Exponential ergodicity of a regime-switching SIS epidemic model with jumps

Applied Mathematics Letters 94 (2019) 133–139

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Applied Mathematics Letters www.elsevier.com/locate/aml

Exponential ergodicity of a regime-switching SIS epidemic model with jumps Yuguo Lin ∗, Yanan Zhao School of Mathematics and Statistics, Beihua University, Jilin 132013, China

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Article history: Received 8 January 2019 Received in revised form 26 February 2019 Accepted 26 February 2019 Available online 5 March 2019

abstract In this paper, we consider a regime-switching SIS epidemic model with jumps. By verifying a Foster–Lyapunov condition, a sufficient condition for the exponential ergodicity is presented. As a by-product, we determine the threshold for the disease to occur or not. © 2019 Elsevier Ltd. All rights reserved.

Keywords: Irreducibility Strong Feller property Exponential ergodicity Extinction Threshold

1. Introduction Zhou and Hethcote [1] proposed a deterministic SIS model with a standard incidence, which takes the form of ⎧ dS(t) βS(t)I(t) ⎪ ⎪ ⎨ dt = Λ − S(t) + I(t) − µS(t) + ϕI(t), (1.1) ⎪ dI(t) βS(t)I(t) ⎪ ⎩ = − (α + µ + ϕ)I(t), dt S(t) + I(t) where S(t) and I(t) denote the number of susceptible and infected individuals at time t respectively; Λ is the rate of susceptible individuals recruited into the population; β is the disease transmission coefficient; µ is the natural death rate; ϕ is the recovery rate; and α is the disease related death rate. According to Vargas-DeLe´ on [2], model (1.1) has two steady states in R2+ : a disease-free steady state E 0 and a unique endemic steady β state E ∗ , and R0 = α+µ+ϕ is the basic reproductive number. If R0 ≤ 1 then E 0 is globally asymptotically stable and if R0 > 1 then E ∗ is globally asymptotically stable. By numerical simulation (Fig. 1, Section 3) we can see that the solution of (1.1) tends to E ∗ very quickly, probably with an exponential convergence rate. However exponential convergence was not theoretically obtained by [1,2]. ∗ Corresponding author. E-mail address: [email protected] (Y. Lin).

https://doi.org/10.1016/j.aml.2019.02.032 0893-9659/© 2019 Elsevier Ltd. All rights reserved.

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Fig. 1. The paths of (1.1). S(0) = 0.8, I(0) = 0.2, Λ = 0.4, β = 0.7, µ = 0.2, α = 0.1, ϕ = 0.2. Clearly R0 = 1.4.

Motivated by this, in this paper we will consider a stochastic analogue of system (1.1). We prove that there exists a threshold Rs such that stochastic system is exponentially ergodic when Rs > 0, and the disease becomes extinct exponentially when Rs < 0. In this sense, our results can provide a deeper insight into the dynamics of SIS epidemic model than those in [1,2]. Up to now, there are lots of literature to study the dynamical behavior of stochastic population models with jumps (see e.g. [3–8]). However exponential ergodicity has not been discussed in those papers. 2. Regime switching SIS model with jumps According to recent literatures environmental noises were usually described by white noise, telegraph noise and L´evy noise. We assume that white noises and jump noises are directly proportional to S(t) and I(t), and telegraph noises are described by a continuous-time Markov chain r(t), t ≥ 0 (see Remark 2.1 for details). Incorporating these noises into (1.1), we can obtain a stochastic SIS epidemic model as follows: ⎧ ( ) β(r(t))S(t)I(t) ⎪ ⎪ − µ(r(t))S(t) + ϕ(r(t))I(t) dt dS(t) = Λ(r(t)) − ⎪ ⎪ ⎪ S(t) + I(t) ⎪ ∫ ⎪ ⎪ ⎪ ⎪ ˜ (dt, dz), ⎪ + σ1 (r(t))S(t)dB1 (t) + C1 (r(t), z)S(t− )N ⎨ Y ( ) (2.1) ⎪ β(r(t))S(t)I(t) ⎪ ⎪ dI(t) = − (α(r(t)) + µ(r(t)) + ϕ(r(t)))I(t) dt ⎪ ⎪ S(t) + I(t) ⎪ ⎪ ∫ ⎪ ⎪ ⎪ ⎪ ˜ (dt, dz), ⎩ + σ2 (r(t))I(t)dB2 (t) + C2 (r(t), z)I(t− )N Y

