key: cord-0988378-774ywcj0
authors: Chen, Yiliang; Wen, Buyu; Teng, Zhidong
title: The global dynamics for a stochastic SIS epidemic model with isolation
date: 2018-02-15
journal: Physica A
DOI: 10.1016/j.physa.2017.11.085
sha: 8e879e1c736a9f254f0e69020f35ac6b76581700
doc_id: 988378
cord_uid: 774ywcj0

In this paper, we investigate the dynamical behavior for a stochastic SIS epidemic model with isolation which is as an important strategy for the elimination of infectious diseases. It is assumed that the stochastic effects manifest themselves mainly as fluctuation in the transmission coefficient, the death rate and the proportional coefficient of the isolation of infective. It is shown that the extinction and persistence in the mean of the model are determined by a threshold value [Formula: see text]. That is, if [Formula: see text] , then disease dies out with probability one, and if [Formula: see text] , then the disease is stochastic persistent in the means with probability one. Furthermore, the existence of a unique stationary distribution is discussed, and the sufficient conditions are established by using the Lyapunov function method. Finally, some numerical examples are carried out to confirm the analytical results.

In this paper, we investigate the dynamical behavior for a stochastic SIS epidemic model with isolation which is as an important strategy for the elimination of infectious diseases. It is assumed that the stochastic effects manifest themselves mainly as fluctuation in the transmission coefficient, the death rate and the proportional coefficient of the isolation of infective. It is shown that the extinction and persistence in the mean of the model are determined by a threshold value R S 0 . That is, if R S 0 < 1, then disease dies out with probability one, and if R S 0 > 1, then the disease is stochastic persistent in the means with probability one. Furthermore, the existence of a unique stationary distribution is discussed, and the sufficient conditions are established by using the Lyapunov function method. Finally, some numerical examples are carried out to confirm the analytical results.

As is well-known, in the theory of epidemiology the quarantine/isolation is an important strategy for the control and elimination of infectious diseases. Such as, in order to control SARS, the Chinese government is the first to use isolation. The various types of classical epidemic models with quarantine/isolation have been investigated in many articles. See, for example [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] and the references cited therein.

Particularly, in [1] , Herbert et al. studied the following SIS epidemic model with isolation ⎧ ⎨ ⎩ S ′ (t) = A − βIS − µS + γ I + ξ Q , I ′ (t) = βIS − (µ + γ + δ + α)I, Q ′ (t) = δI − (µ + ξ + α)Q . (1.1) where S(t) denotes the number of individuals who are susceptible to an infection, I(t) denotes the number of individuals who are infectious but not isolated, Q (t) is the number of individuals who are isolated. A is the recruitment rate of S(t), β is the transmission rate coefficient between compartment S(t) and I(t), µ is natural death rate of S(t), I(t) and Q (t), α is the disease-related death rate of I(t), δ is the proportional coefficient of isolated for the infection, γ and ξ are the rates where individuals recover and return to S(t) from I(t) and Q (t), respectively. All parameters are usually assumed to be nonnegative. In addition, we see that the quarantine/isolation strategies also are introduced and investigated in many practical epidemic model, such as the emerging infectious disease, two-strain avian influenza, childhood diseases, the Middle East respiratory syndrome, Ebola epidemics, Dengue epidemic, H1N1 flu epidemic, Hepatitis B and C, Tuberculosis, etc. See, for example [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] and the references cited therein.

As a matter of fact, epidemic systems are inevitably subjected to environmental white noise. Therefore, the studies for the stochastic epidemic models have more practical significance. In recent years, the stochastic epidemic models with the quarantine and isolation have been investigated in articles [29] [30] [31] [32] . Particularly, in [29] Zhang et al. investigated the dynamics of the deterministic and stochastic SIQS epidemic model with an isolation and nonlinear incidence. The sufficient conditions on the extinction almost surely of the disease and the existence of stationary distribution of the model are established. Zhang et al. in [30] discussed the threshold of a stochastic SIQS epidemic model. The criteria on the extinction and permanence in the mean of global positive solutions with probability one are established. Besides, we also see that the stochastic persistence and the existence of stationary distribution for the various stochastic epidemic models and population models have been widely investigated. Some important recent works can been found in [33] [34] [35] [36] [37] [38] [39] [40] [41] [42] [43] and the references cited therein.

