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In this section, we investigate the VL stability of matrix A defined in (16) to determine the global stability of the endemic equilibrium (E_{*}). The following definitions and preliminary lemmas are necessary prerequisites for this examination.
30,48,49,50. Consider a square matrix, denoted as M, endowed with the property of symmetry and positive (negative) definiteness. In this context, the matrix M is succinctly expressed as (M>0) or ((M<0)).
30,48,49,50. We write a matrix (A_{ntimes n}>0(A_{ntimes n}<0)) if (A_{ntimes n}) is symmetric positive (negative) definite.
30,48,49,50. If a positive diagonal matrix (H_{ntimes n}) exists, such that (HA+A^{T}H^{T}<0) then a non-singular matrix (A_{ntimes n}) is VL stable.
30,48,49,50. If a positive diagonal matrix (H_{ntimes n}) exists, such that (HA+A^{T}H^{T}<0) ((>0)) then, a non singular matrix (A_{ntimes n}) is diagonal stable.
The (2times 2) VL stable matrix is determined by the following lemma.
30,48,49,50. Let (B=begin{bmatrix} B_{11} &{} B_{12} \ B_{21} &{} B_{22}\ end{bmatrix}) is VL stable if and only if (B_{11}<0), (B_{22}<0), and (det (B)=B_{11}B_{22}-B_{12}B_{21}>0).
30,48,49,50. Consider the nonsingular (A_{ntimes n}=[A_{ij}]), where ((nge 2)), the positive diagonal matrix (C_{ntimes n}=textrm{diag}(C_{1},ldots , C_{n})), and (D=A^{-1}) such that
$$begin{aligned} {left{ begin{array}{ll} A_{nn}>0,\ widetilde{C}widetilde{A}+(widetilde{C}widetilde{A})^{T}>0,\ widetilde{C}widetilde{D}+(widetilde{C}widetilde{D})^{T}>0. end{array}right. } end{aligned}$$
Then, for (H_{n}>0), it ensures that the matrix expression (HA + A^{T}H^{T}) is positive definite.
We employ the same methodology used in51 to obtain the necessary outcomes for globally asymptotically.
If (R_{0}<1), the system (2) is globally asymptotically stable at ({E}_{0}=left( frac{b}{sigma },0,0,0,0,0 right)).
Taking the Lyapunov function corresponding to all dependent classes of the model
$$begin{aligned} L=m_1(S-S^*)+m_2(E-E^*)+m_3(I_S-I_S^*)+m_4(I_A-I_A^*)+m_5(R-R^*). end{aligned}$$
Taking derivative with respect to t yields
$$begin{aligned} L^prime =m_1S^prime +m_2E^prime +m_3I_S^prime +m_4I_A^prime +m_5R^prime . end{aligned}$$
After utilizing the values derived from system (2) and performing requisite calculations, we obtain:
$$begin{aligned} L ^prime =& ,m_{1}b+(m_2-m_1)frac{beta _{1}SP}{1+alpha _1P}+(m_2-m_1) frac{beta _2S(I_A+I_S)}{1+alpha _2(I_A+I_S)}+(m_1-m_2)varphi E+(m_4-m_2) upsilon E+(m_3-m_4)eta upsilon E\&+ (m_5-m_3)gamma _S I_S+(m_5-m_4)gamma _A I_A-m_1mu S-m_2mu E -m_3mu I_S-m_3delta I_S-m_4mu I_A-m_4delta I_A-m_5mu R. end{aligned}$$
Using (m_1=m_2=m_3=m_4=m_5=1) in the above equation gives
$$begin{aligned} L^prime= & , b-mu S-mu E-mu I_S-delta I_S-mu I_A-delta I_A-mu R\= & {} -left(mu N+delta (I_{A}+I_{S})-bright)<0. end{aligned}$$
Thus, the system (2) is globally asymptotically stable at (E_{0}), whenever (R_{0}<1). (square)
We propose the following Lyapunov function in order to prove the global stability at (E_{*}).
