Logistic Function Research Articles

Predictive modeling of molds effective elimination by external inactivation sources

Presented paper deals with a novel application of the (nonlinear) logistic equation to model an elimination of microscopic filaments types of fungi-molds from affected materials via different external inactivation techniques. It is shown that if the inactivation rate of the external source is greater than the maximum natural growth rate of mycelium, the mold colony becomes destroyed after a finite time. Otherwise, the mycelium may survive the external attack only at a sufficiently large initial concentration of the inoculum. Theoretically determined growth curves are compared with the experimental data for Aspergillus brasiliensis mold inactivated by using both cold atmospheric plasma (CAP) and UV-germicidal lamp. Model presented in the article may be applied also to other classes of microorganisms (e.g. bacteria).

Artificial neural network-based modeling of Malachite green adsorption onto baru fruit endocarp: insights into equilibrium, kinetic, and thermodynamic behavior

In this study, artificial neural network (ANN) tools were employed to forecast the adsorption capacity of Malachite green (MG) by baru fruit endocarp waste (B@FE) under diverse conditions, including pH, adsorbent dosage, initial dye concentration, contact time, and temperature. Enhanced adsorption efficiency was notably observed under alkaline pH conditions (pH 10). Kinetic analysis indicated that the adsorption process closely followed a pseudo-second-order model, while equilibrium studies revealed the Langmuir isotherm as the most suitable model, estimating a maximum adsorption capacity of 57.85 mg g−1. Furthermore, the chemical adsorption of MG by B@FE was confirmed using the Dubinin–Radushkevich isotherm. Thermodynamic analysis suggested that the adsorption is spontaneous and endothermic. Various ANN architectures were explored, employing different activation functions such as identity, logistic, tanh, and exponential. Based on evaluation metrics like the coefficient of determination (R 2) and root mean square error (RMSE), the optimal network configuration was identified as a 5–11-1 architecture, consisting of five input neurons, eleven hidden neurons, and one output neuron. Notably, the logistic activation function was applied in both the hidden and output layers for this configuration. This study highlights the efficacy of B@FE as an efficient adsorbent for MG removal from aqueous solutions and demonstrates the potential of ANN models in predicting adsorption behavior across varying environmental conditions, emphasizing their utility in this field.

Characterization of solutions of certain class of linear and non-linear shift equations undershared values

In this paper we have considered the generalized form of Pielou Logistic Equation and Riccati Difference equation [6] and characterize the solution of that equation in terms of shared value problem. We have improved and extended the result of Li-Chen in [8].

Microplastics and non-natural cellulosic particles in Spanish bottled drinking water

This investigation explored the presence of microplastics (MPs) and artificial cellulosic particles (ACPs) in commercial water marketed in single use 1.5 L poly(ethylene terephthalate) bottles. In this work we determined a mass concentration of 1.61 (1.10–2.88) µg/L and 1.04 (0.43–1.82) µg/L for MPs and ACPs respectively in five top-selling brands from the Spanish bottled water market. Most MPs consisted of white and transparent polyester and polyethylene particles, while most ACPs were cellulosic fibers likely originating from textiles. The median size of MPs and ACPs was 93 µm (interquartile range 76–130 µm) and 77 µm (interquartile range 60–96 µm), respectively. Particle mass size distributions were fitted to a logistic function, enabling comparisons with other studies. The estimated daily intake of MPs due to the consumption of bottled water falls within the 4–18 ng kg−1 day−1 range, meaning that exposure to plastics through bottled water probably represents a negligible risk to human health. However, it's worth noting that the concentration of plastic found was much higher than that recorded for tap water, which supports the argument in favour of municipal drinking water.

Landscape influences on thermal sensitivity and predicted spatial variability among brook trout streams in the southeastern USA

