a stochastic analysis for a triple delayed siqr epidemic

Journal of Applied Mathematics and Computing (2020) 64:781–805https://doi.org/10.1007/s12190-020-01380-1

ORIG INAL RESEARCH

A stochastic analysis for a triple delayed SIQR epidemicmodel with vaccination and elimination strategies

Mohamed El Fatini1 · Roger Pettersson2 · Idriss Sekkak1 · Regragui Taki3

Received: 21 January 2020 / Revised: 09 May 2020 / Accepted: 08 June 2020 / Published online: 26 June 2020© Korean Society for Informatics and Computational Applied Mathematics 2020

AbstractIn this paper, a delayed SIQR epidemic model with vaccination and elimination hybridstrategies is analysed under a white noise perturbation. We prove the existence andthe uniqueness of a positive solution. Afterwards, we establish a stochastic thresholdRs in order to study the extinction and persistence in mean of the stochastic epidemicsystem. Then we investigate the existence of a stationary distribution for the delayedstochastic model. Finally, some numerical simulations are presented to support ourtheoretical results.

Keywords Extinction · Persistence in mean · Delay · White noise · Epidemic model ·Stationary distribution

Mathematics Subject Classification 92B05 · 60G51 · 60H30 · 60G57

1 Introduction

Governments has always made the public health policy as a priority and adopteddecisions, plans and actions to save human lives from deadly infectious diseases. For

B Mohamed El [email protected]

Roger [email protected]

Idriss [email protected]

Regragui [email protected]

1 Département de Mathématiques, Faculté des Sciences, Université Ibn Tofail, BP 133, Kénitra,Morocco

2 Department of Mathematics, Linnaeus University, 351 95 Växjö, Sweden

3 Chouaib Doukkali University EST Sidi Bennour, El Jadida, Morocco

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http://orcid.org/0000-0002-1179-4366

782 M. E. Fatini et al.

this matter, computational biologists study the dynamics of epidemics to prevent andcontrol the infection from spreading in the population [1,2,6]. Historically, in the 14thcentury, the authorities of the city of Venice have installed a measure of isolationto enter and exit its ports, where every crew in every ship was inspected, once thewhole personal have no symptoms, then they could be cleared out to land. This idea isadopted as a major measure to prevent infectious diseases such as Ebola and Malariafrom spreading. Nowadays, every well equipped hospital has a number of rooms andhalls dedicated for the isolation of infected individuals. In addition, the authoritiescan put their people under quarantine. This decision has a significant impact on thebasic reproduction number, which leads us to the extinction of the infectious disease.Recently, the quarantine measure has proven to be efficient in the extinction of theCOVID19 disease in China, which let many countries to adopt this strategy in absenceof vaccine or a cure to the new Corona virus. In order to understand the effect ofquarantine on the behavior of the epidemics, Heathcote [13] proposed a model withquarantine to describe isolated individuals in the compartmental model followed byother papers such as [5,27].

On the other hand, a new type of delayed stochastic models are proposed to describethe role of time delay in reality which leads to a more complex behavior of the stabilityof the dynamic system. This concept is described as a temporary immunity in [1,10,12]and as a vaccine effect in [8,9]. However, the temporary immunity can impact also thequarantined individuals. Therefore delayed stochastic epidemicmodels are concerningthe persistence in mean, ultimate boundedness and permanence [5] or the asymptoticbehavior around the equilibrium points of the infectious diseases models. In [15], theauthors obtained sufficient conditions for the existence and uniqueness of stationarydistribution for a delayed stochastic differential equations with positivity constraintsand applied theoretical results for biochemical reaction system. Also, Zhang and Yuan[28] used Lyapunov analysis method to investigate the existence of stationary distribu-tion of a stochastic delayed chemostat model. Therefore, it is important to investigatea relatively weak characteristic for a delayed stochastic epidemic model.

In recent years, epidemic models with quarantine for the have been proposed [23,25,26]. On the other hand, for themodelling of the environemental noise. In [3,4,16,20,24,29] perturbed systems with white noise and isolation for population dynamics areconsidered. Liu et al. [22] investigated a SIQR epidemic model with telegraph noise tostudy the influence of isolation for a compartmental SIRmodel, where they establisheda stochastic threshold for the extinction, persistence in mean and obtained sufficientconditions for the existence of positive recurrence of the solutions. By constructing asuitable stochastic Lyapunov function to the epidemic model with regime switching.

Therefore, in order to reflect more the reality we introduced the notion of delay,vaccine and elimination in an SIQR epidemic model. Hence, we propose the followingtriple delayed SIQR epidemic model with vaccination and isolation strategies

S = A − βS(t)I (t) − (μ + p)S(t) + pS(t − τ1)e−μτ1

+ γ I (t − τ2)e−μτ2 + εQ(t − τ3)e

−μτ3 ,

I = βS(t)I (t) − (μ + α1 + δ + γ )I (t),

Q = δ I (t) − (μ + α2 + ε)Q(t),

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A stochastic analysis for a triple delayed SIQR epidemic… 783

R = γ I (t) + pS(t) + εQ(t) − μR(t) − pS(t − τ1)e−μτ1

− γ I (t − τ2)e−μτ2 − εQ(t − τ3)e

−μτ3 , (1)

The compartments susceptible, infected, quarantined or isolated, and recovered aredenoted by S(t), I (t), Q(t) and R(t), respectively. The parameter A represents thepopulation recruitment rate,μ denotes the natural death rate of S, I , Q and R compart-ments, β denotes the transmission coefficient from susceptible to infected individuals,γ describes the recovery rate of the infective individuals, α1 and α2 represents thedeath rate for infected and quarantined individuals because of infection complica-tions, p stands for the proportional coefficient of vaccinated for the susceptible, δ

denotes the rate of infectious individuals who were isolated, ε represents the recov-ered people coming from isolation. The time τ1 > 0 represents the delay for theefficiency of vaccine. The term S(t − τ1)e−μτ1 reflects the fact that some individualsremains susceptible even after the vaccine for a specific time. The time τ2 > 0 is thelength of the immunity period. The term I (t − τ2)e−μτ2 represents the individualswho became susceptible because of the lose of immunity for a specific time. The timeτ3 > 0 denotes the delay for isolated individuals to get back their immunity. The termQ(t − τ3)e−μτ3 represents the individuals coming out from isolation with immunityimpairment. The basic reproduction number of the system (1) is

