Stability analysis and numerical evaluations of a COVID-19 model with vaccination – BMC Medical Research … – BMC Medical Research Methodology
April 28, 2024
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.
Towards this end, we split the time interval ([t_a,t_b]) into (Nge 1) subdomains in the forms
$$begin{aligned} K_n:=[t_{n},t_{n+1}],quad n=0,1,ldots , N-1. end{aligned}$$
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 modelEq. (1) to be in the formEq. (25) as
$$begin{aligned} x^n_{mathcal {J}}(t):=sum limits _{j=0}^{mathcal {J}} omega ^n_j,mathbb {U}_j(t)=varvec{U}_{mathcal {J}}(t),varvec{W}^n_mathcal {J},quad tin K_n, end{aligned}$$
(29)
where we utilized the notations
$$begin{aligned} varvec{W}_{mathcal {J}}^n:=left[ omega ^n_0quad omega ^n_1quad ldots quad omega ^n_{mathcal {J}}right] ^T,quad varvec{U}_{mathcal {J}}(t):=left[ mathbb {U}_0(t)quad mathbb {U}_1(t)quad ldots quad mathbb {U}_J(t)right] , end{aligned}$$
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
$$begin{aligned} x_{mathcal {J}}(t)=sum limits _{n=0}^{N-1} c_n(t),x^n_{mathcal {J}}(t),quad c_n(t):= left{ begin{array}{ll} 0, &{} tnotin K_n,\ 1, &{} tin K_n.\ end{array}right. end{aligned}$$
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 giveninEq. (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).
Our chief aim is to solve the nonlinear COVID-19 systemEq. (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 Bellmans 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.[40,41,42]. 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 modelEq. (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 modelEq. (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 systemEq. (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 systemEq. (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 inEq. (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 systemEq. (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 toEq. (2). We now are able to solve the family of linearized initial-value problemsEqs. (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 ofEq. (29) by utilizing only ((mathcal {J}+1)) SCPSK basis functions, we assume that the eight solutions of systemEq. (32) can be represented in terms ofEq. (29). Thus, we take these solutions at iteration (sge 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 (tin 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 (1le ile 8) at the iteration (sge 1). Also, the vector of SCPSK basis, i.e., (varvec{U}_mathcal {J}(t)) is defined inEq. (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
$$begin{aligned} varvec{Q}_mathcal {J}(t)=left[ 1quad t-t_nquad (t-t_n)^{2}quad ldots quad (t-t_n)^{mathcal {J}}right] . end{aligned}$$
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 inEq. (15). One can also show that (det (varvec{F}_mathcal {J})ne 0) and it is a triangular matrix. It follows that
$$begin{aligned} f_{i,j}:= left{ begin{array}{ll} o_{i,j}, &{} textrm{if}~ ile j,\ 0, &{} textrm{if}~ i> j. end{array}right. end{aligned}$$
We then insert the obtained term (varvec{U}_mathcal {J}(t)) inEq. (35) intoEq. (34). The resulting expansions are
$$begin{aligned} left{ begin{array}{l} S^{n}_{mathcal {J},s}(t)=varvec{Q}_mathcal {J}(t),varvec{F}_mathcal {J},varvec{W}^{n,s}_{mathcal {J},1},quad (S_v)^{n}_{mathcal {J},s}(t)=varvec{Q}_mathcal {J}(t),varvec{F}_mathcal {J},varvec{W}^{n,s}_{mathcal {J},2},\ I^{n}_{mathcal {J},s}(t),=varvec{Q}_mathcal {J}(t),varvec{F}_J,varvec{W}^{n,s}_{mathcal {J},3},quad (I_v)^{n}_{mathcal {J},s}(t)~=varvec{Q}_mathcal {J}(t),varvec{F}_mathcal {J},varvec{W}^{n,s}_{mathcal {J},4},\ R^{n}_{mathcal {J},s}(t) =varvec{Q}_mathcal {J}(t),varvec{F}_mathcal {J},varvec{W}^{n,s}_{mathcal {J},5},quad (R_v)^{n}_{mathcal {J},s}(t)=varvec{Q}_mathcal {J}(t),varvec{F}_mathcal {J},varvec{W}^{n,s}_{mathcal {J},6},\ J^{n}_{mathcal {J},s}(t), =varvec{Q}_mathcal {J}(t),varvec{F}_mathcal {J},varvec{W}^{n,s}_{mathcal {J},7},quad (J_v)^{n}_{mathcal {J},s}(t),=varvec{Q}_mathcal {J}(t),varvec{F}_mathcal {J},varvec{W}^{n,s}_{mathcal {J},8}, end{array}right. tin K_n. end{aligned}$$
(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 inEq. (36).
