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Abstract: The dissemination of cattle brucellosis in Zhejiang province of China can be attributed to the transport of cattle between cities within the province. In this paper,an npatch dynamical model is proposed to study the effect of cattle dispersal on brucellosis spread. Theoretically, we analyze the dynamical behavior of the mutipatch model. For the 2patch submodel, sensitivity analyses of the basic reproduction number R0 and the number of the infectious cattle in term of model parameters are carried out. By numerical analysis, it is obtained that the dispersal of susceptible cattle between patches and the centralization of infected cattle to the large scale patch can alleviate the epidemic and are in favor of the control of disease in the whole region.
Key words: brucellosis; cattle; npatch model; dispersal
CLC number: O 175 Document code: A Article ID: 10005137(2014)05044115
PACS: 87.23.Cc, 45.70.Qj, 89.75.Kd
MSC(2000): 34A34, 37C75, 92D30
Received date: 20140511
Foundation item: This work is supported by the National Natural Science Foundation of China under Grant (11331009,11171314,11147015,11301490);the National Youth Natural Science Foundation (11201434);the Specialized Research Fund for the Doctoral Program of Higher Education(20121420130001);the Research Project Supported by Shanxi Scholarship Council of China (2013-3)
*Corresponding author:Zhen Jin,Professor,Email: [email protected] Introduction
Brucella, one of the world′s major zoonotic pathogens known,causes infectious abortion in animals and Malta Fever in man［1］. Since many kinds of domestic animals,such as sheep, cattle, dogs, pig and so on, can be infected by brucella,brucellosis usually causes economic devastation on a global scale. China is no exception. In Zhejiang province which locates in the southern China, the livestock breeding, dairy, the leather processing industry have gotten great development.A mass of dairy cows, beefs, row fur and other animal byproducts were taken to trade annually. But it has also brought lots of cattle brucellosis infection［2, 3］. In fact, the cow remains an intermittent carrier for years in China［4］. For cattle, transmission of brucella typically occurs through direct contact with brucella carriers or oral contact with aborted foetal material including the bacteria throughout the byre［4, 5］. Bull can spread infection through semen, but often the disease leads to infertility or arthritis. More detailed information about cattle brucellosis can be seen in ［6］. Since Brucellosis caused by brucella is a nonfatal disease, it is often overlooked by the majority of the scientific community. The local government of Zhejiang province has regularly taken detection measures and culled infected cattle immediately. Yet, the data of positive cattle brucellosis in Zhejiang are rising year by year and it has influenced the local economy, even leads to the local prevalence of human brucellosis. From Fig.1, we can see that brucellosis has been spreading from north to south in Zhejiang province.So, one of main reasons of the geographical spread of the disease is the transportation of cattle between cities within Zhejiang province. Cattle transportation can cause cross infection of individuals among different regions. Besides, through vehicles and staff movement, it can also lead to the disperdal of brucella surviving in environment.Therefore, public health officials and scientific community should pay more attention to the transmission of cattle brucellosis.
Analysis of a multipatch dynamical model about cattle brucellosis Dynamical systems method is one of the most useful and important tools in studying biological and epidemiological models［7-13］.Some researches have applied dynamical systems method to study brucellosis［14-17］.In 1994,GonzalezGuzman and Naulin［14］were the first to apply dynamical models to study bovine brucellosis.In 2005,besides transmission within sheep and cattle populations, Zinsstag et al.［17］ considered the transmission to humans in a dynamical model. The livestock are classified into three subclasses: the susceptible, the seropositive and the immunized. In 2009, Xie and Horan［15］ built a simple dynamical model with the susceptible, the infected and the resistant subclasses to discuss brucellosis in the elk and cattle population. In 2010, Ainseba et al.［16］ considered two transmission modes about the ovine brucellosis in their model: direct mode caused by infected individuals and indirect mode related to brucella in the environment. For the transmission of brucellosis in China, there are also some studies［18-22］. Hou et al.［20］ investigated the transmission dynamics of sheep brucellosis in Inner Mongolia Autonomous Region of China. Zhang et al.［21］ and Nie et al.［22］ established dynamical models about dairy cattle brucellosis in Zhejiang and Jilin Provinces, respectively. According to the spatial spread of disease, there are two types of model we can apply:multipatch models［23-30］ and reactiondiffusion models［31-33］.The goal of this paper is to establish an npatch dynamical model to discuss the effects of cattle dispersal and brucella diffusion on the geographical spread of the disease. The article is organized as follows. In Sections 2 , we propose an npatch model about cattle brucellosis with cattle transportation and brucella diffusion, and analyze its dynamical behavior. In section 3, we apply numerical method to discuss the transmission of the disease between two patches under different conditions. In section 4, we give a brief discussion.
