The proposed model architecture
Individuals in the susceptible \(\left( {S_{k}^{c} } \right)\) group are vulnerable to the pandemic diseases where the suspicious \(\left( {S_{k}^{p} } \right)\) ones leave the group (Fig. 1, number 1) and the noninfected \(\left( {I_{k}^{nn} } \right)\) ones rejoin the susceptible group (number 5). The vaccinated \(\left( {V_{k} } \right)\)(number 2), the recovered \(\left( {R_{k} } \right)\) (number 22), and the death \(\left( {D_{k} } \right)\)(number 21) become nonsusceptible \(\left( {S_{k}^{nc} } \right)\)(number 3) and leave the susceptible \(\left( {S_{k}^{c} } \right)\) group where the remained ones constitute the current susceptible \(\left( {S_{k + 1}^{c} } \right)\)(number 4) group. The individuals in the suspicious \(\left( {S_{k}^{p} } \right)\) group, who are tested and/or quarantined, either move to the infected \(\left( {I_{k}^{n} } \right)\)(number 6) group or the noninfected \(\left( {I_{k}^{nn} } \right)\)(number 7) group where some individuals in the infected \(\left( {I_{k}^{n} } \right)\) group can return the suspicious \(\left( {S_{k}^{p} } \right)\) group again. Also, since the infected \(\left( {I_{k}^{n} } \right)\) individuals spread the virus until they are isolated, they act like as an excitation signal \(\left( {I_{k  1}^{na} } \right)\) (number 8) on the suspicious casualties. Individuals in the infected \(\left( {I_{k}^{n} } \right)\) group can be in the hospitalized \(\left( {H_{k} } \right)\)(number 9) group or in the nonhospitalized \(\left( {H_{k}^{n} } \right)\)(number 10) group where the nonhospitalized \(\left( {H_{k} } \right)\) individuals join the recovered \(\left( {R_{k} } \right)\)(number 15) group after a quarantine period. The individuals in the hospitalized \(\left( {H_{k} } \right)\) group can union with the intensive care \(\left( {I_{k}^{t} } \right)\)(number 11), the intubated \(\left( {I_{k}^{b} } \right)\)(number 12), the death \(\left( {D_{k} } \right)\)(number 13), or the recovered \(\left( {R_{k} } \right)\)(number 14) groups. The individuals in the intensive care \(\left( {I_{k}^{t} } \right)\) group can move to the intubated \(\left( {I_{k}^{b} } \right)\)(number 16), the death \(\left( {D_{k} } \right)\)(number 17), or the hospitalized \(\left( {H_{k} } \right)\)(number 18) groups. Similarly, the individuals in the intubated \(\left( {I_{k}^{b} } \right)\) group can join either the intensive care \(\left( {I_{k}^{t} } \right)\)(number 19), or the death \(\left( {D_{k} } \right)\)(number 20) groups. The nonpharmacological policies \(\left( {u_{k} } \right)\)(number 23) and priority and age specific vaccination policy \(\left( {V_{k}^{*} } \right)\)(number 24) act like an external inhibitor on all the casualties at varying rates.
The ScSpInHItIbRDVN model
This section initially formulates the parametric submodels, and then the vaccination and the nonpharmacological policies of the S^{c}S^{p}I^{n}HI^{t}I^{b}RDVN model.
The S^{c}S^{p}I^{n}HI^{t}I^{b}RDVN submodels
This subsection constructs the parametric models of each submodel illustrated in Fig. 1.
The susceptible \(S_{k}^{c}\) submodel
Considering the connections coming in and leaving out the susceptible \(S_{k}^{c}\) group in Fig. 1, one can formulate the \(S_{k}^{c}\) submodel with a difference equation. We can initially write the difference equation of the nonsusceptible \(S_{k}^{nc}\) group shown in Fig. 2 as
$$S_{k}^{nc} = a_{14} R_{k} + a_{15} D_{k} + c_{1} V_{k}$$
(1)
where \(S_{k}^{nc}\) represents the nonsusceptible individuals who have gained immunity and also the individuals who lost their lives, \(R_{k}\) represents the recovered individuals, \(D_{k}\) represents the dead individuals, \(V_{k}\) represents the vaccinated individuals, \(a_{14}\),\(a_{15}\),\(c_{1}\) are the unknown parameters.
