Pedro Roberto de Lima

This text presents three models widely used in epidemiology: the SIS model, the SIR model, and a model for the transmission of toxoplasmosis in cats. The aim is to provide an initiation into the field of mathematical epidemiology by illustrating three common approaches in differential equations: explicit solution, qualitative analysis, and numerical experimentation. The methodology adopted was bibliographic research, which allowed us to present: the explicit solution of the SIS model, identifying conditions for disease persistence or extinction; a qualitative analysis of the SIR model, revealing the typical behavior of epidemic outbreaks; and some numerical experiments for the toxoplasmosis model, highlighting the influence of different parameters on the transmission dynamics. Given the importance of the subject and the level of detail in the exposition, we believe that this text offers a useful reading for students entering the field of epidemiology and/or the analysis of differential equations.

Fazendo uso das equações diferenciais, este texto apresenta uma breve introdução à modelagem matemática com ênfase na etapa da validação – a qual, ao confrontar previsões teóricas com resultados experimentais, mostra o modelo “em funcionamento” se revelando, desta forma, uma etapa bastante interessante para finalidades didáticas.

In this paper, we give a new and more general sufficient condition for exponential stability of thermoelastic Bresse systems with heat flux given by Cattaneo's law acting in shear and longitudinal motion equations. This condition, which we also prove to be necessary in some special cases, is given by a relation between the constants of the system and generalizes the well-known equal wave speed condition.