2 Model and dynamical behavior
There are 11 cities in Zhejiang province, where Hangzhou is the provincial capital. More generally, we propose an epidemic dynamical model with cattle dispersal between n patches. The number of cattle in each patch can be denoted by Ni, i=1,2,…,n. For each patch, the cattle population is divided into three classes:
Fig.1 The distribution of infected dairy cattle in Zhejiang from 2001 to 2010.
(a) 2001. (b)2002. (c)2003. (d) 2004. (e)2005. (f)2006.(g) 2007. (h)2008. (i)2009.(j) 2010.
susceptible,exposed and infective individuals, the numbers of which at time t in i th patch are denoted by Si(t),Ei(t) and Ii(t), respectively. During the infected period, the infected individuals (the exposed and the infectious individuals) discharge brucella into the environment. The quantity of brucella in environment is denoted by Vi(t). Consequently, the susceptible cattle can be infected by contacting with the exposed cattle, the infectious cattle and the brucella in environment. Compared with the infectious individuals, the transmission coefficient of the exposed individuals is relatively smaller. So the auxiliary parameter θ is introduced. The internal relationship of each individual in n patches can be described in the following system and the parameter meanings can be seen in Table 1, where parameters Ai, βi, αi, mi, δi, μi, ri and wi are all positive constants. θ is a parameter whose value is between 0 and 1. aji, bji, cji and dji (j≠i) are nonnegative constants. aii, bii, cii and dii are nonpositive constants.
4 Discussion
For Zhejiang province of China, the recent prevalance of brucellosis in cattle is believed to be caused by the transportation of cattle and brucella between cities in Zhejiang province. In this article, we applied an npatch dynamical model to study the effect of dispersal of cattle and brucella on the spatial transmission of brucellosis. Firstly, we analyzed the dynamical behavior of the model. More specifically, assuming n=2, we carried out the sensitivity analysis of the basic reproduction number and the number of the infectious cattle in term of different parameter values. Finally, it is obtained that the dispersal of the susceptible cattle can relief the spread of brucellosis in the whole region. However, the emigration of the brucella carriers or the diffusion of brucella in patch whose rasing quantity of cattle is larger can increase R0. On the contrary, the emigration of the brucella carriers or the diffusion of brucella in patches where the amount of live cattle is smaller can reduce R0. In summary, the dispersal of the susceptible population of each patch and the centralization of the infected cattle to the patches where the breeding scale is bigger are in favor of the controlling of the disease. References:
［1］ M. L. Boschiroli,V. Foulongne,D.O′Callaghan.Brucellosis:a worldwide zoonosis［J］.Current Opinion in Microbiology,2001,4:58-64.
［2］ W. M. Xu,S. F. Shi,Y. Yang,H. Y. Jin,H. Wang.Epidemic situation and exploration of the prevention and control strategies on Brucellosis in Zhejiang Province［J］.Chinese Rural Health Service Administration,2007,27(3):209-211.
［3］ W. M. Xu,H. Wang,S. J. Zhu,Y. Yang,J. Wang,X. Y. Jin,L. L. Gao,Y. Yang.2000-2007 the human and livestock brucellosis situation and control measure in Zhejiang province［J］.Chin. J. Prev. Med.,2009,43(8):746-747.
［4］ Z. Bercovich.Maintenance of Brucella abortusfree herds:a review with emphasis on the epidemiology and the problems in diagnosing Brucellosis in areas of low prevalence［J］.Vet. Quart.,1998,20:81-88.
