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Threshold dynamics for a cholera epidemic model with periodic transmission rate
Applied Mathematical Modelling 37 (2013) 3093–3101
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Threshold dynamics for a cholera epidemic model with periodic transmission rate q Xue-yong Zhou a,b, Jing-an Cui c,⇑ a
College of Mathematics and Information Science, Xinyang Normal University, Xinyang 464000, Henan, PR China School of Mathematical Sciences, Nanjing Normal University, Nanjing 210046, Jiangsu, PR China c School of Science, Beijing University of Civil Engineering and Architecture, Beijing 100044, PR China b
a r t i c l e
i n f o
Article history: Received 11 December 2011 Received in revised form 14 May 2012 Accepted 12 July 2012 Available online 16 August 2012 Keywords: Cholera model Global stability Uniform persistence Extinction
a b s t r a c t A cholera epidemic model with periodic transmission rate is presented. The basic reproduction number is defined. It is shown that the disease-free equilibrium is globally asymptotically stable and the cholera eventually disappears if the basic reproduction number is less than one. And if the basic reproduction number is greater than one, there exists a positive periodic solution which is globally asymptotically stable. Numerical simulations are provided to illustrate analytical results. Ó 2012 Elsevier Inc. All rights reserved.
1. Introduction Cholera remains a major public health threat in many developing countries around the world. It is an acute intestinal infection caused by the bacterium Vibrio cholerae. Cholera is typically transmitted through water or foods that have been contaminated with fecal matter from a person who is infected with the disease. Cholera can spread rapidly, and epidemics may occur after fecal contamination of food or water supplies [1]. Recently, a number of mathematical models have been developed to help in understanding the dynamics of cholera outbreaks. In 1979, Capasso and Paveri-Fontana [2] presented a mathematical model for cholera epidemic occurred in the European Mediterranean region in 1973. In Capasso’s version, two equations describe the dynamics of infected people in the community and the dynamics of the aquatic population of pathogenic bacteria. After that, several cholera models were formulated and analyzed. In 2001, Codeço extended the cholera model of Capasso and Paveri-Fontana [2] with an additional equation for the susceptible individuals in the host population and explored the role of the aquatic reservoir in the persistence of endemic cholera [3]. Hartley et al. [4] in 2006 extended Codeço’s work to include hyperinfectious vibrios. In Liao and Wang [5], conducted the dynamical analysis (for example, the stability of equilibria of the system) of the deterministic cholera model proposed in [4]. In Zhou and Cui [6], considered a cholera model with vaccination. They analyzed the locally and globally asymptotical stability of the disease-free and endemic equilibria of their system. We may find other mathematical studies on modeling cholera dynamics in references [7–9]. The seasonality of cholera is apparent. Occurrence of cholera is typically seasonal due climatic factors, physical, and biological factor. For example, seasonality of cholera in Bangladesh exhibits two peaks per year and differs from that of other q This work is supported by the National Natural Science Foundation of China (No. 11071011), Funding Project for Academic Human Resources Development in Institutions of Higher Learning Under the Jurisdiction of Beijing Municipality (No. PHR201107123). ⇑ Corresponding author. Tel./fax: +86 010 61209413. E-mail addresses: [email protected] (X.-y. Zhou), [email protected] (J.-a. Cui).
0307-904X/$ - see front matter Ó 2012 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.apm.2012.07.044
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diarrheal diseases [10]. In Pakistan, classical cholera typically increases from November to January and from April to May while in Kolkata, India, seasonal patterns of cholera cases peak in April, May, and June [11]. Seasonality in cholera is seldom considered in the literature of cholera modeling (as well as endemic behavior), even though epidemiological patterns clearly show seasonal signatures and strong correlations with environmental drivers [10]. In [12], Bertuzzo et al. studied how river networks, acting as environmental corridors for pathogens, affect the spreading of cholera epidemics. And the seasonality had been taken into account in this reference. In [13], Mukandavire et al. presented a cholera epidemic model including human-to-human transmission to estimate the reproductive numbers for the 2008–2009 cholera outbreaks in Zimbabwe. But they did not consider the seasonality of cholera. As far as we know, almost no references considered the cholera with both seasonality and human-to-human transmission. In this paper, we extend the model of Mukandavire et al. [13] with periodic transmission rate. We consider the total human population sizes denoted by NðtÞ, which including susceptible individuals SðtÞ, infected individuals IðtÞ and recovered individuals RðtÞ. The pathogen population at time t, is given by BðtÞ. The susceptible human population is increased by births and/or immigration at a constant rate A ð> 0Þ. Natural death occurs in the human classes at a rate l ð> 0Þ. Infected individuals may die due to cholera at a rate d ð> 0Þ. Infective individuals recover with rate constant r ð> 0Þ and then have temporary immunity. The recovered individuals lose immunity and return to the susceptible class at a rate a ð> 0Þ. Infected people con^ ð> 0Þ and the cholera tribute to the concentration of vibrios at a rate n ð> 0Þ. The pathogen population is generated at a rate l ð> 0Þ in the aquatic environment, which in this case, is the set of untreated water conpathogen has a natural death rate l sumed by the population. According to Islam [14], we know that Vibrio cholerae population decay does not necessarily imply >l ^ , and vibrios have a net death rate death but also the transition towards a non-culturable state. Hence, we assume l l ^. d¼l Susceptible individuals acquire cholera infection either by ingesting environmental vibrios from contaminated aquatic reservoirs or through human-to-human transmission resulting from the ingestion of hyperinfectious vibrios at rates SB b1 ðtÞ KþB and b2 ðtÞSI, respectively. Here, K ð> 0Þ is the concentration of vibrios in contaminated water in the environment. b1 ðtÞ and b2 ðtÞ are the rates of ingesting vibrios from the contaminated environment and through human-to-human interaction, respectively. We assume that b1 ðtÞ and b2 ðtÞ are continuous, positive xperiodic functions. The model is given in the following:
8 dSðtÞ SB > ¼ A b1 ðtÞ KþB b2 ðtÞSI lS þ aR; > dt > > > < dIðtÞ ¼ b ðtÞ SB þ b ðtÞSI ðr þ l þ dÞI; 1 2 KþB dt > > dRðtÞ ¼ rI aR lR; > dt > > : dBðtÞ ¼ nI dB: dt
ð1:1Þ
System (1.1) will be discussed under the following nonnegative initial condition
ðSð0Þ; Ið0Þ; Rð0Þ; Bð0ÞÞ 2 R4þ :
ð1:2Þ
