Runge-Kutta Research Articles

A numerical approach to overcome the very-low Reynolds number limitation of the artificial compressibility for incompressible flows

We propose a numerical approach to solve a long-standing challenge which is the applicability of the artificial compressibility (AC) formulation for solving the incompressible Navier—Stokes equations at very-low Reynolds numbers. A wide range of engineering applications involves very-low Reynolds number flows in Micro-ElectroMechanical Systems (MEMS) and in the fields of chemical-, agricultural- and biomedical engineering. It is known that the already existing numerical methods using the AC approach fail to provide physically correct results at very-low Reynolds numbers (Re ≤ 1). To overcome the limitation of the AC method for these engineering applications, we propose a higher-order Neumann-type pressure outflow boundary condition treatment along with their up to fourth-order numerical approximations. We found that the numerical treatment of the pressure at the outlet boundary plays the main role in overcoming the limitation of the AC method at very-low Reynolds numbers (Re << 1). Therefore, we provide numerical evidence on the accuracy of the AC method beyond its previously reported limitations, e.g., the low Reynolds number Oseen flow (Re << 1) is first presented in this work. A third-order explicit total-variation diminishing (TVD) Runge–Kutta scheme has been employed with standard finite difference spatial discretisation schemes for improving the accuracy of the numerical solution. For modelling strongly viscous flows, the Reynolds number ranges from 10−1 to 10−4. Overall, we found that the accuracy limitation of the AC method below Re < 1 can be overcome with an accurate numerical treatment of the outlet pressure boundary condition instead of using high-order schemes in the governing equations. For the investigated Reynolds number range (10−1 ≤ Re ≤ 10−4), the obtained results show that the relative errors were smaller than 1% for the numerical simulations performed on the configurations of both the two- and three-dimensional, straight microfluidic channels. The imposition of high-order derivative Neumann-type pressure outflow boundary conditions reduced the maximum relative errors of the numerical solutions from 85% and 95% to below than 1% at the outlet section of the two- and three-dimensional, straight microfluidic channel flows, respectively. Taking the advantage of the numerical approach proposed here, two- and three-dimensional benchmark problems employed in the current investigation in comparison with analytical solutions available in the literature, clearly demonstrate that the artificial compressibility can be used beyond its previously known constraints for very-low Reynolds number incompressible flows.

Numerical Estimation of Potable Water Production for Single-Slope Solar Stills in the Caspian Region

This study provides a detailed numerical assessment of the productivity of single-slope solar stills across key cities in the Caspian region, including Aktau, Atyrau, Astrakhan, Makhachkala, Baku, Tehran, and Turkmenbashi. Employing a mathematical model based on heat balance equations and the Fourth Order Runge–Kutta method, we simulated the distillation process under various climatic conditions. The results reveal that productivity is significantly influenced by geographic location and local meteorological conditions. Tehran demonstrated the highest productivity across all seasons, with values of 1.75 × 103 kg/(m2·year) and an efficiency of 0.53, due to its optimal solar irradiation and ambient temperature. In contrast, Atyrau and Makhachkala exhibited lower productivity, particularly in colder months, highlighting the effect of ambient temperature on solar still efficiency. The analysis identified the optimal water depth at 2 cm and insulation thickness between 4 and 9 cm for enhancing productivity in continental climates like Aktau. Additionally, the lowest cost of distilled water was USD 0.024 per kilogram in Baku. These findings align with the existing literature, validating the numerical model’s accuracy. Future research will explore integrating solar stills with other renewable and fossil fuel-based technologies.

Enhancing energy capacity of the car’s hood door under excitation frequency via functionally graded triply periodic minimal surface material: Application of machine learning and mathematical simulation

This study investigates the nonlinear dynamic response and vibration characteristics of a car’s hood door, focusing on its energy capacity under various excitation frequencies. The hood door is modeled using a functionally graded triply periodic minimal surface (FG-TPMS) material, which offers superior mechanical properties and lightweight design. The analysis is conducted using a higher-order shear deformation theory (HSDT), which accounts for shear deformations more accurately than classical theories. The incorporation of von Karman nonlinear terms captures the geometric nonlinearity due to large deformations, providing a realistic simulation of the hood door’s behavior under dynamic loads. To solve the complex equations of motion derived from the HSDT and von Karman terms, a fourth-order Runge–Kutta method is employed. This numerical method ensures accurate time integration and stability of the solution. The dynamic response is further analyzed to determine the energy absorption capacity of the hood door material, crucial for safety and durability in automotive applications. Validation of the numerical model is performed using previous published articles and a hybrid machine learning algorithm, which combines data-driven approaches with traditional physics-based models. This hybrid validation enhances the accuracy and reliability of the predicted responses, ensuring that the proposed model can be effectively used in the design and optimization of car components. The results demonstrate the potential of FG-TPMS materials in automotive applications, offering improved energy absorption and vibration control, which are essential for enhancing vehicle safety and performance.

