# A Mathematical Model for Coinfection of Listeriosis and Anthrax Diseases

Authors: Shaibu Osman; Oluwole Daniel Makinde
Journal: International Journal of Mathematics and Mathematical Sciences
 (2018)

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## Abstract

Listeriosis and Anthrax are fatal zoonotic diseases caused by Listeria monocytogene and Bacillus Anthracis, respectively. In this paper, we proposed and analysed a compartmental Listeriosis-Anthrax coinfection model describing the transmission dynamics of Listeriosis and Anthrax epidemic in human population using the stability theory of differential equations. Our model revealed that the disease-free equilibrium of the Anthrax model only is locally stable when the basic reproduction number is less than one. Sensitivity analysis was carried out on the model parameters in order to determine their impact on the disease dynamics. Numerical simulation of the coinfection model was carried out and the results are displayed graphically and discussed. We simulate the Listeriosis-Anthrax coinfection model by varying the human contact rate to see its effects on infected Anthrax population, infected Listeriosis population, and Listeriosis-Anthrax coinfected population.


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## Body

## 1. Introduction

Listeriosis and Anthrax are fatal zoonotic diseases caused by Listeria monocytogene and Bacillus Anthracis, respectively. Listeriosis in infants can be acquired in two forms. Mothers usually acquire it after eating foods that are contaminated with Listeria monocytogenes and can develop sepsis resulting in chorioamnionitis and delivering a septic infant or fetus. Moreover, mothers carrying the pathogens in the gastrointestinal tract can infect the skin and respiratory tract of their babies during childbirth. Listeria monocytogenes are among the commonest pathogens responsible for bacterial meningitis among neonates. Responsible factors for the disease include induced immune suppression linked with HIV infection, hemochromatosis hematologic malignancies, cirrhosis, diabetes, and renal failure with hemodialysis [1].Authors in [2] developed a model for Anthrax transmission but never considered the transmissions in both animal and human populations. Our model is an improvement of the work done by authors in [2, 3]. Both formulated Anthrax models but only concentrated on the disease transmissions in animals cases only. Anthrax disease is caused by bacteria infections and it affects both humans and animals. Our model is an improvement of the two models as we considered Anthrax as a zoonotic disease and also looked at sensitivity analysis and the effects of the contact rate on the disease transmissions.Authors in [4] published a paper on the effectiveness of constant and pulse vaccination policies using SIR model. The analysis of their results under constant vaccination showed that the dynamics of the disease model is similar to the dynamics without vaccination [5, 6]. There are some findings on the spread of zoonotic diseases but a number of these researches focused on the effect of vaccination on the spread and transmission of the diseases as in the case of the authors in [7]. Moreover, authors in [8] investigated a disease transmission model by considering the impact of a protective vaccine and came up with the optimal vaccine coverage threshold required for disease eradication. However, authors in [9] employed optimal control to study a nonlinear SIR epidemic model with a vaccination strategy. Several mathematical modeling techniques have been employed to study the role of optimal control using SIR epidemic model [10–12]. Authors in [13] formulated an SIR epidemic model by considering vaccination as a control measure in their model analysis. Authors in [14] developed a mathematical model for the transmission dynamics of Listeriosis in animal and human populations but did not use optimal control as a control measure in fighting the disease. They divided the animal population into four compartments by introducing the vaccination compartment.Authors in [15] formulated a model and employed optimal control to investigate the impact of chemotherapy on malaria disease with infection immigrants and [16] applied optimal control methods associated with preventing exogenous reinfection based on a exogenous reinfection tuberculosis model. Authors in [17] conducted a research on the identification and reservoirs of pathogens for effective control of sporadic disease and epidemics. Listeria monocytogenes is among the major zoonotic food borne pathogen that is responsible for approximately twenty-eight percent of most food-related deaths in the United States annually and a major cause of serious product recalls worldwide. The dairy farm has been observed as a potential point and reservoir for Listeria monocytogenes.Models are widely used in the study of transmission dynamics of infectious diseases. In recent times, the application of mathematical models in the study of infectious diseases has increased tremendously. Hence the emergence of a branch called mathematical epidemiology. Frequent diagnostic tests, the availability of clinical data, and electronic surveillance have facilitated the applications of mathematical models to critical examining of scientific hypotheses and the design of real-life strategies of controlling diseases [18, 19].Authors in [20] constructed a coinfection model of malaria and cholera diseases with optimal control but never considered sensitivity analysis and analysis of the force of infection. Sensitivity analysis determines the most sensitive parameters to the model and the analysis of the force of infections determines the effects of the contact rate on the disease transmissions.


## 2. Model Formulation

In this section, we divide the model into subcompartments (groups) as shown in Figure1. The total human population Nh is divided into subcompartments consisting of susceptible humans Sh, individuals that are infected with AnthraxIa, individuals that are infected with Listeriosis Il, individuals that are infected with both Anthrax and Listeriosis Ial, and those that have recovered from Anthrax, Listeriosis, and both Anthrax and Listeriosis, respectively, Ra, Rl, and Ral. The total vector population is represented by Nv; this is divided into subcompartments that consist of susceptible animals Sv and animals infected with Anthrax Iv, where Cp is population of carcasses of animals in the soil that may have diet of Anthrax. Carcasses of animals which may have not been properly disposed of have the tendency of generating pathogens. The total vector and human populations are represented as(1)Nh=Sh+Ia+Il+Ial+Ra+Rl+Ral.Nh=Sv+Iv,where π=Cpv/k+Cp.Figure 1
Flowchart for the coinfection model.The concentration of carcasses and ingestion rate are denoted asK and v, respectively. Listeriosis related death rates are m and η, respectively, and Anthrax related death rates are ϕ and n, respectively. Waning immunity rates are given by ω,k, and ψ.α,δ, and σ are the recovery rates, respectively, and τ1-σ are the bi-infected persons who have recovered from Anthrax only. The natural death rates of human and vector populations are μh and μv, respectively, and the modification parameter is given by θ. The coinfected persons who have recovered from Listeriosis are denoted by 1-τ1-σ. This implies that(2)σ+τ1-σ+1-τ1-σ=1.The following differential equations were obtained from the flowchart diagram of the coinfection model in Figure 1:(3)dShdt=Ωh+kRa+ωRl+ψRal-βhIvSh-πSh-μhShdIadt=βhIvSh-πIa-α+μh+ϕIadIldt=πSh-βlIvIl-δ+μh+m+ρIldIaldt=βhIvIl+πIa+σ+μh+η+θIaldRadt=αIa-k+μhRa+1-τγσIaldRldt=δIl-ω+μhRl+1-τ1-γσIaldRaldt=τσIal-ψ+μhRaldCpdt=ρIl+θIal-μbCpdSvdt=Ωv-βvIa+IalSv-μvSvdIvdt=βvIa+cISv-μvIv


## 3. Analysis of Listeriosis Only Model

In this section, only the Listeriosis model is considered in the analysis of the transmission dynamics.(4)dShdt=Ωh+ωRl-πSh-μhShdIldt=πSh-δ+μh+mIldRldt=δIl-ω+μhRldCpdt=ρIl-μbCp


### 3.1. Disease-Free Equilibrium

We obtain the disease-free equilibrium of the Listeriosis only model by setting the system of equations in (4) to zero. At disease-free equilibrium, there are no infections and recovery.(5)Ωh+ωRl-πSh-μhSh=0Sh=Ωhμhξ0l=Sh∗,Il∗,Rl∗,Cp∗=Ωhμh,0,0,0.


