Classification of the asymptotic behavior of a stochastic SIR model

Classification of the asymptotic behavior of a stochastic
SIR model
N. T. Dieu

∗

N.H. Dang,†

N.H. Du,‡

July 30, 2015

Abstract
In this paper, we investigate the asymptotic behavior of a stochastic SIR epidemic
system. We give sufficient conditions for the permanence and ergodicity of the solution
to the system. The conditions obtained in fact are very close to the necessary conditions. We also characterize the support of a unique invariant probability measure and
prove the convergence in total variation norm of transition probability to the invariant
measures. Our results can be considered as an significant improvement of the result
given by Lin, Y. et al. [18].
Keywords. SIR model; Extinction; Permanence; Stationary Distribution; Ergodicity.
Subject Classification. 34C12, 60H10, 92D25.

1

Introduction

Since epidemic models were first introduced by Kermack and McKendrick [13, 14], mathematical models have been used for the purpose of analyzing, predicting the spread and
the control of infectious diseases in host populations (see [1, 3, 4, 13, 14, 17, 19, 22]). One
of classic epidemic models is the SIR (Susceptible-Infected-Removed) model which is suitable for some diseases with permanent immunity such as rubella, whooping cough, measles,
∗

Department of Mathematics,Vinh University, 182 Le Duan, Vinh, Nghe An, Vietnam, This research was supported in part by the Foundation for Science and Technology
Development of Vietnam’s Ministry of Education and Training. No. B2015-27-15. Author would like also
to thank Vietnam Institute for Advance Study in Mathematics (VIASM) for supporting and providing a
fruitful research environment and hospitality.
†
Department of Mathematics, Wayne State University, Detroit, MI 48202, USA,
This research was supported in part by the National Science Foundation
under grant DMS-1207667 and was finished when the author was in the Institute for Advance Study in
Mathematics (VIASM).
‡
Corresponding author: Department of Mathematics, Mechanics and Informatics, Hanoi National University, 334 Nguyen Trai, Thanh Xuan, Hanoi Vietnam,

1


smallpox, etc. In the SIR model, a homogeneous host population is subdevided into three
epidemiologically distinct types of individuals.
• (S), The susceptible class, i.e., the class of those individuals who are capable of contracting the disease and becoming infective,
• (I), the infective class, i.e., the class of those individuals who are capable of transmitting
the disease to others,
• (R), the removed class, i.e., the class of infected individuals who are dead, or have
recovered and are permanently immune, or are isolated.
If we denote by S(t), I(t), R(t) the number of individuals in classes (S),(Z),(R) respectively at
time t, the spread of infection is formulated by the following system of differential equations:


dS(t) = α − βS(t)I(t) − µS)dt
(1.1)
dI(t) = βS(t)I(t) − (µ + ρ + γ)I(t))dt



dR(t) = (γI(t) − µR(t))dt,
where α is the per capita birth rate of the population, µ is the per capita disease-free death
rate and ρ is the excess per capita death rate of infective class, β is the effective per capita
contact rate, γ is per capita recovery rate of the infective individuals. On the other hand, it
is well-recognized that the population is always subject to random factors and we obviously
desire to learn how randomness effects our models. It is therefore of significant importance to
investigate stochastic epidemic models. Jiang et al. [8] investigated the asymptotic behavior
of global positive solution for the non-degenerate stochastic SIR model


dS(t) = α − βS(t)I(t) − µS)dt + σ1 S(t)dB1 (t)
dI(t) = βS(t)I(t) − (µ + ρ + γ)I(t))dt + σ2 I(t)dB3 (t)


dR(t) = (γI(t) − µR(t))dt + σ3 R(t)dB3 (t),

(1.2)

where B1 (t), B2 (t) and B3 (t) are mutually independent Brownian motions, σ1 , σ2 , σ3 are
the intensities of the white noises. This model has been extended to multi-group ones in
[11, 25, 26]. However, in reality, the classes (S), (I), (R) are usually subject to the same
random factors such as temperature, humidity, polution and other extrinsic influences. As
a result, it is more plausible to assume that the random noise perturbing the three classes
is correlated. If we assume that the Brownian motions B1 (t), B2 (t) and B3 (t) are the same,

2


we obtain the following model which has been considered in [18]



dS(t) = α − βS(t)I(t) − µS)dt + σ1 S(t)dB(t)
dI(t) = βS(t)I(t) − (µ + ρ + γ)I(t))dt + σ2 I(t)dB(t)


dR(t) = (γI(t) − µR(t))dt + σ3 R(t)dB(t).

(1.3)

When studying epidemic models, it is naturally important to know whether the models
tend to a disease free state or the disease will survive permanently. For the deterministic
model (1.1), the asymptotic behavior has been classified completely as follows. If λd =
α
βα
− (µ + ρ + γ) ≤ 0 then the population tends to the disease-free equilibrium ( , 0, 0)
µ
µ
while the population approaches an endemic equilibrium in case λd > 0. In [18], the authors
attempted to answer the aforesaid question for the model (1.3) in case σ1 > 0, σ2 > 0. By
using Lyapunov-type functions, they provided some sufficient conditions for extinction or
permanence as well as ergordicity for the solution of system (1.3). Using the same methods,
the extinction and permanence in some different stochastic SIR models have been studied
in [10, 12, 24, 27]. In practice, it is, however, very difficult to find an effective Lyapunov
function in practice so their conditions are restrictive and not close to a necessary condition.
In other words, there has been no classification for stochastic SIR models that is similar to
the deterministic case.
Our main goal in this paper is to provide a sufficient and almost necessary condition for
permanence (as well as ergodicity) and extinction of the disease in the stochastic SIR model
(1.3) in using a value λ, which is similar to λd in the deterministic model. The new method