where the parameters Λ(i), β(i), µ(i), ϕ(i), α(i) and σk (i), i ∈ S are all positive constants; Bk (t) (k = 1, 2) are independent standard Brownian motions defined on (Ω , F , {Ft }t≥0 , P), which is a complete probability space with a filtration {Ft }t≥0 satisfying the usual conditions; X(t− ) denotes the left limit of X(t); N (dt, dz) is a Poisson counting measure with the stationary compensator Π (dz)dt and Π is defined on a measurable ˜ (dt, dz) := N (dt, dz) − Π (dz)dt; Ck : S × Y × Ω → R (k = 1, 2) are subset Y of (0, ∞) with Π (Y) < ∞; N continuous and bounded functions with respect to Π . Throughout we assume that Bk (k = 1, 2), r(t) and N are independent.

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Remark 2.1. The Markov chain r(t) (t ≥ 0) is defined on a finite state space S = {1, 2, . . . , N }. And its generator Γ = (γij )N ×N is given by { γij δ + o(δ) if i ̸= j, P {r(t + δ) = j|r(t) = i} = 1 + γii δ + o(δ) if i = j, ∑ where γij ≥ 0 for i, j ∈ S with j ̸= i and γii = − j̸=i γij for each i ∈ S. Assume Markov chain r(t) is irreducible. Therefore there exists a unique stationary distribution λ = {λ1 , λ2 , . . . , λN } of r(t) which ∑N satisfies λΓ = 0, i=1 λi = 1 and λi > 0, ∀i ∈ S. Throughout make assumptions as follows: (A1) Ck (i, z) > −1, i ∈ S, z ∈ Y, k = 1, 2 and there exists a K > 0 such that ∫ ∫ 1 2 Π (dz) < K, | ln(1 + Ck (i, z))| Π (dz) < K and Y Y 1 + Ck (i, z)

i ∈ S, k = 1, 2.

Under condition (A1), by using the same arguments as in [3, theorem 1], we have that for any initial value (S(0), I(0), r(0)) ∈ R2+ × S, model (2.1) has a unique global positive solution (S(t), I(t), r(t)) ∈ R2+ × S for all t ≥ 0 almost surely. The rest of this paper is organized as follows. In Section 3, we present a sufficient condition for exponential ergodicity; and a detailed proof of main result will be given in Section 4. 3. Exponential ergodicity and numerical simulation For any positive function f (x, k) ≥ 1 defined on R2+ ×S and any signed measure ν(·) defined on B(R2+ ×S), we write ∥ν∥f = sup{|ν(g)| : all measurable functions g satisfying |g(x, k)| ≤ f (x, k) for all (x, k) ∈ R2+ ×S}, where ν(g) denotes the integral of function g with respect to measure ν. Obviously when f ≡ 1, ∥ν∥f is just the total variation norm ∥ν∥. For a function 1 ≤ f < ∞ on R2+ × S, Markov process (S(t), I(t), r(t)) is said to be f -exponentially ergodic if there exist a probability measure π(·), a constant θ < 1 and a finite-valued function Θ(x, k) such that ∥P (t, (x, k), ·) − π(·)∥f ≤ Θ(x, k)θt , for all t ≥ 0 and all (x, k) ∈ R2+ × S. Here P (t, (x, k), ·) is the transition probability of (S(t), I(t), r(t)). Set N ∑ λi Ri , Rs := i=1

where Ri := β(i) − µ(i) − α(i) − ϕ(i) −

σ22 (i) 2

+

∫ Y

(ln(1 + C2 (i, z)) − C2 (i, z))Π (dz), i ∈ S.

