scholarly journals The local stability of a modified multi-strain SIR model for emerging viral strains

PLoS ONE ◽  
2020 ◽  
Vol 15 (12) ◽  
pp. e0243408
Author(s):  
Miguel Fudolig ◽  
Reka Howard
Keyword(s):  
Local Stability ◽  
Reproduction Number ◽  
Equilibrium Points ◽  
Sir Model ◽  
Original Strain ◽  
Viral Strain ◽  
Sir Epidemic Model ◽  
Sir Epidemic ◽  
Cross Immunity ◽  
Disease Free

We study a novel multi-strain SIR epidemic model with selective immunity by vaccination. A newer strain is made to emerge in the population when a preexisting strain has reached equilbrium. We assume that this newer strain does not exhibit cross-immunity with the original strain, hence those who are vaccinated and recovered from the original strain become susceptible to the newer strain. Recent events involving the COVID-19 virus shows that it is possible for a viral strain to emerge from a population at a time when the influenza virus, a well-known virus with a vaccine readily available, is active in a population. We solved for four different equilibrium points and investigated the conditions for existence and local stability. The reproduction number was also determined for the epidemiological model and found to be consistent with the local stability condition for the disease-free equilibrium.

scholarly journals The local stability of a modified multi-strain SIR model for emerging viral strains

2020 ◽  
Author(s):  
Miguel Fudolig ◽  
Reka Howard
Keyword(s):  
Local Stability ◽  
Reproduction Number ◽  
Equilibrium Points ◽  
Sir Model ◽  
Original Strain ◽  
Viral Strain ◽  
Sir Epidemic Model ◽  
Sir Epidemic ◽  
Cross Immunity ◽  
Disease Free

AbstractWe study a novel multi-strain SIR epidemic model with selective immunity by vaccination. A newer strain is made to emerge in the population when a preexisting strain has reached equilbrium. We assume that this newer strain does not exhibit cross-immunity with the original strain, hence those who are vaccinated and recovered from the original strain become susceptible to the newer strain. Recent events involving the COVID-19 virus demonstrates that it is possible for a viral strain to emerge from a population at a time when the influenza virus, a well-known virus with a vaccine readily available for some of its strains, is active in a population. We solved for four different equilibrium points and investigated the conditions for existence and local stability. The reproduction number was also determined for the epidemiological model and found to be consistent with the local stability condition for the disease-free equilibrium.

scholarly journals An SIR epidemic model for COVID-19 spread with fuzzy parameter: the case of Indonesia

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Muhammad Abdy ◽  
Syafruddin Side ◽  
Suwardi Annas ◽  
Wahyuddin Nur ◽  
Wahidah Sanusi
Keyword(s):  
Reproduction Number ◽  
Equilibrium Points ◽  
Sir Model ◽  
Membership Functions ◽  
Sir Epidemic Model ◽  
Sir Epidemic ◽  
Corona Virus ◽  
The Stability ◽  
Fuzzy Parameter ◽  
Fuzzy Parameters

AbstractThe aim of this research is to construct an SIR model for COVID-19 with fuzzy parameters. The SIR model is constructed by considering the factors of vaccination, treatment, obedience in implementing health protocols, and the corona virus-load. Parameters of the infection rate, recovery rate, and death rate due to COVID-19 are constructed as a fuzzy number, and their membership functions are used in the model as fuzzy parameters. The model analysis uses the generation matrix method to obtain the basic reproduction number and the stability of the model’s equilibrium points. Simulation results show that differences in corona virus-loads will also cause differences in the transmission of COVID-19. Likewise, the factors of vaccination and obedience in implementing health protocols have the same effect in slowing or stopping the transmission of COVID-19 in Indonesia.

scholarly journals The Behavior of an SVIR Epidemic Model with Stochastic Perturbation

2014 ◽  
Vol 2014 ◽  
pp. 1-7 ◽  
Author(s):  
Yanan Zhao ◽  
Daqing Jiang
Keyword(s):  
Epidemic Model ◽  
Lyapunov Functions ◽  
Deterministic Model ◽  
Reproduction Number ◽  
Stochastic Perturbation ◽  
Sir Epidemic Model ◽  
Sir Epidemic ◽  
Time Average ◽  
The Difference ◽  
Disease Free

