Adv Syst Sci Appl 2017; 4:46–60
Published online at http://ijassa.ipu.ru/ojs/ijassa/article/view/510

Threshold analysis of a stochastic epidemic model with delay and
temporary immunity

Rui Xue, Fengying Wei ∗

College of Mathematics and Computer Science, Fuzhou University, Fuzhou 350116, P.R. China

Abstract: A stochastic susceptible-infected-recovered model is formulated and investigated
when the temporary immunity is fixed for the population in this paper. The existence and
uniqueness of the global positive solution has been checked with probability one for any initial
value. And the sufficient conditions for the extinction and the persistence of the stochastic
epidemic model with temporary immunity are derived by constructing Lyapunov functions and
the generalized Ito’s formula, where the threshold of the persistence does not depend on the
temporary immunity, while the densities of the infected and recovered are obviously dependent on
the temporary immunity when given a perturbation. Illustrative examples and simulations show
that the perturbations make the properties of the stochastic epidemic model different from the
deterministic one.

Keywords: stochastic epidemic model; delay; persistence; extinction; threshold

1. INTRODUCTION

The susceptible-infected-removed epidemic model is one of the most important models in
epidemiological patterns and disease control. Epidemic models with delay always make
the description of the epidemiological patterns more realistic, more interesting and more
complicated. Some authors investigated the effects of disease latency or immunity, in which
the models are described by systems of ordinary differential equations with delay, for instance
[1–10], and systems without delay [11–15]. Among these recent results we would like to
mention the work by Wen and Yang [10], in which they considered a type of delayed
susceptible-infected-recovered epidemic model with temporary immunity:

Ṡ(t) = Λ− µ1S(t)− βS(t)I(t) + γI(t− τ)e−µ3τ ,

İ(t) = βS(t)I(t)− (µ2 + γ)I(t),

Ṙ(t) = γI(t)− γI(t− τ)e−µ3τ − µ3R(t),

(1.1)

where S(t) is the number of the susceptible individuals to the disease, I(t) represents the
number of the individuals who are infected and R(t) denotes the recovered individuals who
have been removed from the possibility of infection, Λ denotes a constant input of new
members into the population, µ1, µ2, µ3 represent the death rates of the susceptible, the
infected, the recovered, and they assume that µ1 ≤ min{µ2, µ3} from the biological point
of view, β denotes the transmission coefficient between compartments S and I , γ denotes
the recovery rate of the infected individuals, i.e., the rate at which the individuals move from
the infected compartment to the recovered compartment, τ > 0 is the length of immunity

∗Corresponding author: weifengying@fzu.edu.cn

http://ijassa.ipu.ru/ojs/ijassa/article/view/510


THRESHOLD ANALYSIS OF A STOCHASTIC EPIDEMIC MODEL WITH DELAY AND TEMPORARY 47

period. Wen and Yang [10] assume that Λ, µi, β, γ are all positive constants in model (1.1),
and they obtain the expression of the basic reproduction number in terms of parameters, that
is, R0 = βΛ

µ1(µ2+γ)
, which measures how fast the diseases spread in the deterministic model

provided that the epidemics take place.
Some common phenomena in the real world, such as the weather fluctuations, temperature

changes and perturbations caused by human beings, can not been ignored when it comes to
the dynamics of the infectious diseases. From both biological and mathematical perspectives,
the stochastic models have more reasonable patterns and have been investigated by different
approaches presented in the recent literatures [16–20]. Motivated by model (1.1) and the
approaches mentioned in [16–20], we try to investigate several properties of model (1.2)
when transmission coefficient β in (1.1) is perturbed by the white noise β + σξ(t):

dS(t) =
(
Λ− µ1S(t)− βS(t)I(t) + γI(t− τ)e−µ3τ

)
dt− σS(t)I(t)dB(t),

dI(t) =
(
βS(t)I(t)− (µ2 + γ)I(t)

)
dt+ σS(t)I(t)dB(t),

dR(t) =
(
γI(t)− γI(t− τ)e−µ3τ − µ3R(t)

)
dt,

(1.2)

where ξ(t) = dB(t)
dt

, andB(t) is the independent Brownian motion, and σ is the intensity of the
white noise when model (1.2) works on a complete probability space (Ω,F , {Ft}t≥0,P) with
a filtration {Ft}t≥0 satisfying the usual conditions (i.e., it is right continuous and increasing
function while F0 contains all P-null sets).

What we concern about in model (1.2) in this paper are listed below, of course, for any
positive initial value herewith:

• Existence and uniqueness of a global nonnegative solution with probability one;
• The sufficient conditions that guarantees the extinction of the diseases;
• Figuring out the threshold presented in terms of model parameters;
• Demonstrating illustrative examples and their realizations to support the validity of the

main results.

