Developing new techniques for obtaining the threshold of a stochastic SIR epidemic model with 3-dimensional Lévy process
aa r X i v : . [ m a t h . P R ] F e b Developing new techniques for obtaining the threshold of a stochastic SIRepidemic model with 3-dimensional L´evy process
Driss Kiouach ∗ and Yassine Sabbar LPAIS Laboratory, Faculty of Sciences Dhar El Mahraz, Sidi Mohamed Ben Abdellah University, Fez, Morocco.
Abstract
This paper considers the classical SIR epidemic model driven by a multidimensional L´evy jump process.We consecrate to develop a mathematical method to obtain the asymptotic properties of the perturbedmodel. Our method differs from previous approaches by the use of the comparison theorem, mutuallyexclusive possibilities lemma, and some new techniques of the stochastic differential systems. In thisframework, we derive the threshold which can determine the existence of a unique ergodic stationarydistribution or the extinction of the epidemic. Numerical simulations about different perturbations arerealized to confirm the obtained theoretical results.
Keywords:
SIR epidemic model; asymptotic properties; white noise; L´evy jumps; stationary distri-bution; ergodic property.
Mathematics Subject Classification:
1. Introduction
The stochastic systems are largely used in order to describe and control the dissemination of diseasesinto a population [1]. It will continue to be one of the vigorous themes in mathematical biology dueto its significance [2]. The stochastic SIR epidemic model with mass action rate is a standard modelamong many mathematical models that present the first tentative to understand the random transmissionmechanisms of infectious epidemics [3]. Taking the stochastic disturbances into account, the traditionalperturbed SIR epidemic model is described by the following model: dS ( t ) = (cid:0) A − µ S ( t ) − βS ( t ) I ( t ) (cid:1) dt + S ( t ) dH ( t ) ,dI ( t ) = (cid:0) βS ( t ) I ( t ) − ( µ + γ ) I ( t ) (cid:1) dt + I ( t ) dH ( t ) ,dR ( t ) = (cid:0) γI ( t ) − µ R ( t ) (cid:1) dt + R ( t ) dH ( t ) , (1)where H ( t ) = ( H ( t ) , H ( t ) , H ( t )) is a 3-dimensional stochastic process modeling the intensity of randomperturbations of the system. S ( t ) denotes the number of individuals sensitive to the disease, I ( t ) denotesthe number of contagious individuals and R ( t ) denotes the number of recovered individuals with fullimmunity. The positive parameters of the perturbed model (1) are given in the table 1. Before explainingthe aim of our contribution, we first present the following cases:1. Case 1: H ( t ) = 0. The system (1) becomes deterministic which is the object of extensive studies.The equilibrium of (1) is characterized by the basic reproduction number R = βAµ ( µ + γ ) whichis the threshold between the persistence and the extinction of a disease [4]. If R ≤
1, then the ∗ . Corresponding author.E-mail addresses: [email protected] (D. Kiouach), [email protected] (Y. Sabbar) Preprint submitted to . February 24, 2020 ystem (1) has only the disease-free equilibrium P which is globally asymptotically stable; thismeans that the disease will extinct. If R > P will become unstable, therefore there exists aglobally asymptotically stable equilibrium P ∗ ; this means that the disease will persist.Parameters Interpretation A The recruitment rate corresponding to births and immigration. µ The natural mortality rate. β The transmission rate from infected to susceptible individuals. γ The rate of recovering. µ = µ + α The general mortality rate, where α >
Table 1:
Biological meanings of the parameters in model (1).
2. Case 2: H i ( t ) = σ i W i ( t ), ( i = 1 , ,
3) where W i ( t ) ( i = 1 , ,
3) are independent standard Brownianmotions and σ i ( i = 1 , ,
3) are the intensities of environmental white noises [5]. There are numeroussignificant works that analyzed the dynamics of the model (1) with white noises. For instance:(a) In [6], the authors investigate the asymptotic behavior of the model (1) around the disease-freeequilibrium of the deterministic model.(b) In [7], the authors analyze the long-time behavior of the stochastic SIR epidemic model (1).Precisely, they discussed the convergence of densities of the solution in L .3. Case 3: H i ( t ) = σ i W i ( t ) + R t R Z η i ( u ) e N ( dt, du ) where W i ( t ) ( i = 1 , ,
3) are independent Brownianmotions and σ i > i = 1 , ,
3) are their intensities. N is a Poisson counting measure withcompensating martingale e N and characteristic measure ν on a measurable subset Z of (0 , ∞ )satisfying ν ( Z ) < ∞ . W i ( t ) ( i = 1 , ,
