TUBERCULOSIS WITH CONTAMINATION BY THE CONSUMPTION OF UNPASTEURIZED DAIRY PRODUCTS: MATHEMATICAL MODELLING AND NUMERICAL SIMULATIONS

. In this article, we present a mathematical model of transmission tuberculosis which takes account the contamination by consuming unpasteurised dairy products. We have determined the equilibrium points, whose local stability is guaranteed by Lyapunov’s indirect method using the Routh-Hurwitz stability criterion. The results obtained show that, even if there are no bacteria in the environment, the probability of infection with tuberculosis remains high, and this is due to the consumption of unpasteurised dairy products. Finally, we introduce some numerical simulations graphics to validate our results.


INTRODUCTION
Tuberculosis is a disease that has been present since the dawn of humanity. It is a potentially fatal infectious disease caused by a bacterium called Mycobacterium tuberculosis or Koch's bacillus (BK), named by the doctor who discovered it in 1882, Robert Koch . This bacterium is an immobile, straight or slightly curved aerobic bacillus, its average length is 2µm to 4µm for estimates by the World Organization for Animal Health, in some countries up to 10% of human tuberculosis cases are of bovine origin [5].
People infected with the tubercle bacillus have a 5 to 10% risk of developing the disease. This risk is much higher in people with a weakened immune system, especially those living with HIV, suffering from malnutrition or diabetes, smokers... [6]. Tuberculosis then becomes active, contagious and symptomatic. Symptoms such as cough with sometimes bloody sputum, chest pain, weakness, weight loss, fever, and night sweats may remain mild for many months. The disease is remediable provided an early diagnosis, and its treatment is based on combinations of antibiotics given for at least 6 months, sometimes longer, accompanied by patient support from a health worker or trained volunteer. Compliance with the protocol is absolutely necessary, otherwise drug resistance will appear. According to statistics, the diagnosis and treatment of tuberculosis has saved 66 million lives since 2000 [2].
Mathematical models appears to be a good tool for understanding the spread of infectious diseases. Several models exist in the literature to model the transmission of tuberculosis [7], [8].
The first one is built by the statistician Waaler in 1962. He used an unknown function of the number of infectious individuals to formulate the infection rate, and he predicted that the time trend of tuberculosis is improbable to increase, but his linear model did not model all the mechanics of transmission [9]. In 1967, Brogger developed a model based on Waaler's model. He changed the method used for calculating infection rates. His objective was to compare different control strategies such as treating more cases, the vaccination, and mass roentgenography.
In the same year, ReVelle modeled the tuberculosis dynamics using a nonlinear system of ordinary differential equations. His aim is to develop an optimization model to select control strategies that could be carried out at a minimal cost [9]. Incomplete treatment, wrong therapy, and co-infection with other diseases, like HIV, may develop a new form of tuberculosis known as multi-drug resistant. Many models that include this type of tuberculosis have been developed [10], [11]. Most recently, an age-structured tuberculosis model is constructed to look at optimal vaccination strategy problems. The basic reproductive number is calculated and used to study cost related optimization problems [12].
In the present work, we model, analyze and simulate a mathematical model of the dynamics of tuberculosis with contamination by the consumption of unpasteurized dairy products.
The purpose is to study the role of some control measures available in the event of an epidemic. This choice is motivated by an administrative note issued on 9 June 2022, written by the provincial delegate of the Ministry of Health of the Casablanca-Settat region, concerning an upsurge in cases of tuberculosis. He linked this significant increase in cases of tuberculosis to the consumption of dairy products sold by vendors in the streets and around mosques without any respect for hygiene conditions. First, we present the description of the model proposed, then we present its mathematical analysis. Here we show the positivity of the solution and its boundedness, the computation of different equilibrium points and the analysis of their stability.
Numerical simulations are presented and their results are discussed.

