Stochastic regime switching SIS epidemic model with vaccination driven by Lévy noise

Advances in Difference Equations, Dec 2017

We formulate a stochastic SIS epidemic model with vaccination by introducing a Lévy noise and regime switching into the epidemic model. First, we prove that the stochastic model admits a unique global positive solution. Moreover, we study the asymptotic behavior of the stochastic regime switching SIS model with vaccination driven by Lévy noise.

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Stochastic regime switching SIS epidemic model with vaccination driven by Lévy noise

Guo Advances in Difference Equations Stochastic regime switching SIS epidemic model with vaccination driven by Lévy noise Yingjia Guo We formulate a stochastic SIS epidemic model with vaccination by introducing a Lévy noise and regime switching into the epidemic model. First, we prove that the stochastic model admits a unique global positive solution. Moreover, we study the asymptotic behavior of the stochastic regime switching SIS model with vaccination driven by Lévy noise. stochastic SIS epidemic model; vaccination; Lévy noise; Markov switching; global positive solution; asymptotic behavior - between compartments S and I, μ is the natural death rate per capita, p represents the proportional coefficient of vaccinated for the susceptible, γ is the recovery rate, ε is the rate of losing immunity for vaccinated individuals, and α represents the disease-caused death rate of infectious individuals. These parameter values are all nonnegative, and μ, A > . Li and Ma [] analyzed the thresholds, equilibria, and stabilities of the epidemic model (.) of SIS type with vaccination. For system (.), there exists the basic reproduction number R. The asymptotic behavior is globally asymptotically stable convergence to a disease-free equilibrium P(S, I, V) below the threshold R. Otherwise, P is unstable when R > , and there is an endemic equilibrium P∗(S∗, I∗, V ∗), which is globally asymptotically stable. Because of full randomness and stochasticity in real life, many studies have indicated that environmental fluctuations have a huge impact on the transmission of an epidemic [, ]. Zhao and Jiang [] considered the following stochastically perturbed SIS epidemic model with vaccination: ⎧⎪ dS(t) = [( – q)A – βI(t)S(t) – (μ + p)S(t) + γ I(t) + εV (t)] dt ⎪⎪⎪⎨⎪ + σS(t) dB(t), ⎪⎪⎪ dI(t) = [βS(t) – (μ + γ + α)]I(t) dt + σI(t) dB(t), ⎪⎩⎪ dV (t) = [qA + pS(t) – (μ + ε)V (t)] dt + σV (t) dB(t), (.) where Bi(t) (i = , , ) are independent Brownian motions, and σi (i = , , ) are their intensities. They showed that when the perturbations and the disease-related death rate α are small, there is a stationary distribution, and it is ergodic as the reproductive number of the deterministic model R > . If R ≤ , then the solution of model (.) is oscillating around the disease-free equilibrium P. Furthermore, the population may suffer sudden environmental shocks and catastrophes such as climate charges (earthquakes, hurricanes, etc.) and unpredictable disasters. These