AbstractWarming water temperatures as a result of climate change pose a major threat to coldwater organisms. However, the rate of warming is not spatially uniform due to surface‐ground‐water interactions and stream and watershed characteristics. Coldwater habitats that are most resistant to warming serve as thermal refugia and identifying their locations is critical to regional aquatic conservation planning. We quantified the thermal sensitivity of 203 streams providing current and potential habitat for brook trout (Salvelinus fontinalis) across nearly 1000 linear km of their native range in the southern and central Appalachian Mountains region, USA, and characterized their spatial variability with landscape variables available in the National Hydrography Dataset. Using the Bayesian framework, we calculated the maximum slope of the logistic function relating paired weekly mean air temperature and stream temperature as an index of stream thermal sensitivity. Streams differed greatly in thermal sensitivity and those with more resistant water temperature regimes (i.e., thermal refugia) were consistently characterized by southerly latitudes and groundwater input. Landscape variables derived from a principal component analysis explained 16% of the variation in thermal sensitivity, indicating that the existing landscape variables were modestly successful in explaining spatial thermal heterogeneity. Using our model and spatial interpolation, we predicted thermal sensitivity at 8695 stream segments potentially suitable for brook trout in the study region. Thermal refugia were more common southward presumably due to higher elevations, but elsewhere they were also clustered at finer spatial scales. Our analysis informs prioritizing habitat conservation and restoration of this native salmonid and other aquatic organisms that depend on coldwater habitats in a warming world.

Quantitative insights into grapevine anthracnose (Elsinoë ampelina) epidemiology: impact of temperature and leaf age on incubation, lesion development, and sporulation.

Grapevine anthracnose, caused by Elsinoe ampelina, is one of the most devastating diseases for wine and table grapes, particularly in hot, humid regions. This study explores how temperature and leaf age affect incubation and how temperature affects lesion development and sporulation. The influence of temperature and leaf age on incubation period (days) was tested under controlled conditions. Leaves from 1 to 8 days old were inoculated and maintained at temperatures of 5, 10, 15, 20, 25, and 30°C. The time elapsed between inoculation and the emergence of initial lesions was recorded. The effect of temperature on lesion development and sporulation was investigated under vineyard conditions. This was achieved through artificial inoculations, with 17, 11, and 11 inoculations conducted in 2016, 2017, and 2018, respectively. The average incubation period, considering all leaf ages, was 27.50 days at 5°C, 15.10 days at 10°C, 9.70 days at 15°C, 5.90 days at 20°C, 3.70 days at 25°C, and 2.26 days at 30°C. Regardless of temperature, the average incubation period was 3.6, 5.9, 8.3, 9.8, 11.9, 13.4, 15.6, and 17.1 days for leaves 1, 2, 3, 4, 5, 6, 7, and 8 days old respectively. The exponential decay model accurately describes the incubation period as a function of both temperature and leaf age. On average, the relative lesion development (RLD) were 0.00, 0.00, 0.23, 0.47, 0.72, 0.93 0.92, 0.90, 0.94, and 1.0 at 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9 days after inoculation, respectively. The average relative sporulation (RSPO) was 0.03, 0.36, 0.82, 0.96, and 1.0 at 5, 10, 15, 20, and 25 days after inoculation, respectively. Both RDL and RSPO as a function of degree-days (Tbase= 0°C) since inoculation were well described by the logistic function. The rates of change in relative lesion development and relative sporulation were 0.055 and 0.032, respectively. The results of this study provide new quantitative insights into three important stages (monocyclic processes) in the development of grapevine anthracnose caused by E. ampelina.

Quantitative Relationship of Plant Height and Leaf Area Index of Spring Maize under Different Water and Nitrogen Treatments Based on Effective Accumulated Temperature

To optimize the growth management of spring maize, it is essential to understand the dynamics of plant height and leaf area index (LAI) under controlled water and nitrogen supply. This study conducted two-year field experiments (2022–2023) in Karamay, Xinjiang. Three irrigation levels (75%, 100%, and 125% of Crop Evapotranspiration (ETc)) and four nitrogen application rates (0, 93, 186, and 279 kg N/ha) were set. A logistic growth model was fitted using accumulated effective temperature as the independent variable to analyze the growth and development characteristics of spring maize under various water and nitrogen conditions. The results demonstrated that the logistic models, based on relative effective accumulated temperature, had a determination coefficient (R2) of over 0.99 and a Normalized Root Mean Square Error (NRMSE) of less than 10%. Irrigation extended the rapid growth phase of plant height, whereas nitrogen application shortened the time to enter this rapid growth phase and prolonged its duration. Irrigation increased the maximum LAI growth rate and shortened and prolonged the rapid growth phase, while nitrogen extended the duration of the rapid growth phase for LAI. The W2N2 treatment, consisting of 100% ETc irrigation and 186 kg N/ha, was identified as the optimal drip irrigation water–nitrogen combination for spring maize in the study area. Under optimal water and nitrogen supply, both the maximum growth rate and the average growth rate during the rapid growth phase were higher, requiring accumulated effective temperatures of 825.16–845.74 °C·d and 856.68–890.00 °C·d, respectively, to reach these rates. The appropriate water and nitrogen supply significantly enhanced the synergistic promotion of growth and development in spring maize. This study provides a theoretical basis for the quantitative analysis of growth dynamics in summer maize using effective accumulated temperature.

