Antibiotic resistance is a significant public health problem. When resistance genes are being carried on plasmids, the spread can be greatly accelerated. In this paper, the transmission of antibiotic resistance in two types of cells is discussed. A mathematical model is established to describe the dynamics of the transmission of plasmids. The effects of different parameters on the stable solution and sensitivity analysis are studied by numerical simulation. The conclusions show that the concentration of antibiotics must reach a certain level to kill the pathogenic bacteria. If the concentration of antibiotics is not high to a certain extent, the treatment becomes ineffective. If the cost of cells carried on plasmids and the rate of resistance plasmids segregation too high, the drugresistant cells will gradually die out in the system. The rate of horizontal transfer of resistance plasmids is directly related to the spread of drug resistance. With the increase in the horizontal transfer rate of resistance plasmids, cells in the body gradually turn into cells with antibiotic resistance, which causes substantial difficulties in the treatment of diseases.
Keywords
 antibiotic resistance
 plasmids
 horizontal gene transfer
With the development of modern medical technology, a large number of antibiotics have been developed to kill microorganisms that pose a threat to human health. However, the misuse of antibiotics has led to widespread antibiotic resistance, which is now a significant public issue [1]. Antibiotic resistance, also known as drug resistance, refers to the resistance of microorganisms, parasites and tumour cells to the effects of chemotherapy antibiotics [2, 3]. Antibiotic resistance can be divided into two types: acquired antibiotic resistance and natural antibiotic resistance. Pathogens in nature, such as a strain of bacteria, can also have natural resistance, regardless of increasing doses of the antibiotic [4]. Acquired resistance occurs due to specific changes in the DNA (mutation) or through contact with external sources such as plasmids, transposons or integrons [5]. At present, acquired drug resistance is considered as the main cause for the prevalence of antibiotic resistant. Bacterial cells are capable of transferring genes horizontally. When resistance genes are mobile, being carried on plasmids or phages, their spread can be greatly accelerated [6,7]. A plasmid is a small piece of extrachromosomal DNA that may be easily transferred between bacteria via horizontal gene transfer and, in particular, has been implicated in the acquisition of antibiotic resistance genes to many antibiotics [8]. In this paper, we investigate the problem of celltocell transmission of antibiotic resistance caused by plasmids carrying resistance genes.
When antibiotics are applied for a long time, the majority of the sensitive bacteria are killed continuously, and the antibiotic resistant bacteria will keep multiplying in large numbers, so that the resistance rate of the bacteria to the antibiotic keeps rising [2]. The experiments show that the use of antibiotics will promote the transmission of plasmids in the colonies [9]. In fact, the concentration of antibiotics in the environment is not static, but changes continuously during the absorption. We analyse the effect of the drug concentration in the numerical simulation section. The speed at which the bacteria gets wiped out depends on the duration of administration of the antibiotic [10]. Researchers find that the optimal dosing does not necessarily correspond to the continuous administration of the antibiotics [11].
A mathematical model that describes the population dynamics of bacteria exposed to multiple antibiotics simultaneously is formulated [12], assuming the acquisition of resistance is through mutations from antibiotics exposure. In general, the specific mutations that confer resistance to antibiotic control have a cost, which may manifest as a reduction in reproductive capacity or the ability to compete. In the absence of medications, the infection rate of resistant bacteria is less or equal than the infection rate of sensitive bacteria; nevertheless, the population of resistant bacteria could increase to far exceed the population of sensitive bacteria under treatment [13]. Under the condition of decreasing drug concentration, the development of antibiotic resistance may be reversible [14]. This has implications for the management of antibiotic resistance, which can not only be managed by reducing the development of resistance, but also requires the restoring of existing resistance.
Assuming that drug resistance is acquired through mutations and plasmids transmission, a deterministic model for the population dynamics of sensitive and resistant bacteria to multiple bactericidal and bacteriostatic antibiotics is formulated and analysed [15]. Svara and Rankin [16] established an ordinary differential equation model to investigate the dynamics of plasmidcarried antibiotic resistance. By numerical simulations, it is shown that the transmission of plasmids is the key factor influencing plasmidborne antibiotic resistance, and the dosage of antibiotic and the interval between treatments are also important. By employing qualitative analysis of ordinary differential equations, Xu et al. [17] investigate the collective resistance of the bacteria population with resistant horizontal gene transfer under sublethal bactericide pressure. The results show that the possible mechanism of variations in antibiotic susceptibility is the dominance of different bacterial genotypes under sublethal bactericide pressure, rather than persistence, tolerance or resistance. As a matter of fact, this can be seen as one of the innate characteristics of interaction between bacteria and bactericides.
In a systematic review, Quentin et al. [18] searched for mathematical modelling studies that focus on horizontal transfer of antimicrobial resistance (AMR) genes. Their findings highlight the existence of a research gap in the dynamics of transformation and transduction, and the overall public health implications of the horizontal transfer of AMR genes.
Because horizontal transfer of plasmids is an important mechanism, whereby resistance is spread through bacterial populations, we develop a mathematical model to describe this process quantitatively, using it to optimise antimicrobial dosage regimens to minimise resistance development. The transmission of antibiotic resistant plasmids in the coexisting environment of two types of bacteria is discussed in this paper.
In this article, we assume that cells are divided into normal cells and pathogenic bacteria. In the mathematical model we established, the letter ‘a’ represents normal cells and the letter ‘b’ represents pathogenic bacteria. We assume that when antibiotics are used, they only kill the pathogenic bacteria and have no effect on normal cells.
To simplify the model, we assume as follows:
The densitydependent death rates of the two types of cells are equal, and is given by. It means that whether a cell is antibiotic resistant or not has no effect on its natural growth rate or densitydependent death rate.
We assume an antibioticinduced fitness cost, that is, antibiotics can have two effects on a cell: they can either kill the bacteria or they can prevent their reproduction.
To keep our model simple, we assume that genes for resistance against antibiotics are carried by a plasmid rather than by the chromosome.
Due to the absence of resistance genes, the plasmidfree types of pathogenic bacteria are more sensitive to antibiotics. Mathematically, it is reasonable to assume that.
Details of parameters used in our model are given in Table 1.
Model variables and parameters
Density of plasmidfree type of normal cells  
Density of normal cells infected with a resistancecarrying plasmid  
Density of plasmidfree type of pathogenic bacteria  
Density of pathogenic bacteria infected with a resistancecarrying plasmid  
Intrinsic percapita growth rate of cells  
Extrinsic densitydependent death rate of cells  
Total density of cells 

