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

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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Stability Analysis of Mathematical Models of the Dynamics of Spread of Meningitis with the Effects of Vaccination, Campaigns and Treatment

Jurnal Matematika Statistika dan Komputasi ◽

10.20956/jmsk.v17i1.10031 ◽

2020 ◽

Vol 17 (1) ◽

pp. 71-81

Author(s):

Sulma Sulma ◽

Syamsuddin Toaha ◽

Kasbawati Kasbawati

Keyword(s):

Infectious Disease ◽

Equilibrium Point ◽

Equilibrium Points ◽

Vaccination Campaign ◽

Endemic Equilibrium ◽

Asymptotically Stable ◽

Infected People ◽

The Stability ◽

Vaccination Campaigns ◽

The Mathematical Model

Meningitis is an infectious disease caused by bacteria, viruses, and protosoa and has the potential to cause an outbreak. Vaccination and campaign are carried out as an effort to prevent the spread of meningitis, treatment reduces the number of deaths caused by the disease and the number of infected people. This study aims to analyze and determine the stability of equilibrium point of the mathematical model of the spread of meningitis using five compartments namely susceptibles, carriers, infected without symptoms, infected with symptoms, and recovered with the effect of vaccination, campaign, and treatment. The results obtained from the analysis of the model that there are two equilibrium points, namely non endemic and endemic equilibrium points. If a certain condition is met then the non endemic equilibrium point will be asymptotically stable. Numerical simulations show that the spread of disease decreases with the influence of vaccination, campaign, and treatment.

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Analisis Dinamik Model Transmisi COVID-19 dengan Melibatkan Intervensi Karantina

10.31219/osf.io/79hqv ◽

2021 ◽

Author(s):

Resmawan Resmawan ◽

Agusyarif Rezka Nuha ◽

Lailany Yahya

Keyword(s):

Population Dynamics ◽

Stability Analysis ◽

Numerical Simulations ◽

Equilibrium Point ◽

Equilibrium Points ◽

Asymptotically Stable ◽

Human Class ◽

Existence Of Equilibrium ◽

The Stability ◽

Disease Free

This paper discusses the dynamics of COVID-19 transmission by involving quarantine interventions. The model was constructed by involving three classes of infectious causes, namely the exposed human class, asymptotically infected human class, and symptomatic infected human class. Variables were representing quarantine interventions to suppress infection growth were also considered in the model. Furthermore, model analysis is focused on the existence of equilibrium points and numerical simulations to visually showed population dynamics. The constructed model forms the SEAQIR model which has two equilibrium points, namely a disease-free equilibrium point and an endemic equilibrium point. The stability analysis showed that the disease-free equilibrium point was locally asymptotically stable at R0<1 and unstable at R0>1. Numerical simulations showed that increasing interventions in the form of quarantine could contribute to slowing the transmission of COVID-19 so that it is hoped that it can prevent outbreaks in the population.

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Stability Analysis of SEIL Tuberculosis Epidemic Model with Logistic Growth in Susceptible Compartment

ASM Science Journal ◽

10.32802/asmscj.2021.733 ◽

2021 ◽

Vol 16 ◽

pp. 1-9

Author(s):

Joko Harianto

Keyword(s):

Equilibrium Point ◽

Basic Reproduction Number ◽

Reproduction Number ◽

Equilibrium Points ◽

Endemic Equilibrium ◽

Logistic Growth ◽

Asymptotically Stable ◽

Tuberculosis Disease ◽

Disease Free ◽

The Basic Reproduction Number

This article discusses modifications to the SEIL model that involve logistical growth. This model is used to describe the dynamics of the spread of tuberculosis disease in the population. The existence of the model's equilibrium points and its local stability depends on the basic reproduction number. If the basic reproduction number is less than unity, then there is one equilibrium point that is locally asymptotically stable. The equilibrium point is a disease-free equilibrium point. If the basic reproduction number ranges from one to three, then there are two equilibrium points. The two equilibrium points are disease-free equilibrium and endemic equilibrium points. Furthermore, for this case, the endemic equilibrium point is locally asymptotically stable.

