Positivity and Boundedness of Solutionsfor a Stochastic Seasonal EpidemiologicalModel for Respiratory Syncytial Virus(RSV)

Abstract

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. Positividad y acotamiento de soluciones de un modelo epidemiologico estacional estocástico para el virus respiratorio sincitial ResumenEn 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.Palabras clave: Modelo epidemiologico estacional estocástico; virus respiratorio sincitial; modelización matemática; positividad; sistema dinámi-co.

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“…Theorem 2.2 System (2) is said to be stochastically ultimately bounded if for any 𝜖 ∈ (0,1) there exists a positive constant 𝐻 = 𝐻(𝜖) 𝑠 uch that for any initial value (W(0), H(0), W s (0), W r (0)) ∈ Ω the solution 𝑋(𝑡) = (W(0), H(0), W s (0), W r (0)) of (2) has the property lim 𝑛→∞ 𝑠𝑢𝑝 𝕡{|𝑋(𝑡)| ≤ 𝐻(𝜖)} ≥ 1 − 𝜖 [24].…”