# Dynamics of a Nonautonomous Stochastic SIS Epidemic Model with Double Epidemic Hypothesis.

1. IntroductionThe SIS (Susceptible-Infected-Susceptible) model is a basic biological mathematical model describing susceptible and infected epidemic process and is first introduced by Kermack and McKendrick [1]. The SIS model is defined in that individuals start off susceptible, at some stage catch the disease, and after a short infectious period become susceptible again [2]. Therefore, some deterministic SIS epidemic models have been studied by many authors [3-10]. Recently, the authors of [11-13] investigated the epidemic model with double epidemic hypothesis which has two epidemic diseases caused by two different viruses. For example, the deterministic SIS epidemic model with nonlinear saturated incidence rate and double epidemic hypothesis can be expressed as follows [11]:

[mathematical expression not reproducible] (1)

where S(t), [I.sub.1] (t), and [I.sub.2] (t) represent the number of susceptibles and infected individuals with viruses A and B at time t, respectively. The parameters in model (1) have the following meanings: A is the total input susceptible population size, d represents the natural death rate of S, [I.sub.1], and [I.sub.2], [[beta].sub.i ]represents the disease transmission coefficient between compartments S and [I.sub.i] (i = 1,2), [r.sub.1] and [r.sub.2] are the recovery rates of the two diseases, and [[alpha].sub.1] and [[alpha].sub.2] are mortality rates due to diseases, respectively. Functions [[beta].sub.1]S(t) [I.sub.1](t)/([a.sub.1] + [I.sub.2](t)) and [[beta].sub.2]S(t)[I.sub.2](t)/([a.sub.2] + [I.sub.2](t)) represent two different types of saturated incidence rates for the two epidemic diseases [I.sub.1] (t) and [I.sub.2](t). All parameter values are nonnegative.

In the real world, population systems and epidemic systems are inevitably infected by some uncertain environmental disturbances. Hence, many authors have introduced stochastic interferences into differential systems, and the stochastic dynamics of such systems were widely investigated (see [1428]). Moreover, numerous scholars have investigated some stochastic epidemic models (see [29-34]). For example, in [11, 30] they obtained thresholds of the stochastic system which determines the extinction and persistence of the epidemic. Zhang et al. [29] proved that there is a unique ergodic stationary distribution of his model. We assume that environment fluctuations will manifest themselves mainly as fluctuations in the saturated response rate, so that [mathematical expression not reproducible] where B(t) = ([B.sub.1](t), [B.sub.2](t)) is a standard Brownian motion with intensity [[sigma].sub.i] >0 (i = 1,2). Therefore, a stochastic model is described by [11]

[mathematical expression not reproducible] (2)

However, many infectious diseases of human fluctuate over time and often show the seasonal morbidity. Therefore, the existence of periodic solutions of some nonautonomous epidemic models was explored [35-37]. Recently, many scholars focused on nonautonomous stochastic periodic systems. With the development of stochastic differential equations and application of Has'minskii theory, the existence of stochastic periodic solution has been studied [23, 38, 39]. In [23], Zhang et al. considered a nonautonomous stochastic Lotka-Volterra predator-prey model with impulsive effects; they got thresholds for stochastic persistence and extinction of the system. Authors of [38-40] investigated periodic solution of a stochastic nonautonomous epidemic model.

Based on the discussion above, in this paper, we consider a nonautonomous stochastic SIS model with periodic coefficients

[mathematical expression not reproducible] (3)

where the parameter functions A(t), d(t), [[beta].sub.i](t), [[a.sub.i](t), [r.sub.i](t), [[alpha].sub.i](t), [[sigma].sub.i](t) (i = 1,2) are positive, nonconstant, and continuous periodic functions with positive period T.

To the best of our knowledge, there are only few works on research of nonautonomous stochastic epidemic models with nonlinear saturated incidence rate and double epidemic hypothesis. Therefore, based on an autonomous stochastic epidemic model, we propose a nonautonomous stochastic model and investigate the existence of stochastic periodic solution and the extinction of the two epidemic diseases.

