# Optimal Control Strategies for the Infectiology of Brucellosis

Authors: Nkuba Nyerere; Livingstone S. Luboobi; Saul C. Mpeshe; Gabriel M. Shirima
Journal: International Journal of Mathematics and Mathematical Sciences
 (2020)

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## Abstract

Brucellosis is a zoonotic infection caused by Gram-negative bacteria of genusBrucella. The disease is of public health, veterinary, and economic significance in most of the developed and developing countries. Direct contact between susceptible and infective animals or their contaminated products are the two major routes of the disease transmission. In this paper, we investigate the impacts of controls of livestock vaccination, gradual culling through slaughter of seropositive cattle and small ruminants, environmental hygiene and sanitation, and personal protection in humans on the transmission dynamics of Brucellosis. The necessary conditions for an optimal control problem are rigorously analyzed using Pontryagin’s maximum principle. The main ambition is to minimize the spread of brucellosis disease in the community as well as the costs of control strategies. Findings showed that the effective use of livestock vaccination, gradual culling through slaughter of seropositive cattle and small ruminants, environmental hygiene and sanitation, and personal protection in humans have a significant impact in minimizing the disease spread in livestock and human populations. Moreover, cost-effectiveness analysis of the controls showed that the combination of livestock vaccination, gradual culling through slaughter, environmental sanitation, and personal protection in humans has high impact and lower cost of prevention.


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## Body

## 1. Introduction

Brucellosis is a zoonotic infection caused by Gram-negative bacteria of genusBrucella which includes; B. abortus primarly from cattle, B. melitensis from small ruminants, B. suis from swine, and B. canis from dogs [1–4]. It is considered by the Food and Agriculture Organisation (FAO), the World Health Organisation (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, ten brucella species have been identified and primarily named after the features of infection or the animal source. Of these, the following four have moderate-to-significant human pathogenicity: Brucella melitensis (highest pathogenicity), Brucella suis (high pathogenicity), Brucella abortus (moderate pathogenicity), and Brucella canis (moderate pathogenicity) [10–12].Brucellosis is endemic in most of the developing world. It 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 from abortions, sterility, decreased milk production, veterinary fees, and costs of replacement of animals. In animals, brucellosis is transmitted when a susceptible animal ingests contaminated materials such as tissues or discharges from infected animals, while in humans the bacteria is transmitted by ingestion of contaminated unpasteurized milk or other dairy products and direct contact through occupational activities such as farmers, laboratory personnel, abattoir workers, and veterinarians. According to Ducrotoy et al. [14], there are epidemiological situations in which infections of small ruminants by B. abortus occur in areas where they are in contact with cattle and B. melitensis is absent. Ducrotoy et al. [14] suggests that coinfections by two different brucellae are rather unlikely because of the development of immunity in an ongoing infection and, in fact, they have never been convincingly proven.Infected animals exhibit clinical signs that are of economic significance to stakeholders. These signs include reduced fertility, abortion, poor weight gain, lost draught power, and a substantial decline in milk production [13, 15]. Symptoms in human includes continuous or intermittent fever, headache, weakness, profuse sweats, chills, joint pains, aches, and weight loss, as well as devastating complications that leads to miscarriage in pregnant women. Neurological complications, endocarditis, and testicular or bone abscess formation can also occur [16, 17]. The infection can also affect the liver and spleen and may last for longer terms if not treated. Furthermore, the clinical signs of brucellosis in humans present diagnostic difficulties because they overlap with those of typhoid fever, malaria, rheumatic fever, joint diseases, and relapsing fever. Human brucellosis is debilitating and requires prolonged treatment with combination of antibiotics [18].The global burden of human brucellosis remains enormous: The infection causes more than 500,000 new cases per year worldwide. The annual number of reported cases in the United States has dropped significantly to about 100 cases per year due to aggressive 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 [19]. In Africa, brucellosis exists throughout sub-Saharan Africa, but the prevalence is unclear and poorly understood with varying reports from country to country and geographical regions, as well as animal factors [20]. Most African countries have poor socioeconomic status, with people living with and by their livestock, while health networks, surveillance, and vaccination programs are virtually nonexistent. In Tanzania, the first outbreak of brucellosis was reported in Arusha in 1927 [21]. 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%. In humans, the average prevalence varies from 1 to 5% [22]; a recent study by [23] 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 with seroprevalence varying from 0.7 to 20.5% [24, 25]. Despite the WHO, FAO, and OIE efforts and interventions being available, brucellosis continues to pose great economic threat by affecting 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 cost effectiveness if we are to control or eradicate the disease. So far, few studies [10, 26–33] have been developed to analyze dynamics and spread of brucellosis in homogeneous/heterogeneous populations. However, none of these studies had considered the mathematical approach for optimal control and cost effectiveness in reducing or eradicating the disease in cattle, small ruminants, and human populations. In this paper, the dynamics and cost effectiveness of the control strategies for brucellosis using mathematical models are rigorously studied.


## 2. Model Formulation

A mathematical model for the transmission dynamics of brucellosis incorporating the time-dependent controls to some parameters is formulated in this section. Some assumptions used in this section are similar to those in [27, 28], but the time-dependent parameters u1t,u2t,u3t, and u4t make the difference between our previous brucellosis works and the current work. The most important reason for taking preventive and control measures on brucellosis is to minimize the prevalence of the disease, and if possible eradicate it from the population. Particularly, the level of susceptibility of healthy individuals against the infection is minimized by protective measures, whereas the number of infective individuals in the community is reduced by control measures. In this paper, a time-dependent variable u1t is introduced as a control that aims at reducing the number of susceptible animals in the herds and consequently to reduce the disease transmission to the vaccination coverage failure rate 1−u1tλ1. In other words, u1t is the measure of the effectiveness of S19 and Rev1 vaccine for cattle and small ruminants, and 1−u1tλ1 is the failure rate of vaccination for ruminants. That is, if vaccination of ruminant coverage is 100% effective, zero brucellosis incidence may be recorded in that particular region. Thus, u1t aims at reducing the susceptibility level of healthy animals on the disease as well as the disease transmission rate. Based on the fact that there is neither disease-induced deaths nor treatment for infected ruminants, we introduce u2t, a control variable that measures the efficiency of gradual culling of seropositive animal parameters dc and ds for the infected cattle and small ruminants, respectively. Furthermore, gradual culling of seropositive animals targets at curtailing the number of infectious ruminants in the community and consequently reduces the disease transmission rate to u2tdc and u2tds in cattle and small ruminants, respectively. To reduce or eliminate the number of Brucella in the environment, the control variable u3t is introduced as the measure of effectiveness of environmental hygiene and sanitation parameter τ. In particular, environmental hygiene and sanitation refers to proper disposal of placentas and aborted foetuses. A time-dependent control variable u4t is introduced in the model as the measure of the effectiveness of personal protection in humans so that 1−u4tλ1 is the failure rate of the control strategy. In this context, personal protection refers to personal hygiene, protection of the environment, food hygiene (adequate boiling of fresh milk intended for drinking or making other milk products), and adoption to safe working practices including use of personal protection equipments such as gloves, masks, eye wear, and closed footwear when handling potentially infected materials such as aborted foetus, placenta, and gravid uterus practices. Based on the fact that treatment of human brucellosis reduces the risk of disease and that has a very low or negligible transmission blocking effect, the time-dependent control for this parameter is not of much concern. To determine the necessary conditions for optimal impact of incorporated parameters, we use Pontryagin’s Maximum Principle as the method for obtaining the optimal combination of incorporated controls. The aim is to limit and prevent the spread of brucellosis disease and at the same time the cost of administering these controls is minimized.


