Stationary distribution and probability density for a stochastic SEIR-type model of coronavirus (COVID-19) with asymptomatic carriers

Abstract

In this paper, we propose a stochastic SEIR-type model with asymptomatic carriers to describe the propagation mechanism of coronavirus (COVID-19) in the population. Firstly, we show that there exists a unique global positive solution of the stochastic system with any positive initial value. Then we adopt a stochastic Lyapunov function method to establish sufficient conditions for the existence and uniqueness of an ergodic stationary distribution of positive solutions to the stochastic model. Especially, under the same conditions as the existence of a stationary distribution, we obtain the specific form of the probability density around the quasi-endemic equilibrium of the stochastic system. Finally, numerical simulations are introduced to validate the theoretical findings.

1. Introduction

In early December of 2019, a novel coronavirus (COVID-19) disease was reported as a major health hazard by the World Health Organization (WHO) [1]. This coronavirus is a kind of single stranded, enveloped and positive sense virus belonging to the RNA coronaviridae family [2], [3]. At present, this disease has drawn much attention of researchers from all over the world and different areas. To defeat the epidemic, many researchers have made great attempts to develop various mathematical models for the transmission dynamics of this epidemic [1], [4], [5], [6], [7], [8], these models include the SEIR-type model and SEIRS-type model. In these models, the authors assumed that after a latent period, exposed individuals can show symptoms that make them easy to identify and isolate, thus they are not able to transmit the disease. However, exposed individuals can also exhibit asymptomatic after incubation, and they can continue to infect others because asymptomatic and susceptible individuals have the same set of characteristics, there is no effective way to identify them. Therefore, the classical SEIR model or SEIRS model fails to explain the discrepancy by distinguishing between the symptomatic and asymptomatic. Taking this into account, in this paper, we first develop an SEIR-type model to describe the transmission dynamics of COVID-19 which takes the following form:

$$\left\{\begin{array}{c}\dot{S}=\Lambda -\mu S-\beta S(\eta I_A+I_S),\hfill \\ \dot{E}=\beta S(\eta I_A+I_S)-(\mu +\theta )E,\hfill \\ {\dot{I}}_S=p\theta E-(\mu +{\alpha }_S+{\gamma }_S)I_S,\hfill \\ {\dot{I}}_A=(1-p)\theta E-(\mu +{\alpha }_A+{\gamma }_A)I_A,\hfill \\ \dot{R}={\gamma }_S{I}_S+{\gamma }_A{I}_A-\mu R.\hfill \end{array}\right)$$ (1.1)

Here the infectious population $I$ is split into two degenerate infectious groups $I_A$ and $I_S$, $I_A\left(t\right)$ and $I_S\left(t\right)$ denote the undetected asymptomatic and symptomatic members of the infected population at time $t$, respectively. $S\left(t\right)$ denotes the number of individuals who are susceptible to the disease, $E\left(t\right)$ represents the number of exposed (in the latent period) individuals and $R\left(t\right)$ represents the number of individuals who have been infected and then removed from the possibility of being infected again at time $t$, respectively. All parameters are strictly positive and their descriptions are given in Table 1.

Table 1.

Summary of model parameters.

Parameter	Description
$\Lambda$	Recruitment rate of the population
$\mu$	Natural death rate of the population
$\beta$	Transmission or contact rate in response to public health interventions
$\eta$	Asymptomatic infectiousness
$p$	Fraction of infection cases that are symptomatic
$1-p$	Fraction of infection cases that are asymptomatic
$\theta$	Latency rate
${\alpha }_S$	Disease-related death rate of the symptomatic infectious group
${\alpha }_A$	Disease-related death rate of the asymptomatic infectious group
${\gamma }_S$	Symptomatic recovery rate
${\gamma }_A$	Asymptomatic recovery rate

Note that the variable $R$ of system (1.1) does not appear explicitly in the first four equations, which implies that the individuals in the compartment $R$ do not transmit infection. By ignoring the equation for $R$, model (1.1) reduces to the following system:

$$\left\{\begin{array}{c}\dot{S}=\Lambda -\mu S-\beta S(\eta I_A+I_S),\hfill \\ \dot{E}=\beta S(\eta I_A+I_S)-(\mu +\theta )E,\hfill \\ {\dot{I}}_S=p\theta E-(\mu +{\alpha }_S+{\gamma }_S)I_S,\hfill \\ {\dot{I}}_A=(1-p)\theta E-(\mu +{\alpha }_A+{\gamma }_A)I_A.\hfill \end{array}\right)$$ (1.2)

According to the next generation matrix method, the basic reproduction number $R_0$ for system (1.2) is defined by

$$R_0=\frac{p\beta \Lambda \theta }{\mu (\mu +\theta )(\mu +{\alpha }_S+{\gamma }_S)}+\frac{(1-p)\beta \eta \Lambda \theta }{\mu (\mu +\theta )(\mu +{\alpha }_A+{\gamma }_A)},$$

which is used to determine whether the disease occurs or not.

In addition, the corresponding dynamic behavior of system (1.2) is as follows:

$•$ If $R_0\le 1$, system (1.2) has a disease-free equilibrium $P_0(S_0,0,0,0)=(\Lambda /\mu ,0,0,0)$ and it is globally asymptotically stable (GAS) in the positive invariant set $\Omega$, where

$$\Omega ≔\left\{(S,E,I_S,I_A)\in {\mathbb{R}}_{+}^4:S\le \frac{\Lambda }{\mu },S+E+I_S+I_A\le \frac{\Lambda }{\mu }\right\}.$$

$•$ If $R_0>1$, $P_0$ is unstable and there is a unique GAS endemic equilibrium $P^{+}(S^{+},E^{+},I_S^{+},I_A^{+})$ in the interior of $\Omega$, where

$$S^{+}=\frac{\Lambda (1-p)(\mu +{\alpha }_S+{\gamma }_S)}{\beta I_A^{+}[\eta (1-p)(\mu +{\alpha }_S+{\gamma }_S)+p(\mu +{\alpha }_A+{\gamma }_A)]+\mu (1-p)(\mu +{\alpha }_S+{\gamma }_S)},$$

$$E^{+}=\frac{\Lambda \beta I_A^{+}[\eta (1-p)(\mu +{\alpha }_S+{\gamma }_S)+p(\mu +{\alpha }_A+{\gamma }_A)]}{[\beta I_A^{+}(\eta (1-p)(\mu +{\alpha }_S+{\gamma }_S)+p(\mu +{\alpha }_A+{\gamma }_A))+\mu (1-p)(\mu +{\alpha }_S+{\gamma }_S)](\mu +\theta )},$$

$$I_S^{+}=\frac{p{I}_A^{+}(\mu +{\alpha }_A+{\gamma }_A)}{(1-p)(\mu +{\alpha }_S+{\gamma }_S)},$$

and $I_A^{+}$ is the unique positive root of the following equation

$$\frac{(1-p)(\mu +{\alpha }_S+{\gamma }_S)+p(\mu +{\alpha }_A+{\gamma }_A)}{\beta I_A^{+}[\eta (1-p)(\mu +{\alpha }_S+{\gamma }_S)+p(\mu +{\alpha }_A+{\gamma }_A)]+\mu (1-p)(\mu +{\alpha }_S+{\gamma }_S)}-\frac{(\mu +{\alpha }_A+{\gamma }_A)(\mu +\theta )}{(1-p)\Lambda \beta \theta }=0.$$

On the other hand, it is supposed in system (1.2) that the individuals live in a constant environment. However, some parameters involved in epidemic models are always affected by the environmental noise. There has been a growing interest in considering the environmental noise in epidemiology models since it has turned out that the stochastic model can describe biological phenomena and infectious diseases in a more exact way [8], [9], [10], [11], [12], [13], [14], [15], [16]. So far, there are different ways to introduce stochastic perturbations in the model. One of the most important ways is to assume that environmental fluctuations are of white noise type which are proportional to each variable, respectively. Given the above, in the present study it is assumed that the environmental noise is separately proportional to the compartments $S$, $E$, $I_S$ and $I_A$. Then corresponding to the deterministic model (1.2), the stochastic system takes the following form:

$$\left\{\begin{array}{c}dS=[\Lambda -\mu S-\beta S(\eta I_A+I_S)]dt+{\rho }_{1}Sd{B}_1\left(t\right),\hfill \\ dE=[\beta S(\eta I_A+I_S)-(\mu +\theta )E]dt+{\rho }_{2}Ed{B}_2\left(t\right),\hfill \\ d{I}_S=[p\theta E-(\mu +{\alpha }_S+{\gamma }_S)I_S]dt+{\rho }_3{I}_{S}d{B}_3\left(t\right),\hfill \\ d{I}_A=[(1-p)\theta E-(\mu +{\alpha }_A+{\gamma }_A)I_A]dt+{\rho }_4{I}_{A}d{B}_4\left(t\right),\hfill \end{array}\right)$$ (1.3)

where $B_1\left(t\right)$, $B_2\left(t\right)$, $B_3\left(t\right)$ and $B_4\left(t\right)$ are mutually independent standard Brownian motions defined on a complete probability space $(\Omega ,\mathcal{F},{\left\{{\mathcal{F}}_t\right\}}_{t\ge 0},\mathbb{P})$ with a filtration ${\left\{{\mathcal{F}}_t\right\}}_{t\ge 0}$ satisfying the usual conditions [17], ${\rho }_i^2>0$ represent the intensity of white noises ${\dot{B}}_i\left(t\right)$ $(i=1,2,3,4)$, respectively.

Throughout this paper, we introduce the following notations:

$${\mathbb{R}}_{+}^d=\{x={(x_1,\dots ,x_d)}^T\in {\mathbb{R}}^d:x_i>0,1\le i\le d\},{\overline{\mathbb{R}}}_{+}^d=\{x={(x_1,\dots ,x_d)}^T\in {\mathbb{R}}^d:x_i\ge 0,1\le i\le d\},$$

$$x\vee y=max\{x,y\}\text{for any}x,y\in \mathbb{R}.$$

Denote by $C^2({\mathbb{R}}^d;{\overline{\mathbb{R}}}_{+})$ the family of all nonnegative functions $V\left(x\right)$ defined on ${\mathbb{R}}^d$ such that they are continuously twice differentiable in $x$. If $G$ is a vector or matrix, we use the notation $\Vert G\Vert$ to represent its norm and its transpose is defined by $G^T$. For any real numbers $a_i$ $(i=1,\dots ,n)$, we use the notation $\text{diag}(a_1,\dots ,a_n)$ to denote a $n$-order diagonal matrix. If $G$ is an invertible matrix, we use the notation $G^{-1}$ to denote its inverse matrix. If $G$ is a square matrix, its determinant is denoted by $\left|G\right|$. Moreover, if $\mathbb{H}$ and $\mathbb{J}$ are two $d$-dimensional symmetric matrices, we define

$$\mathbb{H}⪰\mathbb{J}:\mathbb{H}-\mathbb{J}\text{is at least a semi-positive definite matrix}.$$ (1.4)

In view of (1.4), we can obtain that the matrix $\mathbb{H}$ is also positive definite if $\mathbb{J}$ is a positive definite matrix.

The organization of this paper is summarized as follows: In Section 2, we verify that there is a unique global positive solution to the stochastic system (1.3) with any positive initial value. In Section 3, we adopt a stochastic Lyapunov function method to establish sufficient conditions for the existence and uniqueness of an ergodic stationary distribution of positive solutions to the stochastic model (1.3). In Section 4, under the same conditions as in Theorem 3.1, we obtain the accurate expression of probability density around the quasi-endemic equilibrium of the stochastic system (1.3), which reflects the strong persistence of the disease. In Section 5, numerical simulations are given to confirm our analytical findings obtained in Sections 3, 4. Finally, a brief conclusion is given to end this paper.

2. Existence and uniqueness of the global positive solution

Before studying the dynamic behavior of an epidemic model, we should ensure that the solution is global and positive. The following theorem guarantees the existence and uniqueness of the global positive solution of system (1.3) with any positive initial value.

Theorem 2.1

For any initial value ${(S\left(0\right),E\left(0\right),I_S\left(0\right),I_A\left(0\right))}^T\in {\mathbb{R}}_{+}^4$ , there exists a unique solution ${(S\left(t\right),E\left(t\right),I_S\left(t\right),I_A\left(t\right))}^T$ to system (1.3) on $t\ge 0$ and the solution will remain in ${\mathbb{R}}_{+}^4$ with probability one, namely, ${(S\left(t\right),E\left(t\right),I_S\left(t\right),I_A\left(t\right))}^T\in {\mathbb{R}}_{+}^4$ for all $t\ge 0$ almost surely (a.s.).

Proof

Note that all the coefficients of system (1.3) are locally Lipschitz continuous, then for any initial value ${(S\left(0\right),E\left(0\right),I_S\left(0\right),I_A\left(0\right))}^T\in {\mathbb{R}}_{+}^4$ there is a unique local solution ${(S\left(t\right),E\left(t\right),I_S\left(t\right),I_A\left(t\right))}^T$ on the interval $[0,{\tau }_e)$, where ${\tau }_e$ is an explosion time [17]. Now we validate this solution is global, i.e., to validate ${\tau }_e=\infty$ a.s. To this end, let $n_0\ge 1$ be sufficiently large such that $S\left(0\right)$, $E\left(0\right)$, $I_S\left(0\right)$ and $I_A\left(0\right)$ all lie within the interval $[1/n_0,n_0]$. For each integer $n\ge n_0$, we define a stopping time by [17]

$${\tau }_n=inf\left\{t\in [0,{\tau }_e):min\{S\left(t\right),E\left(t\right),I_S\left(t\right),I_A\left(t\right)\}\le \frac{1}{n}\text{or}max\{S\left(t\right),E\left(t\right),I_S\left(t\right),I_A\left(t\right)\}\ge n\right\},$$

where throughout this paper we set $inf0̸=\infty$ (commonly, $0̸$ represents the empty set). Apparently, ${\tau }_n$ is increasing as $n\to \infty$. Denote by ${\tau }_{\infty }={lim}_{n\to \infty }{\tau }_n$, whence ${\tau }_{\infty }\le {\tau }_e$ a.s. If ${\tau }_{\infty }=\infty$ a.s. is true, then ${\tau }_e=\infty$ a.s. and ${(S\left(t\right),E\left(t\right),I_S\left(t\right),I_A\left(t\right))}^T\in {\mathbb{R}}_{+}^4$ a.s. for all $t\ge 0$. Namely, to confirm the proof we need to validate ${\tau }_{\infty }=\infty$ a.s. If this statement is false, then there exists a pair of constants $T>0$ and $ϵ\in (0,1)$ such that

$$\mathbb{P}\{{\tau }_{\infty }\le T\}>ϵ.$$

Accordingly, there exists an integer $n_1\ge n_0$ such that

$$\mathbb{P}\{{\tau }_n\le T\}\ge ϵ,\forall n\ge n_1.$$

For any $(S,E,I_S,I_A)\in {\mathbb{R}}_{+}^4$, a nonnegative $C^2$-function $V$ is defined by

$$V(S,E,I_S,I_A)=\left(S-a-aln\frac{S}{a}\right)+(E-1-lnE)+(I_S-1-ln{I}_S)+(I_A-1-ln{I}_A),$$

where $a$ is a positive constant to be chosen later. It is noticed that the nonnegativity of the above function is ensured due to $u-1-lnu\ge 0$ for any $u>0$. Applying Itô’s formula to differentiate $V$, which is shown inAppendix, we have

$$dV(S,E,I_S,I_A)=LV(S,E,I_S,I_A)dt+{\rho }_1(S-a)d{B}_1\left(t\right)+{\rho }_2(E-1)d{B}_2\left(t\right)+{\rho }_3(I_S-1)d{B}_3\left(t\right)+{\rho }_4(I_A-1)d{B}_4\left(t\right),$$

where $LV:{\mathbb{R}}_{+}^4\to \mathbb{R}$ is defined by

$$LV=\Lambda -\mu S-\beta S(\eta I_A+I_S)-\frac{a\Lambda }{S}+a\beta \eta I_A+a\beta I_S+a\left(\mu +\frac{{\rho }_1^2}{2}\right)+\beta S(\eta I_A+I_S)-(\mu +\theta )E-\frac{\beta S(\eta I_A+I_S)}{E}+\mu +\theta +\frac{{\rho }_2^2}{2}+p\theta E-(\mu +{\alpha }_S+{\gamma }_S)I_S-\frac{p\theta E}{I_S}+\mu +{\alpha }_S+{\gamma }_S+\frac{{\rho }_3^2}{2}+(1-p)\theta E-(\mu +{\alpha }_A+{\gamma }_A)I_A-\frac{(1-p)\theta E}{I_A}+\mu +{\alpha }_A+{\gamma }_A+\frac{{\rho }_4^2}{2}\le \Lambda +a\mu +3\mu +\theta +{\alpha }_S+{\alpha }_A+{\gamma }_S+{\gamma }_A+\frac{a{\rho }_1^2}{2}+\frac{{\rho }_2^2}{2}+\frac{{\rho }_3^2}{2}+\frac{{\rho }_4^2}{2}+[a\beta -({\alpha }_S+{\gamma }_S)]I_S+[a\beta \eta -({\alpha }_A+{\gamma }_A)]I_A.$$

Choose $a=min\{({\alpha }_S+{\gamma }_S)/\beta ,({\alpha }_A+{\gamma }_A)/\beta \eta \}$ such that $a\beta -({\alpha }_S+{\gamma }_S)\le 0$ and $a\beta \eta -({\alpha }_A+{\gamma }_A)\le 0$, then we have

$$LV\le \Lambda +a\mu +3\mu +\theta +{\alpha }_S+{\alpha }_A+{\gamma }_S+{\gamma }_A+\frac{a{\rho }_1^2}{2}+\frac{{\rho }_2^2}{2}+\frac{{\rho }_3^2}{2}+\frac{{\rho }_4^2}{2}≔K.$$

Here $K$ is a positive constant which is independent of the variables $S$, $E$, $I_S$ and $I_A$. The remainder of the proof is similar to Theorem 2.1 in the literature [18] and so we omit it here. This completes the proof.

3. Existence of ergodic stationary distribution

In this section, we aim to establish sufficient criteria for the existence and uniqueness of an ergodic stationary distribution of positive solutions to the stochastic system (1.3). We first present some theories about the stationary distribution (see Khasminskii [19]).

Let $X\left(t\right)$ be a regular time-homogeneous Markov process in ${\mathbb{R}}^d$ described by the stochastic differential equation

$$dX\left(t\right)=f\left(X\left(t\right)\right)dt+\sum _{r=1}^k{g}_r\left(X\left(t\right)\right)d{B}_r\left(t\right).$$

The diffusion matrix of the process $X\left(t\right)$ is defined as follows

$$A\left(x\right)=\left(a_{ij}\left(x\right)\right),a_{ij}\left(x\right)=\sum _{r=1}^k{g}_r^i\left(x\right)g_r^j\left(x\right).$$

Lemma 3.1 [19] —

The Markov process $X\left(t\right)$ has a unique ergodic stationary distribution $\pi \left(\cdot \right)$ if there exists a bounded open domain $U\subset {\mathbb{R}}^d$ with regular boundary $\Gamma$ , having the following properties:

$\left(A_1\right)$ In the domain $U$ and some neighborhood thereof, the smallest eigenvalue of the diffusion matrix $A\left(x\right)$ is bounded away from zero.

$\left(A_2\right)$ If $x\in {\mathbb{R}}^d\setminus U$ , the mean time $\tau$ at which a path issuing from $x$ reaches the set $U$ is finite, and ${sup}_{x\in D}{\mathbb{E}}^x\tau <\infty$ for every compact subset $D\subset {\mathbb{R}}^d$ .

Remark 3.1

To validate $A_1$, we only need to prove that the operator $F$ is uniformly elliptical in $U$, where $Fu=b\left(x\right)u_x+\left(\text{tr}\left(A\left(x\right)u_{xx}\right)\right)/2$, i.e., there exists a positive number $\mathbb{M}$ such that

$$\sum _{i,j=1}^d{a}_{ij}\left(x\right){\xi }_i{\xi }_j\ge \mathbb{M}{\Vert \xi \Vert }^2,x\in U,\xi \in {\mathbb{R}}^d$$

(see Chapter 3, p. 103 of [20] and Rayleigh’s principle in [[21], Chapter 6, p. 349]). To prove $A_2$, we need to show that there are some neighborhood $U$ and a nonnegative $C^2$-function such that $LV$ is negative for any ${\mathbb{R}}^d\setminus U$ (for details the readers can refer to [[22], p. 1163]).

Theorem 3.1

If

$$R_0^S=\frac{p\beta \Lambda \theta }{(\mu +\frac{{\rho }_1^2}{2})(\mu +\theta +\frac{{\rho }_2^2}{2})(\mu +{\alpha }_S+{\gamma }_S+\frac{{\rho }_3^2}{2})}+\frac{(1-p)\beta \eta \Lambda \theta }{(\mu +\frac{{\rho }_1^2}{2})(\mu +\theta +\frac{{\rho }_2^2}{2})(\mu +{\alpha }_A+{\gamma }_A+\frac{{\rho }_4^2}{2})}>1,$$

then system (1.3) has a unique stationary distribution $\pi \left(\cdot \right)$ and the ergodicity holds.

Proof

The proof process is divided into two steps: the first step is to prove that the uniform elliptic condition is satisfied, and the second step is to construct a nonnegative Lyapunov function which satisfies the condition $\left(A_2\right)$ of Lemma 3.1.

Step 1. The diffusion matrix of system (1.3) is given by

$$\begin{array}{c}A=\left(\begin{array}{cccc}\hfill {\rho }_1^2{S}^2\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill {\rho }_2^2{E}^2\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill {\rho }_3^2{I}_S^2\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill {\rho }_4^2{I}_A^2\hfill \end{array}\right).\hfill \end{array}$$

Choosing $\mathbb{M}={min}_{{(S,E,I_S,I_A)}^T\in {\overline{U}}_k\subset {\mathbb{R}}_{+}^4}\{{\rho }_1^2{S}^2,{\rho }_2^2{E}^2,{\rho }_3^2{I}_S^2,{\rho }_4^2{I}_A^2\}$, we have

$$\sum _{i,j=1}^4{a}_{ij}(S,E,I_S,I_A){\zeta }_i{\zeta }_j={\rho }_1^2{S}^2{\zeta }_1^2+{\rho }_2^2{E}^2{\zeta }_2^2+{\rho }_3^2{I}_S^2{\zeta }_3^2+{\rho }_4^2{I}_A^2{\zeta }_4^2\ge \mathbb{M}{\Vert \zeta \Vert }^2,$$

for any ${(S,E,I_S,I_A)}^T\in {\overline{U}}_k$ and $\zeta ={({\zeta }_1,{\zeta }_2,{\zeta }_3,{\zeta }_4)}^T\in {\mathbb{R}}^4$, where ${\overline{U}}_k=[1/k,k]\times [1/k,k]\times [1/k,k]\times [1/k,k]$. Accordingly, the condition $A_1$ in Lemma 3.1 holds.

