Stochastic inequalities and applications to dynamics analysis of a novel SIVS epidemic model with jumps

This paper proposes a new nonlinear stochastic SIVS epidemic model with double epidemic hypothesis and Lévy jumps. The main purpose of this paper is to investigate the threshold dynamics of the stochastic SIVS epidemic model. By using the technique of a series of stochastic inequalities, we obtain sufficient conditions for the persistence in mean and extinction of the stochastic system and the threshold which governs the extinction and the spread of the epidemic diseases. Finally, this paper describes the results of numerical simulations investigating the dynamical effects of stochastic disturbance. Our results significantly improve and generalize the corresponding results in recent literatures. The developed theoretical methods and stochastic inequalities technique can be used to investigate the high-dimensional nonlinear stochastic differential systems.


Introduction
Mathematical inequalities are widely used in many fields of mathematical analysis, especially differential systems [-]. Recently, the inequality technique was applied to stochastic differential systems [-], impulsive differential systems [-], and impulsive stochastic differential systems [], thus some new results have been obtained.
As an important factor threatening the safety of human life and property, the investiga- () In system (), βS(t) represents the number of people infected by a patient within a unit time at t. But in reality, the number of people who can be exposed to a patient at a time is limited. To this end, some authors have introduced a saturated infection rate to study the dynamic behavior of the disease [-]. In addition, all creatures on the earth are infected by a variety of environmental noises, of course, the disease is no exception. Motivated by this, some scholars have studied the infection system with environmental noises (such as Brownion noise, Markov noise and Lévy noise) [-]. Meanwhile, populations may be affected by different kinds of infectious diseases at the same time. Therefore, it is of great significance to study the epidemic model with multiple diseases [-].
Recently, Meng et al.
[] considered a novel nonlinear stochastic SIS epidemic model with double epidemic hypothesis as follows: a  +I  (t) -(u + α  + r  )I  (t)) dt + σ  S(t)I  (t) a  +I  (t) dB  (t). () They obtained the threshold of system () for the extinction and the persistence in mean of the epidemic diseases. Based on system (), recently, Zhang et al. [] proposed an SIS system with double epidemic diseases driven by Lévy jumps as follows: a  +I  (t) -β  S(t)I  (t) a  +I  (t) + r  I  (t) + r  I  (t)) dt + σ  S(t) dB  (t) + Z γ  (u)S(t -) N(dt, du), dI  = ( β  S(t)I  (t) a  +I  (t) -(u + α  + r  )I  (t)) dt In model (), the authors discussed in detail the conditions for persistence in mean and extinction of each epidemic disease. Therefore, they discussed the persistence in mean of susceptible individuals under different conditions. The above two studies provide a theoretical basis for the study of infectious diseases. But they just discussed the persistence in mean and extinction of epidemic diseases under different conditions. In real life, however, when an epidemic outbreak occurs, we do not sit idly but take measures to control the spread of the epidemic disease. There are many ways to suppress the spread of a disease, for instance, cut off transmission routes, pay attention to food hygiene, vaccination and so on [, ]. Vaccination is an effective method of preventing infectious diseases and many scientists have explored the effect of vaccination on diseases [-].
Motivated by the above works, in this paper, we propose a stochastic SIVS model with double epidemic diseases and Lévy jumps under vaccination as follows: where S(t), I  (t), I  (t), V (t), respectively, stand for the density of susceptible, infective A, infective B and vaccinated individuals at time t, is a constant input of new numbers into the population, q means a fraction of vaccinated for the newborn, β i is the infection rate coefficient from I i (t) (i = , ) to S(t), respectively. u represents the natural death rate of S(t), I  (t), I  (t), V (t), p is the proportional coefficient of vaccinated for the susceptible, r i , d i is the recovery rate and disease-caused death rate of I i (t), i = , , respectively. δ stands for the rate of losing their immunity for vaccinated individuals, α  and α  are the so-called half-saturation constants, respectively. B(t) = (B  (t), B  (t), B  (t), B  (t)) is a standard Brownian motion with intensity σ i >  (i = , , , ). Throughout this paper, let ( , F, {F } t≥ , P) be a complete probability space with a filtration {F t } t≥ satisfying the usual conditions (i.e. it is increasing and right continuous while F  contains all P-null sets). Function B i (t) (i = , , , ) is a Brownian motion defined on the complete probability space , the intensity of B i (t) is σ i (i = , , , ). N(dt, du) = N(dt, du)λ(du) dt, N is a Poisson counting measure on (, +∞) × Z, λ is the characteristic measure of N on a measurable subset Z, λ(Z) < +∞, γ i (i = , , , ) is bounded and continuous with respect to λ and is B(Z) × F t -measurable. For an integrable function The main purpose of this paper is to investigate the threshold dynamics of the stochastic SIVS epidemic model. In this paper, by using the Lyapunov method and the technique of a series of stochastic inequalities, we obtain sufficient conditions for the persistence in mean and extinction of the stochastic system and the threshold which governs the extinction and the spread of the epidemic diseases. Our results significantly improve and generalize the corresponding results in recent literatures. The developed theoretical methods and stochastic inequalities technique can be used to investigate the high-dimensional nonlinear stochastic differential systems. In Section , we firstly give some lemmas and recall some necessary notations and definitions. Furthermore, we obtain the main results for stochastic disease-free dynamics and stochastic endemic dynamics which imply the extinction and the spread of the epidemic diseases. Finally, this paper gives the conclusions and numerical simulations investigating the dynamical effects of stochastic disturbance.

