Bifurcation analysis of a pair-wise epidemic model on adaptive networks

Jinna Lu; Xiaoguang Zhang; Jinna Lu; Xiaoguang Zhang

doi:10.3934/mbe.2019147

Mathematical Biosciences and Engineering

2019, Volume 16, Issue 4: 2973-2989. doi: 10.3934/mbe.2019147

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Bifurcation analysis of a pair-wise epidemic model on adaptive networks

Jinna Lu ^{1
,
,},
Xiaoguang Zhang ^{2,3
,
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1.
School of Applied Mathematics, Shanxi University of Finance and Economics, Taiyuan 030006, Shanxi, China
2.
School of Mathematical Sciences, Shanxi University, Taiyuan 030006, Shanxi, China
3.
Complex Systems Research Center, Shanxi University, Taiyuan 030006, Shanxi, China

Received: 15 January 2019 Accepted: 14 March 2019 Published: 10 April 2019

The topological structures of complex networks have been playing an important role on the epidemic spreading. There has been several studies of pairwise epidemic models on adaptive networks with Poisson distribution, all of which have shown that the rewiring behaviors can lead to complex dynamics numerically or analytically. However, the triples approximation formula under Poisson distribution overlooked the degree of center node of triples which has dramatic effects on the structures. Therefore in this paper, through a new moment closure incorporating the effect of center node's degree, we study how the topological structures of adaptive networks influences epidemic dynamics. The SIS pairwise epidemic model is first closed by the new triple approximation formula, then we transform the model into an equivalent nondimensionalized three dimensional system. By the qualitative theory and the stability theory of ordinary differential equations, the basic reproduction number R0 of the model is obtained, the existence and stabilities of the equilibria are analyzed. Moreover, we prove that the model exhibits transcritical forward bifurcation, backward bifurcation, saddle-node bifurcation and Hopf bifurcation using the methods of bifurcation theory. In addition, by a numerical example, the normal form of Hopf bifurcation and the first Lyapunov coefficient are derived, which show that a stable limit cycle can bifurcate from the endemic equilibrium with larger epidemicity. Our study show that the adaptive behavior can lead to rich dynamics on epidemic transmission, including oscillation and bistability. Finally the numerical simulations which is consistent with the analytical results above are given.
- epidemic model,
- Hopf bifurcation,
- backward bifurcation,
- bistability
Citation: Jinna Lu, Xiaoguang Zhang. Bifurcation analysis of a pair-wise epidemic model on adaptive networks[J]. Mathematical Biosciences and Engineering, 2019, 16(4): 2973-2989. doi: 10.3934/mbe.2019147

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Abstract

The topological structures of complex networks have been playing an important role on the epidemic spreading. There has been several studies of pairwise epidemic models on adaptive networks with Poisson distribution, all of which have shown that the rewiring behaviors can lead to complex dynamics numerically or analytically. However, the triples approximation formula under Poisson distribution overlooked the degree of center node of triples which has dramatic effects on the structures. Therefore in this paper, through a new moment closure incorporating the effect of center node's degree, we study how the topological structures of adaptive networks influences epidemic dynamics. The SIS pairwise epidemic model is first closed by the new triple approximation formula, then we transform the model into an equivalent nondimensionalized three dimensional system. By the qualitative theory and the stability theory of ordinary differential equations, the basic reproduction number R0 of the model is obtained, the existence and stabilities of the equilibria are analyzed. Moreover, we prove that the model exhibits transcritical forward bifurcation, backward bifurcation, saddle-node bifurcation and Hopf bifurcation using the methods of bifurcation theory. In addition, by a numerical example, the normal form of Hopf bifurcation and the first Lyapunov coefficient are derived, which show that a stable limit cycle can bifurcate from the endemic equilibrium with larger epidemicity. Our study show that the adaptive behavior can lead to rich dynamics on epidemic transmission, including oscillation and bistability. Finally the numerical simulations which is consistent with the analytical results above are given.

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