Deterministic Modeling of the Issue of Dental Caries and Oral Bacterial Growth: A Brief Review

Abstract

Dental caries is a chronic disease that mostly interferes with oral health worldwide. It is caused by acidic bacteria on the enamel surface, mainly Streptococcus mutans, forming an oral biofilm that can be transmitted among people. The spread behavior and population dynamics of bacteria must be studied to control this disease, which can be approached through mathematical modeling. In this review, we aimed to identify the utilization of compartmental models in studying bacterial behavior. The aim was to explore compartmental model utilization and reveal the research gaps. This study was conducted with the PRISMA guidelines for scoping reviews to identify an existing mathematical model to study the phenomenon. Four databases, Scopus, ScienceDirect, PubMed, and Dimensions, were used to search for related studies. Our results showed that compartmental model utilization for studying bacteria’s role in dental caries is sparse and poorly explained. Moreover, the constructed models have not yet considered more intervention efforts. A study is needed to explore this phenomenon by developing a mathematical model considering some situations. When conducted, it will result in more insights into controlling the spread of bacteria to prevent dental caries.

1. Introduction

The quality of individual and community health is a critical national or even global issue. It is represented in Sustainable Development Goals point 3, namely, good health and well-being. In terms of dental health issues, the World Health Organization (WHO) revealed in 2023 that around 3.5 billion people worldwide have oral disease, with three out of four people suffering from oral disease coming from middle-income countries. Moreover, WHO estimates that about two billion people experience permanent dental caries, and 514 million children experience caries of deciduous teeth [1].

Dental caries is a multifactorial disease that deteriorates the tooth structure over time. The disease is highly correlated with conditions involving the tooth structure, oral biofilm—colonized bacteria on the tooth’s surface—and substrates consumed by bacteria, saliva, and genetic factors [2]. The elements affecting the physical and biological risks of dental caries are salivary composition [3], biofilm thickness [4], biofilm maturity [5], overpopulation of cariogenic bacterial [6], lack of fluoride [7], immune system, dental health, and genetic factors [8]. In addition, socioeconomic circumstances [9], education [10], lifestyle [11], environment [12], age [13], and ethnic group [14] are other factors that affect the risk of the disease. The causative factors of individuals suffering dental caries can be illustrated using a Keyes diagram (see Figure 1).

Dental caries arise from dysbiosis in an oral biofilm when the ratio between good and bad bacteria is imbalanced [6]. In detail, dysbiosis in oral biofilms is defined as a state when there is an increase in acid-producing and acid-resistant bacteria, especially S. mutans and Lactobacilli [15]. The oral biofilm is a structured community consisting of various bacteria that attach to the teeth’s surface and significantly affect oral health quality [16]. S. mutans is the bacteria most often discussed in the study of dental caries [17]. Zhang et al. [18] and Zhu et al. [19] revealed that S. mutans is the bacteria that most easily attaches to the surface of the teeth and has an essential role in forming oral biofilms. Overpopulation of S. mutans in oral biofilms results in an imbalance of bacterial varieties and leads to demineralization, promoting the condition of dental caries [20]. Therefore, the growth of S. mutans needs to be studied to develop methods for controlling and preventing oral biofilm formation.

Demineralization is the cessation of the formation of dental minerals in the teeth or the continuous release of dental minerals from the teeth. Continuous demineralization and remineralization occur in each individual’s tooth structure. This process becomes a determining factor in the occurrence of dental caries. The causative factors of demineralization are diet and bacteria in the biofilms, while the causes of remineralization are salivary protection and the use of fluoride [21]. When bacterial metabolism and carbohydrate consumption overwhelm the saliva’s protective ability and use of fluoride, then the tooth tissue is demineralized. As a result, the accumulative loss of good mineral content is higher, followed by the formation of cavities/holes in the teeth. Suppose the cavity is allowed to continue to grow. In that case, the area will become a habitat so that bacteria in the biofilm can adapt to the decrease in pH and the release of mineral ions [3]. Lower pH conditions over a long period are suitable for S. mutans. Peres et al. [22] defined dental caries as the chronic loss of mineral ions from the tooth structure caused by oral biofilms. The pathogenicity of biofilms was identified through two indicators: (1) increased bacterial resistance to antibiotics and (2) bacterial communities that cannot be phagocytosed by inflammatory cells [23].