R0 = βA

(μ + p(1 − e−μτ1))(μ + α1 + δ + γ ). (2)

Next, we establish the following delayed stochastic SIQR epidemic model with vac-cination and elimination strategies

dS = [A − βS(t)I (t) − (μ + p)S(t) + pS(t − τ1)e−μτ1 + γ I (t − τ2)e

−μτ2

+ εQ(t − τ3)e−μτ3 ]dt,+σ1S(t)dB1(t),

d I = [βS(t)I (t) − (μ + α1 + δ + γ )I (t)]dt + σ2 I (t)dB2(t),

dQ = [δ I (t) − (μ + α2 + ε)Q(t)]dt + σ3Q(t)dB3(t),

dR = [γ I (t) + pS(t) + εQ(t) − μR(t) − pS(t − τ1)e−μτ1

− γ I (t − τ2)e−μτ2 − εQ(t − τ3)e

−μτ3 ]dt + σ4R(t)dB4(t), (3)

where, Bi (t) are independent standard Brownian motions defined on a complete prob-ability space (Ω,F ,P) with a filtration {Ft }t≥0 satisfying the usual conditions and σifor i = 1, 2, 3, 4 represent the volatility perturbations.We define the differential operator L, associated with the following general d-dimensional stochastic system

dX(t) = F(t, X(t), X(t − τ)dt + G(t, X(t))dB(t), for all t ≥ −τ, τ ≥ 0, (4)

with the initial condition X(s) = η(s) for s ∈ [−τ, 0], η ∈ C([−τ, 0];Rd+) andη(s) > 0, where F(t, X(t), X(t − τ)) is a function on Rd defined in [−τ,+∞[×R

d ,

G(t, X(t)) is a d × m matrix, F and G are locally Lipschitz functions in x and B(t)

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is an d-dimensional Wiener process. The differential operator L, acts on a functionV ∈ C1,2(C([−τ, 0];Rd+) × [−τ,∞);R+), as follows

LV (t, X) = Vt (t, X) + VX (t, X)F(t, X(t), X(t − τ)

+ 1

2trace

[GT (t, X)VXX (X , t)G(X , t)

].

By Itô’s formula

dV (t, x(t)) = LV (t, X(t))dt + VX (t, X(t))G(t, X(t))dB(t),

where

Vt = ∂V

∂t, Vx =

(∂V

∂x1,

∂V

∂x2, . . . ,

∂V

∂xd

)Vxx =

(∂V 2

∂xi∂x j

)

d×d

.

For any X ∈ R3, the norm |X |, as usual, is given by |X | =

√X21 + X2

2 + X23 .

This work is organized as follows. In Sect. 2, Existence and uniqueness of a globalpositive solution is shown. In Sect. 3, the stochastic threshold is investigated betweenthe extinction and persistence in man. In Sect. 4, we prove sufficient conditions for theexistence of a unique stationary distribution for the delayed stochastic SIQR epidemicmodel. In Sect. 5, numerical simulations are given to support the theoretical results.

2 Existence and uniqueness of the global positive solution

In this section, we prove that the model (3) has a local positive solution. Then weinvestigate the global positivity of the solution.

Throughout this work, we will reduce our stochastic system (3) using the threefirst equations since they do not depend on R(t). The fourth equation can be droppedwithout loss of generality.

Let τ = max{τ1, τ2, τ3}. We denote

R3+ =

{(S, I , Q) ∈ R

3 : S > 0, I > 0, Q > 0}

.

and let C = C([−τ, 0],R3+) be the Banach space of continuous functions mappingfrom the interval [−τ, 0] intoR3+ equipped by the norm ||φ|| = sup−τ≤θ≤0|φ(θ)|. Weset the initial conditions of system (3) to be

S(θ) = φ1(θ), I (θ) = φ2(θ), Q(θ) = φ3(θ),

φi (θ) > 0, θ ∈ [−τ, 0], i = 1, 2, 3,

(φ1, φ2, φ3) ∈ C . (5)

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Theorem 1 For any given initial value (S(0), I (0), Q(0)) ∈ R3+, there is a positive

unique solution (S(t), I (t), Q(t)) of model (3) on t ≥ 0 and the solution will remainin R3+ with probability 1.

Proof Since the coefficients of the system (3) are locally Lipschitz continuous, for anygiven initial value (S(0), I (0), Q(0)) ∈ R

3+ there is a unique local solution positive(S(t), I (t), Q(t)) on t ∈ [τ, τe), where τ = max{τ1, τ2, τ3} and τe is the explosiontime. Now, we show that the solution is global.We have only to prove that τe = ∞ a.s.Consider an ε0 > 0 such that S(0) > ε0, I (0) > ε0, Q(0) > ε0, then we define thestopping time as follows

τε = inf {t ∈ [0, τe) : S(t) ≤ ε, or I (t) ≤ ε, or Q(t) ≤ ε} , ∀ε > 0 such that ε ≤ ε0.

Throughout this paper we set inf ∅ = ∞ (∅ denotes the empty set). It’s clear that, τε

is increasing as ε → 0. Set τ0 = limε→0 τε. Obviously τ0 ≤ τe a.s. If τ0 = ∞ a.s. istrue, then τe = ∞ a.s. and (S(t), I (t), Q(t)) ∈ R

3+ a.s. for t ≥ 0. In other words, tocomplete the proof it is required to show that τ0 = ∞ a.s. If this statement is false,then there exist a pair of constants T > 0 and δ ∈ (0, 1) such that P{τ0 ≤ T } > δ.Thus there is an ε1 > 0 such that

P{τε ≤ T } ≥ δ ∀ε ≤ ε1.

