scholarly journals Two new compartmental epidemiological models and their equilibria

2021 ◽  
Author(s):  
Jonas Balisacan ◽  
Monique Chyba ◽  
Corey Shanbrom
Keyword(s):  
Compartmental Model ◽  
Herd Immunity ◽  
Basic Reproductive Number ◽  
Reproductive Number ◽  
Mathematical Epidemiology ◽  
Epidemiological Models ◽  
Classical Models ◽  
Asymptomatic Individuals ◽  
The Stability ◽  
Disease Free

Compartmental models have long served as important tools in mathematical epidemiology, with their usefulness highlighted by the recent COVID-19 pandemic. However, most of the classical models fail to account for certain features of this disease and others like it, such as the ability of exposed individuals to recover without becoming infectious, or the possibility that asymptomatic individuals can indeed transmit the disease but at a lesser rate than the symptomatic. Furthermore, the rise of new disease variants and the imperfection of vaccines suggest that concept of endemic equilibrium is perhaps more pertinent than that of herd immunity. Here we propose a new compartmental epidemiological model and study its equilibria, characterizing the stability of both the endemic and disease-free equilibria in terms of the basic reproductive number. Moreover, we introduce a second compartmental model, generalizing our first, which accounts for vaccinated individuals, and begin an analysis of its equilibria.

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.

scholarly journals TRANSMISSION DYNAMICS OF DENGUE IN COSTA RICA: THE ROLE OF HOSPITALIZATIONS

2019 ◽  
Vol 27 (1) ◽  
pp. 241-266
Author(s):  
FABIO SANCHEZ ◽  
JORGE ARROYO-ESQUIVEL ◽  
PAOLA VÁSQUEZ
Keyword(s):  
Costa Rica ◽  
Compartmental Model ◽  
Control Strategies ◽  
Transmission Dynamics ◽  
Basic Reproductive Number ◽  
Reproductive Number ◽  
Specific Parameter ◽  
Public Health Officials ◽  
Disease Free

For decades, dengue virus has caused major problems for public health officials in tropical and subtropical countries around the world. We construct a compartmental model that includes the role of hospitalized individuals in the transmission dynamics of dengue in Costa Rica. The basic reproductive number, R0, is computed, as well as a sensitivity analysis on R0 parameters. The global stability of the disease-free equilibrium is established. Numerical simulations under specific parameter scenarios are performed to determine optimal prevention/control strategies.

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.

SVIR epidemic model with age structure in susceptibility, vaccination effects and relapse

2017 ◽  
Vol 82 (5) ◽  
pp. 945-970 ◽  
Author(s):  
Jinliang Wang ◽  
Min Guo ◽  
Shengqiang Liu
Keyword(s):  
Age Structure ◽  
Epidemic Model ◽  
Lyapunov Functional ◽  
Basic Reproductive Number ◽  
Reproductive Number ◽  
Combined Effects ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
The Stability ◽  
Disease Free

Abstract An SVIR epidemic model with continuous age structure in the susceptibility, vaccination effects and relapse is proposed. The asymptotic smoothness, existence of a global attractor, the stability of equilibria and persistence are addressed. It is shown that if the basic reproductive number $\Re_0<1$, then the disease-free equilibrium is globally asymptotically stable. If $\Re_0>1$, the disease is uniformly persistent, and a Lyapunov functional is used to show that the unique endemic equilibrium is globally asymptotically stable. Combined effects of susceptibility age, vaccination age and relapse age on the basic reproductive number are discussed.

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.

scholarly journals An SEIRS Epidemic Model with Immigration and Vertical Transmission

2020 ◽  
pp. 48-53
Author(s):  
Ruksana Shaikh ◽  
Pradeep Porwal ◽  
V. K. Gupta
Keyword(s):  
Vertical Transmission ◽  
Epidemic Model ◽  
Equilibrium Points ◽  
Basic Reproductive Number ◽  
Reproductive Number ◽  
Asymptotically Stable ◽  
Seirs Epidemic Model ◽  
Model Equations ◽  
The Stability ◽  
Disease Free

The study indicates that we should improve the model by introducing the immigration rate in the model to control the spread of disease. An SEIRS epidemic model with Immigration and Vertical Transmission and analyzed the steady state and stability of the equilibrium points. The model equations were solved analytically. The stability of the both equilibrium are proved by Routh-Hurwitz criteria. We see that if the basic reproductive number R0<1 then the disease free equilibrium is locally asymptotically stable and if R0<1 the endemic equilibrium will be locally asymptotically stable.

scholarly journals Dynamical Analysis of a Delayed HIV Virus Dynamic Model with Cell-to-Cell Transmission and Apoptosis of Bystander Cells

