Mathematical Analysis of Compartmental Models with Vaccination and Time-Varying Disease Transmission Rates

Final Defense

Mathematical Analysis of Compartmental Models with Vaccination and Time-Varying Disease Transmission Rates

by Lu Christian S. Ong
PhD Mathematics Candidate

Date: Tuesday, 01 July 2025
Time: 2:00 pm
Venue: SECA 321 (MJR Room)

Advisers:
Mark Anthony C. Tolentino, PhD
Ateneo de Manila University

Timothy Robin Y. Teng, PhD
Ateneo de Manila University

Panelists:
Elvira P. de Lara-Tuprio, PhD (Critic Reader)
Ateneo de Manila University

Jay Michael R. Macalalag, PhD (Critic Reader)
Caraga State University

Timothy Robin Y. Teng, PhD
Ateneo de Manila University

Luis S. Silvestre Jr., PhD
Ateneo de Manila University

Juancho A. Collera, PhD
University of the Philippines Baguio

 

Mathematical models, such as compartmental models, are relevant and useful in analyzing the spread and dynamics of a disease and how human interventions such as vaccinations and quaran- tines affect the disease transmission dynamics. In compartmental models, it is assumed that the total population can be divided into mutually exclusive groups depending on the health status of the individuals.

The transitions between groups are usually described using a system of differential equations and are governed by a set of parameters. In most studies, parameters are held constant over time; how- ever, for infectious diseases like COVID-19 and influenza, the contact rate between individuals varies depending on temporal factors such as season, lockdowns, and/or strict implementations of wearing of face masks and practicing social distancing. Largely affected by the contact rate, the disease transmission rate also changes over time.

In this work, we incorporate time-varying transmission rate/s in a (i) Susceptible-Vaccinated- Exposed-Infectious-Quarantined-Recovered (SVEIQR) compartmental model and a (ii) two-strain SVEIQR model, both described by non-autonomous systems of ordinary differential equations. We then carry out a mathematical analysis of these two models and their solutions. In both models, we considered three types (continuous, general switching, and periodic switching) of disease trans- mission rate. For each model, we first discuss the fundamental properties of the solutions and de- termine the unique equilibrium that corresponds to the disease-free state. Using techniques such as linearization of the system and switched systems theory (i.e., for the case of general or periodic switching transmission rate), we then establish sufficient conditions for the global attractivity of this disease-free equilibrium. We also present several simulations to illustrate our theoretical re- sults and to gain insights regarding the dynamics of solutions in different scenarios.

Key Words: compartmental models, infectious diseases, vaccination, time-varying parameters, reproduction number, global atractivity, linearization, switched systems, periodic