In recent years, financial liberalization and economic integration have continued to develop. However, with the expansion of international financial markets, the frequency and harm of financial risks are also increasing. Comparing the previous situation, we found that global risks always broke out in one country and then spread to other countries through trade, finance and other mechanisms. The process was obviously contagious and did huge harm to infected areas. In our country, the possibility of suffering financial risks and the harm caused by risks have been increasing in recent years. Therefore, the report of the 20th National Congress of the Communist Party of China also pointed out that "it is necessary to improve the financial regulatory system and maintain the bottom line of preventing systemic financial risks." How to effectively monitor and prevent financial risks has become a hot issue studied by scholars.

In order to study the contagion mechanisms of different financial systems and reduce economic losses, many scholars have applied various methods to study financial risks in recent years. For example, some statistical models have been used to empirically study the characteristics of risk contagion among different financial markets in [1–3]. Some scholars have also introduced complex networks into financial systems, studied the topological structures and evolutionary rules of different financial markets, and described financial risk contagion from the perspective of network structure [4–6].

Due to the similarities in the spread of infectious diseases and financial risks [7], the epidemic model has provided new ideas for the study of financial risk contagion. Many scholars have started to use the epidemic model to consider different propagation mechanisms of financial risks. In this regard, Garas et al. [8] first applied a SIR model to study global financial crisis contagion, and obtained 12 central susceptible sources that may trigger the global financial crisis. On the basis of Garas, Toivanen [9] applied a SIR model and simulations to further study the risk contagion among European banks, obtained the relevant factors that easily cause risk contagion. Subsequently, many scholars studied financial risks by establishing different epidemic models. For example, Bucci et al. [10] believed that banks would only be affected by speculation, and infected banks would never develop immunity after recovering from speculation, so they established a susceptible-infected-susceptible (SIS) model of financial risk contagion, which truly describes the contagion process among banks. Since exposed individuals exist in the financial risk system, Chenyu B et al. [11] considered the strictness of government supervision, established a SEIR model by dividing infected institutions into two types, and concluded that supervision stringency affected the stability of system. Xu Kai [12] considered the existence of latent subject (H) based on the SIR model, constructed a SHIR model of associated credit risk contagion, and analyzed the relevant factors that affected associated credit risk contagion. In addition, considering the impact of time delay and the characteristics of Internet P2P lending platforms, Zhao et al. [13] constructed a SEIR model with time delay, used equilibrium and thresholds to analyze the effect of different factors on credit risk contagion behavior. To sum up, we can see that existing research cannot better describe the complex financial systems. Therefore, based on the SEIR model, the financial risk contagion model is improved in this paper.

We arrange the paper as follows: In Section 2, the construction process of the risk contagion model is explained in detail. In Section 3, equilibria stability and Hopf bifurcation analysis are investigated, and the corresponding numerical simulations are given in Section 4. Finally, relevant conclusions are drawn, the theoretical results provide a theoretical basis for governments and enterprises to monitor and prevent financial risks.

In this paper, we construct a risk contagion model by taking into account the prior prevention of regulatory agencies, the self-rescue of individuals, and temporary immunity period. We consider four types of subjects who are susceptible to infect risks (S), exposed to risk shocks and in a latent state (E), infected with risks (I), and recovered from infection (R). Based on [11], the assumptions made in this paper are as follows:

Assumption 1: The system network is not closed but limited, and each financial individual is a node in the network system. Financial individuals enter or exit the system at the same proportion during the risk contagion process.

Assumption 2: There are differences among financial individuals. After being infected, some individuals can become recovered groups through various preventive measures, while some individuals exit the financial system because they cannot acquire immunity.

Assumption 3: All susceptible individuals can establish lending relationships or hold common assets with infected individuals or exposed individuals, which further spread financial risks through connection relationships.

Assumption 4: Exposed individuals can perceive the existence of risks and control the spread of financial risks through their own financing capabilities or external policy support.

Based on the above assumptions, the financial risk contagion model established in this paper is as follows:

$ \begin{equation} \left\{ \begin{aligned} \frac { d S ( t ) } { d t } &= \mu ( 1 - \omega ) T - \beta _ { 1 } S ( t ) E ( t ) - \beta _ { 2 } S ( t ) I ( t ) + \alpha b E ( t ) + \varepsilon R ( t - \tau ) - \mu S ( t ), \\ \frac { d E ( t ) } { d t } &= \beta _ { 1 } S ( t ) E ( t ) + \beta _ { 2 } S ( t ) I ( t ) - \alpha E ( t ) - \mu E ( t ) , \\ \frac { d I ( t ) } { d t } &= \alpha ( 1 - b ) E ( t ) - \sigma I ( t ) - m I ( t ) - \mu I ( t ), \\ \frac { d R ( t ) } { d t } &= \mu \omega T + m I ( t ) - \varepsilon R ( t - \tau ) - \mu R ( t ) , \end{aligned} \right. \end{equation} $

(2.1)

where the initial conditions of Eq.(2.1) are $ S(0)>0, \, E(0)>0, \, I(0)>0, \, R(0)>0 $, and let $ N(0) $ be the total number of individuals when $ t=0 $, i.e.