Step 2. Let

$$\overline{S}=\frac{\Lambda }{\mu +\frac{{\rho }_1^2}{2}},\overline{E}=1,{\overline{I}}_S=\frac{p\theta \overline{E}}{\mu +{\alpha }_S+{\gamma }_S+\frac{{\rho }_3^2}{2}},{\overline{I}}_A=\frac{(1-p)\theta \overline{E}}{\mu +{\alpha }_A+{\gamma }_A+\frac{{\rho }_4^2}{2}},\tilde{S}=\frac{S}{\overline{S}},\tilde{E}=\frac{E}{\overline{E}},{\tilde{I}}_S=\frac{I_S}{{\overline{I}}_S},{\tilde{I}}_A=\frac{I_A}{{\overline{I}}_A}.$$

In view of system (1.3), we have

$$L(-lnS)=-\frac{\Lambda }{S}+\beta \eta I_A+\beta I_S+\mu +\frac{{\rho }_1^2}{2}=-\frac{\Lambda }{\overline{S}}\frac{1}{\tilde{S}}+\mu +\frac{{\rho }_1^2}{2}+\beta \eta I_A+\beta I_S\le -\frac{\Lambda }{\overline{S}}\left(ln\frac{1}{\tilde{S}}+1\right)+\mu +\frac{{\rho }_1^2}{2}+\beta \eta I_A+\beta I_S=\frac{\Lambda }{\overline{S}}ln\tilde{S}+\beta \eta I_A+\beta I_S,$$ (3.1)

where in the inequality we have used the inequality

$$lnx\le x-1\text{for any}x>0.$$

Similarly, we have

$$L(-lnE)=-\frac{\beta \eta S{I}_A}{E}-\frac{\beta S{I}_S}{E}+\mu +\theta +\frac{{\rho }_2^2}{2}=-\frac{\beta \eta \overline{S}{\overline{I}}_A}{\overline{E}}\frac{\tilde{S}{\tilde{I}}_A}{\tilde{E}}-\frac{\beta \overline{S}{\overline{I}}_S}{\overline{E}}\frac{\tilde{S}{\tilde{I}}_S}{\tilde{E}}+\mu +\theta +\frac{{\rho }_2^2}{2}\le -\frac{\beta \eta \overline{S}{\overline{I}}_A}{\overline{E}}\left(ln\frac{\tilde{S}{\tilde{I}}_A}{\tilde{E}}+1\right)-\frac{\beta \overline{S}{\overline{I}}_S}{\overline{E}}\left(ln\frac{\tilde{S}{\tilde{I}}_S}{\tilde{E}}+1\right)+\mu +\theta +\frac{{\rho }_2^2}{2}=-\beta \eta \overline{S}{\overline{I}}_A-\beta \overline{S}{\overline{I}}_S+\mu +\theta +\frac{{\rho }_2^2}{2}-\beta \overline{S}(\eta {\overline{I}}_A+{\overline{I}}_S)ln\tilde{S}+\beta \overline{S}(\eta {\overline{I}}_A+{\overline{I}}_S)ln\tilde{E}-\beta \eta \overline{S}{\overline{I}}_{A}ln{\tilde{I}}_A-\beta \overline{S}{\overline{I}}_{S}ln{\tilde{I}}_S,$$ (3.2)

$$L(-ln{I}_S)=-\frac{p\theta E}{I_S}+\mu +{\alpha }_S+{\gamma }_S+\frac{{\rho }_3^2}{2}=-\frac{p\theta \overline{E}}{{\overline{I}}_S}\frac{\tilde{E}}{{\tilde{I}}_S}+\mu +{\alpha }_S+{\gamma }_S+\frac{{\rho }_3^2}{2}\le -\frac{p\theta \overline{E}}{{\overline{I}}_S}\left(ln\frac{\tilde{E}}{{\tilde{I}}_S}+1\right)+\mu +{\alpha }_S+{\gamma }_S+\frac{{\rho }_3^2}{2}=-\frac{p\theta \overline{E}}{{\overline{I}}_S}ln\tilde{E}+\frac{p\theta \overline{E}}{{\overline{I}}_S}ln{\tilde{I}}_S,$$ (3.3)

and

$$L(-ln{I}_A)=-\frac{(1-p)\theta E}{I_A}+\mu +{\alpha }_A+{\gamma }_A+\frac{{\rho }_4^2}{2}=-\frac{(1-p)\theta \overline{E}}{{\overline{I}}_A}\frac{\tilde{E}}{{\tilde{I}}_A}+\mu +{\alpha }_A+{\gamma }_A+\frac{{\rho }_4^2}{2}\le -\frac{(1-p)\theta \overline{E}}{{\overline{I}}_A}\left(ln\frac{\tilde{E}}{{\tilde{I}}_A}+1\right)+\mu +{\alpha }_A+{\gamma }_A+\frac{{\rho }_4^2}{2}=-\frac{(1-p)\theta \overline{E}}{{\overline{I}}_A}ln\tilde{E}+\frac{(1-p)\theta \overline{E}}{{\overline{I}}_A}ln{\tilde{I}}_A.$$ (3.4)

Define

$$V_1(S,E,I_S,I_A)=-lnE-c_{1}lnS-c_{2}ln{I}_S-c_{3}ln{I}_A+\frac{c_1\beta }{\mu +{\alpha }_S+{\gamma }_S}{I}_S+\frac{c_1\beta \eta }{\mu +{\alpha }_A+{\gamma }_A}{I}_A,$$

where $c_i$ $(i=1,2,3)$ are positive constants which will be determined later. Then from (3.1), (3.2), (3.3), (3.4) it follows that

$$L{V}_1\le -\beta \eta \overline{S}{\overline{I}}_A-\beta \overline{S}{\overline{I}}_S+\mu +\theta +\frac{{\rho }_2^2}{2}-\beta \overline{S}(\eta {\overline{I}}_A+{\overline{I}}_S)ln\tilde{S}+\beta \overline{S}(\eta {\overline{I}}_A+{\overline{I}}_S)ln\tilde{E}-\beta \eta \overline{S}{\overline{I}}_{A}ln{\tilde{I}}_A-\beta \overline{S}{\overline{I}}_{S}ln{\tilde{I}}_S+\frac{c_1\Lambda }{\overline{S}}ln\tilde{S}+c_1\beta \eta I_A+c_1\beta I_S-\frac{c_{2}p\theta \overline{E}}{{\overline{I}}_S}ln\tilde{E}+\frac{c_{2}p\theta \overline{E}}{{\overline{I}}_S}ln{\tilde{I}}_S-\frac{c_3(1-p)\theta \overline{E}}{{\overline{I}}_A}ln\tilde{E}+\frac{c_3(1-p)\theta \overline{E}}{{\overline{I}}_A}ln{\tilde{I}}_A+\frac{c_{1}p\beta \theta }{\mu +{\alpha }_S+{\gamma }_S}E-c_1\beta I_S+\frac{c_1(1-p)\beta \eta \theta }{\mu +{\alpha }_A+{\gamma }_A}E-c_1\beta \eta I_A=-\beta \eta \overline{S}{\overline{I}}_A-\beta \overline{S}{\overline{I}}_S+\mu +\theta +\frac{{\rho }_2^2}{2}+\left(\frac{c_{1}p\beta \theta }{\mu +{\alpha }_S+{\gamma }_S}+\frac{c_1(1-p)\beta \eta \theta }{\mu +{\alpha }_A+{\gamma }_A}\right)E+\left(\frac{c_1\Lambda }{\overline{S}}-\beta \overline{S}(\eta {\overline{I}}_A+{\overline{I}}_S)\right)ln\tilde{S}+\left(\beta \overline{S}(\eta {\overline{I}}_A+{\overline{I}}_S)-\frac{c_{2}p\theta \overline{E}}{{\overline{I}}_S}-\frac{c_3(1-p)\theta \overline{E}}{{\overline{I}}_A}\right)ln\tilde{E}+\left(\frac{c_{2}p\theta \overline{E}}{{\overline{I}}_S}-\beta \overline{S}{\overline{I}}_S\right)ln{\tilde{I}}_S+\left(\frac{c_3(1-p)\theta \overline{E}}{{\overline{I}}_A}-\beta \eta \overline{S}{\overline{I}}_A\right)ln{\tilde{I}}_A.$$

Let

$$\left\{\begin{array}{c}\frac{c_1\Lambda }{\overline{S}}-\beta \overline{S}(\eta {\overline{I}}_A+{\overline{I}}_S)=0,\hfill \\ \frac{c_{2}p\theta \overline{E}}{{\overline{I}}_S}-\beta \overline{S}{\overline{I}}_S=0,\hfill \\ \frac{c_3(1-p)\theta \overline{E}}{{\overline{I}}_A}-\beta \eta \overline{S}{\overline{I}}_A=0,\hfill \end{array}\right)$$

then we obtain

$$\left\{\begin{array}{c}{c}_1=\frac{\beta {\overline{S}}^2(\eta {\overline{I}}_A+{\overline{I}}_S)}{\Lambda },\hfill \\ c_2=\frac{\beta \overline{S}{\overline{I}}_S^2}{p\theta },\hfill \\ c_3=\frac{\beta \eta \overline{S}{\overline{I}}_A^2}{(1-p)\theta }.\hfill \end{array}\right)$$

Therefore

$$L{V}_1\le -\beta \eta \overline{S}{\overline{I}}_A-\beta \overline{S}{\overline{I}}_S+\mu +\theta +\frac{{\rho }_2^2}{2}+\left(\frac{c_{1}p\beta \theta }{\mu +{\alpha }_S+{\gamma }_S}+\frac{c_1(1-p)\beta \eta \theta }{\mu +{\alpha }_A+{\gamma }_A}\right)E=-\left(\mu +\theta +\frac{{\rho }_2^2}{2}\right)(R_0^S-1)+\left(\frac{c_{1}p\beta \theta }{\mu +{\alpha }_S+{\gamma }_S}+\frac{c_1(1-p)\beta \eta \theta }{\mu +{\alpha }_A+{\gamma }_A}\right)E,$$ (3.5)

where

$$R_0^S≔\frac{\beta \eta \overline{S}{\overline{I}}_A+\beta \overline{S}{\overline{I}}_S}{\mu +\theta +\frac{{\rho }_2^2}{2}}=\frac{p\beta \Lambda \theta }{(\mu +\frac{{\rho }_1^2}{2})(\mu +\theta +\frac{{\rho }_2^2}{2})(\mu +{\alpha }_S+{\gamma }_S+\frac{{\rho }_3^2}{2})}+\frac{(1-p)\beta \eta \Lambda \theta }{(\mu +\frac{{\rho }_1^2}{2})(\mu +\theta +\frac{{\rho }_2^2}{2})(\mu +{\alpha }_A+{\gamma }_A+\frac{{\rho }_4^2}{2})}.$$

Next, define

$$V_2\left(S\right)=-lnS,V_3\left(I_S\right)=-ln{I}_S,V_4\left(I_A\right)=-ln{I}_A,V_5(S,E,I_S,I_A)=\frac{1}{\nu +1}{(S+E+I_S+I_A)}^{\nu +1},$$

where $\nu \in (0,2\mu /({\rho }_1^2\vee {\rho }_2^2\vee {\rho }_3^2\vee {\rho }_4^2))$ is a sufficiently small constant. Applying Itô’s formula to $V_2$, $V_3$, $V_4$ and $V_5$, respectively, which is shown inAppendix, we have

$$L{V}_2=-\frac{\Lambda }{S}+\beta \eta I_A+\beta I_S+\mu +\frac{{\rho }_1^2}{2},$$ (3.6)

$$L{V}_3=-\frac{p\theta E}{I_S}+\mu +{\alpha }_S+{\gamma }_S+\frac{{\rho }_3^2}{2},$$ (3.7)

$$L{V}_4=-\frac{(1-p)\theta E}{I_A}+\mu +{\alpha }_A+{\gamma }_A+\frac{{\rho }_4^2}{2},$$ (3.8)

and

$$L{V}_5={(S+E+I_S+I_A)}^{\nu }[\Lambda -\mu (S+E+I_S+I_A)-({\alpha }_S+{\gamma }_S)I_S-({\alpha }_A+{\gamma }_A)I_A]+\frac{\nu }{2}{(S+E+I_S+I_A)}^{\nu -1}\times ({\rho }_1^2{S}^2+{\rho }_2^2{E}^2+{\rho }_3^2{I}_S^2+{\rho }_4^2{I}_A^2)\le \Lambda {(S+E+I_S+I_A)}^{\nu }-\mu {(S+E+I_S+I_A)}^{\nu +1}+\frac{\nu }{2}({\rho }_1^2\vee {\rho }_2^2\vee {\rho }_3^2\vee {\rho }_4^2){(S+E+I_S+I_A)}^{\nu +1}\le J_1-\frac{1}{2}\left(\mu -\frac{\nu }{2}({\rho }_1^2\vee {\rho }_2^2\vee {\rho }_3^2\vee {\rho }_4^2)\right){(S+E+I_S+I_A)}^{\nu +1}\le J_1-\frac{1}{2}\left(\mu -\frac{\nu }{2}({\rho }_1^2\vee {\rho }_2^2\vee {\rho }_3^2\vee {\rho }_4^2)\right)(S^{\nu +1}+E^{\nu +1}+I_S^{\nu +1}+I_A^{\nu +1})=J_1-\frac{\vartheta }{2}(S^{\nu +1}+E^{\nu +1}+I_S^{\nu +1}+I_A^{\nu +1}),$$ (3.9)

where

$$J_1≔\underset{{(S,E,I_S,I_A)}^T\in {\mathbb{R}}_{+}^4}{sup}\left\{\Lambda {(S+E+I_S+I_A)}^{\nu }-\frac{\vartheta }{2}{(S+E+I_S+I_A)}^{\nu +1}\right\}<\infty ,$$

and

$$\vartheta ≔\mu -\frac{\nu }{2}({\rho }_1^2\vee {\rho }_2^2\vee {\rho }_3^2\vee {\rho }_4^2).$$

Define a $C^2$-function $V:{\mathbb{R}}_{+}^4\to \mathbb{R}$ in the following form

$$V(S,E,I_S,I_A)=M{V}_1(S,E,I_S,I_A)+V_2\left(S\right)+V_3\left(I_S\right)+V_4\left(I_A\right)+V_5(S,E,I_S,I_A),$$

where $M$ is a sufficiently large positive constant satisfying the following condition

$$-M\left(\mu +\theta +\frac{{\rho }_2^2}{2}\right)(R_0^S-1)+J_2\le -2,$$ (3.10)

and

$$J_2≔\underset{{(S,E,I_S,I_A)}^T\in {\mathbb{R}}_{+}^4}{sup}\{-\frac{\vartheta }{4}(S^{\nu +1}+E^{\nu +1}+I_S^{\nu +1}+I_A^{\nu +1})+\beta \eta I_A+\beta I_S+J_1+3\mu +{\alpha }_S+{\alpha }_A+{\gamma }_S+{\gamma }_A+\frac{{\rho }_1^2}{2}+\frac{{\rho }_3^2}{2}+\frac{{\rho }_4^2}{2}\}<\infty .$$

In addition, note that $V(S,E,I_S,I_A)$ is not only continuous, but also tends to $\infty$ as ${(S,E,I_S,I_A)}^T$ approaches the boundary of ${\mathbb{R}}_{+}^4$. Therefore, it must have a lower bound and achieve this lower bound at a point ${(S^0,E^0,I_S^0,I_A^0)}^T$ in the interior of ${\mathbb{R}}_{+}^4$. Then we define a $C^2$-function $\overline{V}:{\mathbb{R}}_{+}^4\to {\overline{\mathbb{R}}}_{+}$ as follows

$$\overline{V}(S,E,I_S,I_A)=V(S,E,I_S,I_A)-V(S^0,E^0,I_S^0,I_A^0)=M{V}_1(S,E,I_S,I_A)+V_2\left(S\right)+V_3\left(I_S\right)+V_4\left(I_A\right)+V_5(S,E,I_S,I_A)-V(S^0,E^0,I_S^0,I_A^0).$$

According to (3.5), (3.6), (3.7), (3.8), (3.9), we have

$$L\overline{V}\le -M\left(\mu +\theta +\frac{{\rho }_2^2}{2}\right)(R_0^S-1)+M\left(\frac{c_{1}p\beta \theta }{\mu +{\alpha }_S+{\gamma }_S}+\frac{c_1(1-p)\beta \eta \theta }{\mu +{\alpha }_A+{\gamma }_A}\right)E-\frac{\Lambda }{S}-\frac{p\theta E}{I_S}-\frac{(1-p)\theta E}{I_A}-\frac{\vartheta }{2}(S^{\nu +1}+E^{\nu +1}+I_S^{\nu +1}+I_A^{\nu +1})+\beta \eta I_A+\beta I_S+J_1+3\mu +{\alpha }_S+{\alpha }_A+{\gamma }_S+{\gamma }_A+\frac{{\rho }_1^2}{2}+\frac{{\rho }_3^2}{2}+\frac{{\rho }_4^2}{2}.$$ (3.11)

Now we are in the position to construct a bounded closed domain $U_{ϵ}$ as follows

$$U_{ϵ}=\left\{{(S,E,I_S,I_A)}^T\in {\mathbb{R}}_{+}^4:ϵ\le S\le \frac{1}{ϵ},ϵ\le E\le \frac{1}{ϵ},{ϵ}^2\le I_S\le \frac{1}{{ϵ}^2},{ϵ}^2\le I_A\le \frac{1}{{ϵ}^2}\right\},$$

where $0<ϵ<1$ is a sufficiently small constant. In the set ${\mathbb{R}}_{+}^4\setminus U_{ϵ}$, we can choose $ϵ$ small enough such that the following conditions hold

$$-\frac{\Lambda }{ϵ}+J_3\le -1,$$ (3.12)

$$ϵ\le \frac{1}{M(\frac{c_{1}p\beta \theta }{\mu +{\alpha }_S+{\gamma }_S}+\frac{c_1(1-p)\beta \eta \theta }{\mu +{\alpha }_A+{\gamma }_A})},$$ (3.13)

$$-\frac{p\theta }{ϵ}+J_3\le -1,$$ (3.14)

$$-\frac{(1-p)\theta }{ϵ}+J_3\le -1,$$ (3.15)

$$-\frac{\vartheta }{4{ϵ}^{\nu +1}}+J_3\le -1,$$ (3.16)

$$-\frac{\vartheta }{4{ϵ}^{2(\nu +1)}}+J_3\le -1.$$ (3.17)

Here $J_3$ is a positive constant which is given explicitly in the expression (3.19). For convenience, we can divide ${\mathbb{R}}_{+}^4\setminus U_{ϵ}$ into eight domains,

$$U_1=\{{(S,E,I_S,I_A)}^T\in {\mathbb{R}}_{+}^4:S<ϵ\},U_2=\{{(S,E,I_S,I_A)}^T\in {\mathbb{R}}_{+}^4:E<ϵ\},U_3=\{{(S,E,I_S,I_A)}^T\in {\mathbb{R}}_{+}^4:I_S<{ϵ}^2,E\ge ϵ\},$$

$$U_4=\{{(S,E,I_S,I_A)}^T\in {\mathbb{R}}_{+}^4:I_A<{ϵ}^2,E\ge ϵ\},U_5=\left\{{(S,E,I_S,I_A)}^T\in {\mathbb{R}}_{+}^4:S>\frac{1}{ϵ}\right\},$$

$$U_6=\left\{{(S,E,I_S,I_A)}^T\in {\mathbb{R}}_{+}^4:E>\frac{1}{ϵ}\right\},U_7=\left\{{(S,E,I_S,I_A)}^T\in {\mathbb{R}}_{+}^4:I_S>\frac{1}{{ϵ}^2}\right\},U_8=\left\{{(S,E,I_S,I_A)}^T\in {\mathbb{R}}_{+}^4:I_A>\frac{1}{{ϵ}^2}\right\}.$$

Apparently, ${\mathbb{R}}_{+}^4\setminus U_{ϵ}=U_1\bigcup \dots \bigcup U_8$. Next, we will prove that $L\overline{V}\le -1$ for any ${(S,E,I_S,I_A)}^T\in U_{ϵ}^c$, which is equivalent to proving it on the above eight domains, respectively.

Case 1. For any ${(S,E,I_S,I_A)}^T\in U_1$, according to (3.11), we have

$$L\overline{V}\le -\frac{\Lambda }{S}+M\left(\frac{c_{1}p\beta \theta }{\mu +{\alpha }_S+{\gamma }_S}+\frac{c_1(1-p)\beta \eta \theta }{\mu +{\alpha }_A+{\gamma }_A}\right)E-\frac{\vartheta }{4}(S^{\nu +1}+E^{\nu +1}+I_S^{\nu +1}+I_A^{\nu +1})+\beta \eta I_A+\beta I_S+J_1+3\mu +{\alpha }_S+{\alpha }_A+{\gamma }_S+{\gamma }_A+\frac{{\rho }_1^2}{2}+\frac{{\rho }_3^2}{2}+\frac{{\rho }_4^2}{2}\le -\frac{\Lambda }{S}+J_3\le -\frac{\Lambda }{ϵ}+J_3\le -1,$$ (3.18)

which follows from (3.12) and

$$J_3≔\underset{{(S,E,I_S,I_A)}^T\in {\mathbb{R}}_{+}^4}{sup}\{-\frac{\vartheta }{4}(S^{\nu +1}+E^{\nu +1}+I_S^{\nu +1}+I_A^{\nu +1})+M\left(\frac{c_{1}p\beta \theta }{\mu +{\alpha }_S+{\gamma }_S}+\frac{c_1(1-p)\beta \eta \theta }{\mu +{\alpha }_A+{\gamma }_A}\right)E+\beta \eta I_A+\beta I_S+J_1+3\mu +{\alpha }_S+{\alpha }_A+{\gamma }_S+{\gamma }_A+\frac{{\rho }_1^2}{2}+\frac{{\rho }_3^2}{2}+\frac{{\rho }_4^2}{2}\}<\infty .$$ (3.19)

Case 2. For any ${(S,E,I_S,I_A)}^T\in U_2$, in view of (3.11), we obtain

$$L\overline{V}\le -M\left(\mu +\theta +\frac{{\rho }_2^2}{2}\right)(R_0^S-1)+M\left(\frac{c_{1}p\beta \theta }{\mu +{\alpha }_S+{\gamma }_S}+\frac{c_1(1-p)\beta \eta \theta }{\mu +{\alpha }_A+{\gamma }_A}\right)E-\frac{\vartheta }{4}(S^{\nu +1}+E^{\nu +1}+I_S^{\nu +1}+I_A^{\nu +1})+\beta \eta I_A+\beta I_S+J_1+3\mu +{\alpha }_S+{\alpha }_A+{\gamma }_S+{\gamma }_A+\frac{{\rho }_1^2}{2}+\frac{{\rho }_3^2}{2}+\frac{{\rho }_4^2}{2}\le -M\left(\mu +\theta +\frac{{\rho }_2^2}{2}\right)(R_0^S-1)+M\left(\frac{c_{1}p\beta \theta }{\mu +{\alpha }_S+{\gamma }_S}+\frac{c_1(1-p)\beta \eta \theta }{\mu +{\alpha }_A+{\gamma }_A}\right)E+J_2\le -M\left(\mu +\theta +\frac{{\rho }_2^2}{2}\right)(R_0^S-1)+M\left(\frac{c_{1}p\beta \theta }{\mu +{\alpha }_S+{\gamma }_S}+\frac{c_1(1-p)\beta \eta \theta }{\mu +{\alpha }_A+{\gamma }_A}\right)ϵ+J_2\le -1,$$ (3.20)

which follows from (3.10), (3.13) and

$$J_2≔\underset{{(S,E,I_S,I_A)}^T\in {\mathbb{R}}_{+}^4}{sup}\{-\frac{\vartheta }{4}(S^{\nu +1}+E^{\nu +1}+I_S^{\nu +1}+I_A^{\nu +1})+\beta \eta I_A+\beta I_S+J_1+3\mu +{\alpha }_S+{\alpha }_A+{\gamma }_S+{\gamma }_A+\frac{{\rho }_1^2}{2}+\frac{{\rho }_3^2}{2}+\frac{{\rho }_4^2}{2}\}<\infty .$$

Case 3. For any ${(S,E,I_S,I_A)}^T\in U_3$, by (3.11), we get

$$L\overline{V}\le -\frac{p\theta E}{I_S}+M\left(\frac{c_{1}p\beta \theta }{\mu +{\alpha }_S+{\gamma }_S}+\frac{c_1(1-p)\beta \eta \theta }{\mu +{\alpha }_A+{\gamma }_A}\right)E-\frac{\vartheta }{4}(S^{\nu +1}+E^{\nu +1}+I_S^{\nu +1}+I_A^{\nu +1})+\beta \eta I_A+\beta I_S+J_1+3\mu +{\alpha }_S+{\alpha }_A+{\gamma }_S+{\gamma }_A+\frac{{\rho }_1^2}{2}+\frac{{\rho }_3^2}{2}+\frac{{\rho }_4^2}{2}\le -\frac{p\theta E}{I_S}+J_3\le -\frac{p\theta ϵ}{{ϵ}^2}+J_3\le -\frac{p\theta }{ϵ}+J_3\le -1,$$ (3.21)

which follows from (3.14).

Case 4. For any ${(S,E,I_S,I_A)}^T\in U_4$, by (3.11), we derive

$$L\overline{V}\le -\frac{(1-p)\theta E}{I_A}+M\left(\frac{c_{1}p\beta \theta }{\mu +{\alpha }_S+{\gamma }_S}+\frac{c_1(1-p)\beta \eta \theta }{\mu +{\alpha }_A+{\gamma }_A}\right)E-\frac{\vartheta }{4}(S^{\nu +1}+E^{\nu +1}+I_S^{\nu +1}+I_A^{\nu +1})+\beta \eta I_A+\beta I_S+J_1+3\mu +{\alpha }_S+{\alpha }_A+{\gamma }_S+{\gamma }_A+\frac{{\rho }_1^2}{2}+\frac{{\rho }_3^2}{2}+\frac{{\rho }_4^2}{2}\le -\frac{(1-p)\theta E}{I_A}+J_3\le -\frac{(1-p)\theta ϵ}{{ϵ}^2}+J_3\le -\frac{(1-p)\theta }{ϵ}+J_3\le -1,$$ (3.22)

which follows from (3.15).

Case 5. For any ${(S,E,I_S,I_A)}^T\in U_5$, according to (3.11), we have

$$L\overline{V}\le -\frac{\vartheta }{4}{S}^{\nu +1}+M\left(\frac{c_{1}p\beta \theta }{\mu +{\alpha }_S+{\gamma }_S}+\frac{c_1(1-p)\beta \eta \theta }{\mu +{\alpha }_A+{\gamma }_A}\right)E-\frac{\vartheta }{4}(S^{\nu +1}+E^{\nu +1}+I_S^{\nu +1}+I_A^{\nu +1})+\beta \eta I_A+\beta I_S+J_1+3\mu +{\alpha }_S+{\alpha }_A+{\gamma }_S+{\gamma }_A+\frac{{\rho }_1^2}{2}+\frac{{\rho }_3^2}{2}+\frac{{\rho }_4^2}{2}\le -\frac{\vartheta }{4}{S}^{\nu +1}+J_3\le -\frac{\vartheta }{4{ϵ}^{\nu +1}}+J_3\le -1,$$ (3.23)

which follows from (3.16).

Case 6. For any ${(S,E,I_S,I_A)}^T\in U_6$, in view of (3.11), we obtain

$$L\overline{V}\le -\frac{\vartheta }{4}{E}^{\nu +1}+M\left(\frac{c_{1}p\beta \theta }{\mu +{\alpha }_S+{\gamma }_S}+\frac{c_1(1-p)\beta \eta \theta }{\mu +{\alpha }_A+{\gamma }_A}\right)E-\frac{\vartheta }{4}(S^{\nu +1}+E^{\nu +1}+I_S^{\nu +1}+I_A^{\nu +1})+\beta \eta I_A+\beta I_S+J_1+3\mu +{\alpha }_S+{\alpha }_A+{\gamma }_S+{\gamma }_A+\frac{{\rho }_1^2}{2}+\frac{{\rho }_3^2}{2}+\frac{{\rho }_4^2}{2}\le -\frac{\vartheta }{4}{E}^{\nu +1}+J_3\le -\frac{\vartheta }{4{ϵ}^{\nu +1}}+J_3\le -1,$$ (3.24)

which follows from (3.16).