### 3.2. Basic Reproduction Number

In this section, the concept of the Next-Generation Matrix would be employed in computing the basic reproduction number. Using the theorem in Van den Driessche and Watmough [21] on the Listeriosis model in (4), the basic reproduction number of the Listeriosis only model, R0l, is given by(6)R0l=vρΩhμbμhKδ+μh+m


### 3.3. Existence of the Disease-Free Equilibrium

### 3.4. Endemic Equilibrium

The endemic equilibrium points are computed by setting the system of differential equations in the Listeriosis only model (4) to zero. The endemic equilibrium points are as follows:(7)Sh∗=Ωh+ωRl∗μh+π∗,Il∗=π∗Sh∗δ+μh+m,Rl∗=δIl∗ω+μh,Cp∗=ρIl∗μb.ξ0l=Sh∗,Il∗,Rl∗,Cp∗=Ωh+ωRl∗μh+π∗,π∗Sh∗δ+μh+m,δIl∗ω+μh,ρIl∗μb.(8)Sh∗=Ωh+ωRl∗μh+π∗Il∗=π∗Sh∗δ+μh+mRl∗=δIl∗ω+μhCp∗=ρIl∗μb


### 3.5. Existence of the Endemic Equilibrium

Lemma 1.
The Listeriosis only model has a unique endemic equilibrium if and only if the basic reproduction numberR0l>1.Proof.
The Listeriosis force of infection,π=Cpv/K+Cp, satisfies the polynomial;(9)Pπ∗=Aπ∗2+Bπ∗=0where A=Ωhρω+μh+μbKmω+μh+μhδ+μh+ω, and(10)B=ω+μh1-R0l.By mathematical induction, A>0 and B>0 whenever the basic reproduction number is less than one R0l<1. This implies that π∗=-B/A≤0. In conclusion, the Listeriosis model has no endemic equilibrium and the basic reproductive number is less than one R0l<1.The analysis illustrates the impossibility of backward bifurcation in the Listeriosis model, because there is no existence of endemic equilibrium whenever the basic reproduction number is less than oneR0l<1.


## 3.1. Disease-Free Equilibrium

We obtain the disease-free equilibrium of the Listeriosis only model by setting the system of equations in (4) to zero. At disease-free equilibrium, there are no infections and recovery.(5)Ωh+ωRl-πSh-μhSh=0Sh=Ωhμhξ0l=Sh∗,Il∗,Rl∗,Cp∗=Ωhμh,0,0,0.


## 3.2. Basic Reproduction Number

In this section, the concept of the Next-Generation Matrix would be employed in computing the basic reproduction number. Using the theorem in Van den Driessche and Watmough [21] on the Listeriosis model in (4), the basic reproduction number of the Listeriosis only model, R0l, is given by(6)R0l=vρΩhμbμhKδ+μh+m


## 3.3. Existence of the Disease-Free Equilibrium

## 3.4. Endemic Equilibrium

The endemic equilibrium points are computed by setting the system of differential equations in the Listeriosis only model (4) to zero. The endemic equilibrium points are as follows:(7)Sh∗=Ωh+ωRl∗μh+π∗,Il∗=π∗Sh∗δ+μh+m,Rl∗=δIl∗ω+μh,Cp∗=ρIl∗μb.ξ0l=Sh∗,Il∗,Rl∗,Cp∗=Ωh+ωRl∗μh+π∗,π∗Sh∗δ+μh+m,δIl∗ω+μh,ρIl∗μb.(8)Sh∗=Ωh+ωRl∗μh+π∗Il∗=π∗Sh∗δ+μh+mRl∗=δIl∗ω+μhCp∗=ρIl∗μb


## 3.5. Existence of the Endemic Equilibrium

Lemma 1.
The Listeriosis only model has a unique endemic equilibrium if and only if the basic reproduction numberR0l>1.Proof.
The Listeriosis force of infection,π=Cpv/K+Cp, satisfies the polynomial;(9)Pπ∗=Aπ∗2+Bπ∗=0where A=Ωhρω+μh+μbKmω+μh+μhδ+μh+ω, and(10)B=ω+μh1-R0l.By mathematical induction, A>0 and B>0 whenever the basic reproduction number is less than one R0l<1. This implies that π∗=-B/A≤0. In conclusion, the Listeriosis model has no endemic equilibrium and the basic reproductive number is less than one R0l<1.The analysis illustrates the impossibility of backward bifurcation in the Listeriosis model, because there is no existence of endemic equilibrium whenever the basic reproduction number is less than oneR0l<1.


## 4. Analysis of Anthrax Only Model

In this section, only the Anthrax model is considered in the analysis of the transmission dynamics.(11)dShdt=Ωh+kRa-βhIvSh-μhShdIadt=βIvSh-α+μh+ϕIadRadt=αIa-k+μhRadSvdt=Ωv-βvIaSv-μvSvdIvdt=βvIaSv-μvIv


### 4.1. Disease-Free Equilibrium

The disease-free equilibrium of the Anthrax only model is obtained by setting the system of equations in model (11) to zero. At disease-free equilibrium, there are no infections and recovery.(12)Ωh+kRa-βhIvSh-μhSh=0Sh=Ωhμh.Ωv-βvIaSv-μvSv=0Sv=Ωvμv.(13)ξ0a=Sh∗,Ia∗,Ra∗,Sv∗,Iv∗=Ωhμh,0,0,Ωvμv,0.


### 4.2. Basic Reproduction Number

In this section, the concept of the Next-Generation Matrix would be employed in computing the basic reproduction number. Using the theorem in Van den Driessche and Watmough [21] on the Anthrax model in (11), the basic reproduction number of the Anthrax only model, R0a, is given by(14)R0a=ΩhΩvβhβvμhμv2α+μh+ϕ


### 4.3. Stability of the Disease-Free Equilibrium

Using the next-generation operator concept in Van den Driessche and Watmough [21] on the systems of equations in model (11), the linear stability of the disease-free equilibrium, ξ0a, can be ascertained. The disease-free equilibrium is locally asymptotically stable whenever the basic reproduction number is less than one R0a<1. And it is unstable whenever the basic reproduction number is greater than one R0a>1. The disease-free equilibrium is the state at which there are no infections in the system. At disease-free equilibrium, there are no infections in the system.