$ \begin{equation} N(0) = S ( 0 ) +E(0)+ I ( 0 ) +R(0)> 0. \end{equation} $

(2.2)

The parameters used in Eq.(2.1) and the meanings are shown in the following table.

In order to ensure that Eq.(2.1) has practical significance, in this part we will discuss the positive invariant set of Eq.(2.1) to illustrate its positivity and boundedness.

Proposition 2.1  The positive invariant set of Eq.(2.1) is $ \Omega = \left\{ ( S , E , I , R ) | S , \, E , \, I , \notag \right. \left.R \geq 0 , \, S + E + I + R \leq T \right\} $.

Proof  The initial conditions of (2.1) are $ S(0)>0, \, E(0)>0, \, I(0)>0, \, R(0)>0. $ When $ t \geq 0 $,

$ \begin{aligned} &\frac { d S } { d t } | _ { S = 0 } = \mu ( 1 - \omega ) T + \alpha b E + \varepsilon R ( t - \tau ) > 0, \\ &\frac { d E } { d t } | _ { E = 0 } = \beta _ { 2 } S I > 0, \\ &\frac { d I } { d t } | _ { I = 0 } = \alpha ( 1 - b ) E > 0, \\ &\frac { d R } { d t } | _ { R = 0 } = \mu \omega T + m I > 0.\\ \end{aligned} $

Since $ S(t), \, E(t), \, I(t), \, R(t) $ are continuous functions about $ t $, all solutions in Eq.(2.1) are non-negative when $ t \geq 0 $. So $ \Omega _ { 1 } = \left\{ ( S , E , I , R ) | S , \, E , \, I , \, R \geq 0 \right\} $ is positive and invariant to (2.1).

Then, we set $ N ( t ) = S ( t ) + E ( t ) + I ( t ) + R ( t ) $.

According to (2.1) we can get

$ \begin{equation} \frac { d N } { d t } = \mu T-\mu N-\sigma I \leq \mu T-\mu N . \end{equation} $

(2.3)

So, there is $ 0 \leq N \leq T $.

Therefore, the positive invariant set of Eq.(2.1) is $ \Omega = \left\{ ( S , E , I , R ) | S , \, E , \, I , \, R \geq 0 , \notag \right. \left. 0 \leq S + E + I + R \leq T \right\} . $

In the financial risk system, equilibria of Eq.(2.1) are the key to risk contagion process. Therefore, in this section we will analyze the existence and dynamic characteristics of equilibria in the positive invariant set $ \Omega $.

In this part we will discuss the existence of equilibria in Eq.(2.1).

Due to the non-negative characteristics of Eq.(2.1), we can get that Eq.(2.1) has two equilibria: the risk-free equilibrium $ P_{0}(S_{0}, E_{0}, I_{0}, R_{0}) $ and the risky equilibrium $ P^{*}(S^{*}, E^{*}, \notag I^{*}, R^{*}) $.

First, for the risk-free equilibrium $ P_{0} $, the solution can be obtained as $ S_{0}=\frac { \mu (1- \omega)T+ \varepsilon T } { \varepsilon + \mu } , \notag E_{0}=0, \, I_{0}=0, \, R_{0}=\frac { \mu \omega T} { \varepsilon + \mu } $. Since $ T, \, \varepsilon, \, \mu, \, \varepsilon + \mu>0 $ and $ 0\leq \omega \leq 1 $, $ P_{0} $ exists.

Before analyzing the existence of $ P^{*} $, we first discuss the basic reproduction number $ (\widetilde{R}_{0}) $. The basic reproduction number is a critical indicator that reflects whether risks are contagious among individuals, and is of great significance to our study of risk contagion.

For the calculation of $ \widetilde{R}_{0} $, we use the next generation matrix method in [14] to obtain

$ \begin{equation} \widetilde{R}_{0} = \frac {\beta _ { 1 } S_ { 0 }(\sigma + m+ \mu )+\beta _ { 2 } S_ { 0 }\alpha ( 1 - b ) } { (\alpha + \mu)( \sigma + m+ \mu) } . \end{equation} $

(3.1)

Next we will discuss the existence of $ P^{*} $. According to the expression of $ \widetilde{R}_{0} $, we get that the existence of $ P^{*} $ satisfies the following theorem.

