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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 epidemiological model is proposed to study the transmission dynamics of the herpes virus type 2, a sexually transmitted infectious disease. This model considers two states, susceptible and infected, divides the population into sexes, assumes only heterosexual contacts and includes different transmission rates depending on whether the transmission is woman–man or man–woman. Reported and prevalence series data are retrieved from several sources. We consider the inherent data survey errors and the sensitivity of the diagnosis tests (data uncertainty). To calibrate the model to the available data and their uncertainty, a novel technique is proposed in two steps: (1) the application of the estimation of distribution algorithm (EDA) to find sets of model parameter values close to the data uncertainty and (2) the application of a selection algorithm to get a reduced number of model parameter values whose model outputs capture accurately the data uncertainty. Then, we check its robustness, and we provide a prediction of the evolution of the infected over the next 4 years. From the technical point of view, we conclude that the proposed technique to calibrate probabilistically the model is reliable and robust. Also, it is able to provide confidence intervals for the model parameter values and the predictions. From the medical point of view, the model returns that the transmission woman–man is higher than the man–woman, according to recent literature, and there is a mild increasing trend in the number of infected people over the next years. [ABSTRACT FROM AUTHOR]

This paper presents a methodology to deal with the randomness associated to ecological modelling. Data variability makes it necessary to analyse the impact of random perturbations on the fitted model parameters. We conduct such analysis for the logistic growth model with a certain sigmoid functional form of the carrying capacity, which was proposed in the literature for the study of parasite growth during infection. We show how the probability distributions of the parameters are set via the maximum entropy principle. Then the random variable transformation method allows for computing the density function of the population. [ABSTRACT FROM AUTHOR]

The variability of the data and the incomplete knowledge of the true physics require the incorporation of randomness into the formulation of mathematical models. In this setting, the deterministic numerical methods cannot capture the propagation of the uncertainty from the inputs to the model output. For some problems, such as the Burgers' equation (simplification to understand properties of the Navier–Stokes equations), a small variation in the parameters causes nonnegligible changes in the output. Thus, suitable techniques for uncertainty quantification must be used. The generalized polynomial chaos (gPC) method has been successfully applied to compute the location of the transition layer of the steady‐state solution, when a small uncertainty is incorporated into the boundary. On the contrary, the classical perturbation method does not give reliable results, due to the uncertainty magnitude of the output. We propose a modification of the perturbation method that converges and is comparable with the gPC approach in terms of efficiency and rate of convergence. The method is even applicable when the input random parameters are dependent random variables. [ABSTRACT FROM AUTHOR]

Taking into account the public health importance of the human papillomavirus (HPV) control in the future, it is mandatory to assess the effect of the vaccination campaigns on the control of HPV spread and the associated diseases using reliable mathematical models. We propose a computational random network model with the aim of studying the transmission dynamics of HPV infections. This model reflects the herd immunity effect in the heterosexual network more accurately than the classical models. We perform a sensitivity analysis of the sexual behavior changes consisting of increasing the number of men who have sex with men (MSM), increasing the frequency of the intercourses and increasing the number of sexual partners. We find that large changes in the sexual behavior, in some extent, only have minor effects on the decline of the HPV infections in women and men in the current vaccination campaign in Spain (vaccination of young girls with a coverage of 70%). Therefore, the current vaccination program in Spain is robust for the heterosexuals. However, we cannot say the same for MSM, where they do not benefit by the herd immunity effect of the vaccination of girls, and consequently, the circulation of the virus among them remains unchanged. A consequence of the present study is that the effect of other external factors that may affect the transmission dynamics of the HPV, for instance, the tourism or the immigration, does not influence the protection provided by the current Spanish vaccination program. [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]

The time evolution of microorganisms, such as bacteria, is of great interest in biology. In the article by D. Stanescu et al. [Electronic Transactions on Numerical Analysis, 34, 44–58 (2009)], a logistic model was proposed to model the growth of anaerobic photosynthetic bacteria. In the laboratory experiment, actual data for two species of bacteria were considered: Rhodobacter capsulatus and Chlorobium vibrioforme. In this paper, we suggest a new nonlinear model by assuming that the population growth rate is not proportional to the size of the bacteria population, but to the number of interactions between the microorganisms, and by taking into account the beginning of the death phase in the kinetic curve. Stanescu et al. evaluated the effect of randomness into the model coefficients by using generalized polynomial chaos (gPC) expansions, by setting arbitrary distributions without taking into account the likelihood of the data. By contrast, we utilize a Bayesian inverse approach for parameter estimation to obtain reliable posterior distributions for the random input coefficients in both the logistic and our new model. Since our new model does not possess an explicit solution, we use gPC expansions to construct the Bayesian model and to accelerate the Markov chain Monte Carlo algorithm for the Bayesian inference. [ABSTRACT FROM AUTHOR]

