An SIRVS epidemic model with pulse vaccination strategy

ARTICLE IN PRESS

Journal of Theoretical Biology 250 (2008) 375–381 www.elsevier.com/locate/yjtbi

An SIRVS epidemic model with pulse vaccination strategy$ Tailei Zhang, Zhidong Teng College of Mathematics and System Sciences, Xinjiang University, Urumqi, 830046, People’s Republic of China Received 6 July 2007; received in revised form 24 September 2007; accepted 24 September 2007 Available online 7 November 2007

Abstract The aim of this paper is to analyze an SIRVS epidemic model in which pulse vaccination strategy (PVS) is included. We are interested in finding the basic reproductive number of the model which determine whether or not the disease dies out. The global attractivity of the disease-free periodic solution (DFPS for short) is obtained when the basic reproductive number is less than unity. The disease is permanent when the basic reproductive number is greater than unity, i.e., the epidemic will turn out to endemic. Our results indicate that the disease will go to extinction when the vaccination rate reaches some critical value. r 2007 Elsevier Ltd. All rights reserved. Keywords: Epidemic model; PVS; Global attractivity; Basic reproductive number

1. Introduction Infectious diseases have tremendous influence on human life. Every year, millions of people died of various infectious diseases. Controlling infectious diseases has been an increasingly complex issue in recent year. Pulse vaccination have been testified to be an effective strategy in preventing such viral infectious as rabies, yellow fever, poliovirus, encephalitis B. The strategy of pulse vaccination (PVS) consists of periodical repetitions of impulsive vaccinations in a population, on all the age cohorts (Orsel et al., 2005; Rebecca et al., 2001; Zhang et al., 2006). At each vaccination time a constant fraction of the susceptible population is vaccinated. Theoretical results show that the pulse vaccination strategy can be distinguished from the conventional strategies in leading to disease eradication at relatively low values of vaccination. In a lot of epidemic models (Ma et al., 2004; Liu et al., 1986; Anderson and May, 1992, 1978; Lu et al., 2002; Kermark and Mckendrick, 1927; Diekmann and Heester$ This work was supported by the National Natural Science Foundation of P.R. China (10361004), the Major Project of The Ministry of Education of P.R. China and the Funded by Scientific Research Program of the Higher Education Institution of Xinjiang (XJEDU2004I12). Corresponding author. E-mail addresses: [email protected] (T. Zhang), [email protected] (Z. Teng).

0022-5193/$ - see front matter r 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.jtbi.2007.09.034

beek, 2000), the total population is divided into three groups: the susceptible (S), the infectives (I) and the removed (R). Since Kermack and Mckendrick constructed a system of ODE to study epidemiology in 1927, the method of ‘‘compartment modelling’’ is used until now. Recently, the epidemic models include PVS have been widely studied by a lot of scholars (Gao et al., 2006; Meng et al., 2006; Pang and Chen, 2006). However, the authors assume that the recovered and the vaccinated subjects, that they mix in a unique compartment, have exactly the same length of immunization. These seem to be not accordant with the epidemiology. Your body can become immune to bacteria, viruses and other germs in two ways: by getting a disease (natural immunity), through vaccines (vaccineinduced immunity). Next we will introduce the difference between natural immunity and vaccine-induced immunity which can be found in many books (for example, see (Virgil and Derek, 2005; Barry and Paul-Henri, 2002; Richard et al., 2002). Natural immunity develops after you have been exposed to a certain organism. Your immune system puts into play a complex array of defenses to prevent you from getting sick again from that particular type of virus or bacterium. Exposure to a foreign invader stimulates production of certain white blood cells in your body called B-cells. These B-cells produce plasma cells, which in turn produce antibodies designed specifically to fight that particular