### 4.4. Endemic Equilibrium

The endemic equilibrium points are computed by setting the system of differential equations in the Anthrax only model (11) to zero. The endemic equilibrium points are as follows:(15)Sh=Ωh+kRa∗μh+βhIv∗,Ia∗=βvSh∗Iv∗α+μh+ϕ,Ra∗=αIa∗k+μh,Sv∗=Ωvμv+βvIa∗,Iv∗=βvSv∗Ia∗μv.The endemic equilibrium of the Anthrax only model is given by(16)ξ0a=Sh∗,Ia∗,Ra∗,Sv∗,Iv∗=Ωh+kRa∗μh+βhIv∗,βvSh∗Iv∗α+μh+ϕ,αIa∗k+μh,Ωvμv+βvIa∗,βvSv∗Ia∗μv.(17)ξ0a=Ωh+kRa∗μh+βhIv∗,βvSh∗Iv∗α+μh+ϕ,αIa∗k+μh,Ωvμv+βvIa∗,βvSv∗Ia∗μv.


### 4.5. Existence of the Endemic Equilibrium

Lemma 2.
The Anthrax only model has a unique endemic equilibrium whenever the basic reproduction numberR0a is greater than one R0a>1.Proof.
Considering the endemic equilibrium points of the Anthrax only model,(18)ξ0a=Ωh+kRa∗μh+βhIv∗,βvSh∗Iv∗α+μh+ϕ,αIa∗k+μh,Ωvμv+βvIa∗,βvSv∗Ia∗μv.The endemic equilibrium point satisfies the given polynomial(19)PIa∗=A1Ia∗2+B1Ia∗=0where(20)A1=βvΩvβhkϕ+μhα+k+ϕ+μh+μhk+μhα+ϕ+μhμvand(21)B1=k+μh1-R0a2.By mathematical induction, A1>0 and B1>0 whenever the basic reproduction number is less than one R0a<1. This implies that Ia∗=-B1/A1≤0. In conclusion, the Anthrax only model has no endemic any time the basic reproductive number is less than one R0a<1.The analysis illustrates the impossibility of backward bifurcation in the Anthrax only model. Because there is no existence of endemic equilibrium whenever the basic reproduction number is less than oneR0a<1.


## 4.1. Disease-Free Equilibrium

The disease-free equilibrium of the Anthrax only model is obtained by setting the system of equations in model (11) to zero. At disease-free equilibrium, there are no infections and recovery.(12)Ωh+kRa-βhIvSh-μhSh=0Sh=Ωhμh.Ωv-βvIaSv-μvSv=0Sv=Ωvμv.(13)ξ0a=Sh∗,Ia∗,Ra∗,Sv∗,Iv∗=Ωhμh,0,0,Ωvμv,0.


## 4.2. Basic Reproduction Number

In this section, the concept of the Next-Generation Matrix would be employed in computing the basic reproduction number. Using the theorem in Van den Driessche and Watmough [21] on the Anthrax model in (11), the basic reproduction number of the Anthrax only model, R0a, is given by(14)R0a=ΩhΩvβhβvμhμv2α+μh+ϕ


## 4.3. Stability of the Disease-Free Equilibrium

Using the next-generation operator concept in Van den Driessche and Watmough [21] on the systems of equations in model (11), the linear stability of the disease-free equilibrium, ξ0a, can be ascertained. The disease-free equilibrium is locally asymptotically stable whenever the basic reproduction number is less than one R0a<1. And it is unstable whenever the basic reproduction number is greater than one R0a>1. The disease-free equilibrium is the state at which there are no infections in the system. At disease-free equilibrium, there are no infections in the system.


## 4.4. Endemic Equilibrium

The endemic equilibrium points are computed by setting the system of differential equations in the Anthrax only model (11) to zero. The endemic equilibrium points are as follows:(15)Sh=Ωh+kRa∗μh+βhIv∗,Ia∗=βvSh∗Iv∗α+μh+ϕ,Ra∗=αIa∗k+μh,Sv∗=Ωvμv+βvIa∗,Iv∗=βvSv∗Ia∗μv.The endemic equilibrium of the Anthrax only model is given by(16)ξ0a=Sh∗,Ia∗,Ra∗,Sv∗,Iv∗=Ωh+kRa∗μh+βhIv∗,βvSh∗Iv∗α+μh+ϕ,αIa∗k+μh,Ωvμv+βvIa∗,βvSv∗Ia∗μv.(17)ξ0a=Ωh+kRa∗μh+βhIv∗,βvSh∗Iv∗α+μh+ϕ,αIa∗k+μh,Ωvμv+βvIa∗,βvSv∗Ia∗μv.


## 4.5. Existence of the Endemic Equilibrium

Lemma 2.
The Anthrax only model has a unique endemic equilibrium whenever the basic reproduction numberR0a is greater than one R0a>1.Proof.
Considering the endemic equilibrium points of the Anthrax only model,(18)ξ0a=Ωh+kRa∗μh+βhIv∗,βvSh∗Iv∗α+μh+ϕ,αIa∗k+μh,Ωvμv+βvIa∗,βvSv∗Ia∗μv.The endemic equilibrium point satisfies the given polynomial(19)PIa∗=A1Ia∗2+B1Ia∗=0where(20)A1=βvΩvβhkϕ+μhα+k+ϕ+μh+μhk+μhα+ϕ+μhμvand(21)B1=k+μh1-R0a2.By mathematical induction, A1>0 and B1>0 whenever the basic reproduction number is less than one R0a<1. This implies that Ia∗=-B1/A1≤0. In conclusion, the Anthrax only model has no endemic any time the basic reproductive number is less than one R0a<1.The analysis illustrates the impossibility of backward bifurcation in the Anthrax only model. Because there is no existence of endemic equilibrium whenever the basic reproduction number is less than oneR0a<1.


## 5. Anthrax-Listeriosis Coinfection Model

In this section, the dynamics of the Anthrax-Listeriosis coinfection model in (3) is considered in the analysis of the transmission dynamics.


### 5.1. Disease-Free Equilibrium

The disease-free equilibrium of the Anthrax-Listeriosis model is obtained by setting the system of equations of model (3) to zero. At disease-free equilibrium, there are no infections and recovery.(22)Ωh+kRa+ωRl+ψRal-βhIvSh-πSh-μhSh=0Sh∗=ΩhμhΩv-βvIa+cIalSv-μvSv=0Sv∗=ΩvμvThe disease-free equilibrium is given by(23)ξ0al=Sh∗,Il∗,Ia∗,Ial∗,Rl∗,Ra∗,Ral∗,Cp∗,Sv∗,Iv∗(24)ξ0al=Ωhμh,0,0,0,0,0,0,0,Ωvμv,0


### 5.2. Basic Reproduction Number

The concept of the next-generation operator method in Van den Driessche and Watmough [21] was employed on the system of differential equations in model (3) to compute the basic reproduction number of the Anthrax-Listeriosis coinfection model. The Anthrax-Listeriosis coinfection model has a reproduction number Ral given by(25)Ral=max⁡Ra,Rlwhere Ra and Rl are the basic reproduction numbers of Anthrax and Listeriosis, respectively.(26)Ra=ΩhΩvβhβvμhμv2α+μh+ϕand(27)Rl=vρΩhμbμhKσ+μh+η+θ+θδ+μh+mδ+μh+mσ+μh+η+θTheorem 3.
The disease-free equilibriumξ0al is locally asymptotically stable whenever the basic reproduction number is less than one Ral<1 and unstable otherwise.


