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Threshold dynamics of a malaria transmission model in periodic environment
Commun Nonlinear Sci Numer Simulat 18 (2013) 1288–1303
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Threshold dynamics of a malaria transmission model in periodic environment q Lei Wang, Zhidong Teng ⇑, Tailei Zhang College of Mathematics and System Sciences, Xinjiang University, Urumqi 830046, People’s Republic of China
a r t i c l e
i n f o
Article history: Received 2 February 2012 Received in revised form 8 August 2012 Accepted 5 September 2012 Available online 17 October 2012 Keywords: Malaria transmission Periodic environment Disease-free periodic solution Uniform persistence Extinction Global stability Chaotic attractor Numerical simulation
a b s t r a c t In this paper, we propose a malaria transmission model with periodic environment. The basic reproduction number R0 is computed for the model and it is shown that the disease-free periodic solution of the model is globally asymptotically stable when R0 < 1, that is, the disease goes extinct when R0 < 1, while the disease is uniformly persistent and there is at least one positive periodic solution when R0 > 1. It indicates that R0 is the threshold value determining the extinction and the uniform persistence of the disease. Finally, some examples are given to illustrate the main theoretical results. The numerical simulations show that, when the disease is uniformly persistent, different dynamic behaviors may be found in this model, such as the global attractivity and the chaotic attractor. Ó 2012 Elsevier B.V. All rights reserved.
1. Introduction Malaria remains one of the most prevalent and lethal human infectious diseases in the world. It is estimated that more than 40% of the world’s population-at least 2.4 billion people-are exposed to varying degrees of risk of attaching the disease. The incidence of malaria is approximately 300–500 million clinical cases, resulting in 1 million deaths each year, primarily among young children in Africa, where malaria accounts for 20% of all deaths in children younger than 5 years in 2008 (see [1]). Of all infectious diseases, malaria continues to be one of the biggest contributors to the global disease burdens in terms of suffering and death and keeps on receiving worldwide attention (see [2,3]). Malaria is a protozoan infection of red blood cells cased in human by four species of the genus Plasmodium (Plasmodium falciparum, Plasmodium vivax, Plasmodium ovale and Plasmodium malariae). The malaria parasites are generally transmitted to the human host through the bite of an infected female anopheline mosquito (see [4]). In 1911, Ross initiatively developed a simple mathematical model to understand parasite transmission mechanism of malaria in [5]. Since then, there has been a great deal of work about using mathematical models to study malaria, (for example, see [6–12] and the references cited therein). Recently, Tumwiine et al. [13] proposed an autonomous model describing the
q Supported by the National Natural Science Foundation of China (Grant Nos. 10961022, 11001235, 11201399), the China Postdoctoral Science Foundation (Grant Nos. 20110491750), the Natural Science Foundation of Xinjiang (Grant Nos. 2011211B08) and the Natural Science Foundation of Xinjiang University (Grant Nos. BS100104). ⇑ Corresponding author. E-mail address: [email protected] (Z. Teng).
1007-5704/$ - see front matter Ó 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.cnsns.2012.09.007
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dynamics of malaria transmission, which has a human host and mosquito vector with temporary immunity, and obtained the globally asymptotical stability of the disease-free equilibrium and the endemic equilibrium for this model. In [14], Wei et al. considered an epidemic model of a vector-borne disease with direct transmission and time delay, and showed that the introduction of a time delay in the host-to-vector transmission term can destabilize the system and periodic solutions can arise through Hopf bifurcation. Tumwiine et al. [15] discussed a host-vector model for malaria with infective immigrants, and proved that there is not disease free equilibrium, and the unique endemic equilibrium is globally asymptotically stable if this model is assumed that there is a fraction of infective immigrants. Cai and Li in [16] studied an autonomous vector-host epidemic model with nonlinear incidences, and obtained that the asymptotic behaviors of the model are determined by its basic reproduction number R0 , that is, if R0 6 1,the disease-free equilibrium is globally asymptotically stable, and if R0 > 1, the disease persists and the unique endemic equilibrium is globally asymptotically stable. Malaria often occurs in most tropical and some subtropical regions of the world (see [17]). Environmental and climatic factors play an important role in the geographical distribution and transmission of malaria (see [18–20]). In [20], the authors provided a picture about ENSO-related precipitation anomalies and periodic malaria epidemics, which showed a striking correspondence between periodic malaria epidemics and geographical areas where the ‘‘teleconnections’’ ENSO affect precipitation and temperature. For mosquito population, proper temperature and humidity are more beneficial to give birth and propagate. For example, in temperate climates and in tropical highlands, temperature restricts vector multiplication and the development of the parasite in the mosquito, while in arid climates precipitation restricts mosquito breeding (see [20]). On the other hand, the intensity and pattern of transmission of malaria may vary markedly as a result of variations in altitude or rainfall and other environmental factors, even though within the same country (see [17]). Therefore, malaria fluctuates over time and often exhibits seasonal behaviors, especially in the northern areas. It is meaningful and essential to take account of malaria model with periodic environment. However, up to now, there have been few results about malaria model with periodic environment. In [21], a malaria transmission model with periodic birth rate and age structure for the vector population was presented by Lou and Zhao, and they further showed that the disease would die out if the basic reproduction number R0 < 1, and if the basic reproduction number R0 > 1, there exists at least one positive periodic state and the disease persists, finally they used these analytic results to study the malaria transmission cases in KwaZulu-Natal Province, South Africa. Motivated by the work of Nakata and Kuniya [22], Liu et al. [23] and Yang and Xiao [24], in this paper, we will investigate a malaria transmission model with periodic environment. By applying the way of computing the basic reproduction number for a wide class of compartmental epidemic models in periodic environments given by Wang and Zhao [25], we calculate basic reproduction number R0 for this model, and prove that the disease goes to extinction if R0 < 1, while the disease is uniformly persistent and there is at least one positive periodic solution if R0 > 1, which indicates the basic reproduction number is the threshold value determining whether or not malaria persists. The organization of this paper is as follows. In Section 2, model description and some preliminaries are given. In Section 3, we will compute the basic production number and study the globally asymptotical stability of the disease-free periodic solution and the uniform persistence of the model. In Section 4, some examples and simulations are given to illustrate theoretical results and exhibit different dynamic behaviors, such as the global attractivity (see Fig. 2), the chaotic attractor (see Fig. 3). 2. Model description and preliminaries Based on the transmission mechanism of malaria, we consider host population and vector population are human population and mosquito population, respectively. And we will make the following basic assumptions. (1) A mosquito firstly bites the infected person, then it becomes the infected mosquito. Due to its short lifespan, it cannot recover from the infection. Consequently, we only divide the total mosquito population into two classes: the susceptible and the infected. (2) Because environmental and climatic factors (for example, temperature, humidity, etc.) often have many effects on giving birth and propagating for mosquito population, we assume that the grow rate of the susceptible mosquito is governed by a logistic equation, in which these coefficients are periodic functions on account of seasonal effects. (3) Based on assumption (2), due to mosquito’s periodic birth and death, we assume that transmission rate from mosquito to human and transmission rate from human to mosquito be periodic functions. And both are bilinear incidence rate. (4) For human population, we separate the total human population into three classes: the susceptible, the infective and the recovered. The susceptible individual becomes infectious after being bitten by infective mosquito and can acquire permanent immunity after recovery from disease by curing. (5) For human population, we assume that there are the immigration, natural death and disease-induced death and all newborns are the susceptible. The recovered can become the susceptible again. We further assume that immigration rate, disease-induced death rate, natural death rate and the rate of which the recovered becomes susceptible again are positive constants.
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Under the above assumptions, the following malaria transmission model with periodic environment is proposed.
8 S_R ðtÞ ¼ > > > > > > _ > < IR ðtÞ ¼ S_H ðtÞ ¼ > > > > I_H ðtÞ ¼ > > > : _ RH ðtÞ ¼
SR rðtÞSR ð1 kðtÞ Þ aðtÞSR IH ;
aðtÞSR IH dðtÞIR ; ð1Þ
k bðtÞSH IR dSH þ cRH ; bðtÞSH IR ðd þ l þ rÞIH ; lIH dRH cRH ;
where SR ðtÞ and IR ðtÞ represent the densities of the susceptible and the infected for mosquito population at time t, respectively; SH ðtÞ; IH ðtÞ and RH ðtÞ separately denote the densities of the susceptible, the infective and the recovered for human population at time t; rðtÞ and kðtÞ are the intrinsic growth rate and the carrying capacity of environment for mosquito population at time t, respectively; dðtÞ is the death rate of the infected mosquito, including the natural death rate and disease-induced death rate; k; d; l and r are positive constants, which are the immigration rate, natural death rate, recovered rate and disease-induced death rate for human population, respectively; c is the rate of which the recovered becomes susceptible again; aðtÞ is the transmission rate from mosquito to human, and bðtÞ is the transmission rate from human to mosquito. In view of the biological background of system (1), in this paper we only consider the solution of system (1) starting at t ¼ 0 with initial values:
S0R P 0;
I0R P 0;
S0H P 0;
I0H P 0;
R0H P 0:
ð2Þ
In this paper, for system (1) we always introduce the following assumptions. ðH1 Þ rðtÞ; kðtÞ; dðtÞ; aðtÞ and bðtÞ are continuous and x-periodic functions. ðH2 Þ kðtÞ; dðtÞ > 0 and aðtÞ; bðtÞ P 0 with aðtÞ; bðtÞ X 0 for all t P 0. Rx ðH3 Þ 0 rðtÞdt > 0. When IH ðtÞ 0; RH ðtÞ 0 and IR ðtÞ 0, we can obtain the following two subsystem of system (1)
S_H ðtÞ ¼ k dSH ðtÞ;
ð3Þ
SR : S_R ðtÞ ¼ rðtÞSR 1 kðtÞ
ð4Þ
and
Lemma 1. (a) System (3) has a unique positive globally asymptotically stable equilibrium SH ðtÞ ¼ kd. (b) System (4) allows a globally uniformly attractive positive x-periodic solution SR ðtÞ. Conclusion ðaÞ of Lemma 1 is obvious. Conclusion ðbÞ of Lemma 1 can be easily obtained from Lemma 2 given in [26]. From Lemma 1, we see that system (1) has a disease-free periodic solution E ðtÞ ¼ ðSR ðtÞ; 0; kd ; 0; 0Þ. Let ðRn ; Rnþ Þ be the standard ordered n-dimensional Euclidean space. For u; v 2 Rn , we denote u P v if u v 2 Rnþ ; u > v if u v 2 Rnþ n f0g and u v if u v 2 intRnþ , respectively, where intRnþ denote the interior of Rnþ . Let AðtÞ be a continuous and x-periodic n n matrix function, we consider the following linear system
x_ ¼ AðtÞx:
ð5Þ
Let UA ðtÞ be the fundamental solution matrix of system (5) with initial condition UA ð0Þ ¼ I, where I is n n identity matrix, and let qðUA ðxÞÞ be the spectral radius of matrix UA ðxÞ. Further, we assume that AðtÞ also is cooperative and irreducible, then by the Perron-Frobenius theorem, qðUA ðxÞÞ is the principle eigenvalue of UA ðxÞ in the sense that it is simple and admits an eigenvector m 0. Lemma 2 (See [27]). Let AðtÞ is a continuous, cooperative, irreducible and x-periodic n n matrix function, Then there exists a positive x-periodic function mðtÞ such that xðtÞ ¼ elt mðtÞ is a solution of system (5).
l ¼ x1 ln qðUA ðxÞÞ.
Let X be a metric space with metric d and f : X ! X be a continuous map. For any x 2 X, we denote f n ðxÞ ¼ f ðf n1 ðxÞÞ for any integer n > 1 and f 1 ðxÞ ¼ f ðxÞ. f is said to be compact in X if for any bounded set H X set f ðHÞ ¼ ff ðxÞ : x 2 Hg is precompact in X. f is said to be point dissipative if there is a bounded set B0 X such that for any x 2 X
lim dðf n ðxÞ; B0 Þ ¼ 0:
n!1
For any x0 2 X, the positive semiorbit through x0 is defined by cþ ðx0 Þ ¼ fxn ¼ f n ðx0 Þ; n ¼ 1; 2; . . .g, the negative semiorbit through x0 is defined as a sequence c ðx0 Þ ¼ fxk g satisfying f ðxk1 Þ ¼ xk for integers k 6 0, the omega limit set of cþ ðx0 Þ is defined by xðx0 Þ ¼ fy 2 X: there is a sequence nk ! 1 such that limk!1 xnk ¼ yg and the alpha limit set of c ðx0 Þ is defined by aðx0 Þ ¼ fy 2 X: there is a sequence nk ! 1 such that limk!1 xnk ¼ yg.
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A nonempty set A X is said to be invariant if f ðAÞ # A. A nonempty invariant set M of X is called to be isolated in X if it is the maximal invariant set in a neighborhood of itself. For a nonempty set M of X, set W s ðMÞ :¼ fx 2 X : limn!1 dðf n ðxÞ; MÞ ¼ 0g is called the stable set of M. Let A and B be two isolated invariant sets, set A is said to be chained to set B, written A ! B, if there exists a full orbit though some x R A [ B such that xðxÞ B and aðxÞ A. A finite sequence fM 1 ; . . . ; M k g of isolated invariant sets is called a chain if M 1 ! M 2 ! ! M k , and if M k ¼ M 1 the chain is called a cycle. Let X 0 be a nonempty open set of X. We denote
@X 0 :¼ X n X 0 ;
M @ :¼ fx 2 @X 0 : f n ðxÞ 2 @X 0 for all n P 0g:
Lemma 3. Let f : X ! X be a continuous map. Assume that the following conditions hold: ðC 1 Þ f is compact and point dissipative, and f ðX 0 Þ # X 0 . ðC 2 Þ There exists a finite sequence M ¼ fM 1 ; . . . ; M k g of compact and isolated invariant sets such that. (a) (b) (c) (d)
T Mi M j ¼ ; for any i; j ¼ 1; 2; . . . ; k and i – j; S S XðM@ Þ :¼ x2M@ xðxÞ ki¼1 Mi ; no subset of M forms a cycle in @X 0 ; T W s ðM i Þ X 0 ¼ ; for each 1 6 i 6 k.