［5］ D. C. Blood,O. M. Radostits.Veterinary medicine:A textbook of the diseases of cattle,sheep,pigs,goats,horses.(7th ed.)［M］.London:Balliere Tindall,1989.
［6］ T. England,L. Kelly,R. D. Jones, A. Macmillan, M. Wooldridge.A simulation model of brucellosis spread in British cattle under several testing regimes［J］.Prev. Vet. Med.,2004,63:63-73.
［7］ J. Zhang,Z. Jin,G. Q. Sun,S. G. Ruan.Spatial spread of rabies in China［J］.J. Appl. Analy. Compu.,2012,2(1):111-126.
［8］ J. Q. Li,X. C. Song,F. Y. Gao.Global stability of a virus infection model with two delays and two types of target cells［J］.J. Appl. Analy. Compu.,2012,2(3),281-292.
［9］ K. Hong,P. X. Weng.Stability and traveling waves of diffusive predatorprey model with agestructure and nonlocal effect［J］.J Appl. Analy. Compu.,2012,2(2):173-192.
［10］ J. Zhang,Z. Jin,G. Q. Sun,T. Zhou,S. G. Ruan.Analysis of rabies in China:Transmission dynamics and control［J］.PLoS one,2011,6(7):e20891.
［11］ J. Zhang,Z. Jin,G. Q. Sun,X. D. Sun,S. G. Ruan.Modeling seasonal rabies epidemic in China［J］.Bull. Math. Biol.,2012,74:1226-1251.
［12］ J. Zhang,Z. Jin,G. Q. Sun,X. D. Sun,Y. M. Wang,B. X. Huang.Determination of original infection source of H7N9 avian influenza by dynamical model［J］.Scientific Reports,2014,4:4846.
［13］ M. T. Li,G. Q. Sun,J. Zhang,Z. Jin.Global dynamic behavior of a multigroup cholera model with indirect transmission［J］.Discrete Dynamics in Nature and Society,2013,Article ID 703826.
［14］ J. GonzalezGuzman,R. Naulin.Analysis of a model of bovine brucellosis using singular perturbations［J］.J. Math. Biol.,1994,33:211-223. ［15］ X. Fang,D. H. Richard.Disease and behavioral dynamics for brucellosis control in greater yellowstone area［J］.J. Agr. Resour. Ec.,2009,34(1):11-33.
［16］ B. Ainseba,C. Benosman,P. Magal.A model for ovine brucellosis incorporating direct and indirect transmission［J］.J. Biol. Dynam.,2010,4(1):2-11.
［17］ J. Zinsstag,F. Roth,D. Orkhon,G. ChimedOchir,M. Nansalmaa,J. Kolar,P. Vounatsou.A model of animalhuman brucellosis transmission in Mongolia［J］.Prev. Vet. Med.,2005,69:77-95.
［18］ M. T. Li,G. Q. Sun,Y. F. Wu,J. Zhang,Z. Jin.Transmission dynamics of a multigroup brucellosis model with mixed cross infection in public farm［J］.Applied Mathematics and Computation,2014,237(5):582-594.
［19］ M. T. Li,G.Q. Sun,J. Zhang,Z. Jin,X. D. Sun,Y. M. Wang,B. X. Huang,Y. H. Zheng.Transmission dynamics and control for a Brucellosis model in Hinggan league of Inner Mongolia,China［J］.Mathematical Bioscience and Engineering,(Accepted),2014.
［20］ Q. Hou,X. D. Sun,J. Zhang,Y. J. Liu,Y. M. Wang, Z. Jin.Modeling the transmission dynamics of sheep brucellosis in Inner Mongolia Autonomous Region,China［J］.Math. Biosci.,2013,242:51-58.
［21］ J. Zhang,X. D. Sun,G. Q. Sun,Q. Hou,M. T. Li,B. X Huang,Z. Jin.Prediction and control of brucellosis transmission of dairy cattle in Zhejiang Province,China［M］.In press.