Case 7. For any ${(S,E,I_S,I_A)}^T\in U_7$, by (3.11), we get

$$L\overline{V}\le -\frac{\vartheta }{4}{I}_S^{\nu +1}+M\left(\frac{c_{1}p\beta \theta }{\mu +{\alpha }_S+{\gamma }_S}+\frac{c_1(1-p)\beta \eta \theta }{\mu +{\alpha }_A+{\gamma }_A}\right)E-\frac{\vartheta }{4}(S^{\nu +1}+E^{\nu +1}+I_S^{\nu +1}+I_A^{\nu +1})+\beta \eta I_A+\beta I_S+J_1+3\mu +{\alpha }_S+{\alpha }_A+{\gamma }_S+{\gamma }_A+\frac{{\rho }_1^2}{2}+\frac{{\rho }_3^2}{2}+\frac{{\rho }_4^2}{2}\le -\frac{\vartheta }{4}{I}_S^{\nu +1}+J_3\le -\frac{\vartheta }{4{ϵ}^{2(\nu +1)}}+J_3\le -1,$$ (3.25)

which follows from (3.17).

Case 8. For any ${(S,E,I_S,I_A)}^T\in U_8$, according to (3.11), we derive

$$L\overline{V}\le -\frac{\vartheta }{4}{I}_A^{\nu +1}+M\left(\frac{c_{1}p\beta \theta }{\mu +{\alpha }_S+{\gamma }_S}+\frac{c_1(1-p)\beta \eta \theta }{\mu +{\alpha }_A+{\gamma }_A}\right)E-\frac{\vartheta }{4}(S^{\nu +1}+E^{\nu +1}+I_S^{\nu +1}+I_A^{\nu +1})+\beta \eta I_A+\beta I_S+J_1+3\mu +{\alpha }_S+{\alpha }_A+{\gamma }_S+{\gamma }_A+\frac{{\rho }_1^2}{2}+\frac{{\rho }_3^2}{2}+\frac{{\rho }_4^2}{2}\le -\frac{\vartheta }{4}{I}_A^{\nu +1}+J_3\le -\frac{\vartheta }{4{ϵ}^{2(\nu +1)}}+J_3\le -1,$$ (3.26)

which follows from (3.17).

Thus, according to (3.18), (3.20), (3.21), (3.22), (3.23), (3.24), (3.25), (3.26), we derive that for a sufficiently small $ϵ$,

$$L\overline{V}\le -1\text{for any}{(S,E,I_S,I_A)}^T\in {\mathbb{R}}_{+}^4\setminus U_{ϵ},$$

which implies that the condition $\left(A_2\right)$ in Lemma 3.1 also holds. In view of Lemma 3.1, we obtain that system (1.3) has a unique stationary distribution $\pi \left(\cdot \right)$ and the ergodicity holds. This completes the proof.

4. Probability density for system (1.3)

By Theorem 3.1, we obtain that the global positive solution ${(S\left(t\right),E\left(t\right),I_S\left(t\right),I_A\left(t\right))}^T$ admits a unique ergodic stationary distribution $\pi \left(\cdot \right)$ if $R_0^S>1$. In this section, we will obtain the explicit expression of the probability density of the distribution $\pi \left(\cdot \right)$. Although there are many works that have used the Lyapunov equation (see (4.7) below) to find the probability distribution, such as the papers [23], [24], [25], [26], the monograph of van Kampen [27], and other works. But most of them have used numerical methods, while here we will give an analytical solution which is very interesting since there are few works on it. Firstly, two necessary transformations of system (1.3) should be introduced.

4.1. Two important transformations of system (1.3)

(I) (Logarithmic transformation) Let ${(u_1,u_2,u_3,u_4)}^T={(lnS,lnE,ln{I}_S,ln{I}_A)}^T$. Then by Itô’s formula, which is shown inAppendix, and system (1.3), we have

$$\left\{\begin{array}{c}d{u}_1=\left[\Lambda e^{-u_1}-\beta e^{u_3}-\beta \eta e^{u_4}-\left(\mu +\frac{{\rho }_1^2}{2}\right)\right]dt+{\rho }_{1}d{B}_1\left(t\right),\hfill \\ d{u}_2=\left[\beta e^{u_1+u_3-u_2}+\beta \eta e^{u_1+u_4-u_2}-\left(\mu +\theta +\frac{{\rho }_2^2}{2}\right)\right]dt+{\rho }_{2}d{B}_2\left(t\right),\hfill \\ d{u}_3=\left[p\theta e^{u_2-u_3}-\left(\mu +{\alpha }_S+{\gamma }_S+\frac{{\rho }_3^2}{2}\right)\right]dt+{\rho }_{3}d{B}_3\left(t\right),\hfill \\ d{u}_4=\left[(1-p)\theta e^{u_2-u_4}-\left(\mu +{\alpha }_A+{\gamma }_A+\frac{{\rho }_4^2}{2}\right)\right]dt+{\rho }_{4}d{B}_4\left(t\right).\hfill \end{array}\right)$$ (4.1)

Assume that $R_0^S>1$, there is the quasi-endemic equilibrium $P^{\ast }(S^{\ast },E^{\ast },I_S^{\ast },I_A^{\ast })={(e^{u_1^{\ast }},e^{u_2^{\ast }},e^{u_3^{\ast }},e^{u_4^{\ast }})}^T\in {\mathbb{R}}_{+}^4$ determined by the following equations:

$$\left\{\begin{array}{c}\Lambda e^{-u_1^{\ast }}-\beta e^{u_3^{\ast }}-\beta \eta e^{u_4^{\ast }}-{\mu }_1=0,\hfill \\ \beta e^{u_1^{\ast }+u_3^{\ast }-u_2^{\ast }}+\beta \eta e^{u_1^{\ast }+u_4^{\ast }-u_2^{\ast }}-({\mu }_2+\theta )=0,\hfill \\ p\theta e^{u_2^{\ast }-u_3^{\ast }}-({\mu }_3+{\alpha }_S+{\gamma }_S)=0,\hfill \\ (1-p)\theta e^{u_2^{\ast }-u_4^{\ast }}-({\mu }_4+{\alpha }_A+{\gamma }_A)=0,\hfill \end{array}\right)$$ (4.2)

where ${\mu }_i=\mu +{\rho }_i^2/2$ for any $i=1,2,3,4$. Therefore, in view of (4.2), we can obtain that

$$S^{\ast }=\frac{\Lambda (1-p)({\mu }_3+{\alpha }_S+{\gamma }_S)}{\beta I_A^{\ast }[\eta (1-p)({\mu }_3+{\alpha }_S+{\gamma }_S)+p({\mu }_4+{\alpha }_A+{\gamma }_A)]+{\mu }_1(1-p)({\mu }_3+{\alpha }_S+{\gamma }_S)},$$

$$E^{\ast }=\frac{\Lambda \beta I_A^{\ast }[\eta (1-p)({\mu }_3+{\alpha }_S+{\gamma }_S)+p({\mu }_4+{\alpha }_A+{\gamma }_A)]}{[\beta I_A^{\ast }(\eta (1-p)({\mu }_3+{\alpha }_S+{\gamma }_S)+p({\mu }_4+{\alpha }_A+{\gamma }_A))+{\mu }_1(1-p)({\mu }_3+{\alpha }_S+{\gamma }_S)]({\mu }_2+\theta )},$$

$$I_S^{\ast }=\frac{p{I}_A^{\ast }({\mu }_4+{\alpha }_A+{\gamma }_A)}{(1-p)({\mu }_3+{\alpha }_S+{\gamma }_S)},$$

and $I_A^{\ast }$ is the unique positive root of the following quadratic equation

$$\frac{(1-p)({\mu }_3+{\alpha }_S+{\gamma }_S)+p({\mu }_4+{\alpha }_A+{\gamma }_A)}{\beta I_A^{\ast }[\eta (1-p)({\mu }_3+{\alpha }_S+{\gamma }_S)+p({\mu }_4+{\alpha }_A+{\gamma }_A)]+{\mu }_1(1-p)({\mu }_3+{\alpha }_S+{\gamma }_S)}-\frac{({\mu }_4+{\alpha }_A+{\gamma }_A)({\mu }_2+\theta )}{(1-p)\Lambda \beta \theta }=0.$$

(II) (Equilibrium offset transformation) Let $X={(x_1,x_2,x_3,x_4)}^T={(u_1-u_1^{\ast },u_2-u_2^{\ast },u_3-u_3^{\ast },u_4-u_4^{\ast })}^T$, the corresponding linearized system of (4.1) is as follows

$$\left\{\begin{array}{c}d{x}_1=(-a_{11}{x}_1-a_{13}{x}_3-a_{14}{x}_4)dt+{\rho }_{1}d{B}_1\left(t\right),\hfill \\ d{x}_2=(a_{21}{x}_1-a_{21}{x}_2+a_{23}{x}_3+a_{24}{x}_4)dt+{\rho }_{2}d{B}_2\left(t\right),\hfill \\ d{x}_3=(a_{32}{x}_2-a_{32}{x}_3)dt+{\rho }_{3}d{B}_3\left(t\right),\hfill \\ d{x}_4=(a_{42}{x}_2-a_{42}{x}_4)dt+{\rho }_{4}d{B}_4\left(t\right),\hfill \end{array}\right)$$ (4.3)

where

$$a_{11}=\frac{\Lambda }{S^{\ast }}>0,a_{13}=\beta I_S^{\ast }>0,a_{14}=\beta \eta I_A^{\ast }>0,a_{21}=\frac{\beta S^{\ast }(\eta I_A^{\ast }+I_S^{\ast })}{E^{\ast }}>0,a_{23}=\frac{\beta S^{\ast }{I}_S^{\ast }}{E^{\ast }}>0,$$

$$a_{24}=\frac{\beta \eta S^{\ast }{I}_A^{\ast }}{E^{\ast }}>0,a_{32}=\frac{p\theta E^{\ast }}{I_S^{\ast }}>0,a_{42}=\frac{(1-p)\theta E^{\ast }}{I_A^{\ast }}>0.$$

It is easy to obtain that $a_{13}{a}_{24}-a_{14}{a}_{23}=0$, $a_{13}{a}_{21}-a_{11}{a}_{23}<0$, $a_{14}{a}_{21}-a_{11}{a}_{24}<0$.

Before introducing the corresponding probability density, we still need to introduce an important definition and two necessary lemmas.

Definition 4.1 [28] —

The characteristic polynomial of the square matrix $A_n$ is defined as ${\phi }_{A_n}\left(\lambda \right)={\lambda }^n+a_1{\lambda }^{n-1}+\cdots +a_{n-1}\lambda +a_n$, then $A_n$ is called a Hurwitz matrix if and only if $A_n$ has all negative real-part eigenvalues, i.e., 

$$H_k=\left|\begin{array}{ccccc}\hfill a_1\hfill & \hfill a_3\hfill & \hfill a_5\hfill & \hfill \dots \hfill & \hfill a_{2k-1}\hfill \\ \hfill 1\hfill & \hfill a_2\hfill & \hfill a_4\hfill & \hfill \dots \hfill & \hfill a_{2k-2}\hfill \\ \hfill 0\hfill & \hfill a_1\hfill & \hfill a_3\hfill & \hfill \dots \hfill & \hfill a_{2k-3}\hfill \\ \hfill 0\hfill & \hfill 1\hfill & \hfill a_2\hfill & \hfill \dots \hfill & \hfill a_{2k-4}\hfill \\ \hfill \dots \hfill & \hfill \dots \hfill & \hfill \dots \hfill & \hfill \dots \hfill & \hfill \dots \hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \dots \hfill & \hfill a_k\hfill \end{array}\right|>0,k=1,2,\dots ,n,$$

where the complementary definition is $a_j=0$, $j>n$. Additionally, the corresponding necessary conditions for $A_n$ to be a Hurwitz matrix are as follows

$$\left(i\right)a_j>0,j=1,2,\dots ,n;\left(ii\right)a_i{a}_{i+1}>a_{i-1}{a}_{i+2},i=1,2,\dots ,n-2(a_0=1).$$

Lemma 4.1 [29], [30] —

For the algebraic equation $H_0^2+C_0{\Sigma }_0+{\Sigma }_0{C}_0^T=0$ , where $H_0=\text{diag}(1,0,0,0)$ and ${\Sigma }_0$ is a real symmetric matrix, and the standard matrix

$$C_0=\left(\begin{array}{cccc}\hfill -{\tau }_1\hfill & \hfill -{\tau }_2\hfill & \hfill -{\tau }_3\hfill & \hfill -{\tau }_4\hfill \\ \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill \end{array}\right).$$

If ${\tau }_1>0$ , ${\tau }_3>0$ , ${\tau }_4>0$ and ${\tau }_1{\tau }_2{\tau }_3-{\tau }_3^2-{\tau }_1^2{\tau }_4>0$ , then ${\Sigma }_0$ is a positive definite matrix, where

$${\Sigma }_0=\left(\begin{array}{cccc}\hfill \frac{{\tau }_2{\tau }_3-{\tau }_1{\tau }_4}{2({\tau }_1{\tau }_2{\tau }_3-{\tau }_3^2-{\tau }_1^2{\tau }_4)}\hfill & \hfill 0\hfill & \hfill -\frac{{\tau }_3}{2({\tau }_1{\tau }_2{\tau }_3-{\tau }_3^2-{\tau }_1^2{\tau }_4)}\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill \frac{{\tau }_3}{2({\tau }_1{\tau }_2{\tau }_3-{\tau }_3^2-{\tau }_1^2{\tau }_4)}\hfill & \hfill 0\hfill & \hfill -\frac{{\tau }_1}{2({\tau }_1{\tau }_2{\tau }_3-{\tau }_3^2-{\tau }_1^2{\tau }_4)}\hfill \\ \hfill -\frac{{\tau }_3}{2({\tau }_1{\tau }_2{\tau }_3-{\tau }_3^2-{\tau }_1^2{\tau }_4)}\hfill & \hfill 0\hfill & \hfill \frac{{\tau }_1}{2({\tau }_1{\tau }_2{\tau }_3-{\tau }_3^2-{\tau }_1^2{\tau }_4)}\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill -\frac{{\tau }_1}{2({\tau }_1{\tau }_2{\tau }_3-{\tau }_3^2-{\tau }_1^2{\tau }_4)}\hfill & \hfill 0\hfill & \hfill \frac{{\tau }_1{\tau }_2-{\tau }_3}{2{\tau }_4({\tau }_1{\tau }_2{\tau }_3-{\tau }_3^2-{\tau }_1^2{\tau }_4)}\hfill \end{array}\right).$$

Here $C_0$ in this form is called the standard $R_1$ matrix.

Lemma 4.2 [29] —

For the algebraic equation $H_0^2+{\tilde{C}}_0{\Theta }_0+{\Theta }_0{\tilde{C}}_0^T=0$ , where $H_0=\text{diag}(1,0,0,0)$ , ${\Theta }_0$ is a real symmetric matrix, and the standard matrix

$${\tilde{C}}_0=\left(\begin{array}{cccc}\hfill -l_1\hfill & \hfill -l_2\hfill & \hfill -l_3\hfill & \hfill -l_4\hfill \\ \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -a_{21}\hfill \end{array}\right).$$

If $l_1>0$ , $l_3>0$ and $l_1{l}_2-l_3>0$ , then the matrix ${\Theta }_0$ is semi-positive definite which follows

$${\Theta }_0=\left(\begin{array}{cccc}\hfill \frac{l_2}{2(l_1{l}_2-l_3)}\hfill & \hfill 0\hfill & \hfill -\frac{1}{2(l_1{l}_2-l_3)}\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill \frac{1}{2(l_1{l}_2-l_3)}\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill -\frac{1}{2(l_1{l}_2-l_3)}\hfill & \hfill 0\hfill & \hfill \frac{l_1}{2l_3(l_1{l}_2-l_3)}\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \end{array}\right).$$

Here ${\tilde{C}}_0$ in this form is called the standard $R_2$ matrix.

4.2. Probability density of stationary distribution $\pi \left(\cdot \right)$

Theorem 4.1

If $R_0^S>1$ , then the stationary solution ${(S\left(t\right),E\left(t\right),I_S\left(t\right),I_A\left(t\right))}^T$ of system (1.3) around $P^{\ast }$ follows a unique log-normal probability density $\Phi (S,E,I_S,I_A)$ , which takes the form

$$\Phi (S,E,I_S,I_A)={\left(2\pi \right)}^{-2}{\left|\Sigma \right|}^{-\frac{1}{2}}{e}^{-\frac{1}{2}(ln\frac{S}{S^{\ast }},ln\frac{E}{E^{\ast }},ln\frac{I_S}{I_S^{\ast }},ln\frac{I_A}{I_A^{\ast }}){\Sigma }^{-1}{(ln\frac{S}{S^{\ast }},ln\frac{E}{E^{\ast }},ln\frac{I_S}{I_S^{\ast }},ln\frac{I_A}{I_A^{\ast }})}^T},$$

where the covariance matrix $\Sigma$ is positive definite, and the specific form of $\Sigma$ is given as follows.

(1) If $m_1\ne 0$ , $m_3\ne 0$ and $m_5\ne 0$ .

$•$ If $m_2\ne 0$ , $m_4\ne 0$ and $m_6\ne 0$ , then

$$\Sigma ={\varrho }_1^2{\left(J_2{J}_1\right)}^{-1}{\Sigma }_0{\left[{\left(J_2{J}_1\right)}^{-1}\right]}^T+{\varrho }_2^2{\left(J_7{J}_6{J}_5{J}_4\right)}^{-1}{\Sigma }_0{\left[{\left(J_7{J}_6{J}_5{J}_4\right)}^{-1}\right]}^T+{\varrho }_3^2{\left(J_{13}{J}_{12}{J}_{11}{J}_{10}\right)}^{-1}{\Sigma }_0{\left[{\left(J_{13}{J}_{12}{J}_{11}{J}_{10}\right)}^{-1}\right]}^T+{\varrho }_4^2{\left(J_{19}{J}_{18}{J}_{17}{J}_{16}\right)}^{-1}{\Sigma }_0{\left[{\left(J_{19}{J}_{18}{J}_{17}{J}_{16}\right)}^{-1}\right]}^T;$$

$•$ If $m_2=0$ , $m_4=0$ and $m_6=0$ , then

$$\Sigma ={\varrho }_1^2{\left(J_2{J}_1\right)}^{-1}{\Sigma }_0{\left[{\left(J_2{J}_1\right)}^{-1}\right]}^T+{\tilde{\varrho }}_2^2{\left(J_8{J}_6{J}_5{J}_4\right)}^{-1}{\tilde{\Theta }}_0{\left[{\left(J_8{J}_6{J}_5{J}_4\right)}^{-1}\right]}^T+{\check{\varrho }}_3^2{\left(J_{14}{J}_{12}{J}_{11}{J}_{10}\right)}^{-1}{\check{\Theta }}_0{\left[{\left(J_{14}{J}_{12}{J}_{11}{J}_{10}\right)}^{-1}\right]}^T+{\varpi }_2^2{\left(J_{20}{J}_{18}{J}_{17}{J}_{16}\right)}^{-1}{\overline{\Theta }}_1{\left[{\left(J_{20}{J}_{18}{J}_{17}{J}_{16}\right)}^{-1}\right]}^T;$$

$•$ If $m_2\ne 0$ , $m_4\ne 0$ and $m_6=0$ , then

$$\Sigma ={\varrho }_1^2{\left(J_2{J}_1\right)}^{-1}{\Sigma }_0{\left[{\left(J_2{J}_1\right)}^{-1}\right]}^T+{\varrho }_2^2{\left(J_7{J}_6{J}_5{J}_4\right)}^{-1}{\Sigma }_0{\left[{\left(J_7{J}_6{J}_5{J}_4\right)}^{-1}\right]}^T+{\varrho }_3^2{\left(J_{13}{J}_{12}{J}_{11}{J}_{10}\right)}^{-1}{\Sigma }_0{\left[{\left(J_{13}{J}_{12}{J}_{11}{J}_{10}\right)}^{-1}\right]}^T+{\varpi }_2^2{\left(J_{20}{J}_{18}{J}_{17}{J}_{16}\right)}^{-1}{\overline{\Theta }}_1{\left[{\left(J_{20}{J}_{18}{J}_{17}{J}_{16}\right)}^{-1}\right]}^T;$$

$•$ If $m_2\ne 0$ , $m_4=0$ and $m_6\ne 0$ , then

$$\Sigma ={\varrho }_1^2{\left(J_2{J}_1\right)}^{-1}{\Sigma }_0{\left[{\left(J_2{J}_1\right)}^{-1}\right]}^T+{\varrho }_2^2{\left(J_7{J}_6{J}_5{J}_4\right)}^{-1}{\Sigma }_0{\left[{\left(J_7{J}_6{J}_5{J}_4\right)}^{-1}\right]}^T+{\check{\varrho }}_3^2{\left(J_{14}{J}_{12}{J}_{11}{J}_{10}\right)}^{-1}{\check{\Theta }}_0{\left[{\left(J_{14}{J}_{12}{J}_{11}{J}_{10}\right)}^{-1}\right]}^T+{\varrho }_4^2{\left(J_{19}{J}_{18}{J}_{17}{J}_{16}\right)}^{-1}{\Sigma }_0{\left[{\left(J_{19}{J}_{18}{J}_{17}{J}_{16}\right)}^{-1}\right]}^T;$$

$•$ If $m_2=0$ , $m_4\ne 0$ and $m_6\ne 0$ , then

$$\Sigma ={\varrho }_1^2{\left(J_2{J}_1\right)}^{-1}{\Sigma }_0{\left[{\left(J_2{J}_1\right)}^{-1}\right]}^T+{\tilde{\varrho }}_2^2{\left(J_8{J}_6{J}_5{J}_4\right)}^{-1}{\tilde{\Theta }}_0{\left[{\left(J_8{J}_6{J}_5{J}_4\right)}^{-1}\right]}^T+{\varrho }_3^2{\left(J_{13}{J}_{12}{J}_{11}{J}_{10}\right)}^{-1}{\Sigma }_0{\left[{\left(J_{13}{J}_{12}{J}_{11}{J}_{10}\right)}^{-1}\right]}^T+{\varrho }_4^2{\left(J_{19}{J}_{18}{J}_{17}{J}_{16}\right)}^{-1}{\Sigma }_0{\left[{\left(J_{19}{J}_{18}{J}_{17}{J}_{16}\right)}^{-1}\right]}^T;$$

$•$ If $m_2\ne 0$ , $m_4=0$ and $m_6=0$ , then

$$\Sigma ={\varrho }_1^2{\left(J_2{J}_1\right)}^{-1}{\Sigma }_0{\left[{\left(J_2{J}_1\right)}^{-1}\right]}^T+{\varrho }_2^2{\left(J_7{J}_6{J}_5{J}_4\right)}^{-1}{\Sigma }_0{\left[{\left(J_7{J}_6{J}_5{J}_4\right)}^{-1}\right]}^T+{\check{\varrho }}_3^2{\left(J_{14}{J}_{12}{J}_{11}{J}_{10}\right)}^{-1}{\check{\Theta }}_0{\left[{\left(J_{14}{J}_{12}{J}_{11}{J}_{10}\right)}^{-1}\right]}^T+{\varpi }_2^2{\left(J_{20}{J}_{18}{J}_{17}{J}_{16}\right)}^{-1}{\overline{\Theta }}_1{\left[{\left(J_{20}{J}_{18}{J}_{17}{J}_{16}\right)}^{-1}\right]}^T;$$

$•$ If $m_2=0$ , $m_4\ne 0$ and $m_6=0$ , then

$$\Sigma ={\varrho }_1^2{\left(J_2{J}_1\right)}^{-1}{\Sigma }_0{\left[{\left(J_2{J}_1\right)}^{-1}\right]}^T+{\tilde{\varrho }}_2^2{\left(J_8{J}_6{J}_5{J}_4\right)}^{-1}{\tilde{\Theta }}_0{\left[{\left(J_8{J}_6{J}_5{J}_4\right)}^{-1}\right]}^T+{\varrho }_3^2{\left(J_{13}{J}_{12}{J}_{11}{J}_{10}\right)}^{-1}{\Sigma }_0{\left[{\left(J_{13}{J}_{12}{J}_{11}{J}_{10}\right)}^{-1}\right]}^T+{\varpi }_2^2{\left(J_{20}{J}_{18}{J}_{17}{J}_{16}\right)}^{-1}{\overline{\Theta }}_1{\left[{\left(J_{20}{J}_{18}{J}_{17}{J}_{16}\right)}^{-1}\right]}^T;$$

$•$ If $m_2=0$ , $m_4=0$ and $m_6\ne 0$ , then

$$\Sigma ={\varrho }_1^2{\left(J_2{J}_1\right)}^{-1}{\Sigma }_0{\left[{\left(J_2{J}_1\right)}^{-1}\right]}^T+{\tilde{\varrho }}_2^2{\left(J_8{J}_6{J}_5{J}_4\right)}^{-1}{\tilde{\Theta }}_0{\left[{\left(J_8{J}_6{J}_5{J}_4\right)}^{-1}\right]}^T+{\check{\varrho }}_3^2{\left(J_{14}{J}_{12}{J}_{11}{J}_{10}\right)}^{-1}{\check{\Theta }}_0{\left[{\left(J_{14}{J}_{12}{J}_{11}{J}_{10}\right)}^{-1}\right]}^T+{\varrho }_4^2{\left(J_{19}{J}_{18}{J}_{17}{J}_{16}\right)}^{-1}{\Sigma }_0{\left[{\left(J_{19}{J}_{18}{J}_{17}{J}_{16}\right)}^{-1}\right]}^T.$$

(2) If $m_1=0$ , $m_3=0$ and $m_5=0$ , then

$$\Sigma ={\overline{\varrho }}_1^2{\left(J_3{J}_1\right)}^{-1}{\overline{\Theta }}_0{\left[{\left(J_3{J}_1\right)}^{-1}\right]}^T+{\tilde{\varrho }}_2^2{\left(J_9{J}_5{J}_4\right)}^{-1}{\hat{\Theta }}_0{\left[{\left(J_9{J}_5{J}_4\right)}^{-1}\right]}^T+{\varpi }_1^2{\left(J_{15}{J}_{11}{J}_{10}\right)}^{-1}{\Theta }_1{\left[{\left(J_{15}{J}_{11}{J}_{10}\right)}^{-1}\right]}^T+{\varpi }_3^2{\left(J_{21}{J}_{17}{J}_{16}\right)}^{-1}{\tilde{\Theta }}_1{\left[{\left(J_{21}{J}_{17}{J}_{16}\right)}^{-1}\right]}^T.$$

(3) If $m_1\ne 0$ , $m_3=0$ and $m_5=0$ .

