Let us emphasize that the spectral matrix collocation approach based on the SCPSK may not yield convergence on a long time interval \([t_a,t_b]\). One remedy is to use a large number of bases on the long domains accordingly to reach the desired level of accuracy. Another approach is to divide the given interval into a sequence of subintervals and employ the proposed collocation scheme on each subinterval consequently.
Here, we have
\(t_0:=t_a\) and
\(t_N:=t_b\). The uniform time step is taken as
\(h=t_{n+1}-t_n=(t_b-t_a)/N\). Note that by selecting
\(N=1\), we turn back to the traditional spectral collocation method on the whole domain
\([t_a,t_b]\). Therefore, on each subinterval
\(K_n\) we take the approximate solution of the model Eq. (
1) to be in the form Eq. (
25) as
$$\begin{aligned} x^n_{\mathcal {J}}(t):=\sum \limits _{j=0}^{\mathcal {J}} \omega ^n_j\,\mathbb {U}_j(t)=\varvec{U}_{\mathcal {J}}(t)\,\varvec{W}^n_\mathcal {J},\quad t\in K_n, \end{aligned}$$
(29)
where we utilized the notations
$$\begin{aligned} \varvec{W}_{\mathcal {J}}^n:=\left[ \omega ^n_0\quad \omega ^n_1\quad \ldots \quad \omega ^n_{\mathcal {J}}\right] ^T,\quad \varvec{U}_{\mathcal {J}}(t):=\left[ \mathbb {U}_0(t)\quad \mathbb {U}_1(t)\quad \ldots \quad \mathbb {U}_J(t)\right] , \end{aligned}$$
as the vector of unknown coefficients and the vector of SCPSK bases respectively. Once we get the all local approximate solutions for
\(n=0,1,\ldots ,N-1\), the global approximate solution on the given (large) interval
\([t_a,t_b]\) will be constructed in the form
$$\begin{aligned} x_{\mathcal {J}}(t)=\sum \limits _{n=0}^{N-1} c_n(t)\,x^n_{\mathcal {J}}(t),\quad c_n(t):= \left\{ \begin{array}{ll} 0, &{} t\notin K_n,\\ 1, &{} t\in K_n.\\ \end{array}\right. \end{aligned}$$
In order to collocate a set of
\((\mathcal {J}+1)\) linear equations to be obtained later at some suitable points, we consider the roots of
\(\mathbb {U}_{\mathcal {J}+1}(t)\) on the subinterval
\(K_n\). By modifying the points given in Eq. (
17), we take the collocation nodes as
$$\begin{aligned} t_{\nu ,n}=\frac{1}{2}\left( t_n+t_{n+1}+h\,\cos \left( \frac{\nu \,\pi }{\mathcal {J}+2}\right) \right) ,\quad \nu =1,2,\ldots ,\mathcal {J}+1. \end{aligned}$$
(30)
At the end, we note that in the proposed splitting approach, the given initial conditions of the underlying model problem are prescribed on the first subinterval \(K_0\). Once the approximate solution on \(K_0=[t_0,t_1]\) is determined, we utilize it to assign the initial conditions on the next time interval \(K_1\). To do so, it is sufficient to evaluate the obtained approximation at \(t_1\). We repeat this idea on the next subintervals in order until we arrive at the last subinterval \(K_{N-1}\). Below, we illustrate the main steps of our matrix collocation algorithm on an arbitrary subinterval \(K_n\) for \(n=0,1,\ldots ,N-1\).
The QLM-SCPSK matrix collocation technique
Our chief aim is to solve the nonlinear COVID-19 system Eq. (
1) efficiently by using the spectral method based on SCPSK basis. Towards this end, we first need to get rid of the nonlinearity of the model. This can be done by employing the Bellman’s quasilinearization method (QLM) [
39]. Thus we will get more advantages in terms of running time, especially for large values of
J in comparison to the performance of directly applied collocation methods to nonlinear models, see cf. [
40‐
42]. By combining the idea of QLM and the splitting of the domain we will obtain more gains in terms of accuracy for the approximate solutions of nonlinear model Eq. (
1). Let us first describe the technique of QLM. For more information, we may refer the readers to the above-mentioned works.
