In the field of Dynamical Systems, the study of infectious disease spread serves a crucial role in understanding and mitigating epidemics. This project focuses on investigating the dynamics of disease propagation using the SIS (Susceptible-Infected-Susceptible) model. We enhance the SIS model by incorporating diffusion to explore how simple diseases spread spatially. Applied dynamical system methods, in particular analysis traveling wave solutions, are employed to analyze the behavior of the system.
The goal of this work is proving the existence of traveling wave solutions for the diffusive SIS model. We consider the cases in which diffusion is absent, diffusion is present for only the susceptible members of the population, and when diffusion is present for the entire population but diffusion for infected individuals is greatly restricted. Through our analysis, we find that traveling wave solutions exist in all cases, showing that the disease travels through space like a front, asymptotically approaching two distinct rest states. We also ascertain the stability of each rest state, and that stationary wave solutions may not exist in a physical regime.
This research hopes to contribute to the understanding of epidemiology and the development of effective strategies for disease prevention and mitigation. In future study, this research can be expanded to address cases where the rate of susceptible population diffusion is larger but not significantly larger than infected diffusion, and where the infected population diffuses at a rate higher than the susceptible population. Additionally, it may provide interesting results to extend analysis to adjacent disease models adjusted for diffusion, such as a well-known model known as the SIR model, where infected populations may gain resistance after the disease subsides. Such analysis may prove useful in broader epidemiological applications.
Author(s): Jonathan Waldmann, Priscilla Yinzime
Advisor(s): Anna Ghazaryan, Vahagn Manukian, Department of Mathematics
You must be logged in to post a comment.