Open Markov Chains: Cumulant Dynamics, Fluctuations and Correlations

Disease Spread Model in Structurally Complex Spaces: An Open Markov Chain Approach

Understanding the dynamical behavior of infectious disease propagation within enclosed spaces is crucial for effectively establishing control measures. In this article, we present a modeling approach to analyze the dynamics of individuals in enclosed spaces, where such spaces are comprised of different chambers. Our focus is on capturing the movement of individuals and their infection status using an open Markov chain framework. Unlike ordinary Markov chains, an open Markov chain accounts for individuals entering and leaving the system. We categorize individuals within the system into three different groups: susceptible, carrier, and infected. A discrete-time process is employed to model the behavior of individuals throughout the system. To quantify the risk of infection, we derive a probability function that takes into account the total number of individuals inside the system and the distribution among the different groups. Furthermore, we calculate mathematical expressions for the average number of susceptible, carrier, and infected individuals at each time step. Additionally, we determine mathematical expressions for the mean number and stationary mean populations of these groups. To validate our modeling approach, we compare the theoretical and numerical models proposed in this work.

Open Markov Type Population Models: From Discrete to Continuous Time

We address the problem of finding a natural continuous time Markov type process—in open populations—that best captures the information provided by an open Markov chain in discrete time which is usually the sole possible observation from data. Given the open discrete time Markov chain, we single out two main approaches: In the first one, we consider a calibration procedure of a continuous time Markov process using a transition matrix of a discrete time Markov chain and we show that, when the discrete time transition matrix is embeddable in a continuous time one, the calibration problem has optimal solutions. In the second approach, we consider semi-Markov processes—and open Markov schemes—and we propose a direct extension from the discrete time theory to the continuous time one by using a known structure representation result for semi-Markov processes that decomposes the process as a sum of terms given by the products of the random variables of a discrete time Markov chain by time functions built from an adequate increasing sequence of stopping times.