Theorem 3.1.

If Rs > 0, then (St , It , r(t)) is f -exponentially ergodic.

Theorem 3.2.

If Rs < 0, then the disease It tends to zero exponentially.

Remark 3.1. The proof of Theorem 3.2 is simple. In fact, calculating the Lyapunov exponent of I(t) yields ∑N lim supt→∞ ln I(t) ≤ i=1 λi Ri = Rs < 0. t Remark 3.2. From the expression of Rs , we know that the white noise acting on I(t) plays a positive role in controlling the prevalence of the disease. However, the telegraph noise has the opposite effect. That is, even if the disease persists only in one regime, eventually it will have the opportunity to persist. L´evy jump may play double roles in controlling the disease according to the value of C2 . Note that the white noise and L´evy jump acting on S(t) have no influence on the asymptotic behavior of (2.1).

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Fig. 2. The paths of (2.1). S(0) = 0.9, I(0) = 0.4, R1 = 0.2515, R2 = −0.0748 and Rs = 0.1427.

Example 3.1.) Let Y = {1, 2, 3}, Π (Y) = 2, Π ({1}) = 0.5, Π ({2}) = 1, Π ({3}) = 0.5. Take S = {1, 2}, ( −1 1 Γ = . Obviously, λ = (2/3, 1/3). Moreover, set C1 = C2 and 2 −2 Λ(1) = 0.4, µ(1) = 0.2, α(1) = 0.2, ϕ(1) = 0.2,

β(1) = 0.8, σ1 (1) = 0.1, σ2 (1) = 0.2,

Λ(2) = 0.4, µ(2) = 0.2, α(2) = 0.05, ϕ(2) = 0.25, β(2) = 0.55, σ1 (2) = 0.1, σ2 (2) = 0.3, C2 (1, 1) = C2 (1, 3) = −0.1, C2 (1, 2) = −0.2, C2 (2, 1) = C2 (2, 3) = −0.2, C2 (2, 2) = −0.3. In this case, R1 = 0.2515, R2 = −0.0748 and Rs = 0.1427. The disease will not disappear in regime 1 and become extinct in regime 2. In Fig. 2 we present the curve for the total variation ∥P (t, (x, k), ·) − π(·)∥, which strongly indicates an exponential convergence. 4. The proof of Theorem 3.1 Define the operator A by A V (x, y, i) =

∑ σ22 (i)y 2 ∂ 2 V ∂V σ12 (i)x2 ∂ 2 V ∂V + h + γij V (x, y, j) + + h i,1 i,2 2 ∂x2 2 ∂y 2 ∂x ∂y j∈S ∫ + [V (x + xC1 (i, z), y + yC2 (i, z), i) − V (x, y, i) Y

−

∂V ∂V · xC1 (i, z) − · yC2 (i, z)]Π (dz), ∂x ∂y

β(i)xy and hi,1 (x, y) = Λ(i) − β(i)xy x+y − µ(i)x + ϕ(i)y, hi,2 (x, y) = x+y − (α(i) + µ(i) + ϕ(i))y. According to [9, Theorem 6.3] or [10, Theorem 6.1], in order to prove the exponential ergodicity it suffices to verify the following two conditions:

(i) (S(t), I(t), r(t)) has the strong Feller property and irreducibility; (ii) There is a function V (x, y, i) ∈ C 2 (R2+ × S) such that for some θ, γ > 0, A V (x, y, i) ≤ −θV (x, y, i) + γ,

(x, y, i) ∈ R2+ × S.