$•$ If $m_2\ne 0$ , then

$$\Sigma ={\varrho }_1^2{\left(J_2{J}_1\right)}^{-1}{\Sigma }_0{\left[{\left(J_2{J}_1\right)}^{-1}\right]}^T+{\varrho }_2^2{\left(J_7{J}_6{J}_5{J}_4\right)}^{-1}{\Sigma }_0{\left[{\left(J_7{J}_6{J}_5{J}_4\right)}^{-1}\right]}^T+{\varpi }_1^2{\left(J_{15}{J}_{11}{J}_{10}\right)}^{-1}{\Theta }_1{\left[{\left(J_{15}{J}_{11}{J}_{10}\right)}^{-1}\right]}^T+{\varpi }_3^2{\left(J_{21}{J}_{17}{J}_{16}\right)}^{-1}{\tilde{\Theta }}_1{\left[{\left(J_{21}{J}_{17}{J}_{16}\right)}^{-1}\right]}^T;$$

$•$ If $m_2=0$ , then

$$\Sigma ={\varrho }_1^2{\left(J_2{J}_1\right)}^{-1}{\Sigma }_0{\left[{\left(J_2{J}_1\right)}^{-1}\right]}^T+{\tilde{\varrho }}_2^2{\left(J_8{J}_6{J}_5{J}_4\right)}^{-1}{\tilde{\Theta }}_0{\left[{\left(J_8{J}_6{J}_5{J}_4\right)}^{-1}\right]}^T+{\varpi }_1^2{\left(J_{15}{J}_{11}{J}_{10}\right)}^{-1}{\Theta }_1{\left[{\left(J_{15}{J}_{11}{J}_{10}\right)}^{-1}\right]}^T+{\varpi }_3^2{\left(J_{21}{J}_{17}{J}_{16}\right)}^{-1}{\tilde{\Theta }}_1{\left[{\left(J_{21}{J}_{17}{J}_{16}\right)}^{-1}\right]}^T.$$

(4) If $m_1=0$ , $m_3\ne 0$ and $m_5=0$ .

$•$ If $m_4\ne 0$ , then

$$\Sigma ={\overline{\varrho }}_1^2{\left(J_3{J}_1\right)}^{-1}{\overline{\Theta }}_0{\left[{\left(J_3{J}_1\right)}^{-1}\right]}^T+{\tilde{\varrho }}_2^2{\left(J_9{J}_5{J}_4\right)}^{-1}{\hat{\Theta }}_0{\left[{\left(J_9{J}_5{J}_4\right)}^{-1}\right]}^T+{\varrho }_3^2{\left(J_{13}{J}_{12}{J}_{11}{J}_{10}\right)}^{-1}{\Sigma }_0{\left[{\left(J_{13}{J}_{12}{J}_{11}{J}_{10}\right)}^{-1}\right]}^T+{\varpi }_3^2{\left(J_{21}{J}_{17}{J}_{16}\right)}^{-1}{\tilde{\Theta }}_1{\left[{\left(J_{21}{J}_{17}{J}_{16}\right)}^{-1}\right]}^T;$$

$•$ If $m_4=0$ , then

$$\Sigma ={\overline{\varrho }}_1^2{\left(J_3{J}_1\right)}^{-1}{\overline{\Theta }}_0{\left[{\left(J_3{J}_1\right)}^{-1}\right]}^T+{\tilde{\varrho }}_2^2{\left(J_9{J}_5{J}_4\right)}^{-1}{\hat{\Theta }}_0{\left[{\left(J_9{J}_5{J}_4\right)}^{-1}\right]}^T+{\check{\varrho }}_3^2{\left(J_{14}{J}_{12}{J}_{11}{J}_{10}\right)}^{-1}{\check{\Theta }}_0{\left[{\left(J_{14}{J}_{12}{J}_{11}{J}_{10}\right)}^{-1}\right]}^T+{\varpi }_3^2{\left(J_{21}{J}_{17}{J}_{16}\right)}^{-1}{\tilde{\Theta }}_1{\left[{\left(J_{21}{J}_{17}{J}_{16}\right)}^{-1}\right]}^T.$$

(5) If $m_1=0$ , $m_3=0$ and $m_5\ne 0$ .

$•$ If $m_6\ne 0$ , then

$$\Sigma ={\overline{\varrho }}_1^2{\left(J_3{J}_1\right)}^{-1}{\overline{\Theta }}_0{\left[{\left(J_3{J}_1\right)}^{-1}\right]}^T+{\tilde{\varrho }}_2^2{\left(J_9{J}_5{J}_4\right)}^{-1}{\hat{\Theta }}_0{\left[{\left(J_9{J}_5{J}_4\right)}^{-1}\right]}^T+{\varpi }_1^2{\left(J_{15}{J}_{11}{J}_{10}\right)}^{-1}{\Theta }_1{\left[{\left(J_{15}{J}_{11}{J}_{10}\right)}^{-1}\right]}^T+{\varrho }_4^2{\left(J_{19}{J}_{18}{J}_{17}{J}_{16}\right)}^{-1}{\Sigma }_0{\left[{\left(J_{19}{J}_{18}{J}_{17}{J}_{16}\right)}^{-1}\right]}^T;$$

$•$ If $m_6=0$ , then

$$\Sigma ={\overline{\varrho }}_1^2{\left(J_3{J}_1\right)}^{-1}{\overline{\Theta }}_0{\left[{\left(J_3{J}_1\right)}^{-1}\right]}^T+{\tilde{\varrho }}_2^2{\left(J_9{J}_5{J}_4\right)}^{-1}{\hat{\Theta }}_0{\left[{\left(J_9{J}_5{J}_4\right)}^{-1}\right]}^T+{\varpi }_1^2{\left(J_{15}{J}_{11}{J}_{10}\right)}^{-1}{\Theta }_1{\left[{\left(J_{15}{J}_{11}{J}_{10}\right)}^{-1}\right]}^T+{\varpi }_2^2{\left(J_{20}{J}_{18}{J}_{17}{J}_{16}\right)}^{-1}{\overline{\Theta }}_1{\left[{\left(J_{20}{J}_{18}{J}_{17}{J}_{16}\right)}^{-1}\right]}^T.$$

(6) If $m_1\ne 0$ , $m_3\ne 0$ and $m_5=0$ .

$•$ If $m_2\ne 0$ and $m_4\ne 0$ , then

$$\Sigma ={\varrho }_1^2{\left(J_2{J}_1\right)}^{-1}{\Sigma }_0{\left[{\left(J_2{J}_1\right)}^{-1}\right]}^T+{\varrho }_2^2{\left(J_7{J}_6{J}_5{J}_4\right)}^{-1}{\Sigma }_0{\left[{\left(J_7{J}_6{J}_5{J}_4\right)}^{-1}\right]}^T+{\varrho }_3^2{\left(J_{13}{J}_{12}{J}_{11}{J}_{10}\right)}^{-1}{\Sigma }_0{\left[{\left(J_{13}{J}_{12}{J}_{11}{J}_{10}\right)}^{-1}\right]}^T+{\varpi }_3^2{\left(J_{21}{J}_{17}{J}_{16}\right)}^{-1}{\tilde{\Theta }}_1{\left[{\left(J_{21}{J}_{17}{J}_{16}\right)}^{-1}\right]}^T;$$

$•$ If $m_2\ne 0$ and $m_4=0$ , then

$$\Sigma ={\varrho }_1^2{\left(J_2{J}_1\right)}^{-1}{\Sigma }_0{\left[{\left(J_2{J}_1\right)}^{-1}\right]}^T+{\varrho }_2^2{\left(J_7{J}_6{J}_5{J}_4\right)}^{-1}{\Sigma }_0{\left[{\left(J_7{J}_6{J}_5{J}_4\right)}^{-1}\right]}^T+{\check{\varrho }}_3^2{\left(J_{14}{J}_{12}{J}_{11}{J}_{10}\right)}^{-1}{\check{\Theta }}_0{\left[{\left(J_{14}{J}_{12}{J}_{11}{J}_{10}\right)}^{-1}\right]}^T+{\varpi }_3^2{\left(J_{21}{J}_{17}{J}_{16}\right)}^{-1}{\tilde{\Theta }}_1{\left[{\left(J_{21}{J}_{17}{J}_{16}\right)}^{-1}\right]}^T;$$

$•$ If $m_2=0$ and $m_4\ne 0$ , then

$$\Sigma ={\varrho }_1^2{\left(J_2{J}_1\right)}^{-1}{\Sigma }_0{\left[{\left(J_2{J}_1\right)}^{-1}\right]}^T+{\tilde{\varrho }}_2^2{\left(J_8{J}_6{J}_5{J}_4\right)}^{-1}{\tilde{\Theta }}_0{\left[{\left(J_8{J}_6{J}_5{J}_4\right)}^{-1}\right]}^T+{\varrho }_3^2{\left(J_{13}{J}_{12}{J}_{11}{J}_{10}\right)}^{-1}{\Sigma }_0{\left[{\left(J_{13}{J}_{12}{J}_{11}{J}_{10}\right)}^{-1}\right]}^T+{\varpi }_3^2{\left(J_{21}{J}_{17}{J}_{16}\right)}^{-1}{\tilde{\Theta }}_1{\left[{\left(J_{21}{J}_{17}{J}_{16}\right)}^{-1}\right]}^T;$$

$•$ If $m_2=0$ and $m_4=0$ , then

$$\Sigma ={\varrho }_1^2{\left(J_2{J}_1\right)}^{-1}{\Sigma }_0{\left[{\left(J_2{J}_1\right)}^{-1}\right]}^T+{\tilde{\varrho }}_2^2{\left(J_8{J}_6{J}_5{J}_4\right)}^{-1}{\tilde{\Theta }}_0{\left[{\left(J_8{J}_6{J}_5{J}_4\right)}^{-1}\right]}^T+{\check{\varrho }}_3^2{\left(J_{14}{J}_{12}{J}_{11}{J}_{10}\right)}^{-1}{\check{\Theta }}_0{\left[{\left(J_{14}{J}_{12}{J}_{11}{J}_{10}\right)}^{-1}\right]}^T+{\varpi }_3^2{\left(J_{21}{J}_{17}{J}_{16}\right)}^{-1}{\tilde{\Theta }}_1{\left[{\left(J_{21}{J}_{17}{J}_{16}\right)}^{-1}\right]}^T.$$

(7) If $m_1\ne 0$ , $m_3=0$ and $m_5\ne 0$ .

$•$ If $m_2\ne 0$ and $m_6\ne 0$ , then

$$\Sigma ={\varrho }_1^2{\left(J_2{J}_1\right)}^{-1}{\Sigma }_0{\left[{\left(J_2{J}_1\right)}^{-1}\right]}^T+{\varrho }_2^2{\left(J_7{J}_6{J}_5{J}_4\right)}^{-1}{\Sigma }_0{\left[{\left(J_7{J}_6{J}_5{J}_4\right)}^{-1}\right]}^T+{\varpi }_1^2{\left(J_{15}{J}_{11}{J}_{10}\right)}^{-1}{\Theta }_1{\left[{\left(J_{15}{J}_{11}{J}_{10}\right)}^{-1}\right]}^T+{\varrho }_4^2{\left(J_{19}{J}_{18}{J}_{17}{J}_{16}\right)}^{-1}{\Sigma }_0{\left[{\left(J_{19}{J}_{18}{J}_{17}{J}_{16}\right)}^{-1}\right]}^T;$$

$•$ If $m_2\ne 0$ and $m_6=0$ , then

$$\Sigma ={\varrho }_1^2{\left(J_2{J}_1\right)}^{-1}{\Sigma }_0{\left[{\left(J_2{J}_1\right)}^{-1}\right]}^T+{\varrho }_2^2{\left(J_7{J}_6{J}_5{J}_4\right)}^{-1}{\Sigma }_0{\left[{\left(J_7{J}_6{J}_5{J}_4\right)}^{-1}\right]}^T+{\varpi }_1^2{\left(J_{15}{J}_{11}{J}_{10}\right)}^{-1}{\Theta }_1{\left[{\left(J_{15}{J}_{11}{J}_{10}\right)}^{-1}\right]}^T+{\varpi }_2^2{\left(J_{20}{J}_{18}{J}_{17}{J}_{16}\right)}^{-1}{\overline{\Theta }}_1{\left[{\left(J_{20}{J}_{18}{J}_{17}{J}_{16}\right)}^{-1}\right]}^T;$$

$•$ If $m_2=0$ and $m_6\ne 0$ , then

$$\Sigma ={\varrho }_1^2{\left(J_2{J}_1\right)}^{-1}{\Sigma }_0{\left[{\left(J_2{J}_1\right)}^{-1}\right]}^T+{\tilde{\varrho }}_2^2{\left(J_8{J}_6{J}_5{J}_4\right)}^{-1}{\tilde{\Theta }}_0{\left[{\left(J_8{J}_6{J}_5{J}_4\right)}^{-1}\right]}^T+{\varpi }_1^2{\left(J_{15}{J}_{11}{J}_{10}\right)}^{-1}{\Theta }_1{\left[{\left(J_{15}{J}_{11}{J}_{10}\right)}^{-1}\right]}^T+{\varrho }_4^2{\left(J_{19}{J}_{18}{J}_{17}{J}_{16}\right)}^{-1}{\Sigma }_0{\left[{\left(J_{19}{J}_{18}{J}_{17}{J}_{16}\right)}^{-1}\right]}^T;$$

$•$ If $m_2=0$ and $m_6=0$ , then

$$\Sigma ={\varrho }_1^2{\left(J_2{J}_1\right)}^{-1}{\Sigma }_0{\left[{\left(J_2{J}_1\right)}^{-1}\right]}^T+{\tilde{\varrho }}_2^2{\left(J_8{J}_6{J}_5{J}_4\right)}^{-1}{\tilde{\Theta }}_0{\left[{\left(J_8{J}_6{J}_5{J}_4\right)}^{-1}\right]}^T+{\varpi }_1^2{\left(J_{15}{J}_{11}{J}_{10}\right)}^{-1}{\Theta }_1{\left[{\left(J_{15}{J}_{11}{J}_{10}\right)}^{-1}\right]}^T+{\varpi }_2^2{\left(J_{20}{J}_{18}{J}_{17}{J}_{16}\right)}^{-1}{\overline{\Theta }}_1{\left[{\left(J_{20}{J}_{18}{J}_{17}{J}_{16}\right)}^{-1}\right]}^T.$$

(8) If $m_1=0$ , $m_3\ne 0$ and $m_5\ne 0$ .

$•$ If $m_4\ne 0$ and $m_6\ne 0$ , then

$$\Sigma ={\overline{\varrho }}_1^2{\left(J_3{J}_1\right)}^{-1}{\overline{\Theta }}_0{\left[{\left(J_3{J}_1\right)}^{-1}\right]}^T+{\tilde{\varrho }}_2^2{\left(J_9{J}_5{J}_4\right)}^{-1}{\hat{\Theta }}_0{\left[{\left(J_9{J}_5{J}_4\right)}^{-1}\right]}^T+{\varrho }_3^2{\left(J_{13}{J}_{12}{J}_{11}{J}_{10}\right)}^{-1}{\Sigma }_0{\left[{\left(J_{13}{J}_{12}{J}_{11}{J}_{10}\right)}^{-1}\right]}^T+{\varrho }_4^2{\left(J_{19}{J}_{18}{J}_{17}{J}_{16}\right)}^{-1}{\Sigma }_0{\left[{\left(J_{19}{J}_{18}{J}_{17}{J}_{16}\right)}^{-1}\right]}^T;$$

$•$ If $m_4\ne 0$ and $m_6=0$ , then

$$\Sigma ={\overline{\varrho }}_1^2{\left(J_3{J}_1\right)}^{-1}{\overline{\Theta }}_0{\left[{\left(J_3{J}_1\right)}^{-1}\right]}^T+{\tilde{\varrho }}_2^2{\left(J_9{J}_5{J}_4\right)}^{-1}{\hat{\Theta }}_0{\left[{\left(J_9{J}_5{J}_4\right)}^{-1}\right]}^T+{\varrho }_3^2{\left(J_{13}{J}_{12}{J}_{11}{J}_{10}\right)}^{-1}{\Sigma }_0{\left[{\left(J_{13}{J}_{12}{J}_{11}{J}_{10}\right)}^{-1}\right]}^T+{\varpi }_2^2{\left(J_{20}{J}_{18}{J}_{17}{J}_{16}\right)}^{-1}{\overline{\Theta }}_1{\left[{\left(J_{20}{J}_{18}{J}_{17}{J}_{16}\right)}^{-1}\right]}^T;$$

$•$ If $m_4=0$ and $m_6\ne 0$ , then

$$\Sigma ={\overline{\varrho }}_1^2{\left(J_3{J}_1\right)}^{-1}{\overline{\Theta }}_0{\left[{\left(J_3{J}_1\right)}^{-1}\right]}^T+{\tilde{\varrho }}_2^2{\left(J_9{J}_5{J}_4\right)}^{-1}{\hat{\Theta }}_0{\left[{\left(J_9{J}_5{J}_4\right)}^{-1}\right]}^T+{\check{\varrho }}_3^2{\left(J_{14}{J}_{12}{J}_{11}{J}_{10}\right)}^{-1}{\check{\Theta }}_0{\left[{\left(J_{14}{J}_{12}{J}_{11}{J}_{10}\right)}^{-1}\right]}^T+{\varrho }_4^2{\left(J_{19}{J}_{18}{J}_{17}{J}_{16}\right)}^{-1}{\Sigma }_0{\left[{\left(J_{19}{J}_{18}{J}_{17}{J}_{16}\right)}^{-1}\right]}^T;$$

$•$ If $m_4=0$ and $m_6=0$ , then

$$\Sigma ={\overline{\varrho }}_1^2{\left(J_3{J}_1\right)}^{-1}{\overline{\Theta }}_0{\left[{\left(J_3{J}_1\right)}^{-1}\right]}^T+{\tilde{\varrho }}_2^2{\left(J_9{J}_5{J}_4\right)}^{-1}{\hat{\Theta }}_0{\left[{\left(J_9{J}_5{J}_4\right)}^{-1}\right]}^T+{\check{\varrho }}_3^2{\left(J_{14}{J}_{12}{J}_{11}{J}_{10}\right)}^{-1}{\check{\Theta }}_0{\left[{\left(J_{14}{J}_{12}{J}_{11}{J}_{10}\right)}^{-1}\right]}^T+{\varpi }_2^2{\left(J_{20}{J}_{18}{J}_{17}{J}_{16}\right)}^{-1}{\overline{\Theta }}_1{\left[{\left(J_{20}{J}_{18}{J}_{17}{J}_{16}\right)}^{-1}\right]}^T,$$

with

$$J_1=\left(\begin{array}{cccc}\hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill -\frac{a_{42}}{a_{32}}\hfill & \hfill 1\hfill \end{array}\right),J_3=\left(\begin{array}{cccc}\hfill a_{21}{a}_{32}\hfill & \hfill -a_{32}(a_{21}+a_{32})\hfill & \hfill (a_{23}{a}_{32}+a_{24}{a}_{42}+a_{32}^2)\hfill & \hfill a_{24}{a}_{32}\hfill \\ \hfill 0\hfill & \hfill a_{32}\hfill & \hfill -a_{32}\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \end{array}\right),$$

$$J_2=\left(\begin{array}{cccc}\hfill a_{21}{a}_{32}{m}_1\hfill & \hfill -a_{32}(a_{21}+a_{32}+a_{42})m_1\hfill & \hfill (a_{23}{a}_{32}+a_{24}{a}_{42}+a_{32}^2+a_{32}{a}_{42}+a_{42}^2)m_1\hfill & \hfill a_{24}{a}_{32}{m}_1-a_{42}^3\hfill \\ \hfill 0\hfill & \hfill a_{32}{m}_1\hfill & \hfill -(a_{32}+a_{42})m_1\hfill & \hfill a_{42}^2\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill m_1\hfill & \hfill -a_{42}\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \end{array}\right),$$

$$J_4=\left(\begin{array}{cccc}\hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill \\ \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \end{array}\right),J_5=\left(\begin{array}{cccc}\hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill -\frac{a_{42}}{a_{32}}\hfill & \hfill 0\hfill & \hfill 1\hfill \end{array}\right),J_6=\left(\begin{array}{cccc}\hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{a_{42}(a_{32}-a_{42})}{a_{13}{a}_{32}+a_{14}{a}_{42}}\hfill & \hfill 1\hfill \end{array}\right),$$

$$J_7=\left(\begin{array}{cccc}\hfill -(a_{13}{a}_{32}+a_{14}{a}_{42})m_2\hfill & \hfill j_{7\left(12\right)}\hfill & \hfill j_{7\left(13\right)}\hfill & \hfill j_{7\left(14\right)}\hfill \\ \hfill 0\hfill & \hfill -\frac{a_{13}{a}_{32}+a_{14}{a}_{42}}{a_{32}}{m}_2\hfill & \hfill -(a_{11}+a_{42})m_2\hfill & \hfill j_{7\left(24\right)}\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill m_2\hfill & \hfill \frac{a_{14}{a}_{42}(a_{42}-a_{32})}{a_{13}{a}_{32}+a_{14}{a}_{42}}-a_{42}\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \end{array}\right),$$

$$J_8=\left(\begin{array}{cccc}\hfill -(a_{13}{a}_{32}+a_{14}{a}_{42})\hfill & \hfill a_{11}{a}_{13}+a_{13}{a}_{32}+\frac{a_{14}{a}_{42}(a_{11}+a_{42})}{a_{32}}\hfill & \hfill {\left(a_{11}+\frac{a_{14}{a}_{42}(a_{42}-a_{32})}{a_{13}{a}_{32}+a_{14}{a}_{42}}\right)}^2\hfill & \hfill a_{14}(a_{11}+a_{42})\hfill \\ \hfill 0\hfill & \hfill -\frac{a_{13}{a}_{32}+a_{14}{a}_{42}}{a_{32}}\hfill & \hfill -a_{11}-\frac{a_{14}{a}_{42}(a_{42}-a_{32})}{a_{13}{a}_{32}+a_{14}{a}_{42}}\hfill & \hfill -a_{14}\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \end{array}\right)$$

$$J_9=\left(\begin{array}{cccc}\hfill -(a_{13}{a}_{32}+a_{14}{a}_{42})\hfill & \hfill \left(a_{11}{a}_{13}+a_{13}{a}_{32}+a_{14}{a}_{42}+\frac{a_{11}{a}_{14}{a}_{42}}{a_{32}}\right)\hfill & \hfill a_{11}^2\hfill & \hfill a_{14}(a_{11}+a_{42})\hfill \\ \hfill 0\hfill & \hfill -\frac{a_{13}{a}_{32}+a_{14}{a}_{42}}{a_{32}}\hfill & \hfill -a_{11}\hfill & \hfill -a_{14}\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \end{array}\right),$$

$$J_{10}=\left(\begin{array}{cccc}\hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \\ \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \end{array}\right),J_{11}=\left(\begin{array}{cccc}\hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill \frac{a_{13}}{a_{23}}\hfill & \hfill 0\hfill & \hfill 1\hfill \end{array}\right),J_{12}=\left(\begin{array}{cccc}\hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill -\frac{m_3}{a_{42}}\hfill & \hfill 1\hfill \end{array}\right),$$

$$J_{13}=\left(\begin{array}{cccc}\hfill a_{23}{a}_{42}{m}_4\hfill & \hfill -a_{42}(a_{11}+a_{21}+a_{42})m_4\hfill & \hfill j_{13\left(13\right)}\hfill & \hfill j_{13\left(14\right)}\hfill \\ \hfill 0\hfill & \hfill a_{42}{m}_4\hfill & \hfill \left(\frac{a_{13}{a}_{21}}{a_{23}}-a_{11}-a_{42}\right)m_4\hfill & \hfill \frac{{(a_{13}{a}_{21}-a_{11}{a}_{23})}^2}{a_{23}^2}\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill m_4\hfill & \hfill \frac{a_{13}{a}_{21}-a_{11}{a}_{23}}{a_{23}}\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \end{array}\right),$$

$$J_{14}=\left(\begin{array}{cccc}\hfill a_{23}{a}_{42}\hfill & \hfill -a_{42}\left(a_{21}+a_{42}+\frac{a_{13}{a}_{21}}{a_{23}}\right)\hfill & \hfill a_{24}{a}_{42}+a_{21}{m}_3+a_{42}^2\hfill & \hfill a_{21}{a}_{42}\hfill \\ \hfill 0\hfill & \hfill a_{42}\hfill & \hfill -a_{42}\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \end{array}\right),$$

