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

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 of Listeria monocytogenes from a food processing plant environment is formulated using a system of ordinary differential equations.-emodel 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. -e 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.-e 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. -e results have significant public health implications in the management and containment of any listeriosis disease outbreak.


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 as Listeria 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]. e 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]. e 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. e 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]. e source of this outbreak was traced to a food production company (majorly the meat processing facilities) that manufactures ready-to-eat (RTE) foods [8]. ese are foods that are intended by the producers for direct human consumption without the need for cooking [9].
ere 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]. e 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][12][13][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. e 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. e 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. is paper is arranged as follows. Section 1 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. e paper is concluded in Section 6.

Model Formulation
We develop a mathematical model for listeriosis disease in which the human population is divided into three classes, namely, susceptibles S h (t), infected I h (t), and recovered R h (t). We consider a constant human population over the modelling time. Individuals are recruited into the susceptible class at a rate μ h N and upon infection move into the class I h at a rate λ h (t). After treatment, infected individuals recover at a rate c with no immunity.
ose that recover lose immunity at a rate δ h 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, L(t), by r l and d, respectively. We assume a logistic growth for Listeria with a carrying capacity K L . Uncontaminated workers W n (t) can be infected by Listeria from their working environment and through contaminated food at a rate λ w (t), defined later after the model equations. Infected workers W c (t) are assumed to contaminate uncontaminated food F n (t) at a rate λ f (t). e contaminated food F c (t) is then responsible for the transmission of Listeria to the human population. We assume that human population N(t) comprises of those individuals that do not work in the factories, while W(t) is the population of workers. e population of factory workers is assumed to be constant over the model time. e total amount of food products F(t) that are manufactured is assumed to be constant over the modelling time. We therefore assume a natural mortality rate μ h for all human classes and a removal rate of workers μ w . Contaminated workers can be decontaminated at a rate δ w , through hygiene and disinfectants. We also assume that θ f is the rate of addition of noncontaminated food (rate of food processing) while μ f is the rate of removal of food products. Assuming that we have a constant human population, the total population N(t), the total number of workers W(t), and the total amount of food products produced F(t) at any time t are, respectively, given by (1)

2
International Journal of Mathematics and Mathematical Sciences 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. Figure 1 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 where All parameters for the model system (2) are assumed to be nonnegative for all time t > 0. From (1), we have R h (t) � N(t) − S h (t) − I h (t) and W n (t) � W(t) − W c (t). erefore, the system of equation (2) with the initial conditions given by Set We nondimensionalise (3) and obtain the following: where the nondimensionalised contamination rates are and θ f � (θ f /F), with initial conditions International Journal of Mathematics and Mathematical Sciences

Positivity and Boundedness of Solutions.
e 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. e 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 eorem 1 to ascertain our view.

Theorem 1. For nonnegative initial conditions (9), the solutions
Proof. From the first equation of the system (7), we have that Solving the first-order differential equation (10) gives where s h (0) is the given initial condition. From (11), we see that lim inf us, the solution s h (t) will always be positive for all t ≥ 0. Similarly, from the second equation of model system (7), which yields where i h (0) is the initial condition. Similarly, solving the remaining equations of system (7) and taking limits as t approaches infinity, it can be shown that e biological feasible region (Γ) 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.
e solutions of model system (7) are contained in the region Γ ∈ R 6 + , which is given by for the initial conditions (9) in Γ.

Factory
Human population Listeria Proof.
e total change in human population from the system of equation (7) is given by e solution of (16) is where is the initial condition. Applying Birkhoff and Rota's [20] theorem on differential inequality and the condition, we have that 0 ≤ n ≤ 1 − ϕ(0)e − (μ h +δ h )t so that 1 is the upper bound of n provided that ϕ(0) ≥ 1. us, 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 and its differential inequality is Let m � (r l − 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 l(t) ≤ (m/(r l + A 1 e − mt )). As t ⟶ ∞, l(t) ⟶ (m/r l ). e 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 f n (t) and w c (t) to be respectively. We note that f n (t) ⟶ (θ f /μ f ) and w c (t) ⟶ 1 asymptotically as t ⟶ ∞. From the last equation of the system (7), Substituting λ f � β w 1 Ww c and f n � (θ f /μ f ) which is the boundedness for the noncontaminated food, equation (21) becomes e solution of (23) by the integration factor method is Hence, considering the human population, Listeria, contaminated workers, noncontaminated food products, and contaminated food products, the regions in Γ are invariant and biologically meaningful.

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 with Solving system (25) for the equilibrium points, from the third equation of system (25), we have So l * � 0 or l * � (m/r l ). Note that, for Listeria to exist, then r l > d. Let β 1 � β 1 f * c ; hence, the first and second equations of system (25) become Solving (27) Using the fourth equation, we obtain On the other hand, from the last equation of (25) we have that International Journal of Mathematics and Mathematical Sciences Substituting (30) into the second last equation of (25) and after some algebraic manipulations, we obtain the following quadratic expression: where (32) Note that ] 1 < 0 or ] 1 > 0. When l * � 0, Now, solving for f * c , we obtain Considering the case f * where We thus call R wf the contamination threshold. We thus have the following result on the existence of f * c 2 .

