# A Theoretical Model of Listeriosis Driven by Cross Contamination of Ready-to-Eat Food Products

Authors: C. W. Chukwu; F. Nyabadza
Journal: International Journal of Mathematics and Mathematical Sciences
 (2020)

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## Abstract

Cross contamination that results in food-borne disease outbreaks remains a major problem in processed foods globally. In this paper, a mathematical model that takes into consideration cross contamination ofListeria monocytogenes from a food processing plant environment is formulated using a system of ordinary differential equations. The model has three equilibria: the disease-free equilibrium, Listeria-free equilibrium, and endemic equilibrium points. A contamination threshold ℛwf is determined. Analysis of the model shows that the disease-free equilibrium point is locally stable for ℛwf<1 while the Listeria-free and endemic equilibria are locally stable for ℛwf>1. The time-dependent sensitivity analysis is performed using Latin hypercube sampling to determine model input parameters that significantly affect the severity of the listeriosis. Numerical simulations are carried out, and the results are discussed. The results show that a reduction in the number of contaminated workers and removal of contaminated food products are essential in eliminating the disease in the human population and vice versa. The results have significant public health implications in the management and containment of any listeriosis disease outbreak.


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## Body

## 1. Introduction

Listeria is a Gram-positive facultatively anaerobic bacillus (bacteria) which was discovered by E.G.D. Murray in 1924, and in 1926, it was finally classified asListeria monocytogenes (L. monocytogenes) [1]. Among all the strains of Listeria, Listeria innocua, Listeria seeligeri, Listeria welshimeri, and Listeria ivanovii, only Listeria monocytogenes is pathogenic to both humans and animals [2]. It has an optimum growth temperature ranging between 30°C and 37°C and also at a refrigeration temperature of 4°C [3]. The main primary sources of L. monocytogenes are soil and water. It is then transmitted to animals, plant materials, and then to the food chain [2]. Listeria is one of the most common food-borne pathogens amongst others such as Salmonella, Escherichia coli, Norovirus, and Campylobacter jejuni which pose a great threat to the world population, thereby causing high fatality rates but with a low incidence rate [4]. The bacteria exist naturally in the environment and can be transmitted to food chain either through animals or into the production company by the workers. Transmission of listeriosis is mainly due to food consumption by humans rather than the interaction of the humans and the environment [2]. Epidemiologically, after humans come in contact with Listeria, the causative agent of listeriosis remains in the gastrointestinal tract and survives and develops different strategies to counteract changes in acidity, osmolarity, oxygen tension, or the challenging effects of antimicrobial peptides and bile [5]. In the infected human, L. monocytogenes crosses the epithelium barrier through the transcytosis and invades the mesenteric lymph nodes and then into the blood [6]. Most of the bacteria are then trapped in the liver cells and become part of the circulatory system. Since L. monocytogenes survives and grows even in low temperatures, the survived bacteria replicates in the hepatocytes, thereby recruiting polymorphonuclear cells which lead to hepatocytes lysis. Finally, the bacteria are released, and they circulate in the blood of the host [4]. Furthermore, in the case of pregnant women, the bacteria are transmitted to the unborn foetus either during delivery or through the placenta.The first case of the listeriosis outbreak was diagnosed in South Africa in the Western Cape province in 1977, with an average of 60–80 confirmed cases of the infected humans [7]. Recently, the outbreak occurred again in South Africa between January 2017 and June 2018 as a result of consumption of contaminated food products by consumers [8]. The source of this outbreak was traced to a food production company (majorly the meat processing facilities) that manufactures ready-to-eat (RTE) foods [8]. These are foods that are intended by the producers for direct human consumption without the need for cooking [9]. There were 1049 laboratory confirmed cases across the eight provinces, with Gauteng province having the highest number of infected individuals (58.2%, 611/1049), followed by Western Cape (12.6%, 132/1049) and KwaZulu-Natal (7.6%, 80/1049) provinces [3]. A total of 209 (26.9%) deaths were recorded as a result of the listeriosis disease outbreak [8, 10]. The most affected were neonates (babies less than 28 days old), followed by adults aged 15 to 49 years of age. Listeriosis has also been linked to be the cause of premature labour, death of new born babies, serious illness, meningitis, miscarriages, and stillbirth [3].In recent years, several mathematical models [11–14], predictive microbiology models [15, 16], empirical models [15], and statistical models [17] have been used to model the growth, survival, and death rates of Listeria. None of these models consider the aspect of cross contamination, which is essential in the dynamics of listeriosis in humans. Cross contamination remains a common route for the spread of listeriosis. Cross contamination of L. monocytogenes in fish production plants was considered in [12], cross contamination of L. monocytogenes in pork bowl slicing was considered in [16], surface cross contamination of ready-to-eat meat via slicing operation was considered in [18], and modelling listeriosis disease dynamics in human and vector population was considered in [13]. Membré et al. [15] proposed an emprical model in a single step to describe the growth, survival, and death of L. monocytogenes as a function of temprature, sodium chloride (NaCl), and phenol compounds. The authors in [12] used a differential equation model which was based on the Reed–Frost model to describe transmission of contamination during cross contamination in production plant. The objective of this study is to present a more explicit mathematical model for cross contamination of Listeria occurring in a production plant such as slicing and packaging in the preparation of RTE foods and its impact on humans through consumption of RTE foods.This paper is arranged as follows. Section1 introduces the research paper followed by model formulation in Section 2. Basic properties of the model are presented in Section 3. Model analysis is done in Section 4, and numerical simulations presented in Section 5. The paper is concluded in Section 6.


## 2. Model Formulation

We develop a mathematical model for listeriosis disease in which the human population is divided into three classes, namely, susceptiblesSht, infected Iht, and recovered Rht. We consider a constant human population over the modelling time. Individuals are recruited into the susceptible class at a rate μhN and upon infection move into the class Ih at a rate λht. After treatment, infected individuals recover at a rate γ with no immunity. Those that recover lose immunity at a rate δh and become susceptible again once if they ingest contaminated RTE food [19]. Furthermore, the bacteria can grow or die in its hosts, soil, water, and environment. We denote the growth and death rates of Listeria, Lt, by rl and d, respectively. We assume a logistic growth for Listeria with a carrying capacity KL. Uncontaminated workers Wnt can be infected by Listeria from their working environment and through contaminated food at a rate λwt, defined later after the model equations. Infected workers Wct are assumed to contaminate uncontaminated food Fnt at a rate λft. The contaminated food Fct is then responsible for the transmission of Listeria to the human population. We assume that human population Nt comprises of those individuals that do not work in the factories, while Wt is the population of workers. The population of factory workers is assumed to be constant over the model time. The total amount of food products Ft that are manufactured is assumed to be constant over the modelling time. We therefore assume a natural mortality rate μh for all human classes and a removal rate of workers μw. Contaminated workers can be decontaminated at a rate δw, through hygiene and disinfectants. We also assume that θf is the rate of addition of noncontaminated food (rate of food processing) while μf is the rate of removal of food products. Assuming that we have a constant human population, the total population Nt, the total number of workers Wt, and the total amount of food products produced Ft at any time t are, respectively, given by(1)Nt=Sht+Iht+Rht,Wt=Wnt+Wct,Ft=Fnt+Fct.We assume that the population of workers is constant over the modelling time and contamination of food is only possible through the workers, i.e., no food is manufactured in a contaminated state.Figure1 and the assumptions stated in the model formulations give rise to a nonlinear system of differential equations describing the transmission of listeriosis from RTE food products to the human population given by(2)dShdt=μhN+δhRh−μh+λhSh,dIhdt=λhSh−μh+γIh,dRhdt=γIh−μh+δhRh,dLdt=rlL1−LKL−dL,dWndt=δwWc−λw+μwWn,dWcdt=λwWn−μw+δwWc,dFndt=θf−λf+μfFn,dFcdt=λfFn−μfFc,where(3)λh=βf1Fc,λw=βLL+βf2Fc,λf=βw1Wc.Figure 1
Compartmental model diagram for transmission of Listeria from factory (production plant) to human population. The bold arrows are transition arrows, while the dotted lines depict the effects of the corresponding compartments on the transitions.All parameters for the model system (2) are assumed to be nonnegative for all time t>0. From (1), we have Rht=Nt−Sht−Iht and Wnt=Wt−Wct. Therefore, the system of equation (2) reduces to(4)dShdt=μhN+δhN−Sh−Ih−μh+λhSh,dIhdt=λhSh−μh+γIh,dLdt=rlL1−LKL−dL,dWcdt=λwW−Wc−μw+δwWc,dFndt=θf−λf+μfFn,dFcdt=λfFn−μfFc,with the initial conditions given by(5)Sh0=Sh0>0,Ih0=Ih0≥0,L0=L0≥0,Wc0=Wc0≥0,Fn0=Fn0≥0,Fc0=Fc0≥0.Set(6)sh=ShN,ih=IhN,l=LKL,wc=WcW,fn=FnF,fc=FcF.We nondimensionalise (3) and obtain the following:(7)dshdt=μh+δh1−sh−ih−μh+λ˜hsh,dihdt=λ˜hsh−μh+γih,dldt=rll1−l−dl,dwcdt=λ˜w1−wc−μw+δwwc,dfndt=θ˜f−λ˜f+μffn,dfcdt=λ˜ffn−μffc,where the nondimensionalised contamination rates are(8)λ˜h=βf1Ffc,λ˜w=βLlKL+βf2Ffc,λ˜f=βw1Wwc,and θ˜f=θf/F, with initial conditions(9)sh0=sh0>0,ih0=ih0≥0,l0=l0≥0,wc0=wc0≥0,fn0=fn0≥0,fc0=fc0≥0.


