Stability and bifurcations in a discrete-time epidemic model with vaccination and vital dynamics
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RESEARCH ARTICLE
Open Access
Stability and bifurcations in a discrete‑time epidemic model with vaccination and vital dynamics Mahmood Parsamanesh1, Majid Erfanian2* and Saeed Mehrshad2 *Correspondence: [email protected] 2 Department of Mathematics, Faculty of Science, University of Zabol, Zabol, Iran Full list of author information is available at the end of the article
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 R0 . Then the stability of the equilibria is given in terms of R0 , 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 R0 < 1and R0 > 1 , respectively. Also, the disease-free equilibrium is globally stable when R0 ≤ 1 . The system has a transcritical bifurcation when R0 = 1and 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. Keywords: SIS epidemic model, Discrete-time system, Stability, Lyapunov exponent, Bifurcation
Background The spread of infectious diseases in populations and how to control and eliminate them from the population are important and necessary subjects. Mathematical models are introduced to study what happens when an infection enters in a population, and under which conditions the disease will be wiped out from population or persists in population. The literature about mathematical epidemic models that have been constructed and analyzed for various types of diseases is very rich; see, for example, [1, 2]. Such models can be formulated either as continuous-time models by differential equations or as © The Author(s) 2020. Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material
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