Deterministic Ordinary Differential Equations Research Articles

Experimentally-driven mathematical model to understand the effects of matrix deprivation in breast cancer metastasis.

Normal epithelial cells receive proper signals for growth and survival from attachment to the underlying extracellular matrix (ECM). They perceive detachment from the ECM as a stress and die - a phenomenon termed as 'anoikis'. However, metastatic cancer cells acquire anoikis-resistance and circulate through the blood and lymphatics to seed metastasis. Under normal (adherent) growth conditions, the serine-threonine protein kinase Akt stimulates protein synthesis and cell growth, maintaining an anabolic state in the cancer cell. In contrast, previously we showed that the stress due to matrix deprivation is sensed by yet another serine-threonine kinase, AMP-activated protein kinase (AMPK), that inhibits anabolic pathways while promoting catabolic processes. We illustrated a switch from Akthigh/AMPKlow in adherent condition to AMPKhigh/Aktlow in matrix-detached condition, with consequent metabolic switching from an anabolic to a catabolic state, which aids cancer cell stress-survival. In this study, we utilized these experimental data and developed a deterministic ordinary differential equation (ODE)-based mechanistic mathematical model to mimic attachment-detachment signaling network. To do so, we used the framework of insulin-glucagon signaling with consequent metabolic shifts to capture the pathophysiology of matrix-deprived state in breast cancer cells. Using the developed metastatic breast cancer signaling (MBCS) model, we identified perturbation of several signaling proteins such as IRS, PI3K, PKC, GLUT1, IP3, DAG, PKA, cAMP, and PDE3 upon matrix deprivation. Further, in silico molecular perturbations revealed that several feedback/crosstalks like DAG to PKC, PKC to IRS, S6K1 to IRS, cAMP to PKA, and AMPK to Akt are essential for the metabolic switching in matrix-deprived cancer cells. AMPK knockdown simulations identified a crucial role for AMPK in maintaining these adaptive changes. Thus, this mathematical framework provides insights on attachment-detachment signaling with metabolic adaptations that promote cancer metastasis.

Mathematical modeling predicts that endemics by generalist insects are eradicated if nearly all plants produce constitutive defense

Plants with constitutive defense chemicals exist widely in nature. The phenomenon is backed by abundant data from plant chemical ecology. Sufficient data are also available to conclude that plant defenses act as deterrent and repellent to attacking herbivores, particularly deleterious generalist insects. In the wild, generalist species are usually not endemic, meaning they are not restricted to certain plant species in a region. Therefore, our objective is to inspect theoretically whether evolution of chemical defenses in all plant species eradicate an endemic by any generalist species. The objective is addressed by developing deterministic ordinary differential equations under the following conditions: Plants without constitutive defenses are susceptible to oviposition by generalist insects, while they become defended against generalists by storing chemical defenses. From the models, we explicitly obtain that a generalist-free stable state is only possible if the vast majority of all plant individuals have chemical defenses. The model also allows one to predict the highest possible percentage of undefended plant individuals, which may be considered as free-riders.

Longitudinal single-cell data informs deterministic modelling of inflammatory bowel disease

Single-cell-based methods such as flow cytometry or single-cell mRNA sequencing (scRNA-seq) allow deep molecular and cellular profiling of immunological processes. Despite their high throughput, however, these measurements represent only a snapshot in time. Here, we explore how longitudinal single-cell-based datasets can be used for deterministic ordinary differential equation (ODE)-based modelling to mechanistically describe immune dynamics. We derived longitudinal changes in cell numbers of colonic cell types during inflammatory bowel disease (IBD) from flow cytometry and scRNA-seq data of murine colitis using ODE-based models. Our mathematical model generalised well across different protocols and experimental techniques, and we hypothesised that the estimated model parameters reflect biological processes. We validated this prediction of cellular turnover rates with KI-67 staining and with gene expression information from the scRNA-seq data not used for model fitting. Finally, we tested the translational relevance of the mathematical model by deconvolution of longitudinal bulk mRNA-sequencing data from a cohort of human IBD patients treated with olamkicept. We found that neutrophil depletion may contribute to IBD patients entering remission. The predictive power of IBD deterministic modelling highlights its potential to advance our understanding of immune dynamics in health and disease.