## 3. Basic Properties

### 3.1. Positivity and Boundedness of Solutions

The aim here is to obtain nonnegative solutions, since the necessary conditions under which a differential equation is nonnegative are important in analysing real life problems from which the differential equations are derived. The listeriosis model will be epidemiologically and biologically meaningful if we can establish that all the solutions with nonnegative initial data remain nonnegative at all times, i.e.,t>0 and well posed. We apply Theorem 1 to ascertain our view.Theorem 1.
For nonnegative initial conditions (9), the solutions sht,iht,lt,wct,fnt,fct of the system (7) are all nonnegative for all t≥0.Proof.
From the first equation of the system (7), we have that(10)dshdt≥−δh+μh+λ˜hsh.
Solving the first-order differential equation (10) gives(11)sht≥sh0e−δh+μh+∫λ˜hτdτ>0,where sh0 is the given initial condition. From (11), we see that(12)lim inft⟶∞sht≥0.
Thus, the solutionsht will always be positive for all t≥0. Similarly, from the second equation of model system (7),(13)dihdt≥−μh+γih,which yields(14)iht≥ih0e−μh+γt>0,where ih0 is the initial condition. Similarly, solving the remaining equations of system (7) and taking limits as t approaches infinity, it can be shown that lt>0,wct>0, fnt>0, and fct>0 for all time t>0. This completes the proof.


### 3.2. Invariant Region

The biological feasible regionΓ for the system (7) is in ℝ+6. We show that it is dissipative, that is, all the feasible solutions are uniformly bounded. We thus have the following result.Theorem 2.
The solutions of model system (7) are contained in the region Γ∈ℝ+6, which is given by Γ=sh,ih,l,wc,fn,fc∈ℝ+6:0≤sh+ih≤1,0≤l≤1,0≤wc≤1,0≤fn+fc≤θf˜/μf, for the initial conditions (9) in Γ.Proof.
The total change in human population from the system of equation (7) is given by(15)dndt=μh1−sh−ih+δh1−sh−ih−γih,for n=sh+ih≤1. Let ϕ=sh+ih so that(16)dϕdt=μh1−ϕ+δh1−ϕ−γih≤μh+δh1−ϕ.
The solution of (16) is(17)ϕt=1−ϕ0e−μh+δht,where ϕ0=sh0+ih0 is the initial condition. Applying Birkhoff and Rota’s [20] theorem on differential inequality and the condition, we have that 0≤n≤1−ϕ0e−μh+δht so that 1 is the upper bound of n provided that ϕ0≥1. Thus, every solution to system (16) with initial conditions in Γ remains in Γ for all t>0. Next, we consider the equation of the system (7) describing the transmission of the pathogen (Listeria) from the environment to the factory. We have that(18)dldt−rl−dl=−rll2,and its differential inequality is(19)dldt=rl−dl−rll2≤rl−d.
Letm=rl−d>0. Equation (19) is a Bernoulli’s equation, whose solution is obtained by setting l=y−1. Solving the left hand side of (19), we obtain lt≤m/rl+A1e−mt. As t⟶∞, lt⟶m/rl. The Listeria grows asymptotically and is able to infect the host by causing cross contamination of RTE food in the factory. In a similar approach, we found the solutions for fnt and wct to be(20)fnt=θf˜μf−A2e−μft,wct=1−A3e−λ˜wt,respectively. We note that fnt⟶θf˜/μf and wct⟶1 asymptotically as t⟶∞. From the last equation of the system (7),(21)dfcdt=λ˜ffn−μffc.
Substitutingλ˜f=βw1Wwc and fn=θf˜/μf which is the boundedness for the noncontaminated food, equation (21) becomes(22)dfcdt=βw1Wwcθf˜μf−μffc≤βw1wcθf˜μf−μffc.
Letω=βw1wcθf˜/μf; hence, (22) becomes(23)dfcdt≤ω−μffc.
The solution of (23) by the integration factor method is(24)fct=ωμf−A1e−μft.
Thus,lim supt⟶∞fct≤ω/μf. Hence, considering the human population, Listeria, contaminated workers, noncontaminated food products, and contaminated food products, the regions in Γ are invariant and biologically meaningful.


## 3.1. Positivity and Boundedness of Solutions

The aim here is to obtain nonnegative solutions, since the necessary conditions under which a differential equation is nonnegative are important in analysing real life problems from which the differential equations are derived. The listeriosis model will be epidemiologically and biologically meaningful if we can establish that all the solutions with nonnegative initial data remain nonnegative at all times, i.e.,t>0 and well posed. We apply Theorem 1 to ascertain our view.Theorem 1.
For nonnegative initial conditions (9), the solutions sht,iht,lt,wct,fnt,fct of the system (7) are all nonnegative for all t≥0.Proof.
From the first equation of the system (7), we have that(10)dshdt≥−δh+μh+λ˜hsh.
Solving the first-order differential equation (10) gives(11)sht≥sh0e−δh+μh+∫λ˜hτdτ>0,where sh0 is the given initial condition. From (11), we see that(12)lim inft⟶∞sht≥0.
Thus, the solutionsht will always be positive for all t≥0. Similarly, from the second equation of model system (7),(13)dihdt≥−μh+γih,which yields(14)iht≥ih0e−μh+γt>0,where ih0 is the initial condition. Similarly, solving the remaining equations of system (7) and taking limits as t approaches infinity, it can be shown that lt>0,wct>0, fnt>0, and fct>0 for all time t>0. This completes the proof.