Theorem 3.1  When $ \widetilde{R}_{0}>1 $, the risky equilibrium $ P^{*}(S^{*}, E^{*}, I^{*}, R^{*}) $ of Eq.(2.1) exists; when $ \widetilde{R}_{0}\leq 1 $, $ P^{*} $ does not exist.

Proof  Solving the risky equilibrium of Eq.(2.1), we can get

$ I ^ { * } = \frac {\alpha ( 1 - b )} { \sigma + m+ \mu}E ^ { * } , \; R^ { * } = \frac {mI ^ { * }+\mu \omega T} { \varepsilon + \mu}. $

Substituting $ I ^ { * } = \frac {\alpha ( 1 - b )} { \sigma + m+ \mu}E ^ { * } $ into $ \beta _ { 1 } S^ { * }E ^ { * } + \beta _ { 2 } S^ { * }I ^ { * } - \alpha E ^ { * } - \mu E^ { * }=0, $ we can get

$ \begin{equation} S ^ { * } = \frac {(\alpha + \mu)( \sigma + m+ \mu)} { \beta _ { 1 }(\sigma + m+ \mu)+\beta _ { 2 }\alpha ( 1 - b )} . \end{equation} $

(3.2)

Substituting Eq.(3.2) into $ \mu ( 1 - \omega ) T - \beta _ { 1 } S ^ { * } E ^ { * }- \beta _ { 2 } S ^ { * }I^ { * } + \alpha b E ^ { * } + \varepsilon R ^ { * } - \mu S ^ { * }=0 $, we get

$ \begin{equation} E ^ { * } = \frac {( \sigma + m+ \mu)[ (\varepsilon + \mu)\mu ( 1 - \omega ) T+\varepsilon\mu \omega T-\mu (\varepsilon + \mu)S ^ { * }]} {( \sigma + m+ \mu) (\varepsilon + \mu)(\mu+\alpha ( 1 - b ))+\varepsilon m\alpha ( 1 - b )} . \end{equation} $

(3.3)

$ S ^ { * }>0 $ is obviously established. Next we prove that $ E ^ { * }>0 $.

$ \begin{aligned} &\; \; \; \; \; E ^ { * }>0\\ &\Leftrightarrow (\varepsilon + \mu)\mu ( 1 - \omega ) T+\varepsilon\mu \omega T-\mu (\varepsilon + \mu)S ^ { * }>0\\ &\Leftrightarrow (\varepsilon + \mu)S _{0}- (\varepsilon + \mu)S ^ { * }>0\\ &\Leftrightarrow \beta _ { 1 } S_ { 0 }(\sigma + m+ \mu )+\beta _ { 2 } S_ { 0 }\alpha ( 1 - b )>(\alpha + \mu)( \sigma + m+ \mu)\\ &\Leftrightarrow \widetilde{R}_{0}>1 \end{aligned} $

Therefore, if and only if $ \widetilde{R}_{0}>1 $, the risky equilibrium $ P^{*}(S^{*}, E^{*}, I^{*}, R^{*}) $ exists.

Since the stability of $ P_{0} $ determines whether financial risks will eventually be eliminated, in this part we will analyze the stability of $ P_{0} $ and the analytical conditions for Hopf bifurcation. And the relevant results are shown in the following theorem.

Theorem 3.2  When $ \widetilde{R}_{0}<1 $ and $ \varepsilon \leq \mu $, $ P_{0}(S_{0}, E_{0}, I_{0}, R_{0}) $ of Eq.(2.1) is locally asymptotically stable. When $ \varepsilon > \mu $, Eq.(2.1) undergoes a Hopf bifurcation at $ \tau = \tau _ { 1} $, and when $ \tau \in ( 0 , \tau _ { 1} ) $, $ P_{0} $ is locally asymptotically stable; when $ \tau > \tau _ { 1} $, a cluster of periodic solutions bifurcate from $ P_{0} $.

Proof  For Eq.(2.1), the Jacobian matrix of $ P_{0} $ is

$ J_{0}= \left[ \begin{array} { cccc } { -\mu } & { -\beta _ { 1 } S_ { 0 }+ \alpha b}&{-\beta _ { 2 } S_ { 0 }}&{\varepsilon e^{-\lambda \tau}}\\ { 0} & { \beta _ { 1 } S_ { 0 }- \alpha -\mu}& { \beta _ { 2 } S_ { 0 } } & { 0 }\\ { 0} & {\alpha ( 1 - b )}& { -( \sigma + m+ \mu)} & { 0 }\\ { 0} & { 0}& { m } & { -\mu -\varepsilon e^{-\lambda \tau} }\\ \end{array} \right]. $

Calculating its characteristic equation to get

$ \begin{equation} (\lambda + \mu) (\lambda + \mu+\varepsilon e^{-\lambda \tau})[\lambda^{2}+a_{1}\lambda+a_{2} ] = 0, \end{equation} $

(3.4)

where $ a_{1}=( \sigma + m+ \mu)-(\beta _ { 1 } S_ { 0 }- \alpha -\mu), \, a_{2}=-( \sigma + m+ \mu)(\beta _ { 1 } S_ { 0 }- \alpha -\mu)- \beta _ { 2} S_ { 0 } \alpha (1-b) $.