Testing the intrinsic mechanisms driving the dynamics of Ross River Virus across Australia.

The mechanisms driving dynamics of many epidemiologically important mosquito-borne pathogens are complex, involving combinations of vector and host factors (e.g., species composition and life-history traits), and factors associated with transmission and reporting. Understanding which intrinsic mechanisms contribute most to observed disease dynamics is important, yet often poorly understood. Ross River virus (RRV) is Australia's most important mosquito-borne disease, with variable transmission dynamics across geographic regions. We used deterministic ordinary differential equation models to test mechanisms driving RRV dynamics across major epidemic centers in Brisbane, Darwin, Mandurah, Mildura, Gippsland, Renmark, Murray Bridge, and Coorong. We considered models with up to two vector species (Aedes vigilax, Culex annulirostris, Aedes camptorhynchus, Culex globocoxitus), two reservoir hosts (macropods, possums), seasonal transmission effects, and transmission parameters. We fit models against long-term RRV surveillance data (1991-2017) and used Akaike Information Criterion to select important mechanisms. The combination of two vector species, two reservoir hosts, and seasonal transmission effects explained RRV dynamics best across sites. Estimated vector-human transmission rate (average β = 8.04x10-4per vector per day) was similar despite different dynamics. Models estimate 43% underreporting of RRV infections. Findings enhance understanding of RRV transmission mechanisms, provide disease parameter estimates which can be used to guide future research into public health improvements and offer a basis to evaluate mitigation practices.

Impact of Media Awareness and Use of Face-Masks on Infectious Respiratory Disease

Infectious respiratory diseases have been a threat to our lives and livelihoods. Communities with a higher population density ultimately are at a higher risk of a rapid spread of infectious disease once they occur. Mathematical models have extensively been used to analyse the impact of several mitigation measures to the transmission of infectious diseases. The focus of this study is to determine the impact of media awareness, use of face-masks alongside quarantine and isolation on the transmission dynamics of a respiratory infection where there is no available vaccination or treatment plan. The model is based on a system of deterministic ordinary differential equations (ODE’s). By the use of the next generation matrix (NGM), the basic reproduction is determined. The local stability analysis analysis of the disease free equilibrium (DFE) is obtained by the trace-determinant approach while it’s global stability was determined by the use of the Lyapunov-Krasovskii method. It was established that the local DFE was asymptotically stable while it was unstable globally implying that with application of this intended strategies, the transmission of the disease could be lowered but not eradicated. The importance of media awareness to any campaign towards reduction of transmission is paramount. Effort should be geared towards the public having correct information on a given disease. The impact of face mask is dependent on the efficacy of the face - mask and its compliant and consistent use, these two factors are multiplicative. Numerical simulations were done by the use of Python software and it was established that if quarantine rate and the isolation rate was to be maximised then the disease would be curtailed. The results of this study gives valuable information on the intervention strategies to be applied by the public heath officials.

Distilling identifiable and interpretable dynamic models from biological data.

Mechanistic dynamical models allow us to study the behavior of complex biological systems. They can provide an objective and quantitative understanding that would be difficult to achieve through other means. However, the systematic development of these models is a non-trivial exercise and an open problem in computational biology. Currently, many research efforts are focused on model discovery, i.e. automating the development of interpretable models from data. One of the main frameworks is sparse regression, where the sparse identification of nonlinear dynamics (SINDy) algorithm and its variants have enjoyed great success. SINDy-PI is an extension which allows the discovery of rational nonlinear terms, thus enabling the identification of kinetic functions common in biochemical networks, such as Michaelis-Menten. SINDy-PI also pays special attention to the recovery of parsimonious models (Occam's razor). Here we focus on biological models composed of sets of deterministic nonlinear ordinary differential equations. We present a methodology that, combined with SINDy-PI, allows the automatic discovery of structurally identifiable and observable models which are also mechanistically interpretable. The lack of structural identifiability and observability makes it impossible to uniquely infer parameter and state variables, which can compromise the usefulness of a model by distorting its mechanistic significance and hampering its ability to produce biological insights. We illustrate the performance of our method with six case studies. We find that, despite enforcing sparsity, SINDy-PI sometimes yields models that are unidentifiable. In these cases we show how our method transforms their equations in order to obtain a structurally identifiable and observable model which is also interpretable.

