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On the mathematical modelling and analysis of Listeriosis from contaminated food products

Listeriosis is a foodborne disease caused by a bacterium known as Listeria monocytogenes. Humans can be infected by consuming contaminated food products. A transmission can also occur through contact with infected animals or people, however to a less extent. In this paper, a mathematical model for Listeriosis dynamics was developed. The steady states and their stability of the model system were determined and analyzed. The result shows that the disease free equilibrium is asymptotically stable if the bacteria growth rate is less than its removal rate, and also the growth rate of food contamination is less than its removal rate. It was further observed that we can still have Listeriosis driven by the contaminated food products even if the Listeria bacteria population in the environment is very small. The results indicate that Listeriosis can be effectively controlled by removing contaminated food products, which was the policy adopted by the South African government during the recent Listeriosis outbreak.


Introduction
Listeria monocytogens are pathogens that are found in both raw and processed food products. The consumption of such products causes infections among humans and animals. This pathogen can be characterized as a Gram-positive bacteria [3]. Listeria strives very well in low-temperature areas and unclean environments. Other favourable conditions for their survival include high salt concentration and acidic conditions. Specifically, they are mostly found in soil, unclean water bodies (lakes, rivers, etc.), vegetation, refrigerators, faeces of some animals as well as in foods such as smoked fish, cold meats and soft cheeses. Animals such as cattle and poultry can also carry this bacteria. Due to their presence in soil and vegetation, they easily affect raw foods. Moreover, processed food can be contaminated by the presence of infected raw food and the use of unclean processing materials [15]. Humans can be infected by consuming such contaminated transmission, control and prevention for pregnant and individuals with weak immune systems will help in minimizing the mortality rate of this disease. Moreover, [13] developed a mathematical model to study the effect of the vaccination of animals on the spread of Listeriosis among humans and animals (as vectors of Listeria). In their analysis, they found out that, secondary infections can increase due to the decrease of human and animal death rate as well as animal recovery rate.
Very few mathematical models the role of the environment and contaminated food products have been developed to date. Simple model that compute the role of bacteria and food products in transmission of dynamics of Listeriosis in the human population is formulated with the aim of investigating the role of the few control measures that are available in the event of the out break. The control measure include reduction of the rate of infection, removal of contaminated food product and hygiene in the food manufacturing process. While the model presented in this page represent the simplest caricature of the infection dynamics of Listeriosis, the results have significant influence on the management and qualification of control strategies over the long period of time. We ague that despite its simplest, the model offers significant insights in the modelling of Listeriosis driven by contaminated food products and bacteria from the environment.
The outline of this paper is as follows; Section 1 introduces the research paper followed by a model formulation in Section 2. The model basic properties and analyses are presented in Section 3. Numerical simulations are presented in Section 4 and Section 5 concludes the paper.

Model Formulation
In this section, we develop a mathematical model for Listeriosis dynamics which is divided into three components namely; the human population, Listeria and factory products. The human population comprises of three compartments which are susceptible humans, S(t), infected humans, I(t), and the recovered humans, R(t). Individuals are recruited into susceptible class at a rate proportional to the total human population N (t) so that the recruitment is modelled by of µN, where µ is the natural birth/mortality rate. The susceptible humans can move into infected class either by acquiring Listeriosis through eating contaminated products or through contact with contaminated material from the environment with a force of infection λ h (t), defined after system (2). Furthermore, the infected humans can either die naturally, die of the disease at a rate of δ 1 or recover at a rate of γ and join the recovered class. We are assuming that there is no human to human transmission. Considering the factory dynamics, we are assuming that the amount of food products, F (t) comprises of non-contaminated, F n (t), and contaminated, F c (t) food products. We assume that factory products are manufactured or produced at a rate of δ 2 F . By contact with contaminated surfaces and contaminated products, the non-contaminated products can become contaminated with Listeria bacteria. Non-contaminated factory products become contaminated with a force of infection λ f (t) defined after system (2). The Listeria bacteria in the environment is assumed to grow at a rate of r b , and die at a rate µ b . The growth of the bacteria is assumed to be logistic with a carrying capacity of K. We assume that, at any time, t, the human population and factory food products are constant and respectively given by; (1) The model structure is represented schematically in Figure 1. From the above model description and Figure 1 we obtain the following system of non-linear ordinary differential equations: Figure 1: Model flow chart for Listeriosis disease transmission dynamics. The dotted and solid line shows the contributing factors for the human and food products to become contaminated as well as infection links respectively. where 4 . CC-BY 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review) The copyright holder for this preprint this version posted July 30, 2021. ; https://doi.org/10.1101/2021.07.25.21261099 doi: medRxiv preprint We rescale system (2) by setting and substituting r = n − s − i, to obtain the following system of differential equations: where re-scaled forces of infections arê subject to the following initial conditions All parameters of model system (3) are assumed to be positive at all time t > 0.

Basic Properties
We show that all the solutions of model system (3) has non-negative solutions, and bounded for t > 0, that is remain biologically meaningful into feasible region (Ω).