$$J_{15}=\left(\begin{array}{cccc}\hfill a_{23}{a}_{42}\hfill & \hfill -a_{42}\left(a_{21}+a_{42}+\frac{a_{13}{a}_{21}}{a_{23}}\right)\hfill & \hfill a_{42}(a_{24}+a_{42})\hfill & \hfill a_{21}{a}_{42}\hfill \\ \hfill 0\hfill & \hfill a_{42}\hfill & \hfill -a_{42}\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \end{array}\right),$$

$$J_{16}=\left(\begin{array}{cccc}\hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \\ \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill \\ \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \end{array}\right),J_{17}=\left(\begin{array}{cccc}\hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill \frac{a_{14}}{a_{24}}\hfill & \hfill 0\hfill & \hfill 1\hfill \end{array}\right),J_{18}=\left(\begin{array}{cccc}\hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill -\frac{m_5}{a_{32}}\hfill & \hfill 1\hfill \end{array}\right),$$

$$J_{19}=\left(\begin{array}{cccc}\hfill a_{24}{a}_{32}{m}_6\hfill & \hfill -a_{32}(a_{11}+a_{21}+a_{32})m_6\hfill & \hfill j_{19\left(13\right)}\hfill & \hfill j_{19\left(14\right)}\hfill \\ \hfill 0\hfill & \hfill a_{32}{m}_6\hfill & \hfill \left(\frac{a_{14}{a}_{21}}{a_{24}}-a_{11}-a_{32}\right)m_6\hfill & \hfill \frac{{(a_{14}{a}_{21}-a_{11}{a}_{24})}^2}{a_{24}^2}\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill m_6\hfill & \hfill \frac{a_{14}{a}_{21}-a_{11}{a}_{24}}{a_{24}}\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \end{array}\right),$$

$$J_{20}=\left(\begin{array}{cccc}\hfill a_{24}{a}_{32}\hfill & \hfill -a_{32}\left(a_{21}+a_{32}+\frac{a_{14}{a}_{21}}{a_{24}}\right)\hfill & \hfill a_{23}{a}_{32}+a_{32}^2+a_{21}{m}_5\hfill & \hfill a_{21}{a}_{32}\hfill \\ \hfill 0\hfill & \hfill a_{32}\hfill & \hfill -a_{32}\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \end{array}\right)$$

$$J_{21}=\left(\begin{array}{cccc}\hfill a_{24}{a}_{32}\hfill & \hfill -a_{32}\left(a_{21}+a_{32}+\frac{a_{14}{a}_{21}}{a_{24}}\right)\hfill & \hfill a_{32}(a_{23}+a_{32})\hfill & \hfill a_{21}{a}_{32}\hfill \\ \hfill 0\hfill & \hfill a_{32}\hfill & \hfill -a_{32}\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \end{array}\right),$$

$${\Sigma }_0=\left(\begin{array}{cccc}\hfill \frac{{\tau }_2{\tau }_3-{\tau }_1{\tau }_4}{2({\tau }_1{\tau }_2{\tau }_3-{\tau }_3^2-{\tau }_1^2{\tau }_4)}\hfill & \hfill 0\hfill & \hfill -\frac{{\tau }_3}{2({\tau }_1{\tau }_2{\tau }_3-{\tau }_3^2-{\tau }_1^2{\tau }_4)}\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill \frac{{\tau }_3}{2({\tau }_1{\tau }_2{\tau }_3-{\tau }_3^2-{\tau }_1^2{\tau }_4)}\hfill & \hfill 0\hfill & \hfill -\frac{{\tau }_1}{2({\tau }_1{\tau }_2{\tau }_3-{\tau }_3^2-{\tau }_1^2{\tau }_4)}\hfill \\ \hfill -\frac{{\tau }_3}{2({\tau }_1{\tau }_2{\tau }_3-{\tau }_3^2-{\tau }_1^2{\tau }_4)}\hfill & \hfill 0\hfill & \hfill \frac{{\tau }_1}{2({\tau }_1{\tau }_2{\tau }_3-{\tau }_3^2-{\tau }_1^2{\tau }_4)}\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill -\frac{{\tau }_1}{2({\tau }_1{\tau }_2{\tau }_3-{\tau }_3^2-{\tau }_1^2{\tau }_4)}\hfill & \hfill 0\hfill & \hfill \frac{{\tau }_1{\tau }_2-{\tau }_3}{2{\tau }_4({\tau }_1{\tau }_2{\tau }_3-{\tau }_3^2-{\tau }_1^2{\tau }_4)}\hfill \end{array}\right),$$

$${\overline{\Theta }}_0=\left(\begin{array}{cccc}\hfill \frac{{\overline{l}}_2}{2({\overline{l}}_1{\overline{l}}_2-{\overline{l}}_3)}\hfill & \hfill 0\hfill & \hfill -\frac{1}{2({\overline{l}}_1{\overline{l}}_2-{\overline{l}}_3)}\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill \frac{1}{2({\overline{l}}_1{\overline{l}}_2-{\overline{l}}_3)}\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill -\frac{1}{2({\overline{l}}_1{\overline{l}}_2-{\overline{l}}_3)}\hfill & \hfill 0\hfill & \hfill \frac{{\overline{l}}_1}{2{\overline{l}}_3({\overline{l}}_1{\overline{l}}_2-{\overline{l}}_3)}\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \end{array}\right),$$

$${\tilde{\Theta }}_0=\left(\begin{array}{cccc}\hfill \frac{{\tilde{l}}_2}{2({\tilde{l}}_1{\tilde{l}}_2-{\tilde{l}}_3)}\hfill & \hfill 0\hfill & \hfill -\frac{1}{2({\tilde{l}}_1{\tilde{l}}_2-{\tilde{l}}_3)}\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill \frac{1}{2({\tilde{l}}_1{\tilde{l}}_2-{\tilde{l}}_3)}\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill -\frac{1}{2({\tilde{l}}_1{\tilde{l}}_2-{\tilde{l}}_3)}\hfill & \hfill 0\hfill & \hfill \frac{{\tilde{l}}_1}{2{\tilde{l}}_3({\tilde{l}}_1{\tilde{l}}_2-{\tilde{l}}_3)}\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \end{array}\right),$$

$${\hat{\Theta }}_0=\left(\begin{array}{cccc}\hfill \frac{{\hat{l}}_2}{2({\hat{l}}_1{\hat{l}}_2-{\hat{l}}_3)}\hfill & \hfill 0\hfill & \hfill -\frac{1}{2({\hat{l}}_1{\hat{l}}_2-{\hat{l}}_3)}\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill \frac{1}{2({\hat{l}}_1{\hat{l}}_2-{\hat{l}}_3)}\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill -\frac{1}{2({\hat{l}}_1{\hat{l}}_2-{\hat{l}}_3)}\hfill & \hfill 0\hfill & \hfill \frac{{\hat{l}}_1}{2{\hat{l}}_3({\hat{l}}_1{\hat{l}}_2-{\hat{l}}_3)}\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \end{array}\right),{\check{\Theta }}_0=\left(\begin{array}{cccc}\hfill \frac{{\check{l}}_2}{2({\check{l}}_1{\check{l}}_2-{\check{l}}_3)}\hfill & \hfill 0\hfill & \hfill -\frac{1}{2({\check{l}}_1{\check{l}}_2-{\check{l}}_3)}\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill \frac{1}{2({\check{l}}_1{\check{l}}_2-{\check{l}}_3)}\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill -\frac{1}{2({\check{l}}_1{\check{l}}_2-{\check{l}}_3)}\hfill & \hfill 0\hfill & \hfill \frac{{\check{l}}_1}{2{\check{l}}_3({\check{l}}_1{\check{l}}_2-{\check{l}}_3)}\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \end{array}\right),$$

$${\Theta }_1=\left(\begin{array}{cccc}\hfill \frac{p_2}{2(p_1{p}_2-p_3)}\hfill & \hfill 0\hfill & \hfill -\frac{1}{2(p_1{p}_2-p_3)}\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill \frac{1}{2(p_1{p}_2-p_3)}\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill -\frac{1}{2(p_1{p}_2-p_3)}\hfill & \hfill 0\hfill & \hfill \frac{p_1}{2p_3(p_1{p}_2-p_3)}\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \end{array}\right),$$

$${\overline{\Theta }}_1=\left(\begin{array}{cccc}\hfill \frac{{\overline{p}}_2}{2({\overline{p}}_1{\overline{p}}_2-{\overline{p}}_3)}\hfill & \hfill 0\hfill & \hfill -\frac{1}{2({\overline{p}}_1{\overline{p}}_2-{\overline{p}}_3)}\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill \frac{1}{2({\overline{p}}_1{\overline{p}}_2-{\overline{p}}_3)}\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill -\frac{1}{2({\overline{p}}_1{\overline{p}}_2-{\overline{p}}_3)}\hfill & \hfill 0\hfill & \hfill \frac{{\overline{p}}_1}{2{\overline{p}}_3({\overline{p}}_1{\overline{p}}_2-{\overline{p}}_3)}\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \end{array}\right),$$

$${\tilde{\Theta }}_1=\left(\begin{array}{cccc}\hfill \frac{{\tilde{p}}_2}{2({\tilde{p}}_1{\tilde{p}}_2-{\tilde{p}}_3)}\hfill & \hfill 0\hfill & \hfill -\frac{1}{2({\tilde{p}}_1{\tilde{p}}_2-{\tilde{p}}_3)}\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill \frac{1}{2({\tilde{p}}_1{\tilde{p}}_2-{\tilde{p}}_3)}\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill -\frac{1}{2({\tilde{p}}_1{\tilde{p}}_2-{\tilde{p}}_3)}\hfill & \hfill 0\hfill & \hfill \frac{{\tilde{p}}_1}{2{\tilde{p}}_3({\tilde{p}}_1{\tilde{p}}_2-{\tilde{p}}_3)}\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \end{array}\right),$$

$$j_{7\left(12\right)}=\left(a_{11}{a}_{13}+a_{13}{a}_{42}+a_{13}{a}_{32}+a_{14}{a}_{42}+\frac{a_{14}{a}_{42}(a_{11}+a_{42})}{a_{32}}\right)m_2,$$

$$j_{7\left(13\right)}=(a_{11}+a_{42})\left(a_{11}+\frac{a_{14}{a}_{42}(a_{42}-a_{32})}{a_{13}{a}_{32}+a_{14}{a}_{42}}\right)m_2+{\left(\frac{a_{14}{a}_{42}(a_{42}-a_{32})}{a_{13}{a}_{32}+a_{14}{a}_{42}}-a_{42}\right)}^2{m}_2-a_{14}{m}_2^2,$$

$$j_{7\left(14\right)}=a_{14}\left(a_{11}+2a_{42}-\frac{a_{14}{a}_{42}(a_{42}-a_{32})}{a_{13}{a}_{32}+a_{14}{a}_{42}}\right)m_2+{\left(\frac{a_{14}{a}_{42}(a_{42}-a_{32})}{a_{13}{a}_{32}+a_{14}{a}_{42}}-a_{42}\right)}^3,$$

$$j_{7\left(24\right)}={\left(\frac{a_{14}{a}_{42}(a_{42}-a_{32})}{a_{13}{a}_{32}+a_{14}{a}_{42}}-a_{42}\right)}^2-a_{14}{m}_2,$$

$$j_{13\left(13\right)}=\left(a_{11}^2+a_{11}{a}_{42}+a_{24}{a}_{42}+a_{42}^2-\frac{a_{13}(a_{11}{a}_{21}+a_{21}^2+a_{21}{a}_{42})}{a_{23}}\right)m_4,j_{13\left(14\right)}=\frac{{(a_{13}{a}_{21}-a_{11}{a}_{23})}^3}{a_{23}^3}+a_{21}{a}_{42}{m}_4,$$

$$j_{19\left(13\right)}=\left(a_{11}^2+a_{11}{a}_{32}+a_{23}{a}_{32}+a_{32}^2-\frac{a_{14}{a}_{21}(a_{11}+a_{21}+a_{32})}{a_{24}}\right)m_6,j_{19\left(14\right)}=\frac{{(a_{14}{a}_{21}-a_{11}{a}_{24})}^3}{a_{24}^3}+a_{21}{a}_{32}{m}_6,$$

$$m_1=\frac{a_{42}(a_{32}-a_{42})}{a_{32}},m_2=\frac{a_{42}(a_{11}-a_{42})(a_{42}-a_{32})}{a_{13}{a}_{32}+a_{14}{a}_{42}}+\frac{a_{14}{a}_{42}^2{(a_{42}-a_{32})}^2}{{(a_{13}{a}_{32}+a_{14}{a}_{42})}^2},m_3=\frac{a_{13}(a_{11}-a_{21})}{a_{23}}-\frac{a_{13}^2{a}_{21}}{a_{23}^2},$$

$$m_4=\frac{a_{13}{a}_{21}+a_{23}{a}_{42}-a_{11}{a}_{23}}{a_{23}{a}_{42}}{m}_3,m_5=\frac{a_{14}(a_{11}-a_{21})}{a_{24}}-\frac{a_{14}^2{a}_{21}}{a_{24}^2},m_6=\frac{a_{14}{a}_{21}+a_{24}{a}_{32}-a_{11}{a}_{24}}{a_{24}{a}_{32}}{m}_5,$$

$${\varrho }_1=a_{21}{a}_{32}{m}_1{\rho }_1,{\overline{\varrho }}_1=a_{21}{a}_{32}{\rho }_1,{\varrho }_2=-(a_{13}{a}_{32}+a_{14}{a}_{42})m_2{\rho }_2,{\tilde{\varrho }}_2=-(a_{13}{a}_{32}+a_{14}{a}_{42}){\rho }_2,$$

$${\varrho }_3=a_{23}{a}_{42}{m}_4{\rho }_3,{\check{\varrho }}_3=a_{23}{a}_{42}{\rho }_3,{\varrho }_4=a_{24}{a}_{32}{m}_6{\rho }_4,{\varpi }_1=a_{23}{a}_{42}{\rho }_3,{\varpi }_2=a_{24}{a}_{32}{\rho }_4,{\varpi }_3=a_{24}{a}_{32}{\rho }_4,$$

$${\tau }_1=a_{11}+a_{21}+a_{32}+a_{42},{\tau }_2=a_{11}{a}_{21}+a_{11}{a}_{32}+a_{11}{a}_{42}+a_{21}{a}_{32}+a_{21}{a}_{42}+a_{32}{a}_{42}-a_{23}{a}_{32}-a_{24}{a}_{42},$$

$${\tau }_3=a_{11}{a}_{21}{a}_{32}+a_{11}{a}_{21}{a}_{42}+a_{11}{a}_{32}{a}_{42}+a_{13}{a}_{21}{a}_{32}+a_{14}{a}_{21}{a}_{42}+a_{21}{a}_{32}{a}_{42}-a_{11}{a}_{23}{a}_{32}-a_{11}{a}_{24}{a}_{42}-a_{23}{a}_{32}{a}_{42}-a_{24}{a}_{32}{a}_{42},$$

$${\tau }_4=a_{11}{a}_{21}{a}_{32}{a}_{42}+a_{13}{a}_{21}{a}_{32}{a}_{42}+a_{14}{a}_{21}{a}_{32}{a}_{42}-a_{11}{a}_{23}{a}_{32}{a}_{42}-a_{11}{a}_{24}{a}_{32}{a}_{42},{\overline{l}}_1=a_{11}+a_{21}+a_{32},$$

$${\overline{l}}_2=a_{11}{a}_{21}+a_{11}{a}_{32}+a_{21}{a}_{32}-a_{23}{a}_{32}-a_{24}{a}_{42},{\overline{l}}_3=a_{11}{a}_{21}{a}_{32}+a_{11}{a}_{24}{a}_{42}+a_{13}{a}_{21}{a}_{32}-a_{11}{a}_{23}{a}_{32}-a_{14}{a}_{21}{a}_{42},$$

$${\tilde{l}}_1=a_{21}-a_{11}-\frac{a_{13}{a}_{32}^2+a_{14}{a}_{42}^2}{a_{13}{a}_{32}+a_{14}{a}_{42}},{\tilde{l}}_2=a_{11}{a}_{21}+a_{11}{a}_{32}+\frac{a_{14}{a}_{32}{a}_{42}(a_{42}-a_{32})+a_{21}(a_{13}{a}_{32}^2+a_{14}{a}_{42}^2)}{a_{13}{a}_{32}+a_{14}{a}_{42}}-a_{23}{a}_{32}-a_{24}{a}_{42},$$

$${\tilde{l}}_3=(a_{21}{a}_{32}-a_{23}{a}_{32}-a_{24}{a}_{42})\left(a_{11}+\frac{a_{14}{a}_{42}(a_{42}-a_{32})}{a_{13}{a}_{32}+a_{14}{a}_{42}}\right)+a_{21}(a_{13}{a}_{32}+a_{14}{a}_{42}+a_{42}(a_{42}-a_{32})),$$

$${\hat{l}}_1=a_{11}+a_{21}+a_{32},{\hat{l}}_2=a_{11}{a}_{21}+a_{11}{a}_{32}+a_{21}{a}_{32}-a_{23}{a}_{32}-a_{24}{a}_{42},$$

$${\hat{l}}_3=a_{11}{a}_{21}{a}_{32}+a_{13}{a}_{21}{a}_{32}+a_{14}{a}_{21}{a}_{42}-a_{11}{a}_{23}{a}_{32}-a_{11}{a}_{24}{a}_{42},$$

$${\check{l}}_1=a_{21}+a_{32}+a_{42}+\frac{a_{13}{a}_{21}}{a_{23}},{\check{l}}_2=a_{21}{a}_{32}+a_{21}{a}_{42}+a_{32}{a}_{42}+\frac{a_{13}{a}_{21}}{a_{23}}(a_{32}+a_{42})-a_{23}{a}_{32}-a_{24}{a}_{42}-a_{21}{m}_3,$$

$${\check{l}}_3=a_{32}{a}_{42}\left(\frac{a_{13}{a}_{21}}{a_{23}}+a_{21}-a_{23}-a_{24}-\frac{a_{21}{m}_3}{a_{42}}\right),$$

$$p_1=a_{21}+a_{32}+a_{42}+\frac{a_{13}{a}_{21}}{a_{23}},p_2=\frac{a_{13}{a}_{21}(a_{32}+a_{42})}{a_{23}}+a_{21}{a}_{32}+a_{21}{a}_{42}+a_{32}{a}_{42}-a_{23}{a}_{32}-a_{24}{a}_{42},$$

$$p_3=a_{32}{a}_{42}\left(\frac{a_{13}{a}_{21}}{a_{23}}+a_{21}-a_{23}-a_{24}\right),{\overline{p}}_1=a_{21}+a_{32}+a_{42}+\frac{a_{14}{a}_{21}}{a_{24}},$$

$${\overline{p}}_2=(a_{32}+a_{42})\left(a_{21}+a_{32}+\frac{a_{14}{a}_{21}}{a_{24}}\right)-a_{23}{a}_{32}-a_{32}^2-a_{24}{a}_{42}-a_{21}{m}_5,{\overline{p}}_3=a_{32}{a}_{42}\left(\frac{a_{14}{a}_{21}}{a_{24}}+a_{21}-a_{23}-a_{24}\right)-a_{21}{a}_{42}{m}_5,$$

$${\tilde{p}}_1=a_{21}+a_{32}+a_{42}+\frac{a_{14}{a}_{21}}{a_{24}},{\tilde{p}}_2=a_{21}{a}_{32}+a_{21}{a}_{42}+a_{32}{a}_{42}+\frac{a_{14}{a}_{21}(a_{32}+a_{42})}{a_{24}}-a_{23}{a}_{32}-a_{24}{a}_{42},$$

$${\tilde{p}}_3=a_{32}{a}_{42}\left(\frac{a_{14}{a}_{21}}{a_{24}}+a_{21}-a_{23}-a_{24}\right).$$

Proof

For convenience and simplicity, let $B\left(t\right)={(B_1\left(t\right),B_2\left(t\right),B_3\left(t\right),B_4\left(t\right))}^T$ and

$$C=\left(\begin{array}{cccc}\hfill -a_{11}\hfill & \hfill 0\hfill & \hfill -a_{13}\hfill & \hfill -a_{14}\hfill \\ \hfill a_{21}\hfill & \hfill -a_{21}\hfill & \hfill a_{23}\hfill & \hfill a_{24}\hfill \\ \hfill 0\hfill & \hfill a_{32}\hfill & \hfill -a_{32}\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill a_{42}\hfill & \hfill 0\hfill & \hfill -a_{42}\hfill \end{array}\right),H=\left(\begin{array}{cccc}\hfill {\rho }_1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill {\rho }_2\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill {\rho }_3\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill {\rho }_4\hfill \end{array}\right).$$

With these notations, system (4.3) can be simplified to the following equivalent form

$$dX\left(t\right)=CX\left(t\right)dt+HdB\left(t\right).$$

In the light of the continuous Markov processes theory [31], system (4.3) has a unique density function $\Phi (x_1,x_2,x_3,x_4,t)$, which can be determined by the following four-dimensional Fokker–Planck equation

$$\frac{\partial }{\partial t}\Phi (X\left(t\right),t)+\frac{\partial }{\partial X\left(t\right)}\left[CX\left(t\right)\Phi (X\left(t\right),t)\right]-\sum _{i=1}^4\frac{{\rho }_i^2}{2}\frac{{\partial }^2}{\partial x_i^2}\Phi (X\left(t\right),t)=0.$$ (4.4)

Next, we will give the exact expression of the probability density by solving Eq. (4.4). Note that $\partial \Phi (X\left(t\right),t)/\partial t=0$ under a stationary case, then (4.4) becomes

$$\frac{\partial }{\partial x_1}\left[(-a_{11}{x}_1-a_{13}{x}_3-a_{14}{x}_4)\Phi \right]+\frac{\partial }{\partial x_2}\left[(a_{21}{x}_1-a_{21}{x}_2+a_{23}{x}_3+a_{24}{x}_4)\Phi \right]+\frac{\partial }{\partial x_3}\left[(a_{32}{x}_2-a_{32}{x}_3)\Phi \right]+\frac{\partial }{\partial x_4}\left[(a_{42}{x}_2-a_{42}{x}_4)\Phi \right]-\sum _{i=1}^4\frac{{\rho }_i^2}{2}\frac{{\partial }^2\Phi }{\partial x_i^2}=0.$$ (4.5)

Additionally, since the diffusion matrix $H$ is a constant matrix, then the probability density $\Phi \left(X\right)$ can be described by a Gaussian distribution through the study of Roozen [32], that is,

$$\Phi \left(X\right)=cexp\left\{-\frac{1}{2}XQ{X}^T\right\},$$

where $Q$ is a real symmetric matrix and $c$ is a positive constant satisfying the normalization condition ${\int }_{{\mathbb{R}}^4}\Phi \left(X\right)dX=1$.

Substituting these results into (4.5), we can obtain the constant $c={\left(2\pi \right)}^{-2}{\left|\Sigma \right|}^{-\frac{1}{2}}$ and $Q$ satisfies the following algebraic equation

$$Q{H}^{2}Q+QC+C^{T}Q=0.$$ (4.6)

If the matrix $Q$ is positive definite, hence it is invertible, we define $Q^{-1}=\Sigma$, then the algebraic equation (4.6) can be equivalently rewritten as

$$H^2+C\Sigma +\Sigma C^T=0.$$ (4.7)

By the finite independent superposition principle [31], (4.7) is equivalent to the sum of the following four algebraic sub-equations,

$$H_i^2+C{\Sigma }_i+{\Sigma }_i{C}^T=0,i=1,2,3,4,$$

where $H_1=\text{diag}({\rho }_1,0,0,0)$, $H_2=\text{diag}(0,{\rho }_2,0,0)$, $H_3=\text{diag}(0,0,{\rho }_3,0)$, $H_4=\text{diag}(0,0,0,{\rho }_4)$, and the symmetric matrices ${\Sigma }_i$ $(i=1,2,3,4)$ are their solutions, respectively. Obviously, $\Sigma ={\Sigma }_1+{\Sigma }_2+{\Sigma }_3+{\Sigma }_4$ and $H^2=H_1^2+H_2^2+H_3^2+H_4^2$.

In order to obtain the positive definiteness of $\Sigma$, we define the characteristic equation of matrix $C$ as

$${\phi }_C\left(\lambda \right)={\lambda }^4+{\tau }_1{\lambda }^3+{\tau }_2{\lambda }^2+{\tau }_3\lambda +{\tau }_4=0,$$ (4.8)

where

${\tau }_1=a_{11}+a_{21}+a_{32}+a_{42},{\tau }_2=a_{11}{a}_{21}+a_{11}{a}_{32}+a_{11}{a}_{42}+a_{21}{a}_{32}+a_{21}{a}_{42}+a_{32}{a}_{42}-a_{23}{a}_{32}-a_{24}{a}_{42}$,

${\tau }_3=a_{11}{a}_{21}{a}_{32}+a_{11}{a}_{21}{a}_{42}+a_{11}{a}_{32}{a}_{42}+a_{13}{a}_{21}{a}_{32}+a_{14}{a}_{21}{a}_{42}+a_{21}{a}_{32}{a}_{42}-a_{11}{a}_{23}{a}_{32}-a_{11}{a}_{24}{a}_{42}-a_{23}{a}_{32}{a}_{42}-a_{24}{a}_{32}{a}_{42}$,

${\tau }_4=a_{11}{a}_{21}{a}_{32}{a}_{42}+a_{13}{a}_{21}{a}_{32}{a}_{42}+a_{14}{a}_{21}{a}_{32}{a}_{42}-a_{11}{a}_{23}{a}_{32}{a}_{42}-a_{11}{a}_{24}{a}_{32}{a}_{42}$.

