Modeling Sheep Brucellosis Transmission with a Multi-Stage Model in Changling County of Jilin Province, China

J. Appl. Math. Comput. DOI 10.1007/s12190-015-0901-y

ORIGINAL RESEARCH

Modeling sheep brucellosis transmission with a multi-stage model in Changling County of Jilin Province, China

Qiang Hou1,2 · Xiang-Dong Sun3

Received: 29 January 2015 © Korean Society for Computational and Applied Mathematics 2015

Abstract Brucellosis is one of the major public health problems in Jilin Province of China, especially in Changling County of Jilin Province where at least 95 % of the human brucellosis cases are caused by sheep, which attribute to the large number of sheep kept there and the high positive rate of sheep. In this paper, based on the monitoring data and the characteristics of brucellosis infection in Changling County of Jilin Province, a multi-stage dynamic model is proposed for sheep brucellosis transmission, involving young sheep population and adult sheep population. Firstly, the basic reproduction number R0 is determined and then the dynamic properties of the model is further discussed. By carrying out sensitivity analysis of the basic reproduction number in terms of some parameters, it concluded that the birth rate of sheep, sheep vaccination rate, and the elimination rate of infectious sheep play an important roles in the transmission of brucellosis. After investigating and comparing the effect of vaccination and culling strategies is completed, the results show that vaccinating sheep and culling infectious sheep are two effective and feasible strategies to control the spread of brucellosis in Changling County of Jilin Province, but the latter is more effective than the former.

Keywords Brucellosis · Multi-stage model · Basic reproduction number · Vaccination · Culling

B Qiang Hou [email protected]

1 School of Mathematics and Statistics, Southwest University, Chongqing 400715, People’s Republic of China 2 Department of Mathematics, North University of China, Taiyuan 030051, Shanxi, People’s Republic of China 3 China Animal Health and Epidemiology Center, Qingdao 266032, Shandong, People’s Republic of China 123 Q. Hou, X.-D. Sun

Mathematics Subject Classiﬁcation 34A34 · 34D20 · 92D30

1 Introduction

Brucellosis which is a serious and economically devastating disease can affect many animals, such as sheep, cattle, pig. The disease is caused by bacteria of the genus brucella, of which there are six species: B. abortus, B. melinitis, B. suis, B. ovis, B. canis and B. neotomae [1]. The main transmission sources of human brucellosis include infected livestock and brucella in the environment, and there is no recorded transmission of the infection between humans [2,3]. Therefore, the key to solve the problem of public health is the elimination of animal brucellosis. In humans, mortality is negligible, but the illness can last for several years [4]. In animals, brucellosis mainly affects reproduction and fertility, reduces survival of newborns [5]. There are more than 500,000 new cases that 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 including China [6]. Similar to the situation in many countries and regions, brucellosis is also a serious reemerging disease in China. Now animal brucellosis has been reported in 29 of 32 provinces with some endemic areas remaining. Most of the human brucellosis cases are infected by sheep-type brucella, accounting for 84.5 % of the total cases [7]. In Jilin Province, there are a very large number of sheep, and sheep brucellosis is one of the crucial animal diseases. The incidence of human brucellosis is over 1/10,000 in 2008– 2010 [8], and 95 % or more of the total cases are caused by sheep-type brucella. The most important thing is that the concentration of animal and human brucellosis in Jilin Province mainly resides in several counties. Especially in Changling County of Jilin Province where sheep positive rate (the proportion of infectious sheep) reached 5.5 % in 2012 (data has not been reported) and the incidence of human brucellosis is 3/10,000 in 2008 [9], which is far more than the other counties and cities in Jilin Province. therefore, the present work aims to better understand the dynamics of brucellosis transmission in a dynamic model, and determine the feasible prevention and control measures in Changling County of Jilin Province. Statistical methods have been widely applied to the quantitative study of brucellosis transmission, the research results on the American Yellowstone National Park and the Middle East countries are worthy of attention (see [10–13]). Some dynamic models have been proposed to study the complex dynamics of brucellosis [14–22]. Recently, Hou et al. [23] studied a dynamic model of sheep-to-human brucellosis with indirect transmission in terms of the characteristics and the data of Inner Mongolia of China. But many transmission mechanism and risk factors are not reﬂected in these models, for instance, the impact of birth rate which can be changed by the price and breeding cost of sheep. In order to make a better understanding of the transmission mechanism and the trend of brucellosis in Changling County of Jilin Province, the present paper proposes a multi-stage dynamic model for sheep brucellosis transmission taking the nonlinear birth rate into consideration. it ﬁrstly determines the basic reproduction number R0 and then analyzes the dynamic behaviors of the model, and afterward carries out numerical simulations and the sensitivity analysis of some parameters. Finally, based on the 123 Modeling sheep brucellosis transmission with a multi-stage... analysis and numerical simulations, it discusses the prevention and control strategies in eliminating the spread of brucellosis in Changling County of Jilin Province. The article is organized as follows. In Sect. 2, it proposes the model and determines the basic reproduction number R0. In Sect. 3, it investigates the dynamic properties of the model (1). Data simulations and sensitivity analysis of R0 on some parameters are carried out in Sect. 4. Various control measures and a brief discussion are given in Sect. 5.