introduced in this paper can remove most assumptions in [18] as well as can consider the
case σ1 > 0, σ2 < 0 which has not been taken into consideration in [18]. It is also suitable to
deal with other stochastic variants of (1.1) such as models introduced in [10, 12, 24, 27], etc.
The rest of the paper is arranged as follows. Section 2 derives a threshold that is used
to classify the extinction and permanence of the disease. To establish the desired result,
by considering the dynamics on the boundary, we obtain a threshold λ that enables us to
determine the asymptotic behavior of the solution. In particular, it is shown that if λ < 0, the
disease will decay in an exponential rate. In case λ > 0, the solution converges to a stationary
distribution in total variation. It means that the disease is permanent. The ergodicity of
the solution process is also proved. Finally, Section 3 is devoted to some discussion and
comparison to existing results in literature. Some numerical examples and figures are also
provided to illustrate our results.
3


2

Threshold Between Extinction and Permanence

Let (Ω, F, {Ft }t≥0 , P) be a probability space with the filtration {Ft }t≥0 satisfying the usual
condition, i.e., it is increasing and right continuous while F0 contains all P−null sets. Let
B(t) be an Ft -adapted, Brownian motions. Because the dynamics of class of recover has no
effect on the disease transmission dynamics, we only consider the following system:
dS(t) = [α − βS(t)I(t) − µS(t)]dt + σ1 S(t)dB(t),
dI(t) = [βS(t)I(t) − (µ + ρ + γ)I(t)]dt + σ2 I(t)dB(t).

(2.1)

Assume that σ1 , σ2 = 0. By the symmetry of Brownian motion, without loss of generality, we
suppose throughout this paper that σ1 > 0. Using standard arguments, it can be easily shown

that for any positive initial value (S(0), I(0)) = (u, v) ∈ R2,◦
+ := {(x, y) : x > 0, y > 0}, there
exists uniquely a global solution (S(t), I(t)), t ≥ 0 that remains in R2,◦
+ with probability 1
(see e.g. [8]). To obtain further properties of the solution, we first consider the equation on
the boundary,
dS(t) = (α − µS(t))dt + σ1 S(t)dB(t).

(2.2)

It follows from the comparison theorem ([7, Theorem 1.1, p. 352]) that S(t) ≤ S(t) ∀t ≥ 0
a.s. provided that S(0) = S(0) > 0 and I(0) > 0. By solving the Fokker-Planck equation, it
is shown that the process S(t) has a unique stationary distribution with density given by
f ∗ (x) =
where c1 = µ +

σ12
,a
2
∗

=

2c1
,b
σ12

=

2α

σ12

ba −(a+1) −b
x
ex,
Γ(a)

(2.3)

and Γ(·) is the Gamma function. Since g1 (x) = x−1 , and

g2 (x) = x, are f -integrable (i.e., integrable with respect to the function f ∗ ), by the strong
law of large number we deduce that,
1
t→∞ t

∞

t

S −1 (s)ds =

lim

−∞

0

1
t→∞ t

x−1 f ∗ (x)dx :=
∞

t

lim

xf ∗ (x)dx :=

S(s)ds =
−∞

0

c1
a.s.
α

α
a.s.
µ

(2.4)

(2.5)

Otherwise,
1
1

ln S(t) =
t
t

t

− c1 + αS −1 (s) ds + α
0

B1 (t)
.
t

(2.6)

Consequently,
lim sup
t→∞

1
1
ln S(t) ≤ lim sup ln S(t) = 0 a.s.,
t
t→∞ t
4

(2.7)


and

1
lim sup
t→∞ t

t

S(τ )dτ ≤
0

α
a.s.
µ

(2.8)

To proceed, we estimate E[S(t)]−p . It follows from Itˆo’s formula that S −1 (t) satisfies a
stochastic logistic equation
dS −1 (t) = S −1 (t) (µ + σ12 ) − αS −1 (t) dt − σ1 S −1 (t)dB(t).
It is well known (see e.g. [9, Lemma 2.3, p. 591]) that if S(0) = u ∈ (0, ∞) then
lim sup ES −p (t) ≤ Hp < ∞ ∀ p > 1.

(2.9)

t→∞

We now define the threshold
λ :=

σ2
αβ

− µ+ρ+γ+ 2 .
µ
2

(2.10)

ln I(t)
≤
t
t→∞
λ a.s. and the distribution of S(t) converges weakly to the unique invariant probability mea2,◦
Theorem 2.1. If λ < 0, then for any initial value (S(0), I(0)) ∈ R+
we have lim sup

sure µ∗ with the density f ∗ .
Proof. Let I(t) be the solution to the equation
dI(t) = I(t) − (µ + ρ + γ) + β S(t) dt + σ1 I(t)dB(t),
where S(t) is the solution to (2.2). By comparison theorem, I(t) ≤ I(t) a.s. given that
S(0) = S(0), I(0) = I(0). In view of Itˆo’s formula and the ergodicity of S(t),
lim sup
t→∞