The representation of the susceptible \(S_{k + 1}^{c}\) group in Fig. 2 is
$$S_{k + 1}^{c} =  a_{11} S_{k}^{c}  a_{12} S_{k}^{p} + a_{13} I_{k}^{nn}  S_{k}^{nc}$$
(2)
where \(S_{k}^{c}\) represents the individuals who may be infected and have a lack of immunity, \(S_{k}^{p}\) represents the suspicious individuals, \(I_{k}^{nn}\) represents the noninfected individuals, \(a_{11}\),\(a_{12}\),\(a_{13}\) are the unknown parameters.Substituting Eq. (1) in Eq. (2) yields
$$S_{k + 1}^{c} =  a_{11} S_{k}^{c}  a_{12} S_{k}^{p} + a_{13} I_{k}^{nn}  a_{14} R_{k}  a_{15} D_{k}  c_{1} V_{k}$$
(3)
All the parameters in Eq. (3) are unknown and will be learned from the available data with the RLS algorithm subject to the inequality constraints in the next section.
The next subsection provides the modelling steps of the suspicious \(S_{k}^{p}\) submodel.
The suspicious \(S_{k}^{p}\) submodel
Some of the susceptible \(S_{k}^{c}\) individuals become suspicious \(S_{k}^{p}\) as they exhibit symptoms or contact an infected individual, or return from the regions where the pandemic disease is a threat. These individuals are either tested or quarantined for a time duration. In this paper, we define the suspicious \(S_{k}^{p}\) individuals as the number of the people tested daily. Therefore, the model can predict the number of the required tests in the future. We can represent the \(S_{k}^{p}\) submodel shown in Fig. 3 as
$$\begin{aligned} S_{k + 1}^{p} & =  a_{21} S_{k}^{p} + a_{22} S_{k}^{c}  a_{23} I_{k}^{n}  a_{24} I_{k}^{nn} + a_{25} I_{k  1}^{na} \ldots \\ & \quad {{ \ldots }}  b_{2} u_{k}^{{}}  c_{2} V_{k}^{{S^{p} }} \\ \end{aligned}$$
(4)
where \(I_{k}^{na}\) represents the individuals who can become suspicious again and excitation effect of the infected individuals on the suspicious casualties (related to filiation time), \(u_{k}^{{}}\) is the nonpharmacological policy, \(V_{k}^{{S^{p} }}\) is the vaccination policy, \(a_{21}\), \(a_{22}\), \(a_{23}\), \(a_{24}\), \(a_{25}\) \(b_{2}\), \(c_{2}\) are the parameters.
The next subsection presents the modelling steps of the infected \(I_{k}^{n}\) submodel.
The infected \(I_{k}^{n}\) submodel
Some of the suspicious \(S_{k}^{p}\) individuals becomes infected \(I_{k}^{n}\) where they either become hospitalized \(H_{k}\) or nonhospitalized \(H_{k}^{n}\), who are quarantined for a period of time, as illustrated in Fig. 1. We can formulate its model by considering the corresponding connections in Fig. 4 as
$$I_{k + 1}^{n} =  a_{31} I_{k}^{n} + a_{32} S_{k}^{p}  a_{33} I_{k  1}^{na}  a_{34} H_{k}^{{}}  a_{35} H_{k}^{n}  b_{3} u_{k}  c_{3} V_{k}^{{I^{n} }}$$
(5)
where \(V_{k}^{{I^{n} }}\) is the vaccination policy, \(a_{31}\), \(a_{32}\), \(a_{33}\), \(a_{34}\), \(a_{35}\) \(b_{3}\), \(c_{3}\) are the parameters.
The next subsection introduces the hospitalized \(H_{k}\) submodel.
The hospitalized \(H_{k}\) submodel
Some of the infected \(I_{k}^{n}\) individuals requiring standard treatments join the hospitalized \(H_{k}\) group. The hospitalized \(H_{k}\) individuals can join the intensive care \(I_{k}^{t}\), the intubated \(I_{k}^{b}\), the recovered \(R_{k}\), or the death \(D_{k}\) groups as shown in Fig. 5. We can formulate the hospitalized model as
$$\begin{aligned} H_{k + 1} & =  a_{41} H_{k} + a_{42} I_{k}^{n}  a_{43} I_{k}^{t}  a_{44} I_{k}^{b}  a_{45} R_{k}  a_{46} D_{k} \ldots \\ & \quad \ldots  b_{4} u_{k}  c_{4} V_{k}^{H} \\ \end{aligned}$$
(6)
where \(V_{k}^{H}\) is the vaccination policy, \(a_{41}\), \(a_{42}\), \(a_{43}\), \(a_{44}\), \(a_{45}\), \(a_{46}\), \(b_{4}\), \(c_{4}\) are the parameters,
The next subsection presents the formulation of the intensive care \(I_{k}^{t}\) submodel.