Obviously, any solution of system (1.1) with nonnegative initial condition (1.2) is nonnegative. The remainder of this paper is organized as follows. In the next section, we present the basic reproduction number of system (1.1). In Section 3, we obtain the extinction and uniform presentence of the disease. We discuss the existence and stability of positive xperiodic solution in Section 4. Numerical simulation are provided to validate analytical results in Section 5. The paper ends with a discussion. 2. The basic reproduction number of (1.1) In this section, we introduce the basic reproduction number R0 for system (1.1) according to the general procedure presented in Wang and Zhao [15]. The definition of R0 in periodic environments was firstly developed by Baca€ er and Guernaoui [16]. Theorem 2.1. The solutions ðSðtÞ; IðtÞ; RðtÞ; BðtÞÞ of the model (1.1) are uniformly and ultimately bounded, i.e., there exist an M > 0, and T > 0 such that ðSðtÞ; IðtÞ; RðtÞ; BðtÞÞ 6 ðM; M; M; MÞ, for t P T. Proof. By the first three equations of the system (1.1), we get
dðS þ I þ RÞ ¼ A lðS þ I þ RÞ dI 6 A lðS þ I þ RÞ: dt Hence, there exists t1 > 0 such that S þ I þ R 6 lA, for t P t1 . Then, S 6 lA ; I 6 lA and R 6 lA for t P t 1 . By the fourth equation nA 6 nA of (1.1), we get dB l dB for t P t 1 . Then the comparison theorem implies that there exists T P t 1 such that B 6 ld, for dt t P T. Let M ¼ maxflA ; lnAdg. Then it follows that S 6 M; I 6 M; I 6 M and B 6 M for t P T. Thus the solutions of system (1.1) are uniformly and ultimately bounded. h
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Let ðRn ; Rnþ Þ be the standard ordered ndimensional Euclidean space with a norm k k. For u; v 2 Rn , we denote u P v , if u v 2 Rnþ ; u > v , if u v 2 Rn ; u v , if u v 2 IntðRnþ Þ. Let AðtÞ be a continuous, cooperative, irreducible and x-periodic n n matrix function, UA ðtÞ be the fundamental solution matrix of
dx ¼ AðtÞx: dt
ð2:1Þ
Let qðUA ðxÞÞ be the spectral radius of UA ðxÞ. By Perron–Frobenius theorem, qðUA ðxÞÞ is the principal eigenvalue of UA ðxÞ in the sense that it is simple and admits an eigenvector v 0. The following lemma is useful for the discussion in the next section. 1 Lemma 2.1 ([17]). Let p ¼ x ln qðUA ðxÞÞ. Then there exists a positive x-periodic function v ðtÞ such that ept v ðtÞ is a solution of (2.1). In the following, we will calculate the basic reproduction number of system (1.1). It is easy to see that system (1.1) has exactly one disease-free equilibrium P0 ðS0 ; 0; 0; 0Þ, where S0 ¼ lA. Let x ¼ ðI; B; S; RÞ> . Then model (1.1) can be written as
dx ¼ F ðxÞ VðxÞ; dt where
0 B B B F ðxÞ ¼ B B @
SB b1 ðtÞ KþB þ b2 ðtÞSI
0 0 0
0
1 C C C C; C A
ðr þ l þ dÞI
1
C B dBðtÞ C B C; V þ ðxÞ ¼ B C B SB ðtÞ þ b ðtÞSI þ l S b A @ 1 KþB 2
0
0
C B B nIðtÞ C C V ðxÞ ¼ B C B @ A þ aR A
ða þ lÞR þ
1
rI
and VðxÞ ¼ V ðxÞ V ðxÞ. We can get
FðtÞ ¼
b2 ðtÞS0
b1 ðtÞ SK0
0
0
! ; VðtÞ ¼
rþlþd 0 n
d
:
Let Yðt; sÞ is a 2 2 matrix solution of the system
dYðt; sÞ ¼ VðtÞYðt; sÞ dt for any t 6 s; Yðs; sÞ ¼ I, where I is a 2 2 identity matrix. Let C x be the ordered Banach space of all xperiodic function from R ! R2 , which is equipped with maximum norm k k1 and the positive cone C þ x ¼ f/ 2 C x j/ðtÞ P 0; for any t 2 Rg. Consider the following linear operator L : C x ! C x by
ðL/ÞðtÞ ¼
Z
þ1
Yðt; t aÞFðt aÞ/ðt aÞda 0
for any t 2 R; / 2 C x . Finally, motivated by the concept of next generation matrices introduced in [18], one can define the basic reproduction number R0 of the system (1.1) as the spectral radius of L, i.e.,
R0 ¼ qðLÞ: From the above discussion, we obtain the following result for the local asymptotic stability of the disease free equilibrium P 0 ðS0 ; 0; 0; 0Þ. Theorem 2.2 ([15]). The following statements are valid: (1) R0 ¼ 1 if and only if qðUFV ðxÞÞ ¼ 1; (2) R0 > 1 if and only if qðUFV ðxÞÞ > 1;
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(3) R0 < 1 if and only if qðUFV ðxÞÞ < 1. Thus, the disease-free equilibrium P 0 ðS0 ; 0; 0; 0Þ of (1.1) is asymptotically stable if R0 < 1 and unstable if R0 > 1. 3. Extinction and uniform persistence of the disease In this section, we show that R0 serves as a threshold parameter: when R0 < 1, there exists a globally asymptotically stable disease-free equilibrium P 0 ðS0 ; 0; 0; 0Þ, and when R0 > 1, cholera is persistent in the population. The mathematical analysis is similar to [17]. Theorem 3.1. If R0 < 1, the disease-free equilibrium P0 ðS0 ; 0; 0; 0Þ of (1.1) is globally asymptotically stable and if R0 > 1, then it is unstable.
Proof. From Theorem 2.2, if R0 > 1, then P 0 ðS0 ; 0; 0; 0Þ is unstable and if R0 < 1, then P 0 ðS0 ; 0; 0; 0Þ is locally asymptotically stable. Hence, it is sufficient to show that the global attractivity of P 0 ðS0 ; 0; 0; 0Þ for R0 < 1. Assume that R0 < 1, again from Theorem 2.2, we have qðUFV ðxÞÞ < 1. From the second and the last equations we know that
( dIðtÞ dt dBðtÞ dt
S0 B 6 b1 ðtÞ KþB þ b2 ðtÞS0 I ðr þ l þ dÞI;
¼ nI dB
for t P 0. Consider the following auxiliary system:
( dI
1 ðtÞ dt
dB1 ðtÞ dt
S0 B1 ¼ b1 ðtÞ KþB þ b2 ðtÞS0 I1 ðr þ l þ dÞI1 ; 1
¼ nI1 dB1 :
By Lemma 2.1 and the standard comparison principle, there exists a positive xperiodic function v 1 ðtÞ such that J ðtÞ 6 v 1 ðtÞep t , where J ðtÞ ¼ ðIðtÞ; BðtÞÞ> and p1 ¼ x1 ln qðUFV ðxÞÞ < 0. Then we conclude that limt!1 IðtÞ ¼ 0 and limt!1 BðtÞ ¼ 0, it 1
follows that limt!1 RðtÞ ¼ 0 and limt!1 SðtÞ ¼ lA . Therefore, the disease-free equilibrium P 0 ðS0 ; 0; 0; 0Þ of (1.1) is globally asymptotically stable.
h
Theorem 3.2. If R0 > 1, the system (1.1) is uniformly persistent, i.e., there exists a positive constant e, such that for all initial values ðSð0Þ; Ið0Þ; Rð0Þ; Bð0ÞÞ 2 Rþ IntðRþ Þ Rþ IntðRþ Þ, the solutions of (1.1) satisfies lim inf t!1 ðSðtÞ; IðtÞ; RðtÞ; BðtÞÞ P ðe; e; e; eÞ. Proof. Let us define
X ¼ Rþ4 ;
X 0 ¼ Rþ IntðRþ Þ Rþ IntðRþ Þ;
@X 0 ¼ X n X 0 :
þ þ 0 0 0 0 Define Poincaré map P : Rþ 4 ! R4 , satisfying Pðx Þ ¼ uðx; x Þ; 8x 2 R4 , with uðt; x Þ the unique solution of (1.1) satisfying uð0; x0 Þ ¼ x0 . Firstly, we show that P is uniformly persistent with respect to ðX 0 ; @X 0 Þ. It is easy to see from system (1.1) that X and X 0 are positively invariant. Moreover, @X 0 is a relatively closed set in X. It follows from Theorem 2.1 that solutions of system þ þ (1.1) are uniformly and ultimately bounded. Thus the semiflow P is point dissipative on Rþ 4 , and P : R4 ! R4 is compact. By Theorem 3.4.8 in [19], it then follows that P admits a global attractor, which attracts every bounded set in Rþ 4.
Define
M@ ¼ fðSð0Þ; Ið0Þ; Rð0Þ; Bð0ÞÞ 2 @X 0 : Pm ðS0 ; I0 ; R0 ; B0 Þ 2 @X 0 ; 8m P 0g: Next, we claim that
M@ ¼ fðS; 0; R; 0Þ : S P 0; R P 0g: In fact, it is obvious that
fðS; 0; R; 0Þ : S P 0; R P 0g # M @ : For any ðSð0Þ; Ið0Þ; Rð0Þ; Bð0ÞÞ 2 @X 0 nfðS; 0; R; 0Þ : S P 0; R P 0g, if Ið0Þ ¼ 0; Bð0Þ > 0, it is clear that S > 0; B > 0 for all t > 0, from _ the second equation of (1.1), we have Ið0Þ ¼ b1 ðtÞ Sð0ÞBð0Þ > 0; else if B0 ¼ 0; I0 > 0, then by the last equation of (1.1), we have KþBð0Þ _ Bð0Þ ¼ nIð0Þ > 0. Therefore, ðSðtÞ; IðtÞ; RðtÞ; BðtÞÞ R @X 0 for t > 0 sufficiently small. That is to say, for any ðSð0Þ; Ið0Þ; Rð0Þ; Bð0ÞÞ R fðS; 0; R; 0Þ : S P 0; R P 0g; ðSð0Þ; Ið0Þ; Rð0Þ; Bð0ÞÞ R @X 0 g. This implies that M @ # fðS; 0; R; 0Þ : S P 0; R P 0g. Therefore, M@ ¼ fðS; 0; R; 0Þ : S P 0; R P 0g.