phenomena cannot be modeled by stochastic continuous models. Bao et al. suggested that the non-Gaussian Lévy noise should be suitable for describing these phenomena [–]. They considered stochastic Lotka-Volterra population systems with jumps [] for the first time, and then some important results that reveal that jump processes can bring their effect on the properties of the systems have been reported [–]. There are also many results on the epidemic models with jumps [–]. Chen and Kang [] introduced a Lévy noise into the multistrain SIS epidemic model and investigated its effects on the spread of infectious disease with multiple pathogen strains. However, epidemic models may be perturbed by colored noise, which can cause the system switching from one environmental regime to another []. For example, the transmission rate in winter will be much different from that in summer. Often, the switching between environmental regimes is often memoryless, and the waiting time for the next switching follows the exponential distribution []. Thus, we use a continue-time Markov chain r(t) to model random environments with colored noise. Let r(t) (t ≥ ) be a rightcontinuous Markov chain on the probability space ( , F, P) taking values in a finite state space M = {, , . . . , N } with generator = (γij)N×N , that is, P r(t + δ) = j|r(t) = i = ⎧⎨ γijδ + o(δ) if i = j, ⎩  + γijδ + o(δ) if i = j, where γij ≥  is the transition rate from i to j with i = j whereas γii = – j=i γij. We assume that the Markov chain and Brownian motion are independent. In this paper, we set up a stochastic regime switching SIS model with vaccination driven by a Lévy noise: ⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎧⎪ dS(t) = +–[(σ(μ–((rqr(((ttr))())tS)+()t)p)A(d(rBr((tt)())t)))S–+(tβ)U(+r(Dγt)()(rIr(((ttt))))S,Iu((tt)))S+(t)εN(r((dt)t),Vd u(t)),] dt ⎨ dI(t) = [β(r(t))S(t) – (μ(r(t)) + γ (r(t)) + α(r(t)))]I(t) dt ⎪⎪⎪⎪⎪⎪ + σ(r(t))I(t) dB(t) + U D(r(t), u)I(t)N (dt, du), ⎪⎪⎩⎪⎪⎪⎪⎪⎪ dV (t) = +[qσ(r((tr)()tA))(Vr((tt)))d+Bp((rt()t+))SU(t)D–((rμ(t()r,(ut)))V+(tε)(Nr((td))t),Vd u(t)),] dt where Di(r(t), u) > – (i = , , ), N˜ (dt, du) is the compensated Poisson random measure given by N˜ (dt, du) = N (dt, du) – ν(du) dt, and ν is the characteristic measure of N on a measurable subset U of [, ∞) satisfying ν(du) < ∞. Since stochastic model (.) is perturbed by both Lévy noise and colored noise, its dynamics is an interesting and important question. The paper is organized as follows. In Section , we give some notation and the equivalent form of the studied model. In Section , we study the global positive solution of model (.). In Sections  and , we investigate the asymptotic behavior of the stochastic regime switching SIS model with vaccination driven by a Lévy noise. 