1
1 t
σ2
B(t)
ln I(t) = lim sup
− (µ + ρ + γ + 2 ) + β S(τ ) dτ + σ1
t
t 0

2
t
t→∞
2
αβ
σ
− µ + ρ + γ + 2 = λ < 0 a.s.
=
µ
2

(2.11)

That is, I(t) converges almost surely to 0 at an exponential rate.
For any ε > 0, it follows from (2.11) that there exists t0 > 0 such that P(Ωε ) > 1 − ε
where
Ωε := I(t) ≤ exp

λt
2

∀t ≥ t0 =

ln I(t) ≤

λt
∀t ≥ t0 .
2

2β

λt0
exp
< ε. Let S(t), t ≥ t0 be the solution
λ
2
to (2.2) given that S(t0 ) = S(t0 ). We have from the comparison theorem that P{S(t) ≤
Clearly, we can choose t0 satisfying −

5


S(t) ∀t ≥ t0 } = 1. In view of Itˆo’s formula, for almost ω ∈ Ωε we have
t

0 ≤ ln S(t) − ln S(t) =α
t0
t

≤β
t0

1

−

1
dτ + β
S(τ )

t

I(τ )dτ.
S(τ )
t0
λτ
2β
λt0
λt
exp
dτ = −
exp
− exp
2
λ
2
2

< ε.

As a result,
P{| ln S(t) − ln S(t))| > ε} ≤ 1 − P(Ωε ) < ε ∀t ≥ t0 .

(2.12)

Let ν ∗ be the distribution of a random variable ln X provided that X admits µ∗ as its
distribution. In lieu of proving that the distribution of S(t) converges weakly to µ∗ , we will
claim an equivalent statement that the distribution of ln S(t) converges weakly to ν ∗ . By
the Portmanteau theorem (see [2, Theorem 1, p. 1832]), it is sufficient to prove that for
any g(·) : R → R satisfying |g(x) − g(y)| ≤ |x − y| and |g(x)| < 1 ∀x, y ∈ R, we have
Eg(ln S(t)) → g :=

R

∞
0

g(x)ν ∗ (dx) =

g(ln x)µ∗ (dx).

Since the diffusion (2.2) is non-degenerate, it is well know that the distribution of S(t)
weakly converges to µ∗ as t → ∞ (see e.g. [6]). Thus
lim Eg(ln S(t)) = g.

t→∞

(2.13)

On the one hand,
|Eg(ln S(t)) − g| ≤ |Eg(ln S(t)) − Eg(ln S(t))| + |Eg(ln S(t)) − g|
≤ εP{| ln S(t) − ln S(t)| ≤ ε} + 2P{| ln S(t) − ln S(t)| > ε} + |Eg(ln S(t)) − g|. (2.14)
Applying (2.12) and (2.13) to (2.14) yields
lim sup |Eg(ln S(t)) − g| ≤ 3ε.
t→∞

Since ε is taken arbitrarily, we obtain the desired conclusion. The proof is complete.
Lemma 2.1. For any 0 < p < min{ 2µ
, 2(µ+ρ+γ)
}, there exists a K < ∞ such that
σ2

σ2
1

2

lim sup E[S(t) + I(t)]1+p ≤ K,

(2.15)

t→∞

for any (S(0), I(0)) = (u, v) ∈ R2+ . In particular, when v = 0 we have
lim sup E[S(t)]1+p ≤ K ∀ S(0) = u ∈ (0, ∞).
t→∞

6

(2.16)


Proof. Consider the Lyapunov function V (s, i) = (s + i)1+p . By directly calculating the
differential operator LV (s, i) associated with equation (2.1), we have
(1 + p)p
(s + i)p−1 (σ1 s + σ2 i)2
2
p
p
(µ − σ12 )s2 + (µ + ρ + γ − σ22 )i2 )
2
2

LV (s, i) = (1 + p)(s + i)p (α − µs − (µ + ρ + γ)i) +
= (1 + p)α(s + i)p − (1 + p)(s + i)p−1
+ (2µ + ρ + γ − pσ1 σ2 )si .
It follows from 2µ + ρ + γ >

pσ12
2

+

pσ22
2

≥ pσ1 σ2 that 2µ + ρ + γ − pσ1 σ2 > 0. By choosing a

number K2 satisfying 0 < K2 < min{µ − p2 σ12 , µ + ρ + γ − p2 σ22 } we see that
K1 = sup {LV (s, i) + K2 V (s, i)} < ∞.
s,i∈R+

We obtain
LV (s, i) ≤ K1 − K2 V (s, i) ∀(s, i) ∈ R2+ .
For n ∈ N, define the stopping time
θn = inf{t ≥ 0 : S(t) + I(t) ≥ n}.
In view of Itˆo’s formula, we yield that
eK2 (t∧θn ) V (S(t ∧ θn ), I(t ∧ θn ))
t∧θn

[LV (S(τ ), I(τ )) + K2 V (S(τ ), I(τ ))]eK2 τ dτ

=V (u, v) +
0
t∧θn

eK2 τ [(2 + p)(S(τ ) + I(τ ))1+p (σ1 µS(τ ) + σ2 I(τ ))]dB(τ ).

+
0

t∧θn

K1 eK2 τ dτ

≤V (u, v) +
0
t∧θn

eK2 τ [(2 + p)(S(τ ) + I(τ ))1+p (σ1 µS(τ ) + σ2 I(τ ))]dB(τ ).