Study on the Influence of Sex Ratio on Ecology Based on Improved Lotka-Volterra Model

In a pioneering study of lamprey population dynamics, we employ an enhanced Lotka-Volterra model to illuminate the profound influence of sex ratio shifts on ecosystems. This exploration delves into how such changes affect resource availability, ecosystem equilibrium, and sustainability. We employ dynamic differential equations to simulate population fluctuations among males, females, and predators, incorporating elements like birth rate, mortality rate, competition coefficient, mating success rate, and predation rate. The differential equations are numerically resolved using the Runge-Kutta method, enabling us to analyze population dynamics across various scenarios. Concurrently, we construct Jacobian matrices to appraise ecosystem stability and equilibrium points. The refined Lotka-Volterra model effectively encapsulates the impact of sex ratio variations on reproductive and predation processes, offering profound insights into predator-prey interactions and ecosystem dynamics. Our pivotal discoveries underscore the influence of sex ratio modifications on reproductive capacity, population growth rates, social structure, genetic diversity, and predator-prey relationships. This study emphasizes the criticality of comprehending the repercussions of sex ratio shifts on ecosystems and communities.

Deep cryogenic silicon etching for 3D integrated capacitors: A numerical perspective

One promising approach to increase the capacity density of integral microcapacitors, microsupercapacitors, and microbatteries is three-dimensional structure design, where electrodes are exposed in three dimensions instead of conventional in-plane electrodes. Such structures include nanowires, nanotubes, nanopillars, nanoholes, nanosheets, and nanowalls. In this work, a cryogenic silicon etching process suitable for fabrication of structures with high electrode area is proposed. A numeric model of this process is experimentally calibrated and used for pillar array structure sidewall area optimization. The use of adaptive Runge–Kutta–Fehlberg time integrator allows to achieve almost linear overall computation complexity as a function of simulated etching time, despite the linear increase in conductance computation complexity with depth. A rule for choosing optimal geometric structure parameters under technological constraints is formulated. An optimized trefoil-like structure is proposed, resulting in a total 5.5% increase in sidewall area with respect to the hexagonal array of circular pillars, resulting in 20.33 sidewall area per unit chip area for 30 min long etch or 31.80 for 60 min long etch.

Influence of non-linear thermal radiation on the dynamics of homogeneous and heterogeneous chemical reactions between the cone and the disk

Abstract Purpose The current work presents a theoretical framework to boost heat transmission in a ternary hybrid nanofluid with homogeneous and heterogeneous reactions in the conical gap between the cone and disk apparatus. Furthermore, the impacts of non-linear thermal radiation on the ternary hybrid nanofluid composed of white graphene, diamond, and titanium dioxide dispersed in water are analyzed. Originality/value The combination of cone and disk systems is crucial for designing efficient heat exchange devices in the field of biomedical science for various purposes. For instance, in medical devices, the cone–disk apparatus is used to study the flow and heat transfer characteristics for better design and functionality. Hence, a sincere attempt has been made to study the impact of homogeneous and heterogeneous reactions on the nanofluid flow between the cone and disk in the presence of non-linear thermal radiation. Design/methodology/approach The mathematical model’s governing equations are partial differential equations (PDEs) which are then transformed into non-linear ordinary differential equations through appropriate similarity transformations. These transformed resultant equations are approximated by the Runge–Kutta–Fehlberg fourth/fifth order (RKF45) technique. The influence of essential aspects on the flow field, heat, and mass transfer rates was analyzed using a graphical representation. Findings The interesting part of this research is to discuss the power of parameters in three cases, namely, (1) rotating cone/disk, (2) rotating cone/stationary disk, and (3) stationary cone/rotating disk. Furthermore, the thermal variation of the fluid is analyzed by an artificial neural network with the help of the Levenberg–Marquardt backpropagation algorithm. The regression analysis, mean square error, and error histogram of the neural network are analyzed using this algorithm. From the graph, it is perceived that the flow field climbed up significantly with an increase in the values of radiation parameters in all cases. Also, it is noticed that temperature upsurges significantly by upward values of solid volume fraction of the nanoparticles ( ϕ \phi ).