In this paper, we address the problem of approximating the probability density function of the following random logistic differential equation: P′(t,ω)=A(t,ω)(1−P(t,ω))P(t,ω), t∈[t0,T], P(t0,ω)=P0(ω), where ω is any outcome in the sample space Ω. In the recent contribution [Cortés, JC, et al. Commun Nonlinear Sci Numer Simulat 2019; 72: 121–138], the authors imposed conditions on the diffusion coefficient A(t) and on the initial condition P0 to approximate the density function f1(p,t) of P(t): A(t) is expressed as a Karhunen–Loève expansion with absolutely continuous random coefficients that have certain growth and are independent of the absolutely continuous random variable P0, and the density of P0, fP0, is Lipschitz on (0,1). In this article, we tackle the problem in a different manner, by using probability tools that allow the hypotheses to be less restrictive. We only suppose that A(t) is expanded on L2([t0,T]×Ω), so that we include other expansions such as random power series. We only require absolute continuity for P0, so that A(t) may be discrete or singular, due to a modified version of the random variable transformation technique. For fP0, only almost everywhere continuity and boundedness on (0,1) are needed. We construct an approximating sequence {f1N(p,t)}N=1∞ of density functions in terms of expectations that tends to f1(p,t) pointwise. Numerical examples illustrate our theoretical results. [ABSTRACT FROM AUTHOR]

The study of the dynamics of the size of a population via mathematical modelling is a problem of interest and widely studied. Traditionally, continuous deterministic methods based on differential equations have been used to deal with this problem. However, discrete versions of some models are also available and sometimes more adequate. In this paper, we randomize the Pielou logistic equation in order to include the inherent uncertainty in modelling. Taking advantage of the method of transformation of random variables, we provide a full probabilistic description to the randomized Pielou logistic model via the computation of the probability density functions of the solution stochastic process, the steady state, and the time until a certain level of population is reached. The theoretical results are illustrated by means of two examples: The first one consists of a numerical experiment and the second one shows an application to study the diffusion of a technology using real data. [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]

Population dynamics models consisting of nonlinear difference equations allow us to get a better understanding of the processes involved in epidemiology. Usually, these mathematical models are studied under a deterministic approach. However, in order to take into account the uncertainties associated with the measurements of the model input parameters, a more realistic approach would be to consider these inputs as random variables. In this paper, we study the random time‐discrete epidemiological models SIS, SIR, SIRS, and SEIR using a powerful unified approach based upon the so‐called adaptive generalized polynomial chaos (gPC) technique. The solution to these random difference equations is a stochastic process in discrete time, which represents the number of susceptible, infected, recovered, etc individuals at each time step. We show, via numerical experiments, how adaptive gPC permits quantifying the uncertainty for the solution stochastic process of the aforementioned random time‐discrete epidemiological model and obtaining accurate results at a cheap computational expense. We also highlight how adaptive gPC can be applied in practice, by means of an example using real data. [ABSTRACT FROM AUTHOR]

Cytokinesis is a crucial event in the cell cycle of all living cells. In fungal cells, it requires co-ordinated contraction of an actomyosin ring and synthesis of both plasmatic membrane and a septum structure that will constitute the new cell wall end. Schizosaccharomyces pombe contains four essential putative (1,3)β-d-glucan synthase catalytic subunits, Bgs1p to Bgs4p. Here we examined the function of Bgs1p in septation by studying the lethal phenotypes of bgs1+ shut-off and bgs1Δ cells and demonstrated that Bgs1p is responsible and essential for linear (1,3)β-d-glucan and primary septum formation. bgs1+ shut-off generates a more than 300-fold Bgs1p reduction, but the septa still present large amounts of disorganized linear (1,3)β-d-glucan and partial primary septa. Conversely, both structures are absent in bgs1Δ cells, where there is no Bgs1p. The septum analysis of bgs1+ -repressed cells indicates that linear (1,3)β-d-glucan is necessary but not sufficient for primary septum formation. Linear (1,3)β-d-glucan is the polysaccharide that specifically interacts with the fluorochrome Calcofluor white in fission yeast. We also show that in the absence of Bgs1p abnormal septa are formed, but the cells cannot separate and eventually die. [ABSTRACT FROM AUTHOR]