The intensive care \(I_{k}^{t}\) submodel
Some of the hospitalized \(H_{k}\) individuals move to the intensive care \(I_{k}^{t}\) group where some of them move back to the hospitalized \(H_{k}\) group as shown in Fig. 6. Similarly, some of the intensive care \(I_{k}^{t}\) patients become intubated \(I_{k}^{b}\) where some of them rejoin the intensive care \(I_{k}^{t}\) group, and the rest join the death \(D_{k}\) group. We can construct the intensive care \(I_{k}^{t}\) model as
$$I_{k + 1}^{t} =  a_{51} I_{k}^{t} + a_{52} H_{k}  a_{53} I_{k}^{b}  a_{54} D_{k}  b_{5} u_{k}  c_{5} V_{k}^{{I^{t} }}$$
(7)
where \(V_{k}^{{I^{t} }}\) is the vaccination policy, \(a_{51}\), \(a_{52}\), \(a_{53}\), \(b_{5}\), \(c_{5}\) are the parameters.
The next subsection provides the intubated \(I_{k}^{b}\) submodel.
The intubated \(I_{k}^{b}\) submodel
Some of the hospitalized \(H_{k}\) individuals and the intensive care \(I_{k}^{t}\) patients become intubated \(I_{k}^{b}\) as shown in Fig. 7. A number of the intubated \(I_{k}^{b}\) patients move back to the intensive care \(I_{k}^{t}\) unit while the rest join the death \(D_{k}\) group. We can construct the intubated model as
$$I_{k + 1}^{b} =  a_{61} I_{k}^{b} + a_{62} H_{k} + a_{63} I_{k}^{t}  a_{64} D_{k}  b_{6} u_{k}  c_{6} V_{k}^{{I^{b} }}$$
(8)
where \(V_{k}^{{I^{b} }}\) is the vaccination policy, \(a_{61}\), \(a_{62}\), \(a_{63}\), \(a_{64}\) \(b_{6}\), \(c_{6}\) are the parameters.
The next subsection formulates the recovered \(R_{k}\) submodel.
The recovered \(R_{k}\) submodel
A number of the hospitalized \(H_{k}\) and the nonhospitalized \(H_{k}^{n}\) individuals join the recovered \(R_{k}\) group who become nonsusceptible \(S_{k}^{nc}\) as illustrated in Fig. 8. We can formulate the recovered \(R_{k}\) submodel as
$$\begin{aligned} R_{k + 1} & =  a_{71} R_{k} + a_{72} H_{k} + a_{73} H_{k}^{n}  a_{74} S_{k}^{nc} \ldots \\ & \quad \ldots  b_{7} u_{k}  c_{7} V_{k}^{R} \\ \end{aligned}$$
(9)
where \(V_{k}^{R}\) is the vaccination policy, \(a_{71}\), \(a_{72}\), \(a_{73}\), \(a_{74}\) \(b_{7}\), \(c_{7}\) are the parameters.
The next subsection expresses the death submodel.
The death \(D_{k}\) submodel
Some of the hospitalized \(H_{k}\), the intensive care \(I_{k}^{t}\), and the intubated \(I_{k}^{b}\) individuals join the death \(D_{k}\) group and become nonsusceptible \(S_{k}^{nc}\) as illustrated in Fig. 9. We can form the death \(D_{k}\) model as
$$\begin{aligned} D_{k + 1} & =  a_{81} D_{k} + a_{82} H_{k} + a_{83} I_{k}^{t} + a_{84} I_{k}^{b}  a_{85} S_{k}^{nc} \ldots \\ & \quad \ldots  b_{8} u_{k}  c_{8} V_{k}^{D} \\ \end{aligned}$$
(10)
where \(V_{k}^{D}\) is the vaccination policy, \(a_{81}\), \(a_{82}\), \(a_{83}\), \(a_{84}\), \(a_{85}\) \(b_{8}\), \(c_{8}\) are the parameters.
The next subsection formulates the vaccination policy \(V_{k}^{*}\) and reviews the nonpharmacological \(u_{k}\) policies.