## 3.2. Invariant Region

The biological feasible regionΓ for the system (7) is in ℝ+6. We show that it is dissipative, that is, all the feasible solutions are uniformly bounded. We thus have the following result.Theorem 2.
The solutions of model system (7) are contained in the region Γ∈ℝ+6, which is given by Γ=sh,ih,l,wc,fn,fc∈ℝ+6:0≤sh+ih≤1,0≤l≤1,0≤wc≤1,0≤fn+fc≤θf˜/μf, for the initial conditions (9) in Γ.Proof.
The total change in human population from the system of equation (7) is given by(15)dndt=μh1−sh−ih+δh1−sh−ih−γih,for n=sh+ih≤1. Let ϕ=sh+ih so that(16)dϕdt=μh1−ϕ+δh1−ϕ−γih≤μh+δh1−ϕ.
The solution of (16) is(17)ϕt=1−ϕ0e−μh+δht,where ϕ0=sh0+ih0 is the initial condition. Applying Birkhoff and Rota’s [20] theorem on differential inequality and the condition, we have that 0≤n≤1−ϕ0e−μh+δht so that 1 is the upper bound of n provided that ϕ0≥1. Thus, every solution to system (16) with initial conditions in Γ remains in Γ for all t>0. Next, we consider the equation of the system (7) describing the transmission of the pathogen (Listeria) from the environment to the factory. We have that(18)dldt−rl−dl=−rll2,and its differential inequality is(19)dldt=rl−dl−rll2≤rl−d.
Letm=rl−d>0. Equation (19) is a Bernoulli’s equation, whose solution is obtained by setting l=y−1. Solving the left hand side of (19), we obtain lt≤m/rl+A1e−mt. As t⟶∞, lt⟶m/rl. The Listeria grows asymptotically and is able to infect the host by causing cross contamination of RTE food in the factory. In a similar approach, we found the solutions for fnt and wct to be(20)fnt=θf˜μf−A2e−μft,wct=1−A3e−λ˜wt,respectively. We note that fnt⟶θf˜/μf and wct⟶1 asymptotically as t⟶∞. From the last equation of the system (7),(21)dfcdt=λ˜ffn−μffc.
Substitutingλ˜f=βw1Wwc and fn=θf˜/μf which is the boundedness for the noncontaminated food, equation (21) becomes(22)dfcdt=βw1Wwcθf˜μf−μffc≤βw1wcθf˜μf−μffc.
Letω=βw1wcθf˜/μf; hence, (22) becomes(23)dfcdt≤ω−μffc.
The solution of (23) by the integration factor method is(24)fct=ωμf−A1e−μft.
Thus,lim supt⟶∞fct≤ω/μf. Hence, considering the human population, Listeria, contaminated workers, noncontaminated food products, and contaminated food products, the regions in Γ are invariant and biologically meaningful.


## 4. Model Analysis

### 4.1. Listeriosis Model Equilibria

We solve for the equilibrium points for the system (7), by setting the right side of system (7) to zero, so that(25)μh+δh1−sh∗−ih∗−μh+λ˜hsh∗=0,λ˜hsh∗−μh+γih∗=0,rll∗1−l∗−dl∗=0,λ˜w1−wc∗−μw+δwwc∗=0,θ˜f−λ˜f+μffn∗=0,λ˜ffn∗−μffc∗=0,with β1=βf1F,β2=βLKL,β3=βf2F,β4=βw1W, and m=rl−d. Solving system (25) for the equilibrium points, from the third equation of system (25), we have(26)l∗m−rll∗=0.Sol∗=0 or l∗=m/rl. Note that, for Listeria to exist, then rl>d. Let β˜1=β1fc∗; hence, the first and second equations of system (25) become(27)μh+β˜1sh∗−δh1−sh∗−ih∗−μh=0,β˜1sh∗−μh+γih∗=0.Solving (27) for sh∗ and ih∗ simultaneously, we have(28)sh∗=μh+γμh+δhμh+δhμh+γ+β1˜γ+δh+1,ih∗=β1˜μh+δhμh+δhμh+γ+β1˜γ+δh+1.Using the fourth equation, we obtain(29)β2l∗+β3fc∗1−wc∗−μw+δwwc∗=0,⟹wc∗=β2l∗+β3fc∗β2l∗+β3fc∗+μw+δw.On the other hand, from the last equation of (25) we have that(30)fn∗=μffc∗λ˜f⋅Substituting (30) into the second last equation of (25) and after some algebraic manipulations, we obtain the following quadratic expression:(31)ν2fc∗2+ν1fc∗+ν0=0,where(32)ν0=−β2β4θf˜l∗<0,ν1=−β3β4θf˜+β2β4μfl∗+β2μf2l∗+μwμf2+δwμf2,ν2=β3β4μf+β3μf2>0.Note thatν1<0 or ν1>0.Whenl∗=0,(33)ν2fc∗2+ν1fc∗=0.Now, solving forfc∗, we obtain(34)fc1∗=0orfc2∗=−ν1ν2.Considering the casefc2∗=−ν1/ν2, we have(35)fc2∗=β3β4θf˜−μf2μw−μf2δwβ3β4μf+β3μf2=μfμw+δwβ3β4+μfβ3β4θf˜μf2μw+δw−1=Φ1ℛwf−1,where(36)Φ1=μfμw+δwβ3β4+μf,ℛwf=β3β4θf˜μf2μw+δw.We thus callℛwf the contamination threshold. We thus have the following result on the existence of fc2∗.Lemma 1.
The steady-statefc2∗ exists if and only if ℛwf>1.It is important to note thatℛwf denotes the measure of contamination caused by workers and food products. This is equivalent to the reproduction number ℛ0 in disease modelling [21], where ℛ0 determines whether a disease is established or eradicated. In this case, ℛwf determines whether listeriosis is established or eradicated, i.e., no contamination of food products.Iffc1∗=0, then wc∗=0 and λ˜h=0 which gives ih∗=0 and(37)sh∗=μh+δh−δhih∗δh+μh+λ˜h=1,and fn1∗=θf˜/μf. This results in the disease-free equilibrium (DFE) point E0∗ given by(38)E0∗=sh∗,ih∗,l0∗,fn1∗,wc1∗,fc1∗=1,0,0,0,θf˜μf,0.At DFE, there are no Listeria, contaminated workers, food, and infected and recovered humans. Given that(39)wc∗=β3fc∗β3fc∗+μw+δw,when l∗=0, we have(40)wc2∗=β3β3β4θf˜−μf2μw+δwβ3β3β4θf˜−μf2μw+δw+β3μfμw+δwβ4+μf.Some algebraic manipulations give(41)wc2∗=Φ2ℛwf−1,where Φ2=μf2μw+δw/β4β3θf˜+μfμw+δw. Solving for fn2∗ by substituting (35) and (41) into (30), we obtain(42)fn2∗=β3θf˜+μfμw+δwβ3β4+μf,=Φ3μfβ4ℛwf+1,where Φ3=μfμw+δw/β3β4+μf. We thus have the Listeria-free equilibrium point:(43)E1∗=sh∗,ih∗,l1∗,fn2∗,wc2∗,fc2∗,where(44)sh∗=μh+γμh+δhμh+δhμh+γ+β1Φ1ℛwf−1γ+δh+1,ih∗=β1˜μh+δhμh+δhμh+γ+β1Φ1ℛwf−1γ+δh+1,l1∗=0,wc2∗=Φ2ℛwf−1,fn2∗=Φ3μfβ4ℛwf+1,fc2∗=Φ1ℛwf−1.This implies that as long as we have contaminated workers or food, we will still have the disease in the human population. Thus, removal of contaminated food is essential for disease control. Whenl∗≠0, then(45)ν0=−β2β4θf˜l∗<0,ν1=β2μfl∗β4+μf+μf2μw+δw1−ℛwf,ν2=β3β4μf+β3μf2>0.Solving (31) for fc∗, we obtain(46)fc∗=−ν1±ν12−4ν2ν02ν2⋅Ifν1>0 and ν1<0, we have one positive solution. Thus, irrespective of the sign of ν1, we have one positive root for fc∗, say fc∗+. From (25), we now have(47)λ˜h=β1fc∗+,λ˜w=β2l∗+β3fc∗+,λ˜f=β4wc∗,θf˜=θfF,with l∗=m/rl. Let β1˜+=β1fc∗+, and similarly solving for sh∗+ and ih∗+ simultaneously, we obtain(48)sh∗+=μh+γμh+δhμh+δhμh+γ+β1˜+γ+δh+1,ih∗+=β1˜+μh+δhμh+δhμh+γ+β1˜+γ+δh+1.Ifλ˜w=β2l∗+β3fc∗+, then from the fourth equation of (25), we obtain(49)wc∗+=β2l∗+β3fc∗+β2l∗+β3fc∗++μw+δw⋅Now, settingλ˜f=β4wc∗+ from second last equation of (25), we have(50)η2fc∗+2+η1fc∗++η0=0,where(51)η2=β3β4+μf2>0,η1=−β3β4θf˜+β2μfmrlβ4+μf+μf2μw+δw,η0=−β2β4θf˜mrl<0.Further simplification ofη1 gives the following expression:(52)η1=β2μfmrlβ4+μf+μf2μw+δw1−ℛwf.However, we note that ifℛwf>1, then η1<0, and if ℛwf<1, then η1>0. Hence, solving (50) for fc∗+, we obtain(53)fc∗+=−η1±η12−4η2η02η2⋅Ifη1>0 or η1<0, (53) we will always have one positive root. Therefore, irrespective of the sign of η1, we will always have one positive root. We thus have the following result.Theorem 3.
Model system (7) has a unique endemic equilibrium as long as m>0, i.e., rl>d.Remark 1.
As long as the growth rate of Listeria is greater than the death rate, we will always have listeriosis.
Also, note thatfc∗+ is the positive root of (50). We cannot explicitly write the expression of the roots for fc∗+ due to the intractability of expression (50). Thus, the endemic equilibrium,(54)E2∗=sh∗+,ih∗+,l2∗,wc∗+,fn∗+,is given by(55)sh∗+=μh+γμh+δhμh+δhμh+γ+β1˜+γ+δh+1,ih∗+=β1˜+μh+δhμh+δhμh+γ+β1˜+γ+δh+1,l2∗=mrl,wc∗+=β2l∗+β3fc∗+β2l∗+β3fc∗++μw+δw,fn2∗+=β3θf˜+μfμw+δwβ3β4+μf⋅
We note that at endemic equilibrium, there is infection in the human population as a result of ingestion of contaminated RTE foods. Also, the workers and the food are contaminated.