$$begin{aligned} L=m_1(S-S^*)^2+m_2(E-E^*)^2+m_3(I_S-I_S^*)^2+m_4(I_A-I_A^*)^2+m_5(R-R^*)^2+m_6(P-P^*)^2. end{aligned}$$
Here, (m_{1}), (m_{2}), (m_{3}), (w_{4}), (m_{5}), and (w_{6}) are non-negative constants. By taking the derivative of L with respect to time along the trajectories of the model (8), one has
$$begin{aligned} L^prime =2m_1(S-S^*)S^prime +2m_2(E-E^*)E^prime +2m_3(I_S-I_S^*)I_S^prime +2m_4(I_A-I_A^*)I_A^prime +2m_5(R-R^*)R^prime +2m_6(P-P^*)P^prime . end{aligned}$$
Here
$$begin{aligned}&2m_1(S-S^*)S^prime \&quad =2m_1(S-S^*)bigg{-frac{beta _{1}SP}{1+alpha _{1}P}-frac{beta _{2}S(I_A+I_S)}{1+alpha _{2}(I_A+I_S)}+frac{beta _{1}S^*P^*}{1+alpha _{1}P^*}+ frac{beta _{2}S^*(I_A^*+I_S^*)}{1+alpha _{2}(I_A^*+I_S^*)}+varphi (R-R^*)-mu (S-S^*)bigg}\&quad =2m_1(S-S^*)bigg{-bigg(frac{beta _{1}P}{1+alpha _{1}P}+frac{beta _{2}(I_A+I_S)}{1+alpha _{2}(I_A+I_S)}bigg)S+bigg(frac{beta _{1}P^*}{1+alpha _{1}P^*}+ frac{beta _{2}(I_A^*+I_S^*)}{1+alpha _{2}(I_A^*+I_S^*)}bigg)S^*+varphi (R-R^*)-mu (S-S^*)bigg}\&quad =2m_1(S-S^*)bigg{-textbf{A}S+textbf{B}S^*+varphi (R-R^*)-mu (S-S^*)bigg}\&quad =2m_1(S-S^*)bigg{-textbf{A}S+textbf{A}S^*-textbf{A}S^*+textbf{B}S^*+varphi (R-R^*)-mu (S-S^*)bigg}\&quad =2m_1(S-S^*)bigg{-textbf{A}(S-S^*)-(textbf{A}-textbf{B})S^*+varphi (R-R^*)-mu (S-S^*)bigg}\&quad =2m_1(S-S^*)bigg{-(textbf{A}+mu )(S-S^*)-(textbf{A}-textbf{B})S^*+varphi (E-E^*)bigg}\&quad =2m_1(S-S^*)bigg{-bigg(frac{beta _{1}P}{1+alpha _{1}P}+frac{beta _{2}(I_A+I_S)}{1+alpha _{2}(I_A+I_S)}+mu bigg)(S-S^*) -frac{beta _{1}S^*}{(1+alpha _{1}P)(1+alpha _{1}P^*)}(P-P^*)\&qquad -frac{beta _{2}S^*}{(1+alpha _{2}(I_A+I_S))(1+alpha _{2}(I_A^*+I_S^*))}(I_A-I_A^*) -frac{beta _{2}S^*}{(1+alpha _{2}(I_A+I_S))(1+alpha _{2}(I_A^*+I_S^*))}(I_S-I_S^*)+varphi (R-R^*)bigg}\&quad =2m_1(S-S^*)bigg{-bigg(frac{beta _{1}P}{1+alpha _{1}P}+frac{beta _{2}(I_A+I_S)}{1+alpha _{2}(I_A+I_S)}+mu bigg)(S-S^*)+varphi (R-R^*)\&qquad -frac{beta _{2}S^*}{(1+alpha _{2}(I_A+I_S))(1+alpha _{2}(I_A^*+I_S^*))}(I_S-I_S^*)-frac{beta _{2}S^*}{(1+alpha _{2}(I_A+I_S))(1+alpha _{2}(I_A^*+I_S^*))}(I_A-I_A^*)\&qquad -frac{beta _{1}S^*}{(1+alpha _{1}P)(1+alpha _{1}P^*)}(P-P^*)bigg},\ end{aligned}$$