The vaccination \(V_{k}^{*}\) and nonpharmacological \(u_{k}\) policies
This section firstly introduces the priority and age specific vaccination policy \(V_{k}^{*}\) and reviews the nonpharmacological policies \(u_{k}\) that we have developed recently for the first time in the literature [2].
The priority and age specific vaccination policies \(V_{k}^{*}\)
The \(*\) in the priority and age specific vaccination policy \(V_{k}^{*}\) represents the \(S^{c}\),\(S^{p}\),\(I^{n}\),\(H\),\(I^{t}\),\(R\), and \(D\) in the submodels given by Eqs. from (4) to (10). The priority and age specific vaccination policy basis \(V_{k}^{b}\) is defined in terms of the number of the daily vaccinated people in each group as
$$V_{k}^{b} = \left[ {\begin{array}{*{20}c} {H_{k}^{s} } & {A_{{^{k} }}^{80 + } } & {A_{k}^{65  79} } & {A_{k}^{50  64} } & {A_{k}^{25  49} } & {A_{k}^{15  24} } \\ \end{array} } \right]^{T}$$
(11)
where \(H_{k}^{s}\) is the healthcare staff, \(A_{k}^{80 + }\) is the people age 80 and over, \(A_{k}^{65  79}\) is the people age between 65 and 79, \(A_{k}^{50  64}\) is the people age between 50 and 64, \(A_{k}^{25  49}\) is the people age between 25 and 49, \(A_{k}^{15  24}\) is the people age between 15 and 24.
Since the people age under 15 are not considered for the vaccination, they are not included in the basis \(V_{k}^{b}\). The corresponding weight parameter vector \(w_{k}^{*}\) scales the contribution of the vaccination policy for each submodel. For example, the weight parameter vector for the hospitalized \(w_{k}^{H}\) is
$$w_{k}^{H} = \left[ {\begin{array}{*{20}c} {w_{k}^{s} } & {w_{k}^{80 + } } & {w_{k}^{65  79} } & {w_{k}^{50  64} } & {w_{k}^{25  49} } & {w_{k}^{15  24} } \\ \end{array} } \right]^{T}$$
(12)
where the parameters of the \(w_{k}^{H}\) are \(w_{k}^{s}\) is the percentage of the hospitalized \(H_{k}^{s}\), \(w_{k}^{80 + }\) is the percentage of the hospitalized \(A_{k}^{80 + }\), \(w_{k}^{65  79}\) is the percentage of the hospitalized \(A_{k}^{65  79}\), \(w_{k}^{50  64}\) is the percentage of the hospitalized \(A_{k}^{50  64}\), \(w_{k}^{25  49}\) is the percentage of the hospitalized \(A_{k}^{25  49}\), \(w_{k}^{15  24}\) is the percentage of the hospitalized \(A_{k}^{15  24}\).
Now we can formulate the priority and age specific vaccination policy for the hospitalized \(V_{k}^{H}\) in Eq. (6) as
$$V_{k}^{H} = w_{k}^{{H}{^{T} }} V_{k}^{b}$$
(13)
Similarly, we can construct the priority and age specific vaccination policy \(V_{k}^{*}\) for the other submodels by following the same steps introduced in this section. The next subsection provides the revised nonpharmacological policies \(u_{k}\).
The nonpharmacological policies \(u_{k}\)
The authorities impose various curfews and restrictions to confine the spread of the virus. The most common ones are the curfews on the people age over 65, age under 20, and people with the chronic diseases which have been parametrized in [2] (since there is no available data) as
$$u_{k}^{s} = n^{s} \left( {1  \alpha^{{k  k_{i} }} + \sigma_{k}^{s} } \right)$$
(14)
where \(u_{k}^{s}\) is the response of the curfew (in closed form solution), \(n^{s}\) is the number of the people under the curfew, \(k\) is the number of the days and \(k_{i}\) is the start day of the curfew, \(\alpha\) is the discount factor of the response, where \(\alpha^{k} \approx 0\) for \(\alpha = 0.71\) and \(k = 14\) (quarantine duration), \(\sigma_{k}^{s}\) is the random nonparametric uncertainty in the response.
