MATHEMATICAL MODEL FOR BRUCELLOSIS TRANSMISSION DYNAMICS IN LIVESTOCK AND HUMAN POPULATIONS

1Department of Applied Mathematics and Computational Sciences, Nelson Mandela African Institution of Science and Technology, P. O. Box 447, Arusha, Tanzania 2Department of Mathematics, Informatics and Computational Sciences, Sokoine University of Agriculture, P. O. Box 3038, Morogoro, Tanzania 3Institute of Mathematical Sciences, Strathmore University PO Box 59857-00200, Nairobi, Kenya 4Department of Mathematics, University of Iringa, P. O. Box 200, Iringa, Tanzania 5Department of Global Health and Bio-Medical Sciences, Nelson Mandela African Institution of Science and


INTRODUCTION
Brucellosis is a contagious zoonotic infection caused by Gram-negative bacteria of genus brucella that includes; B. abortus primarly from cattle, B. melitensis from small ruminants, B. suis from swine, and B. canis from dogs [1,2,3,4]. It is considered by the international organizations like Food and Agriculture Organization (FAO), the World Health Organization (WHO) and World Organization for Animal Health (Office International des Epizooties (OIE)) as one of the most widespread zoonoses in the world alongside bovine tuberculosis and rabies [5]. The disease is an ancient one that was described more than 2000 years ago by the Romans [6] and has been known by various names, including Mediterranean fever, Malta fever, gastric remittent fever, bang's disease, crimean fever, gibraltar fever, rock fever, lazybones disease and undulant fever [7].
Brucella bacteria was first isolated in 1887 from an infected individual's blood by a British military medical officer David Bruce and by that reason the disease was named brucellosis to honor his contribution [8]. Furthermore, in 1905 Zamitt carried out an experiment on goats to investigate the origin of human brucellosis, and found that, human brucellosis originates from goats [9]. To date, eight species of brucella have been identified and named primarily for the source animal or features of infection. Of these, the following four have moderateto-significant human pathogenicity: Brucella melitensis (highest pathogenicity), Brucella suis (high pathogenicity), Brucella abortus (moderate pathogenicity), Brucella canis (moderate pathogenicity) [10,11,12].
Brucellosis causes devastating losses to the livestock industry especially small-scale livestock holders, thereby limiting economic growth and hindering access to international markets [13].
The economic importance of the disease is based on the fact that it causes financial losses through abortions, sterility, decreased milk production, veterinary fees and animal replacement costs. In animals, brucellosis is transmitted when a susceptible animal ingest contaminated materials by licking discharges from infected animals and suckling milk from infected dams.
In humans the bacteria is transmitted through ingestion of contaminated raw blood and meat, unpasteurized milk or other dairy products. Furthermore, direct contact with aborted fetuses, vaginal discharges and occupational accidents through needle injection during mass vaccination and during laboratory manipulation may be possible route of brucellosis transmission. In view of this, farmers, laboratory personnels, abattoir workers and veterinarians are at high risk of contracting the disease. According to Ducrotoy et al. [14], there are epidemiological situations in which B. melitensis is absent but infections of small ruminants by B. abortus occur in areas where they are in contact with cattle.
Infected animals exhibit clinical signs that are of economic significance to stakeholders, such as reduced fertility, late term abortion, poor weight gain, lost draught power, and a substantial decline in milk production [13,15]. However, symptoms in human includes; continuous or intermittent fever, headache, weakness, profuse sweats, chills, joint pains, aches, weight loss as well as devastating complications that leads to miscarriage that occurs within the early trimester in pregnant women [16]. Infection may develop into chronic forms that characterised by neurological complications, endocarditis and testicular or bone abscess formation [17,18]. The infection can also affect the liver and spleen, and may last for longer terms if not timely treated.
Furthermore, the clinical signs of brucellosis in human presents diagnostic challenges because they overlap with other febrile conditions such as typhoid fever, malaria, rheumatic fever, joint diseases and relapsing fever. Since human brucellosis is debilitating disease, it requires prolonged treatment with combination of antibiotics [19].
The global burden of human brucellosis remains high and causes more than 500,000 new human cases per year worldwide. The annual number of reported cases in United States has dropped significantly to about 100 cases per year due to stringent animal vaccination programs and milk pasteurization. Most United States cases are now due to the consumption of illegally imported unpasteurized dairy products from Mexico and approximately 60% of human brucellosis cases occur in California and Texas [20].
In Africa, livestock brucellosis exists throughout sub-Saharan Africa, but the prevalence is unclear and poorly understood with varying reports from country to country, geographical regions as well as animal factors [21]. Most African countries have poor socioeconomic status, with people living with and by their livestock, while health networks, surveillance and vaccination programs are virtually non-existent [20]. Livestock brucellosis is a highly prevalent disease in many areas of Tanzania with limited data available regarding its distribution, affected host species and impact. The first outbreak of brucellosis was reported in Arusha in 1927 [22].
Previous surveys in Tanzania have demonstrated the occurrence of the disease in cattle in various production systems, regions and zones with individual animal level seroprevalence varying from 1 to 30% while the average prevalence in humans varies from 1 to 5% [23]. A recent study by [24] shows that brucellosis incidence is moderate in northern Tanzania and suggests that the disease is endemic and an important human health problem in this area. Moreover, human cases had been reported in areas of northern, eastern, lake and western zones of Tanzania with seroprevalence varying from 0.7 to 20.5% [25,26]. Despite the WHO, FAO, OIE efforts and interventions are available, brucellosis continues to pose great economic threat on livelihood and food security in both developed and developing countries from generation to generation. Thus, there is a need to assess the current control strategies and their effectiveness if we are to control or eradicate the disease. So far few studies [10,27,28,29,30,31,32], have been developed to analyze dynamics and spread of brucellosis in a homogeneous/heterogeneous populations. However, none of these studies had considered the mathematical approach to assess the impact of human to human transmission in reducing or eradicating the disease. In this paper, the dynamics and effectiveness of the control strategies for human brucellosis using mathematical models are rigorously studied.