### 5.3. Impact of Listeriosis on Anthrax

In this section, the impact of Listeriosis on Anthrax and vice versa is analysed. This is done by expressing the reproduction number of one in terms of the other by expressing the basic reproduction number of Listeriosis on Anthrax, that is, expressingRl in terms of RafromRa=ΩhΩvβhβv/μhμv2α+μh+ϕ.Solving forμh in the above,(28)μh=-G1Ra+G12Ra2+4G22μvRa,where(29)G1=μvα+ϕandG2=ΩhΩvβhβvAlso, letting(30)G12Ra2+4G2=G3Ra+G4,this implies(31)μh=RaG3-G1+G42μvRaBy substituting μh into the basic reproduction number of Listeriosis Rl,(32)Rl=R0lG4+G3-G1Ra+2σ+η+θμvRa+θG4+G3-G1Ra+2m+δμvRaG4+G3-G1Ra+2σ+η+θμvRawhere the basic reproduction number of Listeriosis only model R0l is given in the relation(33)R0l=vρΩhμbμhKδ+μh+m.Now, taking the partial derivative of Rl with respect to Ra in (32) gives(34)∂Rl∂Ra=2G4θm+δ-σ+η+θμvR0lG4+G3-G1+2σ+η+θμvRa2.If m+δ≥σ+η+θ, the derivative ∂Rl/∂Ra, is strictly positive. Two scenarios can be deduced from the derivative ∂Rl/∂Ra, depending on the values of the parameters:(35)∂Rl∂Ra=0,and∂Rl∂Ra≥0.(1)
If∂Rl/∂Ra=0, it implies that m+δ=σ+η+θ and the epidemiological implication is that Anthrax has no significance effect on the transmission dynamics of Listeriosis.(2)
If∂Rl/∂Ra>0, it implies that m+δ≥σ+η+θ, and the epidemiological implication is that an increase in Anthrax cases would result in an increase Listeriosis cases in the environment. That is Anthrax enhances Listeriosis infections in the environment.However, by expressing the basic reproduction number of Anthrax on Listeriosis, that is expressingRa in terms of Rl,(36)μh=H1-H2Rl+H3Rl2+H4Rl+H52Rl,where(37)H1=1+θR0l,H2=m+δ+σ+η+θH3=σ+η+θ-m-δ,H4=2θ-1m+δ-σ-η-θR0lH1=1+θ2R0l2.By letting(38)H3Rl2+H4Rl+H5=H6Rl+H7,it implies that(39)μh=H6-H2Rl+H7+H12Rl.Therefore,(40)Ra2=4ΩhΩvβhβvRl2H6-H2Rl+H7+H1H7+H1+2α+ϕRl+H6-H2RlμvNow, taking the partial derivative of Ra with respect to Rl in equation (40) gives(41)∂Ra∂Rl=4H7+H1H7+H1+α+ϕ+H6-H2RlΩhΩvβhβvRlH6-H2Rl+H7+H12H7+H1+2α+ϕ+H6-H2Rl2μvIf the partial derivative of Ra with respect to Rl is greater than zero, ∂Ra/∂Rl>0, the biological implication is that an increase in the number of cases of Listeriosis would result in an increase in the number of cases of Anthrax in the environment. Moreover, the impact of Anthrax treatment on Listeriosis can also be analysed by taking the partial derivative of Ra with respect to α, ∂Ra/∂α.(42)∂Ra∂α=-αα+ϕ+μh.Clearly, Ra is a decreasing function of α; the epidemiological implication is that the treatment of Listeriosis would have an impact on the transmission dynamics of Anthrax.


### 5.4. Analysis of Backward Bifurcation

In this section, the phenomenon of backward bifurcation is carried out by employing the center manifold theory on the system of differential equations in model (3). Bifurcation analysis was carried out by employing the center manifold theory in Castillo-Chavez and Song [22]. Considering the human transmission rate βh and v as the bifurcation parameters, it implies that Ra=1 and Rl=1 if and only if(43)βh=βh∗=μhμv2α+ϕ+μhΩhΩvβv,and(44)v=v∗=μbμhKδ+μh+mσ+μh+η+θρΩhσ+μh+η+θ+θm+δ+μh.By considering the following change of variables,(45)Sh=x1,Ia=x2,Il=x3,Ial=x4,Ra=x5,Rl=x6,Ral=x7,Cp=x8,Sv=x9,Iv=x10.This would give the total population as(46)N=x1+x2+x3+x4+x5+x6+x7+x8+x9+x10.By applying vector notation(47)X=x1,x2,x3,x4,x5,x6,x7,x8,x9,x10T.The Anthrax-Listeriosis coinfection model can be expressed as(48)dXdt=FX,whereF=f1,f2,f3,f4,f5,f6,f7,f8,f9,f10T.The following system of differential equations is obtained:(49)dx1dt=Ωh+kx5+ωx6+ψx7-βhx10x1-πx1-μhx1dx2dt=βhx10x1-πx2-α+μh+ϕx2dx3dt=πx1-βlx10x3-δ+μh+m+ρx3dx4dt=βlx10x3+πx2+σ+μh+η+θx4dx5dt=αx2-k+μhx5+1-τγσx4dx6dt=δx3-ω+μhx6+1-τ1-γσx4dx7dt=τσx4-ψ+μhx7dx8dt=ρx3+θx4-μbx8dx9dt=Ωv-βvx2+x4x9-μvx9dx10dt=βvx2+x4x9-μvx10Backward bifurcation is carried out by employing the center manifold theory on the system of differential equations in model (3). This concept involves the computation of the Jacobian of the system of differential equations in (49) at the disease-free equilibrium ξ0. The Jacobian matrix at disease-free equilibrium is given by(50)Jξ0=-μh00J1kωψJ20J30-J40000000J300-J5J1000J200000-J60000000α0J7-J80000000δJ90-J100000000σ00-J1100000ρθ000-μb000-J120-J120000-μv00J120J1200000-μvwhere(51)J1=ρΩhμh,J2=μbδ+μh+mσ+μh+η+θρσ+μh+η+θ+θδ+μh+m,J3=μv3α+ϕ+μhΩvβv,J4=α+ϕ+μh,J5=δ+μh+m,J6=σ+μh+η+θ,J7=1-τγσ,J8=k+μh,J9=1-τ1-γσ,J10=ω+μh,J11=ψ+μhandJ12=Ωvβvμv.Clearly, the Jacobian matrix at disease-free equilibrium has a case of simple zero eigenvalue as well as other eigenvalues with negative real parts. This is an indication that the center manifold theorem is applicable. By applying the center manifold theorem in Castillo-Chavez and Song [22], the left and right eigenvectors of the Jacobian matrix J(ξ0) are computed first. Letting the left and right eigenvector represented by(52)y=y1,y2,y3,y4,y5,y6,y7,y8,y9,y10andw=w1,w2,w3,w4,w5,w6,w7,w8,w9,w10T,respectively, the following were obtained:(53)w1=Kw5μh+w2μv2α+ϕ+μhμh,w2=μv2Ωvβv,w3=w4=w6=w7=w8=0,w5=αμv2Ωvβvk+μh,w9=-w10,w10=1.And(54)y1=y3=y5=y6=y7=y8=y9=0,y2=v10Ωvβvμvα+ϕ+μh,y2=y4,y10=-μvσ+μh+η+θΩvβv.Moreover, by further simplifications, it can be shown that(55)a=τw10μv3σ+μh+η+θΩvβv-2w10βvμv2σ+μh+η+θμhΩvβv+αKμv2σ+μh+η+θμhΩvβvk+μhα+ϕ+μh,b=y2w10Ωhμh>0.It can be deduced that the coefficient b would always be positive. Backward bifurcation will take place in the system of differential equations in (3) if the coefficient a is positive. In conclusion, it implies that the disease-free equilibrium is not globally stable.Figure2 shows the simulation of the coinfection model indicating the phenomenon of backward bifurcation as evidence to the model analysis. This phenomenon usually exists in cases where the disease-free equilibrium and the endemic equilibrium coexist. Epidemically, the implication is that the concept of whenever the basic reproduction number is less than unity, the ability to control the disease is no longer sufficient. Figure 2 confirms the analytical results which shows that endemic equilibrium exists when the basic reproduction number is greater than unity.Figure 2
Simulation of the coinfection model showing the existence of backward bifurcation.