### 4.2. Stability of Equilibria Points

#### 4.2.1. Local Stability of the Disease-Free Equilibrium

We now state the following theorem for the local stability of disease-free equilibrium.Theorem 4.
The equilibrium pointE0∗ is locally asymptotically stable if and only if rl<d and ℛwf<1; otherwise, it is unstable.Proof.
The Jacobian of system (7) is given by the block matrix:(56)where(57)A=−δh+μh+β1fc∗−δh0β1fc∗−μh+γ000rl−2rll∗−d,B=00−β1sh∗00β1sh∗000,C=00β21−wc∗000000,D=−β2l∗+β3fc∗+μw+δw0β31−wc∗−β4fn∗−β4wc∗+μf0β4fn∗β4wc∗−μf⋅
Evaluating at DFEE0∗, we have that(58)where(59)J1E0∗=−δh+μh−δh00−μh+γ000rl−d,J2E0∗=00−β100β1000,J3E0∗=00β2000000,J4E0∗=−μw+δw0β3−β4θf˜μf−μf0β4θf˜μf0−μf⋅
The eigenvalues fromJ1E0∗ are λ1=rl−d<0 provided that rl<d, λ2=−μw+δw, and λ3=−μh+γ. The rest of the eigenvalues are determined from J4E0∗, which is the solution of the cubic equation:(60)λ3+ξ2λ2+ξ1λ+ξ0=0,given that(61)ξ0=−β3β4θfμf+μf3δw+μw,ξ1=−β3β4θf+μf22δw+μf+2μw,ξ2=μf2μf+μw+δw>0.
Further simplification ofξ1 and ξ0 gives(62)ξ1=μf3+2μf2μw+δw1−β3β4θf2μf2μw+δw,=μf3+2μf2μw+δw1−12ℛwf,ξ0=μf2μw+δw1−β3β4θfμf2μw+δwμf,=μf3μw+δw1−ℛwf,respectively. We note that if ℛwf<1, then ξ1>0 and ξ0>0. Next, we show that ξ2ξ1>ξ0 as follows:(63)μf2μf+μw+δwμf3+2μf2μw+δw1−12ℛwf=2μf+μw+δwμf4+2μf3μw+δw1−12ℛwf>μf3μw+δw2−ℛwf>μf3μw+δw1−ℛwf.
Note that2−ℛwf>1−ℛwf if ℛwf<1. Thus, according to Routh–Hurwitz criteria, all the roots of (60) have negative real parts.


#### 4.2.2. Local Stability of the Listeria-Free Equilibrium

We state the following theorem for the local stability of Listeria-free equilibrium.Theorem 5.
The equilibriumE1∗ is locally asymptotically stable if and only if rl<d and ℛwf≥1; otherwise, it is unstable.Proof.
Evaluating the Jacobian of system (7) at E1∗ gives(64)where(65)J1E1∗=−δh+μh+β1fc2∗−δh0β1fc2∗−μh+γ000rl−d,J2E1∗=00−β1sh∗00β1sh∗000,J3E1∗=00β21−wc2∗000000,J4E1∗=−β3fc2∗+μw+δw0β31−wc2∗−β4fn2∗−β4wc2∗+μf0β4fn2∗β4wc2∗−μf.
The eigenvalues fromJ1E1∗ are λ1=rl−d<0 provided that rl<d, which is the case given by Listeria-free equilibrium and the solutions of the quadratic equation:(66)λ2+ζ1λ+ζ0=0,where(67)ζ0=γβ1fc2∗+γδh+β1δhfc2∗+γμh+β1μhfc2∗+δhμh+μh2>0,ζ1=γ+β1fc2∗+δh+2μh>0.
The eigenvalues of (66) have negative real parts since ζ0>0 and ζ1>0 using Routh–Hurwitz criterion. The rest of the eigenvalues are obtained from J4E1∗, which are the solutions to the cubic equation:(68)λ3+ζ4λ2+ζ3λ+ζ2=0,where(69)ζ2=μf+β4wc2∗μfδw+μfμw+μfβ3fc2∗+wc2∗−1β3β4fn2∗,ζ3=μf2δw+μf+2μw+β4wc2∗+2β3fc2∗+β4wc2∗δw+μw+β3fc2∗−β3β4fn2∗1−wc2∗,ζ4=δw+2μf+μw+β4wc2∗+β3fc2∗>0.
We note that ifwc2∗>1, then ζ2>0 and ζ3>0. Next, we show that ζ4ζ3>ζ2 as follows:(70)ζ4ζ3=δw+2μf+μw+β4wc2∗+β3fc2∗2μfδw+μf2+2μfμw+β4wc2∗μf+2β3fc2∗μf+β3β4fn2∗wc2∗−1>ζ2.
Thus, by the Routh–Hurwitz criterion, the eigenvalues of (68) have negative real parts. Thus, E1∗ is locally asymptotically stable.


#### 4.2.3. Local Stability of the Endemic Equilibrium

We state the following theorem for the local stability of endemic equilibrium.Theorem 6.
The equilibrium pointE2∗ is locally asymptotically stable if ℛwf≥1 and unstable otherwise.Proof.
Evaluating the Jacobian of system (7) at E2∗ gives(71)where(72)J1E1∗=−δh+μh+β1fc∗+−δh0β1fc∗+−μh+γ000rl−2rlmrl−d,J2E1∗=00−β1sh∗+00β1sh∗+000,J3E1∗=00β21−wc∗+000000,J4E1∗=−τ0β31−φ−β4δ−β4φ+μf0β4δβ4φ−μf,given that τ=β2m/rl+β3fc∗++μw+δw>0, φ=wc∗+=β2m/rl+β3fc∗+/β2m/rl+β3fc∗++μw+δw>0, and δ=fn∗+>0. The eigenvalues of J1E2∗ are λ3=−rl−d, and the solutions of the quadratic equation are(73)λ2+b1λ+b0=0,where(74)b0=β1fc∗+γ+δh+μh+γδh+μh+μhδh+μh>0,b1=β1fc∗++γ+δh+2μh>0.
The eigenvalues of (73) have negative real parts since from Routh–Hurwitz criterion b0>0 and b1>0. The rest of the eigenvalues are obtained from J4E1∗, which are the solutions to the cubic equation:(75)λ3+b4λ2+b3λ+b2=0,where(76)b2=μfτφβ4+τμf+β3β4δφ−1,b3=τφβ4+μf2τ+φβ4+μf+β3β4δφ−1,b4=τ+φβ4+2μf>0.
We note that ifφ>1, then b2>0 and b3>0. Also, it is clear that b4b3>b2. Therefore, their eigenvalues have negative real parts using the principles of Routh–Hurwitz criterion. Hence, E2∗ is locally stable.


