Modeling the transmission dynamics of sheep brucellosis in Inner Mongolia Autonomous Region, China

Mathematical Biosciences 242 (2013) 51–58

Contents lists available at SciVerse ScienceDirect

Mathematical Biosciences journal homepage: www.elsevier.com/locate/mbs

Modeling the transmission dynamics of sheep brucellosis in Inner Mongolia Autonomous Region, China Qiang Hou a,b, Xiangdong Sun c, Juan Zhang b, Yongjun Liu c, Youming Wang c, Zhen Jin b,⇑ a

School of Mechatronic Engineering, North University of China, Taiyuan, Shan’xi 030051, China Department of Mathematics, North University of China, Taiyuan, Shan’xi 030051, China c China Animal Health and Epidemiology Center, Qingdao, Shan’dong 266032, China b

a r t i c l e

i n f o

Article history: Received 17 May 2012 Received in revised form 6 November 2012 Accepted 30 November 2012 Available online 9 January 2013 Keywords: Brucellosis Dynamic model Stability Basic reproduction number Vaccination

a b s t r a c t Brucellosis is one of the major zoonotic diseases in China, especially in Inner Mongolia where occurs the largest number of human brucellosis cases of Mainland China, which can be attributed to the large number of sheep kept there, since at least 90% of the human brucellosis cases are caused by sheep. Therefore, given the characteristics of the brucellosis infection in Inner Mongolia Autonomous Region, we propose a dynamic model for the sheep-human transmission of brucellosis, involving sheep population, human population and brucella in the environment. We first determine the basic reproduction number R0 and analyze the global stability of the disease-free and endemic equilibrium. Secondly, using the reported human brucellosis data, we carry out numerical simulations and make sensitivity analysis of the basic reproduction number in terms of some parameters. The results show that brucellosis cannot be eradicated even though disinfection rate and vaccination rate of adult sheep are 100%. By investigating and comparing the effect of vaccination, disinfection and eliminating strategies, we find that vaccinating and disinfecting both young and adult sheep are appropriate and effective strategies to control brucellosis in Inner Mongolia of China. Ó 2013 Published by Elsevier Inc.

1. Introduction Brucellosis is one of the world’s major zoonoses, which is caused by various species of Brucella [1]. It can be transmitted to other animals and humans with exposure to infected animals or via ingestion of contaminated water and dust, etc. [2]. Especially, infections among sheep are highly contagious because of the pathogenicity of B. melitensis and close contact caused by the density of the herd. In animals, brucellosis mainly affects reproduction and fertility, and reduces the survival of newborns [3]. In humans, mortality is negligible, but the illness can last for several years [4]. While brucellosis has been controlled with the disease being eradicated or there being low infecting rate in many industrialized countries, more than 500,000 new cases are reported annually around the world and the disease remains endemic in many areas of the world, including Spain, Latin America, the Middle East, parts of Africa, and Asia [2,5]. Similar to the situation in the above-mentioned areas, brucellosis is also a serious reemerging disease in China. Before 1980s, human brucellosis was quite severe. A total of 4.57 million people were examined from 1952 to 1981 with 385,700 sero-positives and positive rate was 8.43% [6]. During 1980s and early 1990s, ⇑ Corresponding author. E-mail address: [email protected] (Z. Jin). 0025-5564/$ - see front matter Ó 2013 Published by Elsevier Inc. http://dx.doi.org/10.1016/j.mbs.2012.11.012

vaccination on animals and other prevention strategies were well implemented, animal positive rate was reduced to 0.55% and the brucellosis got controlled well [6]. With the development of animal husbandry, there was a significant increase in the number of animals from 1987 to 2007 in China. Thereinto, the number of sheep increased twice accordingly [7]. However, vaccination rate dropped from 99.71% to 50% or less [8], which led to sheep positive rate to be 1.49% in 2009, and the new cases of human brucellosis reached 37,734, which was 18.6 times larger than that of human brucellosis in 2000 [9]. Now animal brucellosis has been reported in 29 of 32 provinces with some endemic areas remaining. Most of the human brucellosis patients were infected by sheep-type brucella, accounting for 84.5% of the total cases [10]. Brucellosis is a natural focal disease in Inner Mongolia Autonomous Region of China, which is a serious endemic area of Class I [11]. Inner Mongolia keeps the largest number of sheep in Mainland China, accounting for about 18.2% of the total sheep population [12]. However, sheep vaccination rate there is only 31.6% [13] and human brucellosis cases have been ever increasing. In 2007, new cases were reported in 85 of 102 banners/counties and city districts in Inner Mongolia, and there were still 47 banners/counties and city districts where the incidence of human brucellosis was over 1/10,000 [14]. From 1996 to 2010, 78,246 human brucellosis cases were reported and 90.25% or more of the total cases were caused by sheep-type brucella [15,16], among which 90% of new