［22］ J. Nie,X. D. Sun,G. Q. Sun,J. Zhang, Z. Jin.Modeling the transmission dynamics of dairy cattle brucellosis in Jilin Province,China［M］.In press.
［23］ D. Z. Gao,S. G. Ruan.A multipatch malaria model with logistic growth populations［J］.SIAM J. Appl. Math.,2012,72(3):819-841.
［24］ W. D. Wang,X. Q. Zhao.An epidemic model in a patchy environment［J］.Math. Biosci.,2004,190:97-112.
［25］ D. Z. Gao,S. G. Ruan.An SIS patch model with variable transmission coefficients［J］.Math. Biosci.,2011,232:110-115.
［26］ F. Zhang,X. Q. Zhao.A periodic epidemic model in a patchy environment［J］.J. Math. Anal. Appl.,2007,325:496-516.
［27］ P. Auger,E. Kouokam,G. Sallet,M. Tchuente,B. Tsanou.The RossMacdonald model in a patchy environment［J］.Math. Biosci.,2008,216:123-131.
［28］ E. Kouokam,P. Auger,H. Hbid,M. Tchuente.Effect of the number of patches in a multipatch SIRS model with fast migration on the basic reproduction rate［J］.Acta. Biotheor.,2008,56:75-86.
［29］ W. D. Wang,X. Q. Zhao.An epidemic model with population dispersal and infection period［J］.SIAM J. Appl. Math.,2006,66(4):1454-1472. ［30］ L. J. S. Allen,B. M. Bolker,Y. Lou,A. L. Nevai.Asymptotic profiles of the steady states for an SIS epidemic patch model［J］.SIAM J. Appl. Math.,2007,67:1283-1309.
［31］ Y. Morita.Spectrum comparison for a conserved reactiondiffusion system with a variational property［J］.J. Appl. Analy. Compu.,2012,2(1):57-71.
［32］ H. S. Mahato,M. Bhm.Global existence and uniqueness of solution for a system of semilinear diffusionreaction equations［J］.J. Appl. Analy. Compu.,2012,3(4),357-376.
［33］ L. Z. Li,N. Li,Y. Y. Liu,L. H. Zhang.Existence and uniqueness of a traveling wave front of a model equation in synaptically coupled neuronal networks［J］.J. Appl. Analy. Compu.,2013,3(2),145-167.
［34］ P. Van Den Driessche ,J. Watmough.Reproduction numbers and subthreshold endemic equilibria for compartmental models of disease transmission［J］.Math. Biosci.,2002,180(1-2):29-48.
［35］ H. Thieme.Convergence results and a PoincaréBendixson trichotomy for asymptotically autonomous differential equations［J］.J. Math. Biol.,1992,30:755-763.
［36］ W. M. Hirsch,H. L. Smith,X. Q. Zhao.Chain transitivity attractivity and strong repellors for semidynamical systems［J］.J. Dynam. Differ. Equat.,2001,13(1):107-131.
［37］ H. R. Thieme.Persistence under relaxed pointdissipativity (with application to an endemic model)［J］.SIAM J. Math. Anal.,1993,24:407-435.
［38］ X. Q. Zhao.Uniform persistence and periodic coexistence states in infinitedimensional periodic semiflows with applications［J］.Can. Appl. Math. Quart.,1995,3:473-495.
