Doctoral Thesis
DOI
https://doi.org/10.11606/T.45.2019.tde-25092019-130815
Document
Author
Full name
Vanessa Steindorf
E-mail
Institute/School/College
Knowledge Area
Date of Defense
Published
São Paulo, 2019
Supervisor
Committee
Oliva Filho, Sergio Muniz (President)
Ferreira, Cláudia Pio
Maidana, Norberto Anibal
Peixoto, Pedro da Silva
Wu, Jianhong

Title in English
System of delay differential equations with application in dengue fever
Keywords in English
Distributed delay
Integro-differential equations
Multi-strain model
Temporary immunity
Abstract in English
Dengue fever is endemic in tropical and sub-tropical countries, and some of the important features of Dengue fever spread continue posing challenges for mathematical modelling. We propose a model, namely a system of integro-differential equations, to study a multi-serotype infectious disease. The main purpose is to include and analyse the effect of a general time delay on the model describing the length of the cross immunity protection and the effect of Antibody Dependent Enhancement (ADE), both characteristics of Dengue fever. Analysing the system, we could find the equilibriums in the invariant region. A coexistence endemic equilibrium within the region was proved, even for the asymmetric case. The local stability for the disease free equilibrium and for the boundary endemic equilibriums were proved. We have also results about the stability of the solutions of the system, that is completely determined by the Basic Reproduction Number and by the Invasion Reproduction Number, defined mathematically, as a threshold value for stability. The global dynamics is investigated by constructing suitable Lyapunov functions. Bifurcations structure and the solutions of the system were shown through numerical analysis indicating oscillatory dynamics for specific value of the parameter representing the ADE. The analytical results prove the instability of the coexistence endemic equilibrium, showing complex dynamics. Finally, mortality due to the disease is added to the original system. Analysis and discussions are made for this model as perturbation of the original non-linear system.

Title in Portuguese
Sistemas de equações diferenciais com retardo com aplicação na dengue
Keywords in Portuguese
Equações integro-diferenciais
Modelo de multi sorotipos
Retardo distribuído
Abstract in Portuguese