scholarly journals A TWO-PATCH EPIDEMIC MODEL WITH NONLINEAR REINFECTION

2019 ◽  
Vol 27 (1) ◽  
pp. 23-48
Author(s):  
JUAN G. CALVO ◽  
ALBERTO HERNÁNDEZ ◽  
MASON A. PORTER ◽  
FABIO SANCHEZ
Keyword(s):  
Steady State ◽  
Rural Areas ◽  
Reproduction Number ◽  
Steady States ◽  
Population Heterogeneity ◽  
Population Movement ◽  
Disease Dynamics ◽  
Disease Spreading ◽  
Urban And Rural Areas ◽  
Disease Free

The propagation of infectious diseases and its impact on individuals play a major role in disease dynamics, and it is important to incorporate population heterogeneity into efforts to study diseases. As a simplistic but illustrative example, we examine interactions between urban and rural populations on the dynamics of disease spreading. Using a compartmental framework of susceptible–infected susceptible (SIS) dynamics with some level of immunity, we formulate a model that allows non linear reinfection. We investigate the effects of population movement in a simple scenario: a case with two patches, which allows us to model population movement between urban and rural areas. To study the dynamics of the system, we compute a basic reproduction number for each population (urban and rural). We also compute steady states, determine the local stability of the disease-free steady state, and identify conditions for the existence of endemic steady states. From our analysis and computational experiments, we illustrate that population movement plays an important role in disease dynamics. In some cases, it can be rather beneficial, as it can enlarge the region of stability of a disease-free steady state.

Analysis of a mathematical model for the transmission dynamics of human melioidosis

2020 ◽  
Vol 13 (07) ◽  
pp. 2050062
Author(s):  
Yibeltal Adane Terefe ◽  
Semu Mitiku Kassa
Keyword(s):  
Human Population ◽  
Deterministic Model ◽  
Reproduction Number ◽  
Backward Bifurcation ◽  
Endemic Equilibrium ◽  
Transmission Dynamics ◽  
Asymptotically Stable ◽  
Disease Dynamics ◽  
Number Formula ◽  
Disease Free

A deterministic model for the transmission dynamics of melioidosis disease in human population is designed and analyzed. The model is shown to exhibit the phenomenon of backward bifurcation, where a stable disease-free equilibrium co-exists with a stable endemic equilibrium when the basic reproduction number [Formula: see text] is less than one. It is further shown that the backward bifurcation dynamics is caused by the reinfection of individuals who recovered from the disease and relapse. The existence of backward bifurcation implies that bringing down [Formula: see text] to less than unity is not enough for disease eradication. In the absence of backward bifurcation, the global asymptotic stability of the disease-free equilibrium is shown whenever [Formula: see text]. For [Formula: see text], the existence of at least one locally asymptotically stable endemic equilibrium is shown. Sensitivity analysis of the model, using the parameters relevant to the transmission dynamics of the melioidosis disease, is discussed. Numerical experiments are presented to support the theoretical analysis of the model. In the numerical experimentations, it has been observed that screening and treating individuals in the exposed class has a significant impact on the disease dynamics.

scholarly journals On Oscillatory Pattern of Malaria Dynamics in a Population with Temporary Immunity

2007 ◽  
Vol 8 (3) ◽  
pp. 191-203 ◽  
Author(s):  
J. Tumwiine ◽  
J. Y. T. Mugisha ◽  
L. S. Luboobi
Keyword(s):  
Steady State ◽  
Basic Reproduction Number ◽  
Reproduction Number ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
Oscillatory Pattern ◽  
The Stability ◽  
Disease Free ◽  
The Basic Reproduction Number ◽  
Temporary Immunity

We use a model to study the dynamics of malaria in the human and mosquito population to explain the stability patterns of malaria. The model results show that the disease-free equilibrium is globally asymptotically stable and occurs whenever the basic reproduction number,R0is less than unity. We also note that whenR0>1, the disease-free equilibrium is unstable and the endemic equilibrium is stable. Numerical simulations show that recoveries and temporary immunity keep the populations at oscillation patterns and eventually converge to a steady state.

