Logistic Differential Equation Research Articles

Bayesian analysis of the effect of exosomes in a mouse xenograft model of chronic myeloid leukemia.

The aim of this work is to estimate the effect of Imatinib, exosomes, and Imatinib-exosomes mixture in chronic myeloid leukemia (CML). For this purpose, mathematical models based on Gompertzian and logistic growth differential equations were proposed. The models contained parameters representing the effects of the three components on CML proliferation. Parameters estimation was performed under the Bayesian statistical approach. Experimental data reported in the literature were used, corresponding to four trials of a human leukemia xenograft in BALB/c female rats over a period of forty days. The models were fitted to the following growth dynamics: normal tumor growth, growth with exosomes, growth with Imatinib, and growth with exosomes-Imatinib mixture. For the proposed logistic growth model, it was determined that when using Imatinib treatment the growth rate is 0.93 (95% CrI: 84.33-99.64) slower and reduces the tumor volume to approximately 10% (95% CrI : 8.67-10.81). In the presence of exosome treatment, the growth rate is 0.83 (95% CrI: 1.52-16.59) faster and the tumor volume is expanded by 40% (95% CrI: 25.36-57.28). Finally, in the presence of Imatinib-exosomes mixture treatment, the growth rate is 0.82 (95% CrI: 76.87-88.51) slower and the tumor volume is reduced by 95% (95% CrI: 86.76-99.85). It is concluded that the presence of exosomes partially inactivates the effect of the Imatinib drug on tumor growth in a mouse xenograft model.

Application of logistic differential equation models for early warning of infectious diseases in Jilin Province

BackgroundThere is still a relatively serious disease burden of infectious diseases and the warning time for different infectious diseases before implementation of interventions is important. The logistic differential equation models can be used for predicting early warning of infectious diseases. The aim of this study is to compare the disease fitting effects of the logistic differential equation (LDE) model and the generalized logistic differential equation (GLDE) model for the first time using data on multiple infectious diseases in Jilin Province and to calculate the early warning signals for different types of infectious diseases using these two models in Jilin Province to solve the disease early warning schedule for Jilin Province throughout the year.MethodsCollecting the incidence of 22 infectious diseases in Jilin Province, China. The LDE and GLDE models were used to calculate the recommended warning week (RWW), the epidemic acceleration week (EAW) and warning removed week (WRW) for acute infectious diseases with seasonality, respectively.ResultsFive diseases were selected for analysis based on screening principles: hemorrhagic fever with renal syndrome (HFRS), shigellosis, mumps, Hand, foot and mouth disease (HFMD), and scarlet fever. The GLDE model fitted the above diseases better (0.80 ≤ R2 ≤ 0.94, P < 0. 005) than the LDE model. The estimated warning durations (per year) of the LDE model for the above diseases were: weeks 12–23 and 40–50; weeks 20–36; weeks 15–24 and 43–52; weeks 26–34; and weeks 16–25 and 41–50. While the durations of early warning (per year) estimated by the GLDE model were: weeks 7–24 and 36–51; weeks 13–37; weeks 11–26 and 39–54; weeks 23–35; and weeks 12–26 and 40–50.ConclusionsCompared to the LDE model, the GLDE model provides a better fit to the actual disease incidence data. The RWW appeared to be earlier when estimated with the GLDE model than the LDE model. In addition, the WRW estimated with the GLDE model were more lagged and had a longer warning time.

Selective sweeps in SARS-CoV-2 variant competition

The main mathematical result in this paper is that change of variables in the ordinary differential equation (ODE) for the competition of two infections in a Susceptible-Infected-Removed (SIR) model shows that the fraction of cases due to the new variant satisfies the logistic differential equation, which models selective sweeps. Fitting the logistic to data from the Global Initiative on Sharing All Influenza Data (GISAID) shows that this correctly predicts the rapid turnover from one dominant variant to another. In addition, our fitting gives sensible estimates of the increase in infectivity. These arguments are applicable to any epidemic modeled by SIR equations.

Optimality and sustainability of delayed impulsive harvesting

We consider a logistic differential equation subject to impulsive delayed harvesting, where the deduction information is a function of the population size at the time of one of the previous impulses. A close connection to the dynamics of high-order difference equations is used to conclude that while the inclusion of a delay in the impulsive condition does not impact the optimality of the yield, sustainability may be highly affected and is generally delay-dependent. Maximal and other types of yields are explored, and sharp stability tests are obtained for the model, as well as explicit sufficient conditions. It is also shown that persistence of the solution is not guaranteed for all positive initial conditions, and extinction in finite time is possible, as is illustrated in the simulations.