52

Q. Hou et al. / Mathematical Biosciences 242 (2013) 51–58

cases occurred in 2005–2010, accounting for 40% of the total cases of China. Especially in 2010, this proportion reached 47.2% [17]. Inner Mongolia ranks first in China with the number of brucellosis cases. Several mathematical models have been proposed to research the transmission of brucellosis, such as the models by Al-Talafhah et al., Gonzlez-Guzmn et al., Joly and Messier, Cross et al., Centrell et al., McGiven and Muma et al. [18–24]. However, these studies did not analyze the impact of brucella in the environment on brucellosis transmission. Recently, Zinsstag et al. [25] studied a dynamic model of animal (sheep and cattle)-to-human brucellosis transmission in terms of the characteristics and the data of Mongolia. Al¨nseba et al. [26] proposed an SIC dynamic model for brucellosis transmission in ovine through direct and indirect ways. Taking the brucellosis transmission in Sub-Saharan Africa as an example, Roy et al. [27] proposed an approach of network control to model and optimally control zoonoses. It is worthwhile to pay close attention to the research of animal brucellosis transmission in the Yellowstone Area of the United States using mathematical models (see [28–31]). However, there is no research by now studying the influence of the effectiveness of the vaccination and disinfection on brucellosis prevention and brucella spread, and no particular dynamical model investigating the prevalence of brucellosis in humans and animals in China. In this paper, taking into account multiple transmission ways and main source of brucellosis transmission in Inner Mongolia Autonomous Region of China, we propose a susceptible-exposed-infectious-vaccinated-brucella (SEIVB) dynamic model for the sheep-to-human transmission of brucellosis. We firstly determine the basic reproduction number R0 and analyze the global stability of system, and then we carry out some sensitivity analysis of the basic reproduction number R0 on the segmental parameters and discuss the control strategies of brucellosis infection in Inner Mongolia. The article is structured as follows. In Section 2, we present the model and determine the basic reproduction number R0 . We analyze the global stability of the disease-free equilibrium and the endemic equilibrium in Section 3. In Section 4, The model is used to simulate the human brucellosis in Inner Mongolia, and then numerical simulations and sensitivity analysis of the basic reproduction number R0 in terms of various model parameters are carried out. Various control measures and a brief discussion are given in Section 5.

between the four sheep compartments and three human groups are given in Fig. 1, including the number of the infectious unit BðtÞ. The following are some assumptions and interpretations of our model. (I) In Inner Mongolia, young sheep are not vaccinated or only vaccinated after 6 months of birth or later, so we assume that all young sheep are at the risk of brucellosis in the first year of birth. (II) In Inner Mongolia, S2 (B.suis strain 2) vaccine are used to vaccinate sheep and its immunization protection rate is 80% to 93% [32], so some of the vaccinated individuals still can be infected. (III) In general, the exposed individuals have no clinical manifestations, so the exposed sheep with brucella can not be eliminated. (IV) We assume that exposed sheep have the same infectivity as infected sheep. (V) There is no data reporting human-to-human transmission of brucellosis, so we assume that the rate of human-to-human transmission is zero. (VI) From 2005 to 2010, the human population growth rate is very low in Inner Mongolia, and we assume that the human’s birth rate is equal to the natural mortality. In other words, the total population remains constant. Therefore, the dynamic model is described as the following ordinary differential equations:

8 dS > ¼ A  bSðE þ IÞ  /SB  ðl þ mÞS þ dV; > dt > > > dV > ¼ mS  ðl þ dÞV  bVðE þ IÞ  /VB; > dt > > > dE > > ¼ bðS þ VÞðE þ IÞ þ /ðS þ VÞB  ðr þ lÞE; > dt > > > < dI ¼ rE  ðl þ cÞI; dt dB > ¼ kðE þ IÞ  ðd þ nsÞB; > dt > > > dSh > ¼ bh Sh ðE þ IÞ  /h Sh B þ rh ð1  pÞIah ; > > dt > > > dIah > ¼ bh Sh ðE þ IÞ þ /h Sh B  rh Iah ; > > dt > > : dIch ¼ rh pIah : dt

ð1Þ

The parameters are described in Table 1. Because the last three equations are independent of the first five equations, we can only consider the following model:

8 dS ¼ A  bSðE þ IÞ  /SB  ðl þ mÞS þ dV; > > dt > > > dV > > < dt ¼ mS  ðl þ dÞV  bVðE þ IÞ  /VB; dE ¼ bðS þ VÞðE þ IÞ þ /ðS þ VÞB  ðr þ lÞE; dt > > > dI > dt ¼ rE  ðl þ cÞI; > > > : dB ¼ kðE þ IÞ  ðd þ nsÞB: dt

ð2Þ

From model (2), one can find that 2. Dynamic model and basic reproduction number R0 Based on the facts of brucellosis infection in Inner Mongolia, we classify the sheep population into four compartments: the susceptible compartment SðtÞ, the exposed compartment EðtÞ, the infectious compartment IðtÞ, the vaccinated compartment VðtÞ, and classify the human population into three groups: susceptible Sh ðtÞ, acute infections Iah ðtÞ and chronic infections Ich ðtÞ. Brucellosis is transmitted in animals through the infection of the respiratory tract, oral mucosa, skin and mucous membranes. On the other hand, exposed and infected sheep can shed brucella into the environment due to abortion or sheep secretions, which can survive for several weeks, or even months in the feces or contaminated environment under suitable conditions. Brucella can be harvested by susceptible individuals that become infected depending on the ingested dose, which is different from direct transmission which assumes that brucella is acquired through an infectious contact with an infected individual. Therefore, an infected or exposed sheep generates infection in two ways: the direct and indirect modes of transmission. We define the average number of brucella that is enough for a host to be infected as an infectious unit. Let BðtÞ denote the number of infectious unit in the environment. The connections

dðS þ V þ E þ IÞ ¼ A  lðS þ V þ E þ IÞ  cI 6 A  lðS þ V þ E þ IÞ dt and then it follows that

Fig. 1. Flowchart of sheep brucellosis transmission.