2. EXISTENCE AND UNIQUENESS OF THE POSITIVE SOLUTION

For model (1.2), the sum of three equations gives

d(S(t) + I(t) +R(t))

dt
≤ Λ− µ1(S(t) + I(t) +R(t)), (2.3)

then one can obtain that

S(t) + I(t) +R(t) ≤


Λ

µ1

, S(0) + I(0) +R(0) ≤ Λ

µ1

,

S(0) + I(0) +R(0), S(0) + I(0) +R(0) >
Λ

µ1

.
(2.4)

We denote

N = max

{
Λ

µ1

, S(0) + I(0) +R(0)

}
, (2.5)

therefore, S(t) ≤ N, I(t) ≤ N,R(t) ≤ N . We notice that the first two equations in model
(1.2) do not depend on the third equation, we omit them without loss of generality. Thus, we

Copyright c© 2017 ASSA. Adv Syst Sci Appl (2017)



48 R. XUE, F.Y. WEI

only discuss the simplified model of having two equations:{
dS(t) =

(
Λ− µ1S(t)− βS(t)I(t) + γI(t− τ)e−µ3τ

)
dt− σS(t)I(t)dB(t),

dI(t) =
(
βS(t)I(t)− (µ2 + γ)I(t)

)
dt+ σS(t)I(t)dB(t).

(2.6)

In order to investigate other properties, we firstly study the fundamental property of the
solution of model (2.6). Next we will present the existence and uniqueness of the solution of
model (1.2).
Theorem 2.1:
For any initial value S(0) > 0 and I(ζ) ≥ 0 for all ζ ∈ [−τ, 0) with I(0) > 0, model (2.6)
admits a unique positive solution (S(t), I(t)) on t > 0 and the solution will remain in R2

+

with probability one, that is to say, (S(t), I(t)) ∈ R2
+ for all t > 0 almost surely.

Proof
According to the approach mentioned in [13–15, 21]. The proof will go as follows. Since the
coefficients of model (2.6) satisfy the local Lipschitz conditions, then for any initial value
S(0) > 0 and I(ζ) ≥ 0 for all ζ ∈ [−τ, 0) with I(0) > 0, there is a unique local solution
(S(t), I(t)) on the half-closed interval [−τ, τe), where τe represents the explosion time of
the solution. To verify the solution is global, what we prepare to check that explosion time
τe =∞ holds almost surely. To this end, let k0 ≥ 1 be sufficiently large such that each
component of the initial value (S(0), I(ζ)) always lies within the interval [ 1

k0
, k0]. For each

integer k ≥ k0, let us define the following stopping time

τk = inf

{
t ∈ [0, τe)

∣∣∣∣S(t) /∈
(

1

k
, k

)
or I(t) /∈

(
1

k
, k

)}
. (2.7)

Throughout this paper, we set inf ∅ =∞ (as usual ∅ represents the empty set). Obviously, τk
is an increasing function as k →∞. Let τ∞ be the limit of τk as k →∞. It is obvious that
τ∞ ≤ τe is valid almost surely. In order to complete the proof, we only need to show τ∞ =∞
a.s.. We start the proof by contradiction, if this assertion is false, then there exists a pair of
constants T > 0 and ε ∈ (0, 1) such that

P{τ∞ ≤ T} > ε. (2.8)

Thereby, there is an integer k1 ≥ k0 such that

P{τk ≤ T} ≥ ε for all k ≥ k1. (2.9)

Let us define a C2-function V : R2
+ → R+ as follows:

V (S(t), I(t)) = S(t)− a− a ln
S(t)

a
+ I(t)− 1− ln I(t) + γe−µ3τ

∫ t

t−τ
I(r)dr, (2.10)

where a is a positive constant to be determined later. The nonnegativity of µ− 1− log µ is
obvious for all µ > 0. Let k ≥ k0 and T > 0 be arbitrary positive constants. The application
of Itô’s formula gives

dV (S(t), I(t)) =

(
1− a

S(t)

)
dS(t) +

a

2S2(t)
(dS(t))2 +

(
1− 1

I(t)

)
dI(t)

+
1

2I2(t)
(dI(t))2 + γI(t)e−µ3τdt− γI(t− τ)e−µ3τdt

:= LV (S(t), I(t))dt+ σ(aI(t)− S(t))dB(t),

(2.11)

Copyright c© 2017 ASSA. Adv Syst Sci Appl (2017)



THRESHOLD ANALYSIS OF A STOCHASTIC EPIDEMIC MODEL WITH DELAY AND TEMPORARY 49

where LV (S(t), I(t)) is defined from R2
+ to R and computed as

LV (S(t), I(t))

=

(
1− a

S(t)