Figure 1. Causative factors of dental caries condition based on a Keyes diagram [24].

Figure 1. Causative factors of dental caries condition based on a Keyes diagram [24].

Dental caries can be viewed as an infectious disease based on the findings that S. mutans in oral biofilm can be spread among people. Esra et al. [8] revealed that the S. mutans sampled from parents and children have the same genotype. Furthermore, that study also provides insight into two children sharing a playground who had the bacteria S. mutans with the same genotype. Damle et al. [25] and Ravikumar et al. [26] revealed that the existence of S. mutans has a positive correlation between mother and child. Another fact that strengthens the thought of dental caries as an infectious disease is that the S. mutans genotype is the same among individuals sharing a space. Mattos-Graner et al. [27] and Berkowitz [28] revealed that bacterial transmission of S. mutans can occur between active individuals in the same place. For example, the environment where children play is where they are most at risk of spreading the bacteria S. mutans [29,30]. Therefore, dental caries can be viewed and studied as an infectious disease because the bacteria S. mutans in the oral biofilm that causes dental caries can spread from one individual to another.

Dental caries is a significant cause of other dental and oral health problems. Keep in mind that the impact of bacteria on oral biofilms is not limited to dental and oral health issues. Cavities in the teeth as a result of dental caries can expose the blood flow, so bacteria can enter and begin to spread to other parts of the individual’s body [31]. Further studies revealed findings that similar bacteria were detected in mouths and in the lungs of pneumonia sufferers [32], the hearts of endocarditis patients [33], and the brains of dementia/Alzheimer’s patients [34]. Therefore, dental caries is a disease that causes other chronic diseases. Thus, we need to make an effort to control dental caries and thus prevent other chronic systemic diseases.

Prevention and control of dental caries needs to be achieved to reduce the incidence and risk of chronic diseases derived from dental caries. Pitts et al. [2] emphasized that interventions at the individual and population levels must be conducted optimally and harmoniously. Interventions can be carried out through health policies, laws and regulations, and public health approaches to encourage healthy behavior and influence broader social determinants in the health field [35,36]. Prevention can be achieved by targeting the entire population by fluoridating water and taxing sugar to increase control effectiveness. Furthermore, WHO’s oral health action plan explains the importance of combining oral health prevention programs with other chronic disease prevention programs and policies [1].

Mathematical modeling is a formidable approach to studying how a disease spreads and the involved population dynamics to understand a complex system. This method can be used in testing and comparing different disease-spreading behaviors, considering interventions as strategies to control and prevent the disease. This capability is essential in conducting field trials in a world of limited resources [37]. A mathematical model provides insights for understanding disease conditions quantitatively and allows us to check the hypotheses for understanding their vitality [38].

Constructing a mathematical model for disease transmission needs some assumptions according to the spreading mechanism. A basic compartment-based mathematical model is introduced to describe the disease spread phenomenon [39]. The initial model was frequently used to develop a compartmental model representing a natural phenomenon. For the same purposes, stochastic modeling can be utilized to represent the phenomena while covering natural randomness and its perturbation. However, a limitation of stochastic modeling is the need for complete data for each variable in the research. This means that if we do not have sufficient data for some variables, we must ignore their involvement in the model, and the results will not approach the degree of accuracy they should have. In this situation, compartmental modeling does not need complete data like stochastic modeling. We only need complete information to develop the model, and this is useful when the data is very rare, like the early COVID-19 situation in late 2019. Lately, the results of the compartmental model for infectious disease and population dynamics have been useable for stakeholders and frequently studied in any field [40,41,42,43]. This proves that mathematical model utilization is beneficial in understanding a phenomenon.