3) are independent of N . It assumed that ν is a L´evy measuresuch that e N ( dt, du ) = N ( dt, du ) − ν ( du ) dt . The bounded function η i : Z × Ω → R is B ( Z ) × F t -measurable and continuous with respect to ν . Our work considers the L´evy jumps process case andtreats the following model: dS ( t ) = (cid:0) A − µ S ( t ) − βS ( t ) I ( t ) (cid:1) dt + σ S ( t ) dW ( t ) + R Z η ( u ) S ( t − ) e N ( dt, du ) ,dI ( t ) = (cid:0) βS ( t ) I ( t ) − ( µ + γ ) I ( t ) (cid:1) dt + σ I ( t ) dW ( t ) + R Z η ( u ) I ( t − ) e N ( dt, du ) ,dR ( t ) = (cid:0) γI ( t ) − µ R ( t ) (cid:1) dt + σ R ( t ) dW ( t ) + R Z η ( u ) R ( t − ) e N ( dt, du ) , (2)where S ( t − ), I ( t − ) and R ( t − ) are the left limits of S ( t ), I ( t ) and R ( t ), respectively. The jumpsprocess used to model some unexpected and severe environmental disturbances (tsunami, floods,earthquakes, hurricanes, whirlwinds, etc.) on the disease outbreak.The previous contributions on the dynamic behavior of the model (2) can be summarised as follows:1. In [8], the authors examined how the L´evy noise influences the behavior around the equilibriums.More precisely, they investigated the asymptotic behavior of the model (2) around the disease-freeequilibrium P of the deterministic model as well as the dynamics around the endemic equilibrium P ∗ .2. In [9], the authors proved that the parameter T s = (cid:16) µ + γ (cid:17) − (cid:18) βAµ − σ − Z Z η ( u ) − ln(1 + η ( u )) ν ( du ) (cid:19) is the threshold of the stochastic model (2). More specifically, if T s <
1, the epidemic eventuallyvanishes with probability one; while if T s >
1, the disease persists almost surely.As far as we know, no previous research has investigated the ergodicity of the stochastic system (2).It is of interest to study the long term behavior of the stochastic epidemic model (2) which provides alink between mathematical study, actual diseases, and public health planning. Our contribution aims to2evelop a mathematical method to study the ergodicity of the model (2) as an important asymptoticproperty which means that the stochastic model has a unique stationary distribution that predicts thesurvival of the infected population in the future. Moreover, this work focuses on solving the problemoverlooked by many researchers. For instance, in [10], the authors used the existence of the stationarydistribution of an auxiliary stochastic differential equation for establishing the threshold expression ofthe stochastic chemostat model with L´evy jumps. However, the obtained threshold still unknown due tothe ignorance of the explicit form of the existed stationary distribution. Without using the stationarydistribution of the auxiliary process, we will exploit new techniques in order to obtain the explicit form ofthe threshold which can close the gap left by using the classical method. Further, we employe the Fellerproperty, the mutually exclusive possibilities lemma and the stochastic comparison theorem to prove that T s is the threshold between the existence of the ergodic stationary distribution and the extinction. Itshould be noted that the approach used to prove the ergodicity is different from the Khasminskii methodwidely used in the literature (see for example [11, 12, 13]), and the method used to prove the extinctionis different from that used in [9].Our work is organized as follows. In section 2, we show that there exists a unique global positivesolution to the system (2) with any positive initial value. Under suitable assumptions, the threshold ofthe stochastic model is obtained in section 3. One example is provided to demonstrate our analyticalresults in section 4. Finally, a conclusion is presented to end this paper.
2. Well-posedness of the stochastic model (2)
For the purpose of well analyzing our model (2), it necessary that we make the following standardassumptions:— ( A ) We assume that for a given
K >
0, there exists a constant L K > Z Z | F i ( x, u ) − F i ( y, u ) | ν ( du ) < L K | x − y | , ∀ | x | ∨ | y | ≤ K, where F i ( x, u ) = xη i ( u ) ( i = 1 , , ( A ) ∀ u ∈ Z , we assume that 1 + η i ( u ) >
0, ( i = 1 , ,
3) and R Z (cid:0) η i ( u ) − ln(1 + η i ( u )) (cid:1) ν ( du ) < ∞ .— ( A ) We suppose that exists a constant κ >
0, such that R Z (cid:0) ln(1 + η i ( u )) (cid:1) ν ( du ) ≤ κ < ∞ .— ( A ) We assume that for some p ≥ , χ = µ − (2 p − max { σ , σ } − p ℓ , where ℓ = Z Z (cid:0) (1 + η ( u ) ∨ η ( u )) p − − η ( u ) ∧ η ( u ) (cid:1) ν ( du ) < ∞ . By the assumption ( A ) , the coefficients of the system (2) are locally Lipschitz continuous, then for anyinitial value ( S (0) , I (0) , R (0)) ∈ R there is a unique local solution ( S ( t ) , I ( t ) , R ( t )) on [0 , τ e ), where τ e isthe explosion time. In the following theorem, our goal is to show that the solution is positive and global. Theorem 2.1.