ARTICLE IN PRESS 376

T. Zhang, Z. Teng / Journal of Theoretical Biology 250 (2008) 375–381

invader. These antibodies circulate in your body’s fluids. The next time that invader enters your body, the antibodies will recognize it and destroy it. Once your body has produced a particular antibody, it rapidly produces more antibodies if needed. In addition to the work of B-cells, other white blood cells called macrophages confront and destroy foreign invaders. If your body encounters a germ that it has never been exposed to before, information about the germ is relayed to white blood cells called helper T-cells. These cells aid production of other infectionfighting cells, including memory T-cells. Once you have been exposed to a specific virus or bacterium, the next time you encounter it, antibodies and memory T-cells go to work. They immediately react to the organism, attacking it before disease can develop. Your immune system can recognize and effectively combat thousands of different organisms. Vaccine-induced immunity results after you receive a vaccine. The vaccine triggers your immune system’s infection-fighting ability and memory without exposure to the actual disease-producing germs. A vaccine contains a killed or weakened form or derivative of the infectious germ. When given to a healthy person, the vaccine triggers an immune response. The vaccine makes your body think that it is being invaded by a specific organism, and your immune system goes to work to destroy the invader and prevent it from infecting you again. If you are exposed to a disease for which you have been vaccinated, the invading germs are met by antibodies that will destroy them. The immunity you develop following vaccination is similar to the immunity acquired from natural infection. Several doses of a vaccine may be needed for a full immune response. Some people fail to achieve full immunity to the first doses of a vaccine but respond to later doses. In addition, the immunity provided by some vaccines, such as tetanus and pertussis, isn’t lifelong. Because the immune response may decrease over time, you may need another dose of a vaccine (booster shot) to restore or increase your immunity. Under theses statements, we will consider a model with state variables S, I, R and V that represent the number of susceptible, infected, recovered and vaccinated individuals, respectively. The basic assumptions underlying the dynamics of the system are as follows:  All newborns, denoted by L, join into the susceptible class per unit time.  Natural death rate is a constant per capita rate m.  The mortality induced disease for the infectious individuals is a.  The rate constant for recovery is denoted by g, so that 1=g is the mean time spent in the infective class.  We assume that the impulsive vaccination is applied every t40 years and y 2 ð0; 1Þ denote the proportion of those vaccinated successfully.  Standard epidemiological models use a bilinear incidence rate based on the law of mass action (Ma et al.,

2004; Anderson and May, 1992, 1978), the average number of adequate contacts of an infective and a vaccinated individual per unit time are b and sb, respectively, where b40 and 0psp1. The fraction reflects the effect of reducing the infection rate due to vaccination: s ¼ 0 means that the vaccine is completely effective in preventing infection, while s ¼ 1 means that the vaccine is utterly ineffective, that is, the vaccination need not to be considered as s ¼ 1.  Z is the rate at which the vaccinated individuals return to the susceptible class and 1=Z is the average period of vaccine-induced immunity. Here, we suppose that the recovered acquire the permanent immunity but the vaccinated temporary immunity i.e., the natural immunity is permanent but vaccine-induced immunity temporary. Many diseases have this peculiarity such as Viral Hepatitis A, Chicken pox, Measles, Meningitis, Parotitis, Diphtheria, Rubella, Pertussis, etc. By the method of ‘‘compartment modelling’’, these assumptions lead to a model of the form: 8 > > > > > > > > > > > > > > > > < > > > > > > > > > > > > > > > > :

9 S 0 ðtÞ ¼ L  mSðtÞ  bSðtÞIðtÞ þ ZV ðtÞ; > > > > I 0 ðtÞ ¼ ðbSðtÞ þ sbV ðtÞÞIðtÞ  ðm þ a þ gÞIðtÞ; = R0 ðtÞ ¼ gIðtÞ  mRðtÞ; V 0 ðtÞ ¼ sbV ðtÞIðtÞ  ðm þ ZÞV ðtÞ 9 Sðtþ Þ ¼ ð1  yÞSðtÞ; > > > > = Iðtþ Þ ¼ IðtÞ; ; t ¼ kt; Rðtþ Þ ¼ RðtÞ; > > > > V ðtþ Þ ¼ V ðtÞ þ ySðtÞ: ;

> > > > ;

;

takt; k 2 N;

(1.1) where N ¼ f1; 2; 3; . . .g. Here, NðtÞ ¼ SðtÞ þ IðtÞ þ RðtÞ þ V ðtÞ denotes the total population at time t. The initial condition of (1.1) is given as Sð0þ Þ ¼ S 0 ;

Ið0þ Þ ¼ I 0 ;

Rð0þ Þ ¼ R0 ;

V ð0þ Þ ¼ V 0 , (1.2)

where ðS0 ; I 0 ; R0 ; V 0 Þ 2 R4þ ¼ fðx1 ; x2 ; x3 ; x4 Þ 2 R4 : xi X0; i ¼ 1; 2; 3; 4g.