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Clearly, P0 is one fixed point of P in M @ . If ðSðtÞ; IðtÞ; RðtÞ; BðtÞÞ is a solution of system (1.1) initiating from M @ , it then follows from system (1.1) that SðtÞ ! S0 ; IðtÞ ! 0; RðtÞ ! 0; BðtÞ ! 0 as t ! 1. In the following we shall show that if the invariant set P0 is isolated, then fP0 g is an acyclic covering. To do this, it needs to prove any solution of system (1.1) initiating from M @ will remain into M @ , which can be obtained easily. The isolated invariance of P0 will follow proof. We now show W s ðP 0 Þ \ X 0 ¼ ;. Denote x0 ¼ ðSð0Þ; Ið0Þ; Rð0Þ; Bð0ÞÞ 2 X 0 . By the continuity of solutions with respect to the initial values, 8e 2 ð0; lAÞ, there exists 1 > 0 such that for all x0 2 X 0 with jjx0 P0 jj 6 1, it follows that
jjuðt; x0 Þ uðt; P0 Þjj 6 e;
8t 2 ½0; x:
We will show that lim supt!1 dðP m ðx0 Þ; P0 Þ P 1. If not, then lim supt!1 dðPm ðx0 Þ; P 0 Þ < 1 for some x0 2 X 0 . Without loss of generality, we can assume that lim supt!1 dðPm ðx0 Þ; P0 Þ < 1 for all m > 0. Then we know that
jjuðt; Pm ðx0 ÞÞ uðt; P0 Þjj 6 e;
8t 2 ½0; x:
For any t P 0, let t ¼ mx þ t1 , where t1 2 ½0; x and m ¼ ½xt , which is the greatest integer less than or equal to xt . Then we have
jjuðt; Pm ðx0 ÞÞ uðt; P0 Þjj ¼ jjuðt 1 ; P m ðx0 ÞÞ uðt 1 ; P 0 Þjj 6 e; 0
A
8t 2 ½0; x: A
Set ðSðtÞ; IðtÞ; RðtÞ; BðtÞÞ ¼ uðt; x Þ, it follows that l e 6 S 6 l þ e; 0 6 I 6 e; 0 6 R 6 e and 0 6 B 6 e for t P 0. Then S0 2S0 e S 0 e P SKþ e ¼ K Kþe . Thus, from system (1.1), we obtain KþB
( dIðtÞ
dt dBðtÞ dt
0e P b1 ðtÞðSK0 2S ÞB þ b2 ðtÞðS0 eÞI ðr þ l þ dÞI; Kþe
¼ nI dB:
Set
Me ¼
eb2 ðtÞ
2S0 e b ðtÞ Kþe 1
0
0
! :
By Theorem 2.2, we know that qðUFV ðxÞÞ > 1, then we can choose e > 0 small enough such that qðUFVMe ðxÞÞ > 1. Again by Lemma 2.1 and the standard comparison principle, there exists a positive xperiodic function v 2 ðtÞ such that J ðtÞ P v 2 ðtÞep2 t , where J ðtÞ ¼ ðIðtÞ; BðtÞÞ> and p2 ¼ x1 ln qðUFVMe ðxÞÞ > 0, which implies that limt!1 IðtÞ ¼ 1 and limt!1 BðtÞ ¼ 1, this is a contradiction in M @ converges to P 0 , and hence P0 is acyclic in M @ . By Theorem 1.3.1 and Remark 1.3.1 in [19], we obtain that P is uniformly persistent with respect to ðX 0 ; @X 0 Þ. It follow from Theorem 3.1.1 in [19] that the solution of (1.1) is uniformly persistent. h
4. Periodic solution In this section we shall investigate existence and stability of a positive periodic solution of system (1.1). Theorem 4.1. If R0 > 1, then the system (1.1) admits a positive xperiodic solution which is globally asymptotically stable. Proof. It follows from Theorem 2.1 that solutions of system (1.1) are uniformly and ultimately bounded. Thus the semiflow P is point dissipative on R4þ , and P : R4þ ! R4þ is compact. By Theorem 3.2, P is uniformly persistent with respect to ðX 0 ; @X 0 Þ. b BÞ b 2 IntðR4 Þ. Hence, Then it follows from Theorem 1.3.6 in [19] that the Poincaré map P has a fixed point ðb S; bI; R; þ
b BÞÞ b 2 IntðR4 Þ, for all t > 0. Thus ðb b b uðt; ðb S; bI; R; SðtÞ; bIðtÞ; RðtÞ; BðtÞÞ is a positive xperiodic solution of system (1.1) due to þ the definition of the semiflow P. b ðtÞ ¼ ðb b b Let X SðtÞ; bIðtÞ; RðtÞ; BðtÞÞ be positive xperiodic solution of system (1.1), and XðtÞ ¼ ðSðtÞ; IðtÞ; RðtÞ; BðtÞÞ any solution of system (1.1) initialing from nonnegative initial values (1.2). Define the following Lyapunov function
l b b LðS; I; R; BÞ ¼ jSðtÞ b SðtÞj þ jIðtÞ bIðtÞj þ jRðtÞ RðtÞj þ jBðtÞ BðtÞj: n Using the fact
jxj0 ¼
x0 ; if x P 0 x0 if x < 0
¼ signðxÞx
b B > B; b (2) S > b b B > B; b (3) S > b b and considering the following 16 cases: (1) S > b S; I > bI; R > R; S; I < bI; R > R; S; I > bI; R < R; b b b b b b b b b b b b b b b b B > B; (4) S > S; I > I; R > R; B < B; (5) S > S; I < I; R < R; B > B; (6) S > S; I < I; R > R; B < B; (7) S > S; I > I; R < R;
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b (8) S > b b B < B; b (9) S < b b B > B; b (10) S < b b B > B; b (11) S < b B < B; S; I < bI; R < R; S; I > bI; R > R; S; I < bI; R > R; S; I > bI; b (13) S < b b B > B; b (14) S < b b B < B; b (15) b B > B; b (12) S < b b B < B; S; I < bI; R < R; S; I < bI; R > R; R < R; S; I > bI; R > R; b B < B; b (16) S < b b B < B, b respectively, we can obtain the right-upper derivative Dþ LðtÞ of S bI; R < R; S; I < bI; R < R; LðtÞ following system (1.1):
b b SB
SB b þ signðIðtÞ SÞ þ aðR RÞg b2 ðtÞSI lðS b K þB b b SB SB b þ b2 ðtÞSI ðr þ l þ dÞðI bIÞ b1 ðtÞ SbIg þ signðRðtÞ RðtÞÞfrðI bIÞ bIðtÞÞfb1 ðtÞ b2 ðtÞb b K þB KþB ^ 6 ljS b b þ l signðBðtÞ BðtÞÞfnðI b dl jB Bj: b b Sj djI bIj ljR Rj ða þ lÞðR RÞg bIÞ dðB BÞg n n
Dþ LðS; I; R; BÞ ¼ signðSðtÞ b SðtÞÞfb1 ðtÞ
Let
b KþB
SbI b1 ðtÞ þ b2 ðtÞb
v ¼ minfl; d; dg, then it follows that b þ jB Bjg: b Dþ LðS; I; R; BÞ 6 vfjS b Sj þ jI bIj þ jR Rj
Integrating the above inequality from t to þ1, we obtain
LðtÞ þ v
Z
þ1
t
b þ jB BjÞds b ðjS b Sj þ jI bIj þ jR Rj 6 LðtÞ:
Provided that t > t, it follows that
supv t!1
Z t
þ1
LðtÞ b þ jB BjÞds b ðjS b Sj þ jI bIj þ jR Rj 6 < þ1:
v
Then we have
limjS b Sj ¼ 0;
t!1
limjI bIj ¼ 0;
t!1
b ¼ 0; limjR Rj
t!1
b ¼ 0: limjB Bj
t!1
b b That is to say, the positive period solution ðb SðtÞ; bIðtÞ; RðtÞ; BðtÞÞ is globally asymptotically stable.