According to the expressions of $S^{\ast }$, $E^{\ast }$, $I_S^{\ast }$, $I_A^{\ast }$ and $R_0^S>1$, we can prove that

$${\tau }_1>0,{\tau }_3>0,{\tau }_4>0,{\tau }_1{\tau }_2{\tau }_3-{\tau }_3^2-{\tau }_1^2{\tau }_4>0,$$

which implies that all the roots of the characteristic equation (4.8) have negative real-parts and so the matrix $C$ is a Hurwitz matrix.

Since the matrices $C$ and $C_2$ (see (4.10) below) are similar, in view of the similarity invariant of the characteristic polynomial in linear algebra, we obtain that ${\phi }_C\left(\lambda \right)$ is the similarity invariant of matrix $C$. It is easy to get that ${\tau }_1$, ${\tau }_2$, ${\tau }_3$ and ${\tau }_4$ are also similarity invariants. Thus, there exists a unique standard $R_1$ matrix of $C$.

Now we are in the position to give the specific form of $\Sigma$ and prove its positive definiteness. We divide the proof process into four steps.

Step 1. Consider the algebraic equation

$$H_1^2+C{\Sigma }_1+{\Sigma }_1{C}^T=0.$$ (4.9)

Let $C_1=J_{1}C{J}_1^{-1}$, where the elimination matrix $J_1$ is given by

$$J_1=\left(\begin{array}{cccc}\hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill -\frac{a_{42}}{a_{32}}\hfill & \hfill 1\hfill \end{array}\right).$$

Direct calculation leads to that

$$C_1=\left(\begin{array}{cccc}\hfill -a_{11}\hfill & \hfill 0\hfill & \hfill -\frac{a_{13}{a}_{32}+a_{14}{a}_{42}}{a_{32}}\hfill & \hfill -a_{14}\hfill \\ \hfill a_{21}\hfill & \hfill -a_{21}\hfill & \hfill \frac{a_{23}{a}_{32}+a_{24}{a}_{42}}{a_{32}}\hfill & \hfill a_{24}\hfill \\ \hfill 0\hfill & \hfill a_{32}\hfill & \hfill -a_{32}\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill m_1\hfill & \hfill -a_{42}\hfill \end{array}\right),$$

where

$$m_1=\frac{a_{42}(a_{32}-a_{42})}{a_{32}}.$$

Based on the value of $m_1$, we will consider the following two cases:

(1) $m_1\ne 0$; (2) $m_1=0$.

Case 1. If $m_1\ne 0$, let $C_2=J_2{C}_1{J}_2^{-1}$, where the standardized transformation matrix $J_2$ takes the form

$$J_2=\left(\begin{array}{cccc}\hfill a_{21}{a}_{32}{m}_1\hfill & \hfill -a_{32}(a_{21}+a_{32}+a_{42})m_1\hfill & \hfill (a_{23}{a}_{32}+a_{24}{a}_{42}+a_{32}^2+a_{32}{a}_{42}+a_{42}^2)m_1\hfill & \hfill a_{24}{a}_{32}{m}_1-a_{42}^3\hfill \\ \hfill 0\hfill & \hfill a_{32}{m}_1\hfill & \hfill -(a_{32}+a_{42})m_1\hfill & \hfill a_{42}^2\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill m_1\hfill & \hfill -a_{42}\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \end{array}\right).$$

By direct computation, we have

$$C_2=C_0=\left(\begin{array}{cccc}\hfill -{\tau }_1\hfill & \hfill -{\tau }_2\hfill & \hfill -{\tau }_3\hfill & \hfill -{\tau }_4\hfill \\ \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill \end{array}\right)$$ (4.10)

where ${\tau }_1$, ${\tau }_2$, ${\tau }_3$ and ${\tau }_4$ are the same as above. Moreover, Eq. (4.9) can be equivalently transformed into the following form

$$\left(J_2{J}_1\right)H_1^2{\left(J_2{J}_1\right)}^T+C_2\left[\left(J_2{J}_1\right){\Sigma }_1{\left(J_2{J}_1\right)}^T\right]+\left[\left(J_2{J}_1\right){\Sigma }_1{\left(J_2{J}_1\right)}^T\right]C_2^T=0,$$

i.e., 

$$H_0^2+C_2{\Sigma }_0+{\Sigma }_0{C}_2^T=0,$$

where ${\Sigma }_0={\varrho }_1^{-2}\left(J_2{J}_1\right){\Sigma }_1{\left(J_2{J}_1\right)}^T$, ${\varrho }_1=a_{21}{a}_{32}{m}_1{\rho }_1$. It is noticed that the matrix $C$ has all negative real-part eigenvalues, then $C_2$ is a standard $R_1$ matrix. According to Lemma 4.1, we can obtain that the matrix ${\Sigma }_0$ is positive definite whose explicit form is

$${\Sigma }_0=\left(\begin{array}{cccc}\hfill \frac{{\tau }_2{\tau }_3-{\tau }_1{\tau }_4}{2({\tau }_1{\tau }_2{\tau }_3-{\tau }_3^2-{\tau }_1^2{\tau }_4)}\hfill & \hfill 0\hfill & \hfill -\frac{{\tau }_3}{2({\tau }_1{\tau }_2{\tau }_3-{\tau }_3^2-{\tau }_1^2{\tau }_4)}\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill \frac{{\tau }_3}{2({\tau }_1{\tau }_2{\tau }_3-{\tau }_3^2-{\tau }_1^2{\tau }_4)}\hfill & \hfill 0\hfill & \hfill -\frac{{\tau }_1}{2({\tau }_1{\tau }_2{\tau }_3-{\tau }_3^2-{\tau }_1^2{\tau }_4)}\hfill \\ \hfill -\frac{{\tau }_3}{2({\tau }_1{\tau }_2{\tau }_3-{\tau }_3^2-{\tau }_1^2{\tau }_4)}\hfill & \hfill 0\hfill & \hfill \frac{{\tau }_1}{2({\tau }_1{\tau }_2{\tau }_3-{\tau }_3^2-{\tau }_1^2{\tau }_4)}\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill -\frac{{\tau }_1}{2({\tau }_1{\tau }_2{\tau }_3-{\tau }_3^2-{\tau }_1^2{\tau }_4)}\hfill & \hfill 0\hfill & \hfill \frac{{\tau }_1{\tau }_2-{\tau }_3}{2{\tau }_4({\tau }_1{\tau }_2{\tau }_3-{\tau }_3^2-{\tau }_1^2{\tau }_4)}\hfill \end{array}\right).$$

Therefore, the matrix ${\Sigma }_1={\varrho }_1^2{\left(J_2{J}_1\right)}^{-1}{\Sigma }_0{\left[{\left(J_2{J}_1\right)}^{-1}\right]}^T$ is also positive definite.

Case 2. If $m_1=0$, let $C_3=J_3{C}_1{J}_3^{-1}$, where the standardized transformation matrix $J_3$ takes the form

$$J_3=\left(\begin{array}{cccc}\hfill a_{21}{a}_{32}\hfill & \hfill -a_{32}(a_{21}+a_{32})\hfill & \hfill (a_{23}{a}_{32}+a_{24}{a}_{42}+a_{32}^2)\hfill & \hfill a_{24}{a}_{32}\hfill \\ \hfill 0\hfill & \hfill a_{32}\hfill & \hfill -a_{32}\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \end{array}\right).$$

Then we obtain

$$C_3=\left(\begin{array}{cccc}\hfill -{\overline{l}}_1\hfill & \hfill -{\overline{l}}_2\hfill & \hfill -{\overline{l}}_3\hfill & \hfill -{\overline{l}}_4\hfill \\ \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -a_{42}\hfill \end{array}\right)$$

where

$${\overline{l}}_1=a_{11}+a_{21}+a_{32},{\overline{l}}_2=a_{11}{a}_{21}+a_{11}{a}_{32}+a_{21}{a}_{32}-a_{23}{a}_{32}-a_{24}{a}_{42},$$

$${\overline{l}}_3=a_{11}{a}_{21}{a}_{32}+a_{11}{a}_{24}{a}_{42}+a_{13}{a}_{21}{a}_{32}-a_{11}{a}_{23}{a}_{32}-a_{14}{a}_{21}{a}_{42},{\overline{l}}_4=a_{32}(a_{14}{a}_{21}+a_{24}{a}_{42}-a_{11}{a}_{24}).$$

Then Eq. (4.9) can be transformed into the following equivalent form

$$\left(J_3{J}_1\right)H_1^2{\left(J_3{J}_1\right)}^T+C_3\left[\left(J_3{J}_1\right){\overline{\Theta }}_0{\left(J_3{J}_1\right)}^T\right]+\left[\left(J_3{J}_1\right){\overline{\Theta }}_0{\left(J_3{J}_1\right)}^T\right]C_3^T=0,$$

i.e., 

$$H_0^2+C_3{\overline{\Theta }}_0+{\overline{\Theta }}_0{C}_3^T=0,$$

where ${\overline{\Theta }}_0={\overline{\varrho }}_1^{-2}\left(J_3{J}_1\right){\Sigma }_1{\left(J_3{J}_1\right)}^T$, ${\overline{\varrho }}_1=a_{21}{a}_{32}{\rho }_1$. In view of Lemma 4.2, we get that the matrix ${\overline{\Theta }}_0$ is semi-positive definite whose explicit form is

$${\overline{\Theta }}_0=\left(\begin{array}{cccc}\hfill \frac{{\overline{l}}_2}{2({\overline{l}}_1{\overline{l}}_2-{\overline{l}}_3)}\hfill & \hfill 0\hfill & \hfill -\frac{1}{2({\overline{l}}_1{\overline{l}}_2-{\overline{l}}_3)}\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill \frac{1}{2({\overline{l}}_1{\overline{l}}_2-{\overline{l}}_3)}\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill -\frac{1}{2({\overline{l}}_1{\overline{l}}_2-{\overline{l}}_3)}\hfill & \hfill 0\hfill & \hfill \frac{{\overline{l}}_1}{2{\overline{l}}_3({\overline{l}}_1{\overline{l}}_2-{\overline{l}}_3)}\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \end{array}\right).$$

Thus, the matrix ${\Sigma }_1={\overline{\varrho }}_1^2{\left(J_3{J}_1\right)}^{-1}{\overline{\Theta }}_0{\left[{\left(J_3{J}_1\right)}^{-1}\right]}^T$ is also semi-positive definite and there exists a positive constant ${\nu }_1$ such that

$${\Sigma }_1⪰{\nu }_1\left(\begin{array}{cccc}\hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \end{array}\right).$$

Step 2. Consider the algebraic equation

$$H_2^2+C{\Sigma }_2+{\Sigma }_2{C}^T=0.$$ (4.11)

Let $C_4=J_{4}C{J}_4^{-1}$, where

$$J_4=\left(\begin{array}{cccc}\hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill \\ \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \end{array}\right),C_4=\left(\begin{array}{cccc}\hfill -a_{21}\hfill & \hfill a_{23}\hfill & \hfill a_{21}\hfill & \hfill a_{24}\hfill \\ \hfill a_{32}\hfill & \hfill -a_{32}\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill -a_{13}\hfill & \hfill -a_{11}\hfill & \hfill -a_{14}\hfill \\ \hfill a_{42}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -a_{42}\hfill \end{array}\right).$$

Let $C_5=J_5{C}_4{J}_5^{-1}$, where the elimination matrix $J_5$ takes the form

$$J_5=\left(\begin{array}{cccc}\hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill -\frac{a_{42}}{a_{32}}\hfill & \hfill 0\hfill & \hfill 1\hfill \end{array}\right).$$

Direct calculation leads to that

$$C_5=\left(\begin{array}{cccc}\hfill -a_{21}\hfill & \hfill \frac{a_{23}{a}_{32}+a_{24}{a}_{42}}{a_{32}}\hfill & \hfill a_{21}\hfill & \hfill a_{24}\hfill \\ \hfill a_{32}\hfill & \hfill -a_{32}\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill -\frac{a_{13}{a}_{32}+a_{14}{a}_{42}}{a_{32}}\hfill & \hfill -a_{11}\hfill & \hfill -a_{14}\hfill \\ \hfill 0\hfill & \hfill m_1\hfill & \hfill 0\hfill & \hfill -a_{42}\hfill \end{array}\right),$$

where

$$m_1=\frac{a_{42}(a_{32}-a_{42})}{a_{32}}.$$

In view of the value of $m_1$, we will consider the following two cases:

(1) $m_1\ne 0$; (2) $m_1=0$.

Case 1. If $m_1\ne 0$, let $C_6=J_6{C}_5{J}_6^{-1}$, where the elimination matrix $J_6$ is given by

$$J_6=\left(\begin{array}{cccc}\hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{a_{42}(a_{32}-a_{42})}{a_{13}{a}_{32}+a_{14}{a}_{42}}\hfill & \hfill 1\hfill \end{array}\right).$$

Then

$$C_6=\left(\begin{array}{cccc}\hfill -a_{21}\hfill & \hfill \frac{a_{23}{a}_{32}+a_{24}{a}_{42}}{a_{32}}\hfill & \hfill a_{21}+\frac{a_{24}{a}_{42}(a_{42}-a_{32})}{a_{13}{a}_{32}+a_{14}{a}_{42}}\hfill & \hfill a_{24}\hfill \\ \hfill a_{32}\hfill & \hfill -a_{32}\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill -\frac{a_{13}{a}_{32}+a_{14}{a}_{42}}{a_{32}}\hfill & \hfill -a_{11}-\frac{a_{14}{a}_{42}(a_{42}-a_{32})}{a_{13}{a}_{32}+a_{14}{a}_{42}}\hfill & \hfill -a_{14}\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill m_2\hfill & \hfill \frac{a_{14}{a}_{42}(a_{42}-a_{32})}{a_{13}{a}_{32}+a_{14}{a}_{42}}-a_{42}\hfill \end{array}\right),$$

and

$$m_2=\frac{a_{42}(a_{11}-a_{42})(a_{42}-a_{32})}{a_{13}{a}_{32}+a_{14}{a}_{42}}+\frac{a_{14}{a}_{42}^2{(a_{42}-a_{32})}^2}{{(a_{13}{a}_{32}+a_{14}{a}_{42})}^2}.$$

Considering the following two cases:

(i) $m_2\ne 0$; (ii) $m_2=0$.

Case 1.1. If $m_2\ne 0$, let $C_7=J_7{C}_6{J}_7^{-1}$, where the standardized transformation matrix $J_7$ is given by

$$J_7=\left(\begin{array}{cccc}\hfill -(a_{13}{a}_{32}+a_{14}{a}_{42})m_2\hfill & \hfill j_{7\left(12\right)}\hfill & \hfill j_{7\left(13\right)}\hfill & \hfill j_{7\left(14\right)}\hfill \\ \hfill 0\hfill & \hfill -\frac{a_{13}{a}_{32}+a_{14}{a}_{42}}{a_{32}}{m}_2\hfill & \hfill -(a_{11}+a_{42})m_2\hfill & \hfill j_{7\left(24\right)}\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill m_2\hfill & \hfill \frac{a_{14}{a}_{42}(a_{42}-a_{32})}{a_{13}{a}_{32}+a_{14}{a}_{42}}-a_{42}\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \end{array}\right),$$

and

$$j_{7\left(12\right)}=\left(a_{11}{a}_{13}+a_{13}{a}_{42}+a_{13}{a}_{32}+a_{14}{a}_{42}+\frac{a_{14}{a}_{42}(a_{11}+a_{42})}{a_{32}}\right)m_2,$$

$$j_{7\left(13\right)}=(a_{11}+a_{42})\left(a_{11}+\frac{a_{14}{a}_{42}(a_{42}-a_{32})}{a_{13}{a}_{32}+a_{14}{a}_{42}}\right)m_2+{\left(\frac{a_{14}{a}_{42}(a_{42}-a_{32})}{a_{13}{a}_{32}+a_{14}{a}_{42}}-a_{42}\right)}^2{m}_2-a_{14}{m}_2^2,$$

$$j_{7\left(14\right)}=a_{14}\left(a_{11}+2a_{42}-\frac{a_{14}{a}_{42}(a_{42}-a_{32})}{a_{13}{a}_{32}+a_{14}{a}_{42}}\right)m_2+{\left(\frac{a_{14}{a}_{42}(a_{42}-a_{32})}{a_{13}{a}_{32}+a_{14}{a}_{42}}-a_{42}\right)}^3,$$

$$j_{7\left(24\right)}={\left(\frac{a_{14}{a}_{42}(a_{42}-a_{32})}{a_{13}{a}_{32}+a_{14}{a}_{42}}-a_{42}\right)}^2-a_{14}{m}_2.$$

By direct computation, we have

$$C_7=\left(\begin{array}{cccc}\hfill -{\tau }_1\hfill & \hfill -{\tau }_2\hfill & \hfill -{\tau }_3\hfill & \hfill -{\tau }_4\hfill \\ \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill \end{array}\right)$$

Then Eq. (4.11) can be equivalently transformed into the following form

$$\left(J_7{J}_6{J}_5{J}_4\right)H_2^2{\left(J_7{J}_6{J}_5{J}_4\right)}^T+C_7\left[\left(J_7{J}_6{J}_5{J}_4\right){\Sigma }_2{\left(J_7{J}_6{J}_5{J}_4\right)}^T\right]+\left[\left(J_7{J}_6{J}_5{J}_4\right){\Sigma }_2{\left(J_7{J}_6{J}_5{J}_4\right)}^T\right]C_7^T=0,$$

that is,

$$H_0^2+C_7{\Sigma }_0+{\Sigma }_0{C}_7^T=0.$$

Note that the matrix $C$ has all negative real-part eigenvalues, then $C_7$ is a standard $R_1$ matrix. According to Lemma 4.1, this means that ${\Sigma }_0$ is positive definite, which takes the form

$${\Sigma }_0=\left(\begin{array}{cccc}\hfill \frac{{\tau }_2{\tau }_3-{\tau }_1{\tau }_4}{2({\tau }_1{\tau }_2{\tau }_3-{\tau }_3^2-{\tau }_1^2{\tau }_4)}\hfill & \hfill 0\hfill & \hfill -\frac{{\tau }_3}{2({\tau }_1{\tau }_2{\tau }_3-{\tau }_3^2-{\tau }_1^2{\tau }_4)}\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill \frac{{\tau }_3}{2({\tau }_1{\tau }_2{\tau }_3-{\tau }_3^2-{\tau }_1^2{\tau }_4)}\hfill & \hfill 0\hfill & \hfill -\frac{{\tau }_1}{2({\tau }_1{\tau }_2{\tau }_3-{\tau }_3^2-{\tau }_1^2{\tau }_4)}\hfill \\ \hfill -\frac{{\tau }_3}{2({\tau }_1{\tau }_2{\tau }_3-{\tau }_3^2-{\tau }_1^2{\tau }_4)}\hfill & \hfill 0\hfill & \hfill \frac{{\tau }_1}{2({\tau }_1{\tau }_2{\tau }_3-{\tau }_3^2-{\tau }_1^2{\tau }_4)}\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill -\frac{{\tau }_1}{2({\tau }_1{\tau }_2{\tau }_3-{\tau }_3^2-{\tau }_1^2{\tau }_4)}\hfill & \hfill 0\hfill & \hfill \frac{{\tau }_1{\tau }_2-{\tau }_3}{2{\tau }_4({\tau }_1{\tau }_2{\tau }_3-{\tau }_3^2-{\tau }_1^2{\tau }_4)}\hfill \end{array}\right),$$

where ${\Sigma }_0={\varrho }_2^{-2}\left(J_7{J}_6{J}_5{J}_4\right){\Sigma }_2{\left(J_7{J}_6{J}_5{J}_4\right)}^T$ and ${\varrho }_2=-(a_{13}{a}_{32}+a_{14}{a}_{42})m_2{\rho }_2$. Thus, the matrix ${\Sigma }_2={\varrho }_2^2{\left(J_7{J}_6{J}_5{J}_4\right)}^{-1}{\Sigma }_0{\left[{\left(J_7{J}_6{J}_5{J}_4\right)}^{-1}\right]}^T$ is also positive definite.

Case 1.2. If $m_2=0$, let $C_8=J_8{C}_6{J}_8^{-1}$, where the transformation matrix $J_8$ takes the form

$$J_8=\left(\begin{array}{cccc}\hfill -(a_{13}{a}_{32}+a_{14}{a}_{42})\hfill & \hfill a_{11}{a}_{13}+a_{13}{a}_{32}+\frac{a_{14}{a}_{42}(a_{11}+a_{42})}{a_{32}}\hfill & \hfill {\left(a_{11}+\frac{a_{14}{a}_{42}(a_{42}-a_{32})}{a_{13}{a}_{32}+a_{14}{a}_{42}}\right)}^2\hfill & \hfill a_{14}(a_{11}+a_{42})\hfill \\ \hfill 0\hfill & \hfill -\frac{a_{13}{a}_{32}+a_{14}{a}_{42}}{a_{32}}\hfill & \hfill -a_{11}-\frac{a_{14}{a}_{42}(a_{42}-a_{32})}{a_{13}{a}_{32}+a_{14}{a}_{42}}\hfill & \hfill -a_{14}\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \end{array}\right).$$

By simple computation, we obtain

$$C_8=\left(\begin{array}{cccc}\hfill -{\tilde{l}}_1\hfill & \hfill -{\tilde{l}}_2\hfill & \hfill -{\tilde{l}}_3\hfill & \hfill -{\tilde{l}}_4\hfill \\ \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{a_{14}{a}_{42}(a_{42}-a_{32})}{a_{13}{a}_{32}+a_{14}{a}_{42}}-a_{42}\hfill \end{array}\right),$$

where

$${\tilde{l}}_1=a_{21}-a_{11}-\frac{a_{13}{a}_{32}^2+a_{14}{a}_{42}^2}{a_{13}{a}_{32}+a_{14}{a}_{42}},{\tilde{l}}_2=a_{11}{a}_{21}+a_{11}{a}_{32}+\frac{a_{14}{a}_{32}{a}_{42}(a_{42}-a_{32})+a_{21}(a_{13}{a}_{32}^2+a_{14}{a}_{42}^2)}{a_{13}{a}_{32}+a_{14}{a}_{42}}-a_{23}{a}_{32}-a_{24}{a}_{42},$$

$${\tilde{l}}_3=(a_{21}{a}_{32}-a_{23}{a}_{32}-a_{24}{a}_{42})\left(a_{11}+\frac{a_{14}{a}_{42}(a_{42}-a_{32})}{a_{13}{a}_{32}+a_{14}{a}_{42}}\right)+a_{21}[a_{13}{a}_{32}+a_{14}{a}_{42}+a_{42}(a_{42}-a_{32})],$$

$${\tilde{l}}_4=a_{14}(a_{13}{a}_{32}+a_{14}{a}_{42}+a_{42}^2-a_{23}{a}_{32}-a_{24}{a}_{42})-\frac{a_{13}{a}_{14}{a}_{21}{a}_{32}(a_{42}-a_{32})+a_{14}(a_{13}{a}_{32}^3+a_{14}{a}_{42}^3)}{a_{13}{a}_{32}+a_{14}{a}_{42}}-\frac{a_{13}{a}_{14}{a}_{32}(a_{42}-a_{32})(a_{13}{a}_{32}^2+a_{14}{a}_{42}^2)}{{(a_{13}{a}_{32}+a_{14}{a}_{42})}^2}.$$

So Eq. (4.11) can be equivalently transformed into the following form

$$\left(J_8{J}_6{J}_5{J}_4\right)H_2^2{\left(J_8{J}_6{J}_5{J}_4\right)}^T+C_8\left[\left(J_8{J}_6{J}_5{J}_4\right){\Sigma }_2{\left(J_8{J}_6{J}_5{J}_4\right)}^T\right]+\left[\left(J_8{J}_6{J}_5{J}_4\right){\Sigma }_2{\left(J_8{J}_6{J}_5{J}_4\right)}^T\right]C_8^T=0,$$

i.e., 

$$H_0^2+C_8{\tilde{\Theta }}_0+{\tilde{\Theta }}_0{C}_8^T=0,$$

where ${\tilde{\Theta }}_0={\tilde{\varrho }}_2^{-2}\left(J_8{J}_6{J}_5{J}_4\right){\Sigma }_2{\left(J_8{J}_6{J}_5{J}_4\right)}^T$, ${\tilde{\varrho }}_2=-(a_{13}{a}_{32}+a_{14}{a}_{42}){\rho }_2$. By Lemma 4.2, we obtain that the matrix ${\tilde{\Theta }}_0$ is semi-positive definite whose explicit form is

$${\tilde{\Theta }}_0=\left(\begin{array}{cccc}\hfill \frac{{\tilde{l}}_2}{2({\tilde{l}}_1{\tilde{l}}_2-{\tilde{l}}_3)}\hfill & \hfill 0\hfill & \hfill -\frac{1}{2({\tilde{l}}_1{\tilde{l}}_2-{\tilde{l}}_3)}\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill \frac{1}{2({\tilde{l}}_1{\tilde{l}}_2-{\tilde{l}}_3)}\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill -\frac{1}{2({\tilde{l}}_1{\tilde{l}}_2-{\tilde{l}}_3)}\hfill & \hfill 0\hfill & \hfill \frac{{\tilde{l}}_1}{2{\tilde{l}}_3({\tilde{l}}_1{\tilde{l}}_2-{\tilde{l}}_3)}\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \end{array}\right).$$

Hence, the matrix ${\Sigma }_2={\tilde{\varrho }}_2^2{\left(J_8{J}_6{J}_5{J}_4\right)}^{-1}{\tilde{\Theta }}_0{\left[{\left(J_8{J}_6{J}_5{J}_4\right)}^{-1}\right]}^T$ is also semi-positive definite and there exists a positive constant ${\nu }_2$ such that

$${\Sigma }_2⪰{\nu }_2\left(\begin{array}{cccc}\hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \end{array}\right).$$