An exponential stochastic Runge–Kutta type method of order up to 1.5 for SPDEs of Nemytskii-type

Abstract For the approximation of solutions for stochastic partial differential equations, numerical methods that obtain a high order of convergence and at the same time involve reasonable computational cost are of particular interest. We therefore propose a new numerical method of exponential stochastic Runge–Kutta type that allows for convergence with a temporal order of up to $\frac{3}/{2}$ and that can be combined with several spatial discretizations. The developed family of derivative-free schemes is tailored to stochastic partial differential equations of Nemytskii-type, i.e., with pointwise multiplicative noise operators. We prove the strong convergence of these schemes in the root mean-square sense and present some numerical examples that reveal the theoretical results.

Chaotic behavior and multistability of a tree trunk structure driven by self-and parametric hydrodynamic forces

Abstract This study investigates the chaotic behavior of a tree trunk under dynamic&amp;#xD;wind loads, focusing on control strategies and multistability. We consider time varying wind speeds and analyze a specific case where hydrodynamic drag forces align&amp;#xD;with the flow velocity. The stability of the model’s equilibrium points is analyzed&amp;#xD;theoretically and numerically. Melnikov’s method is employed to identify conditions&amp;#xD;for homoclinic bifurcation. Numerical simulations employing basin of attraction&amp;#xD;confirm the analytical predictions. Our findings show a decrease in the threshold&amp;#xD;for chaos with increasing amplitudes of external excitation, damping coefficient,&amp;#xD;and parametric damping. The global dynamics are explored numerically using a&amp;#xD;fourth-order Runge-Kutta method. When solely subjected to external excitation, the&amp;#xD;system exhibits period doubling bifurcations, multiperiodic oscillations, mixed-mode&amp;#xD;oscillations, and chaos. Conversely, with self- and parametric drag forces, the system&amp;#xD;displays reverse periodic bifurcations, periodic bubbling oscillations, antimonotonicity,&amp;#xD;transient chaos, and chaos. Poincar´e maps analyze the geometric structure of chaotic&amp;#xD;attractors, revealing a strong influence of dimensionless drag parameters. These&amp;#xD;parameters can be manipulated to control or eliminate chaos. Furthermore, the system&amp;#xD;exhibits multistability, with coexisting attractors. Beyond its application in protecting&amp;#xD;infrastructure from wind damage, this research can contribute to ecological balance by&amp;#xD;improving our understanding of tree wind resistance.

Computational analysis of the Covid-19 model using the continuous Galerkin–Petrov scheme

Abstract Epidemiological models feature reliable and valuable insights into the prevention and transmission of life-threatening illnesses. In this study, a novel SIR mathematical model for COVID-19 is formulated and examined. The newly developed model has been thoroughly explored through theoretical analysis and computational methods, specifically the continuous Galerkin–Petrov (cGP) scheme. The next-generation matrix approach was used to calculate the reproduction number ( R 0 ) ({R}_{0}) . Both disease-free equilibrium (DFE) ( E 0 ) ({E}^{0}) and the endemic equilibrium ( E ⁎ ) ({E}^{\ast }) points are derived for the proposed model. The stability analysis of the equilibrium points reveals that ( E 0 ) ({E}^{0}) is locally asymptotically stable when R 0 &lt; 1 {R}_{0}\lt 1 , while E ⁎ {E}^{\ast } is globally asymptotically stable when R 0 &gt; 1 {R}_{0}\gt 1 . We have examined the model’s local stability (LS) and global stability (GS) for endemic equilibrium \text{&amp;#x00A0;} and DFE based on the number ( R 0 ) ({R}_{0}) . To ascertain the dominance of the parameters, we examined the sensitivity of the number ( R 0 ) ({R}_{0}) to parameters and computed sensitivity indices. Additionally, using the fourth-order Runge–Kutta (RK4) and Runge–Kutta–Fehlberg (RK45) techniques implemented in MATLAB, we determined the numerical solutions. Furthermore, the model was solved using the continuous cGP time discretization technique. We implemented a variety of schemes like cGP(2), RK4, and RK45 for the COVID-19 model and presented the numerical and graphical solutions of the model. Furthermore, we compared the results obtained using the above-mentioned schemes and observed that all results overlap with each other. The significant properties of several physical parameters under consideration were discussed. In the end, the computational analysis shows a clear image of the rise and fall in the spread of this disease over time in a specific location.