For any initial value ( S (0) , I (0) , R (0)) ∈ R , there exists a unique positive solution ( S ( t ) , I ( t ) , R ( t )) of the system (2) on t ≥ , and the solution will stay in R almost surely.Proof. We prove that τ e = ∞ a.s. Let ǫ > S (0), I (0), R (0) lie withinthe interval (cid:2) ǫ , ǫ (cid:3) . For each integer ǫ ≥ ǫ , we define the following stopping time: τ ǫ = inf (cid:26) t ∈ [0 , τ e ) : min { S ( t ) , I ( t ) , R ( t ) } ≤ ǫ or max { S ( t ) , I ( t ) , R ( t ) } ≥ ǫ (cid:27) . Evidently, τ ǫ is increasing as ǫ → ∞ . Set τ ∞ = lim ǫ →∞ τ ǫ whence τ ∞ ≤ τ e . If we can prove that τ ∞ = ∞ a.s., then τ e = ∞ and the solution ( S ( t ) , I ( t ) , R ( t )) ∈ R for all t ≥ τ ∞ = ∞ a.s. Suppose the opposite, then there is a pair of positive constants T > k ∈ (0 ,
1) such that P { τ ∞ ≤ T } > k . Hence, there is an integer ǫ ≥ ǫ such that P { τ ǫ ≤ T } ≥ k for all ǫ ≥ ǫ . (3)Define a C -function V : R → R + by V ( S, I, R ) = (cid:18) S − m − m ln Sm (cid:19) + ( I − − ln I ) + ( R − − ln R ) , where α > x − − ln x > x > ≤ t ≤ τ ǫ ∧ T , using Itˆo’s formula, we obtain that dV ( S, I, R ) = LV ( S, I, R ) dt + (cid:16) − mS (cid:17) σ SdW ( t ) + (cid:18) − I (cid:19) σ IdW ( t )+ (cid:18) − R (cid:19) σ RdW ( t ) + Z Z η ( u ) S ( t − ) − m ln(1 + η ( u ))+ η ( u ) I ( t − ) − ln(1 + η ( u ))+ η ( u ) R ( t − ) − ln(1 + η ( u )) e N ( dt, du ) , where, LV ( S, I, R ) = A − µ S − mAS + mβI + mµ − ( µ + γ ) I − βS + ( µ + γ ) + γI − µ R − γ IR + µ + mσ σ σ Z Z mη ( u ) − m ln(1 + η ( u ))+ η ( u ) − ln(1 + η ( u ))+ η ( u ) − ln(1 + η ( u )) ν ( du ) . Then LV ( S, I, R ) ≤ A − µ I + mβI + µ + mµ + µ + γ + mσ σ σ Z Z mη ( u ) − m ln(1 + η ( u ))+ η ( u ) − ln(1 + η ( u ))+ η ( u ) − ln(1 + η ( u )) ν ( du ) . Given the fact that x − ln(1 + x ) ≥ x > A ), we define J = Z Z mη ( u ) − m ln(1 + η ( u ))+ η ( u ) − ln(1 + η ( u ))+ η ( u ) − ln(1 + η ( u )) ν ( du ) . To simplify, we choose m = µ β , then we obtain LV ( S, I, R ) ≤ A − µ I + mβI + µ + mµ + µ + γ + mσ σ σ J ≡ J . Therefore, Z τ ǫ ∧ T dV ( S ( t ) , I ( t ) , R ( t )) ≤ Z τ ǫ ∧ T J dt + Z τ ǫ ∧ T Z Z η ( u ) S ( t − ) − m ln(1 + η ( u ))+ η ( u ) I ( t − ) − ln(1 + η ( u ))+ η ( u ) R ( t − ) − ln(1 + η ( u )) e N ( dt, du ) . Taking expectation yields E V ( S ( τ ǫ ∧ T ) , I ( τ ǫ ∧ T ) , R ( τ ǫ ∧ T )) ≤ V ( S (0) , I (0) , R (0)) + J T. ǫ = { τ ǫ ≤ T } for ǫ ≥ ǫ and by (3), P (Ω ǫ ) ≥ k . For ω ∈ Ω ǫ , there is some component of S ( τ ǫ ), I ( τ ǫ ) and R ( τ ǫ ) equals either ǫ or ǫ . Hence, V ( S ( τ ǫ ) , I ( τ ǫ ) , R ( τ ǫ )) is not less than ǫ − − ln ǫ or ǫ − − ln ǫ . That is V ( S ( τ ǫ ) , I ( τ ǫ ) , R ( τ ǫ )) ≥ ( ǫ − − ln ǫ ) ∧ (cid:18) ǫ − − ln 1 ǫ (cid:19) . Consequently, V ( S (0) , I (0) , R (0)) + J T ≥ E ( Ω ǫ V ( S ( τ ǫ , ω ) , I ( τ ǫ , ω ) , R ( τ ǫ , ω ))) ≥ k (cid:18) ( ǫ − − ln ǫ ) ∧ (cid:18) ǫ − − ln 1 ǫ (cid:19)(cid:19) . Extending ǫ to ∞ leads to the contradiction. Thus, τ ∞ = ∞ a.s. which completes the proof of thetheorem.
3. Threshold analysis of the model (2)
The aim of the following theorem is to determine the threshold for the SDE model (2).
Theorem 3.1.
