
Recently, the area devoted to mathematical epidemiology has attracted much attention. Mathematical formulations have served as models for various infectious diseases. In this regard, mathematical models have also been used to study COVID-19, a threatening disease in present time. This research work is devoted to consider a SEIR (susceptible-exposed-infectious-removed) type mathematical model for investigating COVID-19 alongside a new scenario of fractional calculus. We consider piece-wise fractional order derivatives to investigate the proposed model for qualitative and computational analysis. The results related to the qualitative analysis are studied via using the tools of fixed point approach. In addition, the computational analysis is performed due to a significance of simulation to understand the transmission dynamics of COVID-19 infection in the community. In addition, a numerical scheme based on Newton's polynomials is established to simulate the approximate solutions of the proposed model by using various fractional orders. Additionally, some real data results are also shown in comparison to the numerical results.
Citation: Nadiyah Hussain Alharthi, Mdi Begum Jeelani. Analyzing a SEIR-Type mathematical model of SARS-COVID-19 using piecewise fractional order operators[J]. AIMS Mathematics, 2023, 8(11): 27009-27032. doi: 10.3934/math.20231382
[1] | Ahmed A. El-Deeb, Osama Moaaz, Dumitru Baleanu, Sameh S. Askar . A variety of dynamic α-conformable Steffensen-type inequality on a time scale measure space. AIMS Mathematics, 2022, 7(6): 11382-11398. doi: 10.3934/math.2022635 |
[2] | Mohammed S. El-Khatib, Atta A. K. Abu Hany, Mohammed M. Matar, Manar A. Alqudah, Thabet Abdeljawad . On Cerone's and Bellman's generalization of Steffensen's integral inequality via conformable sense. AIMS Mathematics, 2023, 8(1): 2062-2082. doi: 10.3934/math.2023106 |
[3] | Ahmed A. El-Deeb, Dumitru Baleanu, Nehad Ali Shah, Ahmed Abdeldaim . On some dynamic inequalities of Hilbert's-type on time scales. AIMS Mathematics, 2023, 8(2): 3378-3402. doi: 10.3934/math.2023174 |
[4] | Ahmed A. El-Deeb, Samer D. Makharesh, Sameh S. Askar, Dumitru Baleanu . Bennett-Leindler nabla type inequalities via conformable fractional derivatives on time scales. AIMS Mathematics, 2022, 7(8): 14099-14116. doi: 10.3934/math.2022777 |
[5] | Elkhateeb S. Aly, Y. A. Madani, F. Gassem, A. I. Saied, H. M. Rezk, Wael W. Mohammed . Some dynamic Hardy-type inequalities with negative parameters on time scales nabla calculus. AIMS Mathematics, 2024, 9(2): 5147-5170. doi: 10.3934/math.2024250 |
[6] | Marwa M. Ahmed, Wael S. Hassanein, Marwa Sh. Elsayed, Dumitru Baleanu, Ahmed A. El-Deeb . On Hardy-Hilbert-type inequalities with α-fractional derivatives. AIMS Mathematics, 2023, 8(9): 22097-22111. doi: 10.3934/math.20231126 |
[7] | Bingxian Wang, Mei Xu . Asymptotic behavior of some differential inequalities with mixed delays on time scales and their applications. AIMS Mathematics, 2024, 9(6): 16453-16467. doi: 10.3934/math.2024797 |
[8] | Jian-Mei Shen, Saima Rashid, Muhammad Aslam Noor, Rehana Ashraf, Yu-Ming Chu . Certain novel estimates within fractional calculus theory on time scales. AIMS Mathematics, 2020, 5(6): 6073-6086. doi: 10.3934/math.2020390 |
[9] | Awais Younus, Khizra Bukhsh, Manar A. Alqudah, Thabet Abdeljawad . Generalized exponential function and initial value problem for conformable dynamic equations. AIMS Mathematics, 2022, 7(7): 12050-12076. doi: 10.3934/math.2022670 |
[10] | Haytham M. Rezk, Mohammed Zakarya, Amirah Ayidh I Al-Thaqfan, Maha Ali, Belal A. Glalah . Unveiling new reverse Hilbert-type dynamic inequalities within the framework of Delta calculus on time scales. AIMS Mathematics, 2025, 10(2): 2254-2276. doi: 10.3934/math.2025104 |
Recently, the area devoted to mathematical epidemiology has attracted much attention. Mathematical formulations have served as models for various infectious diseases. In this regard, mathematical models have also been used to study COVID-19, a threatening disease in present time. This research work is devoted to consider a SEIR (susceptible-exposed-infectious-removed) type mathematical model for investigating COVID-19 alongside a new scenario of fractional calculus. We consider piece-wise fractional order derivatives to investigate the proposed model for qualitative and computational analysis. The results related to the qualitative analysis are studied via using the tools of fixed point approach. In addition, the computational analysis is performed due to a significance of simulation to understand the transmission dynamics of COVID-19 infection in the community. In addition, a numerical scheme based on Newton's polynomials is established to simulate the approximate solutions of the proposed model by using various fractional orders. Additionally, some real data results are also shown in comparison to the numerical results.
