Logistic Differential Equation Research Articles

Power-series solution of the L-fractional logistic equation

We consider the L-fractional derivative, which has been proposed in the literature to study fractional differentials in geometry and processes in mechanics. Our context is population growth and epidemiology, for which the use of L-derivatives is motivated by transitions. Using power series, we solve the logistic differential equation model under this fractional derivative. Several conclusions on the method, the derivative, and the singularity of the associated kernel are reached. Fractional Euler numbers, related to the logistic map and the famous Riemann zeta function, are also introduced.

A new solution of the nonlinear fractional logistic differential equations utilizing efficient techniques

The formulation of models and solutions for various physical problems are the primary goals of scientific achievements in engineering and physics. Our paper focuses on using the Caputo fractional derivative operator to solve nonlinear fractional logistic differential equations. In order to solve general nonlinear fractional differential equations, we first introduce a novel numerical methodology termed the Homotopy perturbation transform method. The perturbation approach and the Yang transform method are combined to create the suggested strategy. Second, we introduce a new hybrid method that uses the time-fractional Caputo derivative to approximate and analytically solve nonlinear fractional logistic differential equations. This method combines the Yang transform with the decomposition method. To validate the analysis, we offer three numerical cases of nonlinear fractional logistic differential equations employing the Caputo fractional derivative operator. The resulting solutions exhibit rapid convergence and are presented in series form. In order to verify the efficacy and relevance of the suggested methodologies, the investigated issues were assessed through the implementation of different fractional orders. We examine and show that, under the specified initial conditions, the solution approaches under evaluation are accurate and effective. Graphs in two and three dimensions show the results that were obtained. Numerical simulations are presented to confirm the efficacy of the strategies. The numerical results show that an accurate, reliable, and efficient approximation can be obtained with a minimal number of terms. The results obtained demonstrate that the new analytical solution method is easy to apply and very successful in solving difficult fractional problems that occur in relevant engineering and scientific domains.

How Ecologists Develop the Logistic Model

ABSTRACT The logistic differential equation is ubiquitous in calculus and differential equations textbooks. If the model is developed from first principles in these courses, it is usually done in an abstract mathematical way, rather than in one based in ecology. In this short note, we look at examples of how the model is derived in mathematical texts and compare them to that in an ecology text.

Solution of logistic differential equation in an uncertain environment using neutrosophic numbers

The modeling and forecasting of population dynamics, as well as growth in biological systems more generally, have required the construction of various growth models. This paper presents the logistic growth model, which is a modified version of the exponential growth model and offers a more flexible framework to capture fluctuations and potential deviations from deterministic predictions. This paper aims to advance the model of a logistic growth differential equation within a neutrosophic environment, with a focus on validating the proposed framework by applying it to predict the projected population of India from the year 2001 to 2050. We investigated the neutrosophic solutions and validated them with the exact solution with available projected data on the portal of UNITED NATION. The developed logistic growth differential equation in a neutrosophic environment is visually elucidated through graphical representation, enhancing the clarity and accessibility of the presented model.

A new decomposition transform method for solving nonlinear fractional logistic differential equation

A new decomposition transform method for solving nonlinear fractional logistic differential equation

Modelling the historic total global human population

The historic total global human population dataset is available on Wikipedia and provides an opportunity for modelling with simple models such as the exponential and logistic differential equations for population. Using the per-capita population growth rate (PPGR) predicted by these two models and estimated PPGR from the data, we are able to estimate parameters for the exponential model and discover that we cannot estimate parameters for the logistic model. However, we are able to create a new differential equation that models this data using ideas from the derivation of the logistic model. This is the superexponential model that goes to infinity in finite time. While the initial exponential and superexponential models are good for approximating the population data, neither really matches the PPGRs estimated from the data. We switch to a hybrid model that better matches the estimated PPGRs. It is superexponential for the first portion of the data and logistic for the rest. This hybrid model greatly reduces the modelling error and can extrapolate from the data. We note that the model switches between superexponential and logistic around 1960, the year the FDA-approved oral contraceptives, a fascinating historical tie-in with our modelling.

