biologia - Studia

A MATHEMATICAL MODEL FOR AN EPIDEMIC IN AN OPEN SOCIETY
So, by tending τ to zero we deduce the following system:
.
⎧
⎪
S(
t)
= A[
η(
t)
x(
t)
− β ( t)
S(
t)
I(
t)
− d(
t)
S(
t)]
.
⎨I(
t)
= A[
β ( t)
S(
t)
I(
t)
− d(
t)
I(
t)
−α(
t)
I(
t)]
(*)
.
⎪
⎪x(
t)
= A[
η(
t)
x(
t)
− (1 − h(
t))
α(
t)
I(
t)
− d(
t)
x(
t)]
⎩
.
where A = u(0)
is a constant in the interval (0,1].
.
.
η ( t ) = b(
t)
− g(
t)
e(
t)
b(
t)
− g(
t)
e(
t)
b(
t)
+ c(
t)
f ( t)
+ c(
t)
f ( t)
+ b(
t),
and S ( t)
+ I(
t)
+ R(
t)
= x(
t).
.
Discussion on the compatibility of the model and its applied results
In the system (*) if we put e ( t)
= f ( t)
= c(
t)
= b(
t)
= 0, b(
t)
= d(
t),
and A = 1,
then we conclude the system:
.
⎧
⎪
S(
t)
= −β
( t)
S(
t)
I(
t)
.
⎨I(
t)
= β ( t)
S(
t)
I(
t)
− ( d(
t)
+ α(
t))
I(
t)
.
.
⎪
⎪ x(
t)
= −(1
− h(
t))
α(
t)
I(
t)
⎩
When β ( t),
d(
t)
and α (t)
are constant functions and h ( t)
= 1 then the
above system reduce to the following SIR model (Farkas, 2001, Kermack and
McKendrick, 1927):
.
⎧
⎪
S(
t)
= −βS(
t)
I(
t)
.
⎨I(
t)
= βS(
t)
I(
t)
− ( d + α)
I(
t)
.
.
⎪
⎪ x(
t)
= 0
⎩
Let A = 1 , e(
t)
= f ( t)
= c(
t)
= 0,
and let b(t) and d(t) be two constant
functions. Then we conclude the following generalization of the SIR model with
the vital dynamics (Brauer, 2008, Hethcote, 1976):
.
.
.
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