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The threshold of a stochastic SIVS epidemic model with nonlinear saturated incidence
Physica A 443 (2016) 372–379
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Physica A journal homepage: www.elsevier.com/locate/physa
The threshold of a stochastic SIVS epidemic model with nonlinear saturated incidence Dianli Zhao ∗ , Tiansi Zhang, Sanling Yuan College of Science, University of Shanghai for Science and Technology, Shanghai 200093, China
highlights • This model includes the effects of stochastic perturbations and nonlinear saturated incidence. • When the noise is small, the new threshold parameter is identified. • Large noise will suppress the epidemic from prevailing regardless of the saturated incidence.
article
info
Article history: Received 16 April 2015 Received in revised form 10 August 2015 Available online 9 October 2015 Keywords: Stochastic SIVS epidemic model Saturated incidence Threshold Persistence Extinction
abstract A stochastic version of the SIS epidemic model with vaccination (SIVS) is studied. When the noise is small, the threshold parameter is identified, which determines the extinction and persistence of the epidemic. Besides, the results show that large noise will suppress the epidemic from prevailing regardless of the saturated incidence. The results are illustrated by computer simulations. © 2015 Elsevier B.V. All rights reserved.
1. Introduction The deterministic and stochastic mathematical models are widely used in describing the spread of a disease into a population. Most models are inspired by the works of Anderson and May [1,2], and one of which is the classical SIVS model. Let S (t ) represent the total number of individuals who are susceptible to the disease at time t; I (t ) represent the number of infected individuals who are infectious and can spread the disease by contact with the susceptible individuals, and V (t ) represent the number of individuals who are immune to an infection at time t as a result of vaccination. The SIVS epidemic model can be written as: dS = [(1 − q)Λ − (µ + p) S − β SI + γ I + δ V ] dt , dI = [β SI − (µ + γ + v) I] dt , dV = [qΛ + pS − (µ + δ) V ] dt
(1)
where Λ is a constant input of new members into the population per unit time; q denotes a fraction of vaccinated for newborns; β transmission coefficient between compartments S and I; µ means the natural death rate of S, I, V compartments; p is the proportional coefficient of vaccinated for the susceptible; γ is the recovery rate of infectious
∗
Corresponding author. E-mail address: [email protected] (D. Zhao).
http://dx.doi.org/10.1016/j.physa.2015.09.092 0378-4371/© 2015 Elsevier B.V. All rights reserved.
D. Zhao et al. / Physica A 443 (2016) 372–379
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individuals; v shows the rate of losing their immunity for vaccinated individuals and δ denotes the disease-caused death rate of infectious individuals. Here, we assume that all parameter values are nonnegative and µ, Λ > 0. Li and Ma [3] have proved that system (1) admits a unique disease-free equilibrium P0 = (S0 , 0, V0 ) =
Λ [µ (1 − q) + δ ] Λ [µq + p] , 0, µ (µ + δ + p) µ (µ + δ + p)
and a threshold R0 =
β Λ [µ (1 − q) + δ ] . µ (µ + γ + v) (µ + δ + p)
In order to reflect the actual problem as much as possible, many mathematical researchers and biologists have long been interested in the stochastic effects that may lead to significant changes. As shown in Refs. [4–11], the environmental white noise can prevent the solution of the established model from exploding at a finite time, and sometimes they can change the basic reproduction number of the disease and even can stabilize an unstable system. Inspired by these, Zhao, Jiang and O’Regan [12] present a stochastic SIVS model by assuming that the environmental noise will mainly effect the parameter β such as β → β + σ B˙ (t ), where B(t ) is a standard Brownian motion with intensity σ 2 > 0. The established stochastic version of (1) in Ref. [12] takes the form: dS = [(1 − q)Λ − (µ + p) S − β SI + γ I + δ V ] dt − σ SIdB(t ), dI = [β SI − (µ + γ + v) I] dt + σ SIdB(t ), dV = [qΛ + pS − (µ + δ) V ] dt .
(2)
They find that: σ 2S
β2
2
(H1) if (a) σ 2 > 2(µ+γ +v) , or (b) R0 − 1 < 2(µ+γ0+v) and σ 2 ≤ β S0 , then (S (t ) , I (t ) , V (t )) → (S0 , 0, V0 ); t β σ 2 Λ2 (H2) if R0 − 1 > 2µ2 (µ+γ and σ 2 < S , then lim inft →∞ 1t 0 I (s) ds > 0 a.s. +v) 0
The incidence rate in (2) is bilinear, and several authors have pointed out that the disease transmission process can have a nonlinear incidence rate, see Refs. [6,13,14]. In Ref. [13], Capasso and Serio introduced a nonlinear saturated incidence rate β SI into the epidemic model, where C is a positive constant denoting the infection force of the disease. Xiao and Ruan [14] 1+CI β SI
propose an incidence rate 1+CI 2 . To make model (2) more realistic, we suppose that the incidence rate has a general form: β SI , which satisfies the assumption: ψ(I )
ψ ∈ C [0, ∞)
such that ψ (0) = 1 and
0 ≤ ψ ′ (I ) ≤ L
for some constant L ≥ 0.
(3)
Subsequent to the work of Ref. [12], Lin et al. [15] obtain the existence of stationary distribution of system (2) under assumption that the total population size is a constant. However, they [12,15] did not accurately find the threshold conditions for the stochastic SIVS model that can determine whether the diseases will spread or go extinct. In this paper, we will try to fill the gap by giving the threshold conditions for the following stochastic SIVS epidemic model with general nonlinear saturated incidence:
σ SI β SI + γ I + δ V dt − dB(t ), dS = ( 1 − q ) Λ − + p S − (µ ) ψ (I ) ψ (I ) β SI σ SI dI = − (µ + γ + v) I dt + dB(t ), ψ I ψ ( ) (I ) dV = [qΛ + pS − (µ + δ) V ] dt .
