scholarly journals Global Dynamics of a Susceptible-Infectious-Recovered Epidemic Model with a Generalized Nonmonotone Incidence Rate

2020 ◽  
Author(s):  
Min Lu ◽  
Jicai Huang ◽  
Shigui Ruan ◽  
Pei Yu
Keyword(s):  
Incidence Rate ◽  
Epidemic Model ◽  
Global Dynamics

scholarly journals Stability and Hopf Bifurcation for a Delayed SIR Epidemic Model with Logistic Growth

2013 ◽  
Vol 2013 ◽  
pp. 1-11 ◽  
Author(s):  
Yakui Xue ◽  
Tiantian Li
Keyword(s):  
Incidence Rate ◽  
Epidemic Model ◽  
Threshold Value ◽  
Endemic Equilibrium ◽  
Logistic Growth ◽  
Global Dynamics ◽  
Asymptotically Stable ◽  
Sir Epidemic Model ◽  
Globally Asymptotically Stable ◽  
Sir Epidemic

We study a delayed SIR epidemic model and get the threshold value which determines the global dynamics and outcome of the disease. First of all, for anyτ, we show that the disease-free equilibrium is globally asymptotically stable; whenR0<1, the disease will die out. Directly afterwards, we prove that the endemic equilibrium is locally asymptotically stable for anyτ=0; whenR0>1, the disease will persist. However, for anyτ≠0, the existence conditions for Hopf bifurcations at the endemic equilibrium are obtained. Besides, we compare the delayed SIR epidemic model with nonlinear incidence rate to the one with bilinear incidence rate. At last, numerical simulations are performed to illustrate and verify the conclusions.

scholarly journals Global dynamics and bifurcation analysis of a fractional-order SEIR epidemic model with saturation incidence rate

2021 ◽  
Author(s):  
Parvaiz Ahmad Naik ◽  
Muhammad Bilal Ghori ◽  
Jian Zu ◽  
Zohre Eskandari ◽  
Mehraj-ud-din Naik
Keyword(s):  
Incidence Rate ◽  
Fractional Order ◽  
Epidemic Model ◽  
Disease Transmission ◽  
Disease Status ◽  
Host Population ◽  
Sensitivity Analyses ◽  
Feasible Region ◽  
Global Dynamics ◽  
Globally Stable

The present paper studies a fractional-order SEIR epidemic model for the transmission dynamics of infectious diseases such as HIV and HBV that spreads in the host population. The total host population is considered bounded, and Holling type-II saturation incidence rate is involved as the infection term. Using the proposed SEIR epidemic model, the threshold quantity, namely basic reproduction number R0, is obtained that determines the status of the disease, whether it dies out or persists in the whole population. The model’s analysis shows that two equilibria exist, namely, disease-free equilibrium (DFE) and endemic equilibrium (EE). The global stability of the equilibria is determined using a Lyapunov functional approach. The disease status can be verified based on obtained threshold quantity R0. If R0 < 1, then DFE is globally stable, leading to eradicating the population’s disease. If R0 > 1, a unique EE exists, and that is globally stable under certain conditions in the feasible region. The Caputo type fractional derivative is taken as the fractional operator. The bifurcation and sensitivity analyses are also performed for the proposed model that determines the relative importance of the parameters into disease transmission. The numerical solution of the model is obtained by the generalized Adams- Bashforth-Moulton method. Finally, numerical simulations are performed to illustrate and verify the analytical results.

scholarly journals Global dynamics and bifurcation analysis of a fractional‐order SEIR epidemic model with saturation incidence rate

2022 ◽  
Author(s):  
Muhammad Bilal Ghori ◽  
Parvaiz Ahmad Naik ◽  
Jian Zu ◽  
Zohreh Eskandari ◽  
Mehraj‐ud‐din Naik
Keyword(s):  
Incidence Rate ◽  
Fractional Order ◽  
Bifurcation Analysis ◽  
Epidemic Model ◽  
Global Dynamics

scholarly journals An SIRS Epidemic Model with Vital Dynamics and a Ratio-Dependent Saturation Incidence Rate