$$begin{aligned}&2m_2(E-E^*)E^prime \&quad =2m_2(E-E^*)bigg{bigg(frac{beta _{1}P}{1+alpha _{1}P}+frac{beta _{2}(I_A+I_S)}{1+alpha _{2}(I_A+I_S)}bigg)S-bigg(frac{beta _{1}P^*}{1+alpha _{1}P^*}+ frac{beta _{2}(I_A^*+I_S^*)}{1+alpha _{2}(I_A^*+I_S^*)}bigg)S^*-(mu +upsilon )(E-E^*)bigg}\&quad =2m_2(E-E^*)bigg{textbf{A}S-textbf{B}S^*-(mu +upsilon )(E-E^*)bigg}\&quad =2m_2(E-E^*)bigg{textbf{A}S-textbf{A}S^*+textbf{A}S^*-textbf{B}S^*-(mu +upsilon )(E-E^*)bigg}\&quad =2m_2(E-E^*)bigg{textbf{A}(S-S^*)+(textbf{A}-textbf{B})S^*-(mu +upsilon )(E-E^*)bigg}\&quad =2m_2(E-E^*)bigg{bigg(frac{beta _{1}P}{1+alpha _{1}P}+frac{beta _{2}(I_A+I_S)}{1+alpha _{2}(I_A+I_S)}bigg)(S-S^*)+ frac{beta _{1}S^*}{(1+alpha _{1}P)(1+alpha _{1}P^*)}(P-P^*)\&qquad +frac{beta _{2}S^*}{(1+alpha _{2}(I_A+I_S))(1+alpha _{2}(I_A^*+I_S^*))}(I_A-I_A^*) +frac{beta _{2}S^*}{(1+alpha _{2}(I_A+I_S))(1+alpha _{2}(I_A^*+I_S^*))}(I_S-I_S^*)-(mu +upsilon )(E-E^*)bigg}\&quad =2m_2(E-E^*)bigg{bigg(frac{beta _{1}P}{1+alpha _{1}P}+frac{beta _{2}(I_A+I_S)}{1+alpha _{2}(I_A+I_S)}bigg)(S-S^*)-(mu +upsilon )(E-E^*)\&qquad +frac{beta _{2}S^*}{(1+alpha _{2}(I_A+I_S))(1+alpha _{2}(I_A^*+I_S^*))}(I_S-I_S^*)+frac{beta _{2}S^*}{(1+alpha _{2}(I_A+I_S))(1+alpha _{2}(I_A^*+I_S^*))}(I_A-I_A^*)\&qquad +frac{beta _{1}S^*}{(1+alpha _{1}P)(1+alpha _{1}P^*)}(P-P^*)bigg},\&2m_3(I_S-I_S^*)I_S^prime =2m_3(I_S-I_S^*)bigg{eta upsilon (E-E^*)-(mu +delta +gamma _S+tau _{S})(I_S-I_S^*)bigg},\&2m_4(I_A-I_A^*)I_A^prime =2m_4(I_A-I_A^*)bigg{(1-eta )upsilon (E-E^*)-(mu +delta +gamma _A+tau _{A})(I_A-I_A^*)bigg},\&2m_5(R-R^*)R^prime =2m_5(R-R^*)bigg{gamma _A(I_A-I_A^*)+gamma _S(I_S-I_S^*)-(mu +varphi )(R-R^*)bigg}, quad textrm{and}\&2m_6(P-P^*)P^prime =2m_6(P-P^*)bigg{tau _A(I_A-I_A^*)+tau _S(I_S-I_S^*)-mu _P(P-P^*)bigg}. end{aligned}$$
Using these values in (L^prime), which implies that (L^prime =Z(MA+A^{T}M)Z^{T}.)