Motivated by the works [1, 2, 4, 5, [29] [30] [31] [32] , in this paper as an extension of model (1.1) we firstly assume that the diseaserelated death rates of isolation and no-isolation are different, respectively, denote by α 2 and α 3 . Then, we further define µ 1 = µ, µ 2 = µ+α 2 and µ 3 = µ+α 3 for the convenience. It is clear that µ 1 ≤ min{µ 2 , µ 3 }. Next, we introduce randomness into model (1.1), by replacing the parameters β, µ i (i = 1, 2, 3) and 2, 3, 4, 5) are independent standard Brownian motion defined on some probability space ( , F , P) and parameter σ i > 0 represents the intensity of W i (t). Thus, we establish the following stochastic SIS epidemic model with multi-parameters white noises perturbations and the isolation of infection.

(1.2)

Our purpose in this paper is to study the stochastic extinction and persistence, and the stationary distribution of model (1.2). We will establish a series of sufficient conditions to assure the extinction and persistence in the mean of the model with probability one, and the existence of unique stationary distribution for model (1.2) by using the theory of stochastic processes, the Ito's formula and the Liapunov function method. This paper is organized as follows. In Section 2, we introduce the preliminaries and some useful lemmas. In Section 3, the criteria on the extinction and persistence in the mean with probability one for model (1.2) are stated and proved. In Section 4, the criteria on the existence of a unique stationary distribution for model (1.2) are stated and proved. In Section 5, the numerical examples are carried out to illustrate the main theoretical results.

We denote R 3

As the preliminaries, we give the following lemmas. (1) If R 0 < 1, then model (1.1) has only a disease-free equilibrium E 0 ( A µ , 0, 0), which is globally asymptotically stable. (2) If R 0 > 1, then model (1.1) also has an endemic equilibrium E * (S * , I * , Q * ), which is globally asymptotically stable,

The proof of Lemma 2.1 can be found in [1] . We hence omit it here. That is, solution (S(t), I(t), Q (t)) is defined for all t ≥ 0 and remains in R 3 + with probability one. Lemma 2.2 can be proved by using the similar method given in [29] . Lemma 2.3. Let (S(t), I(t), Q (t)) be the solution of model (1.2) with initial value (S(0), I(0), Q (0)) ∈ R 3 + , then lim sup t→∞ (S(t) + I(t) + Q (t)) < ∞ a.s.

where α 2 = µ 2 − µ 1 ≥ 0 and α 3 = µ 3 − µ 1 ≥ 0. Solving this equation, we further obtain that

Clearly, M(t) is a continuous local martingale with M(0) = 0. Define

where

(1 − e −µ 1 t ) and U(t) = (S(0) + I(0) + Q (0))(1 − e −µ 1 t ). By (2.4) we have S(t) + I(t) + Q (t) ≤ X (t) a.s. for all t ≥ 0. It is clear that A(t) and U(t) are continuous adapted increasing processes on t ≥ 0 with A(0) = U(0) = 0. By Theorem 3.9 in [44] , we obtain that lim t→∞ X (t) < ∞ a.s. Thus, conclusion (2.1) is true.

Since the quadratic variations

by the large number theorem for martingales (See [44, 45] ), we have Similarly, we also have

) , by (2.5) and (2.6), we obtain lim t→∞ ⟨M(t)⟩ = 0. Since 

and

and

(2.10)

Proof. Using Ito's formula, by (2.3) we have dN 2 (t) = LN 2 (t)dt + 2N(t)(σ 2 IdW 2 (t) + σ 3 Q dW 3 (t) + σ 5 SdW 5 (t)), (2.11) where

Integrating (2.11) from 0 to t, we further obtain

(2.12)

Then, dividing t on both sides (2.12), it follows that

where C (t) is given in (2.9). Thus, we finally obtain (2.8).