)(
Λ− µ1S(t)− βS(t)I(t) + γI(t− τ)e−µ3τ

)
+
aσ2

2
I2(t) +

(
1− 1

I(t)

)(
βS(t)I(t)− (µ2 + γ)I(t)

)
+
σ2

2
S2(t) + γI(t)e−µ3τ − γI(t− τ)e−µ3τ

= Λ− µ1S(t)− (µ2 + γ)I(t) + γI(t− τ)e−µ3τ

− aΛ

S(t)
+ µ1a+ aβI(t)− aγe−µ3τ I(t− τ)

S(t)
− βS(t) + µ2 + γ

+γI(t)e−µ3τ − γI(t− τ)e−µ3τ +
σ2

2
(S2(t) + aI2(t))

≤ Λ + µ1a+ µ2 + γ + (aβ + γe−µ3τ − (µ2 + γ))I(t) +
σ2

2
(1 + a)N2.

(2.12)

Choosing a positive constant

a =
µ2 + γ(1− e−µ3τ )

β
(2.13)

such that aβ + γe−µ3τ = µ2 + γ, then we have

LV (S(t), I(t)) ≤ Λ + µ1a+ µ2 + γ +
σ2

2
(1 + a)N2 := K, (2.14)

where K > 0 is a constant. The remainder of the proof follows the similar approach given
in [22].

Remark 2.1:
Theorem 2.1 implies that the stochastic model (1.2) admits a unique solution
(S(t), I(t), R(t)) for any initial value (S(0), I(0), R(0)) ∈ R3

+, I(ζ) ≥ 0 for all ζ ∈ [−τ, 0).
Each component of the solution (S(t), I(t), R(t)) is positive for all t ≥ 0 almost surely.
According to the statement of Theorem 2.1, if the sum of each component with the initial
value is almost surely bounded, say S(0) + I(0) +R(0) ≤ Λ

µ1
, so the region

Γ∗ =

{
(S(t), I(t), R(t)) ∈ R3

+ : S(t) + I(t) +R(t) ≤ Λ

µ1

, t ≥ 0

}
(2.15)

is the positively invariant set of model (1.2).

3. THE SUFFICIENT CONDITIONS OF THE EXTINCTION OF THE DISEASES

Let

R̃0 =
βΛ

µ1(µ2 + γ)
− σ2Λ2

2µ2
1(µ2 + γ)

, (3.16)

which can be presented in terms of the basic reproduction number of the deterministic
model (1.1):

R̃0 = R0 −
σ2Λ2

2µ2
1(µ2 + γ)

. (3.17)

Copyright c© 2017 ASSA. Adv Syst Sci Appl (2017)



50 R. XUE, F.Y. WEI

For the stochastic model (1.2), we will investigate the sufficient conditions of the
extinction of the disease. In other words, the infected individuals will eventually recover from
the attack of the disease, and the capacity of the total population will keep kind of constant in
the long run.
Theorem 3.1:
Let (S(t), I(t), R(t)) be the solution of model (1.2) with initial value (S(0), I(0), R(0)) ∈ Γ∗.
If the intensity of the white noise satisfies

σ2 >
β2

2(µ2 + γ)
(3.18)

or
R̃0 < 1, σ2 ≤ βµ1

Λ
, (3.19)

then the density of the infected declines exponentially

lim sup
t→∞

ln I(t)

t
≤ −(µ2 + γ) +

β2

2σ2
< 0 a.s. (3.20)

or

lim sup
t→∞

ln I(t)

t
≤ (µ2 + γ)(R̃0 − 1) < 0 a.s. (3.21)

respectively; and

lim sup
t→∞

S(t) =
Λ

µ1

a.s.. (3.22)

Proof

Adding up those three equations of model (1.2) gives

d

(
S(t) + I(t) + γe−µ3τ

∫ t

t−τ
I(r)dr

)
= [Λ− µ1S(t)− (µ2 + γ(1− e−µ3τ ))I(t)]dt.

(3.23)
We integrate the above from 0 to t and divided by t, which implies

Λ− µ1〈S(t)〉 − (µ2 + γ(1− e−µ3τ ))〈I(t)〉

=
1

t

(
S(t) + I(t) + γe−µ3τ

∫ t

t−τ
I(r)dr − S(0)− I(0)− γe−µ3τ

∫ 0

−τ
I(r)dr

)
,

(3.24)
and then derives that

〈S(t)〉 =
Λ

µ1

− µ2 + γ(1− e−µ3τ )
µ1

〈I(t)〉+ ϕ(t), (3.25)

where

ϕ(t) =
1

µ1t

(
S(t) + I(t) + γe−µ3τ

∫ t

t−τ
I(r)dr − S(0)− I(0)− γe−µ3τ

∫ 0

−τ
I(r)dr

)
.

