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

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.

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Parameter Estimation and Sensitivity Analysis of Dysentery Diarrhea Epidemic Model

Journal of Applied Mathematics ◽

10.1155/2019/8465747 ◽

2019 ◽

Vol 2019 ◽

pp. 1-13 ◽

Cited By ~ 10

Author(s):

Hailay Weldegiorgis Berhe ◽

Oluwole Daniel Makinde ◽

David Mwangi Theuri

Keyword(s):

Sensitivity Analysis ◽

Reproduction Number ◽

Transmission Rate ◽

Nonlinear Least Squares ◽

Model Parameters ◽

Nonlinear Least Squares Problem ◽

The Stability ◽

Diarrhea Disease ◽

Effective Transmission ◽

Disease Free

In this paper, dysentery diarrhea deterministic compartmental model is proposed. The local and global stability of the disease-free equilibrium is obtained using the stability theory of differential equations. Numerical simulation of the system shows that the backward bifurcation of the endemic equilibrium exists for R0>1. The system is formulated as a standard nonlinear least squares problem to estimate the parameters. The estimated reproduction number, based on the dysentery diarrhea disease data for Ethiopia in 2017, is R0=1.1208. This suggests that elimination of the dysentery disease from Ethiopia is not practical. A graphical method is used to validate the model. Sensitivity analysis is carried out to determine the importance of model parameters in the disease dynamics. It is found out that the reproduction number is the most sensitive to the effective transmission rate of dysentery diarrhea (βh). It is also demonstrated that control of the effective transmission rate is essential to stop the spreading of the disease.

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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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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

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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.

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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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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.

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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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Modeling the impact of temperature on fractional order dengue model with vertical transmission

An International Journal of Optimization and Control Theories & Applications (IJOCTA) ◽

10.11121/ijocta.01.2020.00862 ◽

2020 ◽

Vol 10 (1) ◽

pp. 85-93

Author(s):

Ozlem Defterli

Keyword(s):

Fractional Order ◽

Reproduction Number ◽

Equilibrium Points ◽

Effect Of Temperature ◽

Dengue Disease ◽

Host Interaction ◽

The Stability ◽

The Impact ◽

Disease Free ◽

Dengue Epidemic

A dengue epidemic model with fractional order derivative is formulated to investigate the effect of temperature on the spread of the vector-host transmitted dengue disease. The model consists of system of fractional order differential equations formulated within Caputo fractional operator. The stability of the equilibrium points of the considered dengue model is studied. The corresponding basic reproduction number R_0 is derived and it is proved that if R_0 < 1, the disease-free equilibrium (DFE) is locally asymptotically stable. L1 method is applied to solve the dengue model numerically. Finally, numerical simulations are also presented to illustrate the analytical results showing the influence of thetemperature on the dynamics of the vector-host interaction in dengue epidemics.

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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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Dynamical Transmission and Mathematical Analysis of SIR Measles Epidemiological Model

International Journal of Diabetes & Metabolic Disorders ◽

10.33140/ijdmd/03/03/00002 ◽

2018 ◽

Vol 3 (3) ◽

Keyword(s):

Sensitivity Analysis ◽

Reproduction Number ◽

Threshold Value ◽

Qualitative Behavior ◽

Epidemic Disease ◽

Nonstandard Finite Difference Scheme ◽

The Stability ◽

Disease Free ◽

Nonstandard Finite Difference ◽

The Basic Reproduction Number

In this work, we presented a nonlinear time fractional model of measles in order to understand the outbreaks of this epidemic disease. The stability analysis and sensitivity analysis of the model is provided and the certain threshold value of the basic reproduction number R0 >1, disease free and endemic equilibrium point of the model is also calculated. We develop unconditionally convergent nonstandard finite difference scheme by applying Mickens approach Ø(h)=h+O(h2 ). This method proved to be very efficient technique for solving epidemic models to control the infection diseases. We also discussed the qualitative behavior of the model and numerical simulations are carried out to support the analytic results.

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