This paper is organized as follows. In Section 2, we give some definitions and known results. In Section 3, we prove that system (3) has a unique global positive solution. In Section 4, we present sufficient condition for the existence a nontrivial positive periodic solution of system (3). In Section 5, we obtain the sufficient conditions of system (3) for extinction of the two epidemic diseases. In Section 6, we carry out a series of numerical simulations to illustrate our theoretical findings.

2. Preliminaries

Throughout this paper, let ([OMEGA], F, P) be a complete probability space with a filtration [{[F.sub.t]}.sub.t[greater than or equal to]0] satisfying the usual conditions (i.e., it is increasing and right continuous while [F.sub.0] contains all P-null sets). The function [B.sub.i](t) (i = 1,2,3,4) is defined on this complete probability space.

For simplicity, some notations are given first. If f(t) is an integrable function defined on [0, [infinity]), define [(f).sub.t] = (1/t) [[integral].sup.t.sub.0] f(s)ds, t > 0. If f(t) is a bounded function on [mathematical expression not reproducible].

Here we present some basic theory in stochastic differential equations which are introduced in [41].

In general, consider the /-dimensional stochastic differential equation

dX (t) = f (X (t),t) dt + g(X (t), t) dB (t), t [greater than or equal to] [t.sub.0], (4)

with initial value x([t.sub.0]) = x0 [member of] [R.sup.l]. B(t) stands for a l-dimensional standard Brownian motion defined on the complete probability space ([OMEGA], F, [{[F.sub.t]}.sub.t[greater than or equal to]0], P). Denote by [C.sup.2,1] ([R.sup.l] x [[t.sub.0], [infinity]]; [R.sub.+]) the family of all nonnegative functions V(X, t) defined on [R.sup.l] x[[t.sub.0], [infinity]] such that they are continuously twice differentiable in X and once in t. The differential operator L of (4) is defined by [41]

[mathematical expression not reproducible] (5)

If L acts on a function V [member of] [C.sup.2,1]([R.sup.l] x [[t.sub.0], [infinity]]; [R.sub.+]), then

[mathematical expression not reproducible] (6)

where [mathematical expression not reproducible]. In view of Ito's formula, if X(t) [member of] [R.sup.l], then

dV (X (t),t) = LV (x (t), t) dt + [V.sub.X] (X(t),t)g(X(t),t)dB(t). (7)

Definition 1 (see [42]). A stochastic process [xi](t) = [xi](t, w) (- [infinity] < t < + [infinity]) is said to be periodic with period T if for every finite sequence of numbers [t.sub.1], [t.sub.2],..., [t.sub.n] the joint distribution of random variables [xi] ([t.sub.1] + h), [xi]([t.sub.2] + h),..., [xi]([t.sub.n] + h) is independent of h, where h = kT, k = [+ or -]1, [+ or -] 2,...,

It is shown in [42] that a Markov process x(t) is T-periodic if and only if its transition probability function is T-periodic and the function [P.sub.0] (t, A) = Pjv(t) [member of] A} satisfies the equation

[mathematical expression not reproducible] (8)

Consider the following equation:

[mathematical expression not reproducible] (9)

Lemma 2 (see [42]). Suppose that coefficients of (9) are T-periodic in t and satisfy the condition

[mathematical expression not reproducible] (10)

in every cylinder I x U, where P is a constant. And suppose further that there exists a function V(t, x) [member of] [C.sup.2] in Rd which is T-periodic in t and satisfies the following conditions:

[mathematical expression not reproducible]

(A2) LV(t,x) [less than or equal to] - 1 outside some compact set, where the operator L is given by

[mathematical expression not reproducible] (11)

Then there exists a solution of (9) which is a T-periodic Markov process.