scholarly journals Dynamical Analysis of the SEIB Model for Brucellosis Transmission to the Dairy Cows with Immunological Threshold

Complexity ◽  
2019 ◽  
Vol 2019 ◽  
pp. 1-13
Author(s):  
Wei Zhang ◽  
Juan Zhang ◽  
Yong-Ping Wu ◽  
Li Li
Keyword(s):  
Steady State ◽  
Global Stability ◽  
Dairy Cows ◽  
Immune Cells ◽  
Steady States ◽  
The Body ◽  
Infection Dose ◽  
The One ◽  
Immunological Threshold ◽  
Disease Free

As we all know, bacteria is different from virus which with certain types can be killed by the immune cells in the body. The brucellosis, a bacterial disease, can invade the body by indirect transmission from environment, which has not been researched by combining with immune cells. Considering the effects of immune cells, we put a minimum infection dose of brucellosis invading into the dairy cows as an immunological threshold and get a switch model. In this paper, we accomplish a thorough dynamics analysis of a SEIB switch model. On the one hand, we can get a disease-free and bacteria-free steady state and up to three endemic steady states which may be thoroughly analyzed in different cases of a minimum infection dose in a switch model. On the other hand, we calculate the basic reproduction number R0 and know that the disease-free and bacteria-free steady state is a global stability when R0<1, and the one of the endemic steady state is a conditionally global stability when R0>1. We find that different amounts of R0 may lead to different steady states of brucellosis, and considering the effects of immunology is more serious in mathematics and biology.

scholarly journals Assessing the Potential Impact of Immunity Waning on the Dynamics of COVID-19: An Endemic Model of COVID-19

2021 ◽  
Author(s):  
MUSA RABIU ◽  
Sarafa A. Iyaniwura
Keyword(s):  
Disease Transmission ◽  
Reproduction Number ◽  
Transmission Rate ◽  
Backward Bifurcation ◽  
Disease Dynamics ◽  
Potential Impact ◽  
The Impact ◽  
Induced Immunity ◽  
Disease Free ◽  
Pharmaceutical Interventions

Abstract We developed an endemic model of COVID-19 to assess the impact of vaccination and immunity waning on the dynamics of the disease. Our model exhibits the phenomenon of backward bifurcation and bi-stability, where a stable disease-free equilibrium co-exists with a stable endemic equilibrium. The epidemiological implication of this is that the control reproduction number being less than unity is no longer sufficient to guarantee disease eradication. We showed that this phenomenon could be eliminated by either increasing the vaccine efficacy or by reducing the disease transmission rate (adhering to non-pharmaceutical interventions). Furthermore, we numerically investigated the impacts of vaccination and waning of both vaccine-induced immunity and post-recovery immunity on the disease dynamics. Our simulation results show that the waning of vaccine-induced immunity has more effect on the disease dynamics relative to post-recovery immunity waning, and suggests that more emphasis should be on reducing the waning of vaccine-induced immunity to eradicate COVID-19.

scholarly journals 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

2019 ◽  
Vol 2019 ◽  
pp. 1-22 ◽  
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.

scholarly journals Mathematical modeling of COVID-19 spreading with asymptomatic infected and interacting peoples

2020 ◽  
Author(s):  
Mustapha SERHANI ◽  
Hanane Labbardi
Keyword(s):  
Mathematical Modeling ◽  
Reproduction Number ◽  
Sir Model ◽  
Control Coefficient ◽  
Infected Person ◽  
Containment Control ◽  
Disease Spreading ◽  
The Stability ◽  
Disease Free ◽  
The Basic Reproduction Number

Abstract In this article we propose a modified compartmental (SIR) model describing the transmission of COVID-19 in Morocco. It takes account on the asymptomatic people and the strategies involving hospital isolation of the confirmed infected person, quarantine of people contacting them, and the home containment of all population to restrict mobility. We establish a relationship between the containment control coefficient $c_0$ and the basic reproduction number $\mathcal{R}_0$. Different scenarios are tested with different values of $c_0$, for which the stability of a Disease Free Equilibrium (DFE) point is correlated with the condition linking $\mathcal{R}_0$ and $c_0$. A worst scenario in which the containment is not respected in the same way during the period of confinement leads to several peaks of pandemic. It is shown that the home containment, if lived well, played a crucial role in controlling the disease spreading.