53

Q. Hou et al. / Mathematical Biosciences 242 (2013) 51–58 Table 1 Parameters and their values (unit: year 1 ). Parameter

Value

Interpretation

Source

A

11629200 0:22 0:15 0:4 0:316

The recruitment rate of the sheep population Sheep natural elimination rate Elimination rate caused by brucellosis Sheep loss of vaccination rate Sheep vaccination rate Sheep-to-sheep transmission rate

[A] [A] Fitting [B] [14] Fitting

Invalid vaccination rate Brucella-to-susceptible sheep transmission rate

[B] Fitting Assumption Fitting [C] [C] [C] Fitting

l c d

m

1:48  108 0:18

b



s

1:7  1010 1 15 3:6 0 0

bh

1:58  1010

The rate of clinical outcome of exposed sheep Brucella shedding rate by exposed and infected sheep The decaying rate of brucella in the environment Disinfection times Effective disinfection rate Transmission rate from sheep to humans

/h

1  1011 0:6 0:4

Transmission rate from brucella to humans

Fitting

Transfer rate from acute infections to chronic infections Transfer rate from acute infections to susceptible population

Assumption Assumption

/

r k d n

rh p rh ð1  pÞ

[A] In Inner Mongolia, sheep life span is about 4–5 years and the number of sheep remains about 5:286  107 from 2005 to 2010, so we estimate that the average elimination 1 rate of sheep l is 4:5 ¼ 0:22 and the recruitment rate is 5:286  107  0:22 ¼ 11629200. [B] In Inner Mongolia, vaccine B.suis strain 2 (S2) has been used to control brucellosis. The valid period of S2 is 2–3 years and it can protect 82% of sheep from brucella [32], so we have  ¼ 0:18 and d ¼ 0:4. [C] Brucella’s survival time in manure/dung is one month at 25 °C, one year at 8 °C and two months in the winter [33,34]. Based on the climatic conditions in Inner Mongolia, we estimate that d is about 3.6. Given that the farmers rarely disinfect the sheep, or even never do it, then n and ns is 0.

lim supðS þ V þ E þ IÞ 6

t!1

A

l

;

lim sup B 6

t!1

kA

lðd þ nsÞ

From the last two equations in (4), we can obtain

:

ðl þ cÞI

E¼

So the set

 A X ¼ ðS; V; E; I; BÞ 2 R5þ : S; V; E; I; B P 0; S þ V þ E þ I 6 ;

r

S þ V ¼

kA lðd þ nsÞ

is the positively invariant set for model (2). It is easy to see that the model (2) always has a disease-free equilibrium P0 ¼ ðS0 ; V 0 ; 0; 0; 0Þ, where

S0 ¼

Aðl þ dÞ Am ; V0 ¼ : lðl þ d þ mÞ lðl þ d þ mÞ

ððd þ nsÞb þ k/ÞðS0 þ V 0 Þ R0 ¼ ðd þ nsÞðl þ rÞ

ðd þ nsÞðl þ rÞðl þ cÞ S0 þ  V 0 : ¼ ðl þ c þ rÞððd þ nsÞb þ k/Þ R0

M0 þ M1 I

S¼

M 2 þ M 3 I þ M 4 I2

ð6Þ

mrðd þ nsÞS M6 þ M5 I

¼

;

rðd þ nsÞmðM0 þ M1 IÞ ðM 6 þ M 5 IÞðM2 þ M3 I þ M 4 I2 Þ

;

ð7Þ

where

M0 ¼ Ar2 ðd þ nsÞ2 ðl þ dÞ;

M1

M2 ¼ lr2 ðd þ nsÞ2 ðl þ d þ mÞ;

M4

2

ð3Þ

k/ðS0 þV 0 Þ rbðS0 þV 0 Þ kr/ðS0 þV 0 Þ where RE ¼ bðSl0 þþrV 0 Þ ; REB ¼ ðdþn sÞðlþrÞ ; RI ¼ ðlþcÞðlþrÞ ; RIB ¼ ðdþnsÞðlþcÞðlþrÞ. RE ðRI Þ is the average number of the infected individuals by a single exposed (infectious) individual in a fully susceptible population. REB ðRIB Þ is the average number of the infected individuals by the brucella which are excreted into the environment by the exposed (infectious) individual. Therefore, R0 denotes the total average number of infections. The endemic equilibrium P ðS ; V  ; E ; I ; B Þ of model (2) is determined by equations:

8 A  bSðE þ IÞ  /SB  ðl þ mÞS þ dV ¼ 0; > > > > > > > > mS  ðl þ dÞV  bVðE þ IÞ  /VB ¼ 0; < bðS þ VÞðE þ IÞ þ /ðS þ VÞB  ðr þ lÞE ¼ 0; > > > > > r E  ðl þ cÞI ¼ 0; > > > : kðE þ IÞ  ðd þ nsÞB ¼ 0:

ð5Þ

¼ Arðd þ nsÞðl þ c þ rÞðbðd þ nsÞ þ k/Þ;

ððd þ nsÞrb þ kr/ÞðS0 þ V 0 Þ ðd þ nsÞðl þ cÞðl þ rÞ

¼ RE þ REB þ RI þ RIB ;

kðE þ IÞ kðl þ c þ rÞI ¼ : d þ ns rðd þ nsÞ

Substituting (5) into the first two equations in (4) gives

V¼

According to the next generation matrix formulated in Diekmann et al. and van den Driessche and Watmough [35,36], we define the basic reproduction number of model (2) as