Lemma 3 (see [2], strong law of large numbers). Let M = [{[M.sub.t]}.sub.t] [greater than or equal to] 0 be a real-valued continuous local martingale vanishing at t = 0.

Then

[mathematical expression not reproducible] (12)

and also

[mathematical expression not reproducible] (13)

3. Existence and Uniqueness of the Global Positive Solution

In this section, we prove that system (3) has a unique global positive solution.

Theorem 4. For any initial value (S(0), [I.sub.1](0), [I.sub.2](0)) [member of] [R.sup.3.sub.+], there is a unique positive solution (S(t),[I.sub.1](t),[I.sub.2](t)) of (3) on t [greater than or equal to] 0 and the solution will remain in [R.sup.3.sub.+] with probability one.

Proof. From system (3), we can get

[mathematical expression not reproducible] (14)

Then

[mathematical expression not reproducible] (15)

obviously, we have

[mathematical expression not reproducible] (16)

Since the coefficients of system (3) satisfy the local Lipschitz conditions, then for any given initial value (S(0),[I.sub.1](0), [I.sub.2](0)) [member of] [R.sup.3.sub.+], there is a unique local solution (S(t), [I.sub.1](t), [I.sub.2](t)) on t [member of] [0, [[tau].sub.e]), where [[tau].sub.e] is the explosion time. To demonstrate that this solution is global, we only need to prove that [[tau].sub.e] = [infinity] a.s.

Let [k.sub.0] > 0 be sufficiently large for any initial value S(0), [I.sub.1](0), and [I.sub.2](0) lying within the interval [1 /[k.sub.0],k]. For each integer k [greater than or equal to] [k.sub.0], define the following stopping time:

[mathematical expression not reproducible] (17)

where we set inf [empty set] = [infinity] (as usual [empty set] denotes the empty set). Clearly, [tau].sub.k] is increasing as [mathematical expression not reproducible]; hence [[tau].sub.[infinity]] [less than or equal to] [[tau].sub.k] a.s. Next, we only need to verify [[tau].sub.[infinity]] = [infinity] a.s. If this statement is false, then there exist two constants T > 0 and [epsilon] [member of] (0,1) such that

P {[[tau].sub.[infinity] [less than or equal to] T} > [epsilon]. (18)

Thus there is an integer [k.sub.1] [greater than or equal to] [k.sub.0] such that

P {[[tau].sub.k] [less than or equal to] T} [greater than or equal to] [epsilon], k [greater than or equal to] [k.sub.1]. (19)

Define a [C.sup.2]-function V: [R.sup.3.sub.+] [right arrow] [R.sub.+] as follows:

V(S, [I.sub.1], [I.sub.2]) = S - 1 - ln S + [I.sub.1] -1- ln [I.sub.1] + [I.sub.2] - 1 -ln [I.sub2]; (20)

the nonnegativity of this function can be obtained from

u - 1 - ln u [greater than or equal to] 0, u >0. (21)

Applying Ito's formula yields

[mathematical expression not reproducible] (22)

where

[mathematical expression not reproducible] (23)

where K is a positive constant.

So we have

[mathematical expression not reproducible] (24)

Integrating (24) from 0 to [[tau].sub.k] [??] T and taking expectations on both sides yield

[mathematical expression not reproducible] (25)

Let [[OMEGA].sub.k] = {[[tau].sub.k] [less than or equal to] T}; from inequality (25) we can see that P([[OMEGA].sub.k]) [greater than or equal to] [epsilon]. We have

[mathematical expression not reproducible] (26)

By (25) and (26), one has

[mathematical expression not reproducible] (27)

where 1[[OMEGA]sub.k] is the indicator function of [[OMEGA]sub.k].

Let k [right arrow] [infinity]; we have

[infinity] > V(S(0), [I.sub.1] (0 ), [I.sub.2] (0 )) + KT = [infinity]. (28)

So we obtain [[tau].sub. [infinity]] = [infinity]. The proof is completed.

