The stability in probability of the steady state of the system is proved under suitable conditions on the white noise perturbations

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STABILITY OF STOCHASTIC DELAYED SIR MODEL

GUOTING CHEN

UFR de Math´ematiques, Laboratoire Paul Painlev´e, UMR 8524, Universit´e de Lille 1, 59655 Villeneuve d’Ascq, France

[email protected]

TIECHENG LI

Department of Mathematical Sciences, Tsinghua University, Beijing 100084, P. R. China

[email protected] Received 4 March 2008

A stochastic version of the SIR model is investigated in this paper. The stability in probability of the steady state of the system is proved under suitable conditions on the white noise perturbations. Linearizations of the systems both with and without delay are given and their exponentially mean square stabilities are studied.

Keywords: Stability; SIR model; stochastic differential system; time delay; global solution.

1. Introduction

In the understanding of different scenarios for disease transmissions and behavior of epidemics, many models in the literature represent dynamics of diseases by systems of ordinary differential equations without delay. Recently, several authors propose to include temporal delays in such models which makes them more realistic by allowing to describe the effects of disease latency or immunity [5–7]. One of the main issues in the study of behavior of epidemics is the analysis of the steady states of the model and their stabilities.

A delayed SIR (susceptible, infective and removed) model which incorporates temporary immunity and a general nonlinear incidence rate has the following form (see [15]):

S(t) =˙ µ−µS(t)−φf(I(t))S(t) +γI(t−τ)e^−µτ, I(t) =˙ φf(I(t))S(t)−(γ+µ)I(t),

R(t) =˙ γI(t)−γI(t−τ)e^−µτ−µR(t),

where S(t) is the number of members of a population susceptible to the disease, I(t) is the number of infective members and R(t) is the number of members who have been removed from the possibility of infection through full immunity. The

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nonlinear functionf(I) represents the incidence rate,τ is the length (fixed) of the temporary immunity period.

The functionf is assumed to be of classC²satisfying the following properties:

f(0) = 0; forI≥0, f(I)>0, f(I)<0; lim

I→+∞f(I) =c <∞. (1) Parameters in the system are as follows:µis a natural death rate;γis a recovery rate, i.e. rate with which individuals move from the infected class to the recovered, andφis a recruitment rate from suspectable class to the infected class. Of course these parameters are all positive.

The first two equations in the above system do not depend on the third one, and therefore it can be omitted without loss of generality. Hence, it is enough to consider the system

S(t) =˙ µ−µS(t)−φf(I(t))S(t) +γI(t−τ)e^−µτ,

I(t) =˙ φf(I(t))S(t)−(γ+µ)I(t). (2)

It is easy to see that system (2) always has a disease-free equilibrium (i.e. boundary equilibrium) E₀ = (1,0). Kyrychko and Blyuss [15] proved that system (2) has a unique new endemic equilibrium (i.e. interior equilibrium, in the domainS >0, I >

0) E_τ^∗ = (S_τ^∗, I_τ^∗) if and only if f(0)φ > µ+γ, and investigated the stabilities of these equilibria.

The aim of this paper is to consider a stochastic version of the present model.

For this purpose we first give some basic notations and definitions.

Throughout this paper, we let (Ω,F,{Ft}t≥0, P) be a complete probability space with a filtration {F_t}_t≥0 satisfying the usual conditions (i.e. it is right continuous and F0 contains all P-null sets). Let B₁(t), B₂(t) be one-dimensional standard Brownian motions defined on the probability space, independent from each other. E[·] denotes the expectation of the corresponding random variable.

Letτ ≥ 0 and R₊ = [0,∞). Denote by C([−τ,0],R) and C([−τ,0],R₊) respec- tively the families of continuous functions from [−τ,0] toRandR₊with the norm ϕ= sup_{−τ≤θ≤0}|ϕ(θ)|.

