scholarly journals An Introduction to The Basic Reproduction Number in Mathematical Epidemiology

2018 ◽  
Vol 62 ◽  
pp. 123-138 ◽  
Author(s):  
Antoine Perasso
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Basic Reproduction Number ◽  
Matrix Method ◽  
Reproduction Number ◽  
Mathematical Epidemiology ◽  
Pde Model ◽  
Mathematical Definition ◽  
Next Generation Matrix ◽  
The Basic Reproduction Number

This article introduces the notion of basic reproduction number R0 in mathematical epi-demiology. After an historic reminder describing the steps leading to the statement of its mathematical definition, we explain the next-generation matrix method allowing its calculation in the case of epidemic models described by ordinary differential equations (ODEs). The article then focuses, through four ODEs examples and an infection load structured PDE model, on the usefulness of the R0 to address biological as well mathematical issues.

Analysis of a mathematical model for Dengue-Chikungunya

2017 ◽  
pp. 2933-2940
Author(s):  
Oscar A. Manrique A. ◽  
Dalia M. Munoz P. ◽  
Anibal Munoz L. ◽  
Mauricio Ropero P. ◽  
Steven Raigosa O. ◽  
...  
Keyword(s):  
Dynamical System ◽  
Mathematical Model ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Basic Reproduction Number ◽  
Reproduction Number ◽  
Epidemic Threshold ◽  
Infectious Process ◽  
Linear Ordinary Differential Equations ◽  
Next Generation Matrix

A dynamical system of non-linear ordinary differential equations which describes the Dengue-Chikungunya infectious process is reported. In this model it is considered the presence of two viruses transmitted by the same vector. Taking into account this fact, we have determined the epidemic threshold, basic reproduction number, using the next generation matrix. The simulations of the differential equations system are carried out with the MATLAB software.

A MULTI-GROUP SVEIR EPIDEMIC MODEL WITH DISTRIBUTED DELAY AND VACCINATION

2012 ◽  
Vol 05 (03) ◽  
pp. 1260001 ◽  
Author(s):  
JINLIANG WANG ◽  
YASUHIRO TAKEUCHI ◽  
SHENGQIANG LIU
Keyword(s):  
Basic Reproduction Number ◽  
Reproduction Number ◽  
Distributed Delay ◽  
Necessary Condition ◽  
Global Dynamics ◽  
Vaccination Strategies ◽  
Graph Theoretical Approach ◽  
Number Formula ◽  
Next Generation Matrix ◽  
The Basic Reproduction Number

In this paper, based on a class of multi-group epidemic models of SEIR type with bilinear incidences, we introduce a vaccination compartment, leading to multi-group SVEIR model. We establish that the global dynamics are completely determined by the basic reproduction number [Formula: see text] which is defined by the spectral radius of the next generation matrix. Our proofs of global stability of the equilibria utilize a graph-theoretical approach to the method of Lyapunov functionals. Mathematical results suggest that vaccination is helpful for disease control by decreasing the basic reproduction number. However, there is a necessary condition for successful elimination of disease. If the time for the vaccines to obtain immunity or the possibility for them to be infected before acquiring immunity is neglected in each group, this condition will be satisfied and the disease can always be eradicated by suitable vaccination strategies. This may lead to over evaluation for the effect of vaccination.

scholarly journals Stability of a Mathematical Model of Malaria Transmission with Relapse

2014 ◽  
Vol 2014 ◽  
pp. 1-9 ◽  
Author(s):  
Hai-Feng Huo ◽  
Guang-Ming Qiu
Keyword(s):  
Mathematical Model ◽  
Basic Reproduction Number ◽  
Recovery Rate ◽  
Reproduction Number ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
Next Generation Matrix ◽  
The Government ◽  
Disease Free ◽  
The Basic Reproduction Number

A more realistic mathematical model of malaria is introduced, in which we not only consider the recovered humans return to the susceptible class, but also consider the recovered humans return to the infectious class. The basic reproduction numberR0is calculated by next generation matrix method. It is shown that the disease-free equilibrium is globally asymptotically stable ifR0≤1, and the system is uniformly persistence ifR0>1. Some numerical simulations are also given to explain our analytical results. Our results show that to control and eradicate the malaria, it is very necessary for the government to decrease the relapse rate and increase the recovery rate.