## 5.1. Disease-Free Equilibrium

The disease-free equilibrium of the Anthrax-Listeriosis model is obtained by setting the system of equations of model (3) to zero. At disease-free equilibrium, there are no infections and recovery.(22)Ωh+kRa+ωRl+ψRal-βhIvSh-πSh-μhSh=0Sh∗=ΩhμhΩv-βvIa+cIalSv-μvSv=0Sv∗=ΩvμvThe disease-free equilibrium is given by(23)ξ0al=Sh∗,Il∗,Ia∗,Ial∗,Rl∗,Ra∗,Ral∗,Cp∗,Sv∗,Iv∗(24)ξ0al=Ωhμh,0,0,0,0,0,0,0,Ωvμv,0


## 5.2. Basic Reproduction Number

The concept of the next-generation operator method in Van den Driessche and Watmough [21] was employed on the system of differential equations in model (3) to compute the basic reproduction number of the Anthrax-Listeriosis coinfection model. The Anthrax-Listeriosis coinfection model has a reproduction number Ral given by(25)Ral=max⁡Ra,Rlwhere Ra and Rl are the basic reproduction numbers of Anthrax and Listeriosis, respectively.(26)Ra=ΩhΩvβhβvμhμv2α+μh+ϕand(27)Rl=vρΩhμbμhKσ+μh+η+θ+θδ+μh+mδ+μh+mσ+μh+η+θTheorem 3.
The disease-free equilibriumξ0al is locally asymptotically stable whenever the basic reproduction number is less than one Ral<1 and unstable otherwise.


## 5.3. Impact of Listeriosis on Anthrax

In this section, the impact of Listeriosis on Anthrax and vice versa is analysed. This is done by expressing the reproduction number of one in terms of the other by expressing the basic reproduction number of Listeriosis on Anthrax, that is, expressingRl in terms of RafromRa=ΩhΩvβhβv/μhμv2α+μh+ϕ.Solving forμh in the above,(28)μh=-G1Ra+G12Ra2+4G22μvRa,where(29)G1=μvα+ϕandG2=ΩhΩvβhβvAlso, letting(30)G12Ra2+4G2=G3Ra+G4,this implies(31)μh=RaG3-G1+G42μvRaBy substituting μh into the basic reproduction number of Listeriosis Rl,(32)Rl=R0lG4+G3-G1Ra+2σ+η+θμvRa+θG4+G3-G1Ra+2m+δμvRaG4+G3-G1Ra+2σ+η+θμvRawhere the basic reproduction number of Listeriosis only model R0l is given in the relation(33)R0l=vρΩhμbμhKδ+μh+m.Now, taking the partial derivative of Rl with respect to Ra in (32) gives(34)∂Rl∂Ra=2G4θm+δ-σ+η+θμvR0lG4+G3-G1+2σ+η+θμvRa2.If m+δ≥σ+η+θ, the derivative ∂Rl/∂Ra, is strictly positive. Two scenarios can be deduced from the derivative ∂Rl/∂Ra, depending on the values of the parameters:(35)∂Rl∂Ra=0,and∂Rl∂Ra≥0.(1)
If∂Rl/∂Ra=0, it implies that m+δ=σ+η+θ and the epidemiological implication is that Anthrax has no significance effect on the transmission dynamics of Listeriosis.(2)
If∂Rl/∂Ra>0, it implies that m+δ≥σ+η+θ, and the epidemiological implication is that an increase in Anthrax cases would result in an increase Listeriosis cases in the environment. That is Anthrax enhances Listeriosis infections in the environment.However, by expressing the basic reproduction number of Anthrax on Listeriosis, that is expressingRa in terms of Rl,(36)μh=H1-H2Rl+H3Rl2+H4Rl+H52Rl,where(37)H1=1+θR0l,H2=m+δ+σ+η+θH3=σ+η+θ-m-δ,H4=2θ-1m+δ-σ-η-θR0lH1=1+θ2R0l2.By letting(38)H3Rl2+H4Rl+H5=H6Rl+H7,it implies that(39)μh=H6-H2Rl+H7+H12Rl.Therefore,(40)Ra2=4ΩhΩvβhβvRl2H6-H2Rl+H7+H1H7+H1+2α+ϕRl+H6-H2RlμvNow, taking the partial derivative of Ra with respect to Rl in equation (40) gives(41)∂Ra∂Rl=4H7+H1H7+H1+α+ϕ+H6-H2RlΩhΩvβhβvRlH6-H2Rl+H7+H12H7+H1+2α+ϕ+H6-H2Rl2μvIf the partial derivative of Ra with respect to Rl is greater than zero, ∂Ra/∂Rl>0, the biological implication is that an increase in the number of cases of Listeriosis would result in an increase in the number of cases of Anthrax in the environment. Moreover, the impact of Anthrax treatment on Listeriosis can also be analysed by taking the partial derivative of Ra with respect to α, ∂Ra/∂α.(42)∂Ra∂α=-αα+ϕ+μh.Clearly, Ra is a decreasing function of α; the epidemiological implication is that the treatment of Listeriosis would have an impact on the transmission dynamics of Anthrax.


