Standard Mean-field Model Research Articles

Model of Spread of Infectious Diseases

The Model of infectious diseases continues to develop along with the development of the disease. With the dynamic spread of disease, ongoing research is needed. This study developed the SIR model by taking into account the spread of disease in the presence of Reproductive Number or R0. This study proposes an epidemic model of infectious diseases in dynamic networks for SIRS types, the standard mean-field model is used as a basic framework.

Inferring parameters for a lattice-free model of cell migration and proliferation using experimental data

Collective cell spreading takes place in spatially continuous environments, yet it is often modelled using discrete lattice-based approaches. Here, we use data from a series of cell proliferation assays, with a prostate cancer cell line, to calibrate a spatially continuous individual based model (IBM) of collective cell migration and proliferation. The IBM explicitly accounts for crowding effects by modifying the rate of movement, direction of movement, and the rate of proliferation by accounting for pair-wise interactions. Taking a Bayesian approach we estimate the free parameters in the IBM using rejection sampling on three separate, independent experimental data sets. Since the posterior distributions for each experiment are similar, we perform simulations with parameters sampled from a new posterior distribution generated by combining the three data sets. To explore the predictive power of the calibrated IBM, we forecast the evolution of a fourth experimental data set. Overall, we show how to calibrate a lattice-free IBM to experimental data, and our work highlights the importance of interactions between individuals. Despite great care taken to distribute cells as uniformly as possible experimentally, we find evidence of significant spatial clustering over short distances, suggesting that standard mean-field models could be inappropriate.

Experimental and Modelling Investigation of Monolayer Development with Clustering

Standard differential equation-based models of collective cell behaviour, such as the logistic growth model, invoke a mean-field assumption which is equivalent to assuming that individuals within the population interact with each other in proportion to the average population density. Implementing such assumptions implies that the dynamics of the system are unaffected by spatial structure, such as the formation of patches or clusters within the population. Recent theoretical developments have introduced a class of models, known as moment dynamics models, which aim to account for the dynamics of individuals, pairs of individuals, triplets of individuals, and so on. Such models enable us to describe the dynamics of populations with clustering, however, little progress has been made with regard to applying moment dynamics models to experimental data. Here, we report new experimental results describing the formation of a monolayer of cells using two different cell types: 3T3 fibroblast cells and MDA MB 231 breast cancer cells. Our analysis indicates that the 3T3 fibroblast cells are relatively motile and we observe that the 3T3 fibroblast monolayer forms without clustering. Alternatively, the MDA MB 231 cells are less motile and we observe that the MDA MB 231 monolayer formation is associated with significant clustering. We calibrate a moment dynamics model and a standard mean-field model to both data sets. Our results indicate that the mean-field and moment dynamics models provide similar descriptions of the 3T3 fibroblast monolayer formation whereas these two models give very different predictions for the MDA MD 231 monolayer formation. These outcomes indicate that standard mean-field models of collective cell behaviour are not always appropriate and that care ought to be exercised when implementing such a model.

Selective solubility of E7 components in poly (n-butylacrylate)

Recently, the phase diagram of poly (n-butyl acrylate) and the nematic liquid crystal (LC) mixture E7 has revealed an anomalous behaviour that was attributed to the multicomponent nature of the LC which is made of four cyanoparaphenylenes. The anomalous emergence of a nematic phase at relatively low LC concentration above 60∘C cannot be rationalized in terms of standard meanfield models commonly used to calculate phase diagrams of analogous systems. It was then suspected that the only possibility to explain this observation is to invoke the hypothesis of a selective solubility of the LC components with respect to the polymer. A detailed analysis of the composition of the LC within droplets as a function of the E7 concentration in the samples was made by HPLC. The corresponding measurements indicate unambiguously that E7 undergoes a phase separation in the presence of poly (n-butylacrylate) and exhibits a selective miscibility that is reminiscent of the preferential solvation phenomenon. HPLC chromatograms exhibit peaks that correspond to the four single compounds included in the E7 mixture and can precisely identify those present in the sample droplets. We find that both nature of components and composition depend upon samples under consideration. The overall analysis of data enables us to understand the anomalies revealed by the complete phase diagram obtained by polarized optical microscopy (POM).