Here, (Z=[S-S^*, E-E^*, I_S-I_S^*, I_A-I_A^*, R-R^*, P-P^*]), (M={textbf{diag}}(m_1, m_2, m_3, m_4, m_5, m_6)), and
$$begin{aligned} A= & {} left( begin{array}{ccc} -frac{beta _{1}P}{1+alpha _{1}P}-frac{beta _{2}(I_A+I_S)}{1+alpha _{2}(I_A+I_S)}-mu &{} 0 &{} -frac{beta _{2}S^*}{(1+alpha _{2}(I_A+I_S))(1+alpha _{2}(I_A^*+I_S^*))}\ frac{beta _{1}P}{1+alpha _{1}P}+frac{beta _{2}(I_A+I_S)}{1+alpha _{2}(I_A+I_S)} &{} -(mu +upsilon ) &{} frac{beta _{2}S^*}{(1+alpha _{2}(I_A+I_S))(1+alpha _{2}(I_A^*+I_S^*))}\ 0 &{} eta upsilon &{} -(mu +delta +gamma _S+tau _{S})\ 0 &{} (1-eta )upsilon &{} 0\ 0 &{} 0 &{} gamma _S\ 0 &{} 0 &{} tau _S\ end{array}right. nonumber \{} & {} left. begin{array}{ccc} -frac{beta _{2}S^*}{(1+alpha _{2}(I_A+I_S))(1+alpha _{2}(I_A^*+I_S^*))} &{} varphi &{} -frac{beta _{1}S^*}{(1+alpha _{1}P)(1+alpha _{1}P^*)}\ frac{beta _{2}S^*}{(1+alpha _{2}(I_A+I_S))(1+alpha _{2}(I_A^*+I_S^*))} &{} 0 &{} frac{beta _{1}S^*}{(1+alpha _{1}P)(1+alpha _{1}P^*)}\ 0 &{} 0 &{} 0\ -(mu +delta +gamma _A+tau _{A}) &{} 0 &{} 0\ gamma _A &{} -(mu +varphi ) &{} 0\ tau _A &{} 0 &{} -mu _P end{array}right) . end{aligned}$$
(16)
It is important to note that matrix (A_{(n-1)times (n-1)}) is derived from matrix A by removing its final row and column. In accordance with the works of Zahedi and Kargar48, Shao and Shateyi49, Masoumnezhad et al.50, Chien and Shateyi30, and Parsaei et al.29, we introduce a series of lemmas and theorems to examine the global stability of the EE at (E_{*}).
In order to make the calculation easier, we present the matrix (16) in more simplified form as given below:
$$begin{aligned} A=left( begin{array}{cccccc} -a &{} 0 &{} -b &{} -b &{} c &{} -d\ e &{} -f &{} b &{} b &{} 0 &{} d\ 0 &{} g &{} -h &{} 0 &{} 0 &{} 0\ 0 &{} i &{} 0 &{} -j &{} 0 &{} 0\ 0 &{} 0 &{} k &{} l &{} -m &{} 0\ 0 &{} 0 &{} n &{} p &{} 0 &{} -q\ end{array}right) . end{aligned}$$
(17)
Here, (a=frac{beta _{1}P}{1+alpha _{1}P}+frac{beta _{2}(I_A+I_S)}{1+alpha _{2}(I_A+I_S)}+mu), (b=frac{beta _{2}S^*}{(1+alpha _{2}(I_A+I_S))(1+alpha _{2}(I_A^*+I_S^*))}), (c=varphi), (d=frac{beta _{1}S^*}{(1+alpha _{1}P)(1+alpha _{1}P^*)}), (e=frac{beta _{1}P}{1+alpha _{1}P}+frac{beta _{2}(I_A+I_S)}{1+alpha _{2}(I_A+I_S)}), (f=(mu +upsilon )), (g=eta upsilon), (h=(mu +delta +gamma _S+tau _{S})), (i=(1-eta )upsilon), (j=(mu +delta +gamma _A+tau _{A})), (k=gamma _S), (l=gamma _A), (m=(mu +varphi )), (n=tau _S), (p=tau _A), and (q=mu _P).
It is important to note that all the parameters a,b,c,d,e,f,g,h,i,j,k,l,m,n,p, and q are positive. Throughout this paper, it is important to note that (a-e) and (-a+e) are always equal to (mu) and (-mu) respectively. Moreover, we conclude that all the diagonal elements (the negative values) of the matrix A are negative, which is a good sign for stability. We need to compute the eigenvalues to make a definitive conclusion for the global stability of (E_{*}). But finding the eigenvalues of the matrix A is quite time consuming. Therefore, we introduce a series of lemmas and theorems given below to examine whether all the eigenvalues of the matrix A have negative real parts or not?.
The matrix (A_{6times 6}) defined in (17) is VL stable.