Stability and Optimality Criteria for an SVIR Epidemic Model with Numerical Simulation

The mathematical modeling of infectious diseases plays a vital role in understanding and predicting disease transmission, as underscored by recent global outbreaks; to delve deep into the dynamic of infectious disease considering latent period presciently is inevitable as it bridges the gap between realistic nature and mathematical modeling. This study extended the classical Susceptible–Infected–Recovered (SIR) model by incorporating vaccination strategies during incubation. We introduced multiple time delays to an account incubation period to capture realistic disease dynamics better. The model is formulated as a system of delay differential equations that describe the transmission dynamics of diseases such as polio or COVID-19, or diseases for which vaccination exists. Critical aspects of the study include proving the positivity of the model’s solutions, calculating the basic reproduction number (R0) using next-generation matrix theory, and identifying disease-free and endemic equilibrium points. The local stability of these equilibria is then analyzed using the Routh–Hurwitz criterion. Due to the complexity introduced by the delay components, we examine the stability by studying the roots of a fourth-degree exponential polynomial. The effects of educational campaigns and vaccination efficacy are also investigated as control measures. Furthermore, an optimization problem is formulated, based on Pontryagin’s maximum principle, to minimize the number of infections and associated intervention costs. Numerical simulations of the delay differential equations are conducted, and a modified Runge–Kutta method with delays is used to solve the optimal control problem. Finally, we present a few simulation results to illustrate the analytical findings.

Bifurcation analysis of Van der Pol–Mathieu equation for the dust grain charge with regularized (κ)-distributed electron-ion

This paper presents a novel viewpoint on nonlinear dust-acoustic waves (DAWs) in an unmagnetized collisionless plasma containing regularized [Formula: see text] distributed electrons and ions (ei) and negative dust grain. The nonlinear oscillatory system based on hybridization of the Van der Pol–Mathieu equation (VdPME) is derived by a new technique. By bifurcation analysis of the planar dynamical system (DS), the effects of parameters with the assistance of phase planes and time series of VdPME are studied. After analyzing the equation to identify the resonance region, a fourth-order Runge–Kutta method is used to solve it numerically. We explained the behavior of DA periodic, stable limit cycle, and chaotic limit cycle wave solutions with different parameters. These types of numerical solutions are illustrated in two-dimensional and three-dimensional graphics by changing the rate at which charged dust grain is produced [Formula: see text], as well as waste [Formula: see text] and comparing the results with those of earlier research. A novel bifurcation analysis of VdPE and VdPME is obtained with the effects of the cut-off factor [Formula: see text] of regularized [Formula: see text]-distribution (RKD) distributed ei, the superthermality of ei particles [Formula: see text], the ratio of ion to electron temperature [Formula: see text], and the ratio of dust to electron density [Formula: see text] illustrated. It is noticed that the DAW shows promotion in width and amplitude as the frequency [Formula: see text] increases and it becomes rarefaction as the cut-off parameter [Formula: see text] increases. On the contrary, it becomes compressive under the impact of superthermality [Formula: see text] and [Formula: see text]. The obtained conclusions may help explain and comprehend a variety of applications in experimental plasma containing highly energy regularized [Formula: see text] dispersed ei, as well as in the interstellar medium.

A study on generalized balanced split drift stochastic Runge- Kutta methods for stochastic differential equations

To reduce computational complexity, the balanced numerical approximations of the general split drift stochastic Runge-Kutta methods are analyzed. The primary reasons for considering the numerical approximations of these balanced split stochastic Runge-Kutta methods are their improved stability characteristics and lower mean square error compared to other methods. By balancing the drift and diffusion components, the splitting techniques outperform the mean square error over longer time increments. For Ito multi-dimensional stochastic differential equations, we propose a novel family of balanced universal split stochastic Runge-Kutta procedures. The Kronecker product concept is utilized to derive the mean-square stability conditions. We conduct numerical tests to evaluate these methods against an existing weak order 2 split drift method. Ultimately, a specific numerical example validates the theoretical outcomes of the balanced general split stochastic Runge-Kutta procedures.