The meaningful domain of system (1.1) is   L O ¼ ðS; I; R; V Þ 2 R4þ : S þ I þ R þ V p m and it is easy to prove that O is a positive invariant set. The solution of system (1.1) is a piecewise continuous function F : Rþ ! R4þ , FðtÞ is continuous on ðkt; ðk þ 1Þt, k 2 N and Fðktþ Þ ¼ limt!ktþ FðtÞ exists. In fact, the righthand side of system (1.1) can ensure the existence and uniqueness of solutions of system (1.1). The organization of this paper is as follows: In the next section, using the discrete dynamical system determined by the stroboscopic map, we establish existence of the DFPS and deal with the global attractivity of the DFPS.

ARTICLE IN PRESS T. Zhang, Z. Teng / Journal of Theoretical Biology 250 (2008) 375–381

In Section 3, we will discuss the permanence of the disease. We will give some discussion in Section 4. 2. Global behavior of DFPS One of the most important questions in mathematical studies of epidemics is the possibility of the eradication of disease (see Iwasa et al., 2007). In this section, we analysis the attractivity of the DFPS of system (1.1). Lemma 2.1. Consider the following impulsive differential equation ( 0 u ðtÞ ¼ a  buðtÞ; takt; k 2 N; (2.1) þ uðt Þ ¼ ð1  yÞuðtÞ; t ¼ kt; where a40, b40 and 0oyo1. Then there exists a unique periodic solution of system (2.1) a  a u~ e ðtÞ ¼ þ u  ebðtktÞ ; ktotpðk þ 1Þt b b which is globally asymptotically stable. Here u ¼ abð1  yÞ ð1  ebt Þ=1  ð1  yÞebt . Proof. By calculating, we obtain the solution of the first equation in system (2.1) between two pulses a  a uðtÞ ¼ þ uðktþ Þ  ebðtktÞ ; ktotpðk þ 1Þt. b b Using the second equation of system (2.1), we deduce the stroboscopic map such that a   a uððk þ 1Þtþ Þ ¼ ð1  yÞ þ uðktþ Þ  ebt 9f ðuðktþ ÞÞ. b b where f ðuÞ ¼ ð1  yÞða=b þ ðu  a=bÞebtÞ Þ. It is easy to know that the map f has a unique positive fixed point u ¼

a ð1  yÞð1  ebt Þ . b 1  ð1  yÞebt

Obviously, u satisfies uof ðuÞou as 0ouou and u4f ðuÞ4u as u4u . From Cull (1981), we obtain that u is globally asymptotically stable. It implies that the corresponding periodic solution of system (2.1) a  a u~ e ðtÞ ¼ þ u  ebðtktÞ ; ktotpðk þ 1Þt b b is globally asymptotically stable (see Lakshmikantham et al., 1989). The proof of Lemma 2.1 is completed. & Lemma 2.2. Assume the sequence ftk g satisfies 0pt0 ot1 ot2 o    with limk!1 tk ¼ 1. Let f ðt; xÞ : Rþ ! Rn be quasimonotone nondecreasing in x for each t, and ck ðuÞ 2 C½Rn ; Rn  is nondecreasing in u for k ¼ 1; 2; . . .. Suppose that uðtÞ; vðtÞ 2 PCð½t0 ; 1; Rn Þ satisfy ( þ D uðtÞpf ðt; uðtÞÞ; tXt0 ; uðtþ k 2 N; k Þpck ðuðtk ÞÞ;

(

Dþ vðtÞXf ðt; vðtÞÞ;

tXt0 ;

vðtþ k ÞXck ðvðtk ÞÞ;

k 2 N;

377

Then uðt0 Þpvðt0 Þ implies that uðtÞpvðtÞ for tXt0 . The detailed proofs will appear in Lakshmikanthan et al. (1989, Theorem 1.4.4, pp. 34), we omit it. We first demonstrate the existence of a disease-free solution, in which infectious individuals are entirely absent from the population permanently, i.e. IðtÞ  0 for all tX0. Under this condition, the growth of susceptible individuals, recovered individuals and vaccinated individuals must satisfy the following impulsive system: 9 8 0 S ðtÞ ¼ L  mSðtÞ þ ZV ðtÞ; > > = > > > R0 ðtÞ ¼ mRðtÞ; > ; takt; k 2 N; > > > > ; 0 > > V ðtÞ ¼ ðm þ ZÞV ðtÞ < 9 (2.2) Sðtþ Þ ¼ ð1  yÞSðtÞ; > > > = > > > Rðtþ Þ ¼ RðtÞ; > ; t ¼ kt: > > > > ; þ > : V ðt Þ ¼ V ðtÞ þ ySðtÞ: From the equation N 0 ðtÞ ¼ L  mNðtÞ, we easily obtain limt!1 NðtÞ ¼ L=m. Further, it follows from the second and fifth equations of system (2.2) that limt!1 RðtÞ ¼ 0. Therefore, we have the following limit system of (2.2) V ðtÞ ¼