+
0

Taking expectation to both side, we have
E(eK2 (t∧θn ) V (S(t ∧ θn ), I(t ∧ θn ))) ≤ V (u, v) +

K1 (eK2 (t∧θn ) − 1)
.
K2

By letting n → ∞ we obtain from Fatou’s Lemma that

EeK2 t (V (S(t), I(t))) ≤ V (u, v) +
7

K1 (eK2 t − 1)
.
K2


Dividing both sides by eK2 t and letting t → ∞, we have
lim sup E(V (S(t), I(t))) ≤
t→∞

K1
:= K.
K2

Lemma is proved.
Theorem 2.2. If λ > 0, the process (S(t), I(t)) has an invariant probability measure concentrated on R2,◦
+ .
Proof. Let (S(0), I(0)) = (u, v) ∈ R2,◦
+ . Since 0 ∨ [ln S(t)] ≤ S(t), it follows from (2.15) that
E[ln S(t)] exists (although it may be −∞) and
lim sup
t→∞

E[ln S(t)]
ES(t)
≤ lim sup
= 0.
t

t

(2.17)

E[ln I(t)]
EI(t)
≤ lim sup
= 0.
t
t

(2.18)

Similarly,
lim sup
t→∞

From (2.7) we have
lim sup
t→∞

1
t

t

α
− c1 − βI(τ ) dτ ≤ 0.
S(τ )

E
0

This implies that
lim sup
t→∞

1
t

t

E
0

α
α
α
+E
− c1 − βE[I(τ )] dτ ≤ 0.
−
S(τ ) S(τ )
S(τ )

(2.19)

In view of (2.4) and (2.9), it follows from the uniform integrability of S −1 (t) that
1
t→∞ t

t

lim

α
E

S(τ )

0

− c1 dτ = 0.

(2.20)

Similarly, it follows from (2.5) and (2.16) that
1
t→∞ t

t

lim

E S(τ )]dτ =
0

α
.
µ

(2.21)

From (2.19) and (2.20), we obtain
1
lim sup
t→∞ t

t

E
0

α
α
−
− βE[I(τ )] dτ ≤ 0.
S(τ ) S(τ )

Let m > 0 to be chosen later, not depend on the initial value. We aim to show that
1
lim inf
t→∞ t

t

E[I(τ )]dτ ≥ m.
0

8

(2.22)


If (2.22) does not hold, there exists a sequence {tn }n≥0 with tn → ∞ as n → ∞ such that
tn

1
n→∞ tn
lim

Hence
1
lim sup
tn
n→∞

E[I(τ )]dτ < m.

(2.23)

0

tn

E
0

α
α
−

dτ < βm.
S(τ ) S(τ )

Therefore, for any κ > 0,
1
lim sup
n→∞ tn
1
= lim sup
n→∞ tn
1
≤ lim sup
n→∞ tn

tn

E(S(τ ) − S(τ ))dτ
0
tn

E (S(τ ) − S(τ ))1{S(τ )≤κ} dτ + lim sup
n→∞

0
tn

E
0

tn

1
tn

E (S(τ ) − S(τ ))1{S(τ )≥κ} dτ
0

1
1
1
S(τ )S(τ )1{S(τ )≤κ} dτ +lim sup
−
S(τ ) S(τ )
n→∞ tn
< κ2 βm +

1
1
lim sup
p
κ n→∞ tn

tn

E S(τ )

tn

E S(τ )1{S(τ )≥κ} dτ
0

1+p

dτ <

0

κ2 βm K
+ p.
α
κ

We can choose κ sufficiently large and then choose m sufficiently small such that
and

κ2 βm
α

≤

λ
.
4β

K
κp

≤

λ
4β

Therefore,
lim sup
n→∞

tn

1
tn

E(S(τ ) − S(τ ))dτ ≤
0

λ
,
2β

which in the combination with (2.10) and (2.21) implies that
lim inf
n→∞

B(t)
E ln I(tn )
1 tn
σ2
= lim inf E
− (µ + ρ + γ + 2 ) + βS(τ ) dτ + σ1
n→∞
tn
tn 0

2
t
tn
2
1
σ
= lim E
− (µ + ρ + γ + 2 ) + β S(τ ) dτ
n→∞
tn 0
2
tn
1
− β lim sup
E S(τ ) − S(τ ) dτ
tn 0
n→∞
λ
1 tn
λ
≥λ − β lim sup
E S(τ ) − S(τ ) dτ ≥ λ − β ×
:= > 0.
tn 0
2β
2
n→∞

This contradicts (2.18). As a result, there is a positive constant m > 0 satisfying (2.22).
For 0 <

1
t

< m < H < ∞, H¨older’s inequality yields that

t

E 1{I(τ )≥ } I(τ ) dτ ≤ E
0

1
t

t

1{I(τ )≥ } dτ

p
1+p

0

≤

1
t

1
t

t
1+p

[I(τ )]
0
p
1+p

t

E 1{I(τ )≥
0

9

}

dτ

1
1+p

dτ

1
t

1
1+p

t
1+p

E[I(τ )]
0

dτ

.


