Bifurcation analysis of an SIS epidemic model with saturated incidence rate and saturated treatment function

Zhou, Tingting; Zhang, Weipeng; Lu, Qiuying

doi:10.1016/j.amc.2013.10.020

Cited by 21 publications

(12 citation statements)

References 24 publications

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“…Let n 1 be defined by (28), from Theorems 4.6 and 5.3, we can see that system (5) can exhibit Hopf bifurcation around E 2 (x 2 , y 2 ) when 0 < R 0 < 1.…”

Section: Hopf Bifurcationmentioning

confidence: 98%

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

Qin

Huang

2022

DCDS-B

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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 $\end{document}</tex-math></inline-formula> at which the health care system reaches its capacity such that:(i) When <inline-formula><tex-math id="M7">\begin{document}$ 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 $\end{document}</tex-math></inline-formula> separates disease persistence from disease eradication. (ii) When <inline-formula><tex-math id="M10">\begin{document}$ I_0 < \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.

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“…Let n 1 be defined by (28), from Theorems 4.6 and 5.3, we can see that system (5) can exhibit Hopf bifurcation around E 2 (x 2 , y 2 ) when 0 < R 0 < 1.…”

Section: Hopf Bifurcationmentioning

confidence: 98%

“…See also [17,24,28]. This function is a nonlinear increasing function in I but eventually tends to a saturation level k/α as I increases.…”

mentioning

confidence: 99%

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

Qin

Huang

2022

DCDS-B

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“… 2012 ; Zhou et al. 2014 for additional references). This function is a nonlinear increasing function in I but eventually tends to a saturation level as I increases to infinity.…”

Section: Introductionmentioning

confidence: 99%

An SIRS model with nonmonotone incidence and saturated treatment in a changing environment

2022

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Nonmonotone incidence and saturated treatment are incorporated into an SIRS model under constant and changing environments. The nonmonotone incidence rate describes the psychological or inhibitory effect: when the number of the infected individuals exceeds a certain level, the infection function decreases. The saturated treatment function describes the effect of infected individuals being delayed for treatment due to the limitation of medical resources. In a constant environment, the model undergoes a sequence of bifurcations including backward bifurcation, degenerate Bogdanov-Takens bifurcation of codimension 3, degenerate Hopf bifurcation as the parameters vary, and the model exhibits rich dynamics such as bistability, tristability, multiple periodic orbits, and homoclinic orbits. Moreover, we provide some sufficient conditions to guarantee the global asymptotical stability of the disease-free equilibrium or the unique positive equilibrium. Our results indicate that there exist three critical values and for the treatment rate r : (i) when , the disease will disappear; (ii) when , the disease will persist. In a changing environment, the infective population starts along the stable disease-free state (or an endemic state) and surprisingly continues tracking the unstable disease-free state (or a limit cycle) when the system crosses a bifurcation point, and eventually tends to the stable endemic state (or the stable disease-free state). This transient tracking of the unstable disease-free state when predicts regime shifts that cause the delayed disease outbreak in a changing environment. Furthermore, the disease can disappear in advance (or belatedly) if the rate of environmental change is negative and large (or small). The transient dynamics of an infectious disease heavily depend on the initial infection number and rate or the speed of environmental change. Supplementary Information The online version contains supplementary material available at 10.1007/s00285-022-01787-3.

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“…Later, Wang [4] modified it as a function where treatment rate is proportionate to the infectious population till the treatment capability reaches its maximum limit. Nowadays, a large domain of researchers is dealing with the infectious disease models with a variety of treatment functions [5][6][7][8][9][10][11].…”

Section: Introductionmentioning

confidence: 99%

Dynamical analysis of SEIS model with nonlinear innate immunity and saturated treatment

Jain

Kumar

2021

Eur. Phys. J. Plus

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In this paper, we develop an SEIS model with Holling type II function representing the innate immunity as well as the saturated treatment. We obtain the existence and stability criteria for the equilibrium points. We observe that when the reproduction number is less than unity, the disease-free equilibrium always exists and is locally asymptotically stable. The multiple endemic equilibrium points can exist independent of the basic reproduction number, and the system may experience bistability. We find that the system can encounter backward or forward bifurcation at R 0 = 1, where the contact rate β = β 0 is the bifurcation parameter. Therefore, the disease-free equilibrium may not be globally stable. We deduce the criteria for the presence of Hopf bifurcation where the parameter γ = γ * acts as the bifurcation parameter and the system is a neutrally stable center. We also observe with the aid of a numerical example that a slight perturbation disrupts the neutral stability and the trajectories become either converging or diverging from the equilibrium point. Numerical simulation is performed with the help of MATLAB to justify the findings. We study the effect of nonlinearity of immunity function and the treatment rate on the dynamics of the disease spread. We find that when both are linear, the reproduction number is the same, but the system has a unique endemic equilibrium point that exists for reproduction number greater than unity. We find that there is neither backward bifurcation nor Hopf bifurcation. We also observe that the saturation in treatment enlarges the domain of backward bifurcation making disease eradication an extremely difficult task. The endemic equilibria in the case of saturated treatment may exist far more to the left of the bifurcation parameter β = β 0 . Hence, the nonlinearity of immunity function and treatment function affects the dynamics of an SEIS model highly; therefore, one must be precautious to choose an appropriate function for both while modeling.

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Bifurcation analysis of an SIS epidemic model with saturated incidence rate and saturated treatment function

Cited by 21 publications

References 24 publications

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

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

An SIRS model with nonmonotone incidence and saturated treatment in a changing environment

Dynamical analysis of SEIS model with nonlinear innate immunity and saturated treatment

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