Case 2. If $m_1=0$, let $C_9=J_9{C}_5{J}_9^{-1}$, where the standard transformation matrix $J_9$ is given by

$$J_9=\left(\begin{array}{cccc}\hfill -(a_{13}{a}_{32}+a_{14}{a}_{42})\hfill & \hfill \left(a_{11}{a}_{13}+a_{13}{a}_{32}+a_{14}{a}_{42}+\frac{a_{11}{a}_{14}{a}_{42}}{a_{32}}\right)\hfill & \hfill a_{11}^2\hfill & \hfill a_{14}(a_{11}+a_{42})\hfill \\ \hfill 0\hfill & \hfill -\frac{a_{13}{a}_{32}+a_{14}{a}_{42}}{a_{32}}\hfill & \hfill -a_{11}\hfill & \hfill -a_{14}\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \end{array}\right).$$

By direct calculation, we obtain

$$C_9=\left(\begin{array}{cccc}\hfill -{\hat{l}}_1\hfill & \hfill -{\hat{l}}_2\hfill & \hfill -{\hat{l}}_3\hfill & \hfill -{\hat{l}}_4\hfill \\ \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -a_{42}\hfill \end{array}\right)$$

where

$${\hat{l}}_1=a_{11}+a_{21}+a_{32},{\hat{l}}_2=a_{11}{a}_{21}+a_{11}{a}_{32}+a_{21}{a}_{32}-a_{23}{a}_{32}-a_{24}{a}_{42},$$

$${\hat{l}}_3=a_{11}{a}_{21}{a}_{32}+a_{13}{a}_{21}{a}_{32}+a_{14}{a}_{21}{a}_{42}-a_{11}{a}_{23}{a}_{32}-a_{11}{a}_{24}{a}_{42},$$

$${\hat{l}}_4=a_{13}{a}_{24}{a}_{32}+a_{14}{a}_{21}{a}_{32}+a_{14}{a}_{42}^2-a_{14}{a}_{21}{a}_{42}-a_{14}{a}_{23}{a}_{32}-a_{14}{a}_{32}{a}_{42}.$$

In addition, Eq. (4.11) can be transformed into the following form

$$\left(J_9{J}_5{J}_4\right)H_2^2{\left(J_9{J}_5{J}_4\right)}^T+C_9\left[\left(J_9{J}_5{J}_4\right){\Sigma }_2{\left(J_9{J}_5{J}_4\right)}^T\right]+\left[\left(J_9{J}_5{J}_4\right){\Sigma }_2{\left(J_9{J}_5{J}_4\right)}^T\right]C_9^T=0,$$

that is,

$$H_0^2+C_9{\hat{\Theta }}_0+{\hat{\Theta }}_0{C}_9^T=0,$$

where ${\hat{\Theta }}_0={\tilde{\varrho }}_2^{-2}\left(J_9{J}_5{J}_4\right){\Sigma }_2{\left(J_9{J}_5{J}_4\right)}^T$, ${\tilde{\varrho }}_2=-(a_{13}{a}_{32}+a_{14}{a}_{42}){\rho }_2$. According to Lemma 4.2, it can be concluded that the matrix ${\hat{\Theta }}_0$ is semi-positive definite whose explicit form is

$${\hat{\Theta }}_0=\left(\begin{array}{cccc}\hfill \frac{{\hat{l}}_2}{2({\hat{l}}_1{\hat{l}}_2-{\hat{l}}_3)}\hfill & \hfill 0\hfill & \hfill -\frac{1}{2({\hat{l}}_1{\hat{l}}_2-{\hat{l}}_3)}\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill \frac{1}{2({\hat{l}}_1{\hat{l}}_2-{\hat{l}}_3)}\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill -\frac{1}{2({\hat{l}}_1{\hat{l}}_2-{\hat{l}}_3)}\hfill & \hfill 0\hfill & \hfill \frac{{\hat{l}}_1}{2{\hat{l}}_3({\hat{l}}_1{\hat{l}}_2-{\hat{l}}_3)}\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \end{array}\right).$$

Therefore, the matrix ${\Sigma }_2={\tilde{\varrho }}_2^2{\left(J_9{J}_5{J}_4\right)}^{-1}{\hat{\Theta }}_0{\left[{\left(J_9{J}_5{J}_4\right)}^{-1}\right]}^T$ is also semi-positive definite and there exists a positive constant ${\kappa }_1$ such that

$${\Sigma }_2⪰{\kappa }_1\left(\begin{array}{cccc}\hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \end{array}\right).$$

Step 3. Consider the algebraic equation

$$H_3^2+C{\Sigma }_3+{\Sigma }_3{C}^T=0.$$ (4.12)

Let $C_{10}=J_{10}C{J}_{10}^{-1}$, where the ordering matrix $J_{10}$ takes the form

$$J_{10}=\left(\begin{array}{cccc}\hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \\ \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \end{array}\right).$$

Then we have

$$C_{10}=\left(\begin{array}{cccc}\hfill -a_{32}\hfill & \hfill a_{32}\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill a_{23}\hfill & \hfill -a_{21}\hfill & \hfill a_{24}\hfill & \hfill a_{21}\hfill \\ \hfill 0\hfill & \hfill a_{42}\hfill & \hfill -a_{42}\hfill & \hfill 0\hfill \\ \hfill -a_{13}\hfill & \hfill 0\hfill & \hfill -a_{14}\hfill & \hfill -a_{11}\hfill \end{array}\right).$$

Let $C_{11}=J_{11}{C}_{10}{J}_{11}^{-1}$, where the elimination matrix $J_{11}$ is given by

$$J_{11}=\left(\begin{array}{cccc}\hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill \frac{a_{13}}{a_{23}}\hfill & \hfill 0\hfill & \hfill 1\hfill \end{array}\right).$$

Direct calculation leads to that

$$C_{11}=\left(\begin{array}{cccc}\hfill -a_{32}\hfill & \hfill a_{32}\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill a_{23}\hfill & \hfill -\frac{a_{21}(a_{13}+a_{23})}{a_{23}}\hfill & \hfill a_{24}\hfill & \hfill a_{21}\hfill \\ \hfill 0\hfill & \hfill a_{42}\hfill & \hfill -a_{42}\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill m_3\hfill & \hfill 0\hfill & \hfill \frac{a_{13}{a}_{21}-a_{11}{a}_{23}}{a_{23}}\hfill \end{array}\right),$$

where

$$m_3=\frac{a_{13}(a_{11}-a_{21})}{a_{23}}-\frac{a_{13}^2{a}_{21}}{a_{23}^2}.$$

Next, we will consider the following two conditions:

(1) $m_3\ne 0$; (2) $m_3=0$.

Case 1. If $m_3\ne 0$, let $C_{12}=J_{12}{C}_{11}{J}_{12}^{-1}$, where the elimination matrix $J_{12}$ is given by

$$J_{12}=\left(\begin{array}{cccc}\hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill -\frac{m_3}{a_{42}}\hfill & \hfill 1\hfill \end{array}\right),$$

then

$$C_{12}=\left(\begin{array}{cccc}\hfill -a_{32}\hfill & \hfill a_{32}\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill a_{23}\hfill & \hfill -\frac{a_{21}(a_{13}+a_{23})}{a_{23}}\hfill & \hfill \frac{a_{24}{a}_{42}+a_{21}{m}_3}{a_{42}}\hfill & \hfill a_{21}\hfill \\ \hfill 0\hfill & \hfill a_{42}\hfill & \hfill -a_{42}\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill m_4\hfill & \hfill \frac{a_{13}{a}_{21}-a_{11}{a}_{23}}{a_{23}}\hfill \end{array}\right),$$

where

$$m_4=\frac{a_{13}{a}_{21}+a_{23}{a}_{42}-a_{11}{a}_{23}}{a_{23}{a}_{42}}{m}_3.$$

Considering the following two conditions:

(i) $m_4\ne 0$; (ii) $m_4=0$.

Case 1.1. If $m_4\ne 0$, let $C_{13}=J_{13}{C}_{12}{J}_{13}^{-1}$, where the standardized transformation matrix $J_{13}$ takes the form

$$J_{13}=\left(\begin{array}{cccc}\hfill a_{23}{a}_{42}{m}_4\hfill & \hfill -a_{42}(a_{11}+a_{21}+a_{42})m_4\hfill & \hfill j_{13\left(13\right)}\hfill & \hfill j_{13\left(14\right)}\hfill \\ \hfill 0\hfill & \hfill a_{42}{m}_4\hfill & \hfill \left(\frac{a_{13}{a}_{21}}{a_{23}}-a_{11}-a_{42}\right)m_4\hfill & \hfill \frac{{(a_{13}{a}_{21}-a_{11}{a}_{23})}^2}{a_{23}^2}\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill m_4\hfill & \hfill \frac{a_{13}{a}_{21}-a_{11}{a}_{23}}{a_{23}}\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \end{array}\right),$$

and

$$j_{13\left(13\right)}=\left(a_{11}^2+a_{11}{a}_{42}+a_{24}{a}_{42}+a_{42}^2-\frac{a_{13}(a_{11}{a}_{21}+a_{21}^2+a_{21}{a}_{42})}{a_{23}}\right)m_4,j_{13\left(14\right)}=\frac{{(a_{13}{a}_{21}-a_{11}{a}_{23})}^3}{a_{23}^3}+a_{21}{a}_{42}{m}_4.$$

In view of the uniqueness of the standard $R_1$ matrix, we can conclude that

$$C_{13}=\left(\begin{array}{cccc}\hfill -{\tau }_1\hfill & \hfill -{\tau }_2\hfill & \hfill -{\tau }_3\hfill & \hfill -{\tau }_4\hfill \\ \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill \end{array}\right).$$

By letting ${\Sigma }_0={\varrho }_3^{-2}\left(J_{13}{J}_{12}{J}_{11}{J}_{10}\right){\Sigma }_3{\left(J_{13}{J}_{12}{J}_{11}{J}_{10}\right)}^T$, where ${\varrho }_3=a_{23}{a}_{42}{m}_4{\rho }_3$, then Eq. (4.12) is equivalent to the following form

$$H_0^2+C_{13}{\Sigma }_0+{\Sigma }_0{C}_{13}^T=0.$$

By Lemma 4.1, we can obtain that ${\Sigma }_0$ is positive definite and so the matrix ${\Sigma }_3={\varrho }_3^2{\left(J_{13}{J}_{12}{J}_{11}{J}_{10}\right)}^{-1}{\Sigma }_0{\left[{\left(J_{13}{J}_{12}{J}_{11}{J}_{10}\right)}^{-1}\right]}^T$ is also positive definite.

Case 1.2. If $m_4=0$, let $C_{14}=J_{14}{C}_{12}{J}_{14}^{-1}$, where the standardized transformation matrix $J_{14}$ is given by

$$J_{14}=\left(\begin{array}{cccc}\hfill a_{23}{a}_{42}\hfill & \hfill -a_{42}\left(a_{21}+a_{42}+\frac{a_{13}{a}_{21}}{a_{23}}\right)\hfill & \hfill a_{24}{a}_{42}+a_{21}{m}_3+a_{42}^2\hfill & \hfill a_{21}{a}_{42}\hfill \\ \hfill 0\hfill & \hfill a_{42}\hfill & \hfill -a_{42}\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \end{array}\right),$$

then

$$C_{14}=\left(\begin{array}{cccc}\hfill -{\check{l}}_1\hfill & \hfill -{\check{l}}_2\hfill & \hfill -{\check{l}}_3\hfill & \hfill -{\check{l}}_4\hfill \\ \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{a_{13}{a}_{21}-a_{11}{a}_{23}}{a_{23}}\hfill \end{array}\right)$$

where

$${\check{l}}_1=a_{21}+a_{32}+a_{42}+\frac{a_{13}{a}_{21}}{a_{23}},{\check{l}}_2=a_{21}{a}_{32}+a_{21}{a}_{42}+a_{32}{a}_{42}+\frac{a_{13}{a}_{21}}{a_{23}}(a_{32}+a_{42})-a_{23}{a}_{32}-a_{24}{a}_{42}-a_{21}{m}_3,$$

$${\check{l}}_3=a_{32}{a}_{42}\left(\frac{a_{13}{a}_{21}}{a_{23}}+a_{21}-a_{23}-a_{24}-\frac{a_{21}{m}_3}{a_{42}}\right),{\check{l}}_4=a_{21}{a}_{42}\left(a_{11}-a_{32}-\frac{a_{13}{a}_{21}}{a_{23}}\right).$$

Then we can transform Eq. (4.12) into the following form

$$\left(J_{14}{J}_{12}{J}_{11}{J}_{10}\right)H_3^2{\left(J_{14}{J}_{12}{J}_{11}{J}_{10}\right)}^T+C_{14}\left[\left(J_{14}{J}_{12}{J}_{11}{J}_{10}\right){\Sigma }_3{\left(J_{14}{J}_{12}{J}_{11}{J}_{10}\right)}^T\right]+\left[\left(J_{14}{J}_{12}{J}_{11}{J}_{10}\right){\Sigma }_3{\left(J_{14}{J}_{12}{J}_{11}{J}_{10}\right)}^T\right]C_{14}^T=0,$$

i.e., 

$$H_0^2+C_{14}{\check{\Theta }}_0+{\check{\Theta }}_0{C}_{14}^T=0,$$

where ${\check{\Theta }}_0={\check{\varrho }}_3^{-2}\left(J_{14}{J}_{12}{J}_{11}{J}_{10}\right){\Sigma }_3{\left(J_{14}{J}_{12}{J}_{11}{J}_{10}\right)}^T$, ${\check{\varrho }}_3=a_{23}{a}_{42}{\rho }_3$. By Lemma 4.2, we can conclude that the matrix ${\check{\Theta }}_0$ is semi-positive definite whose explicit form is as follows

$${\check{\Theta }}_0=\left(\begin{array}{cccc}\hfill \frac{{\check{l}}_2}{2({\check{l}}_1{\check{l}}_2-{\check{l}}_3)}\hfill & \hfill 0\hfill & \hfill -\frac{1}{2({\check{l}}_1{\check{l}}_2-{\check{l}}_3)}\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill \frac{1}{2({\check{l}}_1{\check{l}}_2-{\check{l}}_3)}\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill -\frac{1}{2({\check{l}}_1{\check{l}}_2-{\check{l}}_3)}\hfill & \hfill 0\hfill & \hfill \frac{{\check{l}}_1}{2{\check{l}}_3({\check{l}}_1{\check{l}}_2-{\check{l}}_3)}\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \end{array}\right).$$

Hence, the matrix ${\Sigma }_3={\check{\varrho }}_3^2{\left(J_{14}{J}_{12}{J}_{11}{J}_{10}\right)}^{-1}{\check{\Theta }}_0{\left[{\left(J_{14}{J}_{12}{J}_{11}{J}_{10}\right)}^{-1}\right]}^T$ is also semi-positive definite and there exists a positive constant ${\nu }_3$ such that

$${\Sigma }_3⪰{\nu }_3\left(\begin{array}{cccc}\hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \end{array}\right).$$

Case 2. If $m_3=0$, then

$$C_{11}=\left(\begin{array}{cccc}\hfill -a_{32}\hfill & \hfill a_{32}\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill a_{23}\hfill & \hfill -\frac{a_{21}(a_{13}+a_{23})}{a_{23}}\hfill & \hfill a_{24}\hfill & \hfill a_{21}\hfill \\ \hfill 0\hfill & \hfill a_{42}\hfill & \hfill -a_{42}\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{a_{13}{a}_{21}-a_{11}{a}_{23}}{a_{23}}\hfill \end{array}\right).$$

Let $C_{15}=J_{15}{C}_{11}{J}_{15}^{-1}$, where the standardized transformation matrix $J_{15}$ takes the form

$$J_{15}=\left(\begin{array}{cccc}\hfill a_{23}{a}_{42}\hfill & \hfill -a_{42}\left(a_{21}+a_{42}+\frac{a_{13}{a}_{21}}{a_{23}}\right)\hfill & \hfill a_{42}(a_{24}+a_{42})\hfill & \hfill a_{21}{a}_{42}\hfill \\ \hfill 0\hfill & \hfill a_{42}\hfill & \hfill -a_{42}\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \end{array}\right),$$

then

$$C_{15}=\left(\begin{array}{cccc}\hfill -p_1\hfill & \hfill -p_2\hfill & \hfill -p_3\hfill & \hfill -p_4\hfill \\ \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{a_{13}{a}_{21}-a_{11}{a}_{23}}{a_{23}}\hfill \end{array}\right)$$

where

$$p_1=a_{21}+a_{32}+a_{42}+\frac{a_{13}{a}_{21}}{a_{23}},p_2=\frac{a_{13}{a}_{21}(a_{32}+a_{42})}{a_{23}}+a_{21}{a}_{32}+a_{21}{a}_{42}+a_{32}{a}_{42}-a_{23}{a}_{32}-a_{24}{a}_{42},$$

$$p_3=a_{32}{a}_{42}\left(\frac{a_{13}{a}_{21}}{a_{23}}+a_{21}-a_{23}-a_{24}\right),p_4=a_{21}{a}_{42}\left(a_{11}-a_{32}-\frac{a_{13}{a}_{21}}{a_{23}}\right).$$

Then Eq. (4.12) can be transformed into the following equivalent form

$$\left(J_{15}{J}_{11}{J}_{10}\right)H_3^2{\left(J_{15}{J}_{11}{J}_{10}\right)}^T+C_{15}\left[\left(J_{15}{J}_{11}{J}_{10}\right){\Sigma }_3{\left(J_{15}{J}_{11}{J}_{10}\right)}^T\right]+\left[\left(J_{15}{J}_{11}{J}_{10}\right){\Sigma }_3{\left(J_{15}{J}_{11}{J}_{10}\right)}^T\right]C_{15}^T=0,$$

that is,

$$H_0^2+C_{15}{\Theta }_1+{\Theta }_1{C}_{15}^T=0,$$

where ${\Theta }_1={\varpi }_1^{-2}\left(J_{15}{J}_{11}{J}_{10}\right){\Sigma }_3{\left(J_{15}{J}_{11}{J}_{10}\right)}^T$, ${\varpi }_1=a_{23}{a}_{42}{\rho }_3$. From Lemma 4.2 it follows that the matrix ${\Theta }_1$ is semi-positive definite whose explicit form is as follows

$${\Theta }_1=\left(\begin{array}{cccc}\hfill \frac{p_2}{2(p_1{p}_2-p_3)}\hfill & \hfill 0\hfill & \hfill -\frac{1}{2(p_1{p}_2-p_3)}\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill \frac{1}{2(p_1{p}_2-p_3)}\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill -\frac{1}{2(p_1{p}_2-p_3)}\hfill & \hfill 0\hfill & \hfill \frac{p_1}{2p_3(p_1{p}_2-p_3)}\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \end{array}\right).$$

Consequently, the matrix ${\Sigma }_3={\varpi }_1^2{\left(J_{15}{J}_{11}{J}_{10}\right)}^{-1}{\Theta }_1{\left[{\left(J_{15}{J}_{11}{J}_{10}\right)}^{-1}\right]}^T$ is also semi-positive definite and there exists a positive constant ${\kappa }_2$ such that

$${\Sigma }_3⪰{\kappa }_2\left(\begin{array}{cccc}\hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \end{array}\right).$$

Step 4. Consider the algebraic equation

$$H_4^2+C{\Sigma }_4+{\Sigma }_4{C}^T=0.$$ (4.13)

Let $C_{16}=J_{16}C{J}_{16}^{-1}$, where

$$J_{16}=\left(\begin{array}{cccc}\hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \\ \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill \\ \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \end{array}\right),C_{16}=\left(\begin{array}{cccc}\hfill -a_{42}\hfill & \hfill a_{42}\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill a_{24}\hfill & \hfill -a_{21}\hfill & \hfill a_{23}\hfill & \hfill a_{21}\hfill \\ \hfill 0\hfill & \hfill a_{32}\hfill & \hfill -a_{32}\hfill & \hfill 0\hfill \\ \hfill -a_{14}\hfill & \hfill 0\hfill & \hfill -a_{13}\hfill & \hfill -a_{11}\hfill \end{array}\right).$$

Let $C_{17}=J_{17}{C}_{16}{J}_{17}^{-1}$, where the elimination matrix $J_{17}$ is given by

$$J_{17}=\left(\begin{array}{cccc}\hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill \frac{a_{14}}{a_{24}}\hfill & \hfill 0\hfill & \hfill 1\hfill \end{array}\right).$$

Direct calculation leads to that

$$C_{17}=\left(\begin{array}{cccc}\hfill -a_{42}\hfill & \hfill a_{42}\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill a_{24}\hfill & \hfill -\frac{a_{21}(a_{14}+a_{24})}{a_{24}}\hfill & \hfill a_{23}\hfill & \hfill a_{21}\hfill \\ \hfill 0\hfill & \hfill a_{32}\hfill & \hfill -a_{32}\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill m_5\hfill & \hfill 0\hfill & \hfill \frac{a_{14}{a}_{21}-a_{11}{a}_{24}}{a_{24}}\hfill \end{array}\right),$$

where

$$m_5=\frac{a_{14}(a_{11}-a_{21})}{a_{24}}-\frac{a_{14}^2{a}_{21}}{a_{24}^2}.$$

Next, we will consider the following two conditions:

(1) $m_5\ne 0$; (2) $m_5=0$.

Case 1. If $m_5\ne 0$, let $C_{18}=J_{18}{C}_{17}{J}_{18}^{-1}$, where the elimination matrix $J_{18}$ takes the form

$$J_{18}=\left(\begin{array}{cccc}\hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill -\frac{m_5}{a_{32}}\hfill & \hfill 1\hfill \end{array}\right).$$

Then we have

$$C_{18}=\left(\begin{array}{cccc}\hfill -a_{42}\hfill & \hfill a_{42}\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill a_{24}\hfill & \hfill -\frac{a_{21}(a_{14}+a_{24})}{a_{24}}\hfill & \hfill \frac{a_{23}{a}_{32}+a_{21}{m}_5}{a_{32}}\hfill & \hfill a_{21}\hfill \\ \hfill 0\hfill & \hfill a_{32}\hfill & \hfill -a_{32}\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill m_6\hfill & \hfill \frac{a_{14}{a}_{21}-a_{11}{a}_{24}}{a_{24}}\hfill \end{array}\right),$$

where

$$m_6=\frac{a_{14}{a}_{21}+a_{24}{a}_{32}-a_{11}{a}_{24}}{a_{24}{a}_{32}}{m}_5.$$

Considering the following two conditions:

(i) $m_6\ne 0$; (ii) $m_6=0$.

Case 1.1. If $m_6\ne 0$, let $C_{19}=J_{19}{C}_{18}{J}_{19}^{-1}$, where the standardized transformation matrix $J_{19}$ is given by

$$J_{19}=\left(\begin{array}{cccc}\hfill a_{24}{a}_{32}{m}_6\hfill & \hfill -a_{32}(a_{11}+a_{21}+a_{32})m_6\hfill & \hfill j_{17\left(13\right)}\hfill & \hfill j_{17\left(14\right)}\hfill \\ \hfill 0\hfill & \hfill a_{32}{m}_6\hfill & \hfill \left(\frac{a_{14}{a}_{21}}{a_{24}}-a_{11}-a_{32}\right)m_6\hfill & \hfill \frac{{(a_{14}{a}_{21}-a_{11}{a}_{24})}^2}{a_{24}^2}\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill m_6\hfill & \hfill \frac{a_{14}{a}_{21}-a_{11}{a}_{24}}{a_{24}}\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \end{array}\right),$$

and

$$j_{19\left(13\right)}=\left(a_{11}^2+a_{11}{a}_{32}+a_{23}{a}_{32}+a_{32}^2-\frac{a_{14}{a}_{21}(a_{11}+a_{21}+a_{32})}{a_{24}}\right)m_6,j_{19\left(14\right)}=\frac{{(a_{14}{a}_{21}-a_{11}{a}_{24})}^3}{a_{24}^3}+a_{21}{a}_{32}{m}_6.$$

By direct calculation and the uniqueness of the standard $R_1$ matrix, we obtain that

$$C_{19}=\left(\begin{array}{cccc}\hfill -{\tau }_1\hfill & \hfill -{\tau }_2\hfill & \hfill -{\tau }_3\hfill & \hfill -{\tau }_4\hfill \\ \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill \end{array}\right)$$

Then Eq. (4.13) can be equivalently transformed into the following form

$$\left(J_{19}{J}_{18}{J}_{17}{J}_{16}\right)H_4^2{\left(J_{19}{J}_{18}{J}_{17}{J}_{16}\right)}^T+C_{19}\left[\left(J_{19}{J}_{18}{J}_{17}{J}_{16}\right){\Sigma }_4{\left(J_{19}{J}_{18}{J}_{17}{J}_{16}\right)}^T\right]+\left[\left(J_{19}{J}_{18}{J}_{17}{J}_{16}\right){\Sigma }_4{\left(J_{19}{J}_{18}{J}_{17}{J}_{16}\right)}^T\right]C_{19}^T=0,$$

that is,

$$H_0^2+C_{19}{\Sigma }_0+{\Sigma }_0{C}_{19}^T=0,$$

where ${\Sigma }_0={\varrho }_4^{-2}\left(J_{19}{J}_{18}{J}_{17}{J}_{16}\right){\Sigma }_4{\left(J_{19}{J}_{18}{J}_{17}{J}_{16}\right)}^T$ and ${\varrho }_4=a_{24}{a}_{32}{m}_6{\rho }_4$. According to Lemma 4.1, we derive that the matrix ${\Sigma }_0$ is positive definite and hence the matrix ${\Sigma }_4={\varrho }_4^2{\left(J_{19}{J}_{18}{J}_{17}{J}_{16}\right)}^{-1}{\Sigma }_0{\left[{\left(J_{19}{J}_{18}{J}_{17}{J}_{16}\right)}^{-1}\right]}^T$ is also positive definite.

