MARKOV CHAINS MODELS FOR EPIDEMICS
Date
2024-04-19
Authors
Journal Title
Journal ISSN
Volume Title
Publisher
ORCID
Type
Thesis
Degree Level
Masters
Abstract
In this thesis, we review and explore the stochastic models of epidemics developed by researchers in recent
years. These stochastic models encompass both discrete and continuous time Markov chain models, particularly
emphasizing stochastic Susceptible Infected Susceptible (SIS) and Susceptible Infected Recovered (SIR)
models. These models are compared with their deterministic counterparts regarding dynamics, behavior,
and outcomes, assuming a constant population size. The comparison involves quantitative and qualitative
analyses, focusing on the asymptotic dynamics, the mean of the stochastic process versus the deterministic
solution and the differing properties, such as the final size of an epidemic, particularly for when the basic
reproduction number, R0, exceeds 1.
Significant observations include the bimodal nature of probability distributions in stochastic models when
the basic reproduction number, R0, exceeds 1. The two modes correspond respectively to disease elimination
and disease persistence. This highlights the qualitative differences in the asymptotic dynamics between
deterministic and stochastic models. The occurrence of disease elimination in the SIS stochastic models as
time approaches infinity stands in contrast to the SIS deterministic model. Similarly, the potential for disease
elimination during the peak period in the SIR stochastic model contrasts with the SIR deterministic model.
The thesis also investigates the impact of factors like initial infection numbers, basic reproduction number,
and population size on the epidemic’s duration and final size. For the models studied, it is observed that
larger populations lead to longer epidemic durations, and the size of the epidemic increases when the initial
number of infectives increased. Further, the thesis conducts a comparative analysis between stochastic and
deterministic SIR models specifically for a model of COVID-19. A vaccination parameter, presumed to be
100% effective, is introduced to evaluate its impact on the expected time until disease extinction. Findings
reveal that vaccination significantly accelerates the eradication of epidemics.
Overall, this study highlights the crucial role of stochastic models in capturing uncertainties and variations
that real-world epidemics may have, arising from factors like the unpredictable nature of interpersonal contact.
Description
Keywords
Stochastic models, Deterministic models, Markov chain, Asymptotic dynamics, Reproduction number, Disease elimination.
Citation
Degree
Master of Science (M.Sc.)
Department
Mathematics and Statistics
Program
Statistics