2 Preliminaries Here we assume that the Brownian motion and Markov chain are independent. In this paper, we assume that γij >  for i = j, and q(k), A(k), β(k), μ(k), p(k), γ (k), ε(k), and α(k) are all positive constants for each k ∈ M. System (.) can be regarded as the result of a stochastic SIS model with vaccination ⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎧⎪ dS(t) = ++[(γσ–((kkq))(ISk(t()t)))A+d(εBk()k(–)tV)β+((tk))U]Id(Dtt)S(k(t,)u–)S((μt)(Nk)(+dtp, d(ku))),S(t) ⎨ dI(t) = [β(k)S(t) – (μ(k) + γ (k) + α(k))]I(t) dt + σ(k)I(t) dB(t) ⎪⎪⎪⎪⎪⎪ + U D(k, u)I(t)N (dt, du), ⎪⎪⎪⎪⎪⎪⎪⎪⎩ dV (t) = +[q(kU)DA(k(k) ,+up)V(k()tS)N(t)(d–t(,μdu(k),) + ε(k))V (t)] dt + σ(k)V (t) dB(t) switching from one to the others according to the movement of the Markov chain. For the corresponding regime switching SIS model with vaccination of system (.), there exists the disease-free equilibrium Pk(Sk, Ik, Vk) = ( μA((kk)) μμ(k()k()+–εq((kk))+)+p(εk()k) , , μA((kk)) μμ((kk))+qε((kk))++pp((kk)) ) when the threshold Rk ≤  for k ∈ M. Otherwise, there exists an endemic equilibrium Pk∗(Sk∗, Ik∗, Vk∗) such that  – q(k) A(k) = β(k)Ik∗Sk∗ + μ(k) + p(k) Sk∗ – γ (k)Ik∗ – ε(k)Vk∗ (.) (.) and 3 Existence and uniqueness of positive solution Our first concern is whether the solution has a global existence. Moreover, we also consider whether, as a population dynamic model, the value is nonnegative. Therefore, we guarantee the existence of a global positive solution under some assumptions. For the jump diffusion coefficient, we assume that, for each m > , there exists Lm >  such that (H) U |Hi(x, u, k) – Hj(y, u, k)|ν(du) ≤ Lm|x – y| (i = , , , k ∈ M), where H(x, u, k) = D(k, u)S(t), H(x, u, k) = D(k, u)I(t), H(x, u, k) = D(k, u)V (t) with |x| ∨ |y| ≤ m; (H) | log( + Di(k, u))| < ∞ for Di(k, u) > – (i = , , , k ∈ M). Theorem . Let assumptions (H) and (H) hold. Then, for any given initial value (S(), I(), V ()) ∈ R+, there is a unique solution (S(t), I(t), V (t)) of system (.) on t ≥  almost surely, and the solution remains in R+ with probability . Proof Since the drift and the diffusion of system (.) are both locally Lipschitz, for any given initial value (S(), I(), V ()) ∈ R+, there is a unique local solution (S(t), I(t), V (t)) ∈ R+ for any t ∈ [, τe), where τe is the explosion time []. Let η >  be sufficiently large such that τη = inf t ∈ [, τe) : S(t) ∈/  , η , I(t) ∈/ η  , η , or V (t) ∈/ η  , η η . Obviously, τη is increasing as η → ∞, and τ∞ = limη→∞ τη ≤ τe a.s. To show that the solution is global, it suffices to show that τ∞ = ∞ a.s. Consider the following Lyapunov function: W (S, I, V , k) = c(k)(S –  – log S) + c(k)(I –  – log I) + c(k)(V –  – log V ), (.) where ci(k) (i = , , ) are positive constants for all k ∈ M. Set T >  be arbitrary. Then, for any  < t < τη ∧ T , we have dW (S, I, V , k) = LW (S, I, V , k) dt + c(k)σ(k)(S – ) dB(t) + c(k)σ(k)(I – ) dB(t) + c(k)σ(k)(V – ) dB(t) + c(k) D(k, u)S – log  + D(k, u) U + c(k) D(k, u)I – log  + D(k, u) + c(k) D(k, u)V – log  + D(k, u) N˜ (dt, du), (.) where LW (S, I, V , k) ≤ c(k)  – q(k) A(k) + c(k) μ(k) + p(k) + c(k) μ(k) + γ (k) + α(k) + c(k) q(k)A(k) + μ(k) + ε(k) – c(k) – c(k) β(k)SI – c(k) μ(k) + p(k) + c(k)β(k) – c(k)p(k) S – c(k) μ(k) + γ (k) + α(k) – c(k) × γ (k) + β(k) I – c(k) μ(k) + ε(k) – c(k)ε(k) V – c(k)  – q(k) A(k)  S – c(k)γ (k) SI – c(k)ε(k) VS – c(k)q(k)A(k) V – c(k)p(k) VS +  c(k)σ(k)   +  c(k)σ(k) +  c(k)σ(k) + U