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ASIAN JOURNAL OF MATHEMATICS AND APPLICATIONS
Volume 2016, Article ID ama0270, 14 pages
ISSN 2307-7743
http://scienceasia.asia
ANALYSIS STABILITY OF THE NONLINEAR EPIDEMIC MODEL
WITH TEMPORARY IMMUNITY
LAID CHAHRAZED
Abstract. The present paper present a nonlinear mathematical model, which analyzes
the spread and stability of the model epidemic. A population of size N (t) at time t, is
divided into three sub classes, with N (t) = S(t) + I(t) + Q(t); where S(t),I(t), and Q(t)
denote the sizes of the population susceptible to disease, and infectious members, quarantine
members with the possibility of infection through temporary immunity, respectively.This
paper deals with the equilibrium and stability, precisely the global asymptotic stability of
endemic equilibrium under certain conditions on the parameter. Second stochastic stability
and finally the equilibrium and stability of the epidemic model with age.
1. Introduction
Motivated by study the dynamics of the different nonlinear epidemic models in all the
references, the present paper addresses a mathematical model, which examine the evolution
of the infection, reported in the total population.
The spread of the epidemic might naturally dependent on possible contacts between susceptible and infectious.
In this paper, we discuss the equilibrium and stability of the model with temporary immunity and the different positives parameters. We have made the following contributions:
• The equilibrium and stability of the model, we obtain a disease-free equilibrium in
the absence of infection but in the presence of infection, it was a unique positive
endemic equilibrium and we define the basic reproduction number of the infection .
• Second the global asymptotic stability of endemic equilibrium.
• Next, we introduce a Brownian motion to system and we transform it into an Itô
stochastic differential equation.
• Finally study the equilibrium of epidemic model with age.
Key words and phrases. Basic reproduction number; endemic equilibrium; global asymptotic stability;
epidemic model; lyapunov functional; temporary immunity; stochastic stability.
c
2016
Science Asia
1
2
CHAHRAZED
2. Mathematical Model
This paper considers the following epidemic model with temporary immunity:


 Ṡ (t) = ρ + µ − ν − (µ1 + d) S(t) − βS(t)Q(t),
(1)
I˙ (t) = βS(t)Q(t) − (µ2 + d) I(t) − γe−µ2 τ S(t − τ )Q(t − τ ),

 Q̇ (t) = γe−µ2 τ S(t − τ )Q(t − τ ) − (µ + d) Q(t),
3
Consider a population of size N (t) at time t, this population is divided into three subclasses,
with N (t) = S(t) + I(t) + Q(t); where S(t), I(t), and Q(t) denote the sizes of the population
susceptible to disease, and infectious members, quarantine members with the possibility of
infection through temporary immunity, respectively. The positive constants µ1 , µ2 , and
µ3 represent the death rates of susceptible, infectious and quarantine. Biologically, it is
natural to assume that µ1 ≤ min {µ2 , µ3 } . The positive constant d is natural mortality
rate. The positive constants µ represent rate of insidence. The positive constant γ represent
the recovery rate of infection. The positive constant β is the average numbers of contacts
infective for S and I. ρ the positive constant is the parameter of immigration.ν the positive
constant is the parameter of emigration. The term γe−µ2 τ S(t − τ )Q(t − τ ) reflects the fact
that an individual has recovered from infection and still are alive after infectious period τ ,
where τ is the length of immunity period.
The initial condition of (1) is given as.
(2)
S (η) = Φ1 (η) , I (η) = Φ2 (η) , Q (η) = Φ3 (η) ,
− τ ≤ η ≤ 0,
Where Φ = (Φ1 , Φ2 , Φ3 )T ∈ C such that S (η) = Φ1 (η) = Φ1 (0) ≥ 0, I (η) = Φ2 (η) =
Φ2 (0) ≥ 0, Q (η) = Φ3 (η) = Φ3 (0) ≥ 0.
Let C denote the Banach space C([−τ , 0], R3 ) of continuous functions mapping the interval
[−τ , 0] into R3 . With a biological meaning, we further assume that Φi (η) = Φi (0) ≥ 0 for
i = 1, 2, 3.
Consider the system without the parameter of emigrations. Hence system (1) can be
rewritten as