## 4.1. Listeriosis Model Equilibria

We solve for the equilibrium points for the system (7), by setting the right side of system (7) to zero, so that(25)μh+δh1−sh∗−ih∗−μh+λ˜hsh∗=0,λ˜hsh∗−μh+γih∗=0,rll∗1−l∗−dl∗=0,λ˜w1−wc∗−μw+δwwc∗=0,θ˜f−λ˜f+μffn∗=0,λ˜ffn∗−μffc∗=0,with β1=βf1F,β2=βLKL,β3=βf2F,β4=βw1W, and m=rl−d. Solving system (25) for the equilibrium points, from the third equation of system (25), we have(26)l∗m−rll∗=0.Sol∗=0 or l∗=m/rl. Note that, for Listeria to exist, then rl>d. Let β˜1=β1fc∗; hence, the first and second equations of system (25) become(27)μh+β˜1sh∗−δh1−sh∗−ih∗−μh=0,β˜1sh∗−μh+γih∗=0.Solving (27) for sh∗ and ih∗ simultaneously, we have(28)sh∗=μh+γμh+δhμh+δhμh+γ+β1˜γ+δh+1,ih∗=β1˜μh+δhμh+δhμh+γ+β1˜γ+δh+1.Using the fourth equation, we obtain(29)β2l∗+β3fc∗1−wc∗−μw+δwwc∗=0,⟹wc∗=β2l∗+β3fc∗β2l∗+β3fc∗+μw+δw.On the other hand, from the last equation of (25) we have that(30)fn∗=μffc∗λ˜f⋅Substituting (30) into the second last equation of (25) and after some algebraic manipulations, we obtain the following quadratic expression:(31)ν2fc∗2+ν1fc∗+ν0=0,where(32)ν0=−β2β4θf˜l∗<0,ν1=−β3β4θf˜+β2β4μfl∗+β2μf2l∗+μwμf2+δwμf2,ν2=β3β4μf+β3μf2>0.Note thatν1<0 or ν1>0.Whenl∗=0,(33)ν2fc∗2+ν1fc∗=0.Now, solving forfc∗, we obtain(34)fc1∗=0orfc2∗=−ν1ν2.Considering the casefc2∗=−ν1/ν2, we have(35)fc2∗=β3β4θf˜−μf2μw−μf2δwβ3β4μf+β3μf2=μfμw+δwβ3β4+μfβ3β4θf˜μf2μw+δw−1=Φ1ℛwf−1,where(36)Φ1=μfμw+δwβ3β4+μf,ℛwf=β3β4θf˜μf2μw+δw.We thus callℛwf the contamination threshold. We thus have the following result on the existence of fc2∗.Lemma 1.
The steady-statefc2∗ exists if and only if ℛwf>1.It is important to note thatℛwf denotes the measure of contamination caused by workers and food products. This is equivalent to the reproduction number ℛ0 in disease modelling [21], where ℛ0 determines whether a disease is established or eradicated. In this case, ℛwf determines whether listeriosis is established or eradicated, i.e., no contamination of food products.Iffc1∗=0, then wc∗=0 and λ˜h=0 which gives ih∗=0 and(37)sh∗=μh+δh−δhih∗δh+μh+λ˜h=1,and fn1∗=θf˜/μf. This results in the disease-free equilibrium (DFE) point E0∗ given by(38)E0∗=sh∗,ih∗,l0∗,fn1∗,wc1∗,fc1∗=1,0,0,0,θf˜μf,0.At DFE, there are no Listeria, contaminated workers, food, and infected and recovered humans. Given that(39)wc∗=β3fc∗β3fc∗+μw+δw,when l∗=0, we have(40)wc2∗=β3β3β4θf˜−μf2μw+δwβ3β3β4θf˜−μf2μw+δw+β3μfμw+δwβ4+μf.Some algebraic manipulations give(41)wc2∗=Φ2ℛwf−1,where Φ2=μf2μw+δw/β4β3θf˜+μfμw+δw. Solving for fn2∗ by substituting (35) and (41) into (30), we obtain(42)fn2∗=β3θf˜+μfμw+δwβ3β4+μf,=Φ3μfβ4ℛwf+1,where Φ3=μfμw+δw/β3β4+μf. We thus have the Listeria-free equilibrium point:(43)E1∗=sh∗,ih∗,l1∗,fn2∗,wc2∗,fc2∗,where(44)sh∗=μh+γμh+δhμh+δhμh+γ+β1Φ1ℛwf−1γ+δh+1,ih∗=β1˜μh+δhμh+δhμh+γ+β1Φ1ℛwf−1γ+δh+1,l1∗=0,wc2∗=Φ2ℛwf−1,fn2∗=Φ3μfβ4ℛwf+1,fc2∗=Φ1ℛwf−1.This implies that as long as we have contaminated workers or food, we will still have the disease in the human population. Thus, removal of contaminated food is essential for disease control. Whenl∗≠0, then(45)ν0=−β2β4θf˜l∗<0,ν1=β2μfl∗β4+μf+μf2μw+δw1−ℛwf,ν2=β3β4μf+β3μf2>0.Solving (31) for fc∗, we obtain(46)fc∗=−ν1±ν12−4ν2ν02ν2⋅Ifν1>0 and ν1<0, we have one positive solution. Thus, irrespective of the sign of ν1, we have one positive root for fc∗, say fc∗+. From (25), we now have(47)λ˜h=β1fc∗+,λ˜w=β2l∗+β3fc∗+,λ˜f=β4wc∗,θf˜=θfF,with l∗=m/rl. Let β1˜+=β1fc∗+, and similarly solving for sh∗+ and ih∗+ simultaneously, we obtain(48)sh∗+=μh+γμh+δhμh+δhμh+γ+β1˜+γ+δh+1,ih∗+=β1˜+μh+δhμh+δhμh+γ+β1˜+γ+δh+1.Ifλ˜w=β2l∗+β3fc∗+, then from the fourth equation of (25), we obtain(49)wc∗+=β2l∗+β3fc∗+β2l∗+β3fc∗++μw+δw⋅Now, settingλ˜f=β4wc∗+ from second last equation of (25), we have(50)η2fc∗+2+η1fc∗++η0=0,where(51)η2=β3β4+μf2>0,η1=−β3β4θf˜+β2μfmrlβ4+μf+μf2μw+δw,η0=−β2β4θf˜mrl<0.Further simplification ofη1 gives the following expression:(52)η1=β2μfmrlβ4+μf+μf2μw+δw1−ℛwf.However, we note that ifℛwf>1, then η1<0, and if ℛwf<1, then η1>0. Hence, solving (50) for fc∗+, we obtain(53)fc∗+=−η1±η12−4η2η02η2⋅Ifη1>0 or η1<0, (53) we will always have one positive root. Therefore, irrespective of the sign of η1, we will always have one positive root. We thus have the following result.Theorem 3.
Model system (7) has a unique endemic equilibrium as long as m>0, i.e., rl>d.Remark 1.
As long as the growth rate of Listeria is greater than the death rate, we will always have listeriosis.
Also, note thatfc∗+ is the positive root of (50). We cannot explicitly write the expression of the roots for fc∗+ due to the intractability of expression (50). Thus, the endemic equilibrium,(54)E2∗=sh∗+,ih∗+,l2∗,wc∗+,fn∗+,is given by(55)sh∗+=μh+γμh+δhμh+δhμh+γ+β1˜+γ+δh+1,ih∗+=β1˜+μh+δhμh+δhμh+γ+β1˜+γ+δh+1,l2∗=mrl,wc∗+=β2l∗+β3fc∗+β2l∗+β3fc∗++μw+δw,fn2∗+=β3θf˜+μfμw+δwβ3β4+μf⋅
We note that at endemic equilibrium, there is infection in the human population as a result of ingestion of contaminated RTE foods. Also, the workers and the food are contaminated.