Analysis of a diffusive two-strain malaria model with the carrying capacity of the environment for mosquitoes

We propose a malaria model involving the sensitive and resistant strains, which is described by reaction-diffusion equations. The model reflects the scenario that the vector and host populations disperse with distinct diffusion rates, susceptible individuals or vectors cannot be infected by both strains simultaneously, and the vector population satisfies the logistic growth. Our main purpose is to get a threshold type result on the model, especially the interaction effect of the two strains in the presence of spatial structure. To solve this issue, the basic reproduction number (BRN) R0i and invasion reproduction number (IRN) Rˆ0i of each strain (i = 1 and 2 are for the sensitive and resistant strains, respectively) are defined. Furthermore, we investigate the influence of the diffusion rates of populations and vectors on BRNs and IRNs.

Simple mathematical model for predicting COVID-19 outbreaks in Japan based on epidemic waves with a cyclical trend

BackgroundSeveral models have been used to predict outbreaks during the COVID-19 pandemic, with limited success. We developed a simple mathematical model to accurately predict future epidemic waves.MethodsWe used data from the Ministry of Health, Labour and Welfare of Japan for newly confirmed COVID-19 cases. COVID-19 case data were summarized as weekly data, and epidemic waves were visualized and identified. The periodicity of COVID-19 in each prefecture of Japan was confirmed using time-series analysis and the autocorrelation coefficient, which was used to investigate the longer-term pattern of COVID-19 cases. Outcomes using the autocorrelation coefficient were visualized via a correlogram to capture the periodicity of the data. An algorithm for a simple prediction model of the seventh COVID-19 wave in Japan comprised three steps. Step 1: machine learning techniques were used to depict the regression lines for each epidemic wave, denoting the “rising trend line”; Step 2: an exponential function with good fit was identified from data of rising straight lines up to the sixth wave, and the timing of the rise of the seventh wave and speed of its spread were calculated; Step 3: a logistic function was created using the values calculated in Step 2 as coefficients to predict the seventh wave. The accuracy of the model in predicting the seventh wave was confirmed using data up to the sixth wave.ResultsUp to March 31, 2023, the correlation coefficient value was approximately 0.5, indicating significant periodicity. The spread of COVID-19 in Japan was repeated in a cycle of approximately 140 days. Although there was a slight lag in the starting and peak times in our predicted seventh wave compared with the actual epidemic, our developed prediction model had a fairly high degree of accuracy.ConclusionOur newly developed prediction model based on the rising trend line could predict COVID-19 outbreaks up to a few months in advance with high accuracy. The findings of the present study warrant further investigation regarding application to emerging infectious diseases other than COVID-19 in which the epidemic wave has high periodicity.

An intuitionistic fuzzy approach for prey–predator harvesting system with toxicity and time delay

This study presents an intuitionistic fuzzy prey–predator model with time delay, where several toxic substances from external sources contaminate the prey and predator species. The system consists of one prey and one predator, with the logistic growth of prey species and predation terms proportional to prey density. This system incorporates Holling type-II functional response and introduces a time delay in harvesting prey species over a precise size or age. The primary focus of this work is to investigate the impact of age or size-selective delay on the system’s stability within the intuitionistic fuzzy environment. The model employs triangular intuitionistic fuzzy numbers to account for imprecise biological parameters, and it is converted into an intuitionistic fuzzy model using the Hukuhara derivative. The fuzziness of the model is then resolved using Yager’s Ranking method. Without delay, the paper delves into the system’s dynamics, examining local and global stability and bionomic considerations. Numerical simulations are performed using MATLAB to validate the theoretical findings.

The water bimodal inherent structure and the liquid-liquid transition as proposed by the experimental density data.