 Ṡ (t) = ρ + µ − ν − (µ1 + d) S (t) − βS (t) Q (t) ,
(3)
I˙ (t) = βS (t) Q (t) − (µ2 + d) I (t) − γe−µ2 τ S(t − τ )Q(t − τ ),

 Q̇ (t) = γe−µ2 τ S(t − τ )Q(t − τ ) − (µ + d) Q (t) ,
3
With the initial conditions.
(4)
S (η) = Φ1 (η) , I (η) = Φ2 (η) , Q (η) = Φ3 (η) ,
−τ ≤ η ≤ 0,
Where Φ1 (0) ≥ 0, Φ2 (0) ≥ 0, Φ3 (0) ≥ 0,
−τ ≤ η ≺ 0.
3
} is positively invariant set of
The region Ω = {(S, I, Q) ∈ R+ , S + I + Q ≤ N < ρ+µ−ν
µ1 +d
(3).
NONLINEAR EPIDEMIC MODEL WITH TEMPORARY IMMUNITY
3
3. Equilibrium and stability
An equilibrium point of system (3) satisfies


 ρ + µ − ν − (µ1 + d) S − βSQ = 0,
(5)
βkSQ − (µ2 + d) I − γe−µ2 τ S(t − τ )Q(t − τ ) = 0,

 γe−µ2 τ S(t − τ )Q(t − τ ) − (µ + d) Q = 0,
3
We calculate the points of equilibrium in the absence and presence of infection.
In the absence of infection I = 0, the system (5) has a disease-free equilibrium E0 :
T
T ρ + µ − ν
ˆ Q̂ =
E0 = Ŝ, I,
, 0, 0 .
µ1 + d
The eigenvalues can be determined by solving the characteristic equation of the linearization
of (3) near E0 is


− (µ1 + d) − A
0
− β(ρ+µ−ν)
µ1 +d


(ρ+µ−ν)(β−γe−µ2 τ )
=0
(6)
det 
0
− (µ2 + d) − A


µ1 +d
−µ
τ
(ρ+µ−ν)γe 2
0
0
− (µ3 + d) − A
µ +d
1
So the eigenvalues are
A1 = − (µ1 + d) , A2 = − (µ2 + d) .
In order for λ1 , λ2 , to be negative, it is required that.
(ρ + µ − ν) γe−µ2 τ
< µ3 + d
µ1 + d
(7)
Then we define the basic reproduction number of the infection R0 as follows.
(ρ + µ − ν) γe−µ2 τ
R0 =
(µ1 + d) (µ3 + d)
(8)
In the presence of infection I 6= 0, substituting in the system, Ω also contains a unique
positive, endemic equilibrium Eτ∗ = (Sτ∗ , Iτ∗ , Q∗τ )T where

ρ+µ−ν
1
∗


 Sτ = µ1 +dh × R0 ,
i
(µ1 +d)(µ3 +d)
−1 ρ+µ−ν
(9)
Iτ∗ = Rµ 0+d
−
,
R0
β
2


 Q∗ = µ1 +d (R − 1)
τ
β
0
Note that
Nτ∗ =
(ρ + µ − ν) (µ2 − µ1 ) ρ + µ − ν
+
R0 (µ1 + d) (µ2 + d)
µ2 + d
(R0 − 1) (µ1 + d) (µ2 − µ3 )
+
β (µ2 + d)
(10)
So Eτ∗ = (Sτ∗ , Iτ∗ , Q∗τ )T is the unique positive endemic equilibrium point which exists if
R0 > 1.
4
CHAHRAZED
Theorem 1. The disease-free equilibrium E0 is locally asymptotically stable if R0 < 1 and
unstable if R0 > 1.
Theorem 2. With R0 > 1, system (3) has a unique non-trivial equilibrium Eτ∗ is locally
asymptotically stable.
4. Global asymptotic stability of endemic equilibrium
Consider system (3), with introducing the variables,
x(t) = S(t) − Sτ∗ ,
y(t) = I(t) − Iτ∗ ,
z(t) = Q(t) − Q∗τ ,
System (3) is centered at the endemic equilibrium Eτ∗ = (Sτ∗ , Iτ∗ , Q∗τ )T , then