Consider the C2-function V1 : R3+ → R as follows

V1(S, I , Q) = log S(t) + log I (t) + log Q(t) − logφ1(θ) − logφ2(θ) − logφ3(θ),

(6)

Applying Itô’s formula on (6) for all t ∈ [0, τε) and all ω ∈ {τε < T }, we obtain

V1(S, I , Q)

=∫ t

0

[A

S(t)− (μ + p) − β I (t) + pS(s − τ1)e

−μτ1

S(s)+ γ I (s − τ2)e

−μτ2

S(s)

+εQ(s − τ3)e−μτ3

S(s)− σ 2

12

]ds +

∫ t

0

[βS(t) − (μ + α1 + δ + γ ) − σ 2

22

]ds

+∫ t

0

[δ I (s)

Q(s)− (μ + α2 + ε) − σ 2

32

]ds + σ1B1(t) + σ2B2(t) + σ3B3(t)

≥ −∫ t

0

[3μ + p + α1 + α2 + δ + γ + ε + σ 2

1 + σ 22 + σ 2

32

+ β I (s)

]ds + σ1B1(t)

+ σ2B2(t) + σ3B3(t). (7)

According to the stopping time τε, for almost allω in {τε < T } at least one componentof (S(τε), I (τε), Q(τε)) is equal to ε. Thus,

limε→0

V1(S(τε), I (τε), Q(τε)) = −∞ (8)

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Letting t → τε in (7), we obtain

−(3μ + p + α1 + α2 + δ + γ + ε + 1

2(σ 2

1 + σ 22 + σ 2

3 )

)τε

−β

∫ τε

0I (s)ds + σ1B1(τε) + σ2B2(τε) + σ3B3(τε) > −∞. (9)

From (8) and extending ε to zero in (9) contradict our assumption, consequently,τ0 = τe = +∞ a.s.

3 Investigation of a stochastic threshold

In this section, we investigate a stochastic threshold for the extinction and the per-sistence in mean which represent an important issues to study the dynamics of thedisease. Firstly, we will focus on the following lemmas before investigating a stochas-tic threshold of the stochastic system (3).

Lemma 1 Let (S(t), I (t), Q(t)) be a solution of system (3) with any initial value(S(ξ1) ≥ 0, I (ξ2) ≥ 0, Q(ξ3)) ≥ 0 for all ξ1 ∈ [−τ1, 0), ξ2 ∈ [−τ2, 0), ξ3 ∈ [−τ3, 0)with S(0) > 0, I (0) > 0, Q(0) > 0, then

limt→∞

S(t) + I (t) + Q(t) + pS(t − τ1)e−μτ1 + γ I (t − τ2)e−μτ2 + εQ(t − τ3)e−μτ3

t= 0 a.s , (10)

Moreover

limt→∞

S(t)

t= 0, lim

t→∞I (t)

t= 0, lim

t→∞Q(t)

t= 0, lim

t→∞e−μt ∫ t

t−τ1eμs S(s)ds

t= 0

limt→∞

e−μt ∫ tt−τ2

eμs I (s)ds

t= 0, lim

t→∞e−μt ∫ t

t−τ3eμs Q(s)ds

t= 0 a.s.

Proof Let

V1(t) = S(t) + I (t) + Q(t) + pe−μt∫ t

t−τ1

eμs S(s)ds + γ e−μt∫ t

t−τ2

eμs I (s)ds

+ εe−μt∫ t

t−τ3

eμs Q(s)ds.

Define

V2(V1) = (1 + V1)θ ,

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where θ is a positive constant to be determined later. Applying Itô’s formula on V2,we get

dV2(V1) = LV2(V1)dt + θ(1 + V1)θ−1(σ1S(t)dB1(t) + σ2 I (t)dB2(t)

+σ3Q(t)dB3(t)), (11)

where

LV2(V1)= θ(1 + V1)

θ−1[A − μS(t) − (μ + α1)I (t) − (μ + α2)Q(t)

−μpe−μt∫ t

t−τ1

eμs S(s)ds − μγ e−μt∫ t

t−τ2

eμs I (s)ds − μεe−μt∫ t

t−τ3

eμs Q(s)ds]

+ θ(θ − 1)

2(1 + V1)

θ−2(σ 21 S

2 + σ 22 I

2 + σ 23 Q

2)

≤ θ(1 + V1)θ−2 {(1 + V1) [A − μV1 − α1 I (t) − α2Q(t)]

+ θ − 1

2(σ 2

1 S2 + σ 2

2 I2 + σ 2

3 Q2)

}. (12)

Let σ 2 = σ 21 ∨ σ 2

2 ∨ σ 23 . Therefore, we obtain

LV2(V1) ≤ θ(1 + V1)θ−2

{(1 + V1)[A − μV1] +

(θ − 1

2

)σ 2V 2

1

}

≤ θ(1 + V1)θ−2

{−

(μ − σ 2 θ − 1

2

)V 21 + (A − μ)V1 + A

}.

Choose θ > 1 such that μ − (θ−12

)σ 2 := ν > 0 then

L(V2(V1)) ≤ θ(1 + V1)θ−2{−νV 2

1 + (A − μ)V1 + A}.

Yields that

d(V2(V1)) ≤ θ(1 + V1)θ−2[−νV 2

1 + (A − μ)V1 + A]dt+ θ(1 + V1)

θ−1[σ1SdB1(t) + σ2 I dB2(t) + σ3QdB3(t)]. (13)

Therefore, for 0 ≤ k ≤ νθ , we get

E(ekt V2(V1)) = V2(V1(0)) + E∫ t

0L(eksV2(V1(s)))ds, (14)

where

L(ekt V2(V1)) = kekt V2(V1) + ektLV2(V1)≤ θekt (1 + V1)

θ−2{k

θ(1 + V1)

2 − νV 21 + (A − μ)V1 + A

}

123


= θekt (1 + V1)θ−2

(−

(ν − k

θ

)V 21 +

(A − μ + 2k

θ

)V1 + A + k

θ

)

≤ θekt H ,

with

H := supV1∈R+

(1 + V1)θ−2

[−

(ν − k

θ

)V 21 +

(A − μ + 2k

θ

)V1 + A + k

θ

].