Then f is uniformly persistent with respect to ðX 0 ; @X 0 Þ, that is, there exists a constant g > 0 such that lim inf n!1 dðf n ðxÞ; @X 0 Þ P g for all x 2 X 0 . Lemma 3 can be obtained from Theorem 1.1.3, Theorem 1.3.1, Remark 1.3.1 and Theorem 1.3.3 given by Zhao in [28]. Now, based on the assumptions ðH1 Þ ðH3 Þ, we compute the basic reproduction number of system (1) by applying the way given in [25,29] by Wang, Zhao, Driessche and Watmough. Let
0
aðtÞSR IH
0
1
C B B bðtÞSH IR C C B C; B F ðt; xÞ ¼ B 0 C C B 0 A @ 0
dðtÞIR
1
C B B ðd þ l þ rÞIH C C B 2 B SR C V ðt; xÞ ¼ B aðtÞSR IH þ rðtÞ kðtÞ C C B C B @ bðtÞSH IR þ dSH A dRH þ cRH
and
0
0
1
C B 0 C B C B B V ðt; xÞ ¼ B rðtÞSR C C; C B @ k þ cRH A lIH þ
where x ¼ ðIR ; IH ; SR ; SH ; RH ÞT , then system (1) equals to the following form
_ xðtÞ ¼ F ðt; xÞ Vðt; xÞ , f ðt; xðtÞÞ;
ð6Þ
þ
where Vðt; xÞ ¼ V ðt; xÞ V ðt; xÞ. In the following, we will check conditions (A1)–(A7) which are given in [25]. By the expressions of F ðt; xÞ and Vðt; xÞ, we firstly easily see that conditions (A1)–(A5) are satisfied. Obviously, system (6) has disease-free periodic solution x ðtÞ ¼ ð0; 0; SR ðtÞ; kd ; 0Þ. Now, we define
NðtÞ ¼
@fi ðt; x ðtÞÞ ; @xj 36i;j65
where fi ðt; xðtÞÞ and xi are the ith component of f ðt; xðtÞÞ and x, respectively. By simple computations, we can obtain
0 B NðtÞ ¼ @ Since
SR ðtÞ
Z
0
Thus,
x
rðtÞ 2rðtÞ S ðtÞ kðtÞ R
0
0
0
d
c
0
0
ðd þ cÞ
1 C A:
is the globally uniformly attractively x-periodic solution of system (4), it satisfies
S ðtÞ dt ¼ 0: rðtÞ 1 R kðtÞ
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exp
Z
x
rðtÞ
0
Z x Z x 2rðtÞ S ðtÞ S ðtÞ rðtÞ SR ðtÞ dt ¼ exp dt ¼ exp SR ðtÞdt < 1: rðtÞ 1 R R kðtÞ kðtÞ kðtÞ kðtÞ 0 0
Therefore, we finally get qðUN ðxÞÞ < 1, Thus, condition (A6) also holds. Next, we set two 2 2 matrices as follows
FðtÞ ¼
@F i ðt; x ðtÞÞ @xj 16i;j62
and VðtÞ ¼
@V i ðt; x ðtÞÞ @xj 16i;
; j62
where F i ðt; xðtÞÞ and V i ðt; xðtÞÞ are the ith component of F ðt; xðtÞÞ and Vðt; xðtÞÞ, respectively. Then, by simple computations, it follows that
FðtÞ ¼
0
aðtÞSR ðtÞ
bðtÞ kd
0
!
and VðtÞ ¼
dðtÞ 0 : 0 dþlþr
Therefore, from assumption ðH1 Þ, we obtain that qðUV ðxÞÞ < 1. Thus, condition (A7) also holds. Let Yðt; sÞ is the 2 2 matrix solution of the following initial value problem
(
d Yðt; sÞ dt
¼ VðtÞYðt; sÞ for all t P s;
Yðs; sÞ ¼
I:
Let C x be the ordered Banach space of all x-periodic continuous function form R to R2 with the maximum norm k k. The positive cone
C þx ¼ f/ 2 C x : /ðtÞ P 0 for all t 2 Rg: Suppose /ðsÞ 2 C þ x is the initial distribution of infectious individuals in this periodic environment, then FðsÞ/ðsÞ is the rate of new infectious produced by the infected individuals who were introduced at time s, and Yðt; sÞFðsÞ/ðsÞ represents the distributions of those infected individuals who were newly infected at time s and remain in the infected compartment at time t for t P s. It follows that
Z
0
Yðt; sÞFðsÞ/ðsÞds ¼
wðtÞ :¼
Z
þ1
Yðt; t aÞFðt aÞ/ðt aÞda
0
1
denotes the distribution of accumulative new infections at time t produced by all those infected individuals /ðsÞ introduced at previous time to t. We define a linear operator L : C x ! C x as follows
ðL/ÞðtÞ ¼
Z
þ1
Yðt; t aÞFðt aÞ/ðt aÞda for all t 2 R; / 2 C x :
0
Following Wang and Zhao in [25], we call L the next infection operator, and define the basic reproduction number R0 for system (1) by
R0 ¼ qðLÞ; where qðLÞ is the spectral radius of L. Using Theorem 2.2 given in [25], we can obtain the following results on basic reproduction number R0 and the locally asymptotical stability of disease-free periodic solution E ðtÞ for system (1). Lemma 4. (1) On basic reproduction numberR0 , we have (i) R0 ¼ 1if and if onlyqðUFV ðxÞÞ ¼ 1; (ii) R0 > 1if and if onlyqðUFV ðxÞÞ > 1; (iii) R0 < 1if and if onlyqðUFV ðxÞÞ < 1. (2) E ðtÞis locally asymptotically stable ifR0 < 1, and unstable ifR0 > 1. 3. Main results
Theorem 1. LetðSR ðtÞ; IR ðtÞ; SH ðtÞ; IH ðtÞ; RH ðtÞÞbe the solution of system(1)with initial condition(2). Then, we have (a) ðSR ðtÞ; IR ðtÞ; SH ðtÞ; IH ðtÞ; RH ðtÞÞis nonnegative for allt P 0and ultimately bounded. (b) IfS0R > 0; I0R > 0; S0H > 0; I0H > 0andR0H P 0, thenðSR ðtÞ; IR ðtÞ; SH ðtÞ; IH ðtÞ; RH ðtÞÞalso is positive for allt > 0.
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Proof. We firstly prove conclusion ðbÞ. Suppose that ðSR ðtÞ; IR ðtÞ; SH ðtÞ; IH ðtÞ; RH ðtÞÞ is defined for all t 2 ½0; TÞ, where T > 0. Integrating the first equation of system (1) from 0 to t, we have
SR ðtÞ ¼ S0R exp
Z t 0
SR ðsÞ rðsÞ 1 aðsÞIH ðsÞ ds: kðsÞ
ð7Þ
From S0R > 0, we obtain that SR ðtÞ > 0 for all t 2 ½0; TÞ. Suppose that there exists a t1 2 ð0; TÞ such that
minfIR ðt 1 Þ; IH ðt1 Þg ¼ 0: Since I0R > 0 and I0H > 0, we can further assume minfIR ðtÞ; IH ðtÞg > 0 for all t 2 ½0; t 1 Þ. If minfIR ðt1 Þ; IH ðt 1 Þg ¼ IR ðt1 Þ, then from SR ðtÞ > 0 for all t 2 ½0; TÞ, we have
I_R ðtÞ P dðtÞIR ðtÞ for all t 2 ½0; t1 : Hence,
Z 0 ¼ IR ðt1 Þ P I0R exp
t1
dðsÞds > 0;
0
which leads to a contradiction. If minfIR ðt 1 Þ; IH ðt1 Þg ¼ IH ðt 1 Þ, then since
R_ H ðtÞ > ðd þ cÞRH ðtÞ for all t 2 ½0; t 1 Þ; we have
RH ðtÞ > R0H expððd þ cÞtÞ P 0 for all t 2 ð0; t 1 : From the third equation of system (1), we have
S_ H ðtÞ P ðbðtÞIR ðtÞ þ dÞSH ðtÞ for all t 2 ½0; t 1 : Hence,
Z t SH ðtÞ P S0H exp ðbðsÞIR ðsÞ þ dÞds > 0 for all t 2 ½0; t 1 : 0
From this, we further obtain
I_H ðtÞ P ðd þ l þ rÞIH ðtÞ for all t 2 ½0; t 1 ; Hence,
0 ¼ IH ðt 1 Þ P I0H expððd þ l þ rÞt1 Þ > 0; which leads to a contradiction. This shows that IR ðtÞ > 0 and IH ðtÞ > 0 for all t 2 ½0; TÞ. Furthermore, since
R_ H ðtÞ > ðd þ cÞRH ðtÞ for all t 2 ½0; TÞ; we obtain
RH ðtÞ > R0H eðdþcÞt P 0 for all t 2 ð0; TÞ: Further from the third equation of (1) we have
S_ H ðtÞ P ðbðtÞIR ðtÞ þ dÞSH ðtÞ for all t 2 ½0; TÞ: Hence,
Z t SH ðtÞ P S0H exp ðbðsÞIR ðsÞ þ dÞds > 0 for all t 2 ½0; TÞ: 0
This shows that ðSR ðtÞ; IR ðtÞ; SH ðtÞ; IH ðtÞ; RH ðtÞÞ is positive on the interval of existence. This completes the proof of conclusion ðbÞ. Next, we prove conclusion ðaÞ. Firstly, from conclusion ðbÞ and the continuous dependence of solutions of system (1) with respect to initial values, we immediately obtain that ðSR ðtÞ; IR ðtÞ; SH ðtÞ; IH ðtÞ; RH ðtÞÞ with initial condition (2) is nonnegative on the interval of existence. Now, we prove that the interval of existence of ðSR ðtÞ; IR ðtÞ; SH ðtÞ; IH ðtÞ; RH ðtÞÞ is ½0; 1Þ. In fact, if the interval of existence of ðSR ðtÞ; IR ðtÞ; SH ðtÞ; IH ðtÞ; RH ðtÞÞ is a finite interval ½0; TÞ, then we know that ðSR ðtÞ; IR ðtÞ; SH ðtÞ; IH ðtÞ; RH ðtÞÞ is nonbounded on ½0; TÞ. From
S_ R ðtÞ 6 rðtÞSR ðtÞ;
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we obtain
SR ðtÞ 6 S0R exp
Z
t
rðsÞds :
0
Hence, SR ðtÞ is bounded on ½0; TÞ. Let N H ðtÞ ¼ SH ðtÞ þ IH ðtÞ þ RH ðtÞ, from system (1) we have
N_ H ðtÞ ¼ k dNH ðtÞ rIH ðtÞ 6 k; which implies that
NH ðtÞ 6 N0H þ kt: Hence, N H ðtÞ is bounded on ½0; TÞ, which implies that SH ðtÞ, IH ðtÞ and RH ðtÞ are also bounded on ½0; TÞ. Let N R ðtÞ ¼ SR ðtÞ þ IR ðtÞ. From system (1) we have
SR ðtÞ N_ R ðtÞ ¼ rðtÞSR ðtÞ 1 dðtÞIR ðtÞ 6 dðtÞNR ðtÞ þ ðdðtÞ þ rðtÞÞSR ðtÞ 6 dðtÞNR ðtÞ þ M S ; kðtÞ where M S ¼ sup06t
Z s Z Z t Z t NR ðtÞ 6 exp dðsÞds N 0R þ M S exp dðsÞds ds 6 N 0R þ exp 0
0
0
T
dðsÞds M S T:
0
Hence, N R ðtÞ is bounded on ½0; TÞ, which implies that IR ðtÞ also is bounded on ½0; TÞ. This leads to a contradiction. Therefore, we finally have that ðSR ðtÞ; IR ðtÞ; SH ðtÞ; IH ðtÞ; RH ðtÞÞ is defined on ½0; 1Þ. Since for any t P 0
SR ðtÞ ; S_R ðtÞ 6 rðtÞSR ðtÞ 1 kðtÞ by the comparison principle and conclusion ðbÞ of Lemma 1, we can obtain
lim sup SR ðtÞ 6 lim sup SR ðtÞ 6 M S ; t!1
ð8Þ
t!1
where SR ðtÞ is the globally uniformly attractively positive x-periodic solution of system (4) and M S ¼ maxt2½0;x SR ðtÞ. For any constant e0 > 0 there is a T 0 > 0 such that for any t P T 0
SR ðtÞ 6 SR ðtÞ þ e0 6 MS þ e0 : Since for any t P 0 we have
N_ H ðtÞ 6 k dNH ðtÞ and
N_ R ðtÞ 6 dðtÞNR ðtÞ þ ðdðtÞ þ rðtÞÞSR ðtÞ; we obtain
lim sup NH ðtÞ 6 t!1
k d
ð9Þ
and for any t P T 0
N_ R ðtÞ 6 dðtÞNR ðtÞ þ supfdðtÞ þ rðtÞgðM S þ e0 Þ: tP0
Integrating this inequality for any t P T 0 , we obtain
Z s Z t Z t NR ðtÞ 6 exp dðsÞds NR ðT 0 Þ þ ðM S þ e0 ÞsupfdðtÞ þ rðtÞg exp dðsÞds ds : T0
T0
tP0
T0
From this, we further obtain
lim sup NR ðtÞ 6 ðM S þ e0 ÞD; t!1
where D ¼
suptP0 fdðtÞþrðtÞg . inf tP0 dðtÞ
lim sup NR ðtÞ 6 t!1
From the arbitrariness of
MS D:
e0 , we can take e0 ! 0 and finally obtain ð10Þ
Lastly, from (8)–(10), we finally obtain that all solutions of system (1) are ultimately bounded. This completes the proof of conclusion ðaÞ. h
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Remark 1. From (7) we obtain that SR ðtÞ > 0 for all t P 0 when S0R > 0. Therefore, for any small enough constant denote
> 0, we
k d
X ¼ fðSR ; IR ; SH ; IH ; RH Þ : SR > 0; IR P 0; SH P 0; IH P 0; RH P 0; SR þ IR 6 M S D þ ; SH þ IH þ RH 6 þ g; then from Theorem 1, we easily see that X is a positively invariant set with respect to system (1) and also a global attractor of all positive solutions of system (1). Theorem 2. IfR0 < 1, then disease-free periodic solutionE ðtÞof system(1)is globally asymptotically stable. Proof. From Lemma 4, we obtain that if R0 < 1; E ðtÞ is locally asymptotically stable. Now, we will only prove the global attractivity of E ðtÞ for the case R0 < 1. From R0 < 1 and conclusion (iii) of Lemma 4, we have qðUFV ðxÞÞ < 1, then we can choose a small enough constant 2 > 0 such that qðUFVþ2 M ðxÞÞ < 1, where
MðtÞ ¼
0
aðtÞ
bðtÞ
0
:
From (8) and (9), we obtain that for above given constant
k SH ðtÞ 6 þ 2 d
2
there exists a t 1 > 0 such that for all t > t1
and SR ðtÞ 6 SR ðtÞ þ 2 :
From the second and fourth equations of system (1), we obtain that for all t > t1
(
I_R I_H
6 aðtÞðSR ðtÞ þ 2 ÞIH dðtÞIR ; 6 bðtÞ kd þ 2 IR ðd þ l þ rÞIH :
ð11Þ
Considering the following auxiliary system:
8 < eI_
¼ aðtÞðSR ðtÞ þ 2 ÞeI H dðtÞeI R ; ¼ bðtÞ kd þ 2 eI R ðd þ l þ rÞeI H :
R
: e_ IH
For the convenience, we will rewrite it as follows
d dt
eI R eI H
!