MODEL FORMULATION
Human to human brucellosis transmission is possible as indicated in various studies including [16,33,34,35,36]. The possible modes of human to human brucellosis transmission are transplancental, breastfeeding, sexual, blood transfusion and organ transplantation [37]. In this section, we formulate a deterministic mathematical model for the transmission dynamics of brucellosis in domestic small ruminants, cattle and human populations. The model we formulate includes: direct transmission of brucellosis within the cattle, within small ruminants, within humans and from livestock to human, and from the environment to livestock and humans.
Furthermore, susceptible cattle and small ruminants are either vaccinated at some points (pulse vaccination) or remain susceptible. Based on the epidemiological status of individuals, the cattle population at any time t is divided into vaccinated V c (t) , susceptible S c (t) , and infectious I c (t) classes. Similarly, the small ruminant population at any time t is divided into vaccinated V s (t) , susceptible S s (t) , and infectious I s (t) subpopulations while the total human population, N h (t) at any time t is divided into susceptible, S h (t), infected, I h (t) and recovered, R h (t) individuals. Susceptible cattle become infected through direct contact with infected cattle at the rate of β c or through contact with the contaminated environment (indirect transmission) at the rate α c while susceptible small ruminants become infected when they are in contact with infectious small ruminants at the rate of β s or through contact with the contaminated environment at the rate α s . The transmission to humans is expressed as additive contributions of transmissions from infective humans, cattle, small ruminants and contaminated environment.
Appertaining to the fact that it is very difficult to determine the quantity of brucella in environment, we define the average number of brucella that is enough for a host to be infected with brucellosis as an infectious unit and let B(t) to be the number of infectious units in the environment. The incubation period for brucellosis is hardly detected, but individuals at this period can infect the susceptible individuals at the same transmission rate as the infectious individual and discharge the same quantity of brucella into the environment per unit time as in [28]. It is against this background, we assume that individuals in the incubation period and post incubation period are hosted in the same population compartment called infectious class. The interaction within and between the four populations prompts that veterinary surgeons, laboratory assistants, and farmers are predominantly exposed to the brucella bacteria.

Model Assumptions.
In formulation of the model we make the following assumptions: i. The mixing of individuals in each population is homogeneous; ii. There is no direct transmission between cattle and small ruminants; iii. Infected animals shed brucella pathogens in the environment; iv. Livestock seropositivity is life-long lasting; v. Immunized individuals cannot be infected unless their resistance to infection wanes; vi. There is constant natural mortality rate in each of the species; vii. The birth rate for each population is greater than natural mortality rate.
The variables and parameters used in this model are respectively summarized in TABLE 1 and   TABLE 2.

