# Modeling the Transmission Dynamics of Bovine Tuberculosis

Authors: Theresia Shirima Sabini; Jacob Ismail Irunde; Dmitry Kuznetsov
Journal: International Journal of Mathematics and Mathematical Sciences
 (2020)

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## Abstract

Bovine tuberculosis (bTB) is a bacterial and zoonotic disease which is transmitted through consumption of unpasteurized milk and uncooked meat and inhalation of aerosols. In this paper, a deterministic mathematical model is formulated to study the transmission dynamics of bTB in humans and animals. The basic reproduction numberR0 is computed to determine the behavior of the disease. Stability analysis shows that there is a possibility for disease-free equilibrium and endemic equilibrium to coexist when R0 = 1. To determine parameters which drive the dynamics of bTB, we performed sensitivity analysis. The analysis shows that the rate at which dairy products are produced, the rate of transmission of bTB from animal to animal, and the rates at which human acquires bTB from infectious dairy products and animals drive the transmission of bTB. However, the disease decreases as the rate of consumption of dairy products decreases. To control bTB, education campaign, inspection of dairy products, treatment of infected humans, and quarantine of infected animals are recommended.


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## Body

## 1. Introduction

Tuberculosis (TB) is a global health problem which is among the top ten diseases which are leading in causing death for many people [1]. World Health Organization (WHO) has conducted several meetings to discuss strategies on how to end TB but the disease is still claiming lives. According to WHO report of 2018, TB kills 1.3 million people among HIV negative, and there was an addition of 300, 000 people with HIV positive who died with TB. It is estimated that annually 10 million people get ill with TB of which 5.8 million are men, 3.2 million are women, and 1.0 million are children [1]. Africa is reported to have highest number of cases, followed by India, China, and Indonesia with percentage 72%, 27%, 9%, and 8%, respectively [1]. Although TB is a worldwide health problem, other forms of TB such as zoonotic tuberculosis are still neglected.Bovine tuberculosis (bTB) is a bacterial and zoonotic disease, which was originally transmitted to cattle from wild animals especially buffalo and badger, and then spread to other domestic animals like cows, goats, pigs, horses, and sheep [2]. The disease has a great negative economic impact due to slaughtering of infectious animals when they acquire bTB [3]. Also, bTB causes human health problems which sometimes cost lives. It can lead to loss of self-employment for some workers, especially those who depend on livestock keeping as their main source of income [4]. Bovine tuberculosis is transmitted to humans through three main ways, which are, consumption of unpasteurized milk, eating uncooked meat, and inhalation of aerosols [5, 6]. In animals, bTB is transmitted when there is close interaction between uninfected animal and infectious animals [7] and consumption of infectious milk especially during breastfeeding and inhalation of aerosols [8].It is estimated that about 147,000 new cases of bTB in humans were reported, whereby 12,000 people die annually due this disease [2]. In Tanzania, the disease prevalence varies from region to region depending on the number of livestock in a particular place and it ranges from 0.2%–13.3% [6, 9]. The problem of not having surveillance data makes the estimation to be poor and difficult [3]. Places where bTB exist, includes Northern Tanzania (Arusha, Kilimanjaro, and Manyara), dairy farms in Kibaha, and some areas in Morogoro districts [3, 6].Early diagnosis of bTB helps to know the dynamics of the disease and identify ways of controlling the transmission factors before it becomes endemic. Various tools and methods are used to diagnose bTB and some of them include Polymerase chain reaction (PCR) and gene sequencing of culture isolate which is an expensive tool [10, 11], post mortem examination which focused on lymph nodes [12], and single itradermal comparative cervical test (SICCT) [13]. From external examination it shows that there is high possibility of carcasses from slaughtered cattle to contain bTB pathogens [14].Various organizations such as WHO, OIE, and FAO joined together to fight against transmission of bTB in order to eradicate the disease [2]. Though different approaches such as “one health approach, together we can save lives, and secure livelihoods” are used, bTB is still a problem especially in some African countries. Though spread of bTB has been controlled by treating livestock using various medicines, such as pyrazinamide medicine, unfortunately, bTB is still resistant to pyrazinamide medicine because it is used to treat patients with pulmonary TB [2]. Many studies have ignored the role of dairy products in the spread of bTB. This study aims to find the parameters which drive the dynamics of bTB and explore the role played by dairy products in the transmission dynamics of bTB.Studies such as Agusto et al. [15], Liu et al. [16], Mathews et al. [11], and Phepa [17] developed mathematical models to study the transmission dynamics of bTB and its control strategies. Most of these studies did not consider the impacts of dairy products in the transmission of bTB although the products are among the factors which drive the transmission of bTB since they are consumed by large number of people worldwide [18, 19]. These products are among the factors which lead in the spread of bTB as some of the findings reported the products to contain Ramos et al. [20]. This study uses a mathematical model to study transmission dynamics of bTB in human beings and livestock by considering dairy products as a risk factor for the transmission of bTB.This work is organized as follows: materials and methods are presented in the Section2 followed by model analysis, sensitivity analysis, and numerical simulation, and it is concluded with conclusion and recommendation.


## 2. Materials and Methods

### 2.1. Model Formulation

The model is formulated by modifying the tuberculosis model for human and cows in Urumqi, Xinjiang, China which was developed by Liu et al. [16]. The model divides cow population into three groups: Susceptible Sc, Infected Ic, and Quarantine Qc (SIQ) and human population into four groups: Susceptible Sh, Exposed Eh, Infected Ih, and Recovery Rh (SEIR).Our current model includes animal and human populations. Human population is divided into susceptibleSH, exposed EH, and infected IH (SEI) and animal population is divided into susceptible Sa, exposed Ea, and infected Ia compartments. The variable D represents diary products which are produced by infected animals. The proposed model does not include recovery class because it is assumed that there is no natural recovery [21].Susceptible humans are recruited through birth and migration at a rate ΛH, and they acquire bovine tuberculosis latent infection following contacts with infectious human and animals and after consuming dairy products from infectious animals at a rate(1)λH=β1IH+β2Ia+β3DNH.Exposed compartmentEH increases the following latent infection of susceptible humans SH at a rate λH and it decreases due to progression to infectious stage at a rate γH. Infectious humans IH increase at a rate γH and diminish due to disease-induced mortality at a rate αH. All human compartments suffer natural mortality at a rate μH.Susceptible animalsSa are recruited through birth and migration at a rate Λa and acquire bovine tuberculosis latent infection following contacts with infectious humans and animals and after consuming dairy products at a rate(2)λa=β4IH+β5Ia+β6DNa.Exposed animalsEa increase following latent infection of susceptible animals Sa at a rate λa. However, they decrease due to progression to the infectious stage at a rate γa. Infectious animals Ia increase at a rate γa and diminish due to disease-induced mortality at a rate αa. All animal compartments suffer natural mortality at a rate μa.Dairy products are produced by infectious animals at a rateρ and leak out at a rate ω, and susceptible humans consumed dairy products at a rate β3 and susceptible animals at a rate β6.In the model we assume that all humans and animals are susceptible to the disease. Susceptible humanSH contacts bTB when they consume dairy products D such as milk and meat from infected animals; when they inhale aerosols from infected animals and human and have direct contact with dairy product from infected animals through scratches [5]. Susceptible animal acquires infection when they interact with infectious animals and humans, through breastfeeding from infectious animals and inhalation of aerosols. There is constant natural death to both animals and human beings. There is no natural recovery for infected individuals.Figure1 demonstrates the interaction of state variables; Tables 1 and 2 describe state variables and parameters, respectively.Figure 1
Model flow diagram.Table 1
Model variables description.
Symbol
Description
S
H(t)
Number of susceptible human at timet
S
a(t)
Number of susceptible animal at timet
E
H(t)
Number of exposed/latent human beings at timet
E
a(t)
Number of exposed animals at timet
I
H(t)
Number of infected human at timet
I
a(t)
Number of infected animals at timet
D(t)
Dairy products at timetTable 2
Parameters’ descriptions.
Parameter
Descriptions
ΛH
Human recruitment rate
μ
H
Human natural death
γ
H
Progression rate fromEH to IH
α
H
Human death rate due to disease induced
β
1, β2, β3
Humans infection rate fromIH, Ia, and D, respectively
Λa
Animal recruitment rate
μ
a
Animal natural death rate
γ
a
Progression rate fromEa to Ia
α
a
Animals disease induced mortality
ρ
Dairy products production rate
ω
Rate of decaying for unconsumed dairy products
β
4, β5, β6
Animals infection rate fromIH, Ia, and D, respectively


