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, withN(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 sus-ceptible and infectious.
In this paper, we discuss the equilibrium and stability of the model with temporary im-munity 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 Itˆo 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
2. Mathematical Model
This paper considers the following epidemic model with temporary immunity:
(1)
˙
S(t) = ρ+µ−ν−(µ1+d)S(t)−βS(t)Q(t),
˙
I(t) = βS(t)Q(t)−(µ2+d)I(t)−γe−µ2τS(t−τ)Q(t−τ),
˙
Q(t) = γe−µ2τS(t−τ)Q(t−τ)−(µ
3+d)Q(t),
Consider a population of sizeN(t) at timet, this population is divided into three subclasses, withN(t) =S(t) +I(t) +Q(t); whereS(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 µrepresent 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. ρthe positive constant is the parameter of immigration.ν the positive constant is the parameter of emigration. The term γe−µ2τS(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.
LetCdenote the Banach spaceC([−τ ,0],R3) of continuous functions mapping the interval
[−τ ,0] into R3. With a biological meaning, we further assume that Φ
i(η) = Φi(0) ≥ 0 for
i= 1,2,3.
Consider the system without the parameter of emigrations. Hence system (1) can be rewritten as
(3)
˙
S(t) = ρ+µ−ν−(µ1+d)S(t)−βS(t)Q(t),
˙
I(t) = βS(t)Q(t)−(µ2+d)I(t)−γe−µ2τS(t−τ)Q(t−τ),
˙
Q(t) = γe−µ2τS(t−τ)Q(t−τ)−(µ
3+d)Q(t),
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.
The region Ω = {(S, I, Q) ∈ R3
+, S +I +Q ≤ N < ρ+µ−ν
µ1+d } is positively invariant set of
3. Equilibrium and stability
An equilibrium point of system (3) satisfies
(5)
ρ+µ−ν−(µ1+d)S−βSQ= 0,
βkSQ−(µ2+d)I−γe−µ2τS(t−τ)Q(t−τ) = 0, γe−µ2τS(t−τ)Q(t−τ)−(µ
3+d)Q= 0,
We calculate the points of equilibrium in the absence and presence of infection. In the absence of infectionI = 0, the system (5) has a disease-free equilibrium E0:
E0 =
ˆ
S, I,ˆ QˆT =
ρ+µ−ν µ1 +d , 0, 0
T .
The eigenvalues can be determined by solving the characteristic equation of the linearization of (3) near E0 is
(6) det
−(µ1+d)−A 0 −β(ρ+µ−ν)µ
1+d
0 −(µ2+d)−A (ρ+µ−ν)(β−γe
−µ2τ)
µ1+d
0 0 (ρ+µ−ν)γeµ −µ2τ
1+d −(µ3+d)−A
= 0
So the eigenvalues are
A1 =−(µ1+d), A2 =−(µ2+d).
In order for λ1, λ2, to be negative, it is required that.
(7) (ρ+µ−ν)γe
−µ2τ
µ1+d < µ3+d
Then we define the basic reproduction number of the infectionR0 as follows.
(8) R0 =
(ρ+µ−ν)γe−µ2τ
(µ1+d) (µ3+d)
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
(9)
Sτ∗ = ρ+µ−νµ
1+d ×
1 R0, Iτ∗ = R0−1
µ2+d h
ρ+µ−ν
R0 −
(µ1+d)(µ3+d)
β
i , Q∗τ = µ1+d
β (R0−1)
Note that
Nτ∗ = (ρ+µ−ν) (µ2−µ1)
R0(µ1+d) (µ2+d)
+ ρ+µ−ν
µ2+d (10)
+(R0−1) (µ1+d) (µ2−µ3)
β(µ2+d)
So Eτ∗ = (Sτ∗, Iτ∗, Q∗τ)T is the unique positive endemic equilibrium point which exists if
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
(11)
˙
x(t) = [−(µ1+d)−βQ∗τ]x+ [−βSτ∗]z,
˙
y(t) = [(β−γe−µ2τ)Q∗
τ]x+ [−(µ2+d)]y+ [(β−γe−µ2τ)Sτ∗]z,
˙
z(t) = [βγe−µ2τQ∗
τ]x+ [γe −µ2τS∗
τ −(µ3 +d)]z 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 thatS(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
˙
S(t0) = ρ+µ−ν >0.
ForS(t0) = 0, S0 >0, we must have ˙S(t0)<0 which is contradiction. • Assume that I(t0) = 0, then with Eq 2 in the system (1) we have
˙
I(t0) =−γe−µ2τS(t−τ)Q(t−τ)
˙
I(t0) is positive because S(t) and Q(t) solutions of system (1)
are positive for allt >0.