2015 ◽  
Vol 2015 ◽  
pp. 1-9 ◽  
Author(s):  
Xinli Wang
Keyword(s):  
Computer Simulation ◽  
Periodic Solution ◽  
Incidence Rate ◽  
Epidemic Model ◽  
Initial Position ◽  
Mass Action ◽  
Global Dynamics ◽  
Nonlinear Incidence Rate ◽  
Sirs Epidemic Model ◽  
Disease Free

This paper presents an investigation on the dynamics of an epidemic model with vital dynamics and a nonlinear incidence rate of saturated mass action as a function of the ratio of the number of the infectives to that of the susceptibles. The stabilities of the disease-free equilibrium and the endemic equilibrium are first studied. Under the assumption of nonexistence of periodic solution, the global dynamics of the model is established: either the number of infective individuals tends to zero as time evolves or it produces bistability in which there is a region such that the disease will persist if the initial position lies in the region and disappears if the initial position lies outside this region. Computer simulation shows such results.

scholarly journals Global Dynamics of an Epidemic Model with a Non-Monotonic Incidence Rate

2014 ◽  
Vol 10 (2) ◽  
pp. 71-77
Author(s):  
Seema Khekare ◽  
◽  
Sujatha Janardhan
Keyword(s):  
Incidence Rate ◽  
Epidemic Model ◽  
Global Dynamics

scholarly journals Global dynamics and bifurcations in a SIRS epidemic model with a nonmonotone incidence rate and a piecewise-smooth treatment rate

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Qin Pan ◽  
Jicai Huang ◽  
Qihua Huang
Keyword(s):  
Incidence Rate ◽  
Epidemic Model ◽  
Reproduction Number ◽  
Homoclinic Orbits ◽  
Critical Value ◽  
Piecewise Smooth ◽  
Global Dynamics ◽  
Treatment Rate ◽  
Sirs Epidemic Model ◽  
Subcritical Hopf Bifurcation

<p style='text-indent:20px;'>In this paper, we analyze a SIRS epidemic model with a nonmonotone incidence rate and a piecewise-smooth treatment rate. The nonmonotone incidence rate describes the "psychological effect": when the number of the infected individuals (denoted by <inline-formula><tex-math id="M1">\begin{document}$ I $\end{document}</tex-math></inline-formula>) exceeds a certain level, the incidence rate is a decreasing function with respect to <inline-formula><tex-math id="M2">\begin{document}$ I $\end{document}</tex-math></inline-formula>. The piecewise-smooth treatment rate describes the situation where the community has limited medical resources, treatment rises linearly with <inline-formula><tex-math id="M3">\begin{document}$ I $\end{document}</tex-math></inline-formula> until the treatment capacity is reached, after which constant treatment (i.e., the maximum treatment) is taken.Our analysis indicates that there exists a critical value <inline-formula><tex-math id="M4">\begin{document}$ \widetilde{I_0} $\end{document}</tex-math></inline-formula> <inline-formula><tex-math id="M5">\begin{document}$ ( = \frac{b}{d}) $\end{document}</tex-math></inline-formula> for the infective level <inline-formula><tex-math id="M6">\begin{document}$ I_0 $\end{document}</tex-math></inline-formula> at which the health care system reaches its capacity such that:<b>(i)</b> When <inline-formula><tex-math id="M7">\begin{document}$ I_0 \geq \widetilde{I_0} $\end{document}</tex-math></inline-formula>, the transmission dynamics of the model is determined by the basic reproduction number <inline-formula><tex-math id="M8">\begin{document}$ R_0 $\end{document}</tex-math></inline-formula>: <inline-formula><tex-math id="M9">\begin{document}$ R_0 = 1 $\end{document}</tex-math></inline-formula> separates disease persistence from disease eradication. <b>(ii)</b> When <inline-formula><tex-math id="M10">\begin{document}$ I_0 &lt; \widetilde{I_0} $\end{document}</tex-math></inline-formula>, the model exhibits very rich dynamics and bifurcations, such as multiple endemic equilibria, periodic orbits, homoclinic orbits, Bogdanov-Takens bifurcations, and subcritical Hopf bifurcation.</p>

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