L  SðtÞ, m

and 8 < S 0 ðtÞ ¼ L þ ZL  ðm þ ZÞSðtÞ; m : þ Sðt Þ ¼ ð1  yÞSðtÞ;

takt;

(2.3)

t ¼ kt:

According to Lemma 2.1, we know that periodic solution of system (2.3) is of form   L L ðmþZÞðtktÞ  S0 ðtÞ ¼ þ S  ; ktotpðk þ 1Þt e m m (2.4) which is globally asymptotically stable, where S ¼ L=m ð1  yÞð1  eðmþZÞt Þ=1  ð1  yÞeðmþZÞt . Denote V 0 ðtÞ ¼ L=m  S 0 ðtÞ. Obviously, S 0 ðtÞ and V 0 ðtÞ are independent of the initial values. These processes show that the susceptible and vaccinated will approach to S0 ðtÞ and V 0 ðtÞ receptively when the infectives are very few. Substituting S0 ðtÞ and V 0 ðtÞ into the second equation of (1.1), we can assert that I 0 ðtÞ ¼ bS 0 ðtÞ þ sbV 0 ðtÞ  ðm þ a þ gÞ. IðtÞ Hence, the right side periodic function bS 0 ðtÞ þ sbV 0 ðtÞ  ðm þ a þ gÞ can be explain the relative increasing rate per infective individual as the infectives are very few. It follows immediately that we have a intuitionistic conclusion: if the average value of bS 0 ðtÞ þ sbV 0 ðtÞ  ðm þ a þ gÞ is greater

ARTICLE IN PRESS T. Zhang, Z. Teng / Journal of Theoretical Biology 250 (2008) 375–381

378

than 0, then the infectives will not go to extinction. We will give a detailed proof later. According to the above statement, we define the basic reproductive number to be Z t 1 R0 ¼ bS 0 ðtÞ þ sbV 0 ðtÞ dt. tðm þ a þ gÞ 0    L L 1 ð1  eðmþZÞt Þ tb þ bð1  sÞ S  m m mþZ . R0 ¼ tðm þ a þ gÞ

ðbS 0 ðtÞ þ sbV 0 ðtÞ þ ðb þ sbÞ0 Þ  ðm þ a þ gÞ dto0, 0

and nonnegativity of IðtÞ, we have lim IðtÞ ¼ 0.

(2.7)

t!1

N 0 ðtÞXL  mNðtÞ  a1 , R0 ðtÞpg1  mRðtÞ

The biological interpretation of R0 is the average number of the secondary infections produced by a typical infective during the entire period of infectiousness. Theorem 2.1. If R0 o1, then the DFPS ðS0 ðtÞ; 0; 0; V 0 ðtÞÞ of system (1.1) is globally attractive in O Proof. Since R0 o1, we can choose 0 40 small enough such that Z t ðbS0 ðtÞ þ sbV 0 ðtÞ þ ðb þ sbÞ0 Þ  ðm þ a þ gÞ dto0. 0

It follows from the first and fourth equations of system (1.1) that S0 ðtÞ ¼ L  mS  bSI þ ZðN  S  I  RÞpL þ ZL=m  ðm þ ZÞS and V 0 ðtÞp  ðm þ ZÞV . Thus we consider the following comparison impulsive differential system: 8 9 L > 0 = >  ðm þ ZÞxðtÞ; x ðtÞ ¼ L þ Z > > > m ; takt; k 2 N; > > ; > < y0 ðtÞ ¼ ðm þ ZÞyðtÞ ) > > xðtþ Þ ¼ ð1  yÞxðtÞ; > > > ; t ¼ kt: > þ > > : yðt Þ ¼ yðtÞ þ yxðtÞ: (2.5) 0

