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In this paper we investigate the positivity and boundedness of the solution of a stochastic seasonal epidemic model for the respiratory syncytial virus (RSV). The stochasticity in the model is due to fluctuating physical and social environments and is introduced by perturbing the transmission parameter of the seasonal disease. We show the existence and uniqueness of the positive solution of the stochastic seasonal epidemic model which is required in the modeling of populations since all populations must be positive from a biological point of view. In addition, the positivity and boundedness of solutions is important to other nonlinear models that arise in sciences and engineering. Numerical simulations of the stochastic model are performed using the Milstein numerical scheme and are included to support our analytic results. Resumen En este trabajo se investiga la positividad y acotamineto de la solución de un modelo epidemiologico estacional estocástico para el virus respiratorio sincitial (RSV ). La estocasticidad en el modelo se debe a entornos físicos y sociales fluctuantes y se introduce perturbando el parámetro de transmisión de la enfermedad. Se demuestra la existencia y unicidad de la solución positiva del modelo epidemiologico estacional estocástico, lo cual se requiere en el modelado de las poblaciones ya que todas las poblaciones deben ser positivos desde el punto de vista biológico. Adicionalmente, la positividad y la acotación de las soluciones es importante para otros modelos no lineales que se presentan en las ciencias y la ingeniería. Las simulaciones numéricas del modelo estocástico se realizan utilizando el esquema numérico de Milstein y se incluyen para apoyar los resultados analíticos.

A numerical method to solve a general random linear parabolic equation where the diffusion coefficient, source term, boundary and initial conditions include uncertainty, is developed. Diffusion equations arise in many fields of science and engineering, and, in many cases, there are uncertainties due to data that cannot be known, or due to errors in measurements and intrinsic variability. In order to model these uncertainties the corresponding parameters, diffusion coefficient, source term, boundary and initial conditions, are assumed to be random variables with certain probability distributions functions. The proposed method includes finite difference schemes on the space variable and the differential transformation method for the time. In addition, the Monte Carlo method is used to deal with the random variables. The accuracy of the hybrid method is investigated numerically using the closed form solution of the deterministic associated equation. Based on the numerical results, confidence intervals and expected mean values for the solution are obtained. Furthermore, with the proposed hybrid method numerical-analytical solutions are obtained Un método numérico para resolver una ecuación parabólica general aleatoria lineal donde el coeficiente de difusión, el término fuente, las condiciones de contorno e iniciales incluyen la incertidumbre, se ha desarrollado. Ecuaciones de difusión surgen en muchos campos de la ciencia y la ingeniería, y en muchos casos, existen la incertidumbres debido a los datos que no se pueden saber, o debido a errores en las mediciones y la variabilidad intrínseca. Para modelar estas incertidumbres los parámetros correspondientes, coeficiente de difusión, término fuente, condiciones de contorno e iniciales, se suponen que son variables aleatorias con determinadas distribuciones de probabilidad. Basándose en los resultados numéricos, se obtienen los intervalos de confianza y valores medios esperados para la solución. Además, se obtienen con las soluciones numéricas-analíticas del método híbrido propuesto

This research deals with the mathematical modeling of the physical capital diffusion through the borders of the countries. The physical capital is considered an important variable for the economic growth of a country. Here we use an extension of the economic Solow model to describe how the smuggling affects the economic growth of the countries. In this study we rely on a production function that is non-concave instead of the classical Cobb-Douglas production function. In order to model the physical capital diffusion through the borders of the country, we developed a model based on a parabolic partial differential equation that describes the dynamics of physical capital and boundary conditions of Neumann type. Smuggling is present in many borders between countries and may include fuel, machinery and food. This smuggling through the borders is a problematic issue for the country's economies. The smuggling problem usually is related mainly to a non-official exchange rate that is different than the official rate or subsides. Numerical simulations are obtained using an explicit finite difference scheme that shows how the physical capital diffusion through the border of the countries. The study of physical capital is a paramount issue for the economic growth of many countries for the next years. The results show that the dynamics of the physical capital when boundary conditions of Neumann type are different than zero differ from the classical economic behavior observed in the classical spatial Solow model without physical capital flux through the borders of countries. Finally, it can be concluded that avoiding the smuggling through the frontiers is an important factor that affects the economic growth of the countries., 17 pages