scholarly journals Basic reinfection number and backward bifurcation

2021 ◽  
Vol 18 (6) ◽  
pp. 8064-8083
Author(s):  
Baojun Song ◽  
Keyword(s):  
Disease Control ◽  
Basic Reproduction Number ◽  
Reproduction Number ◽  
Backward Bifurcation ◽  
Epidemiological Models ◽  
Super Infection ◽  
Exogenous Reinfection ◽  
The Basic Reproduction Number

<abstract><p>Some epidemiological models exhibit bi-stable dynamics even when the basic reproduction number $ {{{\cal R}_{0}}} $ is below $ 1 $, through a phenomenon known as a backward bifurcation. Causes for this phenomenon include exogenous reinfection, super-infection, relapse, vaccination exercises, heterogeneity among subpopulations, etc. To measure the reinfection forces, this paper defines a second threshold: the basic reinfection number. This number characterizes the type of bifurcation when the basic reproduction number is equal to one. If the basic reinfection number is greater than one, the bifurcation is backward. Otherwise it is forward. The basic reinfection number with the basic reproduction number together gives a complete measure for disease control whenever reinfections (or relapses) matter. We formulate the basic reinfection number for a variety of epidemiological models.</p></abstract>

scholarly journals APLIKASI MODEL EPIDEMIK SEIAR-SEI PADA PENYEBARAN PENYAKIT MALARIA DENGAN INFEKSI ASIMTOMATIK DAN SUPER INFEKSI

2018 ◽  
Vol 15 (2) ◽  
pp. 67
Author(s):  
Stella Maryana Belwawin
Keyword(s):  
Equilibrium Point ◽  
Basic Reproduction Number ◽  
Reproduction Number ◽  
Asymptomatic Infection ◽  
Endemic Equilibrium ◽  
Asymptomatic Infections ◽  
The Stability ◽  
Super Infection ◽  
The Basic Reproduction Number

AbstractThis aim of this study is to determine the point of equilibrium and analyze the stability of SEIAR-SEI model on malaria disease with asymptomatic infection, super infection and the effect of the mosquito's life cycle. This study also aim is to measure the sensitivity of the spread of malaria to the parameters of asymptomatic infections, the rate of treatment, and the rate of birth of mosquitoes through the magnitude of . The method in this research is deductively, through several stage, such as  determination of disease-free equilibrium point and endemic equilibrium point, determination of basic reproduction number (), analyze of the basic reproduction number sensitivity of the spread of malaria to the parameters of asymptomatic infections, the rate of treatment, and the rate of birth of mosquitoes. The endemic equilibrium point was obtained using rule of Descartes. The result show that the change in the value of parameter , , and  has effect on the basic reproduction number (). Treatment factors in the human population influence the elimination of malaria in a population. Whereas asymptomatic infection factors and the birth rate of adult mosquitoes influence the increase in malaria infection. Keywords:  Malaria, asymptomatic infection, super infection, basic reproduction number, rule of descrates. AbstrakPenelitian ini bertujuan menentukan titik keseimbangan dan menganalisis kestabilan dari model SEIAR_SEI pada penyakit malaria dengan pengaruh infeksi asimtomatik, super infeksi, dan siklus hidup nyamuk. Penelitian ini juga bertujuan mengukur tingkat sensitivitas penyebaran penyakit malaria terhadap parameter infeksi asimtomatik, laju pengobatan, serta laju kelahiran nyamuk.melalu besaran .  Metode yang digunakan dalam penelitian ini adalah metode deduktif dengan langkah-langkah : menentukan titik keseimbangan bebas penyakit dan endemik dan menentukan bilangan reproduksi dasar ). Analisis sensitivitas bilangan reproduksi dasar dilakukan terhadap parameter infeksi asimtomatik, pengobatan, dan laju kelahiran nyamuk. Tititk keseimbangan endemik diperoleh dengan aturan descrates. Hasil yang diperoleh menunjukkan parameter , , dan  berpengaruh terhadap bilangan reproduksi dasar (). Faktor pengobatan berpengaruh terhadap eliminasi penyakit malaria. Sedangkan faktor infeksi asimtomatik dan laju kelahiran nyamuk dewasa berpengaruh terhadap peningkatan infeksi penyakit malaria. Kata kunci: Malaria, Infeksi Asimtomatik, Super Infeksi, Bilangan Reproduksi Dasar, Aturan Descrates . 