FORMULATION OF THE EPIDEMIC MODEL
The aim of this article is to study the role of some control measures available in an epidemic.
These measures include reducing the rate of infection, eliminating dairy products contaminated with tuberculosis, and observing hygiene measures in the manufacturing process of these products.
We present, in this article, a mathematical model for the tuberculosis transmission. In this model, we are interested in three main components, namely: the human population, Mycobacterium tuberculosis and dairy products.
The human population comprises three compartments such that at the instant t ≥ 0: S(t) are the susceptible individuals to be infected, I(t) are the infected individuals with tuberculosis, and R(t) are the recovered individuals from tuberculosis. Thus, the size of the human population is The model diagram of tuberculosis transmission taking account consuming dairy products unpasteurised is as follow: Susceptible individuals grow at the rate µ, proportional to the total population N, and die at rate µ, proportional to the susceptible population S. Susceptible individuals can be infected with a force of infection β K , that will be defined later. In fact, there are two modes of transmission of tuberculosis: the direct mode, that is to say when the bacterium is transmitted by close contact between an infected subject and a susceptible host, and the indirect mode, when the transmission takes place through a food such as unpasteurized dairy products. An infected individual may recover at the rate γ, die naturally at the rate µ, or die from tuberculosis at the rate δ .
The bacteria population obey the logistical law with a carrying capacity of K. Its growth rate is r b and its death rate is µ b .
As for dairy products, a distinction is made between non-contaminated products L n (t) and contaminated products L c (t). It is supposed that the dairy products L(t) are produced at the rate α and that the uncontaminated ones become contaminated, either by direct contact with the bacteria or by contact of the contaminated products with a force of infection β L , that will be defined later, and eliminated by a rate of µ l . See Table 1  , and the description of the model, we have the following system of non linear ordinary differential equations: (1) Noted that: • β 1 is the rate at which susceptible individuals become infected by the bacteria in the environnement.
• β 2 is the rate at which susceptible individuals become infected by consuming contaminated dairy products.
• β 3 is the rate at which non contaminated dairy products become infected by contacting the bacteria.
• β 4 is the rate at which non contaminated dairy products become infected by contacting the contaminated one.
To reduce the number of variables, we apply the following change of variables: On the other hand, we have: So, the reduced system is given as: taking into account the following initial conditions: in order to be biologically meaningful.

MATHEMATICAL ANALYSIS
In this section, the model is analysed. We show the positivity and boundedness of the solution and we calculate the equilibrium points.

Positivity and boundedness of the system solution.
The formulated model will be epidemiologically meaningful if all its variables are positive at any time t.
Theorem 3.1. The solutions of the system (s(t), i(t), l c (t), b(t)) for all t ≥ 0 are bounded in the set Ω, which is given by taking into account the following initial conditions: ) be a solution of the system (2) with the previous initial conditions.
From the first equation of the system (2), we can state that: The second equation of the system (2) gives: Since i(0) ≥ 0, then i(t) ≥ 0 for all t ≥ 0.
Also, by setting n = s + i, the variation of the total population is given by: Integrating the inequation (3) from 0 to t, we obtain : where C 0 is a constant.
On the other side, from the equation: Applying integration, we prove that: we get: From the equation that governs the variations of the bacteria, we get: putting: the equation (4) becomes: The solution of the equation (6) is: where k is a constant.
Substituting the expression of z(t) in the equation (5), it yields: As a result, However, the existence of the bacteria requires that its mortality rate must be lower than its growth rate, so µ B r B < 1, and 1 − µ B r B < 1, thus

Equilibrium Points of the system. The equilibrium points of the dynamical system (2)
is obtained by resolving the system: The fourth equation of the system (7) yields: Knowing that ∀t ≥ 0, b(t) ≥ 0, we can state that b 2 exists only if µ B < r B .
• Case 1.1: If b 1 = 0 and l c 1 = 0, the second equation of the system (7) yields i = 0, and its first equation gives s = 1.
So, the first equation of the system (7) gives: Substituting the expression of s 1 in the second equation of the system (7), it yields: . Thus, the endemic equilibrium point with contamination only by consuming unpasteurized dairy products is given by: , 0 • Case 2: the third equation of the system can be written as: with the expression of β L , it becomes: By setting: we get: Thus: We have already demonstrated that l c 3 > 0 for all positive initial condition, so we will choose l c 3 which is in R * + , that it will be noted l + c 3 . Hence, it is easy to get: and from the second equation of the system (7), we obtain: .
Consequently, the endemic equilibrium point is given by: In the next section, we demonstrate the local stability of the different equilibrium points.