## 4.2. Stability of Equilibria Points

### 4.2.1. Local Stability of the Disease-Free Equilibrium

We now state the following theorem for the local stability of disease-free equilibrium.Theorem 4.
The equilibrium pointE0∗ is locally asymptotically stable if and only if rl<d and ℛwf<1; otherwise, it is unstable.Proof.
The Jacobian of system (7) is given by the block matrix:(56)where(57)A=−δh+μh+β1fc∗−δh0β1fc∗−μh+γ000rl−2rll∗−d,B=00−β1sh∗00β1sh∗000,C=00β21−wc∗000000,D=−β2l∗+β3fc∗+μw+δw0β31−wc∗−β4fn∗−β4wc∗+μf0β4fn∗β4wc∗−μf⋅
Evaluating at DFEE0∗, we have that(58)where(59)J1E0∗=−δh+μh−δh00−μh+γ000rl−d,J2E0∗=00−β100β1000,J3E0∗=00β2000000,J4E0∗=−μw+δw0β3−β4θf˜μf−μf0β4θf˜μf0−μf⋅
The eigenvalues fromJ1E0∗ are λ1=rl−d<0 provided that rl<d, λ2=−μw+δw, and λ3=−μh+γ. The rest of the eigenvalues are determined from J4E0∗, which is the solution of the cubic equation:(60)λ3+ξ2λ2+ξ1λ+ξ0=0,given that(61)ξ0=−β3β4θfμf+μf3δw+μw,ξ1=−β3β4θf+μf22δw+μf+2μw,ξ2=μf2μf+μw+δw>0.
Further simplification ofξ1 and ξ0 gives(62)ξ1=μf3+2μf2μw+δw1−β3β4θf2μf2μw+δw,=μf3+2μf2μw+δw1−12ℛwf,ξ0=μf2μw+δw1−β3β4θfμf2μw+δwμf,=μf3μw+δw1−ℛwf,respectively. We note that if ℛwf<1, then ξ1>0 and ξ0>0. Next, we show that ξ2ξ1>ξ0 as follows:(63)μf2μf+μw+δwμf3+2μf2μw+δw1−12ℛwf=2μf+μw+δwμf4+2μf3μw+δw1−12ℛwf>μf3μw+δw2−ℛwf>μf3μw+δw1−ℛwf.
Note that2−ℛwf>1−ℛwf if ℛwf<1. Thus, according to Routh–Hurwitz criteria, all the roots of (60) have negative real parts.


### 4.2.2. Local Stability of the Listeria-Free Equilibrium

We state the following theorem for the local stability of Listeria-free equilibrium.Theorem 5.
The equilibriumE1∗ is locally asymptotically stable if and only if rl<d and ℛwf≥1; otherwise, it is unstable.Proof.
Evaluating the Jacobian of system (7) at E1∗ gives(64)where(65)J1E1∗=−δh+μh+β1fc2∗−δh0β1fc2∗−μh+γ000rl−d,J2E1∗=00−β1sh∗00β1sh∗000,J3E1∗=00β21−wc2∗000000,J4E1∗=−β3fc2∗+μw+δw0β31−wc2∗−β4fn2∗−β4wc2∗+μf0β4fn2∗β4wc2∗−μf.
The eigenvalues fromJ1E1∗ are λ1=rl−d<0 provided that rl<d, which is the case given by Listeria-free equilibrium and the solutions of the quadratic equation:(66)λ2+ζ1λ+ζ0=0,where(67)ζ0=γβ1fc2∗+γδh+β1δhfc2∗+γμh+β1μhfc2∗+δhμh+μh2>0,ζ1=γ+β1fc2∗+δh+2μh>0.
The eigenvalues of (66) have negative real parts since ζ0>0 and ζ1>0 using Routh–Hurwitz criterion. The rest of the eigenvalues are obtained from J4E1∗, which are the solutions to the cubic equation:(68)λ3+ζ4λ2+ζ3λ+ζ2=0,where(69)ζ2=μf+β4wc2∗μfδw+μfμw+μfβ3fc2∗+wc2∗−1β3β4fn2∗,ζ3=μf2δw+μf+2μw+β4wc2∗+2β3fc2∗+β4wc2∗δw+μw+β3fc2∗−β3β4fn2∗1−wc2∗,ζ4=δw+2μf+μw+β4wc2∗+β3fc2∗>0.
We note that ifwc2∗>1, then ζ2>0 and ζ3>0. Next, we show that ζ4ζ3>ζ2 as follows:(70)ζ4ζ3=δw+2μf+μw+β4wc2∗+β3fc2∗2μfδw+μf2+2μfμw+β4wc2∗μf+2β3fc2∗μf+β3β4fn2∗wc2∗−1>ζ2.
Thus, by the Routh–Hurwitz criterion, the eigenvalues of (68) have negative real parts. Thus, E1∗ is locally asymptotically stable.


### 4.2.3. Local Stability of the Endemic Equilibrium

We state the following theorem for the local stability of endemic equilibrium.Theorem 6.
The equilibrium pointE2∗ is locally asymptotically stable if ℛwf≥1 and unstable otherwise.Proof.
Evaluating the Jacobian of system (7) at E2∗ gives(71)where(72)J1E1∗=−δh+μh+β1fc∗+−δh0β1fc∗+−μh+γ000rl−2rlmrl−d,J2E1∗=00−β1sh∗+00β1sh∗+000,J3E1∗=00β21−wc∗+000000,J4E1∗=−τ0β31−φ−β4δ−β4φ+μf0β4δβ4φ−μf,given that τ=β2m/rl+β3fc∗++μw+δw>0, φ=wc∗+=β2m/rl+β3fc∗+/β2m/rl+β3fc∗++μw+δw>0, and δ=fn∗+>0. The eigenvalues of J1E2∗ are λ3=−rl−d, and the solutions of the quadratic equation are(73)λ2+b1λ+b0=0,where(74)b0=β1fc∗+γ+δh+μh+γδh+μh+μhδh+μh>0,b1=β1fc∗++γ+δh+2μh>0.
The eigenvalues of (73) have negative real parts since from Routh–Hurwitz criterion b0>0 and b1>0. The rest of the eigenvalues are obtained from J4E1∗, which are the solutions to the cubic equation:(75)λ3+b4λ2+b3λ+b2=0,where(76)b2=μfτφβ4+τμf+β3β4δφ−1,b3=τφβ4+μf2τ+φβ4+μf+β3β4δφ−1,b4=τ+φβ4+2μf>0.
We note that ifφ>1, then b2>0 and b3>0. Also, it is clear that b4b3>b2. Therefore, their eigenvalues have negative real parts using the principles of Routh–Hurwitz criterion. Hence, E2∗ is locally stable.


## 4.2.1. Local Stability of the Disease-Free Equilibrium

We now state the following theorem for the local stability of disease-free equilibrium.Theorem 4.
The equilibrium pointE0∗ is locally asymptotically stable if and only if rl<d and ℛwf<1; otherwise, it is unstable.Proof.
The Jacobian of system (7) is given by the block matrix:(56)where(57)A=−δh+μh+β1fc∗−δh0β1fc∗−μh+γ000rl−2rll∗−d,B=00−β1sh∗00β1sh∗000,C=00β21−wc∗000000,D=−β2l∗+β3fc∗+μw+δw0β31−wc∗−β4fn∗−β4wc∗+μf0β4fn∗β4wc∗−μf⋅
Evaluating at DFEE0∗, we have that(58)where(59)J1E0∗=−δh+μh−δh00−μh+γ000rl−d,J2E0∗=00−β100β1000,J3E0∗=00β2000000,J4E0∗=−μw+δw0β3−β4θf˜μf−μf0β4θf˜μf0−μf⋅
The eigenvalues fromJ1E0∗ are λ1=rl−d<0 provided that rl<d, λ2=−μw+δw, and λ3=−μh+γ. The rest of the eigenvalues are determined from J4E0∗, which is the solution of the cubic equation:(60)λ3+ξ2λ2+ξ1λ+ξ0=0,given that(61)ξ0=−β3β4θfμf+μf3δw+μw,ξ1=−β3β4θf+μf22δw+μf+2μw,ξ2=μf2μf+μw+δw>0.
Further simplification ofξ1 and ξ0 gives(62)ξ1=μf3+2μf2μw+δw1−β3β4θf2μf2μw+δw,=μf3+2μf2μw+δw1−12ℛwf,ξ0=μf2μw+δw1−β3β4θfμf2μw+δwμf,=μf3μw+δw1−ℛwf,respectively. We note that if ℛwf<1, then ξ1>0 and ξ0>0. Next, we show that ξ2ξ1>ξ0 as follows:(63)μf2μf+μw+δwμf3+2μf2μw+δw1−12ℛwf=2μf+μw+δwμf4+2μf3μw+δw1−12ℛwf>μf3μw+δw2−ℛwf>μf3μw+δw1−ℛwf.
Note that2−ℛwf>1−ℛwf if ℛwf<1. Thus, according to Routh–Hurwitz criteria, all the roots of (60) have negative real parts.