eI ¼ ðFðtÞ VðtÞ þ 2 MðtÞÞ R eI H
! ð12Þ
:
From Lemma 2, it follows that there exists a positive x-periodic function qðtÞ ¼ ðq1 ðtÞ; q2 ðtÞÞT such that ðeI R ðtÞ; eI H ðtÞÞT ¼ el1 t qðtÞ is a solution of system (12), where l1 ¼ x1 lnðqðUFVþ2 M ðxÞÞÞ. Denote JðtÞ ¼ ðIR ðtÞ; IH ðtÞÞT . We can choose a small constant n > 0 such that Jðt 1 Þ 6 nqðt 1 Þ. Then, from (11) the comparison principle implies that
JðtÞ 6 nel1 t qðtÞ for all t > t 1 : By qðUFVþ2 M ðxÞÞ < 1, it follows that
lim IR ðtÞ ¼ 0;
t!1
l1 < 0, then limt!1 JðtÞ ¼ 0, that is,
lim IH ðtÞ ¼ 0:
t!1
Moreover, from the equations of SR ; SH and RH in system (1), we can get
½lim SR ðtÞ ¼ SR ðtÞ; t!1
k lim SH ðtÞ ¼ ; d
t!1
lim RH ðtÞ ¼ 0:
t!1
Hence, disease-free periodic solution E ðtÞ of system (1) is globally attractive. This completes the proof. h Theorem 3. IfR0 > 1, then system(1)is uniformly persistent. That is, there exists a positive constante, such that any solutionðSR ðtÞ; SH ðtÞ; IR ðtÞ; IH ðtÞ; RH ðtÞÞof system(1)with initial conditions(2)satisfies
lim inf ðSR ðtÞ; SH ðtÞ; IR ðtÞ; IH ðtÞ; RH ðtÞÞ P ðe; e; e; e; eÞ: t!1
Proof. From R0 > 1 and conclusion (ii) of Lemma 4, we have qðUFV ðxÞÞ > 1. Then, we can choose a small constant g > 0 such that qðUFVgM ðxÞÞ > 1, where MðtÞ is defined in Theorem 2. From ðH2 Þ and ðH3 Þ, we obtain for any small enough
e>0
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Z
x
½rðtÞ aðtÞe dt > 0:
ð13Þ
0
For this e, we consider the following two perturbed equations
U e ðtÞ eaðtÞU e ðtÞ; U_ e ðtÞ ¼ rðtÞU e ðtÞ 1 kðtÞ
ð14Þ
V_ e ðtÞ ¼ k ebðtÞV e ðtÞ dV e ðtÞ:
ð15Þ
and
Using Lemma 2 given in [26] and Lemma 1 given in [30], from ðH2 Þ and (13), we can get that systems (14) and (15) admit globally uniformly attractive positive x-periodic solutions U e ðtÞ and V e ðtÞ, respectively. By the continuity of solutions with respect to the parameter e, for constant g > 0 given in above, there exists a constant e1 > 0 such that for all 0 < e1 < e and t 2 ½0; x
g
U e1 ðtÞ > SR ðtÞ ; 2
k g V e1 ðtÞ > : d 2
ð16Þ
Define
X ¼ fðSR ; IR ; SH ; IH ; RH Þ : SR > 0; IR P 0; SH P 0; IH P 0; RH P 0g and
X 0 ¼ fðSR ; IR ; SH ; IH ; RH Þ 2 X : IR > 0; IH > 0g: We have
@X 0 ¼ X n X 0 ¼ fðSR ; IR ; SH ; IH ; RH Þ 2 X : IR IH ¼ 0g: From system (1), it is easy to see that X and X 0 are positively invariant, and @X 0 is also a relatively closed set in X. Let P : X ! X be the Poincaré map associated with system (1), that is
Pðx0 Þ ¼ uðx; x0 Þ for all x0 2 X; where uðt; x0 Þ is the unique solution of system (1) satisfying initial condition uð0; x0 Þ ¼ x0 . From the continuity of solutions of system (1) with respect to initial value x0 , we can obtain that P is compact. Moreover, by Theorem 1, we obtain that P is point dissipative on X. Further, we define
M@ ¼ fðS0R ; I0R ; S0H ; I0H ; R0H Þ 2 @X 0 : P m ðS0R ; I0R ; S0H ; I0H ; R0H Þ 2 @X 0 m
where P ¼ PðP
m1
for all m > 0g;
1
Þ for all m > 1 and P ¼ P. Now, we prove
M@ ¼ fðSR ; 0; SH ; 0; 0Þ : SR > 0; SH P 0:g ðS0R ; 0; S0H ; 0; 0Þ,
S0R
ð17Þ S0H
Firstly, for any point where > 0 and P 0, according to the existence and uniqueness of solutions of system (1), we can obtain that ðSR ðtÞ; IR ðtÞ; SH ðtÞ; IH ðtÞ; RH ðtÞÞ with IR ðtÞ 0; IH ðtÞ 0 and RH ðtÞ 0 is the unique solution of system (1) satisfying initial condition ðSR ð0Þ; IR ð0Þ; SH ð0Þ; IH ð0Þ; RH ð0ÞÞ ¼ ðS0R ; 0; S0H ; 0; 0Þ. Therefore, we obtain for any integer m>0
Pm ðS0R ; 0; S0H ; 0; 0Þ 2 fðSR ; 0; SH ; 0; 0Þ : SR > 0; SH P 0g # @X 0 : This shows ðSR ; 0; SH ; 0; 0Þ 2 M @ . Consequently,
fðSR ; 0; SH ; 0; 0Þ : SR > 0; SH P 0g # M @ : On the other hand, if M @ n fðSR ; 0; SH ; 0; 0Þ : SR > 0; SH P 0g – ;, then there exists at least a point ðS0R ; I0R ; S0H ; I0H ; R0H Þ 2 M @ satisfying I0H > 0 or I0R > 0. If I0R ¼ 0 and I0H > 0, then it is clear that from system (1)
IH ðtÞ P I0H eðdþlþrÞt > 0 for all t > 0: From S0R > 0 we can obtain from the first equation of system (1) that SR ðtÞ > 0 for all t > 0. Hence,
Rt Z t Rs IR ðtÞ ¼ I0R þ aðsÞSR ðsÞIH ðsÞe 0 dðuÞdu ds e 0 dðuÞdu > 0 0
for all t > 0. If
I0R
> 0 and I0H ¼ 0, then we have
IR ðtÞ ¼ I0R þ
Z 0
t
Rs
aðsÞSR ðsÞIH ðsÞe
0
dðuÞdu
Rt dðuÞdu ds e 0 >0
L. Wang et al. / Commun Nonlinear Sci Numer Simulat 18 (2013) 1288–1303
1297
for all t > 0. From the third equation of system (1) we have
S_H ðtÞ > ðbðtÞIR þ d þ cÞSH
for all t P 0:
Hence, we further have SH ðtÞ > 0 for all t > 0. Consequently, by the fourth equation of system (1) we have
I_H ðtÞ > ðd þ l þ rÞIH
for all t P 0
and hence IH ðtÞ > 0 for all t > 0. This shows that ðSR ðtÞ; IR ðtÞ; SH ðtÞ; IH ðtÞ; RH ðtÞÞ R @X 0 . Hence, ðS0R ; I0R ; S0H ; I0H ; R0H Þ R M@ which leads to a contradiction. It indicates that M@ # fðSR ; 0; SH ; 0; 0Þ : SR > 0; SH P 0g. Therefore, we finally obtain that claim (17) holds. It is clear that there is a fixed points of P in M @ , which is M 1 ¼ ðSR ð0Þ; 0; kd ; 0; 0Þ. Denote x0 ¼ ðS0R ; I0R ; S0H ; I0H ; R0H Þ 2 X 0 . By the continuity of solutions with respect to the initial value, for above given constant e1 > 0, there exists d0 > 0 such that for all x0 2 X 0 with kx0 M 1 k 6 d0 , it follows that
kuðt; x0 Þ uðt; M1 Þk < e1
for all t 2 ½0; x :
ð18Þ
Now, we prove
lim sup dðPm ðx0 Þ; M 1 Þ P d0 :
ð19Þ
m!1
Suppose the conclusion is not true, then
lim sup dðPm ðx0 Þ; M 1 Þ < d0 m!1
for some x0 2 X 0 . Without loss of generality, we can assume that
dðPm ðx0 Þ; M 1 Þ < d0
for all m P 0:
Further, from (18) we have
kuðt; Pm ðx0 ÞÞ uðt; M 1 Þk < e1
for all m P 0;
0
t 2 ½0; x :
0
For any t P 0, let t ¼ mx þ t ,where t 2 ½0; xÞ and m ¼ ½xt is the greatest integer less than or equal to xt , then we can get
kuðt; x0 Þ uðt; M1 Þk ¼ kuðt 0 ; Pm ðx0 ÞÞ uðt 0 ; M 1 Þk < e1
for all t P 0:
ð20Þ
ðSR ðtÞ; 0; kd ; 0; 0Þ,
0
Since uðt; x Þ ¼ ðSR ðtÞ; IR ðtÞ; SH ðtÞ; IH ðtÞ; RH ðtÞÞ and uðt; M 1 Þ ¼ it follows from (20) that 0 6 IR ðtÞ 6 e1 and 0 6 IH ðtÞ 6 e1 for all t P 0. Then, by the first and third equation of system (1) we get for any t P 0
SR S_R ðtÞ P rðtÞSR 1 e1 aðtÞSR ; kðtÞ and
S_H ðtÞ P k e1 bðtÞSH dSH : By the comparison principle, we obtain for any t P 0
SR ðtÞ P U e1 ðtÞ;
SH ðtÞ P V e1 ðtÞ;
where U e1 ðtÞ and V e1 ðtÞ are the solutions of systems (14) and (15) with parameter e1 satisfying initial conditions U e1 ð0Þ ¼ S0R and V e1 ð0Þ ¼ S0H , respectively. Since systems (14) and (15) with parameter e1 have globally uniformly attractive positive x-periodic solutions U e1 ðtÞ and V e1 ðtÞ, respectively, there exists a t1 > 0 such that
g
U e1 ðtÞ > U e1 ðtÞ ; 2
V e1 ðtÞ > V e1 ðtÞ
g 2
for all t P t 1 :
ð21Þ
Combining (16) and (21), we have
U e1 ðtÞ > SR ðtÞ g;
k V e1 ðtÞ > g for all t P t1 : d
Thus, we finally obtain that for all t P t1
(
I_R I_H
P aðtÞðSR ðtÞ gÞIH dðtÞIR ; P bðtÞ kd g IR ðd þ l þ rÞIH :
Consider the following auxiliary system
ð22Þ
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8 < bI_
¼ aðtÞðSR ðtÞ gÞbI H dðtÞbI R ; ¼ bðtÞ kd g bI R ðd þ l þ rÞbI H :
R
: b_ IH
For the convenience, we will rewrite it as follows
d dt
bI R bI H
!