A novel algorithm for asymptotic stability analysis of some classes of stochastic time-fractional Volterra equations

The article presents a regularization method to study global asymptotic stability of stochastic multi-time scale Volterra equations as an alternative to the algorithms based on Lyapunov functionals. Two different concepts of stability were linked and examined. The central idea of the method is based on a parallelism between the Lyapunov stability and a stochastic version of the input-to-state stability, which is well-known in the control theory of deterministic ordinary differential equations. At the first step the algorithm transforms the given delay equation into a Volterra differential equation with stochastic control, following replacement of the Lyapunov stability of the former with the input-to-state stability of the latter. To estimate the norms of the solutions we use a regularization technique based on the concept of inverse-positive matrices. This algorithm could be extended and applied for stability analysis of new classes of stochastic fractional differential equations and their applications.

Confronting population models with experimental microcosm data: from trajectory matching to state‐space models

AbstractPopulation and community ecology traditionally has a very strong theoretical foundation with well‐known dynamical models, such as the logistic and its variations, and many modifications of the classical Lotka–Volterra predator–prey and interspecific competition models. More and more, these classical models are being confronted with data via fitting to empirical time series for purposes of projections or for estimating model parameters of interest. However, using statistical models to fit theoretical models to data is far from trivial, especially for time series data where subsequent measurements are not independent. This raises the question of whether statistical inferences using pure observation error models, such as simple (non‐)linear regressions, are biased, and whether more elaborate process error models or state‐space models have to be used to address this complexity. In order to help empiricists, especially researchers working with experimental laboratory populations in micro‐ and mesocosms, make informed decisions about the statistical formalism to use, we here compare different error structures one could use when fitting classical deterministic ordinary differential equation (ODE) models to empirical data. We consider a large range of biological scenarios and theoretical models, from single species to community dynamics and trophic interactions. In order to compare the performance of different error structure models, we use both realistically simulated data and empirical data from microcosms in a Bayesian framework. We find that many model parameters can be estimated precisely with an appropriate choice of error structure using pure observation error or state‐space models, if observation errors are not too high. However, Allee effect models are typically hard to identify and state‐space models should be preferred when model complexity increases. Our work shows that, at least in the context of low environmental stochasticity and high quality observations, deterministic models can be used to describe stochastic population dynamics that include process variability and observation error. We discuss when more complex state‐space model formulations may be required for obtaining accurate parameter estimates. Finally, we provide a comprehensive tutorial for fitting these models in R.

Threshold quantities and Lyapunov functions for ordinary differential equations epidemic models with mass action and standard incidence functions

This paper presents a novel algebraic method for the construction of Lyapunov functions to study global stability of the disease-free equilibrium points of deterministic epidemic ordinary differential equation models with mass action and standard incidence functions. The method is named as Jacobian-Determinant method. In our method, a direct algebraic procedure that also relies only on determinant of the Jacobian matrix of the infected subsystem is developed to determine a threshold quantity, R0′ akin to the basic reproduction number, R0 of such class of models. The developed technique is applied on a wide variety of models to construct Lyapunov functions to study the global stability of the infection-free critical points. Further, implementation of our method reveals that the threshold quantity is the same as (or the square) of the basic reproduction numbers as obtained using the next-generation matrix method. It is further observed that even for models that do not use the standard or mass action incidence, the threshold quantity is still related to the basic reproduction numbers as obtained with the next-generation matrix method.