### 2.1. Model Assumptions

Formulation of the optimal control model is guided by 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 shedBrucella pathogens in the environment(iv)
Livestock seropositivity is life-long lasting(v)
Immunized individuals cannot be infected unless they are resistant to infection wanes(vi)
There is a constant natural mortality rate in each of the species(vii)
The birth rate for each population is greater than the natural mortality rateThe compartmental diagram with the time-dependent control strategies is shown in Figure1, whereas the variables and parameters used in this model are, respectively, summarized in Tables 1 and 2.Figure 1
A schematic diagram for direct and indirect transmission of brucellosis in cattle, small ruminant, and human populations. Solid arrows represent transfer of individuals from one subpopulation to another while dotted lines represent interactions that lead to infection.Table 1
Model variables.
Variable
Description
S
h
t
Number of susceptible humans at timet
I
h
t
Number of infected human at timet
R
h
t
Number of recovered humans at timet
S
c
t
Number of susceptible cattle at timet
I
c
t
Number of infected cattle at timet
S
s
t
Number of susceptible small ruminants at timet
I
s
t
Number of infected small ruminants at timet
B
t
Number ofBrucella bacteria load per unit volume in the environment at time tTable 2
Model parameters used in the model and their description.
Parameter
Description
π
c
Per capita cattle birth rate
π
h
Per capita human birth rate
σ
Human recovery rate
μ
h
Per capita human natural death rate
β
c
Within cattle transmission rate
α
c
Brucella from the environment to cattle transmission rate
α
s
Brucella from the environment to small ruminant transmission rate
α
h
Brucella from the environment to human transmission rate
β
c
h
Cattle to human transmission rate
β
s
h
Small ruminants to human transmission rate
π
s
Small ruminants per capita birth rate
β
s
Within small ruminant transmission rate
μ
s
Per capita small ruminant natural death rate
τ
Environmental hygiene and sanitation rate
ε
Decaying rate ofBrucella in the environment
d
c
Culling rate of seropositive cattle
d
s
Culling rate of seropositive small ruminants


### 2.2. Compartmental Flow Diagram for the Disease Dynamics

The interactions between human, cattle, small ruminant populations, andBrucella in the environment are illustrated in Figure 1.The resulting model is shown as a system of differential equations:(1)dScdt=πcNc−1−u1tβcIc+αcB+μcSc,dIcdt=1−u1tβcIc+αcBSc−μc+u2tdcIc,dSsdt=πsNs−1−u1tβsIs+αsB+μsSs,dIsdt=1−u1tβsIs+αsBSs−μs+u2tdsIs,dShdt=πhNh+γRh−1−u4tβhctIc+βhsIs+βhhtIh+αhB+μhSh,dIhdt=1−u4tβhcIc+βhsIs+βhhIh+αhBSh−σ+μh+dhIh,dRhdt=σIh−γ+μhRh,dBdt=ρcIc+ρsIs−ε+u3tτB.


## 2.1. Model Assumptions

Formulation of the optimal control model is guided by 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 shedBrucella pathogens in the environment(iv)
Livestock seropositivity is life-long lasting(v)
Immunized individuals cannot be infected unless they are resistant to infection wanes(vi)
There is a constant natural mortality rate in each of the species(vii)
The birth rate for each population is greater than the natural mortality rateThe compartmental diagram with the time-dependent control strategies is shown in Figure1, whereas the variables and parameters used in this model are, respectively, summarized in Tables 1 and 2.Figure 1
A schematic diagram for direct and indirect transmission of brucellosis in cattle, small ruminant, and human populations. Solid arrows represent transfer of individuals from one subpopulation to another while dotted lines represent interactions that lead to infection.Table 1
Model variables.
Variable
Description
S
h
t
Number of susceptible humans at timet
I
h
t
Number of infected human at timet
R
h
t
Number of recovered humans at timet
S
c
t
Number of susceptible cattle at timet
I
c
t
Number of infected cattle at timet
S
s
t
Number of susceptible small ruminants at timet
I
s
t
Number of infected small ruminants at timet
B
t
Number ofBrucella bacteria load per unit volume in the environment at time tTable 2
Model parameters used in the model and their description.
Parameter
Description
π
c
Per capita cattle birth rate
π
h
Per capita human birth rate
σ
Human recovery rate
μ
h
Per capita human natural death rate
β
c
Within cattle transmission rate
α
c
Brucella from the environment to cattle transmission rate
α
s
Brucella from the environment to small ruminant transmission rate
α
h
Brucella from the environment to human transmission rate
β
c
h
Cattle to human transmission rate
β
s
h
Small ruminants to human transmission rate
π
s
Small ruminants per capita birth rate
β
s
Within small ruminant transmission rate
μ
s
Per capita small ruminant natural death rate
τ
Environmental hygiene and sanitation rate
ε
Decaying rate ofBrucella in the environment
d
c
Culling rate of seropositive cattle
d
s
Culling rate of seropositive small ruminants


## 2.2. Compartmental Flow Diagram for the Disease Dynamics

The interactions between human, cattle, small ruminant populations, andBrucella in the environment are illustrated in Figure 1.The resulting model is shown as a system of differential equations:(1)dScdt=πcNc−1−u1tβcIc+αcB+μcSc,dIcdt=1−u1tβcIc+αcBSc−μc+u2tdcIc,dSsdt=πsNs−1−u1tβsIs+αsB+μsSs,dIsdt=1−u1tβsIs+αsBSs−μs+u2tdsIs,dShdt=πhNh+γRh−1−u4tβhctIc+βhsIs+βhhtIh+αhB+μhSh,dIhdt=1−u4tβhcIc+βhsIs+βhhIh+αhBSh−σ+μh+dhIh,dRhdt=σIh−γ+μhRh,dBdt=ρcIc+ρsIs−ε+u3tτB.