h
5. Numerical simulations From our theoretical results we see that R0 is a threshold parameter to determine whether or not cholera persists in the population. In this section, our numerical simulations will demonstrate the asymptotical behavior of system (1.1) in different cases. The model equations are solved numerically using the ODE solver ode45 in Matlab and results are plotted graphically. 2pt 2pt The is used to solve system (1.1). We select b1 ðtÞ ¼ b10 ð1 þ d10 cos 365 Þ and b2 ðtÞ ¼ b20 ð1 þ d20 cos 365 Þ. And other parameters of the system (1.1) are listed in Table 1. By the approximation method in [21], we can compute the basic reproduction number of (1.1) as
R10
! S0 ðb20 Kd þ b10 nÞ Kðr þ l þ dÞ d210 þ d220 : ’ 1 2p 2 Kðr þ l þ dÞ 2 ð365 Þ þ ðK þ r þ l þ dÞ2
Similarly to Theorems 3.1 and 4.1, we have the following results: (1) If R10 < 1, the disease-free equilibrium P0 ðS0 ; 0; 0; 0Þ of (1.1) is globally asymptotically stable and if R10 > 1, then it is unstable; (2) If R10 > 1, then the system (1.1) admits a positive xperiodic solution which is globally asymptotically stable. Table 1 Estimation of parameters. Parameters
Meaning
Values
Reference
A n
15/day 100 cells/L-per day
Assumed [3]
l
Recruitment rate of susceptible population Contribution of infected individuals to the population of Vibrio cholerae Natural death rate of human
d r d K
Net death rate of Vibrio cholerae Recovery rate Disease-induced death rate Concentration of Vibrio cholerae in water
a
Progression rare of the recovered
5:48 105 /day 0.33/day 0.004/day 0.015/day 109 cells/L 0.025
[20] [3] [1,4] [4] [4] Assumed
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Fig. 1. shows that the disease-free equilibrium P 0 of system (1.1) is globally asymptotically stable when R0 < 1. In this case, the initial values are ðSð0Þ; Ið0Þ; Rð0Þ; Bð0ÞÞ ¼ ð800000; 800; 100; 3000000Þ. (a) Number of susceptible individuals against time; (b) number of infected individuals against time; (c) number of recovered individuals against time; (d) number of pathogen population against time.
Fig. 2. Fig. 2 shows that the system (1.1) exists a periodic solution when R0 > 1. In this case, the initial values are ðSð0Þ; Ið0Þ; Rð0Þ; Bð0ÞÞ ¼ ð8000; 800; 100; 30000Þ. (a) Number of susceptible individuals against time; (b) number of infected individuals against time; (c) number of recovered individuals against time; (d) Number of pathogen population against time.
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Hence, R10 can be used as an approximation to R0 . Case 1: b10 ¼ 0:2143; d10 ¼ 0:5; b20 ¼ 0:00000006; d20 ¼ 0:75. We can obtain R0 ’ 0:5922698711 < 1. The simulation shows that the disease dies out (see Fig. 1). The simulation results are the same as what we got in Theorem 3.1. In this case, the initial values are ðSð0Þ; Ið0Þ; Rð0Þ; Bð0ÞÞ ¼ ð800000; 800; 100; 3000000Þ. Case 2: b10 ¼ 0:2143; d10 ¼ 0:5; b20 ¼ 0:000002; d20 ¼ 0:75. We can obtain R0 ’ 9:788757174 > 1. The results of Theorems 3.2 and 4.1 indicate that system (1.1) has one positive periodic solution and the disease keeps persistent in the population. Fig. 2 confirms this conclusion, and the simulation suggests that in the case where R0 > 1, every solution with nontrivial initial data is asymptotic to a periodic solution (see Fig. 2). In this case, the initial values are ðSð0Þ; Ið0Þ; Rð0Þ; Bð0ÞÞ ¼ ð8000; 800; 100; 30000Þ. 6. Discussion In this paper, we have formulated and analyzed a periodic cholera model. We suppose that the transmission changes periodically as time varies. We obtained a threshold parameter qðUFV ðxÞÞ, which determines the extinction and uniform persistence of the disease. It shows that the disease-free equilibrium is globally asymptotically stable if qðUFV ðxÞÞ < 1, while the disease persists if qðUFV ðxÞÞ > 1 and the system has one positive periodic solution which is globally asymptotically stable. Numerical simulations demonstrate our theoretical results. If both b1 ðtÞ and b2 ðtÞ are constants (denoted by b1 and b2 , respectively), then the system (1.1) is reduced the following
8 dSðtÞ SB > ¼ A b1 KþB b2 SI lS þ aR; > dt > > > < dIðtÞ ¼ b SB þ b SI ðr þ l þ dÞI; 1 KþB 2 dt > dRðtÞ ¼ rI aR lR; > > dt > > : dBðtÞ ¼ nI dB: dt
ð6:1Þ
b2 S0 b1 S0 n A We can obtain the basic reproduction number of the system (6.1) R01 ¼ rþdþ l þ KdðrþlþdÞ, where S0 ¼ l. Similar to [13], we can obtain that the system (6.1) has a disease-free equilibrium P0 ðS0 ; 0; 0; 0Þ. It is globally asymptotically stable whenever R01 < 1. And if R01 > 1, the system (6.1) has a unique endemic equilibrium P ðS ; I ; R ; B Þ, which is globally asymptotically stable whenever R01 > 1. Comparing the results of the system (1.1) and the system (6.1), we can conclude that seasonal fluctuation is an important phenomenon in infectious disease transmission. Mathematical models of cholera with periodic transmission rate can match the actuality of cholera better than the autonomous ones. We can use these analytic results to study the cholera transmission for a certain area, such as Zimbabwe, Uganda, Haiti, China, etc. But we cannot obtain the detailed data. In the future we will contact the Centers for Disease Control and Prevention of some countries which have breakout cholera and use our model practically. Lastly, we can study the complex dynamic behaviors such as bifurcation and chaos of system (1.1) by using the method of Kuznetsov and Piccardi [22]. We leave it in the future.
Acknowledgements We would like to thank the responsible editor, Prof. Turner and the anonymous referees for their careful reading of the original manuscript and their many valuable comments and suggestions that greatly improve the presentation of this work. References [1] T.R. Hendrix, The pathophysiology of cholera, Bull. N.Y. Acad. Med. 47 (10) (1971) 1169–1180. [2] V. Capasso, S.L. Paveri-Fontana, A mathematical model for the 1973 cholera epidemic in the European Mediterranean region, Rev. Epidemiol. Sante Publ. 27 (1979) 121–132. [3] C.T. Codeço, Endemic and epidemic dynamics of cholera: the role of the aquatic reservoir, BMC Infect. Dis. 1 (1) (2001). [4] D.M. Hartley, J.G. Morris, D.L. Smith, Hyperinfectivity: a critical element in the ability of V. cholerae to cause epidemics?, PLoS Med 3 (2006) 63–69. [5] S. Liao, J. Wang, Stability analysis and application of a mathematical cholera model, Math. Biosci. Eng. 8 (3) (2011) 733–752. [6] X.Y. Zhou, J.A. Cui, Modelling and stability analysis for a cholera model with vaccination, Math. Method Appl. Sci. 34 (14) (2011) 1711–1724. [7] I.M. Longini Jr., A. Nizam, M. Ali, M. Yunus, N. Shenvi, J.D. Clemens, Controlling endemic cholera with oral vaccines, PLoS Med. 4 (11) (2007) e336. [8] R.L. Miller Neilan, E. Schaefer, H. Gaff, K.R. Fister, S. Lenhart, Modeling optimal intervention strategies for cholera, Bull. Math. Biol. 72 (2010) 2004– 2018. [9] R.P. Sanches, C.P. Ferreira, R.A. Kraenkel, The role of immunity and seasonality in cholera epidemics, Bull. Math. Biol. 73 (12) (2011) 2916–2931. [10] M. Pascual, M.J. Bouma, A.P. Dobson, Cholera and climate: revisiting the quantitative evidence, Microb. Infect. 4 (2002) 237–245. [11] M. Emch, C. Feldacker, M.S. Islam, M. Ali, Seasonality of cholera from 1974 to 2005: a review of global patterns, Int. J. Health Geographics 7 (2008) 31– 44. [12] E. Bertuzzo, M. Gatto, A. Maritan, S. Azaele, I. Rodiguez-Iturbe, A. Rinaldo, On the space-time evolution of a cholera epidemics, Water Resour. Res. 44 (2008) W01424. [13] Z. Mukandavire, S. Liao, J. Wang, H. Gaff, D.L. Smith, J.G. Morris, Estimating the reproductive numbers for the 2008–2009 cholera outbreaks in Zimbabwe, PNAS 108 (21) (2011) 8767–8772. [14] M.S. Islam, M.A. Miah, M.K. Hasan, R.B. Sack, M.J. Albert, Detection of non-culturable Vibrio cholerae O1 associated with a cyanobacterium from an aquatic environment in Bangladesh, Trans. R. Soc. Trop. Med. Hyg. 88 (1994) 298–299.