The parameter T s is the threshold of the stochastic model (2). That is to say that:1. If T s > , then the stochastic system (2) admits a unique stationary distribution and it has theergodic property for any initial value ( S (0) , I (0) , R (0)) ∈ R .2. If T s < , then the epidemic dies out exponentially with probability one. Before proving the main theorem, we prepare five useful Lemmas. Consider the following subsystem ( dψ ( t ) = ( A − µ ψ ( t )) dt + σ ψ ( t ) dW ( t ) + R Z η ( u ) ψ ( t − ) ˜ N ( dt, du ) ∀ t > ψ (0) = S (0) > . (4) Lemma 3.2. [14] Let ( S ( t ) , I ( t ) , R ( t )) be the positive solution of the system (2) with any given initialcondition ( S (0) , I (0) , R (0)) ∈ R . Let also ψ ( t ) ∈ R + be the solution of the equation (4) with any giveninitial value ψ (0) = S (0) ∈ R + . Then1. lim t →∞ ψ ( t ) t = 0 , lim t →∞ S ( t ) t = 0 , and lim t →∞ I ( t ) t = 0 a.s.2. lim t →∞ R t R Z η ( u ) ψ ( s − ) e N ( ds, du ) t = 0 , lim t →∞ R t R Z η ( u ) S ( s − ) e N ( ds, du ) t = 0 , lim t →∞ R t R Z η ( u ) I ( s − ) e N ( ds, du ) t = 0 a.s. Lemma 3.3.
Let ψ ( t ) be the solution of the system (4) with an initial value ψ (0) ∈ R + . Then, lim t →∞ t Z t ψ ( s ) ds = Aµ a.s. roof. Integrating from 0 to t on both sides of (4) yields ψ ( t ) − ψ (0) t = A − µ t Z t ψ ( s ) ds + σ t Z t ψ ( s ) dW ( s ) + 1 t Z t Z Z η ( u ) ψ ( s − ) e N ( ds, du ) . Clearly, we can derive that1 t Z t ψ ( s ) ds = Aµ + σ µ t Z t ψ ( s − ) dW ( s ) + 1 µ t Z t Z Z η ( u ) ψ ( s − ) e N ( ds, du ) . According to lemma 3.2 and the large number theorem for martingales, we can easily verify thatlim t →∞ t Z t ψ ( s ) ds = Aµ a.s. Lemma 3.4.
Let ( S ( t ) , I ( t ) , R ( t )) be the solution of (2) with initial value ( S (0) , I (0) , R (0)) ∈ R . Then1. E (cid:0) ( S ( t ) + I ( t )) p ( t ) (cid:1) ≤ ( S (0) + I (0)) p e {− pχ t } + χ χ ;2. lim sup t → + ∞ t R t E (cid:0) ( S ( s ) + I ( s )) p (cid:1) ds ≤ χ χ a.s.where χ = sup x> { Ax p − − χ x p } .Proof. Making use of Itˆo’s lemma, we obtain d ( S ( t ) + I ( t )) p ≤ p [ S ( t ) + I ( t )] p − (cid:0) A − µ S ( t ) − ( µ + α + γ ) I ( t ) (cid:1) dt + p (2 p − S ( t ) + I ( t )] p − ( σ S ( t ) + σ I ( t )) dt + 2 p [ S ( t ) + I ( t )] p − ( σ S ( t ) dW ( t ) + σ I ( t ) dW ( t ))+ Z Z [ S ( t ) + I ( t )] p (cid:0) (1 + η ( u ) ∨ η ( u )) p − − η ( u ) ∧ η ( u ) (cid:1) ν ( du ) dt + Z Z [ S ( t ) + I ( t )] p (cid:0) (1 + η ( u ) ∨ η ( u )) p − η ( u ) ∧ η ( u ) (cid:1) ˜ N ( dt, du ) . Then ≤ p [ S ( t ) + I ( t )] p − n A [ S ( t ) + I ( t )] − (cid:16) µ − (2 p − { σ , σ }− p Z Z (cid:0) (1 + η ( u ) ∨ η ( u )) p − − η ( u ) ∧ η ( u ) (cid:1) ν ( du ) (cid:17) [ S ( t ) + I ( t )] o + 2 p [ S ( t ) + I ( t )] p − ( σ S ( t ) dW ( t ) + σ I ( t ) dW ( t ))+ Z Z [ S ( t ) + I ( t )] p (cid:0) (1 + η ( u ) ∨ η ( u )) p − η ( u ) ∧ η ( u ) (cid:1) ˜ N ( dt, du ) . We choose neatly p ≥ such that χ = µ − (2 p − { σ , σ } − p Z Z (cid:0) (1 + η ( u ) ∨ η ( u )) p − − η ( u ) ∧ η ( u ) (cid:1) ν ( du ) > . Hence d ( S ( t ) + I ( t )) p ≤ p [ S ( t ) + I ( t )] p − n χ − χ S ( t ) + I ( t )] p o dt + 2 p [ S ( t ) + I ( t )] p − ( σ S ( t ) dW ( t ) + σ I ( t ) dW ( t ))+ Z Z [ S ( t ) + I ( t )] p (cid:0) (1 + η ( u ) ∨ η ( u )) p − η ( u ) ∧ η ( u ) (cid:1) ˜ N ( dt, du ) .