(4)
Remark 1. The coefficients of system (4) are locally Lipschitz continuous, using the Lyapunov analysis method (see Refs. [5,12]) by defining a C 2 -function W (S , I , V ) = (S − 1 − log S ) + (I − 1 − log I ) + (V − 1 − log V ) for all (S , I , V ) ∈ R3+ , we can prove that the solution of system (4) is positive and global. The proof is similar to Zhao et al. [12], and hence is omitted. Remark 2. By (4), we have d (S + I + V ) = [Λ − µ (S + I + V ) − v I] dt . This admits a unique solution which can be written as follows: S (t ) + I (t ) + V (t ) =
t Λ Λ −µt Λ + S (0) + I (0) + V (0) − e −v e−µ(t −s) I (s) ds ≤ . µ µ µ 0
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D. Zhao et al. / Physica A 443 (2016) 372–379
It is clear that a positive invariant set of system (4) can be denoted by
Λ Γ = (S , I , V ) : S > 0, I > 0, V > 0, S + I + V ≤ a.s. . µ
For simplicity, we define an important parameter
β S0 − σ2 S0 2 = . µ+γ +v 2
RS0
The rest of this paper is organized as follows. In Section 2, we first give sufficient conditions for extinction of the disease due to large noise. Then, for RS0 below unity, we show that the disease goes extinct exponentially in case that the noise is small. In Section 3, the condition RS0 > 1 is shown to be sufficient for the disease to be persistent. Finally, numerical simulations are introduced to support the obtained results. 2. Extinction The following lemma is important for our proof and we get it first. For convenience, in the sequel we denote
⟨x(t )⟩ =
1 t
t
x(s)ds. 0
Lemma 2.1. Let (S (t ), I (t ), V (t )) be the solution of system (4) with initial value (S (0), I (0), V (0)) ∈ Γ . Then S (t ) = S0 + H1 (t ) + G (t )
(5)
where H1 (t ) =
pΛ [µ (1 − q) + δ ]
µ (µ + δ) (µ + δ + p)
e
−(µ+δ+p)t
− V (0) −
qΛ
µ+δ
e
−(µ+δ)t
+
Λ − S (0) + I (0) + V (0) e−µt µ
and G(t ) = −I (t ) − v
t
e−µ(t −s) I (s) ds + p
0
t
t
e−(µ+δ+p)(t −s) I (s)ds + pv
0
e−(µ+δ+p)(t −s)
0
s
e−µ(s−u) I (u) duds.
0
In addition,
⟨S (t )⟩ = S0 −
(µ + v) (µ + δ) ⟨I (t )⟩ − ϕ (t ) µ (µ + δ + p)
(6)
where
ϕ (t ) =
(µ + δ) [S (t ) + I (t ) + V (t ) − S (0) − I (0) − V (0)] − µ [V (t ) − V (0)] . µ (µ + δ + p) t
(7)
Proof. By taking an integration of system (4), we get S (t ) + I (t ) + V (t ) =
Λ − µ
t Λ − S (0) + I (0) + V (0) e−µt − v e−µ(t −s) I (s) ds µ 0
(8)
and V (t ) =
t qΛ + V (0) − e−(µ+δ)t + p S (s) e−(µ+δ)(t −s) ds. µ+δ µ+δ 0 qΛ
(9)
Then combining (8) and (9) gives that S (t ) =
Λ [µ (1 − q) + δ ] − H (t ) − I (t ) − v µ (µ + δ)
t
0
e−µ(t −s) I (s) ds − p
t
S (s) e−(µ+δ)(t −s) ds
0
where H (t ) =
V (0) −
qΛ
µ+δ
e−(µ+δ)t +
Λ − S (0) + I (0) + V (0) e−µt . µ
(10)
D. Zhao et al. / Physica A 443 (2016) 372–379
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We multiply the above equation by e(µ+δ)t and ept , and then rearrange the above equation as
d e
pt
t
S (s) e
(µ+δ)s
(µ+δ)t
t
(µ+δ)s
S (s) e ds dt =e e S (t ) + p 0 Λ [µ (1 − q) + δ ] (µ+δ)t pt e e − H (t )e(µ+δ)t ept − I (t ) e(µ+δ)t ept = µ (µ + δ) t e−µ(t −s) I (s) ds dt . − v e(µ+δ)t ept pt
ds
0
0
Taking an integration shows us that t
S (s) e(µ+δ)s ds =
0
t Λ [µ (1 − q) + δ ] t −p(t −s) (µ+δ)s e e−p(t −s) H (s)e(µ+δ)s ds e ds − µ (µ + δ) 0 0 s t t −p(t −s) (µ+δ)s −p(t −s) (µ+δ)s e e−µ(s−u) I (u) duds. I (s)e e e ds − v − −(µ+δ)t
Multiplying the above equation by e t
S (s) e−(µ+δ)(t −s) ds =
0
0
0
0
, we get
t Λ [µ (1 − q) + δ ] t −(µ+δ+p)(t −s) e ds − e−(µ+δ+p)(t −s) H (s)ds µ (µ + δ) 0 0 t t s −(µ+δ+p)(t −s) −(µ+δ+p)(t −s) − e I (s)ds − v e e−µ(s−u) I (u) duds 0
0
0
t Λ [µ (1 − q) + δ ] = 1 − e−(µ+δ+p)t − e−(µ+δ+p)(t −s) H (s)ds µ (µ + δ) (µ + δ + p) 0 t t s − e−(µ+δ+p)(t −s) I (s)ds − v e−(µ+δ+p)(t −s) e−µ(s−u) I (u) duds. 0
0
(11)
0
By substituting (11) into (10), we obtain (5). Next, we prove (6). From (4), S (t ) + I (t ) + V (t ) − S (0) − I (0) − V (0) t
= Λ − µ ⟨S (t )⟩ − (µ + v) ⟨I (t )⟩ − µ ⟨V (t )⟩
(12)
and V (t ) − V (0) t
= qΛ + p ⟨S (t )⟩ − (µ + δ) ⟨V (t )⟩ .
(13)
Then the desired result is obtained by substituting (13) into (12). The proof is complete.
Theorem 2.2. Let (S (t ), I (t ), V (t )) be the solution of system (4) with initial value (S (0), I (0), V (0)) ∈ Γ . If one of the two assumptions holds (A)
β2
2σ 2
< µ + γ + v, β
(B) S0 ≤ σ 2 and RS0 < 1, then (S (t ) , I (t ) , V (t )) → (S0 , 0, V0 ) a.s. as t → ∞. In addition,
β2 − (µ + γ + v) < 0 a.s. if (A) holds; t I (0) 2σ 2 t →∞ 1 I (t ) lim sup log ≤ (µ + γ + v) RS0 − 1 < 0 a.s. if (B) holds. t I (0) t →∞
lim sup
1
log
I (t )
≤
(14) (15)
Proof. Applying Itô formula of system (4), and taking an integration consequently lead to 1 t
log
I (t ) I (0)
S (t )
σ2
ψ (I (t ))
− (µ + γ + v) −
S (t )
2
M (t ) + ψ (I (t )) t 2 S (t ) σ2 S (t ) M (t ) ≤β − (µ + γ + v) − + . ψ (I (t )) 2 ψ (I (t )) t
=β
2
(16)
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D. Zhao et al. / Physica A 443 (2016) 372–379
σ S (u) dB 0 ψ(I (u))
t
(u) is a martingale with quadratic variation 2 t σ S ( u) 2 Λ 2 ⟨M (t ) , M (t )⟩ = t a.s. du ≤ σ ψ I u µ ( ( )) 0
where M (t ) =
By the strong law of large numbers for martingales, we can show that lim
t →∞
M (t ) t
= 0 a.s.