Model Equations. Based on the assumptions and the inter-relations between the vari-
ables and the parameters shown in FIGURE 1, the transmission dynamics of Brucellosis can be described by the following ordinary differential equations: where, (1) with nonnegative initial data remain nonnegative for all time t ≥ 0. The model system (1) can be expressed in the compact form as: and F is a column vector given by It can be noticed that AX is Meltzer matrix since all of its off diagonal entries are non negative, for all X ∈ R 10 + . Therefore, using the fact that F > 0, the model system (1) is positively invariant in R 10 + , which means that an arbitrary trajectory of the system starting in R 10 + remains in R 10 + forever. In addition, the right hand F is Lipschitz continuous. Thus, a unique maximal solution exists and so: is the feasible region for the model (1). Thus, the model (1) is epidemiologically and mathematically well-posed in the region Ω.

MODEL ANALYSIS
4.1. Disease Free Equilibrium. The Brucellosis free equilibrium point is obtained by setting the right hand side of equations in model system (1) to zero, that is: Let the disease free equilibrium point of Brucellosis model be E 0 . In case there is no disease I c = I s = I h = B = 0 that is, the sum of susceptible and vaccinated populations is equal to total population. There exists a disease free equilibrium (1) where:

The Effective Reproduction Number.
In this subsection, we compute the effective reproduction number for model system (1) using the standard method of the next generation matrix developed in [38,39]. The effective reproduction number, R e is defined as the measure of average number of infections caused by a single infectious individual introduced in a community in which intervention strategies are administered [40]. The magnitude of the effective reproduction number is used to indicate both the risk of an epidemic and effort required to control an infection. When there are no interventions or controls, the number of secondary infections caused by typical infected individual during his entire period of infectiousness is called basic reproduction number, R 0 . Moreover, due to the natural history of some infections, transmissibility is better quantified by the effective reproduction number rather than the basic reproduction number [42]. Considering the system for the infective variables: The effective reproduction number is obtained by taking the spectral radius of the next generation matrix: where E 0 is the brucellosis-free equilibrium point while F i and V i are vectors representing respectively, the rate of appearance of new infection in compartment i and the transfer of infections from compartment i to another, such that: It is important to note that V i is a resultant vector of the two vectors: V + i , defined as the rate of transfer of individuals into compartment i by all other means(e.g births and immigration); and V − i , which is the rate of transfer of individuals out of compartment i (e.g deaths, recovery and emigration). In particular: The Jacobian matrices F of F i and V of V i evaluated at E 0 are respectively: Referring to the infected states with indices i and j, for i, j ∈ [1, 2, 3, 4], the entry F i j is the rate at which individuals in infected state j give rise or produce new infections to individuals in infected state i, in the linearized system. Thus, when there is no new cases produced in infected state i by an individual in infected state j immediately after infection, we have F i j = 0. The inverse of V is found to be: The entry V −1 i j is the average length of time an infected individual spends in compartment j during its lifetime when introduced into the compartment i of disease free equilibrium, assuming that the population remains near the disease free equilibrium and barring reinfection. is the probability that an infected small ruminant will shed brucella into the environment. Moreover, the Next Generation Matrix is calculated to be: where, has a nonnegative eigenvalue. The non-negative eigenvalue is associated with a non-negative eigenvector which represents the distribution of infected individuals that produces the greatest number R e of secondary infections per generation [44]. Thus, the spectral radius for our Next Generation Matrix is: where, The first and the second expressions of equation (7) represents respectively the effective reproduction numbers in the livestock and human populations. It can further be noticed that, the first expression which is independent of the human population represents the threshold transmission dynamics of brucellosis in the cattle and small ruminants populations that was analyzed and discussed in [45]. The fact that human brucellosis significantly reduces work performance of individuals calls for a special interest of investigating the transmission dynamics and controls of human brucellosis. Thus, we focus on brucellosis transmission dynamics within the human population. The effective reproduction number within the human population is found to be: When there is no treatment, σ = 0, we have the within human basic reproduction number which is given by: Besides, brucellosis is a zoonosis; it is transmitted to human from animals, referring to our particular case in the next generation matrix (6) the cattle to human effective reproduction number is intuitively given by: On the other hand, the small ruminants to human effective reproduction number is given by: Moreover, equation (4) indicates that, the transmission of brucellosis in the human population results from human to human transmission, small ruminants to human transmission, cattle to human transmission and environment to human transmission. Thus, if it happens one infected cattle, one infected small ruminant and one infected human are simultaneously introduced in the human population, then the effective human reproduction number is intuitively given by:

Local Stability of the Disease Free Equilibrium.
In this subsection we use the tracedeterminant method to investigate the local stability of the brucellosis free equilibrium point. Proof. We show that, variational matrix J(E 0 ) of the brucellosis model at DFE has a negative trace and positive determinant. The Jacobian matrix for system (1) is given by: where, The trace of the Jacobian matrix J(E 0 )is given by: Thus, the trace of the Jocobian matrix is the less than zero, that is Tr(J(E 0 )) < 0 if: Furthermore, the determinant of matrix J(E 0 ) is computed using Maple 16 Software and is found to be: where, and The determinant of the Jacobian matrix is positive (i.e. J(E 0 ) > 0) if:

Global Stability of the Disease-Free Equilibrium.
In this section, we analyze the global stability of the disease-free equilibrium point by applying the [47] approach. We write model system (1) in the form: where X s is the vector representing the non-transmitting compartments and X i is the vector representing the transmitting components. The DFE is globally asymptotically stable if A has real negative eigenvalues and A 2 is a Metzler matrix (i.e. the off-diagonal elements of A 2 are non-negative). From model system (1) we have: We need to check whether a matrix A for the non-transmitting compartments has real negative eigenvalues and that A 2 is a Metzler matrix. From the equation for non-transmitting compartments in (1) we have: Appertaining the fact that all model parameters and variables are non-negative, it is evident that Proof. We construction an explicit Lyapunov function for model system (1) using [48,49,50,51,52] approach as it is useful to most of the sophisticated compartmental epidemiological models. In this approach, we construct Lyapunov functions of the form: where a i is a properly selected positive constant, x i is the population of the i th compartment and x * i is the equilibrium level. We define the Lyapunov function candidate V for model system (1) as: where A 1 , A 2 , A 3 , A 4 , A 5 , A 6 and A 7 are positive constants. The time derivative of the Lyapunov function L is given by: where, Equation (12) can be written as: where, F is the balance of the right hand terms of equation (12). Following the approach of [29,48,49,51,50,52], F is a non-positive function for S c ,V c , I c , S s ,V s , I s , S h , I h , R h , B ≥ 0.
Therefore, if R e > 1, model system (1) has a unique endemic equilibrium point E * which is globally asymptotically stable.

NUMERICAL SIMULATIONS
This section presents numerical simulations of model system (1)   Similarly, from FIGURE 3a we see that effective environmental hygiene and sanitation controls the transmission route of brucellosis from contaminated environment to human population. However, the ruminants to human effective reproduction number does not reduce to less than unit due to the fact that direct contact between infective cattle or small ruminants is not effectively controlled. In addition FIGURE 3b illustrates that, human treatment has a significant  On the other hand, FIGURE 4b illustrates that, recovered humans increases with the increase in treatment rate. This implies that, in order to minimize or eliminate the prevalence of brucellosis in the human population, measures should be taken to control the disease in animals as well as eliminating the disease in humans through treatment.

CONCLUSION
This paper aimed at formulating and analyzing a mathematical model to investigate the impacts of different control parameters to the transmission dynamics of brucellosis in the human and animal populations. We focused on livestock vaccination, gradual culling of ruminants through slaughter, environmental hygiene and sanitation, and human treatment. Analytical solutions as well as numerical simulations reveals that human brucellosis can be prevented or controlled only if the prevalence in both ruminants and humans can be controlled. Moreover, prevention of human brucellosis largely depends on prevention of the disease in domestic animals. In view of that, the effective control of brucellosis needs cooperation between public health and animal health sectors.

CONFLICT OF INTERESTS
The author(s) declare that there is no conflict of interests.