In view of ðxðtÞ þ yðtÞÞ ¼ L þ ZL=m  ðm þ ZÞðxðtÞ þ yðtÞÞ and Lemma 2.1, we see that the periodic solution of system (2.5) is ðS0 ðtÞ; V 0 ðtÞÞ which is globally asymptotically stable. Let ðSðtÞ; IðtÞ; RðtÞ; V ðtÞÞ be a solution of system (1.1) with initial value (1.2) and Sð0þ Þ ¼ S0 40; V ð0þ Þ ¼ V 0 X0, ðxðtÞ; yðtÞÞ be the solution of system (2.5) with initial value xð0þ Þ ¼ S0 ; yð0þ Þ ¼ V 0 . By Lemma 2.2, there exists an integer k1 40 such that SðtÞpxðtÞoS0 ðtÞ þ 0 , ð2:6Þ

for all ktotpðk þ 1Þt and kXk1 . Furthermore, from the second equation of system (1.1) and (2.6), we obtain I 0 ðtÞ ¼ ðbSðtÞ þ sbV ðtÞÞIðtÞ  ðm þ a þ gÞIðtÞ pðbS0 ðtÞ þ sbV 0 ðtÞ þ ðb þ sbÞ0 ÞIðtÞ  ðm þ a þ gÞIðtÞ for all tXkt and k4k1 .

t

Therefore, for any sufficiently small 1 2 ð0; 1Þ, there exists an integer k2 Xk1 satisfying IðtÞo1 for all t4k2 t. This gives

That is

V ðtÞpyðtÞoV 0 ðtÞ þ 0

According to

Z

for all t4k2 t. By comparison, there is an integer k3 Xk2 such that NðtÞX

L  a1 1  m 2

and

RðtÞp

g1 1 þ m 2

(2.8)

for all t4k3 t. As 1 can be arbitrarily small and lim supt!1 NðtÞpL=m, we have lim NðtÞ ¼

t!1

L m

and

lim RðtÞ ¼ 0.

t!1

(2.9)

Write K ¼ bL=m þ Zð2 þ a=m þ g=mÞ. Therefore, the first equation of (1.1) yields S 0 ðtÞ ¼ L  mSðtÞ  bSðtÞIðtÞ þ ZðNðtÞ  SðtÞ  IðtÞ  RðtÞÞ   L X L þ Z  K1  ðm þ ZÞSðtÞ m for all t4k3 t. Consider the following comparison impulsive differential equation, 8   > < uðtÞ ¼ L þ ZL  K1  ðm þ ZÞu0 ðtÞ; takt; m (2.10) > : uðtþ Þ ¼ ð1  yÞuðtÞ; t ¼ kt: In view of Lemma 2.1, we know that the periodic solution of system (2.10) is as follows S1 ðtÞ ¼ G þ ðG  GÞeðmþZÞðtktÞ ;

ktotpðk þ 1Þt

which is globally asymptotically stable, where G¼

L þ Z Lm  K1 mþZ

;

G ¼ G

ð1  yÞð1  eðmþZÞt Þ . 1  ð1  yÞeðmþZÞt

According to the comparison theorem of impulsive differential equations, there exists an integer k4 4k3 such that SðtÞ4S 1 ðtÞ  1 ;

ktotpðk þ 1Þt; k4k4 .

(2.11)

Because 0 and 1 can be arbitrarily small, it follows from (2.6), (2.7), (2.9) and (2.11) that lim SðtÞ ¼ S 0 ðtÞ;

t!1

lim V ðtÞ ¼ V 0 ðtÞ.

t!1

(2.12)

Finally, it follows from (2.7), (2.9) and (2.12) that the DFPS ðS 0 ðtÞ; 0; 0; V 0 ðtÞÞ of system (1.1) is globally attractive. The proof of Theorem 2.1 is complete. &

ARTICLE IN PRESS T. Zhang, Z. Teng / Journal of Theoretical Biology 250 (2008) 375–381

We would like to solve the critical vaccination proportion, i.e. the value of y0 such that R0 ðy0 Þ ¼ 1. Here R0 ðy0 Þ stands for R0 in which y is replaced by y0 . It is easy to compute   L tðm þ ZÞ b  ðm þ a þ gÞ m  3 . y0 ¼ 2 L tðm þ ZÞ b  ðm þ a þ gÞ 6 7 L m 7 bð1  sÞ  eðmþZÞt 6 4 5 ðmþZÞt 1e m (2.13) By dR0 =dyo0, we know R0 o1 when and only when y4y0 . According to Theorem 2.1, it is easily to obtain the following result.

379

where S ¼ L þ ZLm  K 1 =m þ Zð1  yÞð1  eðmþZÞt Þ=1 ðmþZÞt ð1  yÞe . It follows that S  ðtÞ uniformly tends to S0 ðtÞ as  tends 0. By R0 41, we can choose 40 small enough such that Z t bS  ðtÞ þ sbV 0 ðtÞ  K 3   ðm þ a þ gÞ dt40. (3.1) 0

We first prove lim sup IðtÞX.

