biologia - Studia

M. R. MOLAEI
the infection and their contacts to the others have no any effect to the transmission.
We denote the number of offspring and dead people at time t by B(t) and D(t)
respectively. b(t) and d(t) are the per capita birth and the mortality rate at time t
respectively. In fact if we choose a discrete small time τ as a unit time, then the
per capita birth and the mortality rate at time t can be approximated by
B(
t +τ ) − B(
t)
D(
t +τ ) − D(
t)
and
respectively.
x(
t)
x(
t)
e(t) is the rate of those who leave the society at time t. If g(t) = 1-d(t) and if there is
no any infection then x(t)g(t) is the number of those survive at time t and
x(t)g(t)e(t) is the number of those who leave the society at time t. x(t)g(t)e(t)b(t) is
the number of those who are born at time t by the people who leave the society and
take their offspring with themselves.
We denote the rate of those who enter the society at time t by c(t), and we
denote their per capita birth by f(t). If there is no any infection then x(t)c(t)f(t) is the
number of those who are born by the individuals who enter the society.
An average member of the population at time t makes contact to transmit infection
is denoted by β ( t)
x(
t).
For t ∈ R , and small positive number τ we define h(t) and α (t)
by
R(
t +τ ) − R(
t)
I(
t +τ ) − I(
t)
and
respectively. By choosing a function u (τ )
x(
t)
x(
t)
with t < u( τ ) < t + τ we can find the following approximations:
S( t + τ ) − S(
t)
b(
u)
− b(
t)
g(
u)
e(
u)
b(
u)
− s(
t)
e(
t)
b(
t)
≅ x(
t)
− x(
t)
+
τ
τ
τ
c(
u)
f ( u)
− c(
t)
f ( t)
b(
t)
x(
t)
u(
t)
β ( t)
S(
t)
I(
t)
u(
t)
d(
t)
S(
t)
u(
τ )
x(
t)
+
−
−
τ
τ
τ
τ
and
I(
t + τ ) − I(
t)
β ( t)
S(
t)
I(
t)
u(
τ ) d(
t)
I(
t)
u(
τ ) α(
t)
I(
t)
u(
τ )
≅
−
−
.
τ
τ
τ
τ
Moreover,
x( t + τ ) − x(
t)
b(
u)
− b(
t)
g(
u)
e(
u)
b(
u)
− s(
t)
e(
t)
b(
t)
≅ x(
t)
− x(
t)
τ
τ
τ
c(
u)
f ( u)
− c(
t)
f ( t)
b(
t)
x(
t)
u(
τ ) (1 − h(
t))
α(
t)
I(
t)
u(
τ ) d(
t)
x(
t)
u(
τ )
+ x(
t)
+
−
−
.
τ
τ
τ
τ
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