(17)
If (A) holds, then from (16) we have 1 t
log
I (t ) I (0)
≤
M (t ) β2 − (µ + γ + v) + . 2 2σ t
(18)
If (B) holds, then by (16) we have 1 t
log
I (t ) I (0)
≤β
S (t )
ψ (I (t ))
+ ϕ (t ) − (µ + γ + v) −
σ2 2
S (t )
ψ (I (t ))
2 M (t ) + ϕ (t ) dt − Φ (t ) +
(19)
t
where
Φ (t ) = βϕ (t ) −
σ2
S (t )
2
2
ψ (I (t ))
ϕ (t ) + ϕ (t ) . 2
β
In view of the assumption S0 ≤ σ 2 and S (t )
ψ (I (t ))
+ ϕ (t ) ≤ ⟨S (t )⟩ + ϕ (t ) = S0 −
(µ + v) (µ + δ) ⟨I (t )⟩ ≤ S0 , µ (µ + δ + p)
then it follows from (19) that 1 t
log
I (t ) I (0)
≤ β S0 − (µ + γ + v) −
σ2 2
M (t )
S0 2 − Φ (t ) +
t
M (t ) . = (µ + γ + v) RS0 − 1 − Φ (t ) +
(20)
t
ϕ(t )
Since (S (0), I (0), V (0)) ∈ Γ means (S (t ), I (t ), V (t )) ∈ Γ a.s., then limt →∞ t = 0, which implies that limt →∞ a.s. By (17) (18) and (20), the infective will decay exponentially to zero, namely
Φ (t ) t
=0
lim I (t ) = 0 a.s.
t →∞
Now, let us prove the assertion S (t ) → S0 and V (t ) → V0 . According to (5) and L’Hospital’s rule, we get lim S (t ) = S0 + lim H1 (t ) + lim G (t )
t →∞
t →∞
t →∞
µ+δ v = S0 − 1+ lim I (t ) µ+δ+p µ t →∞ = S0 a.s. and then by use of (8) lim V (t ) = lim
t →∞
t →∞
The proof is complete.
Λ − S (t ) − I (t ) − H0 e−µt − v µ
t
e−µ(t −s) I (s) ds
0
=
Λ [µq + p] µ (µ + δ + p)
a.s.
Remark 3. Theorem 2.2 shows that the disease will die out with probability one under suitable conditions. From (A), large noise will suppress the epidemic from prevailing. When the noise is small, it is of interest to see from (B) that RS0 < 1 is sufficient for the infectious class to be extinct for all kinds of the incidence rate under the assumption (3). Thus, the previously-known result on extinction is generalized. 3. Persistence Definition 3.1. System (4) is said to be persistent in the mean, if lim inf ⟨x(t )⟩ ≥ χ t →∞
a.s. for some constant χ > 0.
D. Zhao et al. / Physica A 443 (2016) 372–379
377
Theorem 3.2. Let (S (t ), I (t ), V (t )) be the solution of system (4) with initial value (S (0), I (0), V (0)) ∈ Γ . If RS0 > 1,
(21)
then the disease will be persistent in mean, namely, lim inf
t
t →∞
t
1
I (s) ds ≥ η RS0 − 1 > 0 a.s.,
(22)
0
where
η=
LΛ + β (µ + v) + σ
2
µ (µ + γ + v) 2 +4 Λ + vµ3Λ + µ
µ+v S0 p µ+γ +v
p2 Λ µ(µ+δ+p)2
+
(pv)2 Λ µ3 (µ+δ+p)2
.
Proof. By use of Itô formula to system (4) and Lemma 2.1, it yields by (3) 1 t
log
I
=β
S (t )
2
S (t )
+
˜ (t ) M
ψ (I (t )) t ˜ (t ) σ2 2 M S (t ) [ψ (I (t )) − ψ (0)] ≥ β ⟨S (t )⟩ − (µ + γ + v) − S (t ) − β + 2 ψ (I (t )) t
I0
ψ (I (t ))
− (µ + γ + v) −
σ2
≥ β S0 − (µ + γ + v) −
σ 2 S0 2 2
2
−L
˜ (t ) Λ M ⟨I (t )⟩ + K (t ) + , µ t
(23)
where K (t ) = β − σ 2 S0 (⟨H1 (t )⟩ + ⟨G(t )⟩) − σ 2
˜ (t ) = and M that
t 0
σ S (u) dB(u) such that limt →∞
⟨H1 (t )⟩ =
1
t
˜ (t ) M t
pΛ [µ (1 − q) + δ ]
H12 (t ) + G2 (t ) ,
= 0 a.s. due to the strong law of large numbers for martingales. Compute
e−(µ+δ+p)s −
V (0) −
qΛ
e−(µ+δ)s − H0 e−µs ds
µ (µ + δ) (µ + δ + p) µ+δ 1 − e−(µ+δ)t 1 − e−(µ+δ+p)t V (0) qΛ H0 1 − e−µt − − − = 2 2 t µ+δ t µ t µ (µ + δ) (µ + δ + p) (µ + δ) → 0 as t → ∞, u u 1 t ⟨G(t )⟩ = −I ( u ) − v e−µ(u−s) I (s) ds + p e−(µ+δ+p)(u−s) I (s)ds t 0 0 0 s u + pv e−(µ+δ+p)(u−s) e−µ(s−w) I (w) dw ds du 0 0 v t u −µ(u−s) 1 t I (u) du − ≥ − e I (s) dsdu t
0
pΛ [µ (1 − q) + δ ]
t
t
0
0
0
µ+v ⟨I (t )⟩ , ≥ − µ t u t p pv −(µ+δ+p)(u−s) ⟨G(t )⟩ ≤ e I (s)dsdu + ≤
t 0 0 p (µ + v)
µ+δ+p t
3 H12 (t ) ≤
t
0
u
e
−(µ+δ+p)(u−s)
0
s
e−µ(s−w) I (w) dw dsdu 0
⟨I (t )⟩ , pΛ [µ (1 − q) + δ ]
2
qΛ
e−2(µ+δ+p)s + V (0) − µ (µ + δ) (µ + δ + p) µ+δ 2 2 3 pΛ [µ (1 − q) + δ ] qΛ ≤ + V (0) − + H02 t µ (µ + δ) (µ + δ + p) µ+δ → 0 as t → ∞ t
0
2
e−2(µ+δ)s + H02 e−2µs ds
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D. Zhao et al. / Physica A 443 (2016) 372–379
and
G2 (t ) ≤
u
−µ(u−s)
t I (u) + v e 0 t 0 + (pv)2 2
2
4
I (s) ds
2 +p
2
u
e
−(µ+δ+p)(u−s)
2
I (s)ds
0 u
e−(µ+δ+p)(u−s)
s
e−µ(s−w) I (w) dw ds
2
du
0
0
u Λ p2 Λ v 2 Λ u −µ(u−s) −(µ+δ+p)(u−s) t e I s ds + e I ( s ) ds I ( u) + ( ) µ 4 µ2 0 µ (µ + δ + p) 0 du ≤ s u t 0 (pv)2 Λ −µ(s−w) −(µ+δ+p)(u−s) I (w) dw ds e e + 2 µ (µ + δ + p) 0 0 Λ v2 Λ p2 Λ (pv)2 Λ ⟨I (t )⟩ ≤ 4 + 3 + + 3 µ µ µ (µ + δ + p)2 µ (µ + δ + p)2 =: Q ⟨I (t )⟩ .