∗
∗

 ẋ(t) = [− (µ1 + d) − βQτ ] x + [−βSτ ] z,
(11)
ẏ(t) = [(β − γe−µ2 τ ) Q∗τ ] x + [− (µ2 + d)] y + [(β − γe−µ2 τ ) Sτ∗ ] z,

 ż(t) = [βγe−µ2 τ Q∗ ] x + [γe−µ2 τ S ∗ − (µ + d)] z
3
τ
τ
Lemma 3. Let
S (s) = S (0) > 0, I (s) = I (0) ≥ 0 for all s ∈ [−τ , 0] and Q (0) > 0.
S(t), I(t) and Q(t) solutions of system (3) are positive for all t > 0.
Proof. For contraduction there exists the first time t0 , such that S (t0 ) Q (t0 ) = 0.
• Assume that S (t0 ) = 0, then Q (t) ≥ 0 for all t ∈ [0, t0 ].With Eq 1 in the system (1)
we have
Ṡ (t0 ) = ρ + µ − ν > 0.
For S (t0 ) = 0, S0 > 0, we must have Ṡ (t0 ) < 0 which is contradiction.
• Assume that I (t0 ) = 0, then with Eq 2 in the system (1) we have
I˙ (t0 ) = −γe−µ2 τ S(t − τ )Q(t − τ )
I˙ (t0 ) is positive because S(t) and Q(t) solutions of system (1)
are positive for all t > 0.
• For I (t0 ) = 0, I > 0, we must have I˙ (t0 ) < 0 which is contradiction.
• Assume that Q (t0 ) = 0, then S (t) ≥ 0 for all t ∈ [0, t0 ]. with Eq 3 in the system (3)
we have
Q̇ (t0 ) = γe−µ2 τ S(t − τ )Q(t − τ ),
Rt0 −µ (t0 −s)
Q (t0 ) = γ
e 2
S(s)Q(s)ds.
t0 −τ
S (s) > 0, S (s) > 0 for all t ∈ [0, t0 ) . We have γ
Rt0
e−µ2 (t0 −s) S(s)Q(s)ds > 0, and
t0 −τ
Q (t0 ) = 0, which is contradiction.
NONLINEAR EPIDEMIC MODEL WITH TEMPORARY IMMUNITY
5
Lemma 4. Let
S (s) = S0 > 0, Q (s) = Q0 > 0 for all s ∈ [−τ , 0] and Q0 > 0.
Then
S (t) ≤ max
ρ+µ−ν
, S0 + I0 + Q0
µ1 + d
=M
Proof. We have
N (t) = S(t) + I(t) + Q(t),
For R0 < 1 the solutions S(t), I(t) and Q(t) approach the disease free equilibrium as
t → ∞.
With Eq 2 in the system (3) we have I˙ ≤ − (µ2 + d) I, hence if µ2 + d < 0,
lim I(t) = 0,
t→∞
With Eq 3 in the system (3) we have.
lim Q(t) = 0,
t→∞
With Eq 1 in the system (3) we obtain Ṡ = ρ + µ − ν − (µ1 + d) S.
lim S(t) =
ρ+µ−ν
,
µ1 + d
lim N (t) =
ρ+µ−ν
.
µ1 + d
t→∞
Hence.
t→∞
From lemma1, S(t), I(t) and Q(t) solutions of system (1) are positive.
S(t) ≤
ρ+µ−ν
, for allt ≤ 0.
µ1 + d
Suppose that
N (0) ≤
ρ+µ−ν
ρ+µ−ν
, then N (t) ≤
µ1 + d
µ1 + d
On the contrary
If N (0) > ρ+µ−ν
then N (t) < N (0), and S(t) < N (0) for all t > 0.
µ +d
1
Theorem 5. Let S (s) = S0 > 0, Q (s) = Q0 > 0 for all s ∈ [−τ , 0] and Q0 > 0. Eτ∗ is
globally asymptotically stable for all τ


+3ωQ∗τ
1
ln ωM
,


γ
2ω(µ1 +d)


