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

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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).
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