Online inference of lane changing events for connected and automated vehicle applications with analytical logistic diffusion stochastic differential equation

Inferences of the lane changing behaviors (including lane changing initiation moment LCIM, cross-lane-mark moment CLMM, and lane changing duration LCD) of the surrounding vehicles and taking actions to avoid collision are essential for the safe operation of connected and autonomous vehicles (CAVs). The majority of current models rely on data-driven methods that need to be trained before deployment. Besides, an analytical and stochastic formulation of the lateral model which can generate CLMM and LCD distribution and adjust the parameters online is still lacking. To the best of our knowledge, currently, no method can provide an analytically stochastic lateral movement that explicitly considers the system noise and simultaneously outputs the information of lane changing initiation moment, cross-lane-mark moment, and lane changing duration in a real-time manner. To fill such a gap, a stochastic lateral trajectory framework with parsimonious parameters is established and an online simultaneous inference of LCIM, CLMM, and LCD is developed. The proposed method is tested with the highD and NGSIM datasets. The results show that 1) computational efficiency-wise, the algorithm takes milliseconds to run, which suggests promising prospects for field deployment; 2) the false negative and false positive errors of the LCIM inference are as low as 1.5%, indicating that the method is robust to the stochastic noise; 3) LCIM inference is insensitive to the lane width, which means the model is error tolerant when such information is lacking; and 4) the error of CLMM inference is within 2 sec, while most of the error of LCD locates within 4 sec, suggesting satisfactory performance.

On the generalized logistic random differential equation: Theoretical analysis and numerical simulations with real-world data

Based on the previous literature about the random logistic and Gompertz models, the aim of this paper is to extend the investigations to the generalized logistic differential equation in the random setting. First, this is done by rigorously constructing its solution in two different ways, namely, the sample-path approach and the mean-square calculus. Secondly, the probability density function at each time instant is derived in two ways: by applying the random variable transformation technique and by solving the associated Liouville’s partial differential equation. It is also proved that both the stochastic solution and its density function converge, under specific conditions, to the corresponding solution and density function of the logistic and Gompertz models, respectively. The investigation finishes showing some examples, where a number of computational techniques are combined to construct reliable approximations of the probability density of the stochastic solution. In particular, we show, step-by-step, how our findings can be applied to a real-world problem.

Reliable and efficient parameter estimation using approximate continuum limit descriptions of stochastic models

Stochastic individual-based mathematical models are attractive for modelling biological phenomena because they naturally capture the stochasticity and variability that is often evident in biological data. Such models also allow us to track the motion of individuals within the population of interest. Unfortunately, capturing this microscopic detail means that simulation and parameter inference can become computationally expensive. One approach for overcoming this computational limitation is to coarse-grain the stochastic model to provide an approximate continuum model that can be solved using far less computational effort. However, coarse-grained continuum models can be biased or inaccurate, particularly for certain parameter regimes. In this work, we combine stochastic and continuum mathematical models in the context of lattice-based models of two-dimensional cell biology experiments by demonstrating how to simulate two commonly used experiments: cell proliferation assays and barrier assays. Our approach involves building a simple statistical model of the discrepancy between the expensive stochastic model and the associated computationally inexpensive coarse-grained continuum model. We form this statistical model based on a limited number of expensive stochastic model evaluations at design points sampled from a user-chosen distribution, corresponding to a computer experiment design problem. With straightforward design point selection schemes, we show that using the statistical model of the discrepancy in tandem with the computationally inexpensive continuum model allows us to carry out prediction and inference while correcting for biases and inaccuracies due to the continuum approximation. We demonstrate this approach by simulating a proliferation assay, where the continuum limit model is the well-known logistic ordinary differential equation, as well as a barrier assay where the continuum limit model is closely related to the well-known Fisher-KPP partial differential equation. We construct an approximate likelihood function for parameter inference, both with and without discrepancy correction terms. Using maximum likelihood estimation, we provide point estimates of the unknown parameters, and use the profile likelihood to characterise the uncertainty in these estimates and form approximate confidence intervals. For the range of inference problems considered, working with the continuum limit model alone leads to biased parameter estimation and confidence intervals with poor coverage. In contrast, incorporating correction terms arising from the statistical model of the model discrepancy allows us to recover the parameters accurately with minimal computational overhead. The main tradeoff is that the associated confidence intervals are typically broader, reflecting the additional uncertainty introduced by the approximation process. All algorithms required to replicate the results in this work are written in the open source Julia language and are available at GitHub.