Complexity ◽  
2020 ◽  
Vol 2020 ◽  
pp. 1-12
Author(s):  
Tongqian Zhang ◽  
Junling Wang ◽  
Yi Song ◽  
Zhichao Jiang
Keyword(s):  
Time Delay ◽  
Basic Reproductive Number ◽  
Dynamical Analysis ◽  
Reproductive Number ◽  
Globally Asymptotically Stable ◽  
Local Hopf Bifurcation ◽  
Hiv Virus ◽  
Bystander Cells ◽  
The Stability ◽  
Disease Free

In this paper, a delayed viral dynamical model that considers two different transmission methods of the virus and apoptosis of bystander cells is proposed and investigated. The basic reproductive number R0 of the model is derived. Based on the basic reproductive number, we prove that the disease-free equilibrium E0 is globally asymptotically stable for R0<1 by constructing suitable Lyapunov functional. For R0>1, by regarding the time delay as bifurcation parameter, the existence of local Hopf bifurcation is investigated. The results show that time delay can change the stability of endemic equilibrium and cause periodic oscillations. Finally, we give some numerical simulations to illustrate the theoretical findings.

scholarly journals Effect of Rainfall for the Dynamical Transmission Model of the Dengue Disease in Thailand

2017 ◽  
Vol 2017 ◽  
pp. 1-17 ◽  
Author(s):  
Pratchaya Chanprasopchai ◽  
Puntani Pongsumpun ◽  
I. Ming Tang
Keyword(s):  
Equilibrium State ◽  
Basic Reproductive Number ◽  
Transmission Model ◽  
Reproductive Number ◽  
Dengue Disease ◽  
The Stability ◽  
Disease Free ◽  
Endemic Equilibrium State ◽  
Stability Of The Solution

The SEIR (Susceptible-Exposed-Infected-Recovered) model is used to describe the transmission of dengue virus. The main contribution is determining the role of the rainfall in Thailand in the model. The transmission of dengue disease is assumed to depend on the nature of the rainfall in Thailand. We analyze the dynamic transmission of dengue disease. The stability of the solution of the model is analyzed. It is investigated by using the Routh-Hurwitz criteria. We find two equilibrium states: a disease-free state and an endemic equilibrium state. The basic reproductive number (R0) is obtained, which indicates the stability of each equilibrium state. Numerical results taking into account the rainfall are obtained and they are seen to correspond to the analytical results.

scholarly journals A Stochastic TB Model for a Crowded Environment

2018 ◽  
Vol 2018 ◽  
pp. 1-8 ◽  
Author(s):  
Sibaliwe Maku Vyambwera ◽  
Peter Witbooi
Keyword(s):  
Compartmental Model ◽  
Deterministic Model ◽  
Reproduction Number ◽  
Stochastic Perturbation ◽  
Equation Model ◽  
Ordinary Differential Equation Model ◽  
Differential Equation Model ◽  
The Stability ◽  
Crowded Environments ◽  
Disease Free

We propose a stochastic compartmental model for the population dynamics of tuberculosis. The model is applicable to crowded environments such as for people in high density camps or in prisons. We start off with a known ordinary differential equation model, and we impose stochastic perturbation. We prove the existence and uniqueness of positive solutions of a stochastic model. We introduce an invariant generalizing the basic reproduction number and prove the stability of the disease-free equilibrium when it is below unity or slightly higher than unity and the perturbation is small. Our main theorem implies that the stochastic perturbation enhances stability of the disease-free equilibrium of the underlying deterministic model. Finally, we perform some simulations to illustrate the analytical findings and the utility of the model.

scholarly journals Estimation of the probability of reinfection with COVID-19 coronavirus by the SEIRUS model

2020 ◽  
Author(s):  
Victor Alexander Okhuese
Keyword(s):  
Recovery Rate ◽  
Death Rate ◽  
Basic Reproductive Number ◽  
Reproductive Number ◽  
Chain Reaction ◽  
Health Community ◽  
Polymerase Chain ◽  
Model Equations ◽  
Disease Free ◽  
Pcr Test

AbstractWith sensitivity of the Polymerase Chain Reaction (PCR) test used to detect the presence of the virus in the human host, the global health community has been able to record a great number of recovered population. Therefore, in a bid to answer a burning question of reinfection in the recovered class, the model equations which exhibits the disease-free equilibrium (E0) state for COVID-19 coronavirus was developed in this study and was discovered to both exist as well as satisfy the criteria for a locally or globally asymptotic stability with a basic reproductive number R0 = 0 for and endemic situation. Hence, there is a chance of no secondary reinfections from the recovered population as the rate of incidence of the recovered population vanishes, that is, B = 0.Furthermore, numerical simulations were carried to complement the analytical results in investigating the effect of the implementation of quarantine and observatory procedures has on the projection of the further spread of the virus globally. Result shows that the proportion of infected population in the absence of curative vaccination will continue to grow globally meanwhile the recovery rate will continue slowly which therefore means that the ratio of infection to recovery rate will determine the death rate that is recorded globally and most significant for this study is the rate of reinfection by the recovered population which will decline to zero over time as the virus is cleared clinically from the system of the recovered class.

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