Obviously (-A_{66}>0). Consider (C=-widetilde{A}) be a (5times 5) matrix that is produced by eliminating the final row and column (-A). Utilizing Lemma 4.2, we can demonstrate the diagonal stability of matrices (C=-widetilde{A}) and (mathbb{C}=-widetilde{A^{-1}}). Let examine the matrix (C=-widetilde{A}) and (mathbb{C}=-widetilde{A^{-1}}), from (17) we obtain
$$begin{aligned} C=left( begin{array}{ccccc} a &{} 0 &{} b &{} b &{} -c\ -e &{} f &{} -b &{} -b &{} 0\ 0 &{} -g &{} h &{} 0 &{} 0\ 0 &{} -i &{} 0 &{} j &{} 0\ 0 &{} 0 &{} -k &{} -l &{} m end{array}right) , quad textrm{and}quad ,mathbb{C}=frac{1}{det (-A)}left( begin{array}{ccccc} mathbf{{c}_{11}} &{} mathbf{c_{12}} &{} mathbf{c_{13}} &{} mathbf{c_{14}} &{} mathbf{c_{15}}\ mathbf{{c}_{21}} &{} mathbf{c_{22}} &{} mathbf{c_{23}} &{} mathbf{c_{24}} &{} mathbf{c_{25}}\ mathbf{{c}_{31}} &{} mathbf{c_{32}} &{} mathbf{c_{33}} &{} mathbf{c_{34}} &{} mathbf{c_{35}}\ mathbf{{c}_{41}} &{} mathbf{c_{42}} &{} mathbf{c_{43}} &{} mathbf{c_{44}} &{} mathbf{c_{45}}\ mathbf{{c}_{51}} &{} mathbf{c_{52}} &{} mathbf{c_{53}} &{} mathbf{c_{54}} &{} mathbf{c_{55}}\ end{array}right) . end{aligned}$$
Here
$$begin{aligned} det (-A)=-d (a - e) m (g j n +h i p) - left(c e (g j k +h i l) + (b (a - e) h i + (b (a - e) g - a f h) j) mright) q<0, end{aligned}$$
and
$$begin{aligned} mathbf{c_{11}}= & {} -mleft(d g j n + d h i p + b h i q + b g j q - f h j qright),\ mathbf{c_{12}}= & {} -d m (g j n + h i p) + left(c (g j k + h i l) - b (h i + g j) mright) q,\ mathbf{c_{13}}= & {} -f j m (d n + b q) + c left(d i (l n - k p) + (f j k + b i (-k + l)) qright),\ mathbf{c_{14}}= & {} -f h m (d p + b q) + c left(d g (-l n + k p) + (b g (k - l) + f h l) qright),\ mathbf{c_{15}}= & {} -c left(d g j n + d h i p + b h i q + b g j q - f h j qright),\ mathbf{c_{21}}= & {} e h j m q, quad mathbf{c_{22}}=a h j m q,\ mathbf{c_{23}}= & {} d (a - e) j m n + j left(c e k + b (a - e) mright) q,\ mathbf{c_{24}}= & {} d (a - e) h m p + h left(c e l + b (a - e) mright) q,\ mathbf{c_{25}}= & {} c e h j q,quad mathbf{c_{31}}=e g j m q,\ mathbf{c_{32}}= & {} a g j m q,\ mathbf{c_{33}}= & {} -a m left(d i p + b i q - f j qright) + e i left(d m p - c l q + b m qright),\ mathbf{c_{34}}= & {} d (a - e) g m p + g left(c e l + b (a - e) mright) q,\ mathbf{c_{35}}= & {} c e g j q,quad mathbf{c_{41}}=e h i m q,quad mathbf{c_{42}}=a h i m q,\ mathbf{c_{43}}= & {} d (a - e) i m n + i left(c e k + b (a - e) mright) q,\ mathbf{c_{44}}= & {} -a m left(d g n + b g q - f h qright) + e g left(d m n - c k q + b m qright),\ mathbf{c_{45}}= & {} c e h i q,\ mathbf{c_{51}}= & {} e left(g j k + h i lright) q,\ mathbf{c_{52}}= & {} a left(g j k + h i lright) q, \ mathbf{c_{53}}= & {} d (a - e) i (l n - k p) + left(b e i (k - l) + a (f j k + b i (-k + l))right) q,\ mathbf{c_{54}}= & {} d (-a + e) g (l n - k p) + left(b e g (-k + l) + a (b g (k - l) + f h l)right) q,\ mathbf{c_{55}}= & {} d (-a + e) (g j n + h i p) + left(b (-a + e) h i + left(-a b g + b e g + a f hright) jright) q. end{aligned}$$
Since
$$begin{aligned} det (-A)=-d (a - e) m (g j n +h i p) - left(c e (g j k +h i l) + left(b (a - e) h i + (b (a - e) g - a f h) jright) mright) q<0, end{aligned}$$