By reformulating the original COVID-19 model Eq. (
1) in a compact form we get
$$\begin{aligned} \frac{d}{dt} \varvec{z}(t)=\varvec{G}(t,\varvec{z}(t)), \end{aligned}$$
(31)
where
$$\begin{aligned} \varvec{z}(t)=\left[ \begin{array}{c} S(t)\\ S_v(t)\\ I(t)\\ I_v(t)\\ R(t)\\ R_v(t)\\ J(t)\\ J_v(t) \end{array}\right] ,\quad \varvec{G}(t,\varvec{z}(t))=\left[ \begin{array}{c} g_1(t)\\ g_2(t)\\ g_3(t)\\ g_4(t)\\ g_5(t)\\ g_6(t)\\ g_7(t)\\ g_8(t) \end{array}\right] = \left[ \begin{array}{c} \Lambda - \beta S(I+I_v)- (\lambda +\mu ) S+ \theta _1 R\\ -\beta ' S_v(I+ I_v)+ \theta _2R_v+ \lambda S- (\delta +\mu ) S_v\\ \beta S(I+ I_v)- (\gamma _1+\alpha _1+\mu ) I \\ \beta ' S_v(I+ I_v)- (\gamma _2+\alpha _2+\mu )I_v\\ \gamma _1 I-(\theta _1+\mu ) R+ \eta _1 J \\ \gamma _2 I_v- (\theta _2+\mu ) R_v+ \eta _2J_v+ \delta S_v\\ \alpha _1 I- (\eta _1+\mu _1)J\\ \alpha _2I_v- (\eta _2+\mu _2) J_v \end{array}\right] . \end{aligned}$$
To begin the QLM process, we assume
\(\varvec{z}_0(t)\) is available as an initial rough approximation for the solution
\(\varvec{z}(t)\) of the COVID-19 system Eq. (
31). Through an iterative manner, the QLM procedure reads as follows
$$\begin{aligned} \frac{d}{dt}\varvec{z}_{s}(t)\approx \varvec{G}(t,\varvec{z}_{s-1}(t))+\varvec{G}_{\varvec{z}}(t,\varvec{z}_{s-1}(t))\,\left( \varvec{z}_{s}(t)-\varvec{z}_{s-1}(t)\right) ,\quad s=1,2,\ldots . \end{aligned}$$
Here, the notation
\(\varvec{G}_{\varvec{z}}\) stands for the Jacobian matrix of the COVID-19 system Eq. (
31), which is of size 8 by 8. By performing some calculations we reach the linearized equivalent model form as
$$\begin{aligned} \frac{d}{dt}\varvec{z}_{s}(t)+\varvec{M}_{s-1}(t)\,\varvec{z}_{s}(t)=\varvec{r}_{s-1}(t),\qquad s=1,2,\ldots , \end{aligned}$$
(32)
where
\(\varvec{M}_{s-1}(t):=\varvec{J}(S_{s-1}(t), (S_v)_{s-1}(t), I_{s-1}(t), (I_v)_{s-1}(t))\) as the Jacobian matrix
\(\varvec{J}\) previously constructed in Eq. (
7). Also we have
$$\begin{aligned} \varvec{z}_{s}(t)= \left[ \begin{array}{c} S_{s-1}(t)\\ (S_v)_{s-1}(t)\\ I_{s-1}(t)\\ (I_v)_{s-1}(t)\\ R_{s-1}(t)\\ (R_v)_{s-1}(t)\\ J_{s-1}(t)\\ (J_v)_{s-1}(t) \end{array}\right] ,\quad \varvec{r}_{s-1}(t)= \left[ \begin{array}{c} \Lambda +\beta \,S_{s-1}(t)\Big (I_{s-1}(t)+(I_v)_{s-1}(t)\Big )\\ \beta '\,(S_v)_{s-1}(t)\Big (I_{s-1}(t)+(I_v)_{s-1}(t)\Big )\\ -\beta \,S_{s-1}(t)\Big (I_{s-1}(t)+(I_v)_{s-1}(t)\Big )\\ -\beta '\,(S_v)_{s-1}(t)\Big (I_{s-1}(t)+(I_v)_{s-1}(t)\Big )\\ 0\\ 0\\ 0\\ 0 \end{array}\right] . \end{aligned}$$
Along with the system Eq. (
32) the initial conditions
$$\begin{aligned} \varvec{z}_{s}(0)=\left[ \begin{array}{cccccccc} S_0&S_{v0}&I_0&I_{v0}&R_0&R_{v0}&J_0&J_{v0} \end{array}\right] ^T, \end{aligned}$$
(33)
are given due to Eq. (
2). We now are able to solve the family of linearized initial-value problems Eqs. (
32)-(
33) numerically by our proposed matrix collocation method on an arbitrary (long) domain
\([t_a,t_b]\). For this purpose and for clarity of exposition, we restrict our illustrations to a local subinterval
\(K_n\) for
\(n=0,1,\ldots ,N-1\).