• ForI(t0) = 0, I >0, we must have ˙I(t0)<0 which is contradiction.
• Assume that Q(t0) = 0, thenS(t)≥0 for all t∈[0, t0].with Eq 3 in the system (3)
we have
˙
Q(t0) = γe−µ2τS(t−τ)Q(t−τ),
Q(t0) = γ t0 R
t0−τ
e−µ2(t0−s)S(s)Q(s)ds.
S(s) > 0, S(s) > 0 for all t ∈ [0, t0). We have γ t0 R
t0−τ
e−µ2(t0−s)S(s)Q(s)ds > 0, and Q(t0) = 0, which is contradiction.
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
µ1+d , S0 +I0+Q0
=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 ifµ2+d <0,
lim
t→∞I(t) = 0,
With Eq 3 in the system (3) we have.
lim
t→∞Q(t) = 0,
With Eq 1 in the system (3) we obtain ˙S =ρ+µ−ν−(µ1 +d)S.
lim
t→∞S(t) =
ρ+µ−ν µ1+d ,
Hence.
lim
t→∞N(t) =
ρ+µ−ν µ1+d .
From lemma1, S(t), I(t) and Q(t) solutions of system (1) are positive.
S(t)≤ ρ+µ−ν
µ1+d ,for allt≤0.
Suppose that
N(0) ≤ ρ+µ−ν
µ1 +d , then N(t)≤
ρ+µ−ν µ1+d
On the contrary IfN(0) > ρ+µ−νµ
1+d then N(t)< N(0), and S(t)< N(0) for all t >0.
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
τ >max
1 γln
ωM+3ωQ∗
τ
2ω(µ1+d) ,
1 γln
ωM+3ωQ∗τ
2ω(µ2+d)+(µ2+d)−βM,
1 γln
Q∗τ−3ωM
2(µ3+d)−βM
Where
M = max
ρ+µ−ν
µ1+d , S0+I0+Q0
,
ω = βQ
∗ τ
µ1+µ2 + 2d
Proof. We consider system (3).
Let us introduce the functional
V (x, y, z) = 1
2ω(x+y)
2
+1 2 y
2+z2 ,
The derivative ˙V (x, y, z) is ˙
V (x, y, z) = ω(x+y) ( ˙x+ ˙y) +yy˙ +zz˙
= ω(x+y)
"
(−(µ1+d)−βQ∗τ)x−βSτ∗z+ (β−γe−µ2τ)Q∗
τx −(µ2+d)y+ (β−γe−µ2τ)S∗
τz
#
+y β−γe−µ2τQ∗
τx−(µ2+d)y+ β−γe −µ2τ
Sτ∗z+ +zβγe−µ2τQ∗
τx+ γe −µ2τS∗
τ −(µ3+d)
z
= −ω(µ1+d)x2−[(ω+ 1) (µ2 +d)]y2
− γe−µ2τS∗
τ −(µ3+d)
z2
+ [βQ∗τ −ω(µ1+d)−ω(µ2+d)]xy
+βSτ∗yz−
ωQ∗τγe−µ2τxx(t−τ)
−(ω+ 1)Q∗τγe−µ2τyx(t−τ)
+Q∗τγe−µ2τzx(t−τ)−ωS∗
τγe
−µ2τxz(t−τ) −(ω+ 1)Sτ∗γe−µ2τyz(t−τ) +S∗
τγe
−µ2τzz(t−τ).