Modelling the leaping cycle by modified Lotka-Volterra equations with applications to technology and safety

ABSTRACT A bounded version of the Lotka – Volterra model based on logistic differential equations is suggested to emulate cyclic behaviour in complex evolutionary and other systems. The model provides reasonable approximations for the so-called leaping cycles and other oscillational processes when the main state variables are limited by physical constraints. The model represents a basic model-building block that can be used in system dynamics and other frameworks and applications that require fundamental understanding and modelling of principal features of complex cyclic processes. Two application areas – technological progress and industrial safety – are specifically considered here. The model is shown to emulate techno-economic cycles reasonably well and give an appropriate qualitative description of risk and safety variations in the industrial environment. Simulations using the suggested model are quantitatively compared with the results of agent-based simulations. While a simple model cannot match all details of complex agent-based simulations of the risk and benefit dilemma, the bounded Lotka – Volterra model still gives a reasonable approximation of the process. The conceptual implications of the model are, nevertheless, significant, pointing to common cyclic mechanisms taking place in complex evolutionary systems that may belong to different branches of science.

Data-Driven Deep Learning Neural Networks for Predicting the Number of Individuals Infected by COVID-19 Omicron Variant.

Infectious disease epidemics are challenging for medical and public health practitioners. They require prompt treatment, but it is challenging to recognize and define epidemics in real time. Knowing the prediction of an infectious disease epidemic can evaluate and prevent the disease's impact. Mathematical models of epidemics that work in real time are important tools for preventing disease, and data-driven deep learning enables practical algorithms for identifying parameters in mathematical models. In this paper, the SIR model was reduced to a logistic differential equation involving a constant parameter and a time-dependent function. The time-dependent function leads to constant, rational, and birational models. These models use several constant parameters from the available data to predict the time and number of people reported to be infected with the COVID-19 Omicron variant. Two out of these three models, rational and birational, provide accurate predictions for countries that practice strict mitigation measures, but fail to provide accurate predictions for countries that practice partial mitigation measures. Therefore, we introduce a time-series model based on neural networks to predict the time and number of people reported to be infected with the COVID-19 Omicron variant in a given country that practices both partial and strict mitigation measures. A logistics-informed neural network algorithm was also introduced. This algorithm takes as input the daily and cumulative number of people who are reported to be infected with the COVID-19 Omicron variant in the given country. The algorithm helps determine the analytical solution involving several constant parameters for each model from the available data. The accuracy of these models is demonstrated using error metrics on Omicron variant data for Portugal, Italy, and China. Our findings demonstrate that the constant model could not accurately predict the daily or cumulative infections of the COVID-19 Omicron variant in the observed country because of the long series of existing data of the epidemics. However, the rational and birational models accurately predicted cumulative infections in countries adopting strict mitigation measures, but they fell short in predicting the daily infections. Furthermore, both models performed poorly in countries with partial mitigation measures. Notably, the time-series model stood out for its versatility, effectively predicting both daily and cumulative infections in countries irrespective of the stringency of their mitigation measures.

Validation of computational models to characterize cumulative intake curves from video-coded meals.

Observational coding of eating behaviors (e.g., bites, eating rate) captures behavioral characteristics but is limited in its ability to capture dynamic patterns (e.g., temporal changes) across a meal. While the Universal Eating Monitor captures dynamic patterns of eating through cumulative intake curves, it is not commonly used in children due to strict behavioral protocols. Therefore, the objective of this study was to test the ability of computational models to characterize cumulative intake curves from video-coded meals without the use of continuous meal weight measurement. Cumulative intake curves were estimated using Kisslieff's Quadratic model and Thomas's logistic ordinary differential equation (LODE) model. To test if cumulative intake curves could be characterized from video-coded meals, three different types of data were simulated: (1) Constant Bite: simplified cumulative intake data; (2) Variable Bite: continuously measured meal weight data; and (3) Bite Measurement Error: video-coded meals that require the use of average bite size rather than measured bite size. Performance did not differ by condition, which was assessed by examining model parameter recovery, goodness of fit, and prediction error. Therefore, the additional error incurred by using average bite size as one would with video-coded meals did not impact the ability to accurately estimate cumulative intake curves. While the Quadratic and LODE models were comparable in their ability to characterize cumulative intake curves, the LODE model parameters were more distinct than the Quadradic model. Greater distinctness suggests the LODE model may be more sensitive to individual differences in cumulative intake curves. Characterizing cumulative intake curves from video-coded meals expands our ability to capture dynamic patterns of eating behaviors in populations that are less amenable to strict protocols such as children and individuals with disordered eating. This will improve our ability to identify patterns of eating behavior associated with overconsumption and provide new opportunities for treatment.