Probabilistic solvers enable a straight-forward exploration of numerical uncertainty in neuroscience models

Understanding neural computation on the mechanistic level requires models of neurons and neuronal networks. To analyze such models one typically has to solve coupled ordinary differential equations (ODEs), which describe the dynamics of the underlying neural system. These ODEs are solved numerically with deterministic ODE solvers that yield single solutions with either no, or only a global scalar error indicator on precision. It can therefore be challenging to estimate the effect of numerical uncertainty on quantities of interest, such as spike-times and the number of spikes. To overcome this problem, we propose to use recently developed sampling-based probabilistic solvers, which are able to quantify such numerical uncertainties. They neither require detailed insights into the kinetics of the models, nor are they difficult to implement. We show that numerical uncertainty can affect the outcome of typical neuroscience simulations, e.g. jittering spikes by milliseconds or even adding or removing individual spikes from simulations altogether, and demonstrate that probabilistic solvers reveal these numerical uncertainties with only moderate computational overhead.

Estimation of optimal antiviral stockpile for a novel influenza pandemic

BackgroundA stockpile of antiviral drugs is important for mitigating a novel influenza pandemic. Recently, intervention strategies against such a pandemic have improved significantly, affecting the required size and composition of the stockpile. Our goal is to estimate the optimal ratio of conventional to newer antiviral drugs. MethodWe estimated epidemic parameters from daily-case data about H1N1pdm09 in the Republic of Korea, and used a deterministic ordinary differential equation model and stochastic simulation to predict the number of patients in a future pandemic. We considered an antiviral stockpile containing neuraminidase inhibitors (NAI) and a new drug, cap-dependent endonuclease inhibitor (CENI), seeking the optimum ratio of the two drugs under different epidemiological and economic assumptions. ResultsWith an effective reproductive number of 1.36, the expected cumulative cases did not exceed 30 % of the population in all vaccination scenarios. If the non-pharmaceutical intervention strategy is intensified and the effective reproductive number is decreased to 1.29, a 20 % antiviral stockpile of the population is sufficient. Assuming that CENI is prescribed for 10 % of patients, the expected total number of cases is decreased from 30 % to approximately 25 % of the population. If the cost of CENI is triple that of NAI, no expenditures beyond the current budget are necessary; if it is quintuple, expenditures increase by 17 %. ConclusionStockpiling CENI reduces the number of patients by reducing the infectious period. However, the government needs to consider the cost-effective stockpile ratio of such new drugs. This will depend not only on the cost of the drugs, but on factors difficult to anticipate, such as the transmissibility of the virus, the time needed for vaccine development, and (especially) the emergence of resistance. If this information can be estimated, our model can be used to obtain the optimum.

Phase Portraits and Bounded and Singular Traveling Wave Solution of Stochastic Nonlinear Biswas–Arshed Equation

The main purpose of the current paper is to study the phase portraits and bounded and singular traveling wave solution of the stochastic nonlinear Biswas–Arshed equation by using the “three‐step method” of Professor Li’s method together with the phase orbit of planar dynamical system. Firstly, by employing the traveling wave transformation, the stochastic nonlinear Biswas–Arshed equation is simplified into deterministic nonlinear ordinary differential equation. Secondly, phase portraits of the stochastic nonlinear Biswas–Arshed equation are plotted by analyzing the planar dynamic system of the nonlinear ordinary differential equation. Finally, the bounded and singular traveling wave solutions of the stochastic nonlinear Biswas–Arshed equation are constructed.

Mathematical modeling of the dynamics of maize streak virus disease (MSVD)

<abstract><p>In this paper, a deterministic ordinary differential equations model for the transmission dynamics of maize streak virus disease (MSVD) in maize plants is proposed and analyzed qualitatively. Using the next generation matrix approach, the basic reproduction number, $ \mathcal{R}_{0} $ with respect to the MSVD free equilibrium is found to be $ 4.7065\approx5 $. The conditions for local stability of the disease-free equilibrium and endemic equilibrium points were established. From the results, the disease-free equilibrium point was found to be unstable whenever $ \mathcal{R}_{0} > 1 $ and the endemic equilibrium point was found to be locally asymptotically stable whenever $ \mathcal{R}_{0} > 1 $. The sensitivity indices of various parameters with respect to the MSVD eradication or spreading were determined. It was found that $ b\; , \beta_{{1}}\; , \beta_{{11}}\; , $ and $ \beta_{{2}} $ are the parameters that are directly related to $ \mathcal{R}_{0} $, and $ H_{2}, \mu, \mu_{{1}} $ and $ \gamma $ are inversely related to $ \mathcal{R}_{0} $. Numerical simulation was performed and displayed graphically to justify the analytical results.</p></abstract>

Stochastic dynamics, large deviation principle, and nonequilibrium thermodynamics.