bI ¼ ðFðtÞ VðtÞ gMðtÞÞ R bI H
! ð23Þ
:
From Lemma 2, it follows that there exists a positive x-periodic function pðtÞ ¼ ðp1 ðtÞ; p2 ðtÞÞT such that ðbI R ðtÞ; bI H ðtÞÞ ¼ el2 t pðtÞ is a solution of system (23), where l2 ¼ x1 lnðqðUFVgM ðxÞÞÞ. Since JðtÞ2 intR2þ , where JðtÞ ¼ ðIR ðtÞ; IH ðtÞÞT , we can select a small constant a > 0 such that Jðt 1 Þ > apðt 1 Þ. Then, by (22) and the comparison principle, we can obtain that
JðtÞ P ael2 t pðtÞ for all t P t 1 : By qðUFVgM ðxÞÞ > 1, it follows that
limIR ðtÞ ¼ 1;
l2 > 0, then limt!1 JðtÞ ¼ 1, that is,
limIH ðtÞ ¼ 1;
t!1
t!1
which is a contradiction with 0 6 IR ðtÞ 6 e1 and 0 6 IH ðtÞ 6 e1 . T Therefore, claim (19) holds. This shows W s ðM 1 Þ X 0 ¼ ;. From Lemma 1, we can obtain that fM 1 g is globally attractive in M@ , that is, each orbit in M @ converges to fM 1 g. Hence, fM 1 g is isolated in M @ , and hence in X. Furthermore, fM 1 g also is invariant and fM1 g does not form a cycle in M @ , and hence in @X 0 . Therefore, by Lemma 3, we finally obtain that P is uniformly persistent with respect to ðX 0 ; @X 0 Þ. Finally, from Theorem 3.1.1 given in [28] we further obtain that all solutions of system (1) is uniformly persistent with respect to ðX 0 ; @X 0 Þ. Furthermore, from the last equation of system (1) we can directly obtain that RH in system (1) also is uniformly persistent. This completes the proof. h As a consequence of Theorem 3, from the main results given in [31] on the existence of positive periodic solutions for general population dynamical systems, we have the following result. Corollary 1. IfR0 > 1, then system(1)admits at least a positive x-periodic solution. Remark 2. By Lemma 4, we can see that in the realistic applications, if we want to verify R0 is greater or less than unity, it is suffice to verify qðUFV ðxÞÞ is greater or less than unity. However, from the expression of qðUFV ðxÞÞ we see that it is very difficult to calculate it for system (1) because it is periodic system, let alone extending it to higher dimension systems, so we want to further continue our work to find similar integral conditions in [32], which are tend to more easily verify and can be applicable to more general non-autonomous system. Remark 3. When system (1) degenerates an into the autonomous case with rðtÞ r; kðtÞ k; dðtÞ d; aðtÞ a and bðtÞ b for all t P 0, we obtain SR ðtÞ ¼ k and
FðtÞ ¼
0
ak
b kd
0
!
and VðtÞ ¼
d
0
0 dþlþr
:
Using the method given by Driessche and Watmough given in [29] (see also Lemma 2.2 (ii) in [25]), we obtain basic reproduction number
R0 ¼ qðFV 1 Þ ¼
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi kkab a k b : ¼ k ddðl þ d þ rÞ d d ðl þ d þ rÞ
The biological meaning of R0 can be interpreted as follows, when the total number of human population reaches stable state kd and everyone is the susceptible, the number of the new infected human produced by each infected mosquito (though the bite of the infected mosquito) over its expected infectious period is ad kd, and when the total number of mosquito population achieves the stable state k and every mosquito is the susceptible, the number of the new infected mosquito produced by each b infected people over his/her expected infectious period is k lþdþ r. The square root arises from the two ‘generations’ required for an infected people or mosquito to ‘reproduce’ itself. As a consequence of Theorem 2, Theorem 3 and Corollary 1, then we can obtain the following result.
Corollary 2. For autonomous malaria transmission system(1)we have the following conclusions.
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(i) If R0 < 1, then disease-free equilibrium ðk; 0; kd ; 0; 0Þ is globally asymptotically stable. (ii) If R0 > 1, then there exists at least one positive endemic equilibrium and the disease is uniformly persistent.
Remark 4. In this paper, both transmission rate from mosquito to human and transmission rate from human to mosquito are bilinear incidence rate. Bilinear incidence rate bases on the law of mass action. However, an infectious individual (or mosquito) can contact a finite number of mosquitoes (or individuals) per unit time in a large population. The standard incidence rate seems more reasonable than bilinear incidence rate. Therefore, in our future work, we further discuss the dynamic behaviors for a malaria model with standard incidence in a periodic environment. 4. Numerical simulations In this section, we give some examples and numerical simulations to confirm the above theoretical analysis.
a
b
1.4 1.2
2.5
1
2
0.8
I (t)
3
R
R
S (t)
3.5
1.5
0.6
1
0.4
0.5
0.2
0 0
c
50
100
150
200
t
250
0 0
300
9
d
8
150
t
200
6
4
I (t)
6
H
H
S (t)
100
5
7
5
3 2
4
1
3 2 0
50
200
400
t
600
e
800
0 0
1000
50
100
150
t
200
250
300
8 7 6
H
R (t)
5 4 3 2 1 0 0
500
t
1000
1500
Fig. 1. Global stability of disease-free periodic solution of system (1) with parameters in Example 1, when the basic reproduction number R0 0:66319 < 1, which shows that the disease dies out.
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Example 1. Take k ¼ 0:05; d ¼ 0:006; l ¼ 0:1; r ¼ 0, rðtÞ ¼ 3 þ 0:1 sin p6 t ; kðtÞ ¼ 2 þ 0:3 cos 16 pt ; dðtÞ ¼ 1 þ cos p6 t , 1 1 aðtÞ ¼ 0:02 2 þ 0:03 sin 6 pt and bðtÞ ¼ 0:01 7 þ 0:06 sin 6 pt in system (1). Clearly, (H1)–(H3) hold, and by numerical calculations, we obtain basic reproduction number R0 0:66319 < 1. Then, by Theorem 2, disease-free periodic solution ðSR ðtÞ; 0; kd ; 0; 0Þ of system (1) is globally asymptotically stable, which indicates that the disease tend to be extinct. See numerical simulations (a)–(e) of Fig. 1.
Example 2. In Example 1, we keep some parameters unchanged, and only adjust the values of two transmission rates aðtÞ and bðtÞ. Let aðtÞ ¼ 0:2 2 þ 0:03 sinð16 ptÞ and bðtÞ ¼ 0:011 7 þ 0:06 sin 16 pt in system (1). By numerical calculations, we obtain basic reproduction number R0 2:2002 > 1. Then, by Theorem 3, system (1) with these parameters is uniformly persistent. The corresponding numerical simulations are given in Fig. 2(a)–(e), and these figures demonstrate that the disease will tend towards periodic oscillation along with time passing. It means that there exists a periodic solution, which is in accordance with the obtained conclusion in Corollary 1, and it is seemed that this periodic solution is globally attractive.
a
3.5
5
b
4.5
3
4
2.5
3.5 3
I (t)
R
R
S (t)
2 1.5
2 1.5
1
1
0.5 0 0
0.5 100
200
t
300
400
0 0
500
7
6
5
5
4
4
I (t)
6
H
3
2
1
1 100
200
t
300
e
400
0 0
500
200
100
200
t
300
400
500
300
400
500
3
2
0 0
100
7
d
H
S (t)
c
2.5
t
12 10
H
R (t)
8 6 4 2 0 0
100
200
300
400
t
500
600
700
800
Fig. 2. Uniform persistence of system (1) with parameters in Example 2, when the basic reproduction number R0 2:2002 > 1.
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a
2.5
b
1.6 1.4
2
1.2 1
I (t)
0.8
R
R
S (t)
1.5
1
0.6 0.4
0.5
0.2 0
c
0
100
200
t
300
400
0 0
500
2.1
d
2
I (t)
1.7
200
0.5
1
300
400
500
300
400
500
1.5
2
2.5
0.7
0.4
H
H
S (t)
100
t
0.5
1.8
1.6
0.3 0.2
1.5
0.1
1.4 1.3 0
100
200
t
300
400
0 0
500
f
1.1
t
1.6 1.4
1
1.2
0.9
1 0.8
R
0.8
I
H
R (t)
200
0.6
1.9
e
100
0.7
0.6
0.6
0.4
0.5
0.2
0.4 0
100
200
t
300
400
0 0
500
S
R
g 1.4
R
H
1.2 1 0.8 0.6 0.4 0.8
0.6
0.4
IH
0.2
0
1.4
1.6
1.8
2
2.2
SH
Fig. 3. Chaotic behavior of system (1) with parameters in Example 3, when the basic reproduction number R0 1:5472 > 1.
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Remark 5. It is well known that transmission rate plays an important role in an epidemic model, by comparing values of aðtÞ and bðtÞ in Example 1 with those of Example 2, we can know that if we decrease values of aðtÞ and bðtÞ, especially aðtÞ, i.e., to reduce the transmission rate from mosquito to human by adopting some protecting matures, such as spraying with medical insecticide, use of mosquito nets, vaccine manufacture, etc., the malaria disease will eventually be controlled and eliminated from the population. Example 3. From Fig. 2(a)–(e), we can obtain it is seemed that the positive periodic solution is globally attractive. However, when we take the following parameters with k ¼ 0:099, d ¼ 0:0346; l ¼ 0:1; r ¼ 0; rðtÞ ¼ 3 þ 0:1 sin p6 t ; kðtÞ ¼ 2 þ 0:3 1 p 1 cos pt , dðtÞ ¼ 1:35 þ 0:01 cos 6 t , aðtÞ ¼ 3:899 2 þ 0:03 sin 6 pt and bðtÞ ¼ 0:21 ð0:03þ 0:46169 sin 16 pt þ 0:1 cos 1 6 pt Þ in system (1), then by numerical calculations we still obtain basic reproduction number R0 1:5472 > 1. Then, by The6 orem 3, system (1) with these parameters is uniformly persistent, as shown in numerical simulations of Fig. 3(a)–(e), which not only illustrate the validity of the proposed results, but also display the interesting complex dynamic behaviors, that is, there is not periodic oscillation along with time passing, and from (f) and (g) in Fig. 3, it can be obviously seen that there is a strange chaotic attractor, which may contribute to the better understanding about reasons that complex chaotic behaviors can cause a high risk of the uncertain prevalence for the disease due to the unpredictability. Remark 6. In [33], Bai, Zou and Zhang have analyzed an SIR model with a season contact rate and a staged treatment strategy and established the existence of multiple periodic solutions by continuation theorem. And numerical simulations demonstrate the coexistence of two stable periodic solutions and unstable periodic solution. In this paper, numerical simulations (g) and (h) of Fig. 3 in Example 3 show system (1) may appear to a strange chaotic attractor. All these interestingly abundant and complex dynamic behaviors reveal more complicated properties for a periodic epidemic model than the corresponding autonomous model.
5. Conclusion In this paper, we investigate the dynamic behaviors of a malaria model in periodic environment and obtain that dynamic behaviors of this model is determined by its basic reproduction number R0 . That is, if R0 < 1,then the disease-free periodic solution is globally asymptotically stable, and if R0 > 1, then the disease is uniformly persistent and there is at least one positive periodic solution. Some numerical simulations are carried out to support theoretical analysis of the research, and the last simulation result suggests that there may be interesting dynamic behaviors in this model-a strange chaotic attractor, which displays the complexity of periodic epidemic model. Furthermore, chaos may cause the disease approaching to the uncontrollable state due to the unpredictability. Thus, how to control chaos in an epidemic model is very important, which needs further our investigation. Moreover, pulse vaccination and the quarantine measures have been testified to be an effective strategy in preventing viral infectious disease. The strategy of pulse vaccination (PVS) consists of periodic repetitions of impulsive vaccinations in a population, on all the age cohorts. Therefore, we can further continue our work about the effect of pulse vaccination, the effect of personal protection measures, etc., on the dynamic behaviors of malaria disease. References [1] World Health Organization, World Malaria Day: save lives though local action 2011. Available at:
L. Wang et al. / Commun Nonlinear Sci Numer Simulat 18 (2013) 1288–1303 [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33]
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Lou Y, Zhao X. A climated-based malaria transmission model with structured vector population. SIAM J Appl Math 2010;70:2023–44. Nakata Y, Kuniya T. Global dynamics of a class of SEIRS epidemic models in a periodic environment. J Math Anal Appl 2010;363:230–7. Liu L, Zhao X, Zhou Y. A tuberculosis model with seasonality. Bull Math Biol 2010;72:931–52. Yang Y, Xiao Y. Threshold dynamics for an HIV model in periodic environments. J Math Anal Appl 2010;361:59–68. Wang W, Zhao X. Threshold dynamics for compartmental epidemic models in periodic environments. J Dyn Diff Equat 2008;20:699–717. Teng Z, Li Z. Permanence and asymptotic behavior of the N-species nonautonomous Lotka–Volterra competitive systems. Comput Math Appl 2000;39:107–16. Zhang F, Zhao X. A periodic epidemic model in a patchy environment. J Math Anal Appl 2007;325:496–516. Zhao X. Dynamical systems in population biology. New York: Springer-Verlag; 2003. Van Den Driessche P, Watmough J. Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission. Math Biosci 2002;180:29–48. Teng Z, Liu Y, Zhang L. Persistence and extinction of disease in non-autonomous SIRS epidemic models with disease-induced mortality. Nonlinear Anal 2008;69:2599–614. Teng Z, Chen L. The positive periodic solutions of periodic Kolmogorove type systems with delays. Acta Math Appl Sin 1999;22:446–56 (in Chinese). Zhang T, Teng Z. On a nonautonomous SEIRS model in epidemiology. Bull Math Biol 2007;69(8):2537–59. Bai Z, Zhou Y, Zhang T. Existence of multiple periodic solutions for an SIR model with seasonality. Nonlinear Anal 2011;74:3548–55.