Obviously, $ \lambda_{1} =- \mu $ has a negative real part.

For

$ \begin{equation} \lambda^{2}+a_{1}\lambda+a_{2} = 0 . \end{equation} $

(3.5)

From Eq.(3.1), we can obtain when $ \widetilde{R}_{0}<1 $, then $ \beta _ { 1 } S_ { 0 }<\alpha +\mu $, thus $ a_{1}>0 $; when $ \widetilde{R}_{0}<1 $, then $ \beta _ { 1 } S_ { 0 }(\sigma + m+ \mu )+\beta _ { 2 } S_ { 0 }\alpha ( 1 - b )<(\alpha + \mu)( \sigma + m+ \mu) $, thus $ a_{2}>0 $.

Therefore, $ a_{1}a_{2}>0 $ is established. According to the Routh-Hurwitz criteria, all roots of Eq.(3.5) have negative real parts.

For

$ \begin{equation} \lambda+\mu+\varepsilon e^{-\lambda \tau}=0. \end{equation} $

(3.6)

Let $ \lambda = i \eta( \eta > 0 ) $ be the root of Eq.(3.6), then substituting it into Eq.(3.6) we obtain

$ \begin{equation} i\eta+\mu+\varepsilon\cos\eta\tau-i\varepsilon\sin\eta\tau=0. \end{equation} $

(3.7)

Separating the real and imaginary parts of Eq.(3.7) we can get

$ \begin{equation} \begin{cases} \eta-\varepsilon\sin\eta\tau=0, \\ \mu+\varepsilon\cos\eta\tau=0. \end{cases} \end{equation} $

(3.8)

From Eq.(3.8), we can see that $ \eta^{2}=\varepsilon^{2}-\mu^{2} $, so $ \eta $ exists when $ \varepsilon>\mu $.

When $ \varepsilon>\mu $, the time delay corresponding to Eq.(3.7) is

$ \begin{equation} \tau _ { 1 } ^{(j)}= \frac { 1 } { \eta } \arccos (-\frac {\mu } {\varepsilon } ) + \frac { 2 j \pi } { \eta } , j=0, 1, 2, ... \end{equation} $

(3.9)

Let $ \tau _ { 1 }=\min \left\{ \tau _ { 1 } ^ { ( j ) } \right\} , j=0, 1, 2, ... $, then differentiating two sides of Eq.(3.6) with respect to $ \tau $, we can obtain

$ \begin{equation} \left[ \frac { d \lambda } { d \tau } \right] ^ { - 1 } = -\frac {1} { \lambda ( \lambda +\mu) } - \frac { \tau } { \lambda } . \end{equation} $

(3.10)

And then

$ \begin{equation} R e \left[ \frac { d \lambda } { d \tau } \right]^ { - 1 } _ { \tau =\tau _ { 1 } }=\frac {1 } { \eta^{2} + \mu^{2} } \neq 0. \end{equation} $

(3.11)

Therefore, using the Hopf bifurcation theorem in [15], we can get that the Hopf bifurcation occurs in Eq.(2.1) when $ \tau=\tau _ { 1 } $.

To sum up, we can conclude that when $ \widetilde{R}_{0}<1 $ and $ \varepsilon \leq \mu $, the risk-free equilibrium of Eq.(2.1) is asymptotically stable, and risks will eventually disappear; when $ \widetilde{R}_{0}>1 $, the risk-free equilibrium $ P_{0} $ will lose stability, and the risky equilibrium $ P^{*} $ will occur.

Before studying the stability of $ P^{*} $ and the analytical conditions for Hopf bifurcation, we first discuss the roots of the fourth degree polynomial.

For

$ \begin{equation} v ^ { 4 } + l _ { 11 } v ^ { 3 } + l _ { 1 2 } v ^ { 2 } + l _ { 13 } v + l _ { 14 } = 0 . \end{equation} $

(3.12)

We assume that $ h ( v ) = v ^ { 4 } + l _ { 11 } v ^ { 3 } + l _ { 1 2 } v ^ { 2 } + l _ { 13 } v + l _ { 14 }, \, p _ { 1 } = \frac { l _ { 1 2 } } { 2 } - \frac { 3 } { 1 6 } l _ { 11 } ^ { 2 } , \, q _ { 1 } = \frac { l _ { 11 }^ { 3 } } { 3 2 } - \frac {l _ { 11 }l _ { 12 } } { 8 } + l _ { 13 } , \, D = ( \frac { q _ { 1 } } { 2 } ) ^ { 2 } + ( \frac { p _ { 1 } } { 3 } ) ^ { 3 } $, then the roots of Eq.(3.12) satisfy the following lemma.