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Analisis Dinamik Model Transmisi COVID-19 dengan Melibatkan Intervensi Karantina

Jambura Journal of Mathematics ◽

10.34312/jjom.v3i1.8699 ◽

2021 ◽

Vol 3 (1) ◽

pp. 66-79

Author(s):

Resmawan Resmawan ◽

Agusyarif Rezka Nuha ◽

Lailany Yahya

Keyword(s):

Population Dynamics ◽

Numerical Simulations ◽

Equilibrium Point ◽

Equilibrium Points ◽

Asymptotically Stable ◽

Human Class ◽

Existence Of Equilibrium ◽

Visual Model ◽

The Stability ◽

Disease Free

ABSTRAKMakalah ini membahas dinamika transmisi COVID-19 dengan melibatkan intervensi karantina. Model dikonstruksi dengan melibatkan tiga kelas penyebab infeksi, yaitu kelas manusia terpapar, kelas manusia terinfeksi tanpa gejala klinis, dan kelas manusia terinfeksi disertai gejala klinis. Variabel yang merepresentasikan intervensi karantina untuk menekan pertumbuhan infeksi juga dipertimbangkan pada model. Selanjutnya, analisis model difokuskan pada eksistensi titik kesetimbangan dan simulasi numerik untuk menunjukkan dinamika populasi secara visual. Model yang dikonstruksi membentuk model SEAQIR yang memiliki dua titik kesetimbangan, yaitu titik kesetimbangan bebas penyakit dan titik kesetimbangan endemik. Analisis kestabilan menunjukkan bahwa titik kesetimbangan bebas penyakit bersifat stabil asimtotik lokal pada saat R01 dan tidak stabil pada saat R01. Simulasi numerik menunjukkan bahwa peningkatan intervensi berupa karantina dapat berkontribusi memperlambat transmisi COVID-19 sehingga diharapkan dapat mencegah terjadinya wabah pada populasi.ABSTRACTThis paper discusses the dynamics of COVID-19 transmission by involving quarantine interventions. The model was constructed by involving three classes of infectious causes, namely the exposed human class, asymptotically infected human class, and symptomatic infected human class. Variables were representing quarantine interventions to suppress infection growth were also considered in the model. Furthermore, model analysis is focused on the existence of equilibrium points and numerical simulations to visually showed population dynamics. The constructed model forms the SEAQIR model which has two equilibrium points, namely a disease-free equilibrium point and an endemic equilibrium point. The stability analysis showed that the disease-free equilibrium point was locally asymptotically stable at R01 and unstable at R01. Numerical simulations showed that increasing interventions in the form of quarantine could contribute to slowing the transmission of COVID-19 so that it is hoped that it can prevent outbreaks in the population.

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DINAMIKA LOKAL MODEL EPIDEMI SVIR DENGAN IMIGRASI PADA KOMPARTEMEN VAKSINASI

BAREKENG JURNAL ILMU MATEMATIKA DAN TERAPAN ◽

10.30598/barekengvol14iss2pp297-304 ◽

2020 ◽

Vol 14 (2) ◽

pp. 297-304

Author(s):

Joko Harianto ◽

Titik Suparwati ◽

Inda Puspita Sari

Keyword(s):

Equilibrium Point ◽

Basic Reproduction Number ◽

Reproduction Number ◽

Equilibrium Points ◽

Asymptotically Stable ◽

Mathematical Epidemiology ◽

Unique Equilibrium ◽

The Stability ◽

Disease Free ◽

The Basic Reproduction Number

Abstrak Artikel ini termasuk dalam ruang lingkup matematika epidemiologi. Tujuan ditulisnya artikel ini untuk mendeskripsikan dinamika lokal penyebaran suatu penyakit dengan beberapa asumsi yang diberikan. Dalam pembahasan, dianalisis titik ekuilibrium model epidemi SVIR dengan adanya imigrasi pada kompartemen vaksinasi. Dengan langkah pertama, model SVIR diformulasikan, kemudian titik ekuilibriumnya ditentukan, selanjutnya, bilangan reproduksi dasar ditentukan. Pada akhirnya, kestabilan titik ekuilibirum yang bergantung pada bilangan reproduksi dasar ditentukan secara eksplisit. Hasilnya adalah jika bilangan reproduksi dasar kurang dari satu maka terdapat satu titik ekuilbirum dan titik ekuilbrium tersebut stabil asimtotik lokal. Hal ini berarti bahwa dalam kondisi tersebut penyakit akan cenderung menghilang dalam populasi. Sebaliknya, jika bilangan reproduksi dasar lebih dari satu, maka terdapat dua titik ekuilibrium. Dalam kondisi ini, titik ekuilibrium endemik stabil asimtotik lokal dan titik ekuilibrium bebas penyakit tidak stabil. Hal ini berarti bahwa dalam kondisi tersebut penyakit akan tetap ada dalam populasi. Kata Kunci : Model SVIR, Stabil Asimtotik Lokal Abstract This article is included in the scope of mathematical epidemiology. The purpose of this article is to describe the dynamics of the spread of disease with some assumptions given. In this paper, we present an epidemic SVIR model with the presence of immigration in the vaccine compartment. First, we formulate the SVIR model, then the equilibrium point is determined, furthermore, the basic reproduction number is determined. In the end, the stability of the equilibrium point is determined depending on the number of basic reproduction. The result is that if the basic reproduction number is less than one then there is a unique equilibrium point and the equilibrium point is locally asymptotically stable. This means that in those conditions the disease will tend to disappear in the population. Conversely, if the basic reproduction number is more than one, then there are two equilibrium points. The endemic equilibrium point is locally asymptotically stable and the disease-free equilibrium point is unstable. This means that in those conditions the disease will remain in the population. Keywords: SVIR Model, Locally Asymptotically stable.