4. Existence of Nontrivial T-Periodic Solution

In this section, we verify that system (3) admits at least one nontrivial positive T-periodic solution. Define

[mathematical expression not reproducible] (29)

Theorem 5. When [mathematical expression not reproducible], then there exists a nontrivial positive T-periodic solution of system (3).

Proof. Define a [C.sup.2]-function [bar.V] : [0, [infinity]) x [R.sup.3.sub.+] [right arrow] [R.sub.+]:

[mathematical expression not reproducible] (30)

where

[mathematical expression not reproducible] (31)

and 0 < [theta] < min{1, [d.sup.l]/([[sigma].sup.2u.sub.1] + [[sigma].sup.2u.sub.2])}; M is a sufficiently large positive constant and satisfies the following conditions:

[mathematical expression not reproducible] (32)

where

[mathematical expression not reproducible] (33)

where

[mathematical expression not reproducible] (34)

Next we prove that condition ([A.sub.1]) in Lemma 2 holds. It is easy to check that [bar.V](t, S, [I.sub.1], [I.sub.2]) is a T-periodic function in t and satisfies

[mathematical expression not reproducible], (35)

where [U.sub.k] = (1/k,k) x (1/k,k) x (1/k,k) and k > 1 is a sufficiently large number.

By Ito's formula, we obtain

[mathematical expression not reproducible] (36)

where

[mathematical expression not reproducible] (37)

Note that [mathematical expression not reproducible] hold; then

L([V.sub.1]) [less than or equal to] [R.sub.0] (t). (38)

Define the T-periodic function w(t) satisfying

w' (t) = [([R.sub.0]).sub.T] - [R.sub.0] (7). (39)

So

[mathematical expression not reproducible] (40)

Applying Ito's formula, we can also have

[mathematical expression not reproducible] (41)

where

[mathematical expression not reproducible] (42)

Therefore

[mathematical expression not reproducible] (43)

Now, we are in the position to construct a compact subset U such that [A.sub.2] in Lemma 2 holds. Define the following bounded closed set:

[mathematical expression not reproducible] (44)

where [epsilon] >0 is a sufficiently small number. In the set [R.sup.3.sub.+]\U, we can choose [member of] sufficiently small such that

-M[lambda] - [A.sup.l]/[epsilon] + C [less than or equal to] -1, (45)

[mathematical expression not reproducible], (46)

[mathematical expression not reproducible], (47)

[mathematical expression not reproducible], (48)

[mathematical expression not reproducible], (49)

[mathematical expression not reproducible], (50)

where C, D, E, G, and H are positive constants which can be found from the following inequations (52), (54), (56), (59), and (61), respectively. For the sake of convenience, we divide into six domains,

[mathematical expression not reproducible] (51)

Next we will prove that L[bar.V](S, [I.sub.1], [I.sub.2]) [less than or equal to] - 1 on [R.sup.3+\U, which is equivalent to proving it on the above six domains.

Case 1. If (S, [I.sub.1],[I.sub.2]) [member of] [U.sub.1], one can see that

[mathematical expression not reproducible] (52)

where

[mathematical expression not reproducible] (53)

According to (45), we have L[bar.V] [less than or equal to] - 1 for all (S, [I.sub.1], [I.sub.2]) [member of] [U.sub.1].

Case 2. If (S, [I.sub.1], [I.sub.2]) [member of] [U.sub.2], one can get that

[mathematical expression not reproducible] (54)

where

[mathematical expression not reproducible] (55)

In view of (46), we can obtain that L[bar.V] [less than or equal to] - 1 for all (S, [I.sub.1], [I.sub.2]) [member of] [U.sub.2].

Case 3. If (S, [I.sub.1],[I.sub.2]) [member of] [U.sub.3],we have

[mathematical expression not reproducible] (56)

where

[mathematical expression not reproducible] (57)

By (47), we can conclude that L[bar.V] [less than or equal to] - 1 for all (S, [I.sub.1], [I.sub.2]) [member of] [U.sub.3].

