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This paper studies the updated estimation method for estimating the transmission rate changes over time. The models for the population dynamics under SEIR epidemic models with stochastic perturbations are analysed the dynamics of the COVID-19 pandemic in Bogotá, Colombia. We performed computational experiments to interpret COVID-19 dynamics using actual data for the proposed models. We estimate the model parameters and updated their estimates for reported infected and recovered data.

In this paper, we discuss the basic reproduction number of stochastic epidemic models with random perturbations. We define the basic reproduction number in epidemic models by using the integral of a function or survival function. We study the systems of stochastic differential equations for SIR, SIS, and SEIR models and their stability analysis. Some results on deterministic epidemic models are also obtained. We give the numerical conditions for which the disease-free equilibrium point is asymptotically stable.

An easily accessible, real-world approach to probability and stochastic processes Introduction to Probability and Stochastic Processes with Applications presents a clear, easy-to-understand treatment of probability and stochastic processes, providing readers with a solid foundation they can build upon throughout their careers. With an emphasis on applications in engineering, applied sciences, business and finance, statistics, mathematics, and operations research, the book features numerous real-world examples that illustrate how random phenomena occur in nature and how to use probabilistic techniques to accurately model these phenomena. The authors discuss a broad range of topics, from the basic concepts of probability to advanced topics for further study, including Itô integrals, martingales, and sigma algebras. Additional topical coverage includes: Distributions of discrete and continuous random variables frequently used in applications Random vectors, conditional probability, expectation, and multivariate normal distributions The laws of large numbers, limit theorems, and convergence of sequences of random variables Stochastic processes and related applications, particularly in queueing systems Financial mathematics, including pricing methods such as risk-neutral valuation and the Black-Scholes formula Extensive appendices containing a review of the requisite mathematics and tables of standard distributions for use in applications are provided, and plentiful exercises, problems, and solutions are found throughout. Also, a related website features additional exercises with solutions and supplementary material for classroom use. Introduction to Probability and Stochastic Processes with Applications is an ideal book for probability courses at the upper-undergraduate level. The book is also a valuable reference for researchers and practitioners in the fields of engineering, operations research, and computer science who conduct data analysis to make decisions in their everyday work.

In this work, the pricing problem of a variable annuity (VA) contract embedded with a guaranteed lifelong withdrawal benefit (GLWB) rider has been considered. VAs are annuities whose value is linked with a sub-account fund consisting of bonds and equities. The GLWB rider provides a series of regular payments to the policyholder during the policy duration when he is alive irrespective of the portfolio performance. Also, the remaining fund value is given to his nominee, at the time of death of the policyholder. The appropriate modelling of fund plays a crucial role in the pricing of VA products. In the literature, several authors model the fund value in a VA contract using a geometric Brownian motion (GBM) model with a constant variance. However, in real life, the financial assets returns are not Normal distributed. The returns have non-zero skewness, high kurtosis, and leverage effect. This paper proposes a discrete-time model for annuity pricing using generalized autoregressive conditional heteroscedastic (GARCH) models, which overcome the limitations of the GBM model. The proposed model is analyzed with numerical illustration along with sensitivity analysis.

Two stochastic models to study the course of the transient behaviour of the total infectivity present in an infinite population of susceptible individuals are developed. The conditional intensity function of the contagion comprises two components: one is due to the external sources only and the other is the contribution of each of the infected persons which is nonstationary in nature. The statistical characteristics of the number of infected individuals at any time are explicitly obtained. Estimation of the model parameters is also indicated.

Mixtures of symmetric distributions, in particular normal mixtures as a tool in statistical modeling, have been widely studied. In recent years, mixtures of asymmetric distributions have emerged as a top contender for analyzing statistical data. Tukey’s g family of generalized distributions depend on the parameters, namely, g , which controls the skewness. This paper presents the probability density function (pdf) associated with a mixture of Tukey’s g family of generalized distributions. The mixture of this class of skewed distributions is a generalization of Tukey’s g family of distributions. In this paper, we calculate a closed form expression for the density and distribution of the mixture of two Tukey’s g families of generalized distributions, which allows us to easily compute probabilities, moments, and related measures. This class of distributions contains the mixture of Log-symmetric distributions as a special case.

Dengue remains a major public health challenge in Colombia, with Valle del Cauca experiencing recurrent outbreaks characterized by seasonal fluctuations and long-term variability. Understanding the transmission dynamics of Dengue across age groups is critical for targeted interventions. In this study, we developed an age-structured stochastic host-vector model, incorporating a compartmental SIR-SI framework within a stochastic differential equation (SDE) approach. The population is stratified into youths (0-17 years) and adults (18 years and older), enabling analysis of age-specific infection and recovery patterns. Simulations and forecasts were performed using the Euler-Maruyama method, informed by fixed parameters from the literature, estimated disease-specific parameters, and epidemiological data from Colombia's Public Health Surveillance System (SIVIGILA) spanning 2013-2023. Additionally, a Seasonal Autoregressive Integrated Moving Average (SARIMA) model was employed as a complementary approach to capture and forecast monthly Dengue incidence. Our results highlighted distinct epidemic patterns across age groups, the higher infection burden among adults, and the complementary roles of mechanistic SDE modeling and SARIMA forecasting for surveillance and control planning.

This paper studies the updated estimation method for estimating the transmission rate changes over time. The models for the population dynamics under SEIR epidemic models with stochastic perturbations are analysed the dynamics of the COVID-19 pandemic in Bogotá, Colombia. We performed computational experiments to interpret COVID-19 dynamics using actual data for the proposed models. We estimate the model parameters and updated their estimates for reported infected and recovered data.

Climatic factors exert a substantial influence on both biotic and abiotic components of marine ecosystems, significantly affecting the abundance and spatial distribution of fish species. In this study, we introduced a stochastic modeling framework, grounded in stochastic differential equations (SDEs), to analyze the temporal dynamics of sea surface temperature and its relationship with the abundance of Mahi Mahi ( ) in a region of the Colombian Pacific coast. Model parameters such as sea surface temperature, fish stock, and catch per unit effort for the period 2000 to 2012 were estimated using the maximum likelihood method, implemented via the Euler-Maruyama numerical scheme. The model's performance was assessed using empirical data through numerical simulation, cross-validation, and sensitivity analysis, demonstrating its applicability and robustness in capturing key ecological dynamics.