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This study is aimed at building and analysing a SIRS model and also simulating the model to predict the number of dengue fever cases. Methods applied for this model are building the SIRS model by modifying the SIR model, analysing the SIRS model using the Lyapunov function to prove three theorems (the existence, the free disease, and the endemic status of dengue fever), and simulating the SIRS model using the number of dengue case data in South Sulawesi by Maple. The results obtained are the SIRS model of dengue fever transmission, stability analysis, global stability, and the value of the basic reproduction number R 0 . The simulation done for the dengue fever case in South Sulawesi found the basic reproduction number R 0 = 26.47609 > 1 ; it means that South Sulawesi is in the endemic stage of transmission for dengue fever disease. Simulation of the SIRS model for dengue fever can predict the number of dengue cases in South Sulawesi that could be a recommendation for the government in an effort to prevent the number of dengue fever cases.

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The aims of this study are: to build a SIPA model on the spread of HIV/AIDS; analyse and simulation of SIPA model and to predict the spread of HIV/AIDS. An applied mathematics for Analysis of the SIPA model in case of HIV/AIDS spreading using the Jacobi matrix method to obtain eigenvalues in two conditions, namely endemic and disease-free, while the simulation model uses Maple with initial value data in the form of assumptions represented in research. The research result are the mathematical SIPA model of HIV/AIDS spreading which is a system of differential equations. The analysis of the model gives the value of the disease-free equilibrium point and the asymptotically stable endemic equilibrium point. The results also found that the basic reproduction number was R0=0.0067 for disease-free conditions and R0=2.7944 for endemic conditions indicating the condition of HIV/AIDS spreading cases in the population. The simulation results found that there is a very significant difference between the numbers of AIDS populations when free from disease and during endemic conditions, so that attention is needed for the government to be able to tackle the spread of HIV/AIDS.

Rubella disease is an infectious disease transmitted through the respiratory tract caused by a virus. In some cases, there are diseases that can enter an endemic condition, which is a condition in which the outbreak of a disease in a certain area over a long period of time. This condition can be modeled mathematically by using certain assumptions which will then seek for analytical and numerical solutions. This type of research is a literature study. This study examines the theory and application of the Runge-Kutta method in analyzing the spread of Rubella with the effect of vaccination. The Runge-Kutta method is widely used in solving ordinary differential equations and is more accurate than the Euler method. In this study, two methods were used to analyze it, namely the Runge-Kutta Order 4 and Order 5 methods which were used to analyze and compare the numerical results obtained, and the aim of the study was to obtain and interpret mathematically a mathematical model of the spread of Rubella’s disease with the effect of vaccination. and comparing the results obtained from the numerical solution using the Runge-Kutta Order 4 and Runge-Kutta Order 5 methods simulated with Maple software 13. From the results of this study, the results obtained in the 1st iteration of the numerical solution model for Rubella’s disease using the Runge-Kutta Order 4 is the value of 푆1 = 8771655, E 1 = 142, 퐼1 = 38, and R 1 = 30. While the results in the first iteration of the numerical solution model for Rubella’s disease using the Runge-Kutta Order 5 method were that the values of 푆1 = 8771759, E 1 = 142, 퐼1 = 37, and R 1 = 30 were obtained. So it can be concluded that the Order 5 Runge-Kutta is better than the Order 4 Runge-Kutta in predicting the state rate of the Rubella disease case population in the Suspectible, Exposed, Infected, Recovered, Suspected (SEIRS) model. Because the image / graph of Order 5 Runge-Kutta is more like / closer to the results of the analytical simulation using maple 13 software, which is as many as 27 infected individuals.

The increasing number of cases and the development of new variants of the Covid-19 virus globally including the territory of Indonesia, especially in the province of South Sulawesi are increasingly worrying and need to be prevented. Therefore, this study aims to develop a SEIR model on the spread of Covid-19 with vaccination control, optimal control analysis, stability analysis and numerical simulation of the SEIR model on the spread of Covid-19 in South Sulawesi. This study uses the SEIR epidemic model to predict the spread of Covid-19 in South Sulawesi Province with parameters such as birth rate, cure rate, mortality rate, interaction rate and vaccination. The SEIR model was chosen because it is one of the basic methods in the epidemiological model.  The method used to build the model is a time delay model by considering the vaccination factor as a model parameter, model analysis using the next generation matrix method to determine the basic reproduction number and stability of the Covid-19 distribution model in South Sulawesi. Numerical model simulation using secondary data on the number of Covid-19 cases in South Sulawesi starting in 2021 which was obtained from the South Sulawesi Provincial Health Office. The results obtained are model analysis provides evidence of the existence of optimal control in the model. Based on the results obtained, it can also be seen that vaccination greatly influences the spread of Covid-19 in South Sulawesi, so that awareness is needed for the people of South Sulawesi to follow the government's recommendation to vaccinate to prevent or reduce the rate of transmission of Covid-19 in South Sulawesi.

Objectives: The aim of this study is to obtain SIRI model for dengue fever (DHF) transmission, conduct analysis, and simulation of SIRI model in disease-free and endemic and also to predict the number of DHF cases. Methods/statistical analysis: Dengue fever is caused by a virus carried by the Aedes aegypti and Aedes albopictus mosquitoes, the SIRI model is a modification of the SIR model. Analysis of the SIRI model use the Lyapunov function method, then the data used in the simulation are assuming to show two possible dengue status are disease free and endemic status. The simulation also using the number of dengue case in Makassar city for showing the status of dengue fever transmission in Makassar city. Simulation models using Maple software are to predict the number of dengue cases in the following months. Findings: The results of this study are the SIRI model of the transmission of dengue fever with variables that have recovered can be re-infected with dengue fever, analysis of the SIRI model of dengue transmission provides information that the equation system in the SIRI model which is asymptotically stable, it means that dengue cases always exist at a certain time and certain region. The simulation results of the SIRI model in this study illustrate the number of dengue cases in the following months. While the first simulation found the basic reproduction number is R 0 = 0.0366 ≤ 1 this means that dengue transmission is at an alarming stage, but the second simulation finds the basic reproduction number R 0 = 31.2733 > 1, this means that, a person infected with dengue causes eight individuals will be infected with dengue fever, so that it is in the endemic stage, and the last simulation using data of the number of dengue case in Makassar city found = 1, that means, Makassar city is a free disease case for dengue fever transmission. Application/ improvements: SIRI model for DHF transmission is a mathematical health application. Model analysis guarantees existence, disease-free Article Type: Article

The aim of this study was to obtain a SIRS model solution for the spread of dengue fever (DF) using the 4th order Runge-Kutta method. The data used was data on the number of dengue cases in Makassar city. The method used is the 4th order Runge Kutta Method. The results of this study are numerical solutions of the SIRS model on the spread of DF in Makassar city using the Runge Kutta method; Model simulation analysis shown that the estimated number of dengue cases in Makassar city, so that the government can take steps to prevent the spread of dengue in Makassar city.