Contents lists available at SciVerse ScienceDirect
Commun Nonlinear Sci Numer Simulat journal homepage: www.elsevier.com/locate/cnsns
Threshold dynamics of a malaria transmission model in periodic environment q Lei Wang, Zhidong Teng ⇑, Tailei Zhang College of Mathematics and System Sciences, Xinjiang University, Urumqi 830046, People’s Republic of China
a r t i c l e
i n f o
Article history: Received 2 February 2012 Received in revised form 8 August 2012 Accepted 5 September 2012 Available online 17 October 2012 Keywords: Malaria transmission Periodic environment Disease-free periodic solution Uniform persistence Extinction Global stability Chaotic attractor Numerical simulation
a b s t r a c t In this paper, we propose a malaria transmission model with periodic environment. The basic reproduction number R0 is computed for the model and it is shown that the disease-free periodic solution of the model is globally asymptotically stable when R0 < 1, that is, the disease goes extinct when R0 < 1, while the disease is uniformly persistent and there is at least one positive periodic solution when R0 > 1. It indicates that R0 is the threshold value determining the extinction and the uniform persistence of the disease. Finally, some examples are given to illustrate the main theoretical results. The numerical simulations show that, when the disease is uniformly persistent, different dynamic behaviors may be found in this model, such as the global attractivity and the chaotic attractor. Ó 2012 Elsevier B.V. All rights reserved.
1. Introduction Malaria remains one of the most prevalent and lethal human infectious diseases in the world. It is estimated that more than 40% of the world’s population-at least 2.4 billion people-are exposed to varying degrees of risk of attaching the disease. The incidence of malaria is approximately 300–500 million clinical cases, resulting in 1 million deaths each year, primarily among young children in Africa, where malaria accounts for 20% of all deaths in children younger than 5 years in 2008 (see [1]). Of all infectious diseases, malaria continues to be one of the biggest contributors to the global disease burdens in terms of suffering and death and keeps on receiving worldwide attention (see [2,3]). Malaria is a protozoan infection of red blood cells cased in human by four species of the genus Plasmodium (Plasmodium falciparum, Plasmodium vivax, Plasmodium ovale and Plasmodium malariae). The malaria parasites are generally transmitted to the human host through the bite of an infected female anopheline mosquito (see [4]). In 1911, Ross initiatively developed a simple mathematical model to understand parasite transmission mechanism of malaria in [5]. Since then, there has been a great deal of work about using mathematical models to study malaria, (for example, see [6–12] and the references cited therein). Recently, Tumwiine et al. [13] proposed an autonomous model describing the
q Supported by the National Natural Science Foundation of China (Grant Nos. 10961022, 11001235, 11201399), the China Postdoctoral Science Foundation (Grant Nos. 20110491750), the Natural Science Foundation of Xinjiang (Grant Nos. 2011211B08) and the Natural Science Foundation of Xinjiang University (Grant Nos. BS100104). ⇑ Corresponding author. E-mail address: [email protected] (Z. Teng).
1007-5704/$ - see front matter Ó 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.cnsns.2012.09.007
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dynamics of malaria transmission, which has a human host and mosquito vector with temporary immunity, and obtained the globally asymptotical stability of the disease-free equilibrium and the endemic equilibrium for this model. In [14], Wei et al. considered an epidemic model of a vector-borne disease with direct transmission and time delay, and showed that the introduction of a time delay in the host-to-vector transmission term can destabilize the system and periodic solutions can arise through Hopf bifurcation. Tumwiine et al. [15] discussed a host-vector model for malaria with infective immigrants, and proved that there is not disease free equilibrium, and the unique endemic equilibrium is globally asymptotically stable if this model is assumed that there is a fraction of infective immigrants. Cai and Li in [16] studied an autonomous vector-host epidemic model with nonlinear incidences, and obtained that the asymptotic behaviors of the model are determined by its basic reproduction number R0 , that is, if R0 6 1,the disease-free equilibrium is globally asymptotically stable, and if R0 > 1, the disease persists and the unique endemic equilibrium is globally asymptotically stable. Malaria often occurs in most tropical and some subtropical regions of the world (see [17]). Environmental and climatic factors play an important role in the geographical distribution and transmission of malaria (see [18–20]). In [20], the authors provided a picture about ENSO-related precipitation anomalies and periodic malaria epidemics, which showed a striking correspondence between periodic malaria epidemics and geographical areas where the ‘‘teleconnections’’ ENSO affect precipitation and temperature. For mosquito population, proper temperature and humidity are more beneficial to give birth and propagate. For example, in temperate climates and in tropical highlands, temperature restricts vector multiplication and the development of the parasite in the mosquito, while in arid climates precipitation restricts mosquito breeding (see [20]). On the other hand, the intensity and pattern of transmission of malaria may vary markedly as a result of variations in altitude or rainfall and other environmental factors, even though within the same country (see [17]). Therefore, malaria fluctuates over time and often exhibits seasonal behaviors, especially in the northern areas. It is meaningful and essential to take account of malaria model with periodic environment. However, up to now, there have been few results about malaria model with periodic environment. In [21], a malaria transmission model with periodic birth rate and age structure for the vector population was presented by Lou and Zhao, and they further showed that the disease would die out if the basic reproduction number R0 < 1, and if the basic reproduction number R0 > 1, there exists at least one positive periodic state and the disease persists, finally they used these analytic results to study the malaria transmission cases in KwaZulu-Natal Province, South Africa. Motivated by the work of Nakata and Kuniya [22], Liu et al. [23] and Yang and Xiao [24], in this paper, we will investigate a malaria transmission model with periodic environment. By applying the way of computing the basic reproduction number for a wide class of compartmental epidemic models in periodic environments given by Wang and Zhao [25], we calculate basic reproduction number R0 for this model, and prove that the disease goes to extinction if R0 < 1, while the disease is uniformly persistent and there is at least one positive periodic solution if R0 > 1, which indicates the basic reproduction number is the threshold value determining whether or not malaria persists. The organization of this paper is as follows. In Section 2, model description and some preliminaries are given. In Section 3, we will compute the basic production number and study the globally asymptotical stability of the disease-free periodic solution and the uniform persistence of the model. In Section 4, some examples and simulations are given to illustrate theoretical results and exhibit different dynamic behaviors, such as the global attractivity (see Fig. 2), the chaotic attractor (see Fig. 3). 2. Model description and preliminaries Based on the transmission mechanism of malaria, we consider host population and vector population are human population and mosquito population, respectively. And we will make the following basic assumptions. (1) A mosquito firstly bites the infected person, then it becomes the infected mosquito. Due to its short lifespan, it cannot recover from the infection. Consequently, we only divide the total mosquito population into two classes: the susceptible and the infected. (2) Because environmental and climatic factors (for example, temperature, humidity, etc.) often have many effects on giving birth and propagating for mosquito population, we assume that the grow rate of the susceptible mosquito is governed by a logistic equation, in which these coefficients are periodic functions on account of seasonal effects. (3) Based on assumption (2), due to mosquito’s periodic birth and death, we assume that transmission rate from mosquito to human and transmission rate from human to mosquito be periodic functions. And both are bilinear incidence rate. (4) For human population, we separate the total human population into three classes: the susceptible, the infective and the recovered. The susceptible individual becomes infectious after being bitten by infective mosquito and can acquire permanent immunity after recovery from disease by curing. (5) For human population, we assume that there are the immigration, natural death and disease-induced death and all newborns are the susceptible. The recovered can become the susceptible again. We further assume that immigration rate, disease-induced death rate, natural death rate and the rate of which the recovered becomes susceptible again are positive constants.
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Under the above assumptions, the following malaria transmission model with periodic environment is proposed.
8 S_R ðtÞ ¼ > > > > > > _ > < IR ðtÞ ¼ S_H ðtÞ ¼ > > > > I_H ðtÞ ¼ > > > : _ RH ðtÞ ¼
SR rðtÞSR ð1 kðtÞ Þ aðtÞSR IH ;
aðtÞSR IH dðtÞIR ; ð1Þ
k bðtÞSH IR dSH þ cRH ; bðtÞSH IR ðd þ l þ rÞIH ; lIH dRH cRH ;
where SR ðtÞ and IR ðtÞ represent the densities of the susceptible and the infected for mosquito population at time t, respectively; SH ðtÞ; IH ðtÞ and RH ðtÞ separately denote the densities of the susceptible, the infective and the recovered for human population at time t; rðtÞ and kðtÞ are the intrinsic growth rate and the carrying capacity of environment for mosquito population at time t, respectively; dðtÞ is the death rate of the infected mosquito, including the natural death rate and disease-induced death rate; k; d; l and r are positive constants, which are the immigration rate, natural death rate, recovered rate and disease-induced death rate for human population, respectively; c is the rate of which the recovered becomes susceptible again; aðtÞ is the transmission rate from mosquito to human, and bðtÞ is the transmission rate from human to mosquito. In view of the biological background of system (1), in this paper we only consider the solution of system (1) starting at t ¼ 0 with initial values:
S0R P 0;
I0R P 0;
S0H P 0;
I0H P 0;
R0H P 0:
ð2Þ
In this paper, for system (1) we always introduce the following assumptions. ðH1 Þ rðtÞ; kðtÞ; dðtÞ; aðtÞ and bðtÞ are continuous and x-periodic functions. ðH2 Þ kðtÞ; dðtÞ > 0 and aðtÞ; bðtÞ P 0 with aðtÞ; bðtÞ X 0 for all t P 0. Rx ðH3 Þ 0 rðtÞdt > 0. When IH ðtÞ 0; RH ðtÞ 0 and IR ðtÞ 0, we can obtain the following two subsystem of system (1)
S_H ðtÞ ¼ k dSH ðtÞ;
ð3Þ
SR : S_R ðtÞ ¼ rðtÞSR 1 kðtÞ
ð4Þ
and
Lemma 1. (a) System (3) has a unique positive globally asymptotically stable equilibrium SH ðtÞ ¼ kd. (b) System (4) allows a globally uniformly attractive positive x-periodic solution SR ðtÞ. Conclusion ðaÞ of Lemma 1 is obvious. Conclusion ðbÞ of Lemma 1 can be easily obtained from Lemma 2 given in [26]. From Lemma 1, we see that system (1) has a disease-free periodic solution E ðtÞ ¼ ðSR ðtÞ; 0; kd ; 0; 0Þ. Let ðRn ; Rnþ Þ be the standard ordered n-dimensional Euclidean space. For u; v 2 Rn , we denote u P v if u v 2 Rnþ ; u > v if u v 2 Rnþ n f0g and u v if u v 2 intRnþ , respectively, where intRnþ denote the interior of Rnþ . Let AðtÞ be a continuous and x-periodic n n matrix function, we consider the following linear system
x_ ¼ AðtÞx:
ð5Þ
Let UA ðtÞ be the fundamental solution matrix of system (5) with initial condition UA ð0Þ ¼ I, where I is n n identity matrix, and let qðUA ðxÞÞ be the spectral radius of matrix UA ðxÞ. Further, we assume that AðtÞ also is cooperative and irreducible, then by the Perron-Frobenius theorem, qðUA ðxÞÞ is the principle eigenvalue of UA ðxÞ in the sense that it is simple and admits an eigenvector m 0. Lemma 2 (See [27]). Let AðtÞ is a continuous, cooperative, irreducible and x-periodic n n matrix function, Then there exists a positive x-periodic function mðtÞ such that xðtÞ ¼ elt mðtÞ is a solution of system (5).
l ¼ x1 ln qðUA ðxÞÞ.
Let X be a metric space with metric d and f : X ! X be a continuous map. For any x 2 X, we denote f n ðxÞ ¼ f ðf n1 ðxÞÞ for any integer n > 1 and f 1 ðxÞ ¼ f ðxÞ. f is said to be compact in X if for any bounded set H X set f ðHÞ ¼ ff ðxÞ : x 2 Hg is precompact in X. f is said to be point dissipative if there is a bounded set B0 X such that for any x 2 X
lim dðf n ðxÞ; B0 Þ ¼ 0:
n!1
For any x0 2 X, the positive semiorbit through x0 is defined by cþ ðx0 Þ ¼ fxn ¼ f n ðx0 Þ; n ¼ 1; 2; . . .g, the negative semiorbit through x0 is defined as a sequence c ðx0 Þ ¼ fxk g satisfying f ðxk1 Þ ¼ xk for integers k 6 0, the omega limit set of cþ ðx0 Þ is defined by xðx0 Þ ¼ fy 2 X: there is a sequence nk ! 1 such that limk!1 xnk ¼ yg and the alpha limit set of c ðx0 Þ is defined by aðx0 Þ ¼ fy 2 X: there is a sequence nk ! 1 such that limk!1 xnk ¼ yg.