Nonlinear deterministic dynamic analysis of a GPL micropipe conveying pulsating laminar flow

In several industries, the handling of fluid-filled micropipes is common. New-brand structures are increasingly being manufactured using advanced materials. This article tackles one of the crucial dynamic analyses that an advanced fluid-filled micropipe encounters. The dynamics of a graphene nanoplatelets-reinforced micropipe (GPL micropipe) under pulsating laminar flow for the first time is examined. The Euler-Bernoulli beam model follows the modified couple stress theory (MCST) and von-Karman’s nonlinear strain to formulate the problem. The impression of the flow profile exerted by a real flow as a consequence of fluid viscosity, is taken into account. The plug flow model results are compared to those regarding real flow consideration. Using the method of multiple scales (MMS), we determine the instability area and the steady state response of the GPL micropipe subjected to the principal parametric resonance of one of its modes due to pulsating laminar flow by applying this method to the discretized couple nonlinear gyroscopic governing equations. The Floquet theory treats the linear equations and fourth-order Runge–Kutta (RK4) method attacks nonlinear ones to validate MMS findings. It is confirmed that the 0.3 % addition of the GPL to matrix phase significantly enhances its advanced attributes, making it a highly effective material. The GPL reinforcement phase in the FGO pattern increases the critical flow velocity that exhibits the micropipe to static instability by 37.98 % , and critical flow stimulation amplitude fraction by 152.19 % , while declines instability area bandwidth by 36.95 % , and steady state response by 67.17 % relative to a non-reinforced micropipe. A denser flow degrades the corresponding values by 18.40 % , 42.59 % , 41.67 % , and 227.28 % in comparison to a lighter fluid. The declared findings provide crucial insights into the key design elements affecting significantly the functioning of fluid-filled GPL micropipes system, and regulate future research and expand openings for industry applications.

A new approach to determine occupational accident dynamics by using ordinary differential equations based on SIR model

The motivation of this study is to develop and establish an occupational accident dynamical model (OA model) based on Susceptible-Infected-Recovered framework. In order to investigate the dynamics of the OA model, monthly occupational accident data from Turkey between 2013 and 2020 has been selected as dataset. The OA model is defined by a coupled first-order ordinary nonlinear differential equation with four variables. In addition, the relationships between these variables are described with ten parameters. The OA model’s characterization of the equilibrium points is analyzed by investigating the behaviors of these points according to the eigenvalues derived from the Jacobian matrix. Also, the stability of these points is obtained according to the eigenvalues. These results show the behavior of the system near equilibrium points. After that, the reproduction number is computed by using the next-generation matrix method. The calculated reproduction number for given parameters reveals that the OA model is unstable. The OA model is numerically solved in 96 steps, with a time interval of 1 month, using the ODE45 Matlab routine based on the explicit Runge–Kutta algorithm. In addition, a modified OA model is developed by adding the Occupational Health and Safety (OHS) re-training parameter to the OA model to observe a reducing effect on occupational accident numbers. The main results of this study provide a new approach about the future estimation of the number of occupational accidents. Furthermore, through the comparison of numerical results from both models, the study demonstrates that national safety policies, particularly those enhancing the efficacy of OHS training, can effectively mitigate accidents.

Boundary Treatment for High-Order IMEX Runge–Kutta Local Discontinuous Galerkin Schemes for Multidimensional Nonlinear Parabolic PDEs

Boundary Treatment for High-Order IMEX Runge–Kutta Local Discontinuous Galerkin Schemes for Multidimensional Nonlinear Parabolic PDEs

Parallel-in-Time Solver for the All-at-Once Runge–Kutta Discretization

Parallel-in-Time Solver for the All-at-Once Runge–Kutta Discretization

Numerical simulation of an unsteady ternary hybrid nanofluid along a convectively heated curved stretching sheet: A Tiwari-Das model