Positivity of Solutions
The positivity of solutions to our model is given by the following theorem.
and b(t) of system (3) are non-negative for all time t ≥ 0.
Proof. Let (s(t), i(t), f c (t), b(t)) be a solution of the system (3) with the given initial conditions. From the first equation of system (3), we have that is the differential inequality from (4). Integration yields . CC-BY 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.

(which was not certified by peer review)
The copyright holder for this preprint this version posted July 30, 2021. ; https://doi.org/10.1101/2021.07.25.21261099 doi: medRxiv preprint where s(0) is the given initial condition for s(t). From the second equation of model equation (3) we have the differential inequality so that The remaining equations yields Thus, the solutions (s(t), i(t), f c (t), b(t)) will remain positive in Ω at all time t ≥ 0.

Feasible Region
The feasible region of our model is captured by the following theorem.
Theorem 2. Consider the biologically feasible region given by The solution of the system (3) with the non-negative initial condition are bounded for all t ≥ 0 in the biologically feasible region, (Ω).
Proof. The total change in the human population is given by where n = (s + i) ≤ 1. However in the absence of mortality due to human Listeriosis infections, equation (6) reduces to whose solutions yields where C 1 is a constant. We note that as t → ∞, then n(t) → 1.
Considering the contaminated food product given by last equation of system (3), we have that whose solution yields . CC-BY 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.

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where φ 2 is a constant. Note that as t → ∞ we have f c (t) → 1. Furthermore, for the bacteria population, considering the third equation of system (3), we have the Bernoulli's equation The solution of equation (8) we obtain So, we note that for the Listeria to exist its death rate (µ b ) must be less than the growth rate (r b ), which implies that, 0 ≤ b ≤ 1. Hence (s, i, f c , b) are all bounded in region Ω, and are biologically feasible, which complete the proof.

Model Steady States
We determine the steady states of system (3) by setting the right hand side to zero as follows From the third equation of the system (10), we have that Now from the last equation of the system (10) we havê . CC-BY 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.

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The copyright holder for this preprint this version posted July 30, 2021. ; and this yields , which implies that from the second equation of (10). Now substituting into first equation of the system (10) we have Therefore, we have a disease free state (DFS) denoted by From the second equation of the system (10) we havê Substituting equation (11) into the first equation of the system (10) we have .
This results in the bacteria free state (BFS) 8 . CC-BY 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.

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The copyright holder for this preprint this version posted July 30, 2021. ; https://doi.org/10.1101/2021.07.25.21261099 doi: medRxiv preprint Similarly from the last equation of the system (10), we have that after some algebraic simplification, where Solving equation (12) we have, Since 0 < 0 and 2 > 0 then 2 0 < 0, and this implies that f * c 3 has one positive solution, say f * + c 3 irrespective of the conditions imposed on 1 . Now from the second equation of the system (10), we have Further, from the first equation of the system (10), we have This give .

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Stability Analyses
In this section we prove the local stability of the steady states by using the Jaccobian matrix linearized at each steady state.

Local Stability of the DFS
Proof. In order to determine the local stability of the disease free steady state we evaluate the Jaccobian matrix at E * 0 which gives Therefore, the eigenvalues are; Since all parameters are positive, it is clear that λ 1 and λ 2 are negative. However, λ 3 and λ 4 are negative if and only if R f > 1 and r b < µ b respectively. Therefore, E * 0 , is locally asymptotically stable if R f > 1 and r b < µ b .
Remark 2. The growth rate of food contamination must be less than its removal and the growth rate of bacteria must be less than its removal (natural death rate of bacteria).

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Local Stability of the BFS
Theorem 4. The bacteria free state E * 1 is locally asymptotically stable if r b < µ b and R f < 1.
Proof. We evaluate the Jacobian matrix at E * 1 and obtain The eigenvalues from J 3 (E 1 ) are; Further, the eigenvalues from J 1 (E 1 ) is given by the characteristic polynomial where Since ρ 1 > 0 and ρ 0 > 0, the eigenvalues of equation (15) have negative real parts by the Routh-Hurwitz stability criterion. Therefore, the bacteria free state is locally asymptotically stable if and only if µ b > r b and R f < 1.

Local Stability of the ES
Theorem 5. The endemic state E * 2 is locally asymptotically stable if r b < (2r b b * 2 + µ b ) and

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The copyright holder for this preprint this version posted July 30, 2021. ; Proof. We evaluate the Jacobian matrix at E * 2 to obtain The eigenvalues from J 3 (E 2 ) are; Further, the eigenvalues of matrix J 1 (E 2 ) is given by the characteristic polynomial where So, equation (16) can be rewritten as where φ 1 = η 1 η 2 > 0 and φ 0 = η 0 η 2 > 0. Since φ 1 > 0 and φ 0 > 0, The eigenvalues of equation (17) have negative real parts by the Routh-Hurwitz stability criterion. Therefore, the endemic state is locally asymptotically stable if r b < (2r b b * 2 + µ b ) andβ 2 J * < 1.