(Zhenzhen Feng)
Key words: brucellosis; cattle; npatch model; dispersal
CLC number: O 175 Document code: A Article ID: 10005137(2014)05044115
PACS: 87.23.Cc, 45.70.Qj, 89.75.Kd
MSC(2000): 34A34, 37C75, 92D30
Received date: 20140511
Foundation item: This work is supported by the National Natural Science Foundation of China under Grant (11331009,11171314,11147015,11301490);the National Youth Natural Science Foundation (11201434);the Specialized Research Fund for the Doctoral Program of Higher Education(20121420130001);the Research Project Supported by Shanxi Scholarship Council of China (2013-3)
*Corresponding author:Zhen Jin,Professor,Email: [email protected] Introduction
Brucella, one of the world′s major zoonotic pathogens known,causes infectious abortion in animals and Malta Fever in man［1］. Since many kinds of domestic animals,such as sheep, cattle, dogs, pig and so on, can be infected by brucella,brucellosis usually causes economic devastation on a global scale. China is no exception. In Zhejiang province which locates in the southern China, the livestock breeding, dairy, the leather processing industry have gotten great development.A mass of dairy cows, beefs, row fur and other animal byproducts were taken to trade annually. But it has also brought lots of cattle brucellosis infection［2, 3］. In fact, the cow remains an intermittent carrier for years in China［4］. For cattle, transmission of brucella typically occurs through direct contact with brucella carriers or oral contact with aborted foetal material including the bacteria throughout the byre［4, 5］. Bull can spread infection through semen, but often the disease leads to infertility or arthritis. More detailed information about cattle brucellosis can be seen in ［6］. Since Brucellosis caused by brucella is a nonfatal disease, it is often overlooked by the majority of the scientific community. The local government of Zhejiang province has regularly taken detection measures and culled infected cattle immediately. Yet, the data of positive cattle brucellosis in Zhejiang are rising year by year and it has influenced the local economy, even leads to the local prevalence of human brucellosis. From Fig.1, we can see that brucellosis has been spreading from north to south in Zhejiang province.So, one of main reasons of the geographical spread of the disease is the transportation of cattle between cities within Zhejiang province. Cattle transportation can cause cross infection of individuals among different regions. Besides, through vehicles and staff movement, it can also lead to the disperdal of brucella surviving in environment.Therefore, public health officials and scientific community should pay more attention to the transmission of cattle brucellosis.
Analysis of a multipatch dynamical model about cattle brucellosis Dynamical systems method is one of the most useful and important tools in studying biological and epidemiological models［7-13］.Some researches have applied dynamical systems method to study brucellosis［14-17］.In 1994,GonzalezGuzman and Naulin［14］were the first to apply dynamical models to study bovine brucellosis.In 2005,besides transmission within sheep and cattle populations, Zinsstag et al.［17］ considered the transmission to humans in a dynamical model. The livestock are classified into three subclasses: the susceptible, the seropositive and the immunized. In 2009, Xie and Horan［15］ built a simple dynamical model with the susceptible, the infected and the resistant subclasses to discuss brucellosis in the elk and cattle population. In 2010, Ainseba et al.［16］ considered two transmission modes about the ovine brucellosis in their model: direct mode caused by infected individuals and indirect mode related to brucella in the environment. For the transmission of brucellosis in China, there are also some studies［18-22］. Hou et al.［20］ investigated the transmission dynamics of sheep brucellosis in Inner Mongolia Autonomous Region of China. Zhang et al.［21］ and Nie et al.［22］ established dynamical models about dairy cattle brucellosis in Zhejiang and Jilin Provinces, respectively. According to the spatial spread of disease, there are two types of model we can apply:multipatch models［23-30］ and reactiondiffusion models［31-33］.The goal of this paper is to establish an npatch dynamical model to discuss the effects of cattle dispersal and brucella diffusion on the geographical spread of the disease. The article is organized as follows. In Sections 2 , we propose an npatch model about cattle brucellosis with cattle transportation and brucella diffusion, and analyze its dynamical behavior. In section 3, we apply numerical method to discuss the transmission of the disease between two patches under different conditions. In section 4, we give a brief discussion.
2 Model and dynamical behavior
There are 11 cities in Zhejiang province, where Hangzhou is the provincial capital. More generally, we propose an epidemic dynamical model with cattle dispersal between n patches. The number of cattle in each patch can be denoted by Ni, i=1,2,…,n. For each patch, the cattle population is divided into three classes:
Fig.1 The distribution of infected dairy cattle in Zhejiang from 2001 to 2010.