ωM +3ωQ∗τ
1
ln
,
τ > max
γ
2ω(µ2 +d)+(µ2 +d)−βM 

∗ −3ωM


1
ln 2(µQτ+d)−βM
γ
3
6
CHAHRAZED
Where
ρ+µ−ν
M = max
, S0 + I0 + Q0 ,
µ1 + d
βQ∗τ
ω =
µ1 + µ2 + 2d
Proof. We consider system (3) .
Let us introduce the functional
1 2
1
y + z2 ,
V (x, y, z) = ω (x + y)2 +
2
2
The derivative V̇ (x, y, z) is
V̇ (x, y, z) = ω (x + y) (ẋ + ẏ) + y ẏ + z ż
"
#
(− (µ1 + d) − βQ∗τ ) x − βSτ∗ z + (β − γe−µ2 τ ) Q∗τ x
= ω (x + y)
− (µ2 + d) y + (β − γe−µ2 τ ) Sτ∗ z
+y β − γe−µ2 τ Q∗τ x − (µ2 + d) y + β − γe−µ2 τ Sτ∗ z +
+z βγe−µ2 τ Q∗τ x + γe−µ2 τ Sτ∗ − (µ3 + d) z
= −ω (µ1 + d) x2 − [(ω + 1) (µ2 + d)] y 2
− γe−µ2 τ Sτ∗ − (µ3 + d) z 2
+ [βQ∗τ − ω (µ1 + d) − ω (µ2 + d)] xy
+βSτ∗ yz − ωQ∗τ γe−µ2 τ xx(t − τ )
− (ω + 1) Q∗τ γe−µ2 τ yx(t − τ )
+Q∗τ γe−µ2 τ zx(t − τ ) − ωSτ∗ γe−µ2 τ xz(t − τ )
− (ω + 1) Sτ∗ γe−µ2 τ yz(t − τ ) + Sτ∗ γe−µ2 τ zz(t − τ ).
By lemma 2 we have S(t) ≤ M for all t ≥ 0 and ω is an arbitrary real constant choosing
as follows
βQ∗τ
ω=
µ1 + µ2 + 2d
V̇ (x, y, z) ≤ −ω (µ1 + d) x2 − [(ω + 1) (µ2 + d)] y 2
− (µ3 + d) z 2
+βM yz − ωQ∗τ γe−µ2 τ xx(t − τ )
− (ω + 1) Q∗τ γe−µ2 τ yx(t − τ )
+Q∗τ γe−µ2 τ zx(t − τ ) − ωM γe−µ2 τ xz(t − τ )
− (ω + 1) M γe−µ2 τ yz(t − τ )
+M γe−µ2 τ zz(t − τ ).
NONLINEAR EPIDEMIC MODEL WITH TEMPORARY IMMUNITY
Applying Cauchy-Chwartz inequality; we obtain:
V̇ (x, y, z) ≤ −ω (µ1 + d) x2 − [(ω + 1) (µ2 + d)] y 2
− (µ3 + d) z 2
1
− ωQ∗τ γe−µ2 τ x2 + x2 (t − τ )
2
1
− (ω + 1) Q∗τ γe−µ2 τ y 2 + x2 (t − τ )
2
1 ∗ −µ2 τ 2
+ Qτ γe
z + x2 (t − τ )
2