Main results
The main purpose of this paper is to investigate the threshold dynamics of the stochastic SIVS epidemic model. In this section, by using the technique of a series of stochastic inequalities, we obtain sufficient conditions for the persistence in mean and extinction of the stochastic system and the threshold which governs the extinction and the spread of epidemic diseases.

Preliminary knowledge
For the sake of notational simplicity, we define Throughout this paper, suppose that the following two assumptions hold.
Assumption . The following hold: Remark . This assumption means that the intensities of Lévy noises are not infinite.
Assumption . Suppose that there exists some >  such that the following inequality holds: The species X(t) is said to be persistent in mean if lim t→+∞ X(t) * > .
The following elementary inequality will be used frequently in the sequel.
Lemma . (Burkholder-Davis-Gundy inequality []) Let g ∈ L  (R + ; R d×m ). For any t ≥ , define Then, for every p > , there exist two positive constants c p , C p such that where c p , C p only depend on p.

Lemma . (Hölder inequality [])
For any a i , b i ∈ R and k ≥ , if p, q >  and  p +  q = , the following inequality holds:

Lemma . (Doob's martingale inequality []) Let X be a submartingale taking nonnegative real values, either in discrete or continuous time. That is, for all times s and t with s < t,
Then, for any constant C > , where P denotes the probability measure on the sample space of the stochastic process X : [, T] × → [, +∞) and E denotes the expected value with respect to the probability measure P.

Lemma . ([, ]) Assume that X(t) ∈ R + is an Itô's-Lévy process of the form
Then the generalized Itô's formula with Lévy jumps is given by We have the following conclusions.
Proof Define Applying the generalized Itô's formula to Q(X), we have where Choose a positive constant >  that satisfies For any constant k satisfying k ∈ (, b ), one has Integrating from  to t and taking expectation on both sides of (), we have Easily, one has By Lemma ., applying the Burkholder-Davis-Gundy inequality, integrating equation () from  to t, and for an arbitrarily small positive constant δ, one has Choose a positive constant δ that satisfies Combining it with equation (), one has Applying the arbitrariness of κ X >  and Lemma . for Chebyshev's inequality, one obtains holds for all but finitely many k. Therefore, for any positive constant k ≥ k  and almost all ω ∈ , there is k  (ω) such that equation () holds. Thus, for almost all ω ∈ , once conditions k ≥ k  and kδ ≤ t ≤ (k + )δ hold, then we have Taking the limit superior on both sides of equation () and applying the arbitrariness of Easily, for any satisfying That is to say, for any constant τ satisfying  < τ <  - , there is a constant N = N(ω), and once condition t ≥ N holds, then we have This completes the proof.

Lemma . For any initial value
) of model () has the following property: Applying Lemma . for the Burkholder-Davis-Gundy inequality and Lemma . for Hölder's inequality, one has For any constant κ X  > , applying Lemma . for Doob's martingale inequality, one obtains Applying the Borel-Cantelli lemma, one has Taking the limit superior on both sides of equation () and applying the arbitrariness of κ X > , one has That is to say, for any constant τ satisfying  < τ <   - , there is a constant N = N(ω), and once t ≥ N , w ∈ τ holds, then we have Dividing both sides of equation () by t and taking the limit superior, we have Combining it with lim inf t→∞ Similarly, one obtains This completes the proof.