## 3. Model Properties

### 3.1. Invariant Region

In this section, we investigate whether model variables have biological interpretation and have a unique bounded solution that exists for all the time. That is solutions of model system (1) with nonnegative initial data remain nonnegative for all time t≥0. We apply the approach in [27, 28] to the optimal control model (1). Model system (1) can be expressed in the compact form as follows:(2)dXdt=MX+F,where(3)M=−d000000001−u1λ1−d300000000−d100000001−u1λ2−d400000000−d2−μs+u2ds0000001−u4λ3−σ+μh+dh0000000σ−γ+μh00ρc0ρs000−ε+u3τ,with(4)d0=1−u1λ1+μc,d1=1−u1λ1+μs,d2=1−u4λ3+μh,d3=μc+u2dc,d4=μs+u2ds,X=Sc,Ic,Ss,Is,Sh,Ih,Rh,B,and F is a column vector given by(5)F=πcNc0,0,πsNs0,0,πhNh0,0,0,0T.It can be noticed thatMX is Meltzer matrix since all of its off-diagonal entries are nonnegative, for all X∈ℝ+8. Therefore, using the fact that F>0, the model system (1) is positively invariant in ℝ+8, which means that an arbitrary trajectory of the system starting in ℝ+8 remains in ℝ+8 forever. In addition, the right hand F is Lipschitz continuous. Thus, a unique maximal solution exists and so(6)Ω=Sc,Ic,Ss,Is,Sh,Ih,Rh,B≥0∈ℝ+8,is the feasible region for model (1). Thus, model (1) is epidemiologically and mathematically well posed in the region Ω.


## 3.1. Invariant Region

In this section, we investigate whether model variables have biological interpretation and have a unique bounded solution that exists for all the time. That is solutions of model system (1) with nonnegative initial data remain nonnegative for all time t≥0. We apply the approach in [27, 28] to the optimal control model (1). Model system (1) can be expressed in the compact form as follows:(2)dXdt=MX+F,where(3)M=−d000000001−u1λ1−d300000000−d100000001−u1λ2−d400000000−d2−μs+u2ds0000001−u4λ3−σ+μh+dh0000000σ−γ+μh00ρc0ρs000−ε+u3τ,with(4)d0=1−u1λ1+μc,d1=1−u1λ1+μs,d2=1−u4λ3+μh,d3=μc+u2dc,d4=μs+u2ds,X=Sc,Ic,Ss,Is,Sh,Ih,Rh,B,and F is a column vector given by(5)F=πcNc0,0,πsNs0,0,πhNh0,0,0,0T.It can be noticed thatMX is Meltzer matrix since all of its off-diagonal entries are nonnegative, for all X∈ℝ+8. Therefore, using the fact that F>0, the model system (1) is positively invariant in ℝ+8, which means that an arbitrary trajectory of the system starting in ℝ+8 remains in ℝ+8 forever. In addition, the right hand F is Lipschitz continuous. Thus, a unique maximal solution exists and so(6)Ω=Sc,Ic,Ss,Is,Sh,Ih,Rh,B≥0∈ℝ+8,is the feasible region for model (1). Thus, model (1) is epidemiologically and mathematically well posed in the region Ω.


## 4. Model Analysis

### 4.1. Disease Free Equilibrium

The Brucellosis free equilibrium pointE0 for the constant controls case was computed by setting the right-hand side of equations in model system (1) to zero, and it was found to be(7)E0=Sc0,0,Ss0,0,Sh0,0,0,0,where(8)Sc0=πcNc0μc,Ss0=πsNs0μs,Sh0=πhNh0μh.


### 4.2. The Effective Reproduction Number

Computation of the effective reproduction numberRe for model system (1) using the standard method of the next generation matrix developed by Diekmann et al. [34, 35] is carried out in this section. Re 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 [36]. 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, R0. Upon computation, the effective reproduction number was found to be(9)Re=maxR11+R22+R22−R112+4R12R212,1−u4βhhπhNh0μhσ+μh+dh,where(10)R11=1−u1ε+u3τβc+αcρcπcNc0μcμc+u2dcε+u3τ,R12=αcρsπcNc0μcμs+u2dsε+u3τ,R22=1−u1ε+u2τβs+αsρsπsNs0μsμs+u2dsε+u3τ,R21=αsρcπsNs0μsμc+u2dcε+u3τ.The first and the second expressions of equation (9) represent, respectively, the effective reproduction numbers in the livestock and human populations. When there are no controls (u1=u2=u3=0) for the disease in both human population and livestock, the effective reproduction is reduced to the basic reproduction number given by(11)R0=maxR110+R220+R220−R1102+4R120R2102,βhhπhNh0μhμh+dh,where(12)R110=εβc+αcρcπcNc0μc2ε,R120=αcρsπcNc0μcμsε,R220=εβs+αsρsπsNs0μs2ε,R210=αsρcπsNs0μsμcε,as in [27, 28]. Based on the fact that human-to-human transmission is less than within livestock transmission [37–42] and that environmental contamination by humans is negligible, the effective reproduction number and basic reproductive number are, respectively, given by(13)Re=R11+R22+R22−R112+4R12R212,and(14)R0=R110+R220+R220−R1102+4R120R2102.The numerical simulations for the comparison between effective or control reproductive number and basic reproductive number with respect to variations in some parameters are illustrated in Figures2 and 3. Parameter values used for the simulations are presented in Table 3.Figure 2
Variations in the reproduction number with respect to changes in effective contact rate in livestock.
(a)
(b)Figure 3
Variations in the reproduction number with respect to changes of the consumption rate ofBrucella from the environment by livestock.
(a)
(b)Table 3
Model parameter values.
Parameter
Value
Unit
π
c
0.3
year
−
1
β
c
0.0011
year
−
1
μ
c
0.25
year
−
1
d
c
0.35
year
−
1
α
c
0.00035
year
−
1
ρ
c
10
year
−
1
ϵ
8
year
−
1
τ
12
year
−
1
π
s
0.4
year
−
1
β
s
0.001
year
−
1
μ
s
0.35
year
−
1
d
s
0.4
year
−
1
α
s
0.00032
year
−
1
ρ
s
15
year
−
1Figure2 shows that both R0 and Re increases with the increase in effective contact rate between livestock (direct transmission). For instance, when within cattle effective contact rate βc is 0.3, the control reproduction number and basic reproduction are, respectively, 0.3 and 1.4. On the contrary, if the within small ruminants effective contact rate βs is 0.3, R0 and Re are 1.2 and 0.25, respectively.Similarly, Figure3 shows that both R0 and Re increases with the increase in consumption rate of Brucella from the contaminated environment by livestock (indirect transmission). In particular, when Brucella consumption rate by cattle αc is 0.3, R0 and Re are, respectively, 0.1 and 1.1. On the contrary, if the small ruminants consumption rate of Brucella from the environment αs is 0.3, R0 and Re are 2 and 1.1, respectively. More importantly, Figure 3(b) reveals that αs>0.3 leads to Re>1 (disease persistence) and that small ruminants are more susceptible to the contaminated environment than cattle. The possible reasons include small ruminant density and herd turnover due to births and introduction of new animals, as pointed in [43].Generally, the controlsu1, u2, and u3 have high impact on Re by keeping it always less than R0.