### 2.2. Model Equations

From Figure1, we have the following system of differential equations:(3a)dSHdt=ΛH−β1IH+β2Ia+β3DNHSH−μHSH,(3b)dEHdt=β1IH+β2Ia+β3DNHSH−γH+μHEH,(3c)dIHdt=γHEH−μH+αHIH,(3d)dSadt=Λa−β4IH+β5Ia+β6DNaSa−μaSa,(3e)dEadt=β4IH+β5Ia+β6DNaSa−γa+μaEa,(3f)dIadt=γaEa−μa+αaIa,(3g)dDdt=ρIa−ωD,subject to their initial conditions: SH(0) = 0, EH(0) = 0, IH(0) = 0, Sa(0) = 0, Ea(0) = 0, Ia(0) = 0, and D(0) = 0.


## 2.1. Model Formulation

The model is formulated by modifying the tuberculosis model for human and cows in Urumqi, Xinjiang, China which was developed by Liu et al. [16]. The model divides cow population into three groups: Susceptible Sc, Infected Ic, and Quarantine Qc (SIQ) and human population into four groups: Susceptible Sh, Exposed Eh, Infected Ih, and Recovery Rh (SEIR).Our current model includes animal and human populations. Human population is divided into susceptibleSH, exposed EH, and infected IH (SEI) and animal population is divided into susceptible Sa, exposed Ea, and infected Ia compartments. The variable D represents diary products which are produced by infected animals. The proposed model does not include recovery class because it is assumed that there is no natural recovery [21].Susceptible humans are recruited through birth and migration at a rate ΛH, and they acquire bovine tuberculosis latent infection following contacts with infectious human and animals and after consuming dairy products from infectious animals at a rate(1)λH=β1IH+β2Ia+β3DNH.Exposed compartmentEH increases the following latent infection of susceptible humans SH at a rate λH and it decreases due to progression to infectious stage at a rate γH. Infectious humans IH increase at a rate γH and diminish due to disease-induced mortality at a rate αH. All human compartments suffer natural mortality at a rate μH.Susceptible animalsSa are recruited through birth and migration at a rate Λa and acquire bovine tuberculosis latent infection following contacts with infectious humans and animals and after consuming dairy products at a rate(2)λa=β4IH+β5Ia+β6DNa.Exposed animalsEa increase following latent infection of susceptible animals Sa at a rate λa. However, they decrease due to progression to the infectious stage at a rate γa. Infectious animals Ia increase at a rate γa and diminish due to disease-induced mortality at a rate αa. All animal compartments suffer natural mortality at a rate μa.Dairy products are produced by infectious animals at a rateρ and leak out at a rate ω, and susceptible humans consumed dairy products at a rate β3 and susceptible animals at a rate β6.In the model we assume that all humans and animals are susceptible to the disease. Susceptible humanSH contacts bTB when they consume dairy products D such as milk and meat from infected animals; when they inhale aerosols from infected animals and human and have direct contact with dairy product from infected animals through scratches [5]. Susceptible animal acquires infection when they interact with infectious animals and humans, through breastfeeding from infectious animals and inhalation of aerosols. There is constant natural death to both animals and human beings. There is no natural recovery for infected individuals.Figure1 demonstrates the interaction of state variables; Tables 1 and 2 describe state variables and parameters, respectively.Figure 1
Model flow diagram.Table 1
Model variables description.
Symbol
Description
S
H(t)
Number of susceptible human at timet
S
a(t)
Number of susceptible animal at timet
E
H(t)
Number of exposed/latent human beings at timet
E
a(t)
Number of exposed animals at timet
I
H(t)
Number of infected human at timet
I
a(t)
Number of infected animals at timet
D(t)
Dairy products at timetTable 2
Parameters’ descriptions.
Parameter
Descriptions
ΛH
Human recruitment rate
μ
H
Human natural death
γ
H
Progression rate fromEH to IH
α
H
Human death rate due to disease induced
β
1, β2, β3
Humans infection rate fromIH, Ia, and D, respectively
Λa
Animal recruitment rate
μ
a
Animal natural death rate
γ
a
Progression rate fromEa to Ia
α
a
Animals disease induced mortality
ρ
Dairy products production rate
ω
Rate of decaying for unconsumed dairy products
β
4, β5, β6
Animals infection rate fromIH, Ia, and D, respectively


## 2.2. Model Equations

From Figure1, we have the following system of differential equations:(3a)dSHdt=ΛH−β1IH+β2Ia+β3DNHSH−μHSH,(3b)dEHdt=β1IH+β2Ia+β3DNHSH−γH+μHEH,(3c)dIHdt=γHEH−μH+αHIH,(3d)dSadt=Λa−β4IH+β5Ia+β6DNaSa−μaSa,(3e)dEadt=β4IH+β5Ia+β6DNaSa−γa+μaEa,(3f)dIadt=γaEa−μa+αaIa,(3g)dDdt=ρIa−ωD,subject to their initial conditions: SH(0) = 0, EH(0) = 0, IH(0) = 0, Sa(0) = 0, Ea(0) = 0, Ia(0) = 0, and D(0) = 0.


## 3. Model Analysis

To determine whether the model is mathematically and epidemiologically meaningful, we find invariant region and test positivity of solutions. The model is biologically and mathematically meaningful, when its solutions are positive and bounded.


### 3.1. Invariant Region

Invariant region shows the feasibility of the model solutions. To study the feasibility of the model solutions, we denote humans and animal populations byNH and Na, respectively.Beginning with human population, we have(4)NH=SH+EH+IH,dNHdt=ΛH−μHNH−αHIH.Solving (4), we obtain(5)NH0≤ΛHμH+NH0−ΛHμHe−μHt.Analysis ofNH consider two cases(6)whenNH0>ΛHμHand whenNH0<ΛHμH.For all two cases, we have(7)NHt≤ΛHμH≤ΛHμH+NH0−ΛHμHe−μHt,NHt≤ΛHμH+NH0−ΛHμHe−μHt≤ΛHμH.Ast ⟶ ∞,(8)0≤NH≤ΛHμH.When we apply the same procedure for animals population, ast ⟶ ∞,(9)0≤Na≤Λaμa.SinceNa≤Λa/μa, then Ia≤Λa/μa.For the case of dairy products, whent ⟶ ∞ we have(10)Dt≤Λaμaρω.Therefore, we can see that model (3) is a positive invariant in the region(11)Z=SH,EH,IH,Sa,Ea,Ia,D∈R+7:0≤NH≤ΛHμH;0≤Na≤Λaμa;0≤D≤Λaμaρω.


