MATHEMATICAL ANALYSIS OF THE ENDEMIC EQUILIBRIUM OF MALARIAHYGIENE MODEL

2021 ◽  
Vol 5 (10) ◽  
Author(s):  
Oluwafemi Temidayo J. ◽  
Azuaba E. ◽  
Lasisi N. O.
Keyword(s):  
Numerical Simulation ◽  
Mathematical Model ◽  
Equilibrium Point ◽  
Reproduction Number ◽  
Endemic Equilibrium ◽  
The Mathematical Model ◽  
Globally Stable ◽  
Well Posed ◽  
The Basic Reproduction Number ◽  
Invariant Principle

In this study, we analyzed the endemic equilibrium point of a malaria-hygiene mathematical model. We prove that the mathematical model is biological and meaningfully well-posed. We also compute the basic reproduction number using the next generation method. Stability analysis of the endemic equilibrium point show that the point is locally stable if reproduction number is greater that unity and globally stable by the Lasalle’s invariant principle. Numerical simulation to show the dynamics of the compartment at various hygiene rate was carried out.

scholarly journals A Mathematical Model to Investigate the Transmission of COVID-19 in the Kingdom of Saudi Arabia

2020 ◽  
Vol 2020 ◽  
pp. 1-13 ◽  
Author(s):  
Fehaid Salem Alshammari
Keyword(s):  
Numerical Simulation ◽  
Mathematical Model ◽  
Saudi Arabia ◽  
Reproduction Number ◽  
Real Data ◽  
Kingdom Of Saudi Arabia ◽  
Globally Stable ◽  
Disease Free ◽  
The Basic Reproduction Number ◽  
Good Agreement

Since the first confirmed case of SARS-CoV-2 coronavirus (COVID-19) on March 02, 2020, Saudi Arabia has not reported quite a rapid COVD-19 spread as seen in America and many European countries. Possible causes include the spread of asymptomatic COVID-19 cases. To characterize the transmission of COVID-19 in Saudi Arabia, a susceptible, exposed, symptomatic, asymptomatic, hospitalized, and recovered dynamical model was formulated, and a basic analysis of the model is presented including model positivity, boundedness, and stability around the disease-free equilibrium. It is found that the model is locally and globally stable around the disease-free equilibrium when R 0 < 1 . The model parameterized from COVID-19 confirmed cases reported by the Ministry of Health in Saudi Arabia (MOH) from March 02 till April 14, while some parameters are estimated from the literature. The numerical simulation showed that the model predicted infected curve is in good agreement with the real data of COVID-19-infected cases. An analytical expression of the basic reproduction number R 0 is obtained, and the numerical value is estimated as R 0 ≈ 2.7 .

scholarly journals Pemodelan Matematika SEIRS Pada Penyebaran Penyakit Malaria di Kabupaten Mimika

2021 ◽  
Vol 4 (1) ◽  
pp. 21
Author(s):  
Hisyam Ihsan ◽  
Syafruddin Side ◽  
Musdalifa Pagga
Keyword(s):  
Mathematical Model ◽  
Equilibrium Point ◽  
Basic Reproduction Number ◽  
Reproduction Number ◽  
District Health Office ◽  
District Health ◽  
Simulation Results ◽  
The Mathematical Model ◽  
Malaria Model ◽  
The Basic Reproduction Number