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W.D. Wang, X.Q. Zhao, Threshold dynamics for compartmental epidemic models in periodic environments, J. Dyn. Differ. Equat. 20 (3) (2008) 699–717. N. Bacaër, S. Guernaoui, The epidemic threshold of vector-borne diseases with seasonality, J. Math. Biol. 53 (2006) 421–436. F. Zhang, X.Q. Zhao, A periodic epidemic model in a patchy environment, J. Math. Anal. Appl. 325 (1) (2007) 496–516. P. van den Driessche, J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosci. 180 (2002) 29–48. X.Q. Zhao, Dynamical Systems in Population Biology, CMS Books Math./Ouvrages Math. SMC, vol. 16, Springer-Verlag, New York, 2003. Z. Mukandavire, W. Garira, Sex-structured HIV/AIDS model to analyse the effects of condom use with application to Zimbabwe, J. Math. Biol. 54 (2007) 669–699. N. Bacaër, Approximation of the basic reproduction number R0 for vector-borne diseases with a periodic vector population, Bull. Math. Biol. 69 (2007) 1067–1091. Y.A. Kuznetsov, C. Piccardi, Bifurcation analysis of periodic SEIR and SIR epidemic models, J. Math. Biol. 32 (1994) 109–121.
Contents lists available at SciVerse ScienceDirect
Applied Mathematical Modelling journal homepage: www.elsevier.com/locate/apm
Threshold dynamics for a cholera epidemic model with periodic transmission rate q Xue-yong Zhou a,b, Jing-an Cui c,⇑ a
College of Mathematics and Information Science, Xinyang Normal University, Xinyang 464000, Henan, PR China School of Mathematical Sciences, Nanjing Normal University, Nanjing 210046, Jiangsu, PR China c School of Science, Beijing University of Civil Engineering and Architecture, Beijing 100044, PR China b
a r t i c l e
i n f o
Article history: Received 11 December 2011 Received in revised form 14 May 2012 Accepted 12 July 2012 Available online 16 August 2012 Keywords: Cholera model Global stability Uniform persistence Extinction
a b s t r a c t A cholera epidemic model with periodic transmission rate is presented. The basic reproduction number is defined. It is shown that the disease-free equilibrium is globally asymptotically stable and the cholera eventually disappears if the basic reproduction number is less than one. And if the basic reproduction number is greater than one, there exists a positive periodic solution which is globally asymptotically stable. Numerical simulations are provided to illustrate analytical results. Ó 2012 Elsevier Inc. All rights reserved.
1. Introduction Cholera remains a major public health threat in many developing countries around the world. It is an acute intestinal infection caused by the bacterium Vibrio cholerae. Cholera is typically transmitted through water or foods that have been contaminated with fecal matter from a person who is infected with the disease. Cholera can spread rapidly, and epidemics may occur after fecal contamination of food or water supplies [1]. Recently, a number of mathematical models have been developed to help in understanding the dynamics of cholera outbreaks. In 1979, Capasso and Paveri-Fontana [2] presented a mathematical model for cholera epidemic occurred in the European Mediterranean region in 1973. In Capasso’s version, two equations describe the dynamics of infected people in the community and the dynamics of the aquatic population of pathogenic bacteria. After that, several cholera models were formulated and analyzed. In 2001, Codeço extended the cholera model of Capasso and Paveri-Fontana [2] with an additional equation for the susceptible individuals in the host population and explored the role of the aquatic reservoir in the persistence of endemic cholera [3]. Hartley et al. [4] in 2006 extended Codeço’s work to include hyperinfectious vibrios. In Liao and Wang [5], conducted the dynamical analysis (for example, the stability of equilibria of the system) of the deterministic cholera model proposed in [4]. In Zhou and Cui [6], considered a cholera model with vaccination. They analyzed the locally and globally asymptotical stability of the disease-free and endemic equilibria of their system. We may find other mathematical studies on modeling cholera dynamics in references [7–9]. The seasonality of cholera is apparent. Occurrence of cholera is typically seasonal due climatic factors, physical, and biological factor. For example, seasonality of cholera in Bangladesh exhibits two peaks per year and differs from that of other q This work is supported by the National Natural Science Foundation of China (No. 11071011), Funding Project for Academic Human Resources Development in Institutions of Higher Learning Under the Jurisdiction of Beijing Municipality (No. PHR201107123). ⇑ Corresponding author. Tel./fax: +86 010 61209413. E-mail addresses: [email protected] (X.-y. Zhou), [email protected] (J.-a. Cui).
0307-904X/$ - see front matter Ó 2012 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.apm.2012.07.044
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diarrheal diseases [10]. In Pakistan, classical cholera typically increases from November to January and from April to May while in Kolkata, India, seasonal patterns of cholera cases peak in April, May, and June [11]. Seasonality in cholera is seldom considered in the literature of cholera modeling (as well as endemic behavior), even though epidemiological patterns clearly show seasonal signatures and strong correlations with environmental drivers [10]. In [12], Bertuzzo et al. studied how river networks, acting as environmental corridors for pathogens, affect the spreading of cholera epidemics. And the seasonality had been taken into account in this reference. In [13], Mukandavire et al. presented a cholera epidemic model including human-to-human transmission to estimate the reproductive numbers for the 2008–2009 cholera outbreaks in Zimbabwe. But they did not consider the seasonality of cholera. As far as we know, almost no references considered the cholera with both seasonality and human-to-human transmission. In this paper, we extend the model of Mukandavire et al. [13] with periodic transmission rate. We consider the total human population sizes denoted by NðtÞ, which including susceptible individuals SðtÞ, infected individuals IðtÞ and recovered individuals RðtÞ. The pathogen population at time t, is given by BðtÞ. The susceptible human population is increased by births and/or immigration at a constant rate A ð> 0Þ. Natural death occurs in the human classes at a rate l ð> 0Þ. Infected individuals may die due to cholera at a rate d ð> 0Þ. Infective individuals recover with rate constant r ð> 0Þ and then have temporary immunity. The recovered individuals lose immunity and return to the susceptible class at a rate a ð> 0Þ. Infected people con^ ð> 0Þ and the cholera tribute to the concentration of vibrios at a rate n ð> 0Þ. The pathogen population is generated at a rate l ð> 0Þ in the aquatic environment, which in this case, is the set of untreated water conpathogen has a natural death rate l sumed by the population. According to Islam [14], we know that Vibrio cholerae population decay does not necessarily imply >l ^ , and vibrios have a net death rate death but also the transition towards a non-culturable state. Hence, we assume l l ^. d¼l Susceptible individuals acquire cholera infection either by ingesting environmental vibrios from contaminated aquatic reservoirs or through human-to-human transmission resulting from the ingestion of hyperinfectious vibrios at rates SB b1 ðtÞ KþB and b2 ðtÞSI, respectively. Here, K ð> 0Þ is the concentration of vibrios in contaminated water in the environment. b1 ðtÞ and b2 ðtÞ are the rates of ingesting vibrios from the contaminated environment and through human-to-human interaction, respectively. We assume that b1 ðtÞ and b2 ðtÞ are continuous, positive xperiodic functions. The model is given in the following:
8 dSðtÞ SB > ¼ A b1 ðtÞ KþB b2 ðtÞSI lS þ aR; > dt > > > < dIðtÞ ¼ b ðtÞ SB þ b ðtÞSI ðr þ l þ dÞI; 1 2 KþB dt > > dRðtÞ ¼ rI aR lR; > dt > > : dBðtÞ ¼ nI dB: dt
ð1:1Þ
System (1.1) will be discussed under the following nonnegative initial condition
ðSð0Þ; Ið0Þ; Rð0Þ; Bð0ÞÞ 2 R4þ :
ð1:2Þ
Obviously, any solution of system (1.1) with nonnegative initial condition (1.2) is nonnegative. The remainder of this paper is organized as follows. In the next section, we present the basic reproduction number of system (1.1). In Section 3, we obtain the extinction and uniform presentence of the disease. We discuss the existence and stability of positive xperiodic solution in Section 4. Numerical simulation are provided to validate analytical results in Section 5. The paper ends with a discussion. 2. The basic reproduction number of (1.1) In this section, we introduce the basic reproduction number R0 for system (1.1) according to the general procedure presented in Wang and Zhao [15]. The definition of R0 in periodic environments was firstly developed by Baca€ er and Guernaoui [16]. Theorem 2.1. The solutions ðSðtÞ; IðtÞ; RðtÞ; BðtÞÞ of the model (1.1) are uniformly and ultimately bounded, i.e., there exist an M > 0, and T > 0 such that ðSðtÞ; IðtÞ; RðtÞ; BðtÞÞ 6 ðM; M; M; MÞ, for t P T. Proof. By the first three equations of the system (1.1), we get
dðS þ I þ RÞ ¼ A lðS þ I þ RÞ dI 6 A lðS þ I þ RÞ: dt Hence, there exists t1 > 0 such that S þ I þ R 6 lA, for t P t1 . Then, S 6 lA ; I 6 lA and R 6 lA for t P t 1 . By the fourth equation nA 6 nA of (1.1), we get dB l dB for t P t 1 . Then the comparison theorem implies that there exists T P t 1 such that B 6 ld, for dt t P T. Let M ¼ maxflA ; lnAdg. Then it follows that S 6 M; I 6 M; I 6 M and B 6 M for t P T. Thus the solutions of system (1.1) are uniformly and ultimately bounded. h
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Let ðRn ; Rnþ Þ be the standard ordered ndimensional Euclidean space with a norm k k. For u; v 2 Rn , we denote u P v , if u v 2 Rnþ ; u > v , if u v 2 Rn ; u v , if u v 2 IntðRnþ Þ. Let AðtÞ be a continuous, cooperative, irreducible and x-periodic n n matrix function, UA ðtÞ be the fundamental solution matrix of
dx ¼ AðtÞx: dt
ð2:1Þ
Let qðUA ðxÞÞ be the spectral radius of UA ðxÞ. By Perron–Frobenius theorem, qðUA ðxÞÞ is the principal eigenvalue of UA ðxÞ in the sense that it is simple and admits an eigenvector v 0. The following lemma is useful for the discussion in the next section. 1 Lemma 2.1 ([17]). Let p ¼ x ln qðUA ðxÞÞ. Then there exists a positive x-periodic function v ðtÞ such that ept v ðtÞ is a solution of (2.1). In the following, we will calculate the basic reproduction number of system (1.1). It is easy to see that system (1.1) has exactly one disease-free equilibrium P0 ðS0 ; 0; 0; 0Þ, where S0 ¼ lA. Let x ¼ ðI; B; S; RÞ> . Then model (1.1) can be written as
dx ¼ F ðxÞ VðxÞ; dt where
0 B B B F ðxÞ ¼ B B @
SB b1 ðtÞ KþB þ b2 ðtÞSI
0 0 0
0
1 C C C C; C A
ðr þ l þ dÞI
1
C B dBðtÞ C B C; V þ ðxÞ ¼ B C B SB ðtÞ þ b ðtÞSI þ l S b A @ 1 KþB 2
0
0
C B B nIðtÞ C C V ðxÞ ¼ B C B @ A þ aR A
ða þ lÞR þ
1
rI
and VðxÞ ¼ V ðxÞ V ðxÞ. We can get
FðtÞ ¼
b2 ðtÞS0
b1 ðtÞ SK0
0
0
! ; VðtÞ ¼
rþlþd 0 n
d
:
Let Yðt; sÞ is a 2 2 matrix solution of the system
dYðt; sÞ ¼ VðtÞYðt; sÞ dt for any t 6 s; Yðs; sÞ ¼ I, where I is a 2 2 identity matrix. Let C x be the ordered Banach space of all xperiodic function from R ! R2 , which is equipped with maximum norm k k1 and the positive cone C þ x ¼ f/ 2 C x j/ðtÞ P 0; for any t 2 Rg. Consider the following linear operator L : C x ! C x by
ðL/ÞðtÞ ¼
Z
þ1
Yðt; t aÞFðt aÞ/ðt aÞda 0
for any t 2 R; / 2 C x . Finally, motivated by the concept of next generation matrices introduced in [18], one can define the basic reproduction number R0 of the system (1.1) as the spectral radius of L, i.e.,
R0 ¼ qðLÞ: From the above discussion, we obtain the following result for the local asymptotic stability of the disease free equilibrium P 0 ðS0 ; 0; 0; 0Þ. Theorem 2.2 ([15]). The following statements are valid: (1) R0 ¼ 1 if and only if qðUFV ðxÞÞ ¼ 1; (2) R0 > 1 if and only if qðUFV ðxÞÞ > 1;
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(3) R0 < 1 if and only if qðUFV ðxÞÞ < 1. Thus, the disease-free equilibrium P 0 ðS0 ; 0; 0; 0Þ of (1.1) is asymptotically stable if R0 < 1 and unstable if R0 > 1. 3. Extinction and uniform persistence of the disease In this section, we show that R0 serves as a threshold parameter: when R0 < 1, there exists a globally asymptotically stable disease-free equilibrium P 0 ðS0 ; 0; 0; 0Þ, and when R0 > 1, cholera is persistent in the population. The mathematical analysis is similar to [17]. Theorem 3.1. If R0 < 1, the disease-free equilibrium P0 ðS0 ; 0; 0; 0Þ of (1.1) is globally asymptotically stable and if R0 > 1, then it is unstable.
Proof. From Theorem 2.2, if R0 > 1, then P 0 ðS0 ; 0; 0; 0Þ is unstable and if R0 < 1, then P 0 ðS0 ; 0; 0; 0Þ is locally asymptotically stable. Hence, it is sufficient to show that the global attractivity of P 0 ðS0 ; 0; 0; 0Þ for R0 < 1. Assume that R0 < 1, again from Theorem 2.2, we have qðUFV ðxÞÞ < 1. From the second and the last equations we know that
( dIðtÞ dt dBðtÞ dt
S0 B 6 b1 ðtÞ KþB þ b2 ðtÞS0 I ðr þ l þ dÞI;
¼ nI dB
for t P 0. Consider the following auxiliary system:
( dI
1 ðtÞ dt
dB1 ðtÞ dt
S0 B1 ¼ b1 ðtÞ KþB þ b2 ðtÞS0 I1 ðr þ l þ dÞI1 ; 1
¼ nI1 dB1 :
By Lemma 2.1 and the standard comparison principle, there exists a positive xperiodic function v 1 ðtÞ such that J ðtÞ 6 v 1 ðtÞep t , where J ðtÞ ¼ ðIðtÞ; BðtÞÞ> and p1 ¼ x1 ln qðUFV ðxÞÞ < 0. Then we conclude that limt!1 IðtÞ ¼ 0 and limt!1 BðtÞ ¼ 0, it 1
follows that limt!1 RðtÞ ¼ 0 and limt!1 SðtÞ ¼ lA . Therefore, the disease-free equilibrium P 0 ðS0 ; 0; 0; 0Þ of (1.1) is globally asymptotically stable.
h
Theorem 3.2. If R0 > 1, the system (1.1) is uniformly persistent, i.e., there exists a positive constant e, such that for all initial values ðSð0Þ; Ið0Þ; Rð0Þ; Bð0ÞÞ 2 Rþ IntðRþ Þ Rþ IntðRþ Þ, the solutions of (1.1) satisfies lim inf t!1 ðSðtÞ; IðtÞ; RðtÞ; BðtÞÞ P ðe; e; e; eÞ. Proof. Let us define
X ¼ Rþ4 ;
X 0 ¼ Rþ IntðRþ Þ Rþ IntðRþ Þ;
@X 0 ¼ X n X 0 :
þ þ 0 0 0 0 Define Poincaré map P : Rþ 4 ! R4 , satisfying Pðx Þ ¼ uðx; x Þ; 8x 2 R4 , with uðt; x Þ the unique solution of (1.1) satisfying uð0; x0 Þ ¼ x0 . Firstly, we show that P is uniformly persistent with respect to ðX 0 ; @X 0 Þ. It is easy to see from system (1.1) that X and X 0 are positively invariant. Moreover, @X 0 is a relatively closed set in X. It follows from Theorem 2.1 that solutions of system þ þ (1.1) are uniformly and ultimately bounded. Thus the semiflow P is point dissipative on Rþ 4 , and P : R4 ! R4 is compact. By Theorem 3.4.8 in [19], it then follows that P admits a global attractor, which attracts every bounded set in Rþ 4.