### 4.3. Local Stability of the Equilibria

In this section, the trace-determinant method is employed to investigate the local stability of the Brucellosis free equilibrium point for the model system (1).Theorem 1.
The disease-free equilibrium for the Brucellosis model system (1) is locally asymptotically stable if R0<1 and unstable if R0>1.Proof.
We show that the variational matrixJE0 of the brucellosis free model system has a negative trace and positive determinant. The Jacobian matrix for system (1) is given by(15)JE0=−1−u1μc−1−u1βcSc000000−a0−a100000a00−a2−a3000−b000b1000b0−1−u4βhcSh00−b2−b3−b4γ−c01−u4βhcSh00b20c10c00000σ−γ+μh00ρc0ρs000−r,where(16)a=1−u1αcSc0,a1=1−u1βcSc0−μc+u2dc,a2=1−u1μs,a3=1−u1βsSs0,b=1−u1αsSs0,b1=1−u1βsSs0−μs+u2ds,b2=1−u4βhsSh0,b3=1−u4μh,b4=1−u4βhhSh0,c1=βhhSh0−σ+μh+dh,r=ε+u3τ.
Direct computation of the Jacobian matrixJE0 gives(17)TrJE0=−μc+u2dc1−1−u1βcSc0μc+u2dc−μs+u2ds1−1−u1βsSs0μs+u2ds−σ+μh+dh1−1−u4βhhSh0σ+μh+dh−1−u1μc+μs+1−u4μh−γ+μh+ε+u3τ.
Thus, the trace of the Jacobian matrix is less than zero, that is,TrJE0<0 if(18)1−u1βcSc0μc+u2dc<1,1−u1βsSs0μc+u2ds<1,1−u4βhhSh0σ+μh+dh<1.
On the contrary, the determinant of matrixJE0 is computed using Maple 16 Software and was found to be(19)DetJE0=Q01−RhQ11−Rc+μs+u2ds1−RsQ2+Q31−Rc,where(20)Rh=1−u4βhhSh0σ+μh+dh,Rs=1−u1βsSs0μs+u2ds,Rc=1−u1βcSc0μc+u2dc,Q0=1−u121−u4γ+μhσ+μh+dhμcμsμh,Q1=1−u1μc+u2dcαsρsSs0,Q2=1−u1αcρcSc0,Q3=ε+u3τμc+u2dc.
It follows thatDetJE0>0 if Rh<1,Rc<1, and Rs<1. Thus, the Brucellosis free equilibrium for model system (1) is locally asymptotically stable if Rh<1,Rc<1, and Rs<1. Local stability of the Brucellosis free equilibrium suggests local stability of the endemic equilibrium for the reverse condition [28].


## 4.1. Disease Free Equilibrium

The Brucellosis free equilibrium pointE0 for the constant controls case was computed by setting the right-hand side of equations in model system (1) to zero, and it was found to be(7)E0=Sc0,0,Ss0,0,Sh0,0,0,0,where(8)Sc0=πcNc0μc,Ss0=πsNs0μs,Sh0=πhNh0μh.


## 4.2. The Effective Reproduction Number

Computation of the effective reproduction numberRe for model system (1) using the standard method of the next generation matrix developed by Diekmann et al. [34, 35] is carried out in this section. Re 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 [36]. 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, R0. Upon computation, the effective reproduction number was found to be(9)Re=maxR11+R22+R22−R112+4R12R212,1−u4βhhπhNh0μhσ+μh+dh,where(10)R11=1−u1ε+u3τβc+αcρcπcNc0μcμc+u2dcε+u3τ,R12=αcρsπcNc0μcμs+u2dsε+u3τ,R22=1−u1ε+u2τβs+αsρsπsNs0μsμs+u2dsε+u3τ,R21=αsρcπsNs0μsμc+u2dcε+u3τ.The first and the second expressions of equation (9) represent, respectively, the effective reproduction numbers in the livestock and human populations. When there are no controls (u1=u2=u3=0) for the disease in both human population and livestock, the effective reproduction is reduced to the basic reproduction number given by(11)R0=maxR110+R220+R220−R1102+4R120R2102,βhhπhNh0μhμh+dh,where(12)R110=εβc+αcρcπcNc0μc2ε,R120=αcρsπcNc0μcμsε,R220=εβs+αsρsπsNs0μs2ε,R210=αsρcπsNs0μsμcε,as in [27, 28]. Based on the fact that human-to-human transmission is less than within livestock transmission [37–42] and that environmental contamination by humans is negligible, the effective reproduction number and basic reproductive number are, respectively, given by(13)Re=R11+R22+R22−R112+4R12R212,and(14)R0=R110+R220+R220−R1102+4R120R2102.The numerical simulations for the comparison between effective or control reproductive number and basic reproductive number with respect to variations in some parameters are illustrated in Figures2 and 3. Parameter values used for the simulations are presented in Table 3.Figure 2
Variations in the reproduction number with respect to changes in effective contact rate in livestock.
(a)
(b)Figure 3
Variations in the reproduction number with respect to changes of the consumption rate ofBrucella from the environment by livestock.
(a)
(b)Table 3
Model parameter values.
Parameter
Value
Unit
π
c
0.3
year
−
1
β
c
0.0011
year
−
1
μ
c
0.25
year
−
1
d
c
0.35
year
−
1
α
c
0.00035
year
−
1
ρ
c
10
year
−
1
ϵ
8
year
−
1
τ
12
year
−
1
π
s
0.4
year
−
1
β
s
0.001
year
−
1
μ
s
0.35
year
−
1
d
s
0.4
year
−
1
α
s
0.00032
year
−
1
ρ
s
15
year
−
1Figure2 shows that both R0 and Re increases with the increase in effective contact rate between livestock (direct transmission). For instance, when within cattle effective contact rate βc is 0.3, the control reproduction number and basic reproduction are, respectively, 0.3 and 1.4. On the contrary, if the within small ruminants effective contact rate βs is 0.3, R0 and Re are 1.2 and 0.25, respectively.Similarly, Figure3 shows that both R0 and Re increases with the increase in consumption rate of Brucella from the contaminated environment by livestock (indirect transmission). In particular, when Brucella consumption rate by cattle αc is 0.3, R0 and Re are, respectively, 0.1 and 1.1. On the contrary, if the small ruminants consumption rate of Brucella from the environment αs is 0.3, R0 and Re are 2 and 1.1, respectively. More importantly, Figure 3(b) reveals that αs>0.3 leads to Re>1 (disease persistence) and that small ruminants are more susceptible to the contaminated environment than cattle. The possible reasons include small ruminant density and herd turnover due to births and introduction of new animals, as pointed in [43].Generally, the controlsu1, u2, and u3 have high impact on Re by keeping it always less than R0.


