Lie integrable cases of the simplified multistrain/two-stream model for tuberculosis and dengue fever

We apply the techniques of Lie's symmetry analysis to a caricature of the simplified multistrain model of Castillo-Chavez and Feng [C. Castillo-Chavez, Z. Feng, To treat or not to treat: The case of tuberculosis, J. Math. Biol. 35 (1997) 629–656] for the transmission of tuberculosis and the coupled two-stream vector-based model of Feng and Velasco-Hernández [Z. Feng, J.X. Velasco-Hernández, Competitive exclusion in a vector-host model for the dengue fever, J. Math. Biol. 35 (1997) 523–544] to identify the combinations of parameters which lead to the existence of nontrivial symmetries. In particular we identify those combinations which lead to the possibility of the linearization of the system and provide the corresponding solutions. Many instances of additional symmetry are analyzed.