Case 1.2. If $m_6=0$, let $C_{20}=J_{20}{C}_{18}{J}_{20}^{-1}$, where the transformation matrix $J_{20}$ takes the form

$$J_{20}=\left(\begin{array}{cccc}\hfill a_{24}{a}_{32}\hfill & \hfill -a_{32}\left(a_{21}+a_{32}+\frac{a_{14}{a}_{21}}{a_{24}}\right)\hfill & \hfill a_{23}{a}_{32}+a_{32}^2+a_{21}{m}_5\hfill & \hfill a_{21}{a}_{32}\hfill \\ \hfill 0\hfill & \hfill a_{32}\hfill & \hfill -a_{32}\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \end{array}\right)$$

Direct calculation leads to that

$$C_{20}=\left(\begin{array}{cccc}\hfill -{\overline{p}}_1\hfill & \hfill -{\overline{p}}_2\hfill & \hfill -{\overline{p}}_3\hfill & \hfill -{\overline{p}}_4\hfill \\ \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{a_{14}{a}_{21}-a_{11}{a}_{24}}{a_{24}}\hfill \end{array}\right),$$

where

$${\overline{p}}_1=a_{21}+a_{32}+a_{42}+\frac{a_{14}{a}_{21}}{a_{24}},{\overline{p}}_2=(a_{32}+a_{42})\left(a_{21}+a_{32}+\frac{a_{14}{a}_{21}}{a_{24}}\right)-a_{23}{a}_{32}-a_{32}^2-a_{24}{a}_{42}-a_{21}{m}_5,$$

$${\overline{p}}_3=a_{32}{a}_{42}\left(\frac{a_{14}{a}_{21}}{a_{24}}+a_{21}-a_{23}-a_{24}\right)-a_{21}{a}_{42}{m}_5,{\overline{p}}_4=a_{21}{a}_{32}\left(a_{11}-a_{42}-\frac{a_{14}{a}_{21}}{a_{24}}\right).$$

Then we can transform Eq. (4.13) into the following equivalent form

$$\left(J_{20}{J}_{18}{J}_{17}{J}_{16}\right)H_4^2{\left(J_{20}{J}_{18}{J}_{17}{J}_{16}\right)}^T+C_{20}\left[\left(J_{20}{J}_{18}{J}_{17}{J}_{16}\right){\Sigma }_4{\left(J_{20}{J}_{18}{J}_{17}{J}_{16}\right)}^T\right]+\left[\left(J_{20}{J}_{18}{J}_{17}{J}_{16}\right){\Sigma }_4{\left(J_{20}{J}_{18}{J}_{17}{J}_{16}\right)}^T\right]C_{20}^T=0,$$

that is,

$$H_0^2+C_{20}{\overline{\Theta }}_1+{\overline{\Theta }}_1{C}_{20}^T=0,$$

where ${\overline{\Theta }}_1={\varpi }_2^{-2}\left(J_{20}{J}_{18}{J}_{17}{J}_{16}\right){\Sigma }_4{\left(J_{20}{J}_{18}{J}_{17}{J}_{16}\right)}^T$ and ${\varpi }_2=a_{24}{a}_{32}{\rho }_4$. In view of Lemma 4.2, we get that the matrix ${\overline{\Theta }}_1$ is semi-positive definite whose explicit form is as follows

$${\overline{\Theta }}_1=\left(\begin{array}{cccc}\hfill \frac{{\overline{p}}_2}{2({\overline{p}}_1{\overline{p}}_2-{\overline{p}}_3)}\hfill & \hfill 0\hfill & \hfill -\frac{1}{2({\overline{p}}_1{\overline{p}}_2-{\overline{p}}_3)}\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill \frac{1}{2({\overline{p}}_1{\overline{p}}_2-{\overline{p}}_3)}\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill -\frac{1}{2({\overline{p}}_1{\overline{p}}_2-{\overline{p}}_3)}\hfill & \hfill 0\hfill & \hfill \frac{{\overline{p}}_1}{2{\overline{p}}_3({\overline{p}}_1{\overline{p}}_2-{\overline{p}}_3)}\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \end{array}\right).$$

Accordingly, the matrix ${\Sigma }_4={\varpi }_2^2{\left(J_{20}{J}_{18}{J}_{17}{J}_{16}\right)}^{-1}{\overline{\Theta }}_1{\left[{\left(J_{20}{J}_{18}{J}_{17}{J}_{16}\right)}^{-1}\right]}^T$ is also semi-positive definite and there exists a positive constant ${\nu }_4$ such that

$${\Sigma }_4⪰{\nu }_4\left(\begin{array}{cccc}\hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \end{array}\right).$$

Case 2. If $m_5=0$, then

$$C_{17}=\left(\begin{array}{cccc}\hfill -a_{42}\hfill & \hfill a_{42}\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill a_{24}\hfill & \hfill -\frac{a_{21}(a_{14}+a_{24})}{a_{24}}\hfill & \hfill a_{23}\hfill & \hfill a_{21}\hfill \\ \hfill 0\hfill & \hfill a_{32}\hfill & \hfill -a_{32}\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{a_{14}{a}_{21}-a_{11}{a}_{24}}{a_{24}}\hfill \end{array}\right).$$

Let $C_{21}=J_{21}{C}_{17}{J}_{21}^{-1}$, where the standardized transformation matrix $J_{21}$ is given by

$$J_{21}=\left(\begin{array}{cccc}\hfill a_{24}{a}_{32}\hfill & \hfill -a_{32}\left(a_{21}+a_{32}+\frac{a_{14}{a}_{21}}{a_{24}}\right)\hfill & \hfill a_{32}(a_{23}+a_{32})\hfill & \hfill a_{21}{a}_{32}\hfill \\ \hfill 0\hfill & \hfill a_{32}\hfill & \hfill -a_{32}\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \end{array}\right),$$

then

$$C_{21}=\left(\begin{array}{cccc}\hfill -{\tilde{p}}_1\hfill & \hfill -{\tilde{p}}_2\hfill & \hfill -{\tilde{p}}_3\hfill & \hfill -{\tilde{p}}_4\hfill \\ \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{a_{14}{a}_{21}-a_{11}{a}_{24}}{a_{24}}\hfill \end{array}\right)$$

where

$${\tilde{p}}_1=a_{21}+a_{32}+a_{42}+\frac{a_{14}{a}_{21}}{a_{24}},{\tilde{p}}_2=a_{21}{a}_{32}+a_{21}{a}_{42}+a_{32}{a}_{42}+\frac{a_{14}{a}_{21}(a_{32}+a_{42})}{a_{24}}-a_{23}{a}_{32}-a_{24}{a}_{42},$$

$${\tilde{p}}_3=a_{32}{a}_{42}\left(\frac{a_{14}{a}_{21}}{a_{24}}+a_{21}-a_{23}-a_{24}\right),{\tilde{p}}_4=a_{21}{a}_{32}\left(a_{11}-a_{42}-\frac{a_{14}{a}_{21}}{a_{24}}\right).$$

Then Eq. (4.13) can be transformed into the following equivalent form

$$\left(J_{21}{J}_{17}{J}_{16}\right)H_4^2{\left(J_{21}{J}_{17}{J}_{16}\right)}^T+C_{21}\left[\left(J_{21}{J}_{17}{J}_{16}\right){\Sigma }_4{\left(J_{21}{J}_{17}{J}_{16}\right)}^T\right]+\left[\left(J_{21}{J}_{17}{J}_{16}\right){\Sigma }_4{\left(J_{21}{J}_{17}{J}_{16}\right)}^T\right]C_{21}^T=0,$$

that is,

$$H_0^2+C_{21}{\tilde{\Theta }}_1+{\tilde{\Theta }}_1{C}_{21}^T=0,$$

where ${\tilde{\Theta }}_1={\varpi }_3^{-2}\left(J_{21}{J}_{17}{J}_{16}\right){\Sigma }_4{\left(J_{21}{J}_{17}{J}_{16}\right)}^T$, ${\varpi }_3=a_{24}{a}_{32}{\rho }_4$. In view of Lemma 4.2, we obtain that the matrix ${\tilde{\Theta }}_1$ is semi-positive definite whose explicit form is as follows

$${\tilde{\Theta }}_1=\left(\begin{array}{cccc}\hfill \frac{{\tilde{p}}_2}{2({\tilde{p}}_1{\tilde{p}}_2-{\tilde{p}}_3)}\hfill & \hfill 0\hfill & \hfill -\frac{1}{2({\tilde{p}}_1{\tilde{p}}_2-{\tilde{p}}_3)}\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill \frac{1}{2({\tilde{p}}_1{\tilde{p}}_2-{\tilde{p}}_3)}\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill -\frac{1}{2({\tilde{p}}_1{\tilde{p}}_2-{\tilde{p}}_3)}\hfill & \hfill 0\hfill & \hfill \frac{{\tilde{p}}_1}{2{\tilde{p}}_3({\tilde{p}}_1{\tilde{p}}_2-{\tilde{p}}_3)}\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \end{array}\right).$$

Thus, the matrix ${\Sigma }_4={\varpi }_3^2{\left(J_{21}{J}_{17}{J}_{16}\right)}^{-1}{\tilde{\Theta }}_1{\left[{\left(J_{21}{J}_{17}{J}_{16}\right)}^{-1}\right]}^T$ is also semi-positive definite and there exists a positive constant ${\kappa }_3$ such that

$${\Sigma }_4⪰{\kappa }_3\left(\begin{array}{cccc}\hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \end{array}\right).$$

Now we are in the position to prove the matrix $\Sigma ={\Sigma }_1+{\Sigma }_2+{\Sigma }_3+{\Sigma }_4$ in Eq. (4.7) is positive definite. If $m_1=0$, $m_3=0$ and $m_5=0$, then the covariance matrix

$$\Sigma ={\Sigma }_1+{\Sigma }_2+{\Sigma }_3+{\Sigma }_4⪰{\nu }_1\left(\begin{array}{cccc}\hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \end{array}\right)+{\kappa }_1\left(\begin{array}{cccc}\hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \end{array}\right)+{\kappa }_2\left(\begin{array}{cccc}\hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \end{array}\right)+{\kappa }_3\left(\begin{array}{cccc}\hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \end{array}\right)⪰({\nu }_1\wedge {\kappa }_1\wedge {\kappa }_2\wedge {\kappa }_3)\left(\begin{array}{cccc}\hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \end{array}\right).$$

Since the parameters ${\nu }_1$ and ${\kappa }_i$ $(i=1,2,3)$ are positive, so the matrix $\Sigma$ is positive definite. Similarly, we can prove that in other cases, the covariance matrix $\Sigma$ is also positive definite. Hence, according to the relationship between systems (4.1), (4.3), we get that the stationary distribution $\pi \left(\cdot \right)$ around $P^{\ast }$ follows a unique log-normal probability density $\Phi (S,E,I_S,I_A)$, which takes the form

$$\Phi (S,E,I_S,I_A)={\left(2\pi \right)}^{-2}{\left|\Sigma \right|}^{-\frac{1}{2}}{e}^{-\frac{1}{2}(ln\frac{S}{S^{\ast }},ln\frac{E}{E^{\ast }},ln\frac{I_S}{I_S^{\ast }},ln\frac{I_A}{I_A^{\ast }}){\Sigma }^{-1}{(ln\frac{S}{S^{\ast }},ln\frac{E}{E^{\ast }},ln\frac{I_S}{I_S^{\ast }},ln\frac{I_A}{I_A^{\ast }})}^T},$$

where the specific form of $\Sigma$ can be determined by the above discussion. This completes the proof.

5. Numerical simulations

In this section, numerical simulations are carried out to verify the theoretical results. For the stochastic system (1.3), we adopt the Milstein higher-order method developed in [33] and the discretization form of the stochastic system is given by:

$$\left\{\begin{array}{c}{S}^{k+1}=S^k+[\Lambda -\mu S^k-\beta S^k(\eta I_A^k+I_S^k)]\Delta t+S^k\left[{\rho }_1\sqrt{\Delta t}{\varpi }_{1,k}+\frac{{\rho }_1^2}{2}({\varpi }_{1,k}^2-1)\Delta t\right],\hfill \\ E^{k+1}=E^k+[\beta S^k(\eta I_A^k+I_S^k)-(\mu +\theta )E^k]\Delta t+E^k\left[{\rho }_2\sqrt{\Delta t}{\varpi }_{2,k}+\frac{{\rho }_2^2}{2}({\varpi }_{2,k}^2-1)\Delta t\right],\hfill \\ I_S^{k+1}=I_S^k+[p\theta E^k-(\mu +{\alpha }_S+{\gamma }_S)I_S^k]\Delta t+I_S^k\left[{\rho }_3\sqrt{\Delta t}{\varpi }_{3,k}+\frac{{\rho }_3^2}{2}({\varpi }_{3,k}^2-1)\Delta t\right],\hfill \\ I_A^{k+1}=I_A^k+[(1-p)\theta E^k-(\mu +{\alpha }_A+{\gamma }_A)I_A^k]\Delta t+I_A^k\left[{\rho }_4\sqrt{\Delta t}{\varpi }_{4,k}+\frac{{\rho }_4^2}{2}({\varpi }_{4,k}^2-1)\Delta t\right],\hfill \end{array}\right)$$

where the time step $\Delta t>0$, ${\rho }_i^2$ denote the intensity of white noises, ${\varpi }_{i,k}$ $(i=1,2,3,4;k=1,2,\dots ,n)$ are mutually independent Gaussian random variables following the distribution $N(0,1)$. Based on the realistic parameter values in the literature, the parameter values in all the simulations are chosen from Table 2.

Table 2.

List of parameters.

Parameter	Units	Range	References
$\Lambda$	${\text{day}}^{-1}$	0.05812	[34]
$\mu$	${\text{day}}^{-1}$	$[\frac{83}{3650000},\frac{1}{80.3\times 365}]$	CIAc
$\beta$	${\text{day}}^{-1}$	0.4417	[6]
$\eta$	${\text{day}}^{-1}$	0.75	CDCa
$p$	${\text{day}}^{-1}$	0.5	Assumed
$\theta$	${\text{day}}^{-1}$	$[\frac{1}{14},\frac{1}{2}]$	[35], [36], [37]
${\alpha }_S$	${\text{day}}^{-1}$	0.05	Estimated
${\alpha }_A$	${\text{day}}^{-1}$	0.04	Estimated
${\gamma }_S$	${\text{day}}^{-1}$	0.24	Assumed
${\gamma }_A$	${\text{day}}^{-1}$	$\frac{1}{10.5}$	[8]

^ahttps://www.cdc.gov/coronavirus/2019-ncov/hcp/planning-scenarios.html$♯$ Table 2, accessed 03.06.2021.

${}^b$https://www/cia/gov/the-world-factbook/countries/united-states/, accessed 03.07.2021.

^cCIA World Factbook, https://www.cia.gov/library/publications/the-world-factbook/, accessed 09.29.2020.

Next, by numerical simulations, we mainly pay attention to validate two aspects:

(i) there is a unique ergodic stationary distribution if the condition $R_0^S>1$ holds;

(ii) the existence of the probability density.

Example 5.1

In order to obtain the existence of an ergodic stationary distribution numerically, we choose $\mu =83/3650000$, $\theta =1/2$, ${\sigma }_1={\sigma }_2={\sigma }_3={\sigma }_4=0.15$ and the other parameter values are given in Table 2. Direct calculation leads to that

$$R_0^S=\frac{p\beta \Lambda \theta }{(\mu +\frac{{\rho }_1^2}{2})(\mu +\theta +\frac{{\rho }_2^2}{2})(\mu +{\alpha }_S+{\gamma }_S+\frac{{\rho }_3^2}{2})}+\frac{(1-p)\beta \eta \Lambda \theta }{(\mu +\frac{{\rho }_1^2}{2})(\mu +\theta +\frac{{\rho }_2^2}{2})(\mu +{\alpha }_A+{\gamma }_A+\frac{{\rho }_4^2}{2})}\approx 9.3965>1.$$

In other words, the condition of Theorem 3.1 is satisfied. By Theorem 3.1, we obtain that system (1.3) has a unique ergodic stationary distribution $\pi \left(\cdot \right)$ which shows that the disease is persistent a.s. Fig. 1 confirms this.

Fig. 1.

The left column shows the time series diagrams of the susceptible population, the exposed population, the asymptomatic population and the symptomatic population in the stochastic model (1.3) and their corresponding deterministic model (1.2) with $\mu =83/3650000$, $\theta =1/2$, ${\sigma }_1={\sigma }_2={\sigma }_3={\sigma }_4=0.15$. The right column displays the marginal density function and frequency histogram of each population.

Example 5.2

To show the existence of the probability density around the quasi-endemic equilibrium, we choose $\mu =83/3650000$, $\theta =1/2$, ${\sigma }_1={\sigma }_2={\sigma }_3={\sigma }_4=0.15$ and the other parameter values are shown in Table 2. By direct calculation, we obtain $P^{\ast }={(S^{\ast },E^{\ast },I_S^{\ast },I_A^{\ast })}^T={(0.5035,0.1162,0.1002,0.2148)}^T$ and

$$R_0^S=\frac{p\beta \Lambda \theta }{(\mu +\frac{{\rho }_1^2}{2})(\mu +\theta +\frac{{\rho }_2^2}{2})(\mu +{\alpha }_S+{\gamma }_S+\frac{{\rho }_3^2}{2})}+\frac{(1-p)\beta \eta \Lambda \theta }{(\mu +\frac{{\rho }_1^2}{2})(\mu +\theta +\frac{{\rho }_2^2}{2})(\mu +{\alpha }_A+{\gamma }_A+\frac{{\rho }_4^2}{2})}\approx 9.3965>1.$$

That is to say, the condition of Theorem 4.1 holds. Hence, system (1.3) has a log-normal probability density around the quasi-endemic equilibrium $P^{\ast }$. Furthermore, we have $m_1=0.0932055>0$, $m_2=-0.08528<0$, $m_3=-0.586066<0$, $m_4=0.03658>0$, $m_5=-1.8849<0$ and $m_6=0.37849>0$. Thus, according to the first case of Theorem 4.1, we calculate the specific expression of the covariance matrix $\Sigma$,

$$\Sigma ={\varrho }_1^2{\left(J_2{J}_1\right)}^{-1}{\Sigma }_0{\left[{\left(J_2{J}_1\right)}^{-1}\right]}^T+{\varrho }_2^2{\left(J_7{J}_6{J}_5{J}_4\right)}^{-1}{\Sigma }_0{\left[{\left(J_7{J}_6{J}_5{J}_4\right)}^{-1}\right]}^T+{\varrho }_3^2{\left(J_{13}{J}_{12}{J}_{11}{J}_{10}\right)}^{-1}{\Sigma }_0{\left[{\left(J_{13}{J}_{12}{J}_{11}{J}_{10}\right)}^{-1}\right]}^T+{\varrho }_4^2{\left(J_{19}{J}_{18}{J}_{17}{J}_{16}\right)}^{-1}{\Sigma }_0{\left[{\left(J_{19}{J}_{18}{J}_{17}{J}_{16}\right)}^{-1}\right]}^T=\left(\begin{array}{cccc}\hfill 0.1440\hfill & \hfill 0.04626\hfill & \hfill -0.003752\hfill & \hfill -0.07315\hfill \\ \hfill 0.04626\hfill & \hfill 0.2227\hfill & \hfill 0.2044\hfill & \hfill 0.1576\hfill \\ \hfill -0.003752\hfill & \hfill 0.2044\hfill & \hfill 0.2303\hfill & \hfill 0.1687\hfill \\ \hfill -0.07315\hfill & \hfill 0.1576\hfill & \hfill 0.1687\hfill & \hfill 0.2407\hfill \end{array}\right),$$

and the corresponding probability density $\Phi (S,E,I_S,I_A)$ is as follows

$$\Phi (S,E,I_S,I_A)={\left(2\pi \right)}^{-2}{\left|\Sigma \right|}^{-\frac{1}{2}}{e}^{-\frac{1}{2}(ln\frac{S}{S^{\ast }},ln\frac{E}{E^{\ast }},ln\frac{I_S}{I_S^{\ast }},ln\frac{I_A}{I_A^{\ast }}){\Sigma }^{-1}{(ln\frac{S}{S^{\ast }},ln\frac{E}{E^{\ast }},ln\frac{I_S}{I_S^{\ast }},ln\frac{I_A}{I_A^{\ast }})}^T}=4.4792exp\left\{-\frac{1}{2}\left(ln\frac{S}{0.5035},ln\frac{E}{0.1162},ln\frac{I_S}{0.1002},ln\frac{I_A}{0.2148}\right){\Sigma }^{-1}{\left(ln\frac{S}{0.5035},ln\frac{E}{0.1162},ln\frac{I_S}{0.1002},ln\frac{I_A}{0.2148}\right)}^T\right\}.$$

Fig. 2 shows this.

Fig. 2.

Numerical simulations for: (i) the frequency histogram fitting density curves of $S$, $E$, $I_S$ and $I_A$ of system (1.3) with 50 000 iteration points, respectively. (ii) The marginal probability densities of $S$, $E$, $I_S$ and $I_A$ of system (1.3). All of the parameter values are the same as in Fig. 1.

6. Conclusion

In this paper, we formulate and analyze a stochastic SEIR-type model which is used to describe the transmission dynamics of COVID-19 in the population. Firstly, we prove that system (1.3) has a unique global positive solution with any given positive initial value. Then we use a stochastic Lyapunov function method to obtain sufficient criteria for the existence and uniqueness of an ergodic stationary distribution, which is a probability distribution with some invariant properties. In particular, under the same conditions as the existence of a stationary distribution, we get the exact expression of the probability density, which is a function that describes the probability of the output value of the random variable around the quasi-endemic equilibrium of system (1.3). Mathematically, the existence of a stationary distribution implies the weak stability in stochastic sense while the existence of the probability density of system (1.3) is more in-depth and specific than that of the stationary distribution. Biologically, the existence of a stationary distribution and probability density indicates the persistence and coexistence of all individuals.

Numerically, based on the actual parameter values in the existing literature, we obtain two important results: (i) small environmental noise makes each population fluctuate very little and hence it can retain some stochastic weak stability to some extent; (ii) we get the specific expression of the probability density around the quasi-endemic equilibrium of system (1.3).

On the other hand, there are still many important topics worthy of further study. For example, in this paper, we assume that the compartment $R$ acquires permanent immunity and cannot be infected by the infectious individuals. However, in fact, some people who had been infected COVID-19 may be infected once again if they are in close contact with someone who has COVID-19 and thus they will become susceptible. With that in mind, using an SEIRS-type model to describe the transmission dynamics of COVID-19 may be more proper. But for the SEIRS-type model, because of its high dimension, when we calculate the local probability density, the discussion will become more complicated. In addition, it is also interesting to study the effects of other types of random perturbations (such as nonlinear perturbations, Poisson jumps et al.) on COVID-19 models. So far as we know, there is little literature to analyze five-dimensional epidemic models with nonlinear perturbations [38], [39], [40] since there are many obstacles to solving the corresponding Fokker–Planck equation due to the limitations of mathematical methods. We look forward to fully addressing it in the near future. The relevant work is now underway.

CRediT authorship contribution statement

Qun Liu: Conceptualization, Methodology, Software, Writing – original draft, Formal analysis, Supervision, Writing – review & editing, Funding acquisition.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Appendix. Preliminaries of SDEs

Here we give some basic theories of stochastic differential equations (see [17] for a detailed introduction).

Consider a $d$-dimensional stochastic differential equation

$$dX\left(t\right)=f(X\left(t\right),t)dt+g(X\left(t\right),t)dB\left(t\right)\text{for}t\ge t_0,$$ (A.1)

with the initial value $X\left(t_0\right)=X_0\in {\mathbb{R}}^d$, where $B\left(t\right)$ is a $d$-dimensional standard Brownian motion defined on the complete probability space $(\Omega ,\mathcal{F},{\left\{{\mathcal{F}}_t\right\}}_{t\ge 0},\mathbb{P})$. Let $C^{2,1}({\mathbb{R}}^d\times [t_0,\infty ];{\overline{\mathbb{R}}}_{+})$ be the family of all nonnegative functions $V(X,t)$ defined on ${\mathbb{R}}^d\times [t_0,\infty ]$ such that they are continuously twice differentiable in $X$ and once in $t$. The differential operator $L$ related to Eq. (A.1) is defined by [17]

$$L=\frac{\partial }{\partial t}+\sum _{i=1}^d{f}_i(X,t)\frac{\partial }{\partial X_i}+\frac{1}{2}\sum _{i,j=1}^d{\left[g^T(X,t)g(X,t)\right]}_{ij}\frac{{\partial }^2}{\partial X_i\partial X_j}.$$ (A.2)

By operating $L$ on a function $V\in C^{2,1}({\mathbb{R}}^d\times [t_0,\infty ];{\overline{\mathbb{R}}}_{+})$, we have

$$LV(X,t)=V_t(X,t)+V_X(X,t)f(X,t)+\frac{1}{2}\text{trace}\left[g^T(X,t)V_{XX}(X,t)g(X,t)\right],$$

where $V_t=\partial V/\partial t$, $V_X=(\partial V/\partial X_1,\dots ,\partial V/\partial X_d)$ and $V_{XX}={({\partial }^{2}V/\left(\partial X_i\partial X_j\right))}_{d\times d}$. If $X\left(t\right)\in {\mathbb{R}}^d$, then the Itô’s formula is described as follows:

$$dV(X\left(t\right),t)=LV(X\left(t\right),t)dt+V_X(X\left(t\right),t)g(X\left(t\right),t)dB\left(t\right).$$

Data availability

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