The bulk water density data are studied in a very large temperature-pressure range, from stable liquid to glass in the frame of water polymorphism. Because this thermodynamic variable evidences a crossover T*, above which the hydrogen bond (HB) is unable to arrange tetrahedral networks, the T-dependence of their isobars was considered. Such an analysis also shows pressure, P*, around which their thermal behaviors are completely different: concave below P* (with maxima and minima) and convex above (without extremes). Having ρ's measured values of the bimodal structures of the liquid phase, HDL (ρHDL), made of not-bonded monomers (ρNHB) and partially bonded dimers plus trimers (ρNHB), and LDL tetramers (ρLDL) the isobars of the relative distributions [W(P, T)] of the three species (WLDL, WPHB, and WNHB) have been evaluated. The results were studied by means of a logistic function (LF) that details the evolutions of the relative polpulations of the water LDL and HDL phases by decreasing T (for the isobars, in the range of 0.1-400 MPa). The LFs analysis obtained by proposing a full connection between liquid water from its supercooled metastable phase to the stable up to the boiling temperature identifies the Widom line quite satisfactorily and fully supports the presence of the liquid-liquid critical point in the deep supercooled region, located at about 190K and in the range 200 > P > 180 MPa.

Classification of Intracranial Hemorrhage (CT) Images Using CNN- LSTM Method and Image-Based GLCM Features

In this study, a hybrid approach is utilized, which combines image-based grey-level co-occurrence matrix attributes and transformed-based attributes. The combination of attribute-based techniques is more accurate in categorizing cerebral hemorrhages in CT images when compared with the transformed attribute-based and image-attribute-based approaches. Deep learning methods, particularly long short-term memory (LSTM), have gained popularity in natural language processing applications like text and sentiment analysis. This paper presents an in-depth deep-learning system that is fully automated. The purpose of the system is to organize radiological data and identify cases of intracranial hemorrhage (ICH) for diagnosis. The structure of the proposed DL model includes 1D convolution neural networks (CNN), long short-term memory (LSTM) units, and a logistic function. A significant dataset consisting of 12,852 radiological reports of head computed tomography (CT) was used to train and assess these components.

On generalized logistic equations with non-local term of feedback control type

We consider a problem of finding functions λ(x),u(x) such that −Δpu=λf(x,u,∇u)−g(x,u) in Ω and u=0 on ∂Ω, λ(x)∈F(x,u(x)) a.e. in Ω. Assume that, the single-valued functions f,g and the multi-valued function F satisfy certain growth conditions. Using the fixed point index theory for multi-valued operators with non-necessary convex values, we prove the existence of one or two nontrivial non-negative solutions of the problem.

Methods for comparing theoretical models parameterized with field data using biological criteria and Sobol analysis

Prey–predator models are frequently developed to investigate trophic webs and to predict the population dynamics of prey and predators. However, the parameters of these models are often implemented without empirical data and may even be chosen arbitrarily. Commonly, only a few parameters are tested regarding their sensitivity and it is rare to read about the comparison between different prey–predator models (i.e. predation function structure). Here, we propose a method to compare four prey–predator models designed for two populations. We then apply this method to select the more biologically plausible one to model a simplified agricultural trophic system, including one predator compartment (the red fox Vulpes vulpes) and one prey group compartment (small mammals). These models are based on four Holling functional responses for the predation interaction and take the prey intrinsic growth rate into account through a Verhulst logistic function. Most parameters’ values (like attack rates or growth rates) were calculated from field data or based on literature review. We then used Sobol indices to conduct parameter exploration around mean parameter values to investigate and compare the model dynamics responses. Our first results showed that under our assumptions, the two most relevant models for our case study are the saturated Holling I and II models. We were also able to discriminate that among the 6 scaled parameters that vary, the model outputs are particularly sensitive to four of them (κ: saturation rate of the environment, Tr1: characteristic intrinsic decay time of the predators, Tc: characteristic growth time of the predator via the predation on the prey and λ: saturation rate of a predator’s stomach per time unit) and much less sensitive to two others (Tr2: characteristic intrinsic growth time of the prey and Ta: characteristic decay time of the prey due to the predation). These first encouraging results open the way for the next step, which will be to adapt this model construction to more complex prey–predator systems, with several predator and/or several prey compartments.