The Weighted Complementarity Problem (WCP), which is to find a pair of (x,s,y)∈Rn×Rn×Rm such that
x⩾0,s⩾0,xs=w,F(x,s,y)=0, | (1.1) |
where, F:R2n+m→Rn+m is a continuously differentiable function, w∈Rn+ is the given weight vector, xs:=x∘s is the componentwise product of the vectors x and s. When w=0, WCP (1.1) reduces to the classical Nonlinear Complementarity Problem (NCP). At present, there are many effective algorithms [1,2,3,4,5] that can solve NCP. For examples, Newton method [1], Quasi-Newton method [2], L-M method [3,4], Neural-Networks method [5] etc. If
F(x,s,y)=Px+Qs+Ry−a, | (1.2) |
problem (1.1) is the Linear Weighted Complementarity Problem (LWCP) studied in this paper, which is to find a pair of (x,s,y)∈Rn×Rn×Rm such that
x⩾0,s⩾0,xs=w,Px+Qs+Ry=a, | (1.3) |
where, P∈R(n+m)×n,Q∈R(n+m)×n,R∈R(n+m)×m,a∈Rn+m, are given matrices and vector. In addition, when
F(x,s,y)=(∇f(x)−s−ATyAx−b), | (1.4) |
problem (1.1) is the perturbed Karush-Kuhn-Tucker(KKT) condition for the following Nonlinear Programming(NLP)
minf(x),s.t.Ax=b,x≥0. | (1.5) |
Problem(1.3) was introduced by Potra [6] in 2012 and has been widely studied for its important applications in management, market equilibrium, etc. Many equilibrium problems can also be transformed into LWCP to solve, such as the famous Fisher market equilibrium problem [7], and the quadratic programming and weighted center problem [6].
In recent years, many effective algorithms have been proposed to solve problem (1.1) or (1.3) [8,9,10,11,12,13]. For examples, Chi et al. [9] proposed the full-Newton step infeasible interior-point method for solving LWCP. Zhang et al. [12] proposed the smoothing Newton type method for solving LWCP. Tang et al. [13] proposed the nonmonotone L-M method for NWCP. The interior point method depends on the choice of initial value. The classical Newton method needs the positive definite of Hessian matrix, otherwise, it is difficult to guarantee that the Newton direction is descending. The L-M method does not depend on the choice of initial values, nor does it require the positive definiteness of the Hessian matrix. Therefore, this paper mainly considers using L-M method to solve problem (1.3). Motivated by [13], we consider using a nonmonotone L-M method to solve LWCP.
LWCP is a more general complementary model. For the solution of this model, we hope to use the WCP functions obtained by the extension of NCP functions. However, due to the existence of weighting term, not all NCP functions can be directly extended to WCP functions. For NCP function in the form of FB function, many scholars have extended it to WCP function. In this paper, motivated by the smoothed penalty function for [14], we construct a smoothng function for WCP. And then use L-M method to approximate the equivalent reconstruction equations of problem (1.3). The comparison experiment of random generation shows the feasibility and effectiveness of our algorithm.
The following notations will be used throughout this paper. The superscript T denotes transpose. R denotes real numbers, Rn represents the set of all n dimensional real column vectors. The matrix I denotes the identity matrix, and ‖ denotes 2-norm. All vectors in this article are column vectors.
In this section, we study a class of complementary functions with participation weights and discuss its properties. Based on this weighted complementary function, the equivalent reconstruction equations of problem (1.3) are given.
Definition 2.1. For a fixed c \geqslant 0 , a function \phi :{R^2} \to R is called a weighted complementarity function [13], if it satisfies
{\phi ^c}(a, b) = 0 \Leftrightarrow a \geqslant 0, b \geqslant 0, ab = c. | (2.1) |
When c = 0 , {\phi ^c}(a, b) reduces to the NCP function.