By lemma 2 we haveS(t)≤M for allt ≥0 andωis an arbitrary real constant choosing as follows
ω = βQ
∗ τ
µ1+µ2+ 2d
˙
V (x, y, z) ≤ −ω(µ1+d)x2−[(ω+ 1) (µ2+d)]y2
−(µ3+d)z2
+βM yz−
ωQ∗τγe−µ2τxx(t−τ)
−(ω+ 1)Q∗τγe−µ2τyx(t−τ)
+Q∗τγe−µ2τzx(t−τ)−ωM γe−µ2τxz(t−τ)
−(ω+ 1)M γe−µ2τyz(t−τ)
Applying Cauchy-Chwartz inequality; we obtain:
˙
V (x, y, z) ≤ −ω(µ1+d)x2−[(ω+ 1) (µ2+d)]y2
−(µ3+d)z2
−1 2ωQ
∗ τγe
−µ2τx2+x2(t−τ)
−1
2(ω+ 1)Q
∗ τγe
−µ2τ
y2+x2(t−τ)
+1 2Q
∗ τγe
−µ2τ
z2+x2(t−τ)
−1
2ωM γe
−µ2τ
x2+z2(t−τ)
−1
2(ω+ 1)M γe
−µ2τy2+z2(t−τ)
+1 2M γe
−µ2τ
z2+z2(t−τ)
+1 2βM
y2+z2,
≤
−ω(µ1+d)− 1 2ωγe
−µ2τ(M +Q∗ τ)
x2
+
1
2βM−(ω+ 1) (µ2+d)− 1
2(ω+ 1)γe
−µ2τ(M+Q∗ τ)
y2
+
1
2βM−(µ3+d) + 1 2γe
−µ2τ(M +Q∗ τ)
z2
−
ωQ∗τγe−µ2τx2(t−τ)−
1
2(3ω+ 1)γM e
−µ2τ
z2(t−τ)
Choose the Lyapunov functional
V (xt, yt, zt) = V (x, y, z)−ωQ∗τγe −µ2τ Rt
t−τ
x2(θ)dθ
−1
2(3ω+ 1)γM e
−µ2τ Rt
t−τ
Then
˙
V (xt, yt, zt) = V˙ (x, y, z)−ωQ∗τγe
−µ2τx2(t)
+ωQ∗τγe−µ2τx2(t−τ)
−1
2(3ω+ 1)γM e
−µ2τ
z2(t)
+1
2(3ω+ 1)γM e
−µ2τz2(t−τ)
≤
−ω(µ1+d)−1 2ωγe
−µ2τ
(M +Q∗τ)
x2
+
"
1
2βM −(ω+ 1) (µ2+d) −1
2(ω+ 1)γe
−µ2τ(M +Q∗ τ) # y2 + 1
2βM −(µ3+d) + 1 2γe
−µ2τ(M +Q∗ τ)
z2
−
ωQ∗τγe−µ2τx2+ωQ∗
τγe
−µ2τx2(t−τ) −
1
2(3ω+ 1)γM e
−µ2τ
z2
+1
2(3ω+ 1)γM e
−µ2τ
z2(t−τ),
≤
−ω(µ1+d)−1 2ωγe
−µ2τ(M +Q∗ τ) x2 + 1
2βM −(ω+ 1) (µ2+d)− 1
2(ω+ 1)γe
−µ2τ(M +Q∗
τ) y2 + 1
2βM −(µ3+d) + 1 2γe
−µ2τ(M +Q∗ τ)
z2
−
ωQ∗τγe−µ2τx2−
1
2(3ω+ 1)γM e
−µ2τ
z2
Therefore
˙
V (xt, yt, zt) ≤ −
ω(µ1+d) + 1 2ωγe
−µ2τ(M + 3Q∗ τ)
x2
−
"
(ω+ 1) (µ2+d)− 1 2βM
+12(ω+ 1)γe−µ2τ(M +Q∗
τ)
# y2
−
"
(µ3+d)− 1 2βM −1
2γe
−µ2τ(Q∗
τ −3ωM)
While the above inequality is always negative provided that
τ > max
1 γln
ωM+3ωQ∗τ
2ω(µ1+d)
,γ1ln ωM+3ωQ∗τ
2ω(µ2+d)+(µ2+d)−βM,
1 γln
Q∗
τ−3ωM
2(µ3+d)−βM
With application of the Lyapunov-LaSalle type theorem in [10]
lim
t→∞x(t) = 0, t→∞lim y(t) = 0, t→∞lim z(t) = 0.
5. Stochastic Stability
We limit ourselves here to perturbing only the contact rate so we replaceβ byβ+σW(t), whereW(t) is white noise (Brownian motion). The system (3) is transformed to the fol-lowing Itˆo stochastic differential equations, with γ0 =γe−µ2τ
(12)
dS= [ρ+µ−ν−(µ1+d)S−βSQ]−σSQdW, dI = [βSQ−(µ2+d)I−γ0SQ] +σSQdW, dQ= [γ0SQ−(µ3+d)Q],
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
N(t)≤ ρ+µ−ν
µ1+d +
N0−
ρ µ1 +d
, for all t ≥0.
Since (S0, I0, Q0)∈ Ω, then
(13) N(t)≤ ρ+µ−ν
µ1+d , for all t ≥0.
There exist ε0 >0, such thatS0 > ε0 >0, I0 > ε0 >0 andQ0 > ε0 >0.