þ

B¼

Substituting (5) into the third equation in (4) gives

l

B6

;

¼ ðl þ c þ rÞ ðbðd þ nsÞ þ k/Þ2 ; M3 ¼ rðd þ nsÞðl þ c þ rÞðbðd þ nsÞ þ k/Þððl þ mÞ þ l þ dÞ; M5 ¼ ðl þ c þ rÞðbðd þ nsÞ þ k/Þ; Direct calculation for I P 0 shows

dSðIÞ M1 M2  M0 M3  2M 0 M 4 I  M1 M 4 I2 ¼ <0 dI ðM 2 þ M 3 I þ M4 I2 Þ2 and then S and V is monotonically decreasing for I > 0. Combining Eqs. (5) and (6), we can define the equation:

f ðIÞ , S þ V  ð4Þ

M 6 ¼ rðd þ nsÞðl þ dÞ:

S0 þ  V 0 ¼ 0: R0

From the Eq. (7), we can find that the function f ðIÞ is also monotonically decreasing for I > 0. As the function f ð0Þ ¼ S0 þ V 0  S0 þR0 V 0 > 0 and f ðlAÞ < 0 when R0 > 1, the model (2) has a unique endemic equilibrium P ðS ; V  ; E ; I ; B Þ.

54

Q. Hou et al. / Mathematical Biosciences 242 (2013) 51–58

3. The global stability of the disease-free and endemic equilibrium In this section, we investigate the global stability of disease-free equilibrium and the endemic equilibrium of model (2). It is important for us to understand the extinction and persistence of the disease. Theorem 1. If R0 6 1, then the disease-free equilibrium P 0 of system (2) is the global asymptotically stable in X. Proof. For the disease-free equilibrium P0 , model (2) can be rewritten as follows:

8       dS > ¼ S A 1S  S10 þ d VS  VS00  bðE þ IÞ  /B ; > > dt > >     > > S > dV S > < dt ¼ V m V  V00  bðE þ IÞ  /B ;

8 ds



 ¼ s SA 1s  1 þ dVS vs  1  bE ðe  1Þ  bI ði  1Þ  /B ðb  1Þ ; > dt > > 



> > dv > ¼ v mVS vs  1  bE ðe  1Þ  bI ði  1Þ  /B ðb  1Þ ; > dt > >

    > < de ¼ bS eðs  1Þ þ bV  eðv  1Þ þ bS I e si  1 þ bV I e v i  1

dt e e E E



    > > þ /SEB e sbe  1 þ /VE B e veb  1 ; > > >  > di > dt ¼ rIE iðei  1Þ; > > >



: db k  e ¼ B b E b  1 þ I bi  1 : dt ð9Þ It is easy to find that model (9) has a unique endemic equilibrium P ð1; 1; 1; 1; 1Þ, and that the global stability of P is the same as that of P , so we will investigate the global stability of P  instead of P . Defining the Lyapunov function

Lðs; v ; e; i; bÞ ¼ S ðs  1  ln sÞ þ V  ðv  1  ln v Þ þ E ðe  1  ln eÞ

dE ¼ ðbðE þ IÞ þ /BÞðS0 þ V 0 þ S þ V  S0  V 0 Þ  ðr þ lÞE; > dt > > > > dI > ¼ rE  ðl þ cÞI; > dt > > : dB ¼ kðE þ IÞ  ðd þ nsÞB: dt

ð8Þ Since R0 6 1, so function

ððdþnsÞbþk/ÞðS0 þV 0 Þ ðdþnsÞðlþcÞ

< 1, we define the Lyapunov

S V þ V  V 0  V 0 ln þE S0 V0 /ðS0 þ V 0 Þ þ B d þ ns ðd þ nsÞðl þ rÞ  ððd þ nsÞb þ k/ÞðS0 þ V 0 Þ I: þ ðd þ nsÞr

LðS; V; E; I; BÞ ¼ S  S0  S0 ln

Then the derivative of L along solutions of model (8) is

      dL 1 1 V V0 S S0 þ dðS  S0 Þ þ mðV  V 0 Þ ¼ AðS  S0 Þ    dt S S0 S S0 V V0 þ bðS0 þ V 0 ÞðE þ IÞ þ /ðS0 þ V 0 ÞB  ðr þ lÞE ðd þ nsÞðl þ rÞ  ððd þ nsÞb þ k/ÞðS0 þ V 0 Þ E ðd þ nsÞ ðd þ nsÞðl þ rÞ  ððd þ nsÞb þ k/ÞðS0 þ V 0 Þ ðl þ cÞI  ðd þ nsÞr k/ðS0 þ V 0 Þ ðE þ IÞ  /ðS0 þ V 0 ÞB þ d þ ns ðl þ cÞðl þ rÞ S0 ¼ ðR0  1ÞI þ 2A þ dV 0 þ mS0  A r S S S0 V0 V ðl þ cÞðl þ rÞ ¼ ðR0  1ÞI  lV 0  lS0  dV  mS S0 V0 S V r     V S0 V 0 S VS0 V 0 S þ dV 0 2  þ lV 0 3     V 0 S VS0 V 0 S VS0   S S0 6 0: þ l S0 2   S0 S þ

