A Discrete Numerical Solution of The SIR Model with Horizontal and Vertical Transmission

The aim of this paper is to is to generalize the SIR model with horizontal and vertical transmission. In this paper, we develop the discrete version of the model. We use Euler method to approximate numerical solution of the model. We found two equilibrium points, that is disease free and endemic equilibrium points. The existence of these points depend on basic reproduction number R0. We found that if R0 < 1 then only disease free equilibrium points exists, while both points exists when R0 > 1. We also found that the stability of these equilibrium points depend on the value of step-size h. Some numerical experiments were presented as illustration.

Download Full-text

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.

Download Full-text

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

10.21203/rs.3.rs-28704/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

Download Full-text

A study on the AH1N1/09 influenza transmission model with the fractional Caputo–Fabrizio derivative

Advances in Difference Equations ◽

10.1186/s13662-020-02945-x ◽

2020 ◽

Vol 2020 (1) ◽

Author(s):

Shahram Rezapour ◽

Hakimeh Mohammadi

Keyword(s):

Fractional Order ◽

Fixed Point Theory ◽

Equilibrium Points ◽

Point Theory ◽

Euler Method ◽

Transmission Model ◽

Order Model ◽

Influenza Transmission ◽

The Stability ◽

Disease Free

Abstract We study the SEIR epidemic model for the spread of AH1N1 influenza using the Caputo–Fabrizio fractional-order derivative. The reproduction number of system and equilibrium points are calculated, and the stability of the disease-free equilibrium point is investigated. We prove the existence of solution for the model by using fixed point theory. Using the fractional Euler method, we get an approximate solution to the model. In the numerical section, we present a simulation to examine the system, in which we calculate equilibrium points of the system and examine the behavior of the resulting functions at the equilibrium points. By calculating the results of the model for different fractional order, we examine the effect of the derivative order on the behavior of the resulting functions and obtained numerical values. We also calculate the results of the integer-order model and examine their differences with the results of the fractional-order model.

Download Full-text

Complex mathematical SIR model for spreading of COVID-19 virus with Mittag-Leffler kernel

Advances in Difference Equations ◽

10.1186/s13662-021-03470-1 ◽

2021 ◽

Vol 2021 (1) ◽

Author(s):

F. Talay Akyildiz ◽

Fehaid Salem Alshammari

Keyword(s):

Fractional Order ◽

Basic Reproduction Number ◽

Reproduction Number ◽

Equilibrium Points ◽

Sir Model ◽

Endemic Equilibrium ◽

Asymptotically Stable ◽

Exponential Kernel ◽

Disease Free ◽

The Basic Reproduction Number

AbstractThis paper investigates a new model on coronavirus-19 disease (COVID-19), that is complex fractional SIR epidemic model with a nonstandard nonlinear incidence rate and a recovery, where derivative operator with Mittag-Leffler kernel in the Caputo sense (ABC). The model has two equilibrium points when the basic reproduction number $R_{0} > 1$ R 0 > 1 ; a disease-free equilibrium $E_{0}$ E 0 and a disease endemic equilibrium $E_{1}$ E 1 . The disease-free equilibrium stage is locally and globally asymptotically stable when the basic reproduction number $R_{0} <1$ R 0 < 1 , we show that the endemic equilibrium state is locally asymptotically stable if $R_{0} > 1$ R 0 > 1 . We also prove the existence and uniqueness of the solution for the Atangana–Baleanu SIR model by using a fixed-point method. Since the Atangana–Baleanu fractional derivative gives better precise results to the derivative with exponential kernel because of having fractional order, hence, it is a generalized form of the derivative with exponential kernel. The numerical simulations are explored for various values of the fractional order. Finally, the effect of the ABC fractional-order derivative on suspected and infected individuals carefully is examined and compared with the real data.

Download Full-text

Mathematical Analysis of a Diarrhoea Model in the Presence of Vaccination and Treatment Waves with Sensitivity Analysis

Journal of Applied Sciences and Environmental Management ◽

10.4314/jasem.v25i7.2 ◽

2021 ◽

Vol 25 (7) ◽

pp. 1107-1114

Author(s):

E.I. Akinola ◽

B.E. Awoyemi ◽

I.A. Olopade ◽

O.D. Falowo ◽

T.O. Akinwumi

Keyword(s):

Sensitivity Analysis ◽

Mathematical Analysis ◽

Reproduction Number ◽

Equilibrium Points ◽

Runge Kutta Method ◽

Modelling Techniques ◽

Uniqueness And Existence ◽

The Stability ◽

Order Four ◽

Disease Free

In this study, the diarrhoea model is developed based on basic mathematical modelling techniques leading to a system (five compartmental model) of ordinary differential equations (ODEs). Mathematical analysis of the model is then carried out on the uniqueness and existence of the model to know the region where the model is epidemiologically feasible. The equilibrium points of the model and the stability of the disease-free state were also derived by finding the reproduction number. We then progressed to running a global sensitivity analysis on the reproduction number with respect to all the parameters in it, and four (4) parameters were found sensitive. The work was concluded with numerical simulations on Maple 18 using Runge-Kutta method of order four (4) where the values of six (6) parameters present in the model were each varied successively while all other parameters were held constant so as to know the behaviour and effect of the varied parameter on how diarrhoea spreads in the population. The results from the sensitivity analysis and simulations were found to be in sync.