2 Dynamic model and basic reproduction number

Among animals, transmission dynamics of brucellosis are complicated due to multiple interaction. On one hand, infected animal has an incubation period, during which many animal brucellosis are hardly infectious. Furthermore, the exposed animals are difficult to be found through serological test, which is an important risk factor for the spread of brucellosis. On the other hand, brucella infection causes disease primarily in adults, young sheep may be infected but generally show only a weak and transient serological response, so young sheep infectivity can be neglected. In addition, the treatment of infected animals is rarely attempted because of the high cost. Therefore, the standard control measures applied to brucellosis are vaccination, test and culling. It is noteworthy that these risk factors and intervention measures play an important role in the prevalence of brucellosis. Therefore, sheep population is classified into five compartments: the susceptible compartment S1(t) (young sheep), the susceptible compartment S2(t) (adult or sexually mature sheep), the vaccinated compartment V (t), the exposed compartment E(t), the infectious compartment I (t). If the total population at time t is denoted by N(t), then

N(t) = S1(t) + S2(t) + V (t) + E(t) + I (t).

The infectious class I represents that sheep have clinical features or are infectious carriers which can be found by serological test. It should be noted that, infectious animal can shed brucella into the environment, and brucella can be harvested by susceptible individuals that become infected, but indirect transmission (environmental transmission) plays a relatively small role on the spread of brucellosis [23]. In addition, in the estimation of model parameters, ignoring the indirect infection only leads to the increase of direct infection rate [24], it implies that brucella infection which is indirect transmission can be ignored in the application of the model. In Changling County, sheep sexual maturation span is about 6 months, The recruit- ment of susceptible sheep (1–6 months) is generated by birth and restocking from other regions. Since the birth rate can be affected by the price of sheep which is determined by the number of sheep and the number of E(t) is far less than that of S2(t) + V (t),it ( + ) is assumed that the birth rate is b S2 V . The sheep which come from other regions 1+τ(S2+V ) are mainly restocked into large breeding units, so the ﬂuctuation of restocking rate is relatively small over a given price range, and is assumed to be a constant A.The population of susceptible sheep is reduced by the transfer rate σ1 S1 from class S1 to class S2 and the output rate dS1 from Changling County to other areas. By elimination, i.e. natural death, at a rate μ1 S1, one can obtain 123 Q. Hou, X.-D. Sun

dS1 b(S2 + V ) = A + − μ1 S1 − dS1 − σ1 S1. dt 1 + τ(S2 + V )

The epidemiological class S2(t) is increased at the rate σ1 S1 and sheep loss of vaccination rate δV . it is decreased because of contact transmission rate βS2 I and the vaccination rate θ S2. By natural elimination rate at a rate μ2 S2, so that

dS 2 = σ S − βS I − μ S − θ S + δV. dt 1 1 2 2 2 2

In China, S2 (B. suis strain 2) vaccine are used to vaccinate sheep and its immu- nization protection rate is 80–93 % [23], so some of the vaccinated individuals still can be infected. Therefore, the vaccinated compartment V (t) is increased at the rate θ S2, it is reduced by loss of vaccination rate δV , infection at the rate βVIand natural elimination rate μ2V , then

dV = θ S − βVI− (μ + δ)V. dt 2 2

The exposed sheep E is increased by the infection of susceptible sheep at the rate β I (V + S2). This population is decreased by the transfer rate σ2 E from exposed sheep to infectious sheep and by natural elimination rate at a rate μ2 E. Thus,

dE = β(V + S )I − (μ + σ )E. dt 2 2 2

The population of infectious sheep is increased by the transfer rate σ2 E. This population is decreased following the elimination (natural elimination and the elimination caused by disease) at the rate μ2 I + cI. Hence

dI = σ E − μ I − cI. dt 2 2

In summary, the model for the transmission dynamics of sheep brucellosis is given by the following deterministic system (the ﬂowchart of sheep brucellosis transmission is depicted in Fig. 1): ⎧ ( + ) ⎪ dS1 b S2 V ⎪ = A + +τ( + ) − (μ1 + d + σ1)S1, ⎪ dt 1 S2 V ⎪ dS2 = σ − β − θ − μ + δ , ⎨ dt 1 S1 S2 I S2 2 S2 V dV = θ − β − (μ + δ) , (1) ⎪ dt S2 VI 2 V ⎪ ⎪ dE = β(V + S )I − (μ + σ )E, ⎩⎪ dt 2 2 2 dI = σ − (μ + ) . dt 2 E 2 c I 123 Modeling sheep brucellosis transmission with a multi-stage...

