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# A hierarchical parametric analysis on Hopf bifurcation of an epidemic model

• * Corresponding author: Pei Yu
• A common task in studying nonlinear dynamical systems is to derive the conditions on stability and bifurcations, which becomes difficulty when the system contain multiple parameters. In particular, finding the explicit conditions under which Hopf bifurcation can occur is not easy and becomes very involved even for simple models. In this paper, an epidemic model is presented to illustrate how to use a hierarchical parametric analysis for bifurcation study, in particular to demonstrate how to choose proper parameters as bifurcation parameters, how to deal with other "control" parameters, and how to derive the conditions on stability and Hopf bifurcation, which are explicitly expressed in terms of system parameters.

Mathematics Subject Classification: Primary: 34C07, 34C23; Secondary: 34D20.

 Citation: • • Figure 1.  Simulation of the epidemic model (2) for $m = 2$, $n = \frac{1}{6}$, $\varepsilon = \frac{1}{2}$ and $k = \frac{13}{100}$, showing non-well-posedness solution preperty

Figure 2.  (a) bifurcation diagram for the epidemic model (2) projected on the $k$-$Y_2$ plane with $m = 2, \, n = \frac{1}{6}, \, \varepsilon = \frac{1}{2}$, corresponding to Case (1b) in Theorem 3.1 having two Hopf critical points, with $k_{\rm T} = 0.097222$, $k_{\rm H_-} = 0.110329$ and $k_{\rm H_+} = 0.149040$; and (b) simulations of three stable limit cycles, starting from the initial point $(X, Y) = (0.25, 5)$, for three values of $k$ (marked by the red circles on the $k$-axis): $k = 0.112$ (blue color), $k = 0.13$ (red color), $k = 0.147$ (green color)

Figure 3.  (a) bifurcation diagram for the epidemic model (2) projected on the $k$-$Y_2$ plane with $m \! = \! 2, \, n \! = \! \frac{2}{5}, \, \varepsilon \! = \! 2$, corresponding to the Case (2c)(i) in Theorem 3.1 having one Hopf critical point, with $k_{\rm SN} \! = \!0.215089$, $k^* \! = \!0.228381$, $k_{\rm H_+} \! = \! 0.235367$ and $k_{\rm T} \! = \! 0.28$; (b) simulated phase portrait with $k \! = \! 0.222$, showing the bistable phenomenon with two stable equilibria ${\rm E_1}$ and ${\rm E_{2-}}$; (c) simulated phase portrait with $k \! = \! 0.232$, showing the bistable phenomenon with the stable equilibrium ${\rm E_1}$ and a stable limit cycle; and (d) simulated phase portrait with $k \! = \! 0.25$, showing the bistable phenomenon with two stable equilibria ${\rm E_1}$ and ${\rm E_{2-}}$. The three values of $k$ are marked by the circles on the $k$-axis in Figure 3(a)

Figure 4.  (a) bifurcation diagram for the epidemic model (2) projected on the $k$-$Y_2$ plane with $m = 2, \, n = \frac{3}{4}, \, \varepsilon = 8$, corresponding to the Case (2c)(ii) in Theorem 3.1 having one Hopf critical point, with $k_{\rm SN} = 0.233287$, $k_{\rm H_+} = 0.233834$, $k^* \! = \! 0.240547$, and $k_{\rm T} = 0.656250$; (b) simulated phase portrait with $k \! = \! 0.2336$, showing the stable node ${\rm E_1}$ and the unstable focus ${\rm E_{2-}}$; (c) simulated phase portrait with $k \! = \! 0.234$, showing the stable node ${\rm E_1}$ and an unstable limit cycle; and (d) simulated phase portrait with $k \! = \! 0.24$, showing the bistable phenomenon with two stable equilibria ${\rm E_1}$ and ${\rm E_{2-}}$. The three values of $k$ are marked by the circles on the $k$-axis in Figure 4(a)

Figure 5.  (a) bifurcation diagram for the epidemic model (2) projected on the $k$-$Y_2$ plane with $m = 2, \, n = \frac{1}{3}, \, \varepsilon = \frac{5}{4}$, corresponding to the Case (2d) in Theorem 3.1 having two Hopf critical points, with $k^* \! = \! 0.197348$, $k_{\rm SN} = 0.201638$, $k_{\rm H_-} = 0.202731$, $k_{\rm H_+} = 0.221025$ and $k_{\rm T} = 0.222222$; (b) simulated phase portrait with $k \! = \! 0.202$, showing the bistable phenomenon with two stable equilibria ${\rm E_1}$ and ${\rm E_{2-}}$; (c) simulated phase portrait with $k \! = \! 0.205$, showing the bistable phenomenon with the stable equilibrium ${\rm E_1}$ and a stable limit cycle; and (d) simulated phase portrait with $k \! = \! 0.22$, showing the bistable phenomenon with the stable equilibrium ${\rm E_1}$ and a stable limit cycle. The three values of $k$ are marked by the circles on the $k$-axis in Figure 5(a)

Figure 6.  Bifurcation diagram for the epidemic model (2) on the $k$-$\varepsilon$ parameter plane with $m = 2, \, n = \frac{5}{11}$, where SN, T, H, BT and GH denote the saddle-node, transcritical, Hopf, Bogdanov-Takens and generalized Hopf bifurcations, respectively

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