(3.26)
The strong law of large numbers and the fact S(0) + I(0) ≤ Λ

µ1
show that

lim
t→∞

ϕ(t) = 0 a.s.. (3.27)

Copyright c© 2017 ASSA. Adv Syst Sci Appl (2017)



THRESHOLD ANALYSIS OF A STOCHASTIC EPIDEMIC MODEL WITH DELAY AND TEMPORARY 51

For the second equation of model (1.2), the generalized Itô’s formula leads to

d ln I(t) =

(
βS(t)− (µ2 + γ)− σ2S2(t)

2

)
dt+ σS(t)dB(t). (3.28)

Integrating (3.28) from 0 to t and divided by t gives

ln I(t)

t
= β〈S(t)〉 − (µ2 + γ)− σ2

2
〈S2(t)〉+

σ

t

∫ t

0

S(r)dB(r) +
ln I(0)

t
. (3.29)

We denote

M(t) := σ

∫ t

0

S(r)dB(r), (3.30)

where M(t) is a local continuous martingale with M(0) = 0. Moreover,

lim sup
t→∞

〈M(t),M(t)〉t
t

≤ σ2Λ2

µ2
1

<∞ a.s.. (3.31)

The strong law of large numbers for martingales presented in [23] thus implies that

lim sup
t→∞

M(t)

t
= 0 a.s.. (3.32)

If σ2 > β2

2(µ2+γ)
, rewriting (3.29) gives

ln I(t)

t
= −σ

2

2t

∫ t

0

(
S(r)− β

σ2

)2

dr − (µ2 + γ) +
β2

2σ2
+
σ

t

∫ t

0

S(r)dB(r) +
ln I(0)

t

≤ −(µ2 + γ) +
β2

2σ2
+
σ

t

∫ t

0

S(r)dB(r) +
ln I(0)

t
.

(3.33)
Taking superior limit on both sides of (3.33), then we get that

lim sup
t→∞

ln I(t)

t
≤ −(µ2 + γ) +

β2

2σ2
< 0 a.s.. (3.34)

Copyright c© 2017 ASSA. Adv Syst Sci Appl (2017)



52 R. XUE, F.Y. WEI

If R̃0 < 1 and σ2 ≤ βµ1
Λ

hold, the Schwarz inequality and substituting (3.25) into (3.29)
yield that

ln I(t)

t
≤ β〈S(t)〉 − (µ2 + γ)− σ2

2
〈S(t)〉2 +

σ

t

∫ t

0

S(r)dB(r) +
ln I(0)

t

=
βΛ

µ1

− (µ2 + γ)− β(µ2 + γ(1− e−µ3τ ))
µ1

〈I(t)〉+ βϕ(t)

+
σ

t

∫ t

0

S(r)dB(r)− σ2

2

[
Λ

µ1

− µ2 + γ(1− e−µ3τ )
µ1

〈I(t)〉+ ϕ(t)

]2

=
βΛ

µ1

− (µ2 + γ)− β(µ2 + γ(1− e−µ3τ ))
µ1

〈I(t)〉+ βϕ(t)

+
σ

t

∫ t

0

S(r)dB(r)− σ2

2

[
Λ2

µ2
1

+

(
µ2 + γ(1− e−µ3τ )

µ1

〈I(t)〉
)2

+ϕ2(t)− 2Λ(µ2 + γ(1− e−µ3τ ))
µ2

1

〈I(t)〉

+
2Λ− 2(µ2 + γ(1− e−µ3τ ))〈I(t)〉

µ1

ϕ(t)

]2

,

(3.35)

which gives that

ln I(t)

t
≤ βΛ

µ1

− σ2Λ2

2µ2
1

− µ2 + γ(1− e−µ3τ )
µ1

(
β − σ2Λ

µ1

)
〈I(t)〉

−(µ2 + γ) + ψ(t),

(3.36)

where

ψ(t) = βϕ(t) +
σ

t

∫ t

0

S(r)dB(r)

−σ
2

2

[
ϕ2(t) +

2Λ− 2(µ2 + γ(1− e−µ3τ ))〈I(t)〉
µ1

ϕ(t)

]
.

(3.37)

It is easy to check that
lim
t→∞

ψ(t) = 0 a.s.. (3.38)

We take superior limit on both sides of (3.35) and derive that

lim sup
t→∞

ln I(t)

t
≤ (µ2 + γ)(R̃0 − 1) < 0 a.s.. (3.39)

Further, the density of the infected individuals declines to zero, and the density of the
susceptible individuals keeps constant in a long time run, mathematically speaking, these two
expressions

lim
t→∞

I(t) = 0, lim
t→∞

S(t) =
Λ

µ1

(3.40)

hold almost surely when condition (3.18) or condition (3.19) is satisfied.