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Forecast analysis of the epidemics trend of COVID-19 in the United States by a generalized fractional-order SEIR model

10.1101/2020.04.24.20078493 ◽

2020 ◽

Cited By ~ 5

Author(s):

Conghui Xu ◽

Yongguang Yu ◽

YangQuan Chen ◽

Zhenzhen Lu

Keyword(s):

United States ◽

Equilibrium Point ◽

Fractional Order ◽

Equilibrium Points ◽

The United States ◽

Endemic Equilibrium ◽

Qualitative Properties ◽

Prediction Ability ◽

Seir Model ◽

Disease Free

AbstractIn this paper, a generalized fractional-order SEIR model is proposed, denoted by SEIQRP model, which has a basic guiding significance for the prediction of the possible outbreak of infectious diseases like COVID-19 and other insect diseases in the future. Firstly, some qualitative properties of the model are analyzed. The basic reproduction number R0 is derived. When R0 < 1, the disease-free equilibrium point is unique and locally asymptotically stable. When R0 > 1, the endemic equilibrium point is also unique. Furthermore, some conditions are established to ensure the local asymptotic stability of disease-free and endemic equilibrium points. The trend of COVID-19 spread in the United States is predicted. Considering the influence of the individual behavior and government mitigation measurement, a modified SEIQRP model is proposed, defined as SEIQRPD model. According to the real data of the United States, it is found that our improved model has a better prediction ability for the epidemic trend in the next two weeks. Hence, the epidemic trend of the United States in the next two weeks is investigated, and the peak of isolated cases are predicted. The modified SEIQRP model successfully capture the development process of COVID-19, which provides an important reference for understanding the trend of the outbreak.

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Mathematical Modelling on Double Quarantine Process in the Spread and Stability of Covid-19

10.21203/rs.3.rs-40189/v1 ◽

2020 ◽

Author(s):

Jangyadatta Behera ◽

Aswin Kumar Rauta ◽

Yerra Shankar Rao ◽

Sairam Patnaik

Keyword(s):

Basic Reproduction Number ◽

Reproduction Number ◽

Equilibrium Points ◽

Endemic Equilibrium ◽

Asymptotically Stable ◽

Globally Asymptotically Stable ◽

Analytical Result ◽

The Stability ◽

The Impact ◽

Disease Free

Abstract In this paper, a mathematical model is proposed on the spread and control of corona virus disease2019 (COVID19) to ascertain the impact of pre quarantine for suspected individuals having travel history ,immigrants and new born cases in the susceptible class following the lockdown or shutdown rules and adopted the post quarantine process for infected class. Set of nonlinear ordinary differential equations (ODEs) are generated and parameters like natural mortality rate, rate of COVID-19 induced death, rate of immigrants, rate of transmission and recovery rate are integrated in the scheme. A detailed analysis of this model is conducted analytically and numerically. The local and global stability of the disease is discussed mathematically with the help of Basic Reproduction Number. The ODEs are solved numerically with the help of Runge-Kutta 4th order method and graphs are drawn using MATLAB software to validate the analytical result with numerical simulation. It is found that both results are in good agreement with the results available in the existing literatures. The stability analysis is performed for both disease free equilibrium and endemic equilibrium points. The theorems based on Routh-Hurwitz criteria and Lyapunov function are proved .It is found that the system is locally asymptotically stable at disease free and endemic equilibrium points for basic reproduction number less than one and globally asymptotically stable for basic reproduction number greater than one. Finding of this study suggest that COVID-19 would remain pandemic with the progress of time but would be stable in the long-term if the pre and post quarantine policy for asymptomatic and symptomatic individuals are implemented effectively followed by social distancing, lockdown and containment.

