On the global stability of the endemic state in an epidemic model with vaccination

Parsamanesh, Mahmood; Farnoosh, Rahman

doi:10.1007/s40096-018-0271-3

Cited by 9 publications

(8 citation statements)

References 18 publications

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“…The model described in Fig. 1 can be stated as a continuous-time model by the following system of ordinary differential equations (see [ 28 , 29 ]): We see that and therefore the population size is constant. Similar to the discrete-time model, we get the following two-dimensional system by substituting and omitting variable V from the system: Now in this section, we discretize and analyze model ( 9 ) by using the forward Euler method.…”

Section: Resultsmentioning

confidence: 99%

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Stability and bifurcations in a discrete-time epidemic model with vaccination and vital dynamics

2020

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Background The spread of infectious diseases is so important that changes the demography of the population. Therefore, prevention and intervention measures are essential to control and eliminate the disease. Among the drug and non-drug interventions, vaccination is a powerful strategy to preserve the population from infection. Mathematical models are useful to study the behavior of an infection when it enters a population and to investigate under which conditions it will be wiped out or continued. Results A discrete-time SIS epidemic model is introduced that includes a vaccination program. Some basic properties of this model are obtained; such as the equilibria and the basic reproduction number $$\mathcal {R}_0$$ R 0 . Then the stability of the equilibria is given in terms of $$\mathcal {R}_0$$ R 0 , and the bifurcations of the model are studied. By applying the forward Euler method on the continuous version of the model, a discretized model is obtained and analyzed. Conclusion It is proven that the disease-free equilibrium and endemic equilibrium are stable if $$\mathcal {R}_0<1$$ R 0 < 1 and $$\mathcal {R}_0>1$$ R 0 > 1 , respectively. Also, the disease-free equilibrium is globally stable when $$\mathcal {R}_0\le 1$$ R 0 ≤ 1 . The system has a transcritical bifurcation when $$\mathcal {R}_0=1$$ R 0 = 1 and it might also have period-doubling bifurcation. The sufficient conditions for the stability of equilibria in the discretized model are established. The numerical discussions verify the theoretical results.

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Section: Resultsmentioning

confidence: 99%

“…The model described in Fig. 1 can be stated as a continuous-time model by the following system of ordinary differential equations (see [28,29]):…”

Section: The Model Obtained By the Forward Euler Discretizationmentioning

confidence: 99%

Stability and bifurcations in a discrete-time epidemic model with vaccination and vital dynamics

2020

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“…According to Equation (23), Equation (24), and Lemma 3 in reference [30], it can be known that the local dynamics of model ( 1) around E 0 are determined by a and b. So, for b > 0, if there is M > 1, then a < 0; if there is M < 1, then a > 0.…”

Section: Advances In Mathematical Physicsmentioning

confidence: 99%

“…Vaccination strategies cannot be developed without taking into account vaccine availability, vaccination rates, efficacy, and the actual situation of loss of natural immune response and viral mutation [27][28][29]. In addition, Parsamanesh et al established a series of epidemiological models on vaccination and theoretically demonstrated the local and global stability of the disease-free equilibrium point and the endemic equilibrium point, as well as various possible bifurcations [30][31][32][33][34][35].…”

Section: Introductionmentioning

confidence: 99%

Dynamic Analysis of a VSEIR Model with Vaccination Efficacy and Immune Decline

Han

Hua

Lin

et al. 2022

Advances in Mathematical Physics

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Vaccination is an effective way to prevent the spread of infectious diseases. In this study, we formulate a VSEIR mathematical model to explore the effects of vaccination rate, vaccine efficacy, and immune decline on the COVID-19 transmission. The existence and stability criteria of equilibrium states were determined by analyzing the model. Model analysis was performed. One of the interesting phenomena involved in this issue is that diseases may or may not die out when the basic reproduction number falls below unity (i.e., a backward bifurcation may exist and cause multistability). The disease eventually becomes endemic in the population when the basic reproduction number exceeds one. By comparing different vaccination rates, vaccine efficacy, and infection rate factors, the diseases can be eliminated, not only by vaccines but also by strict protective measures. In addition, we used the COVID-19 number of reported cases in Xiamen in September 2021 to fit the model, and the model and the reported data were well matched.

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“…The model described in Figure 1 can be stated as a continuous-time model by the following system of ordinary differential equations (see [12,13]):…”

Section: The Model Obtained By the Forward Euler Discretizationmentioning

confidence: 99%

Stability and Bifurcations in a Discrete-Time Epidemic Model with Vaccination and Vital Dynamics

Parsamanesh

Erfanian

Mehrshad

2020

Preprint

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BackgroundThe spread of infectious diseases is such important that changes the demography of the population. Therefore, prevention and intervention measures are essential to control and eliminate the disease. Among the drug and non-drug interventions, vaccination is a powerful strategy to preserve the population from infection. Mathematical models are useful to study the behavior of an infection when it enters a population and investigate under which conditions it will be wiped out or continued.ResultsA discrete-time SIS epidemic model is introduced that includes a vaccination program. Some basic properties of this model are obtained; such as the equilibria and the basic reproduction number $\mathcal{R}_0$ . Then the stability of the equilibria is given in terms of $\mathcal{R}_0$ , and moreover, the bifurcations of the model are studied. By applying the forward Euler method on the continuous version of model, a discretized model is obtained and analyzed. ConclusionIt is proved that the disease-free equilibrium and endemic equilibrium are stable if $\mathcal{R}_0<1$ and $\mathcal{R}_0>1$ , respectively. The system has a transcritical bifurcation when $\mathcal{R}_0=1$ and it might also have period-doubling bifurcation. The sufficient conditions for the stability of equilibria in the discretized model are established. The numerical discussions verify the theoretical results.

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On the global stability of the endemic state in an epidemic model with vaccination

Cited by 9 publications

References 18 publications

Stability and bifurcations in a discrete-time epidemic model with vaccination and vital dynamics

Stability and bifurcations in a discrete-time epidemic model with vaccination and vital dynamics

Dynamic Analysis of a VSEIR Model with Vaccination Efficacy and Immune Decline

Stability and Bifurcations in a Discrete-Time Epidemic Model with Vaccination and Vital Dynamics

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