## 4.2.2. Local Stability of the Listeria-Free Equilibrium

We state the following theorem for the local stability of Listeria-free equilibrium.Theorem 5.
The equilibriumE1∗ is locally asymptotically stable if and only if rl<d and ℛwf≥1; otherwise, it is unstable.Proof.
Evaluating the Jacobian of system (7) at E1∗ gives(64)where(65)J1E1∗=−δh+μh+β1fc2∗−δh0β1fc2∗−μh+γ000rl−d,J2E1∗=00−β1sh∗00β1sh∗000,J3E1∗=00β21−wc2∗000000,J4E1∗=−β3fc2∗+μw+δw0β31−wc2∗−β4fn2∗−β4wc2∗+μf0β4fn2∗β4wc2∗−μf.
The eigenvalues fromJ1E1∗ are λ1=rl−d<0 provided that rl<d, which is the case given by Listeria-free equilibrium and the solutions of the quadratic equation:(66)λ2+ζ1λ+ζ0=0,where(67)ζ0=γβ1fc2∗+γδh+β1δhfc2∗+γμh+β1μhfc2∗+δhμh+μh2>0,ζ1=γ+β1fc2∗+δh+2μh>0.
The eigenvalues of (66) have negative real parts since ζ0>0 and ζ1>0 using Routh–Hurwitz criterion. The rest of the eigenvalues are obtained from J4E1∗, which are the solutions to the cubic equation:(68)λ3+ζ4λ2+ζ3λ+ζ2=0,where(69)ζ2=μf+β4wc2∗μfδw+μfμw+μfβ3fc2∗+wc2∗−1β3β4fn2∗,ζ3=μf2δw+μf+2μw+β4wc2∗+2β3fc2∗+β4wc2∗δw+μw+β3fc2∗−β3β4fn2∗1−wc2∗,ζ4=δw+2μf+μw+β4wc2∗+β3fc2∗>0.
We note that ifwc2∗>1, then ζ2>0 and ζ3>0. Next, we show that ζ4ζ3>ζ2 as follows:(70)ζ4ζ3=δw+2μf+μw+β4wc2∗+β3fc2∗2μfδw+μf2+2μfμw+β4wc2∗μf+2β3fc2∗μf+β3β4fn2∗wc2∗−1>ζ2.
Thus, by the Routh–Hurwitz criterion, the eigenvalues of (68) have negative real parts. Thus, E1∗ is locally asymptotically stable.


## 4.2.3. Local Stability of the Endemic Equilibrium

We state the following theorem for the local stability of endemic equilibrium.Theorem 6.
The equilibrium pointE2∗ is locally asymptotically stable if ℛwf≥1 and unstable otherwise.Proof.
Evaluating the Jacobian of system (7) at E2∗ gives(71)where(72)J1E1∗=−δh+μh+β1fc∗+−δh0β1fc∗+−μh+γ000rl−2rlmrl−d,J2E1∗=00−β1sh∗+00β1sh∗+000,J3E1∗=00β21−wc∗+000000,J4E1∗=−τ0β31−φ−β4δ−β4φ+μf0β4δβ4φ−μf,given that τ=β2m/rl+β3fc∗++μw+δw>0, φ=wc∗+=β2m/rl+β3fc∗+/β2m/rl+β3fc∗++μw+δw>0, and δ=fn∗+>0. The eigenvalues of J1E2∗ are λ3=−rl−d, and the solutions of the quadratic equation are(73)λ2+b1λ+b0=0,where(74)b0=β1fc∗+γ+δh+μh+γδh+μh+μhδh+μh>0,b1=β1fc∗++γ+δh+2μh>0.
The eigenvalues of (73) have negative real parts since from Routh–Hurwitz criterion b0>0 and b1>0. The rest of the eigenvalues are obtained from J4E1∗, which are the solutions to the cubic equation:(75)λ3+b4λ2+b3λ+b2=0,where(76)b2=μfτφβ4+τμf+β3β4δφ−1,b3=τφβ4+μf2τ+φβ4+μf+β3β4δφ−1,b4=τ+φβ4+2μf>0.
We note that ifφ>1, then b2>0 and b3>0. Also, it is clear that b4b3>b2. Therefore, their eigenvalues have negative real parts using the principles of Routh–Hurwitz criterion. Hence, E2∗ is locally stable.


## 5. Numerical Simulations

### 5.1. Parameter Estimation

In this section, we present numerical simulations of the model system (7) which was done using the sets of parameters values given in Table 1. There are very few mathematical models done on L. monocytogenes; hence, parameter values cannot be easily found from the literature. We thus estimate the parameter values. We have used fourth-order Runge–Kutta numerical scheme in Matlab with a unit time step 1 to carry out the simulations with the initial values: sh0=0.6, ih0=0.2, l0=0.2, wc0=0.3, fn=0.8, and fc=0.2. The initial conditions are hypothetically chosen and are for illustrative purposes only and are not meant to represent any real scenario. This is useful for confirming some of the analytical results in this paper.Table 1
Parameter values used for simulations.
Parameter
Symbol
Point value (day−1)
Mortality of humans
μ
h
0.02/365
Removal rate of workers
μ
w
0.00274
Removal rate of food products
μ
f
0.0099
Recovery rate of humans
γ
0.034
Recovery rate of workers
δ
w
0.0008
Rate of lost of immunity by humans
δ
h
0.2
Growth rate of listeriosis
r
l
0.32
Death rate of listeriosis
d
0.25
Listeria carrying capacity
K
L
0.008
Food contamination rate by contaminated workers
β
w
1
0.0048
Contact rate of listeriosis by humans
β
f
1
0.008
Workers contamination rate by contaminated food
β
f
2
0.00056
Contact rate of contaminated workers by Listeria
β
L
0.7
Food products
F
10
Workers
W
5
Rate of food processing
θ
f
0.055


### 5.2. Sensitivity Analysis

In this section, we used the sensitivity analysis method known as the Latin hypercube sampling [22], which is an uncertainty analysis method to determine which parameters of the model are responsible for generating variability in the value of the model output. This method calculates the partial rank correlation coefficients (PRCCs) given an input value to the model. We did our simulation using 1000 runs and all the parameters as an input. The parameters that are significant to our model are depicted in the tornado plot (Figure 2). The parameters kL,βL, and βf2 were found to have PRCCs >0 while δw and μw have PRCCs <0 as shown in the scatter plots of Figures 3 and 4, respectively. This has also been shown in Figure 2. Thus, an increase in kl,βL, or βf2 will increase the listeriosis disease epidemic and the decrease in δw or μw will decrease the disease variability.Figure 2
Tornado plots showing PRCCs of different parameter values driving the listeriosis disease.Figure 3
(a) Scatter plot of parameter carrying capacity of ListeriakL with positive PRCC. (b) Scatter plot of parameter contact rate of Listeria by workers βL with positive PRCC. (c) Scatter plot of parameter worker contamination rate from contaminated food products βf2 with positive PRCC.
(a)
(b)
(c)Figure 4
(a) Scatter plot of parameter removal rate of contaminated workersδw with negative PRCC. (b) Scatter plot of parameter removal rate of workers μw with negative PRCC.
(a)
(b)


### 5.3. Effects of Parametersβf2,μf,δw, and βw1 on Contamination Threshold ℛwf

In this section, we investigate the effect of varying parameters onℛwf. Figure 5(a) shows that as the number of contamination rate of workers by the contaminated food products increases, the number of ℛwf increases; this will result in invasion of the listeriosis disease epidemic. On the other hand, the increase in the removal of the food product μf decreases the values of the contamination threshold ℛwf, which will cause the disease to be eradicated (see Figure 5(a)). Similarly, the increase in the removal of the number of contaminated workers δw decreases ℛwf, which means that the rate of progression of the contamination of food products by the workers in the factory decreases, and hence the epidemic is contained as depicted by Figure 5(b). Furthermore, increase in the rate of food contamination by the workers βw1 increases the rate at which humans get infected with the disease due to high value of ℛwf>1, as a result of more food products getting contaminated in the production plant.Figure 5
(a) 3D plot showing the effect ofβf2 and removal of food products μf on ℛwf. (b) 3D plot showing the effect of removal of workers δw and rate of contamination of food products by workers βw1 on ℛwf.
(a)
(b)


