Sustained oscillations in simple epidemic models
Abstract Book of the XXIII IUPAP International Conference on Statistical Physics, Genova, Italy, (9-13 July 2007)Abstract
Cycles are a very striking behaviour of prey-predator systems also seen in a variety of other
host-enemy systems - a case in point is the well documented patterns of recurrent epidemics
of measles and other childhood diseases in the pre-vaccination era. Recently, an exceedingly
simple and general mechanism of resonant amplification of demographic stochasticity has
been proposed to describe the cycling behaviour of prey-predator systems 3. The resonant
mechanism is generic for a class of stochastic systems that includes the majority of the
classical models of diseases that confer either lifelong or temporary total immunity 2, and
goes a long way in describing realistic patterns of the recurrent epidemics of measles and
pertussis, in the absence of external forcing terms 1.
Here we consider the simplest epidemiological unforced model, the Susceptible-Infective-
Recovered-Susceptible (SIRS) model, where immunity is lost after a given time, and suggest
that it may exhibit two types of cycling behaviour, i.e. in a certain range of parameters
the model exhibits resonant amplification of stochastic fluctuations as in 3 and in a narrow
range of parameters, that is relevant in the epidemiological context, the limit cycles
persist for infinite populations. We analyse the dynamical transition (or bifurcation) that
separates the stochastic resonance regime from the oscillatory phase using an uncorrelated
pair approximation. Analogous behaviour is found in the related demographic Susceptible-
Infective-Recovered (dSIR) model 4, where the pool of susceptibles is replenished through
births.
We provide further evidence of the phase diagram based on the analytical pair approximation
with results for the power spectra calculated through Monte Carlo simulations of the
individually based model interacting through a regular random graph (RRG) with 4 links
per node, for large population sizes N.The latter exhibits distinctive behaviour in a region
of parameter space, characterized by a low rate of replenishment of susceptibles, consistent
with the presence of stable cycles in the limit of infinite populations. We also relate our
findings for the presence of an oscillatory phase with those previously published for the same
model on a square lattice 5.
Finally, we discuss results for various prey-predator models where cycling behaviour, in
the limit of infinite systems, may also be observed 3, 6.
References
1) A. J. McKane, T. J. Newman Phys Rev. Lett. 94, 218102 (2005).\\
2) R. M. Anderson and R. M. May, Infectious Diseases of Humans: Dynamics and Control
(Oxford University Press, Oxford, 1992).\\
3) J. Verdasca, M. Telo da Gama, A. Nunes, N. R. Bernardino, J. M. Pacheco, and M. C. Gomes,
J Theoret. Biol. 233, 553 (2005).\\
4) Benoit J, Nunes A and Telo da Gama M 2006 Eur. Phys. J. B 50 177.\\
5) J. Joo and J. L. Lebowitz, Phys. Rev. E 70, 036114 (pages 9) (2004), q-bio/0404035.\\
6) T. Antal and M. Droz Phys. Rev. E 63, 056119 (2001)