Modeling synergistic effects by using general Hill-type response surfaces describing drug interactions

The classification of effects caused by mixtures of agents as synergistic, antagonistic or additive depends critically on the reference model of ’null interaction’. Two main approaches to describe co-operative effects are currently in use, the Additive Dose (ADM) or concentration addition (CA) and the Multiplicative Survival (MSM) or independent action (IA) models. Recently we proposed an approach which describes ’zero-interaction’ surfaces based on the only requirement that simultaneous administration of different drugs leads to Hill-type response surfaces, which are solutions of the underlying logistic differential equations. No further assumptions, neither on mechanisms of action nor on limitations of parameter combinations are required. This defines—and limits—the application range of our approach. Resting on the same principle, we extend this ansatz in the present paper in order to describe deviations from the reference surface by generalized Hill-type functions. To this end we introduce two types of parameters, perturbations of the pure drug Hill-parameters and interaction parameters that account for n-tuple interactions between all components of a mixture. The resulting ‘full-interaction’ response surface is a valid solution of the basic partial differential equation (PDE), satisfying appropriate boundary conditions. This is true irrespective of its actual functional form, as within our framework the number of parameters is not fixed. We start by fitting the experimental data to the ‘full-interaction’ model with the maximum possible number of parameters. Guided by the fit-statistics, we then gradually remove insignificant parameters until the optimum response surface model is obtained. The ’full-interaction’ Hill response surface ansatz can be applied to mixtures of n compounds with arbitrary Hill parameters including those describing baseline effects. Synergy surfaces, i.e., differences between full- and null-interaction models, are used to identify dose-combinations showing peak synergies. We apply our approach to binary and ternary examples from the literature, which range from mixtures behaving according to the null-interaction model to those showing strong synergistic or antagonistic effects. By comparing ’null-’ and ’full-response’ surfaces we identify those dose-combinations that lead to maximum synergistic or antagonistic effects. In one example we identify both synergistic and antagonistic effects simlutaneously, depending on the dose-ratio of the components. In addition we show that often the number of parameters necessary to describe the response can be reduced without significantly affecting the accuracy. This facilitates an analysis of the synergistic effects by focussing on the main factors causing the deviations from ’null-interaction’.

Fractional Euler numbers and generalized proportional fractional logistic differential equation

We solve a logistic differential equation for generalized proportional Caputo fractional derivative. The solution is found as a fractional power series. The coefficients of that power series are related to the Euler polynomials and Euler numbers as well as to the sequence of Euler’s fractional numbers recently introduced. Some numerical approximations are presented to show the good approximations obtained by truncating the fractional power series. This generalizes previous cases including the Caputo fractional logistic differential equation and Euler’s numbers.

Positive piecewise continuous quasi-periodic solutions to logistic impulsive differential equations

ABSTRACT To prove the existence of piecewise continuous solutions to a logistic quasi-periodic differential system with impulses (whose coefficients have rationally independent periods), this system is divided into a differential equation and a difference equation. The quasi-periodicity of a function is proved by showing that this function is the uniform limit of a series of trigonometric polynomials with a finite total number of frequencies. The asymptotically stable quasi-periodic positive and piecewise continuous solution is proved to exist and to be unique. Quasi-periodic variation of the environment leads to a quasi-periodic growth of the population size in the sense that the rationally independent frequencies of the system are also frequencies of the quasi-periodic solution. The positive solutions have a repeated behavior similar to that of the quasi-periodic solution for a sufficiently long time due to asymptotical stability. The separation of the continuous-discrete system into a differential equation and a difference equation is a method of proving the existence of a quasi-periodic solution with perturbed coefficients of the impulsive system.

An Easy-to-Use Public Health-Driven Method (the Generalized Logistic Differential Equation Model) Accurately Simulated COVID-19 Epidemic in Wuhan and Correctly Determined the Early Warning Time.

