Boundedness and Stability Properties of Solutions of Mathematical Model of Measles.

2021 ◽  
Vol 52 (1) ◽  
pp. 91-112
Author(s):  
Babatunde Sunday Ogundare ◽  
James Akingbade
Keyword(s):  
Mathematical Model ◽  
Asymptotic Stability ◽  
Global Asymptotic Stability ◽  
Endemic Equilibrium ◽  
Basic Reproductive Number ◽  
Reproductive Number ◽  
Jacobian Determinant ◽  
Stability Of Solutions ◽  
Open Population ◽  
Disease Free

In this paper, asymptotic stability and global asymptotic stability of solutions to a deterministic and compartmental mathematical model of measles infection is considered using the ideas of the Jacobian determinant as well as the second method of Lyapunov, criteria/conditions that guaranteed asymptotic stability of disease free equilibrium and endemic equilibrium were established. Also the basic reproductive number $R_0$ was obtained. The results in this work compliments existing work and provided further information in controlling the disease in an open population.

A MODEL FOR VECTOR TRANSMITTED DISEASES WITH SATURATION INCIDENCE

2001 ◽  
Vol 09 (04) ◽  
pp. 235-245 ◽  
Author(s):  
LOURDES ESTEVA ◽  
MARIANO MATIAS
Keyword(s):  
Endemic Equilibrium ◽  
Saturation Effect ◽  
Basic Reproductive Number ◽  
Infected Host ◽  
Reproductive Number ◽  
The Stability ◽  
Stability Results ◽  
Disease Free

A model for a disease that is transmitted by vectors is formulated. All newborns are assumed susceptible, and human and vector populations are assumed to be constant. The model assumes a saturation effect in the incidences due to the response of the vector to change in the susceptible and infected host densities. Stability of the disease free equilibrium and existence, uniqueness and stability of the endemic equilibrium is investigated. The stability results are given in terms of the basic reproductive number R0.

scholarly journals Dynamical Behavior of a New Epidemiological Model

2014 ◽  
Vol 2014 ◽  
pp. 1-9
Author(s):  
Zizi Wang ◽  
Zhiming Guo
Keyword(s):  
Time Delay ◽  
Dynamical Behavior ◽  
Endemic Equilibrium ◽  
Epidemiological Model ◽  
Basic Reproductive Number ◽  
Reproductive Number ◽  
Sufficient Condition ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
Disease Free

A new epidemiological model is introduced with nonlinear incidence, in which the infected disease may lose infectiousness and then evolves to a chronic noninfectious disease when the infected disease has not been cured for a certain timeτ. The existence, uniqueness, and stability of the disease-free equilibrium and endemic equilibrium are discussed. The basic reproductive numberR0is given. The model is studied in two cases: with and without time delay. For the model without time delay, the disease-free equilibrium is globally asymptotically stable provided thatR0≤1; ifR0>1, then there exists a unique endemic equilibrium, and it is globally asymptotically stable. For the model with time delay, a sufficient condition is given to ensure that the disease-free equilibrium is locally asymptotically stable. Hopf bifurcation in endemic equilibrium with respect to the timeτis also addressed.

MODELING THE SCHISTOSOMIASIS ON THE ISLETS IN NANJING

2012 ◽  
Vol 05 (04) ◽  
pp. 1250037 ◽  
Author(s):  
LONGXING QI ◽  
JING-AN CUI ◽  
YUAN GAO ◽  
HUAIPING ZHU
Keyword(s):  
Differential Equations ◽  
Global Stability ◽  
Field Study ◽  
Rattus Norvegicus ◽  
Compartmental Model ◽  
Endemic Equilibrium ◽  
Basic Reproductive Number ◽  
Reproductive Number ◽  
Parasitic Diseases ◽  
Disease Free

A compartmental model is established for schistosomiasis infection in Qianzhou and Zimuzhou, two islets in the center of Yangtzi River near Nanjing, P. R. China. The model consists of five differential equations about the susceptible and infected subpopulations of mammalian Rattus norvegicus and Oncomelania snails. We calculate the basic reproductive number R0 and discuss the global stability of the disease free equilibrium and the unique endemic equilibrium when it exists. The dynamics of the model can be characterized in terms of the basic reproductive number. The parameters in the model are estimated based on the data from the field study of the Nanjing Institute of Parasitic Diseases. Our analysis shows that in a natural isolated area where schistosomiasis is endemic, killing snails is more effective than killing Rattus norvegicus for the control of schistosomiasis.