(a) 2001. (b)2002. (c)2003. (d) 2004. (e)2005. (f)2006.(g) 2007. (h)2008. (i)2009.(j) 2010.
susceptible,exposed and infective individuals, the numbers of which at time t in i th patch are denoted by Si(t),Ei(t) and Ii(t), respectively. During the infected period, the infected individuals (the exposed and the infectious individuals) discharge brucella into the environment. The quantity of brucella in environment is denoted by Vi(t). Consequently, the susceptible cattle can be infected by contacting with the exposed cattle, the infectious cattle and the brucella in environment. Compared with the infectious individuals, the transmission coefficient of the exposed individuals is relatively smaller. So the auxiliary parameter θ is introduced. The internal relationship of each individual in n patches can be described in the following system and the parameter meanings can be seen in Table 1, where parameters Ai, βi, αi, mi, δi, μi, ri and wi are all positive constants. θ is a parameter whose value is between 0 and 1. aji, bji, cji and dji (j≠i) are nonnegative constants. aii, bii, cii and dii are nonpositive constants.
4 Discussion
For Zhejiang province of China, the recent prevalance of brucellosis in cattle is believed to be caused by the transportation of cattle and brucella between cities in Zhejiang province. In this article, we applied an npatch dynamical model to study the effect of dispersal of cattle and brucella on the spatial transmission of brucellosis. Firstly, we analyzed the dynamical behavior of the model. More specifically, assuming n=2, we carried out the sensitivity analysis of the basic reproduction number and the number of the infectious cattle in term of different parameter values. Finally, it is obtained that the dispersal of the susceptible cattle can relief the spread of brucellosis in the whole region. However, the emigration of the brucella carriers or the diffusion of brucella in patch whose rasing quantity of cattle is larger can increase R0. On the contrary, the emigration of the brucella carriers or the diffusion of brucella in patches where the amount of live cattle is smaller can reduce R0. In summary, the dispersal of the susceptible population of each patch and the centralization of the infected cattle to the patches where the breeding scale is bigger are in favor of the controlling of the disease. References:
［1］ M. L. Boschiroli,V. Foulongne,D.O′Callaghan.Brucellosis:a worldwide zoonosis［J］.Current Opinion in Microbiology,2001,4:58-64.
［2］ W. M. Xu,S. F. Shi,Y. Yang,H. Y. Jin,H. Wang.Epidemic situation and exploration of the prevention and control strategies on Brucellosis in Zhejiang Province［J］.Chinese Rural Health Service Administration,2007,27(3):209-211.
［3］ W. M. Xu,H. Wang,S. J. Zhu,Y. Yang,J. Wang,X. Y. Jin,L. L. Gao,Y. Yang.2000-2007 the human and livestock brucellosis situation and control measure in Zhejiang province［J］.Chin. J. Prev. Med.,2009,43(8):746-747.
［4］ Z. Bercovich.Maintenance of Brucella abortusfree herds:a review with emphasis on the epidemiology and the problems in diagnosing Brucellosis in areas of low prevalence［J］.Vet. Quart.,1998,20:81-88.
［5］ D. C. Blood,O. M. Radostits.Veterinary medicine:A textbook of the diseases of cattle,sheep,pigs,goats,horses.(7th ed.)［M］.London:Balliere Tindall,1989.
［6］ T. England,L. Kelly,R. D. Jones, A. Macmillan, M. Wooldridge.A simulation model of brucellosis spread in British cattle under several testing regimes［J］.Prev. Vet. Med.,2004,63:63-73.
［7］ J. Zhang,Z. Jin,G. Q. Sun,S. G. Ruan.Spatial spread of rabies in China［J］.J. Appl. Analy. Compu.,2012,2(1):111-126.
［8］ J. Q. Li,X. C. Song,F. Y. Gao.Global stability of a virus infection model with two delays and two types of target cells［J］.J. Appl. Analy. Compu.,2012,2(3),281-292.
［9］ K. Hong,P. X. Weng.Stability and traveling waves of diffusive predatorprey model with agestructure and nonlocal effect［J］.J Appl. Analy. Compu.,2012,2(2):173-192.
［10］ J. Zhang,Z. Jin,G. Q. Sun,T. Zhou,S. G. Ruan.Analysis of rabies in China:Transmission dynamics and control［J］.PLoS one,2011,6(7):e20891.