c(k) D(k, u) – log  + D(k, u) + c(k) D(k, u) – log  + D(k, u) + c(k) D(k, u) – log  + D(k, u) ν(du) Choose + c(k) μ(k) + ε(k) > c(k)ε(k), and c(k) μ(k) + γ (k) + α(k) > c(k) γ (k) + β(k) for k ∈ M. By Assumption (H) and the inequality Di(k, u) – log( + Di(k, u)) ≥  for Di(k, u) > –, we have LW ≤ c(k)  – q(k) A(k) + c(k) μ(k) + p(k) + c(k) μ(k) + γ (k) + α(k)   + c(k) q(k)A(k) + μ(k) + ε(k) +  c(k)σ(k) +  c(k)σ(k)  +  c(k)σ(k) + K + N l= γklV (S, I, V , l), where K = max≤i≤{ U ci(k)(Di(k, u) – log( + Di(k, u)))ν(du)}. Let cˇ = max{ ccii((kl)) :  ≤ i ≤ ,  ≤ l, k ≤ N }. Then, for any l, k ∈ M, we get W (S, I, V , l) ≤ cˇ c(k)(S –  – log S) + c(k)(I –  – log I) + c(k)(V –  – log V ) Therefore N l= = cˇW (S, I, V , k). N |γkl| W (S, I, V , k), and thus where LW (S, I, V , k) := K¯ + γklW (S, I, V , k) K˜ = max K¯ , cˇ N , K¯ = c(k)  – q(k) A(k) + c(k) μ(k) + p(k) + c(k) μ(k) + γ (k) + α(k)    + c(k) q(k)A(k) + μ(k) + ε(k) +  c(k)σ(k) +  c(k)σ(k) +  c(k)σ(k) + K . Integrating both sides of (.) from  to τη ∧ T and taking expectation yield EW S(τη ∧ T ), I(τη ∧ T ), V (τη ∧ T ), r(τη ∧ T )  τη∧T ≥ m≤ii≤n ci(k)(η –  – log η), ci(k) η –  + log η P(τη ≤ T ), where {τη∧T} is the indicator function of {τη ∧ T }. Letting η → ∞ implies P(τ∞ ≤ T ) = . P(τ∞ = ∞) = . By the arbitrariness of T we can see that Thus, the proof is complete. 4 Asymptotic behavior around Pk0 It is clear that Pk(Sk, , Vk) is the solution for the corresponding regime switching SIS model with vaccination of system (.), which is called the disease-free equilibrium. If Rk ≤ , then Pk is globally asymptotically stable. This means that the disease will disappear  after some period of time. Whereas for the stochastic regime switching SIS model with vaccination driven by a Lévy noise, Pk is no longer a disease-free equilibrium. Thus for the solution of system (.), what kind of changes will appear around Pk? In this section, we study the asymptotic behavior around Pk. Theorem . Let (S(t), I(t), V (t)) be the solution of system (.) with any initial value (S(), I(), V ()) ∈ R+. Suppose that Rk ≤  and the following conditions are satisfied: μ(k) (μ(kε)(k+)p(k)) +  >  + a(k) σ(k) +  + a(k) μ(k) + α(k)  + a(k) >  σ(k)  + a(k) +  + a(k) where limt→s∞up t E  t  ≤ M˜ a(k) = M˜ = min μ(k) εμ(k(k)) , μ(k) > σ(k) +  D(k, u)ν(du). U S(s) – Sk + I(s) + V (s) – Vk  ds σ(k) Sk   + a(k) + σ(k) Vk  , (μ(kε)(k+)p(k)) +  –  + a(k) σ(k) –  + a(k) D(k, u)ν(du), U μ(k) + α(k)  + a(k) –  σ(k)  + a(k) –  + a(k) U D(k, u)ν(du), μ(k) – σ(k) –  D(k, u)ν(du) . U Proof First, by the change of variables system (.) can be rewritten as follows: x(t) = S(t) – Sk, y(t) = I(t), z(t) = V (t) – Vk ⎧⎪ dx(t) = [–β(k)(x(t) + Sk)y(t) – (μ(k) + p(k))x(t) + γ (k)y(t) + ε(k)z(t)] dt ⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪ dy(t) = [+βσ(k()kx)((tx)y(t()t)++S(kβ)(dkB)S(kt)–+(μU(kD)+(kγ,(uk))(x+(αt)(+k)S))ky)(Nt)(]ddtt, du), ⎪⎪⎪⎪⎩⎪⎪⎪⎪⎪⎪⎪ dz(t) = ++[pσ(kU)(Dxk()ty()(kt–,)ud(μ)B((zk((t)t))+++εV(kUk))D)Nz(((tdk)],td,udt)yu+()t.