scholarly journals A Mathematical Model for Coinfection of Listeriosis and Anthrax Diseases

2018 ◽  
Vol 2018 ◽  
pp. 1-14 ◽  
Author(s):  
Shaibu Osman ◽  
Oluwole Daniel Makinde
Keyword(s):  
Human Population ◽  
Reproduction Number ◽  
Zoonotic Diseases ◽  
Transmission Dynamics ◽  
Model Parameters ◽  
Contact Rate ◽  
Disease Dynamics ◽  
Human Contact ◽  
The Stability ◽  
Disease Free

Listeriosis and Anthrax are fatal zoonotic diseases caused by Listeria monocytogene and Bacillus Anthracis, respectively. In this paper, we proposed and analysed a compartmental Listeriosis-Anthrax coinfection model describing the transmission dynamics of Listeriosis and Anthrax epidemic in human population using the stability theory of differential equations. Our model revealed that the disease-free equilibrium of the Anthrax model only is locally stable when the basic reproduction number is less than one. Sensitivity analysis was carried out on the model parameters in order to determine their impact on the disease dynamics. Numerical simulation of the coinfection model was carried out and the results are displayed graphically and discussed. We simulate the Listeriosis-Anthrax coinfection model by varying the human contact rate to see its effects on infected Anthrax population, infected Listeriosis population, and Listeriosis-Anthrax coinfected population.

scholarly journals Global Dynamics of an SEIR Model with the Age of Infection and Vaccination

Mathematics ◽  
2021 ◽  
Vol 9 (18) ◽  
pp. 2195
Author(s):  
Huaixing Li ◽  
Jiaoyan Wang
Keyword(s):  
Local Stability ◽  
Reproduction Number ◽  
Steady States ◽  
Global Dynamics ◽  
Seir Model ◽  
Age Of Infection ◽  
The Stability ◽  
Theoretical Results ◽  
Disease Free ◽  
The Basic Reproduction Number

This paper is concerned with the stability of a SEIR (susceptible-exposed-infectious-recovered) model with the age of infection and vaccination. Firstly, we prove the positivity, boundedness, and asymptotic smoothness of the solutions. Next, the existence and local stability of disease-free and endemic steady states are shown. The basic reproduction number R0 is introduced. Furthermore, the global stability of the disease-free and endemic steady states is derived. Numerical simulations are shown to illustrate our theoretical results.

scholarly journals Mathematical Modelling of Tungiasis Disease Dynamics Incorporating Hygiene as a Control Strategy

2019 ◽  
pp. 1-8
Author(s):  
Faith Kabura Mbuthia ◽  
Isaac Chepkwony
Keyword(s):  
Mathematical Modelling ◽  
Control Strategy ◽  
Basic Reproduction Number ◽  
Reproduction Number ◽  
Equilibrium Points ◽  
Asymptotically Stable ◽  
Disease Dynamics ◽  
Intensive Research ◽  
The Disabled ◽  
Disease Free

Tungiasis is a disease that mostly affects the children,the disabled,alcoholics and the aged in Kenya and other parts of the world.Despite the intensive research that has been doneon tungiasis disease,the disease remains a threat in Muranga County.In this research, weformulated a model which is mathematical in nature and derived a system of ordinary differentialequations from it,which we used to study the dynamics of tungiasis disease, incorporating properhygiene as a control measure.The basic reproduction number, R0, is calculated using the nextgeneration matrix.We determined the equilibrium points of the model and also carried out theirstability analysis. From stability, both disease free equilibrium and endemic equilibrium points of the model were found to be locally asymptotically stable when R0 < 1 and R0 > 1 respectively.Numerical simulation of the model carried out showed that effective proper hygiene leads to afaster decrease in the spread of tungiasis.

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.

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