Therefore, the equality dL ¼ 0 holds if and only if S ¼ S0 ; V ¼ V 0 or dt I¼ 0. Since P0 is the only invariant set of model (1) in ðS; V; E; I; BÞ 2 X : dL ¼ 0 , thus the disease-free equilibrium P0 is dt global asymptotic stability in X by LaSalle’s Invariance Principle. h In the next subsection, we analyze the global asymptotic stability of the endemic equilibrium. Let

s¼

S ; S

v¼

V ; V

e¼

E ; E

i¼

I ; I

b¼

B : B

The model (2) is transformed into the following form

þ

ðS þ V  ÞI2 ðbE þ bI þ /B Þ ði  1  ln iÞ rE ðE þ I Þ

þ

/ðS þ V  ÞB2 ðb  1  ln bÞ: kðE þ I Þ

For the equilibrium state P  , we have the following equations:

8 A þ dV  ¼ bS ðE þ I Þ þ /S B þ ðl þ mÞS ; > > > >        > > < mS ¼ ðl þ dÞV þ bV ðE þ I Þ þ /V B ;        bðS þ V ÞðE þ I Þ þ /ðS þ V ÞB ¼ ðr þ lÞE ; > > > rE ¼ ðl þ cÞI ; > > > : kðE þ I Þ ¼ ðd þ nsÞB :

ð10Þ

Then differentiating L with respect to t along solution curves of model (9) gives

   v  dL 1 ¼ ðs  1Þ A  1 þ dV   1  bS E ðe  1Þ  bS I ði  1Þ dt s s  s     /S B ðb  1ÞÞ þ ðv  1Þ mS  1  bV  E ðe  1Þ

v

 bV  I ði  1Þ  /V  B ðb  1ÞÞ    si 1 þ ðe  1Þ bS E ðs  1Þ þ bV  E ðv  1Þ þ bS I e       v i sb v b  1 þ /S B  1 þ /V  B 1 þ bV  I e e e e  ðS þ V  ÞI ðbE þ bI þ /B Þ ði  1Þ  1 þ E þ I  i         /ðS þ V ÞB  e  i ðb  1Þ E  1 þ I  1 þ b b E þ I  ¼ 2A þ dV  þ mS þ M1 þ /ðS þ V  ÞB  ðA þ dV   bS I  /S B A v s  mS Þs  ðmS  dV   bV  I  /V  B Þv   dV   mS s s v e e i si vi  M 1  M 2 E  M 2 I  bS I  bV  I i b b e e   sb   vb  /S B  /V B ; e e 

















þbI þ/B Þ V ÞB where M 1 ¼ ðS þV ÞI EðbE , M2 ¼ /ðS Eþ þI .   þI From the Eq. (10), we can obtain

dL A ¼ 2A þ dV  þ mS þ M 1 þ /ðS þ V  ÞB  ðlS þ bS E Þs  dt s e  s    v e  i  ðlV þ bV E Þv  dV  mS  M 2 E  M 2 I  M 1 b b i s v   si   sb   vi   vb  bS I  bV I  /S B  /V B : e e e e After some algebraic manipulations, we have

55

Q. Hou et al. / Mathematical Biosciences 242 (2013) 51–58

Remark 1. Biologically speaking, Theorem 1 shows that elimination of brucellosis is independent of initial sizes of the populations, and the disease will be eliminated only if the epidemiological threshold R0 6 1. From Theorem 2, we can see that when R0 > 1, the disease persists in the sheep population no matter how large the initial population is.

4

3

x 10

2.5

2 Iah

4. The application of the model to the prevention of brucellosis transmission in Inner Mongolia 1.5

1

0.5 2005

2006

2007

2008 2009 t(year)

2010

2011

2012

Fig. 2. Simulation of human brucellosis cases over time in Inner Mongolia of China. The smooth curve represents the solution Iah of model (1) and the stars are the reported data of human cases.

   dL 1 v s ¼ ðlS þ bS E Þ 2  s  þ dV  2   dt s s v     1 e si 1 s      þ ðlV þ bV E Þ 3   v  þ bS I 3    s i e s v     /S B E 1 e sb /S B I 1 e i sb þ   4    þ   3   s b e s i b e E þI E þI       1 s v i e  /V B I 1 s e i vb þ bV  I 4     þ   5      s v e i s v i b e E þI   /V  B E 1 s e vb 6 0: þ   4    s v b e E þI The equality dL ¼ 0 holds only for s ¼ v ¼ 1; e ¼ i ¼ b, which cordt   ¼ ðS; V; E; I; BÞ : S ¼ S ; V ¼ V  ; E ¼ responds to the set X E  I ¼ BB g  X. This means that P is the maximum invariant set of I  , and then the endemic equilibrium P is global model (1) in X asymptotic stability in X. According to the above discussion, we can derive the following theorem: Theorem 2. For system (2), if R0 > 1, then the endemic equilibrium P  exists and is the global asymptotic stability.