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Mathematical Modeling and Analysis of Transmission Dynamics and Control of Schistosomiasis

Journal of Applied Mathematics ◽

10.1155/2021/6653796 ◽

2021 ◽

Vol 2021 ◽

pp. 1-20

Author(s):

Ebrima Kanyi ◽

Ayodeji Sunday Afolabi ◽

Nelson Owuor Onyango

Keyword(s):

Equilibrium Point ◽

Control Strategies ◽

Equilibrium Points ◽

Threshold Value ◽

Endemic Equilibrium ◽

Transmission Dynamics ◽

Asymptotically Stable ◽

Globally Asymptotically Stable ◽

Modeling And Analysis ◽

Disease Free

This paper presents a mathematical model that describes the transmission dynamics of schistosomiasis for humans, snails, and the free living miracidia and cercariae. The model incorporates the treated compartment and a preventive factor due to water sanitation and hygiene (WASH) for the human subpopulation. A qualitative analysis was performed to examine the invariant regions, positivity of solutions, and disease equilibrium points together with their stabilities. The basic reproduction number, R 0 , is computed and used as a threshold value to determine the existence and stability of the equilibrium points. It is established that, under a specific condition, the disease-free equilibrium exists and there is a unique endemic equilibrium when R 0 > 1 . It is shown that the disease-free equilibrium point is both locally and globally asymptotically stable provided R 0 < 1 , and the unique endemic equilibrium point is locally asymptotically stable whenever R 0 > 1 using the concept of the Center Manifold Theory. A numerical simulation carried out showed that at R 0 = 1 , the model exhibits a forward bifurcation which, thus, validates the analytic results. Numerical analyses of the control strategies were performed and discussed. Further, a sensitivity analysis of R 0 was carried out to determine the contribution of the main parameters towards the die out of the disease. Finally, the effects that these parameters have on the infected humans were numerically examined, and the results indicated that combined application of treatment and WASH will be effective in eradicating schistosomiasis.

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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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A FRACTIONAL ORDER HIV/AIDS MODEL USING CAPUTO-FABRIZIO OPERATOR

African Journal of Infectious Diseases ◽

10.21010/ajid.v15i2s.1 ◽

2021 ◽

Vol 15 (2s) ◽

pp. 1-18

Author(s):

Ebenezar Nkemjika Unaegbu ◽

Ifeanyi Sunday Onah ◽

Moses Oladotun Oyesanya

Keyword(s):

Equilibrium Point ◽

Fractional Order ◽

Numerical Solutions ◽

Fixed Point Theory ◽

Equilibrium Points ◽

Endemic Equilibrium ◽

Existence Of A Solution ◽

Aids Model ◽

Disease Free ◽

Hiv Aids

Background: HIV is a virus that is directed at destroying the human immune system thereby exposing the human body to the risk of been affected by other common illnesses and if it is not treated, it generates a more chronic illness called AIDS. Materials and Methods: In this paper, we employed the fixed-point theory in developing the uniqueness and existence of a solution of fractional order HIV/AIDS model having Caputo-Fabrizio operator. This approach adopted in this work is not conventional when solving biological models by fractional derivatives. Results: The results showed that the model has two equilibrium points namely, disease-free, and endemic equilibrium points, respectively. We showed conditions necessitating the existence of the endemic equilibrium point and showed that the disease-free equilibrium point is locally asymptotically stable. We also tested the stability of our solution using the iterative Laplace transform method on our model which was also shown stable agreeing with the disease-free equilibrium. Conclusions: Numerical simulations of our model showed clear comparison with our analytical results. The numerical solutions show that given fractional operator like the Caputo-Fabrizio operator, it is less noisy and plays a major role in making a precise decision and gives room (‘freedom’) to use data of specific patients as the model can be easily adjusted to accommodate this, as it a better fit for the patients’ data and provide meaningful predictions. Finally, the result showed the advantage of using fractional order derivative in the analysis of the dynamics of HIV/AIDS over the classical case.

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