biologia - Studia

M. R. MOLAEI
.
⎧
⎪
S(
t)
= bx(
t)
− β ( t)
S(
t)
I(
t)
− d(
t)
S(
t)
.
⎨I(
t)
= β ( t)
S(
t)
I(
t)
− d(
t)
I(
t)
− α(
t)
I(
t)
.
.
⎪
⎪x(
t)
= bx(
t)
− (1 − h(
t))
α(
t)
I(
t)
− d(
t)
x(
t)
⎩
In fact when α ( t),
β ( t)
and d(t) are constant functions then we deduce the
usual SIR model with the vital dynamics.
d(
t)
+ α(
t)
.
In the system (*) if S(
t)
= then I ( t)
= 0.
So I (t)
is a
β ( t)
constant such as c. If we put S(t) in the equation
.
S(
t)
= A[
η ( t)
x(
t)
− β ( t)
S(
t)
I(
t)
− d(
t)
S(
t)]
then we can find the following relation between x(t) and d(t):
.
.
( d(
t)
+ α ( t))
β ( t)
− β ( t)(
d(
t)
+ α(
t))
x( t)
=
2
Aβ
( t)
2
d ( t)
d(
t)
α(
t)
+ ( d(
t)
+ α ( t))
c + + .
β ( t)
β ( t)
The equation:
.
x(
t)
= A[
η ( t)
x(
t)
− (1 − h(
t))
α(
t)
I(
t)
− d(
t)
x(
t)]
determines the recovery rate h(t) by:
.
.
x(
t)
x(
t)(
d(
t)
−η(
t))
h(
t)
= 1+
+
.
α(
t)
Ac cα
( t)
So by this recovery rate we can sure that there is no any epidemic of the
d(
t)
+ α(
t)
.
disease. Moreover if S(
t)
< then I ( t)
< 0 and there is no epidemic,
β ( t)
d(
t)
+ α(
t)
but if S(
t)
> then there is an epidemic of the disease.
β ( t)
64
Conclusion
In this paper we try to introduce a new mathematical model which can
apply for the diseases which are carried by passengers such as influenza. The recovery
rate is the main biological result of this model. The consideration of this model
from the mathematical viewpoint is also a topic for further research.