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Viruses are organisms that need to infect a host cell in order to reproduce. The new viruses leave the infected cell and look for other susceptible cells to infect. The mathematical models for virus propagation are very similar to population and epidemic models, and involve a relatively large number of parameters. These parameters are very difficult to establish with accuracy, while variability in the cell and virus populations and measurement errors are also to be expected. To deal with this issue, we consider the parameters to be random variables with given distributions. We use a non-intrusive variant of the polynomial chaos method to obtain statistics from the differential equations of two different virus models. The equations to be solved remain the same as in the deterministic case; thus no new computer codes need to be developed. Some examples are presented.

Biofilms are colonies of bacteria that attach to surfaces by producing extracellular polymer substances. They may cause serious infections in humans and animals, and also cause problems in hydraulic machinery. In this paper we model the growth of a biofilm established on a medical implant. We assume that the biofilm's growth is given by a logistic reaction term with the growth rate being a random variable with a given distribution. This way we take into account the variability in the bacterial populations, and the measurement and experimental errors. The diffusion coefficient of the microbes is also taken to be random. A stochastic spectral representation of the parameters and the unknown stochastic process is used, together with the polynomial chaos method, to obtain a system of partial differential equations, which is integrated numerically to obtain the evolution of the mean and higher-order moments with respect to time. Some examples are presented.

Mathematical models are very important in epidemiology. Many of the models are given by differential equations and most consider that the parameters are deterministic variables. But in practice, these parameters have large variability that depends on the measurement method and its inherent error, on differences in the actual population sample size used, as well as other factors that are difficult to account for. In this paper the parameters that appear in SIR and SIRS epidemic model are considered random variables with specified distributions. A stochastic spectral representation of the parameters is used, together with the polynomial chaos method, to obtain a system of differential equations, which is integrated numerically to obtain the evolution of the mean and higher-order moments with respect to time.

In the mathematical modeling of population growth, and in particular of bacterial growth, parameters are either measured directly or determined by curve fitting. These parameters have large variability that depends on the experimental method and its inherent error, on differences in the actual population sample size used, as well as other factors that are difficult to account for. In this work the parameters that appear in the Monod kinetics growth model are considered random variables with specified distributions. A stochastic spectral representation of the parameters is used, together with the polynomial chaos method, to obtain a system of differential equations, which is integrated numerically to obtain the evolution of the mean and higher-order moments with respect to time.