Accordingly to (14), we get

E[ekt (1 + V1(t))θ ] ≤ (1 + V1(0))

θ + θH

kekt . (15)

Knowing that V1(t) is continuous, yields that there is a positive constant K such that

E[(1 + V1(t))θ ] ≤ K for all t ≥ 0. (16)

Using (13) and according to Burkholder–Davis–Gundy inequality, for a sufficientlysmall � > 0, k = 1, 2, . . . and for a positive constant c1, we have

E

[sup

k�≤t≤(k+1)�(1 + V1(t))

θ

]≤ E[(1 + V1(k�))θ + Υ1 + Υ2,

in which

Υ1 = E

[sup

k�≤t≤(k+1)�

∣∣∣∣∫ t

k�θ(1 + V1(s))

θ−2[−νV1(s)

2 + (A − μ)V1(s) + A]ds

∣∣∣∣]

≤ c1E

[∫ (k+1)�

k�(1 + V1(s))

θds

]

≤ c1�E

[sup

k�≤t≤(k+1)�(1 + V1(t))

θ

],

and

Υ2 = E

[sup

k�≤t≤(k+1)�

∣∣∣∣∫ t

k�θ(1 + V1(s))

θ−1(σ1S(s)dB1(s) + σ2 I (s)dB2(s) + σ3Q(s)dB3(s)

∣∣∣∣]

≤ √32E

[∫ (k+1)�

k�θ2(1 + V1(s))

2(θ−1)(σ 21 S

2(s) + σ 22 I

2(s) + σ3Q2(s))ds

] 12

≤ √32θσ�

12 E

[sup

k�≤t≤(k+1)�(1 + V1(t))

θ

].

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Thus

E

[sup

k�≤t≤(k+1)�(1 + V1(t))

θ

]

≤ E[(1 + V1(k�))θ ] + [c1� + √32θσδ

12 ]E

[sup

k�≤t≤(k+1)�(1 + V1(t))

θ

].

We choose � > 0 as c1� + √32θσ�

12 ≤ 1

2 and from (16) we have

E

[sup

k�≤t≤(k+1)�(1 + V1)

θ

]≤ 2K .

Choosing εu > 0 as an arbitrary number and applying Chebyshev’s inequality, we get

P

{sup

k�≤t≤(k+1)�(1 + V1)

θ > (k�)1+εu

}≤

E

[sup

k�≤t≤(k+1)�(1 + V1(t))θ

]

(k�)1+εu

≤ 2K

(k�)1+εu.

For almost all σ ∈ Ω and by Borel–Cantelli’s lemma, we have

supk�≤t≤(k+1)�

(1 + V1(t))θ ≤ (k�)1+εu , (17)

holds for all finite k. Therefore, there exists for almost all ω ∈ Ω , a random integerk0(ω), where (17) is satisfied for k ≥ k0. Thus, for almost all ω ∈ Ω , if k ≥ k0 andk� ≤ t ≤ (k + 1)�, we have

log(1 + V1(t))θ

log t≤ (1 + εu) log(k�))

log(k�)= 1 + εu .

Then

lim supt→∞

log(1 + V1(t))θ

log t≤ 1 + εu a.s

Let εu → 0, then

lim supt→∞

log(1 + V1(t))θ

log t≤ 1 a.s.

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Hence

lim supt→∞

log V2(t)

log t≤ lim sup

t→∞log(1 + V1(t))θ

log t≤ 1

θ, a.s

That is to say, for any small 0 < ξ < 1 − 1θ, there exists a constant T = T (ω) and a

set Ωξ as P(Ωξ ) ≥ 1 − ξ, and for t ≥ T , ω ∈ Ωξ, we get

log V1(t) ≤(1

θ+ ξ

)log t

Therefore

lim supt→∞

V1(t)

t≤ lim sup

t→∞t1θ+ξ

t= 0,

where together with the positivity of the solution, we get

limt→∞

V1(t)

t= 0 a.s

In which, we get (10). Finally, we obtain

limt→∞

S(t)

t= lim

t→∞I (t)

t= lim

t→∞Q(t)

t= lim

t→∞e−μt

∫ tt−τ1

eμs S(s)ds

t=

limt→∞

e−μt∫ tt−τ1

eμs I (s)ds

t,

limt→∞

e−μt∫ tt−τ1

eμs Q(s)ds

t= 0 a.s. Hence, this completes the proof.

Lemma 2 Let (S(t), I (t), Q(t)) be the solution of system (3) With any given initialcondition S(ξ1) ≥ 0, I (ξ2) ≥ 0 and Q(ξ3) ≥ 0 for all ξ1 ∈ [−τ1, 0), ξ2 ∈ [−τ2, 0)and ξ3 ∈ [−τ3, 0) with S(0), I (0) and Q(0) > 0 then

limt→∞

∫ t0 S(s)dB1(s)

t= lim

t→∞

∫ t0 I (s)dB2(s)

t= lim

t→∞

∫ t0 Q(s)dB3(s)

t= 0. (18)

Proof Let 1 < θ < 1+ 2μσ

and denote Xi (t) = ∫ t0 xi (s)dBi (t) with i = {1, 2, 3} and

xi (t) ∈ {S(t), I (t), Q(t)}.According to theBurkholder–Davis–Gundy inequality (Theorem7.3 [21]),wehave

E

[sup

0≤s≤t|Xi (s)|θ

]≤ Cθ E

[∫ t

0x2i (r)dr

] θ2 ≤ Cθ t

θ2 E

[sup

0≤r≤tx2i (r)

] θ2

≤ 2MiCθ tθ2 , (19)

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where Mi are positive constants. Now, let εXi be an arbitrary positive constant, fori = {1; 2; 3}, Applying Doob’s martingale inequality (Theorem 3.8 [21]) yields that

P

{ω : sup

k≤t≤(k+1)|Xi (t)| > (k)

12+εXi + θ

2

}≤ E[|Xi (k + 1)|θ ]

(k)12+εXi + θ

2

≤ 2MiCθ (k + 1)θ2

(k)12+εXi + θ

2

.

Thus, by Borel–Cantelli lemma, we get that for almost all ω ∈ Ω

supk≤t≤(k+1)

|Xi (s)|θ ≤ (k)12+εXi + θ

2 . (20)

Verified for all finite k. Therefore, there exists a positive random integer k0(ω), foralmost all ω ∈ Ω , where (20) satisfied whenever k ≥ k0. Thus, for almost all ω ∈ Ω ,if k ≥ k0 and k ≤ t ≤ (k + 1),

log |Xi (t)|θlog t

≤( 12 + εXi + θ

2

)log(k)

log(k)= 1

2+ εXi + θ

2.

Then

lim supt→∞

log |Xi (t)|log t

≤12 + εXi + θ

2

θ.

Letting εXi → 0, we get

limt→∞

log |Xi (t)|log t

≤ 1 + θ

2θ= 1

2+ 1

2θ.