Taking the integration for the third equation of model (1.2) yields

(2.13)

Dividing t on both sides of Eq. (2.13), we have

(2.14)

Integrating (2.3) from 0 to t, and then dividing t on both sides, we have

Consequently,

) .

(2.15) By substituting (2.14) into (2.15), we obtain

where K (t) is given in (2.10). Thus, we finally obtain (2.7). This completes the proof. □ Lemma 2.5. Assume that functions Y ∈ C (R + × , R + ) and Z ∈ C (R + × , R + ) satisfies lim t→∞

Lemma 2.5 can be found in Liu et al. [46] .

Define

.

is stochastic persistent in the mean with probability one.

Proof. Applying Ito's formula, we have

(3.1)

Integrating (3.1) from 0 to t and then dividing t on both sides, we have 

On the other hand, from (2.2) in Lemma 2.3 we have that for any enough small ε > 0 there is a T > 0 such that

for all t ≥ T . By substituting (2.14) and (3.6) into (3.5), we obtain for all t ≥ T

where

] .

(3.8)

By the large number theorem for martingales, Lemmas 2.3 and 2.4, we have from (2.9), (2.10) and (3.8)

Therefore, from (3.9) and the arbitrariness of ε we finally obtain

.

From the first equation of model (1.2), we easily obtain

by the large number theorem for martingales we have lim t→∞ 

From the third equation of model (1.2), we directly have

Ids a.s.

Hence, we further have

This shows that model (1.2) is persistent in the mean with probability one. This completes the proof. □ Remark 3.1. It is unfortunate that in Theorem 3.1 σ 5 = 0 is assumed. From the proof of Theorem 3.1 we see that this assumption only is used to deal with the term ⟨S 2 (t)⟩ in (3.3) . Therefore, an interesting open problem is to establish a similar result like Theorem 3.1 for model (1.2) in σ 5 > 0.

In Theorem 3.1 we only obtain the persistence in the mean of model (1.2). However, as a consequence of Theorem 3.1 we have the following result on the permanence in the mean for the disease in model (1.2).

0 > 1, and σ 1 = 0 or σ 2 = 0 and σ 3 = 0, then the disease I(t) is permanent in the mean with probability one.

In fact, when σ 1 = 0 or σ 2 = 0 and σ 3 = 0, from (3.11) we have B = β( µ 2 µ 1 + δµ 3 µ 1 (µ 3 +ξ ) ), which is independent for L. Therefore, by Theorem 3.1, we obtain from (3.10) that

which implies that the disease I(t) is permanent in the mean with probability one.

Remark 3.2. From the above Corollary 3.1, we can propose an important open problem. That is, when R S 0 > 1, σ 1 > 0 and σ 2 > 0 or σ 3 > 0, whether we can establish the permanence in the mean of the disease I for model (1.2) . An example will be given in Section 5 to show that the result can hold.

Proof. Applying Ito's formula, directly computing, we have

(3.13)

Integrating (3.13) and then dividing t yields

From (2.14), we further have

where

By the large number theorem for martingales and Lemma 2.3, we have lim t→∞ B(t) = 0 a.s. Therefore, by Lemma 2.5 we finally can obtain that

Furthermore, from (2.14) we can obtain

and from (2.7) we further obtain

This completes the proof. □ Remark 3.3. Particularly, when σ i = 0 (i = 1, 2, 3, 4, 5) and µ 2 = µ 3 = µ 1 +α, then the stochastic model (1.2) degenerates into the deterministic model (1.1). We also have R S 0 = R 0 = βA µ 1 (δ+γ +µ 1 +α) . From Theorem 3.2, when R 0 > 1 we can obtain that for any solution (S(t), I(t), Q (t)) of model (1.1) with initial value (S(0), + . Suppose that one of the following two conditions holds:

and lim sup 

By the large number theorem for martingales, Lemmas 2.3 and 2.4, we have lim t→∞ (t) = 0 a.s. Therefore, we finally obtain lim sup

ln

Thus, we also have lim sup

From (3.14) and (3.15), there is a constant m > 0 such that for almost all ω ∈ there exists a T 0 = T 0 (ω) > 0, when t ≥ T 0 one has I(t, ω) ≤ e −mt . Without loss of generality, we assume that I(t, ω) ≤ e −mt for all t ≥ 0. It follows that from