The basic reproduction number

2012 ◽  
Author(s):  
Odo Diekmann ◽  
Hans Heesterbeek ◽  
Tom Britton
Keyword(s):  
Infectious Disease ◽  
Basic Reproduction Number ◽  
Reproduction Number ◽  
Infected Individual ◽  
Vital Role ◽  
Point Of View ◽  
Epidemiological Models ◽  
Disease Epidemiology ◽  
Infectious Disease Epidemiology ◽  
The Basic Reproduction Number

The basic reproduction number (or ratio) R₀ is arguably the most important quantity in infectious disease epidemiology. It is among the quantities most urgently estimated for infectious diseases in outbreak situations, and its value provides insight when designing control interventions for established infections. From a theoretical point of view R₀ plays a vital role in the analysis of, and consequent insight from, infectious disease models. There is hardly a paper on dynamic epidemiological models in the literature where R₀ does not play a role. R₀ is defined as the average number of new cases of an infection caused by one typical infected individual, in a population consisting of susceptibles only. This chapter shows how R₀ can be characterized mathematically and provides detailed examples of its calculation in terms of parameters of epidemiological models, culminating in a set of algorithms (or “recipes”) for the calculation for compartmental epidemic systems.

scholarly journals A Malaria Model with Two Delays

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
Hui Wan ◽  
Jing-an Cui
Keyword(s):  
Basic Reproduction Number ◽  
Time Delays ◽  
Reproduction Number ◽  
Backward Bifurcation ◽  
Transmission Model ◽  
Two Delays ◽  
Malaria Model ◽  
The Basic Reproduction Number

A transmission model of malaria with two delays is formulated. We calculate the basic reproduction numberR0for the model. It is shown that the basic reproduction number is a decreasing function of two time delays. The existence of the equilibria is studied. Our results suggest that the model undergoes a backward bifurcation, which implies that bringing the basic reproduction number below 1 is not enough to eradicate malaria.

scholarly journals Modeling the Dynamics of Coronavirus Disease Pandemic Coupled with Fear Epidemics

2021 ◽  
Vol 2021 ◽  
pp. 1-9
Author(s):  
Saul C. Mpeshe ◽  
Nkuba Nyerere
Keyword(s):  
Disease Control ◽  
Numerical Simulations ◽  
Basic Reproduction Number ◽  
Recovery Rate ◽  
Reproduction Number ◽  
The Third ◽  
Modeling Approach ◽  
Second Wave ◽  
The Basic Reproduction Number ◽  
Initial Transmission

A modeling approach to investigate the dynamics of COVID-19 epidemics coupled with fear is presented in this paper. The basic reproduction number R 0 is computed and employed in analysing the effect of initial transmission and the conditions for disease control or eradication. Numerical simulations show that whenever there is an outbreak coupled with fear, the disease is likely to persist in the first two months, and after that, it will start to slow down as the recovery rate from fear increases. An increase in the number of recovered individuals lead to a rise in the number of susceptibles and consequently set off a second wave of infection in the third month of the epidemic.

scholarly journals Analysis of a Patch Model for the Dynamical Transmission of Echinococcosis

2014 ◽  
Vol 2014 ◽  
pp. 1-13
Author(s):  
Kai Wang ◽  
Xueliang Zhang ◽  
Zhidong Teng ◽  
Lei Wang ◽  
Liping Zhang
Keyword(s):  
Disease Control ◽  
Basic Reproduction Number ◽  
Reproduction Number ◽  
Endemic Equilibrium ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
Patch Model ◽  
Disease Free ◽  
The Basic Reproduction Number

A patch model for echinococcosis due to dogs migration is proposed to explore the effect of dogs migration among patches on the spread of echinococcosis. We firstly define the basic reproduction numberR0. The mathematical results show that the dynamics of the model can be completely determined byR0. IfR0<1, the disease-free equilibrium is globally asymptotically stable. WhenR0>1, the model is permanence and endemic equilibrium is globally asymptotically stable. According to the simulations, it is shown that the larger diffusion of dogs from the lower epidemic areas to the higher prevalence areas can intensify the spread of echinococcosis. However, the larger diffusion of dogs from the higher prevalence areas to the lower epidemic areas can reduce the spread and is beneficial for disease control.

scholarly journals Backward Bifurcation of an Epidemic Model with Infectious Force in Infected and Immune Period and Treatment