Those together with (23) lead to the desired result: lim inf ⟨I (t )⟩ ≥ t →∞
The proof is complete.
µ (µ + γ + v) RS0 − 1 a.s. µ+v LΛ + β (µ + v) + σ 2 S0 p µ+γ +v + Q
Remark 4. If ψ (I ) = 1, system (4) reduces to system (2). Let L = 0 in (3), then Theorem 3.2 holds. If ψ (I ) = 1 + CI, let L = C in (3), then Theorem 3.2 holds. If ψ (I ) = 1 + CI 2 , let L = 2C Λ in (3), then Theorem 3.2 also holds. µ β
Remark 5. Comparing with (H2), Theorem 3.2 not only releases the assumption σ 2 < S , but also expands the value range 0 of R0 . In this sense, Theorem 3.2 improves the conditions given by Zhao et al. in Ref. [12]. β
Remark 6. From Theorems 2.2 and 3.2, we see that when the noise is so small such that S0 ≤ σ 2 , then the value of RS0 that is below 1 or above 1 will lead to the disease to go extinct or prevail. Therefore, RS0 = threshold value of the stochastic system (4).
2 β S0 − σ2 S0 2
µ+γ +v
may be considered as the
4. Computer simulation and conclusion In this section, computer simulation of the path of S (t ), I (t ), V (t ) for the stochastic SIVS epidemic model (4) is given by using the EM method with step size 1t = 0.001 and initial value (S (0), I (0), V (0)) = (1.8, 0.6, 1.0). Let us suppose that Λ = 0.4, q = 0.8, β = 0.4, µ = 0.1, p = 0.7, v = 0.2, γ = 0.3, δ = 0.6. Let ψ(I ) ≡ 1 for comparisons. The only difference between conditions of Fig. 1(a)–(d) is that the value of σ is different. In Fig. 1(a), we choose σ ≡ 0. Compute that R0 = 1.181. Then by Ref. [3], P0 is unstable and there is an endemic equilibrium which is globally asymptotically stable, as shown in Fig. 1(a). σ Λ = 0.731 < 1. By the work of Ref. [12], we are not In Fig. 1(b), we choose σ = 0.15. Compute that R0 − 2µ2 (µ+γ +v) sure that the disease will be persistent or not. However, RS0 = 1.0927 > 1, by Theorem 3.2 established in this report, 2
limt →∞
1 t
t 0
2
I (s) ds > 0 a.s. Fig. 1(b) confirms these.
In Fig. 1(c), we choose σ = 0.4. Then RS0 = 0.7626 < 1 and σ 2 = 0.16 < 0.2259 = (S (t ) , I (t ) , V (t )) → (1.771, 0, 2.2285) a.s. as t → ∞.
β S0
, by (B) of Theorem 2.2, we have
β2
In Fig. 1(d), we choose σ = 0.8. Then 2σ 2 = 0.125 < 0.6 = µ + γ + v , by (A) of Theorem 2.2, we have (S (t ) , I (t ) , V (t )) → (1.771, 0, 2.2285) a.s. as t → ∞. 2
In this report, the threshold parameter RS0 =
β S0 − σ2 S0 2 µ+γ +v
is identified. Obviously, RS0 → R0 if σ → 0. When the noise is
small, Theorems 2.2 and 3.2 show that the disease will be extinct if RS0 < 1, and the disease will be persistent in mean if RS0 > 1. These results have been illustrated by computer simulations. It is also interesting to see from Fig. 1(a)–(d) that large noise intensity has the effect of suppressing the epidemic, which means the white noises can change the disease dynamics. Acknowledgments The authors would like to thank the anonymous referees for very helpful comments and suggestions. This work was supported by NSFC (Nos. 11271260, 11501148), The Hujiang Foundation of China (B14005) and Shanghai Leading Academic Discipline Project (No. XTKX2012).
D. Zhao et al. / Physica A 443 (2016) 372–379
(a) Time.
(b) Time.
(c) Time.
(d) Time.
379
Fig. 1. Simulations for the SDE model (4) have been done with different noise parameter values. (a) is with σ (t ) = 0; (b) is with σ (t ) = 0.15; (c) is with σ (t ) = 0.4; (d) is with σ (t ) = 0.8.
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15]
R.M. Anderson, R.M. May, Population biology of infectious disease: Part I, Nature 280 (1979) 361–376. R.M. May, R.M. Anderson, Population biology of infectious disease: Part II, Nature 280 (1979) 455–461. J. Li, Z. Ma, Global analysis of SIS epidemic models with variable total population size, Math. Comput. Modelling 39 (2004) 1231–1242. E. Tornatore, S.M. Buccellato, P. Vetro, Stability of a stochastic SIR system, Physica A 354 (2005) 111–126. A. Gray, D. Greenhalgh, L. Hu, X. Mao, J. Pan, A stochastic differential equation SIS epidemic model, SIAM J. Appl. Math. 71 (2011) 876–902. A. Lahrouz, L. Omari, Extinction and stationary distribution of a stochastic SIRS epidemic model with non-linear incidence, Statist. Probab. Lett. 83 (2013) 960–968. P.J. Witbooi, Stability of an SEIR epidemic model with independent stochastic perturbations, Physica A 392 (2013) 4928–4936. C. Ji, D. Jiang, Threshold behaviour of a stochastic SIR model, Appl. Math. Model. 38 (2014) 5067–5079. Y. Zhao, D. Jiang, The threshold of a stochastic SIRS epidemic model with saturated incidence, Appl. Math. Lett. 34 (2014) 90–93. E. Tornatore, P. Vetro, S.M. Buccellato, SIVR epidemic model with stochastic perturbation, Neural Comput. Appl. 24 (2014) 309–315. H. Schurz, K. Tosun, Stochastic asymptotic stability of SIR model with vare diffusion rates, J. Dynam. Differential Equations 27 (2015) 69–82. Y. Zhao, D. Jiang, D. O’Regan, The extinction and persistence of the stochastic SIS epidemic model with vaccination, Physica A 392 (2013) 4916–4927. V. Capasso, G. Serio, A generalization of the Kermack–McKendrick deterministic epidemic model, Math. Biosci. 42 (1978) 43–61. D. Xiao, S. Ruan, Global analysis of an epidemic model with nontone incidence rate, Math. Biosci. 208 (2007) 419–429. Y. Lin, D. Jiang, S. Wang, Stationary distribution of a stochastic SIS epidemic model with vaccination, Physica A 394 (2014) 187–197.
