Physica A 389 (2010) 1151–1157 Contents lists available at ScienceDirect Physica A journal homepage: www.elsevier.com/locate/physa Modal series solution for an epidemic model L. Acedo a , Gilberto González-Parra a,b,∗ , Abraham J. Arenas a,c a Instituto de Matemática Multidisciplinar, Universidad Politécnica de Valencia Edif. 8G, 2 o , 46022 Valencia, Spain b Departamento Cálculo, Universidad de los Andes, Mérida, Venezuela c Departamento de Matemáticas y Estadística, Universidad de Córdoba, Montería, Colombia article info Article history: Received 6 July 2009 Received in revised form 1 November 2009 Available online 13 November 2009 Keywords: SIRS epidemic model Exact solution Modal expansion infinite series abstract In this article, we generalize a recently proposed method to obtain an exact general solution for the classical Susceptible, Infected, Recovered and Susceptible (SIRS ) epidemic mathematical model. This generalization is based upon the nonlinear coupling of two frequencies in an infinite modal series solution. It is shown that these series provide a nonstandard approach in order to obtain an accurate analytical solution for the classical SIRS epidemic model. Numerical results of the SIRS epidemic model for real and complex frequencies are included in order to test the validity and reliability of the method. This method could be applied to a wide class of models in physics, chemistry or engineering. © 2009 Elsevier B.V. All rights reserved. 1. Introduction At the beginning of the second quarter of the past century, Kermack and McKendrick [1] proposed a system of ordinary differential equations to determine the time evolution of the susceptible, infected and recovered populations of individuals exposed to an infectious disease. This model is generally known by the acronym SIR and similar names have been given to other compartmental models such as SIS , SIRS , SEIR, etc., by disposing the different populations considered in the models in the same order as the flow from one compartment to another takes place. The importance acquired throughout the past and present centuries by this simple model cannot be underestimated. This is probably the most widely used model in Mathematical Epidemiology with applications to many epidemics such as: the bubonic plague [2], measles [3], cholera [4], rubella [5], respiratory syncytial virus [6,7], hepatitis [8], influenza [9] and many other. Random perturbations has also been added to these models [10]. From another point of view, this model has also been used to illustrate Fokker-Planck relaxation to an equilibrium point in Statistical Physics [11]. A connection with the kinetic of reactions can also be easily developed, if we consider S and I as two different chemical species which react as follows: S + I → I + I . Thus, relaxation towards chemical equilibrium can also be described by very similar models which incorporate the binary reaction term SI into the dynamics. In order to make reliable predictions, numerical integration methods such as Runge–Kutta can be efficiently implemented virtually in any computer language. Nevertheless, exact solutions play a very important role in any nonlinear theory: Onsager’s solution of 2D Ising model [12], the Schwarzschild and Kerr solutions of Einstein’s Field Equations [13] exemplify the creation of new concepts such as criticality or Black Holes which were disclosed more easily; thanks to these solutions. Closed-form expressions for infected and susceptible individuals are known time ago for the SI epidemic model [14]. Recently, some authors have also found special solutions of the SIR model by means of modal series [15,16]. We used modal series in terms of a single frequency to obtain a solution for a particular set of initial conditions [16]. The homotopy perturbation method provides a similar approach, but oscillations in the transient regime are never observed because a single real time scale is taken into account [15]. On the other hand, many numerical methods have been presented for solving epidemics models [17]. ∗ Corresponding author at: Departamento Cálculo, Universidad de los Andes, Mérida, Venezuela. E-mail addresses: luiacrod@imm.upv.es (L. Acedo), gcarlos@ula.ve (G. González-Parra), aarenas@sinu.unicordoba.edu.co (A.J. Arenas). 0378-4371/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.physa.2009.11.003