System of delay differential equations with application in dengue fever

• Autor(a) principal: Steindorf, Vanessa
• Data de Publicação: 2019
• Tipo de documento: Tese
• Idioma: eng
• Título da fonte: Biblioteca Digital de Teses e Dissertações da USP
• Texto Completo: http://www.teses.usp.br/teses/disponiveis/45/45132/tde-25092019-130815/
• topic: Antibody Dependent Enhancement (ADE); Antibody Dependent Enhancement - ADE; Distributed delay; Equações integro-diferenciais; Imunidade temporária cruzada; Integro-differential equations; Modelo de multi sorotipos; Multi-strain model; Retardo distribuído; Temporary immunity

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.