THE ROLE OF ANTIBIOTICS AND PROBIOTICS SUPPLEMENTS ON THE STABILITY OF GUT FLORA BACTERIA INTERACTIONS

: Dyspepsia is a significant public health issue that affects the entire world population. In this work, we formulate and analyze a deterministic model for the population dynamics of Gut bacteria in the presence of antibiotics and Probiotic supplements. All the possible equilibria and their local stability are obtained. The global stability around the positive equilibrium point is established. Numerical simulations back up our analytical findings and show the temporal dynamics of gut microorganisms


INTRODUCTION
The human large intestine consists not just of cells but is also an ecosystem such as "gut flora," one of the trillion different types of bacteria. The majority of these microorganisms have advantages. A portion of the small and large intestines is home to friendly bacteria. Bacteria cannot develop in the stomach's acidic environment. The body's gut microorganisms play a variety of functions. For instance, gastrointestinal bacteria create the vitamins and 12 . Limit the spread of dangerous organisms. Toxins are broken down in the big intestine. Break down the fiber and some carbohydrates and sugars in meals that cannot be absorbed. Enzymes made by bacteria break down the carbohydrates in plant cell walls.
Without these bacteria, most of the nutritional content of plant matter would be lost. These facilitate the digestion of plant meals like spinach [1]. Some of the human gut's microorganisms are pathogens that can cause sickness. Other bacteria are beneficial and give a wide range of health advantages. Bacteria in the gut play a crucial part in digestion by assisting your body in breaking down food and absorbing nutrients [2]. It is an integral part of the human body. Gut bacteria are essential to human health by delivering crucial nutrients, generating vitamin K, aiding cellulose digestion, and stimulating angiogenesis and enteric nerve activity. Due to the alteration in their composition brought on by the gut ecosystem's aberrant alterations by disease, antibiotics, ageing, stress, lifestyle choices, and poor dietary practices can also be potentially dangerous. Numerous chronic disorders, such as cancer, inflammatory bowel disease, autism, and obesity, can be brought on by dysbiosis of the gut bacterial communities [3]- [4]. We are also nourishing the microorganisms in our guts when we eat. These bacteria like feasting on proteins, carbs and milk sugars just as we do. The human and the bacteria in our gut gain through this symbiotic eating connection [5].
Recent research has illuminated the collateral damage that antibiotics due to the bacteria in the human gut. These medications have been proven to have immediate and occasionally long-lasting impacts that affect the human gut bacteria's taxonomic, genomic, and functional capabilities. While increasing and decreasing the membership of particular native taxa, broad-spectrum antibiotics reduce bacterial diversity [6].
Probiotics are living bacteria that exist naturally in the human body. Our body is continually infected with both healthy and dangerous microorganisms. When you acquire an illness, more nasty bacteria enter your system, throwing your system off balance. Good bacteria aid in the elimination of excess harmful bacteria, restoring equilibrium. Probiotic supplements are a way to 3 STABILITY ANALYSIS OF PATHOGENIC AND PROBIOTIC BACTERIA supply your body with beneficial microorganisms which help the human body digest food [7].
Mathematical modelling is a procedure that involves translating issues from the actual world into mathematical terms, solving them in a symbolic system, and then testing the results in the original system. Mathematical models such as biology, ecosystem, and physics are utilized in the natural sciences. Extensive studies have used mathematical modelling to solve problems describing a system, investigate the consequences of various elements, and forecast behavior [1], [8]- [14] [4].
In this paper, the mathematical model describes the interplay of bacteria in the human gut in the presence of antibiotic and Probiotic supplements is offered. The rest of the article is set as follows.
The structure of the proposed model is described in Sect. 2. The existence of the possible equilibria is shown in Sects. 3. The stability property of the equilibria is investigated in Sects. 4. Some examples and their numeric simulations are presented in Sect. 5 to show the feasibility of the main results. We end this paper with a brief discussion.

MATHEMATICAL MODELLING
Suppose an ecosystem in the large intestine contains two symbiotic bacteria. Let 1 ( ) and

PARAMETER EXPLANATION
Growth rate of 1 .
Growth rate of 2 .
Carrying capacity of 1 and 2 .
The rate of effectiveness of Probiotic supplements.
The transfer rate of good bacteria to harmful bacteria is due to mutations of good bacteria exposed to antibiotics.
The rate of eliminating good bacteria by an antibiotic.
The rate of eliminating harmful bacteria by antibiotics.
The natural death rate of 1 .
The natural death rate of 2 .
The elimination rate of harmful bacteria by the immune system.
The constant intake of non-decomposing toxins in the large intestine.
The natural degradation of non-decomposing toxins in the large intestine.
The increased rate of non-decomposing toxins due to a large amount or quantity of harmful bacteria.
The decreased rate of non-decomposing toxins due to a large amount or quantity of good bacteria.
The concentration rate of antibiotics.
The degradation rate of antibiotics.

EXISTENCE OF EQUILIBRIA
System (1) has twelve non-negative equilibrium points, namely: and, * = 0 exists when positive, the following would be the case: (2)  , and ̂1 = − 1 1̂2 + ( 1 + 0 − 1 ) 1 . Clearly, ̂1 > 0 if the following condition holds: It should also be noted that for ̂2 > 0 to be positive, one of the following conditions must be the case:  Using Descartes's rule of sign [15], Eq. (14) has a unique positive root 2 ′ if one of the following conditions is satisfied: 1. > 0 and < 0,
Further, for 1 ′ to be positive, the following must be the case

STABILITY ANALYSIS
This section explores the local stability behavior of the system (1) 's equilibrium points.
The Jacobin matrix of system (1) at any point, say ( 1 , 2 , , ), can be written as: Consequently, the following is obtained.
Then, the eigenvalues of ( 10 ) are given by That 10 is a locally asymptotical stable point if: 11. The Jacobian matrix at 11 = ( 1 = , 2 = , 0, * ) can be written as: 12. The Jacobian matrix at 12 = ( 1 * , 2 * , * , * ) can be written as: Now, Using the Lyapunov approach [16], this section delves into the requirements that must be met for the global stability property of the system to exist (1) at the positive equilibrium point.
are satisfied. ➢ Case 3: the behavior of the system (1) with probiotic supplementation and antibiotic

CONCLUSION
A model consisting of good bacteria, harmful bacteria, toxins and antibiotics in the large intestine has been studied. The terms of probiotic supplementation of the good bacteria have been included.
The theoretical analysis of the proposed mathematical model shows the existing conditions of the twelve non-negative equilibrium points. Based on the Routh-Hurwitz stability criteria, the positive equilibria 12 = ( 1 * , 2 * , * , * ) Showed asymptotically stable behavior under certain conditions.
Further, using the Lyapunov method, the appropriate states that guarantee the global stability of the positive equilibria have been established. According to the numerical simulation results, system movement always happens around the positive equilibria if the system stability conditions are met.
In contrast, the absence of probiotic supplements will increase toxins in the large intestine. That means the beneficial bacteria play a role in digestion and the enhancement of nutrient absorption.

CONFLICT OF INTERESTS
The author(s) declare that there is no conflict of interests.