σ)N(k(d)(tz, (dtu))+, Vk) dB(t) Consider the C function W (x, y, z, k) = a(k)y + a(k) (x + y) +  (x + y + z), (.) where LW = – a(k) μ(k) + p(k) + μ(k) x – a(k) +  μ(k) + α(k) y – μ(k)z + a(k) β(k)Sk – μ(k) + α(k) + γ (k) y + a(k)β(k) – a(k) μ(k) + α(k) + p(k) – μ(k) + α(k) xy + a(k)ε(k) – μ(k) + α(k) yz + a(k)ε(k) – μ(k) xz + a(k) U + U +  σ(k) a(k) +  x + Sk  +  σ(k) a(k) +  y +  σ(k) z + Vk  (D(k, u)x + D(k, u)y ν(du) (D(k, u)x + D(k, u)y + D(k, u)z ν(du) + – εμ(k(k)) +  U D(k, u)ν(du) x (.) μ(k)  + ε(k) μ(k) ε(k) +  U Sk  Denote aˇ = max{ aaii((kl)) ,  ≤ i ≤ ,  ≤ l, k ≤ N }. Then γklW (x, y, z, l) ≤ aˇ |γkl| W (x, y, z, k) l= := MW (x, y, z, k), LW ≤ – μ(k) (μ(kε)(k+)p(k)) +  –  + εμ(k(k)) σ(k) μ(k) ε(k) +  Integrating both sides of (.) from  to t and taking expectation, we obtain μ(k) (μ(kε)(k+)p(k)) +  –  + εμ(k(k)) σ(k) ≤ W x(), y(), z(), r() – – + – εμ(k(k)) +   εμ(k(k)) +  εμ(k(k)) +  U U U U U D(k, u)ν(du) y + μ(k) – σ(k) –  D(k, u)ν(du) z ds μ(k) + σ(k) Vk  + σ(k)  + ε(k) Sk  t + ME  t W x(s), y(s), z(s), r(s) ds – E t μ(k) (μ(kε)(k+)p(k)) D(k, u)ν(du) x(s) μ(k) μ(k) + α(k) –  σ(k)  + ε(k) D(k, u)ν(du) y + μ(k) – σ(k) –  D(k, u)ν(du) z ds μ(k) + σ(k) Vk  + σ(k)  + ε(k) Sk  t eMt, which implies that lim sup  E t→∞ t  ≤ M˜  t S(s) – Sk  + I(s) + V (s) – Vk  ds σ(k) Vk  + σ(k)  + εμ(k(k)) Sk  , where M˜ are defined as in Theorem .. This completes the proof. Remark . From Theorem . we can see that the solution of system (.) will oscillate around the disease-free equilibrium Pk(Sk, , Vk) under some conditions. The lower the vibration intensity of the Lévy noise, the nearer the solution of the stochastic SIS model (.) to the disease-free equilibrium Pk. Hence the disease will die out. Besides, if Di(k, u) =  (i = , , ), then Theorem . shows the asymptotic behavior of the stochastic regime switching SIS model with vaccination. Then the solution will oscillate around the diseasefree equilibrium where the intensity is relevant to the values of σi(k) (i = , , ). 5 Asymptotic behavior around Pk∗ In this section, we assume that Rk > . Then Pk∗ is the endemic equilibrium of the corresponding regime switching SIS model with vaccination for system (.). But it is no longer an endemic equilibrium of system (.). Similarly, we also expect to find out whether or not the solution goes around Pk∗. We get the following result. Theorem . Let (S(t), I(t), V (t)) be the solution of system (.) with any initial value (S(), I(), V ()) ∈ R+. Suppose that Rk >  and the following conditions are satisfied: μ(k) μ(k) + p(k) + μ(k) >  σ(k)  + b(k) + ε(k) εμ(k(k)) +  μ(k) + p(k) >  σ(k)  + εμ(k(k)) + εμ(k(k)) +  εμ(k(k)) +  U D(k, u)ν(du), U D(k, u)ν(du) and Then where μ(k) >  σ(k) +   U ≤ αα˜((kk)) , α(k) = εμ(k(k)) μ(k) + p(k) + μ(k) –  σ(k)  + b(k)  – b(k) +  U D(k, u)ν(du), α(k) = μ(k) + α(k)  + εμ(k(k)) –  σ(k)  + b(k)  – b(k) +  α(k) = μ(k) –  σ(k) – α(k) = εμ(k(k)) ν(k) + p(k) + μ(k), μ(k) α(k) = μ(k) + α(k)  + ε(k) , U   U α˜ (k) = min αi(k) . ≤i≤ ×  σ(k)  + b(k) + b(k) +    + α(k) μ(k) μ(k) + α(k)  + ε(k)  ×  σ(k)  + b(k) + b(k) +  U U D(k, u) Sk∗  D(k, u)ν(du) I∗  k + αμ((kk))  σ(k) +   U D(k, u)ν(du) Vk∗  +  b(k)σ(k)Ik∗ D(k, u) – log  + D(k, u) ν(du), (.) Proof