6n the other hand, we have d ( S ( t ) + I ( t )) p e pχ t = pχ [ S ( t ) + I ( t )] p e pχ t + e pχ t d ( S ( t ) + I ( t )) p ≤ pχ e pχ t + e pχ t p ( S ( t ) + I ( t )) p − ( σ S ( t ) dW ( t ) + σ I ( t ) dW ( t ))+ Z Z e pχ t [ S ( t ) + I ( t )] p (cid:0) (1 + η ( u ) ∨ η ( u )) p − η ( u ) ∧ η ( u ) (cid:1) ˜ N ( dt, du ) . Then by taking integrations and taking the expectations, we get( S ( t ) + I ( t )) p ≤ ( S (0) + I (0)) p e − pχ t + 2 pχ Z t e pχ ( t − s ) ds ≤ ( S (0) + I (0)) p e − pχ t + 2 χ χ . Obviously, we obtainlim sup t → + ∞ t Z t E ( S ( t ) + I ( t )) p ( u )) du ≤ ( S (0) + I (0)) p lim sup t → + ∞ t Z t e − pχ u du + 2 χ χ = 2 χ χ . Lemma 3.5. [10] Let h ( t ) > , k ( t ) ≥ and G ( t ) be functions on [0 , + ∞ ) , c ≥ and d > be constants,such that lim t →∞ G ( t ) t = 0 and ln h ( t ) ≤ ct + k ( t ) − d Z t h ( s ) ds + G ( t ) . If k ( t ) is a non-decreasing function, then lim sup t →∞ t (cid:18) − k ( t ) + d Z t h ( s ) ds (cid:19) ≤ c. Lemma 3.6 ([15]) . Let X ( t ) ∈ R n be a stochastic Feller process, then either an ergodic probabilitymeasure exists, or lim t →∞ sup ν t Z t Z P ( u, x, Σ) ν ( dx ) du = 0 , for any compact set Σ ∈ R n , (5) where the supremum is taken over all initial distributions ν on R d and P ( t, x, Σ) is the probability for X ( t ) ∈ Σ with X (0) = x ∈ R n .Proof of Theorem 3.1. Similar to the proof of Lemma 3.2. in [16], we briefly verify the Feller property ofthe SDE model (2). The main purpose of the next analysis is to prove that (5) is impossible.Applying Itˆo’s formula gives d ln I ( t ) = (cid:16) βS ( t ) − ( µ + γ ) − σ − Z Z η ( u ) − ln(1 + η ( u )) ν ( du ) (cid:17) dt + σ dW ( t ) + Z Z ln(1 + η ( u )) ˜ N ( dt, du ) . (6)7herefore d n ln I ( t ) − βµ (cid:0) ψ ( t ) − S ( t ) (cid:1)o = (cid:16) βS ( t ) − ( µ + γ ) − σ − Z Z η ( u ) − ln(1 + η ( u )) ν ( du ) (cid:17) dt − βµ (cid:16) − µ ( ψ ( t ) − S ( t )) + βS ( t ) I ( t ) (cid:17) dt + σ dW ( t ) − βµ ( ψ ( t ) − S ( t )) dW ( t ) + Z Z ln(1 + η ( u )) ˜ N ( dt, du ) − βµ Z Z η ( u )( ψ ( t ) − S ( t )) ˜ N ( dt, du ) . Hence d n ln I ( t ) − βµ (cid:0) ψ ( t ) − S ( t ) (cid:1)o = (cid:16) βψ ( t ) − ( µ + γ ) − σ − Z Z η ( u ) − ln(1 + η ( u )) ν ( du ) (cid:17) dt − β S ( t ) I ( t ) µ dt + σ dW ( t ) − βµ ( ψ ( t ) − S ( t )) dW ( t )+ Z Z ln(1 + η ( u )) ˜ N ( dt, du ) − βµ Z Z η ( u )( ψ ( t ) − S ( t )) ˜ N ( dt, du ) . (7)Integrating from 0 to t on both sides of (7) yieldsln I ( t ) I (0) − βµ ( ψ ( t ) − S ( t )) + βµ ( ψ (0) − S (0))= Z t βψ ( s ) ds − (cid:16) ( µ + γ ) + σ Z Z η ( u ) − ln(1 + η ( u )) ν ( du ) (cid:17) − β µ Z t S ( s ) I ( s ) ds + σ W ( t ) − βµ Z t ( ψ ( s ) − S ( s )) dW ( s )+ Z t Z Z ln(1 + η ( u )) ˜ N ( ds, du ) − βµ Z t Z Z η ( u )( ψ ( s ) − S ( s )) ˜ N ( ds, du ) . Then we have Z t βS ( s ) I ( s ) ds = µ β Z t βψ ( s ) ds − µ β (cid:16) ( µ + γ ) + σ Z Z η ( u ) − ln(1 + η ( u )) ν ( du ) (cid:17) + ( ψ ( t ) − S ( t )) − ( ψ (0) − S (0)) − µ β ln I ( t ) I (0) + µ σ β W ( t ) − Z t ( ψ ( s ) − S ( s )) dW ( s ) + µ β Z t Z Z ln(1 + η ( u )) ˜ N ( ds, du ) − Z t Z Z η ( u )( ψ ( s ) − S ( s )) ˜ N ( ds, du ) . (8)Let M ( t ) = µ σ β W ( t ) − Z t ( ψ ( s ) − S ( s )) dW ( s )+ µ β Z t Z Z ln(1 + η ( u )) ˜ N ( ds, du ) − Z t Z Z η ( u )( ψ ( s ) − S ( s )) ˜ N ( ds, du ) .