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A nonempty set A X is said to be invariant if f ðAÞ # A. A nonempty invariant set M of X is called to be isolated in X if it is the maximal invariant set in a neighborhood of itself. For a nonempty set M of X, set W s ðMÞ :¼ fx 2 X : limn!1 dðf n ðxÞ; MÞ ¼ 0g is called the stable set of M. Let A and B be two isolated invariant sets, set A is said to be chained to set B, written A ! B, if there exists a full orbit though some x R A [ B such that xðxÞ B and aðxÞ A. A finite sequence fM 1 ; . . . ; M k g of isolated invariant sets is called a chain if M 1 ! M 2 ! ! M k , and if M k ¼ M 1 the chain is called a cycle. Let X 0 be a nonempty open set of X. We denote
@X 0 :¼ X n X 0 ;
M @ :¼ fx 2 @X 0 : f n ðxÞ 2 @X 0 for all n P 0g:
Lemma 3. Let f : X ! X be a continuous map. Assume that the following conditions hold: ðC 1 Þ f is compact and point dissipative, and f ðX 0 Þ # X 0 . ðC 2 Þ There exists a finite sequence M ¼ fM 1 ; . . . ; M k g of compact and isolated invariant sets such that. (a) (b) (c) (d)
T Mi M j ¼ ; for any i; j ¼ 1; 2; . . . ; k and i – j; S S XðM@ Þ :¼ x2M@ xðxÞ ki¼1 Mi ; no subset of M forms a cycle in @X 0 ; T W s ðM i Þ X 0 ¼ ; for each 1 6 i 6 k.
Then f is uniformly persistent with respect to ðX 0 ; @X 0 Þ, that is, there exists a constant g > 0 such that lim inf n!1 dðf n ðxÞ; @X 0 Þ P g for all x 2 X 0 . Lemma 3 can be obtained from Theorem 1.1.3, Theorem 1.3.1, Remark 1.3.1 and Theorem 1.3.3 given by Zhao in [28]. Now, based on the assumptions ðH1 Þ ðH3 Þ, we compute the basic reproduction number of system (1) by applying the way given in [25,29] by Wang, Zhao, Driessche and Watmough. Let
0
aðtÞSR IH
0
1
C B B bðtÞSH IR C C B C; B F ðt; xÞ ¼ B 0 C C B 0 A @ 0
dðtÞIR
1
C B B ðd þ l þ rÞIH C C B 2 B SR C V ðt; xÞ ¼ B aðtÞSR IH þ rðtÞ kðtÞ C C B C B @ bðtÞSH IR þ dSH A dRH þ cRH
and
0
0
1
C B 0 C B C B B V ðt; xÞ ¼ B rðtÞSR C C; C B @ k þ cRH A lIH þ
where x ¼ ðIR ; IH ; SR ; SH ; RH ÞT , then system (1) equals to the following form
_ xðtÞ ¼ F ðt; xÞ Vðt; xÞ , f ðt; xðtÞÞ;
ð6Þ
þ
where Vðt; xÞ ¼ V ðt; xÞ V ðt; xÞ. In the following, we will check conditions (A1)–(A7) which are given in [25]. By the expressions of F ðt; xÞ and Vðt; xÞ, we firstly easily see that conditions (A1)–(A5) are satisfied. Obviously, system (6) has disease-free periodic solution x ðtÞ ¼ ð0; 0; SR ðtÞ; kd ; 0Þ. Now, we define
NðtÞ ¼
@fi ðt; x ðtÞÞ ; @xj 36i;j65
where fi ðt; xðtÞÞ and xi are the ith component of f ðt; xðtÞÞ and x, respectively. By simple computations, we can obtain
0 B NðtÞ ¼ @ Since
SR ðtÞ
Z
0
Thus,
x
rðtÞ 2rðtÞ S ðtÞ kðtÞ R
0
0
0
d
c
0
0
ðd þ cÞ
1 C A:
is the globally uniformly attractively x-periodic solution of system (4), it satisfies
S ðtÞ dt ¼ 0: rðtÞ 1 R kðtÞ
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exp
Z
x
rðtÞ
0
Z x Z x 2rðtÞ S ðtÞ S ðtÞ rðtÞ SR ðtÞ dt ¼ exp dt ¼ exp SR ðtÞdt < 1: rðtÞ 1 R R kðtÞ kðtÞ kðtÞ kðtÞ 0 0
Therefore, we finally get qðUN ðxÞÞ < 1, Thus, condition (A6) also holds. Next, we set two 2 2 matrices as follows
FðtÞ ¼
@F i ðt; x ðtÞÞ @xj 16i;j62
and VðtÞ ¼
@V i ðt; x ðtÞÞ @xj 16i;
; j62
where F i ðt; xðtÞÞ and V i ðt; xðtÞÞ are the ith component of F ðt; xðtÞÞ and Vðt; xðtÞÞ, respectively. Then, by simple computations, it follows that
FðtÞ ¼
0
aðtÞSR ðtÞ
bðtÞ kd
0
!
and VðtÞ ¼
dðtÞ 0 : 0 dþlþr
Therefore, from assumption ðH1 Þ, we obtain that qðUV ðxÞÞ < 1. Thus, condition (A7) also holds. Let Yðt; sÞ is the 2 2 matrix solution of the following initial value problem
(
d Yðt; sÞ dt
¼ VðtÞYðt; sÞ for all t P s;
Yðs; sÞ ¼
I:
Let C x be the ordered Banach space of all x-periodic continuous function form R to R2 with the maximum norm k k. The positive cone
C þx ¼ f/ 2 C x : /ðtÞ P 0 for all t 2 Rg: Suppose /ðsÞ 2 C þ x is the initial distribution of infectious individuals in this periodic environment, then FðsÞ/ðsÞ is the rate of new infectious produced by the infected individuals who were introduced at time s, and Yðt; sÞFðsÞ/ðsÞ represents the distributions of those infected individuals who were newly infected at time s and remain in the infected compartment at time t for t P s. It follows that
Z
0
Yðt; sÞFðsÞ/ðsÞds ¼
wðtÞ :¼
Z
þ1
Yðt; t aÞFðt aÞ/ðt aÞda
0
1
denotes the distribution of accumulative new infections at time t produced by all those infected individuals /ðsÞ introduced at previous time to t. We define a linear operator L : C x ! C x as follows
ðL/ÞðtÞ ¼
Z
þ1
Yðt; t aÞFðt aÞ/ðt aÞda for all t 2 R; / 2 C x :
0
Following Wang and Zhao in [25], we call L the next infection operator, and define the basic reproduction number R0 for system (1) by
R0 ¼ qðLÞ; where qðLÞ is the spectral radius of L. Using Theorem 2.2 given in [25], we can obtain the following results on basic reproduction number R0 and the locally asymptotical stability of disease-free periodic solution E ðtÞ for system (1). Lemma 4. (1) On basic reproduction numberR0 , we have (i) R0 ¼ 1if and if onlyqðUFV ðxÞÞ ¼ 1; (ii) R0 > 1if and if onlyqðUFV ðxÞÞ > 1; (iii) R0 < 1if and if onlyqðUFV ðxÞÞ < 1. (2) E ðtÞis locally asymptotically stable ifR0 < 1, and unstable ifR0 > 1. 3. Main results
Theorem 1. LetðSR ðtÞ; IR ðtÞ; SH ðtÞ; IH ðtÞ; RH ðtÞÞbe the solution of system(1)with initial condition(2). Then, we have (a) ðSR ðtÞ; IR ðtÞ; SH ðtÞ; IH ðtÞ; RH ðtÞÞis nonnegative for allt P 0and ultimately bounded. (b) IfS0R > 0; I0R > 0; S0H > 0; I0H > 0andR0H P 0, thenðSR ðtÞ; IR ðtÞ; SH ðtÞ; IH ðtÞ; RH ðtÞÞalso is positive for allt > 0.
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Proof. We firstly prove conclusion ðbÞ. Suppose that ðSR ðtÞ; IR ðtÞ; SH ðtÞ; IH ðtÞ; RH ðtÞÞ is defined for all t 2 ½0; TÞ, where T > 0. Integrating the first equation of system (1) from 0 to t, we have
SR ðtÞ ¼ S0R exp
Z t 0
SR ðsÞ rðsÞ 1 aðsÞIH ðsÞ ds: kðsÞ
ð7Þ
From S0R > 0, we obtain that SR ðtÞ > 0 for all t 2 ½0; TÞ. Suppose that there exists a t1 2 ð0; TÞ such that
minfIR ðt 1 Þ; IH ðt1 Þg ¼ 0: Since I0R > 0 and I0H > 0, we can further assume minfIR ðtÞ; IH ðtÞg > 0 for all t 2 ½0; t 1 Þ. If minfIR ðt1 Þ; IH ðt 1 Þg ¼ IR ðt1 Þ, then from SR ðtÞ > 0 for all t 2 ½0; TÞ, we have
I_R ðtÞ P dðtÞIR ðtÞ for all t 2 ½0; t1 : Hence,
Z 0 ¼ IR ðt1 Þ P I0R exp
t1
dðsÞds > 0;
0
which leads to a contradiction. If minfIR ðt 1 Þ; IH ðt1 Þg ¼ IH ðt 1 Þ, then since
R_ H ðtÞ > ðd þ cÞRH ðtÞ for all t 2 ½0; t 1 Þ; we have
RH ðtÞ > R0H expððd þ cÞtÞ P 0 for all t 2 ð0; t 1 : From the third equation of system (1), we have
S_ H ðtÞ P ðbðtÞIR ðtÞ þ dÞSH ðtÞ for all t 2 ½0; t 1 : Hence,
Z t SH ðtÞ P S0H exp ðbðsÞIR ðsÞ þ dÞds > 0 for all t 2 ½0; t 1 : 0
From this, we further obtain
I_H ðtÞ P ðd þ l þ rÞIH ðtÞ for all t 2 ½0; t 1 ; Hence,
0 ¼ IH ðt 1 Þ P I0H expððd þ l þ rÞt1 Þ > 0; which leads to a contradiction. This shows that IR ðtÞ > 0 and IH ðtÞ > 0 for all t 2 ½0; TÞ. Furthermore, since
R_ H ðtÞ > ðd þ cÞRH ðtÞ for all t 2 ½0; TÞ; we obtain
RH ðtÞ > R0H eðdþcÞt P 0 for all t 2 ð0; TÞ: Further from the third equation of (1) we have
S_ H ðtÞ P ðbðtÞIR ðtÞ þ dÞSH ðtÞ for all t 2 ½0; TÞ: Hence,
Z t SH ðtÞ P S0H exp ðbðsÞIR ðsÞ þ dÞds > 0 for all t 2 ½0; TÞ: 0
This shows that ðSR ðtÞ; IR ðtÞ; SH ðtÞ; IH ðtÞ; RH ðtÞÞ is positive on the interval of existence. This completes the proof of conclusion ðbÞ. Next, we prove conclusion ðaÞ. Firstly, from conclusion ðbÞ and the continuous dependence of solutions of system (1) with respect to initial values, we immediately obtain that ðSR ðtÞ; IR ðtÞ; SH ðtÞ; IH ðtÞ; RH ðtÞÞ with initial condition (2) is nonnegative on the interval of existence. Now, we prove that the interval of existence of ðSR ðtÞ; IR ðtÞ; SH ðtÞ; IH ðtÞ; RH ðtÞÞ is ½0; 1Þ. In fact, if the interval of existence of ðSR ðtÞ; IR ðtÞ; SH ðtÞ; IH ðtÞ; RH ðtÞÞ is a finite interval ½0; TÞ, then we know that ðSR ðtÞ; IR ðtÞ; SH ðtÞ; IH ðtÞ; RH ðtÞÞ is nonbounded on ½0; TÞ. From
S_ R ðtÞ 6 rðtÞSR ðtÞ;
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we obtain
SR ðtÞ 6 S0R exp
Z
t
rðsÞds :
0
Hence, SR ðtÞ is bounded on ½0; TÞ. Let N H ðtÞ ¼ SH ðtÞ þ IH ðtÞ þ RH ðtÞ, from system (1) we have
N_ H ðtÞ ¼ k dNH ðtÞ rIH ðtÞ 6 k; which implies that
NH ðtÞ 6 N0H þ kt: Hence, N H ðtÞ is bounded on ½0; TÞ, which implies that SH ðtÞ, IH ðtÞ and RH ðtÞ are also bounded on ½0; TÞ. Let N R ðtÞ ¼ SR ðtÞ þ IR ðtÞ. From system (1) we have
SR ðtÞ N_ R ðtÞ ¼ rðtÞSR ðtÞ 1 dðtÞIR ðtÞ 6 dðtÞNR ðtÞ þ ðdðtÞ þ rðtÞÞSR ðtÞ 6 dðtÞNR ðtÞ þ M S ; kðtÞ where M S ¼ sup06t
Z s Z Z t Z t NR ðtÞ 6 exp dðsÞds N 0R þ M S exp dðsÞds ds 6 N 0R þ exp 0
0
0
T
dðsÞds M S T:
0
Hence, N R ðtÞ is bounded on ½0; TÞ, which implies that IR ðtÞ also is bounded on ½0; TÞ. This leads to a contradiction. Therefore, we finally have that ðSR ðtÞ; IR ðtÞ; SH ðtÞ; IH ðtÞ; RH ðtÞÞ is defined on ½0; 1Þ. Since for any t P 0
SR ðtÞ ; S_R ðtÞ 6 rðtÞSR ðtÞ 1 kðtÞ by the comparison principle and conclusion ðbÞ of Lemma 1, we can obtain
lim sup SR ðtÞ 6 lim sup SR ðtÞ 6 M S ; t!1
ð8Þ
t!1
where SR ðtÞ is the globally uniformly attractively positive x-periodic solution of system (4) and M S ¼ maxt2½0;x SR ðtÞ. For any constant e0 > 0 there is a T 0 > 0 such that for any t P T 0
SR ðtÞ 6 SR ðtÞ þ e0 6 MS þ e0 : Since for any t P 0 we have
N_ H ðtÞ 6 k dNH ðtÞ and
N_ R ðtÞ 6 dðtÞNR ðtÞ þ ðdðtÞ þ rðtÞÞSR ðtÞ; we obtain
lim sup NH ðtÞ 6 t!1
k d
ð9Þ
and for any t P T 0
N_ R ðtÞ 6 dðtÞNR ðtÞ þ supfdðtÞ þ rðtÞgðM S þ e0 Þ: tP0
Integrating this inequality for any t P T 0 , we obtain
Z s Z t Z t NR ðtÞ 6 exp dðsÞds NR ðT 0 Þ þ ðM S þ e0 ÞsupfdðtÞ þ rðtÞg exp dðsÞds ds : T0
T0
tP0
T0
From this, we further obtain
lim sup NR ðtÞ 6 ðM S þ e0 ÞD; t!1
where D ¼
suptP0 fdðtÞþrðtÞg . inf tP0 dðtÞ
lim sup NR ðtÞ 6 t!1
From the arbitrariness of
MS D:
e0 , we can take e0 ! 0 and finally obtain ð10Þ
Lastly, from (8)–(10), we finally obtain that all solutions of system (1) are ultimately bounded. This completes the proof of conclusion ðaÞ. h
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Remark 1. From (7) we obtain that SR ðtÞ > 0 for all t P 0 when S0R > 0. Therefore, for any small enough constant denote
> 0, we
k d
X ¼ fðSR ; IR ; SH ; IH ; RH Þ : SR > 0; IR P 0; SH P 0; IH P 0; RH P 0; SR þ IR 6 M S D þ ; SH þ IH þ RH 6 þ g; then from Theorem 1, we easily see that X is a positively invariant set with respect to system (1) and also a global attractor of all positive solutions of system (1). Theorem 2. IfR0 < 1, then disease-free periodic solutionE ðtÞof system(1)is globally asymptotically stable. Proof. From Lemma 4, we obtain that if R0 < 1; E ðtÞ is locally asymptotically stable. Now, we will only prove the global attractivity of E ðtÞ for the case R0 < 1. From R0 < 1 and conclusion (iii) of Lemma 4, we have qðUFV ðxÞÞ < 1, then we can choose a small enough constant 2 > 0 such that qðUFVþ2 M ðxÞÞ < 1, where
MðtÞ ¼
0
aðtÞ
bðtÞ
0
:
From (8) and (9), we obtain that for above given constant
k SH ðtÞ 6 þ 2 d
2
there exists a t 1 > 0 such that for all t > t1
and SR ðtÞ 6 SR ðtÞ þ 2 :
From the second and fourth equations of system (1), we obtain that for all t > t1
(
I_R I_H
6 aðtÞðSR ðtÞ þ 2 ÞIH dðtÞIR ; 6 bðtÞ kd þ 2 IR ðd þ l þ rÞIH :
ð11Þ
Considering the following auxiliary system:
8 < eI_
¼ aðtÞðSR ðtÞ þ 2 ÞeI H dðtÞeI R ; ¼ bðtÞ kd þ 2 eI R ðd þ l þ rÞeI H :
R
: e_ IH
For the convenience, we will rewrite it as follows
d dt
eI R eI H
!
