biologia - Studia

STUDIA UNIVERSITATIS BABEŞ – BOLYAI, BIOLOGIA, LV, 1, 2010, p. 61-65
A MATHEMATICAL MODEL FOR AN EPIDEMIC
IN AN OPEN SOCIETY
MOHAMMAD REZA MOLAEI 1
SUMMARY. In this paper a mathematical model for an infectious disease
in an open society is introduced. The parameters of this model are time
dependent, and it is an extension of the susceptible, infectious and removed
(SIR) models. This model has this ability to determine the speed of
recovery rate to protect the society from an epidemic.
Keywords: mortality rate, recovery rate, per capita birth, SIR model
Introduction
Theory of infectious disease epidemics is based on mathematical models
for the epidemics. Beside its applications in epidemiology it brings many bright
mathematical results in dynamical systems (Bauch and Earn, 2003, Earn, et al.,
2000, Korobeinikov and Maini, 2004). In the susceptible, infectious and removed
(SIR) model and its extensions, the society is closed and its parameters such as per
capita birth and mortality rate are constant numbers. In this paper we consider an
open society. It means a society in which individuals enter it and leave it. Moreover
its parameters are time dependent. For example the influenza infection in which
passengers have essential role in its infection and this model can apply for it. In the
next section we introduce this model and then we show that it is an extension of the
previous models (Farkas, 2001). Moreover we find a recovery rate as a function of
a time as a limitation for the speed of recovery to protect the society from an
epidemic of a disease.
The model
We denote the number of individuals at time t by x(t). S(t) is the number of
individuals who have no immunity to the infectious agent. We denote the number
of individuals who are infected at time t and can transmit the infection to the
susceptible individuals by I(t). R(t) is the number of individuals who are immune to
1 Department of Mathematics, University of Kerman (Shahid Bahonar), 76169-14111, Kerman, Iran.
E-mail: mrmolaei@mail.uk.ac.ir