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Physica A journal homepage: www.elsevier.com/locate/physa
The threshold of a stochastic SIVS epidemic model with nonlinear saturated incidence Dianli Zhao ∗ , Tiansi Zhang, Sanling Yuan College of Science, University of Shanghai for Science and Technology, Shanghai 200093, China
highlights • This model includes the effects of stochastic perturbations and nonlinear saturated incidence. • When the noise is small, the new threshold parameter is identified. • Large noise will suppress the epidemic from prevailing regardless of the saturated incidence.
article
info
Article history: Received 16 April 2015 Received in revised form 10 August 2015 Available online 9 October 2015 Keywords: Stochastic SIVS epidemic model Saturated incidence Threshold Persistence Extinction
abstract A stochastic version of the SIS epidemic model with vaccination (SIVS) is studied. When the noise is small, the threshold parameter is identified, which determines the extinction and persistence of the epidemic. Besides, the results show that large noise will suppress the epidemic from prevailing regardless of the saturated incidence. The results are illustrated by computer simulations. © 2015 Elsevier B.V. All rights reserved.
1. Introduction The deterministic and stochastic mathematical models are widely used in describing the spread of a disease into a population. Most models are inspired by the works of Anderson and May [1,2], and one of which is the classical SIVS model. Let S (t ) represent the total number of individuals who are susceptible to the disease at time t; I (t ) represent the number of infected individuals who are infectious and can spread the disease by contact with the susceptible individuals, and V (t ) represent the number of individuals who are immune to an infection at time t as a result of vaccination. The SIVS epidemic model can be written as: dS = [(1 − q)Λ − (µ + p) S − β SI + γ I + δ V ] dt , dI = [β SI − (µ + γ + v) I] dt , dV = [qΛ + pS − (µ + δ) V ] dt
(1)
where Λ is a constant input of new members into the population per unit time; q denotes a fraction of vaccinated for newborns; β transmission coefficient between compartments S and I; µ means the natural death rate of S, I, V compartments; p is the proportional coefficient of vaccinated for the susceptible; γ is the recovery rate of infectious
∗
Corresponding author. E-mail address: [email protected] (D. Zhao).
http://dx.doi.org/10.1016/j.physa.2015.09.092 0378-4371/© 2015 Elsevier B.V. All rights reserved.
D. Zhao et al. / Physica A 443 (2016) 372–379
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individuals; v shows the rate of losing their immunity for vaccinated individuals and δ denotes the disease-caused death rate of infectious individuals. Here, we assume that all parameter values are nonnegative and µ, Λ > 0. Li and Ma [3] have proved that system (1) admits a unique disease-free equilibrium P0 = (S0 , 0, V0 ) =
Λ [µ (1 − q) + δ ] Λ [µq + p] , 0, µ (µ + δ + p) µ (µ + δ + p)
and a threshold R0 =
β Λ [µ (1 − q) + δ ] . µ (µ + γ + v) (µ + δ + p)
In order to reflect the actual problem as much as possible, many mathematical researchers and biologists have long been interested in the stochastic effects that may lead to significant changes. As shown in Refs. [4–11], the environmental white noise can prevent the solution of the established model from exploding at a finite time, and sometimes they can change the basic reproduction number of the disease and even can stabilize an unstable system. Inspired by these, Zhao, Jiang and O’Regan [12] present a stochastic SIVS model by assuming that the environmental noise will mainly effect the parameter β such as β → β + σ B˙ (t ), where B(t ) is a standard Brownian motion with intensity σ 2 > 0. The established stochastic version of (1) in Ref. [12] takes the form: dS = [(1 − q)Λ − (µ + p) S − β SI + γ I + δ V ] dt − σ SIdB(t ), dI = [β SI − (µ + γ + v) I] dt + σ SIdB(t ), dV = [qΛ + pS − (µ + δ) V ] dt .
(2)
They find that: σ 2S
β2
2
(H1) if (a) σ 2 > 2(µ+γ +v) , or (b) R0 − 1 < 2(µ+γ0+v) and σ 2 ≤ β S0 , then (S (t ) , I (t ) , V (t )) → (S0 , 0, V0 ); t β σ 2 Λ2 (H2) if R0 − 1 > 2µ2 (µ+γ and σ 2 < S , then lim inft →∞ 1t 0 I (s) ds > 0 a.s. +v) 0
The incidence rate in (2) is bilinear, and several authors have pointed out that the disease transmission process can have a nonlinear incidence rate, see Refs. [6,13,14]. In Ref. [13], Capasso and Serio introduced a nonlinear saturated incidence rate β SI into the epidemic model, where C is a positive constant denoting the infection force of the disease. Xiao and Ruan [14] 1+CI β SI
propose an incidence rate 1+CI 2 . To make model (2) more realistic, we suppose that the incidence rate has a general form: β SI , which satisfies the assumption: ψ(I )
ψ ∈ C [0, ∞)
such that ψ (0) = 1 and
0 ≤ ψ ′ (I ) ≤ L
for some constant L ≥ 0.
(3)
Subsequent to the work of Ref. [12], Lin et al. [15] obtain the existence of stationary distribution of system (2) under assumption that the total population size is a constant. However, they [12,15] did not accurately find the threshold conditions for the stochastic SIVS model that can determine whether the diseases will spread or go extinct. In this paper, we will try to fill the gap by giving the threshold conditions for the following stochastic SIVS epidemic model with general nonlinear saturated incidence:
σ SI β SI + γ I + δ V dt − dB(t ), dS = ( 1 − q ) Λ − + p S − (µ ) ψ (I ) ψ (I ) β SI σ SI dI = − (µ + γ + v) I dt + dB(t ), ψ I ψ ( ) (I ) dV = [qΛ + pS − (µ + δ) V ] dt .