Compartmental modeling is frequently used to study disease transmission or population dynamics phenomena. In the field of disease transmission, models were constructed to understand the dynamics of COVID-19 [44], dengue [45], typhoid [46], HIV-AIDS [47], etc. Whereas in population dynamic studies, this approach was able to represent the population growth of animals [48] or bacteria [49] in a region. In addition, various possible intervention efforts were considered in developing a model of disease transmission and population dynamics to obtain more insights into the phenomenon. For instance, the effect of education [50] and social distancing [44] efforts in controlling disease spread as well as the use of antibacterial [51] methods in the growth of bacteria populations. However, compartmental modeling is not well-utilized in studying dental health issues. This topic seems to have escaped researcher’s attention and is not of interest. Furthermore, Kumar et al. [52] briefly stated that no compartmental model exists in the literature for studying the tooth cavity spreading phenomenon. They claimed their model is the first ever constructed model to understand this issue. This motivated us to explore the development of compartmental model utilization in studying dental health issues, especially in dental caries or tooth cavity conditions, in this review.

This review article explores the utilization of compartmental modeling for studying dental health issues. We processed a dataset containing collected papers from databases using the PRISMA method. This method guided us in selecting the articles that meet the criteria to obtain suitable pieces for this review. We elaborated on the insights from the existing model on the chosen issue and revealed the research gaps. This review sheds light on the history of compartmental modeling for this issue and identifies areas for further study. In addition, we confirmed the statement from Kumar et al. [52].

2. Materials and Methods

Our review began with determining the keywords to search published articles indexed in four databases: Scopus, Science Direct, PubMed, and Dimensions in the middle of September 2023. The keywords set used was in the form of (“MATHEMATICAL MODEL” OR “MATHEMATICAL MODELING” OR “MATHEMATICAL MODELLING”) AND (“ORAL BIOFILM” OR “TOOTH CAVITY” OR “TEETH CAVITY” OR “DENTAL CARIES”). The collected papers were limited to research articles and written in English. We did not restrict the publication time to obtain published papers from whenever and filtered as much research as has been conducted within this remarkable scope. We identified 103 articles from Scopus, 209 from Science Direct, 35 from PubMed, and 71 from Dimensions.

The method used is the Preferred Reporting Item for Systematic Review and Meta-Analyses, also known as the PRISMA method. It is a procedure for obtaining results in articles that match our review protocol to be studied by conducting some selection processes. We elaborate on the primary process following the steps in [53]:

Step 1:

Remove duplicated articles and identify whole papers that meet the review protocol.

Step 2:

Screen the papers that pass the first protocol terms. This step allows us to determine the documents that meet the criterion in each step, resulting in the most relevant articles for our review. Generally, it is separated into several stages, as needed.

Step 3:

Collect the suitable studies for further review and analysis.

The authors [54] revealed that the method of PRISMA is an analysis employed to identify elements of articles in a database. It allows us to perform a brief review based on the formulated research question to obtain insights into the topic by identifying, selecting, and critically assessing the related literature following the above steps. Meanwhile, we outline our review protocol as follows:

• The article studies a compartmental-based model utilizing dynamical system theory.
• The compartmental model is constructed in an ordinary differential equations system (ODEs) or delay differential equations system (DDEs).
• The papers utilize the constructed model to study a population dynamics issue.
• The population dynamics studied in the selected paper represent the dental health issue in epidemiology or bacteria population growth terms.

3. Results and Discussion

3.1. Output of PRISMA Method

The PRISMA method was conducted in semi-automatic and manual steps. The semi-automatic action was employed to screen the existence of duplicate papers in a database against the others. In our study, the result of the Scopus database was selected as the primary reference for checking the possibility of duplicate articles in the results of the Science Direct, PubMed, and Dimensions databases. We obtained 323 unique articles after mixing the results from the three other databases without duplicate articles. Next, the manual action was a process of identifying the articles that met our review protocol after a brief understanding. This process was divided into two steps. Firstly, we examined the articles related to our review study based on their titles, abstracts, and keywords. Secondly, the selected papers were read comprehensively to determine their relationship with our review protocol.