We consider a stochastic version of model (2) with perturbations of white noise type. The perturbations are supposed to be directly proportional to both the sizes S(t) andI(t), and do not change the interior equilibrium. By this way, the system in consideration is of the form

dS(t) = [µ−µS(t)−φS(t)f(I(t)) +γI(t−τ)e^−µτ]dt +S(t)g₁(S(t), I(t), τ)dB₁(t),

dI(t) = [φS(t)f(I(t))−(γ+µ)I(t)]dt+I(t)g₂(S(t), I(t), τ)dB₂(t),

(3) whereg₁, g₂are functions of classC² onR³and verify the following conditions:

g₁(S_τ^∗, I_τ^∗, τ) = 0, g₂(S_τ^∗, I_τ^∗, τ) = 0 for all τ≥0. (4) Definition 1. The equilibrium in the following is assumed to be (0,0).

(i) The equilibrium (0,0) of a two-dimensional system of stochastic differential equations is exponentially mean square stable, if there is a pair of constants

K >0 andν >0 such that

E[u²₁(t) +u²₂(t)]≤Ke^−νtE[ξ²₁+ξ₂²] for all t≥0,

where (u₁(t),u₂(t)) is the solution to the system with initial condition (ξ₁, ξ₂) which is aR²-valuedF0-measurable random variable satisfying

E[ξ₁²+ξ₂²]<∞.

(ii) The equilibrium (0,0) of a two-dimensional system of stochastic delayed dif- ferential equations is exponentially mean square stable, if there is a pair of constantsK >0 andν >0 such that

E[u²₁(t) +u²₂(t)]≤Ke^−νtE[ξ₁²+ξ₂²] for all t≥0,

where (u₁(t), u₂(t)) is the solution to the system with initial condition (ξ₁, ξ₂), andξ₁, ξ₂ areC([−τ,0],R)-valuedF0-measurable random variable satisfying

E[ξ₁²+ξ₂²]<∞.

(iii) The equilibrium (0,0) of a two-dimensional stochastic delayed system is sto- chastically stable or stable in probability, if for every pair of ε ∈ (0,1) and r >0, there exists aδ >0 such that

P{u²₁(t) +u²₂(t)< r² for allt≥0} ≥1−ε,

wheneverξ₁²+ξ₂²< δ²,where (u₁(t), u₂(t)) is the solution of the system with initial condition (ξ₁, ξ₂), andξ₁, ξ₂∈C([−τ,0],R).

Stochastic differential delay equations, introduced by Itˆo and Nisio [12] in the 1960s, has attracted much attention in the last decade. However, this field is still in its infancy. For example, conditions for the stability of generalized linear stochastic differential delay equation with constant coefficients, are not known [17, 21]. For a specific SIR model with distributed delay, the stability of the boundary equilibrium (1,0,0) is studied in [23].

In this paper, we shall study the stability of the endemic equilibrium E_τ^∗ = (S_τ^∗, I_τ^∗) of system (3) under the basic assumptions:µ >0, γ >0, τ ≥0, f(0)φ >

µ+γ,f satisfies (1),g₁, g₂ verify (4). We first prove in Sec. 2 that for any positive initial condition, system (3) admits a unique global solution and it remains positive for allt≥0. In Sec. 3, we consider the linearization of system (3) near the endemic equilibrium and its stability. The stability of stochastic SIR model (3) is studied in Sec. 4. Finally we consider a particular case where f(I) =I/(1 +I) which is suggested in [15].

2. Positive and Global Solutions

Consider model (3). As the statesS and I of the system are the sizes of the sus- ceptible members and the infective members respectively, they should be positive.

Moreover, in order for a stochastic differential equation to have a unique global

solution (i.e. no explosion in a finite time) for any given initial value, the coeffi- cients of the equation are generally required to satisfy the linear growth condition and local Lipschitz condition (see, for example, [8, 19]). However, the coefficients of Eq. (3) do not satisfy the linear growth condition, though they are locally Lipschitz continuous, so the solution of Eq. (3) may explode at a finite time. In this section we shall show that the solution of Eq. (3) with positive initial data remains positive for allt >0.

The function f(I) is usually supposed to be defined for allI≥0, since we are interested in solutions withI≥0. Taking into account condition (1), we can extend f by definingf(I) =f(0)I+f(0)I²/2 for I≤0 so thatf is defined in Randf is still of classC²in R.

