Stochastic Modelling of Infectious Disease Dynamics in Open Populations with External Infections
Date
2025-01-22
Journal Title
Journal ISSN
Volume Title
Publisher
ORCID
Type
Thesis
Degree Level
Masters
Abstract
This thesis investigates stochastic models based on birth-death and birth-death-immigration processes to analyze the dynamics of infectious disease spread in open populations. A key focus is how these models capture the inherent randomness in disease transmission. The study begins with a review of the deterministic SIR model, highlighting its foundational quantities and its limitations, particularly its assumption of a closed population and its deterministic nature. To address these limitations, first, a stochastic birth-death process for modelling disease dynamics in open populations is reviewed.
Further, the investigations extend to stochastic models incorporating external infections, exploring two scenarios: (1) infected individuals enter the population at a constant rate regardless of the current infection count, and (2) external infections are only introduced when the infected population drops to zero. These two scenarios are interesting because they capture distinct real-world dynamics: continuous external infections or the re-introduction of infections only after an extinction. For these birth-death-immigration models, this thesis reviews derivations of time-dependent probability distributions, expected values, and variances of infected individuals, providing detailed clarifications of key steps. A notable feature is the emergence of a generalized negative binomial distribution, characterized by a long tail and high variability, reflecting the potential for extreme epidemic scenarios. Simulation paths and percentile curves illustrate the high variability in the infection dynamics process and demonstrate how infection dynamics are influenced by initial conditions. Additionally, the study examines the effects of non-homogeneous internal and external infection rates on the expected number of infections. This includes a novel analysis of the implications of simultaneously varying both the internal and external infection rates with time. Results reveal how prompt interventions can mitigate outbreaks and reduce the number of infections.
Overall, this research underscores the importance of stochastic modelling for understanding epidemic dynamics in open populations. By incorporating factors such as external infections, these models provide valuable insights into the uncertainties and variations in disease spread, offering a useful framework for studying epidemics in real-world scenarios.
Description
Keywords
Stochastic Modelling, Birth-Death Model, Birth-Death-Immigration Model, Non-Homogeneous Birth-Death-Immigration Model, Simulation
Citation
Degree
Master of Science (M.Sc.)
Department
Mathematics and Statistics
Program
Statistics