#### 3.1.1. Positivity of Solution

Theorem 1.
Let the initial variable of model (3) beSH(0) > 0, EH(0) > 0, IH(0) > 0, Sa(0) > 0, Ea(0) > 0, Ia(0) > 0, and D > 0; then, the solutions SH > 0, EH > 0, IH > 0, Sa > 0, Ea > 0, Ia, and D > 0 are positive ∀t > 0.Proof:.
Let us consider equation (3a) of the model system (3), which is(12)dSHdt=ΛH−β1IH+β2Ia+β3DNHSH−μHSH,dSHdt≥−β1IH+β2Ia+β3DNHSH−μHSH.
By separating variables (12) and integrating, we obtain(13)dSHSH≥−β1IH+β2Ia+β3DNH+μHdt,∫dSHSH≥−∫0tβ1IHs+β2Ias+β3DsNHs+μHds,lnSH≥−∫0tβ1IHs+β2Ias+β3DsNHs+μHds+C,SH≥Ce∫0t−β1IHs+β2Ias+β3DsNHs+μHds.
At initial condition, we obtain(14)SH≥SH0e∫0t−β1IHs+β2Ias+β3DsNHs+μHds.
Following the same approach for the remaining variables (3b)–(3g) of the model system (3), we obtain(15)EH≥EH0e−γH+μHt,IH≥CeμH+αHt,Sa≥Sa0e∫0t−β4IHs+β5Ias+β6DsNas+μads,Ea≥Ea0e−γa+μat,Ia≥Ia0e−μa+αat,D≥D0e−ωt.
Since(16)SH>0,EH>0,IH>0,Sa>0,Ea>0,Ia>0andD>0,then the model solutions are positive ∀t > 0.
Model (3) is mathematically and epidemiologically meaningful; therefore, we can consider the flow generated by the model for analysis.


### 3.2. Disease-Free Equilibrium (DFE)

The disease-free equilibrium point is the state when there is no disease in the population. When there is no bTB in human and animal populations, the disease-free equilibrium is given by(17)DF0=SH,EH,IH,Sa,Ea,Ia,D=ΛHμH,0,0,Λaμa,0,0,0.


#### 3.2.1. The Basic Reproduction NumberR0

The basic reproduction number refers to the average number of new cases that a single infectious individual causes when introduced into an entirely susceptible population [22]. It determines whether the disease persists or clears out in the population. When the basic reproduction number R0 < 1, the disease clears out in the population and it persists if the basic reproduction number R0 > 1. This is because when an infectious individual is introduced into an entirely susceptible population, he/she infects more than one individual, hence the disease persists [23, 24].To compute the basic reproduction numberR0, we use the next generation matrix method where new infections and transfer terms are considered [22, 24]. If bTB new infectious and transfer terms are denoted by Fi and Vi, respectively, then the basic reproduction number R0 is given as the maximum eigenvalue. That is,(18)R0=ρFV−1,where(19)F=∂Fi∂XjDF0andV=∂Vi∂XjDF0.From the model system (3), the basic reproduction numberR0 is given by(20)R0=β1γH/μH+γHμH+αH+γaωβ5+ρβ6/γa+μaμa+αaω2+β1γH/μH+γHμH+αH−γaωβ5+ρβ6/γa+μaμa+αaω2+4ce4,where ce=γHγaωβ2+ρβ3γH+μHαH+μHγa+μaω.The terms1/γH+μH and 1/γa+μa in (20) are the average periods an individual human and animal spends in their corresponding exposed classes, 1/αH+μH and 1/αa+μa are the average periods an infectious human and animal spend in their infectious classes, β1γH/γH+μHαH+μH and β4γH/γH+μHαH+μH are the proportions of infected humans that develop bTB and move from exposed class to infectious class after contacting infectious humans and animals, respectively, ωβ5+ρβ6γa/γa+μaαa+μaω is the sum of proportions of infected animals that progress from exposed class to infectious class after coming into contact with infectious animals and after consuming infectious dairy products, and γHγaωβ2+ρβ3/γH+μHαH+μHγa+μaαa+μaω is the sum of proportions of infected humans who develop bTB by contacting infectious animals and after consuming infectious dairy products.


### 3.3. Sensitivity Analysis of Basic Reproduction Number (R0)

Sensitivity analysis ofR0 helps to understand the effect of each parameter on the model output and their influence in the spread of disease in the population [25, 26]. We perform sensitivity analysis of R0 by using normalized forward sensitivity analysis index as used by Chitnis et al. [27] and Silva and Torres [26]. A normalized forward sensitivity index of variable β with respect to basic reproduction number R0 is defined as(21)ΥβR0=∂R0∂β×βR0.Using estimated parameters and from related literature, sensitivity index of each parameter with respect to basic reproduction numberR0 is computed and summarized in Table 3.Table 3
Sensitivity indices forR0.
Parameter
Index value
β
1
0.0271
β
2
0.0530
β
3
0.1177
β
4
0.1708
β
5
0.3601
β
6
0.2713
γ
H
0.0892
μ
H
−0.2728
α
H
−0.0144
γ
a
0.1671
α
a
−0.5793
μ
a
−0.3898
ρ
0.3890
ω
−0.3890Sensitivity analysis shows that human infection rates due to consumption of dairy productsβ3 and contact rate with infected animals β2, animal infection rates due to contact with infectious animals β5, and consumption of diary product β6 drive the dynamics of bTB. Generally, the most sensitive parameter is the rate of producing dairy products ρ. As dairy products increases by 10%, the basic reproduction number R0 increases by 3.89%. However, when animal mortality rate due to disease αa, natural death rate for animals μa, human disease-induced death rate αH, natural death rate for humans μH, and dairy products decaying rate ω increases, and the basic reproduction number R0 decrease consequently.


### 3.4. Stability Analysis for Disease-Free Equilibrium (DFE)

To determine local stability of disease-free equilibrium, we use the linearization method where trace and determinant are used. Disease-free equilibrium is said to be locally asymptotically stable if the eigenvalues of matrix (22) are negative or have a negative real part. Linearization of the model system (3) gives the matrix(22)J=−μH0−β100−β2−β30−μH−γHβ100β2β30γH−μH−αH000000−β4−μa0−β5−β600β40−μa−γaβ5β60000γa−μa−αa000000ρ−ω.Matrix (22) has negative eigenvalues −μH and −μa. Matrix (22) now reduces to(23)K=−μH−γHβ10β2β3γH−μH−αH0000β4−μa−γaβ5β600γa−μa−αa0000ρ−ω.We analyze matrixK by using trace tr and determinant det. Disease-free equilibrium is locally stable if trace is negative tr(K) < 0 and determinant is positive det(K) > 0. From (23), trace of the matrix K is given by(24)trK=−μH+γH+μH+γH+μa+γa+μa+αa+ω,trK<0,and det(K) is given by(25)detK=γH+μHαH+μHωβ5γa+γH+μHμH+αHρβ6γa+γa+μaαa+μaωβ1γH+ωβ2β4γHγa+ρβ3β4γHγa−ρβ1β6γHγa−γH+μHαH+μHγa+μaαa+μaω−ωβ1β5γHγa,det(K) > 0 if(26)β1γHμH+γHμH+αH+γaωβ5+ρβ6ωγa+μaμa+αa+β4γHωβ2γa+ρβ3γaωγH+μHαH+μHγa+μaαa+μa−γHγaβ1ωβ5+ρβ6ωγH+μHαH+μHγa+μaαa+μa>1.Theorem 1.
The disease-free equilibrium is locally asymptotically stable if condition (26) holds and R0 < 1.
However, the disease-free equilibrium may not be globally asymptotically stable due to the possibility of model (3) to undergo backward bifurcation whenR0 = 1. We analyze backward bifurcation in Section 3.5.