The screening was based on the relevance of each article’s title, abstract, and keyword, producing 119 related articles. Next, we employed manual filtering by reading the text thoroughly. The whole article can be read using the skimming method to obtain a comprehensive identification for selecting ODEs or DDEs for population dynamics in dental health issues. In addition, the skimming method was used to present a brief understanding of the selected articles. We obtained only three papers that matched our review protocol. The selected papers were divided into two issues, namely, epidemiology and bacterial population growth in oral biofilm. Moreover, we explored each article to gain information about the constructed model and its results. This also reveals the gap in exploring how we can develop mathematical studies for dental health issues. In brief, the selection process is shown in Figure 2.

Figure 2 shows that some papers were excluded due to the lack of relevance of the title, abstract, and keywords. Unless the title, abstract, and keywords are relevant to the study, we explain why they should be eliminated from further review. As we can see in selecting a related full paper, the reasons articles were excluded from our research are written briefly. All of the articles that met the requirements of title, abstract, and keyword relevance were skimmed to justify the suitability of the article content with the goal of this review. This process identified several articles that did not provide what we needed in our study.

First, we checked whether the article explains a compartmental-based model and found 93 articles did not fulfill this. For instance, authors [55,56] employed deep learning methods and neural network architecture to study earlier dental caries detection. Other papers studied dental caries formation due to socioeconomic inequality [57] and oral biofilm dysbiosis [58] using a statistical approach to solve the problem.

Next, the collected documents were checked to determine whether the compartmental-based model is constructed in ODEs or DDEs or not. We found that 16 articles were not in ODEs or DDEs but in a partial differential equations system (PDEs). For example, the study [59,60] explored the progression of dental caries and bacterial behavior in oral biofilm using PDEs. Further, the ten articles that constructed a compartmental-based model using ODEs or DDEs determined whether studying a dental caries phenomenon was population dynamics or not. We found that six of ten papers did not the ODEs or DDEs to investigate a population dynamic. Still, saliva concentration keeps the remineralization ability [61] and dental caries formation due to xylitol crystallization [62] and synthesis of galactooligosaccharides and transgalactosylation [63].

In an early final step selection, we have four articles that employ ODEs or DDEs to study a dental health issue as a population dynamics view of human-to-human transmission or bacterial population growth in oral biofilm. In obtaining a comprehensive understanding from the four articles, we found that the study [49] explored bacterial population growth but not in an oral biofilm ecosystem. Hence, we eliminated that study from our review. Finally, the processes resulted in three articles that fulfilled our protocols. However, not accessing any other journal databases such as Web of Science (WoS), EMBASE, or Cochrane is a limitation of this study. Therefore, this research leaves room for any other potential author who has access to any other databases to extend this study further.

We divided the three collected articles based on their issues, namely (1) epidemiology and (2) bacterial population growth in oral biofilm. In detail, the first issue was studied by Kumar et al. [52]. At the same time, the second issue was explored by Shen et al. [64] and Jing et al. [51]. As a piece of early information, the study conducted in [51] is the development of the study in [64] but in a simplified view. We elaborate on their research in the following subsections.

3.2. Elaboration of Selected Articles

3.2.1. Elaboration of the First Human-to-Human Transmission Dental Health Issue

Kumar et al. [52] viewed the tooth cavity as an infectious disease among people. The issue was interpreted as a DDEs with five compartments regarding the conditions of each subpopulation. In detail, susceptible compartment $S\left(t\right)$ represents all tooth cavity susceptible individuals at time $t$, and exposed compartment $E\left(t\right)$ defines the number of people who feel some early symptoms of a cavity, such as holes, tooth pain, black marks, etc. Compartment $I\left(t\right)$ means those humans who are infectious, and compartment $T\left(t\right)$ represents the number of individuals undergoing serious dental treatment in medical facilities. The last compartment, $R\left(t\right)$, defines all recovered individuals from the tooth cavity. We rewrite the constructed model as follows:

susceptible subpopulation dynamic	(1)
$\frac{dS\left(t\right)}{dt}=\varphi -\mathsf{\Psi }S\left(t\right)I\left(t\right)-wS\left(t\right),$
exposed subpopulation dynamic
$\frac{dE\left(t\right)}{dt}=\mathsf{\Psi }S\left(t\right)I\left(t\right)-\gamma \mathsf{\Psi }S\left(t-\tau \right)I\left(t-\tau \right)e^{-w\tau }-wE\left(t\right)-rE\left(t\right) ,$
infectious subpopulation dynamic
$\frac{dI\left(t\right)}{dt}=\gamma \mathsf{\Psi }S\left(t-\tau \right)I\left(t-\tau \right)e^{-w\tau }-pI\left(t\right)-qI\left(t\right)-rI\left(t\right)-wI\left(t\right),$
treated subpopulation dynamic
$\frac{dT\left(t\right)}{dt}=qI\left(t\right)-\psi T\left(t\right)-wT\left(t\right),$
recovered subpopulation dynamic
$\frac{dR\left(t\right)}{dt}=rI\left(t\right)+rE\left(t\right)+\psi T\left(t\right)-wR\left(t\right).$

The total population size is formulated as $N\left(t\right)=S\left(t\right)+E\left(t\right)+I\left(t\right)+T\left(t\right)+R\left(t\right)\forall t\ge 0.$

As we can see in system (1), the authors used some parameters to construct the model. The parameter $\varphi$ defines the recruitment rate of new susceptible individuals, $\mathsf{\Psi }$ represents the transmission rate from infectious individuals that causes a transition from susceptible into an exposed subpopulation, and $w$ is the transition rate of those people who are not aware of the tooth cavity anymore. The transition rate from exposed to an infection subpopulation is defined as a parameter. $\gamma$. The parameter $r$ represents the recovery rate due to homemade treatments. The exciting aspect of their study is the assumption of considering society’s behavior, such as ignoring a tooth cavity after knowing the symptoms and defining it as a parameter $p$. The parameter $q$ represents the rate of medical treatment, which is the transition from infectious to a treated subpopulation. The recovery rate due to the medical treatment is interpreted as a parameter $\psi$. The parameter $\tau$ represents the delay in time $t$ regarding the latent period of cavity progression. It represents the period of forming the cavity since the first attachment of S. mutans at time $t$. As we see in model (1), the authors consider the transition from the exposed to infection compartment is due to the multiplication of $\gamma$ with the transmission case in the past as far as $\tau$ that is still alive since they incorporate the exponential function as the survival ability due to the existence of a natural death rate over the delay time.

3.2.2. Elaboration of the Bacterial Population Growth Model Studies in Oral Biofilm

Shen et al. [64] studied the reactive kinetics of oral biofilm formation, considering the existence of antimicrobial agents through a mathematical modeling approach. The model was developed in ODEs and consists of the bacterial population and its supportive components. First, the authors divided the bacterial population into three phenotypes, namely the susceptible (which means the portion of bacteria sensitive to the antimicrobial agents), the persisters (which defines the portion of persistent bacteria in the presence of the antimicrobial agents), and the dead bacteria. Their volume portions were denoted as S, P, and D, respectively. In addition, the authors were concerned with the portion of the extracellular polymeric substances (EPS) and solvent in the oral biofilm, which are denoted as E and T. Hence, the material mixture constituting the oral biofilm can be written as follows:

$$S+P+D+E+T=1.$$

(2)

The parameters needed to construct the model were calibrated based on their experiment. The experimental results showed that the bacteria would have a lag phase once transferred into new treatment circumstances. To deal with this physiological state and regulate the lag process of oral biofilm formation, the authors considered a functional component such as a growth factor to regulate cell proliferation. It determined the necessary proteins, signal molecules, or extracellular DNA produced in the lag phase, which affect the proliferation and quorum sensing molecules synchronization in a later stage [65,66,67]. The authors believe that molecules of the growth factor, nutrient, antimicrobial, and quorum sensing portions are small compared with the other portion comprising the biomass of the oral biofilm. Therefore, the mass of the four molecules is disregarded in the model. However, the authors still consider their existence in the biofilm formation since the chemical and biological effects are prominent and thereby retained. Hence, the concentration of nutrients, quorum sensing, antimicrobial agents, and growth factors are denoted by $C\left(t\right), H\left(t\right), A\left(t\right),$ and $Q\left(t\right)$, respectively. Briefly, their constructed model can be written as follows:

susceptible bacteria proportion	(3)
$\frac{dS}{dt}=c_2\frac{Q^2}{k_Q^2+Q^2}\frac{k_{12}^2}{A^2+k_{12}^2}\frac{C}{C+k_2}\left(1-\frac{S}{S_{max}}\right)S-b_{sp}S+b_{ps}P-r_{bs}S-\frac{c_3\gamma A}{A+k_3}S,$
persister bacteria proportion
$\frac{dP}{dt}=c_4\frac{Q^2}{k_Q^2+Q^2}\frac{k_{12}^2}{A^2+k_{12}^2}\frac{C}{C+k_2}\left(1-\frac{P}{P_{max}}\right)P+b_{sp}S-b_{ps}P-\frac{c_{12}\gamma A}{A+k_3}P,$
dead bacteria proportion
$\frac{dD}{dt}=\left(r_{bs}+c_3\frac{\gamma A}{A+k_3}\right)S+\frac{c_{12}\gamma A}{A+k_3}P-r_{dp}\frac{k_{13}}{A+k_{13}}\frac{D}{D+k_{15}}D,$
EPS proportion
$\frac{dE}{dt}=c_5\left(S+P\right)\frac{C}{C+k_2}\frac{H^2}{H^2+k_9^2}\left(1-\frac{E}{E_{max}}\right)+r_{dp}\frac{k_{13}}{A+k_{13}}\frac{D}{D+k_{15}}D,$
nutrition concentration
$\frac{dC}{dt}=-c_7\left(S+P\right)\frac{C}{C+k_2},$
antimicrobial agents
$\frac{dA}{dt}=-c_8\left(S+P\right)\frac{A}{A+k_8}-r_{a}A,$
quorum sensing concentration
$\frac{dH}{dt}=c_{A}S\frac{Q^2}{k_Q^2+Q^2}\left(1-\frac{H}{H_{max}}\right),$
growth factor concentration
$\frac{dQ}{dt}=c_{q}S\left(1-\frac{Q}{Q_{max}}\right),$
slow-penetration factor
$\gamma =\frac{1}{T+\frac{E}{D_{pr}}}\frac{2\left(T+E\right)}{2\left(S+P+D\right)}.$

Note that the solvent portion of the oral biofilm biomass is not explicitly written and can be determined by calculating $T=1-\left(S+P+D+E\right)$. The definition of each parameter is elaborated in

Table 1. Definition of each parameter used in system (3).

Parameter	Definition
$b_{sp}$	The transition rate from susceptible to persister portion.
$b_{ps}$	The transition rate from persister to susceptible portion.
$r_{bs}$	The natural death rate of susceptible bacteria.
$c_3$	The death rate of susceptible bacteria is due to the application of antimicrobial agents.
$c_{12}$	The persister bacteria death rate is due to the application of antimicrobial agents.
$S_{max}$	The carrying capacity for susceptible bacteria.
$P_{max}$	The carrying capacity for persister bacteria.
$k_{12},k_3$	The rate of half-saturation for Hill-type reactive kinetics.
$r_{dp}$	The rate of maximum breakdown for dead bacteria.
$k_{13},k_{15}$	The rate of half-saturation for Hill-type switch function.
$c_5$	The rate of maximum EPS production.
$k_2,k_9$	The rate of half-saturation for Hill functions.
$c_7,c_8$	The rate of maximum consumption for nutrient and antimicrobial agents.
$c_A$	The rate of maximum production for quorum sensing.
$c_q$	The rate of maximum production for growth factors.
$k_8$	The rate of half saturation.
$r_a$	The rate of antimicrobial agent natural decay.
$H_{max}$	The carrying capacity for quorum sensing.
$Q_{max}$	The carrying capacity for growth factor.

.

The first constructed model for oral biofilm formation was an output of research with an experimental approach in a laboratory. According to the observation results, the model is formulated in a complex system with many factors. It is interpreted by various parameters involved in the model. In addition, considering functional components in the model is the proper approach to handling the lag phase. The authors viewed it as an output of their experimental research. Hence, the model is no longer analyzed mathematically. At the same time, we can obtain insights into bacterial growth behavior by analyzing the model using mathematical theories. Over time, the model is reconstructed in observation research by Jing et al. [51].