1.88 1.86

1.8

1.84

1.7

1.82 0

1.9

1.6

R

0

b

a

2

R

In this section, based on the reported human brucellosis data in Inner Mongolia, using least square method, we carry out parameter estimation of model (1) and make some sensitivity analysis on some parameters. From the bulletin of China’s major zoonotic epidemics, we can obtain the data on human brucellosis cases. However, data monitoring can not be easily conducted since there are too many sheep and little monitoring budget. Therefore, we can only rely on the existing facts to make some rational assumptions to fit data. The values of parameters are listed in Table 1. For the initial values needed for fitting data, we can directly obtain Sh ð0Þ ¼ 2:384  107 [37]. In Inner Mongolia, there are 44.5 and 53.18 million sheep exposed to brucellosis in 2004 and 2005 respectively. Before 2011, each adult sheep is only vaccinated once a year, and the vaccination rate is about 31.6%. Hence we estimate that the number of vaccinated sheep is 8.44 million (44:5  0:316 ð1  0:4Þ), that is, Vð0Þ ¼ 8:44  106 . Based on actual survey, the true prevalence of sheep is about 2.5% in 2005 (unpublished data), and we estimate Ið0Þ ¼ 1:33  106 ; Eð0Þ ¼ 0; Sð0Þ ¼ 4:341  107 . We use the data fitting to obtain Bð0Þ ¼ 6  106 . Using model (1), we evaluate the human brucellosis data in Inner Mongolia from 2005 to 2010 and make a prediction about the trend of human brucellosis infection. Fig. 2 shows that the simulation of our model with reasonable parameter values provides a good match with the data on infected human brucellosis cases in Inner Mongolia from 2005 to 2010. Moreover, with the current control measures, it can be predicted by our model that the human brucellosis cases will continue increasing in the next few years. With the simulated parameter values, we estimate that R0 ¼ 1:8, which indicates that human brucellosis will persist in Inner Mongolia with the current control and prevention measures. Fig. 3(a) depicts the influence of m (adult sheep vaccination rate) on R0 . Though vaccinating susceptible sheep is an effective

1.8

1.5

1.78

1.4

1.76

1.3

1.74

0.2

0.3

0.4

0.5

0.6 ν

0.7

0.8

0.9

1

1.72

2

Fig. 3. R0 in terms of m and d.

4

6

8

10 d

12

14

16

18

56

Q. Hou et al. / Mathematical Biosciences 242 (2013) 51–58

a

2.8

b

1.81

2.6

1.8

2.4

1.79

2.2

1.78

2 0

R

R

0

1.77 1.8

1.76 1.6 1.75

1.4

1.74

1.2

1.73

1 0.8

0

0.1

0.2

0.3

0.4 c

0.5

0.6

0.7

1.72

0.8

0

10

20

30

40

50

nτ

Fig. 4. R0 in terms of c and ns.

a

4

5.5

x 10

5

3.5

3.5 ah

4

3

3

2.5

2.5

2

2

1.5

1.5

1

1 0

2

4

nτ=0, ν=0.316 nτ=2, ν=0.6 nτ=6, ν=0.8 nτ=25, ν=1

4.5

4

0.5

x 10

5

I

I

ah

4.5

6 t (year)

b

4

5.5

nτ=0, ν=0.316 nτ=0, ν=0.6 nτ=0, ν=0.8 nτ=0, ν=1

8

10

12

0.5

0

2

4

6 t (year)

8

10

12

Fig. 5. Simulation of the number of human infection on variable m and ns.

measure to decrease R0 ; R0 cannot become less than one even if the vaccination rate m is 100%. From Fig. 3(b) and 4(b), we can see that the effect of decaying rate of brucella d and disinfection rate ns is relatively little on R0 . Fig. 4(a) represents the relationship between R0 and the mortality caused by brucellosis, which shows that the influence of parameter c on R0 is greater than parameters ns and m. As Fig. 5 shows, the more disinfection and vaccination are given, the less infection will be caused. However, human brucellosis cases cannot ultimately be reduced below the initial cases even though disinfection rate and vaccination rate are 100%. Thus, we can conclude that under current situation, eliminating infected sheep in time can effectively decrease R0 below 1 and is the most effective strategy to eradicate brucellosis in Inner Mongolia. However, giving only one vaccination to adult sheep a year is far from enough to eradicate brucellosis.

5. Discussion As a zoonotic and chronic disease, brucellosis has become a serious public health problem which can not be ignored in China. In this paper, based on the characteristics of the brucellosis

infection in Inner Mongolia of China, we proposed an SEIVB dynamic model for the sheep-human transmission of brucellosis. By validating our model, we can conclude that human brucellosis cases would gradually increase in the next a couple of years, which implies that brucellosis cannot be eradicated with the current strategies. By carrying out some sensitivity analyses of the basic reproduction number on various parameters, we find that the vaccination rate of adult sheep and eliminating rate of infected sheep play important roles in the prevention of brucellosis. However, from Fig. 5(a), we see that although acute infection cases can be reduced by giving vaccination to adult sheep, brucellosis cannot be eventually eradicated if vaccination to adult sheep is only once a year. Furthermore, from Fig. 6(a), we can see that, even if vaccination is given twice a year and the vaccination rate is 100% (which is almost impossible), R0 is relatively close to one (R0 ¼ 0:93), which means that brucellosis will continue persisting a very long time in Inner Mongolia. Therefore, vaccinating adult sheep alone is not adequate to eradicate brucellosis. Eliminating infected sheep can effectively control brucellosis, but this results in a huge financial burden on the herdsman in Inner Mongolia, so the strategy to cull infectious animals is difficult to carry out for Inner Mongolia Autonomous Region government.