Theorem 1. Let notations and assumptions be as above. Let the initial data be S(0, ω) =ϕ₁(0) >0 and I(θ, ω) = ϕ₂(θ) ∈C([−τ,0],R₊) with ϕ₂(0) >0. Then there is a unique solution (S(t, ω), I(t, ω)) to system (3) for all t ≥ 0 and the solution is positive for allt >0with probability1,namelyS(t, ω)>0andI(t, ω)>

0 for allt≥0 almost surely. Moreover, for allt≥0, E[S(t) +I(t)]≤1 +e^−µt

ϕ₁(0) +ϕ₂(0) +γ ₀

−τ

e^µθϕ₂(θ)dθ

. (5)

Proof. Since the coefficients of system (3) are locally Lipschitz continuous, for any given initial data S(θ, ω) = ϕ₁(θ), I(θ, ω) = ϕ₂(θ), θ ∈ [−τ,0], with ϕ_i ∈ C([−τ,0];R), i= 1,2, there is a unique maximal local solution (S(t, ω), I(t, ω)) on t ∈ [−τ, τ_e(ω)), where τ_e(ω) is the explosion time (cf. Appleby and Mao [1], Theorem A.2; Mao [18], Theorem 2.2, p. 95).

From now on we assume that ϕ_i ∈C([−τ,0];R₊), ϕ_i(0)>0, i= 1,2. Firstly, we show thatS(t) andI(t) are positive for allt∈(0, τ_e(ω)) almost surely. Define the stopping time

t₊= sup{t∈(0, τ_e):S|_[0,t]>0 andI|_[0,t]>0}.

We need to show that t₊ =τ_e a.s. To see this, we assume thatP{t₊ < τ_e} >0, to the contrary. Itˆo’s formula shows that, for almost all ω ∈ {t₊ < τ_e} and all t∈[0, t₊),

lnS(t) + lnI(t)−lnϕ₁(0)−lnϕ₂(0)

= _t

0

µ

S(s)−µ−φf(I(s)) +γI(s−τ)e^−µτ

S(s) −1

2g₁(S(s), I(s), τ)²

ds

+ _t

0

φS(s)f(I(s))

I(s) −(γ+µ)−1

2g₂(S(s), I(s), τ)²

ds

+ _t

0 g₁(S(s), I(s), τ)dB₁(s) + _t

0 g₂(S(s), I(s), τ)dB₂(s)

≥ _t

0

−(γ+ 2µ)−φf(I(s))−1

2g₁(S(s), I(s), τ)²−1

2g₂(S(s), I(s), τ)²

ds

+ _t

0

g₁(S(s), I(s), τ)dB₁(s) + _t

0

g₂(S(s), I(s), τ)dB₂(s).

It is easy to see that, for almost allω in {t₊ < τ_e}, S(t) and I(t) are positive on [0, t₊) andS(t₊)I(t₊) = 0, hence

ttlim+

[lnS(t) + lnI(t)] =−∞. By the above inequality,

−∞ ≥ _t+

0

−(γ+ 2µ)−φf(I(s))−1

2g₁(S(s), I(s), τ)²−1

2g₂(S(s), I(s), τ)²

ds

+ _t+

0

g₁(S(s), I(s), τ)dB₁(s) + _t+

0

g₂(S(s), I(s), τ)dB₂(s), (6) which is a contradiction since the right-hand side of the above inequality is finite, so we must therefore havet₊=τ_ea.s.

For each integerk greater than or equal to ϕ₁(0) +ϕ₂(0), define the stopping time

τ_k = sup

t∈[0, τ_e) : (S+I)|_[0,t] ≤k .

Clearly, τ_k is increasing as k → ∞. Set τ_∞ = lim_k→∞τ_k, whence τ_∞ ≤ τ_e a.s.

Now we show that P{τ_∞ =τ_e} = 0. To see this, we assume to the contrary that P{τ_∞ < τ_e} >0. Itˆo’s formula shows that, for almost all ω ∈ {τ_∞ < τ_e} and all k≥ϕ₁(0) +ϕ₂(0),

e^µτk(S(τ_k) +I(τ_k))

=ϕ₁(0) +ϕ₂(0) + _τk

0

e^µs[µ+γI(s−τ)e^−µτ−γI(s)]ds +

_τk

0

e^µs[S(s)g₁(S(s), I(s), τ)dB₁(s) +I(s)g₂(S(s), I(s), τ)dB₂(s)].

