evolutionary algorithms, which are utilized to search the optimal structures and parameters of differential equation models. This investigation could provide a comprehensive understanding of differential equation models, and lead to the discovery of novel differential equation models.

Population Dynamics and the Functional Response

2021 ◽  
pp. 47-54
Author(s):  
John P. DeLong
Keyword(s):  
Differential Equation ◽  
Population Dynamics ◽  
Functional Response ◽  
Dynamic Data ◽  
Predator Prey ◽  
Differential Equation Models ◽  
Starting Point

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.

scholarly journals Heterogeneity matters: Contact structure and individual variation shape epidemic dynamics

PLoS ONE ◽  
2021 ◽  
Vol 16 (7) ◽  
pp. e0250050
Author(s):  
Gerrit Großmann ◽  
Michael Backenköhler ◽  
Verena Wolf
Keyword(s):  
Viral Load ◽  
Herd Immunity ◽  
Dynamical Evolution ◽  
Stochastic Simulations ◽  
Contact Network ◽  
Initial Increase ◽  
Epidemic Dynamics ◽  
Ode Models ◽  
Point Estimates ◽  
Model Complex

In the recent COVID-19 pandemic, mathematical modeling constitutes an important tool to evaluate the prospective effectiveness of non-pharmaceutical interventions (NPIs) and to guide policy-making. Most research is, however, centered around characterizing the epidemic based on point estimates like the average infectiousness or the average number of contacts. In this work, we use stochastic simulations to investigate the consequences of a population’s heterogeneity regarding connectivity and individual viral load levels. Therefore, we translate a COVID-19 ODE model to a stochastic multi-agent system. We use contact networks to model complex interaction structures and a probabilistic infection rate to model individual viral load variation. We observe a large dependency of the dispersion and dynamical evolution on the population’s heterogeneity that is not adequately captured by point estimates, for instance, used in ODE models. In particular, models that assume the same clinical and transmission parameters may lead to different conclusions, depending on different types of heterogeneity in the population. For instance, the existence of hubs in the contact network leads to an initial increase of dispersion and the effective reproduction number, but to a lower herd immunity threshold (HIT) compared to homogeneous populations or a population where the heterogeneity stems solely from individual infectivity variations.

scholarly journals The “Destabilizing” Effect of Cannibalism in a Spatially Explicit Three-Species Age Structured Predator-Prey Model

Complexity ◽  
2017 ◽  
Vol 2017 ◽  
pp. 1-15 ◽  
Author(s):  
Aladeen Al Basheer ◽  
Jingjing Lyu ◽  
Adom Giffin ◽  
Rana D. Parshad
Keyword(s):  
Turing Instability ◽  
The Other ◽  
Spatially Explicit ◽  
Ode Model ◽  
Predator Prey ◽  
Classic Work ◽  
Predator Prey Model ◽  
Pde Model ◽  
Ode Models ◽  
Age Structured

Cannibalism, the act of killing and consumption of conspecifics, is generally considered to be a stabilising process in ODE models of predator-prey systems. On the other hand, Sun et al. were the first to show that cannibalism can cause Turing instability, in the classical Rosenzweig-McArthur two-species PDE model, which is an impossibility without cannibalism. Magnússon’s classic work is the first to show that cannibalism in a structured three-species predator-prey ODE model can actually be destabilising. In the current manuscript we consider the PDE form of the three-species model proposed in Magnússon’s classic work. We prove that, in the absence of cannibalism, Turing instability is an impossibility in this model, forany range of parameters. However, theinclusionof cannibalism can cause Turing instability. Thus, to the best of our knowledge, we report thefirstcannibalism induced Turing instability result, in spatially explicit three-species age structured predator-prey systems. We also show that, in the classical ODE model proposed by Magnússon, cannibalism can act as a life boat mechanism,for the prey.

scholarly journals Why ODE models for COVID-19 fail: Heterogeneity shapes epidemic dynamics

2021 ◽  
Author(s):  
Gerrit Großmann ◽  
Michael Backenköhler ◽  
Verena Wolf
Keyword(s):  
Viral Load ◽  
Herd Immunity ◽  
Age Groups ◽  
Stochastic Simulations ◽  
Contact Network ◽  
Initial Increase ◽  
Epidemic Dynamics ◽  
Ode Models ◽  
Point Estimates ◽  
Model Complex