Lemma 3.1 [16]  Let $ v _ { 1 } , v _ { 2 } , v _ { 3 } $ be the roots of $ h' ( v ) =0 $, and then

(1) when $ l _ { 14 }<0 $, Eq.(3.12) has at least one positive root;

(2) when $ l _ { 14 }\geq 0 $:

(ⅰ) if $ D\geq 0 $, Eq.(3.12) has positive roots if and only if $ v _ { 1 } > 0 $ and $ h ( v _ { 1 } ) < 0 $;

(ⅱ) if $ D<0 $, Eq.(3.12) has positive roots if and only if there is at least one $ v ^ { * } \in \left\{ v _ { 1 } , v _ { 2 } , v _ { 3 } \right\} $ such that $ v ^ { * } > 0 $ and $ h ( v ^ { * } ) \leq 0 $.

Then, through Lemma 3.1 and related analysis, we can conclude the following theorem.

Theorem 3.3  For the conditions:

$ (H_{10}) $ $ \widetilde{R}_{0}>1 $;

$ (H_{11}) $ One of the following cases holds:

$ a:l _ { 14 }<0 $;

$ b:l _ { 14 }\geq 0 $, $ D\geq 0 $ and $ v _ { 1 } > 0 $, $ h ( v _ { 1 } ) < 0 $;

$ c:l _ { 14 }\geq 0 $, $ D< 0 $ and there exists $ v ^ { * } \in \left\{ v _ { 1 } , v _ { 2 } , v _ { 3 } \right\} $ such that $ v ^ { * } > 0 $, $ h ( v ^ { * } ) \leq 0 $;

$ (H_{12}) $ $ h^ { \prime } ( v _ { 1 0 } ) \neq 0 $.

If $ (H_{10})-(H_{12}) $ hold, then when $ \tau \in ( 0 , \tau _ { 2 } ) $, $ P^{*} $ is locally asymptotically stable; when $ \tau = \tau _ { 2 } $, Eq.(2.1) undergoes a Hopf bifurcation and a cluster of periodic solutions bifurcate from $ P^{*} $.

Proof  First, we linearize Eq.(2.1) at $ P^{*} $ to get

$ \begin{equation} \left\{ \begin{aligned} \frac { d \hat{S} ( t ) } { d t } &= a_{11} \hat{S} ( t )+a_{12} \hat{E} ( t )+a_{13}\hat{I} ( t ) +a_{14} \hat{R} ( t - \tau ) , \\ \frac { d \hat{E} ( t ) } { d t } &= a_{21} \hat{S} ( t )+a_{22} \hat{E} ( t )+a_{23}\hat{I} ( t ) , \\ \frac { d\hat{I} ( t ) } { d t } &= a_{31} \hat{E} ( t )+a_{32}\hat{I} ( t ), \\ \frac { d\hat{R} ( t ) } { d t } &= a_{41}\hat{I} ( t ) +a_{42}\hat{R} ( t )+a_{43} \hat{R} ( t - \tau ) , \end{aligned} \right. \end{equation} $

(3.13)

where

$ \begin{aligned} &\hat{S}=S-S^{ * }, \, \hat{E}=E-E^{ * }, \, \hat{I}=I-I^{ * }, \, \hat{R}=R-R^{ * }.\\ &a _ { 1 1 } = - ( \beta _ { 1 } E^{ * } + \beta _ { 2 } I^{ * } + \mu ) , \, a _ { 1 2 } = - \beta _ { 1 } S^{ * }+\alpha b , \, a _ { 1 3 } = - \beta _ { 2 } S^{ * } , \, a _ { 1 4 } = \varepsilon , \\ &a _ { 2 1 } = \beta _ { 1 } E^{ * } + \beta _ { 2 } I^{ * } , \, a _ { 2 2 } = \beta _ { 1 } S^{ * }-\alpha -\mu , \, a _ { 2 3 } = \beta _ { 2 } S^{ * } , \\ &a _ { 31 } = \alpha (1-b), \, a _ { 32 } = -( \sigma + m+ \mu) , \, a _ { 4 1 } = m, \, a _ { 4 2 } = -\mu , \, a _ { 4 3 } =-\varepsilon . \end{aligned} $