Hence,

where

It is clear that

Consider H 3 (t), choose the constants η 0 > 0 and ε 0 > 0 such that

Since lim t→∞ W 3 (t) t = 0, without loss of generality, we assume |W 3 (t)| ≤ ε 0 t for all t ≥ 0. Let H * 3 (t) = e ηt H 3 (t), then we have ⟨H *

By the large number theorem for martingales, we have lim t→∞ H * 3 (t) t = 0. For any small enough ε > 0, we can obtain

Hence, lim t→∞

By Lemma 2.3, the large number theorem of martingales, lim t→∞ I(t) = 0 a.s. and lim t→∞ Q (t) = 0 a.s., we have lim t→∞ ⟨SI⟩ = 0, lim t→∞ ⟨I⟩ = 0, lim t→∞ ⟨Q ⟩ = 0, lim t→∞ − (µ 2 + δ + γ + 1 2 σ 2 2 + 1 2 σ 2 4 ) > 0 and β < A µ 1 σ 2 1 , whether we also can obtain the extinction of the disease I with probability one for model (1.2 ). An example is given in Section 5 to show that the result can hold. 

Therefore, as a consequence of Theorem 3.3, we have the following corollary. , I(t) and Q (t) almost surely exponentially converge to zero.

In this section, we study the existence of unique stationary distribution of model (1.2) . Before giving the main results, we introduce the following lemma.

Let x(t) be a regular temporally homogeneous Markov process in R d described by the stochastic differential equation

. . , σ d r (x)) and B r (t) (r = 1, 2, . . . , k)) are independent standard Brownian motions defined on some probability space ( , F , P). The diffusion matrix for Eq. (4.1) is defined as 

where f (x) is a function integrable with respect to the measure µ 1 . [47, 48] ). To validate condition (ii), it is sufficient to show that there is a nonnegative C 2 -function V (x) and a bounded domain U ⊂ R d with regular boundary such that for some constant k > 0 one has LV (x) < −k for all x ∈ R d \ U (See [49] ). When in model (1.2) there is not any stochastic perturbation, that is σ i = 0 (i = 1, 2, 3, 4, 5), then model (1.2) degenerates into the following deterministic model LetR 0 = βA µ 1 (δ+γ +µ 2 ) . We can prove that whenR 0 > 1 then model (4.2) has a unique endemic equilibrium (S * , I * , R * ), where

Define the constants

Now, on the existence and uniqueness of stationary distribution for model (1.2) we have the following result. Proof. Define the Lyapunov function as follows.

By computing, we have

Therefore, we have LV (S, I, Q ) =a 1 LV 1 (Q ) + a 2 LV 2 (I) + a 3 LV 3 (S, I) + LV 4 (S, I, Q ),

lie in the positive zone of R 3 + . Hence, there exists a constant C > 0 and a compact set K ⊂ R 3 + such that for any

Thus, we finally have 

where |ζ | = (ζ 2 1 + ζ 2 2 + ζ 

(4.5)

We will give a new conclusion on the existence of unique stationary distribution for model (4.5) . Define the constant

. Theorem 4.2. Assume thatR 0 > 1. Then model (4.5) has a unique stationary distribution and the ergodic property.

whit θ is a constant satisfying 0 < θ < 

Therefore, the differential operator L acting on the V yields

Now, we construct a compact subset D such that the condition (ii) in Lemma 4.1 holds. Define the bounded closed set

where ε i (i = 1, 2, 3) are small enough positive constants, which will be determined later.

For convenience, we divide R 3

We will prove that LV (S, I, Q ) ≤ −1 on R 3 + \ D, which is equivalent to show it on the above six domains. Case 1. If (S, I, Q ) ∈ D 1 , we can obtain

We choose a constant ε 1 > 0 small enough such that − A ε 1 Case 3. If (S, I, Q ) ∈ D 3 , we can obtain 

Choose a constant ε 1 > 0 small enough such that − 1 2 µ * ( 1 

Choose a constant ε 2 > 0 small enough such that − 1 2 µ * ( 1 

Choose a constant ε 3 > 0 small enough such that − 1 2 µ * ( 1 

In this section, we further analyze the stochastic model (1.2) by means of the numerical examples.

