Model matematika SMEIUR pada penyebaran penyakit campak dengan faktor pengobatan

This study discusses the spread of measles in a mathematical model. Mathematical modeling is not only limited to the world of mathematics but can also be applied in the health sector. Measles is a disease with a high transmission rate. The spread of measles in this model was modified by adding the treated population and the treatment parameters of the exposed population. In this article, we examine the equilibrium points in the SMEIUR mathematical model and perform stability analysis and numerical simulations. In this study, two equilibrium points were obtained, namely the disease-free and endemic equilibrium point. After getting the equilibrium point, an analysis is carried out to find the stability of the model. Furthermore, the simulation produces a stable disease-free equilibrium point at conditions R01 and a stable endemic equilibrium point at conditions R01. In this study, a numerical simulation was carried out to see population dynamics by varying the parameter values. The simulation results show that to reduce the spread of measles, it is necessary to increase the rate of advanced immunization, the rate of the infected population undergoing treatment, and the proportion of individuals who are treated cured.

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Analisis Kestabilan Model SEIR Penyebaran Penyakit Campak dengan Pengaruh Imunisasi dan Vaksin MR

Jurnal Matematika Statistika dan Komputasi ◽

10.20956/jmsk.v16i1.6594 ◽

2019 ◽

Vol 16 (1) ◽

pp. 107

Author(s):

Willyam Daniel Sihotang ◽

Ceria Clara Simbolon ◽

July Hartiny ◽

Desrinawati Tindaon ◽

Lasker Pangarapan Sinaga

Keyword(s):

Infectious Disease ◽

Equilibrium Point ◽

Numerical Solutions ◽

Equilibrium Points ◽

Endemic Equilibrium ◽

Asymptotically Stable ◽

Seir Model ◽

Rate Of Spread ◽

The Stability ◽

Disease Free

Measles is a contagious infectious disease caused by a virus and has the potential to cause an outbreak. Immunization and vaccination are carried out as an effort to prevent the spread of measles. This study aims to analyze and determine the stability of the SEIR model on the spread of measles with the influence of immunization and MR vaccines. The results obtained from model analysis, namely there are two disease free and endemic equilibrium points. If the conditions are met, the measles-free equilibrium point will be asymptotically stable and the measles endemic equilibrium point will be stable. Numerical solutions show a decrease in the rate of spread of measles due to the effect of immunization and the addition of MR vaccines.

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On a Discrete SEIR Epidemic Model with Exposed Infectivity, Feedback Vaccination and Partial Delayed Re-Susceptibility

Mathematics ◽

10.3390/math9050520 ◽

2021 ◽

Vol 9 (5) ◽

pp. 520

Author(s):

Manuel De la Sen ◽

Santiago Alonso-Quesada ◽

Asier Ibeas

Keyword(s):

Equilibrium Point ◽

Epidemic Model ◽

Transmission Rate ◽

Equilibrium Points ◽

Endemic Equilibrium ◽

Asymptotically Stable ◽

Globally Asymptotically Stable ◽

Control Gain ◽

Disease Free ◽

Partial Immunity

A new discrete Susceptible-Exposed-Infectious-Recovered (SEIR) epidemic model is proposed, and its properties of non-negativity and (both local and global) asymptotic stability of the solution sequence vector on the first orthant of the state-space are discussed. The calculation of the disease-free and the endemic equilibrium points is also performed. The model has the following main characteristics: (a) the exposed subpopulation is infective, as it is the infectious one, but their respective transmission rates may be distinct; (b) a feedback vaccination control law on the Susceptible is incorporated; and (c) the model is subject to delayed partial re-susceptibility in the sense that a partial immunity loss in the recovered individuals happens after a certain delay. In this way, a portion of formerly recovered individuals along a range of previous samples is incorporated again to the susceptible subpopulation. The rate of loss of partial immunity of the considered range of previous samples may be, in general, distinct for the various samples. It is found that the endemic equilibrium point is not reachable in the transmission rate range of values, which makes the disease-free one to be globally asymptotically stable. The critical transmission rate which confers to only one of the equilibrium points the property of being asymptotically stable (respectively below or beyond its value) is linked to the unity basic reproduction number and makes both equilibrium points to be coincident. In parallel, the endemic equilibrium point is reachable and globally asymptotically stable in the range for which the disease-free equilibrium point is unstable. It is also discussed the relevance of both the vaccination effort and the re-susceptibility level in the modification of the disease-free equilibrium point compared to its reached component values in their absence. The influences of the limit control gain and equilibrium re-susceptibility level in the reached endemic state are also explicitly made viewable for their interpretation from the endemic equilibrium components. Some simulation examples are tested and discussed by using disease parameterizations of COVID-19.