IntroductionModeling on infectious diseases is significant to facilitate public health policymaking. There are two main mathematical methods that can be used for the simulation of the epidemic and prediction of optimal early warning timing: the logistic differential equation (LDE) model and the more complex generalized logistic differential equation (GLDE) model. This study aimed to compare and analyze these two models.MethodsWe collected data on (coronavirus disease 2019) COVID-19 and four other infectious diseases and classified the data into four categories: different transmission routes, different epidemic intensities, different time scales, and different regions, using R2 to compare and analyze the goodness-of-fit of LDE and GLDE models.ResultsBoth models fitted the epidemic curves well, and all results were statistically significant. The R2 test value of COVID-19 was 0.924 (p < 0.001) fitted by the GLDE model and 0.916 (p < 0.001) fitted by the LDE model. The R2 test value varied between 0.793 and 0.966 fitted by the GLDE model and varied between 0.594 and 0.922 fitted by the LDE model for diseases with different transmission routes. The R2 test values varied between 0.853 and 0.939 fitted by the GLDE model and varied from 0.687 to 0.769 fitted by the LDE model for diseases with different prevalence intensities. The R2 test value varied between 0.706 and 0.917 fitted by the GLDE model and varied between 0.410 and 0.898 fitted by the LDE model for diseases with different time scales. The GLDE model also performed better with nation-level data with the R2 test values between 0.897 and 0.970 vs. 0.731 and 0.953 that fitted by the LDE model. Both models could characterize the patterns of the epidemics well and calculate the acceleration weeks.ConclusionThe GLDE model provides more accurate goodness-of-fit to the data than the LDE model. The GLDE model is able to handle asymmetric data by introducing shape parameters that allow it to fit data with various distributions. The LDE model provides an earlier epidemic acceleration week than the GLDE model. We conclude that the GLDE model is more advantageous in asymmetric infectious disease data simulation.

Property of Gompertz data through convex of upper limit estimated with logistic difference equation

Property of Gompertz data through convex of upper limit estimated with logistic difference equation

Inference for a discretized stochastic logistic differential equation and its application to biological growth

In this paper, we present a method to adjust a stochastic logistic differential equation (SLDE) to a set of highly sparse real data. We assume that the SLDE have two unknown parameters to be estimated. We calculate the Maximum Likelihood Estimator (MLE) to estimate the intrinsic growth rate. We prove that the MLE is strongly consistent and asymptotically normal. For estimating the diffusion parameter, the quadratic variation of the data is used. We validate our method with several types of simulated data. For more realistic cases in which we observe discretizations of the solution, we use diffusion bridges and the stochastic expectation-maximization algorithm to estimate the parameters. Furthermore, when we observe only one point for each path for a given number of trajectories we were still able to estimate the parameters of the SLDE. As far as we know, this is the first attempt to fit stochastic differential equations (SDEs) to these types of data. Finally, we apply our method to real data coming from fishery. The proposed adjustment method can be applied to other examples of SDEs and is highly applicable in several areas of science, especially in situations of sparse data.

Applications of logistic and generalized logistic difference equations in economics: AK model

In this paper, we demonstrate how logistic equation and generalized logistic equation can be applied in teaching economics with the aim that students easily grasp the AK model. To this end, the AK model is firstly modified into three versions, after which the first and the second model are reduced to the generalized logistic equations. The purpose of this paper is to present how through applying the mathematical modifications of mentioned versions of AK models, they become more cognizable to students, which should result by more deeply understanding of well-known economic models.

Studying the Logistic Model

Several studies have used the logistic equation to model the growth of cancer cell populations 1 as seen in Eq. ( 1 ). This has included correlated multiplicative, [Formula: see text] and additive, [Formula: see text], noise terms. These noise terms can affect the growth rate, [Formula: see text], and death rate, [Formula: see text], of tumor cells and can be induced from factors such as radiotherapy or other cancer treatments. Depending on the intensity of the noise the terms, the fluctuations can induce a phase transition. Noise-induced transitions of nonlinear stochastic systems have applications in the fields of physics, chemistry and biology. [Formula: see text] We study the logistic differential equation with a multiplicative noise term before and at phase transition. Computational methods used to investigate this cancer cell model include a Diffusion Entropy Analysis method and a waiting time distribution method. 2 , 3 , 4 DEA will establish the scaling of a simulated series without altering the data through detrending. We hypothesize the treatment that causes a phase transition in the logistic model will induce tumor extinction and management. Understanding how to better evaluate and study cancer cell growth models will assist in assessing the efficacy of cancer treatments. Future work will include running simulations with a modified DEA method that includes the use of stripes. 2 For better statistics, the code will be adopted to run ensembles of simulated data instead of a single series. Generating and analyzing these large datasets can be computationally expensive. Through multiprocessing and the use of a supercomputer, we believe these computational limitations can be overcome.