That is to say, for any small 0 < χ1 < 12− 1

2θ . There exist a positive constant T = T (ω)

and set Ωχ1 as P(Ωχ1) ≥ 1 − χ1 and for t ≥ T , ω ∈ Ωχ1 , we get

ln |Xi | ≤(

1

2θ+ χ1

)ln t,

so

lim supt→∞

|Xi (t)|t

≤ lim supt→∞

t12θ +χ1

t= 0,

Which together with lim inf t→∞ |Xi (t)|t ≥ 0, we get

limt→∞

|Xi (t)|t

= 0 a.s.

123


yields that

limt→∞

Xi (t)

t= 0 a.s.

Therefore, we get (18). This finishes the proof.

3.1 Extinction

The main preoccupation in the study of dynamical behavior of epidemic models ishow to control the disease dynamics in order to die out in long term. In this section,we shall establish sufficient conditions for extinction of the disease in the stochasticmodel (3). In the sequel, we set 〈 f (t)〉 = 1

t

∫ t0 f (u)du, and we will investigate the

behaviour of the stochastic epidemic model (3) according to a following stochasticthreshold

Rs = R0 − σ 22

2(μ + α1 + δ + γ ). (21)

Theorem 2 Let (S(t), I (t), Q(t)) be a solution of system (3) with any initial valueS(ξ1) ≥ 0, I (ξ1) ≥ 0 and Q(ξ1) ≥ 0 for all ξ1 ∈ [−τ1, 0), ξ2 ∈ [−τ2, 0) andξ3 ∈ [−τ3, 0) with S(0) > 0, I (0) > 0 and Q(0) > 0. IfRs < 1 then

lim supt→∞

log(I (t))

t≤ (μ + α1 + δ + γ )(Rs − 1) < 0, a.s. (22)

Moreover,

limt→∞〈S(t)〉 = A

μ + p(1 − e−μτ2)a.s (23)

and

limt→∞〈Q(t)〉 = 0 a.s. (24)

Proof We have

d(S(t) + I (t) + εe−μτ3

μ + α2 + εQ(t) + pe−μτ1

∫ t

t−τ1

S(s)ds + γ e−μτ2

∫ t

t−τ2

I (s)ds + εe−μτ3

∫ t

t−τ3

Q(s)ds)

=[A − (μ + p(1 − e−μτ1 ))S(t) −

(μ + α1 + γ (1 − e−μτ2 ) + δ − δεe−μτ3

μ + α2 + ε

)I (t)

]dt + σ1SdB1(t)

+ σ2 I dB2(t) + εe−μτ3

μ + α2 + εQ(t)dB3(t)

= [A − (μ + p(1 − e−μτ1 ))S(t)

]dt + σ1S(t)dB1(t) + σ2 I (t)dB2(t) + σ3εe−μτ3

μ + α2 + εQ(t)dB3(t)

123


− (μ + α1 + γ (1 − e−μτ2 ) + δ)(μ + α2) + ε(μ + α1 + γ (1 − e−μτ2 ) + δ(1 − e−μτ3 ))

μ + α2 + εI (t)dt . (25)

Then, we can get

S(t) + I (t) + εe−μτ3

μ+α2+εQ(t) + pe−μτ1

∫ tt−τ1

S(s)ds + γ e−μτ2∫ tt−τ2

I (s)ds + εe−μτ3∫ tt−τ3

Q(s)ds

t

− S(0) + I (0) + εe−μτ3

μ+α2+εQ(0) + pe−μτ1

∫ 0−τ1

S(s)ds + γ e−μτ2∫ 0−τ2

I (s)ds + εe−μτ3∫ 0−τ3

Q(s)ds

t

= A − (μ + p(1 − e−μτ1 ))〈S(t)〉 + σ1

tS(t)dB1(t) + σ2

tI (t)dB2(t) + σ3εe−μτ3

(μ + α2 + ε)tQ(t)dB3(t)

−[

(μ + α1 + γ (1 − e−μτ2 ) + δ)(μ + α2) + ε(μ + α1 + γ (1 − e−μτ2 ) + δ(1 − e−μτ3 ))

μ + α2 + ε

]〈I (t)〉.

Thus

〈S(t)〉 =A −

[(μ+α1+γ (1−e−μτ2 )+δ)(μ+α2)+ε(μ+α1+γ (1−e−μτ2 )+δ(1−e−μτ3 ))

μ+α2+ε

]〈I (t)〉

(μ + p(1 − e−μτ1 ))− φ(t), (26)

where

φ(t) = S(t) + I (t) + εe−μτ3

μ+α2+εQ(t) + pe−μτ1

∫ tt−τ1

S(s)ds + γ e−μτ2∫ tt−τ2

I (s)ds + εe−μτ3∫ tt−τ3

Q(s)ds

(μ + p(1 − e−μτ1 ))t

− S(0) + I (0) + εe−μτ3

μ+α2+εQ(0) + pe−μτ1

∫ 0−τ1

S(s)ds + γ e−μτ2∫ 0−τ2

I (s)ds + εe−μτ3∫ 0−τ3

Q(s)ds

(μ + p(1 − e−μτ1 ))t

+ 1

(μ + p(1 − e−μτ1 ))

(σ1

tS(t)dB1(t) + σ2

tI (t)dB2(t) + σ3εe−μτ3

(μ + α2 + ε)tQ(t)dB3(t)

). (27)

From the strong law of large numbers and lemma 2,we have

limt→∞ φ(t) = 0 a.s. (28)

Applying Itô’s formula, we get

d log(I (t)) =[βS(t) −

(μ + α1 + δ + γ + σ 2

2

2

)]dt + σ2dB2(t). (29)

From (26) and (29) we get

log I (t)

t≤ βA

μ + p(1 − e−μτ1)−

(μ + α1 + δ + γ + σ 2

2

2

)

−βφ(t) + σ2B2(t)

t+ log(I (0))

t.

123


By the law of large number for martingales [21] and (28), it follows that for Rs < 1,we obtain

lim supt→∞

log I (t)

t≤ (μ + α1 + δ + γ )(Rs − 1) < 0 a.s,

which leads to

limt→∞ I (t) = 0 a.s. (30)

From (26)

limt→∞〈S(t)〉 = A

μ + p(1 − e−μτ1 )

− (μ + α1 + γ (1 − e−μτ2 ) + δ)(μ + α2) + ε(μ + α1 + γ (1 − e−μτ2 ) + δ(1 − e−μτ3 ))

(μ + α2 + ε)(μ + p(1 − e−μτ1 ))limt→∞〈I (t)〉

= A

μ + p(1 − e−μτ1 )a.s.