8e know that R t R Z ln(1 + η ( u )) ˜ N ( ds, du ) is a local martingale with quadratic variation D Z t Z Z ln(1 + η ( u )) ˜ N ( ds, du ) , Z t Z Z ln(1 + η ( u )) ˜ N ( ds, du ) E = (cid:16) Z Z (cid:0) ln(1 + η ( u ) (cid:1) ν ( du ) (cid:17) t. By using the Strong Low of Large Numbers, we get lim t → + ∞ M ( t ) t = 0, a.s.From the system (2), we obtain d ( S ( t ) + I ( t )) = (cid:0) A − µ S ( t ) − ( µ + γ ) I ( t ) (cid:1) dt + σ S ( t ) dW ( t )+ σ I ( t ) dW ( t ) + Z Z (cid:0) η ( u ) S ( t − ) + η ( u ) I ( t − ) (cid:1) ˜ N ( ds, du ) . (9)Applying Itˆo’s formula to the equality (9) gives that d ln (cid:18) S ( t ) + I ( t ) (cid:19) = − AS ( t ) + I ( t ) + µ S ( t ) + ( µ + γ ) I ( t ) S ( t ) + I ( t ) + σ S ( t ) + σ I ( t )2( S ( t ) + I ( t )) − Z Z (cid:18) ln (1 + η ( u )) S ( t ) + (1 + η ( u )) I ( t ) S ( t ) + I ( t ) − η ( u ) S ( t ) + η I ( t ) S ( t ) + I ( t ) (cid:19) ν ( du ) − σ S ( t ) S ( t ) + I ( t ) dW ( t ) − σ I ( t ) S ( t ) + I ( t ) dW ( t ) − Z Z ln (1 + η ( u )) S ( t ) + (1 + η ( u )) I ( t ) S ( t ) + I ( t ) ˜ N ( dt, du ) . Taking integration, we getln (cid:18) S ( t ) + I ( t ) (cid:19) = ln (cid:18) S (0) + I (0) (cid:19) − A Z t S ( s ) + I ( s ) ds + M ( t ) + M ( t ) , where M ( t ) = Z t µ S ( s ) + ( µ + γ ) I ( s ) S ( s ) + I ( s ) ds + Z t σ S ( s ) + σ I ( s )2( S ( s ) + I ( s )) ds − Z t Z Z (cid:18) ln (1 + η ( u )) S ( s ) + (1 + η ( u )) I ( s ) S ( s ) + I ( s ) − η ( u ) S ( s ) + η I ( s ) S ( s ) + I ( s ) (cid:19) ν ( du ) ds, and M ( t ) = − Z t σ S ( s ) S ( s ) + I ( s ) dW ( s ) − Z t σ I ( s ) S ( s ) + I ( s ) dW ( s ) − Z t Z Z ln (1 + η ( u )) S ( s ) + (1 + η ( u )) I ( s ) S ( s ) + I ( s ) ˜ N ( ds, du ) . By lemma 3.5, we get lim sup t → + ∞ t (cid:18) Z t AS ( s ) + I ( s ) ds − M ( t ) (cid:19) ≤ t → + ∞ t ln (cid:0) S ( t ) + I ( t ) (cid:1) ≤ t → + ∞ t ln I ( t ) I (0) ≤ lim sup t → + ∞ t ln (cid:0) S ( t ) + I ( t ) (cid:1) I (0) ≤ t → + ∞ t Z t βS ( s ) I ( s ) du ≥ µ β (cid:18) lim inf t → + ∞ t Z t βψ ( s ) ds − (cid:16) ( µ + γ ) + σ Z Z η ( u ) − ln(1 + η ( u )) ν ( du ) (cid:17)(cid:19) = µ β (cid:18) lim t → + ∞ t Z t βψ ( s ) ds − (cid:16) ( µ + γ ) + σ Z Z η ( u ) − ln(1 + η ( u )) ν ( du ) (cid:17)(cid:19) = µ β ( T s − > = { ( S, I, R ) ∈ R | S ≥ ǫ, and , I ≥ ǫ } ,Ω = { ( S, I, R ) ∈ R | S ≤ ǫ } and Ω = { ( S, I, R ) ∈ R | I ≤ ǫ } where ǫ > t → + ∞ t Z t E (cid:16) βS ( s ) I ( s ) Ω (cid:17) ds ≥ lim inf t → + ∞ t Z t E (cid:16) βS ( s ) I ( s ) (cid:17) ds − lim sup t → + ∞ t Z t E (cid:16) βS ( s ) I ( s ) Ω (cid:17) ds − lim sup t → + ∞ t Z t E (cid:16) βS ( s ) I ( s ) Ω (cid:17) ds ≥ µ β ( T s − − βǫ lim sup t → + ∞ t Z t E ( I ( s )) ds − βǫ lim sup t → + ∞ t Z t E ( S ( s )) ds. By lemma 3.4, we see thatlim inf t → + ∞ t Z t E (cid:16) βS ( s ) I ( s ) Ω (cid:17) ds ≥ µ β ( T s − − Aβǫµ − ℓ . We can choose ǫ ≤ µ β A ( µ − ℓ )( T s − t → + ∞ t Z t E (cid:16) βS ( u ) I ( u ) Ω (cid:17) du ≥ µ β ( T s − > q = a > < p = a a − , µ − (2 p − max { σ , σ } − p ℓ > q + p = 1. By utilizing the Young inequality xy ≤ x p p + y q q for all x , y >