AbstractIn the recent COVID-19 pandemic, mathematical modeling constitutes an important tool to evaluate the prospective effectiveness of non-pharmaceutical interventions (NPIs) and to guide policy-making. Most research is, however, centered around characterizing the epidemic based on point estimates like the average infectiousness or the average number of contacts.In this work, we use stochastic simulations to investigate the consequences of a population’s heterogeneity regarding connectivity and individual viral load levels.Therefore, we translate a COVID-19 ODE model to a stochastic multi-agent system. We use contact networks to model complex interaction structures and a probabilistic infection rate to model individual viral load variation.We observe a large dependency of the dispersion and dynamical evolution on the population’s heterogeneity that is not adequately captured by point estimates, for instance, used in ODE models. In particular, models that assume the same clinical and transmission parameters may lead to different conclusions, depending on different types of heterogeneity in the population. For instance, the existence of hubs in the contact network leads to an initial increase of dispersion and the effective reproduction number, but to a lower herd immunity threshold (HIT) compared to homogeneous populations or a population where the heterogeneity stems solely from individual infectivity variations.Author summaryComputational modeling can support decision-making in the face of pandemics like COVID-19. Models help to understand transmission data and predict important epidemiological properties (e.g., When will herd immunity be reached?). They can also examine the effectiveness of certain measures, and—to a limited extent—extrapolate the dynamics under specific assumptions. In all these cases, the heterogeneity of the population plays an important role. For instance, it is known that connectivity differences in (and among) age groups influence the dynamics of epidemic propagation. Here we focus on two types of differences among individuals: their social interactions and on how infectious they are. We show that only considering population averages (e.g., What is the average number of contacts of an individual?) may lead to misleading conclusions, because the individual differences (such as those related to the epidemic (over-)dispersion) play an important role in shaping the epidemic dynamics. Many commonly used model classes, such as SEIR-type ODE compartmental models, ignore differences within a population to a large extent. This omission bears the potential of misleading conclusions.

scholarly journals Identifiability analysis for stochastic differential equation models in systems biology

2020 ◽  
Vol 17 (173) ◽  
pp. 20200652
Author(s):  
Alexander P. Browning ◽  
David J. Warne ◽  
Kevin Burrage ◽  
Ruth E. Baker ◽  
Matthew J. Simpson
Keyword(s):  
Differential Equation ◽  
Stochastic Differential Equation ◽  
Synthetic Data ◽  
Parameter Estimates ◽  
Practical Case ◽  
Identifiability Analysis ◽  
Parameter Identifiability ◽  
Ode Models ◽  
Differential Equation Models ◽  
Reliable Parameter

Mathematical models are routinely calibrated to experimental data, with goals ranging from building predictive models to quantifying parameters that cannot be measured. Whether or not reliable parameter estimates are obtainable from the available data can easily be overlooked. Such issues of parameter identifiability have important ramifications for both the predictive power of a model, and the mechanistic insight that can be obtained. Identifiability analysis is well-established for deterministic, ordinary differential equation (ODE) models, but there are no commonly adopted methods for analysing identifiability in stochastic models. We provide an accessible introduction to identifiability analysis and demonstrate how existing ideas for analysis of ODE models can be applied to stochastic differential equation (SDE) models through four practical case studies. To assess structural identifiability , we study ODEs that describe the statistical moments of the stochastic process using open-source software tools. Using practically motivated synthetic data and Markov chain Monte Carlo methods, we assess parameter identifiability in the context of available data. Our analysis shows that SDE models can often extract more information about parameters than deterministic descriptions. All code used to perform the analysis is available on Github .

scholarly journals The SAMI Galaxy Survey: Bayesian inference for gas disc kinematics using a hierarchical Gaussian mixture model

2019 ◽  
Vol 485 (3) ◽  
pp. 4024-4044 ◽  
Author(s):  
Mathew R Varidel ◽  
Scott M Croom ◽  
Geraint F Lewis ◽  
Brendon J Brewer ◽  
Enrico M Di Teodoro ◽  
...  
Keyword(s):  
Gaussian Mixture Model ◽  
Mixture Model ◽  
Velocity Dispersion ◽  
Likelihood Function ◽  
Gaussian Mixture ◽  
Relative Difference ◽  
Gaussian Component ◽  
Mean Velocity ◽  
Gas Kinematics ◽  
Model Complex

Abstract We present a novel Bayesian method, referred to as blobby3d, to infer gas kinematics that mitigates the effects of beam smearing for observations using integral field spectroscopy. The method is robust for regularly rotating galaxies despite substructure in the gas distribution. Modelling the gas substructure within the disc is achieved by using a hierarchical Gaussian mixture model. To account for beam smearing effects, we construct a modelled cube that is then convolved per wavelength slice by the seeing, before calculating the likelihood function. We show that our method can model complex gas substructure including clumps and spiral arms. We also show that kinematic asymmetries can be observed after beam smearing for regularly rotating galaxies with asymmetries only introduced in the spatial distribution of the gas. We present findings for our method applied to a sample of 20 star-forming galaxies from the SAMI Galaxy Survey. We estimate the global H α gas velocity dispersion for our sample to be in the range $\bar{\sigma }_v \sim$[7, 30] km s−1. The relative difference between our approach and estimates using the single Gaussian component fits per spaxel is $\Delta \bar{\sigma }_v / \bar{\sigma }_v = - 0.29 \pm 0.18$ for the H α flux-weighted mean velocity dispersion.

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