From the third equation of the system, it follows that

Q(t) − Q(0)

t= δ〈I (t)〉 − (μ + α2 + ε)〈Q(t)〉 + σ3

t

∫ t

0Q(s)dB3(s).

Thus, it follows from Lemma 2 and (30) that

limt→∞〈Q(t)〉 = 0 a.s.

This finishes the proof.

3.2 Persistence inmean

In this section, to study the persistence of the disease, we will establish sufficientconditions to fulfill the conditions in the definition of persistence in mean in [14] , wealso need the following lemma presented in [14].

Lemma 3 Let g ∈ C([0,∞) × Ω, (0,∞)) and G ∈ C([0,∞) × Ω,R) such thatlimt→∞ G(t)

t = 0 a.s. If for all t ≥ 0

ln g(t) ≥ λ0t − λ

∫ t

0g(s)ds + G(t) a.s.

Then

lim inft→∞ 〈g(t)〉 ≥ λ0

λa.s,

where λ0 ≥ 0 and λ > 0 are two real numbers.

123


Theorem 3 Let (S(t), I (t), Q(t)) be the solution of system (3) with any initial valueS(ξ1) ≥ 0, I (ξ1) ≥ 0 and Q(ξ1) ≥ 0 for all ξ1 ∈ [−τ1, 0), ξ2 ∈ [−τ2, 0) andξ3 ∈ [−τ3, 0) with S(0) > 0, I (0) > 0 and Q(0) > 0. IfRs > 1 then

lim inft→∞ 〈I (t)〉 = I ∗ > 0,

where

I ∗ = (μ + α2 + ε)(μ + p(1 − e−μτ1 ))(μ + α1 + γ + δ)(Rs − 1)

β[(μ + α1 + γ (1 − e−μτ2 ) + δ)(μ + α2) + ε(μ + α1 + γ (1 − e−μτ2 ) + δ(1 − e−μτ3 ))] ,

lim supt→∞

〈S(t)〉 = μ + α1 + γ + δ + σ 222

β

and

lim inft→∞ 〈Q(t)〉 = δ I ∗

(μ + α2 + ε)> 0.

Proof It follows from (26) and (29) that

log I (t) =(

βA

μ + p(1 − e−μτ1 )−

(μ + α1 + γ + δ + σ 2

2

2

))t

− β[(μ + α1 + γ (1 − e−μτ2 ) + δ)(μ + α2) + ε(μ + α1 + γ (1 − e−μτ2 ) + δ(1 − e−μτ3 ))](μ + α2 + ε)(μ + p(1 − e−μτ1 ))

〈I (t)〉t+ σ2B2(t) + log I (0) − βtφ(t).

Therefore, from (28) and lemma 3 , we get

lim inft→∞ 〈I (t)〉 = I ∗. (31)

Thus

lim supt→∞

〈S(t)〉 = μ + α1 + γ + δ + σ 222

βa.s.

From the third equation of the system, we get

Q(t) − Q(0)

t= δ〈I (t)〉 − (μ + α2 + ε)〈Q(t)〉 + σ3

t

∫ t

0Q(s)dB3(s). (32)

By the vertue of lemma 2 and (31)

lim inft→∞ 〈Q(t)〉 = δ

μ + α2 + εlim inft→∞ 〈I (t)〉

= δ I ∗

(μ + α2 + ε)a.s.

123


Therefore the conditions in the definition of persistence in mean [14] are verified. Thiscompletes the proof.

4 The existence of stationary distribution

The ergodicity is one of the most important proprieties which will result that the infec-tious diseasewill survive in a population, whichmeans a relativelyweak characteristic.In the following, wewill give a definition of the stationary distribution of the stochasticdelayed systems. Then, we will discuss the existence of stationary distribution and theergodicity of the delayed stochastic system (3) by constructing a suitable Lyapunovfunctional and using the stochastic Lyapunov analysis methods.

We recall that for each t ≥ 0 and probability measure μ on (C([−τ, 0];Rd+),

M[−τ, 0]), where M[−τ, 0] is the associated Borel σ -algebra in [−τ, 0], considerthe probability measure μPt on (C([−τ, 0];Rd+),M[−τ, 0]) defined by

(μPt )(Δ) =∫

C([−τ,0];Rd+

) Pt (x,Δ)μ(dx), for Δ ∈ M[−τ, 0]. (33)

Definition 1 Stationary Distribution [15]A stationary distribution for (4) is a probability measure π on (C([−τ, 0];Rd+),

M[−τ,0]) such that (π Pt )(Δ) = π(Δ) for all t ≥ 0 and Δ ∈ M[−τ,0].

Theorem 4 LetRs > 1, then for any initial conditions (5), stochastic delayed system(3) admits a stationary distribution π(.), and the solution of system (3) is ergodic.

Proof The diffusion matrix of the stochastic delayed SIQR model (3)

A(S, I , Q) =⎡⎣

σ 21 S

2 0 00 σ 2

2 I2 0

0 0 σ 23 Q

2

⎤⎦ . (34)

Let Γ be any bounded domain in R3+, then there exists a positive constant

L0 = min{σ 21 S

2, σ 22 I

2, σ 23 Q

2, (S, I , Q) ∈ Γ }.

such that

Σ3i, j=1ai j (S, I , Q)ξiξ j = σ 2

1 S2ξ21 + σ 2

2 I2ξ22 + σ 2

3 Q2ξ23

≥ L0|ξ |2, (S, I , Q) ∈ Γ , ξ = (ξ1, ξ2, ξ3) ∈ R3.

This leads to verify the first condition, where the smallest eigenvalue of the diffusionmatrix A(S, I , Q) is bounded away from zero.