Pattern formation and delay-induced instability in a Leslie–Gower type prey–predator system with Smith growth function

In this article, we have investigated the spatiotemporal dynamics and delay-induced instability of a Leslie–Gower type prey–predator model under the influence of environmental toxicants with Smith growth function. This growth function is more realistic than logistic growth as it better describes the growth of the biological population. It has been used where the growth limitations are based on the proportion of available resources not utilized. A few works of Smith’s growth models are reported in the literature. Therefore, spatiotemporal dynamics and pattern formation with delay effect remain an exciting area of research, which motivates the present work. This work has studied two types of dynamical systems: (i) an ordinary differential temporal system with time delay and (ii) a reaction–diffusion system with time delay. The existence of equilibrium points and their stability conditions are discussed. Hopf bifurcation emerges in both proposed systems with respect to delay parameter. The stability and direction of Hopf bifurcation and delay-diffusion-driven instability have been investigated for the reaction–diffusion system. Numerical simulation is performed to support the analytical results and theorems. Moreover, the existence of Hopf and delay-induced instability are proved numerically. Interesting one-dimensional regular and irregular stripe patterns are obtained for increased values of the time delay parameter. Also, the presence of natural toxicants has a negative impact on the growth of prey–predator species.

Intracellular oxygen transient quantification in vivo during Ultra-High Dose Rate (UHDR) FLASH radiotherapy.

Large, rapid extracellular oxygen transients (ΔpO2) have been measured in vivo during Ultra-High Dose Rate (UHDR) radiotherapy; however, it has been unclear if this matches intracellular oxygen levels. Here, the endogenously produced Protoporphyrin IX (PpIX) delayed fluorescence (DF) signal was measured as an intracellular in vivo oxygen sensor to quantify these transients, with direct comparison to extracellular pO2. Intracellular ΔpO2 is closer to the cellular DNA, the site of major radiobiological damage and therefore should help elucidate radiochemical mechanisms of the FLASH effect and potentially be translated to human tissue measurement. Protoporphyrin IX (PpIX) was induced in mouse skin through intraperitoneal injection of 250mg/kg of aminolevulinic acid. The animals were also administered a 50 µL intradermal injection of 10 µM Oxyphor G4 (PdG4) for phosphorescence lifetime pO2 measurement. Paired oxygen transients were quantified on leg or flank tissues, while delivering 10 MeV electrons in 3 µs pulses at 360 Hz for a total dose of 10-28Gy. Transient reductions in pO2 were quantifiable in both PpIX delayed fluorescence and Oxyphor phosphorescence, corresponding to intracellular and extracellular pO2 values, respectively. Reponses were quantified for 10, 22, and 28 Gy doses, with ΔpO2 being proportional to dose on average. The ΔpO2 values were dependent upon initial pO2 in a logistic function. The average and standard deviations in ΔpO2 per dose were 0.56±0.18 mmHg/Gy and 0.43±0.06 mmHg/Gy for PpIX and Oxyphor, respectively, for initial pO2>20 mmHg. While there was large variability in the individual animal measurements of ΔpO2, the average values demonstrated a direct and proportional correlation between intracellular and extracellular pO2 changes, following a linear 1:1 relationship. A fundamentally new approach to measure intracellular oxygen depletion in living tissue showed that ΔpO2 transients seen during UHDR-RT matched those taken using extracellular oxygen measurement. This approach could be translated to humans to quantify intracellular ΔpO2. The measurement of these transients could potentially allow estimation of intracellular reactive oxygen species production.

Modeling Bellman-error with logistic distribution with applications in reinforcement learning

In modern Reinforcement Learning (RL) approaches, optimizing the Bellman error is a critical element across various algorithms, notably in deep Q-Learning and related methodologies. Traditional approaches predominantly employ the mean-squared Bellman error (MSELoss) as the standard loss function. However, the assumption of Bellman errors following the Gaussian distribution may oversimplify the nuanced characteristics of RL applications. In this work, we revisit the distribution of Bellman error in RL training, demonstrating that it tends to follow the Logistic distribution rather than the commonly assumed Normal distribution. We propose replacing MSELoss with a Logistic maximum likelihood function (LLoss) and rigorously test this hypothesis through extensive numerical experiments across diverse online and offline RL environments. Our findings consistently show that integrating the Logistic correction into the loss functions of various baseline RL methods leads to superior performance compared to their MSE counterparts. Additionally, we employ Kolmogorov–Smirnov tests to substantiate that the Logistic distribution offers a more accurate fit for approximating Bellman errors. This study also offers a novel theoretical contribution by establishing a clear connection between the distribution of Bellman error and the practice of proportional reward scaling, a common technique for performance enhancement in RL. Moreover, we explore the sample-accuracy trade-off involved in approximating the Logistic distribution, leveraging the Bias–Variance decomposition to mitigate excessive computational resources. The theoretical and empirical insights presented in this study lay a significant foundation for future research, potentially advancing methodologies, and understanding in RL, particularly in the distribution-based optimization of Bellman error.