Parameter Values
In this section, we simulate our model for Listeriosis in order to determine the time series trajectories of the disease in human population. The system of equations were solved by using Julia and Matlab software over a period of time using the estimated parameter values in Table  2. We hypothetically chose the parameter values in such a way that they yield the results of the analyses of the steady states. The parameter values in Table 2 are all assumed since there are very few mathematical models on human Listeriosis disease infection and in literature. A summary descriptions of the parameters used in the model are shown in Table 2 Figure 2(a) depicts the graph of the equilibrium point E * 0 in which all the infected compartments tends to zero. Therefore this shows the stability of the disease free equilibrium point. We can infer that this is due to the high death rate, µ b = 0.4, of the bacteria as compared to their growth rate r b = 0.18. So there are less infections caused by the bacteria. Moreover, the rate of removal of contaminated food, δ 2 = 0.5, is higher as compared to the growth rate of food contamination, β 2 = 0.006. Figure 2(b) depicts the graph of the equilibrium point E * 1 . We can observe from this graph that the bacteria population goes to extinct since their growth rate is less that their death rate. However, Listeriosis infections exist in the population due to presence of contaminated food. This can be attributed to the fact that the rate of removal of contaminated food, δ 2 = 0.1, is less as compared to the growth rate of food contamination,β 2 = 0.16. Figure 2(c) shows the graph of the endemic equilibrium point E * 2 in which the contaminated products, the bacteria in the environment and the human population are all non-zero. Hence we can observe that the disease will persist.

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The copyright holder for this preprint this version posted July 30, 2021.  Table 2

Sensitivity Analysis
Global sensitivity analysis is a statistical tool used to determine the sensitivity and uncertainly of epidemiological parameters. Here, we employ the method of Latin-hyper cube sampling as in [24] to determine model parameters driving Listeriosis subject to the model under investigation. According to [24], for any modelling exercise, we can carry out sensitivity at a time point of interest or over a range of time frames to determine the sensitivity of the model parameters. Since humans can devolve fully Listeriosis infections between 1 to 90 days after consuming contaminated food products contaminated by Listeria. We carry out the simulation implemented in Matlab with a time step of 1 and between 1 to 90 days, [25], with a time average of 45 days as it is assumed to be an average time an infected person with Listeria should have to develop the symptom of the disease. The simulation result gives the partial Rank Correlation Coefficient's (PRCCC) and P-values shown in Table 2 with the Tornado plot showing different parameters sensitivity values as shown in Figure 3. We observe in Figure 3, that, the parametersβ 3 and r b are strongly positively correlated while µ b is strongly negatively correlated.. Hence, the increase 14 .
CC-BY 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.

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The copyright holder for this preprint this version posted July 30, 2021. ; https://doi.org/10.1101/2021.07.25.21261099 doi: medRxiv preprint in the food contamination rateβ 3 and the bacteria growth r b will impact the disease variability leading to more infected humans in the community affected by Listeriosis. An increase in the removal rate of bacteria, µ b , will certainly result in fewer human infections.   Figure 4(a) depicts the behaviour of contamination threshold R f against the rate of contamination of food products,β 2 , the graph shows that as the rate of contamination of food products, β 2 , increase then the contamination threshold R f decrease. Figure 4(b) depicts the behaviour of contamination threshold, R f , against the disposal rate δ 2 , the graph shows that as disposing rate δ 2 increase then the contamination threshold, R f , also increase.   Figure  5(b) shows how infectious human decrease as the bacterial natural death (removal) rate µ b increases.
(a) (b) Figure 5(c) shows how infectious humans decrease as Listeria bacteria infection rate from the environment decrease. Figure 5(d) shows how infectious human decrease as the rate of food contamination β 4 , decreases.
16 . CC-BY 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review) The copyright holder for this preprint this version posted July 30, 2021.

Conclusion
In this paper, we formulate a mathematical model to eludidate the contribution of contaminated food products and bacteria in the environment on the spread of Listeriosis. The model steady states were determined and their stabilities determined. The model has three steady states, the disease free steady statesE 0 in which their in only the susceptible population while the remaining state variables are zeros. While this is mathematically plausible this steady state may not be realistic as the bacteria will always be present in the environment. The second steady state is bacteria free steady state E 1 in which the is driven by only contaminated food products. This steady state is driven by contamination within the food manufacturing environment. The third steady is the endemic state in which all the state variable are no-zero. This steady state depicts a scenario in which the environment, the human population and food products are active in spread of Listeriosis.
The model simulation are carried out to investigate the role of interventions in the spread of Listeriosis. The most important interventions in the removal of contaminated food product, modelled by the parameters δ 2 . Sensitivity analysis is also carried to determine the most significant parameters that influence the transmission dynamics. It is through these parameters that interventions can be designed and quantified.

Data Availability
No data used for this research work

Conflicts of Interest
Authors declare no conflict of interest.

Funding Statement
No funding or grants utilised for this research work.

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