### 3.5. Bifurcation Analysis

To determine the possibility of model (3) to undergo backward bifurcation, we rename the state variablesSH, EH, IH, Sa, Ea, Ia, and D to be x1, x2, x3, x4, x5, x6, and x7, respectively, where NH = x1 + x2 + x3 and Na = x4 + x5 + x7. By introducing the vector notations X=x1,x2,x3,x4,x5,x6,x7T, the model system (3) is now written as dX/dt=FX, where FX=f1,f2,f3,f4,f5,f6,f7T. The model system (3) is then rewritten as follows:(27)dx1dt=f1=ΛH−β1x3+β2x6+β3x7x1+x2+x3x1−μHx1,dx2dt=f2=β1x3+β2x6+β3x7x1+x2+x3x1−γH+μHx2,dx3dt=f3=γHx2−μH+αHx3,dx4dt=f4=Λa−β4x3+β5x6+β6x7x4+x5+x6x4−μax4,dx5dt=f5=β4x3+β5x6+β6x7x4+x5+x6x4−γa+μax5,dx6dt=f6=γax5−μa+αax6,dx7dt=f7=ρx6−μDx7.The Jacobian of system (27) at disease-free equilibrium is given by(28)J=−μH0−β100−β2−β30−μH−γHβ100β2β30γH−μH−αH000000−β4−μa0−β5−β600β40−μa−γaβ5β60000γa−μa−αa000000ρ−ω.To determine whether system (27) undergoes backward bifurcation at R0 = 1, we adopt Theorem 1 in Castillo-Chavez and Song [28], and it is restated as follows.Theorem 2.
Consider the following general system of ordinary differential equations with a parameterβ∗.dx/dt=fx,β∗,f:ℜ×ℜn↦ℜn, and f∈C2ℜn×ℜ, where 0 is an equilibrium point of the system (that is, f(0, β∗) ≡ 0 ∀β∗ and(1)
A=Dxf0,0=∂fi/∂xj0,0 is a linearization matrix of the system around the equilibrium 0 with β∗ at 0.(2)
Zero is a simple eigenvalue of A and all other eigenvalues of A have negative real parts.(3)
Matrix A has a right eigenvectorsw and left eigenvectors v corresponding to the zero eigenvalues.
Letfk be the kth component of f and(29)a=∑k,i,j=1nvkwiwj∂2fk∂xi∂xj0,0,b==∑k,i=1nvkwi∂2fk∂xi∂β∗0,0.Then, the local dynamics of the system around the equilibrium point is totally determined by the signs ofa and b. Particularly, if a > 0 and b > 0 then a backward bifurcation occurs at β∗ = 0.The local dynamics at (27) around 0 are totally determined by signs of a and b.(i)
a > 0 and b > 0. When β∗ < 0 with |β∗| << 1, 0 is locally asymptotically stable, and there exists a positive unstable equilibrium; when 0 < β∗ << 1, 0 is unstable and there exists a negative and locally asymptotically stable equilibrium.(ii)
a < 0 and b < 0. When β∗ < 0 with |β∗| << 1, 0 is unstable; when 0 < β∗ << 1, 0 is locally asymptotically stable, and there exists a positive unstable equilibrium.(iii)
a > 0 and b < 0. When β∗ < 0 with |β∗| << 1, 0 is unstable, and there exists a locally asymptotically stable negative equilibrium; when 0 < β∗ << 1, 0 is stable, and a positive unstable equilibrium appears.(iv)
a < 0 and b > 0. When β∗ changes from negative to positive, 0 changes its stability from stable to unstable. Correspondingly, a negative unstable equilibrium becomes positive and locally asymptotically stable.Letβ3 = β∗ be a bifurcation parameter when R0 = 1. Now, solving for β3 = β∗ when R0 = 1, we obtain(30)β3=β∗=M2−β4γHγH+μHαH+μH−γaρβ6+ωβ2γa+μaαa+μa2+Mβ1γHγH+μHαH+μH−γaρβ6+ωβ2γa+μaαa+μa2.where(31)M=γa+μaαa+μaγH+μHαH+μHωγHγaρβ4.From (28), right eigenvectors w=wiT, where i = 1, 2, …, 7 are(32)w5=ωαa+μaw7γaρ,w6=ωw7ρ,w3=ωγa+μaαa+μa−γaωβ5+ρβ6γaρβ4w7,w4=ωγa+μaαa+μa−2ωβ5+ρβ6γaγaμaρw7,w1=−ωM1+ωβ5+ρβ6+γaβ4ωβ2+ρβ3β4γaμHρw7,w2=ωγa+μaαa+μa−γaωβ5+ρβ6αH+μHγHγaβ4ρw7,for whichw7>0is free right eigenvector,whereM1=γa+μaαa+μa.The left eigenvectorsv=viT, where i = 1, 2, …, 7 are(33)v1=v4=0,v2=γHv3γH+μH,v5=γH+μHαH+μH−γHβ2β4γH+μHv3,v6=γa+μaγH+μHαH+μH−γHβ2γaβ4γH+μHv3,v7=γHβ3β4+β6γH+μHαH+μH−γHβ2ωγH+μHβ4v3,for whichv3>0is free left eigenvector.


#### 3.5.1. Computation ofa

From the model system (4), the associated nonzero partial derivatives of F at disease free equilibrium are given by(34)∂2f2∂x32=−2β1μHΛH,∂2f2∂x2∂x3=−β1μHΛH,∂2f2∂x2∂x6=−β2μHΛH,∂2f2∂x2∂x7=−β∗μHΛH,∂2f2∂x3∂x6=−β2μHΛH,∂2f2∂x3∂x7=−β∗μHΛH,∂2f5∂x62=−2β5μaΛa,∂2f5∂x3∂x6=−β4μaΛa,∂2f5∂x5∂x6=−β5μaΛa,∂2f5∂x5∂x7=−β6μaΛa,∂2f5∂x6∂x7=−β6μaΛa,∂2f5∂x3∂x5=−2β4μaΛa.Sincev1=v2=0, it follows that(35)a=v2∑i,j=1nwiwj∂2f2∂xi∂xj+v5∑i,j=1nwiwj∂2f5∂xi∂xj.To computea, we substitute the partial derivatives into (35) to obtain(36)a=2μHΛHw3M2β1γaωβ5+ρβ6ΛHγHΛaγaρβ4w7v2−2μHΛaw3M1β1ωM1+γaβ4ωβ2+ρβ3ΛHγHΛaγaρβ4w7v2−2ΛHγHωβ4μaw5+w6M1ΛHγHΛaγaρβ4w7v5,whereM2=αH+μH+γH.To analyze the sign ofa, we consider two cases.Case I:a < 0 if(37)γaωβ5+ρβ6ωγa+μaαa+μa<1ΛHγHμaωβ4w5+w6M1v5+ΛaμHw3ωM1β1+ωβ2+ρβ3γaβ4M1v2ΛHγaμHM2β1ωβ5+ρβ6w3v2>1.Case II:a > 0 if(38)γaωβ5+ρβ6ωγa+μaαa+μa>1ΛHγHμaωβ4w5+w6M1v5+ΛaμHw3ωM1β1+ωβ2+ρβ3γaβ4M1v2ΛHγaμHM2β1ωβ5+ρβ6w3v2<1.