Case 4. If (S, [I.sub.1], [I.sub.2]) [member of] [U.sub.4], one can derive that

[mathematical expression not reproducible] (58)

Together with (48), we can deduce that L[bar.V] [greater than or equal to] -1 for all (S, [I.sub.1], [I.sub.2]) [member of] [U.sub.4].

Case 5. If (S, [I.sub.1], [I.sub.2]) [member of] [U.sub.5], it follows that

[mathematical expression not reproducible] (59)

where

[mathematical expression not reproducible] (60)

By virtue of (49), we can deduce that L[bar.V] [less than or equal to] -1 for all (5. [I.sub.1], [I.sub.2]) [member of] [U.sub.5].

Case 6. If (S, [I.sub.1], [I.sub.2]) [member of] [U.sub.6], we obtain

[mathematical expression not reproducible] (61)

where

[mathematical expression not reproducible] (62)

It follows from (50) that L[bar.V] [less than or equal to] -1 for all (S, [I.sub.1], [I.sub.2]) [member of] [U.sub.6].

Clearly, one can see from (52), (54), (56), (58), (59), and (61) that, for a sufficiently small e,

L[bar.V](S,[I.sub.1], [I.sub.2]) [less than or equal to] -1, (S, [I.sub.1], [I.sub.2]) [member of] [R.sup.3.sub.+]\U. (63)

Hence [A.sub.2] in Lemma 2 is satisfied. This completes the proof of Theorem 5.

5. Extinction

In this section, we investigate the conditions for the extinction of the two infectious diseases of system (3).

Let

[mathematical expression not reproducible] (64)

Theorem 6. Let (S(t), [I.sub.1](t), [I.sub.2](t)) be a solution of system (3) with initial value (S(0), [I.sub.1](0), [I.sub.2](0)) [member of] [R.sup.3.sub.+].

Then if

[mathematical expression not reproducible] (65)

or [mathematical expression not reproducible] (66)

hold, the two infectious diseases of system (3) go to extinction a.s.; that is,

[mathematical expression not reproducible]. (67)

Proof. Applying Ito's formula to system (3), we have

[mathematical expression not reproducible] (68)

Case (i). Integrating (68) from 0 to 7 and dividing 7 on both sides, we obtain

[mathematical expression not reproducible] (69)

Case (ii). Integrating (68) from 0 to t first and then dividing by t on both sides yield

[mathematical expression not reproducible] (70)

where [mathematical expression not reproducible] which is a local continuous martingale with [M.sub.i](0) = 0. By Lemma 3, we have

[mathematical expression not reproducible] (71)

Taking the limit superior of both sides of (69) leads to

[mathematical expression not reproducible] (72)

which implies [mathematical expression not reproducible]

Taking the superior limit of both sides of (70) leads to

[mathematical expression not reproducible], (73)

which implies [lim.sub.t[right arrow][infinity]] [I.sub.i](t) = 0, (=1,2. This completes the proof.

Remark 7. Theorem 6 shows that the two diseases will die out if the white noise disturbance is large or the white noise disturbance is not large and [R.sub.i] < 1. When [mathematical expression not reproducible], the two infectious diseases of system (3) die out almost surely; that is to say, large white noise stochastic disturbance can lead to the two epidemics being extinct.

6. Numerical Simulations

Now we introduce some numerical simulations examples which illustrate our theoretical results.

Example 8. In model (3), let

[mathematical expression not reproducible] (74)

Note that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] hold; that is, the conditions of Theorem 5 hold. Hence, system (3) has a positive periodic solution with T =1. Figure 1(a) shows the periodicity of the nonautonomous stochastic model (3) with [[sigma].sub.1] = 0 and [[sigma].sub.2] = 0. Figure 1(b) shows that solution of the nonautonomous stochastic model (3) with the initial value (S(t), [I.sub.1], (t), [I.sub.2](t)) = (0.3,0.15,0.15) tends to a periodic orbit in the sense of joint distribution.