Fig. 1 Flowchart of sheep brucellosis transmission

From model (1), one can ﬁnd that

d(S1 + S2 + V + E + I ) b(S2 + V ) = A + − (μ1 + d)S1 dt 1 + τ(S2 + V ) − μ2(S2 + V + E + I ) − cI b ≤ A + − (μ + d)S − μ (S + V + E + I ) τ 1 1 2 2 b ≤ A + − μ(S + S + V + E + I ), τ 1 2

where μ = min{μ1 + d,μ2}.Itfollowsthat

Aτ + b lim sup(S1 + S2 + V + E + I ) ≤ . t→∞ μτ

So the set Aτ + b = (S , S , V, E, I ) ∈ R5 : S + S + V + E + I ≤ 1 2 + 1 2 μτ is the positively invariant set for model (1). It is evident that model (1) has a disease-free = ( 0, 0, 0, , ) equilibrium P0 S1 S2 V 0 0 , which satisﬁes ⎧ ( + ) ⎨ A + b S2 V = (μ + d + σ )S , 1+τ(S2+V ) 1 1 1 σ + δ = θ + μ , ⎩ 1 S1 V S2 2 S2 (2) θ S2 = μ2V + δV, 123 Q. Hou, X.-D. Sun where 2 M0 + M + 4Aμ2τσ1(μ1 + d + σ1) 0 = 0 , S1 2τσ1(μ1 + d + σ1) σ (μ + δ)S0 θ S0 0 = 1 2 1 , 0 = 2 , S2 V μ2(μ2 + θ + δ) μ2 + δ and M0 = (Aτ + b)σ1 − μ2(μ1 + d + σ1). According to the next generation matrix formulated in Diekmann et al. [25], the basic reproduction number is deﬁned by

βσ ( 0 + 0) 2 V S2 R0 = . (3) (μ2 + c)(μ2 + σ2)

3 Dynamic properties of model (1)

In this section, the local stability of the disease-free equilibrium is analyzed and then the existence and local stability of the endemic equilibrium of model (1) is studied, and ﬁnally the global stability of equilibria under certain conditions is investigated. It is important for us to understand the extinction and persistence of the disease.

3.1 The local stability of disease-free equilibrium

For the disease-free equilibrium of model (1), one can establish the following result:

= ( 0, 0, 0, , ) Theorem 3.1 The disease-free equilibrium P0 S1 S2 V 0 0 of model (1)is locally asymptotically stable if R0 < 1, and unstable if R0 > 1.

Proof The Jacobian matrix J0 at P0 is given by ⎛ ⎞ b b −(d + μ1 + σ1) 00 +τ 0+ 0 2 +τ 0+ 0 2 ⎜ 1 S2 V 1 S2 V ⎟ ⎜ σ −(θ + μ )δ −β 0 ⎟ ⎜ 1 2 0 S2 ⎟ J = ⎜ 0 ⎟ . 0 ⎜ 0 θ −(μ2 + δ) 0 −βV ⎟ ⎝ −(μ + σ )β 0 + 0 ⎠ 0002 2 V S2 000σ2 −(μ2 + c)

The characteristic equation of Jacobian matrix J:

3 2 2 (λ + a1λ + a2λ + a3)(λ + b1λ + b2) = 0, where

a1 = 2μ2 + θ + δ + d + μ1 + σ1, bσ1 a2 = (μ2 + θ + δ)(μ1 + μ2 + d + σ1) + μ2(μ1 + d + σ1) − , +τ( 0+ 0) 2 1 S2 V 123 Modeling sheep brucellosis transmission with a multi-stage...

bσ1 a3 = (μ2 + θ + δ) μ2(μ1 + d + σ1) − , +τ( 0+ 0) 2 1 S2 V = μ + + σ , b1 2 2 c 2 = (μ + )(μ + σ ) − σ β 0 + 0 . b2 2 c 2 2 2 V S2

It follows from (2) that

0 0 bσ1 S + V μ (d + μ + σ ) S0 + V 0 = 2 + Aσ . 2 1 1 2 + τ 0 + 0 1 1 S2 V

Thus,

bσ1 bσ1 μ2(d + μ1 + σ1)> > . 1 + τ S0 + V 0 + τ 0 + 0 2 2 1 S2 V

It is easy to see that a2, a3, a1a2 − a3, b1 > 0 and b2 > 0ifR0 < 1. According to Hurwitz Criterion, the real part of the eigenvalues are negative and then the disease- free equilibrium P0 of model (1) is local asymptotic stability. Moreover, if R0 > 1, b2 < 0, there is a positive eigenvalue. Therefore, the disease-free equilibrium P0 is unstable if R0 > 1.

Remark Biologically speaking, R0 represents the average number of secondary infec- tions produced by a typical infectious individual in a community. The epidemiological implication of this result is that if R0 < 1, the inﬂux of a few infectious individuals will not generate large outbreaks (and the disease will die out). Disease outbreak will occur if R0 > 1.

3.2 The existence and local stability of endemic equilibrium

∗( ∗, ∗, ∗, ∗, ∗) The possible endemic equilibrium P S1 S2 V E I of model (1) is derived by the following equations: ⎧ b(S2+V ) ⎪ A + = μ1 S1 + dS1 + σ1 S1, ⎪ 1+τ(S2+V ) ⎪ ⎨ σ1 S1 + δV = βS2 I + θ S2 + μ2 S2, β(V + S )I = (μ + σ )E, (4) ⎪ 2 2 2 ⎪ θ = μ + δ + β , ⎩⎪ S2 2V V VI σ2 E = μ2 I + cI.