Define
M@ ¼ fðSð0Þ; Ið0Þ; Rð0Þ; Bð0ÞÞ 2 @X 0 : Pm ðS0 ; I0 ; R0 ; B0 Þ 2 @X 0 ; 8m P 0g: Next, we claim that
M@ ¼ fðS; 0; R; 0Þ : S P 0; R P 0g: In fact, it is obvious that
fðS; 0; R; 0Þ : S P 0; R P 0g # M @ : For any ðSð0Þ; Ið0Þ; Rð0Þ; Bð0ÞÞ 2 @X 0 nfðS; 0; R; 0Þ : S P 0; R P 0g, if Ið0Þ ¼ 0; Bð0Þ > 0, it is clear that S > 0; B > 0 for all t > 0, from _ the second equation of (1.1), we have Ið0Þ ¼ b1 ðtÞ Sð0ÞBð0Þ > 0; else if B0 ¼ 0; I0 > 0, then by the last equation of (1.1), we have KþBð0Þ _ Bð0Þ ¼ nIð0Þ > 0. Therefore, ðSðtÞ; IðtÞ; RðtÞ; BðtÞÞ R @X 0 for t > 0 sufficiently small. That is to say, for any ðSð0Þ; Ið0Þ; Rð0Þ; Bð0ÞÞ R fðS; 0; R; 0Þ : S P 0; R P 0g; ðSð0Þ; Ið0Þ; Rð0Þ; Bð0ÞÞ R @X 0 g. This implies that M @ # fðS; 0; R; 0Þ : S P 0; R P 0g. Therefore, M@ ¼ fðS; 0; R; 0Þ : S P 0; R P 0g.
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Clearly, P0 is one fixed point of P in M @ . If ðSðtÞ; IðtÞ; RðtÞ; BðtÞÞ is a solution of system (1.1) initiating from M @ , it then follows from system (1.1) that SðtÞ ! S0 ; IðtÞ ! 0; RðtÞ ! 0; BðtÞ ! 0 as t ! 1. In the following we shall show that if the invariant set P0 is isolated, then fP0 g is an acyclic covering. To do this, it needs to prove any solution of system (1.1) initiating from M @ will remain into M @ , which can be obtained easily. The isolated invariance of P0 will follow proof. We now show W s ðP 0 Þ \ X 0 ¼ ;. Denote x0 ¼ ðSð0Þ; Ið0Þ; Rð0Þ; Bð0ÞÞ 2 X 0 . By the continuity of solutions with respect to the initial values, 8e 2 ð0; lAÞ, there exists 1 > 0 such that for all x0 2 X 0 with jjx0 P0 jj 6 1, it follows that
jjuðt; x0 Þ uðt; P0 Þjj 6 e;
8t 2 ½0; x:
We will show that lim supt!1 dðP m ðx0 Þ; P0 Þ P 1. If not, then lim supt!1 dðPm ðx0 Þ; P 0 Þ < 1 for some x0 2 X 0 . Without loss of generality, we can assume that lim supt!1 dðPm ðx0 Þ; P0 Þ < 1 for all m > 0. Then we know that
jjuðt; Pm ðx0 ÞÞ uðt; P0 Þjj 6 e;
8t 2 ½0; x:
For any t P 0, let t ¼ mx þ t1 , where t1 2 ½0; x and m ¼ ½xt , which is the greatest integer less than or equal to xt . Then we have
jjuðt; Pm ðx0 ÞÞ uðt; P0 Þjj ¼ jjuðt 1 ; P m ðx0 ÞÞ uðt 1 ; P 0 Þjj 6 e; 0
A
8t 2 ½0; x: A
Set ðSðtÞ; IðtÞ; RðtÞ; BðtÞÞ ¼ uðt; x Þ, it follows that l e 6 S 6 l þ e; 0 6 I 6 e; 0 6 R 6 e and 0 6 B 6 e for t P 0. Then S0 2S0 e S 0 e P SKþ e ¼ K Kþe . Thus, from system (1.1), we obtain KþB
( dIðtÞ
dt dBðtÞ dt
0e P b1 ðtÞðSK0 2S ÞB þ b2 ðtÞðS0 eÞI ðr þ l þ dÞI; Kþe
¼ nI dB:
Set
Me ¼
eb2 ðtÞ
2S0 e b ðtÞ Kþe 1
0
0
! :
By Theorem 2.2, we know that qðUFV ðxÞÞ > 1, then we can choose e > 0 small enough such that qðUFVMe ðxÞÞ > 1. Again by Lemma 2.1 and the standard comparison principle, there exists a positive xperiodic function v 2 ðtÞ such that J ðtÞ P v 2 ðtÞep2 t , where J ðtÞ ¼ ðIðtÞ; BðtÞÞ> and p2 ¼ x1 ln qðUFVMe ðxÞÞ > 0, which implies that limt!1 IðtÞ ¼ 1 and limt!1 BðtÞ ¼ 1, this is a contradiction in M @ converges to P 0 , and hence P0 is acyclic in M @ . By Theorem 1.3.1 and Remark 1.3.1 in [19], we obtain that P is uniformly persistent with respect to ðX 0 ; @X 0 Þ. It follow from Theorem 3.1.1 in [19] that the solution of (1.1) is uniformly persistent. h
4. Periodic solution In this section we shall investigate existence and stability of a positive periodic solution of system (1.1). Theorem 4.1. If R0 > 1, then the system (1.1) admits a positive xperiodic solution which is globally asymptotically stable. Proof. It follows from Theorem 2.1 that solutions of system (1.1) are uniformly and ultimately bounded. Thus the semiflow P is point dissipative on R4þ , and P : R4þ ! R4þ is compact. By Theorem 3.2, P is uniformly persistent with respect to ðX 0 ; @X 0 Þ. b BÞ b 2 IntðR4 Þ. Hence, Then it follows from Theorem 1.3.6 in [19] that the Poincaré map P has a fixed point ðb S; bI; R; þ
b BÞÞ b 2 IntðR4 Þ, for all t > 0. Thus ðb b b uðt; ðb S; bI; R; SðtÞ; bIðtÞ; RðtÞ; BðtÞÞ is a positive xperiodic solution of system (1.1) due to þ the definition of the semiflow P. b ðtÞ ¼ ðb b b Let X SðtÞ; bIðtÞ; RðtÞ; BðtÞÞ be positive xperiodic solution of system (1.1), and XðtÞ ¼ ðSðtÞ; IðtÞ; RðtÞ; BðtÞÞ any solution of system (1.1) initialing from nonnegative initial values (1.2). Define the following Lyapunov function
l b b LðS; I; R; BÞ ¼ jSðtÞ b SðtÞj þ jIðtÞ bIðtÞj þ jRðtÞ RðtÞj þ jBðtÞ BðtÞj: n Using the fact
jxj0 ¼
x0 ; if x P 0 x0 if x < 0
¼ signðxÞx
b B > B; b (2) S > b b B > B; b (3) S > b b and considering the following 16 cases: (1) S > b S; I > bI; R > R; S; I < bI; R > R; S; I > bI; R < R; b b b b b b b b b b b b b b b b B > B; (4) S > S; I > I; R > R; B < B; (5) S > S; I < I; R < R; B > B; (6) S > S; I < I; R > R; B < B; (7) S > S; I > I; R < R;
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b (8) S > b b B < B; b (9) S < b b B > B; b (10) S < b b B > B; b (11) S < b B < B; S; I < bI; R < R; S; I > bI; R > R; S; I < bI; R > R; S; I > bI; b (13) S < b b B > B; b (14) S < b b B < B; b (15) b B > B; b (12) S < b b B < B; S; I < bI; R < R; S; I < bI; R > R; R < R; S; I > bI; R > R; b B < B; b (16) S < b b B < B, b respectively, we can obtain the right-upper derivative Dþ LðtÞ of S bI; R < R; S; I < bI; R < R; LðtÞ following system (1.1):
b b SB
SB b þ signðIðtÞ SÞ þ aðR RÞg b2 ðtÞSI lðS b K þB b b SB SB b þ b2 ðtÞSI ðr þ l þ dÞðI bIÞ b1 ðtÞ SbIg þ signðRðtÞ RðtÞÞfrðI bIÞ bIðtÞÞfb1 ðtÞ b2 ðtÞb b K þB KþB ^ 6 ljS b b þ l signðBðtÞ BðtÞÞfnðI b dl jB Bj: b b Sj djI bIj ljR Rj ða þ lÞðR RÞg bIÞ dðB BÞg n n
Dþ LðS; I; R; BÞ ¼ signðSðtÞ b SðtÞÞfb1 ðtÞ
Let
b KþB
SbI b1 ðtÞ þ b2 ðtÞb
v ¼ minfl; d; dg, then it follows that b þ jB Bjg: b Dþ LðS; I; R; BÞ 6 vfjS b Sj þ jI bIj þ jR Rj
Integrating the above inequality from t to þ1, we obtain
LðtÞ þ v
Z
þ1
t
b þ jB BjÞds b ðjS b Sj þ jI bIj þ jR Rj 6 LðtÞ:
Provided that t > t, it follows that
supv t!1
Z t
þ1
LðtÞ b þ jB BjÞds b ðjS b Sj þ jI bIj þ jR Rj 6 < þ1:
v
Then we have
limjS b Sj ¼ 0;
t!1
limjI bIj ¼ 0;
t!1
b ¼ 0; limjR Rj
t!1
b ¼ 0: limjB Bj
t!1
b b That is to say, the positive period solution ðb SðtÞ; bIðtÞ; RðtÞ; BðtÞÞ is globally asymptotically stable.