(4)
Remark 1. The coefficients of system (4) are locally Lipschitz continuous, using the Lyapunov analysis method (see Refs. [5,12]) by defining a C 2 -function W (S , I , V ) = (S − 1 − log S ) + (I − 1 − log I ) + (V − 1 − log V ) for all (S , I , V ) ∈ R3+ , we can prove that the solution of system (4) is positive and global. The proof is similar to Zhao et al. [12], and hence is omitted. Remark 2. By (4), we have d (S + I + V ) = [Λ − µ (S + I + V ) − v I] dt . This admits a unique solution which can be written as follows: S (t ) + I (t ) + V (t ) =
t Λ Λ −µt Λ + S (0) + I (0) + V (0) − e −v e−µ(t −s) I (s) ds ≤ . µ µ µ 0
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D. Zhao et al. / Physica A 443 (2016) 372–379
It is clear that a positive invariant set of system (4) can be denoted by
Λ Γ = (S , I , V ) : S > 0, I > 0, V > 0, S + I + V ≤ a.s. . µ
For simplicity, we define an important parameter
β S0 − σ2 S0 2 = . µ+γ +v 2
RS0
The rest of this paper is organized as follows. In Section 2, we first give sufficient conditions for extinction of the disease due to large noise. Then, for RS0 below unity, we show that the disease goes extinct exponentially in case that the noise is small. In Section 3, the condition RS0 > 1 is shown to be sufficient for the disease to be persistent. Finally, numerical simulations are introduced to support the obtained results. 2. Extinction The following lemma is important for our proof and we get it first. For convenience, in the sequel we denote
⟨x(t )⟩ =
1 t
t
x(s)ds. 0
Lemma 2.1. Let (S (t ), I (t ), V (t )) be the solution of system (4) with initial value (S (0), I (0), V (0)) ∈ Γ . Then S (t ) = S0 + H1 (t ) + G (t )
(5)
where H1 (t ) =
pΛ [µ (1 − q) + δ ]
µ (µ + δ) (µ + δ + p)
e
−(µ+δ+p)t
− V (0) −
qΛ
µ+δ
e
−(µ+δ)t
+
Λ − S (0) + I (0) + V (0) e−µt µ
and G(t ) = −I (t ) − v
t
e−µ(t −s) I (s) ds + p
0
t
t
e−(µ+δ+p)(t −s) I (s)ds + pv
0
e−(µ+δ+p)(t −s)
0
s
e−µ(s−u) I (u) duds.
0
In addition,
⟨S (t )⟩ = S0 −
(µ + v) (µ + δ) ⟨I (t )⟩ − ϕ (t ) µ (µ + δ + p)
(6)
where
ϕ (t ) =
(µ + δ) [S (t ) + I (t ) + V (t ) − S (0) − I (0) − V (0)] − µ [V (t ) − V (0)] . µ (µ + δ + p) t
(7)
Proof. By taking an integration of system (4), we get S (t ) + I (t ) + V (t ) =
Λ − µ
t Λ − S (0) + I (0) + V (0) e−µt − v e−µ(t −s) I (s) ds µ 0
(8)
and V (t ) =
t qΛ + V (0) − e−(µ+δ)t + p S (s) e−(µ+δ)(t −s) ds. µ+δ µ+δ 0 qΛ
(9)
Then combining (8) and (9) gives that S (t ) =
Λ [µ (1 − q) + δ ] − H (t ) − I (t ) − v µ (µ + δ)
t
0
e−µ(t −s) I (s) ds − p
t
S (s) e−(µ+δ)(t −s) ds
0
where H (t ) =
V (0) −
qΛ
µ+δ
e−(µ+δ)t +
Λ − S (0) + I (0) + V (0) e−µt . µ
(10)
D. Zhao et al. / Physica A 443 (2016) 372–379
375
We multiply the above equation by e(µ+δ)t and ept , and then rearrange the above equation as
d e
pt
t
S (s) e
(µ+δ)s
(µ+δ)t
t
(µ+δ)s
S (s) e ds dt =e e S (t ) + p 0 Λ [µ (1 − q) + δ ] (µ+δ)t pt e e − H (t )e(µ+δ)t ept − I (t ) e(µ+δ)t ept = µ (µ + δ) t e−µ(t −s) I (s) ds dt . − v e(µ+δ)t ept pt
ds
0
0
Taking an integration shows us that t
S (s) e(µ+δ)s ds =
0
t Λ [µ (1 − q) + δ ] t −p(t −s) (µ+δ)s e e−p(t −s) H (s)e(µ+δ)s ds e ds − µ (µ + δ) 0 0 s t t −p(t −s) (µ+δ)s −p(t −s) (µ+δ)s e e−µ(s−u) I (u) duds. I (s)e e e ds − v − −(µ+δ)t
Multiplying the above equation by e t
S (s) e−(µ+δ)(t −s) ds =
0
0
0
0
, we get
t Λ [µ (1 − q) + δ ] t −(µ+δ+p)(t −s) e ds − e−(µ+δ+p)(t −s) H (s)ds µ (µ + δ) 0 0 t t s −(µ+δ+p)(t −s) −(µ+δ+p)(t −s) − e I (s)ds − v e e−µ(s−u) I (u) duds 0
0
0
t Λ [µ (1 − q) + δ ] = 1 − e−(µ+δ+p)t − e−(µ+δ+p)(t −s) H (s)ds µ (µ + δ) (µ + δ + p) 0 t t s − e−(µ+δ+p)(t −s) I (s)ds − v e−(µ+δ+p)(t −s) e−µ(s−u) I (u) duds. 0
0
(11)
0
By substituting (11) into (10), we obtain (5). Next, we prove (6). From (4), S (t ) + I (t ) + V (t ) − S (0) − I (0) − V (0) t
= Λ − µ ⟨S (t )⟩ − (µ + v) ⟨I (t )⟩ − µ ⟨V (t )⟩
(12)
and V (t ) − V (0) t
= qΛ + p ⟨S (t )⟩ − (µ + δ) ⟨V (t )⟩ .
(13)
Then the desired result is obtained by substituting (13) into (12). The proof is complete.
Theorem 2.2. Let (S (t ), I (t ), V (t )) be the solution of system (4) with initial value (S (0), I (0), V (0)) ∈ Γ . If one of the two assumptions holds (A)
β2
2σ 2
< µ + γ + v, β
(B) S0 ≤ σ 2 and RS0 < 1, then (S (t ) , I (t ) , V (t )) → (S0 , 0, V0 ) a.s. as t → ∞. In addition,
β2 − (µ + γ + v) < 0 a.s. if (A) holds; t I (0) 2σ 2 t →∞ 1 I (t ) lim sup log ≤ (µ + γ + v) RS0 − 1 < 0 a.s. if (B) holds. t I (0) t →∞
lim sup
1
log
I (t )
≤
(14) (15)
Proof. Applying Itô formula of system (4), and taking an integration consequently lead to 1 t
log
I (t ) I (0)
S (t )
σ2
ψ (I (t ))
− (µ + γ + v) −
S (t )
2
M (t ) + ψ (I (t )) t 2 S (t ) σ2 S (t ) M (t ) ≤β − (µ + γ + v) − + . ψ (I (t )) 2 ψ (I (t )) t
=β
2
(16)
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D. Zhao et al. / Physica A 443 (2016) 372–379
σ S (u) dB 0 ψ(I (u))
t
(u) is a martingale with quadratic variation 2 t σ S ( u) 2 Λ 2 ⟨M (t ) , M (t )⟩ = t a.s. du ≤ σ ψ I u µ ( ( )) 0
where M (t ) =
By the strong law of large numbers for martingales, we can show that lim
t →∞
M (t ) t
= 0 a.s.