Lemma 1.
e steady-state f * c 2 exists if and only if R wf > 1.
It is important to note that R wf denotes the measure of contamination caused by workers and food products. is is equivalent to the reproduction number (R 0 ) in disease modelling [21], where R 0 determines whether a disease is established or eradicated. In this case, R wf determines whether listeriosis is established or eradicated, i.e., no contamination of food products. If and f * n 1 � (θ f /μ f ). is results in the disease-free equilibrium (DFE) point E * 0 given by At DFE, there are no Listeria, contaminated workers, food, and infected and recovered humans. Given that when l * � 0, we have Some algebraic manipulations give where . Solving for f n * 2 by substituting (35) and (41) into (30), we obtain where Φ 3 � (μ f (μ w + δ w )/β 3 (β 4 + μ f )). We thus have the Listeria-free equilibrium point: where is implies that as long as we have contaminated workers or food, we will still have the disease in the human population. us, removal of contaminated food is essential for disease control. When l * ≠ 0, then (45) If ] 1 > 0 and ] 1 < 0, we have one positive solution. us, irrespective of the sign of ] 1 , we have one positive root for f * c , say f * + c . From (25), we now have with l * � (m/r l ). Let β 1 + � β 1 f * + c , and similarly solving for s * + h and i * + h simultaneously, we obtain If λ w � β 2 l * + β 3 f * + c , then from the fourth equation of (25), we obtain Now, setting λ f � β 4 w * + c from second last equation of (25), we have where Further simplification of η 1 gives the following expression: However, we note that if R wf > 1, then η 1 < 0, and if R wf < 1, then η 1 > 0. Hence, solving (50) for f * + c , we obtain If η 1 > 0 or η 1 < 0, (53) we will always have one positive root. erefore, irrespective of the sign of η 1 , we will always have one positive root. We thus have the following result. (7) has a unique endemic equilibrium as long as m > 0, i.e., r l > d.

Remark 1.
As long as the growth rate of Listeria is greater than the death rate, we will always have listeriosis.
Also, note that f * + c is the positive root of (50). We cannot explicitly write the expression of the roots for f * + c due to the intractability of expression (50).
us, the endemic equilibrium, is given by 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.

Local Stability of the Disease-Free Equilibrium.
We now state the following theorem for the local stability of disease-free equilibrium.

Theorem 4.
e equilibrium point E * 0 is locally asymptotically stable if and only if r l < d and R wf < 1; otherwise, it is unstable.
Proof. e Jacobian of system (7) is given by the block matrix: where Evaluating at DFE (E * 0 ), we have that e eigenvalues from J 1 (E * 0 ) are λ 1 � (r l − d) < 0 provided that r l < d, λ 2 � − (μ w + δ w ), and λ 3 � − (μ h + c). e rest of the eigenvalues are determined from J 4 (E * 0 ), which is the solution of the cubic equation: given that

(61)
Further simplification of ξ 1 and ξ 0 gives respectively. We note that if R wf < 1, then ξ 1 > 0 and ξ 0 > 0. Next, we show that ξ 2 ξ 1 > ξ 0 as follows: δ � f * + n > 0. e eigenvalues of J 1 (E * 2 ) are λ 3 � − (r l − d), and the solutions of the quadratic equation are e eigenvalues of (73) have negative real parts since from Routh-Hurwitz criterion b 0 > 0 and b 1 > 0. e rest of the eigenvalues are obtained from J 4 (E * 1 ), which are the solutions to the cubic equation: We note that if φ > 1, then b 2 > 0 and b 3 > 0. Also, it is clear that b 4 b 3 > b 2 . erefore, their eigenvalues have negative real parts using the principles of Routh-Hurwitz criterion. Hence, E * 2 is locally stable.

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. ere 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

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. is 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. e parameters that are significant to our model are depicted in the tornado plot ( Figure 2). e parameters k L , β L , and β f 2 were found to have PRCCs >0 while δ w and μ w have PRCCs <0 as shown in the scatter plots of Figures 3  and 4, respectively. is has also been shown in Figure 2.
us, an increase in k l , β L , or β f 2 will increase the listeriosis disease epidemic and the decrease in δ w or μ w will decrease the disease variability.

5.3.
Effects of Parameters β f 2 , μ f , δ w , and β w 1 on Contamination reshold (R wf ). In this section, we investigate the effect of varying parameters on R wf . Figure 5(a) shows that as the number of contamination rate of workers by the contaminated food products increases, the number of R wf increases; this will result in invasion of the listeriosis disease epidemic. On the other hand, the increase in the removal of the food product (μ f ) decreases the values of the contamination threshold (R wf ), which will cause the disease to be eradicated (see Figure 5(a)). Similarly, the increase in the removal of the number of contaminated workers (δ w ) decreases R wf , 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 (β w 1 ) increases the rate at which humans get infected with the disease due to high value of R wf > 1, as a result of more food products getting contaminated in the production plant.

5.4.
Effects of Varying β f 2 , β w 1 , and δ w 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 products (β f 2 ) 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 β f 2 increases R wf as shown in Figure 6(a). Also, an increase in removal of the contaminated workers results in a decrease in the values of R wf , 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 (β w 1 ) increases R wf and the fractions of infected humans. erefore, the disease can never be eradicated (see Figure 6(b)).

Conclusion
is 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. e analytical results of the model have shown that there exists a domain where the model is epidemiologically and mathematically well posed. e model contamination threshold (R wf ) was determined during the process of solving for the equilibrium points. ree equilibria were found analytically, which are the disease-free (E * 0 ), the Listeria-free (E * 1 ), and the endemic (E * 2 ) points. e analytical method has shown that the disease-free equilibrium is asymmetrically locally stable for R wf < 1 and asymptotically locally stable for R wf > 1 for both Listeria-free and endemic equilibrium points, respectively. e 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 (K L ), workers' contamination rate by Listeria (β L ), workers' contamination rate by contaminated food products (β f 2 ), removal rate of contaminated workers (δ w ), and removal rate of workers (μ w ) 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. ird, 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.

Data Availability
No data were used to support this study.

Conflicts of Interest
e authors declare that there are no conflicts of interest.