Hence,

lim inf
t→∞

1
t

lim inf

t

E 1{I(τ )≥

}

dτ ≥

t→∞

0

t
0

1
t

E 1{I(τ )≥ } I(τ ) dτ

lim sup 1t
≥K

1
lim inf
t→∞ t

− p1

1+p
p

t→∞
t

t
0

E[I(τ )]1+p dτ

1+p
p

E[I(τ )]dτ −

1
p

1

≥ K − p (m − )

1+p
p

> 0. (2.24)

0

We also have from (2.15) that the following inequality holds with probability 1.
lim sup
t→∞

1
t

t

E1{S(τ )+I(τ )≥H} dτ ≤
0

1
H 1+p

lim sup
t→∞

1
t

t

E(S(τ ) + I(τ ))1+p dτ ≤
0

It follows from (2.24) and (2.25) that we can choose H sufficiently large and

K
. (2.25)
H 1+p
sufficiently

small such that
1
lim inf
t→∞ t

t

E1{(S(τ ),I(τ ))∈D} dτ ≥

0

(m − )
K

1
p

1+p
p

−

K
> 0,
H 1+p

(2.26)

where D = {(s, i) : i ≥ , s+i ≤ H}. By virtue of the invariance of M = {(s, i) : s ≥ 0, i > 0}
under equation (2.1), we can consider the Markov process (S(t), I(t)) on the state space M.
It is easy to show that (S(t), I(t)) has the Feller property. Thus, in view of inequality (2.26)
and the compactness of D in M, we implies that there is an invariant probability measure π ∗
on M (see [23] or [20]). Since I(t) → 0 provided that S(0) = 0, limt→∞ P (t, (0, I(0)), K) = 0
for all compact set K ⊂ M. Thus, we must have π ∗ ({(0, i) : i > 0}) = 0, equivalently
2,◦
∗
π ∗ (R2,◦
+ ) = 1. Furthermore, by the invariance of R+ , we derive that π is an invariant
2,◦

probability measure of (S(t), I(t)) on R+
.

To obtain properties of π ∗ , we first rewrite equation (2.1) in Stratonovich’s form
dS(t) = [α − c1 S(t) − βS(t)I(t)]dt + σ1 S(t) ◦ dB(t),
dI(t) = [−c2 I(t) + βS(t)I(t)]dt + σ2 I(t) ◦ dB(t).
where c1 = µ +

σ12
, c2
2

=µ+ρ+γ+

σ22
.
2

(2.27)

Denote by (S s,i (t), I s,i (t)) the solution to (2.1) with

initial value (s, i) and let P (t, (s, i), ·) be its transition probabilities. Put
A(x, y) =

α − c1 x − βxy
−c2 y + βxy

and B(x, y) =

σ1 x
σ2 y

.

To proceed, we first recall the notion of Lie bracket. If Φ(x, y) = (Φ1 , Φ2 ) and Ψ(x, y) =
(Ψ1 , Ψ2 ) are vector fields on R2 then the Lie bracket [Φ, Ψ] is a vector field given by
[Φ, Ψ]j (x, y) = Φ1

∂Ψj
∂Φj
∂Ψj
∂Φj
(x, y) − Ψ1
(x, y) + Φ2
(x, y) − Ψ2
(x, y) , j = 1, 2.
∂x
∂x
∂y
∂y
10


Denote by L(x, y) the Lie algebra generated by A(x, y), B(x, y) and L0 (x, y) the ideal in
L(x, y) generated by B. We have the following lemma.
Lemma 2.2. For σ1 > 0, σ2 = 0, the H¨ormader condition holds for the diffusion (2.27). To
2,◦
be more precise, we have dimL0 (x, y) = 2 at every (x, y) ∈ R+
or equivalently, the set of

vectors B, [A, B], [A, [A, B]], [B, [A, B]], . . . spans R2 at every (x, y) ∈ R2,◦
+ .
Proof. This lemma has been proved in [18] for the case σ2 > 0. Assume that r = −

σ2
> 0.
σ1

It is easy to obtain
C :=

1
B(x, y) =
σ1

x
,
−ry

D :=[A, C](x, y) =

α − rβxy
,
−βxy

E :=[C, D](x, y) =

−α + r2 βxy
,

−βxyn

F :=[C, E](x, y) =

α − r3 βxy
.
−βxy

Since det(D, F ) = 0 only if r2 = 1 or r = 1 (since r > 0). When r = 1, solving det(D, E) = 0
obtains βxy = α which implies
det(C, D) =

x
0
−y −α

= 0.

2,◦
As a result, B, D, E, F span R2 for all (x, y) ∈ R+
. The lemma is proved.

In order to describe the support of the invariant measure π ∗ and to prove the ergodicity
of (2.1), we need to investigate the following control system on R2,◦
u˙ φ (t) = σ1 uφ (t)φ(t) + α − βuφ (t)vφ (t) − c1 uφ (t),
v˙ φ (t) = σ2 vφ (t)φ(t) + βuφ (t)vφ (t) − c2 vφ (t),

(2.28)

where φ is taken from the set of piecewise continuous real valued functions defined on R+ .