## 5.4. Analysis of Backward Bifurcation

In this section, the phenomenon of backward bifurcation is carried out by employing the center manifold theory on the system of differential equations in model (3). Bifurcation analysis was carried out by employing the center manifold theory in Castillo-Chavez and Song [22]. Considering the human transmission rate βh and v as the bifurcation parameters, it implies that Ra=1 and Rl=1 if and only if(43)βh=βh∗=μhμv2α+ϕ+μhΩhΩvβv,and(44)v=v∗=μbμhKδ+μh+mσ+μh+η+θρΩhσ+μh+η+θ+θm+δ+μh.By considering the following change of variables,(45)Sh=x1,Ia=x2,Il=x3,Ial=x4,Ra=x5,Rl=x6,Ral=x7,Cp=x8,Sv=x9,Iv=x10.This would give the total population as(46)N=x1+x2+x3+x4+x5+x6+x7+x8+x9+x10.By applying vector notation(47)X=x1,x2,x3,x4,x5,x6,x7,x8,x9,x10T.The Anthrax-Listeriosis coinfection model can be expressed as(48)dXdt=FX,whereF=f1,f2,f3,f4,f5,f6,f7,f8,f9,f10T.The following system of differential equations is obtained:(49)dx1dt=Ωh+kx5+ωx6+ψx7-βhx10x1-πx1-μhx1dx2dt=βhx10x1-πx2-α+μh+ϕx2dx3dt=πx1-βlx10x3-δ+μh+m+ρx3dx4dt=βlx10x3+πx2+σ+μh+η+θx4dx5dt=αx2-k+μhx5+1-τγσx4dx6dt=δx3-ω+μhx6+1-τ1-γσx4dx7dt=τσx4-ψ+μhx7dx8dt=ρx3+θx4-μbx8dx9dt=Ωv-βvx2+x4x9-μvx9dx10dt=βvx2+x4x9-μvx10Backward bifurcation is carried out by employing the center manifold theory on the system of differential equations in model (3). This concept involves the computation of the Jacobian of the system of differential equations in (49) at the disease-free equilibrium ξ0. The Jacobian matrix at disease-free equilibrium is given by(50)Jξ0=-μh00J1kωψJ20J30-J40000000J300-J5J1000J200000-J60000000α0J7-J80000000δJ90-J100000000σ00-J1100000ρθ000-μb000-J120-J120000-μv00J120J1200000-μvwhere(51)J1=ρΩhμh,J2=μbδ+μh+mσ+μh+η+θρσ+μh+η+θ+θδ+μh+m,J3=μv3α+ϕ+μhΩvβv,J4=α+ϕ+μh,J5=δ+μh+m,J6=σ+μh+η+θ,J7=1-τγσ,J8=k+μh,J9=1-τ1-γσ,J10=ω+μh,J11=ψ+μhandJ12=Ωvβvμv.Clearly, the Jacobian matrix at disease-free equilibrium has a case of simple zero eigenvalue as well as other eigenvalues with negative real parts. This is an indication that the center manifold theorem is applicable. By applying the center manifold theorem in Castillo-Chavez and Song [22], the left and right eigenvectors of the Jacobian matrix J(ξ0) are computed first. Letting the left and right eigenvector represented by(52)y=y1,y2,y3,y4,y5,y6,y7,y8,y9,y10andw=w1,w2,w3,w4,w5,w6,w7,w8,w9,w10T,respectively, the following were obtained:(53)w1=Kw5μh+w2μv2α+ϕ+μhμh,w2=μv2Ωvβv,w3=w4=w6=w7=w8=0,w5=αμv2Ωvβvk+μh,w9=-w10,w10=1.And(54)y1=y3=y5=y6=y7=y8=y9=0,y2=v10Ωvβvμvα+ϕ+μh,y2=y4,y10=-μvσ+μh+η+θΩvβv.Moreover, by further simplifications, it can be shown that(55)a=τw10μv3σ+μh+η+θΩvβv-2w10βvμv2σ+μh+η+θμhΩvβv+αKμv2σ+μh+η+θμhΩvβvk+μhα+ϕ+μh,b=y2w10Ωhμh>0.It can be deduced that the coefficient b would always be positive. Backward bifurcation will take place in the system of differential equations in (3) if the coefficient a is positive. In conclusion, it implies that the disease-free equilibrium is not globally stable.Figure2 shows the simulation of the coinfection model indicating the phenomenon of backward bifurcation as evidence to the model analysis. This phenomenon usually exists in cases where the disease-free equilibrium and the endemic equilibrium coexist. Epidemically, the implication is that the concept of whenever the basic reproduction number is less than unity, the ability to control the disease is no longer sufficient. Figure 2 confirms the analytical results which shows that endemic equilibrium exists when the basic reproduction number is greater than unity.Figure 2
Simulation of the coinfection model showing the existence of backward bifurcation.


## 6. Sensitivity Analysis of the Coinfection Model

In this section, we performed the sensitivity analysis of the basic reproduction number of the coinfection model to each of the parameter values. This is to determine the significance or contribution of each parameter on the basic reproduction number. The sensitivity index of the basic reproduction numberR0 to a parameter P is given by the relation(56)ΠPR0=∂R0∂PPR0.Sensitivity analysis of the basic reproduction number of Anthrax R0a and Listeriosis R0a to each of the parameter values was computed separately, since the basic reproduction number of the coinfection model is usually(57)R0=max⁡R0a,R0l.


### 6.1. Sensitivity Indices ofR0a

In this section, we derive the sensitivity ofR0a, to each of the parameters. Table 1 shows the detailed sensitivity indices of the basic reproduction number of Anthrax R0a to each of the parameter values. From the values in Table 1, it can be observed that the most sensitive parameters are human transmission rate, vector transmission rate, human recruitment rate, and vector recruitment rate. Since the basic reproduction number is less than one, increasing the human recruitment rate by 10% would increase the basic reproduction number of Anthrax by 12.164%. However, decreasing the human recruitment rate by 10% would decrease the basic reproduction number of Anthrax by 12.164%. Moreover, decreasing human and vector transmission rates by 10% would decrease the basic reproduction number of Anthrax by 1.216% and 0.243%, respectively. However, increasing human and vector transmission rates by 10% would increase the basic reproduction number of Anthrax by 1.216% and 0.243%, respectively. The sensitivity analysis determines the contribution of each parameter to the basic reproduction number. This is an improvement of the work done by authors in [2, 3].Table 1
Sensitivity indices ofR0a to each of the parameter values.
Parameter Description Sensitivity Index Ωh Human recruitment rate 1.2164 Ωv Vector recruitment rate 0.2433 βh Human transmission rate 0.1216 βv Vector transmission rate 0.0243 α Anthrax recovery rate - 0.0037 μh Human natural death rate - 0.0122 μv Vector natural death rate - 0.0061 ϕ Anthrax related death rate - 0.0065 θ Modification parameter 3.42913 ∗ 1 0 - 6