Nanofluids are smart fluids, which are very useful for heat and mass transfer enhancement. They are very much used in industrial, transportation, biomedical, and electronics fields. In the present research, we examine the nanofluid flow across a curved stretching sheet (CSS). Also, this work’s noteworthy originality is its discussion of the impacts of the physical parameters on the ternary hybrid nanofluid flow and heat transfer. Furthermore, the simulation considers nanofluids with water as the base fluid. The Koo–Kleinstreuer–Li (KKL) model examines the viscosity and effective thermal conductivity of fluid flow with suspended nanoparticles. The governing equations of the momentum and the temperature are transformed into ordinary differential equations (ODEs) by similarity transformations. These equations are then solved using a Secant iteration method combined with a Runge–Kutta–Fehlberg (RKF) integration scheme with a shooting technique. Through graphs and tables, the influences of the relevant non-dimensional parameters are presented and analyzed. The findings show that an increase in the curvature parameter and the Biot number is to increase the temperature gradient. The Streamline profiles for various values of the magnetic parameter are also analyzed through figures. We believe that this research has significant implications in biomedical devices and pharmaceutical fluid preparation in biomedical sciences. Some of them are high temperature and cooling processes, paints, space technology, medicines, cosmetics, conductive coatings, bio-sensors, and to name a few.

Entropy generation in Casson nanofluid flow over a lubricated wall in the presence of induced magnetic field and activation energy

ABSTRACT The research on entropy generation in a mechanical system is quite imperative owing to its immense applications in the production of advanced cooling devices and medical appliances. The basic need on this research is how to optimize the irreversibilities within the systems and channelize that energy towards the benefit. This study elucidates the entropy generation analysis of a Casson nanofluid near a stagnation point over a lubricated vertical wall. A thin film of shear-thinning liquid with power law index of 0.5 is employed on the wall to ensure the lubrication. The effect of induced magnetic field and the chemical reaction stimulated by the activation energy is also incorporated within the flow. The non-linear PDEs of the substantive problem along with the compatible interfacial and boundary conditions are reverted into ODEs with the aid of suitable similarity transformation, and then cracked numerically using the sixth-order Runge Kutta method assisted with Nachtsheim–Swigert shooting practice. One of the significant findings reveals that the fluid friction irreversibilities in the entropy generation are diminished with the magnetic parameter and the Casson parameter, whereas the thermal irreversibility predominates over the total entropy generation for the escalating values of the activation energy parameter.

On a posteriori error estimation for Runge–Kutta discontinuous Galerkin methods for linear hyperbolic problems

AbstractA posteriori bounds for the error measured in various norms for a standard second‐order explicit‐in‐time Runge–Kutta discontinuous Galerkin (RKDG) discretization of a one‐dimensional (in space) linear transport problem are derived. The proof is based on a novel space‐time polynomial reconstruction, hinging on high‐order temporal reconstructions for continuous and discontinuous Galerkin time‐stepping methods. Of particular interest is the question of error estimation under dynamic mesh modification. The theoretical findings are tested by numerical experiments.

SACR EPIDEMIC MODEL FOR THE SPREAD OF HEPATITIS B DISEASE BY CONSIDERING VERTICAL TRANSMISSION

Hepatitis B is an infectious disease that causes inflammation of the liver due to infection with the Hepatitis B virus. Hepatitis B is divided into two phases: the acute phase and the chronic phase. Hepatitis B virus (HBV) can be prevented through vaccination and treatment of susceptible and infected individuals. The spread of the virus can be modeled using mathematical modeling of epidemics. In this study, the model used consists of four classes, namely vulnerable individuals (S), acute individuals (A), chronic individuals (C), and recovered individuals (R). The purpose of this study is to explain the formation of the Hepatitis B disease epidemic model, analyze the stability of the model, perform simulations, and conduct parameter sensitivity analysis on the basic reproductive number. The result of this study is the construction of an epidemic model of the spread of hepatitis B disease in the form of a SACR model. This model takes into account the transmission that occurs not only through interactions between susceptible individuals and chronic individuals but also through the birth process, which occurs in chronic subpopulations because babies born can be chronically infected (vertical transmission from mother to baby). The model produces two equilibrium points, the disease-free equilibrium and the endemic equilibrium. Both points were analyzed for stability using the linearization method and were found to be asymptotically stable. Furthermore, the model simulation was carried out using the fourth-order Runge-Kutta method and sensitivity analysis of the basic reproduction number. From the results obtained, it can be concluded that the spread of hepatitis B disease can be minimized by reducing contact between susceptible and chronic individuals, increasing treatment of chronic individuals, and increasing the number of vaccinated individuals in susceptible populations.