By examining the deterministic limit of a general ε-dependent generator for Markovian dynamics, which includes the continuous Fokker-Planck equations and discrete chemical master equations as two special cases, the intrinsic connections among mesoscopic stochastic dynamics, deterministic ordinary differential equations or partial differential equations, large deviation rate functions, and macroscopic thermodynamic potentials are established. Our result not only solves the long-lasting question of the origin of the entropy function in classical irreversible thermodynamics, but also reveals an emergent feature that arises automatically during the deterministic limit, through its large deviation rate function, with both time-reversible dynamics equipped with a Hamiltonian function and time-irreversible dynamics equipped with an entropy function.

The systems biology simulation core library.

SummaryStudying biological systems generally relies on computational modeling and simulation, e.g., model-driven discovery and hypothesis testing. Progress in standardization efforts led to the development of interrelated file formats to exchange and reuse models in systems biology, such as SBML, the Simulation Experiment Description Markup Language (SED-ML) or the Open Modeling EXchange format. Conducting simulation experiments based on these formats requires efficient and reusable implementations to make them accessible to the broader scientific community and to ensure the reproducibility of the results. The Systems Biology Simulation Core Library (SBSCL) provides interpreters and solvers for these standards as a versatile open-source API in JavaTM. The library simulates even complex bio-models and supports deterministic Ordinary Differential Equations; Stochastic Differential Equations; constraint-based analyses; recent SBML and SED-ML versions; exchange of results, and visualization of in silico experiments; open modeling exchange formats (COMBINE archives); hierarchically structured models; and compatibility with standard testing systems, including the Systems Biology Test Suite and published models from the BioModels and BiGG databases.Availability and implementationSBSCL is freely available at https://draeger-lab.github.io/SBSCL/ and via Maven Central.Supplementary information Supplementary data are available at Bioinformatics online.

Push-forward method for piecewise deterministic biochemical simulations

A biochemical network can be simulated by a set of ordinary differential equations (ODE) under well-stirred reactor conditions, for large numbers of molecules, and frequent reactions. This is no longer a robust representation when some molecular species are in small numbers and reactions changing them are infrequent. In this case, discrete stochastic events trigger changes of the smooth deterministic dynamics of the biochemical network. Piecewise-deterministic Markov processes (PDMP) are well adapted for describing such situations. Although PDMP models are now well established in biology, these models remain computationally challenging. Previously we have introduced the push-forward method to compute how the probability measure is spread by the deterministic ODE flow of PDMPs, through the use of analytic expressions of the corresponding semigroup. In this paper we provide a more general simulation algorithm that works also for non-integrable systems. The method can be used for biochemical simulations with applications in fundamental biology, biotechnology and biocomputing. This work is an extended version of the work presented at the conference CMSB2019.

Combinatorial mathematical modelling approaches to interrogate rear retraction dynamics in 3D cell migration.