1
− ωM γe−µ2 τ x2 + z 2 (t − τ )
2
1
− (ω + 1) M γe−µ2 τ y 2 + z 2 (t − τ )
2
1
+ M γe−µ2 τ z 2 + z 2 (t − τ )
2
1
+ βM y 2 + z 2 ,
2
1
−µ2 τ
∗
(M + Qτ ) x2
≤ −ω (µ1 + d) − ωγe
2
1
1
−µ2 τ
∗
(M + Qτ ) y 2
+ βM − (ω + 1) (µ2 + d) − (ω + 1) γe
2
2
1
1 −µ2 τ
∗
+ βM − (µ3 + d) + γe
(M + Qτ ) z 2
2
2
∗ −µ τ 2
1
−µ
τ
− ωQτ γe 2 x (t − τ ) −
(3ω + 1) γM e 2 z 2 (t − τ )
2
Choose the Lyapunov functional
Rt
V (xt , yt , zt ) = V (x, y, z) − ωQ∗τ γe−µ2 τ
x2 (θ)dθ
t−τ
Rt
1
− (3ω + 1) γM e−µ2 τ z 2 (θ)dθ
2
t−τ
7
8
CHAHRAZED
Then
V̇ (xt , yt , zt ) = V̇ (x, y, z) − ωQ∗τ γe−µ2 τ x2 (t)
+ωQ∗τ γe−µ2 τ x2 (t − τ )
1
− (3ω + 1) γM e−µ2 τ z 2 (t)
2
1
+ (3ω + 1) γM e−µ2 τ z 2 (t − τ )
2
1
−µ2 τ
∗
(M + Qτ ) x2
≤ −ω (µ1 + d) − ωγe
2
"
#
1
βM
−
(ω
+
1)
(µ
+
d)
2
2
+
y2
1
−µ2 τ
∗
− 2 (ω + 1) γe
(M + Qτ )
1
1 −µ2 τ
∗
+ βM − (µ3 + d) + γe
(M + Qτ ) z 2
2
2
− ωQ∗τ γe−µ2 τ x2 + ωQ∗τ γe−µ2 τ x2 (t − τ )
1
−µ2 τ
(3ω + 1) γM e
z2
−
2
1
+ (3ω + 1) γM e−µ2 τ z 2 (t − τ ),
2
1
−µ2 τ
∗
≤ −ω (µ1 + d) − ωγe
(M + Qτ ) x2
2
1
1
−µ2 τ
∗
+ βM − (ω + 1) (µ2 + d) − (ω + 1) γe
(M + Qτ ) y 2
2
2
1 −µ2 τ
1
∗
(M + Qτ ) z 2
+ βM − (µ3 + d) + γe
2
2
∗ −µ τ 2
1
−µ2 τ
2
(3ω + 1) γM e
z2
− ωQτ γe
x −
2
Therefore
1
−µ2 τ
∗
V̇ (xt , yt , zt ) ≤ − ω (µ1 + d) + ωγe
(M + 3Qτ ) x2
2
"
#
(ω + 1) (µ2 + d) − 21 βM
−
y2
1
−µ2 τ
∗
+ 2 (ω + 1) γe
(M + Qτ )
"
#
(µ3 + d) − 21 βM
z2.
−
− 12 γe−µ2 τ (Q∗τ − 3ωM )
NONLINEAR EPIDEMIC MODEL WITH TEMPORARY IMMUNITY
While the above inequality is always negative provided that