We discuss a stochastic SIR epidemic model with vaccination. We investigate the asymptotic behavior according to the perturbation and the reproduction numberR0. We deduce the globally asymptotic stability of the disease-free equilibrium whenR0≤ 1and the perturbation is small, which means that the disease will die out. WhenR0>1, we derive that the disease will prevail, which is measured through the difference between the solution and the endemic equilibrium of the deterministic model in time average. The key to our analysis is choosing appropriate Lyapunov functions.

scholarly journals Complex mathematical SIR model for spreading of COVID-19 virus with Mittag-Leffler kernel

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
F. Talay Akyildiz ◽  
Fehaid Salem Alshammari
Keyword(s):  
Fractional Order ◽  
Basic Reproduction Number ◽  
Reproduction Number ◽  
Equilibrium Points ◽  
Sir Model ◽  
Endemic Equilibrium ◽  
Asymptotically Stable ◽  
Exponential Kernel ◽  
Disease Free ◽  
The Basic Reproduction Number

AbstractThis paper investigates a new model on coronavirus-19 disease (COVID-19), that is complex fractional SIR epidemic model with a nonstandard nonlinear incidence rate and a recovery, where derivative operator with Mittag-Leffler kernel in the Caputo sense (ABC). The model has two equilibrium points when the basic reproduction number $R_{0} > 1$ R 0 > 1 ; a disease-free equilibrium $E_{0}$ E 0 and a disease endemic equilibrium $E_{1}$ E 1 . The disease-free equilibrium stage is locally and globally asymptotically stable when the basic reproduction number $R_{0} <1$ R 0 < 1 , we show that the endemic equilibrium state is locally asymptotically stable if $R_{0} > 1$ R 0 > 1 . We also prove the existence and uniqueness of the solution for the Atangana–Baleanu SIR model by using a fixed-point method. Since the Atangana–Baleanu fractional derivative gives better precise results to the derivative with exponential kernel because of having fractional order, hence, it is a generalized form of the derivative with exponential kernel. The numerical simulations are explored for various values of the fractional order. Finally, the effect of the ABC fractional-order derivative on suspected and infected individuals carefully is examined and compared with the real data.

scholarly journals Global stability for a discrete SIR epidemic model with delay in the general incidence function

2019 ◽  
Vol 8 (2) ◽  
pp. 32
Author(s):  
Guiro Aboudramane ◽  
Dramane Ouedraogo ◽  
Harouna Ouedraogo
Keyword(s):  
Epidemic Model ◽  
Reproduction Number ◽  
Asymptotically Stable ◽  
Sir Epidemic Model ◽  
Step Size ◽  
Globally Asymptotically Stable ◽  
Incidence Function ◽  
Sir Epidemic ◽  
Disease Free ◽  
Boundedness Of Solution

In this paper, we construct a backward difference scheme for a class of general SIR epidemic model with general incidence function f. We use the step size h > 0, for the discretization. The dynamical properties are investigated (positivity and the boundedness of solution). By constructing the Lyapunov function, under the conditions that function f satisfies some assumptions. The global stabilities of equilibria are obtained. If the basic reproduction number R0<1, the disease-free equilibrium is globally asymptotically stable. If R0>1, the endemic equilibrium is globally asymptotically stable.

scholarly journals Kestabilan Model Epidemi Seir dengan Matriks Hurwitz

2016 ◽  
Vol 11 (2) ◽  
pp. 74
Author(s):  
Roni Tri Putra ◽  
Sukatik - ◽  
Sri Nita
Keyword(s):  
Basic Reproduction Number ◽  
Epidemic Model ◽  
Local Stability ◽  
Reproduction Number ◽  
Equilibrium Points ◽  
Asymptotically Stable ◽  
Stability Of Equilibrium ◽  
Disease Free ◽  
The Basic Reproduction Number

In this paper, it will be studied local stability of equilibrium points of  a SEIR epidemic model with infectious force in latent, infected and immune period. From the model it will be found investigated the existence and its stability of points its equilibrium by Hurwitz matrices. The local stability of equilibrium points is depending on the value of the basic reproduction number  If   the disease free equilibrium is local asymptotically stable.