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Mathematical Model of the Transmission Dynamics of Novel Corona Virus (COVID-19) Pandamic Disease with Optimal Control

International Journal of Mathematical Models and Methods in Applied Sciences ◽

10.46300/9101.2021.15.27 ◽

2021 ◽

Vol 15 ◽

pp. 195-214

Author(s):

Getachew Beyecha Batu ◽

Eshetu Dadi Gurmu

Keyword(s):

Mathematical Model ◽

Reproduction Number ◽

Equilibrium Points ◽

Endemic Equilibrium ◽

Transmission Dynamics ◽

Relative Significance ◽

Corona Virus ◽

Next Generation Matrix ◽

The Stability ◽

Disease Free

In this paper, we have developed a deterministic mathematical model that discribe the transmission dynamics of novel corona virus with prevention control. The disease free and endemic equilibrium point of the model were calculated and its stability analysis were prformed. The reproduction number R0 of the model which determine the persistence of the disease or not was calculated by using next generation matrix and also used to determine the stability of the disease free and endemic equilibrium points which exists conditionally. Furthermore, sensitivity analysis of the model was performed on the parameters in the equation of reproduction to determine their relative significance on the transmission dynamics of COVID- 19 pandemic disease. Finally the simulations were carried out using MATLAB R2015b with ode45 solver. The simulation results illustrated that applying prevention control can successfully reduces the transmission dynamic of COVID-19 infectious disease.

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On an SEIR Epidemic Model with Vaccination of Newborns and Periodic Impulsive Vaccination with Eventual On-Line Adapted Vaccination Strategies to the Varying Levels of the Susceptible Subpopulation

Applied Sciences ◽

10.3390/app10228296 ◽

2020 ◽

Vol 10 (22) ◽

pp. 8296 ◽

Cited By ~ 1

Author(s):

Malen Etxeberria-Etxaniz ◽

Santiago Alonso-Quesada ◽

Manuel De la Sen

Keyword(s):

Equilibrium Point ◽

Basic Reproduction Number ◽

Epidemic Model ◽

Disease Transmission ◽

Reproduction Number ◽

Equilibrium Points ◽

Endemic Equilibrium ◽

Impulsive Vaccination ◽

Disease Free ◽

The Basic Reproduction Number

This paper investigates a susceptible-exposed-infectious-recovered (SEIR) epidemic model with demography under two vaccination effort strategies. Firstly, the model is investigated under vaccination of newborns, which is fact in a direct action on the recruitment level of the model. Secondly, it is investigated under a periodic impulsive vaccination on the susceptible in the sense that the vaccination impulses are concentrated in practice in very short time intervals around a set of impulsive time instants subject to constant inter-vaccination periods. Both strategies can be adapted, if desired, to the time-varying levels of susceptible in the sense that the control efforts be increased as those susceptible levels increase. The model is discussed in terms of suitable properties like the positivity of the solutions, the existence and allocation of equilibrium points, and stability concerns related to the values of the basic reproduction number. It is proven that the basic reproduction number lies below unity, so that the disease-free equilibrium point is asymptotically stable for larger values of the disease transmission rates under vaccination controls compared to the case of absence of vaccination. It is also proven that the endemic equilibrium point is not reachable if the disease-free one is stable and that the disease-free equilibrium point is unstable if the reproduction number exceeds unity while the endemic equilibrium point is stable. Several numerical results are investigated for both vaccination rules with the option of adapting through ime the corresponding efforts to the levels of susceptibility. Such simulation examples are performed under parameterizations related to the current SARS-COVID 19 pandemic.

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A Mathematical Model of a Tuberculosis Transmission Dynamics Incorporating First and Second Line Treatment

Journal of Applied Sciences and Environmental Management ◽

10.4314/jasem.v24i5.29 ◽

2020 ◽

Vol 24 (5) ◽

pp. 917-922

Author(s):

J. Andrawus ◽

F.Y. Eguda ◽

I.G. Usman ◽

S.I. Maiwa ◽

I.M. Dibal ◽

...