Example 9. Choose the parameters in model (3) as follows:

[mathematical expression not reproducible] (75)

Note that [mathematical expression not reproducible]. Therefore, conditions (i) of Theorem 6 hold. Then the two infectious diseases will go to extinction. Figure 2(a) shows that one of two diseases in the deterministic SIS epidemic model is extinct and the other is persistent. Figure 2(b) shows that the two diseases will die out under the large white noise disturbance of model (3).

Example 10. Choose the parameters in model (3) as follows:

[mathematical expression not reproducible] (76)

Note that [mathematical expression not reproducible]. That is, conditions (ii) of Theorem 6 hold. Then the two infectious diseases will go to extinction. Figure 3(a) shows that one of two diseases in the deterministic SIS epidemic model is extinct and the other is persistent without the white noises. Figure 3(b) shows that the two diseases will die out under a small white noise disturbance of model (3).

7. Discussion and Conclusions

This paper explores the existence of nontrivial positive T-periodic solution of a nonautonomous stochastic SIS epidemic model with nonlinear growth rate and double epidemic hypothesis. By constructing a suitable stochastic Lyapunov function, we establish sufficient conditions for the existence of nontrivial positive T-periodic solution of system (3). Furthermore, the sufficient conditions for the extinction of the two diseases are obtained. Our results are given as follows:

[mathematical expression not reproducible] hold, the SIS model has at least one nontrivial positive T-periodic solution, where

[mathematical expression not reproducible] (77)

(2) If [mathematical expression not reproducible], the two infectious diseases go extinct.

(3) If [mathematical expression not reproducible], the two infectious diseases also go extinct, where

[mathematical expression not reproducible] (78)

Some interesting questions deserve further investigation. On the one hand, we may explore some realistic but complex models, such as considering the effects of impulsive or delay perturbations on system (3). On the other hand, we can concern the dynamics of a nonautonomous stochastic SIS epidemic model with two infectious diseases driven by Levy jumps. What is more, we can also investigate the nonautonomous stochastic SIS epidemic model with two infectious diseases by a continuous time Markov chain. We will investigate these cases in our future work.

https://doi.org/10.1155/2017/4861391

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (11371230), Joint Innovative Center for Safe and Effective Mining Technology and Equipment of Coal Resources, Shandong Province, the SDUST Research Fund (2014TDJH102), and Shandong Provincial Natural Science Foundation, China (ZR2015AQ001).

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Haokun Qi, (1) Lidan Liu, (1) and Xinzhu Meng (1,2)

(1) College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao 266590, China

(2) State Key Laboratory of Mining Disaster Prevention and Control Co-Founded by Shandong Province and the Ministry of Science and Technology, Shandong University of Science and Technology, Qingdao 266590, China

Correspondence should be addressed to Xinzhu Meng; mxz721106@sdust.edu.cn

Received 25 August 2017; Accepted 11 October 2017; Published 8 November 2017

Caption: Figure 1: The solution (S(t),[I.sub.1](t), [I.sub.2](t)) = (1,0.2,0.2) to the nonautonomous stochastic model (3).

Caption: Figure 2: The solution (S(t), [I.sub.1](t), [I.sub.2](t)) = (0.3,0.25,0.25) to the nonautonomous stochastic model (3).

Caption: Figure 3: The solution (S(t),[I.sub.1] (t), [I.sub.2](t)) = (0.3,0.25,0.25) to the nonautonomous stochastic model (3).

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Title Annotation: | Research Article; Susceptible-Infected-Susceptible |
---|---|

Author: | Qi, Haokun; Liu, Lidan; Meng, Xinzhu |

Publication: | Complexity |

Article Type: | Report |

Geographic Code: | 1USA |

Date: | Jan 1, 2017 |

Words: | 5087 |

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