| Citation: |
Shumin Zhou, Yunxian Dai, Hongyan Wang. STABILITY AND HOPF BIFURCATION ANALYSIS OF A NETWORKED SIR EPIDEMIC MODEL WITH TWO DELAYS AND DELAY DEPENDENT PARAMETERS[J]. Journal of Applied Analysis & Computation, 2025, 15(4): 1996-2026. doi: 10.11948/20240382
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STABILITY AND HOPF BIFURCATION ANALYSIS OF A NETWORKED SIR EPIDEMIC MODEL WITH TWO DELAYS AND DELAY DEPENDENT PARAMETERS
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Abstract
The spread of infectious diseases is generally influenced by the random contact of different individuals in uneven spatial structure. To describe this contact effect, network is introduced into a two delays SIR epidemic model with incubation period delay and temporary immunity delay. Due to the existence of the temporary immunity term, the characteristic equation of epidemic model has two delays and the parameters depend on one of them. We prove the stability of the disease-free equilibrium and the endemic equilibrium. We additionally obtain the stability switching curves to study the stability switching properties of the endemic equilibrium on the two delays plane when two delays change simultaneously, and further discuss the existence of Hopf bifurcation. The stability and the direction of the Hopf bifurcation are investigated with the normal form method and center manifold theorem. To illustrate our theoretical conclusions visually, we performed numerical simulation on a small-world Watts-Strogatz graph.
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References
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Shumin Zhou, Yunxian Dai, Hongyan Wang. STABILITY AND HOPF BIFURCATION ANALYSIS OF A NETWORKED SIR EPIDEMIC MODEL WITH TWO DELAYS AND DELAY DEPENDENT PARAMETERS[J]. Journal of Applied Analysis & Computation, 2025, 15(4): 1996-2026. doi: 10.11948/20240382
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Figure 1.
Watts-Strogatz network WS(
$ m $ $ r $ $ p_1 $ -
Figure 2.
Positive equilibrium
$ E_2(10.4348, 0.1517) $ $ \tau_1=0, \tau_2=0 $ $ (S_0, I_0)=(0.9114, 0.0710) $ -
Figure 3.
Positive equilibrium
$ E_2(9.3356, 0.0658) $ $ \tau_1=\tau_2=15 $ $ (S_0, I_0)=(0.9114, 0.0710) $ -
Figure 4.
Positive equilibrium
$ E_2(9.2648, 0.0602) $ $ \tau_1=\tau_2=19 $ $ (S_0, I_0)=(0.9114, 0.0710) $ -
Figure 5.
Positive equilibrium
$ E_2(9.9372, 0.1128) $ $ \tau_1=0, \tau_2=3 $ $ (S_0, I_0)=(0.9114, 0.0710) $ -
Figure 6.
(a) The image of
$ Q_i(\omega_j, \tau_2) $ $ i=1, 2, 3 $ $ \omega_j\in (0, 2.5], \tau_2\in [0, 60] $ $ \Omega $ $ C $ -
Figure 7.
(a) The stability switching curves
$ \mathcal{T} $ -
Figure 8.
Partial enlargements of Ⅰ, Ⅱ, Ⅲ of Figure 6(b) and cross direction.
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Figure 9.
Positive equilibrium
$ E_2( 10.0438, 0.1211) $ $ (\tau_1, \tau_2)=P_1(1.4, 2.15) $ $ (S_0, I_0)=(0.9114, 0.0710) $ -
Figure 10.
Positive equilibrium
$ E_2( 9.9849, 0.1165) $ $ (\tau_1, \tau_2)=P_2(0.4, 2.6) $ $ (S_0, I_0)=(0.9114, 0.0710) $ -
Figure 11.
Positive equilibrium
$ E_2( 10.0039, 0.1180) $ $ (\tau_1, \tau_2)=P_3(0.07, 2.45) $ $ (S_0, I_0)=(0.9114, 0.0710) $ -
Figure 12.
Positive equilibrium
$ E_2(10.0235, 0.1195) $ $ (\tau_1, \tau_2)=P_4(3.8, 2.3) $ $ (S_0, I_0)=(0.9114, 0.0710) $ -
Figure 13.
Positive equilibrium
$ E_2(10.0259, 0.1197) $ $ (\tau_1, \tau_2)=P_5(3.57, 2.282) $ $ (S_0, I_0)=(0.9114, 0.0710) $ -
Figure 14.
Positive equilibrium
$ E_2(10.0272, 0.1198) $ $ (\tau_1, \tau_2)=P_6(7.4, 2.272) $ $ (S_0, I_0)=(0.9114, 0.0710) $ -
Figure 15.
Positive equilibrium
$ E_2(10.0271, 0.1197) $ $ (\tau_1, \tau_2)=P_7(7, 2.272) $ $ (S_0, I_0)=(0.9114, 0.0710) $