scholarly journals Government control in increasing vehicles taxpayer compliance

2021 ◽  
Vol 26 ◽  
pp. 502-513
Author(s):  
Diah Anggeraini Hasri ◽  
Zulkieflimansyah Zulkieflimansyah ◽  
Muhammad Nurjihadi ◽  
Nova Adhitya Ananda ◽  
Lukmanul Hakim
Keyword(s):  
Mathematical Model ◽  
Regression Analysis ◽  
Differential Equations ◽  
Basic Reproduction Number ◽  
Reproduction Number ◽  
Sensitivity Index ◽  
Control Parameters ◽  
Government Control ◽  
Analysis Tools ◽  
The Basic Reproduction Number

This study aims to determine the dynamics of taxpayer compliance from time to time by using a mathematical model. This study uses two analysis tools, namely differential equations, to create a model of taxpayer compliance and Moderated Regression Analysis to determine the effect of moderating government control on increasing taxpayer compliance. This study indicates that government control can reduce the number of non-compliant taxpayers by looking at the sensitivity index. The results of the sensitivity index of government control parameters can reduce the basic reproduction number. Statistically, it is also proven that the moderation of government control can strengthen the effect of awareness on taxpayer compliance by 82.5%.

scholarly journals Mathematical Assessment of the Dynamical Model of Smoking Tobacco Epidemic in Bangladesh

2020 ◽  
pp. 36-48
Author(s):  
Sk. Abdus Samad ◽  
Md. Tusberul Islam ◽  
Sayed Toufiq Hossain Tomal ◽  
MHA Biswas
Keyword(s):  
Numerical Simulation ◽  
Mathematical Model ◽  
Stability Analysis ◽  
Differential Equations ◽  
Basic Reproduction Number ◽  
Conversion Rate ◽  
Reproduction Number ◽  
The World ◽  
Tobacco Epidemic ◽  
The Basic Reproduction Number

Bangladesh is one of the largest tobacco users in the world being troubled by smoking related issues. In this paper we consider a compartmental mathematical model of smoking in which the population is divided into five compartments: susceptible, expose, smokers, temporary quitters and permanent quitters described by ordinary differential equations. We study by including the conversion rate from light smoker to permanent quit smokers. The basic reproduction number R0 has been derived and then we found two euilibria of the model one of them is smoking-free and other of them is smoking-present. We establish the positivity, boundedness of the solutions and perform stability analysis of the model. To decrease the smoking propensity in Bangladesh we perform numerical simulation for various estimations of parameters which offer understanding to give up smoking and how they influence the smoker and exposed class. This model gives us legitimate thought regarding the explanations for the spread of smoking in Bangladesh.

scholarly journals DINAMIKA LOKAL MODEL EPIDEMI SVIR DENGAN IMIGRASI PADA KOMPARTEMEN VAKSINASI

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.

scholarly journals Mathematical Analysis of COVID-19 Transmission Dynamics with a Case Study of Nigeria and its Computer Simulation

2020 ◽  
Author(s):  
B. C. Agbata ◽  
Ogala Emmanuel ◽  
Tenuche Bashir ◽  
Obeng-Denteh William
Keyword(s):  
Basic Reproduction Number ◽  
Reproduction Number ◽  
Transmission Dynamics ◽  
Government Policies ◽  
Asymptotically Stable ◽  
Deadly Disease ◽  
Next Generation Matrix ◽  
Disease Free ◽  
The Basic Reproduction Number

AbstractIn this article, we formulated a mathematical model for the spread of the COVID-19 disease and we introduced quarantined and isolated compartments. The next generation matrix method was adopted to compute the basic reproduction number (R0) in order to assess the transmission dynamics of the COVID-19 deadly disease. Stability analysis of the disease free equilibrium is investigated based on the basic reproduction number and the result shows that it is locally and asymptotically stable for R0 less than 1. Numerical calculation of the basic reproduction number revealed that R0 < 1 which means that the disease can be eradicated from Nigeria. The study shows that isolation, quarantine and other government policies like social distancing and lockdown are the best approaches to control the pernicious nature of COVID-19 pandemic.

scholarly journals Parameterization and Forecasting of Childhood Pneumonia Model Using Least Square Approximation, Lagrange Polynomial and Monte Carlo Simulation

2020 ◽  
pp. 102-114
Author(s):  
Cyrus Gitonga Ngari ◽  
Dominic Makaa Kitavi
Keyword(s):  
Monte Carlo Simulation ◽  
Monte Carlo ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Basic Reproduction Number ◽  
Total Population ◽  
Reproduction Number ◽  
Research Result ◽  
Least Square ◽  
Under Five