Effect of pollution on dynamics of SIR model with treatment

2015 ◽  
Vol 08 (06) ◽  
pp. 1550083 ◽  
Author(s):  
Sudipa Chauhan ◽  
Sumit Kaur Bhatia ◽  
Surbhi Gupta
Keyword(s):  
Reproduction Number ◽  
Sir Model ◽  
Backward Bifurcation ◽  
Global Dynamics ◽  
Sir Epidemic Model ◽  
Disease Eradication ◽  
Sir Epidemic ◽  
Local And Global Dynamics ◽  
High Level ◽  
Very High

In this paper, an SIR epidemic model with treatment affected by pollution is proposed. The existence, local and global dynamics of the model are studied. It is shown that backward bifurcation occurs at R0 < 1 and p0 < 1 because of insufficient capacity of treatment. It is also found that due to pollution the number of infective has gone to a very high level. As a result, backward bifurcation occurs for R0 < 1, even when p0 > 1. Further, there exist bistable endemic equilibria for a very low capacity for R0 > 1. Thus, we found that disease can be eradicated for R0 < 1 only by increasing the capacity to a sufficiently high level. Persistence of endemicity of the system is obtained and the mathematical results suggest that the basic reproduction number is insufficient for disease eradication. Numerical simulations are presented to illustrate the results obtained.

Qualitative behavior of a discrete SIR epidemic model

2016 ◽  
Vol 09 (06) ◽  
pp. 1650092 ◽  
Author(s):  
Qamar Din
Keyword(s):  
Asymptotic Stability ◽  
Discrete Time ◽  
Epidemic Model ◽  
Equilibrium Points ◽  
Comparison Method ◽  
Qualitative Behavior ◽  
Theoretical Discussion ◽  
Sir Epidemic Model ◽  
Sir Epidemic ◽  
Disease Free

In this paper, we study the qualitative behavior of a discrete-time epidemic model. More precisely, we investigate equilibrium points, asymptotic stability of both disease-free equilibrium and the endemic equilibrium. Furthermore, by using comparison method, we obtain the global stability of these equilibrium points under certain parametric conditions. Some illustrative examples are provided to support our theoretical discussion.

The asymptotic behavior and ergodicity of stochastically perturbed SVIR epidemic model

2016 ◽  
Vol 09 (03) ◽  
pp. 1650042 ◽  
Author(s):  
Yanan Zhao ◽  
Daqing Jiang
Keyword(s):  
Epidemic Model ◽  
Lyapunov Functions ◽  
Deterministic Model ◽  
Reproduction Number ◽  
Dynamical Behavior ◽  
Standard Technique ◽  
Sir Epidemic Model ◽  
Sir Epidemic ◽  
Number Formula ◽  
Disease Free

In this paper, we introduce stochasticity into an SIR epidemic model with vaccination. The stochasticity in the model is a standard technique in stochastic population modeling. When the perturbations are small, by the method of stochastic Lyapunov functions, we carry out a detailed analysis on the dynamical behavior of the stochastic model regarding of the basic reproduction number [Formula: see text]. If [Formula: see text], the solution of the model is oscillating around a steady state, which is the disease-free equilibrium of the corresponding deterministic model. If [Formula: see text], there is a stationary distribution and the solution has the ergodic property, which means that the disease will prevail.

scholarly journals A Discrete Numerical Solution of The SIR Model with Horizontal and Vertical Transmission

CAUCHY ◽  
2019 ◽  
Vol 6 (1) ◽  
pp. 1
Author(s):  
Trija Fayeldi
Keyword(s):  
Numerical Solution ◽  
Vertical Transmission ◽  
Reproduction Number ◽  
Equilibrium Points ◽  
Sir Model ◽  
Euler Method ◽  
Discrete Version ◽  
Step Size ◽  
The Stability ◽  
Disease Free

The aim of this paper is to is to generalize the SIR model with horizontal and vertical transmission. In this paper, we develop the discrete version of the model. We use Euler method to approximate numerical solution of the model. We found two equilibrium points, that is disease free and endemic equilibrium points. The existence of these points depend on basic reproduction number <em>R</em><sub>0</sub>. We found that if <em>R</em><sub>0</sub> <span style="text-decoration-line: underline;">&lt;</span> 1 then only disease free equilibrium points exists, while both points exists when <em>R</em><sub>0</sub> &gt; 1. We also found that the stability of these equilibrium points depend on the value of step-size <em>h</em>. Some numerical experiments were presented as illustration.

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