(17)
If (A) holds, then from (16) we have 1 t
log
I (t ) I (0)
≤
M (t ) β2 − (µ + γ + v) + . 2 2σ t
(18)
If (B) holds, then by (16) we have 1 t
log
I (t ) I (0)
≤β
S (t )
ψ (I (t ))
+ ϕ (t ) − (µ + γ + v) −
σ2 2
S (t )
ψ (I (t ))
2 M (t ) + ϕ (t ) dt − Φ (t ) +
(19)
t
where
Φ (t ) = βϕ (t ) −
σ2
S (t )
2
2
ψ (I (t ))
ϕ (t ) + ϕ (t ) . 2
β
In view of the assumption S0 ≤ σ 2 and S (t )
ψ (I (t ))
+ ϕ (t ) ≤ ⟨S (t )⟩ + ϕ (t ) = S0 −
(µ + v) (µ + δ) ⟨I (t )⟩ ≤ S0 , µ (µ + δ + p)
then it follows from (19) that 1 t
log
I (t ) I (0)
≤ β S0 − (µ + γ + v) −
σ2 2
M (t )
S0 2 − Φ (t ) +
t
M (t ) . = (µ + γ + v) RS0 − 1 − Φ (t ) +
(20)
t
ϕ(t )
Since (S (0), I (0), V (0)) ∈ Γ means (S (t ), I (t ), V (t )) ∈ Γ a.s., then limt →∞ t = 0, which implies that limt →∞ a.s. By (17) (18) and (20), the infective will decay exponentially to zero, namely
Φ (t ) t
=0
lim I (t ) = 0 a.s.
t →∞
Now, let us prove the assertion S (t ) → S0 and V (t ) → V0 . According to (5) and L’Hospital’s rule, we get lim S (t ) = S0 + lim H1 (t ) + lim G (t )
t →∞
t →∞
t →∞
µ+δ v = S0 − 1+ lim I (t ) µ+δ+p µ t →∞ = S0 a.s. and then by use of (8) lim V (t ) = lim
t →∞
t →∞
The proof is complete.
Λ − S (t ) − I (t ) − H0 e−µt − v µ
t
e−µ(t −s) I (s) ds
0
=
Λ [µq + p] µ (µ + δ + p)
a.s.
Remark 3. Theorem 2.2 shows that the disease will die out with probability one under suitable conditions. From (A), large noise will suppress the epidemic from prevailing. When the noise is small, it is of interest to see from (B) that RS0 < 1 is sufficient for the infectious class to be extinct for all kinds of the incidence rate under the assumption (3). Thus, the previously-known result on extinction is generalized. 3. Persistence Definition 3.1. System (4) is said to be persistent in the mean, if lim inf ⟨x(t )⟩ ≥ χ t →∞
a.s. for some constant χ > 0.
D. Zhao et al. / Physica A 443 (2016) 372–379
377
Theorem 3.2. Let (S (t ), I (t ), V (t )) be the solution of system (4) with initial value (S (0), I (0), V (0)) ∈ Γ . If RS0 > 1,
(21)
then the disease will be persistent in mean, namely, lim inf
t
t →∞
t
1
I (s) ds ≥ η RS0 − 1 > 0 a.s.,
(22)
0
where
η=
LΛ + β (µ + v) + σ
2
µ (µ + γ + v) 2 +4 Λ + vµ3Λ + µ
µ+v S0 p µ+γ +v
p2 Λ µ(µ+δ+p)2
+
(pv)2 Λ µ3 (µ+δ+p)2
.
Proof. By use of Itô formula to system (4) and Lemma 2.1, it yields by (3) 1 t
log
I
=β
S (t )
2
S (t )
+
˜ (t ) M
ψ (I (t )) t ˜ (t ) σ2 2 M S (t ) [ψ (I (t )) − ψ (0)] ≥ β ⟨S (t )⟩ − (µ + γ + v) − S (t ) − β + 2 ψ (I (t )) t
I0
ψ (I (t ))
− (µ + γ + v) −
σ2
≥ β S0 − (µ + γ + v) −
σ 2 S0 2 2
2
−L
˜ (t ) Λ M ⟨I (t )⟩ + K (t ) + , µ t
(23)
where K (t ) = β − σ 2 S0 (⟨H1 (t )⟩ + ⟨G(t )⟩) − σ 2
˜ (t ) = and M that
t 0
σ S (u) dB(u) such that limt →∞
⟨H1 (t )⟩ =
1
t
˜ (t ) M t
pΛ [µ (1 − q) + δ ]
H12 (t ) + G2 (t ) ,
= 0 a.s. due to the strong law of large numbers for martingales. Compute
e−(µ+δ+p)s −
V (0) −
qΛ
e−(µ+δ)s − H0 e−µs ds
µ (µ + δ) (µ + δ + p) µ+δ 1 − e−(µ+δ)t 1 − e−(µ+δ+p)t V (0) qΛ H0 1 − e−µt − − − = 2 2 t µ+δ t µ t µ (µ + δ) (µ + δ + p) (µ + δ) → 0 as t → ∞, u u 1 t ⟨G(t )⟩ = −I ( u ) − v e−µ(u−s) I (s) ds + p e−(µ+δ+p)(u−s) I (s)ds t 0 0 0 s u + pv e−(µ+δ+p)(u−s) e−µ(s−w) I (w) dw ds du 0 0 v t u −µ(u−s) 1 t I (u) du − ≥ − e I (s) dsdu t
0
pΛ [µ (1 − q) + δ ]
t
t
0
0
0
µ+v ⟨I (t )⟩ , ≥ − µ t u t p pv −(µ+δ+p)(u−s) ⟨G(t )⟩ ≤ e I (s)dsdu + ≤
t 0 0 p (µ + v)
µ+δ+p t
3 H12 (t ) ≤
t
0
u
e
−(µ+δ+p)(u−s)
0
s
e−µ(s−w) I (w) dw dsdu 0
⟨I (t )⟩ , pΛ [µ (1 − q) + δ ]
2
qΛ
e−2(µ+δ+p)s + V (0) − µ (µ + δ) (µ + δ + p) µ+δ 2 2 3 pΛ [µ (1 − q) + δ ] qΛ ≤ + V (0) − + H02 t µ (µ + δ) (µ + δ + p) µ+δ → 0 as t → ∞ t
0
2
e−2(µ+δ)s + H02 e−2µs ds
378
D. Zhao et al. / Physica A 443 (2016) 372–379
and
G2 (t ) ≤
u
−µ(u−s)
t I (u) + v e 0 t 0 + (pv)2 2
2
4
I (s) ds
2 +p
2
u
e
−(µ+δ+p)(u−s)
2
I (s)ds
0 u
e−(µ+δ+p)(u−s)
s
e−µ(s−w) I (w) dw ds
2
du
0
0
u Λ p2 Λ v 2 Λ u −µ(u−s) −(µ+δ+p)(u−s) t e I s ds + e I ( s ) ds I ( u) + ( ) µ 4 µ2 0 µ (µ + δ + p) 0 du ≤ s u t 0 (pv)2 Λ −µ(s−w) −(µ+δ+p)(u−s) I (w) dw ds e e + 2 µ (µ + δ + p) 0 0 Λ v2 Λ p2 Λ (pv)2 Λ ⟨I (t )⟩ ≤ 4 + 3 + + 3 µ µ µ (µ + δ + p)2 µ (µ + δ + p)2 =: Q ⟨I (t )⟩ .