h
5. Numerical simulations From our theoretical results we see that R0 is a threshold parameter to determine whether or not cholera persists in the population. In this section, our numerical simulations will demonstrate the asymptotical behavior of system (1.1) in different cases. The model equations are solved numerically using the ODE solver ode45 in Matlab and results are plotted graphically. 2pt 2pt The is used to solve system (1.1). We select b1 ðtÞ ¼ b10 ð1 þ d10 cos 365 Þ and b2 ðtÞ ¼ b20 ð1 þ d20 cos 365 Þ. And other parameters of the system (1.1) are listed in Table 1. By the approximation method in [21], we can compute the basic reproduction number of (1.1) as
R10
! S0 ðb20 Kd þ b10 nÞ Kðr þ l þ dÞ d210 þ d220 : ’ 1 2p 2 Kðr þ l þ dÞ 2 ð365 Þ þ ðK þ r þ l þ dÞ2
Similarly to Theorems 3.1 and 4.1, we have the following results: (1) If R10 < 1, the disease-free equilibrium P0 ðS0 ; 0; 0; 0Þ of (1.1) is globally asymptotically stable and if R10 > 1, then it is unstable; (2) If R10 > 1, then the system (1.1) admits a positive xperiodic solution which is globally asymptotically stable. Table 1 Estimation of parameters. Parameters
Meaning
Values
Reference
A n
15/day 100 cells/L-per day
Assumed [3]
l
Recruitment rate of susceptible population Contribution of infected individuals to the population of Vibrio cholerae Natural death rate of human
d r d K
Net death rate of Vibrio cholerae Recovery rate Disease-induced death rate Concentration of Vibrio cholerae in water
a
Progression rare of the recovered
5:48 105 /day 0.33/day 0.004/day 0.015/day 109 cells/L 0.025
[20] [3] [1,4] [4] [4] Assumed
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Fig. 1. shows that the disease-free equilibrium P 0 of system (1.1) is globally asymptotically stable when R0 < 1. In this case, the initial values are ðSð0Þ; Ið0Þ; Rð0Þ; Bð0ÞÞ ¼ ð800000; 800; 100; 3000000Þ. (a) Number of susceptible individuals against time; (b) number of infected individuals against time; (c) number of recovered individuals against time; (d) number of pathogen population against time.
Fig. 2. Fig. 2 shows that the system (1.1) exists a periodic solution when R0 > 1. In this case, the initial values are ðSð0Þ; Ið0Þ; Rð0Þ; Bð0ÞÞ ¼ ð8000; 800; 100; 30000Þ. (a) Number of susceptible individuals against time; (b) number of infected individuals against time; (c) number of recovered individuals against time; (d) Number of pathogen population against time.
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Hence, R10 can be used as an approximation to R0 . Case 1: b10 ¼ 0:2143; d10 ¼ 0:5; b20 ¼ 0:00000006; d20 ¼ 0:75. We can obtain R0 ’ 0:5922698711 < 1. The simulation shows that the disease dies out (see Fig. 1). The simulation results are the same as what we got in Theorem 3.1. In this case, the initial values are ðSð0Þ; Ið0Þ; Rð0Þ; Bð0ÞÞ ¼ ð800000; 800; 100; 3000000Þ. Case 2: b10 ¼ 0:2143; d10 ¼ 0:5; b20 ¼ 0:000002; d20 ¼ 0:75. We can obtain R0 ’ 9:788757174 > 1. The results of Theorems 3.2 and 4.1 indicate that system (1.1) has one positive periodic solution and the disease keeps persistent in the population. Fig. 2 confirms this conclusion, and the simulation suggests that in the case where R0 > 1, every solution with nontrivial initial data is asymptotic to a periodic solution (see Fig. 2). In this case, the initial values are ðSð0Þ; Ið0Þ; Rð0Þ; Bð0ÞÞ ¼ ð8000; 800; 100; 30000Þ. 6. Discussion In this paper, we have formulated and analyzed a periodic cholera model. We suppose that the transmission changes periodically as time varies. We obtained a threshold parameter qðUFV ðxÞÞ, which determines the extinction and uniform persistence of the disease. It shows that the disease-free equilibrium is globally asymptotically stable if qðUFV ðxÞÞ < 1, while the disease persists if qðUFV ðxÞÞ > 1 and the system has one positive periodic solution which is globally asymptotically stable. Numerical simulations demonstrate our theoretical results. If both b1 ðtÞ and b2 ðtÞ are constants (denoted by b1 and b2 , respectively), then the system (1.1) is reduced the following
8 dSðtÞ SB > ¼ A b1 KþB b2 SI lS þ aR; > dt > > > < dIðtÞ ¼ b SB þ b SI ðr þ l þ dÞI; 1 KþB 2 dt > dRðtÞ ¼ rI aR lR; > > dt > > : dBðtÞ ¼ nI dB: dt
ð6:1Þ
b2 S0 b1 S0 n A We can obtain the basic reproduction number of the system (6.1) R01 ¼ rþdþ l þ KdðrþlþdÞ, where S0 ¼ l. Similar to [13], we can obtain that the system (6.1) has a disease-free equilibrium P0 ðS0 ; 0; 0; 0Þ. It is globally asymptotically stable whenever R01 < 1. And if R01 > 1, the system (6.1) has a unique endemic equilibrium P ðS ; I ; R ; B Þ, which is globally asymptotically stable whenever R01 > 1. Comparing the results of the system (1.1) and the system (6.1), we can conclude that seasonal fluctuation is an important phenomenon in infectious disease transmission. Mathematical models of cholera with periodic transmission rate can match the actuality of cholera better than the autonomous ones. We can use these analytic results to study the cholera transmission for a certain area, such as Zimbabwe, Uganda, Haiti, China, etc. But we cannot obtain the detailed data. In the future we will contact the Centers for Disease Control and Prevention of some countries which have breakout cholera and use our model practically. Lastly, we can study the complex dynamic behaviors such as bifurcation and chaos of system (1.1) by using the method of Kuznetsov and Piccardi [22]. We leave it in the future.
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