Eur. Phys. J. B 81, 363-371 (2011)
https://doi.org/10.1140/epjb/e2011-20054-9

Oscillations in SIRS model with distributed delays

¹ Instituto de Física, Universidade Federal do Rio Grande do Sul, Caixa Postal 15051, 90501-970 Porto Alegre, RS, Brazil
² Centro Atómico Bariloche, CONICET and Instituto Balseiro, 8400 S. C. de Bariloche, Argentina

Corresponding author: ^a marfcg@gmail.com

Received: 19 January 2011
Revised: 29 March 2011
Published online: 4 May 2011

Abstract

The ubiquity of oscillations in epidemics presents a long standing challenge for the formulation of epidemic models. Whether they are external and seasonally driven, or arise from the intrinsic dynamics is an open problem. It is known that fixed time delays destabilize the steady state solution of the standard SIRS model, giving rise to stable oscillations for certain parameters values. In this contribution, starting from the classical SIRS model, we make a general treatment of the recovery and loss of immunity terms. We present oscillation diagrams (amplitude and period) in terms of the parameters of the model, showing how oscillations can be destabilized by the shape of the distributions of the two characteristic (infectious and immune) times. The formulation is made in terms of delay equations which are both numerically integrated and linearized. Results from simulations are included showing where they support the linear analysis and explaining why not where they do not. Considerations and comparison with real diseases are presented along.