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We develop a full randomization of the classical hyper‐logistic growth model by obtaining closed‐form expressions for relevant quantities of interest, such as the first probability density function of its solution, the time until a given fixed population is reached, and the population at the inflection point. These results are obtained under very general hypotheses on the distributions of the random model parameters by taking extensive advantage of the so‐called random variable transformation method. To illustrate the practical implications of our findings, we apply them to model the growth of multicellular tumor spheroids using empirical data. In this context, we explore two methodologies—the Bayesian approach and the random least mean square method—aimed at effectively addressing the challenge of assigning appropriate distributions to model parameters. This ensures that probabilistic fits accurately capture the inherent uncertainties of tumor growth dynamics. Finally, we notably show that the results obtained using both approaches in the randomized hyper‐logistic model align closely with each other, surpassing those yielded by the randomized logistic model. [ABSTRACT FROM AUTHOR]

We study, from a probabilistic standpoint, a full randomization of nonhomogeneous second‐order linear difference equations assuming that its data (initial conditions, coefficients, and forcing term) are random variables. Our analysis consists of computing the so‐called first probability density function of the solution, which is a stochastic process, and then analyzing the stability of the solution assuming that all data have an arbitrary joint probability density function. To achieve these goals, we take extensive advantage of the so‐called random variable transformation technique. The theoretical results extend their deterministic counterpart, and then, they have many applications in real‐world problems where uncertainty plays a key role. Our findings are first illustrated by means of several numerical examples, where different simulations are carried out, and, second, by means of a model belonging to biomathematics. [ABSTRACT FROM AUTHOR]

An open problem in diabetes clinical practice is determining where and how much insulin should be administered to a person with diabetes (PwD) and how many carbohydrates they should eat to maintain blood glucose levels at healthy safe levels. Here, we propose the use of a minimal model describing the glucose dynamics of PwD. Using glucose Pwd's data, we calibrate the minimal model considering the uncertainty due to errors in glucose measurement, finding the model parameter values that best reproduce the current glucose levels. Then, all the possible combinations of insulin administration and carbohydrate intake are analysed with the aim of maintaining the glucose at safe levels during the following hours. The resulting procedure is tested with data from two real persons with scenarios of the most typical situations. We expect to apply this procedure in more complex models to help the physicians to give suitable recommendations to PwD. [ABSTRACT FROM AUTHOR]

The Baranyi--Roberts model describes the dynamics of the volumetric densities of two interacting cell populations. We randomize this model by considering that the initial conditions are random variables whose distributions are determined by using sample data and the principle of maximum entropy. Subsequenly, we obtain the Liouville--Gibbs partial differential equation for the probability density function of the two-dimensional solution stochastic process. Because the exact solution of this equation is unaffordable, we use a finite volume scheme to numerically approximate the aforementioned probability density function. From this key information, we design an optimization procedure in order to determine the best growth rates of the Baranyi--Roberts model, so that the expectation of the numerical solution is as close as possible to the sample data. The results evidence good fitting that allows for performing reliable predictions. [ABSTRACT FROM AUTHOR]

In this paper, we deal with the randomized generalized diffusion equation with delay: ut(t, x) = a2uxx(t, x) + b2uxx(t − τ, x), t > τ, 0 ≤ x ≤ l; u(t,0)=u(t,l)=0, t ≥ 0; u(t,x)=φ(t,x), 0 ≤ t ≤ τ, 0 ≤ x ≤ l. Here, τ > 0 and l > 0 are constant. The coefficients a2 and b2 are nonnegative random variables, and the initial condition φ(t, x) and the solution u(t, x) are random fields. The separation of variables method develops a formal series solution. We prove that the series satisfies the delay diffusion problem in the random Lebesgue sense rigorously. By truncating the series, the expectation and the variance of the random‐field solution can be approximated. [ABSTRACT FROM AUTHOR]

This paper deals with the randomized heat equation defined on a general bounded interval [L1, L2] and with nonhomogeneous boundary conditions. The solution is a stochastic process that can be related, via changes of variable, with the solution stochastic process of the random heat equation defined on [0,1] with homogeneous boundary conditions. Results in the extant literature establish conditions under which the probability density function of the solution process to the random heat equation on [0,1] with homogeneous boundary conditions can be approximated. Via the changes of variable and the Random Variable Transformation technique, we set mild conditions under which the probability density function of the solution process to the random heat equation on a general bounded interval [L1, L2] and with nonhomogeneous boundary conditions can be approximated uniformly or pointwise. Furthermore, we provide sufficient conditions in order that the expectation and the variance of the solution stochastic process can be computed from the proposed approximations of the probability density function. Numerical examples are performed in the case that the initial condition process has a certain Karhunen‐Loève expansion, being Gaussian and non‐Gaussian. [ABSTRACT FROM AUTHOR]

Recently, the transmission dynamics of the Human Papillomavirus (HPV) has been studied. In previous works, we have designed and implemented a computational model (agent-based simulation model) where the contagion of the HPV is described on a network of lifetime sexual partners. The run of a single simulation of this computational model, composed of a network with 500 000 nodes, takes about one hour and a half. In addition to set an adequate model, finding out the model parameters that best fit the proposed model to the available data of prevalence is a crucial goal. Taking into account that the necessary number of simulations to perform the calibration of the model may be very high, the aforementioned goal may become unaffordable. In this paper, we present a procedure to fit the proposed HPV model to the available data and the design of an asynchronous version of the Particle Swarm Optimization (PSO) algorithm adapted to the distributed computing environment. In the process, the number of particles used in PSO should be set carefully looking for a compromise between quality of the solutions and computation time. Another feature of the procedure presented here is that we want to capture the intrinsic uncertainty in the data (data come from a survey) when calibrating the model. To do so, we also propose the design of an algorithm to select the model parameter sets obtained during the calibration that best capture the data uncertainty. • We use a non-deterministic model which make necessary to treat uncertainty. • The objective function is not the typical differentiable and closed-form function. • We use the determination of an appropriate measure of goodness-of-fit. • We find model parameter values matching the figures reported in medical studies. • Our method is reliable and reproducible. • The best parallel implementation is found to obtain good solutions efficiently. [ABSTRACT FROM AUTHOR]

This paper aims at extending a previous contribution dealing with the random autonomous-homogeneous linear differential equation with discrete delay τ > 0 , by adding a random forcing term f (t) that varies with time: x ′ (t) = a x (t) + b x (t − τ) + f (t) , t ≥ 0 , with initial condition x (t) = g (t) , − τ ≤ t ≤ 0 . The coefficients a and b are assumed to be random variables, while the forcing term f (t) and the initial condition g (t) are stochastic processes on their respective time domains. The equation is regarded in the Lebesgue space L p of random variables with finite p-th moment. The deterministic solution constructed with the method of steps and the method of variation of constants, which involves the delayed exponential function, is proved to be an L p -solution, under certain assumptions on the random data. This proof requires the extension of the deterministic Leibniz's integral rule for differentiation to the random scenario. Finally, we also prove that, when the delay τ tends to 0, the random delay equation tends in L p to a random equation with no delay. Numerical experiments illustrate how our methodology permits determining the main statistics of the solution process, thereby allowing for uncertainty quantification. [ABSTRACT FROM AUTHOR]