(3.2)

t!1

Suppose that (3.2) is not true, then there exists a T 1 X0 such that IðtÞo for all tXT 1 . From Eq. (1.1), we have N 0 ðtÞ ¼ L  mN  aIXL  a  mN, R0 ðtÞ ¼ gI  mRpg  mR

Corollary 2.1. The DFPS ðS0 ðtÞ; 0; 0; V 0 ðtÞÞ of system (1.1) is globally attractive provided that y4y0 .

for all tXT 1 . Hence, there exists a T 2 XT 1 , for any tXT 2 , we can obtain

Remark 2.1. Theorem 2.1 determines the global attractivity of disease in O for the case R0 o1. Its epidemiological implication is that the infectious population vanishes i.e. the disease dies out. Corollary 2.1 implies that the disease will disappear if the pulse vaccination rate is larger than y0 . Hence, we theoretically testify that pulse vaccination is an effective strategy in preventing disease.

NðtÞX

3. Permanence In this section, we say the disease becomes endemic if the infectious population persists above a certain positive level for a long period. Definition 3.1. (i) Infective I is said to be strong persistent if lim inf t!1 IðtÞ40 for any solution ðSðtÞ; IðtÞ; RðtÞ; V ðtÞÞ of system (1.1) with initial value Sð0þ Þ40; Ið0þ Þ40; Rð0þ ÞX0; V ð0þ ÞX0. (ii) Infective I is said to be permanent, if there are constants M4m40 such that mp lim inf IðtÞp lim sup IðtÞpM t!1

t!1

for any solution ðSðtÞ; IðtÞ; RðtÞ; V ðtÞÞ of system (1.1). Theorem 3.1. If R0 41, then the infective I is permanent. Proof. For the convenience of the reader we write   L aþg aþg , K1 ¼ b þ Z 2 þ ; K2 ¼ 3 þ m m m K 3 ¼ bð1 þ sK 2 Þ

and

RðtÞp

ge  þ . m 2

(3.3)

We conclude from the first equation of (1.1) that, as tXT 2 , S 0 ðtÞ ¼ L  mS  bSI þ ZðN  S  I  RÞ, L XL þ Z  ðm þ ZÞS  K 1 . m We consider the following impulsive auxiliary equation: 8 < u0 ðtÞ ¼ L þ ZL  ðm þ ZÞu  K 1 ; takt; k 2 N; m : þ uðt Þ ¼ ð1  yÞuðtÞ; t ¼ kt: (3.4) Lemma 2.1 shows that (3.4) has a unique periodic solution S ðtÞ which is globally asymptotically stable. By comparison, there exists a k1 satisfying SðtÞXS  ðtÞ   for all ktptpðk þ 1Þt and kXk1 . On the other hand, the following system: ( 0 S ðtÞpL þ Z Lm  ðm þ ZÞSðtÞ; takt; Sðtþ Þ ¼ ð1  yÞSðtÞ;

t ¼ kt

implies that there exists a k2 such that SðtÞpS 0 ðtÞ þ  for all ktptpðk þ 1Þt and kXk2 . Choose T 3 ¼ maxfT 2 ; k1 t; k2 tg, then V ðtÞ ¼ NðtÞ  SðtÞ  IðtÞ  RðtÞXV 0 ðtÞ  K 2  holds for all tXT 3 . By the second equation of (1.1), we obtain I 0 ðtÞXðbS  ðtÞ þ sbV 0 ðtÞ  K 3 ÞIðtÞ  ðm þ a þ gÞIðtÞ

and 0 1 L L L þ Z  K 1 B L þ Z  K 1 C m m  C S  ðtÞ ¼  þB S  @ A mþZ mþZ eðmþZÞðtktÞ ;

L  a   m 2

ktotpðk þ 1Þt,

for all tXT 3 . Integrating the above inequality from T 3 to t gives IðtÞXIðT 3 Þ

Z

t

 exp T3

 bS ðtÞ þ sbV 0 ðtÞ  K 3   ðm þ a þ gÞ dt .