Jing et al. [51] conducted a laboratory experiment to observe the bacterial behavior and calibrate the parameter. Next, the model was reconstructed in ODEs but in a simple form by minimizing the phenotype classification of bacteria and reducing the involved factors. The bacterial population was divided into two subpopulations, namely live and dead bacteria. Their volume portions are denoted respectively as $L$ and $D$. The volume portion of the EPS and solvent are also considered and are denoted respectively as $E$ and $T$. Therefore, the material mixture forming the oral biofilm can be expressed as follows:

$$L+D+E+T=1.$$

(4)

Experimental results [51] indicated the biofilms would reach maturity in about the third week. In addition, the study [51,64] pointed out that the mature biofilms need a long period to reform after antibacterial treatment. To model the nonlinear phenomenon of biofilm reformation, the authors also viewed growth factors as a functional component that regulates cell proliferation. Moreover, nutrition concentration is assumed to be a constant value in reducing the functional components for oral biofilm formation. Hence, only three functional responses, such as antimicrobial agents $\left(A\right)$, quorum sensing molecules $\left(H\right)$, and growth factors $\left(Q\right)$, are involved in their model. Hence, the reconstructed model can be rewritten as follows:

live bacteria proportion	(5)
$\frac{dL}{dt}=c_2\frac{Q^2}{Q^2+k_q^2}\left(1-\frac{L}{L_{max}}\right)L-r_{bs}L-c_3\gamma AL,$
dead bacteria proportion
$\frac{dD}{dt}=r_{bs}L+c_3\gamma AL-r_{dp}\frac{k_{13}}{k_{13}+A}D,$
EPS proportion
$\frac{dE}{dt}=\left(c_5\frac{H^2}{H^2+k_9^2}L+r_{dp}\frac{k_{13}}{k_{13}+A}D\right)\left(1-\frac{E}{E_{max}}\right),$
antimicrobial agents
$\frac{dA}{dt}=-c_{8}AL-r_{a}A,$
quorum sensing molecules
$\frac{dH}{dt}=c_a\frac{Q^2}{Q^2+k_q^2}L\left(1-\frac{H}{H_{max}}\right),$
growth factors
$\frac{dQ}{dt}=c_{q}L\left(1-\frac{Q}{Q_{max}}\right),$
slow-penetration factors
$\gamma =\frac{1}{T+\frac{E}{D_{pr}}}\frac{2\left(T+E\right)}{2\left(L+D\right)}.$

Remember that in the reconstructed model, the solvent portion of the oral biofilm biomass is also not explicitly written and can be estimated by calculating $T=1-\left(L+D+E\right)$. Each parameter definition is elaborated in

Table 2. Definition of each parameter used in system (5).

Parameter	Definition
$c_2$	The live cells proliferation rate.
$k_q,k_9$	The constant of Hill-function.
$r_{bs}$	The live cell natural death rate.
$c_3$	The live cell death rate is due to the application of an antibacterial agent.
$r_{dp}$	The degradation rate of dead bacteria.
$k_{13}$	The rate of half-saturation for Hill-type switch function.
$c_5$	The EPS production rate is due to live bacteria.
$E_{max}$	The carrying capacity for the EPS portion.
$c_8$	The killing rate of the bacteria by antibacterial agents.
$r_a$	The degradation rate of antibacterial agent.
$c_a$	The growth rate of quorum sensing molecules.
$c_q$	The growth rate of growth factors.
$H_{max}$	The carrying capacity for quorum sensing molecules.
$Q_{max}$	The carrying capacity of growth factors.
$D_{pr}$	The relative diffusivity of EPS.

.

The reconstructed model represents a biofilm renewal in a more straightforward mathematical form than the first. In addition, the lag process of biofilm recovery is also interpreted as a delay in mathematical modeling of the nonlinear phenomenon, which is suitable for representing the condition. The authors confidently stated that the reconstructed model performs as well or even better than the model in [64]. Furthermore, the authors also performed numerical simulations to figure out the population dynamics of the bacterial population in the oral biofilm. Nevertheless, as the model is an output of its study, it is not analyzed mathematically to obtain population dynamics insights through mathematics interpretation.