The characteristic equation of Eq.(3.13) at $ P^{*} $ is

$ \begin{equation} \lambda ^ { 4 } + c _ { 1 }\lambda ^ { 3 } + c _ { 2 } \lambda ^ { 2 } + c _ { 3 } \lambda + c _ { 4 } + (d _ { 1 }\lambda ^ { 3 } + d _ { 2 } \lambda ^ { 2 } + d _ { 3 } \lambda + d _ { 4 }) e ^ { - \lambda \tau} = 0 , \end{equation} $

(3.14)

where

$ \begin{aligned} &b_{1}=a_{32}+a_{22}, \, b_{2}=a_{32}a_{22}-a_{23}a_{31}, \\ &c _ { 1 } = -b_{1}-a_{42}-a_{11}, \, c _ { 2 } = b_{2}+b_{1}(a_{42}+a_{11})+a_{42}a_{11}-a_{21}a_{12}, \\ &c_{3}=-b_{2}(a_{42}+a_{11})-b_{1}a_{42}a_{11}+a_{21}a_{12}(a_{42}+a_{32})-a_{21}a_{31}a_{13}, \\ &c_{4}=b_{2}a_{42}a_{11}-a_{21}a_{12}a_{42}a_{32}+a_{21}a_{31}a_{13}a_{42}, \\ &d _ { 1 } = -a_{43}, \, d _ { 2} = b_{1}a_{43}+a_{43}a_{11}, \, d _ {3} = -b_{2}a_{43}-b_{1}a_{43}a_{11}+a_{21}a_{12}a_{43}, \\ &d _ {4} = b_{2}a_{11}a_{43}-a_{21}a_{12}a_{32}a_{43}+a_{21}a_{31}a_{13}a_{43}-a_{21}a_{31}a_{41}a_{14}. \end{aligned} $

Let $ \lambda = i \theta( \theta > 0 ) $ be the root of Eq.(3.14), then substituting it into Eq.(3.14) we obtain

$ \begin{equation} \theta ^ { 4 } - i c _ { 1 } \theta ^ { 3 } - c _ { 2} \theta ^ { 2 } + i c _ { 3 } \theta + c _ { 4 } + ( - i d _ { 1 } \theta ^ { 3 } - d _ { 2 } \theta^ { 2 } + i d _ { 3 } \theta+ d _ { 4 } ) ( \cos \theta\tau - i \sin \theta\tau ) = 0 . \end{equation} $

(3.15)

Separating the real and imaginary parts of Eq.(3.15) we can get

$ \begin{equation} \begin{cases} E _ { 1 1 } ( \theta ) \cos \theta\tau + E _ { 1 2 } ( \theta ) \sin\theta\tau = E _ { 1 3 } ( \theta) , \\ E _ { 1 2 } ( \theta ) \cos \theta\tau - E _ { 1 1 } ( \theta ) \sin\theta\tau = E _ { 1 4 } ( \theta ) , \end{cases} \end{equation} $

(3.16)

where

$ E _ { 11 } ( \theta ) = d _ { 4 } - d _ { 2 }\theta ^ { 2 } , \, E _ { 12 } ( \theta ) = d _ { 3 }\theta - d _ { 1 }\theta ^ { 3 } , \, E _ { 13 } ( \theta ) = - \theta ^ { 4 }+c _ { 2 }\theta ^ { 2 }-c _ { 4 } , \, E _ { 14 } ( \theta ) = c _ { 1 }\theta ^ { 3 }- c _ { 3 }\theta . $

Simplifying Eq.(3.16) we can get the following equation about $ \theta $:

$ \begin{equation} \theta ^ { 8 } + l _ { 11 } \theta ^ { 6 } + l _ { 1 2 } \theta^ { 4 } + l _ { 13 } \theta ^ { 2 } + l _ { 14 } = 0, \end{equation} $

(3.17)

where

$ l _ { 11 }=c _ { 1 }^ { 2 }-2c _ { 2 }-d _ { 1 }^ { 2 }, \, l _ { 12 }=2c _ { 4 }+c _ { 2 }^ { 2 }-2c _ { 1 }c _ { 3 }-d _ { 2 }^ { 2 }+2d _ { 1 }d _ { 3 }, \, l _ { 13 }=c _ { 3 }^ { 2 }-2c _ { 2 }c _ { 4 }+2d _ { 2 }d _ { 4 }-d _ { 3 }^ { 2 }, \, l _ { 14 }=c _ { 4 }^ { 2 }-d _ { 4 }^ { 2 }. $

Let $ \theta ^ { 2 }=v $, then Eq.(3.17) can be reduced to

$ \begin{equation} v ^ { 4 } + l _ { 11 } v ^ { 3 } + l _ { 1 2 } v ^ { 2 } + l _ { 13 } v + l _ { 14 } = 0 . \end{equation} $

(3.18)

