Hopf and forward bifurcation of an integer and fractional-order SIR epidemic model with logistic growth of the susceptib
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Hopf and forward bifurcation of an integer and fractional-order SIR epidemic model with logistic growth of the susceptible individuals M. H. Akrami1
· A. Atabaigi2
Received: 21 January 2020 © Korean Society for Informatics and Computational Applied Mathematics 2020
Abstract This paper deals with the qualitative behavior of an integer and fractional-order SIR epidemic model with logistic growth of the susceptible individuals. Firstly, the positivity and boundedness of solutions for integer-order model are proved. The basic reproduction number R0 is driven and it is shown that the disease-free equilibrium is globally asymptotically stable if R0 < 1 in integer-order model. Using the methods of bifurcations theory, it is proved that the integer-order model exhibits forward bifurcation and Hopf bifurcation. Next, with the aim of the stability theory of fractional-order systems, some conditions, which can guarantee the local stability of the fractionalorder model, are developed and occurrence of forward and Hopf bifurcations in this model are studied. Lastly, numerical simulations are illustrated to support the theoretical results and a comparison between the integer and fractional-order systems is presented. Keywords SIR epidemic model · Logistic growth · Fractional-order derivative · Forward bifurcation · Hopf bifurcation Mathematics Subject Classification 37N25 · 26A33 · 34C23
1 Introduction Currently, the study and analysis of biomathematical models is one of the most important topics in interdisciplinary sciences [3]. There are many articles on population models, infectious and epidemic diseases in literatures, for example see [6,14,25,29] and references therein. Various epidemiological models have been introduced by some
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M. H. Akrami [email protected]
1
Department of Mathematics, Yazd University, Yazd, Iran
2
Department of Mathematics, Razi University, Kermanshah, Iran
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M. H. Akrami, A. Atabaigi
authors and have been extensively studied, which lead to great advances in the control and prevention of diseases. In the last decade, the modelling of epidemic diseases has made a lot of progress. Some authors have added several restrictions to their models, such as restrictions on the capacity of treatment, medicine, medical resource, hospital beds and so on. For instance, Zhou and Fan [33] investigated the dynamics and bifurcation of an SIR epidemic model with limited medical resources. Shan and Zhu [25] studied the bifurcations and complex dynamics of an SIR epidemic model with the impact of the number of hospital beds. We recall that in SIR models the total population is divided into three groups susceptible (S), infective (I ) and recovered individuals (R). Several researchers by presenting different functions for treatment and incidence rate, introduced different epidemic SIR models. For example, Capasso and Serio [4] incorporated the saturated incidence rate g(I )S into the epidemic model, where the infectious force g(I ) is a function of an infected individual. Liu et al. [18,19] considered κ Il
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