Download 1 The SIRS Model

James Malm
A Mini-Essay on Epidemiological Modelling of Influenza
AIMS December 2004
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The SIRS Model
Dynamical systems are employed in numerous circumstances to model infectious diseases. These
models lead to a better understanding of the causes, distribution and control of diseases. Public
health officials use these models in decision making. A number of interesting consequences can be
understood by constructing simple models using ordinary differential equations. In this mini-essay,
we will model the disease, influenza, using the SIRS model. Influenza is a contagious disease that is
caused by the influenza virus. It attacks the respiratory tract in humans (nose, throat, and lungs).
Most people who get influenza will recover in one to two weeks. In developing a model for influenza,
we need to choose variables to represent the quantities of interest. We will do this by dividing the
host population into three distinct classes: susceptible, infective and removed.
The susceptible class refers to the people who have not yet caught influenza. Individuals who currently
have influenza make up the infective class. The people who have been removed from the influenzainteracting population by recovery constitute the removed class. We denote the number of individuals
in the susceptible, infective and removed class by S, I and R respectively.
λR
PSfrag replacements
S
βIS
I
γI
R
Figure 1: The SIRS flow diagram
The rate at which the susceptible class changes is equal to the rate at which infection takes place plus
the rate at which people lose immunity. Infection occurs when the disease is passed from an infective
individual to a susceptible individual. The number of susceptible-infective contacts is proportional to
the product of S and I. Of these contacts, a proportion will catch the disease. Also, the rate at which
individuals lose immunity is proportional to the number of people in the removed class. Therefore,
the rate of change in the susceptible population is given by:
dS
= −βSI + λR,
dt
(1)
where βI is the force of infection and λ is the per capita rate at which people in the removed class lose
immunity. The first term in (1) negative because during infection the number of susceptible people
decreases. The rate at which infection is taking place is called the Incidence [Gross, 2004]. Members
of the susceptible class who become infected increase the number in the infective class. The rate at
which people leave the susceptible class is equal to the rate at which they join the infective class.
We also need to consider the number of people recovering from influenza in our analysis. We will
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assume that the number of people leaving the infective class for the removed class is some proportion
of the infective class size. If we let γ denote the proportion of people leaving the infective class for the
removed class, the rate of change of the infective class will be given by:
dI
= βSI − γI.
dt
(2)
R increases at a rate proportional to the number of infectives entering the class. It decreases at a rate
proportional to its size when a member loses immunity and joins the susceptible class. Therefore, the
rate of change in the removed population is given by:
dR
= γI − λR.
dt
(3)
Collecting the three equations we have:













1.1
dS
dt
= −βSI + λR
dI
dt
=
βSI − γI
dR
dt
=
γI − λR.
(4)
Determination of the steady states
The total population S + I + R is constant. We can find the steady states for the system of differential
equations in (4) by setting all three derivatives to zero. Solving the equations we obtain:

R












=
βSI
λ
γI
= βSI
R
=
(5)
γI
λ .
The second equation gives I = 0 or S = βγ . Since we need S + I + R = 0, we find the trivial steady
state with I = 0:
(S ∗ , I ∗ , R∗ ) = (N, 0, 0).
(6)
If we set S =
γ
β
and R =
γI
λ
we obtain:
N =S+I +R=
I=
γI
γ
+I +
,
β
λ
λ(βN − γ)
,
β(λ + γ)
(7)
(8)
which leads to another steady state
S∗ =
γ
,
β
I∗ =
λ(βN − γ)
,
β(λ + γ)
R∗ =
γ(βN − γ)
.
β(λ + γ)
(9)
In the first equilibrium state (6), the whole population is healthy and the disease, influenza, is eradicated. In the second equilibrium state (9), the community consist of some constant proportions of
each class, provided (S ∗ , I ∗ , R∗ ) are all positive quantities. This is the case if βN
γ > 1.
2
βN
γ
is called the basic reproduction number (R 0 ) and it is defined as the average number of infectives
produced when one infective individual is introduced into a completely susceptible population. This
important threshold effect was discovered by Kermack and McKendrick [Anderson and May, 1979].
References
R. M. Anderson and R. M. May. Population biology of infectious disease:part i. Nature, 280:361–367,
1979.
K. Gross. Lecture notes on epidemiological modelling. Epi-AIMS, 2004.
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