Mathematical Modeling of the Transmission Dynamics of Gumboro Disease

. Gumboro disease is a viral poultry disease that causes immune suppression on the infected birds leading to poor production, mortality, and exposure to secondary infections, hence a major threat in the poultry industry worldwide. A mathematical model of the transmission dynamics of Gumboro disease is developed in this paper having four compartments of chicken population and one compartment of Gumboro pathogen population. The basic reproduction number R og is derived, and the dynamical behaviors of both the disease-free equilibrium (DFE) and endemic equilibrium are analyzed using the ordinary di ﬀ erential equation theory. From the analysis, we found that the system exhibits an asymptotic stable DFE whenever R og < 1 and an asymptotic stable EE whenever R og > 1 . The numerical simulation to verify the theoretical results was carried out using MATLAB ode45 solver, and the results were found to be consistent with the theoretical ﬁ ndings.


Introduction
Infectious bursal disease (IBD) popularly known as Gumboro is a viral poultry diseases that cause high morbidity and mortality, hence a major threat to the poultry industry due to high economic losses associated with it worldwide.Gumboro disease is associated with clinical disease symptoms such as depression, watery diarrhea, ruffled feathers, and dehydration [1].
Gumboro disease affects mostly young chickens around 3-6 weeks of age.Gumboro virus is extremely difficult to eradicate as it is hardy and can live in a great range of environmental conditions, and it is transmitted from one bird to another through feacal-oral route [2].
Generally, the poultry sector plays an important role in the growth of the economy as well as in poverty reduction.According to the Agricultural Sector Development Strategy 2010, in Kenya, each year, about 20 tonnes of poultry meat worth 3.5 billion Kenyan shillings and 1.3 billion eggs worth 9.7 billion Kenyan shillings are produced.This increased production of poultry is necessitated by the increased demand for quality protein especially in developing countries [3].
The IBD virus was observed 40 years ago with Kenya's first case reported in 1991 in commercial birds on the Kenyan coast; the disease has remained to be a great threat to the commercial poultry industry not only in Kenya but also in the whole world [4].
Mathematical modeling over the years has become a very important tool that is used in the prediction, assessment, and control of various outbreaks.A number of these models that describe the impacts of preventive and control strategies on the transmission dynamics of various poultry infectious diseases have been developed.A study to investigate the impacts of quarantine and vaccination in controlling avian influenza disease was done by [5], and the results established that combining quarantine with vaccination is an effective strategy for control of the disease.
A mathematical model of Newcastle disease with optimal control having five compartments was formulated by [6].The findings of the study showed that in the absence of control measures, the number of infected bird increased significantly and reduced significantly in the presence of control measure implying that the control measures were effective methods of controlling the disease.
Several mathematical models have been done to describe the dynamics of Gumboro disease; for instance, [7] formulated a model to investigate the impact of the environment in the spread of Gumboro infections while [8] described a model to describe the effects of vaccination and biosecurity measures in controlling Gumboro infection.
Although several Gumboro models have been developed, to the best of our knowledge, a Gumboro model with pathogen compartment has not been developed.In this paper, we formulate a Gumboro model capturing the pathogen compartment to study the dynamics of Gumboro disease.

Model Formulation
A model with Gumboro pathogen population N g and the chicken population N c is developed in this research.Thus, the total population at a given time t is N t = N c t + N g t .Gumboro pathogen population N g has one compartment consisting of concentration of Gumboro virus in the environment C V .The chicken population N c is grouped into four compartments which consist of susceptible chicken S, birds that are at early stages of infection with Gumboro E g , birds that are in acute stages of infection with Gumboro I g , and those that will recover from Gumboro disease in both early and acute stages of infections simultaneously R. The model assumes that the bird population recruitment rate to the susceptible compartment will be Λ.The susceptible birds are infected with Gumboro at the rate of ωC V /τ 2 + C V .
Where ω is the contact rate of susceptible birds with an IBD virus-contaminated environment, τ 2 is the IBD virus concentration in the environment with a 50% probability of Gumboro infections.All bird populations experience natural death at the rate η.Additionally, they die from Gumboro at the rate of μ.The Gumboro-infected birds in both stages of infection shed the virus to the environment at the rates ω 1−2 which die at the rate α 2 , while σ g and δ g are recovery rates for birds at both early and acute stages of infection.Birds at the early stages of Gumboro infections move to the acute stages of infection at the rates ϕ g .

2.
1. Model Assumptions.The model assumptions are as follows: 1.The bird species that are infected with Gumboro is chicken.
2. Chickens are recruited into the system by birth or immigration.

Model Flow Chart and Equations.
From Figure 1, the following equations are developed:

Basic Properties of the Model
In this section, we discuss the positivity and boundedness of the solutions of the model.