It is easy to see thatS(τ_k) +I(τ_k) =k a.s. Lettingk→ ∞leads to

∞=ϕ₁(0) +ϕ₂(0) + _τ∞

0

e^µs[µ+γI(s−τ)e^−µτ−γI(s)]ds +

_τ∞

0

e^µs[S(s)g₁(S(s), I(s), τ)dB₁(s) +I(s)g₂(S(s), I(s), τ)dB₂(s)], which is a contradiction since the right-hand side of the above inequality is finite, so we must therefore haveτ_∞=τ_e a.s.

Next we show thatP{τ_e<∞, τ_∞<∞}= 0. LetT >0 be arbitrary. We have by Itˆo’s formula that

1_{τe<∞}e^µ(τk∧T)[S(τ_k∧T) +I(τ_k∧T)]−1_{τe<∞}(ϕ₁(0) +ϕ₂(0))

= 1_{τe<∞}

_τk∧T

0

e^µs[µ+γI(s−τ)e^−µτ −γI(s)]ds

+ 1_{τe<∞}

_τk∧T

0

e^µs[S(s)g₁(S(s), I(s), τ)dB₁(s) +I(s)g₂(S(s), I(s), τ)dB₂(s)], where 1_{·}means the indicator function of the corresponding sets. Taking expecta- tions, we obtain

E[1_{τe<∞}e^µ(τk∧T)(S(τ_k∧T) +I(τ_k∧T))]

≤ϕ₁(0) +ϕ₂(0) +E

1_{τe<∞}

_τk∧T

0

e^µs[µ+γI(s−τ)e^−µτ−γI(s)]ds

. (7) Obviously,

1_{τe<∞}e^µ(τk∧T)[S(τ_k∧T) +I(τ_k∧T)]≥1_{{τe<∞, τk≤T}}[S(τ_k) +I(τ_k)]

= 1_{{τe<∞, τk≤T}}k.

Compute E

1_{τe<∞}

_τk∧T

0

e^µs[µ+γI(s−τ)e^−µτ −γI(s)]ds

=E

1_{τe<∞}

e^µ(τk∧T)−1 +γ

_{(τk∧T)−τ}

−τ

e^µsI(s)ds−γ _τk∧T

0

e^µsI(s)ds

=E

1_{τe<∞}

e^µ(τk∧T)−1 +γ ₀

−τ

e^µsI(s)ds−γ _τk∧T

τk∧T−τ

e^µsI(s)ds

≤e^µT +γ ₀

−τ

e^µθϕ₂(θ)dθ.

Substituting these into (7) gives

P{τ_e<∞, τ_k ≤T}k≤ϕ₁(0) +ϕ₂(0) +e^µT +γ ₀

−τ

e^µθϕ₂(θ)dθ.

Lettingk→ ∞leads to lim_k→∞P{τ_e<∞, τ_k ≤T}= 0 and hence P{τ_e<∞, τ_∞≤T}= 0.

SinceT >0 is arbitrary, we then have

P{τ_e<∞, τ_∞<∞}= 0.

Now by using the relation

{τ_e<∞}={τ_e<∞, τ_∞=∞} ∪ {τ_e<∞, τ_∞<∞}

⊆ {τ_∞=τ_e} ∪ {τ_e<∞, τ_∞<∞}, we obtainP{τ_e<∞}= 0. HenceP{τ_e=∞}= 1.

Finally we show the inequality (5). In the same way as above, we have for all t≥0 that

E[e^µt(S(t) +I(t))]

=ϕ₁(0) +ϕ₂(0) +E _t

0

e^µs[µ+γI(s−τ)e^−µτ−γI(s)]ds

≤ϕ₁(0) +ϕ₂(0) +e^µt+γ ₀

−τ

e^µθϕ₂(θ)dθ, which yields the inequality (5). The theorem is thus proved.