From the last two equations in (4), we have

θ S (μ + c)I V = 2 , E = 2 . (5) μ2 + δ + β I σ2 123 Q. Hou, X.-D. Sun

Substituting (5) into the second and third equations in (4)give 1 δθS2 S1 = βS2 I + (μ2 + θ)S2 − , (6) σ1 μ2 + δ + β I and

(μ + σ )(μ + c) S = F (I ) = 2 2 2 , (7) 2 1 θ σ β + 1 2 μ2+δ+β I which is strictly increasing for I ≥ 0. Substituting (5) and (6) into the ﬁrst equation in (4)gives

τ 2 + ( − ( τ + )σ ) − σ = , M1 M2 M3 S2 M1 M2 A b 1 M3 S2 A 1 0 where δθ M1 = μ1 + d + σ1, M2 = β I + μ2 + θ − > 0, μ2 + δ + β I θ M3 = 1 + . μ2 + δ + β I

It can be concluded that

(Aτ + b)σ M − M M S = F (I ) = 1 3 1 2 2 2 τ 2 M1 M2 M3 ((Aτ + b)σ M − M M )2 + 4Aτσ M M M + 1 3 1 2 1 1 2 3 . (8) 2τ M1 M2 M3

Since the function (Aτ + b)σ1 M3 − M1 M2 is strictly monotonically decreasing on I , it is easy to verify that S2 is also strictly monotonically decreasing when (Aτ + b)σ1 M3 − M1 M2 ≥ 0. For (Aτ +b)σ1 M3 < M1 M2,letX = (Aτ +b)σ1 M3 − M1 M2, Y = 4Aτσ1 M1 M2 M3, Z = 2τ M1 M2 M3. Equation (8) can be written as follows: √ 2 X + X + Y Y 2Aσ1 S2 = F2(I ) = = √ = √ , Z Z X 2 + Y − X X 2 + Y − X it follows that

lim F2(I ) = 0. I →+∞

2 −X, X and Y is strictly monotonically increasing on I ,soS2(I ) is strictly monotonically decreasing for (Aτ + b)σ1 M3 < M1 M2 and then F2(I ) is strictly decreasing for I ≥ 0. 123 Modeling sheep brucellosis transmission with a multi-stage...

Since F1 is strictly increasing and F2 is strictly decreasing, the curves deﬁned by = ( ) = ( ) ( ∗, ∗) ∗ > , ∗ > S2 F1 I and S2 F2 I have a common point S2 I with S2 0 I 0on condition that if F1(0)

(μ + σ )(μ + c) F (0) = 2 2 2 < F (0) = S0. 1 θ 2 2 σ β + 1 2 μ2+δ

Using the basic reproduction number of model (1) deﬁned in (3), this condition can be rewritten as R0 > 1. Therefore, if R0 > 1, model (1) has a unique positive solution ∗ = ( ∗, ∗, ∗, ∗, ∗) P S1 S2 V E I .

Theorem 3.2 the endemic equilibrium P∗ of model (1) is locally asymptotically stable for R0 > 1, but close to 1.

Proof Here, we use the central manifold theory to establish the local stability of endemic equilibrium taking β as bifurcation parameter. A critical value of bifurcation parameter β at R0 = 1 is given as

(μ + )(μ + σ ) β = 2 c 2 2 . c σ 0 + 0 2 V S2

It can be easily veriﬁed that the Jacobian J0 at β = β0 has a right eigenvector (cor- T responding to the zero eigenvalue) given by W = (w1,w2,w3,w4,w5) , where w = 1 and w = 1 which satisfy the following equation: 4 σ2 5 μ2+c

⎧ b b ⎪ −(μ1 + d + σ1)w1 + w2 + w3 = 0, ⎪ +τ 0+ 0 2 +τ 0+ 0 2 ⎨ 1 S1 V 1 S1 V β 0 c S2 σ1w1 − (θ + μ2)w2 + δw3 = μ + , ⎪ 2 c ⎩ β 0 θw − (μ + δ)w = c V . 2 2 3 μ2+c

The direct calculation shows w1 < 0 and w2 + w3 < 0. The left eigenvector σ (μ + )(μ +σ ) V = (v ,v ,v ,v ,v )T = 2 2 c 2 2 (0, 0, 0, 1 , 1 ) satisfy the equality 1 2 3 4 5 2μ2+σ2+c μ2+σ2 σ2 V · W = 1. Using the notations x1 = S1, x2 = S2, x3 = V, x4 = E, x5 = I ,the model (1) can be written as follows: ⎧ dx1 b(x2+x3) ⎪ = f1 = A + +τ( + ) − (μ1 + d + σ1)x1, ⎪ dt 1 x2 x3 ⎪ dx2 = = σ − β − θ − μ + δ , ⎨ dt f2 1x1 x2x5 x2 2x2 x3 dx3 = = θ − β − (μ + δ) , ⎪ dt f3 x2 x3x5 2 x3 ⎪ ⎪ dx4 = f = β(x + x )x − (μ + σ )x , ⎩⎪ dt 4 3 2 5 2 2 4 dx5 = = σ − (μ + ) . dt f5 2x4 2 c x5 123 Q. Hou, X.-D. Sun