In view of Eq. (
29) by utilizing only (
\(\mathcal {J}+1\)) SCPSK basis functions, we assume that the eight solutions of system Eq. (
32) can be represented in terms of Eq. (
29). Thus, we take these solutions at iteration
\(s\ge 1\) as
$$\begin{aligned} \left\{ \begin{array}{l} S^{n}_{\mathcal {J},s}(t)=\sum _{j=0}^{\mathcal {J}}\omega ^{n,s}_{j,1}\,\mathbb {U}_j(t)=\varvec{U}_{\mathcal {J}}(t)\,\varvec{W}^{n,s}_{\mathcal {J},1},\quad (S_v)^{n}_{\mathcal {J},s}(t)=\sum _{j=0}^{\mathcal {J}}\omega ^{n,s}_{j,2}\,\mathbb {U}_j(t)=\varvec{U}_{\mathcal {J}}(t)\,\varvec{W}^{n,s}_{\mathcal {J},2},\\ I^{n}_{\mathcal {J},s}(t)\,=\sum _{j=0}^{\mathcal {J}}\omega ^{n,s}_{j,3}\,\mathbb {U}_j(t)=\varvec{U}_{\mathcal {J}}(t)\,\varvec{W}^{n,s}_{\mathcal {J},3},\quad (I_v)^{n}_{\mathcal {J},s}(t)~=\sum _{j=0}^{\mathcal {J}}\omega ^{n,s}_{j,4}\,\mathbb {U}_j(t)=\varvec{U}_{\mathcal {J}}(t)\,\varvec{W}^{n,s}_{\mathcal {J},4},\\ R^{n}_{\mathcal {J},s}(t)=\sum _{j=0}^{\mathcal {J}}\omega ^{n,s}_{j,5}\,\mathbb {U}_j(t)=\varvec{U}_{\mathcal {J}}(t)\,\varvec{W}^{n,s}_{\mathcal {J},5},\quad (R_v)^{n}_{\mathcal {J},s}(t)=\sum _{j=0}^{\mathcal {J}}\omega ^{n,s}_{j,6}\,\mathbb {U}_j(t)=\varvec{U}_{\mathcal {J}}(t)\,\varvec{W}^{n,s}_{\mathcal {J},6},\\ J^{n}_{\mathcal {J},s}(t)\,=\sum _{j=0}^{\mathcal {J}}\omega ^{n,s}_{j,7}\,\mathbb {U}_j(t)=\varvec{U}_{J}(t)\,\varvec{W}^{n,s}_{\mathcal {J},7},\quad (J_v)^{n}_{\mathcal {J},s}(t)\,=\sum _{j=0}^{\mathcal {J}}\omega ^{n,s}_{j,8}\,\mathbb {U}_j(t)=\varvec{U}_{\mathcal {J}}(t)\,\varvec{W}^{n,s}_{\mathcal {J},8},\\ \end{array}\right. \end{aligned}$$
(34)
for
\(t\in K_n\). Moreover, by
\(\varvec{W}^{n,s}_{\mathcal {J},i}= \left[ \begin{array}{cccc} \omega ^{n,s}_{0,i}&\omega ^{n,s}_{1,i}&\dots&\omega ^{n,s}_{\mathcal {J},i} \end{array}\right] ^T\) we denote the vectors of unknowns for
\(1\le i\le 8\) at the iteration
\(s\ge 1\). Also, the vector of SCPSK basis, i.e.,
\(\varvec{U}_\mathcal {J}(t)\) is defined in Eq. (
29). We next provide a decomposition for
\(\varvec{U}_\mathcal {J}(t)\) given by
$$\begin{aligned} \varvec{U}_\mathcal {J}(t)=\varvec{Q}_\mathcal {J}(t)\,\varvec{F}_\mathcal {J}. \end{aligned}$$
(35)
Here, the vector
\(\varvec{Q}_\mathcal {J}(t)\) including the powers of
\((t-t_n)\) introduced by
$$\begin{aligned} \varvec{Q}_\mathcal {J}(t)=\left[ 1\quad t-t_n\quad (t-t_n)^{2}\quad \ldots \quad (t-t_n)^{\mathcal {J}}\right] . \end{aligned}$$