3. Linearized Systems and Their Stabilities 3.1. Linearization

We now study the linearization of system (3) at the endemic equilibrium E_τ^∗ = (S_τ^∗, I_τ^∗) and then study the stabilities of the linearized systems with and without delay. Throughout the paper we shall always assume that the basic hypotheses given in the Introduction are satisfied. Recall that in this case system (3) has a unique interior equilibriumE_τ^∗= (S_τ^∗, I_τ^∗).

According to the conditions onf(I), one can write

f(I) =a_τ+b_τ(I−I_τ^∗) +F(I−I_τ^∗, τ), (8) where F represents terms of order≥2 inI−I_τ^∗, that isF(0, τ) =F_I(0, τ) = 0 for allτ. The point (S_τ^∗, I_τ^∗) being an equilibrium of (3) means that

µ−µS_τ^∗−φS_τ^∗f(I_τ^∗) +γI_τ^∗e^−µτ = 0, φS_τ^∗f(I_τ^∗)−(γ+µ)I_τ^∗= 0, or

µ−µS_τ^∗−φa_τS_τ^∗+γI_τ^∗e^−µτ = 0, φa_τS_τ^∗−(γ+µ)I_τ^∗= 0.

We introduce new variablesu₁=S−S_τ^∗ andu₂=I−I_τ^∗. Under the conditions (4), one can write

g₁(S, I, τ) =σ₁₁(τ)u₁+σ₁₂(τ)u₂+ ˜g₁(u₁, u₂, τ), g₂(S, I, τ) =σ₂₁(τ)u₁+σ₂₂(τ)u₂+ ˜g₂(u₁, u₂, τ),

where ˜g₁(u₁, u₂, τ) and ˜g₂(u₁, u₂, τ) represent terms of order≥2 inu₁, u₂.

Now system (3) can be rewritten in the following form:

du₁(t) = [(−µ−φa_τ)u₁(t)−φb_τS_τ^∗u₂(t) +u₂(t−τ)γe^−µτ+ ˜F₁(u₁, u₂, τ)]dt + (S_τ^∗(σ₁₁(τ)u₁(t) +σ₁₂(τ)u₂(t)) + ˜G₁(u₁, u₂, τ))dB₁(t),

du₂(t) = [φa_τu₁(t) + (φb_τS_τ^∗−γ−µ)u₂(t) + ˜F₂(u₁, u₂, τ)]dt + (I_τ^∗(σ₂₁(τ)u₁(t) +σ₂₂(τ)u₂(t)) + ˜G₂(u₁, u₂, τ))dB₂(t),

(9)

where ˜F₁(u₁, u₂, τ), ˜F₂(u₁, u₂, τ), ˜G₁(u₁, u₂, τ) and ˜G₂(u₁, u₂, τ) represent terms of order≥2 inu₁, u₂.

Along with system (9) we consider the linearized system

du₁(t) = [(−µ−φa_τ)u₁(t)−φb_τS_τ^∗u₂(t) +u₂(t−τ)γe^−µτ]dt +S^∗_τ[σ₁₁(τ)u₁(t) +σ₁₂(τ)u₂(t)]dB₁(t),

du₂(t) = [φa_τu₁(t) + (φb_τS_τ^∗−γ−µ)u₂(t)]dt +I_τ^∗[σ₂₁(τ)u₁(t) +σ₂₂(τ)u₂(t)]dB₂(t),

(10)

and the auxiliary linear system without delay

du₁(t) = [(−µ−φa₀)u₁(t) + (γ−φb₀S^∗₀)u₂(t)]dt +S₀^∗[σ₁₁(0)u₁(t) +σ₁₂(0)u₂(t)]dB₁(t), du₂(t) = [φa₀u₁(t) + (φb₀S₀^∗−γ−µ)u₂(t)]dt

+I₀^∗[σ₂₁(0)u₁(t) +σ₂₂(0)u₂(t)]dB₂(t).

(11)

Before studying the stabilities of systems (10) and (11), we give a lemma.

Lemma 1. We assume the basic assumptions given in the Introduction. Define p_τ = 2µ(γ+µ−φb_τS_τ^∗) +φa_τ(γ+ 2µ−φb_τS_τ^∗)

φ²a_τ² ,

m_τ = 4p_τ(µ+φa_τ)(µ+γ−φb_τS_τ^∗)−[φb_τS_τ^∗−γ−p_τφa_τ]². Then we have

β_τ=µ+γ−φb_τS_τ^∗>0, p_τ>0, m_τ >0.