ARTICLE IN PRESS T. Zhang, Z. Teng / Journal of Theoretical Biology 250 (2008) 375–381

380

It follows from (3.1) that IðtÞ ! 1 as t ! 1. This is a contradiction. In the following, we will prove that there is a constant v40 such that lim inf IðtÞ4v

(3.5)

t!1

for any solution ðSðtÞ; IðtÞ; RðtÞ; V ðtÞÞ of system (1.1) with initial value Sð0þ Þ40; Ið0þ Þ40; Rð0þ ÞX0; V ð0þ ÞX0. In fact, from (3.1), we can obtain there are positive constants P and r such that Z tþx bS  ðtÞ þ sbV 0 ðtÞ  K 3   ðm þ a þ gÞ dt4r (3.6) t

for all tX0 and xXP. If (3.5) is not true, then there is a sequence of initial value X n ¼ ðSn ; I n ; Rn ; V n Þ 2 R4þ ðn ¼ 1; 2; . . .Þ such that  lim inf Iðt; X n Þo 2 ; n ¼ 1; 2; . . . . t!1 n From (3.2), for every n there are two time sequences ftðnÞ q g and fsðnÞ g satisfying q ðnÞ ðnÞ ðnÞ ðnÞ ðnÞ 0osðnÞ 1 ot1 os2 ot2 o    osq otq o   

and limq!1 sðnÞ q ¼ 1 such that   IðtðnÞ IðsðnÞ q ; X nÞ ¼ ; q ; X nÞ ¼ 2 n n and   ðnÞ for all t 2 ðsðnÞ oIðt; X n Þo q ; tq Þ. n2 n Since I 0 ðt; X n ÞX  ðm þ a þ gÞIðt; X n Þ, integrating it from ðnÞ sðnÞ q to tq yields ðnÞ ðnÞ ðnÞ IðtðnÞ q ; X n ÞXIðsq ; X n Þ expfðm þ a þ gÞðtq  sq Þg.

ln n !1 mþaþg

as n ! 1.

In the same manner we can see that there exists a T  which is independent of any n and q such that SðtÞXS ðtÞ   ðnÞ  and V ðtÞXV 0 ðtÞ  K 2  for all t 2 ðsðnÞ q þ T ; tq Þ (if  ðnÞ ðnÞ ðnÞ tq  sq 4T ). Choose n large enough makes tq  sðnÞ q 4 T  þ P . We finally have  ¼ IðtðnÞ q ; X nÞ n2  ¼ IðsðnÞ q þ T ; X nÞ (Z ðnÞ ) tq ðbSðtÞ þ sbV ðtÞ  ðm þ a þ gÞ dt  exp sqðnÞ þT 

(Z

 exp n2  4 2. n

)

tqðnÞ

X

sqðnÞ þT 

Remark 3.1. Corollary 3.1 indicates that a small pulse vaccination rate will imply the permanence of the disease, that is to say, the disease spreads around and generates an endemic ultimately. Remark 3.2. Theorems 2.1 and 3.1 suggest that the basic reproductive number R0 unquestionably decides when the disease will die out. Eradication of the disease need to make the vaccination rate at least attain the critical value y0 . Unfortunate, the global dynamical behavior of the system is not obtained when R0 41. This question is at present far from being solved. We leave it in our future work. Remark 3.3. From (2.13), we know that y0 p0 or y0 41 is possible. For example, if we fix m ¼ L ¼ 1=70, t ¼ 1, a ¼ 0, g ¼ Z ¼ s ¼ 0:1, then y0 ¼ 0:0158, 0.5131 and 1.0776 when b ¼ 0:1, 0.5 and 0.8, respectively. However, the vaccination rate y is between 0 and 1. Corollaries 2.1 and 3.1 shows that the disease will disappear when y0 p0 and be permanent when y0 41. Here, we give some biological explanations. If y0 o0, then the disease will always go to extinction even if no pulse vaccination. If y0 41, then the disease will become endemic in the sense of permanence even if 100% pulse vaccination. These show that we not only control vaccination rate but also other parameters such as contact rate b, recovery rate g and so on for eradication of the disease, i.e. only using pulse vaccination is not enough, we must combine other measure to control the disease. 4. Discussion

Consequently,   ðnÞ X expfðm þ a þ gÞðtðnÞ q  sq Þg. 2 n n which implies ðnÞ tðnÞ q  sq X

Corollary 3.1. Assume that yoy0 , then the disease is permanent.