GLOBAL STABILITY OF AN HIV/AIDS MODEL WITH STOCHASTIC PERTURBATION

2011 ◽  
Vol 04 (02) ◽  
pp. 349-358 ◽  
Author(s):  
Junyuan Yang ◽  
Xiaoyan Wang ◽  
Xuezhi Li
Keyword(s):  
Stochastic Perturbation ◽  
Endemic Equilibrium ◽  
Basic Reproductive Number ◽  
Reproductive Number ◽  
Asymptotically Stable ◽  
Ode Model ◽  
Globally Asymptotically Stable ◽  
Stochastic Perturbations ◽  
Hiv Model ◽  
Disease Free

In this paper, we investigate the dynamic behavior of an HIV model with stochastic perturbation. Firstly, in ODE model, the disease-free equilibrium E0 is globally asymptotically stable if the basic reproductive number R0 < 1. When R0 > 1, the endemic equilibrium E* is globally asymptotically stable. Secondly, the criterion for robustness of the system is established under stochastic perturbations. The conditions of stochastic stability of the endemic equilibrium E* are obtained. Finally, we simulate our analytical results.

scholarly journals SEIPR-Mathematical Model of the Pneumonia Spreading in Toddlers with Immunization and Treatment Effects

2020 ◽  
Vol 17 (2) ◽  
pp. 202-218
Author(s):  
Rusniwati S. Imran ◽  
Resmawan Resmawan ◽  
Novianita Achmad ◽  
Agusyarif Rezka Nuha
Keyword(s):  
Mathematical Model ◽  
Equilibrium Point ◽  
Equilibrium Points ◽  
Treatment Success ◽  
Stable State ◽  
Threshold Value ◽  
Endemic Equilibrium ◽  
Basic Reproductive Number ◽  
Disease Spread ◽  
Reproductive Number

This research discussed the SEIPR mathematical model on the spread of pneumonia among children under five years old. The development of the model was done by considering factors of immunization and treatment factors, in an effort to reduce the rate of spread of pneumonia. In this research, mathematical model construction, stability analysis, and numerical simulation were carried out to see the dynamics of pneumonia cases in the population. The model analysis produces two equilibrium points, which are the equilibrium point without the disease, the endemic equilibrium point, and the basic reproduction number ( ) as the threshold value for disease spread. The point of equilibrium without disease reaches a stable state at the moment , which indicates that pneumonia will disappear from the population, while the endemic equilibrium point reaches a stable state at that time , which indicates that the disease will spread in the population. Furthermore, numerical simulations show that increasing the rate parameters of infected individuals undergoing treatment ( ), the treatment success rate ( ), and the immunization proportion ( ), could suppress the basic reproductive number so that control of the disease spread rate can be accelerated.

scholarly journals The Impact of Vaccination on Covid-19 Disease Transmission Patterns in a Human Population: A Theoretical Analysis

2021 ◽  
pp. 1-14
Author(s):  
A. B. Okrinya ◽  
C. N. Timinibife
Keyword(s):  
Mathematical Model ◽  
Disease Transmission ◽  
Human Population ◽  
Index Case ◽  
Basic Reproductive Number ◽  
Reproductive Number ◽  
Disease Eradication ◽  
Free Region ◽  
The Impact ◽  
Disease Free

We construct a Mathematical model that describes the effect of vaccination on the dynamics of the transmission of COVID-19 disease in a human population. The model is a system of ordinary differential equations that describes the evolution of humans in a range of Covid-19 states due to emergence of an index case in a disease free region. The analysis of the model shows that effective vaccination can lead to disease eradication, where in the disease free state is locally asymptomatically stable if the basic reproductive number, and unstable when The numerical simulations suggests the use of other social measures alongside  vaccination in order to avert the possibility of the disease  becoming endemic.

scholarly journals Stability, Bifurcation, and a Pair of Conserved Quantities in a Simple Epidemic System with Reinfection for the Spread of Diseases Caused by Coronaviruses

2021 ◽  
Vol 2021 ◽  
pp. 1-18
Author(s):  
Jorge Fernando Camacho ◽  
Cruz Vargas-De-León
Keyword(s):  
Numerical Solutions ◽  
Endemic Equilibrium ◽  
Basic Reproductive Number ◽  
Reproductive Number ◽  
Nonlinear Ordinary Differential Equations ◽  
Conserved Quantities ◽  
Asymptotically Stable ◽  
Infinite Line ◽  
Disease Free ◽  
Significant Mortality