Proof. Sinceφf(I_τ^∗)S_τ^∗−(µ+γ)I_τ^∗= 0, we have that, by the definition of b_τ, β_τ =µ+γ−φb_τS_τ^∗= (µ+γ)[f(I_τ^∗)−f(I_τ^∗)I_τ^∗]

f(I_τ^∗) , by the Lagrange Mean Value Theorem,

f(I_τ^∗)−f(I_τ^∗)I_τ^∗=f(0) +f(I₁)I_τ^∗−f(I_τ^∗)I_τ^∗=f(0) +f(I₂)(I₁−I_τ^∗)I_τ^∗, where 0< I₁ < I₂ < I_τ^∗. Thus, by the conditions (1), we getµ+γ−φb_τS_τ^∗ >0.

Thereforep_τ is positive. Straightforward computations now lead to 4p_τ(µ+φa_τ)(µ+γ−φb_τS_τ^∗)−[φb_τS_τ^∗−γ−p_τφa_τ]²

= 4µ(µ+φa_τ)(γ+µ−φb_τS^∗_τ)(µ+γ−φb_τS_τ^∗+φa_τ)

φ²a²_τ ,

which is positive too. The lemma is thus proved.

3.2. Stability of the linearized system without delay

We first study the stability of the linearized system without delay (11).

Proposition 1. Let notations and assumptions be as above. Ifσ_ij =σ_ij(0)satisfy the following conditions:

2(µ+φa₀)−S₀^∗2σ₁₁² −p₀I₀^∗2σ₂₁² >0,

[2(µ+φa₀)−S^∗₀²σ₁₁² −p₀I₀^∗2σ²₂₁][2p₀β₀−S₀^∗2σ²₁₂−p₀I₀^∗2σ₂₂² ]

−[φb₀S₀^∗−γ−p₀φa₀−S₀^∗2σ₁₁σ₁₂−p₀I₀^∗2σ₂₁σ₂₂]²>0,

(12) then the trivial solution of system(11)is exponentially mean square stable.

Proof. One first remarks that, by using Lemma 1, the conditions (12) are satisfied by σ_ij(0) = 0. So by continuity, the conditions are also satisfied for at least small σ_ij(0).

We will prove the result by constructing a quadratic formu^tQuas a (stochastic) Lyapunov function for system (11), whereQ = (q_ij) is a positive definite matrix.

Let

A= (a_ij) =

−µ−φa₀ γ−φb₀S₀^∗ φa₀ φb₀S₀^∗−γ−µ

. Then the characteristic equation ofAis

(λ+µ) (λ+µ+γ+φa₀−φb₀S₀^∗) = 0,

hence−µandφb₀S₀^∗−(µ+γ+φa₀) are the two eigenvalues ofA, which are negative.

Now consider the matrixC defined by

C= (c_ij) =−[A^tQ+QA], (13)

and determineQsuch thatC is also a positive-definite matrix.

In order to do so we setq₁₂=q₂₁= 0. Then

c₁₁=−2q₁₁a₁₁, c₁₂=c₂₁=−q₁₁a₁₂−q₂₂a₂₁, c₂₂=−2q₂₂a₂₂ and

c₁₁c₂₂−c²₁₂= 4q₁₁q₂₂a₁₁a₂₂−(q₁₁a₁₂+q₂₂a₂₁)²

=−a²₂₁

q₂₂−q₁₁2a₁₁a₂₂−a₁₂a₂₁ a²₂₁

₂

+4q₁₁² a₁₁a₂₂(a₁₁a₂₂−a₁₂a₂₁)

a²₂₁ .