Positivity of the Solutions of the Model
Theorem 1.There exists a nonnegative solution set S, E g , I g , R, C V t of model ( 1) for all t > 0 given that the initial conditions S 0 > 0, E g 0 ≥ 0, From the first equation of system (1), we have Equation (2) can be expressed as Separating the variables, we obtain Integrating on both sides of Equation (4), we have ln From Equation (5), it is clear that S 0 = c for t = 0. Therefore, S t > S 0 e − γ g +η t and as t ⟶ ∞, we have S t > 0∀t > 0.
Also from the second equation of system (1) Solving the above equation by separation of variables, we have Upon integrating on both sides, we have ln E g t ≥ − σ g + ϕ g + η + μ t + c: ⇒ln E g t > ce − σ g +ϕ g +η+μ t 6 Clearly for t = 0, c = E g 0 .Thus, Equation ( 6) becomes E t ≥ E g 0 e − σ g +ϕ g +η+μ t and as t ⟶ ∞, we have Applying the same method to other equations of the system (1), we get

Boundedness of the Solutions of the Model
+ be a feasible region in which the solutions of the total population are bounded, where Ω C is the feasible region of the solutions of the bird population and that of the Gumboro pathogen population is given by Ω C V .We show that the solutions of the system (1) are bounded in the feasible region.The total bird population is N C given by Thus, from system (1), we have When the birds are not infected, Equation ( 7) reduces to Upon solving Equation ( 8), we get And taking limits as t ⟶ ∞, we have Thus, the bird population is bounded in

Journal of Applied Mathematics
Considering the last equation of system (1), that is Equation (10), we have Upon reduction of Equation ( 11), we have By using the integrating factor, we solve Equation ( 12) to get Taking the limit of Equation ( 13), as t tends to infinity, gives N g ≤ Λ ω 1 + ω 2 /ηα 2 .Thus, the Gumboro population is bounded in the region Since the bird population and Gumboro pathogen population are bounded, then the model will be analyzed in a suitable feasible region

Analysis of the Model
and setting the righthand side of the equations of the system (1) equal to zero, then solving the resulting system of equations.Hence, we get Reproduction Number for Gumboro Model.The reproduction number R 0g is described as the average number of secondary cases that result from an average initial case in a completely susceptible population in [9].In our situation, the reproduction number R 0g is the average number of secondary cases resulting from a typical Gumboro infection case in a completely uninfected population.R 0g is found using the Next Generation Matrix approach by [10].Let the rates of new infections in class j be denoted by f j , while the rates of chicken transfers into and out of class j are rep-resented by v j .The Next Generation Matrix is given by FV −1 , where F and V are the Jacobian matrices of the vectors f j and v j , respectively, at E * g .The following equations capture the infected population.
From the system (14), we have Also, from the system (14), we have By definition of F and V, we have Using Mathematica software, the inverse of V is given by Journal of Applied Mathematics Thus which implies that the basic reproduction number, R 0g , for the Gumboro model is given by The DFE E * g is locally asymptotically stable if all the real parts of the eigenvalues of the Jacobian matrix of the system (1) at the DFE E * g are all negative.
The Jacobian matrix of the system (1) at the DFE E * g is given by From the Jacobian matrix (15), we have the following characteristic equation: In view of Equation ( 16), λ 1 = −η and λ 2 = −η are the eigenvalues of the Jacobian matrix (15).The other three eigenvalues can be obtained from the following reduced characteristic equation: where It is clear that According to Routh-Hurwitz criteria, Equation (17) has roots with negative real parts if a 0 > 0, a 1 > 0, a 2 > 0, a 3 > 0 and a 1 a 2 − a 0 a 3 > 0. Thus, the DFE of the system of Equation (1) is locally asymptotically stable if the following theorem holds.
Theorem 2. The DFE of the system of Equation ( 1) is locally asymptotically stable when R 0g < 1 and unstable otherwise.

Global Stability of DFE.
In this section, we use the Castillo-Chavez theorem in [11] to analyse the disease-free state's global asymptotic stability.To begin, the system (1) must be expressed in the following format: where X g = S, R and Z g = E g , I g , C V .Uninfected individuals are represented by the components of X g ∈ ℝ, while infected ones are represented by the components of Z g ∈ ℝ.
The system's DFE now becomes E * g = X * , 0 , X * g = Λ/η , 0 .The following two conditions must be met to provide global asymptotic stability.
1. dX g /dt = F X g , 0 , X g is globally asymptotically stable (GAS) where G = D Z g H X * g , 0 is an M-matrix (the off-diagonal components of G are nonnegative) and Ω represents the region where the model makes biological sense.The following theorem holds if the system (18) meets the aforementioned two conditions.Theorem 3. The DFE E * g = X * g , 0 is a GAS equilibrium of system (18) provided that R 0g < 1 and the assumptions in Equation ( 19) are met.Proof 2. From Theorem 2, the DFE (E * g ) is locally asymptotically stable when R 0g < 1.Consider From Equation (20), we have and it follows that Thus, the first and the second conditions in Equation ( 19) are satisfied since and H X g , Z g ≥ 0, respectively.Therefore, E * g has a global asymptotic stability V and setting the right-hand side of the equations of the system (1) equal to zero, we get Explicitly solving for values of S * * ,E * * g ,I * * g ,R * * ,C * * V , we obtain 6 Journal of Applied Mathematics 24 Thus, a positive

