Four time-dependent dengue transmission models are considered in order to fit the incidence data from the City of Campinas, Brazil, recorded from October 1^st 1995 to September 30^th 2012. The entomological parameters are allowed to depend on temperature and precipitation, while the carrying capacity and the hatching of eggs depend only on precipitation. The whole period of incidence of dengue is split into four periods, due to the fact that the model is formulated considering the circulation of only one serotype. Dengue transmission parameters from human to mosquito and mosquito to human are fitted for each one of the periods. The time varying partial and overall effective reproduction numbers are obtained to explain the incidence of dengue provided by the models.

We develop a mathematical model to describe the role of antibody-dependent enhancement (ADE) in heterologous secondary infections, assuming that antibodies specific to primary dengue virus (DENV) infection are being produced by immunological memory. The model has a virus-free equilibrium (VFE) and a unique virus-presence equilibrium (VPE). VFE is asymptotically stable when VPE is unstable; and unstable, otherwise. Additionally, there is an asymptotic attractor (not a fixed point) due to the fact that the model assumes unbounded increase in memory cells. In the analysis of the model, ADE must be accounted in the initial stage of infection (a window of time of few days), period of time elapsed from the heterologous infection until the immune system mounting an effective response against the secondary infection. We apply the results yielded by model to evaluate ADE phenomonon in heterologous DENV infection. We also associate the possible occurrence of severe dengue with huge viremia mediated by ADE phenomenon.

We analysed a epidemiological model with varying populations of susceptible, carriers, infectious and recovered (SCIR) and a general non-linear incidence rate of the form $f(S)[g(C)+h(I)]$. We show that this model exhibits two positive equilibriums: the disease-free and disease equilibrium. We proved using the Lyapunov direct method that these two equilibriums are globally asymptotically stable under some sufficient conditions over the functions f, g, h.