3.3. Revealing the Gap for Further Study

Deterministic models are essential in studying bacterial transmission and population growth dynamics. They represent the observed phenomenon by constructing compartments according to each condition of the subpopulation considering related interventions. A compartmental modeling study is essential for understanding the phenomenon and its intervention measures. Constructing appropriate compartmental structures to deal with the actual condition is the basis and premise of making dynamic models work. Hence, we need to learn about how the bacteria spread and theoretically, their capability to grow in an oral biofilm.

After reviewing the ongoing dental health issues modeling studies, we concluded that the compartmental-based model has not been well-utilized. We only found three related articles on this topic. This is a very small number amidst the growing use of compartment-based models in studying the phenomena of disease spread or population growth. In addition, the conducted study has several limitations, such as the intervention involved and the mathematical analysis. At the same time, both give a more representative model and more insights into understanding the long-term behavior of the phenomenon by interpreting the mathematical result.

The constructed model in [52] has some limitations. We have already summarized the limitations and its potential development for further study. First, the involvement of compartment $E$ must be reviewed. In the context of dental caries, the transmission of bacteria can occur even before the onset of tooth cavities when individuals in compartment $E$ transfer the bacteria via saliva to susceptible individuals. Therefore, it is crucial to consider the transmission risk due to the contact between $E$ and $S$, as it may significantly affect the number of $E$ and $I$ in the future. Secondly, the parameter $p$ must be reviewed since it leaves an ambiguous definition. When infected people address concerns about their dental health, it increases the transmission risk. This may strongly impact the higher number of $E$ and $I$ in the future. Hence, we recommend eliminating the $p$ or creating another compartment, which is defined as the group of people who are riskier in transmitting the bacteria. Third, Kumar et al. only considered curative action and no preventive action in the model. We recommend involving preventive action, which may cover the conducted public health education effort, and exploring how dental caries cases may be reduced by preventing transmission risk.

The studies of bacterial dynamics also have several limitations. First, the model constructed in [51,64] only represents the phenomenon of biofilm recovery in laboratory observations. It can be viewed by the formulation of nutrient concentration (C) in [64] and antimicrobial agent (A) in [51,64]. No parameter can be interpreted as replenishment of the concentration and agent. In actual conditions, both can be increased naturally and still exist due to daily activity. For instance, the nutrient concentration can increase due to carbohydrate or sucrose consumption [16,58], while an antimicrobial agent can be obtained naturally through consuming tea [6], propolis [68], or cranberry [69]. Lastly, mathematically, we need to conduct a mathematical analysis to obtain an interpretable solution for understanding the long-term behavior. This results in more insights into exploring the phenomenon through a mathematical interpretation. In addition, a sensitivity analysis can be conducted to understand which factors are essential in studying biofilm reformation. This fulfills the aim of Jing et al. in [51] to investigate the most crucial functional components of the model obtained due to the calibration process.

4. Conclusions

We comprehensively reviewed the utilization of compartmental-based modeling for dental health issues. We used four databases to collect existing published articles and obtain articles related to our topic. The PRISMA method was used to identify articles that meet our criterion in this review. We found that the compartmental-based model was only explored in three studies, one epidemiological study and two bacterial population growth studies. As we elaborated in Section 3, each study has advantages and limitations. The study’s limitations led to the emergence of the need for further study, both in terms of modeling the issue and conducting analyses. For instance, unsolved problems such as how the functional components in system (3) and (5) represent the actual conditions of nutrients and antimicrobial concentrations, and also, uninvolved factors such as preventive action in system (1) that can be considered further. Our study revealed the development and the limitations and provided essential insights for further constructing compartmental-based models on this topic. This may be useful when creating a reasonable transmission model or bacterial population growth model involving any factor not considered in the existing study. In this article, we also provide several suggestions that can be considered in remodeling the existing model as guidance for further research. In addition, we recommend that researchers who are focusing on stochastic modeling start exploring dental caries studies and the potential transmission of its bacteria. We found only two articles that used stochastic modeling for dental caries studies at the individual level [70,71], and no one used it for dental caries studies at the population level.