The other common precaution is the curfews on the weekends and holidays, which has a transient ascent part as
$$\begin{aligned} u_{i,k}^{wh} & = n^{wh} \left( {1  \alpha^{k  i} + \sigma_{i,k}^{wh} } \right)\delta_{i,k} \ldots \\ & \quad \quad ...{\text{for }}\left\{ {\begin{array}{*{20}l} {\delta_{i,k} = 1} \hfill & {\left\{ {\begin{array}{*{20}c} {{\text{Curfew}}\,{\text{at}}\,ith\,{\text{day}}} \\ {i \le k \le i + 6} \\ \end{array} } \right.} \hfill \\ {\delta_{i,k} = 0} \hfill & {{\text{Otherwise}}} \hfill \\ \end{array} } \right. \\ \end{aligned}$$
(15)
where \(u_{i,k}^{wh}\) is the response of the curfews on the weekends and holidays, \(n^{wh}\) is the number of the people under the curfews on the weekends and holidays, \(\sigma_{i,k}^{wh}\) is the random uncertainty in the response.
Its transient descent part is modelled as
$$\begin{aligned} u_{i,k}^{wh} & = n^{wh} \left( {\alpha^{k  i} + \sigma_{i,k}^{wh} } \right)\delta_{i,k} \ldots \\ & \quad \quad \ldots {\text{for}}\left\{ {\begin{array}{*{20}l} {\delta_{i,k} = 1} \hfill & {\left\{ {\begin{array}{*{20}c} {{\text{Curfew}}\,{\text{at}}\,ith\,{\text{day}}} \\ {i + 7 \le k \le i + 14} \\ \end{array} } \right.} \hfill \\ {\delta_{i,k} = 0} \hfill & {{\text{Otherwise}}} \hfill \\ \end{array} } \right. \\ \end{aligned}$$
(16)
The overall response \(u_{k}^{wh}\) is
$$u_{k}^{wh} = \sum\limits_{i = k  14}^{k} {u_{i,k}^{wh} }$$
(17)
In terms of the closure of the schools and universities, it is not a curfew as it only hinders mass gatherings of the students; hence, they can come together in smaller groups. Therefore, the response has a transient ascent part as in Eq. (15) and transient descent part as in Eq. (16). These parts are essentially for removing the negative impacts of the schools being open. Then an uncertain saturated part \(u_{sat}\) represents the small gatherings after the closure of the schools. After the transient ascent and descent parts, the saturated part can be represented as
$$u_{k}^{su} = n^{su} \left( {u_{sat} + \sigma_{k}^{su} } \right)\quad {\text{for}}\quad k = k_{i}^{su} , \ldots ,k_{n}$$
(18)
where \(n^{su}\) is the number of the students, \(\sigma_{k}^{su}\) is the random uncertainty in the response, \(k_{n} = k_{i}^{su} + k_{n}^{su}\) where \(k_{i}^{su}\) is the start day and \(k_{n}^{su}\) is the duration of the closure.
Comparison of the prediction models
One can summarize the main advantages of the constructed S^{c}S^{p}I^{n}HI^{t}I^{b}RDVN model over the wellknown models such as the SIR, SEIR models in terms of the solution and analysis as

It has difference equations rather than the differential equations. Therefore, it can be solved iteratively without requiring an ordinary differential equation solver.

It has coupled and linear dynamics instead of the slightly coupled nonlinear dynamics. Thus, the mathematical analysis of the parametric model is straightforward.

Its unknown parameters are learned from the reported data by using the wellknown multidimensional optimization approaches rather than the single dimensional statistical approaches.
The next section forms the RLS approach with the inequality constraints to learn the unknown parameters of the S^{c}S^{p}I^{n}HI^{t}I^{b}RDVN model.
The constrained RLS algorithm
In this paper, the constrained optimization is considered for two reasons: The first one is that the submodels have certain parameter structures together with the corresponding parameter signs and the second reason is to reflect the contributions of the data having huge magnitude differences (for example, while the susceptible \(S_{k}^{c}\) group covers millions of the individuals, the hospitalized \(H_{k}\) group covers only thousands of them). In this section, initially we will divide the optimization problem in terms of the estimated submodel casualties (outputs) and the real casualties. Then, the RLS algorithm with the inequality constraints are modified to learn the unknown parameters.