Selective Solubility of E7 Components in Poly(n-butylacrylate)

Recently, the phase diagram of poly(n-butyl acrylate) and the nematic liquid crystal (LC) mixture E7 has revealed an anomalous behaviour that was attributed to the multicomponent nature of the LC which is made of four cyanoparaphenylenes. The anomalous emergence of a nematic phase at relatively low LC concentration above 60°C cannot be rationalized in terms of standard meanfield models commonly used to calculate phase diagrams of analogous systems. It was then suspected that the only possibility to explain this observation is to invoke the hypothesis of a selective solubility of the LC components with respect to the polymer. A detailed analysis of the composition of the LC within droplets as a function of the E7 concentration in the samples was made by HPLC. The corresponding measurements indicate unambigously that E7 undergoes a phase separation in the presence of poly(n-butylacrylate) and exhibits a selective miscibility that is reminiscent of the preferential solvation phenomenon. HPLC chromatograms exhibit peaks that correspond to the four single compounds included in the E7 mixture and can precisely identify those present in the sample droplets. We find that both nature of components and composition depend upon samples under consideration. The overall analysis of data enables us to understand the anomalies revealed by the complete phase diagram.

Nonlinear Fokker–Planck equation exhibiting bifurcation phenomena and generalized thermostatistics

A nonlinear Fokker–Planck equation exhibiting bifurcation phenomena is proposed within the framework of generalized thermostatistics. The nonlinearity responsible for the occurrence of bifurcation of solutions is assumed to be of the form appearing in the standard mean field model. A Liapunov function is defined that takes the form of free energy involving generalized entropies of Tsallis and an H-theorem is proved to show that the free energy, which is bounded below, continues to decrease until the system approaches one of the equilibrium distributions. The H-theorem ensures, instead of uniqueness of the equilibrium distribution, global stability of the system in that either one of multisolutions must be approached for large times. Local stability analysis is conducted and the second-order variation of the Liapunov function is computed to find its relevant part whose sign governs stability of the equilibrium distribution of the system. The case with a bistable potential is investigated, as an example of confirming the theory, to give the bifurcation diagram displaying the order parameter as a function of the coefficient of the nonlinear diffusion term.

Observable dependence of fluctuation-dissipation relations and effective temperatures.

We study the nonequilibrium fluctuation-dissipation theorem (FDT) in the glass phase of Bouchaud's trap model. We incorporate an arbitrary observable m and obtain its correlation and response functions in closed form. A limiting nonequilibrium FDT plot is approached at long times for most choices of m. In contrast to standard mean field models, however, the shape of the plot depends nontrivially on the observable, and its slope varies continuously even though there is a single scaling of relaxation times with age. Nonequilibrium FDT plots can therefore not be used to define a meaningful effective temperature T(eff) in this model. Consequences for the wider applicability of an FDT-derived T(eff) are discussed.

THE ETIOLOGY OF RICKETS

<h3>Abstract</h3> Collective cell spreading takes place in spatially continuous environments, yet it is often modelled using discrete lattice-based approaches. Here, we use data from a series of cell proliferation assays, with a prostate cancer cell line, to calibrate a spatially continuous individual based model (IBM) of collective cell migration and proliferation. The IBM explicitly accounts for crowding effects by modifying the rate of movement, direction of movement, and the rate of proliferation by accounting for pair-wise interactions. Taking a Bayesian approach we estimate the free parameters in the IBM using rejection sampling on three separate, independent experimental data sets. Since the posterior distributions for each experiment are similar, we perform simulations with parameters sampled from a new posterior distribution generated by combining the three data sets. To explore the predictive power of the calibrated IBM, we forecast the evolution of a fourth experimental data set. Overall, we show how to calibrate a lattice-free IBM to experimental data, and our work highlights the importance of interactions between individuals. Despite great care taken to distribute cells as uniformly as possible experimentally, we find evidence of significant spatial clustering over short distances, suggesting that standard mean-field models could be inappropriate.