J. Math. Biol. DOI 10.1007/s00285-017-1130-9 Mathematical Biology Chaotic dynamics in the seasonally forced SIR epidemic model Pablo G. Barrientos 1 · J. Ángel Rodríguez 2 · Alfonso Ruiz-Herrera 2 Received: 12 December 2016 / Revised: 8 April 2017 © Springer-Verlag Berlin Heidelberg 2017 Abstract We prove analytically the existence of chaotic dynamics in the forced SIR model. Although numerical experiments have already suggested that this model can exhibit chaotic dynamics, a rigorous proof (without computer-aided) was not given before. Under seasonality in the transmission rate, the coexistence of low birth and mortality rates with high recovery and transmission rates produces infinitely many periodic and aperiodic patterns together with sensitive dependence on the initial con- ditions. Keywords Stretching along paths · Seasonality · SIR model · Sensitive dependence on the initial conditions · Chaos Mathematics Subject Classification 92B05 · 37F99 1 Introduction Seasonal forces, that include from climatic factors to human phenomena (e.g. school schedules), play an important role in the transmission and dynamical behaviour of most infectious diseases (Diedrichs et al. 2014; Earn et al. 2000; Rebelo et al. 2012; B Alfonso Ruiz-Herrera ruizalfonso@uniovi.es Pablo G. Barrientos pgbarrientos@id.uff.br J. Ángel Rodríguez jarodriguez@uniovi.es 1 Instituto de Matemática e Estatística, Universidade Federal Fluminense, Niterói, Brazil 2 Departamento de Matemáticas, Universidad de Oviedo, Oviedo, Spain 123