Rate of horizontal transfer of resistance plasmid from infected normal cells  
Rate of horizontal transfer of resistance plasmid from infected pathogenic bacteria  
Rate of the resistance plasmid segregation from normal cells  
Rate of the resistance plasmid segregation from pathogenic bacteria  
Rate of the resistance plasmid segregation from pathogenic bacteria  
Cost of infected normal cells carried on plasmid  
Cost of antibiotic resistance when pathogenic bacteria is carried on resistance plasmid  
Concentration of antibiotic  
Death rate of plasmidfree type of pathogenic bacteria due to antibiotics  
Death rate of pathogenic bacteria with resistancecarrying plasmid due to antibiotics 
Assuming logistic growth, with a growth rate of
For the sake of simplicity, let
Boundary equilibria and conditions
Always exist  

Always exist 



It is too complicated to express 

It is too complicated to express 
The boundary equilibria are analysed using stability theory. Through analysis, we can draw the following conclusions.
The proofs of Theorems 1–3 are as follows.
For
For
For
We built numerical simulations to investigate how parameters affect the variation of system (1). To attain a more realistic model, we assume that the antibiotic degrade at an exponential rate [19]:
When using an antibiotic to treat the disease caused by pathogenic bacteria, the concentration of the drug must reach a certain level; otherwise, it cannot kill the pathogenic bacteria. In other words, if the concentration of the antibiotic is not high to some extent, it becomes ineffective at treating the disease. The drug concentration over time is marked by a blue dotted curve in each figure for clearer emphasis. The four other lines show the density variations of cells. As we know from Eq. (2), this means that the smaller the
Here, the plots are calculated by running the simulation for a number of parameter values for over 500 time steps. Parameters used are:
With the increase of
As for the parameter of cost, when cells are carrying plasmids, the corresponding number of cells infected with resistancecarrying plasmids decreases with the increase of cost value. In Figure 4(a), when
Figure 5 shows the role of parameters involved in the segregation rate of plasmids. We can see from Figure 5 that the ability of resistancecarried plasmids to spread is reduced with the increasing segregation rate value
The goals of sensitivity analysis with respect to random perturbations of the model parameters is as follows:
To show how robust the simplified uncomplicated bacteria resistance model is in relation to perturbed parameter values.
To explore which parameters the system is more sensitive to understand the key process and bacteria resistance system mechanisms.
The sensitivity of drug degradation rates is considered in this section. Figure 6 shows the effect of drug degradation rates on the population of bacteria. The values of
The sensitivity of the parameter
The sensitivity of the parameter
The sensitivity of segregation rate
In Figures 9 and 10, curves of the same colour show the variations of the proportion of the bacteria under the same set of parameters. As we can see from the figures, the effect of
It is shown that the concentration of antibiotic drugs must reach a certain level to kill the pathogenic bacteria. If the concentration of antibiotic is not high to some extent, it becomes an ineffective treatment. The rate of horizontal transfer of resistance plasmids is directly related to the spread of drug resistance. With the increase in the horizontal transfer of resistance plasmids, cells in the body gradually turn into cells with antibiotic resistance, which cause great difficulties in the treatment of diseases.
In addition, the risks associated with the development of antibiotic resistance in populations are often latent and undetectable. However, when people get sick, it will be much harder for them to heal because of antibiotic resistance. Therefore, the potential threat of drug resistance to people's health cannot be ignored.
Boundary equilibria and conditions
Always exist  

Always exist 



It is too complicated to express 

It is too complicated to express 
Model variables and parameters
Density of plasmidfree type of normal cells  
Density of normal cells infected with a resistancecarrying plasmid  
Density of plasmidfree type of pathogenic bacteria  
Density of pathogenic bacteria infected with a resistancecarrying plasmid  
Intrinsic percapita growth rate of cells  
Extrinsic densitydependent death rate of cells  
Total density of cells 

Rate of horizontal transfer of resistance plasmid from infected normal cells  
Rate of horizontal transfer of resistance plasmid from infected pathogenic bacteria  
Rate of the resistance plasmid segregation from normal cells  
Rate of the resistance plasmid segregation from pathogenic bacteria  
Rate of the resistance plasmid segregation from pathogenic bacteria  
Cost of infected normal cells carried on plasmid  
Cost of antibiotic resistance when pathogenic bacteria is carried on resistance plasmid  
Concentration of antibiotic  
Death rate of plasmidfree type of pathogenic bacteria due to antibiotics  
Death rate of pathogenic bacteria with resistancecarrying plasmid due to antibiotics 
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