eI ¼ ðFðtÞ VðtÞ þ 2 MðtÞÞ R eI H
! ð12Þ
:
From Lemma 2, it follows that there exists a positive x-periodic function qðtÞ ¼ ðq1 ðtÞ; q2 ðtÞÞT such that ðeI R ðtÞ; eI H ðtÞÞT ¼ el1 t qðtÞ is a solution of system (12), where l1 ¼ x1 lnðqðUFVþ2 M ðxÞÞÞ. Denote JðtÞ ¼ ðIR ðtÞ; IH ðtÞÞT . We can choose a small constant n > 0 such that Jðt 1 Þ 6 nqðt 1 Þ. Then, from (11) the comparison principle implies that
JðtÞ 6 nel1 t qðtÞ for all t > t 1 : By qðUFVþ2 M ðxÞÞ < 1, it follows that
lim IR ðtÞ ¼ 0;
t!1
l1 < 0, then limt!1 JðtÞ ¼ 0, that is,
lim IH ðtÞ ¼ 0:
t!1
Moreover, from the equations of SR ; SH and RH in system (1), we can get
½lim SR ðtÞ ¼ SR ðtÞ; t!1
k lim SH ðtÞ ¼ ; d
t!1
lim RH ðtÞ ¼ 0:
t!1
Hence, disease-free periodic solution E ðtÞ of system (1) is globally attractive. This completes the proof. h Theorem 3. IfR0 > 1, then system(1)is uniformly persistent. That is, there exists a positive constante, such that any solutionðSR ðtÞ; SH ðtÞ; IR ðtÞ; IH ðtÞ; RH ðtÞÞof system(1)with initial conditions(2)satisfies
lim inf ðSR ðtÞ; SH ðtÞ; IR ðtÞ; IH ðtÞ; RH ðtÞÞ P ðe; e; e; e; eÞ: t!1
Proof. From R0 > 1 and conclusion (ii) of Lemma 4, we have qðUFV ðxÞÞ > 1. Then, we can choose a small constant g > 0 such that qðUFVgM ðxÞÞ > 1, where MðtÞ is defined in Theorem 2. From ðH2 Þ and ðH3 Þ, we obtain for any small enough
e>0
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Z
x
½rðtÞ aðtÞe dt > 0:
ð13Þ
0
For this e, we consider the following two perturbed equations
U e ðtÞ eaðtÞU e ðtÞ; U_ e ðtÞ ¼ rðtÞU e ðtÞ 1 kðtÞ
ð14Þ
V_ e ðtÞ ¼ k ebðtÞV e ðtÞ dV e ðtÞ:
ð15Þ
and
Using Lemma 2 given in [26] and Lemma 1 given in [30], from ðH2 Þ and (13), we can get that systems (14) and (15) admit globally uniformly attractive positive x-periodic solutions U e ðtÞ and V e ðtÞ, respectively. By the continuity of solutions with respect to the parameter e, for constant g > 0 given in above, there exists a constant e1 > 0 such that for all 0 < e1 < e and t 2 ½0; x
g
U e1 ðtÞ > SR ðtÞ ; 2
k g V e1 ðtÞ > : d 2
ð16Þ
Define
X ¼ fðSR ; IR ; SH ; IH ; RH Þ : SR > 0; IR P 0; SH P 0; IH P 0; RH P 0g and
X 0 ¼ fðSR ; IR ; SH ; IH ; RH Þ 2 X : IR > 0; IH > 0g: We have
@X 0 ¼ X n X 0 ¼ fðSR ; IR ; SH ; IH ; RH Þ 2 X : IR IH ¼ 0g: From system (1), it is easy to see that X and X 0 are positively invariant, and @X 0 is also a relatively closed set in X. Let P : X ! X be the Poincaré map associated with system (1), that is
Pðx0 Þ ¼ uðx; x0 Þ for all x0 2 X; where uðt; x0 Þ is the unique solution of system (1) satisfying initial condition uð0; x0 Þ ¼ x0 . From the continuity of solutions of system (1) with respect to initial value x0 , we can obtain that P is compact. Moreover, by Theorem 1, we obtain that P is point dissipative on X. Further, we define
M@ ¼ fðS0R ; I0R ; S0H ; I0H ; R0H Þ 2 @X 0 : P m ðS0R ; I0R ; S0H ; I0H ; R0H Þ 2 @X 0 m
where P ¼ PðP
m1
for all m > 0g;
1
Þ for all m > 1 and P ¼ P. Now, we prove
M@ ¼ fðSR ; 0; SH ; 0; 0Þ : SR > 0; SH P 0:g ðS0R ; 0; S0H ; 0; 0Þ,
S0R
ð17Þ S0H
Firstly, for any point where > 0 and P 0, according to the existence and uniqueness of solutions of system (1), we can obtain that ðSR ðtÞ; IR ðtÞ; SH ðtÞ; IH ðtÞ; RH ðtÞÞ with IR ðtÞ 0; IH ðtÞ 0 and RH ðtÞ 0 is the unique solution of system (1) satisfying initial condition ðSR ð0Þ; IR ð0Þ; SH ð0Þ; IH ð0Þ; RH ð0ÞÞ ¼ ðS0R ; 0; S0H ; 0; 0Þ. Therefore, we obtain for any integer m>0
Pm ðS0R ; 0; S0H ; 0; 0Þ 2 fðSR ; 0; SH ; 0; 0Þ : SR > 0; SH P 0g # @X 0 : This shows ðSR ; 0; SH ; 0; 0Þ 2 M @ . Consequently,
fðSR ; 0; SH ; 0; 0Þ : SR > 0; SH P 0g # M @ : On the other hand, if M @ n fðSR ; 0; SH ; 0; 0Þ : SR > 0; SH P 0g – ;, then there exists at least a point ðS0R ; I0R ; S0H ; I0H ; R0H Þ 2 M @ satisfying I0H > 0 or I0R > 0. If I0R ¼ 0 and I0H > 0, then it is clear that from system (1)
IH ðtÞ P I0H eðdþlþrÞt > 0 for all t > 0: From S0R > 0 we can obtain from the first equation of system (1) that SR ðtÞ > 0 for all t > 0. Hence,
Rt Z t Rs IR ðtÞ ¼ I0R þ aðsÞSR ðsÞIH ðsÞe 0 dðuÞdu ds e 0 dðuÞdu > 0 0
for all t > 0. If
I0R
> 0 and I0H ¼ 0, then we have
IR ðtÞ ¼ I0R þ
Z 0
t
Rs
aðsÞSR ðsÞIH ðsÞe
0
dðuÞdu
Rt dðuÞdu ds e 0 >0
L. Wang et al. / Commun Nonlinear Sci Numer Simulat 18 (2013) 1288–1303
1297
for all t > 0. From the third equation of system (1) we have
S_H ðtÞ > ðbðtÞIR þ d þ cÞSH
for all t P 0:
Hence, we further have SH ðtÞ > 0 for all t > 0. Consequently, by the fourth equation of system (1) we have
I_H ðtÞ > ðd þ l þ rÞIH
for all t P 0
and hence IH ðtÞ > 0 for all t > 0. This shows that ðSR ðtÞ; IR ðtÞ; SH ðtÞ; IH ðtÞ; RH ðtÞÞ R @X 0 . Hence, ðS0R ; I0R ; S0H ; I0H ; R0H Þ R M@ which leads to a contradiction. It indicates that M@ # fðSR ; 0; SH ; 0; 0Þ : SR > 0; SH P 0g. Therefore, we finally obtain that claim (17) holds. It is clear that there is a fixed points of P in M @ , which is M 1 ¼ ðSR ð0Þ; 0; kd ; 0; 0Þ. Denote x0 ¼ ðS0R ; I0R ; S0H ; I0H ; R0H Þ 2 X 0 . By the continuity of solutions with respect to the initial value, for above given constant e1 > 0, there exists d0 > 0 such that for all x0 2 X 0 with kx0 M 1 k 6 d0 , it follows that
kuðt; x0 Þ uðt; M1 Þk < e1
for all t 2 ½0; x :
ð18Þ
Now, we prove
lim sup dðPm ðx0 Þ; M 1 Þ P d0 :
ð19Þ
m!1
Suppose the conclusion is not true, then
lim sup dðPm ðx0 Þ; M 1 Þ < d0 m!1
for some x0 2 X 0 . Without loss of generality, we can assume that
dðPm ðx0 Þ; M 1 Þ < d0
for all m P 0:
Further, from (18) we have
kuðt; Pm ðx0 ÞÞ uðt; M 1 Þk < e1
for all m P 0;
0
t 2 ½0; x :
0
For any t P 0, let t ¼ mx þ t ,where t 2 ½0; xÞ and m ¼ ½xt is the greatest integer less than or equal to xt , then we can get
kuðt; x0 Þ uðt; M1 Þk ¼ kuðt 0 ; Pm ðx0 ÞÞ uðt 0 ; M 1 Þk < e1
for all t P 0:
ð20Þ
ðSR ðtÞ; 0; kd ; 0; 0Þ,
0
Since uðt; x Þ ¼ ðSR ðtÞ; IR ðtÞ; SH ðtÞ; IH ðtÞ; RH ðtÞÞ and uðt; M 1 Þ ¼ it follows from (20) that 0 6 IR ðtÞ 6 e1 and 0 6 IH ðtÞ 6 e1 for all t P 0. Then, by the first and third equation of system (1) we get for any t P 0
SR S_R ðtÞ P rðtÞSR 1 e1 aðtÞSR ; kðtÞ and
S_H ðtÞ P k e1 bðtÞSH dSH : By the comparison principle, we obtain for any t P 0
SR ðtÞ P U e1 ðtÞ;
SH ðtÞ P V e1 ðtÞ;
where U e1 ðtÞ and V e1 ðtÞ are the solutions of systems (14) and (15) with parameter e1 satisfying initial conditions U e1 ð0Þ ¼ S0R and V e1 ð0Þ ¼ S0H , respectively. Since systems (14) and (15) with parameter e1 have globally uniformly attractive positive x-periodic solutions U e1 ðtÞ and V e1 ðtÞ, respectively, there exists a t1 > 0 such that
g
U e1 ðtÞ > U e1 ðtÞ ; 2
V e1 ðtÞ > V e1 ðtÞ
g 2
for all t P t 1 :
ð21Þ
Combining (16) and (21), we have
U e1 ðtÞ > SR ðtÞ g;
k V e1 ðtÞ > g for all t P t1 : d
Thus, we finally obtain that for all t P t1
(
I_R I_H
P aðtÞðSR ðtÞ gÞIH dðtÞIR ; P bðtÞ kd g IR ðd þ l þ rÞIH :
Consider the following auxiliary system
ð22Þ
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8 < bI_
¼ aðtÞðSR ðtÞ gÞbI H dðtÞbI R ; ¼ bðtÞ kd g bI R ðd þ l þ rÞbI H :
R
: b_ IH
For the convenience, we will rewrite it as follows
d dt
bI R bI H
!