0, we getlim inf t → + ∞ t Z t E (cid:0) βS ( u ) I ( u ) Ω (cid:1) du ≤ lim inf t → + ∞ t Z t E (cid:18) p − ( ηβS ( u ) I ( u )) p + q − η − q Ω (cid:19) du ≤ lim inf t → + ∞ t Z t E (cid:0) q − η − q Ω (cid:1) du + p − ( ηβ ) p lim sup t → + ∞ t Z t E (cid:0) ( S ( u ) + I ( u )) p (cid:1) du, where η is a positive constant satisfying η p ≤ pµ χ β − ( p +1) χ ( T s − .
10y lemma 3.4 and (10), we deduce thatlim inf t → + ∞ t Z t E ( Ω ) du ≥ qη q µ β ( T s − − χ η p β p pχ ! ≥ µ qη q β ( T s − > = { ( S, I, R ) ∈ R | S ≥ ζ, or , I ≥ ζ } and Σ = { ( S, I, R ) ∈ R | ǫ ≤ S ≤ ζ, and , ǫ ≤ I ≤ ζ } where ζ > E ( Ω ) ≤ P ( S ( t ) ≥ ζ ) + P ( I ( t ) ≥ ζ ) ≤ ζ E ( S ( t ) + I ( t )) ≤ ζ (cid:18) Aµ − ℓ + (cid:0) S (0) + I (0) (cid:1)(cid:19) . Choosing ζ ≤ µ qη q β ( T s − (cid:16) Aµ − ℓ + (cid:0) S (0) + I (0) (cid:1)(cid:17) − . We thus obtainlim sup t → + ∞ t Z t E ( Ω ) du ≤ µ qη q β ( T s − . According to (11), one can derive thatlim inf t → + ∞ t Z t E ( Σ ) du ≥ lim inf t → + ∞ t Z t E ( Ω ) du − lim sup t → + ∞ t Z t E ( Ω ) du ≥ µ qη q β T s > ⊂ R such thatlim inf t → + ∞ t Z t P ( u, ( S (0) , I (0) , R (0)) , Σ) du ≥ µ qη q β ( T s − > . (12)By (12), we show that (5) is unverifiable. Applying similar arguments to those in [16, 17], we show theuniqueness of the ergodic stationary distribution of our model (2).Now, we will prove that if T s <
1, we have the exticntion of the disease. In view of (6) and lamma3.3 , we get thatlim sup t →∞ t ln I ( t ) I (0) = β lim sup t →∞ Z t S ( s ) ds − (cid:16) ( µ + γ ) + σ Z Z η ( u ) − ln(1 + η ( u )) ν ( du ) (cid:17) ≤ β lim t →∞ Z t ψ ( s ) ds − (cid:16) ( µ + γ ) + σ Z Z η ( u ) − ln(1 + η ( u )) ν ( du ) (cid:17) = ( µ − γ ) (cid:16) T s − (cid:17) <
4. Example
In this section, we will validate our theoretical result with the help of numerical simulations takingparameters from the theoretical data mentioned in the table 2. We numerically simulate the solution ofthe system (2) with initial value ( S (0) , I (0) , R (0)) = (0 . , . , . A The recruitment rate 0.09 µ The natural mortality rate 0.05 β The transmission rate 0.06 γ The recovered rate 0.01 µ The general mortality 0.09
Table 2:
Some theoretical parameter values of the model (2). (700) D e n s i t y I(700) D e n s i t y (a) The left figure is the stationary distribution for S(t), the right picture is the stationary distribution I(t).
R(700) D e n s i t y time S(t)I(t)R(t) (b)
The left figure is the stationary distribution for R(t), the right picture is the trajectory of the solution.
Figure 1:
The numerical illustration of obtained results in the theorem 3.1.
12e have chosen the stochastic fluctuations intensities σ = 0 . σ = 0 .