57

Q. Hou et al. / Mathematical Biosciences 242 (2013) 51–58

a

b 2.5

2.6 2.4 2.2

2

2

R

R0

0

1.8 1.5

1.6 1.4 1

1.2 1 0.8

0

0.2

0.4

0.6

0.8

1

ν

1.2

1.4

1.6

1.8

0.5

2

0

0.2

0.4

0.6

0.8

1

ν

1.2

1.4

1.6

1.8

2

Fig. 6. (a) R0 in terms of m with adult sheep vaccinated. (b) R0 in terms of m with young and adult sheep vaccinated.

a

4

3

x 10

2.5

ν=0.316, nτ=0 ν=0.86, nτ=2 ν=1.2, nτ=6 ν=1.8, nτ=25

2

Iah

Iah

x 10

2.5

2

1.5

1.5

1

1

0.5

0.5

0 2005

b

4

3

ν=0.316 ν=0.86 ν=1.2 ν=1.8

2010

2015

2020

2025

2030

0 2005

2010

t (year)

2015

2020

2025

2030

t (year)

Fig. 7. The solid line is the solution curve of model (1) from 2005 to 2011. (a) Fixed all other parameters except m, simulation and prediction of human infection on variable m (0.86, 1.2 and 1.8) with all sheep vaccinated from 2011 to 2030 in Inner Mongolia; (b) Fixed all other parameters except m and ns, simulation and prediction of human infection on variable m and ns ((0.86, 2), (1.2, 6), (1.8, 25)) with all sheep vaccinated from 2011 to 2030 in Inner Mongolia.

While brucella infection causes disease primarily in adults (sexually mature), young animals may be infected but generally show only a weak and transient serological response. Generally speaking, young sheep turn sexually mature within 3–6 months. Therefore, immune young sheep timely in the first year of birth is very necessary. In 2011, the government of Inner Mongolia Autonomous Region began to give vaccination to all sheep twice a year, which implied that part of the recruitment of sheep population would be classified into the vaccinated compartment rather than the susceptible compartment. That is, in the model (1), the number of young sheep to be classified into the vaccinated compartment is Am. Fig. 6(b) indicates that R0 will be below 1 if the vaccination rate is more than 86% every year, and if the vaccination rate reaches 1.8 (vaccinating twice a year), R0 will be reduced to 0.64. Fig. 7(a) shows that human brucellosis cases will first increase and then reduce even though the vaccination rate is greater than 0.86. Simultaneously, Fig. 7(a) reveals that the larger the vaccination rate to all sheep is, the smaller public health hazard will be. Comparing Fig. 7(a) and (b), we can find that combining the strategy of disinfection with vaccination is more effective than vaccination alone. To sum up, only when both animal brucellosis surveillance and prevention are emphasized and strengthened, human brucellosis

epidemic in Inner Mongolia of China could be effectively controlled. We also can claim that our model can be applied to study the brucellosis spread in other regions, such as the provinces of Shanxi, Heilongjiang, Jilin, etc. where brucellosis is also prevalent. Meanwhile, we cannot deny the limitations of the research, which may influence the generality of the model. Firstly, the influence of season was not taken into account, while brucellosis may have different infection rates and different transmission ways in different seasons. Secondly, regular sheep brucellosis surveillance has not been carried out for some reasons in China, so the data such as the positive rate of sheep was limited. We leave these for further research. Acknowledgments This research was partially supported by the National Natural Science Foundation of China (11171314), the National Youth Science Foundation (11201434), Bairen Project for Basic Research (2010011007), International and Technical Cooperation Project (2010081005) of Shanxi Province, Youth Foundation (2011021001-2) of Shanxi Province, Special Fund for Agro-scientific Research in the Public Interest (200903055).

58

Q. Hou et al. / Mathematical Biosciences 242 (2013) 51–58

References [1] M.L. Boschiroli, V. Foulogne, D. O’Callaghan, Brucellosis: a worldwide zoonosis, Curr. Opin. Microbiol. 1 (2001) 58. [2] G. Pappas, N. Akritidis, M. Bosilkovski, E. Tsianos, Brucellosis, N. Engl. J. Med. 352 (2005) 2325. [3] M.M.H. Sewell, D.W. Brocklesby, Animal Diseases in the Tropics, fourth ed., Baillie’re Tindall, London, 1990. [4] M.M. Madkour, Madkours Brucellosis, Springer Verlag, Heidelberg, Berlin, 2001. [5] M.J. Corbel, Brucellosis: an overview, Emerg. Infect. Dis. 3 (1997) 213. [6] S.Y. Zhang, S.L. Gang, G.I. Zhang, The analysis of epidemic state of brucellosis in China, Chin. J. Control. Endem. Dis. 12 (1997) 1 (in Chinese). [7] G.M. Wang, L.J. Chi, S.C. Ma, Z.H. Su, Y. Chen, L. Yang, Prevalence features and cause analysis of livestock brucelloses in our country, China Anim. Health Inspect. 7 (2010) 62 (in Chinese). [8] S.L. Jiang, S.Y. Zhang, the recent situations of national and the suggestions of enhancement prevention, Chin. J. Control Endem. Dis. 2 (1997) 93 (in Chinese). [9] T.F. Man, D.L. Wang, B.Y. Cui, Y. Wang, F. Ding, T.F. Li, H. Sun, R.J. Wei, Q. Li, Z.J. Feng, W.W. Yin, Analysis on surveillance data of brucellosis in China, 2009, Dis. Surv. 12 (2010) 944 (in Chinese). [10] D.Q. Shang, D.L. Xiao, J.M. Yin, Epidemiology and control of brucellosis in China, Vet. Microbiol. 69 (2002) 77. [11] S.Y. Zhang, D. Zhu, S.Y. Jiang, 50-year review on prevention and control of brucellosis in China, Chin. J. Control Endem. Dis. 5 (2003) 347 (in Chinese). [12] CAHY, China animal husbandry yearbook, China Agriculture Press, Beijing, 2005–2010. [13] W.J. Chen, B.Y. Cui, Q.H. Zhang, H.R. Jin, R.P. Wei, Y.H. Ma, Analysis of epidemic characteristics on brucellosis in Inner Mongolia, Chin. J. Control Endem. Dis. 1 (2008) 56 (in Chinese). [14] J.C. Mi, Q.H. Zhang, R.P. Wei, L.T. Song, Z. Zheng, The epidemiological characteristics of human brucellosis in Inner Mongolia, Chin. J. Control Endem. Dis. 1 (2010) 34 (in Chinese). [15] D.R. Pu, L.Y. Li, H.Y. Zhao, B.Y. Cui, Epidemic situation analysis of human brucellosis in Inner Mongolia during 1952 to 2007, Chin. J. Control Endem. Dis. 4 (2009) 420 (in Chinese). [16] Q.H. Zhang, Z. Zheng, L.T. Song, Strains of brucella virulence and epidemic in Inner Mongolia Autonomous Region, Chin. J. Control Endem. Dis. 2 (2004) 104 (in Chinese). [17] MOHC, Ministry of Health of the People’s Republic of China, 2011, Bulletins, . [18] A.H. Al-Talafhah, S.Q. Lafi, Y. Al-Tarazi, Epidemiology of ovine brucellosis in Awassi sheep and goats in northern Jordan, Prev. Vet. Med. 60 (2003) 297. [19] J. González-Guzmán, R. Naulin, Analysis of a model of bovine brucellosis using singular perturbations, J. Math. Biol. 33 (1994) 211. [20] Damien O. Joly, F. Messier, The effect of bovine tuberculosis and brucellosis on reproduction and survival of wood bison in Wood Buffalo National Park, J. Anim. Ecol. 74 (3) (2005) 543. [21] Paul C. Cross, Dennis M. Heisey, Brandon M. Scurlock, William H. Edwards, Michael R. Ebinger, Angela Brennan, Mapping brucellosis increases relative to