By setting alsoq₁₁= 1 andq₂₂= (2a₁₁a₂₂−a₁₂a₂₁)/a²₂₁, we getq₂₂=p₀>0, and c₁₁= 2µ+ 2φa₀>0,

c₁₂=c₂₁=−2(γ+µ−φb₀S₀^∗)(µ+φa₀)

φa₀ ,

c₂₂= 2p₀(γ+µ−φb₀S₀^∗),

c₁₁c₂₂−c²₁₂= 4µ(µ+φa₀)(γ+µ−φb₀S₀^∗)(µ+γ−φb₀S₀^∗+φa₀)

φ²a²₀ >0.

Hence the matricesQandC are positive definite, and they satisfy Eq. (13).

Define a smooth functionV :R²−→R₊ by

V(u₁, u₂) =u²₁+p₀u²₂. (14) By the conditions (12) we know that the matricesH = (H_ij) with

H₁₁= 2(µ+φa₀)−S₀^∗2σ₁₁² −p₀I₀^∗2σ₂₁² ,

H₁₂=H₂₁=φb₀S₀^∗−γ−p₀φa₀−S₀^∗2σ₁₁σ₁₂−p₀I₀^∗2σ₂₁σ₂₂, H₂₂= 2p₀(µ+γ−φb₀S₀^∗)−S₀^∗2σ²₁₂−p₀I₀^∗2σ²₂₂

and

G=

H₁₁ H₁₂/√p₀ H₂₁/√p₀ H₂₂/p₀

are positive definite. Therefore, for allu= (u₁, u₂)^t∈R², u^tHu= (u₁,√p₀u₂)G

u₁

√p₀u₂

≤λ_max(G)(u²₁+p₀u²₂) =λ_max(G)V(u₁, u₂), (15) similarly,

u^tHu≥λ_min(G)V(u₁, u₂), (16) whereλ_max(·) andλ_min(·) are the maximum and minimum eigenvalues of the cor- responding matrices.

Letν₁=λ_min(G), then Itˆo’s formula states e^ν1tV(u₁(t), u₂(t))−V(ξ₁, ξ₂)

= _t

0

e^ν1s[ν₁V(u₁(s), u₂(s)) +L₀V(u₁(s), u₂(s))]ds +

_t

0

e^ν1s[2S^∗₀(σ₁₁u²₁(s) +σ₁₂u₁(s)u₂(s))dB₁(s)

+ 2p₀I₀^∗(σ₂₁u₁(s)u₂(s) +σ₂₂u²₂(s))dB₂(s)], (17) where the operatorL₀V :R²−→Ris defined by

L₀V(u₁, u₂) = 2u₁[(−µ−φa₀)u₁+ (γ−φb₀S₀^∗)u₂] +S₀^∗2(σ₁₁u₁+σ₁₂u₂)² + 2p₀u₂[φa₀u₁+ (φb₀S₀^∗−µ−γ)u₂] +p₀I₀^∗2(σ₂₁u₁+σ₂₂u₂)².

(18) One then has

L₀V(u₁, u₂) =−H₁₁u²₁−2H₁₂u₁u₂−H₂₂u²₂=−u^tHu≤ −ν₁V(u₁, u₂), where inequalities (16) was used in the last one. Taking expectations on both sides of (17) and using the above inequality we obtain

e^ν1tE[V(u₁(t), u₂(t))]−E[V(ξ₁, ξ₂)]≤0, i.e.

E[u²₁(t) +p₀u²₂(t)]≤e^−ν1tE[ξ₁²+p₀ξ²₂],

which means that the trivial solution of system (11) is exponentially mean square stable. The proposition is thus proved.

Ifσ₁₂=σ₂₁= 0, then conditions (12) is reduced to

2(µ+φa₀)−S₀^∗2σ₁₁² >0, [2(µ+φa₀)−S₀^∗2σ²₁₁][2p₀β₀−p₀I₀^∗2σ₂₂² ]−[φb₀S^∗₀−γ−p₀φa₀]² >0.

The domain ofσ₁₁, σ₂₂ for the stability of the system in this special case is shown in the following figure.

3.3. Stability of the linearized system with delay

Now we carry over the above proposition to the linearized system with delay (10).