Remark 3.1:
Wen and Yang [10] had shown that the disease-free equilibrium E0( Λ

µ1
, 0, 0) of the

deterministic model (1.1) was globally asymptotically stable if the basic reproduction number
R0 < 1 (i.e., the disease disappeared under the condition R0 < 1). While for the stochastic

Copyright c© 2017 ASSA. Adv Syst Sci Appl (2017)



THRESHOLD ANALYSIS OF A STOCHASTIC EPIDEMIC MODEL WITH DELAY AND TEMPORARY 53

model (1.2), we obtain that the disease disappears when conditions R̃0 < 1 and σ2 ≤ βµ1
Λ

are
being satisfied in this chapter. This indicates that the condition of extinction to the stochastic
model (1.2) is weaker than that of the corresponding deterministic model (1.1).

4. THE THRESHOLD FOR THE PERSISTENCE OF THE DISEASES

Theorem 4.1:
Let (S(t), I(t), R(t)) be any solution of model (1.2), (S(0), I(0), R(0)) ∈ Γ∗ be any initial
value with I(ζ) ≥ 0 for all ζ ∈ [−τ, 0). We assume that R̃0 > 1, and also assume that the
intensity of the white noise satisfies

σ2 ≤ βµ1

Λ
, (4.41)

then the densities of the infected and the recovered individuals have the following properties:

Ĩ∗ ≤ lim inf
t→∞

〈I(t)〉 ≤ lim sup
t→∞

〈I(t)〉 ≤ Ĩ∗ a.s. (4.42)

and
R̃∗ ≤ lim inf

t→∞
〈R(t)〉 ≤ lim sup

t→∞
〈R(t)〉 ≤ R̃∗ a.s., (4.43)

where

Ĩ∗ =
µ1(µ2 + γ)

β(µ2 + γ(1− e−µ3τ ))
(R̃0 − 1), R̃∗ =

γ(1− e−µ3τ )
µ3

Ĩ∗, (4.44)

and

Ĩ∗ =
µ2

1(µ2 + γ)

(βµ1 − σ2Λ)(µ2 + γ(1− e−µ3τ ))
(R̃0 − 1), R̃∗ =

γ(1− e−µ3τ )
µ3

Ĩ∗. (4.45)

Proof

From (3.28), the second equation of model (1.2) implies that

d ln I(t) ≥
(
βS(t)− µ2 − γ −

σ2Λ2

2µ2

)
dt+ σS(t)dB(t), (4.46)

integrating (4.46) on both sides and substituting (3.25) into the integration, which derive that

ln I(t)− ln I(0)

t
≥ β〈S(t)〉 − µ2 − γ −

σ2Λ2

2µ2
+
σ

t

∫ t

0

S(r)dB(r)

=
βΛ

µ1

−
(
µ2 + γ +

σ2Λ2

2µ2

)
− β(µ2 + γ(1− e−µ3τ ))

µ1

〈I(t)〉

+βϕ(t) +
σ

t

∫ t

0

S(r)dB(r).

(4.47)

Inequality (4.47) can be rewritten as

〈I(t)〉 ≥ µ1

β(µ2 + γ(1− e−µ3τ ))

(
βΛ

µ1

−
(
µ2 + γ +

σ2Λ2

2µ2

)
− ln I(t)− ln I(0)

t
+ βϕ(t) +

σ

t

∫ t

0

S(r)dB(r)

)
.

(4.48)

Copyright c© 2017 ASSA. Adv Syst Sci Appl (2017)



54 R. XUE, F.Y. WEI

According to (2.15), we have

−∞ < ln I(t) < ln
Λ

µ1

, lim
t→∞

σ

t

∫ t

0

S(r)dB(r) = 0, (4.49)

and
lim
t→∞

ϕ(t) = 0. (4.50)

Taking the inferior limit on both sides of (4.48), we have

lim inf
t→∞

〈I(t)〉 ≥ µ1(µ2 + γ)

β(µ2 + γ(1− e−µ3τ ))
(R̃0 − 1) := Ĩ∗. (4.51)

On the other hand, integrating the first equation of (4.46) from 0 to t on both sides yields

ln I(t)− ln I(0)

t
= β〈S(t)〉 − µ2 − γ −

σ2

2
〈S2(t)〉+

σ

t

∫ t

0

S(r)dB(r). (4.52)

We rewrite (3.35), then we get that

〈I(t)〉 ≤ µ2
1

(µ2 + γ(1− e−µ3τ ))(βµ1 − σ2Λ)

×
(
βΛ

µ1

−
(
µ2 + γ +

σ2Λ2

2µ2
1

)
− ln I(t)− ln(0)

t
+ ψ(t)

)
,

(4.53)

taking the superior limit on both sides of above equation, then one can obtain that

lim sup
t→∞

〈I(t)〉 ≤ µ2
1

(µ2 + γ(1− e−µ3τ ))(βµ1 − σ2Λ)
(R̃0 − 1) := Ĩ∗. (4.54)