123


Next, we construct a C2− function V (Xt , t) with Xt = (S(t), I (t), Q(t)) and aclosed set Uε ∈ R

3+ such that supX∈R3+\UεLV < −M < 0, with M is a positive

constant and

μ − mσ 2

2> 0, (35)

where m is a positive constant.We define a Lyapunov functional as follows

V (Xt , t) = MV1 + V2 + V3, (36)

where

V1 = −(log I + β

u + p(1 − e−μτ1)

(S + I + pe−μτ1

∫ t

t−τ1

S(s)ds

)),

V2 = − log S − log Q,

V3 = 1

m + 1

(S + I + Q + pe−μt

∫ t

t−τ1

eμs S(s)ds

+γ e−μt∫ t

t−τ2

eμs I (s)ds + εe−μt∫ t

t−τ3

eμs Q(s)ds

)m+1

,

and M > 0 is a constant large enough verifying

− M λ + D ≤ −1, (37)

with the terms λ and D defined later.Moreover, V (Xt , t) is a continuous function and have aminimumpoint (S0, I0, Q0)

in the interior of R+3 . Therefore, Let V : R3+ → R+ be a nonnegative function such

that

V = V (S, I , Q) − V (S0, I0, Q0).

Applying Itô’s formula on V1, we obtain

LV1 ≤ − βS +(

μ + α1 + γ + δ + σ 22

2

)

− Aβ

μ + p(1 − e−μτ1)+ βS + β(μ + α1 + γ + δ)

μ + p(1 − e−μτ1)I

:= − (μ + α1 + γ + δ)(Rs − 1) + β(μ + α1 + γ + δ)

μ + p(1 − e−μτ1)I

:= − λ + β(μ + α1 + γ + δ)

μ + p(1 − e−μτ1)I ,

123


where λ = (μ+α1 + γ + δ)(Rs − 1) > 0, Using Itó’s formula on V2 and V3, we get

LV2 = − A

S+ β I +

(μ + p + σ 2

1

2

)− pS(t − τ1)e−μτ1

S(t)

−γ I (t − τ2)e−μτ2

S(t)− εQ(t − τ3)e−μτ1

S(t)

−ε I (t)

Q(t)+ μ + α2 + ε + σ 2

3

2,

LV3 = V m3 (A − μ(S + I + Q) − α1 I − α2Q − μpe−μt

∫ t

t−τ1

eμs S(s)ds

−μγ e−μt∫ t

t−τ2

eμs I (s)ds − μεe−μt∫ t

t−τ3

eμs Q(s)ds)

+m

2V m−13 (σ 2

1 S2 + σ 2

2 I2 + σ 2

3 Q2)

≤ V m3 (A − μV3) + mσ 2

2V m+13

≤ B − 1

2

(μ − mσ 2

2

)(Sm+1 + Im+1 + Qm+1),

where

B = sup(S,I ,Q)∈R3+

{AV m

1 − 1

2

(μ − mσ 2

2

)V m+11 − 1

2

(μ − mσ 2

2

) (pe−μt

∫ t

t−τ1

e−μs S(s)ds

+ γ e−μt∫ t

t−τ2

e−μs I (s)ds + εe−μt∫ t

t−τ3

e−μs Q(s)ds

)}. (38)

Hence

LV ≤ −M λ + Mβ(μ + α1 + δ + γ )

μ + p(1 − e−μτ1)I − A

S− δ I (t)

Q(t)

+B + β I + 2μ + p + α2 + ε + σ 21 + σ 2

3

2

−1

2

[μ − mσ 2

2

](Sm+1 + Im+1 + Qm+1). (39)

Define a bounded set such that

Uε ={(S, I , Q) ∈ R

3+, ε ≤ S ≤ 1

ε, ε2 ≤ I ≤ 1

ε, ε3 ≤ Q ≤ 1

ε3

}, (40)

where 0 < ε < 1 is a sufficiently small and such that

−min(A, γ )

ε+ D < −1, (41)

123


−1

4

[μ − mσ 2

2

]εm+1 + E < −1, (42)

−1

4

[μ − mσ 2

2

]εm+1 + F < −1, (43)

−1

4

[μ − mσ 2

2

]εm+1 + G < −1, (44)

with D, E, F and G are positive constants, where the expressions are introduced afterin the presented cases. Knowing that for a sufficiently small ε . We divideR3+\Uε intosix domains, such that

U1 = {(S, I , Q) ∈ R3+, 0 < S < ε}, U2 = {(S, I , Q) ∈ R

3+, 0 < I < ε},U3 = {(S, I , Q) ∈ R

3+, ε2 < I , 0 < Q < ε3},U4 =

{(S, I , Q) ∈ R

3+, S >1

ε

}, U5 =

{(S, I , Q) ∈ R

3+, I >1

ε

},

U6 ={(S, I , Q) ∈ R

3+, Q >1

ε3

}.

In the following, we should prove that LV ≤ −1 on R3+\Uε, it means verifying it onthe above six domains.

Case 1 If (S, I , Q) ∈ U1, then

LV ≤ − A

S+ M

β(μ + α1 + α + γ )

μ + p(1 − e−μτ1)I − 1

2

[μ − mσ 2

2

](Sm+1 + Im+1 + Qm+1) + B

+β I + 2μ + p + α2 + εσ 21 + σ 2

3

2

≤ − A

ε+ D, (45)

where

D = sup(S,I ,Q)∈R3+

{M

β(μ + α1 + δ + γ )

μ + p(1 − e−μτ1)I − 1

2

[μ − mσ 2

2

]

(Sm+1 + Im+1 + Qm+1) + β I + B + 2μ + p + α2 + ε + σ 21 + σ 3

2

2

}.(46)

According to (41), one can get that for a sufficiently small ε

LV ≤ −1 for any (S, I , Q) ∈ U1. (47)

123


Case 2 If (S, I , Q) ∈ U2, we get

LV ≤ −Mλ + Mβ(μ + α1 + α + γ )

μ + p(1 − e−μτ1)I − 1

2

[μ − mσ 2

2

](Sm+1 + Im+1 + Qm+1)

+B + β I + 2μ + p + α2 + ε + σ 21 + σ 2

3

2≤ −M λ + D. (48)

According to (37), one can get that for a sufficiently small ε

LV ≤ −1 for any (S, I , Q) ∈ U2. (49)

Case 3 When (S, I , Q) ∈ U3, it follows that

LV ≤ −δ I

Q+ M

β(μ + α1 + δ + γ )

μ + p(1 − e−μτ1)I − 1

2

[μ − mσ 2

2

](Sm+1 + Im+1 + Qm+1)

+β I + B + 2μ + p + α2 + ε + σ 21 + σ 2

3

2

≤ − δ

ε+ D. (50)

By (41), we conclude that LV ≤ −1 on U3Case 4 When (S, I , Q) ∈ U4, we get

LV ≤ −1

4

[μ − mσ 2

2

]Sm+1 + E

≤ −1

4

[μ − mσ 2

2

]εm+1 + E, (51)

where

E = sup(S,I ,Q)∈R3+

{−1

4

[μ − mσ 2

2

]Sm+1 + Mβ(μ + α1 + δ + γ )

μ + p(1 − e−μτ1)I + β I

+B + 2μ + p + α2 + ε − 1

2

[μ − mσ 2

2

](Im+1 + Qm+1) + σ 2

1 + σ 23

2

}.