Evaluation of the Number of Visits to Chinese Medical Institutions Using a Logistic Differential Equation Model

In this study, we established a two-dimensional logistic differential equation model to study the number of visits in Chinese PHCIs and hospitals based on the behavior of patients. We determine the model's equilibrium points and analyze their stability and then use China medical services data to fit the unknown parameters of the model. Finally, the sensitivity of model parameters is evaluated to determine the parameters that are susceptible to influence the system. The results indicate that the system corresponds to the zero-equilibrium point, the boundary equilibrium point, and the positive equilibrium point under different parameter conditions. We found that, in order to substantially increase visits to PHCIs, efforts should be made to improve PHCI comprehensive capacity and maximum service capacity.

Fractional-Order Logistic Differential Equation with Mittag–Leffler-Type Kernel

In this paper, we consider the Prabhakar fractional logistic differential equation. By using appropriate limit relations, we recover some other logistic differential equations, giving representations of each solution in terms of a formal power series. Some numerical approximations are implemented by using truncated series.

Solving differential equations via artificial neural networks: Findings and failures in a model problem

In this work, we discuss some pitfalls when solving differential equations with neural networks. Due to the highly nonlinear cost functional, local minima might be approximated by which functions may be obtained, that do not solve the problem. The main reason for these failures is a sensitivity on initial guesses for the nonlinear iteration. We apply known algorithms and corresponding implementations, including code snippets, and present an example and counter example for the logistic differential equations. These findings are further substantiated with variations in collocation points and learning rates.

An alternative delayed population growth difference equation model.

We propose an alternative delayed population growth difference equation model based on a modification of the Beverton-Holt recurrence, assuming a delay only in the growth contribution that takes into account that those individuals that die during the delay, do not contribute to growth. The model introduced differs from a delayed logistic difference equation, known as the delayed Pielou or delayed Beverton-Holt model, that was formulated as a discretization of the Hutchinson model. The analysis of our delayed difference equation model identifies a critical delay threshold. If the time delay exceeds this threshold, the model predicts that the population will go extinct for all non-negative initial conditions. If the delay is below this threshold, the population survives and its size converges to a positive globally asymptotically stable equilibrium that is decreasing in size as the delay increases. We show global asymptotic stability of the positive equilibrium using two different techniques. For one set of parameter values, a contraction mapping result is applied, while the proof for the remaining set of parameter values, relies on showing that the map is eventually componentwise monotone.

Meromorphic solutions of delay differential equations related to logistic type and generalizations

Let {bj}j=1k be meromorphic functions, and let w be admissible meromorphic solutions of delay differential equationw′(z)=w(z)[P(z,w(z))Q(z,w(z))+∑j=1kbj(z)w(z−cj)] with distinct delays c1,…,ck∈C∖{0}, where the two nonzero polynomials P(z,w(z)) and Q(z,w(z)) in w with meromorphic coefficients are prime each other. We obtain that if limsupr→∞log⁡T(r,w)r=0, thendegw⁡(P/Q)≤k+2. Furthermore, if Q(z,w(z)) has at least one nonzero root, then degw⁡(P)=degw⁡(Q)+1≤k+2; if all roots of Q(z,w(z)) are nonzero, then degw⁡(P)=degw⁡(Q)+1≤k+1; if degw⁡(Q)=0, then degw⁡(P)≤1.In particular, whenever degw⁡(Q)=0 and degw⁡(P)≤1 and without the growth condition, any admissible meromorphic solution of the above delay differential equation (called Lenhart-Travis' type logistic delay differential equation) with reduced form can not be an entire function w satisfying N‾(r,1w)=O(N(r,1w)); while if all coefficients are rational functions, then the condition N‾(r,1w)=O(N(r,1w)) can be omitted. Furthermore, any admissible meromorphic solution of the logistic delay differential equation (that is, for the simplest special case where k=1 and degw⁡(P/Q)=0) satisfies that N(r,w) and T(r,w) have the same growth category. Some examples support our results.