[22]

[23] [24]

[25]

[26] [27]

[28]

[29]

[30] [31]

[32]

[33]

[34] [35]

[36]

[37]

elk density using hierarchical Bayesian models, PLoS One 5 (4) (2010) e10322, http://dx.doi.org/10.1371/journal.pone.0010322. R.S. Cantrell, C. Cosner, W.F. Fagan, Brucellosis, boties, and brainworms: the impact of edge habitats on pathogen transmission and species extinction, J. Math. Biol. 42 (2001) 95. J. McGiven, The improved specificity of bovine brucellosis testing in Great Britain, Res. Vet. Sci. 84 (2008) 38. J.B. Muma, N. Toft, J. Oloya, A. Lund, K. Nielsen, K. Samui, E. Skjerve, Evaluation of three serological tests for brucellosis in naturally infected cattle using latent class analysis, Vet. Microbiol. 125 (2007) 187. J. Zinsstag, F. Roth, D. Orkhon, G. Chimed-Ochir, M. Nansalmaa, J. Kolar, P. Vounatsou, A model of animal-human brucellosis transmission in Mongolia, Prev. Vet. Med. 69 (2005) 77. B. A1¨nseba, C. Benosman, P. Magal, A model for ovine brucellosis incorporating direct and indirect transmission, J. Biol. Dyn. 4 (1) (2010) 2. S. Roy, Terry F. McElwain, Y. Wan, A network control theory approach to modeling and optimal control of zoonoses: case study of brucellosis transmission in sub-Saharan Africa, PLoS Negl. Trop. Dis. 5 (10) (2011) e1259, http://dx.doi.org/10.1371/journal.pntd.0001259. John J. Treanor, Chris Geremia, Philip H. Crowley, John J. Cox, Patrick J. White, Rick L. Wallen, Douglas W. Blanton, Estimating probabilities of active brucellosis infection in Yellowstone Bison through quantitative serology and tissue culture, J. Appl. Ecol. 48 (6) (2011) 1324. Chris Geremia, P.J. White, Rick L. Wallen, Fred G.R. Watson, John J. Treanor, John Borkowski, Christopher S. Potter, Robert L. Crabtree, Predicting bison migration out of Yellowstone National Park using Bayesian models, PLoS One 6 (2) (2011) e16848, http://dx.doi.org/10.1371/journal.pone.0016848. Andrew Dobson, Mary Meagher, The population dynamics of brucellosis in the Yellowstone National Park, Ecology 77 (1996) 1026. F. Xie, Richard D. Horan, Disease and behavioral dynamics for brucellosis control in elk and cattle in the greater Yellowstone Area, J. Agr. Resour. Econ. 34 (2009) 11. T.Z. Sun, Z.R. Wu, X.Y. Pang, Prevention measures and countermeasures on brucellosis in Inner Mongolia, Neimenggu Prevent. Med. 1 (2000) 37 (in Chinese). G.G. Alton, The epidemiology of brucella melitensis in sheep and goats, in: J.M. Verger, M. Plommet (Eds.), Brucella melitensis, a CEC seminar, Martinus Nijoff, Dordrecht, Boston, Lancaster, 1985, p. 187. P. Nicoletti, The epidemiology of bovine brucellosis, Adv. Vet. Sci. Comput. Med. 24 (1980) 69. O. Diekmann, J.A.P. Heesterbeek, J.A.J. Metz, On the definition and the computation of the basic reproduction ratio R0 in models for infectious diseases in heterogeneous populations, J. Math. Biol. 28 (1990) 365. P. van den Driessche, J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosci. 180 (2002) 29. IMBS, Inner Mongolia autonomous region bureau of statistics, population and its composition at the year-end, 2008, .