Define the function W : R+ → R+ by W (S, I, V , k) = b(k) I – Ik∗ – I∗ log k I I∗ k + b(k)  S – S∗ + I – I∗ k k  +   S – S∗ + I – I∗ + V – V ∗ k k k  , where b(k) and b(k) are positive constants to be determined later. Applying Itô’s formula, we get +   + b(k) σ(k)S + σ(k)I +  σ(k)V  +  b(k)σ(k)Ik∗ I – Ik∗  +  σ(k)  + b(k) + b(k) +  U U D(k, u)ν(du) S D(k, u)ν(du) I U N l= – μ(k) V – Vk∗  +  σ(k) +   U  D(k, u)ν(du) V  +  b(k)σ(k)Ik∗ ≤ –α(k) S – αα((kk)) Sk∗  – α(k) I – αα((kk)) Ik∗  – α(k) V – αμ((kk)) Vk∗  + α(k) + γklW (S, I, V , l), where αi(k) (i = , . . . , ) are defined as in Theorem .. Furthermore, we define bˇ = max{ bbii((kl)) , i = , ,  ≤ k, l ≤ N }, and there exists a constant B¯ such that N l= γklW (S, I, V , l) ≤ bˇ |γkl| W (S, I, V , k) N l= := B¯W (S, I, V , k). Integrating both sides of (.) from  to t and taking expectation, we have  ≤ EW S(t), I(t), V (t), r(t)  t LW dτ = W S(), I(), V (), r() – E α(k) S – αα((kk)) Sk∗  + α(k) I – αα((kk)) Ik∗  dτ + α(k)t + B¯E W S(τ ), I(τ ), V (τ ), r(τ ) dτ  t α(k) S – αα((kk)) Sk∗  + α(k) I – αα((kk)) Ik∗  dτ + α(k)t eB¯ t. Therefore, we get where α˜ (k) = min≤i≤{αi(k)}. The theorem is proved. Remark . According to Theorem ., we obtain that the solution of model (.) fluctuates around a certain level relevant to Ek∗( αα((kk)) Sk∗, αα((kk)) Ik∗, αμ((kk)) Vk∗), σi(k) (i = , , ), and Di(k, u) (i = , , ). With the values of σi(k) and Di(k, u) (i = , , ) decreasing, the solution of system (.) will be close to Ek∗, and the difference between (S(t), I(t), V (t)) and Ek∗ will also decrease. Besides, if Di(k, u) =  (i = , , ), Theorem . shows that the solution of the stochastic regime switching SIS model with vaccination will fluctuate around a certain level relevant to Ek∗ and σi(k) (i = , , ). 6 Conclusions We present a stochastic regime switching SIS epidemic model with vaccination driven by a Lévy noise. Based on this model, we analyze the existence and uniqueness of its global positive solution. We also discuss the asymptotic behavior of the solution to this model around the disease-free equilibrium Pk and the endemic equilibrium Pk∗. The asymptotic behavior of solutions to SDEs is very important. From the view of applications, investigating the stability in distribution is a more interesting question in stochastic population systems. Bao et al. [, ] did a pioneering work in this area. After that, many results on the stochastic population models with jumps have been reported [, , –]. Acknowledgements The author thanks Prof. Y Li for his valuable discussion. We would like to thank the editor and anonymous referees for their valuable comments and suggestions on this paper. The work is supported by the Department of Education of Jilin Province 13th Five-Year Plan to support scientific research projects (JJKH20170025KJ). Competing interests The author declares that they have no competing interests. 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Yingjia Guo. Stochastic regime switching SIS epidemic model with vaccination driven by Lévy noise, Advances in Difference Equations, 375,