and
$$begin{aligned} mathbf{c_{55}}=d (-a + e) (g j n + h i p) + left(b (-a + e) h i + ((-a + e)b g + a f h) jright) q. end{aligned}$$
Therefore
$$begin{aligned} mathbb{C}_{55}=frac{-left(mu (d g j n + d h i p + b h i q + b g j q) - a f h j qright)}{-mu m (d g j n + d h i p + b h i q + b g j q)-(c e (g j k + h i l) - a f h j m) q}, end{aligned}$$
which implies that
$$begin{aligned} mathbb{C}_{55}=frac{mu left(d g j n + d h i p + b h i q + b g j qright) - a f h j q}{mu m left(d g j n + d h i p + b h i q + b g j qright)+left(c e (g j k + h i l) - a f h j mright) q}>0. end{aligned}$$
Using the information provided in Lemma 4.2, we assert and demonstrate that (C=-widetilde{A}) and (mathbb{C}=-widetilde{A^{-1}}) exhibit diagonal stability, thereby establishing the VL stability of the matrix A. (square)
In accordance with Theorem 4.6, the matrix (C=-widetilde{A}) is affirmed to possess diagonal stability.
As (C_{55}>0), using Lemma 4.2, we must demonstrate the diagonal stability of the reduced matrix (D=widetilde{C}) and its inverse (mathbb{D}=widetilde{C^{-1}}) to complete the proof. Thus the matrix D can be obtained from the matrix C given in Theorem 4.6, we have
$$begin{aligned} D=left( begin{array}{cccc} a &{} 0 &{} b &{} b\ -e &{} f &{} -b &{} -b\ 0 &{} -g &{} h &{} 0\ 0 &{} -i &{} 0 &{} j end{array}right) , quad textrm{and}quad ,,mathbb{D}=frac{1}{det (C)}left( begin{array}{cccc} mathbf{d_{11}} &{} mathbf{d_{12}} &{} mathbf{d_{13}} &{} mathbf{d_{14}}\ mathbf{d_{21}} &{} mathbf{d_{22}} &{} mathbf{d_{23}} &{} mathbf{d_{24}}\ mathbf{d_{31}} &{} mathbf{d_{32}} &{} mathbf{d_{33}} &{} mathbf{d_{34}}\ mathbf{d_{41}} &{} mathbf{d_{42}} &{} mathbf{d_{43}} &{} mathbf{d_{44}} end{array}right) , end{aligned}$$
here
$$begin{aligned} det (C)=-c e (g j k + h i l) + left(b (-a + e) h i + ((-a + e)b g + a f h) jright) m<0, end{aligned}$$
and
$$begin{aligned} mathbf{d_{11}}= & {} -(b h i + b g j - f h j) m,\ mathbf{d_{12}}= & {} c (g j k + h i l) - b (h i + g j) m,\ mathbf{d_{13}}= & {} c f j k - b left(c i (k - l) + f j mright), \ mathbf{d_{14}}= & {} b c g (k - l) + f h (c l - b m),\ mathbf{d_{21}}= & {} e h j m, mathbf{d_{22}}=a h j m,\ mathbf{d_{23}}= & {} j left(c e k + b (a - e) mright),\ mathbf{d_{24}}= & {} h left(c e l + b (a - e) mright),\ mathbf{d_{31}}= & {} e g j m, mathbf{d_{32}}=a g j m,\ mathbf{d_{33}}= & {} -c e i l + left((-a + e)b i + a f jright) m,\ mathbf{d_{34}}= & {} g left(c e l + b (a - e) mright),\ mathbf{d_{41}}= & {} e h i m, mathbf{d_{42}}=a h i m,\ mathbf{d_{43}}= & {} i left(c e k + b (a - e) mright),\ mathbf{d_{44}}= & {} -c e g k + left((-a + e)b g + a f hright) m. end{aligned}$$
Since
$$begin{aligned} det (C)=-c e (g j k + h i l) + left(b (-a + e) h i + left((-a + e) b g + a f hright) jright) m, end{aligned}$$
and
$$begin{aligned} mathbf{d_{44}}=-c e g k + left((-a + e)b g + a f hright) m. end{aligned}$$
Therefore
$$begin{aligned} mathbb{D}_{44}=frac{c e g k +mu b g m - a f h m}{ce(gjk+hil)+bmu (hi+gj)m-afhjm}>0. end{aligned}$$
(square)
The matrix (mathbb{C}=-widetilde{A^{-1}}) defined in Theorem 4.6 is diagonal stable.