(4.1)

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First, we prove condition (i). For any fixed i ∈ S, let (Si0 (t), Ii0 (t)) be Markov process governed by ) ( ⎧ β(i)Si0 (t)Ii0 (t) 0 0 0 ⎪ ⎪ − µ(i)S (t) + ϕ(i)I (t) dt + σ1 (i)Si0 (t)dB1 (t), dS (t) = Λ(i) − ⎨ i i i Si0 (t) + Ii0 (t) ) ( (4.2) ⎪ β(i)Si0 (t)Ii0 (t) ⎪ 0 0 ⎩ dIi0 (t) = − (α(i) + µ(i) + ϕ(i))I (t) dt + σ (i)I (t)dB (t), 2 2 i i Si0 (t) + Ii0 (t) with initial condition (Si0 (0), Ii0 (0)) ∈ R2+ . Let u0i (t) = ln Si0 (t) and vi0 (t) = ln Ii0 (t). Then (4.2) becomes ( ) ⎧ vi0 (t) 2 0 0 0 ⎪ β(i)e σ (i) ⎪ du0 (t) = Λ(i)e−ui (t) − ⎪ − µ(i) − 1 + ϕ(i)evi (t)−ui (t) dt + σ1 (i)dB1 (t), ⎪ 0 0 ⎨ i 2 eui (t) + evi (t) ( ) 0 ⎪ ⎪ σ22 (i) β(i)eui (t) 0 ⎪ ⎪ dvi (t) = − (α(i) + µ(i) + ϕ(i)) − dt + σ2 (i)dB2 (t). ⎩ 0 0 2 eui (t) + evi (t)

(4.3)

Obviously the diffusion matrix of system (4.3) satisfies the uniformly elliptic condition. Hence, the transition probability of (u0i (t), vi0 (t)) has a positive smooth density on R+ × R2 × R2 . Consequently, the diffusion (Si0 (t), Ii0 (t)) has the strong Feller property and a positive transition probability density defined on R+ ×R2+ × R2+ (with respect to the Lebesgue measure). According to [11, Theorem 5.2] we know that (S(t), I(t), r(t)) has the strong Feller property. In addition, irreducibility of (S(t), I(t), r(t)) on R2+ × S can be easily obtained by the positivity of the transition density. Next, we will verify the condition (ii). It is easy to check that under condition (A1), ∫ ∫ 1 −p lim ((1 + C2 (i, z)) − 1)Π (dz) = − ln(1 + C2 (i, z))Π (dz). p→0+ p Y Y ˜ s = ∑N λi R ˜ i > 0, where Take 0 < p < 1 small enough such that R i=1 ∫ ∫ 2 ˜ i = β(i) − µ(i) − α(i) − ϕ(i) − (p + 1)σ2 (i) − C2 (i, z)Π (dz) − 1 ((1 + C2 (i, z))−p − 1)Π (dz). R 2 p Y Y According to [12, Lemma 2.3], linear equation ( s ) ˜ 1, . . . , R ˜ N )′ − Γ x = R ˜ ,...,R ˜s ′ (R

(4.4)

has a unique positive solution, denoted by ϖ = (ϖ1 , . . . , ϖN )′ . Define a nonnegative C 2 -function V as follows: V (x, y, i) = q(x + y − log x) + (1 + pϖi ) · y −p , (x, y, i) ∈ R2+ × S, ˜i + R ˜s > where p > 0, q > 0 satisfying 0 < 1 + pϖi < 2, pϖi R

˜s R 2 ,

i ∈ S and

q · min{Λ(i)} > 4p · max{β(i)}. i∈S

i∈S

(4.5)

Denote V1 = x + y, V2 = − log x and V3 = (1 + pϖi ) · y −p . Then A V = q(A V1 + A V2 ) + A V3 . By direct calculation, we obtain A V1 = Λ(i) − µ(i)x − (α(i) + µ(i))y ≤ Λ(i) − µ(i)(x + y), ∫ Λ(i) β(i)y y σ12 (i) A V2 = − + + µ(i) − ϕ(i) + + (C1 (i, z) − ln(1 + C1 (i, z)))Π (dz) x x+y x 2 Y ∫ Λ(i) σ12 (i) ≤− + β(i) + µ(i) + + (C1 (i, z) − ln(1 + C1 (i, z)))Π (dz), x 2 Y