The last equation of model (1.2) yields

〈R(t)〉 =
γ

µ3

〈I(t)〉 − γe−µ3τ

µ3

〈I(t− τ)〉 − R(t)−R(0)

µ3t
, (4.55)

then

lim inf
t→∞

〈R(t)〉 =
γ(1− e−µ3τ )

µ3

lim inf
t→∞

〈I(t)〉 ≥ γ(1− e−µ3τ )
µ3

Ĩ∗ = R̃∗, (4.56)

lim sup
t→∞

〈R(t)〉 =
γ(1− e−µ3τ )

µ3

lim sup
t→∞

〈I(t)〉 ≤ γ(1− e−µ3τ )
µ3

Ĩ∗ = R̃∗. (4.57)

Remark 4.1:
Both Theorem 2 and Theorem 3 have the common condition σ2 ≤ βµ1

Λ
therewith. While the

opposite properties of these two theorems depend on the value of R̃0, that is, R̃0 < 1 indicates
the extinction of the disease and R̃0 > 1 means the persistence of the disease. The expression
R̃0 plays the role of threshold of model (1.2).

Copyright c© 2017 ASSA. Adv Syst Sci Appl (2017)



THRESHOLD ANALYSIS OF A STOCHASTIC EPIDEMIC MODEL WITH DELAY AND TEMPORARY 55

5. ILLUSTRATIVE EXAMPLES AND THEIR REALIZATIONS

Example 1
Let the parameters of model (1.2) be Λ = 0.2, µ1 = µ2 = µ3 = 0.2, β = 0.8, γ =

0.5, σ = 0.6, and the initial value be (S(0), I(0), R(0)) = (0.1, 0.6, 0.3).
We can verify that

R̃0 = R0 −
σ2Λ2

2µ2
1(µ2 + γ)

= 0.8857 < 1, σ2 = 0.36 ≤ βµ1

Λ
= 0.8. (5.58)

Condition (3.19) of Theorem 1.2 is satisfied, the solution of model (1.2) has the properties as
follows:

lim sup
t→∞

ln I(t)

t
≤ (µ2 + γ)(R̃0 − 1) = −0.08 < 0 a.s. (5.59)

and
lim sup
t→∞

S(t) =
Λ

µ1

= 1 a.s. (5.60)

Figure 5.1 indicates that the infected vanishes exponentially, and the susceptible individuals
reach their maximum when given a long time run.

Obviously, the basic reproduction number of the deterministic model (1.1) can be
computed as R0 = 1.1429 > 1 in this case. And the endemic equilibrium E∗ of model (1.1)
is globally asymptotically stable according to Theorem 5.2 in [10].

We conclude that in the case of medium perturbation, say σ = 0.6, the density of the
susceptible approaches one almost surely, and is much higher than that of the deterministic
model. Compared with the deterministic model, the density of the infected declines fast to
zero with the exponential rate −0.08 at early time scale 1.5× 104 days. And the density of
the recovered is somehow affected by the infected and ends up at zero at 2.5× 104 days
instead of persistence for the deterministic model.

Example 2
We keep the initial value and other parameters same as shown in Example 1 except for

σ = 0.9. Here

σ2 ≥ β2

2(µ2 + γ)
= 0.4571, (5.61)

and condition (3.18) of Theorem 1.2 is being satisfied, then the solution of model (1.2) admits
the following property:

lim sup
t→∞

ln I(t)

t
≤ −(µ2 + γ) +

β2

2σ2
= −0.3349 < 0 a.s. (5.62)

The corresponding simulations would be shown in Figure 5.2 to support the main results of
Theorem 1.2 we got in the previous section.

Under large perturbation, say σ = 0.9 in this case, we find that the density of the
susceptible behaves the similar dynamics. While the curve of the infected shows more sharper
than that in Example 1, and still decays exponentially with a larger rate−0.3349 at early time
scale 1× 104 days. And the density of the recovered is close to zero at 2× 104 days compared
with the deterministic model.

Example 3
Let the intensity of the white noise be σ = 0.1, and the initial value and other parameters

be the same as shown in Example 1. Here

R̃0 = R0 −
σ2Λ2

2µ2
1(µ2 + γ)

= 1.1357 > 1, σ2 = 0.01 ≤ βµ1

Λ
= 0.8. (5.63)

Copyright c© 2017 ASSA. Adv Syst Sci Appl (2017)



56 R. XUE, F.Y. WEI

0 0.5 1 1.5 2 2.5 3 3.5 4

t,days ×104

0

0.2

0.4

0.6

0.8

1

1.2

The Stochastic System

S(t)
I(t)
R(t)

0 0.5 1 1.5 2 2.5 3 3.5 4

t,days ×104

0

0.2

0.4

0.6

0.8

1

1.2

The Deterministic System

S(t)
I(t)
R(t)

Fig. 5.1. Realizations for model (1.2) with R̃0 < 1 and model (1.1) with R0 > 1 respectively.