### 6.2. Sensitivity Indices ofR0l

In this section, we derive the sensitivity ofR0l to each of the parameters. The detailed sensitivity indices of the basic reproduction number of Listeriosis R0l to each of the parameter values are shown in Table 2. We observe from the values in Table 2 that the most sensitive parameters are bacteria ingestion rate, Listeriosis related death, human recruitment rate, and Listeriosis contribution to environment. Decreasing the human recruitment rate by 10% would cause a decrease in the basic reproduction number of Listeriosis by 0.201487%. However, increasing the human recruitment rate by 10% would cause an increase in the basic reproduction number of Listeriosis by 0.201487%. Moreover, decreasing Listeriosis contribution to environment and bacteria ingestion rate by 10% would cause a decrease in the basic reproduction number of Listeriosis. Increasing Listeriosis contribution to environment and bacteria ingestion rate by 10% would cause an increase in the basic reproduction number of Listeriosis.Table 2
Sensitivity indices ofR0l to each of the parameter values.
Parameter Description Sensitivity Index Ωh Human recruitment rate 0.0201487 σ Co-infected human recovery rate - 5.41441 ∗ 1 0 - 6 μh Human natural death rate - 0.00014638 η Listeriosis death rate among co-infected - 5.41441 ∗ 1 0 - 6 θ Modification parameter 3.42913 ∗ 1 0 - 6 v Bacteria ingestion rate 0.0000402975 ρ Listeriosis contribution to environment 0.0000309981 K Concentration of carcasses - 2.01487 ∗ 1 0 - 10 δ Listeriosis recovery rate - 0.0000402218 μb Carcasses mortality rate - 0.0080595 m Listeriosis related death - 0.0000402218


## 6.1. Sensitivity Indices ofR0a

In this section, we derive the sensitivity ofR0a, to each of the parameters. Table 1 shows the detailed sensitivity indices of the basic reproduction number of Anthrax R0a to each of the parameter values. From the values in Table 1, it can be observed that the most sensitive parameters are human transmission rate, vector transmission rate, human recruitment rate, and vector recruitment rate. Since the basic reproduction number is less than one, increasing the human recruitment rate by 10% would increase the basic reproduction number of Anthrax by 12.164%. However, decreasing the human recruitment rate by 10% would decrease the basic reproduction number of Anthrax by 12.164%. Moreover, decreasing human and vector transmission rates by 10% would decrease the basic reproduction number of Anthrax by 1.216% and 0.243%, respectively. However, increasing human and vector transmission rates by 10% would increase the basic reproduction number of Anthrax by 1.216% and 0.243%, respectively. The sensitivity analysis determines the contribution of each parameter to the basic reproduction number. This is an improvement of the work done by authors in [2, 3].Table 1
Sensitivity indices ofR0a to each of the parameter values.
Parameter Description Sensitivity Index Ωh Human recruitment rate 1.2164 Ωv Vector recruitment rate 0.2433 βh Human transmission rate 0.1216 βv Vector transmission rate 0.0243 α Anthrax recovery rate - 0.0037 μh Human natural death rate - 0.0122 μv Vector natural death rate - 0.0061 ϕ Anthrax related death rate - 0.0065 θ Modification parameter 3.42913 ∗ 1 0 - 6


## 6.2. Sensitivity Indices ofR0l

In this section, we derive the sensitivity ofR0l to each of the parameters. The detailed sensitivity indices of the basic reproduction number of Listeriosis R0l to each of the parameter values are shown in Table 2. We observe from the values in Table 2 that the most sensitive parameters are bacteria ingestion rate, Listeriosis related death, human recruitment rate, and Listeriosis contribution to environment. Decreasing the human recruitment rate by 10% would cause a decrease in the basic reproduction number of Listeriosis by 0.201487%. However, increasing the human recruitment rate by 10% would cause an increase in the basic reproduction number of Listeriosis by 0.201487%. Moreover, decreasing Listeriosis contribution to environment and bacteria ingestion rate by 10% would cause a decrease in the basic reproduction number of Listeriosis. Increasing Listeriosis contribution to environment and bacteria ingestion rate by 10% would cause an increase in the basic reproduction number of Listeriosis.Table 2
Sensitivity indices ofR0l to each of the parameter values.
Parameter Description Sensitivity Index Ωh Human recruitment rate 0.0201487 σ Co-infected human recovery rate - 5.41441 ∗ 1 0 - 6 μh Human natural death rate - 0.00014638 η Listeriosis death rate among co-infected - 5.41441 ∗ 1 0 - 6 θ Modification parameter 3.42913 ∗ 1 0 - 6 v Bacteria ingestion rate 0.0000402975 ρ Listeriosis contribution to environment 0.0000309981 K Concentration of carcasses - 2.01487 ∗ 1 0 - 10 δ Listeriosis recovery rate - 0.0000402218 μb Carcasses mortality rate - 0.0080595 m Listeriosis related death - 0.0000402218


## 7. Numerical Methods and Results

In this section, we carried out the numerical simulations of the coinfection model to illustrate the results of the qualitative analysis of the model which has already been performed. The variable and parameter values in Table3 were used in the simulation of the coinfection model in (3). For the purposes of illustrations, we assumed some of the parameter values. Table 3 shows the detailed description of parameters and values that were used in the simulations of model (3). We used a Range-Kutta fourth-order scheme in the numerical solutions of the system of differential equations in model (3) by using matlab program.Table 3
Variable and parameter values of the coinfection model.
Parameter Description Value Reference ϕ Anthrax related death rate 0.2 (Health line, Dec., 2015) m Listeriosis related death rate 0.2 Adak et al., 2002. q Anthrax death rate among co-infected 0.04 assumed η Listeriosis death rate among co-infected 0.08 assumed βh Human transmission rate 0.01 [23] βv Vector transmission rate 0.05 assumed k Anthrax waning immunity 0.02 assumed μv Vector natural death rate 0.0004 [23] Ωh Human recruitment rate 0.001 assumed Ωv Vector recruitment rate 0.005 [23] α Anthrax recovery rate 0.33 [24] δ Listeriosis recovery rate 0.002 assumed ψ Anthrax-Listeriosis waning immunity 0.07 assumed ρ Listeriosis contribution to environment 0.65 assumed σ Co-infected recovery rate 0.005 assumed μb Bacteria death rate 0.0025 assumed μh Human natural death rate 0.2 [23] ω Listeriosis waning immunity 0.001 assumed θ Modification parameter 0.45 assumed ε Co-infected who recover from Anthrax only 0.025 assumed K Concentration of carcasses 10000 [20] v Bacteria ingestion rate 0.5 [20]


### 7.1. Simulation of Model Showing the Effects of Increasing Force of Infection on Infectious Anthrax and Listeriosis Populations Only

In this section, we simulate the system of differential equations in model (3) by varying the human contact rate to see its effects on infected Anthrax population, infected Listeriosis population, and Anthrax-Listeriosis coinfected population. This was done by setting the values of human contact rate as βh=0.01,βh=0.02,βh=0.03, and βh=0.04. Figure 3 shows an increase in the infected Anthrax population as the value of contact rate increases. Moreover, as the value of the human contact rate decreases, the number of Anthrax infected population decreases with time. However, an increase or decrease in the human contact rate increases or decreases the Listeriosis infected population with time as confirmed in Figure 4. The number of Anthrax-Listeriosis coinfected population shows a sharp reduction in the number of individuals infected with both diseases but the there is an increase in the number of infectious population as shown in Figure 5. An increase or decrease in the human contact rate shows an increase or decrease in the number of Anthrax-Listeriosis coinfected population as indicated in Figure 5. Analysis of force of infection gives a better understanding of the effects of the contact rate which was not considered by the work of authors in [2, 3, 20].Figure 3
Simulation showing the effects of increasing the force of infection on Anthrax infected population only.Figure 4
Simulation showing the effects of increasing the force of infection on Anthrax infected population only.Figure 5
Simulation showing the effects of increasing the force of infection on Anthrax infected population only.