2012 ◽  
Vol 2012 ◽  
pp. 1-14 ◽  
Author(s):  
Yakui Xue ◽  
Junfeng Wang
Keyword(s):  
Basic Reproduction Number ◽  
Epidemic Model ◽  
Reproduction Number ◽  
Backward Bifurcation ◽  
Disease Spread ◽  
Treatment Rate ◽  
Stability Of Equilibria ◽  
Existence And Stability ◽  
The Basic Reproduction Number

An epidemic model with infectious force in infected and immune period and treatment rate of infectious individuals is proposed to understand the effect of the capacity for treatment of infective on the disease spread. It is assumed that treatment rate is proportional to the number of infective below the capacity and is constant when the number of infective is greater than the capacity. It is proved that the existence and stability of equilibria for the model is not only related to the basic reproduction number but also the capacity for treatment of infective. It is found that a backward bifurcation occurs if the capacity is small. It is also found that there exist bistable endemic equilibria if the capacity is low.

scholarly journals Tuberculosis Model with Exogenous Reinfection and Treatment

2020 ◽  
pp. 50-65
Author(s):  
Ebenezer Appiagyei ◽  
Mojeeb Al-Rahman El-Nor Osman ◽  
Isaac Kwasi Adu
Keyword(s):  
Global Stability ◽  
Numerical Simulations ◽  
Basic Reproduction Number ◽  
Multiple Equilibria ◽  
Reproduction Number ◽  
Tuberculosis Model ◽  
The Stability ◽  
Disease Free ◽  
Exogenous Reinfection ◽  
The Basic Reproduction Number

The epidemiology of tuberculosis model, which includes exogenous reinfection and treatment, is analyzed. We consider both the diagnosed and the undiagnosed individuals as infectious. Since tuberculosis is treatable, we also included the treatment for diagnosed infective. We presented the basic reproduction number and the stability of the disease-free and endemic equilibria. We also analyzed the model when the exogenous re-infection is set to zero. The exogenous re-infection is shown to be capable of supporting multiple equilibria. Lastly, we presented the global stability and numerical simulations of the model.

Backward Bifurcation in a Cholera Model: A Case Study of Outbreak in Zimbabwe and Haiti

2017 ◽  
Vol 27 (11) ◽  
pp. 1750170
Author(s):  
Sandeep Sharma ◽  
Nitu Kumari
Keyword(s):  
Basic Reproduction Number ◽  
Deterministic Model ◽  
Reproduction Number ◽  
Numerical Study ◽  
Equilibrium Points ◽  
Geometric Approach ◽  
Backward Bifurcation ◽  
Global Dynamics ◽  
Proposed Model ◽  
The Basic Reproduction Number

In this paper, a nonlinear deterministic model is proposed with a saturated treatment function. The expression of the basic reproduction number for the proposed model was obtained. The global dynamics of the proposed model was studied using the basic reproduction number and theory of dynamical systems. It is observed that proposed model exhibits backward bifurcation as multiple endemic equilibrium points exist when [Formula: see text]. The existence of backward bifurcation implies that making [Formula: see text] is not enough for disease eradication. This, in turn, makes it difficult to control the spread of cholera in the community. We also obtain a unique endemic equilibria when [Formula: see text]. The global stability of unique endemic equilibria is performed using the geometric approach. An extensive numerical study is performed to support our analytical results. Finally, we investigate two major cholera outbreaks, Zimbabwe (2008–09) and Haiti (2010), with the help of the present study.

scholarly journals Estimating the basic reproduction number at the beginning of an outbreak under incomplete data

2021 ◽  
Author(s):  
Sawitree Boonpatcharanon ◽  
Jane M Heffernan ◽  
Hanna Jankowski
Keyword(s):  
Basic Reproduction Number ◽  
Incomplete Data ◽  
Reproduction Number ◽  
Respiratory Viruses ◽  
Model Structure ◽  
Epidemiological Models ◽  
Early Stages ◽  
The Basic Reproduction Number

We compare different methods of estimating the basic reproduction number, R0, focusing on the early stages of an epidemic, and considering weekly reports of new infecteds. We study three standard epidemiological models: SIR, SEIR, and SEAIR and examine the sensitivity of the estimators to the model structure. As some methods are developed assuming specific epidemiological models, our work adds a study of their performance in both the well- and miss-specified settings. We focus on parameters matching various types of respiratory viruses, although the general approach is easily extendable to other scenarios.

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