By Theorem 4.1, the following property of model (1.2) holds:

0.0817 ≤ lim inf
t→∞

〈I(t)〉 ≤ lim sup
t→∞

〈I(t)〉 ≤ 0.0827 a.s. (5.64)

If we keep the initial value and other parameters same as shown in Example 1 except for
µ2 = 0.3, γ = 0.4. We easily check that Examples 1 and 2 still keep the same conclusions.
While the persistence level of model (1.2) is lower than that in (5.64) by Theorem 4.1. That
is,

0.0638 ≤ lim inf
t→∞

〈I(t)〉 ≤ lim sup
t→∞

〈I(t)〉 ≤ 0.0646 a.s. (5.65)

Figure 5.3 reveals that the prevalence of the disease takes place under small perturbation
of the white noise. The properties of the solutions for the stochastic model (1.2) and the
deterministic model (1.1) demonstrate the similar behaviors when set a small perturbation.

We would like to conclude that the density levels for the susceptible, the infected and the
recovered are all alike when given a small perturbation environment. Especially, the higher
the recovery rate for the infected individuals, the more the infected, such as, the density

Copyright c© 2017 ASSA. Adv Syst Sci Appl (2017)



THRESHOLD ANALYSIS OF A STOCHASTIC EPIDEMIC MODEL WITH DELAY AND TEMPORARY 57

0 0.5 1 1.5 2 2.5 3 3.5 4

t,days ×104

0

0.2

0.4

0.6

0.8

1

1.2

The Stochastic System

S(t)
I(t)
R(t)

0 0.5 1 1.5 2 2.5 3 3.5 4

t,days ×104

0

0.2

0.4

0.6

0.8

1

1.2

The Deterministic System

S(t)
I(t)
R(t)

Fig. 5.2. Realizations for model (1.2) with σ = 0.9 and model (1.1) with R0 > 1 respectively.

of the infected varies from [0.0817, 0.0827] when γ = 0.5 (the green line in Figure 5.4) to
[0.0638, 0.0646] when γ = 0.4 (the blue line in Figure 5.4).

6. CONCLUSIONS

In this paper, we work on the susceptible-infected-recovered model, where the individuals
stayed in the recovered compartment finally lost temporary immunity returned to the
susceptible compartment.

The research results of this paper demonstrate that the existence and uniqueness of the
global positive solution of model (1.2) has nothing to do with the temporary immunity due
to the construction of Lyapunov function (2.10) therewith. While no matter how big the
temporary immunity period is, the sufficient condition of the extinction of the diseases merely
depends on the parameters of model (1.2), say condition (3.18) or (3.19) is Λ, µ1, µ2, β, γ, σ-
dependent, and τ -dependent instead in Theorem 2.

Copyright c© 2017 ASSA. Adv Syst Sci Appl (2017)



58 R. XUE, F.Y. WEI

0 0.5 1 1.5 2 2.5 3 3.5 4

t,days ×104

0

0.2

0.4

0.6

0.8

1

1.2

The Stochastic System

S(t)
I(t)
R(t)

0 0.5 1 1.5 2 2.5 3 3.5 4

t,days ×104

0

0.2

0.4

0.6

0.8

1

1.2

The Deterministic System

S(t)
I(t)
R(t)

Fig. 5.3. Realizations for model (1.2) with σ = 0.1 and model (1.1) with R0 > 1 respectively.

We therefore conclude that, in Theorem 3, the expressions of Ĩ∗ and Ĩ∗ are the inverse
functions of factor 1− e−µ3τ , where R̃0 is a τ -independent expression; and R̃∗ and R̃∗ are
respectively the saturated functions of 1− e−µ3τ . The illustrative examples have shown that
the larger the perturbation, the sharper the infected individuals decline exponentially, the
earlier the recovered individuals meet extinction.

ACKNOWLEDGEMENTS

This work is supported by the National Natural Science Foundation of China (Grants
No. 11201075 and 61773122), Natural Science Foundation of Fujian Province of China
(Grant No. 2016J01015).

Copyright c© 2017 ASSA. Adv Syst Sci Appl (2017)



THRESHOLD ANALYSIS OF A STOCHASTIC EPIDEMIC MODEL WITH DELAY AND TEMPORARY 59

0 0.5 1 1.5 2 2.5 3 3.5 4

t,days ×104

0

0.2

0.4

0.6

0.8

1

1.2

The Stochastic System

I(t)
I(t)

Fig. 5.4. Realizations for model (1.2) with γ = 0.5 (the green) and γ = 0.4 (the blue) respectively.