It can be obtained

5 2 ∂ fk a = vkwi w j = βc(1 + )v4w5(w2 + w3)<0, ∂xi ∂x j k,i, j=1 5 ∂2 = v w fk = v w 0 + 0 > . b k i 4 5 V S2 0 ∂xi ∂β k,i=1

Since a < 0 and b > 0atβ = βc, using Theorem 4.1 and Remark 1 stated in [26], a transcritical bifurcation occurs at R0 = 1 and the endemic equilibrium is locally asymptotically stable if R0 > 1.

3.3 The global stability of equilibria

In this subsection, we analyze the global stability of equilibria for a special case: τ = 0 and b <μ2 (the solutions of model (1) are uniformly bounded). In this case, σ (μ +δ) σ θ model (1) has a disease-free equilibrium P = A , A 1 2 , A 1 , 0, 0 , where 0 π0 π0π1 π0π1 π0 = μ2(μ1 + d + σ1) − bσ1 and π1 = μ2(μ2 + θ + δ); meanwhile there is an ∗ endemic equilibrium P if R0 > 1 which satisﬁes ⎧ ⎪ A + b(S2 + V ) = μ1 S1 + dS1 + σ1 S1, ⎪ ⎨ σ1 S1 + δV = βS2 I + θ S2 + μ2 S2, β( + ) = (μ + σ ) , ⎪ V S2 I 2 2 E ⎪ θ = μ + δ + β , ⎩⎪ S2 2V V VI σ2 E = μ2 I + cI.

Moreover, it is easy to verify that the set A = (S , S , V, E, I ) ∈ R5 : S + S + V + E + I ≤ 1 2 + 1 2 μ

is the positively invariant set for model (1), where μ = min{μ1 + d,μ2 − b,μ2}. Furthermore, R0 is still deﬁned as the form (3). Based on the above analysis, one can state the following theorems:

Theorem 3.3 Assume that the parameters τ = 0, b <μ2 holds. If R0 ≤ 1,the disease-free equilibrium P0 of model (1) is globally asymptotically stable.

Proof Deﬁne a Lyapunov function L as follows:

= − 0 − 0 S1 + − 0 − 0 S2 L S1 S1 S1 ln 0 S2 S2 S2 ln 0 S1 S2 V μ + σ + V − V 0 − V 0 + E + 2 2 I. ln 0 V σ2 123 Modeling sheep brucellosis transmission with a multi-stage...

The derivative of L is given by dL S0 dS S0 dS V 0 dV dE μ + σ dI = 1 − 1 1 + 1 − 2 2 + 1 − + + 2 2 dt S1 dt S2 dt V dt dt σ2 dt S0 A + b S0 + V 0 = − 1 + ( + ) − 2 1 A b S2 V 0 S1 S1 S 1 S0 σ S0 + δV 0 + − 2 σ − β + δ − 1 1 1 1 S1 S2 I V 0 S2 S2 S2 V 0 θ S0 + 1 − θ S − βVI− 2 V V 2 V 0 (μ + σ )(μ + c) + β (V + S ) I − 2 2 2 I 2 σ 2 S S0 S S S0 S0 = (μ + ) 0 − 1 − 1 + (μ − ) 0 − 2 − 1 2 − 1 1 d S1 2 0 2 b S2 3 0 0 S S1 S S S S1 1 2 1 2 0 0 0 0 VS S V S1 S S S2 + δV 0 − 2 − 2 + bS0 − 2 − 1 2 0 0 2 2 0 0 S2V VS S S S S 2 1 2 1 2 0 0 0 S V S V S1 S + μ V 0 − bV 0 − 1 − − 2 − 2 2 4 0 0 0 S1 V VS S S 2 1 2 0 0 0 VS S V S1 S + bV 0 − 1 − 2 − 2 3 0 0 0 S1V VS2 S2 S1 (μ2 + σ2)(μ2 + c) + (R0 − 1)I σ2 ≤ 0.

dL = = 0, = 0, = 0 = The equality dt 0 holds if and only if S1 S1 S2 S2 V V and either R0 1 = {( , , , , ) ∈ : dL = } or I 0. Since P0 is the only invariant set in S1 S2 V E I dt 0 ,the disease-free equilibrium P0 is globally asymptotically stable by LaSalle’s Invariance Principle.

Theorem 3.4 Assume that the parameters τ = 0, b <μ2 holds. If R0 > 1,the ∗ = ( ∗, ∗, ∗, ∗, ∗) endemic equilibrium P S1 S2 V E I of model (1) is globally asymptotically stable.