Let (uφ (t, u, v), vφ (t, u, v)) be the solution to equation (2.28) with control φ and initial value
2
(u, v). Denote by O1+ (u, v) the reachable set from (u, v) ∈ R2,◦
+ , that is the set of (u , v ) ∈ R

such that there exists a t ≥ 0 and a control φ(·) satisfying uφ (t, u, v) = u , vφ (t, u, v) = v .
We now recall some concepts introduced in [16]. Let X be a subset of R2 satisfying the
property that for any w1 , w2 ∈ X, we have w2 ∈ O1+ (w1 ). Then there is a unique maximal
set Y ⊃ X such that this property still holds for Y . Such Y is called a control set. A control
set W is said to be invariant if O1+ (w) ⊂ W for all w ∈ W .
11


Putting r :=

−σ2
σ1

and zφ (t) = urφ (t)vφ (t), we have an equivalent system
u˙ φ (t) = σ1 φ(t)uφ (t) + g(uφ (t), zφ (t)),
z˙φ (t) = h(uφ (t), zφ (t)),

(2.29)

where
g(u, z) = −c1 eu + α − βzu1−r ,
and
h(u, z) = u−r z − (c1 r + c2 )ur + βu1+r + αrur−1 − βrz .
2,◦
Denote by O2+ (u, z) the set of (u , z ) ∈ R+

such that there is t > 0 and a control φ(·) such

that uφ (t, u, z) = u , zφ (t, u, v) = z .
Lemma 2.3. For the control system (2.28), the following claims hold
1. For any u0 , u1 , z0 > 0 and ρ > 0, there exists a control φ and T > 0 such that
uφ (T, u0 , z0 ) = u1 , |zφ (T, u0 , z0 ) − z0 | < ρ.
2. For any 0 < z0 < z1 , there is a u0 > 0, a control φ, and T > 0 such that zφ (T, u0 , z0 ) =
z1 and that uφ (t, u0 , z0 ) = u0 ∀ 0 ≤ t ≤ T .
3. Let d∗ = inf {−(c1 r + c2 )ur + βu1+r + αrur−1 }.
u>0

(a) If d∗ ≤ 0 then for any z0 > z1 , there is u0 > 0, a control φ, and T > 0 such that
zφ (T, u0 , z0 ) = z1 and that uφ (t, u0 , z0 ) = u0 ∀ 0 ≤ t ≤ T .
d∗
. If c∗ < z1 < z0 , there is u0 > 0 and a
(b) Suppose that d∗ > 0 and z0 > c∗ :=
βr
control φ and T > 0 such that zφ (T, u0 , z0 ) = z1 and that uφ (t, u0 , z0 ) = u0 ∀ 0 ≤
t ≤ T . However, there is no control φ and T > 0 such that zφ (T, u0 , z0 ) < c∗ .
Proof. Suppose that u0 < u1 and let ρ1 = sup{|g(u, z)|, |h(u, z)| : u0 ≤ u ≤ u1 , |z − z0 | ≤ ρ}.
σ1 ρ2 u0
We choose φ(t) ≡ ρ2 with
−1 ρ ≥ u1 −u0 . It is easy to check that with this control,
ρ1
there is 0 ≤ T ≤ ρρ1 such that uφ (T, u0 , z0 ) = u1 , |zφ (T, u0 , z0 ) − z0 | < ρ. If u0 > u1 , we can
construct φ(t) similarly. Then the item 1 is proved.
Now, by choosing u0 to be sufficiently large, there is a ρ3 > 0 such that h(u0 , z) > ρ3 ∀z0 ≤
z ≤ z1 . This property, combining with (2.29), implies the existence of a feedback control φ
and T > 0 satisfying that zφ (T, u0 , z0 ) = z1 and that uφ (t, u0 , z0 ) = u0 , ∀ 0 ≤ t ≤ T .
12

We now prove item 3. If r < 0 then lim − (c1 r + c2 )ur + βu1+r + αrur−1 ] = −∞
u→0

r

and lim − (c1 r + c2 )u + βu
u→0

1+r

r−1

+ αru

] = 0 if r > 1. As a result, d∗ ≤ 0 if r ∈
/ (0, 1]

which implies that for any z0 > z1 , we choose u0 such that supz∈[z1 ,z0 ] h(u0 , z) < 0, which
implies that there is a feedback control φ and T > 0 satisfying zφ (T, u0 , z0 ) = z1 and
uφ (t, u0 , z0 ) = u0 ∀0 ≤ t ≤ T .
+ αrur−1
= d∗ . If d∗ ≤ 0, then
If r ∈ (0, 1] there exists u0 such that −(c1 r + c2 )ur0 + βu1+r
0
0
2
for any z0 > z1 > 0 we have supz∈[z1 ,z0 ] h(u0 , z) ≤ u−r
0 supz∈[z1 ,z0 ] {−βrz } < 0 which implies

the desired claim.
Consider the remaining case when r ∈ (0, 1] and d∗ > 0. First, assume c∗ < z1 < z0 . Let
u0 satisfy −(c1 r + c2 )ur0 + βu1+r
+ αru0 er−1 = d∗ = βrc∗ . Hence
0
sup {h(u0 , z)} =u−r
sup
0
z∈[z1 ,z0 ]

z − (c1 r + c2 )ur0 + βu1+r
+ αrur−1
− βrz
0
0

z∈[z1 ,z0 ]
∗
= − βru−r
0 z1 (c − z1 ) < 0.