Consider the matrix (mathbb{C}) in 4.6, we see that (mathbb{C}_{55}>0), Using Lemma 4.2, we must demonstrate that the modified matrix (E=widetilde{mathbb{C}}) and its inverse (mathbb{E}=widetilde{mathbb{C}^{-1}}) exhibit diagonal stability. This establishes the completion of the proof. Thus the matrix D can be obtained from the matrix (mathbb{C}) given in Theorem 4.6, we have
$$begin{aligned} E=frac{1}{det (-A)}left( begin{array}{cccc} mathbf{{c}_{11}} &{} mathbf{c_{12}} &{} mathbf{c_{13}} &{} mathbf{c_{14}}\ mathbf{{c}_{21}} &{} mathbf{c_{22}} &{} mathbf{c_{23}} &{} mathbf{c_{24}}\ mathbf{{c}_{31}} &{} mathbf{c_{32}} &{} mathbf{c_{33}} &{} mathbf{c_{34}}\ mathbf{{c}_{41}} &{} mathbf{c_{42}} &{} mathbf{c_{43}} &{} mathbf{c_{44}}\ end{array}right) ,,quad textrm{and}quad ,mathbb{E}=frac{1}{det (mathbb{C})}left( begin{array}{cccc} mathbf{{e}_{11}} &{} mathbf{e_{12}} &{} mathbf{e_{13}} &{} mathbf{e_{14}}\ mathbf{{e}_{21}} &{} mathbf{e_{22}} &{} mathbf{e_{23}} &{} mathbf{e_{24}}\ mathbf{{e}_{31}} &{} mathbf{e_{32}} &{} mathbf{e_{33}} &{} mathbf{e_{34}}\ mathbf{{e}_{41}} &{} mathbf{e_{42}} &{} mathbf{e_{43}} &{} mathbf{e_{44}}\ end{array}right) . end{aligned}$$
Here
$$begin{aligned} det (mathbb{C})&=frac{qell ^{4}}{det (-A)}<0, end{aligned}$$
and
$$begin{aligned} mathbf{e_{11}}= & {} -a qell ^{3}, mathbf{e_{12}}=0, mathbf{e_{13}}=-(d n + b q)ell ^{3},\ mathbf{e_{14}}= & {} -(d p + b q)ell ^{3}, mathbf{e_{21}}=e qell ^{3}, mathbf{e_{22}}=-f qell ^{3},\ mathbf{e_{23}}= & {} (d n + b q)ell ^{3}, mathbf{e_{24}}=(d p + b q)ell ^{3}, mathbf{e_{31}}=0,\ mathbf{e_{32}}= & {} g q ell ^{3}, mathbf{e_{33}}=-h qell ^{3}, mathbf{e_{34}}=0, mathbf{e_{41}}=0,\ mathbf{e_{42}}= & {} i qell ^{3}, mathbf{e_{43}}=0, mathbf{e_{44}}=-j qell ^{3}. end{aligned}$$
Here
$$begin{aligned} ell =left[d (a - e) m (g j n + h i p) + left{c e (g j k + h i l) + left(b (a - e) h i + (b (a - e) g - a f h) jright) mright} qright] end{aligned}$$
Since
$$begin{aligned} det (mathbb{C})&=frac{qell ^{4}}{det (-A)}, end{aligned}$$
and
$$begin{aligned} mathbf{e_{44}}=-j q ell ^{3}. end{aligned}$$
Therefore
$$begin{aligned} mathbb{E}_{44}=frac{-j}{det (-A)ell }>0, end{aligned}$$
where (det (-A)<0) and (ell >0). (square)
The matrix D, as described in Lemma 4.3 and denoted by (widetilde{C}), exhibits diagonal stability.