#### 3.5.2. Computation ofb

Recall from (33), since v1=v4=0 b becomes(39)b=v2∑i=1nwi∂2fk∂xi∂β∗0,0,b=v2w7∂2f2∂x7∂β∗,b=γHw7v3γH+μH>0.From the computation ofa and b, we can establish the following results.Theorem 3.
If condition (38) holds, bTB undergoes backward bifurcation at R0 = 1. This implies that if a > 0, b > 0 when β∗ < 0, DFE is locally asymptotically stable and there exist a positive unstable equilibrium and when β∗ > 0, and DFE is unstable and there exist a negative and locally asymptotically stable equilibrium.


## 3.1. Invariant Region

Invariant region shows the feasibility of the model solutions. To study the feasibility of the model solutions, we denote humans and animal populations byNH and Na, respectively.Beginning with human population, we have(4)NH=SH+EH+IH,dNHdt=ΛH−μHNH−αHIH.Solving (4), we obtain(5)NH0≤ΛHμH+NH0−ΛHμHe−μHt.Analysis ofNH consider two cases(6)whenNH0>ΛHμHand whenNH0<ΛHμH.For all two cases, we have(7)NHt≤ΛHμH≤ΛHμH+NH0−ΛHμHe−μHt,NHt≤ΛHμH+NH0−ΛHμHe−μHt≤ΛHμH.Ast ⟶ ∞,(8)0≤NH≤ΛHμH.When we apply the same procedure for animals population, ast ⟶ ∞,(9)0≤Na≤Λaμa.SinceNa≤Λa/μa, then Ia≤Λa/μa.For the case of dairy products, whent ⟶ ∞ we have(10)Dt≤Λaμaρω.Therefore, we can see that model (3) is a positive invariant in the region(11)Z=SH,EH,IH,Sa,Ea,Ia,D∈R+7:0≤NH≤ΛHμH;0≤Na≤Λaμa;0≤D≤Λaμaρω.


### 3.1.1. Positivity of Solution

Theorem 1.
Let the initial variable of model (3) beSH(0) > 0, EH(0) > 0, IH(0) > 0, Sa(0) > 0, Ea(0) > 0, Ia(0) > 0, and D > 0; then, the solutions SH > 0, EH > 0, IH > 0, Sa > 0, Ea > 0, Ia, and D > 0 are positive ∀t > 0.Proof:.
Let us consider equation (3a) of the model system (3), which is(12)dSHdt=ΛH−β1IH+β2Ia+β3DNHSH−μHSH,dSHdt≥−β1IH+β2Ia+β3DNHSH−μHSH.
By separating variables (12) and integrating, we obtain(13)dSHSH≥−β1IH+β2Ia+β3DNH+μHdt,∫dSHSH≥−∫0tβ1IHs+β2Ias+β3DsNHs+μHds,lnSH≥−∫0tβ1IHs+β2Ias+β3DsNHs+μHds+C,SH≥Ce∫0t−β1IHs+β2Ias+β3DsNHs+μHds.
At initial condition, we obtain(14)SH≥SH0e∫0t−β1IHs+β2Ias+β3DsNHs+μHds.
Following the same approach for the remaining variables (3b)–(3g) of the model system (3), we obtain(15)EH≥EH0e−γH+μHt,IH≥CeμH+αHt,Sa≥Sa0e∫0t−β4IHs+β5Ias+β6DsNas+μads,Ea≥Ea0e−γa+μat,Ia≥Ia0e−μa+αat,D≥D0e−ωt.
Since(16)SH>0,EH>0,IH>0,Sa>0,Ea>0,Ia>0andD>0,then the model solutions are positive ∀t > 0.
Model (3) is mathematically and epidemiologically meaningful; therefore, we can consider the flow generated by the model for analysis.


## 3.1.1. Positivity of Solution

Theorem 1.
Let the initial variable of model (3) beSH(0) > 0, EH(0) > 0, IH(0) > 0, Sa(0) > 0, Ea(0) > 0, Ia(0) > 0, and D > 0; then, the solutions SH > 0, EH > 0, IH > 0, Sa > 0, Ea > 0, Ia, and D > 0 are positive ∀t > 0.Proof:.
Let us consider equation (3a) of the model system (3), which is(12)dSHdt=ΛH−β1IH+β2Ia+β3DNHSH−μHSH,dSHdt≥−β1IH+β2Ia+β3DNHSH−μHSH.
By separating variables (12) and integrating, we obtain(13)dSHSH≥−β1IH+β2Ia+β3DNH+μHdt,∫dSHSH≥−∫0tβ1IHs+β2Ias+β3DsNHs+μHds,lnSH≥−∫0tβ1IHs+β2Ias+β3DsNHs+μHds+C,SH≥Ce∫0t−β1IHs+β2Ias+β3DsNHs+μHds.
At initial condition, we obtain(14)SH≥SH0e∫0t−β1IHs+β2Ias+β3DsNHs+μHds.
Following the same approach for the remaining variables (3b)–(3g) of the model system (3), we obtain(15)EH≥EH0e−γH+μHt,IH≥CeμH+αHt,Sa≥Sa0e∫0t−β4IHs+β5Ias+β6DsNas+μads,Ea≥Ea0e−γa+μat,Ia≥Ia0e−μa+αat,D≥D0e−ωt.
Since(16)SH>0,EH>0,IH>0,Sa>0,Ea>0,Ia>0andD>0,then the model solutions are positive ∀t > 0.
Model (3) is mathematically and epidemiologically meaningful; therefore, we can consider the flow generated by the model for analysis.


## 3.2. Disease-Free Equilibrium (DFE)

The disease-free equilibrium point is the state when there is no disease in the population. When there is no bTB in human and animal populations, the disease-free equilibrium is given by(17)DF0=SH,EH,IH,Sa,Ea,Ia,D=ΛHμH,0,0,Λaμa,0,0,0.