+3ωQ∗τ
1
ln ωM

γ
2ω(µ1 +d)

ωM +3ωQ∗τ
,
τ > max , γ1 ln 2ω(µ2 +d)+(µ
2 +d)−βM


Q∗τ −3ωM
1
ln 2(µ +d)−βM
γ
9





3
With application of the Lyapunov-LaSalle type theorem in [10]
lim x(t) = 0,
t→∞
lim y(t) = 0,
t→∞
lim z(t) = 0.
t→∞
5. Stochastic Stability
We limit ourselves here to perturbing only the contact rate so we replace β by β + σW (t),
whereW (t) is white noise (Brownian motion). The system (3) is transformed to the following Itô stochastic differential equations, with γ 0 = γe−µ2 τ


 dS = [ρ + µ − ν − (µ1 + d) S − βSQ] − σSQdW,
(12)
dI = [βSQ − (µ2 + d) I − γ 0 SQ] + σSQdW,

 dQ = [γ SQ − (µ + d) Q] ,
0
3
In this section, we will proof, under some conditions, that E0 is globally exponentially
mean square and almost surely stable, and for this purpose, we need the following Theorem
Theorem 6. The set Ω is almost surely invariant by the stochastic system (12). Thus if
(S0 , I0 , Q0 ) ∈ Ω , then P [(S, I, Q) ∈ Ω] = 1.
Proof. The system (12) implies that dN ≤ [ρ + µ − ν − (µ1 + d) N ] dt, then we have
ρ+µ−ν
ρ
N (t) ≤
+ N0 −
, for all t ≥ 0.
µ1 + d
µ1 + d
Since (S0 , I0 , Q0 ) ∈ Ω, then
N (t) ≤
(13)
ρ+µ−ν
, for all t ≥ 0.
µ1 + d
There exist ε0 > 0, such that S0 > ε0 > 0, I0 > ε0 > 0 and Q0 > ε0 > 0.
Considering
υ ε = inf {t ≥ 0, S (t) ≤ ε or I (t) ≤ ε or Q (t) ≤ ε, } , for ε ≤ ε0 ,
υ = lim υ ε = inf {t ≥ 0, S (t) ≤ 0 or I (t) ≤ 0 or Q (t) ≤ 0, }
t→0
Let
V (t) = log
ρ+µ−ν
ρ+µ−ν
ρ+µ−ν
+ log
+ log
.
(µ1 + d) S (t)
(µ1 + d) I (t)
(µ1 + d) Q (t)
(14)
10
CHAHRAZED
Then, using Itô formula we have, for all t ≥ 0 and T ∈ [0, t ∧ υ ε ],
"
#
− ρ+µ−ν
S
dV (T ) =
dT + σQdW
+ (µ1 + d) + βQ + 12 I 2
"
#
SQ
SQ
(γ 0 − β) I
dW
+
dT
+
σ
I
+ (µ2 + d) + 12 S 2
+ [−γ 0 S + (µ3 + d)] dT,
"
µ1 + µ2 + µ3 + 3d
(15)
dV (T ) ≤
+βQ + 21 I 2 + 12 S 2
h
i
ρ+µ−ν
With (13) , we have S, I and Q ∈ 0, (µ +d)
1
Let
#
dT + σ
Q
(I − S) dW
I
L = µ1 + µ2 + µ3 + 3d
2
ρ+µ−ν
ρ+µ−ν
,
+
+β
µ1 + d
µ1 + d
Q
f (I) =
,
I
(16)
We remplace (16) into (15) , we obtain
dV (T ) ≤ LdT + σ (I (T ) − S (T )) f (I (T )) dW,
(17)
Then
RT
V (T ) ≤ LT + σ f (I (u)) (I (u) − S (u)) dW (u) ,
(18)
0
RT
With proposition 7.6 in [6] , σ f (I (u)) (I (u) − S (u)) dW (u) is mean zero process then,
0
E (V (T )) ≤ LT
(19)
for all t ≥ 0 and T ∈ [0, t ∧ υ ε ] ,
ρ
S (t ∧ υ ε ) , I (t ∧ υ ε ) , and Q (t ∧ υ ε ) ∈ 0,
,
(µ1 + d)
Then
E (V (t ∧ υ ε )) ≤ L (t ∧ υ ε ) ≤ Lt,
V (t ∧ υ ε ) ≥ 0,
(20)
E (V (t ∧ υ ε )) ≥ E (V (t)) × κ[υε ≤t] ≥ P (υ ε ≤ t) log
Where κ[υε ≤t] is the indicator function of a subset [υ ε ≤ t] ,
ρ+µ−ν
(µ1 + d) ε
NONLINEAR EPIDEMIC MODEL WITH TEMPORARY IMMUNITY
11
Combining (19) , and (20) , we obtain
(21)
P (υ ε ≤ t) ≤
Lt
ρ+µ−ν
log (µ
+d)ε
, for all t ≥ 0;
1
for all t ≥ 0, and ε → 0, we obtain P (υ ≤ t) = 0;
From where
P (υ ≤ ∞) = 0
6. The Model with Age
The age distributions of the numbers in the classes are denoted by S(a, t), I(a, t), and
Q(a, t), denote the sizes of the population susceptible to disease, and infectious members,
quarantine members with the possibility of infection through temporary immunity, respectively of age a, at time t, d(a) is the age-specific death rate,
The system of partial equations for the age distributions is

∂S
∂S

 ∂t + ∂a = − (µ1 + d(a)) S(a, t) + β 1 (t)S(a, t),
∂I
∂I
(22)
+ ∂a
= −β 1 (t)S(a, t) − (µ2 + d(a)) I(a, t) + γ 1 (t − τ )S(a, t − τ ),
∂t