In this paper, we study a modified SIRI model without vital dynamics, based on a system of nonlinear ordinary differential equations, for epidemics that exhibit partial immunity after infection, reinfection, and disease-induced death. This model can be applied to study epidemics caused by SARS-CoV, MERS-CoV, and SARS-CoV-2 coronaviruses, since there is the possibility that, in diseases caused by these pathogens, individuals recovered from the infection have a decrease in their immunity and can be reinfected. On the other hand, it is known that, in populations infected by these coronaviruses, individuals with comorbidities or older people have significant mortality rates or deaths induced by the disease. By means of qualitative methods, we prove that such system has an endemic equilibrium and an infinite line of nonhyperbolic disease-free equilibria, we determine the local and global stability of these equilibria, and we also show that it has no periodic orbits. Furthermore, we calculate the basic reproductive number R 0 and find that the system exhibits a forward bifurcation: disease-free equilibria are stable when R 0 < 1 / σ and unstable when R 0 > 1 / σ , while the endemic equilibrium consist of an asymptotically stable upper branch that appears from R 0 > 1 / σ , σ being the rate that quantifies reinfection. We also show that this system has two conserved quantities. Additionally, we show some of the most representative numerical solutions of this system.

scholarly journals A TB Model with Infectivity in Latent Period and Imperfect Treatment

2012 ◽  
Vol 2012 ◽  
pp. 1-19 ◽  
Author(s):  
Juan Wang ◽  
Sha-Sha Gao ◽  
Xue-Zhi Li
Keyword(s):  
Asymptotic Stability ◽  
Numerical Simulations ◽  
Latent Period ◽  
Global Asymptotic Stability ◽  
Endemic Equilibrium ◽  
Epidemiological Model ◽  
Free State ◽  
Local Asymptotic Stability ◽  
Sustained Oscillations ◽  
Disease Free

An epidemiological model of TB with infectivity in latent period and imperfect treatment is introduced. As presented, sustained oscillations are not possible and the endemic proportions either approach the disease-free equilibrium or an endemic equilibrium. The expanded model that stratified the infectious individuals according to their time-since-infectionθis also carried out. The global asymptotic stability of the infection-free state is established as well as local asymptotic stability of the endemic equilibrium. At the end, numerical simulations are presented to illustrate the results.

scholarly journals Modelling and Simulating a Transmission of COVID-19 Disease: Niger Republic Case

2020 ◽  
Vol 13 (3) ◽  
pp. 549-566
Author(s):  
Abba Mahamane Oumarou ◽  
Saley Bisso
Keyword(s):  
Matrix Method ◽  
Equilibrium Points ◽  
Endemic Equilibrium ◽  
Basic Reproductive Number ◽  
Reproductive Number ◽  
Contact Rate ◽  
Asymptotically Stable ◽  
Next Generation Matrix ◽  
The Impact ◽  
Disease Free

This paper focuses on the dynamics of spreads of a coronavirus disease (Covid-19).Through this paper, we study the impact of a contact rate in the transmission of the disease. We determine the basic reproductive number R0, by using the next generation matrix method. We also determine the Disease Free Equilibrium and Endemic Equilibrium points of our model. We prove that the Disease Free Equilibrium is asymptotically stable if R0 < 1 and unstable if R0 > 1. The asymptotical stability of Endemic Equilibrium is also establish. Numerical simulations are made to show the impact of contact rate in the spread of disease.

scholarly journals Mathematical Analysis and Stochastic Stability of Nonlinear Epidemic Model with Incidence Rate

2020 ◽  
pp. 8-19
Author(s):  
Laid Chahrazed
Keyword(s):  
Incidence Rate ◽  
Epidemic Model ◽  
Basic Reproductive Number ◽  
Reproductive Number ◽  
Globally Asymptotically Stable ◽  
Saturated Incidence Rate ◽  
Saturated Incidence ◽  
The Stability ◽  
Disease Free ◽  
Temporary Immunity

In this work, we consider a nonlinear epidemic model with temporary immunity and saturated incidence rate. Size N(t) at time t, is divided into three sub classes, with N(t)=S(t)+I(t)+Q(t); where S(t), I(t) and Q(t) denote the sizes of the population susceptible to disease, infectious and quarantine members with the possibility of infection through temporary immunity, respectively. We have made the following contributions: The local stabilities of the infection-free equilibrium and endemic equilibrium are; analyzed, respectively. The stability of a disease-free equilibrium and the existence of other nontrivial equilibria can be determine by the ratio called the basic reproductive number, This paper study the reduce model with replace S with N, which does not have non-trivial periodic orbits with conditions. The endemic -disease point is globally asymptotically stable if R0 ˃1; and study some proprieties of equilibrium with theorems under some conditions. Finally the stochastic stabilities with the proof of some theorems. In this work, we have used the different references cited in different studies and especially the writing of the non-linear epidemic mathematical model with [1-7]. We have used the other references for the study the different stability and other sections with [8-26]; and sometimes the previous references.

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