Local Stability of Endemic Equilibrium.
We use the center manifold theory as described in [12] to assess the stability of the endemic equilibrium, E * * g , because evaluating the eigenvalues of the Jacobian matrix of system (1) at the endemic equilibrium is complicated.Theorem 4.1 in [12] outlines the procedure of analyzing the local stability of endemic equilibrium in a nutshell.The coefficients, a and b of the normal form, are two key quantities in expressing the system's dynamics on the center manifold theory as described in Theorem 4.1 in [12].According to part (iv) of Theorem 4.1 in [12], if a < 0 and b > 0, the endemic equilibrium E * * g is locally asymptotically stable for R 0g > 1 but close to 1.

Theorem 5. The endemic equilibrium E * *
g is locally asymptotically stable if R 0g > 1 Proof 4. By using the center manifold theory, Theorem 4.1 in [12], we rename the variables in the system (1) as S = x 1 , T .Further, by using X = x 1 , x 2 , x 3 , x 4 , x 5 T , the system (1) can be written in the form dX/dt = F X , with T , as follows: Suppose that ω = ω * is a bifurcation parameter when R 0g = 1, solving for ω * for R 0g = 1 from we have The Jacobian matrix of the system (25) at E * g with ω = ω * is given as The Jacobian matrix (27) has zero eigenvalues close to ω = ω * ; thus, the center manifold theory is utilized to examine the dynamics of the system.Let w = w 1 , w 2 , w 3 , w 4 , w 5 T , a right eigenvector associated with the Jacobian matrix (Equation (27)), be close to ω = ω * , then Solving system (28), we get T , a left eigenvector associated with the Jacobian matrix (Equation ( 27)), be close to ω = ω * , such that Solving system (30), we obtain Using the formula described in [12], we compute a and b: To get the bifurcation coefficient a, we first obtain the nonzero partial derivatives of the model system (25) evaluated at E * g , ω * .Thus, it is evident that Using Equations ( 29) and (31), Equation (33) can be written as For the coefficient b, we have the following nonzero partial derivatives of the model system (25) evaluated at E * g , ω * : Using Equations ( 29) and (31), Equation (35) can be written as Since a < 0 and b > 0, it follows that the endemic equilibrium E * * g is locally asymptotically stable if R 0g > 1.

Global Stability of the Endemic Equilibrium Point
Theorem 6.The endemic equilibrium point E * * g of the system (1) is GAS if R 0g > 1. g is investigated [13].It follows from Dulac's multiplier, 1/SE g I g RC V , that Due to the fact that Ω is positively invariant and the endemic equilibrium exists whenever R 0g > 1, there are no

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Journal of Applied Mathematics periodic orbits in Ω according to Dulac's criterion.From the Poincaré-Bendixson theorem, it follows that all solutions of the limiting system originating in Ω remain in Ω for all time t.The absence of periodic orbits suggests that whenever R 0g > 1, the special endemic equilibrium of the Gumboro model is GAS.

Numerical Simulation
Numerical simulations to prove the theoretical results for the Gumboro disease mathematical model were carried out using MATLAB ode 45 solver.This was made possible by the use of some parameters in the literature and others that were estimated or assumed as shown in Table 1.
From Figure 2, it was shown that the number of birds at early stages of infection, acute stages of infection, and the pathogen population converges to zero while the susceptible birds' population tends to a constant λ/η when R 0g < 1 which implies that whenever Gumboro disease dies out, only the susceptible chicken would remain.Figure 3 also shows that birds at early and acute stages of infection together with the pathogen population tend towards the endemic equilibrium point when R 0g > 1 while the susceptible birds converge to zero indicating that Gumboro disease remains endemic.From Figure 4, it was shown that birds at early stages of infection increase with the increase in the contact rate of susceptible to Gumboro disease-contaminated environment.

Conclusion
A mathematical model of Gumboro disease transmission dynamics was formulated in this paper.The disease-free and endemic equilibrium points were determined, and the reproduction number was derived.The findings showed that Gumboro disease dies out whenever R 0g < 1 and persists in the chicken population whenever R 0g > 1.Also, it was realized that minimizing the contact rate of chicken to a contaminated environment lowers cases of infections in a population.These numerical simulation findings were found to be in harmony with the theoretical stability analysis results.

4. 1 .
Disease-Free Equilibrium (DFE) Point.The DFE of the system (1) is computed by letting