According to Lemma 3.1, we can get that when the condition $ (H_{11}) $ holds, Eq.(3.18) will have at least one positive root. Without loss of generality, we assume that Eq.(3.18) has four positive roots, denoted $ v _ { 11 }, \, v _ { 12 }, \, v _ { 13 } $ and $ v _ { 14 } $ respectively, then Eq.(3.17) has four positive roots $ \theta _ { k } = \sqrt { v _ { 1 k } }, \, k=1, 2, 3, 4. $

Therefore, for each fixed $ \theta _ { k } $, the time delay corresponding to Eq.(3.15) is

$ \begin{equation} \tau _ { 2 k }^{(j)} = \frac { 1 } { \theta _ { k } } \arccos \left[ \frac { E _ { 1 1 } ( \theta _ { k } ) E _ { 13 } ( \theta _ { k } ) + E _ { 1 2 } ( \theta _ { k } ) E _ { 14 } ( \theta _ { k } ) } { E _ { 1 1 } ^ { 2 } ( \theta _ { k } ) + E _ { 1 2 } ^ { 2 } ( \theta _ { k } ) } \right] + \frac { 2 j \pi } { \theta _ { k } } , \, k=1, 2, 3, 4.\, j=0, 1, 2, ... \end{equation} $

(3.19)

Let $ \tau _ { 2 } = \min \left\{ \tau _ { 2 k } ^ { ( 0 ) } \right\} , \, v _ { 1 0 } = \theta _ { 10 } ^ { 2 }, \, \theta _ { 10 } = \theta _ { k } |_{\tau = \tau _ { 2 }} , \, k = 1 , 2 , 3, 4 . $

Differentiating two sides of Eq.(3.14) with respect to $ \tau $, we can obtain

$ \begin{equation} \left[ \frac { d \lambda } { d \tau } \right] ^ { - 1 } = -\frac {4\lambda ^ { 3 } +3c_{1} \lambda ^ { 2 } + 2 c _ { 2 } \lambda + c _ { 3 } } { \lambda ( \lambda ^ { 4 } + c _ { 1 } \lambda ^ { 3 } + c _ { 2 } \lambda^ { 2 } + c _ { 3 } \lambda+c_{4}) } + \frac { 3d _ { 1 } \lambda^ { 2 }+2 d _ { 2 } \lambda + d _ { 3 } } { \lambda ( d _ { 1 }\lambda ^ { 3 }+ d _ { 2 } \lambda ^ { 2 } + d _ { 3 } \lambda + d _ { 4 } ) } - \frac { \tau } { \lambda } . \end{equation} $

(3.20)

And then

$ \begin{equation} R e \left[ \frac { d \lambda } { d \tau } \right]^ { - 1 } _ { \tau=\tau _ { 2 } } =\frac { h ^ { \prime } ( v _ { 1 0 } ) } { E _ { 1 1 } ^ { 2 } ( \theta _ { 10 }) + E _ { 1 2 } ^ { 2 } ( \theta _ { 10 }) } . \end{equation} $

(3.21)

So when the condition $ (H_{12}):h^ { \prime } ( v _ { 1 0 } ) \neq 0 $ holds, $ R e \left[ \frac { d \lambda } { d \tau } \right] _ { \tau =\tau _ { 2 }}\neq 0 $.

Based on above discussion and the Hopf bifurcation theorem in [15], we can get that the Hopf bifurcation occurs when $ \tau=\tau _ { 2 } $.

Therefore, the critical value of temporary immunity period is not infinite, but there is a threshold. When temporary immunity time is within the critical value, the risk will eventually stabilize, and when it exceeds the critical value, the system will eventually become unstable.

In this section, we will give specific parameter values to support the theoretical analysis in Section 3.

From Theorem 3.2, we conclude that when $ \widetilde{R}_{0}<1 $ and $ \varepsilon \leq \mu $, $ P_{0}(S_{0}, E_{0}, I_{0}, R_{0}) $ is locally asymptotically stable. Therefore, we take $ \mu=\varepsilon=0.05, \; \omega=0.8 , \; T=6, \; \beta_{1}=0.01, \; \beta_{2}=0.02, \; \alpha=0.06, \; b=0.5 , \; \sigma=0.02 , \; m=0.08 $ here. The calculation shows that $ \widetilde{R}_{0}=0.3535<1 $, and the system only has a risk-free equilibrium $ P_{0}(3.6, 0, 0, 2.4) $. At this time, the number of infected individuals will continue to decrease, which is shown in Fig. 2, financial risks will eventually disappear.