REFERENCES

1. Kyrychko Y. & Blyuss K. (2005) Global properties of a delayed SIR model with
temporary immunity and nonlinear incidence rate. Nonlinear Anal: Real World Appl.,
6, 495–507.

2. Xu R. & Ma Z. (2010) Global stability of a delayed SEIRS epidemic model with
saturation incidence rate. Nonlinear Anal: Real World Appl., 61, 229–239.

3. Muroya Y., Enatsu Y. & Nakata Y. (2011) Global stability of a delayed SIRS epidemic
model with a non-monotonic incidence rate. J. Math. Anal. Appl., 377, 1–14.

4. Lahrouz A. (2015) Dynamics of a delayed epidemic model with varying immunity
period and nonlinear transmission. Int. J. Biomath., 8, 1550027.

5. Ma W., Song M. & Takeuchi Y. (2004) Global stability of an SIR epidemic model with
time delay. Appl. Math. Lett., 17, 1141–1145.

6. Guo H. & Li M. (2006) Global dynamics of a staged progression model for infectious
diseases. Math. Biosci. Eng., 3 (3), 513–525.

7. Beretta E. & Kuang Y. (2001) Modeling and analysis of a marine bacteriophage
infection with latency period. Nonlinear Anal: Real World Appl., 2, 35–74.

8. Beretta E. & Takeuchi Y. (1995) Global stability of an SIR epidemic model with time
delays. J. Math. Biol., 33, 250–260.

9. Beretta E. & Takeuchi Y. (1997) Convergence results in SIR epidemic models with
varying population sizes. Nonlinear Anal: Theory Methods Appl., 28, 1909–1921.

10. Wen L. & Yang X. Global stability of a delayed SIRS model with temporary immunity.
Chaos Solitons Fractals, 38, 221–226.

11. Melnichenko O.A. & Romanyukha A.A. (2008) A model of tuberculosis epidemiology:
estimation of parameters and analysis of factors influencing the dynamics of an epidemic
process. Russian J. Numer. Anal. Math. Model., 23 (1), 63–75.

12. Melnichenko O.A. & Romanyukha A.A. (2009) A model of tuberculosis epidemiology:
Data analysis and estimation of parameters, Math. Models Comput. Simul., 1 (4),
428–444.

13. Liu J.M., Wei F.Y. Dynamics of stochastic SEIS epidemic model with varying
population size. Physica A. 2016; 464: pp. 241-250.

14. Wei F.Y., Liu J.M. Long-time behavior of a stochastic epidemic model with varying
population size. Physica A. 2017; 470: pp. 146-153.

Copyright c© 2017 ASSA. Adv Syst Sci Appl (2017)



60 R. XUE, F.Y. WEI

15. Chen L.H., Wei F.Y. Persistence and distribution of a stochastic susceptible-infected-
recovered epidemic model with varying population size. Physica A. 2017; 483: pp. 386-
397.

16. Dalal N., Greenhalgh D., Mao X. A stochastic model of AIDS and condom use. J. Math.
Anal. Appl. 2007; 325: pp. 36-53.

17. Lahrouz A., Omari L., Kiouach D., Belmaati A. Complete global stability for an SIRS
epidemic model with generalized non-linear incidence and vaccination. Appl. Math.
Comput. 2012; 218 (11): pp. 6519-6525.

18. Zhang X., Jiang D., Alsaedi A., Hayat T. Stationary distribution of stochastic SIS
epidemic model with vaccination under regime switching. Appl. Math. Lett. 2016; 59:
pp. 87-93.

19. Yu J., Jiang D., Shi N. Global stability of two-group SIR model with random
perturbation. J. Math. Anal. Appl. 2009; 360: pp. 235-244.

20. Xue R., Wei F.Y. Persistence and extinction of a stochastic SIS epidemic model with
double epidemic hypothesis. Ann. Appl. Math. 2017; 33(1): pp. 77-89.

21. Dalal N., Greenhalgh D., Mao X. A stochastic model of AIDS and condom use. J. Math.
Anal. Appl. 2007; 325: pp. 36-53.

22. Mao X., Marion G., Renshaw E. Environmental Brownian noise suppresses explosions
in population dynamics. Stoch. Process Appl. 2002; 97; pp. 95-110.

23. Mao X. Stochastic Differential Equations and Applications (2nd ed.), Horwood,
Chichester, UK, 2007.

Copyright c© 2017 ASSA. Adv Syst Sci Appl (2017)


	Introduction
	Existence and uniqueness of the positive solution
	The sufficient conditions of the extinction of the diseases
	The threshold for the persistence of the diseases
	Illustrative examples and their realizations
	Conclusions