Considering
υε = inf{t≥0, S(t)≤ε orI(t)≤ε or Q(t)≤ε,}, for ε≤ε0, (14)
υ = lim
t→0υε = inf{t≥0, S(t)≤0 or I(t)≤0 or Q(t)≤0,}
Let
V (t) = log ρ+µ−ν
(µ1+d)S(t) + log
ρ+µ−ν
(µ1+d)I(t)+ log
ρ+µ−ν
Then, using Itˆo formula we have, for allt ≥0 and T ∈[0, t∧υε],
dV (T) =
"
−ρ+µ−νS
+ (µ1+d) +βQ+ 12I2
#
dT +σQdW
+
"
(γ0−β)SQI + (µ2+d) + 12S2
#
dT +σSQ I dW
+ [−γ0S+ (µ3+d)]dT,
(15) dV (T)≤
"
µ1+µ2+µ3+ 3d
+βQ+12I2 +1 2S
2
#
dT +σQ
I (I −S)dW
With (13), we have S, I and Q∈h0,ρ+µ−ν(µ 1+d) i
Let
L = µ1+µ2+µ3+ 3d (16)
+βρ+µ−ν µ1+d +
ρ+µ−ν µ1+d
2 ,
f(I) = Q
I ,
We remplace (16) into (15), we obtain
(17) dV (T)≤LdT +σ(I(T)−S(T))f(I(T))dW,
Then
(18) V (T)≤LT +σ
T
R
0
f(I(u)) (I(u)−S(u))dW(u),
With proposition 7.6 in [6], σ
T
R
0
f(I(u)) (I(u)−S(u))dW(u) is mean zero process then,
(19) E(V (T))≤LT
for all t≥0 and T ∈[0, t∧υε],
S(t∧υε), I(t∧υε), and Q(t∧υε)∈
0, ρ
(µ1 +d)
,
Then
E(V (t∧υε))≤L(t∧υε)≤Lt,
V (t∧υε)≥0,
(20) E(V (t∧υε))≥E(V (t))×κ[υε≤t]≥P (υε ≤t) log
ρ+µ−ν
(µ1+d)ε
Combining (19), and (20), we obtain
(21) P(υε ≤t)≤
Lt
log(µρ+µ−ν
1+d)ε
, for all t≥0;
for all t≥0, and ε→0, we obtainP (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, respec-tively of age a,at time t, d(a) is the age-specific death rate,
The system of partial equations for the age distributions is
(22)
∂S ∂t +
∂S
∂a =−(µ1+d(a))S(a, t) +β1(t)S(a, t), ∂I
∂t + ∂I
∂a =−β1(t)S(a, t)−(µ2+d(a))I(a, t) +γ1(t−τ)S(a, t−τ), ∂Q
∂t + ∂Q
∂a =−γ1(t−τ)S(a, t−τ)−(µ3+d(a))Q(a, t),
With
β1(t) = −βRQ(a, t)da (23)
γ1(t−τ) = −γR
e−µ2τQ(a, t−τ)da
6.1. Equilibrium and stability. Assume that sub population does not depend on the time when the system (22) is written as follows
(24)
dS
da = (β1−µ1−d(a))S(a), dI
da = (γ1−β1)S(a)−(µ2 +d(a))I(a), dQ
da =−γ1S(a)−(µ3+d(a))Q(a),
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) S(a) = S1e−(µ1−β1)a Φ(a),
(27) I(a) = I1Φ(a)e−µ2a−
(γ1−β1)S1Φ(a)
µ1−β1−µ2 e
−(µ1−β1)a−e−µ2a ,
(28) Q(a) =Q1Φ(a)e−µ3a−
γ1S1Φ(a)
µ1−β1−µ3 e
Where
(29) Φ(a) = exp −R
d(a)da
The system (24) has the unique positive equilibrium point P1,
P1 =
ˆ
S1, Iˆ1, Qˆ1
T
= (0, 0, 0)T .
We calculate the Jacobian matrix according to the system (24) with P1
J(P1) =
β1−µ1−d(a) 0 0
λ−γ0 −(µ2+d(a)) 0 −γ0 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) nearP1 is
(31) det
β1−µ1−d(a)−A 0 0
λ−γ0 −(µ2+d(a))−A 0
−γ0 0 −(µ3+d(a))−A
= 0
So the eigenvalues are
A1 =β1−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
β1 < µ1+d(a)
The basic reproduction number R0 is defined as the total number of infected
popula-tion in the resulting sub-infected populapopula-tion where almost all of the uninfected. The basic reproduction number of the infection R0 is defined as follows:
(32) R0 =
β1 µ1+d(a)
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 1The disease-free equilibriumP1 is locally asymptotically stable ifR0 <1 and
unstable ifR0 >1.
Let (27),so
(34) I(a)≤
I1Φ(a)−
(γ1−β1)S1Φ(a)
µ1−β1−µ2
e−m1a, m
1 = min{µ1−β1, µ2}
IfR0 <1,i(a) converges to zero.
Let (28),so
(35) Q(a) =
Q1Φ(a)−
γ1S1Φ(a)
µ1−β1−µ3
e−m2a, m
1 = min{µ1−β1, µ3}
IfR0 <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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