(52)

By virtue of (42), we get that LV ≤ −1 for all (S, I , Q) ∈ U4Case 5When (S, I , Q) ∈ U5, we get

LV ≤ −1

4

[μ − mσ 2

2

]Im+1 + F

123


≤ −1

4

[μ − mσ 2

2

]εm+1 + F, (53)

where

F = sup(S,I ,Q)∈R3+

{−1

4

[μ − mσ 2

2

]Im+1 + Mβ(μ + α1 + δ + γ )

μ + p(1 − e−μτ1)I + β I

+ B + 2μ + p + α2 + ε − 1

2

[μ − mσ 2

2

](Sm+1 + Qm+1) + σ 2

1 + σ 23

2

}.

(54)

By virtue of (43), we get that LV ≤ −1 for all (S, I , Q) ∈ U5Case 6 When (S, I , Q) ∈ U6, we get

LV ≤ −1

4

[μ − mσ 2

2

]Qm+1 + F

≤ −1

4

[μ − mσ 2

2

]ε3m+3 + F, (55)

where

F = sup(S,I ,Q)∈R3+

{−1

4

[μ − mσ 2

2

]Qm+1 + Mβ(μ + α1 + δ + γ )

μ + p(1 − e−μτ1)I + β I

+ B + 2μ + p + α2 + ε − 1

2

[μ − mσ 2

2

](Sm+1 + Im+1) + σ 2

1 + σ 23

2

}.

(56)

It follows from (44), we get that LV ≤ −1 for all (S, I , Q) ∈ U6. Hence, from (45),(48), (50), (51), (53) and (55), we get that for a sufficiently small ε

LV (S, I , Q) ≤ −1 for all (S, I , Q) ∈ R3+\Uε .

This means that if the solution (S, I , Q) ∈ R3+\Uε of the delayed stochastic epidemic

model (3), the mean time τx at which a path issuing from Xt reaches the setU is finite,and supX∈K Exτ < ∞ for every compact K ⊂ R

3+.In addition, Theorem 1 shows that the stochastic epidemic model has a unique

global positive solution and by the vertue of Theorem 3.9 in [21]. The solution ofthe stochastic delayed system (3) is bounded. Therefore, according to [15], theseproperties imply that Theorem 2.2.1 is verified. Hence, we can obtain that the delayedstochastic system (3) is ergodic and admits a unique stationary distribution.

123


5 Numerical simulations

In order to simulate our theoretical results for Theorems 2 and 3, we illustrate thepaths of the delayed deterministic epidemicmodel (1) and delayed stochastic epidemicmodel (3) using the Euler–Maruyama method to investigate numerically the resultson the stochastic thresholdRs .

Example 1 In this simulated example, choosing the initial value (S(0), I (0), Q(0)) =(5, 15, 5), and the parameters values are A = 1, μ = 0.09, β = 0.18, γ =0.55, δ = 0.44, ε = 0.6, p = 0.2, σ1 = 0.2, σ2 = 0.85, σ3 = 0.5, α1 = 0.4, α2 =0.02, and τ1 = τ2 = τ3 = 0.5. We can calculate easily the basic reproduction rateR0 = 1.2309 and the stochastic thresholdRs = 0.9868 < 1. According to Theorem 2the disease will go to extinction. In Fig. 1, the extinction of the disease is well observedin the illustration of the delayed stochastic system trajectories.

Example 2 In this simulated example, choosing the initial value (S(0), I (0), Q(0)) =(10, 0.1, 0.1), we keep the same parameters as Example 1 and we change μ =0.09, β = 0.39, γ = 0.55, σ2 = σ3 = 0.4. We alleviate the quarantine strategyby reducing the quarantined individuals rate to ε = 0.3. We can calculate easily thebasic reproduction rate R0 = 2.9679 and the stochastic threshold Rs = 2.9077 > 1.As observed in Fig. 2 the disease will persist in mean which support the conclusionof Theorem 3.

0 2000 4000 6000 8000 100002468

10121416

S(t)

Susceptibles

0 2000 4000 6000 8000 100000

5

10

15

20

I(t)

Infected

0 2000 4000 6000 8000 1000002468

10

Q(t)

Quarantine

DeterministicStochastic

Fig. 1 Trajectories of stochastic and deterministic systems for example 1

123


0 2000 4000 6000 8000 1000002468

101214

S(t)

Susceptibles

0 2000 4000 6000 8000 100000

2

4

6

8

I(t)

Infected

0 2000 4000 6000 8000 100000

0.51

1.52

2.53

Q(t)

Quarantine

DeterministicStochastic

Fig. 2 Trajectories of stochastic and deterministic systems for Example 2

6 Conclusion

Since the quarantine has proven significant results in the control of the infectiousdiseases in a population such as the case of the recent pandemic COVID19 case.Therefore, we investigated in this work the behaviour of a delayed stochastic epidemicmodel with isolation, vaccination, elimination, temporary immunity.

In fact, we have studied the dynamics of a SIQR epidemic model including thenotion of delay to describe the time efficiency of vaccine proposed in [9], the temporaryloss of immunity [1] and the temporary loss of immunity after a quarantine is illustratedusing the delay for quarantined individuals.Wepresent a stochastic thresholdRs whichis used to establish a sufficient condition for the extinction, persistence in mean andthe existence of stationary distribution.

Acknowledgements The authors are very grateful to the Editor and the Reviewers for their helpful andconstructive comments and suggestions. The authors are also thankful to the laboratoryMAD (Managementde l’ agriculture Durable) of EST Sidi Bannour, the Faculty of sciences, Ibn Tofail University, Kenitra andLinnaeus University, V ax j o for their help and support.

123


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a stochastic analysis for a triple delayed siqr epidemic

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