Example 5.1. In model (1.2) we take the parameters A = 2.5, β = 0.08, µ 1 = 0.1, σ 1 = 0.06, σ 2 = 0.7, σ 3 = 0.2, σ 4 = 0.6, σ 5 = 0.1, γ = 0.16, ξ = 0.1, µ 2 = 0.2, µ 3 = 0.2 and δ = 0.1. We obtain by computing R S 0 = 0.9783 < 1, β = 0.08 < A µ 1 σ 2 1 = 0.09, β 2 2σ 2 1 − (µ 2 + δ + γ + 1 2 (σ 2 2 + σ 2 4 )) = 0.0039 > 0. Therefore, Theorem 3.3 is not applicable.

However, from the numerical simulations given in Fig. 1 , we can see that the infective I(t) and isolation Q (t) in model (1.2) are extinct with probability one, and the susceptible S(t) in model (1.2) is permanent in the mean with probability one. Example 5.2. In model (1.2), we take the parameters A = 2, β = 0.1, µ 1 = 0.1, σ 1 = 0.01, σ 2 = 0.12, σ 3 = 0.001, σ 4 = 0.14, σ 5 = 0.01, γ = 0.1, ξ = 0.05, µ 2 = 0.11, µ 3 = 0.22 and δ = 0.11. We obtain R S 0 = 6.1344 > 1. From the numerical simulations given in Fig. 2 , we can see that the infective I(t), isolation Q (t) and susceptible S(t) in model (1.2) are not only persistent in the mean with probability one, but also permanent in the mean with probability one. shows that Theorem 4.1 is not applicable. But, from the numerical simulations given in Fig. 3 , we can see that the solution (S(t), I(t), Q (t)) of model (1.2) still has a unique stationary distribution.

Example 5.4. In model (1.2), we take the parameters A = 0.9, β = 0.1, µ 1 = 0.1, µ 2 = 0.12, α 3 = 0.11 σ 1 = 0.01, σ 2 = 0.12, σ 3 = 0.001, σ 3 = 0.14, σ 4 = 0.001, σ 5 = 0.7 γ = 0.1, ξ = 0.05, and δ = 0.11. We obtain R S 0 = 2.6932 > 1, R 0 = 0.7736 < 1. This shows that Theorem 4.2 is not applicable. But, from the numerical simulations given in Fig. 4 , we can see that the solutions of model (1.2) (S(t), I(t), Q (t)) may not exist the stationary distribution.

In this paper, we have investigated the global dynamics for a stochastic SIS epidemic model with isolation of the infection. The stochastic effects are assumed as the fluctuations in the transmission coefficient, disease-related rate and the proportional coefficient of isolated of infection. The research given in this paper shows that the extinction and persistence in the mean of the model are determined by a threshold value R S 0 . Concretely, we have proved that if R S 0 < 1 then disease dies out with probability one (Theorem 3.3), if R S 0 > 1, then the model is stochastic persistent or permanent in the means with probability one (Theorems 3.1 and 3.2). Furthermore, we also established the sufficient conditions for the existence of a unique stationary distribution (Theorems 4.1 and 4.2) by constructing the new suitable Lyapunov function. Particularly, we also see that the researches given in this paper extend the results on the global stability of the disease-free and endemic equilibria for the corresponding deterministic model given in Lemma 2.1. We see that, in order to deal with the isolation term for the stochastic SIS epidemic model, some novel interesting research techniques are proposed. They are presented in Lemma 2.4 and the proofs of Theorems 3.1-3.3 and 4.2. In addition, we also see that there are still many problems for the considered model. These problems have been shown in Remarks 3.1, 3.2, 3.4, 3.5 and 4.3, which are interesting and valuable to be further investigated in the future.

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This research is supported by the Natural Science Foundation of Xinjiang (Grant Nos. 2016D03022).