## 4.3. Local Stability of the Equilibria

In this section, the trace-determinant method is employed to investigate the local stability of the Brucellosis free equilibrium point for the model system (1).Theorem 1.
The disease-free equilibrium for the Brucellosis model system (1) is locally asymptotically stable if R0<1 and unstable if R0>1.Proof.
We show that the variational matrixJE0 of the brucellosis free model system has a negative trace and positive determinant. The Jacobian matrix for system (1) is given by(15)JE0=−1−u1μc−1−u1βcSc000000−a0−a100000a00−a2−a3000−b000b1000b0−1−u4βhcSh00−b2−b3−b4γ−c01−u4βhcSh00b20c10c00000σ−γ+μh00ρc0ρs000−r,where(16)a=1−u1αcSc0,a1=1−u1βcSc0−μc+u2dc,a2=1−u1μs,a3=1−u1βsSs0,b=1−u1αsSs0,b1=1−u1βsSs0−μs+u2ds,b2=1−u4βhsSh0,b3=1−u4μh,b4=1−u4βhhSh0,c1=βhhSh0−σ+μh+dh,r=ε+u3τ.
Direct computation of the Jacobian matrixJE0 gives(17)TrJE0=−μc+u2dc1−1−u1βcSc0μc+u2dc−μs+u2ds1−1−u1βsSs0μs+u2ds−σ+μh+dh1−1−u4βhhSh0σ+μh+dh−1−u1μc+μs+1−u4μh−γ+μh+ε+u3τ.
Thus, the trace of the Jacobian matrix is less than zero, that is,TrJE0<0 if(18)1−u1βcSc0μc+u2dc<1,1−u1βsSs0μc+u2ds<1,1−u4βhhSh0σ+μh+dh<1.
On the contrary, the determinant of matrixJE0 is computed using Maple 16 Software and was found to be(19)DetJE0=Q01−RhQ11−Rc+μs+u2ds1−RsQ2+Q31−Rc,where(20)Rh=1−u4βhhSh0σ+μh+dh,Rs=1−u1βsSs0μs+u2ds,Rc=1−u1βcSc0μc+u2dc,Q0=1−u121−u4γ+μhσ+μh+dhμcμsμh,Q1=1−u1μc+u2dcαsρsSs0,Q2=1−u1αcρcSc0,Q3=ε+u3τμc+u2dc.
It follows thatDetJE0>0 if Rh<1,Rc<1, and Rs<1. Thus, the Brucellosis free equilibrium for model system (1) is locally asymptotically stable if Rh<1,Rc<1, and Rs<1. Local stability of the Brucellosis free equilibrium suggests local stability of the endemic equilibrium for the reverse condition [28].


## 5. Optimal Control

To investigate the optimal level of efforts that would be required to control brucellosis infections, we first formulate the objective functionJ to be minimized subject to the number of ruminants and human infections and cost of applying the controls:(21)J=∫0tfA1Ic+A2Is+A3B+A4Ih+B1u122+B2u222+B3u322+B4u422dt,where A1,A2,A3, and A4 are positive weight constants of infected cattle, infected small ruminants, Brucella in the environment, and infected human classes, respectively. Furthermore, the constants B1,B2,B3, and B4 are positive weights which balance the cost factors associated with control strategies u1,u2,u3, and u4, respectively. More importantly, the cost of each control strategy is assumed to be nonlinear and takes the quadratic form that is B1u12/2 is the cost of control strategy associated with vaccination of ruminants, B2u22/2 is the cost associated with gradual culling of seropositive animals strategy, B3u32/2 is the cost associated with environmental hygiene and sanitation, and B4u42/2 is the cost associated with personal protections in humans.With the objective functionJu1,u2,u3,u4, our goal is to minimize the number of infected ruminants and humans, while minimizing the cost of controls, u1t,u2t,u3t, and u4t. We seek an optimal control u1∗t,u2∗t,u3∗t, and u4∗t such that(22)Ju1∗,u2∗,u3∗,u4∗=minJu1,u2,u3,u4u1,u2,u3,u4∈u,where u=u1,u2,u3,u4 such that u1,u2,u3, and u4 are measurable with 0≤u1≤1, 0≤u2≤1, 0≤u3≤1, and 0≤u4≤1, for t∈0,tf is the control set.


## 6. Existence of an Optimal Control

Theorem 2.
There exists an optimal control setu1∗,u2∗,u3∗,u4∗∈u with corresponding nonnegative states Sc,Ic,Ss,Is,Sh,Ih,Rh that minimize the objective functional Ju1t,u2t,u3t,u4t.Proof.
The positivity and uniform boundedness of the state variables as well as the controls on0,tf entail the existence of a minimizing sequence:(23)Ju1nt,u2nt,u3nt,u4nt,such that(24)limn⟶∞Ju1nt,u2nt,u3nt,u4nt=infu1nt,u2nt,u3nt,u4nt∈uJu1nt,u2nt,u3nt,u4nt.
The boundedness of all the state and control variables implies that all the derivatives of the state variables are also bounded. If the corresponding sequence of state variables is denoted bySc,Ic,Ss,Is,Sh,Ih,Rh,B, then all state variables are Lipschitz continuous with the same Lipschitz constant. This implies that the sequence Sc,Ic,Ss,Is,Sh,Ih,Rh is uniformly equicontinuous in 0,tf. Following the approach in [44], the state sequence has a subsequence that converges uniformly to Sc,Ic,Ss,Is,Sh,Ih,Rh,B in 0,tf. In addition, we can establish that the control sequence un=Scn,Icn,Ssn,Isn,Shn,Ihn,Rhn,Bn has a subsequence that converges weakly in L20,tf. Let u1∗,u2∗,u3∗,u4∗∈u be such that uin⟶ui∗ weakly in L20,tf for i=1,2,3,4. Applying the lower semicontinuity of norms in weak L2, we have(25)ui∗L22≤limn⟶∞infuintL22,i=1,2,3,4.
This means that(26)Ju1∗,u2∗,u3∗,u4∗≤limn⟶∞∫0tfA1Icn+A2Is2+A3Ihn+A4B4n+B1u1n2+B2u2n2+B3u3n2+B4u4n2dt.
Thus, there exists a set of controlsu1∗,u2∗,u3∗,u4∗ that minimizes our objective functional Ju1,u2,u3,u4.