Next, we take $ \varepsilon=0.05, \; \mu=0.02 $, and keep other parameters unchanged. At this time $ \varepsilon > \mu $, we can get $ \widetilde{R}_{0}=0.6364<1 $, and the system only has a risk-free equilibrium $ P_{0}(4.6286, 0, 0, 1.3714) $. According to Eq.(3.8), we can conclude that $ \eta=0.0458 $ exists, which satisfies the conditions for Hopf bifurcation, and the time delay critical value $ \tau_{ 1}=43.2576 $.

Fig. 3 shows the case of $ \tau=37.6<\tau_{ 1} $. It can be seen that the system produces damped oscillation and finally converges to $ P_{0} $. This means that under these parameter values, the risk system will eventually stabilize and the risk will disappear.

Fig. 4 shows the case of $ \tau=44>\tau_{ 1} $. It can be seen that when $ \tau $ passes through the critical value $ \tau_{ 1 } $, the Hopf bifurcation occurs and a cluster of periodic solutions bifurcate from $ P_{0} $. In addition, according to the definition of $ \widetilde{R}_{0} $, when $ \widetilde{R}_{0}<1 $, the maximum number of individuals that can be infected by an infectious individual is less than 1, so when $ \widetilde{R}_{0}<1 $ the risk can always be gradually eliminated.

According to Lyapunov's first method, we use the linearized model Eq.(3.13) to study the local stability of Eq.(2.1) near $ P^{*} $.

We conclude that $ P^{*}(S^{*}, E^{*}, I^{*}, R^{*}) $ exists only when $ \widetilde{R}_{0}>1 $ in Theorem 3.1. Therefore, we take $ \mu=0.02, \; \varepsilon=0.6, \; \omega=0.5 , \; T=500, \; \beta_{1}=0.07, \; \beta_{2}=0.08, \; \alpha=0.1, \; b=0.3 , \; \sigma=0.02, \; m=0.6 $. The calculation shows that $ \widetilde{R}_{0}=322.8327>1 $. At this time, the system has a risky equilibrium $ P^{*}(1.5238, 370.2347, 40.4944, 47.2527) $. In addition, we can get $ l_{14}=-0.0733<0 $ and $ h^ { \prime } ( v _ { 2 0 } )=282.5254 \neq 0 $, so the conditions $ (H_{10})-(H_{12}) $ in Theorem 3.3 hold. From the above parameters, we can get $ \theta_{ 1 0 }=0.6277 $, so the corresponding time delay critical value $ \tau_{ 2 }=2.6425 $.

We take $ \tau=2.5<2.6425 $ in Fig. 5, it can be seen that $ P^{*} $ is locally asymptotically stable. But when $ \tau $ passes through the critical value 2.6425, $ P^{*} $ will lose stability and a cluster of periodic solutions will bifurcate from $ P^{*} $. Therefore, we take $ \tau=2.65>2.6425 $, and the corresponding result is shown in Fig. 6. Furthermore, the simulation results also confirmed the similarity of local stability between Eq.(2.1) and Eq.(3.13).

In this part we will test the effect of preventive intensity $ (\omega) $ and self-rescue capabilities $ (b) $ on the number of $ I(t) $. From Eq.(3.1) we can conclude that both $ \omega $ and $ b $ will effect the value of $ R_{0} $ and the speed of risk propagation.

In addition, we can see from Fig. 7 that when risks spread in the system, the improvement of self-rescue capabilities can significantly reduce the number of $ I(t) $, and even cause a negative growth in $ I(t) $. In contrast, the prior prevention of regulatory agencies has little effect in reducing the number of $ I(t) $, which shows the restriction of relevant policies cannot fundamentally solve risk infection in the operation of financial systems.

In this paper, we comprehensively consider the prior prevention of regulatory agencies, the self-rescue of individuals, and the temporary immunity period in financial risk contagion, build a dynamic model to describe the contagion process of financial risks. It has been shown that the temporary immunity time delay has a significant impact on the stability of risk contagion model. When relevant parameters satisfy certain conditions, the system will exist a critical value $ \tau_{ i} $ for immunity period. If the immunity period is shorter than $ \tau_{ i} $, the model will be stable, but if the immunity period is longer than $ \tau_{ i} $, the model will undergo Hopf bifurcation. This illustrates that when risks are transmitted among individuals, recovered groups do not gain permanent immunity, but rather there exists a critical value that allows the system to operate smoothly.

Furthermore, we derived the conditions for the disappearance and persistence of risks, demonstrated that the system was stable and risks would eventually disappear when $ \widetilde{R}_{0}<1 $. If $ \widetilde{R}_{0}>1 $, risks will persist. This provides theoretical support for government and enterprises to effectively monitor risks. Finally, it was found that when risks spread in a financial system, the prior prevention of regulatory agencies can only serve as an auxiliary role. However, the self-rescue capabilities of individuals have a significant impact on the number of infections. Therefore, individuals should actively utilize their own resources to counteract risks when infected with risks.