Bifurcation Analysis of a Diffusive SIR Model with Saturated Treatment in a Heterogeneous Environment

In this paper, we propose a diffusive SIR model with general incidence rate, saturated treatment rate and spatially heterogeneous diffusion coefficients. We first prove the global existence of bounded solutions for the model and compute the basic reproduction number. We study the local and global stabilities of the disease-free equilibrium and the uniform persistence. In the case when the diffusion rate of infected individuals is constant, we carry out a bifurcation analysis of equilibria by considering the maximal treatment rate as the bifurcation parameter. Finally, we perform some numerical simulations, which show that the solutions to our model present periodic oscillations for certain values of the parameters.

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Qualitative analysis and optimal control of an SIR model with logistic growth, non-monotonic incidence and saturated treatment

Mathematical Modelling of Natural Phenomena ◽

10.1051/mmnp/2021004 ◽

2021 ◽

Author(s):

Jayanta Kumar Ghosh ◽

Prahlad Majumdar ◽

Uttam Ghosh

Keyword(s):

Optimal Control ◽

Basic Reproduction Number ◽

Reproduction Number ◽

Transmission Rate ◽

Sir Model ◽

Logistic Growth ◽

Treatment Rate ◽

Globally Asymptotically Stable ◽

Susceptible Population ◽

Saturated Treatment

This paper describes an SIR model with logistic growth rate of susceptible population, non-monotonic incidence rate and saturated treatment rate. The existence and stability analysis of equilibria have been investigated. It has been shown that the disease free equilibrium point ( DFE ) is globally asymptotically stable if the basic reproduction number is less than unity and the transmission rate of infection less than some threshold. The system exhibits the transcritical bifurcation at DFE with respect to the cure rate. We have also found the condition for occurring the backward bifurcation, which implies the value of basic reproduction number less than unity is not enough to eradicate the disease. Stability or instability of different endemic equilibria has been shown analytically. The system also experiences the saddle-node and Hopf bifurcation. The existence of Bogdanov - Takens bifurcation ( BT ) of co-dimension 2 has been investigated which has also been shown through numerical simulations. Here we have used two control functions, one is vaccination control and other is treatment control. We have solved the optimal control problem both analytically and numerically. Finally, the efficiency analysis has been used to determine the best control strategy among vaccination and treatment.

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Seasonality Impact on the Transmission Dynamics of Tuberculosis

Computational and Mathematical Methods in Medicine ◽

10.1155/2016/8713924 ◽

2016 ◽

Vol 2016 ◽

pp. 1-12 ◽

Cited By ~ 2

Author(s):

Yali Yang ◽

Chenping Guo ◽

Luju Liu ◽

Tianhua Zhang ◽

Weiping Liu

Keyword(s):

Basic Reproduction Number ◽

Reproduction Number ◽

Autonomous Systems ◽

Positive Periodic Solution ◽

Transmission Dynamics ◽

Uniform Persistence ◽

Globally Asymptotically Stable ◽

The Impact ◽

Disease Free ◽

The Basic Reproduction Number

The statistical data of monthly pulmonary tuberculosis (TB) incidence cases from January 2004 to December 2012 show the seasonality fluctuations in Shaanxi of China. A seasonality TB epidemic model with periodic varying contact rate, reactivation rate, and disease-induced death rate is proposed to explore the impact of seasonality on the transmission dynamics of TB. Simulations show that the basic reproduction number of time-averaged autonomous systems may underestimate or overestimate infection risks in some cases, which may be up to the value of period. The basic reproduction number of the seasonality model is appropriately given, which determines the extinction and uniform persistence of TB disease. If it is less than one, then the disease-free equilibrium is globally asymptotically stable; if it is greater than one, the system at least has a positive periodic solution and the disease will persist. Moreover, numerical simulations demonstrate these theorem results.

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A VECTOR-HOST EPIDEMIC MODEL WITH HOST MIGRATION

International Journal of Biomathematics ◽

10.1142/s1793524511001179 ◽

2011 ◽

Vol 04 (01) ◽

pp. 93-108

Author(s):

QINGKAI KONG ◽

ZHIPENG QIU ◽

YUN ZOU

Keyword(s):

Epidemic Model ◽

Reproduction Number ◽

Threshold Value ◽

Endemic Equilibrium ◽

Uniform Persistence ◽

Asymptotically Stable ◽

Host Disease ◽

Disease Free

The host migration is one of the important elements that cause the worldwide diffusion and outbreak of many vector-host diseases. In this paper, we formulate a patchy model to investigate the effect of host migration between two patches on the spread of a vector-host disease. The results of the paper show that the reproduction number R0 is a threshold value that determines the uniform persistence and extinction of the disease. If the reproduction number R0 < 1 the disease free equilibrium (DFE) is locally asymptotically stable. If the reproduction number R0 > 1 then the DFE is unstable and the system is uniformly persistent. It is also shown that a unique endemic equilibrium, which exists when R0 > 1, is locally stable if both regions are identical.