Those together with (23) lead to the desired result: lim inf ⟨I (t )⟩ ≥ t →∞
The proof is complete.
µ (µ + γ + v) RS0 − 1 a.s. µ+v LΛ + β (µ + v) + σ 2 S0 p µ+γ +v + Q
Remark 4. If ψ (I ) = 1, system (4) reduces to system (2). Let L = 0 in (3), then Theorem 3.2 holds. If ψ (I ) = 1 + CI, let L = C in (3), then Theorem 3.2 holds. If ψ (I ) = 1 + CI 2 , let L = 2C Λ in (3), then Theorem 3.2 also holds. µ β
Remark 5. Comparing with (H2), Theorem 3.2 not only releases the assumption σ 2 < S , but also expands the value range 0 of R0 . In this sense, Theorem 3.2 improves the conditions given by Zhao et al. in Ref. [12]. β
Remark 6. From Theorems 2.2 and 3.2, we see that when the noise is so small such that S0 ≤ σ 2 , then the value of RS0 that is below 1 or above 1 will lead to the disease to go extinct or prevail. Therefore, RS0 = threshold value of the stochastic system (4).
2 β S0 − σ2 S0 2
µ+γ +v
may be considered as the
4. Computer simulation and conclusion In this section, computer simulation of the path of S (t ), I (t ), V (t ) for the stochastic SIVS epidemic model (4) is given by using the EM method with step size 1t = 0.001 and initial value (S (0), I (0), V (0)) = (1.8, 0.6, 1.0). Let us suppose that Λ = 0.4, q = 0.8, β = 0.4, µ = 0.1, p = 0.7, v = 0.2, γ = 0.3, δ = 0.6. Let ψ(I ) ≡ 1 for comparisons. The only difference between conditions of Fig. 1(a)–(d) is that the value of σ is different. In Fig. 1(a), we choose σ ≡ 0. Compute that R0 = 1.181. Then by Ref. [3], P0 is unstable and there is an endemic equilibrium which is globally asymptotically stable, as shown in Fig. 1(a). σ Λ = 0.731 < 1. By the work of Ref. [12], we are not In Fig. 1(b), we choose σ = 0.15. Compute that R0 − 2µ2 (µ+γ +v) sure that the disease will be persistent or not. However, RS0 = 1.0927 > 1, by Theorem 3.2 established in this report, 2
limt →∞
1 t
t 0
2
I (s) ds > 0 a.s. Fig. 1(b) confirms these.
In Fig. 1(c), we choose σ = 0.4. Then RS0 = 0.7626 < 1 and σ 2 = 0.16 < 0.2259 = (S (t ) , I (t ) , V (t )) → (1.771, 0, 2.2285) a.s. as t → ∞.
β S0
, by (B) of Theorem 2.2, we have
β2
In Fig. 1(d), we choose σ = 0.8. Then 2σ 2 = 0.125 < 0.6 = µ + γ + v , by (A) of Theorem 2.2, we have (S (t ) , I (t ) , V (t )) → (1.771, 0, 2.2285) a.s. as t → ∞. 2
In this report, the threshold parameter RS0 =
β S0 − σ2 S0 2 µ+γ +v
is identified. Obviously, RS0 → R0 if σ → 0. When the noise is
small, Theorems 2.2 and 3.2 show that the disease will be extinct if RS0 < 1, and the disease will be persistent in mean if RS0 > 1. These results have been illustrated by computer simulations. It is also interesting to see from Fig. 1(a)–(d) that large noise intensity has the effect of suppressing the epidemic, which means the white noises can change the disease dynamics. Acknowledgments The authors would like to thank the anonymous referees for very helpful comments and suggestions. This work was supported by NSFC (Nos. 11271260, 11501148), The Hujiang Foundation of China (B14005) and Shanghai Leading Academic Discipline Project (No. XTKX2012).
D. Zhao et al. / Physica A 443 (2016) 372–379
(a) Time.
(b) Time.
(c) Time.
(d) Time.
379
Fig. 1. Simulations for the SDE model (4) have been done with different noise parameter values. (a) is with σ (t ) = 0; (b) is with σ (t ) = 0.15; (c) is with σ (t ) = 0.4; (d) is with σ (t ) = 0.8.
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15]
R.M. Anderson, R.M. May, Population biology of infectious disease: Part I, Nature 280 (1979) 361–376. R.M. May, R.M. Anderson, Population biology of infectious disease: Part II, Nature 280 (1979) 455–461. J. Li, Z. Ma, Global analysis of SIS epidemic models with variable total population size, Math. Comput. Modelling 39 (2004) 1231–1242. E. Tornatore, S.M. Buccellato, P. Vetro, Stability of a stochastic SIR system, Physica A 354 (2005) 111–126. A. Gray, D. Greenhalgh, L. Hu, X. Mao, J. Pan, A stochastic differential equation SIS epidemic model, SIAM J. Appl. Math. 71 (2011) 876–902. A. Lahrouz, L. Omari, Extinction and stationary distribution of a stochastic SIRS epidemic model with non-linear incidence, Statist. Probab. Lett. 83 (2013) 960–968. P.J. Witbooi, Stability of an SEIR epidemic model with independent stochastic perturbations, Physica A 392 (2013) 4928–4936. C. Ji, D. Jiang, Threshold behaviour of a stochastic SIR model, Appl. Math. Model. 38 (2014) 5067–5079. Y. Zhao, D. Jiang, The threshold of a stochastic SIRS epidemic model with saturated incidence, Appl. Math. Lett. 34 (2014) 90–93. E. Tornatore, P. Vetro, S.M. Buccellato, SIVR epidemic model with stochastic perturbation, Neural Comput. Appl. 24 (2014) 309–315. H. Schurz, K. Tosun, Stochastic asymptotic stability of SIR model with vare diffusion rates, J. Dynam. Differential Equations 27 (2015) 69–82. Y. Zhao, D. Jiang, D. O’Regan, The extinction and persistence of the stochastic SIS epidemic model with vaccination, Physica A 392 (2013) 4916–4927. V. Capasso, G. Serio, A generalization of the Kermack–McKendrick deterministic epidemic model, Math. Biosci. 42 (1978) 43–61. D. Xiao, S. Ruan, Global analysis of an epidemic model with nontone incidence rate, Math. Biosci. 208 (2007) 419–429. Y. Lin, D. Jiang, S. Wang, Stationary distribution of a stochastic SIS epidemic model with vaccination, Physica A 394 (2014) 187–197.