The next object is the matrix
\(\varvec{F}_\mathcal {J}=(f_{i,j})_{i,j=0}^{\mathcal {J}}\) of size
\((\mathcal {J}+1)\times (\mathcal {J}+1)\). The entries of the latter matrix are given in Eq. (
15). One can also show that
\(\det (\varvec{F}_\mathcal {J})\ne 0\) and it is a triangular matrix. It follows that
$$\begin{aligned} f_{i,j}:= \left\{ \begin{array}{ll} o_{i,j}, &{} \textrm{if}~ i\le j,\\ 0, &{} \textrm{if}~ i> j. \end{array}\right. \end{aligned}$$
We then insert the obtained term
\(\varvec{U}_\mathcal {J}(t)\) in Eq. (
35) into Eq. (
34). The resulting expansions are
$$\begin{aligned} \left\{ \begin{array}{l} S^{n}_{\mathcal {J},s}(t)=\varvec{Q}_\mathcal {J}(t)\,\varvec{F}_\mathcal {J}\,\varvec{W}^{n,s}_{\mathcal {J},1},\quad (S_v)^{n}_{\mathcal {J},s}(t)=\varvec{Q}_\mathcal {J}(t)\,\varvec{F}_\mathcal {J}\,\varvec{W}^{n,s}_{\mathcal {J},2},\\ I^{n}_{\mathcal {J},s}(t)\,=\varvec{Q}_\mathcal {J}(t)\,\varvec{F}_J\,\varvec{W}^{n,s}_{\mathcal {J},3},\quad (I_v)^{n}_{\mathcal {J},s}(t)~=\varvec{Q}_\mathcal {J}(t)\,\varvec{F}_\mathcal {J}\,\varvec{W}^{n,s}_{\mathcal {J},4},\\ R^{n}_{\mathcal {J},s}(t) =\varvec{Q}_\mathcal {J}(t)\,\varvec{F}_\mathcal {J}\,\varvec{W}^{n,s}_{\mathcal {J},5},\quad (R_v)^{n}_{\mathcal {J},s}(t)=\varvec{Q}_\mathcal {J}(t)\,\varvec{F}_\mathcal {J}\,\varvec{W}^{n,s}_{\mathcal {J},6},\\ J^{n}_{\mathcal {J},s}(t)\, =\varvec{Q}_\mathcal {J}(t)\,\varvec{F}_\mathcal {J}\,\varvec{W}^{n,s}_{\mathcal {J},7},\quad (J_v)^{n}_{\mathcal {J},s}(t)\,=\varvec{Q}_\mathcal {J}(t)\,\varvec{F}_\mathcal {J}\,\varvec{W}^{n,s}_{\mathcal {J},8}, \end{array}\right. t\in K_n. \end{aligned}$$
(36)
We then proceed by nothing that the derivative of the vector
\(\varvec{Q}_\mathcal {J}(t)\) can be stated in terms of itself. A vivid calculation reveals that
$$\begin{aligned} \dot{\varvec{Q}}_{\mathcal {J}}(t)=\varvec{Q}_{\mathcal {J}}(t)\,\varvec{D}_\mathcal {J},\quad \varvec{D}_\mathcal {J}=\left[ \begin{array}{lllll} 0 &{} 1 &{} 0 &{}\ldots &{} 0\\ 0 &{} 0 &{} 2 &{}\ldots &{} 0\\ \vdots &{} \vdots &{} \ddots &{}\vdots &{} \vdots \\ 0 &{} 0 &{} 0 &{}\ddots &{} \mathcal {J}\\ 0 &{} 0 &{} 0 &{} \ldots &{} 0 \end{array}\right] _{(\mathcal {J}+1)\times (\mathcal {J}+1)}. \end{aligned}$$
(37)
From this relation, we are able to derive a matrix forms of the derivatives of the unknown solutions in Eq. (
36).