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[ ] ∑ β(i)x (p + 1)σ22 (i) A V3 = p(1 + pϖi )y − + µ(i) + α(i) + ϕ(i) + + py −p γij ϖj x+y 2 j∈S ∫ + (1 + pϖi )y −p ((1 + C2 (i, z))−p − 1 + pC2 (i, z))Π (dz) Y ⎤ ⎡ ∑ ˜ i + (1 + pϖi ) β(i)y + γij ϖj ⎦ = py −p ⎣−(1 + pϖi )R x+y j∈S ] [ β(i)y s −p ˜ ˜ ˜ + Ri − R (where (4.4) is used) = py −(1 + pϖi )Ri + (1 + pϖi ) x+y 1−p ˜i + R ˜ s )y −p + p(1 + pϖi )β(i) y = −p(pϖi R x+y y 1−p p ˜ s −p . ≤ − R y + 2pβ(i) 2 x+y −p

Hence,

( ) Λ(i) y 1−p p ˜ s −p A V ≤ q −µ(i)(x + y) − + Ki − R y + 2pβ(i) , x 2 x+y { ∫ σ 2 (i) µ(1) where Ki = Λ(i) + β(i) + µ(i) + 12 + Y (C1 (i, z) − ln(1 + C1 (i, z)))Π (dz). Take 0 < ε < min 12 , Λ(1) ,..., } µ(N ) y 1−p 1 1 Λ(N ) . In view of inequalities x − 1 + log x ≥ 0 (x > 0), x+y ≤ 1 + x (0 < y < ∞) and (4.5), it follows that ( ) (1 − ε)Λ(i) 1 p ˜ s −p A V ≤q −µ(i)(x + y) − + εΛ(i) log x + Ki − R y + 2pβ(i)(1 + ) x 2 x 2pβ(i) q(1 − ε)Λ(i) p ˜ s −p y + 2pβ(i) + − =q (−µ(i)(x + y) + εΛ(i) log x + Ki ) − R 2 x x p ˜ s −p ≤q (−µ(i)(x + y) + εΛ(i) log x + Ki ) − R y + 2pβ(i) 2 p ˜ s −p ≤qεΛ(i)(−x − y + log x) − R y + qKi + 2pβ(i). 2 ˜s

pR Take θ = mini∈S {εΛ(i), 2(1+pϖ ) }, γ = maxi∈S {qKi + 2pβ(i)}. Thus (4.1) holds. The proof is completed. i

Acknowledgments The authors would like to thank the anonymous reviewers for careful reading of our manuscript and providing such valuable suggestions, which make our manuscript more rigorous and readable. References [1] J. Zhou, H.W. Hethcote, Population size dependent incidence in models for diseases without immunity, J. Math. Biol. 32 (1994) 809–834. [2] C. Vargas-De-Le´ on, Stability analysis of a SIS epidemic model with standard incidence, Foro-Red-Mat: Rev. Electr´ on. Conten. Mat. 28 (2011) 1–11. [3] X. Zhang, K. Wang, Stochastic SIR model with jumps, Appl. Math. Lett. 26 (2013) 867–874. [4] Y. Zhou, W. Zhang, Threshold of a stochastic SIR epidemic model with L´ evy jumps, Phys. A 446 (2016) 204–216. [5] X. Zhang, D. Jiang, T. Hayat, et al., Dynamics of a stochastic SIS model with double epidemic diseases driven by L´ evy jumps, Phys. A 471 (2017) 767–777. [6] Y. Guo, Stochastic regime switching SIR model driven by L´ evy noise, Phys. A 479 (2017) 1–11. [7] J. Bao, X. Mao, G. Yin, C. Yuan, Competitive Lotka–Volterra population dynamics with jumps, Nonlinear Anal. 74 (2011) 6601–6616. [8] M. Liu, Y. Zhu, Stationary distribution and ergodicity of a stochastic hybrid competition model with L´ evy jumps, Nonlinear Anal. Hybrid Syst. 30 (2018) 225–239. [9] F. Xi, On the stability of jump-diffusions with Markovian switching, J. Math. Anal. Appl. 341 (2008) 588–600.

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