Thus, there is a feedback control φ and T > 0 satisfying zφ (T, u0 , z0 ) = z1 and uφ (t, u0 , z0 ) =
u0 ∀0 ≤ t ≤ T . The final assertion follows from the fact that h(u, c∗ ) ≥ 0 for all u ∈ R.
Proposition 2.1. The control system (2.28) has only one invariant control set C. If d∗ ≤ 0,
C = R2,◦ . If d∗ > 0, C = {(u, v) : ur v ≥ c∗ }.
Proof. If d∗ ≤ 0, it follows from items 1, 2, and item 3a of Lemma 2.3 that for any
(u1 , z1 ), (u2 , z2 ) ∈ R2 , (u2 , z2 ) ∈ O2+ (u1 , z1 ). Hence, for any (u1 , v1 ), (u2 , v2 ) ∈ R2 , we have
(u2 , v2 ) ∈ O1+ (u1 , v1 ). This implies that R2 is an unique invariant control set. If d∗ > 0, items
1, 2 and 3b of Lemma 2.3 imply that O2+ (u, z) ⊃ {(u , z ) : z ≥ c∗ } for all (u, v) ∈ R2 and

∗
O2+ (u, z) = {(u , z ) : z ≥ c∗ } for all u > 0, z ≥ c∗ . As a result, {(u, z) ∈ R2,◦
+ : z ≥ c } is
r
∗
a unique invariant control set for (2.29). In conclusion, C := {(u, z) ∈ R2,◦
+ : u v ≥ c } is a

unique invariant control set for (2.28).
Note that if λ > 0, there is an invariant probability measure π ∗ of the process (S(t), I(t)).
Since there is only one invariant control set C, it follows from Lemma 2.2 that π ∗ is the unique
invariant probability measure with support C. Moreover, for any initial value (S(0), I(0)) =
(u, v) ∈ C and a π ∗ -integrable function f we have
P

1
T →∞ T

T

lim

f (u , v )π ∗ (du , dv ) = 1.

f S(t), I(t) dt =
R2,◦
+

0

13

(2.30)


These results are proved in [16]. Moreover, it follows from [6, Proposition 5.1]
lim P (t, (u, v), ·) − π ∗ (·) = 0 ∀(u, v) ∈ C,

t→∞

(2.31)

where P (t, (u, v), ·) is the transition probability of the Markov process (S(t), I(t)) and

·

is the total variation norm.
2,◦
We aim to prove that (2.30) and (2.31) hold for all (u, v) ∈ R+
. We need only consider
∗
the case d∗ > 0 since C = R2,◦
+ in case d ≤ 0.

Proposition 2.2. Suppose that d∗ > 0. Then, for any initial value (S(0), I(0)) = (u, v) ∈
2,◦
R+
, we have τCu,v < ∞ almost surely with τCu,v = inf{t > 0 : (S(t), I(t)) ∈ C}.

Proof. Note that d∗ > 0 only if r ∈ (0, 1]. We have τCu,v = inf{t : βrS r (t)I(t) ≥ d∗ }. Let

Z(t) = ln[I(t)S r (t)]. By virtue of Itˆo’s formula, we obtain
dZ(t) = [−c3 + βS(t) + αrS −1 (t) − rβI(t)]dt
= S −r (t)[−c3 S r (t) + βS 1+r (t) + αrS −1+r (t) − βrS r (t)I(t)]dt,
where c3 = c1 r + c2 > 0. Suppose that P{τCu,v < ∞} < 1, then there exists even Ω∗ such that
P(Ω∗ ) > 0 and τCu,v = ∞ in Ω∗ . If ω ∈ Ω∗ , then
d∗ − βrS r (t)I(t) ≥ 0 ∀t ≥ 0.
Thus
dZ(t) ≥ 0 ∀t ≥ 0.

(2.32)

On the other hand,
lim −c3 xr + βx1+r + αrx−1+r = +∞ if r ∈ (0, 1),

x→0

and
lim −c3 xr + βx1+r + αrx−1+r = αr > d∗ if r = 1.

x→0

Hence, for any r ∈ (0, 1], there exists m > 0, δ ∗ > 0 such that
inf {−c3 xr + βx1+r + αrx−1+r } > d∗ + δ ∗ .

0≤x≤m

It follows from (2.32) and (2.33) that for almost all ω ∈ Ω∗
dZ(t) ≥ δ ∗ S −r (t)1{S(t)≤m} dt ≥ δ ∗ S −r (t)1{S(t)≤m} dt,
14

(2.33)


where S(t) is the solution to (2.2) with S(0) = eu . By the ergodicity of S(t) we have
m

t

1
t→∞ t

s−r f ∗ (s)ds := m > 0, a.s.

[S(τ )]−r 1{S(τ )≤m} dτ =

lim

0

0

It implies that
t

[S(τ )]−r 1{S(τ )≤m} dτ = +∞, a.s.

lim

t→∞

0

∗

Therefore, for almost ω ∈ Ω ,
t

[S(τ )]−r 1{S(τ )≤m} dτ = +∞, a.s.

lim Z(t) ≥ lim

t→∞

t→∞

0

which contradict the assumption that for all t ≥ 0, eZ(t) <

d∗
βr

in Ω∗ . Thus, P{τCu,v < ∞} = 1.

The proof is complete.
Since we have already shown that for any initial value (S(0), I(0)) = (u, v) ∈ R2,◦ ,
(S(t), I(t)) eventually enters C. So we obtain the following theorem.
Theorem 2.3. Suppose σ1 , σ1 = 0, λ > 0. Then, (2.27) has a unique invariant probability
measure π ∗ with support C. For any π ∗ -integrable function f , and initial value (S(0), I(0)) =
(u, v) ∈ R2,◦

+ , we have
1
P lim
T →∞ T

T

f (u , v )π ∗ (du , dv ) = 1.

f S(t), I(t) dt =
R2,◦
+

0

Moreover, the transition probability P (t, (u, v), ·) converges to π ∗ (·) in total variation.
Proof. The assertions can be proved using (2.30), (2.31), and Propositions 2.2.