08 and σ = 0 .
01. Furthermore,we assume that η ( u ) = 0 . η ( u ) = 0 . η ( u ) = 0 . Z = (0 , ∞ ) and ν ( Z ) = 1. Then, T s =1 . >
1. From figure 1, we show the existence of the unique stationary distributions for S ( t ), I ( t ) and R ( t ) of model (2) at t = 700, where the smooth curves are the probability density functions of S ( t ), I ( t )and R ( t ), respectively (see figure 1 (a) and (b)-left). Now, we choose A = 0 .
08, Then, T s = 0 . < I ( t ) will tend to zero exponentially with probability one (see figure 1 (b)-right).
5. Conclusion
The dissemination of the epidemic diseases presents a global issue that concerns decision-makers toelude deaths and deterioration of economies. Many scientists are motivated to understand and suggestthe ways for diminishing the epidemic dissemination. The first generation proposed the deterministicmodels that showed a lack of realism due to the neglecting of environmental perturbations. Recentstudies present a deep understanding of the process of outbreak diseases by taking into account theirrandom aspect. This contribution presents new techniques to analyze the threshold of a stochastic SIRepidemic model with L´evy jumps. We have based on the following new techniques:1. The calculation of the temporary average of a solution of (4) instead of the classic method basedon the explicit form of the stationary distribution in the model (4).2. The use of Feller property and mutually exclusive possibilities lemma for proving the ergodicity ofthe model (2).According to the above techniques, our analysis leads to establish the threshold parameter for the existenceof an ergodic stationary distribution and the extinction of the disease.
ReferencesReferences [1] W. O. Kermack and A. G. McKendrick, “A contribution to the mathematical theory of epidemics,”
Proceedings of The Royal Society A Mathematical Physical and Engineering Sciences , vol. 115,no. 772, pp. 700–721, 1927.[2] E. Beretta, T. Hara, and W. Ma, “Global asymptotic stability of an SIR epidemic model withdistributed time delay,”
Nonlinear Analysis , vol. 47, pp. 4107–4115, 2001.[3] H. Guo, M. Li, and Z. S. Shuai, “Global stability of the endemic equilibrium of multigroup SIRepidemic models,”
Canadian Applied Mathematics Quarterly , vol. 14, pp. 259–284, 2006.[4] M. Roy and R. D. Holt, “Effects of predation on host-pathogen dynamics in SIR models,”
TheoreticalPopulation Biology , vol. 73, pp. 319–331, 2008.[5] L. Allen, “An introduction to stochastic epidemic models,”
Mathematical Epidemiology , vol. 144,pp. 81–130, 2008.[6] C. Ji, D. Jiang, and N. Shi, “Asymptotic behavior of global positive solution to a stochastic SIRmodel,”
Applied Mathematical Modelling , vol. 45, pp. 221–232, 2011.[7] Y. Lin, D. Jiang, and P. Xia, “Long-time behavior of a stochastic SIR model,”
Applied Mathematicsand Computation , vol. 236, pp. 1–9, 2014.[8] X. Zhang and K. Wang, “Stochastic SIR model with jumps,”
Applied Mathematics letters , vol. 826,pp. 867–874, 2013. 139] Y. Zhou and W. Zhang, “Threshold of a stochastic SIR epidemic model with Levy jumps,”
PhysicaA , vol. 446, pp. 204–2016, 2016.[10] D. Zhao, S. Yuan, and H. Liu, “Stochastic dynamics of the delayed chemostat with Levy noises,”
International Journal of Biomathematics , vol. 12, no. 5, 2019.[11] Q. Yang, D. Jiang, N. Shi, and C. Ji, “The ergodicity and extinction of stochastically perturbedSIR and SEIR epidemic models with saturated incidence,”
Journal of Mathematical Analysis andApplications , vol. 388, pp. 248–271, 2012.[12] Y. Zhang, K. Fan, S. Gao, and S. Chen, “A remark on stationary distribution of a stochastic SIRepidemic model with double saturated rates,”
Applied Mathematics Letters , vol. 76, pp. 46–52, 2018.[13] Y. Lin, D. Jiang, and M. Jin, “Stationary distribution of a stochastic SIR model with saturatedincidence and its asymptotic stability,”
Acta Mathematica Scientia , vol. 35, no. 3, pp. 619–629,2015.[14] Y. Zhou, S. Yuan, and D. Zhao, “Threshold behavior of a stochastic SIS model with Levy jumps,”
Discrete Dynamics in Nature and Society , vol. 275, pp. 255–267, 2016.[15] L. Stettner, “On the existence and uniqueness of invariant measure for continuous-time markovprocesses,”
Technical Report, LCDS, Brown University, province, RI , pp. 18–86, 1986.[16] J. Tong, Z. Zhang, and J. Bao, “The stationary distribution of the facultative population model witha degenerate noise,”
Statistics and Probability Letters , vol. 83, no. 14, pp. 655–664, 2013.[17] R. Khasminskii, “Stochastic stability of differential equations,”