It is clear that (D_{44}>0). By applying Lemma 4.2, our task is to demonstrate the diagonal stability of the reduced matrix (F=widetilde{D}) and its inverse (mathbb{F}=widetilde{D^{-1}}), thereby completing the proof. Thus from Lemma 4.3, we have
$$begin{aligned} F=left( begin{array}{ccc} a &{} 0 &{} b\ -e &{} f &{} -b\ 0 &{} -g &{} h end{array}right) , quad textrm{and}quad ,,mathbb{F}=frac{1}{det ({D})}left( begin{array}{ccc} mathbf{f_{11}} &{} mathbf{f_{12}} &{} mathbf{f_{13}}\ mathbf{f_{21}} &{} mathbf{f_{22}} &{} mathbf{f_{23}}\ mathbf{f_{31}} &{} mathbf{f_{32}} &{} mathbf{f_{33}} end{array}right) . end{aligned}$$
Here
$$begin{aligned} det (D)=b (-a + e) h i + left((-a + e)b g + a f hright) j>0 (textrm{see} textrm{Proposition} 4.1 (iii)), end{aligned}$$
and (mathbf{f_{11}}=f h j - b (h i + g j)>0 (textrm{see} textrm{Proposition} 4.1 (iv))), (mathbf{f_{12}}=-b (h i + g j)), (mathbf{f_{13}}=b f j), (mathbf{f_{21}}=e h j), (mathbf{f_{22}}=a h j), (mathbf{f_{23}}=b (a - e) j), (mathbf{f_{31}}=e g j), (mathbf{f_{32}}=a g j), (mathbf{f_{33}}=(-a + e) b i + a f j>0 (textrm{see} textrm{Proposition} 4.1 (v))).
Since (det (D)>0), and (mathbf{f_{33}}>0), therefore (mathbb{F}_{33}>0). (square)
The matrix (mathbb{D}), as defined in Lemma 4.3 denoted by (widetilde{C^{-1}}), exhibits diagonal stability.
It is obvious that (mathbb{D}_{44}>0), by employing Lemma 4.2, our task is to demonstrate the diagonal stability of a modified matrix (G=widetilde{mathbb{D}}) and its inverse (mathbb{G}=widetilde{mathbb{D}^{-1}}). This verification serves as the completion of the proof. Thus from Lemma 4.3, we have
$$begin{aligned} G=frac{1}{det (C)}left( begin{array}{ccc} mathbf{d_{11}} &{} mathbf{d_{12}} &{} mathbf{d_{13}}\ mathbf{d_{21}} &{} mathbf{d_{22}} &{} mathbf{d_{23}}\ mathbf{d_{31}} &{} mathbf{d_{32}} &{} mathbf{d_{33}} end{array}right) , quad textrm{and}quad ,mathbb{G}=frac{1}{det (mathbb{D})}left( begin{array}{ccc} mathbf{g_{11}} &{} mathbf{g_{12}} &{} mathbf{g_{13}}\ mathbf{g_{21}} &{} mathbf{g_{22}} &{} mathbf{g_{23}}\ mathbf{g_{31}} &{} mathbf{g_{32}} &{} mathbf{g_{33}} end{array}right) . end{aligned}$$
Here
$$begin{aligned} det (mathbb{D})&=frac{-mell _{1}^{3}}{det (C)}>0, end{aligned}$$
and
$$begin{aligned} mathbf{g_{11}}= & a mell _{1}^{2}, mathbf{g_{12}}=0, mathbf{g_{13}}=-(c k - b m)ell _{1}^{2}, mathbf{g_{21}}=-e mell _{1}^{2}, mathbf{g_{22}}=f mell _{1}^{2}, ,mathbf{g_{23}}= & {} -b mell _{1}^{2}, mathbf{g_{31}}=0, mathbf{g_{32}}=-g mell _{1}^{2}, mathbf{g_{33}}=h mell _{1}^{2}. end{aligned}$$
Here
$$begin{aligned} ell _{1}=left[c e (g j k + h i l) + left{b (a - e) h i + left(b (a - e) g - a f hright) jright} mright] end{aligned}$$
Read more from the original source:
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