Keyword(s):

Mathematical Model ◽

Equilibrium Point ◽

Reproduction Number ◽

Endemic Equilibrium ◽

Transmission Dynamics ◽

Asymptotically Stable ◽

Line Treatment ◽

Second Line ◽

Tuberculosis Transmission ◽

Disease Free

This paper presents a new mathematical model of a tuberculosis transmission dynamics incorporating first and second line treatment. We calculated a control reproduction number which plays a vital role in biomathematics. The model consists of two equilibrium points namely disease free equilibrium and endemic equilibrium point, it has been shown that the disease free equilibrium point was locally asymptotically stable if thecontrol reproduction number is less than one and also the endemic equilibrium point was locally asymptotically stable if the control reproduction number is greater than one. Numerical simulation was carried out which supported the analytical results. Keywords: Mathematical Model, Biomathematics, Reproduction Number, Disease Free Equilibrium, Endemic Equilibrium Point

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Pengembangan Model Matematika SIRD (Susceptibles-Infected-Recovery-Deaths) Pada Penyebaran Virus Ebola

Jurnal Fourier ◽

10.14421/fourier.2016.51.23-34 ◽

2016 ◽

Vol 5 (1) ◽

pp. 23

Author(s):

Endah Purwati ◽

Sugiyanto Sugiyanto

Keyword(s):

Equilibrium Point ◽

Basic Reproduction Number ◽

Direct Contact ◽

Ebola Virus ◽

Reproduction Number ◽

Body Fluids ◽

Endemic Equilibrium ◽

The Stability ◽

Disease Free ◽

The Basic Reproduction Number

Ebola is a deadly disease caused by a virus and is spread through direct contact with blood or body fluids such as urine, feces, breast milk, saliva and semen. In this case, direct contact means that the blood or body fluids of patients were directly touching the nose, eyes, mouth, or a wound someone open. In this paper examined two mathematical models SIRD (Susceptibles-Infected-Recovery-Deaths) the spread of the Ebola virus in the human population. Both the mathematical model SIRD on the spread of the Ebola virus is a model by Abdon A. and Emile F. D. G. and research development model. This study was conducted to determine the point of disease-free equilibrium and endemic equilibrium point and stability analysis of the dots, knowing the value of the basic reproduction number (R0) and a simulation model using Matlab software version 6.1.0.450. From the analysis of the two models, obtained the same point for disease-free equilibrium point with the stability of different points and different points for endemic equilibrium point with the stability of both the same point and the same value to the value of the basic reproduction number (R0). After simulating the model using Matlab software version 6.1.0.450, it can be seen changes in the behavior of the population at any time.

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Analisis Model Matematika Penyebaran Demam Berdarah Dengue dengan Fungsi Lyapunov

Journal of Mathematics, Computations, and Statistics ◽

10.35580/jmathcos.v1i2.9188 ◽

2019 ◽

Vol 1 (2) ◽

pp. 125

Author(s):

Syafruddin Side ◽

Ahmad Zaki ◽

Nurwahidah Sari

Keyword(s):