$$\begin{aligned} \left\{ \begin{array}{l} \dot{S}^{n}_{\mathcal {J},s}(t)=\varvec{Q}_\mathcal {J}(t)\,\varvec{D}_\mathcal {J}\,\varvec{F}_\mathcal {J}\,\varvec{W}^{n,s}_{\mathcal {J},1},\quad (\dot{S}_v)^{n}_{\mathcal {J},s}(t)=\varvec{Q}_\mathcal {J}(t)\,\varvec{D}_\mathcal {J}\,\varvec{F}_\mathcal {J}\,\varvec{W}^{n,s}_{\mathcal {J},2},\\ \dot{I}^{n}_{\mathcal {J},s}(t)\,=\varvec{Q}_\mathcal {J}(t)\,\varvec{D}_\mathcal {J}\,\varvec{F}_\mathcal {J}\,\varvec{W}^{n,s}_{\mathcal {J},3},\quad (\dot{I}_v)^{n}_{\mathcal {J},s}(t)~=\varvec{Q}_\mathcal {J}(t)\,\varvec{D}_\mathcal {J}\,\varvec{F}_\mathcal {J}\,\varvec{W}^{n,s}_{\mathcal {J},4},\\ \dot{R}^{n}_{\mathcal {J},s}(t)=\varvec{Q}_\mathcal {J}(t)\,\varvec{D}_\mathcal {J}\,\varvec{F}_\mathcal {J}\,\varvec{W}^{n,s}_{\mathcal {J},5},\quad (\dot{R}_v)^{n}_{\mathcal {J},s}(t)=\varvec{Q}_\mathcal {J}(t)\,\varvec{D}_\mathcal {J}\,\varvec{F}_\mathcal {J}\,\varvec{W}^{n,s}_{\mathcal {J},6},\\ \dot{J}^{n}_{\mathcal {J},s}(t)\,=\varvec{Q}_\mathcal {J}(t)\,\varvec{D}_\mathcal {J}\,\varvec{F}_\mathcal {J}\,\varvec{W}^{n,s}_{\mathcal {J},7},\quad (\dot{J}_v)^{n}_{\mathcal {J},s}(t)\,=\varvec{Q}_\mathcal {J}(t)\,\varvec{D}_\mathcal {J}\,\varvec{F}_\mathcal {J}\,\varvec{W}^{n,s}_{\mathcal {J},8}, \end{array}\right. t\in K_n. \end{aligned}$$
(38)
The exact solutions of the linearized system Eq. (
32) can be written in a vectorized form as
$$\begin{aligned} \varvec{z}_s(t)\approx \varvec{z}^n_{\mathcal {J},s}(t):= \left[ \begin{array}{l} S^{n}_{\mathcal {J},s}(t)\\ (S_v)^{n}_{\mathcal {J},s}(t)\\ I^{n}_{\mathcal {J},s}(t)\\ (I_v)^{n}_{\mathcal {J},s}(t)\\ R^{n}_{\mathcal {J},s}(t)\\ (R_v)^{n}_{\mathcal {J},s}(t)\\ J^{n}_{\mathcal {J},s}(t)\\ (J_v)^{n}_{\mathcal {J},s}(t) \end{array}\right] ,\quad \dot{\varvec{z}}_s(t)\approx \frac{d}{dt}\varvec{z}^n_{\mathcal {J},s}(t):= \left[ \begin{array}{l} \dot{S}^{n}_{\mathcal {J},s}(t)\\ (\dot{S}_v)^{n}_{\mathcal {J},s}(t)\\ \dot{I}^{n}_{\mathcal {J},s}(t)\\ (\dot{I}_v)^{n}_{\mathcal {J},s}(t)\\ \dot{R}^{n}_{\mathcal {J},s}(t)\\ (\dot{R}_v)^{n}_{\mathcal {J},s}(t)\\ \dot{J}^{n}_{\mathcal {J},s}(t)\\ (\dot{J}_v)^{n}_{\mathcal {J},s}(t) \end{array}\right] . \end{aligned}$$
(39)
We next introduce the following block diagonal matrices of dimensions
\(8(\mathcal {J}+1)\times 8(\mathcal {J}+1)\) as
$$\begin{aligned} \widehat{\varvec{Q}}(t){} & {} =\mathrm {{\textbf {Diag}}} \left( \begin{array}{cccccccc} \varvec{Q}_\mathcal {J}(t)&\varvec{Q}_\mathcal {J}(t)&\varvec{Q}_\mathcal {J}(t)&\varvec{Q}_\mathcal {J}(t)&\varvec{Q}_\mathcal {J}(t)&\varvec{Q}_\mathcal {J}(t)&\varvec{Q}_\mathcal {J}(t)&\varvec{Q}_\mathcal {J}(t) \end{array}\right) ,\\ \widehat{\varvec{D}}{} & {} =\mathrm {{\textbf {Diag}}} \left( \begin{array}{cccccccc} \varvec{D}_\mathcal {J}&\varvec{D}_\mathcal {J}&\varvec{D}_\mathcal {J}&\varvec{D}_\mathcal {J}&\varvec{D}_\mathcal {J}&\varvec{D}_\mathcal {J}&\varvec{D}_\mathcal {J}&\varvec{D}_\mathcal {J} \end{array}\right) ,\\ \widehat{\varvec{F}}{} & {} =\mathrm {{\textbf {Diag}}} \left( \begin{array}{cccccccc} \varvec{F}_\mathcal {J}&\varvec{F}_\mathcal {J}&\varvec{F}_\mathcal {J}&\varvec{F}_\mathcal {J}&\varvec{F}_\mathcal {J}&\varvec{F}_\mathcal {J}&\varvec{F}_\mathcal {J}&\varvec{F}_\mathcal {J} \end{array}\right) . \end{aligned}$$
By the aid of the former definitions, the matrix formats of
\(\varvec{z}^n_{\mathcal {J},s}(t)\) and
\(\dot{\varvec{z}}^n_{\mathcal {J},s}(t)\) will rewrite concisely as
$$\begin{aligned} \varvec{z}^n_{\mathcal {J},s}(t)=\widehat{\varvec{Q}}(t)\,\widehat{\varvec{F}}\,\varvec{W}^n,\quad \dot{\varvec{z}}^n_{\mathcal {J},s}(t)=\widehat{\varvec{Q}}(t)\,\widehat{\varvec{F}}\,\widehat{\varvec{D}}\,\varvec{W}^n. \end{aligned}$$
(40)
Here,
\(\varvec{W}^n\) is the successive vector of eight previously defined vector of unknowns
$$\begin{aligned} \varvec{W}^n=\left[ \begin{array}{cccc} \varvec{W}^{n,s}_{\mathcal {J},1}&\varvec{W}^{n,s}_{\mathcal {J},2}&\ldots&\varvec{W}^{n,s}_{\mathcal {J},8} \end{array}\right] ^T. \end{aligned}$$
We now can collocate the linearized Eq. (
32) at the zeros of SCPSK given in Eq. (
17) on the subdomain
\(K_n\). We get
$$\begin{aligned} \frac{d}{dt}\varvec{z}_{s}(t_{\nu ,n})+\varvec{M}_{s-1}(t_{\nu ,n})\,\varvec{z}_{s}(t_{\nu ,n})=\varvec{r}_{s-1}(t_{\nu ,n}),\qquad \nu =1,2,\ldots ,\mathcal {J}, \end{aligned}$$
(41)
for
\(s=1,2,\ldots\). Denote the coefficient matrix by
\(\widehat{\varvec{M}}^n_{s-1}\) and the right-hand-side vector as
\(\widehat{\varvec{R}}^n_{s-1}\). These are defined by
$$\begin{aligned} \widehat{\varvec{M}}^n_{s-1}= \left[ \begin{array}{cccc} \varvec{M}_{s-1}(t_{0,n})&{}\textbf{0}&{}\ldots &{}\textbf{0}\\ \textbf{0}&{}\varvec{M}_{s-1}(t_{1,n})&{}\ldots &{}\textbf{0}\\ \vdots &{}\vdots &{}\ddots &{}\vdots \\ \textbf{0}&{}\textbf{0}&{}\ldots &{}\varvec{M}_{s-1}(t_{\mathcal {J},n}) \end{array}\right] ,\quad \widehat{\varvec{R}}^n_{s-1}= \left[ \begin{array}{c} \varvec{r}_{s-1}(t_{0,n})\\ \varvec{r}_{s-1}(t_{1,n})\\ \vdots \\ \varvec{r}_{s-1}(t_{\mathcal {J},n}) \end{array}\right] . \end{aligned}$$
Let us define further the vectors of unknowns as
$$\begin{aligned} \dot{\varvec{Z}}^n_s= \left[ \begin{array}{c} \dot{\varvec{z}}_{s}(t_{0,n})\\ \dot{\varvec{z}}_{s}(t_{1,n})\\ \vdots \\ \dot{\varvec{z}}_{s}(t_{\mathcal {J},n}) \end{array}\right] ,\quad \varvec{Z}^n_s= \left[ \begin{array}{c} \dot{\varvec{z}}_{s}(t_{0,n})\\ \dot{\varvec{z}}_{s}(t_{1,n})\\ \vdots \\ \dot{\varvec{z}}_{s}(t_{\mathcal {J},n}) \end{array}\right] . \end{aligned}$$
