Mathematical Modelling of Infectious Diseases using Delay Differential Equations

• Main Author: Macalalag, Jay Michael
• Format: text
• Published: Archīum Ateneo 2019
• Subjects: Lyapunov functional, LaSalle’s Invariance Principle, local stability, global stability, Hopf bifurcation, delay differential equation, tuberculosis, dengue, HIV/AIDS
• Online Access: https://archium.ateneo.edu/theses-dissertations/447
• Institution: Ateneo De Manila University

This paper introduces compartmental models with time delays for the transmis- sion of tuberculosis, dengue, and human immunodeficiency virus (HIV)/acquired im- munodeficiency syndrome (AIDS). A Susceptible-Exposed-Infectious-Treated (SEIT) compartmental model for tuberculosis transmission, with the incorporation of latent and treatment period time delays, is studied. The incubation periods for humans and mosquitoes, together with the presence of human awareness and vector controls are in- coporated in a Susceptible-Infected-Recovered Susceptible-Infected (SIR-SI) human- vector model for dengue transmission. For HIV/AIDS transmission, a compartmental model with time delays on media coverage and vertical transmission, is introduced and analyzed. The analysis of each model starts with establishing the existence, uniqueness, non- negativity and boundedness of solutions. Equilibrium points are then computed and the basic reproductive numbers are determined. The global stability properties of equilib- rium points are established via Lyapunov functionals and LaSalle’s Invariance Principle. Numerical simulations are presented to support theoretical results of the study.