Lyapunov Function ◽

Equilibrium Point ◽

Dengue Fever ◽

Reproduction Number ◽

Equilibrium Points ◽

Secondary Data ◽

Hemorrhagic Fever ◽

Endemic Equilibrium ◽

Sirs Model ◽

Disease Free

Abstrak. Artike lini adalah penelitian teori dan terapan. Artikelini bertujuan untuk membahas mengenai model matematika SIRS untuk penyebaran Demam Berdarah Dengue. Data yang digunakanadalah data sekunder jumlah penderita penyakit Demam Berdarah Dengue dari Side pada tahun 2014. Pembahasan di mulai dari membangun model matematika SIRS penyakit Demam Berdarah Dengue, menentukan eksistensi model SIRS menggunakan fungsi Lyapunov, penentuan titik ekuilibrium, kemudian mencari analisis kestabilan titik ekuilibrium menggunakan fungsi Lyapunov, menentukan nilai bilangan reproduksi dasar , membuat simulasi model, dan menginterpretasikannya. Dalam artikel ini diperoleh model matematika SIRS untuk penyakit Demam Berdarah Dengue, eksistensi model SIRS, dua titik ekuilibrium bebas penyakit dan endemik dari model SIRS, kestabilan global keseimbangan bebas penyakit dan endemik dari model SIRS dengan nilai bilangan reproduksi dasar , ini menunjukkan bahwa penyakit Demam Berdarah Dengue berstatus epidemik.Kata Kunci: Model Matematika, Penyebaran Penyakit, Demam Berdarah Dengue, Model SIRS, Fungsi LyapunovAbstract. This paper is theorethical and applied research. This paper aims to discus about SIRS mathematical models for the spread of dengue fever. The data used is a secondary data about the number of people with dengue fever disease from Side (2014). The discussion start from constructing SIRS models of dengue fever disease, determining the existence of SIRS models using Lyapunov function, determining equilibrium point, then looking for stability analysis of equilibrium point using Lyapunov function, determining reproduction number , making models simulation, and interpreting it. In this paper, we obtained mathemathical models of SIRS for dengue fever disease, existence of SIRS models, disease-free and endemic equilibrium points of SIRS models, global stability of disease-free and endemic equilibrium of SIRS models with basic reproduction number , it shows that dengue fever disease is epidemic status. , This shows that Dengue Hemorrhagic Fever is an epidemic.Keyword: Mathematical Model, Spread of Disease, Dengue Fever, SIRS Model, Lyapunov Function

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Model matematika penyebaran COVID-19 dengan penggunaan masker kesehatan dan karantina

Jambura Journal of Biomathematics (JJBM) ◽

10.34312/jjbm.v2i2.10483 ◽

2021 ◽

Vol 2 (2) ◽

pp. 68-79

Author(s):

Muhammad Manaqib ◽

Irma Fauziah ◽

Eti Hartati

Keyword(s):

Numerical Simulations ◽

Equilibrium Point ◽

Basic Reproduction Number ◽

Reproduction Number ◽

Equilibrium Points ◽

Endemic Equilibrium ◽

Reproduction Numbers ◽

Geometric Picture ◽

Disease Free ◽

The Basic Reproduction Number

This study developed a model for the spread of COVID-19 disease using the SIR model which was added by a health mask and quarantine for infected individuals. The population is divided into six subpopulations, namely the subpopulation susceptible without a health mask, susceptible using a health mask, infected without using a health mask, infected using a health mask, quarantine for infected individuals, and the subpopulation to recover. The results obtained two equilibrium points, namely the disease-free equilibrium point and the endemic equilibrium point, and the basic reproduction number (R0). The existence of a disease-free equilibrium point is unconditional, whereas an endemic equilibrium point exists if the basic reproduction number is more than one. Stability analysis of the local asymptotically stable disease-free equilibrium point when the basic reproduction number is less than one. Furthermore, numerical simulations are carried out to provide a geometric picture related to the results that have been analyzed. The results of numerical simulations support the results of the analysis obtained. Finally, the sensitivity analysis of the basic reproduction numbers carried out obtained four parameters that dominantly affect the basic reproduction number, namely the rate of contact of susceptible individuals with infection, the rate of health mask use, the rate of health mask release, and the rate of quarantine for infected individuals.

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Analysis of the SARS-CoV-2 Outbreak in Northern Cyprus using a Mathematical Model

10.21203/rs.3.rs-28213/v1 ◽

2020 ◽

Author(s):

Tamer Sanlidag ◽

Nazife Sultanoglu ◽

Bilgen Kaymakamzade ◽

Evren Hincal ◽

Murat Sayan ◽

...

Keyword(s):

Mathematical Model ◽

Basic Reproduction Number ◽

Reproduction Number ◽

Equilibrium Points ◽

Real Data ◽

Northern Cyprus ◽

The Stability ◽

Disease Free ◽

The Basic Reproduction Number

Abstract The present study studied the dynamics of SARS-CoV-2 in Northern-Cyprus (NC) by using real data and a designed mathematical model. The model consisted of two equilibrium points, which were disease-free and epidemic. The stability of the equilibrium points was determined by the magnitude of the basic reproduction number (𝑹𝟎). If 𝑹𝟎 < 1, the disease eventually disappears, if 𝑹𝟎 ≥ 1, the presence of an epidemic is stated. 𝑹𝟎 has been calculated patient zero, with a range of 2.38 to 0.65. Currently, the 𝑹𝟎 for NC was found to be 0.65, indicating that NC is free from the SARS-CoV-2 epidemic.

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