### 3.2.1. The Basic Reproduction NumberR0

The basic reproduction number refers to the average number of new cases that a single infectious individual causes when introduced into an entirely susceptible population [22]. It determines whether the disease persists or clears out in the population. When the basic reproduction number R0 < 1, the disease clears out in the population and it persists if the basic reproduction number R0 > 1. This is because when an infectious individual is introduced into an entirely susceptible population, he/she infects more than one individual, hence the disease persists [23, 24].To compute the basic reproduction numberR0, we use the next generation matrix method where new infections and transfer terms are considered [22, 24]. If bTB new infectious and transfer terms are denoted by Fi and Vi, respectively, then the basic reproduction number R0 is given as the maximum eigenvalue. That is,(18)R0=ρFV−1,where(19)F=∂Fi∂XjDF0andV=∂Vi∂XjDF0.From the model system (3), the basic reproduction numberR0 is given by(20)R0=β1γH/μH+γHμH+αH+γaωβ5+ρβ6/γa+μaμa+αaω2+β1γH/μH+γHμH+αH−γaωβ5+ρβ6/γa+μaμa+αaω2+4ce4,where ce=γHγaωβ2+ρβ3γH+μHαH+μHγa+μaω.The terms1/γH+μH and 1/γa+μa in (20) are the average periods an individual human and animal spends in their corresponding exposed classes, 1/αH+μH and 1/αa+μa are the average periods an infectious human and animal spend in their infectious classes, β1γH/γH+μHαH+μH and β4γH/γH+μHαH+μH are the proportions of infected humans that develop bTB and move from exposed class to infectious class after contacting infectious humans and animals, respectively, ωβ5+ρβ6γa/γa+μaαa+μaω is the sum of proportions of infected animals that progress from exposed class to infectious class after coming into contact with infectious animals and after consuming infectious dairy products, and γHγaωβ2+ρβ3/γH+μHαH+μHγa+μaαa+μaω is the sum of proportions of infected humans who develop bTB by contacting infectious animals and after consuming infectious dairy products.


## 3.2.1. The Basic Reproduction NumberR0

The basic reproduction number refers to the average number of new cases that a single infectious individual causes when introduced into an entirely susceptible population [22]. It determines whether the disease persists or clears out in the population. When the basic reproduction number R0 < 1, the disease clears out in the population and it persists if the basic reproduction number R0 > 1. This is because when an infectious individual is introduced into an entirely susceptible population, he/she infects more than one individual, hence the disease persists [23, 24].To compute the basic reproduction numberR0, we use the next generation matrix method where new infections and transfer terms are considered [22, 24]. If bTB new infectious and transfer terms are denoted by Fi and Vi, respectively, then the basic reproduction number R0 is given as the maximum eigenvalue. That is,(18)R0=ρFV−1,where(19)F=∂Fi∂XjDF0andV=∂Vi∂XjDF0.From the model system (3), the basic reproduction numberR0 is given by(20)R0=β1γH/μH+γHμH+αH+γaωβ5+ρβ6/γa+μaμa+αaω2+β1γH/μH+γHμH+αH−γaωβ5+ρβ6/γa+μaμa+αaω2+4ce4,where ce=γHγaωβ2+ρβ3γH+μHαH+μHγa+μaω.The terms1/γH+μH and 1/γa+μa in (20) are the average periods an individual human and animal spends in their corresponding exposed classes, 1/αH+μH and 1/αa+μa are the average periods an infectious human and animal spend in their infectious classes, β1γH/γH+μHαH+μH and β4γH/γH+μHαH+μH are the proportions of infected humans that develop bTB and move from exposed class to infectious class after contacting infectious humans and animals, respectively, ωβ5+ρβ6γa/γa+μaαa+μaω is the sum of proportions of infected animals that progress from exposed class to infectious class after coming into contact with infectious animals and after consuming infectious dairy products, and γHγaωβ2+ρβ3/γH+μHαH+μHγa+μaαa+μaω is the sum of proportions of infected humans who develop bTB by contacting infectious animals and after consuming infectious dairy products.


## 3.3. Sensitivity Analysis of Basic Reproduction Number (R0)

Sensitivity analysis ofR0 helps to understand the effect of each parameter on the model output and their influence in the spread of disease in the population [25, 26]. We perform sensitivity analysis of R0 by using normalized forward sensitivity analysis index as used by Chitnis et al. [27] and Silva and Torres [26]. A normalized forward sensitivity index of variable β with respect to basic reproduction number R0 is defined as(21)ΥβR0=∂R0∂β×βR0.Using estimated parameters and from related literature, sensitivity index of each parameter with respect to basic reproduction numberR0 is computed and summarized in Table 3.Table 3
Sensitivity indices forR0.
Parameter
Index value
β
1
0.0271
β
2
0.0530
β
3
0.1177
β
4
0.1708
β
5
0.3601
β
6
0.2713
γ
H
0.0892
μ
H
−0.2728
α
H
−0.0144
γ
a
0.1671
α
a
−0.5793
μ
a
−0.3898
ρ
0.3890
ω
−0.3890Sensitivity analysis shows that human infection rates due to consumption of dairy productsβ3 and contact rate with infected animals β2, animal infection rates due to contact with infectious animals β5, and consumption of diary product β6 drive the dynamics of bTB. Generally, the most sensitive parameter is the rate of producing dairy products ρ. As dairy products increases by 10%, the basic reproduction number R0 increases by 3.89%. However, when animal mortality rate due to disease αa, natural death rate for animals μa, human disease-induced death rate αH, natural death rate for humans μH, and dairy products decaying rate ω increases, and the basic reproduction number R0 decrease consequently.


## 3.4. Stability Analysis for Disease-Free Equilibrium (DFE)

To determine local stability of disease-free equilibrium, we use the linearization method where trace and determinant are used. Disease-free equilibrium is said to be locally asymptotically stable if the eigenvalues of matrix (22) are negative or have a negative real part. Linearization of the model system (3) gives the matrix(22)J=−μH0−β100−β2−β30−μH−γHβ100β2β30γH−μH−αH000000−β4−μa0−β5−β600β40−μa−γaβ5β60000γa−μa−αa000000ρ−ω.Matrix (22) has negative eigenvalues −μH and −μa. Matrix (22) now reduces to(23)K=−μH−γHβ10β2β3γH−μH−αH0000β4−μa−γaβ5β600γa−μa−αa0000ρ−ω.We analyze matrixK by using trace tr and determinant det. Disease-free equilibrium is locally stable if trace is negative tr(K) < 0 and determinant is positive det(K) > 0. From (23), trace of the matrix K is given by(24)trK=−μH+γH+μH+γH+μa+γa+μa+αa+ω,trK<0,and det(K) is given by(25)detK=γH+μHαH+μHωβ5γa+γH+μHμH+αHρβ6γa+γa+μaαa+μaωβ1γH+ωβ2β4γHγa+ρβ3β4γHγa−ρβ1β6γHγa−γH+μHαH+μHγa+μaαa+μaω−ωβ1β5γHγa,det(K) > 0 if(26)β1γHμH+γHμH+αH+γaωβ5+ρβ6ωγa+μaμa+αa+β4γHωβ2γa+ρβ3γaωγH+μHαH+μHγa+μaαa+μa−γHγaβ1ωβ5+ρβ6ωγH+μHαH+μHγa+μaαa+μa>1.Theorem 1.
The disease-free equilibrium is locally asymptotically stable if condition (26) holds and R0 < 1.
However, the disease-free equilibrium may not be globally asymptotically stable due to the possibility of model (3) to undergo backward bifurcation whenR0 = 1. We analyze backward bifurcation in Section 3.5.