$$begin{aligned} left{ begin{array}{l} dot{S}^{n}_{mathcal {J},s}(t)=varvec{Q}_mathcal {J}(t),varvec{D}_mathcal {J},varvec{F}_mathcal {J},varvec{W}^{n,s}_{mathcal {J},1},quad (dot{S}_v)^{n}_{mathcal {J},s}(t)=varvec{Q}_mathcal {J}(t),varvec{D}_mathcal {J},varvec{F}_mathcal {J},varvec{W}^{n,s}_{mathcal {J},2},\ dot{I}^{n}_{mathcal {J},s}(t),=varvec{Q}_mathcal {J}(t),varvec{D}_mathcal {J},varvec{F}_mathcal {J},varvec{W}^{n,s}_{mathcal {J},3},quad (dot{I}_v)^{n}_{mathcal {J},s}(t)~=varvec{Q}_mathcal {J}(t),varvec{D}_mathcal {J},varvec{F}_mathcal {J},varvec{W}^{n,s}_{mathcal {J},4},\ dot{R}^{n}_{mathcal {J},s}(t)=varvec{Q}_mathcal {J}(t),varvec{D}_mathcal {J},varvec{F}_mathcal {J},varvec{W}^{n,s}_{mathcal {J},5},quad (dot{R}_v)^{n}_{mathcal {J},s}(t)=varvec{Q}_mathcal {J}(t),varvec{D}_mathcal {J},varvec{F}_mathcal {J},varvec{W}^{n,s}_{mathcal {J},6},\ dot{J}^{n}_{mathcal {J},s}(t),=varvec{Q}_mathcal {J}(t),varvec{D}_mathcal {J},varvec{F}_mathcal {J},varvec{W}^{n,s}_{mathcal {J},7},quad (dot{J}_v)^{n}_{mathcal {J},s}(t),=varvec{Q}_mathcal {J}(t),varvec{D}_mathcal {J},varvec{F}_mathcal {J},varvec{W}^{n,s}_{mathcal {J},8}, end{array}right. tin K_n. end{aligned}$$
(38)
The exact solutions of the linearized systemEq. (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 inEq. (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})inEq. (42) in the matrix representation forms. The proof is easy by just considering the definitions of the involved matrices and vectors inEq. (40).
If two vectors (varvec{z}^n_{mathcal {J},s}(t)) and (dot{varvec{z}}^n_{mathcal {J},s}(t)) inEq. (40) computed at the collocation pointsEq. (30), we arrive at the next matrix forms
$$begin{aligned} varvec{Z}^n_s=bar{widehat{varvec{Q}}},widehat{varvec{F}},varvec{W}^n,qquad dot{varvec{Z}}^n_s=bar{widehat{varvec{Q}}},widehat{varvec{F}},widehat{varvec{D}},varvec{W}^n, end{aligned}$$
(43)
where the matrix (bar{widehat{varvec{Q}}}) is given by
$$begin{aligned} bar{widehat{varvec{Q}}}=[widehat{varvec{Q}}(t_{0,n})quad widehat{varvec{Q}}(t_{1,n})quad ldots quad widehat{varvec{Q}}(t_{mathcal {J},n}) ]^T. end{aligned}$$
Moreover, two matrices (widehat{varvec{Q}}, widehat{varvec{F}}) are defined inEq. (40). Similarly, the vector (varvec{W}^n) is given inEq. (40).
By turning to relationEq. (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
$$begin{aligned} varvec{B}_n,varvec{W}^n=widehat{varvec{R}}^n_{s-1}, quad textrm{or}quad left[ varvec{B}_n;widehat{varvec{R}}^n_{s-1}right] ,quad sge 1,~0le nle N-1, end{aligned}$$
(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 conditionsinEq. (2) and add them intoEq. (44). So, the next task is to constitute the matrix representation ofEq. (2). Let us approach (trightarrow 0) in the first relation ofEq. (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 modelEq. (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 conditionsinEq. (2) will be used to find the corresponding approximations for the systemEq. (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 (0le nle N-1).