Cell migration in 3D microenvironments is a complex process which depends on the coordinated activity of leading edge protrusive force and rear retraction in a push-pull mechanism. While the potentiation of protrusions has been widely studied, the precise signalling and mechanical events that lead to retraction of the cell rear are much less well understood, particularly in physiological 3D extra-cellular matrix (ECM). We previously discovered that rear retraction in fast moving cells is a highly dynamic process involving the precise spatiotemporal interplay of mechanosensing by caveolae and signalling through RhoA. To further interrogate the dynamics of rear retraction, we have adopted three distinct mathematical modelling approaches here based on (i) Boolean logic, (ii) deterministic kinetic ordinary differential equations (ODEs) and (iii) stochastic simulations. The aims of this multi-faceted approach are twofold: firstly to derive new biological insight into cell rear dynamics via generation of testable hypotheses and predictions; and secondly to compare and contrast the distinct modelling approaches when used to describe the same, relatively under-studied system. Overall, our modelling approaches complement each other, suggesting that such a multi-faceted approach is more informative than methods based on a single modelling technique to interrogate biological systems. Whilst Boolean logic was not able to fully recapitulate the complexity of rear retraction signalling, an ODE model could make plausible population level predictions. Stochastic simulations added a further level of complexity by accurately mimicking previous experimental findings and acting as a single cell simulator. Our approach highlighted the unanticipated role for CDK1 in rear retraction, a prediction we confirmed experimentally. Moreover, our models led to a novel prediction regarding the potential existence of a ‘set point’ in local stiffness gradients that promotes polarisation and rapid rear retraction.

An efficient spectral method for the numerical solution to some classes of stochastic differential equations

We consider a new approach for the numerical approximation to some classes of stochastic differential equations driven by white noise. The proposed method shares some features with the stochastic collocation techniques, and in particular, it takes advantage of the assumption of smoothness of the functional to be approximated, to achieve fast convergence. The solution to the stochastic differential equation (SDE) is represented by means of Lagrange polynomials. The coefficients of the polynomial basis are functions of time, and they can be computed by solving a system of deterministic ordinary differential equations. Another motivation of this work lies in the novelty of the numerical scheme that does not belong to classical techniques to solve SDEs. Numerical examples are presented to illustrate the accuracy and the efficiency of the proposed method. In particular, we observe a spectral convergence for the mean and the variance of the solution.

Mathematical assessment of the impact of human-antibodies on sporogony during the within-mosquito dynamics of Plasmodium falciparum parasites

We develop and analyze a deterministic ordinary differential equation mathematical model for the within-mosquito dynamics of the Plasmodium falciparum malaria parasite. Our model takes into account the action and effect of blood resident human-antibodies, ingested by the mosquito during a blood meal from humans, in inhibiting gamete fertilization. The model also captures subsequent developmental processes that lead to the different forms of the parasite within the mosquito. Continuous functions are used to model the switching transition from oocyst to sporozoites as well as human antibody density variations within the mosquito gut are proposed and used. In sum, our model integrates the developmental stages of the parasite within the mosquito such as gametogenesis, fertilization and sporogenesis culminating in the formation of sporozoites. Quantitative and qualitative analyses including a sensitivity analysis for influential parameters are performed. We quantify the average sporozoite load produced at the end of the within-mosquito malaria parasite’s developmental stages. Our analysis shows that an increase in the efficiency of the ingested human antibodies in inhibiting fertilization within the mosquito’s gut results in lowering the density of oocysts and hence sporozoites that are eventually produced by each mosquito vector. So, it is possible to control and limit oocysts development and hence sporozoites development within a mosquito by boosting the efficiency of antibodies as a pathway to the development of transmission-blocking vaccines which could potentially reduce oocysts prevalence among mosquitoes and hence reduce the transmission potential from mosquitoes to human.

Dodge and survive: Modeling the predatory nature of dodgeball.

The analysis of games and sports as complex systems can give insights into the dynamics of human competition and has been proven useful in soccer, basketball, and other professional sports. In this paper, we present a model for dodgeball, a popular sport in U.S. schools, and analyze it using an ordinary differential equation (ODE) compartmental model and stochastic agent-based game simulations. The ODE model reveals a rich landscape with different game dynamics occurring depending on the strategies used by the teams, which can in some cases be mapped to scenarios in competitive species models. Stochastic agent-based game simulations confirm and complement the predictions of the deterministic ODE models. In some scenarios, game victory can be interpreted as a noise-driven escape from the basin of attraction of a stable fixed point, resulting in extremely long games when the number of players is large. Using the ODE and agent-based models, we construct a strategy to increase the probability of winning.