Despite a study by [1] proposing a simple model of under five years pneumonia, doubt lingers regarding its reliability, sufficiency and validity. The research question is whether the model is valid for use or not?  The objectives of this study were to: incorporate exit rate from under five-year age bracket in the model, use Kenya data to parameterize the model, taking into account the uncertainties and finally to predict the dynamics of pneumonia. The model was rescaled through nondimensionalization. Data was fitted using theory of general solutions of nonlinear Ordinary differential equations, numerical differentiation using Lagrange polynomials and least square approximation method. Uncertainties due to disparities and round off errors were simulated using Monte Carlo simulation. Predictions of dynamics of pneumonia were carried out using MATLAB inbuilt ode solvers. Excel software was used to predict dynamics of discrete ordinary differential equations and to fit data. The basic reproduction number (  and effective reproduction number ( ) were obtained as  Iteration of uncertainties on R was carried out 1000 times by Monte Carlo simulation. The maximum and minimum R were obtained as 90 and 55, respectively. Using MATLAB software and effective reproduction number, the ratio of infective class to the total population and the ratio of class under treatment to the total population will remain constant at 0.095 and 0.2297 respectively for the years 2021, 2022 and 2023. Research result indicted that it is more effective and efficient to use effective reproduction number ( ) than basic reproduction number (  in mathematical modelling of Infectious diseases whenever study focuses on proportion of population. On basis of large absolute errors in fitting data to model, findings cast doubt on model formulation and/or observed data.

scholarly journals Dynamical Analysis of a Mathematical Model of COVID-19 Spreading on Networks

2021 ◽  
Vol 8 ◽  
Author(s):  
Wang Li ◽  
Xinjie Fu ◽  
Yongzheng Sun ◽  
Maoxing Liu
Keyword(s):  
Mathematical Model ◽  
Basic Reproduction Number ◽  
Reproduction Number ◽  
Dynamical Analysis ◽  
Asymptotically Stable ◽  
Effective Strategies ◽  
Disease Spreading ◽  
Next Generation Matrix ◽  
Disease Free ◽  
The Basic Reproduction Number

In this article, an SEAIRS model of COVID-19 epidemic on networks is established and analyzed. Following the method of the next-generation matrix, we derive the basic reproduction number R0, and it shows that the asymptomatic infector plays an important role in disease spreading. We analytically show that the disease-free equilibrium E0 is asymptotically stable if R0≤1; moreover, the effects of various quarantine strategies are investigated and compared by numerical simulations. The results obtained are informative for us to further understand the asymptomatic infector in COVID-19 propagation and get some effective strategies to control the disease.

scholarly journals Fractional SIR-Model for Estimating Transmission Dynamics of COVID-19 in India

J ◽  
2021 ◽  
Vol 4 (2) ◽  
pp. 86-100
Author(s):  
Nita H. Shah ◽  
Ankush H. Suthar ◽  
Ekta N. Jayswal ◽  
Ankit Sikarwar
Keyword(s):  
Basic Reproduction Number ◽  
Reproduction Number ◽  
Transmission Rate ◽  
Sir Model ◽  
Time Dependent ◽  
Contact Rate ◽  
Distribution Maps ◽  
Fundamental Parameters ◽  
Different Order ◽  
The Basic Reproduction Number

In this article, a time-dependent susceptible-infected-recovered (SIR) model is constructed to investigate the transmission rate of COVID-19 in various regions of India. The model included the fundamental parameters on which the transmission rate of the infection is dependent, like the population density, contact rate, recovery rate, and intensity of the infection in the respective region. Looking at the great diversity in different geographic locations in India, we determined to calculate the basic reproduction number for all Indian districts based on the COVID-19 data till 7 July 2020. By preparing district-wise spatial distribution maps with the help of ArcGIS 10.2, the model was employed to show the effect of complete lockdown on the transmission rate of the COVID-19 infection in Indian districts. Moreover, with the model's transformation to the fractional ordered dynamical system, we found that the nature of the proposed SIR model is different for the different order of the systems. The sensitivity analysis of the basic reproduction number is done graphically which forecasts the change in the transmission rate of COVID-19 infection with change in different parameters. In the numerical simulation section, oscillations and variations in the model compartments are shown for two different situations, with and without lockdown.

Sign in / Sign up

Export Citation Format

Share Document