## 3.5. Bifurcation Analysis

To determine the possibility of model (3) to undergo backward bifurcation, we rename the state variablesSH, EH, IH, Sa, Ea, Ia, and D to be x1, x2, x3, x4, x5, x6, and x7, respectively, where NH = x1 + x2 + x3 and Na = x4 + x5 + x7. By introducing the vector notations X=x1,x2,x3,x4,x5,x6,x7T, the model system (3) is now written as dX/dt=FX, where FX=f1,f2,f3,f4,f5,f6,f7T. The model system (3) is then rewritten as follows:(27)dx1dt=f1=ΛH−β1x3+β2x6+β3x7x1+x2+x3x1−μHx1,dx2dt=f2=β1x3+β2x6+β3x7x1+x2+x3x1−γH+μHx2,dx3dt=f3=γHx2−μH+αHx3,dx4dt=f4=Λa−β4x3+β5x6+β6x7x4+x5+x6x4−μax4,dx5dt=f5=β4x3+β5x6+β6x7x4+x5+x6x4−γa+μax5,dx6dt=f6=γax5−μa+αax6,dx7dt=f7=ρx6−μDx7.The Jacobian of system (27) at disease-free equilibrium is given by(28)J=−μH0−β100−β2−β30−μH−γHβ100β2β30γH−μH−αH000000−β4−μa0−β5−β600β40−μa−γaβ5β60000γa−μa−αa000000ρ−ω.To determine whether system (27) undergoes backward bifurcation at R0 = 1, we adopt Theorem 1 in Castillo-Chavez and Song [28], and it is restated as follows.Theorem 2.
Consider the following general system of ordinary differential equations with a parameterβ∗.dx/dt=fx,β∗,f:ℜ×ℜn↦ℜn, and f∈C2ℜn×ℜ, where 0 is an equilibrium point of the system (that is, f(0, β∗) ≡ 0 ∀β∗ and(1)
A=Dxf0,0=∂fi/∂xj0,0 is a linearization matrix of the system around the equilibrium 0 with β∗ at 0.(2)
Zero is a simple eigenvalue of A and all other eigenvalues of A have negative real parts.(3)
Matrix A has a right eigenvectorsw and left eigenvectors v corresponding to the zero eigenvalues.
Letfk be the kth component of f and(29)a=∑k,i,j=1nvkwiwj∂2fk∂xi∂xj0,0,b==∑k,i=1nvkwi∂2fk∂xi∂β∗0,0.Then, the local dynamics of the system around the equilibrium point is totally determined by the signs ofa and b. Particularly, if a > 0 and b > 0 then a backward bifurcation occurs at β∗ = 0.The local dynamics at (27) around 0 are totally determined by signs of a and b.(i)
a > 0 and b > 0. When β∗ < 0 with |β∗| << 1, 0 is locally asymptotically stable, and there exists a positive unstable equilibrium; when 0 < β∗ << 1, 0 is unstable and there exists a negative and locally asymptotically stable equilibrium.(ii)
a < 0 and b < 0. When β∗ < 0 with |β∗| << 1, 0 is unstable; when 0 < β∗ << 1, 0 is locally asymptotically stable, and there exists a positive unstable equilibrium.(iii)
a > 0 and b < 0. When β∗ < 0 with |β∗| << 1, 0 is unstable, and there exists a locally asymptotically stable negative equilibrium; when 0 < β∗ << 1, 0 is stable, and a positive unstable equilibrium appears.(iv)
a < 0 and b > 0. When β∗ changes from negative to positive, 0 changes its stability from stable to unstable. Correspondingly, a negative unstable equilibrium becomes positive and locally asymptotically stable.Letβ3 = β∗ be a bifurcation parameter when R0 = 1. Now, solving for β3 = β∗ when R0 = 1, we obtain(30)β3=β∗=M2−β4γHγH+μHαH+μH−γaρβ6+ωβ2γa+μaαa+μa2+Mβ1γHγH+μHαH+μH−γaρβ6+ωβ2γa+μaαa+μa2.where(31)M=γa+μaαa+μaγH+μHαH+μHωγHγaρβ4.From (28), right eigenvectors w=wiT, where i = 1, 2, …, 7 are(32)w5=ωαa+μaw7γaρ,w6=ωw7ρ,w3=ωγa+μaαa+μa−γaωβ5+ρβ6γaρβ4w7,w4=ωγa+μaαa+μa−2ωβ5+ρβ6γaγaμaρw7,w1=−ωM1+ωβ5+ρβ6+γaβ4ωβ2+ρβ3β4γaμHρw7,w2=ωγa+μaαa+μa−γaωβ5+ρβ6αH+μHγHγaβ4ρw7,for whichw7>0is free right eigenvector,whereM1=γa+μaαa+μa.The left eigenvectorsv=viT, where i = 1, 2, …, 7 are(33)v1=v4=0,v2=γHv3γH+μH,v5=γH+μHαH+μH−γHβ2β4γH+μHv3,v6=γa+μaγH+μHαH+μH−γHβ2γaβ4γH+μHv3,v7=γHβ3β4+β6γH+μHαH+μH−γHβ2ωγH+μHβ4v3,for whichv3>0is free left eigenvector.


### 3.5.1. Computation ofa

From the model system (4), the associated nonzero partial derivatives of F at disease free equilibrium are given by(34)∂2f2∂x32=−2β1μHΛH,∂2f2∂x2∂x3=−β1μHΛH,∂2f2∂x2∂x6=−β2μHΛH,∂2f2∂x2∂x7=−β∗μHΛH,∂2f2∂x3∂x6=−β2μHΛH,∂2f2∂x3∂x7=−β∗μHΛH,∂2f5∂x62=−2β5μaΛa,∂2f5∂x3∂x6=−β4μaΛa,∂2f5∂x5∂x6=−β5μaΛa,∂2f5∂x5∂x7=−β6μaΛa,∂2f5∂x6∂x7=−β6μaΛa,∂2f5∂x3∂x5=−2β4μaΛa.Sincev1=v2=0, it follows that(35)a=v2∑i,j=1nwiwj∂2f2∂xi∂xj+v5∑i,j=1nwiwj∂2f5∂xi∂xj.To computea, we substitute the partial derivatives into (35) to obtain(36)a=2μHΛHw3M2β1γaωβ5+ρβ6ΛHγHΛaγaρβ4w7v2−2μHΛaw3M1β1ωM1+γaβ4ωβ2+ρβ3ΛHγHΛaγaρβ4w7v2−2ΛHγHωβ4μaw5+w6M1ΛHγHΛaγaρβ4w7v5,whereM2=αH+μH+γH.To analyze the sign ofa, we consider two cases.Case I:a < 0 if(37)γaωβ5+ρβ6ωγa+μaαa+μa<1ΛHγHμaωβ4w5+w6M1v5+ΛaμHw3ωM1β1+ωβ2+ρβ3γaβ4M1v2ΛHγaμHM2β1ωβ5+ρβ6w3v2>1.Case II:a > 0 if(38)γaωβ5+ρβ6ωγa+μaαa+μa>1ΛHγHμaωβ4w5+w6M1v5+ΛaμHw3ωM1β1+ωβ2+ρβ3γaβ4M1v2ΛHγaμHM2β1ωβ5+ρβ6w3v2<1.


### 3.5.2. Computation ofb

Recall from (33), since v1=v4=0 b becomes(39)b=v2∑i=1nwi∂2fk∂xi∂β∗0,0,b=v2w7∂2f2∂x7∂β∗,b=γHw7v3γH+μH>0.From the computation ofa and b, we can establish the following results.Theorem 3.
If condition (38) holds, bTB undergoes backward bifurcation at R0 = 1. This implies that if a > 0, b > 0 when β∗ < 0, DFE is locally asymptotically stable and there exist a positive unstable equilibrium and when β∗ > 0, and DFE is unstable and there exist a negative and locally asymptotically stable equilibrium.