ðbS 0 ðtÞ þ sbV 0 ðtÞ  K 3   ðm þ a þ gÞ dt

This leads to a contradiction. Thus, we finally prove that inequality (3.5) is true. This completes the proof. &

In this paper, we have analyzed the dynamical behavior of an SIRVS epidemic model with PVS. It should be noted that in the theory of dynamical systems, pulse vaccination corresponds to a system of differential equations with impulses. We obtained the basic reproductive number R0 which determines whether the disease will go to extinction or not. From Cull (1981), we obtain that u is globally asymptotically stable. Using the stroboscopic maps and comparison theorem for ODE and impulsive ODE, we have also established the global attractivity of the DFPS and the permanence of the disease. Our results indicate that a large vaccinated rate will lead to eradication of the disease. Remark 3.3 shows that only using pulse vaccination is not enough, we must combine other measure to control the disease. Furthermore, we obtained a useful formula (2.13) which is the critical value of the proportion of the susceptible vaccinated successfully to lead to extinction of the disease. If we use our model to study these disease: Viral Hepatitis A, Chicken pox, Measles, Meningitis, Parotitis, Diphtheria, Rubella, Pertussis etc, some proportions of vaccination can be solved, which guarantee the eradication of the disease.

ARTICLE IN PRESS T. Zhang, Z. Teng / Journal of Theoretical Biology 250 (2008) 375–381

Acknowledgements The authors wish to express their thanks to editor and two anonymous referees for their careful reading of the original manuscript and their many valuable comments and suggestions that greatly improved the presentation of this work. References Anderson, R.M., May, R.M., 1978. Regulation and stability of host–parasite population interactious II: Destabilizing process. J. Anim. Ecol. 47, 219–267. Anderson, R.M., May, R.M., 1992. Infectious Disease of Humans, Dynamical and Control. Oxford University Press, Oxford. Barry, B., Paul-Henri, L., 2002. The Vaccine Book. Academic Press, New York. Cull, P., 1981. Global stability for population models. Bull. Math. Biol. 43, 47–58. Diekmann, O., Heesterbeek, J.A.P., 2000. Mathematical Epidemiology of Infectious Diseases: Model Building, Analysis and Interpretation. Wiley, New York, Chichester. Gao, S., Chen, L., Nieto, J., Torres, A., 2006. Analysis of a delayed epidemic model with pulse vaccination and saturation incidence. Vaccine 24, 6037–6045. Iwasa, Y., Sato, K., Takeuchi, Y., 2007. Mathematical studies of dynamics and evolution of infectious diseases. In: Takeuchi, Y., Iwasa, Y., Sato, K. (Eds.), Mathematics for life science and medicine. Springer, Berlin, pp. 1–4 (Chapter 1).

381

Kermark, M.D., Mckendrick, A.G., 1927. Contributions to the mathematical theory of epidemics. Part I Proc. Roy. Soc. A 115, 700–721. Lakshmikantham, V., Bainov, D.D., Simeonov, P.S., 1989. Theory of Impulsive Differential Equations. World Scientific, Singapore. Liu, W.M., Levin, S.A., Iwasa, Y., 1986. Influence of nonlinear incidence transmission rates upon the behavior of SIRS epidemic models. J. Math. Biol. 23, 187–204. Lu, Z., Chi, X., Chen, L., 2002. The effect of constant and pulse vaccination on SIR epidemic model with horizontal and vertical transmission. Math. Comput. Model. 36, 1039–1057. Ma, Z., Zhou, Y., Wang, W., Jin, Z., 2004. Mathematical Modelling and Research of Epidemic Dynamical Systems. Science Press, Beijing (in Chinese). Meng, X., Chen, L., Chen, H., 2006. Two profitless delays for the SEIRS epidemic disease model with nonlinear incidence and pulse vaccination. Appl. Math. Comput. 2006; doi:10.1016/j.amc.2006.07.124. Orsel, K., Dekker, A., Bouma, A., et al., 2005. Vaccination against foot and mouth disease reduces virus transmission in groups of calves. Vaccine 23, 4887–4894. Pang, G., Chen, L., 2006. A delayed SIRS epidemic model with pulse vaccination. Chaos, Solitons and Fractals 34, 1629–1635. Rebecca, J., Eva, M., Hakon, S., Haaheim, L., 2001. The effect of zanamivir treatment on the early immune response to influenza vaccination. Vaccine 19, 4743–4749. Richard, A.G., et al., 2002. Immunology. W.H. Freeman, New York. Virgil, S., Derek, O., 2005. Immunopotentiators in Modern Vaccine. Academic Press, New York. Zhang, Y., Peng, B., Deng, H., et al., 2006. Anti-HBV immune response by electroporation mediated DNA vaccination. Vaccine 24, 897–903.