## 7. Characterization of Optimal Control

In this section, we derive necessary conditions for an optimal control and formulate an optimality system that characterizes the optimal control using upper and lower bound technique. The necessary condition is that an optimal control problem must satisfy Pontryagin’s maximum principle [45]. The principle converts system (1) and equation (21) into a problem of minimizing pointwise a Hamiltonian H, with respect to u1,u2,u3, and u4 defined by(27)H=A1Ic+A2Is+A3B+A4Ih+B1u122+B2u222+B3u322+B4u422+λScπcNc−1−u1βcIc+αcB+μcSc+λIc1−u1βcIc+αcBSc−μc+u2Ic+λSsπsNs−1−u1βsIs+αsB+μsSs+λIs1−u1βsIs+αsBSs−μs+u2Is+λBρcIc+ρsIs−ε+u3τB+λShπhNh+γRh−1−u4βhcIc+βhsIs+βhhIh+αhB+μhSh+λIh1−u4βhcIc+βhsIs+βhhIh+αhBSh−σ+μh+dhIh+λRhσIh−γ+μh+dhRh,where λSc, λIc, λSs, λIs, λB, λSh, λIh, and λRh are the adjoint or costate variables.Applying Pontryagin’s maximum principle [45] and the existence result for the optimal control [46], we obtain the following.Theorem 3.
For optimal tetra controlsu1∗,u2∗,u3∗, and u4∗ that minimizes Ju1,u2,u3,u4 over u, there exist adjoint variables λSc, λIc, λSs, λIs, λB, λSh, λIh, and λRh, satisfying(28)dλScdt=1−u1βcIc+αcBλSc−λIc+μcλSc,dλIcdt=−A1+1−u1λSc−λIcβcSc+λIcμc+u2dc+1−u4λSh−λIhβhcSh−ρcλB,dλSsdt=1−u1βsIs+αsBλSs−λIs+μsλSs,dλIsdt=−A2+1−u1λSs−λIsβsSs+λIsμs+u2ds+1−u4λSh−λIhβhsSh−λBρs,dλShdt=1−u4βhcIc+βhsIs+βhhIh+αhBλSh−λIh+μhλSh,dλIhdt=−A4+1−u4λSh−λIhβhhSh+λIhσ+μh+dh−σλRh,dλRhdt=γλRh−λSh+μhλRh,dλBdt=−A3+1−u1λSc−λIcαcSc+1−u1λSs−λIsαsSs+1−u4λSh−λIhαhSh+λBε+u3τ,with transversality conditions:(29)λSctf=λIctf=λSstf=λIstf=λBtf=λShtf=λIhtf=λRhtf=0.
The following characterization holds on the interior of the control setu:(30)u1∗=max0,min1,1B1βcIc+αcBλIc−λScSc+βsIs+αsBλIs−λSsSs,u2∗=max0,min1,1B2dcIcλIc+dsIsλIsSc,u3∗=max0,min1,1B3λBB,u4∗=max0,min1,1B4βhcIc+βhsIs+βhhIh+αhBλIh−λShSh,where λSc, λIc, λSs, λIs, λB, λSh, λIh, and λRh are solutions of equation (29).Proof.
The form of adjoint (or costate) system (28) and transversality conditions (29) are standard results from Pontryagin’s Maximum Principle [45]. To obtain the costate system (28), the partial derivatives of the Hamiltonian (H) equation (27) with respect to each state variable are computed as follows:(31)dλScdt=−∂H∂Sc,λSctf=0,⋮dλRhdt=−∂H∂Rh,λRhtf=0.
The optimality equation (30) is obtained by finding the partial derivative of the Hamiltonian equation (27) with respect to each control variable and solving for the optimal values of ui∗ where the derivative vanishes. That is, ∂H/∂ui=0 for i=1,2,3,4.
Solving forui∗ subject to the constraints gives the characterization equation (30). Next, we discuss the numerical solutions of the optimality system and the corresponding results of varying the optimal controls u1,u2,u3, and u4, the parameter choices, and the interpretations from various cases.


## 8. Numerical Simulations

In this section, we analyze numerically the optimal control strategies for the brucellosis transmission in model system (27). The controls of interest are vaccination of susceptible ruminants, gradual culling of seropositive ruminants, environmental hygiene and sanitation, and personal protection in humans. The optimal control solution is obtained by solving the optimality system which consists of the state system (1) and the adjoint system (28). We start by solving the state equations with a guess for the controls over the simulated time using the fourth-order Runge–Kutta iterative scheme method. The adjoint equations are solved by the backward fourth-order Runge–Kutta scheme using the current iterations solutions of the state equations because of the transversality conditions (31). Furthermore, the controls are updated by using a convex combination of the previous controls and the value from the characterizations (30). This process is repeated and the iterations are stopped if the values of the unknowns at the previous iterations are very close to the ones at the present iteration [47]. Based on the fact that brucellosis is endemic in most of the sub-Saharan Africa countries and that one control cannot stop the disease transmission, we investigate the impacts of combining at least three control strategies in a period of six years. Moreover, the computation of real weights of the objective function is very involving and needs a lot of information. In view of the aforesaid, the weights of the objective function are theoretically chosen to be A1=15, A2=20, A3=5, A4=10, B1=15, B2=10, B3=10, and B4=10 just to concede the control strategies proposed in this paper, and the parameter values used are in Table 3. The initial state variables are chosen as Sc0=200,Ic0=10,Ss0=200,Is0=5,Sh0=50,Ih0=10, Rh0=5,andB0=100.The parameter values used in our computations are mainly from [3], a literature similar to this work.