Lemma . For any initial value
Proof The proof is similar to Refs. [, ] by defining Q(S, I  , I  , V ) = S - -ln S + I  -ln I  + I  - -ln I  + V - -ln V , and hence is omitted.

Stochastic disease-free dynamics
Theorem . Suppose that conditions R  <  and R  <  hold. Then, for any initial value (S(), I  (), I  (), V ()) ∈ R  + , the solution (S(t), I  (t), I  (t), V (t)) of model () has the following property: That is to say, the two epidemic diseases go to extinct almost surely.
Proof By equation (), one has Dividing both sides of equation () by t and integrating over the time interval  to t yield Applying Lemmas . and ., we obtain that lim t→+∞ (t) =  a.s. () Applying the generalized Itô's formula in Lemma . to α  ln I  (t) + I  (t) yields Dividing both sides of equation () by t, integrating over the time interval  to t and taking the limit, one obtains that Combining equations () and (), one obtains where Similarly, applying the generalized Itô's formula in Lemma . to α  ln I  (t) + I  (t) yields where Applying Lemmas . and ., we obtain that That is to say, By equation (), one has Dividing both sides of equation () by t, integrating over the time interval t =  to t and taking the limit, one obtains that Applying (), (), Lemmas . and ., we have This completes the proof.

Stochastic endemic dynamics
Theorem . For any initial value (S(), I  (), I  (), V ()) ∈ R  + , the solution (S(t), I  (t), I  (t), V (t)) of model () has the following property: (i) If R  >  and R  < , then the epidemic disease I  (t) is persistent in mean and I  (t) goes extinct, i.e. lim t→∞ I  (t) = R  ϒ  > , lim t→∞ I  (t) =  a.s. Moreover, (ii) If R  <  and R  > , then the epidemic disease I  (t) goes extinct and Proof Case (i): From equation () we have where From Theorem ., when R  <  one has Therefore, there exists an arbitrarily small constant ε >  such that when t is large enough, we have I  (t) < ε. Applying this into equation () leads to Applying Lemma . and the arbitrariness of ε, we obtain Applying (), (), (), Lemmas . and . into equation (), we have where From Theorem ., when R  <  one has Therefore, there exists an arbitrarily small constant ε >  such that when t is large enough, we have I  (t) < ε. Applying this into equation () leads to Applying Lemma . and the arbitrariness of ε, we obtain Applying equations (), (), () into equation (), we obtain that Applying (), (), (), Lemmas . and . into equation (), we have This completes the proof.

Theorem . Suppose that conditions R  >  and R
there exists an arbitrarily small constant ε >  such that when t is large enough, we have From equation () and equation (), when t is large enough, one has () Since ϒ  R  < ϒ  R  and ϒ  ϒ  > ϒ  ϒ  , taking the limit superior of both sides of equation (), applying equation () and the arbitrariness of ε, we have That is to say, From equation (), there exists an arbitrarily small constant ε >  such that when t is large enough, we have Applying equation () into equation (), one obtains that By using Lemma . and the arbitrariness of ε, we obtain that Similarly, one obtains Applying equations (), (), () and () leads to Applying () and () into equation (), we obtain that Applying (), (), Lemmas . and . into equation (), we have This completes the proof.

Conclusions and numerical simulations
In this paper, we propose a novel stochastic epidemic system with double epidemic diseases under vaccination. By using stochastic differential equation theory, we study the persistence in mean and extinction of the two diseases.
That is to say, the two epidemic diseases go to extinct almost surely. II. Stochastic endemic dynamics (i) If one of the following conditions holds: That is to say, the epidemic disease I  (t) is persistent in mean and I  (t) is extinct. (ii) If one of the following conditions hold: That is to say, the epidemic disease I  (t) is extinct and I  (t) is persistent in mean.
That is to say, the epidemic diseases I  and I  are persistent in mean.
In [, ], Meng and Chang et al. obtained the lower boundedness of the persistence in mean for I  and I  as follows: where m * is a positive constant. However, this paper proves that I  and I  have their own limit, that is, Thus this paper contains and significantly improves the results for persistence in mean in [, ]. The developed theoretical methods can be used to investigate the high-dimensional nonlinear stochastic differential systems.