Download Full-text

A Study of the psychological impact of media using modified Beddington-DeAngelis function in the SIR model for contagious diseases

Psychology and Education Journal ◽

10.17762/pae.v58i1.1200 ◽

2021 ◽

Vol 58 (1) ◽

pp. 3008-3015

Author(s):

Gurpreet Singh Tuteja

Keyword(s):

Dynamical System ◽

Jacobian Matrix ◽

Reproduction Number ◽

Equilibrium Points ◽

Psychological Impact ◽

Transmission Function ◽

Sir Model ◽

Transmission Dynamics ◽

Contagious Diseases ◽

The Stability

In this paper, we study mathematically the psychological effect of media on the transmission dynamics of contagious diseases [1]. The SIR model based on compartment theory[18] consisting of three compartments: susceptible, infected and recovered with the transmission function as modified Beddington-DeAngelis function including a parameter governing media awareness is considered. The governing differential equations are defined for the dynamical system. The reproduction number 𝑅0, of the model, is calculated using the Jacobian matrix method [12] and is found to depend on m (parameter controlling media) and δ (a measure of inhibition due to awareness of infected). The stability of the dynamical system at the equilibrium points is discussed. The numerical solution is obtained by varying the introduced parameters of the above-said function and analysed graphically.

Download Full-text

Mathematical modelling of pneumonia dynamics of children under the age of five

10.21203/rs.3.rs-194578/v1 ◽

2021 ◽

Author(s):

Idowu Kabir Oluwatobi ◽

Erinle-Ibrahim L.M

Keyword(s):

Mathematical Modelling ◽

Reproduction Number ◽

Equilibrium Points ◽

Control Measures ◽

Transmission Dynamics ◽

Natural Immunity ◽

Paper Work ◽

Effect Of Treatment ◽

The Stability ◽

Disease Free

Abstract This paper work was designed to study the effect of treatment on the transmission of pneumonia infection. When studying the transmission dynamics of infectious diseases with an objective of suggesting control measures, it is important to consider the stability of equilibrium points. In this paper, basic reproduction number, effective reproduction number, existences and stability of the equilibrium point were established.Using Lyaponov function we discovered that the disease free equilibrium is unstable. The results are presented in graphs and it is discovered that the spread of the infection will be greatly affected by the rate of treatment and natural immunity.

Download Full-text

The local stability of a modified multi-strain SIR model for emerging viral strains

PLoS ONE ◽

10.1371/journal.pone.0243408 ◽

2020 ◽

Vol 15 (12) ◽

pp. e0243408

Author(s):

Miguel Fudolig ◽

Reka Howard

Keyword(s):

Local Stability ◽

Reproduction Number ◽

Equilibrium Points ◽

Sir Model ◽

Original Strain ◽

Viral Strain ◽

Sir Epidemic Model ◽

Sir Epidemic ◽

Cross Immunity ◽

Disease Free

We study a novel multi-strain SIR epidemic model with selective immunity by vaccination. A newer strain is made to emerge in the population when a preexisting strain has reached equilbrium. We assume that this newer strain does not exhibit cross-immunity with the original strain, hence those who are vaccinated and recovered from the original strain become susceptible to the newer strain. Recent events involving the COVID-19 virus shows that it is possible for a viral strain to emerge from a population at a time when the influenza virus, a well-known virus with a vaccine readily available, is active in a population. We solved for four different equilibrium points and investigated the conditions for existence and local stability. The reproduction number was also determined for the epidemiological model and found to be consistent with the local stability condition for the disease-free equilibrium.

Download Full-text

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.

Download Full-text

Some Formal Results on Positivity, Stability, and Endemic Steady-State Attainability Based on Linear Algebraic Tools for a Class of Epidemic Models with Eventual Incommensurate Delays

Discrete Dynamics in Nature and Society ◽

10.1155/2019/8959681 ◽

2019 ◽

Vol 2019 ◽

pp. 1-22 ◽

Cited By ~ 6

Author(s):

M. De la Sen ◽

R. Nistal ◽

S. Alonso-Quesada ◽

A. Ibeas

Keyword(s):

Steady State ◽

Reproduction Number ◽

Equilibrium Points ◽

Epidemic Models ◽

Formal Description ◽

Related Research ◽

Formal Study ◽

Complete State ◽

The Stability ◽

Disease Free

A formal description of typical compartmental epidemic models obtained is presented by splitting the state into an infective substate, or infective compartment, and a noninfective substate, or noninfective compartment. A general formal study to obtain the reproduction number and discuss the positivity and stability properties of equilibrium points is proposed and formally discussed. Such a study unifies previous related research and it is based on linear algebraic tools to investigate the positivity and the stability of the linearized dynamics around the disease-free and endemic equilibrium points. To this end, the complete state vector is split into the dynamically coupled infective and noninfective compartments each one containing the corresponding state components. The study is then extended to the case of commensurate internal delays when all the delays are integer multiples of a base delay. Two auxiliary delay-free systems are defined related to the linearization processes around the equilibrium points which correspond to the zero delay, i.e., delay-free, and infinity delay cases. Those auxiliary systems are used to formulate stability and positivity properties independently of the delay sizes. Some examples are discussed to the light of the developed formal study.

Download Full-text