Let notations be as in Lemma 1. DenoteG_τ = (G^ij_τ) with G¹¹_τ = 2(µ+φa_τ)−S_τ^∗2σ₁₁² −p_τI_τ^∗2σ₂₁² ,

G¹²_τ =G²¹_τ = (φb_τS_τ^∗−γ−p_τφa_τ−S_τ^∗2σ₁₁σ₁₂−p_τI_τ^∗2σ₂₁σ₂₂)/√ p_τ, G²²_τ = (2p_τ(γ+µ−φb_τS_τ^∗)−S_τ^∗2σ₁₂² −p_τI_τ^∗2σ₂₂² )/p_τ

and

J_τ=









φ²a²_τ φa_τ(φb_τS_τ^∗−γ)

√p_τ

φa_τ(φb_τS_τ^∗−γ)

√p_τ

(φb_τS_τ^∗−γ)² p_τ







,

where σ_ij =σ_ij(τ). Since detJ_τ = 0, we haveλ_min(J_τ) = 0 and λ_max(J_τ) =φ²a²_τ+ (φb_τS_τ^∗−γ)²/p_τ.

σ₁₁ σ₂₂

Fig. 1. Domain of stability forσ11, σ22whenσ12=σ21= 0.

Proposition 2. Let notations be as above and assume that the basic assumptions are satisfied. Ifτ andσ_ij =σ_ij(τ)satisfy the following conditions







G¹¹_τ >0, G¹¹_τ G²²_τ −(G¹²_τ )²>0, λ_min(G_τ)> γτ(1 +λ_max(J_τ)) =γτ

1 +φ²a²_τ+(φb_τS_τ^∗−γ)² p_τ

,

(19)

then the trivial solution of system(10)is exponentially mean square stable.

Proof. One first remarks that the conditions in (19) are satisfied byσ_ij(0) =τ= 0.

So by continuity, the conditions are also satisfied for at least smallτ, σ_ij(τ).

Now for all (u₁, u₂)∈R², (u₁,√p_τu₂)G_τ

u₁

√p_τu₂

≥λ_min(G_τ)(u²₁+p_τu²₂).

Chooseν₂>0 such thatλ_min(G_τ)−ν₂> γτ[1 +λ_max(J_τ)e^ν2τ], and define a smooth functionU :R²→R₊ by

U(u₁, u₂) =u²₁+p_τu²₂. (20) Then Itˆo’s formula states that

e^ν2tU(u₁(t), u₂(t))−U(u₁(0), u₂(0))

= _t

0

e^ν2s[ν₂U(u₁(s), u₂(s)) +L₁U(u₁(s), u₂(s))]ds +

_t

0

e^ν2s[2S_τ^∗u₁(s)(σ₁₁u₁(s) +σ₁₂u₂(s))dB₁(s)

+ 2p_τI_τ^∗u₂(s)(σ₂₁u₁(s) +σ₂₂u₂(s))dB₂(s)], (21) whereL₁ is an operator defined as follows.

L₁U(u₁(s), u₂(s)) = 2u₁(s)[(−µ−φa_τ)u₁(s)−φb_τS_τ^∗u₂(s) +u₂(s−τ)γe^−µτ] + 2p_τu₂(s)[φa_τu₁(s) + (φb_τS_τ^∗−γ−µ)u₂(s)]

+S_τ^∗2(σ₁₁u₁(s) +σ₁₂u₂(s))²+p_τI_τ^∗2(σ₂₁u₁(s) +σ₂₂u₂(s))². (22) Straightforward computations lead to

L₁U(u₁(s), u₂(s))

= [2(−µ−φa_τ) +S_τ^∗2σ₁₁² +p_τI_τ^∗2σ₂₁² ]u²₁(s)

+ 2(γ−φb_τS_τ^∗+p_τφa_τ+S_τ^∗2σ₁₁σ₁₂+p_τI_τ^∗2σ₂₁σ₂₂)u₁(s)u₂(s) + (2p_τ(φb_τS_τ^∗−γ−µ) +S_τ^∗2σ²₁₂+p_τI_τ^∗2σ²₂₂)u²₂(s)

+ 2γu₁(s)[u₂(s−τ)e^−µτ−u₂(s)]

=−(u₁(s), √p_τu₂(s))G_τ

u₁(s)

√p_τu₂(s)

+ 2γu₁(s)[u₂(s−τ)e^−µτ−u₂(s)].