LOCAL STABILITY OF THE EQUILIBRIUM POINTS
Using the theorem of Poincarré-Lyapunov in [13], [14], an equilibrium point is locally asymptotically stable if and only if all the eigenvalues of the Jacobian matrix of the dynamical system evaluated at this equilibrium point are strictly negatives. In this regard, we will use the Routh-Hurwitz criterion for the second-degree polynomial [13]. It shows that if all the coefficients of the characteristic polynomial are strictly positives, then it does not admit a positive root.
Proposition 4.1. The non-endemic equilibrium point P 0 is locally asymptotically stable if β 4 < µ l and r B < µ B .
Proof 4.1. The Jacobian matrix of the dynamical system (2) at P = (s, i, l c , b) is given by: The Jacobian matrix on the point P 0 is given by: Its eigenvalues are: Since all the parameters of the system are strictly positives, λ 1 and λ 2 are strictly negatives.
Therefore, The nonendemic equilibrium point P 0 is locally asymptotically stable if and only if λ 3 and λ 4 are strictly negatives, that is if β 4 < µ l and r B < µ B .
Proposition 4.2. The endemic equilibrium point with contamination only by consuming unpasteurized dairy products P 1 is locally asymptotically stable if and only if µ l < β 4 and r B < µ B .
Proof 4.2. The Jacobian matrix on the point P 1 is given by: Putting:   such as: The characteristic polynomial associated to the matrix J 1 (P 1 ) is: It's clear that the coefficients of P(λ ) are strictly positives. Thus, by Routh-Hurwitz Criterion, its eigenvalues have a strictly negative real part.
As for J 4 (P 1 ), its eigenvalues are given by: Proof 4.3. The Jacobian matrix on the point P 2 is given by: Putting:   such as: The characteristic polynomial associated to the matrix J 1 (E 2 ) is given by: where: Since all the coefficients of P(λ ) are strictly positives, then P(λ ) does not have positive roots.
Therfore, by Routh-Hurwitz Criterion, its eigenvalues have a strictly negative real part.
As for the matrix J 4 (P 2 ), its eigenvalues are: where: C = β 3 b 2 + µ l + 2 β 4 l + c 3 is a positive constant, and λ 2 = r B − µ B − 2r B b 2 . So, λ 1 < 0 if and only if β 4 < C, and λ 2 < 0 if and only if r B < 2r B b 2 + µ B . Thus, the endemic equilibrium point P 2 is locally asymptotically stable if and only if β 4 < C and r B < 2r B b 2 + µ B .

NUMERICAL SIMULATION
Numerical simulations for the tuberculosis model given by the system (2) are done using Matlab, whose objective is the description of the behavior of the system solutions over time, and the affirmation of the results obtained.
For the simulation of the nonendemic equilibrium point, we start with initial conditons: s 0 = 0.4, i 0 = 0.1, l c 0 = 0.2, b 0 = 0.4. We will run simulation, in an interval of 150 days, and we obtain the numerical solution for the system (2). We defined the values of the parameters in Table 2 Table 3.

CONCLUSION
We have presented in this article a compartmental mathematical model of the dynamics of tuberculosis in Morocco, that took in consideration the infection by consumption of unpasteurised dairy products. We have calculate the equilibrium points and studied their local stability using eigenvalues analysis. In the end, numerical simulations are presented to illustrate the results obtained. This research confirms that the elimination of unpasteurized dairy products allows the reduction in the rate of infection by tuberculosis, and consequently the number of people infected with this bacillus. In addition, it is recommended to know some practices aimed at ensuring food safety, including washing hands regularly, separating raw dairy products from cooked dairy products, and the pasteurization which remains the most effective measure to prevent the food transmission of pathogens, including M. Bovis, to humans.

CONFLICT OF INTERESTS
The authors declare that there is no conflict of interests.