Consequently, the system of Eq. (
41) can be stated briefly as
$$\begin{aligned} \dot{\varvec{Z}}^{n}_{s}+\widehat{\varvec{M}}^n_{s-1}\,\varvec{Z}^n_s=\widehat{\varvec{R}}^n_{s-1},\quad n=0,1,\ldots ,N-1, \end{aligned}$$
(42)
and with
\(s=1,2,\ldots\). Before we talk about the fundamental matrix equation, we need to state two vectors
\(\varvec{Z}^n_s\) and
\(\dot{\varvec{Z}}^{n}_{s}\) in Eq. (
42) in the matrix representation forms. The proof is easy by just considering the definitions of the involved matrices and vectors in Eq. (
40).
By turning to relation Eq. (
40) we substitute the derived matrix formats into it. Precisely speaking, after replacing
\(\varvec{Z}^n_s\) and
\(\dot{\varvec{Z}}^n_s\) we gain the so-called fundamental matrix equation (FME) of the form
$$\begin{aligned} \varvec{B}_n\,\varvec{W}^n=\widehat{\varvec{R}}^n_{s-1}, \quad \textrm{or}\quad \left[ \varvec{B}_n;\widehat{\varvec{R}}^n_{s-1}\right] ,\quad s\ge 1,~0\le n\le N-1, \end{aligned}$$
(44)
where
$$\begin{aligned} \varvec{B}_n:=\bar{\widehat{\varvec{Q}}}\,\widehat{\varvec{F}}+\widehat{\varvec{M}}^n_{s-1}\,\bar{\widehat{\varvec{Q}}}\,\widehat{\varvec{F}}\,\widehat{\varvec{D}}. \end{aligned}$$
To complete the process of QLM-SCPSK approach, it is necessary to implement the initial conditions in Eq. (
2) and add them into Eq. (
44). So, the next task is to constitute the matrix representation of Eq. (
2). Let us approach
\(t\rightarrow 0\) in the first relation of Eq. (
40). It gives us
$$\begin{aligned} \varvec{B}_{0,n}\,\varvec{W}^n=\widehat{\varvec{R}}^n_{s-1,0},\qquad \varvec{B}_{0,n}:=\widehat{\varvec{Q}}(0)\,\widehat{\varvec{F}},\quad \widehat{\varvec{R}}^n_{s-1,0}=\left[ \begin{array}{cccccccc} S_0&S_{v0}&I_0&I_{v0}&R_0&R_{v0}&J_0&J_{v0} \end{array}\right] ^T. \end{aligned}$$
We then replace eight rows of the augmented matrix
\([\varvec{B}_n;\widehat{\varvec{R}}^n_{s-1}]\) by the already obtained row matrix
\([\varvec{B}_{0,n};\widehat{\varvec{R}}^n_{s-1,0}]\). Denote the modified FME by
$$\begin{aligned} \check{\varvec{B}_{n}}\,\varvec{W}^n=\check{\textbf{R}}^n_{s-1},\quad \textrm{or} \quad \left[ \check{\varvec{B}_{n}};\check{\textbf{R}}^n_{s-1}\right] . \end{aligned}$$
(45)
This implies that the solution of the model Eq. (
1) is obtainable on each subdomain
\(K_n\) by iterating
\(n=0,1,\ldots ,N-1\). On
\(K_0\) as the first subdomain, the given initial conditions in Eq. (
2) will be used to find the corresponding approximations for the system Eq. (
1). Hence, this approximate solutions on
\(K_0\) evaluated at the starting point of
\(K_1\) will be utilized for the initial conditions on
\(K_1\). By repeating this process we acquire all approximations on all
\(K_n\) for
\(0\le n\le N-1\).