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International Conference on Complex Systems (ICCS2006)

The SIR model with delay

Marcelo Ferreira da Costa Gomes
Instituto de Física - Universidade Federal do Rio Grande do

Sebastián Gonçalves
Instituto de Física - Universidade Federal do Rio Grande do Sul

     Full text: Not available
     Last modified: May 19, 2006

The theoretical study of epidemics and its dynamics have attracted the attention
of mathematicians for decades. Recently, with the introduction of small-world
and scale free networks, many physicists have devoted to their study as
well. However, basic aspect of the classical approach, namely the
Susceptible-Infected- Removed or SIR model, deserves a bit of discussion. For
example the solution of the differential equation of the SIR model (mean field
approach) do not agree with the numerical solution of the model, obtained by
probabilistic cellular automaton techniques. In this paper we address the
possible origins of that differences. Modifying the way the removal or death
rate ---due to the disease--- is taken into account in the SIR differential
equations, we reconcile both solutions. We show that if the standard SIR
equations, with the removed term as a radioactive decay rate, are replaced by
SIR equations with delay, where each infected individual is removed at a
specific time after being infected, the dynamics for the infected and
susceptible agree with the numerical implementation of the model. That is a good
indication that the later form (SIR equation with delay) should be the correct
way to deal with that part of the SIR model. Besides we applied the method to
more than one infection period, where a probability distribution defines a
specific infection time for each individual. Even in the general case, the
agreement between the differential equations with delay and the exact numerical
implementation is excellent.

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