Generally, finding the true solutions of the COVID-19 systemEq. (1) is not possible practically. In this case, the residual error functions (REFs) help us to measure the quality of approximations obtained by the QLM-SCPSK technique. Once we calculate the eight approximations by the illustrated method, we substitute them into the model systemEq. (1). In fact, the REFs are defined as the difference between the left-hand side and the right-hand side of the considered equation. On the subdomain (K_n) we set the REFs as
$$begin{aligned}{} & {} mathbb {R}_{1,mathcal {J}}^{n}(t):=left| dot{S}^{n}_{mathcal {J},s}(t)-Lambda +beta S^{n}_{mathcal {J},s}(t)L^n_{mathcal {J},s}(t)+(lambda +mu ) S^{n}_{mathcal {J},s}(t)- theta _1 R^{n}_{mathcal {J},s}(t)right| cong 0, nonumber \{} & {} mathbb {R}_{2,mathcal {J}}^{n}(t):=left| (dot{S_v})^{n}_{mathcal {J},s}(t)+beta ' (S_v)^{n}_{mathcal {J},s}(t)L^n_{mathcal {J},s}(t)- theta _2(R_v)^{n}_{mathcal {J},s}(t)- lambda S^{n}_{mathcal {J},s}(t)+ (delta +mu ) (S_v)^{n}_{mathcal {J},s}(t)right| cong 0, nonumber \{} & {} mathbb {R}_{3,mathcal {J}}^{n}(t):=left| dot{I}^{n}_{mathcal {J},s}(t)-beta S^{n}_{mathcal {J},s}(t)L^n_{mathcal {J},s}(t)+ (gamma _1+alpha _1+mu ) I^{n}_{mathcal {J},s}(t)right| cong 0, nonumber \{} & {} mathbb {R}_{4,mathcal {J}}^{n}(t):=left| (dot{I_v})^{n}_{mathcal {J},s}(t)- beta ' (S_v)^{n}_{mathcal {J},s}(t)L^n_{mathcal {J},s}(t)+ (gamma _2+alpha _2+mu )(I_v)^{n}_{mathcal {J},s}(t)right| cong 0, nonumber \{} & {} mathbb {R}_{5,mathcal {J}}^{n}(t):=left| dot{R}^{n}_{mathcal {J},s}(t) - gamma _1 I^{n}_{mathcal {J},s}(t)+(theta _1+mu ) R^{n}_{mathcal {J},s}(t)- eta _1 J^{n}_{mathcal {J},s}(t)right| cong 0, nonumber \{} & {} mathbb {R}_{6,mathcal {J}}^{n}(t):=left| (dot{R_v})^{n}_{mathcal {J},s}(t)- gamma _2 (I_v)^{n}_{mathcal {J},s}(t)+ (theta _2+mu ) (R_v)^{n}_{mathcal {J},s}(t)- eta _2(J_v)^{n}_{mathcal {J},s}(t)- delta (S_v)^{n}_{mathcal {J},s}(t)right| cong 0, nonumber \{} & {} mathbb {R}_{7,mathcal {J}}^{n}(t):=left| dot{J}^{n}_{mathcal {J},s}(t) -alpha _1 I^{n}_{mathcal {J},s}(t)+ (eta _1+mu _1)J^{n}_{mathcal {J},s}(t)right| cong 0, nonumber \{} & {} mathbb {R}_{8,mathcal {J}}^{n}(t):=left| (dot{J_v})^{n}_{mathcal {J},s}(t)- alpha _2(I_v)^{n}_{mathcal {J},s}(t) +(eta _2+mu _2) (J_v)^{n}_{mathcal {J},s}(t)right| cong 0, end{aligned}$$
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for a fixed iteration number s and we have defined (L^n_{mathcal {J},s}:=I^{n}_{mathcal {J},s}(t)+ (I_v)^{n}_{mathcal {J},s}(t)) for brevity.
Analogously, at the fixed iteration s, the numerical order of convergence associated with the obtained REFs can be defined in the infinity norm. These are given by
$$begin{aligned} L^{infty }_{ell }equiv L^{infty }_{ell }(mathcal {J}):=max _{0le nle N-1}left( max _{tin K_n},left| mathbb {R}_{ell ,mathcal {J}}^{n}(t)right| right) ,quad ell =1,2,ldots ,8. end{aligned}$$
Therefore, the convergence order (Co) for each solution is defined by
$$begin{aligned} textrm{Co}_{mathcal {J}}^{ell }:=log _2left( frac{L^{infty }_{ell }(mathcal {J})}{L^{infty }_{ell }(2mathcal {J})}right) ,quad ell =1,2,ldots ,8. end{aligned}$$
(47)
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Stability analysis and numerical evaluations of a COVID-19 model with vaccination - BMC Medical Research ... - BMC Medical Research Methodology