### 8.1. Strategy A: Optimal Vaccination, Gradual Culling, and Environmental Sanitation

Under this strategy, the effectiveness of vaccination of ruminant controlu1, gradual culling of seropositive ruminants u2, and environmental hygiene and sanitation u3 are used to minimize the objective function J, whereas personal protection in humans control u4 is set to zero. Figure 4 illustrates the trends of the infective classes.Figure 4
Dynamics of brucellosis with optimal vaccination, gradual culling of seropostive ruminants, and environmental hygiene and sanitation.
(a)
(b)
(c)
(d)Figure4 shows that combination of the three control strategies leads to eradication of brucellosis from cattle in four years, small ruminants in two years, and humans in between two to three years. In case of no controls, the infective populations grow exponentially.


### 8.2. Strategy B: Optimal Vaccination, Gradual Culling, and Personal Protection

In this strategy, ruminant vaccinationu1, gradual culling of seropositive ruminants u2, and personal protection in humans u4 are used to optimize the objective function J while the environmental hygiene sanitation control u3 is set to zero. Figure 5 illustrates the variations in the infective classes.Figure 5
Dynamics of brucellosis with optimal vaccination, gradual culling of seropositive ruminants, and personal protection controls.
(a)
(b)
(c)
(d)Figure5 shows that the cattle population attains its disease equilibrium in less than four years, whereas the disease can be eradicated from the small ruminants and humans in less than two years if the combination of the three controls is applied. In case of no control, the infected populations grow exponentially.


### 8.3. Strategy C: Optimal Vaccination, Environmental Hygiene, and Personal Protection

In this strategy, ruminant vaccination controlu1, environmental sanitation u3, and personal protection u4 are used to optimize the objective function J while gradual culling of seropostive ruminants control u2 is set to zero. Figure 6 illustrates the variations in the seropositive populations.Figure 6
Dynamics of brucellosis with optimal vaccination, environmental hygiene, and personal protection controls.
(a)
(b)
(c)
(d)It can be seen from Figure6 that the combination of the three control strategies leads to elimination of the infective humans and consequently a disease-free equilibrium is attained within a two years period. Furthermore, the small ruminant population attains its disease-free equilibrium point in five years and that of cattle will attain its equilibrium point in more than six years. In case of no controls, the number of infective individuals increases.


### 8.4. Strategy D: Optimal Gradual Culling, Environmental Hygiene, and Personal Protection

In this strategy, gradual culling of seropostive ruminantsu2, environmental hygiene and sanitation u3, and personal protection u4 are used to minimize the objective function J, whereas ruminants vaccination u1 is set to zero. Figure 7 illustrates the variations in the seropositive populations.Figure 7
Dynamics of brucellosis with optimal gradual culling of seropostive ruminants, environmental sanitation, and personal protection controls.
(a)
(b)
(c)
(d)It can be seen from Figure7 that optimal implementation of gradual culling of seropostive ruminants, environmental hygiene, and personal protection reduces the number of infected human to zero in a period of less than two years, whereas the infected ruminants does not go to zero in a period of more than six years. This implies that implementation the three interventions under consideration does not make the ruminants population attain their disease-free equilibrium points. Thus, this strategy is not mathematically recommended.


### 8.5. Strategy E: Optimal Vaccination, Gradual Culling, Environmental Sanitation, and Personal Protection

With this strategy, the combination of the four control strategies under consideration, ruminants vaccination,u1, gradual culling of seropositive ruminants, u2, environmental hygiene and sanitation, u3, and personal protection, u4, are used to optimize the objective function J. Figure 8 illustrates the variations in the infective populations.Figure 8
Dynamics of brucellosis with optimal vaccination, gradual culling of seropositive ruminants, environmental sanitation, and personal protection controls.
(a)
(b)
(c)
(d)Figure8 shows that due to the combination of the four control strategies, the number of infected small ruminants and humans decreases to zero in less than two years while that of infective cattle reduces to zero in less than four years. In case of no controls, the number of infective populations grows exponentially.


## 8.1. Strategy A: Optimal Vaccination, Gradual Culling, and Environmental Sanitation

Under this strategy, the effectiveness of vaccination of ruminant controlu1, gradual culling of seropositive ruminants u2, and environmental hygiene and sanitation u3 are used to minimize the objective function J, whereas personal protection in humans control u4 is set to zero. Figure 4 illustrates the trends of the infective classes.Figure 4
Dynamics of brucellosis with optimal vaccination, gradual culling of seropostive ruminants, and environmental hygiene and sanitation.
(a)
(b)
(c)
(d)Figure4 shows that combination of the three control strategies leads to eradication of brucellosis from cattle in four years, small ruminants in two years, and humans in between two to three years. In case of no controls, the infective populations grow exponentially.


## 8.2. Strategy B: Optimal Vaccination, Gradual Culling, and Personal Protection

In this strategy, ruminant vaccinationu1, gradual culling of seropositive ruminants u2, and personal protection in humans u4 are used to optimize the objective function J while the environmental hygiene sanitation control u3 is set to zero. Figure 5 illustrates the variations in the infective classes.Figure 5
Dynamics of brucellosis with optimal vaccination, gradual culling of seropositive ruminants, and personal protection controls.
(a)
(b)
(c)
(d)Figure5 shows that the cattle population attains its disease equilibrium in less than four years, whereas the disease can be eradicated from the small ruminants and humans in less than two years if the combination of the three controls is applied. In case of no control, the infected populations grow exponentially.


## 8.3. Strategy C: Optimal Vaccination, Environmental Hygiene, and Personal Protection

In this strategy, ruminant vaccination controlu1, environmental sanitation u3, and personal protection u4 are used to optimize the objective function J while gradual culling of seropostive ruminants control u2 is set to zero. Figure 6 illustrates the variations in the seropositive populations.Figure 6
Dynamics of brucellosis with optimal vaccination, environmental hygiene, and personal protection controls.
(a)
(b)
(c)
(d)It can be seen from Figure6 that the combination of the three control strategies leads to elimination of the infective humans and consequently a disease-free equilibrium is attained within a two years period. Furthermore, the small ruminant population attains its disease-free equilibrium point in five years and that of cattle will attain its equilibrium point in more than six years. In case of no controls, the number of infective individuals increases.