The estimated submodels
We can represent the estimated submodels \(\hat{y}_{k}^{*}\) in terms of the known basis vector \(b_{k}^{*}\) and the unknown parameter vector \(w_{k}^{*}\), where the \(*\) is denoted for the \(S^{c}\),\(S^{p}\),\(I^{n}\),\(H\),\(I^{t}\),\(R\), and \(D\) in the submodels given by Eqs. from (3) to (10) as
$$\hat{y}_{k}^{*} = w_{k}^{{*^{T} }} b_{k}^{*}$$
(19)
For example, the basis \(b_{k}^{{S^{c} }}\) of the estimated susceptible \(\hat{y}_{k}^{{S^{c} }}\) submodel is formed with respect to the left hand side of Eq. (3) as
$$b_{k}^{{S^{c} }} = \left[ {\begin{array}{*{20}c} {  S_{k}^{c} } & {  S_{k}^{p} } & {I_{k}^{nn} } & {  R_{k} } & {  D_{k} } & {  V_{k} } \\ \end{array} } \right]^{T}$$
(20)
And the corresponding unknown parameter vector \(w_{k}^{{S^{c} }}\) of the estimated susceptible \(\hat{y}_{k}^{{S^{c} }}\) submodel with respect to the right hand side of Eq. (3) is
$$w_{k}^{{S^{p} }} = \left[ {\begin{array}{*{20}c} {a_{11} } & {a_{12} } & {a_{13} } & {a_{14} } & {a_{15} } & {c_{1} } \\ \end{array} } \right]$$
(21)
The other estimated submodels, their bases and parameter vectors are formed by following the same procedures as in Eqs. (19), (20), and (21), respectively. The next subsection introduces the modified RLS algorithm with the inequality constraints to learn the unknown parameter vectors \(w_{k}^{*}\).
Learning the Unknown Parameters with the Constrained RLS
The reported casualties are the outputs of the S^{c}S^{p}I^{n}HI^{t}I^{b}RDVN submodels and we call them as the real outputs \(y_{k}^{*}\). For example, the real output of the susceptible submodel is the left hand side of Eq. (3), which is \(S_{k + 1}^{c}\). The objective function is constructed with the 2norm of the instant estimation error defined as
$$\begin{array}{*{20}l} {\mathop {{\text{min}}}\limits_{{w_{k}^{*} }} } \hfill & {\frac{1}{2}\left\ {y_{k}^{*}  \hat{y}_{k}^{*} } \right\_{2} } \hfill \\ {{\text{subject}}\,{\text{to}}} \hfill & {\left\ {w_{k}^{*} } \right\_{2} \le \alpha } \hfill \\ \end{array}$$
(22)
where \(\alpha\) is the inequality constraints which are the lower bound of the parameters. We can construct the Lagrange multipliers used for solving the optimization problems as
$$L\left( {w_{k}^{*} ,\lambda } \right) = \frac{1}{2}\left\ {y_{k}^{*}  w_{k}^{{*^{T} }} b_{k}^{*} } \right\_{2}^{2} + \frac{\lambda }{2}\left( {\left\ {w_{k}^{*} } \right\_{2}^{2}  \alpha_{2}^{2} } \right)$$
(23)
Getting partial derivative of \(L\left( {w_{k}^{*} ,\lambda } \right)\) with respect to the \(w_{k}^{*}\) yields
$$\left( {b_{k}^{{*^{T} }} b_{k}^{*} + \lambda } \right)w_{k}^{*} = b_{k}^{{*^{T} }} y_{k}^{*}$$
(24)
Getting partial derivative of \(L\left( {w_{k}^{*} ,\lambda } \right)\) with respect to the \(\lambda\) gives
$$\left\ {w_{k}^{*} } \right\_{2}^{2}  \alpha^{2} = 0$$
(25)
Reorganizing Eq. (24) as \(w_{k}^{*}\) is on the left and the rest are on the right, and then substituting it in Eq. (25) yields
$$\left( {\frac{{b_{k}^{*} y_{k}^{*} }}{{b_{k}^{{*^{T} }} b_{k}^{*} + \lambda }}} \right)^{2}  \alpha^{2} = 0$$
(26)
The Lagrange multiplier \(\lambda\) from Eq. (26) is obtained as
$$\lambda = \left( {b_{k}^{{*^{T} }} y_{k}^{*}  \alpha b_{k}^{{*^{T} }} b_{k}^{*} } \right)/\alpha$$
(27)
Then by reinserting Eq. (27) into Eq. (24), the unknown parameter vector \(w_{k}^{*}\) can be attained. The next section extensively analyses the S^{c}S^{p}I^{n}HI^{t}I^{b}RDVN model.