Proof Deﬁne the following function:

= − ∗ − ∗ S1 + − ∗ − ∗ S2 L1 S1 S1 S1 ln ∗ S2 S2 S2 ln ∗ S1 S2 ∗ ∗ V ∗ ∗ E + V − V − V ln + E − E − E ln . V ∗ E∗ 123 Q. Hou, X.-D. Sun

Differentiating L1 along solution curves of system (1), it follows that dL S∗ dS S∗ dS V ∗ dV E∗ dE 1 = 1 − 1 1 + 1 − 2 2 + 1 − + 1 − dt S1 dt S2 dt V dt E dt ∗ + ∗ + ∗ S1 A b S2 V = 1 − A + b (S2 + V ) − ∗ S1 S1 S1 ∗ ∗ ∗ S σ1 S + δV ∗ + 1 − 2 σ S − βS I + δV − 1 − β I S S 1 1 2 S∗ 2 2 2 ∗ θ S∗ + − V θ − β − 2 − β ∗ 1 S2 VI ∗ I V V V E∗ + 1 − (β (V + S ) I − (μ + σ) E) E 2 S∗ = + ∗ + ∗ + σ ∗ + δ ∗ + θ ∗ − 1 − ∗ S2 − ∗ V 2A b S2 V 1 S1 V S2 A bS1 bS1 S1 S1 S1 − (μ + ) ∗ S1 − (μ − ) ∗ S2 − δ ∗ V − σ ∗ S1 − θ ∗ S2 1 d S1 ∗ 2 b S2 ∗ S2 1 S2 V S1 S2 S2 S2 V V − (μ − b)V ∗ + β I V ∗ + S∗ 2 V ∗ 2 E∗ − (μ2 + σ2) E − β I (S2 + V ) E S S∗ S S S∗ S∗ = (μ + ) ∗ − 1 − 1 + (μ − ) ∗ − 2 − 1 2 − 1 1 d S1 2 ∗ 2 b S2 3 ∗ ∗ S1 S1 S2 S1 S2 S1 ∗ ∗ ∗ ∗ ∗ VS S V ∗ S1 S S S2 + δV 2 − 2 − 2 + bS 2 − 2 − 1 S V ∗ VS∗ 2 S∗ S S S∗ 2 2 1 2 1 2 ∗ ∗ ∗ ∗ ∗ S V S V S1 S + μ V − bV 4 − 1 − − 2 − 2 2 S V ∗ VS∗ S∗ S 1 2 1 2 VS∗ ∗ S S∗ + ∗ − 1 − S2V − 1 2 bV 3 ∗ ∗ ∗ S1V VS2 S2 S1 ∗ ∗ ∗ ∗ ∗ I S S1 S E S IE + βS I 3 + − 1 − 2 − − 2 2 I ∗ S S S∗ E∗ S∗ I ∗ E 1 2 1 2 S∗ S S∗ ∗ ∗ + β ∗ ∗ + I − 1 − 1 2 − S2V − E − VIE V I 4 ∗ ∗ ∗ ∗ ∗ ∗ I S1 S2 S1 S2 V E V I E Deﬁne

∗ ∗ I L = I − I − I ln . 2 I ∗

Then the derivative of L2 along positive solutions of system (1)is ∗ dL ∗ E I EI 2 = σ E − − + 1 . dt E∗ I ∗ E∗ I 123 Modeling sheep brucellosis transmission with a multi-stage...

We consider the function ν(x) = 1 − x + ln x, which is nonpositive for x > 0 and ν(x) = 0 if and only if x = 1. We obtain

∗ ∗ ∗ I S S1 S E S IE 3 + − 1 − 2 − − 2 I ∗ S S S∗ E∗ S∗ I ∗ E 1 2 1 2 S∗ S S∗ ∗ = ν 1 + ν 1 2 + ν S2 IE ∗ ∗ ∗ S1 S2 S1 S2 I E I I E E + − ln − + ln (9) I ∗ I ∗ E∗ E∗ and

∗ ∗ ∗ ∗ I S S1 S S V E VIE 4 + − 1 − 2 − 2 − − I ∗ S S S∗ S∗V E∗ V ∗ I ∗ E 1 2 1 2 S∗ S S∗ ∗ ∗ = ν 1 + ν 1 2 + ν S2V + ν VIE ∗ ∗ ∗ ∗ S1 S2 S1 S2 V V I E I I E E + − ln − + ln . (10) I ∗ I ∗ E∗ E∗

Similarly, one can obtain

∗ E − I − EI + ∗ ∗ ∗ 1 E I E I EI∗ E E I I = ν + − ln − + ln . (11) E∗ I E∗ E∗ I ∗ I ∗

Deﬁne a Lyapunov functional L as

β V ∗ + S∗ I ∗ = + 2 . L L1 ∗ L2 σ2 E

Calculating the derivative of L along positive solutions of system (1), it follows that from (9)–(11)

