Explaining all published articles on the typhoid disease transmission model was carried out. It has been conducted to understand how Salmonella is transmitted among humans and vectors with variation interventions to control the spread of the typhoid disease. Specific objectives were to (1) identify the model developed, (2) describe the studies, and (3) identify the interventions of the model. It systemically searched and reviewed Dimension, Scopus, and ScienceDirect databases from 2013 through to 2022 for articles that studied the spread of typhoid fever through a compartmental mathematical model. This study obtained 111 unique articles from three databases, resulting in 23 articles corresponding to the created terms. All the articles were elaborated on to identify their identities for more explanation. Various interventions were considered in the model of each article, are identified, and then summarized to find out the opportunities for model development in future works. The whole article’s content was identified and outlined regarding how mathematics plays a role in model analysis and study of typhoid disease spread with various interventions. The study of mathematical modeling for typhoid disease transmission can be developed on analysis and creating the model with direct and indirect interventions to the human population for further work.

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.

In this paper, we constructed and analyzed a modified SIR-type model for the epidemic problem which considering waning-immunity class in the population, with the nonlinear incidence and recovery rates. Depending on the basic reproduction ratio, we investigated conditions for both non-endemic and co-existing cases. The RouthHurwitz criteria is used to verify the local stability of equilibria, whereas for the global stability, the suitable Lyapunov function is selected to analyze the behavioral stability for each equilibria. Moreover, we studied the optimal control problem for this case. Numerically, we give some simulations to support our analitycal findings.

Dental caries is an example of an oral infectious disease that affects many people worldwide, but it is not well studied in deterministic mathematical modeling. Therefore, we are interested in studying the dynamics of tooth cavity disease using a deterministic modeling approach. We propose a delay differential equation system (DDEs) to describe the phenomenon. The breakthrough of the constructed model is the formulation of the recovery rate as a saturation function constrained by healthcare capacity and the plausibility of caries reformation. In addition, we consider two controls, such as a health campaign and a post-treatment intervention. The mathematical analysis yields equilibrium solutions and their stability, which is determined by the basic reproduction number $\left({\mathfrak{R}}_0\right)$. Furthermore, backward bifurcation occurs as the medical facility’s capacity decreases, driven by an increasing infectious population. The sensitivity analysis results indicate that both considered controls are the most influential parameters. The optimal control problem is formulated using the Pontryagin Maximum Principle to obtain an optimal solution in suppressing the number of caries formation cases. At the end, a numerical simulation shows that interventions reduce the risk of transmission and suppress the number of infectious individuals. The constructed model has excellent future potential, such as generating a function for relapse cases or other preventive actions into an optimal control problem.