 ∂Q + ∂Q = −γ (t − τ )S(a, t − τ ) − (µ + d(a)) Q(a, t),
1
3
∂t
∂a
With
R
β 1 (t) = −β Q(a, t)da
R
γ 1 (t − τ ) = −γ e−µ2 τ Q(a, t − τ )da
(23)
6.1. Equilibrium and stability. Assume that sub population does not depend on the time
when the system (22) is written as follows

dS

 da = (β 1 − µ1 − d(a)) S(a),
dI
(24)
= (γ 1 − β 1 ) S(a) − (µ2 + d(a)) I(a),
da

 dQ = −γ S(a) − (µ + d(a)) Q(a),
1
3
da
The initial condition of (24) is given as
(25)
S(0) = S1 ,
I(0) = I1 ,
Q(0) = Q1
Differential equations of the system (24) are solved with different methods of resolutions and
with (25), so
(26)
(27)
(28)
S(a) = S1 e−(µ1 −β 1 )a Φ(a),
I(a) = I1 Φ(a)e−µ2 a −
(γ 1 − β 1 ) S1 Φ(a) −(µ1 −β 1 )a
e
− e−µ2 a ,
µ1 − β 1 − µ2
Q(a) = Q1 Φ(a)e−µ3 a −
γ 1 S1 Φ(a)
e−(µ1 −β 1 )a − e−µ3 a
µ1 − β 1 − µ3
12
CHAHRAZED
Where
R
Φ(a) = exp − d(a)da
(29)
The system (24) has the unique positive equilibrium point P1 ,
T
P1 = Ŝ1 , Iˆ1 , Q̂1 = (0, 0, 0)T .
We calculate the Jacobian matrix according to the system (24) with P1


β 1 − µ1 − d(a)
0
0


J (P1 ) = 
λ − γ0
− (µ2 + d(a))
0

−γ 0
0
− (µ3 + d(a))
The epidemic is locally asymptotically stable if and only if all eigenvalues of the Jacobian
matrix J (P1 ) have negative real part. The eigenvalues can be determined by solving the
characteristic equation of the linearization of (25) near P1 is


β 1 − µ1 − d(a) − A
0
0


(31)
det 
λ − γ0
− (µ2 + d(a)) − A
0
=0
−γ 0
0
− (µ3 + d(a)) − A
So the eigenvalues are
A1 = β 1 − µ1 − d(a), A2 = − (µ2 + d(a)) , A3 = − (µ3 + d(a))
In order to A1 , A2 ,and A3 will be negative, it is required that
β 1 < µ1 + d(a)
The basic reproduction number R0 is defined as the total number of infected population in the resulting sub-infected population where almost all of the uninfected. The basic
reproduction number of the infection R0 is defined as follows:
(32)
β1
µ1 + d(a)
R0 =
The time during which people remain infective is defined as
T =
1
µ1 + d(a)
The doubling time td of the epidemic can be obtained as
(33)
td =
(ln 2) T
R0 − 1
Theorem 1 The disease-free equilibrium P1 is locally asymptotically stable if R0 < 1 and
unstable if R0 > 1.
Let (26) , so if R0 < 1 then µ1 − β 1 > 0, so S(a) converges to zero.
NONLINEAR EPIDEMIC MODEL WITH TEMPORARY IMMUNITY
13
Let (27) , so
(34)
(γ 1 − β 1 ) S1 Φ(a) −m1 a
I(a) ≤ I1 Φ(a) −
e
, m1 = min {µ1 − β 1 , µ2 }
µ1 − β 1 − µ2
If R0 < 1, i(a) converges to zero.
Let (28) , so
γ 1 S1 Φ(a)
(35)
Q(a) = Q1 Φ(a) −
e−m2 a , m1 = min {µ1 − β 1 , µ3 }
µ1 − β 1 − µ3
If R0 < 1, Q(a) converges to zero.
Conclusion 7. This paper addresses a epidemic model with temporary immunity, whenever
the quarantine individuals will return to the susceptible. The endemic equilibrium is globally
asymptotically stable, then under some conditions, study the stochastic stability. Finally, the
equilibrium and stability of the epidemic model with age.
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Department of Mathematics, Faclty of Exacte sciences,, University des Freres Mentouri,
Algeria