β V ∗ + S∗ I ∗ dL = dL1 + 2 dL2 ≤ . ∗ 0 dt dt σ2 E dt

dL = = ∗, = ∗, = ∗, = ∗, = The equality dt 0 holds if and only if S1 S1 S2 S2 V V E E I ∗ {( , , , , ) ∈ : dL = } I . The maximum invariant set in S1 S2 V E I dt 0 is the singleton P∗. Therefore, the endemic equilibrium P∗ is globally asymptotically stable in the interior of . 123 Q. Hou, X.-D. Sun

Table 1 The reported data in Changling County of Jilin Province, China

Year 2008 2009 2010 2011 2012

Young sheep (1–6 months) 44,524 65,474 71,199 73,232 73,421 The output number of young sheep – 11,784 29,176 34,436 36,420 The input number of young sheep 73,523 75,833 77,535 78,199 77,080 The positive rate (adult sheep) (%) 6.9 5.1 4.6 4.8 5.5

4 The application of the model to the prediction of sheep brucellosis in Changling County of Jilin Province

In this section, based on the reported sheep brucellosis data in Changling County of Jilin Province, China, we carry out parameter estimation of model (1) and make some sensitivity analysis on some parameters by least square method. From China Animal Health and Epidemiology Center, we can obtain the data on adult sheep positive rate of Changling County, the number of young sheep (1–6 months) and the output (input) number of young sheep from 2009 (2008) to 2011. The data on sheep are listed in Table 1 (where “——” indicates missing data). However, the data can not be acquired easily since there are very few monitoring budget. Thus, we rely on the existing facts to make some rational assumptions or data fitting. The values of parameters are listed in Table 2. There are 108,021 and 127,404 adult sheep exposed to brucellosis in 2008 and 2009 respectively in Changling County. Therefore, the infectious sheep is 108021 × 0.069 = 7454 and 127404 × 0.051 = 6497, then ( ) = ( ) = 6497−(1−0.4)×7454 = ( ) = , ( ) = I 0 7454 and E 0 3.4 595. So S1 0 44 524 and S2 0 99, 972. The vaccination rate of adult sheep is 0, so V (0) = 0. Using model (1), we evaluate sheep brucellosis data in Changling County from 2008 to 2012 and make a prediction about the trend of sheep brucellosis infection. Figure 2 shows that the simulation of our model with reasonable parameter values provides a good match with the data on sheep in Changling County from 2008 to 2012. Figure 2a shows the relationship between the simulation of our model and the data which represents the number of young sheep (1–6 moths). Figure 2b shows that the simulation of positive rate of adult sheep over time in Changling County. We estimate that the basic reproduction number R0 = 3 from 2008 to 2012, which indicates that sheep brucellosis will persist in Changling County under the current prevention and control measures. If we fix all parameters except θ (the vaccination rate of adult sheep) and c (the elimination rate caused by brucellosis), the basic reproduction number R0 decreases with θ and c increases. Figure 3a depicts the plots of R0 in terms of θ. We can see that vaccinating susceptible sheep is an effective measure to decrease R0, and R0 becomes less than one if only the vaccination rate is greater than 2.8. Figure 3b represents the relationship between R0 and the mortality rate caused by brucellosis c, it shows that the influence of parameter c on R0 is greater than that of parameter θ. Therefore, we can conclude that eliminating infectious sheep and vaccinating susceptible sheep in time can effectively decrease R0 below 1, but eliminating infectious sheep is the most effective strategy to eradicate brucellosis in Changling County. 123 Modeling sheep brucellosis transmission with a multi-stage...

− Table 2 Parameters and their values (Unit: year 1)

Parameter Value Interpretation Source

A 76434 The input number of young sheep [A] b 1.5 The natural birth rate of adult sheep [B] − τ 1.582 × 10 5 The extent of the birth being delayed Fitting

μ1 0.1 Young sheep natural mortality rate [A] σ1 1.06 Transfer rate from young sheep to [A] adult sheep d 0.84 The output number of young sheep [A]

μ2 0.25 The elimination rate of adult sheep [C] δ 0.4 Sheep loss of vaccination rate [21] θ 0 Adult sheep vaccination rate [B] − β 3.844 × 10 6 Sheep-to-sheep transmission rate Fittting 0.18 Ineffective vaccination rate [21]

σ2 3.4 Transfer rate from exposed sheep to [C] infected sheep c 0.15 The elimination rate caused by [21] brucellosis

[A] In Changling County, the output rate of young sheep is 0.256, 0.446, 0.484 and 0.497, then d = 0.256+0.446+0.484+0.497 × 12 ≈ . μ = . × 12 = . 4 6 0 84. The survival rate of the lamb is 0.95, so 1 0 05 6 0 1and σ = ( − . − . ) × 12 ≈ . 1 1 0 05 0 42 6 1 06. The average input number of young sheep is 76,434 every year, so A = 76434 [B] The birth rate of adult sheep is about 1.5–1.6 every year, we assume that b = 1.5. There are few prevention and control measures to eliminate sheep brucellosis before 2012, so we assume that the vaccination rate of adult sheep is 0 [C] Adult sheep life span is about 4 years, so we estimate that μ2 is 0.25. Since the incubation period may range from 2 weeks to 7 months [27], it can not be accurately determined, we assume that the average time σ = 12 = . is 3.5 months and then 2 3.5 3 4

