The Stiffness Phenomena for the Epidemiological SIR Model: a Numerical Approach

  • Marline Ilha da Silva Federal Institute of Rio Grande do Sul
  • Joice Chaves Marques Federal Univ. of Rio Grande, Inst. of Mathematics, Statistics and Physics
  • Adelaida Otazu Conza National University of the Altiplano
  • Adriano De Cezaro Federal Univ. of Rio Grande, Inst. of Mathematics, Statistics and Physics
  • Ana Carla Ferreira Nicola Gomes Federal Univ. of Rio Grande, Inst. of Mathematics, Statistics and Physics
Keywords: Stiffness, SIR model, Numerical Methods


Mathematical models are among the most successful strategies for predicting the dynamics of a disease spreading in a population. Among them, the so-called compartmental models, where the total population is proportionally divided into compartments, are widely used. The SIR model (Susceptible-Infected-Recovered) is one of them, where the dynamics between the compartments follows a system of nonlinear differential equations. As a result of the non-linearity of the SIR dynamics, it has no analytical solution. Therefore, some numerical methods must be used to obtain an approximate solution. In this contribution, we present simulated scenarios for the SIR model showing its stiffness, a phenomenon that implies the necessity of a small step size choice in the numerical approximation. The numerical results, in particular, show that the stiffness phenomenon increases with higher transmission rates and lower birth and mortality rates . We compare the numerical solutions and errors for the SIR model using explicit Euler, Runge Kutta, and the semi-implicit Rosenbrock methods and analyze the numerical implications of the stiffness on them. As a result, we conclude that any accurate numerical solution of the SIR model will depend on an appropriately chosen numerical method and the time step, in terms of the values of the parameters.



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How to Cite
M. Ilha da Silva, J. Marques, A. Conza, A. De Cezaro, and A. C. Gomes, “The Stiffness Phenomena for the Epidemiological SIR Model: a Numerical Approach”, LAJC, vol. 10, no. 2, pp. 32-45, Jul. 2023.
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