Abstrak. Penelitian ini  bertujuan untuk membangun model penyebaran pada penyakit malaria tipe SEIRS (Susceptible-Exposed- Infected- Recovered- Susceptible) dengan menambahkan parameter penanganan(pengobatan) pada kelas Exposed dan asumsi bahwa manusia yang pulih dapat rentan kembali terkena penyakit malaria. Model ini dibagi menjadi empat kelas yaitu, rentan, terinfeksi tapi belum aktif, terinfeksi, dan sembuh. Data yang digunakan adalah data jumlah penderita penyakit malaria dari Dinas Kesehatan Kabupaten Mimika tahun 2018. Model matematika tipe SEIRS digunakan untuk menentukan titik equilibrium. Berdasarkan hasil simulasi dari model SEIRS diperoleh bilangan reproduksi dasar  sebesar 0,09 yang menandakan bahwa penyebaran penyakit malaria tidak menyebabkan orang lain terkena penyakit malaria.Kata Kunci: Titik Equilibrium, Bilangan Reproduksi Dasar, Malaria, Model SEIRSAbstract. This research aims to build a model of the spread of malaria diseases type SEIRS (Susceptible-Exposed-Infected-Recovered-Susceptible) by adding treatment parameters (treatment) in the Exposed class and the assumption that humans who recover can be vulnerable to malaria again. This model is divided into four classes namely, vulnerable, infected but not yet active, infected, and cured. The data used are data on the number of malaria sufferers from the Mimika District Health Office in 2018. The mathematical model of the type SEIRS is used to determine the equilibrium point. Based on the simulation results of the SEIRS model, the basic reproduction number (R0) of 0.09 indicates that the spread of malaria does not cause others to contract malaria.Keywords: Equilibrium Point, Basic Reproductive Numbers, Malaria, SEIRS Model

scholarly journals Mathematical Analysis of an Immune-structured Hikungunya Transmission Model

2019 ◽  
Vol 12 (4) ◽  
pp. 1533-1552
Author(s):  
Kambire Famane ◽  
Gouba Elisée ◽  
Tao Sadou ◽  
Blaise Some
Keyword(s):  
Numerical Simulation ◽  
Asymptotic Stability ◽  
Deterministic Model ◽  
Reproduction Number ◽  
Transmission Model ◽  
Acquired Immunity ◽  
Local Asymptotic Stability ◽  
Well Posed ◽  
Disease Free ◽  
The Basic Reproduction Number

In this paper, we have formulated a new deterministic model to describe the dynamics of the spread of chikunguya between humans and mosquitoes populations. This model takes into account the variation in mortality of humans and mosquitoes due to other causes than chikungunya disease, the decay of acquired immunity and the immune sytem boosting. From the analysis, itappears that the model is well posed from the mathematical and epidemiological standpoint. The existence of a single disease free equilibrium has been proved. An explicit formula, depending on the parameters of the model, has been obtained for the basic reproduction number R0 which is used in epidemiology. The local asymptotic stability of the disease free equilibrium has been proved. The numerical simulation of the model has confirmed the local asymptotic stability of the diseasefree equilbrium and the existence of endmic equilibrium. The varying effects of the immunity parameters has been analyzed numerically in order to provide better conditions for reducing the transmission of the disease.

scholarly journals APLIKASI MODEL EPIDEMIK SEIAR-SEI PADA PENYEBARAN PENYAKIT MALARIA DENGAN INFEKSI ASIMTOMATIK DAN SUPER INFEKSI

2018 ◽  
Vol 15 (2) ◽  
pp. 67
Author(s):  
Stella Maryana Belwawin
Keyword(s):  
Equilibrium Point ◽  
Basic Reproduction Number ◽  
Reproduction Number ◽  
Asymptomatic Infection ◽  
Endemic Equilibrium ◽  
Asymptomatic Infections ◽  
The Stability ◽  
Super Infection ◽  
The Basic Reproduction Number