4 a x 10 b 7.5 0.075

7 0.07

6.5 0.065

6 0.06 5.5

0.055 5

The positive rate of adult sheep 0.05 4.5 The number of young sheep (1−6 months) 4 0.045 2008 2009 2010 2011 2012 2008 2009 2010 2011 2012 t (year) t (year) Fig. 2 Simulation of the number of young sheep (1–6 months) and the positive rate (the proportion of infectious sheep) of adult sheep over time in Changling County of Jilin Province, China. The smooth curve represents the solution of model (1) and the stars are the reported data of sheep

123 Q. Hou, X.-D. Sun

a 3 b 5

4.5

2.5 4

3.5 2 3 0 0 R R 2.5 1.5 2

1.5 1

0.5 0.5 0 1 2 3 4 5 6 0 0.5 1 1.5 2 θ c

Fig. 3 R0 in terms of θ and c

4 b 0.7 a τ=0.000015821 −−−−−− τ 0.6 =0.0001582 τ=0.001582 3.5

0.5

3 0.4 0 R 0.3 2.5

0.2

2 The positive rate of adult sheep τ=0.00001582 0.1

1.5 0 0 1 2 3 4 5 6 2008 τ −4 t (year) x 10

Fig. 4 R0 in terms of τ and simulation of the positive rate of adult sheep on variable τ

Figure 4 shows the relationship between the basic reproduction number R0 and parameter τ.FromFig.4a, b, we can ﬁnd that R0 is reduced on τ and the larger parameter τ is, the smaller inﬂuence on R0 will be. We further conclude that the price and breeding cost of sheep has a certain impact on brucellosis transmission if there are no extreme cases (e.g., the price of sheep is very low in a long time, but it is almost impossible), but the extinction of the disease rarely depends on these factors.

5 Conclusions and discussions

As a zoonotic and chronic disease, brucellosis which has caused serious economic and health problems in the world can not be ignored. Based on the connections between the characteristics of brucellosis infection and some risk factors of sheep brucellosis in Changling County of Jilin Province, China, this paper proposed a multi-stage dynamic model for the spread of sheep brucellosis. it deﬁned the basic reproduction number R0, and subsequently analyzed the dynamic behaviors of the model (1). Finally, it applied the model to simulate and predict brucellosis transmission in Changling County. By validating our model, we can conclude that sheep brucellosis cannot be eradicated under the current situation. After carrying out some sensitivity analysis of R0 on various parameters, we ﬁnd that the vaccination rate of adult sheep and eliminating 123 Modeling sheep brucellosis transmission with a multi-stage...

a 0.07 b 0.07 θ=0 c=0 −−−−−− −−−−−− 0.06 0.06 θ=1.8 (R =1.2) c=0.7 (R =1.26) 0 0 θ=2.8 (R =1.0) c=0.95 (R =1.0) 0 0.05 0 θ=4.8 (R =0.83) c=1.5 (R =0.68) 0.05 0 0

0.04 0.04 0.03

0.03 0.02 The positive rate of adult sheep 0.02 The positive rate of adult sheep 0.01

0.01 0 2008 2008 t (year) t (year) Fig. 5 The blue line is the solution curve of model (1) from 2008 to 2012. a Fixed all other parameters except θ, the prediction of positive rate on variable θ (1.8, 2.8 and 4.8) with adult sheep vaccinated in Changling County; b Fixed all other parameters except c, the prediction of positive rate on variable c (0.7, 0.95, 1.5) with culling of infectious sheep. (Color figure online) rate of infectious sheep play important roles in the prevention of brucellosis. Figure 5 reveals that the larger the elimination rate c and vaccination rate θ are, the less infection will be. However, the positive rate of adult sheep increases at first, and then decreases no matter how large the vaccination rate is. Furthermore, sheep brucellosis will persist for a very long time in Changling County even if the vaccination rate is relatively large. Comparing Fig. 5a, b, culling infectious sheep can significantly reduce the positive rate within a short time, which means that sheep brucellosis cases are becoming less at the same time, and the economic losses caused by brucellosis will be cut. There are some limitations in this study. Firstly, the incubation period of animal brucellosis ranges from 2 weeks to 7 months and the exact time is difficult to be determined, so the value of the parameter σ2 can not be estimated accurately. Secondly, the influence of season is not included in the model, while brucellosis may have different infection rates in different seasons. Thirdly, the fluctuation of restocking rate is very small in a short time, but it must be taken into account if the price or the breeding cost of sheep has a great change. Moreover, we estimate the parameters by the positive rate of sheep, but more detailed data need to be collected so that the model parameters can be measured or estimated more accurately. Finally, the global stability of the endemic equilibrium has not been analyzed only when R0 is greater than 1. We leave these for future consideration.

Acknowledgments We would like to thank anonymous reviewers for their helpful comments which improved the presentation of this work. This research is partially supported by TianYuan Special Foundation of the National Natural Science Foundation of China (11426208), YouthFoundation (2015021018) of Shanxi Province.

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