Journal of Mathematical Biology https://doi.org/10.1007/s00285-019-01342-7 Mathematical Biology Chaos analysis and explicit series solutions to the seasonally forced SIR epidemic model Jorge Duarte 1,2 · Cristina Januário 1,3 · Nuno Martins 2 · Svitlana Rogovchenko 4 · Yuriy Rogovchenko 5 Received: 26 August 2018 / Revised: 27 January 2019 © Springer-Verlag GmbH Germany, part of Springer Nature 2019 Abstract Despite numerous studies of epidemiological systems, the role of seasonality in the recurrent epidemics is not entirely understood. During certain periods of the year incidence rates of a number of endemic infectious diseases may fluctuate dramat- ically. This influences the dynamics of mathematical models describing the spread of infection and often leads to chaotic oscillations. In this paper, we are concerned with a generalization of a classical Susceptible–Infected–Recovered epidemic model which accounts for seasonal effects. Combining numerical and analytic techniques, we gain new insights into the complex dynamics of a recurrent disease influenced by the seasonality. Computation of the Lyapunov spectrum allows us to identify different chaotic regimes, determine the fractal dimension and estimate the predictability of the appearance of attractors in the system. Applying the homotopy analysis method, we obtain series solutions to the original nonautonomous SIR model with a high level of accuracy and use these approximations to analyze the dynamics of the system. The efficiency of the method is guaranteed by the optimal choice of an auxiliary control parameter which ensures the rapid convergence of the series to the exact solution of the forced SIR epidemic model. Keywords Explicit solutions · SIR epidemic model · Seasonal fluctuations · Chaotic behavior 1 Introduction Outbreaks of infectious diseases threaten public health; an uncontrolled spread of infections creates serious problems for the economic and social development of our society. The impact of infectious diseases can be significantly minimized through B Jorge Duarte jduarte@adm.isel.pt Extended author information available on the last page of the article 123