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Bifurcation analysis of an SIR epidemic model with saturated treatment function

10.32920/ryerson.14663469 ◽

2021 ◽

Author(s):

Phuc Ngo

Keyword(s):

Epidemic Models ◽

Stable Node ◽

Sir Epidemic Model ◽

Treatment Rate ◽

Globally Asymptotically Stable ◽

Sir Epidemic ◽

Treatment Function ◽

Sir Epidemic Models ◽

Saturated Treatment ◽

Disease Free

In this thesis we investigate the dynamics and bifurcation of SIR epidemic models with horizontal and vertical transmissions and saturated treatment rate. It is proved that such SIR epidemic models always have positive disease free equilibria and also have three positive epidemic equilibria. The ranges of the parameters related in the model were found under which the equilibria of the models are positive. By applying the qualitative theory of planar systems, it is shown the disease free equilibria is a saddle, stable node and globally asymptotically stable. Furthermore, it is also shown that the interior equilibria are saddle, saddle node or saddle point.

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Backward and Hopf bifurcation analysis of an SEIRS COVID-19 epidemic model with saturated incidence and saturated treatment response

10.1101/2020.08.28.20183723 ◽

2020 ◽

Author(s):

David A. Oluyori ◽

Ángel G. C. Pérez ◽

Victor A. Okhuese ◽

Muhammad Akram

Keyword(s):

Hopf Bifurcation ◽

Local Stability ◽

Complex Dynamics ◽

Reproduction Number ◽

Backward Bifurcation ◽

Treatment Measures ◽

Saturated Incidence ◽

Second Wave ◽

Saturated Treatment ◽

Disease Free

AbstractIn this work, we further the investigation of an SEIRS model to study the dynamics of the Coronavirus Disease 2019 pandemic. We derive the basic reproduction number R0 and study the local stability of the disease-free and endemic states. Since the condition R0 < 1 for our model does not determine if the disease will die out, we consider the backward bifurcation and Hopf bifurcation to understand the dynamics of the disease at the occurrence of a second wave and the kind of treatment measures needed to curtail it. Our results show that the limited availability of medical resources favours the emergence of complex dynamics that complicates the control of the outbreak.

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

Advances in Difference Equations ◽

10.1186/s13662-021-03470-1 ◽

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.

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

PLoS ONE ◽

10.1371/journal.pone.0243408 ◽

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.

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On a Stochastic SEIS Model with Treatment Rate of Latent Population

Abstract and Applied Analysis ◽

10.1155/2014/894842 ◽

2014 ◽

Vol 2014 ◽

pp. 1-16

Author(s):

Shujing Gao ◽

Yanfei Dai ◽

Yan Zhang ◽

Yujiang Liu

Keyword(s):

Asymptotic Behavior ◽

Deterministic Model ◽

Reproduction Number ◽

Deterministic System ◽

Treatment Rate ◽

Initial Value ◽

Asymptotic Dynamics ◽

Disease Free ◽

The Basic Reproduction Number

The asymptotic dynamics of a stochastic SEIS epidemic model with treatment rate of latent population is investigated. First, we show that the system provides a unique positive global solution starting from the positive initial value. Then, the long-term asymptotic behavior of the model is studied: ifR0, which is called the basic reproduction number of the corresponding deterministic model, is not more than unity, the solution of the model is oscillating around the disease-free equilibrium of the corresponding deterministic system, whereas ifR0is larger than unity, we show how the solution spirals around the endemic equilibrium of deterministic system under certain parametric restrictions. Finally, numerical simulations are carried out to support our theoretical findings.

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Threshold dynamics of reaction–diffusion partial differential equations model of Ebola virus disease

International Journal of Biomathematics ◽

10.1142/s1793524518501085 ◽

2018 ◽

Vol 11 (08) ◽

pp. 1850108 ◽

Cited By ~ 3

Author(s):

Kazuo Yamazaki

Keyword(s):

Basic Reproduction Number ◽

Diffusion Coefficients ◽

Ebola Virus ◽

Reproduction Number ◽

Virus Disease ◽

Reaction Diffusion ◽

Technical Difficulty ◽

Threshold Dynamics ◽

Uniform Persistence ◽

Disease Free

We study the reaction–diffusion Ebola PDE model that consists of equations that govern the evolution of susceptible, infected, recovered and deceased human individuals, as well as Ebola virus pathogens in the environment, with diffusive terms in all except the equation of the deceased human individuals. Under the setting of a spatial domain that is bounded, we prove the global well-posedness of the system; in contrast to the previous work on similar models such as cholera, avian influenza, malaria and dengue fever, diffusion coefficients may be different. Moreover, we derive its basic reproduction number, and under the condition that the diffusion coefficients of the susceptible and infected hosts are same, we prove the global stability of the disease-free-equilibrium, and uniform persistence in cases when the basic reproduction number lies beneath and above one, respectively. Again, we do not require that the diffusion coefficients of the recovered hosts be the same as the diffusion coefficients of the susceptible and infected hosts, in contrast to previous work on other models of infectious diseases. Another technical difficulty in our model is that the solution semiflow is not compact due to the lack of diffusion in the equation of the deceased human individuals; we overcome this difficulty using functional analysis techniques concerning Kuratowski measure of non-compactness.

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