## 8.4. Strategy D: Optimal Gradual Culling, Environmental Hygiene, and Personal Protection

In this strategy, gradual culling of seropostive ruminantsu2, environmental hygiene and sanitation u3, and personal protection u4 are used to minimize the objective function J, whereas ruminants vaccination u1 is set to zero. Figure 7 illustrates the variations in the seropositive populations.Figure 7
Dynamics of brucellosis with optimal gradual culling of seropostive ruminants, environmental sanitation, and personal protection controls.
(a)
(b)
(c)
(d)It can be seen from Figure7 that optimal implementation of gradual culling of seropostive ruminants, environmental hygiene, and personal protection reduces the number of infected human to zero in a period of less than two years, whereas the infected ruminants does not go to zero in a period of more than six years. This implies that implementation the three interventions under consideration does not make the ruminants population attain their disease-free equilibrium points. Thus, this strategy is not mathematically recommended.


## 8.5. Strategy E: Optimal Vaccination, Gradual Culling, Environmental Sanitation, and Personal Protection

With this strategy, the combination of the four control strategies under consideration, ruminants vaccination,u1, gradual culling of seropositive ruminants, u2, environmental hygiene and sanitation, u3, and personal protection, u4, are used to optimize the objective function J. Figure 8 illustrates the variations in the infective populations.Figure 8
Dynamics of brucellosis with optimal vaccination, gradual culling of seropositive ruminants, environmental sanitation, and personal protection controls.
(a)
(b)
(c)
(d)Figure8 shows that due to the combination of the four control strategies, the number of infected small ruminants and humans decreases to zero in less than two years while that of infective cattle reduces to zero in less than four years. In case of no controls, the number of infective populations grows exponentially.


## 9. Cost-Effectiveness Analysis

In this section, the cost effectiveness of the control strategies A, B, C, D, and E is analyzed using the Incremental Cost-Effectiveness Ratio (ICER), as described in [48–51]. The choice of the method is based on the fact that it allows the comparison between the cost effectiveness of combination of at least two of the controls. In particular, we compare two competing intervention strategies incrementally; one intervention is being compared with the next less-effective alternative. ICER is defined by the difference in cost between two possible interventions divided by the difference in their outcome, given that they compete for the same resource. Mathematically,(32)ICER forX=Cost of interventionX−Cost of interventionYEffect of interventionX−Effect of interventionY,where X and Y are the two intervention strategies being compared in this case, and the effect or benefits in health status are measured in terms of quality-adjusted life years (QALYs) gained or lost. In this approach, alternatives that are more expensive and less effective are excluded. Following the numerical simulation results of model (1), we rank the five strategies in order of increasing effectiveness measured as total infections averted and compute the ICER for every two competing strategies, and the results are presented in Table 4.Table 4
ICER for alternative control strategies A, B, C, D, and E.
Strategy
Total infection averted
Total cost ($)
ICER
Strategy A
12953
4795.7
0.3702
Strategy B
19789
4368
−0.0625
Strategy D
20346
45535
73.9075
Strategy C
21409
13169
−30.4478
Strategy E
24832
4047
−2.6649From Table4, we see that strategy D is the most expensive and less effective strategy; thus, we exclude it from the set of alternatives so it does not consume limited resources. The recalculated ICER for remaining four alternatives is in Table 5.Table 5
ICER for alternative control strategies A, B, C, and E.
Strategy
Total infection averted
Total cost ($)
ICER
Strategy A
12953
4795.7
0.3702
Strategy B
19789
4368
−0.0625
Strategy C
21409
13169
5.4324
Strategy E
24832
4047
−2.6649Table5 shows that strategy C is the most expensive and less effective strategy; thus, we exclude it from the set of alternatives so it does not consume limited resources. Table 6 presents the recalculated ICER for remaining three alternatives.Table 6
ICER for alternative control strategies A, B, and E.
Strategy
Total infection averted
Total cost ($)
ICER
Strategy A
12953
4795.7
0.3702
Strategy B
19789
4368
−0.0625
Strategy E
24832
4047
−0.0638The comparison between strategies A and B shows a cost saving of $0.0625 for strategy B over strategy A. The lower ICER for strategy B indicates that strategy A is strongly dominated; that is, strategy A is more costly and less effective than strategy B. Therefore, we exclude strategy A from the set of alternatives so it does not consume limited resources. The recalculated ICER is shown in Table7.Table 7
ICER for alternative control strategies B and E.
Strategy
Total infection averted
Total cost ($)
ICER
Strategy B
19789
4368
0.2208
Strategy E
24832
4047
−0.0638Moreover, Table7 shows a cost saving of $0.0638 for strategy E over strategy B. In this case, the incremental cost is negative and the incremental effect is positive (south-east quadrant of the cost-effectiveness plane), and the intervention is unequivocally cost effective (it is dominant, achieving better outcomes at lower cost) [48–50]. The lower ICER for strategy E indicates that strategy B is more costly and less effective than strategy E; in other words, strategy E is more effective and saves money compared with the strategy B. Hence, strategy B is excluded from the set of alternatives so it does not consume limited resources. Thus, strategy E which is the combination of vaccination, gradual culling through slaughter of seropositive cattle and small ruminants, environmental sanitation, and personal protection in humans has the least ICER and therefore is more cost-effective strategy than all alternatives under consideration.


## 10. Conclusion

This paper aimed at formulating and analyzing a mathematical model for the impacts of different control options to the transmission dynamics of Brucellosis. We focused on livestock vaccination; gradual culling through slaughter of seropositive cattle and small ruminants; environmental hygiene and sanitation; and personal protection in humans. Pontryagin’s Maximum Principle approach and incremental cost-effectiveness ratio were, respectively, used to analyze the optimal control problem and the cost effectiveness of the control strategies. Findings in both optimal control and cost-effectiveness analysis revealed that combination of vaccination, gradual culling of seropositive cattle and small ruminants, environmental hygiene, and personal protection in humans is unequivocally best control strategy as it has high impact with lower cost of controlling the disease. It was revealed further that strategy B which is the combination of vaccination, gradual culling through slaughter of seropositive cattle and small ruminants, and personal protection in humans is the second cost-effective strategy, followed by strategy A which is the combination of vaccination, gradual culling of seropositive cattle and small ruminants, and environmental hygiene. Furthermore, strategy C which is the combination of vaccination of livestock, environmental hygiene, and personal protection in humans is less effective in controlling the disease while strategy D is the least among the alternatives and cannot be recommended for implementation as it has lower impact and higher cost. Critical analysis of the four control options showed that vaccination of susceptible livestock and gradual culling of seropostive cattle and small ruminants are the key parameter for any design of a control option. This paper recommends that, for effective and efficiency brucellosis transmission control, the combination of vaccination, gradual culling of seropositive cattle and small ruminants, environmental hygiene, and personal protection in humans should be adopted.



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