On Solutions of Fractional Logistic Differential Equations

On Solutions of Fractional Logistic Differential Equations

Implicit solution for logistic Caputo-Fabrizio fractional differential equation with Allee effect

In the current research work, the fractional version of logistic differential model is studied with Allee effect which includes a threshold population. It is also studied with initial conditions. The definition of the fractional derivative utilised in this research is the Caputo-Fabrizio. In order to give the solution of the posed problem by an implicit representation, a method based on fractional derivative properties is used.

Modeling the biological growth with a random logistic differential equation

We modeled biological growth using a random differential equation (RDE), where the initial condition is a random variable, and the growth rate is a suitable stochastic process. These assumptions let us obtain a model that represents well the random growth process observed in nature, where only a few individuals of the population reach the maximal size of the species, and the growth curve for every individual behaves randomly. Since we assumed that the initial condition is a random variable, we assigned a priori density, and we performed Bayesian inference to update the initial condition’s density of the RDE. The Karhunen–Loeve expansion was then used to approximate the random coefficient of the RDE. Then, using the RDE’s approximations, we estimated the density f(p, t). Finally, we fitted this model to the biological growth of the giant electric ray (or Cortez electric ray) Narcine entemedor. Simulations of the solution of the random logistic equation were performed to construct a curve that describes the solutions’ mean for each time. As a result, we estimated confidence intervals for the mean growth that described reasonably well the observed data. We fit the proposed model with a training dataset, and the model is tested with a different dataset. The model selection is performed with the square of the errors.

Verhulst Discrete Logistic Growth

Summary In undergraduate mathematics classes, the most common discrete version of logistic growth is defined by the difference equation . While this is a natural analog of the logistic differential equation, and while in many cases it produces results similar to those of the continuous model, it can also give rise to chaotic behavior. This paper derives in a natural way an alternative discrete logistic model, defined by the Verhulst difference equation, with several noteworthy properties. For example the Verhulst equation has closed form solutions given by continuous logistic curves and never leads to chaotic behavior. Our development of the Verhulst equation also provides a beautiful example of the formulation-application-refinement cycle of mathematical modeling. For these and other reasons, the Verhulst equation deserves a place in the undergraduate curriculum alongside the more familiar logistic difference equation given above.

A generalized feedback control model for the logistic differential equation

A generalized feedback control model for the logistic differential equation

A POWER SERIES METHOD FOR THE FUZZY FRACTIONAL LOGISTIC DIFFERENTIAL EQUATION

Power series, as an important means to analyze functions in different complex settings, are employed in various applied areas to solve differential equations and nonlinear problems and provide the assessment of intervals of convergence. Accordingly, the fuzzy logistic differential equation using the Caputo operator has been studied in this paper. Accordingly, the fuzzy logistic differential equation using the Caputo operator has been studied in this paper. The generalized Hukuhara difference and the generalized Hukuhara derivative are also used, and a power series representation is proposed for the solution of the fuzzy fractional logistic equation. Afterward, power series coefficients are obtained using a recursive formula. Finally, numerical experiments and illustrated results of the computations are presented to allow for more realistic decisions reflecting high complexity and underlying uncertainty. Thus, the numerical computations in our study reveal the effectiveness and accuracy of the power series method. Therefore, it is found that the fuzzy solution converges to the deterministic solution when uncertainty decreases, and, based on the technical analyses, it has been demonstrated that the results obtained are more fundamental in preventing geometric growth in nonlinear phenomena where uncertainties emerge due to impreciseness and inexactness.

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.