Simulation of microbial growth based on Euler’s method

Microorganisms such as bacteria, fungi and yeast produce valuable metabolites when they are grown in suitable culture conditions. The cultivation condition affects the cell growth, metabolism, and product production in a sophisticated and nonlinear way. Therefore, in this research, the growth of Lactococcus lactis NZ9000 in response to the growth conditions was simulated using different growth models. The objective was to simulate the effect of temperature, agitation speed, carbon and nitrogen sources, on the cell growth using the exponential model, logistic and Monod equations. All equations were solved according to the Euler’s method using MATLAB R2021a for simulation. The experimental data used for the simulation were from literature. The accuracy of the model was expressed as percentage relative error between the maximum value of experimental and simulated data. Simulation results shows that the optimum conditions for cell growth was achieved at temperature 27°C, agitation speed of 100 rpm with glucose and peptone as the carbon and nitrogen sources respectively equation. The maximum cell concentration by logistic equation gives the lowest percentage error of 6.40% and 0.33% for the effect of temperature and agitation respectively. While Monod equation give the closest accuracy of 1.84% and 7.11% for carbon and nitrogen sources respectively. Thus, it was shown that the complexity of the microorganism growth was able to be simulated using suitable model such as logistic equation with the lowest relative error.

Bifurcation analysis and chaos for a double-strains HIV coinfection model with intracellular delays, saturated incidence and Logistic growth

In this paper, a class of virus-to-cell HIV model with intracellular delays, saturated incidence and Logistic growth is proposed to characterize the interaction between two types HIV strains, i.e., wild-type and drug-resistant strains. First, a series of threshold criteria on the locally and globally asymptotic stability of (infection-free, dominant, coexistence) equilibria are discussed based on the basic reproduction number R0. Furthermore, a detailed Hopf bifurcation analysis is performed on the coexistence equilibrium using two delays as bifurcation parameters. We find that the Hopf bifurcations induced by double-strains are evidently different and more complicated than that of single strain, the former switches from stability (periodic branches) to un-stability (chaos) more frequently and earlier than the latter since double-strains would yield more pairs of imaginary roots in the characteristic equations. Meanwhile, the total viral load of double-strains would be higher than that of single-strain as well. The emergence of drug resistance imposes either negative or positive influences on the survival of wild-type strain, which would further facilitate the transmission of HIV.

Prediction of Individual Disease Progression Including Parameter Uncertainty in Rare Neurodegenerative Diseases: The Example of Autosomal-Recessive Spastic Ataxia Charlevoix Saguenay (ARSACS)

The aim of this study was to develop a model to predict individual subject disease trajectories including parameter uncertainty and accounting for missing data in rare neurological diseases, showcased by the ultra-rare disease Autosomal-Recessive Spastic Ataxia Charlevoix Saguenay (ARSACS). We modelled the change in SARA (Scale for Assessment and Rating of Ataxia) score versus Time Since Onset of symptoms using non-linear mixed effect models for a population of 173 patients with ARSACS included in the prospective real-world multicenter Autosomal Recessive Cerebellar Ataxia (ARCA) registry. We used the Multivariate Imputation Chained Equation (MICE) algorithm to impute missing covariates, and a covariate selection procedure with a pooled p-value to account for the multiply imputed data sets. We then investigated the impact of covariates and population parameter uncertainty on the prediction of the individual trajectories up to 5 years after their last visit. A four-parameter logistic function was selected. Men were estimated to have a 25% lower SARA score at disease onset and a moderately higher maximum SARA score, and time to progression (T50) was estimated to be 35% lower in patients with age of onset over 15 years. The population disease progression rate started slowly at 0.1 points per year peaking to a maximum of 0.8 points per year (at 36.8 years since onset of symptoms). The prediction intervals for SARA scores 5 years after the last visit were large (median 7.4 points, Q1-Q3: 6.4–8.5); their size was mostly driven by individual parameter uncertainty and individual disease progression rate at that time.Graphical