## 3.5.1. Computation ofa

From the model system (4), the associated nonzero partial derivatives of F at disease free equilibrium are given by(34)∂2f2∂x32=−2β1μHΛH,∂2f2∂x2∂x3=−β1μHΛH,∂2f2∂x2∂x6=−β2μHΛH,∂2f2∂x2∂x7=−β∗μHΛH,∂2f2∂x3∂x6=−β2μHΛH,∂2f2∂x3∂x7=−β∗μHΛH,∂2f5∂x62=−2β5μaΛa,∂2f5∂x3∂x6=−β4μaΛa,∂2f5∂x5∂x6=−β5μaΛa,∂2f5∂x5∂x7=−β6μaΛa,∂2f5∂x6∂x7=−β6μaΛa,∂2f5∂x3∂x5=−2β4μaΛa.Sincev1=v2=0, it follows that(35)a=v2∑i,j=1nwiwj∂2f2∂xi∂xj+v5∑i,j=1nwiwj∂2f5∂xi∂xj.To computea, we substitute the partial derivatives into (35) to obtain(36)a=2μHΛHw3M2β1γaωβ5+ρβ6ΛHγHΛaγaρβ4w7v2−2μHΛaw3M1β1ωM1+γaβ4ωβ2+ρβ3ΛHγHΛaγaρβ4w7v2−2ΛHγHωβ4μaw5+w6M1ΛHγHΛaγaρβ4w7v5,whereM2=αH+μH+γH.To analyze the sign ofa, we consider two cases.Case I:a < 0 if(37)γaωβ5+ρβ6ωγa+μaαa+μa<1ΛHγHμaωβ4w5+w6M1v5+ΛaμHw3ωM1β1+ωβ2+ρβ3γaβ4M1v2ΛHγaμHM2β1ωβ5+ρβ6w3v2>1.Case II:a > 0 if(38)γaωβ5+ρβ6ωγa+μaαa+μa>1ΛHγHμaωβ4w5+w6M1v5+ΛaμHw3ωM1β1+ωβ2+ρβ3γaβ4M1v2ΛHγaμHM2β1ωβ5+ρβ6w3v2<1.


## 3.5.2. Computation ofb

Recall from (33), since v1=v4=0 b becomes(39)b=v2∑i=1nwi∂2fk∂xi∂β∗0,0,b=v2w7∂2f2∂x7∂β∗,b=γHw7v3γH+μH>0.From the computation ofa and b, we can establish the following results.Theorem 3.
If condition (38) holds, bTB undergoes backward bifurcation at R0 = 1. This implies that if a > 0, b > 0 when β∗ < 0, DFE is locally asymptotically stable and there exist a positive unstable equilibrium and when β∗ > 0, and DFE is unstable and there exist a negative and locally asymptotically stable equilibrium.


## 4. Numerical Simulation

In this section, we discuss the dynamics of bTB in humans and animal population by considering parameters which drive the transmission dynamics of bTB. We use estimated parameters and some from the related literature as summarized in Table4.Table 4
Parameter values of the model system3.1.
Parameter
Interpretation
Valueyr−1
Source
ΛH
Human recruitment rate
36
[16]
β
1
Human infection rate from infected human
0.35
Estimated
β
2
Human infection rate from infected animals
0.55
[29]
β
3
Human infection rate from infected dairy products
0.999
Estimated
β
4
Rate of cow infected via human
0.25
Estimated
β
5
Rate of cow infected via animal
0.6
[15]
Λa
Animal recruitment rate
200
[15]
β
6
Rate of animals infected via dairy products
0.34
Estimated
μ
H
Human natural death rate
0.01
[16]
α
H
Human death rate due to disease induced
0.139
[16]
γ
a
Progression rate fromEa to Ia
0.18
[30]
α
a
Animal death due to disease induced
0.0304
[15]
γ
H
Progression rate fromEH to IH
0.18
Estimated
μ
a
Animal natural death rate
0.05
Estimated
ρ
Dairy production rate
0.69
Estimated
ω
Rate of decaying dairy products
0.4
EstimatedSusceptible humans and animals decrease after acquiring bTB when they come into contact with infectious humans and animals and after consuming infectious dairy products, as shown in Figure2. However, infectious classes increase as individuals from susceptible class acquire bTB and move to exposed class and then to infectious class.Figure 2
Dynamics of bTB in (a) human population and (b) animal population.
(a)
(b)Figures3(a) and 3(b) show the variations of susceptible humans and animals as rates of consuming dairy products increase. As dairy products from infectious animals increase, their consumption increases thus increasing the bTB infection rate for susceptible humans and animals.Figure 3
Variation of the dairy production rate into susceptible (a) humans and (b) animals with change in time.
(a)
(b)As we increase the rate of producing infectious dairy products, susceptible classes decrease and infectious classes increase. This is due to the fact that the more dairy products are produced the more susceptible humans and animals consume the products and get infection, thus replenish infectious classes, as demonstrated in Figure4.Figure 4
Variation of the dairy production rate into infected (a) humans and (b) animals with change in time.
(a)
(b)Figure5 shows the effects of varying human and animal transmission rates from infectious animals. If effective control measures are not taken to contain the transmission, infectious class increases as humans and animals come in contact with infectious animals. When the rate of contact increases, susceptible human and animal classes decrease, as shown in Figure 5.Figure 5
Variation of humans and animals infection rates from infectious animals to susceptible humans and animals classes. (a) Susceptible humans. (b) Susceptible animals.
(a)
(b)Infectious human and animals classes increase over time as we vary infection rates. This is due to the fact that the more interaction between susceptible humans and animals with infectious animals increases infection rate, as shown in Figure6.Figure 6
Variation of humans and animals infection rates from infectious animals to infectious human and animals classes, respectively. (a) Infected humans. (b) Infected animals.
(a)
(b)Figure7 shows the impacts of transmission rates from infectious dairy products β3 and β6 to susceptible human and animal classes. The graphs show that the increase in production of infectious dairy products increases proportionally to the transmission rates. As rates of transmission β3 and β6 increase, susceptible humans and animals decrease, as shown in Figure 7.Figure 7
The impact of rates of transmission form dairy products to susceptible (a) humans and (b) animals.
(a)
(b)Figures8(a) and 8(b) show how the increase in the transmission rate from dairy products to susceptible humans β3 and animals β6 decrease susceptible classes, which then increase infectious classes.Figure 8
The impact of rates of transmission form dairy products to infected (a) humans and (b) animals.
(a)
(b)


## 5. Conclusion and Recommendations

A deterministic model for transmission dynamics of bTB is developed and analyzed to determine parameters that drive the disease. We computed basic reproduction numberR0 and compute the sensitivity index of each parameter with respect to R0. Analysis shows that the animal infection rate from infectious animals β5, production of infectious dairy products ρ, human infection rate from dairy products β3, and humans infection rate from infectious animals β2 drive the dynamics of bTB. Stability of equilibrium states was investigated, disease-free equilibrium DFE is locally asymptotically stable when the basic reproduction number R0 < 1. However, both disease-free and endemic equilibria are not globally stable due to possibility of the model to undergo backward bifurcation when the basic reproduction number R0 = 1. To contain the disease, control strategies should target to reduce animal infection rate, production of infectious dairy products, infection rate from dairy products, and infection rate from infectious animals. We recommend quarantine of infected animals, inspection of meat, pasteurizing of milk, and education campaign to reduce contact between humans and animals.



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*Source: 7424075-2020-03-30.xml*