bI ¼ ðFðtÞ VðtÞ gMðtÞÞ R bI H
! ð23Þ
:
From Lemma 2, it follows that there exists a positive x-periodic function pðtÞ ¼ ðp1 ðtÞ; p2 ðtÞÞT such that ðbI R ðtÞ; bI H ðtÞÞ ¼ el2 t pðtÞ is a solution of system (23), where l2 ¼ x1 lnðqðUFVgM ðxÞÞÞ. Since JðtÞ2 intR2þ , where JðtÞ ¼ ðIR ðtÞ; IH ðtÞÞT , we can select a small constant a > 0 such that Jðt 1 Þ > apðt 1 Þ. Then, by (22) and the comparison principle, we can obtain that
JðtÞ P ael2 t pðtÞ for all t P t 1 : By qðUFVgM ðxÞÞ > 1, it follows that
limIR ðtÞ ¼ 1;
l2 > 0, then limt!1 JðtÞ ¼ 1, that is,
limIH ðtÞ ¼ 1;
t!1
t!1
which is a contradiction with 0 6 IR ðtÞ 6 e1 and 0 6 IH ðtÞ 6 e1 . T Therefore, claim (19) holds. This shows W s ðM 1 Þ X 0 ¼ ;. From Lemma 1, we can obtain that fM 1 g is globally attractive in M@ , that is, each orbit in M @ converges to fM 1 g. Hence, fM 1 g is isolated in M @ , and hence in X. Furthermore, fM 1 g also is invariant and fM1 g does not form a cycle in M @ , and hence in @X 0 . Therefore, by Lemma 3, we finally obtain that P is uniformly persistent with respect to ðX 0 ; @X 0 Þ. Finally, from Theorem 3.1.1 given in [28] we further obtain that all solutions of system (1) is uniformly persistent with respect to ðX 0 ; @X 0 Þ. Furthermore, from the last equation of system (1) we can directly obtain that RH in system (1) also is uniformly persistent. This completes the proof. h As a consequence of Theorem 3, from the main results given in [31] on the existence of positive periodic solutions for general population dynamical systems, we have the following result. Corollary 1. IfR0 > 1, then system(1)admits at least a positive x-periodic solution. Remark 2. By Lemma 4, we can see that in the realistic applications, if we want to verify R0 is greater or less than unity, it is suffice to verify qðUFV ðxÞÞ is greater or less than unity. However, from the expression of qðUFV ðxÞÞ we see that it is very difficult to calculate it for system (1) because it is periodic system, let alone extending it to higher dimension systems, so we want to further continue our work to find similar integral conditions in [32], which are tend to more easily verify and can be applicable to more general non-autonomous system. Remark 3. When system (1) degenerates an into the autonomous case with rðtÞ r; kðtÞ k; dðtÞ d; aðtÞ a and bðtÞ b for all t P 0, we obtain SR ðtÞ ¼ k and
FðtÞ ¼
0
ak
b kd
0
!
and VðtÞ ¼
d
0
0 dþlþr
:
Using the method given by Driessche and Watmough given in [29] (see also Lemma 2.2 (ii) in [25]), we obtain basic reproduction number
R0 ¼ qðFV 1 Þ ¼
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi kkab a k b : ¼ k ddðl þ d þ rÞ d d ðl þ d þ rÞ
The biological meaning of R0 can be interpreted as follows, when the total number of human population reaches stable state kd and everyone is the susceptible, the number of the new infected human produced by each infected mosquito (though the bite of the infected mosquito) over its expected infectious period is ad kd, and when the total number of mosquito population achieves the stable state k and every mosquito is the susceptible, the number of the new infected mosquito produced by each b infected people over his/her expected infectious period is k lþdþ r. The square root arises from the two ‘generations’ required for an infected people or mosquito to ‘reproduce’ itself. As a consequence of Theorem 2, Theorem 3 and Corollary 1, then we can obtain the following result.
Corollary 2. For autonomous malaria transmission system(1)we have the following conclusions.
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L. Wang et al. / Commun Nonlinear Sci Numer Simulat 18 (2013) 1288–1303
(i) If R0 < 1, then disease-free equilibrium ðk; 0; kd ; 0; 0Þ is globally asymptotically stable. (ii) If R0 > 1, then there exists at least one positive endemic equilibrium and the disease is uniformly persistent.
Remark 4. In this paper, both transmission rate from mosquito to human and transmission rate from human to mosquito are bilinear incidence rate. Bilinear incidence rate bases on the law of mass action. However, an infectious individual (or mosquito) can contact a finite number of mosquitoes (or individuals) per unit time in a large population. The standard incidence rate seems more reasonable than bilinear incidence rate. Therefore, in our future work, we further discuss the dynamic behaviors for a malaria model with standard incidence in a periodic environment. 4. Numerical simulations In this section, we give some examples and numerical simulations to confirm the above theoretical analysis.
a
b
1.4 1.2
2.5
1
2
0.8
I (t)
3
R
R
S (t)
3.5
1.5
0.6
1
0.4
0.5
0.2
0 0
c
50
100
150
200
t
250
0 0
300
9
d
8
150
t
200
6
4
I (t)
6
H
H
S (t)
100
5
7
5
3 2
4
1
3 2 0
50
200
400
t
600
e
800
0 0
1000
50
100
150
t
200
250
300
8 7 6
H
R (t)
5 4 3 2 1 0 0
500
t
1000
1500
Fig. 1. Global stability of disease-free periodic solution of system (1) with parameters in Example 1, when the basic reproduction number R0 0:66319 < 1, which shows that the disease dies out.
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Example 1. Take k ¼ 0:05; d ¼ 0:006; l ¼ 0:1; r ¼ 0, rðtÞ ¼ 3 þ 0:1 sin p6 t ; kðtÞ ¼ 2 þ 0:3 cos 16 pt ; dðtÞ ¼ 1 þ cos p6 t , 1 1 aðtÞ ¼ 0:02 2 þ 0:03 sin 6 pt and bðtÞ ¼ 0:01 7 þ 0:06 sin 6 pt in system (1). Clearly, (H1)–(H3) hold, and by numerical calculations, we obtain basic reproduction number R0 0:66319 < 1. Then, by Theorem 2, disease-free periodic solution ðSR ðtÞ; 0; kd ; 0; 0Þ of system (1) is globally asymptotically stable, which indicates that the disease tend to be extinct. See numerical simulations (a)–(e) of Fig. 1.
Example 2. In Example 1, we keep some parameters unchanged, and only adjust the values of two transmission rates aðtÞ and bðtÞ. Let aðtÞ ¼ 0:2 2 þ 0:03 sinð16 ptÞ and bðtÞ ¼ 0:011 7 þ 0:06 sin 16 pt in system (1). By numerical calculations, we obtain basic reproduction number R0 2:2002 > 1. Then, by Theorem 3, system (1) with these parameters is uniformly persistent. The corresponding numerical simulations are given in Fig. 2(a)–(e), and these figures demonstrate that the disease will tend towards periodic oscillation along with time passing. It means that there exists a periodic solution, which is in accordance with the obtained conclusion in Corollary 1, and it is seemed that this periodic solution is globally attractive.
a
3.5
5
b
4.5
3
4
2.5
3.5 3
I (t)
R
R
S (t)
2 1.5
2 1.5
1
1
0.5 0 0
0.5 100
200
t
300
400
0 0
500
7
6
5
5
4
4
I (t)
6
H
3
2
1
1 100
200
t
300
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R (t)
8 6 4 2 0 0
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t
500
600
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Fig. 2. Uniform persistence of system (1) with parameters in Example 2, when the basic reproduction number R0 2:2002 > 1.
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a
2.5
b
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I (t)
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1.2 1 0.8 0.6 0.4 0.8
0.6
0.4
IH
0.2
0
1.4
1.6
1.8
2
2.2
SH
Fig. 3. Chaotic behavior of system (1) with parameters in Example 3, when the basic reproduction number R0 1:5472 > 1.
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Remark 5. It is well known that transmission rate plays an important role in an epidemic model, by comparing values of aðtÞ and bðtÞ in Example 1 with those of Example 2, we can know that if we decrease values of aðtÞ and bðtÞ, especially aðtÞ, i.e., to reduce the transmission rate from mosquito to human by adopting some protecting matures, such as spraying with medical insecticide, use of mosquito nets, vaccine manufacture, etc., the malaria disease will eventually be controlled and eliminated from the population. Example 3. From Fig. 2(a)–(e), we can obtain it is seemed that the positive periodic solution is globally attractive. However, when we take the following parameters with k ¼ 0:099, d ¼ 0:0346; l ¼ 0:1; r ¼ 0; rðtÞ ¼ 3 þ 0:1 sin p6 t ; kðtÞ ¼ 2 þ 0:3 1 p 1 cos pt , dðtÞ ¼ 1:35 þ 0:01 cos 6 t , aðtÞ ¼ 3:899 2 þ 0:03 sin 6 pt and bðtÞ ¼ 0:21 ð0:03þ 0:46169 sin 16 pt þ 0:1 cos 1 6 pt Þ in system (1), then by numerical calculations we still obtain basic reproduction number R0 1:5472 > 1. Then, by The6 orem 3, system (1) with these parameters is uniformly persistent, as shown in numerical simulations of Fig. 3(a)–(e), which not only illustrate the validity of the proposed results, but also display the interesting complex dynamic behaviors, that is, there is not periodic oscillation along with time passing, and from (f) and (g) in Fig. 3, it can be obviously seen that there is a strange chaotic attractor, which may contribute to the better understanding about reasons that complex chaotic behaviors can cause a high risk of the uncertain prevalence for the disease due to the unpredictability. Remark 6. In [33], Bai, Zou and Zhang have analyzed an SIR model with a season contact rate and a staged treatment strategy and established the existence of multiple periodic solutions by continuation theorem. And numerical simulations demonstrate the coexistence of two stable periodic solutions and unstable periodic solution. In this paper, numerical simulations (g) and (h) of Fig. 3 in Example 3 show system (1) may appear to a strange chaotic attractor. All these interestingly abundant and complex dynamic behaviors reveal more complicated properties for a periodic epidemic model than the corresponding autonomous model.
5. Conclusion In this paper, we investigate the dynamic behaviors of a malaria model in periodic environment and obtain that dynamic behaviors of this model is determined by its basic reproduction number R0 . That is, if R0 < 1,then the disease-free periodic solution is globally asymptotically stable, and if R0 > 1, then the disease is uniformly persistent and there is at least one positive periodic solution. Some numerical simulations are carried out to support theoretical analysis of the research, and the last simulation result suggests that there may be interesting dynamic behaviors in this model-a strange chaotic attractor, which displays the complexity of periodic epidemic model. Furthermore, chaos may cause the disease approaching to the uncontrollable state due to the unpredictability. Thus, how to control chaos in an epidemic model is very important, which needs further our investigation. Moreover, pulse vaccination and the quarantine measures have been testified to be an effective strategy in preventing viral infectious disease. The strategy of pulse vaccination (PVS) consists of periodic repetitions of impulsive vaccinations in a population, on all the age cohorts. Therefore, we can further continue our work about the effect of pulse vaccination, the effect of personal protection measures, etc., on the dynamic behaviors of malaria disease. References [1] World Health Organization, World Malaria Day: save lives though local action 2011. Available at:
L. Wang et al. / Commun Nonlinear Sci Numer Simulat 18 (2013) 1288–1303 [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33]
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Lou Y, Zhao X. A climated-based malaria transmission model with structured vector population. SIAM J Appl Math 2010;70:2023–44. Nakata Y, Kuniya T. Global dynamics of a class of SEIRS epidemic models in a periodic environment. J Math Anal Appl 2010;363:230–7. Liu L, Zhao X, Zhou Y. A tuberculosis model with seasonality. Bull Math Biol 2010;72:931–52. Yang Y, Xiao Y. Threshold dynamics for an HIV model in periodic environments. J Math Anal Appl 2010;361:59–68. Wang W, Zhao X. Threshold dynamics for compartmental epidemic models in periodic environments. J Dyn Diff Equat 2008;20:699–717. Teng Z, Li Z. Permanence and asymptotic behavior of the N-species nonautonomous Lotka–Volterra competitive systems. Comput Math Appl 2000;39:107–16. Zhang F, Zhao X. A periodic epidemic model in a patchy environment. J Math Anal Appl 2007;325:496–516. Zhao X. Dynamical systems in population biology. New York: Springer-Verlag; 2003. Van Den Driessche P, Watmough J. Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission. Math Biosci 2002;180:29–48. Teng Z, Liu Y, Zhang L. Persistence and extinction of disease in non-autonomous SIRS epidemic models with disease-induced mortality. Nonlinear Anal 2008;69:2599–614. Teng Z, Chen L. The positive periodic solutions of periodic Kolmogorove type systems with delays. Acta Math Appl Sin 1999;22:446–56 (in Chinese). Zhang T, Teng Z. On a nonautonomous SEIRS model in epidemiology. Bull Math Biol 2007;69(8):2537–59. Bai Z, Zhou Y, Zhang T. Existence of multiple periodic solutions for an SIR model with seasonality. Nonlinear Anal 2011;74:3548–55.