3

Discussion and Numerical Examples

We have shown that the extinction and permanence of the disease in a stochastic SIR model
can be determined by the sign of a threshold value λ. Only the critical case λ = 0 is not
studied in this paper. To illustrate the significance of our results, let us compare our results
with those in [18].
Theorem 3.1. [18, Theorem 3.1] Assume that σ1 > 0, σ2 > 0. Let (ξ(t), η(t)) be a solution
of system (2.27). If µ > σ12 , µ + ρ + γ > σ22 , R0 > 1 and
δ < min

µ2
(µ + ρ + γ)2 ∗ 2
∗2
S
,
I
,
µ − σ12
µ + ρ + γ − σ22
15


then there exists a stationary distribution π ∗ for the Markov process (ξ(t), η(t)) which is the
limit in total variation of transition probability P (t, (u, v), ·). Here
δ=
S∗ =

µσ12 ∗ 2
(µ + ρ + γ)σ22 ∗ 2 (µ + ρ + γ ∗ 2
S
+
I +
I σ2 ,
µ − σ12
µ + ρ + γ − σ22
2β

µ+ρ+γ ∗
α
µ

βα
, I =
− ; R0 =
.
β
µ+ρ+γ β
µ(µ + ρ + γ)

By straightforward calculations or by arguments in Section 4 of [5] we can show that
their conditions are much more restrictive than the condition λ > 0. Moreover, it should be
noted that Theorem 2.1 is the same as Lemma 3.5 in [18]. However, the weak convergence of
S(t) to µ∗ was not be proved carefully in that paper. For this reason, we re-stated Theorem
2.1 and provided a rigorous proof in this paper.
Moreover, the estimates in Theorems 2.1 and 2.2 still hold for the non-degenerate model
(1.2). Note that for a non-degenerate diffusion, the existence of an invariant probability
measure implies the ergodicity of the diffusion as well as the convergence in total variation
of the transition probability to the invariant measure (see [6, 15]). As a result we have the
following theorem for the model (1.2).
Theorem 3.2. Let (S(t), I(t)) be the solution to (1.2) with initial value (S(0), I(0)) ∈ R2,◦
+ .
Define λ as (2.10).
If λ < 0, then lim I(t) = 0 a.s. and the distribution of S(t) converges weakly to µ∗ ,
t→∞

which has the density (2.3).
If λ > 0, the solution process (S(t), I(t)) has a unique invariant probability measure
ϕ∗ whose support is R2,◦
+ . Moreover, the transition probability P (t, (u, v), ·) of (S(t), I(t))
converges to ϕ∗ (·) in total variation and for any ϕ∗ -integrable function f , we have
1

t→∞ t

t

f S(u), I(u) du =

P lim

0

R2,◦
+

2,◦
f (u , v )ϕ∗ (du , dv ) = 1 ∀(u, v) ∈ R+
.

It should be emphasized that our techniques can be also used to improve results in
[10, 12, 24, 27].
Let us finish this paper by providing some numerical examples.
Example 3.1. Consider (2.1) with parameters α = 20, β = 4, µ = 1, ρ = 10, γ = 1, σ1 =
1, σ2 = −1. Direct calculation shows that λ = 67.5 > 0, d∗ = 7.75 > 0, c∗ = 1.9375. As a
result of Theorem 2.2 that (2.1) has a unique invariant probability measure π ∗ whose support
16


is {(S, I) : S ≥

1.9375
}.

I

Consequently, the strong law of large numbers and the convergence in

total variation norm of the transition probability hold. A sample path of solution to (2.1) is
illustrated by Figures 1, while the phase portrait in Figure 2 demonstrates that the support
of π ∗ lies above and includes the curve S =

c∗
I

=

1.9375
I

as well as the empirical density of π ∗ .

Figure 1: Trajectories of S(t), I(t) in Example 3.1.

Figure 2: Phase portait of (2.1); the boundary I =
empirical density of π ∗ in Example 3.1.

1.9375
S

of the support of π ∗ and the

Example 3.2. Consider (2.1) with parameters α = 50, β = 5, µ = 3, ρ = 2.5, γ = 4.5,
σ1 = 4.3 σ2 = 0.5. For these parameters, the conditions in Theorem 3.1 are not satisfied.

We obtain λ = 73.2083 > 0, d∗ = −∞. As a result of Theorem 2.2 that (2.1) has a unique
2,◦
invariant probability measure π ∗ whose support is R+
. Consequently, the strong law of large

numbers and the convergence in total variation norm of the transition probability hold. A
17


sample path of solution to (2.1) is illustrated by Figures 3, while the phase portrait in Figure
4 demonstrates that the support of π ∗ and the empirical density of π ∗ .

Figure 3: Trajectories of S(t), I(t) in Example 3.2.

Figure 4: Phase portrait of (2.1) and the support of π ∗ in Example 3.2.
and the empirical density of π ∗ .
Example 3.3. Consider (2.1) with parameters α = 5, β = 5, µ = 4, ρ = 1, γ = 1, σ1 = 2,
σ2 = −1. By calculation, λ = −1.75 < 0 As a result of Theorem 2.1 that I(t) → 0 a.s. as
t → ∞. This claim is justified in Figures 5. This means, population will eventually have no
disease.

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18


Figure 5: Trajectories of S(t), I(t) in Example 3.3.
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21