### 5.4. Effects of Varyingβf2,βw1, and δw on the Dynamics of Listeriosis

We did simulations with some of the model parameters to find out its effect on the listeriosis disease dynamics. We varied the rate of contamination workers by the food productsβf2 against the number of infected humans. It was found that the increase in the contamination rate of workers by contaminated food products results in the increase in the fraction of infected humans. Hence, the increase in βf2 increases ℛwf as shown in Figure 6(a). Also, an increase in removal of the contaminated workers results in a decrease in the values of ℛwf, and hence less fraction of humans get infected, as depicted in Figure 6(c). Simulations also reveal that an increase in the rate of contamination of food by contaminated workers βw1 increases ℛwf and the fractions of infected humans. Therefore, the disease can never be eradicated (see Figure 6(b)).Figure 6
(a) Effects of contaminated Foodβf2 on the fractions of infected humans. (b) Effect rate of food contamination by contaminated workers βw1 on the fractions of infected humans. (c) Effect of removal of contaminated workers δw on the fractions of infected humans.
(a)
(b)
(c)


## 5.1. Parameter Estimation

In this section, we present numerical simulations of the model system (7) which was done using the sets of parameters values given in Table 1. There are very few mathematical models done on L. monocytogenes; hence, parameter values cannot be easily found from the literature. We thus estimate the parameter values. We have used fourth-order Runge–Kutta numerical scheme in Matlab with a unit time step 1 to carry out the simulations with the initial values: sh0=0.6, ih0=0.2, l0=0.2, wc0=0.3, fn=0.8, and fc=0.2. The initial conditions are hypothetically chosen and are for illustrative purposes only and are not meant to represent any real scenario. This is useful for confirming some of the analytical results in this paper.Table 1
Parameter values used for simulations.
Parameter
Symbol
Point value (day−1)
Mortality of humans
μ
h
0.02/365
Removal rate of workers
μ
w
0.00274
Removal rate of food products
μ
f
0.0099
Recovery rate of humans
γ
0.034
Recovery rate of workers
δ
w
0.0008
Rate of lost of immunity by humans
δ
h
0.2
Growth rate of listeriosis
r
l
0.32
Death rate of listeriosis
d
0.25
Listeria carrying capacity
K
L
0.008
Food contamination rate by contaminated workers
β
w
1
0.0048
Contact rate of listeriosis by humans
β
f
1
0.008
Workers contamination rate by contaminated food
β
f
2
0.00056
Contact rate of contaminated workers by Listeria
β
L
0.7
Food products
F
10
Workers
W
5
Rate of food processing
θ
f
0.055


## 5.2. Sensitivity Analysis

In this section, we used the sensitivity analysis method known as the Latin hypercube sampling [22], which is an uncertainty analysis method to determine which parameters of the model are responsible for generating variability in the value of the model output. This method calculates the partial rank correlation coefficients (PRCCs) given an input value to the model. We did our simulation using 1000 runs and all the parameters as an input. The parameters that are significant to our model are depicted in the tornado plot (Figure 2). The parameters kL,βL, and βf2 were found to have PRCCs >0 while δw and μw have PRCCs <0 as shown in the scatter plots of Figures 3 and 4, respectively. This has also been shown in Figure 2. Thus, an increase in kl,βL, or βf2 will increase the listeriosis disease epidemic and the decrease in δw or μw will decrease the disease variability.Figure 2
Tornado plots showing PRCCs of different parameter values driving the listeriosis disease.Figure 3
(a) Scatter plot of parameter carrying capacity of ListeriakL with positive PRCC. (b) Scatter plot of parameter contact rate of Listeria by workers βL with positive PRCC. (c) Scatter plot of parameter worker contamination rate from contaminated food products βf2 with positive PRCC.
(a)
(b)
(c)Figure 4
(a) Scatter plot of parameter removal rate of contaminated workersδw with negative PRCC. (b) Scatter plot of parameter removal rate of workers μw with negative PRCC.
(a)
(b)


## 5.3. Effects of Parametersβf2,μf,δw, and βw1 on Contamination Threshold ℛwf

In this section, we investigate the effect of varying parameters onℛwf. Figure 5(a) shows that as the number of contamination rate of workers by the contaminated food products increases, the number of ℛwf increases; this will result in invasion of the listeriosis disease epidemic. On the other hand, the increase in the removal of the food product μf decreases the values of the contamination threshold ℛwf, which will cause the disease to be eradicated (see Figure 5(a)). Similarly, the increase in the removal of the number of contaminated workers δw decreases ℛwf, which means that the rate of progression of the contamination of food products by the workers in the factory decreases, and hence the epidemic is contained as depicted by Figure 5(b). Furthermore, increase in the rate of food contamination by the workers βw1 increases the rate at which humans get infected with the disease due to high value of ℛwf>1, as a result of more food products getting contaminated in the production plant.Figure 5
(a) 3D plot showing the effect ofβf2 and removal of food products μf on ℛwf. (b) 3D plot showing the effect of removal of workers δw and rate of contamination of food products by workers βw1 on ℛwf.
(a)
(b)


## 5.4. Effects of Varyingβf2,βw1, and δw on the Dynamics of Listeriosis

We did simulations with some of the model parameters to find out its effect on the listeriosis disease dynamics. We varied the rate of contamination workers by the food productsβf2 against the number of infected humans. It was found that the increase in the contamination rate of workers by contaminated food products results in the increase in the fraction of infected humans. Hence, the increase in βf2 increases ℛwf as shown in Figure 6(a). Also, an increase in removal of the contaminated workers results in a decrease in the values of ℛwf, and hence less fraction of humans get infected, as depicted in Figure 6(c). Simulations also reveal that an increase in the rate of contamination of food by contaminated workers βw1 increases ℛwf and the fractions of infected humans. Therefore, the disease can never be eradicated (see Figure 6(b)).Figure 6
(a) Effects of contaminated Foodβf2 on the fractions of infected humans. (b) Effect rate of food contamination by contaminated workers βw1 on the fractions of infected humans. (c) Effect of removal of contaminated workers δw on the fractions of infected humans.
(a)
(b)
(c)


## 6. Conclusion

This paper presents a mathematical model for cross contamination of the listeriosis disease, due to the consumption of ready-to-eat food products from production plant by humans. The analytical results of the model have shown that there exists a domain where the model is epidemiologically and mathematically well posed. The model contamination thresholdℛwf was determined during the process of solving for the equilibrium points. Three equilibria were found analytically, which are the disease-free E0∗, the Listeria-free E1∗, and the endemic E2∗ points. The analytical method has shown that the disease-free equilibrium is asymmetrically locally stable for ℛwf<1 and asymptotically locally stable for ℛwf>1 for both Listeria-free and endemic equilibrium points, respectively. The Jacobian matrix of system (7) was calculated, and the principle of Routh–Hurwitz criterion was used to deduce the local stability of the equilibria points in our analytical results. Sensitivity analysis shows that carrying capacity of Listeria KL, workers’ contamination rate by Listeria βL, workers’ contamination rate by contaminated food products βf2, removal rate of contaminated workers δw, and removal rate of workers μw are the most sensitive parameters of the model system (7), since they have high significant PRCC values. Further numerical simulations reveal that increase in the number of contaminated food products and the rate of contamination of food by contaminated workers increases the contamination of listeriosis by humans.While the model presented in this paper presents some interesting aspects on cross contamination and Listeria infection dynamics, it has some aspects that require further examination and modelling. First, our population structure could have a nonsusceptible group of individuals who can be infected by listeriosis. However, in this work, we do not consider this particular group of individuals and this could be an interesting aspect to consider. Second, the model does not consider the fact that many people do not eat processed meats, and thus a fraction of the population should be considered as susceptible. Third, it is documented that the disease mainly affects the young, pregnant women, and the elderly. It will thus be prudent to consider an age structured model for such a disease. Fourth, different people, mount different levels of immune response to the disease and hence, an in-host model could be used to model how different people respond to the disease upon infection. Lastly, the model assumptions on the constant population of humans and workers could be relaxed to obtain a more accurate prediction of the model output with respect to the infection. Despite these weaknesses, the model still presents some interesting results on listeriosis dynamics and can easily be used to influence policy, especially if it is fitted to epidemiological data.



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*Source: 9207403-2020-03-09.xml*