### 7.2. Simulation of Model Showing Infected Anthrax, Listeriosis, and Coinfected Populations

In this section, we simulate the model (3) to see the behaviour of Anthrax infected population, Listeriosis infected population, and Anthrax-Listeriosis coinfected population. Figure 6 shows an increase in the number of Anthrax infected individuals and a sharp increase in the number of Listeriosis infected individuals. Figure 7 shows a sharp reduction in the number of Anthrax-Listeriosis coinfected population from the beginning and it increases steadily at a point in time. Since the number of susceptible human populations increases in the system with time, there are higher chances of individuals being infected with Anthrax, Listeriosis, and Anthrax-Listeriosis coinfection. This is because the concept of mass action was one of the assumptions that was incorporated in our model.Figure 6
Simulation showing infected Anthrax and infected Listeriosis population.Figure 7
Simulation of model showing Anthrax-Listeriosis coinfection population.


### 7.3. Simulation of Model Showing Susceptible Human Bacteria Populations

In this section, we simulate model (3), to observe the behaviour of the susceptible human population and how the bacteria (carcasses) growth behaves with time in the epidemics. Figure 8 shows an increase in both the susceptible and bacteria growth. An increase in the number of susceptible from the beginning confirms the increase in the number of Anthrax infection and Listeriosis infection in Figure 6. The increase in the number of susceptible human populations could be attributed to our model being an open system.Figure 8
Simulation of model showing susceptible human and bacteria populations.


## 7.1. Simulation of Model Showing the Effects of Increasing Force of Infection on Infectious Anthrax and Listeriosis Populations Only

In this section, we simulate the system of differential equations in model (3) by varying the human contact rate to see its effects on infected Anthrax population, infected Listeriosis population, and Anthrax-Listeriosis coinfected population. This was done by setting the values of human contact rate as βh=0.01,βh=0.02,βh=0.03, and βh=0.04. Figure 3 shows an increase in the infected Anthrax population as the value of contact rate increases. Moreover, as the value of the human contact rate decreases, the number of Anthrax infected population decreases with time. However, an increase or decrease in the human contact rate increases or decreases the Listeriosis infected population with time as confirmed in Figure 4. The number of Anthrax-Listeriosis coinfected population shows a sharp reduction in the number of individuals infected with both diseases but the there is an increase in the number of infectious population as shown in Figure 5. An increase or decrease in the human contact rate shows an increase or decrease in the number of Anthrax-Listeriosis coinfected population as indicated in Figure 5. Analysis of force of infection gives a better understanding of the effects of the contact rate which was not considered by the work of authors in [2, 3, 20].Figure 3
Simulation showing the effects of increasing the force of infection on Anthrax infected population only.Figure 4
Simulation showing the effects of increasing the force of infection on Anthrax infected population only.Figure 5
Simulation showing the effects of increasing the force of infection on Anthrax infected population only.


## 7.2. Simulation of Model Showing Infected Anthrax, Listeriosis, and Coinfected Populations

In this section, we simulate the model (3) to see the behaviour of Anthrax infected population, Listeriosis infected population, and Anthrax-Listeriosis coinfected population. Figure 6 shows an increase in the number of Anthrax infected individuals and a sharp increase in the number of Listeriosis infected individuals. Figure 7 shows a sharp reduction in the number of Anthrax-Listeriosis coinfected population from the beginning and it increases steadily at a point in time. Since the number of susceptible human populations increases in the system with time, there are higher chances of individuals being infected with Anthrax, Listeriosis, and Anthrax-Listeriosis coinfection. This is because the concept of mass action was one of the assumptions that was incorporated in our model.Figure 6
Simulation showing infected Anthrax and infected Listeriosis population.Figure 7
Simulation of model showing Anthrax-Listeriosis coinfection population.


## 7.3. Simulation of Model Showing Susceptible Human Bacteria Populations

In this section, we simulate model (3), to observe the behaviour of the susceptible human population and how the bacteria (carcasses) growth behaves with time in the epidemics. Figure 8 shows an increase in both the susceptible and bacteria growth. An increase in the number of susceptible from the beginning confirms the increase in the number of Anthrax infection and Listeriosis infection in Figure 6. The increase in the number of susceptible human populations could be attributed to our model being an open system.Figure 8
Simulation of model showing susceptible human and bacteria populations.


## 8. Conclusion

In this paper, we analysed the transmission dynamics of Anthrax-Listeriosis coinfection model. The compartmental model was analysed qualitatively and quantitatively to fully understand the transmission mechanism of Anthrax-Listeriosis coinfection. Our model revealed that the disease-free equilibrium of the Anthrax model only is locally stable when the basic reproduction number is less than one and a unique endemic equilibrium whenever the basic reproduction number is greater than one. The disease-free equilibrium of the Listeriosis model only is locally stable when the basic reproduction number is less than one and a unique endemic equilibrium whenever the basic reproduction number is greater than one. Our model analysis also reveals that the disease-free equilibrium of the Anthrax-Listeriosis coinfection model is locally stable whenever the basic reproduction number is less than one. The phenomenon of backward bifurcation was exhibited by our model. The biological implication is that the idea of the model been locally stable whenever the reproduction number is less than unity and unstable otherwise does not apply. This means that the Anthrax-Listeriosis coinfection model shows a case of coexistence of the disease-free equilibrium and the endemic equilibrium whenever the basic reproduction number is less than one.We performed the sensitivity analysis of the basic reproductive number to each of the parameters to determine which parameter is more sensitive. The sensitivity indices of the basic reproduction number of Anthrax to each of the parameter values revealed that the most sensitive parameters are human transmission rate, vector transmission rate, human recruitment rate, and vector recruitment rate. Since the basic reproduction number is less than one, increasing the human recruitment rate would increase the basic reproduction number. This analysis is an improvement of the work done by [2, 3]. They considered the dynamics of Anthrax in animal population but never considered sensitivity analysis to determine the most sensitive parameter to the model.The sensitivity indices of the basic reproduction number of Listeriosis to each of the parameter values shows that the most sensitive parameters are bacteria ingestion rate, Listeriosis related death, human recruitment rate, and Listeriosis contribution to environment.We simulate the Anthrax-Listeriosis coinfection model by varying the human contact rate to see its effects on infected Anthrax population, infected Listeriosis population, and Anthrax-Listeriosis coinfected population. This analysis is an improvement of the work done by authors in [2, 3, 20]. Our simulation shows an increase in the infected Anthrax population, an increase the number Listeriosis infected population, and an increase in the number of Anthrax-Listeriosis coinfected population as the value of the human contact rate increases.



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*Source: 1725671-2018-08-02.xml*