Disease extinction and persistence in a discrete-time sis epidemic model with vaccination and varying population size

Filomat ◽  
2017 ◽  
Vol 31 (15) ◽  
pp. 4735-4747 ◽  
Author(s):  
Rahman Farnoosh ◽  
Mahmood Parsamanesh
Keyword(s):  
Population Size ◽  
Discrete Time ◽  
Epidemic Model ◽  
Jacobian Matrix ◽  
Sufficient Conditions ◽  
Period Doubling ◽  
System Of Difference Equations ◽  
Sis Epidemic Model ◽  
Necessary And Sufficient ◽  
Sis Epidemic

A discrete-time SIS epidemic model with vaccination is introduced and formulated by a system of difference equations. Some necessary and sufficient conditions for asymptotic stability of the equilibria are obtained. Furthermore, a sufficient condition is also presented. Next, bifurcations of the model including transcritical bifurcation, period-doubling bifurcation, and the Neimark-Sacker bifurcation are considered. In addition, these issues will be studied for the corresponding model with constant population size. Dynamics of the model are also studied and compared in detail with those found theoretically by using bifurcation diagrams, analysis of eigenvalues of the Jacobian matrix, Lyapunov exponents and solutions of the models in some examples.

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

2020 ◽  
Vol 21 (1) ◽  
Author(s):  
Mahmood Parsamanesh ◽  
Majid Erfanian ◽  
Saeed Mehrshad
Keyword(s):  
Discrete Time ◽  
Epidemic Model ◽  
Sufficient Conditions ◽  
Period Doubling ◽  
Euler Method ◽  
Continuous Version ◽  
Sis Epidemic Model ◽  
The Stability ◽  
Disease Free ◽  
Discretized Model

Abstract 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.

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

2020 ◽  
Author(s):  
Mahmood Parsamanesh ◽  
Majid Erfanian ◽  
Saeed Mehrshad
Keyword(s):  
Discrete Time ◽  
Epidemic Model ◽  
Sufficient Conditions ◽  
Period Doubling ◽  
Euler Method ◽  
Continuous Version ◽  
Sis Epidemic Model ◽  
Intervention Measures ◽  
The Stability ◽  
Discretized Model

Abstract 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.

Pattern Dynamics of an SIS Epidemic Model with Nonlocal Delay

2019 ◽  
Vol 29 (02) ◽  
pp. 1950027 ◽  
Author(s):  
Zun-Guang Guo ◽  
Li-Peng Song ◽  
Gui-Quan Sun ◽  
Can Li ◽  
Zhen Jin
Keyword(s):  
Epidemic Model ◽  
Sufficient Conditions ◽  
Average Density ◽  
Reaction Diffusion Equation ◽  
Spatial Average ◽  
Stripe Pattern ◽  
Sis Epidemic Model ◽  
Nonlocal Delay ◽  
Pattern Dynamics ◽  
Sis Epidemic

In this paper, we study an SIS epidemic model with nonlocal delay based on reaction–diffusion equation. The spatiotemporal distribution of the model solution is studied in detail, and sufficient conditions for the occurrence of the Turing pattern were obtained using the analysis of Turing instability. It was found that the delay not only prohibited the spread of infectious disease, but also had great effects on the spatial steady-state patterns. More specifically, the spatial average density of the infected populations will decrease, as well the width of the stripe pattern will increase as delay increases. When the delay increases to a certain value, the stripe pattern changes to the mixed pattern. The results in this work provide new theoretical guidance for the prevention and treatment of infectious diseases.

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