［11］ J. Zhang,Z. Jin,G. Q. Sun,X. D. Sun,S. G. Ruan.Modeling seasonal rabies epidemic in China［J］.Bull. Math. Biol.,2012,74:1226-1251.
［12］ J. Zhang,Z. Jin,G. Q. Sun,X. D. Sun,Y. M. Wang,B. X. Huang.Determination of original infection source of H7N9 avian influenza by dynamical model［J］.Scientific Reports,2014,4:4846.
［13］ M. T. Li,G. Q. Sun,J. Zhang,Z. Jin.Global dynamic behavior of a multigroup cholera model with indirect transmission［J］.Discrete Dynamics in Nature and Society,2013,Article ID 703826.
［14］ J. GonzalezGuzman,R. Naulin.Analysis of a model of bovine brucellosis using singular perturbations［J］.J. Math. Biol.,1994,33:211-223. ［15］ X. Fang,D. H. Richard.Disease and behavioral dynamics for brucellosis control in greater yellowstone area［J］.J. Agr. Resour. Ec.,2009,34(1):11-33.
［16］ B. Ainseba,C. Benosman,P. Magal.A model for ovine brucellosis incorporating direct and indirect transmission［J］.J. Biol. Dynam.,2010,4(1):2-11.
［17］ J. Zinsstag,F. Roth,D. Orkhon,G. ChimedOchir,M. Nansalmaa,J. Kolar,P. Vounatsou.A model of animalhuman brucellosis transmission in Mongolia［J］.Prev. Vet. Med.,2005,69:77-95.
［18］ M. T. Li,G. Q. Sun,Y. F. Wu,J. Zhang,Z. Jin.Transmission dynamics of a multigroup brucellosis model with mixed cross infection in public farm［J］.Applied Mathematics and Computation,2014,237(5):582-594.
［19］ M. T. Li,G.Q. Sun,J. Zhang,Z. Jin,X. D. Sun,Y. M. Wang,B. X. Huang,Y. H. Zheng.Transmission dynamics and control for a Brucellosis model in Hinggan league of Inner Mongolia,China［J］.Mathematical Bioscience and Engineering,(Accepted),2014.
［20］ Q. Hou,X. D. Sun,J. Zhang,Y. J. Liu,Y. M. Wang, Z. Jin.Modeling the transmission dynamics of sheep brucellosis in Inner Mongolia Autonomous Region,China［J］.Math. Biosci.,2013,242:51-58.
［21］ J. Zhang,X. D. Sun,G. Q. Sun,Q. Hou,M. T. Li,B. X Huang,Z. Jin.Prediction and control of brucellosis transmission of dairy cattle in Zhejiang Province,China［M］.In press.
［22］ J. Nie,X. D. Sun,G. Q. Sun,J. Zhang, Z. Jin.Modeling the transmission dynamics of dairy cattle brucellosis in Jilin Province,China［M］.In press.
［23］ D. Z. Gao,S. G. Ruan.A multipatch malaria model with logistic growth populations［J］.SIAM J. Appl. Math.,2012,72(3):819-841.
［24］ W. D. Wang,X. Q. Zhao.An epidemic model in a patchy environment［J］.Math. Biosci.,2004,190:97-112.
［25］ D. Z. Gao,S. G. Ruan.An SIS patch model with variable transmission coefficients［J］.Math. Biosci.,2011,232:110-115.
［26］ F. Zhang,X. Q. Zhao.A periodic epidemic model in a patchy environment［J］.J. Math. Anal. Appl.,2007,325:496-516.
［27］ P. Auger,E. Kouokam,G. Sallet,M. Tchuente,B. Tsanou.The RossMacdonald model in a patchy environment［J］.Math. Biosci.,2008,216:123-131.
［28］ E. Kouokam,P. Auger,H. Hbid,M. Tchuente.Effect of the number of patches in a multipatch SIRS model with fast migration on the basic reproduction rate［J］.Acta. Biotheor.,2008,56:75-86.
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(Zhenzhen Feng)