AbstractThis aim of this study is to determine the point of equilibrium and analyze the stability of SEIAR-SEI model on malaria disease with asymptomatic infection, super infection and the effect of the mosquito's life cycle. This study also aim is to measure the sensitivity of the spread of malaria to the parameters of asymptomatic infections, the rate of treatment, and the rate of birth of mosquitoes through the magnitude of . The method in this research is deductively, through several stage, such as  determination of disease-free equilibrium point and endemic equilibrium point, determination of basic reproduction number (), analyze of the basic reproduction number sensitivity of the spread of malaria to the parameters of asymptomatic infections, the rate of treatment, and the rate of birth of mosquitoes. The endemic equilibrium point was obtained using rule of Descartes. The result show that the change in the value of parameter , , and  has effect on the basic reproduction number (). Treatment factors in the human population influence the elimination of malaria in a population. Whereas asymptomatic infection factors and the birth rate of adult mosquitoes influence the increase in malaria infection. Keywords:  Malaria, asymptomatic infection, super infection, basic reproduction number, rule of descrates. AbstrakPenelitian ini bertujuan menentukan titik keseimbangan dan menganalisis kestabilan dari model SEIAR_SEI pada penyakit malaria dengan pengaruh infeksi asimtomatik, super infeksi, dan siklus hidup nyamuk. Penelitian ini juga bertujuan mengukur tingkat sensitivitas penyebaran penyakit malaria terhadap parameter infeksi asimtomatik, laju pengobatan, serta laju kelahiran nyamuk.melalu besaran .  Metode yang digunakan dalam penelitian ini adalah metode deduktif dengan langkah-langkah : menentukan titik keseimbangan bebas penyakit dan endemik dan menentukan bilangan reproduksi dasar ). Analisis sensitivitas bilangan reproduksi dasar dilakukan terhadap parameter infeksi asimtomatik, pengobatan, dan laju kelahiran nyamuk. Tititk keseimbangan endemik diperoleh dengan aturan descrates. Hasil yang diperoleh menunjukkan parameter , , dan  berpengaruh terhadap bilangan reproduksi dasar (). Faktor pengobatan berpengaruh terhadap eliminasi penyakit malaria. Sedangkan faktor infeksi asimtomatik dan laju kelahiran nyamuk dewasa berpengaruh terhadap peningkatan infeksi penyakit malaria. Kata kunci: Malaria, Infeksi Asimtomatik, Super Infeksi, Bilangan Reproduksi Dasar, Aturan Descrates . 

scholarly journals 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

2020 ◽  
Vol 10 (22) ◽  
pp. 8296 ◽  
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.

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

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

THE DYNAMICS OF A DISCRETE SEIT MODEL WITH AGE AND INFECTION AGE STRUCTURES

2012 ◽  
Vol 05 (03) ◽  
pp. 1260004 ◽  
Author(s):  
HUI CAO ◽  
YANNI XIAO ◽  
YICANG ZHOU
Keyword(s):  
Reproduction Number ◽  
Endemic Equilibrium ◽  
Disease Spread ◽  
Threshold Parameter ◽  
Infection Age ◽  
The Stability ◽  
Globally Stable ◽  
Disease Free ◽  
The Basic Reproduction Number ◽  
Age Structures

Age and infection age have significant influence on the transmission of infectious diseases, such as HIV/AIDS and TB. A discrete SEIT model with age and infection age structures is formulated to investigate the dynamics of the disease spread. The basic reproduction number R0 is defined and used as the threshold parameter to characterize the disease extinction or persistence. It is shown that the disease-free equilibrium is globally stable if R0 < 1, and it is unstable if R0 > 1. When R0 > 1, there exists an endemic equilibrium, and the disease is uniformly persistent. The stability of the endemic equilibrium is investigated numerically.

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

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.

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

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.

scholarly journals Local Dynamics of an SVIR Epidemic Model with Logistic Growth

CAUCHY ◽  
2020 ◽  
Vol 6 (3) ◽  
pp. 122-132
Author(s):  
Joko Harianto ◽  
Inda Puspita Sari
Keyword(s):  
Stability Analysis ◽  
Equilibrium Point ◽  
Basic Reproduction Number ◽  
Local Stability ◽  
Reproduction Number ◽  
Endemic Equilibrium ◽  
Logistic Growth ◽  
Asymptotically Stable ◽  
Local Stability Analysis ◽  
The Basic Reproduction Number

Discussion of local stability analysis of SVIR models in this article is included in the scope of applied mathematics. The purpose of this discussion was to provide results of local stability analysis that had not been discussed in some articles related to the SVIR model. The SVIR models discussed in this article involve logistics growth in the vaccinated compartment. The results obtained, i.e. if the basic reproduction number less than one and m is positive, then there is one equilibrium point i.e. E0 is locally asymptotically stable. In the field of epidemiology, this means that the disease will disappear from the population. However, if the basic reproduction number more than one and b1 more than b, then there are two equilibrium points i.e. disease-free equilibrium point denoted by E0 and the endemic equilibrium point denoted by E1*. In this case the endemic equilibrium point E1* is locally asymptotically stable. In the field of epidemiology, this means that the disease will remain in the population. The numerical simulation supports these results.

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