In this paper, to solve the LWCP (1.3), we hope to use the WCP functions obtained by the extension of NCP functions. However, due to the existence of weighting term, not all NCP functions can be directly generalized to WCP functions. For example, the two piecewise NCP functions given in [2]:
\phi \left( {a, b} \right) = \left\{ {\begin{array}{*{20}{c}} {3a - \left( {\frac{{{a^2}}}{b}} \right), b \geqslant a > 0, or3b > - a \geqslant 0;} \\ {3a - \left( {\frac{{{b^2}}}{a}} \right), a > b > 0, or3a > - b \geqslant 0;} \\ {9a + 9b, else.} \end{array}} \right. | (2.2) |
\phi \left( {a, b} \right) = \left\{ {\begin{array}{*{20}{c}} {{k^2}a, b \geqslant k\left| a \right|;} \\ {2kb - \left( {\frac{{{b^2}}}{a}} \right), a > \frac{{\left| b \right|}}{k};} \\ {2{k^2}a + 2kb + \left( {\frac{{{b^2}}}{a}} \right), a < - \frac{{\left| b \right|}}{k};} \\ {{k^2}a + 4kb, b \leqslant - k\left| a \right|.} \end{array}} \right. | (2.3) |
For FB function, many scholars have extended it to WCP function. For example, Liu et al. [11] based on the symmetric disturbance FB function in [15] constructed:
{\phi _c}(\mu , a, b) = \left( {1 + \mu } \right)\left( {a + b} \right) - \sqrt {{{\left( {a + \mu b} \right)}^2} + {{\left( {\mu a + b} \right)}^2} + 2c + 2{\mu ^2}} , | (2.4) |
where, c is a given nonnegative vector.
Zhang[12] proposed:
{\phi _\theta }(\mu , a, b, c) = \sqrt {{a^2} + {b^2} - 2\theta ab + 2\left( {1 + \theta } \right)c + 2\mu } - a - b, | (2.5) |
where, \theta \in \left( { - 1, 1} \right], c is a given nonnegative vector.
In addition, [13] provides another smooth function:
\phi _{_{\tau , q}}^c(a, b) = {\left( {a + b} \right)^q} - {\left( {\sqrt {{a^2} + {b^2} + \left( {\tau - 2} \right)ab + \left( {4 - \tau } \right)c} } \right)^q}, | (2.6) |
where, c is a given nonnegative vector, \tau \in \left[ {0, 4} \right) is a constant, q > 1 is an odd integer. Compared with (2.4) and (2.5), (2.6) does not need to introduce the smoothing factor \mu . By controlling the value of q , smoothing can be achieved. This smoothing method will be used to smooth the new WCP function given below.
\phi _\tau ^c(a, b) = a + b - \sqrt {\tau {{(a - b)}^2} + (1 - \tau )({a^2} + {b^2}) + 2(1 + \tau )c} , | (2.7) |
where, c is a given nonnegative vector, \tau \in \left[ {0, 1} \right] is a constant.
Since Eq (2.7) is not smooth, we make the following smoothing treatment:
\phi _{\tau , q}^c(a, b) = {(a + b)^q} - {(\sqrt {\tau {{\left( {a - b} \right)}^2} + \left( {1 - \tau } \right)\left( {{a^2} + {b^2}} \right) + 2(1 + \tau )c} )^q}, | (2.8) |
where, c is a given nonnegative vector, \tau \in \left[ {0, 1} \right] is a constant, q > 1 is an odd integer.
Theorem 2.1. Let \phi _{\tau , q}^c be defined by (2.8) with \tau \in \left[ {0, 1} \right] and q > 1 being a positive odd interger. Then \phi _\tau ^q is a family of WCP functions, i.e.,
\phi _{\tau , q}^c(a, b) = 0 \Leftrightarrow a \geqslant 0, b \geqslant 0, ab = c. | (2.9) |
Proof. Since for any \alpha , \beta \in R and any positive odd interger q , there is {\alpha ^q} = {\beta ^q} \Leftrightarrow \alpha = \beta . So we have
\begin{array}{l} \phi _{\tau , q}^c(a, b) = 0 \Leftrightarrow {(a + b)^q} = {(\sqrt {\tau {{(a - b)}^2} + (1 - \tau )({a^2} + {b^2}) + 2(1 + \tau )c} )^q} \\ \;\; \;\;\;\;\;\;\;\; \Leftrightarrow a + b = \sqrt {\tau {{(a - b)}^2} + (1 - \tau )({a^2} + {b^2}) + 2(1 + \tau )c} \\ \;\; \;\;\;\;\;\;\;\; \Leftrightarrow \phi _\tau ^c(a, b) = 0. \end{array} | (2.10) |
That is to say, we only need to prove that \phi _\tau ^c(a, b) is a family of WCP functions. On the one hand, we fist suppose that \forall a, b \in R satisfy, \phi _\tau ^c(a, b) = 0 i.e.,
\sqrt {\tau {{(a - b)}^2} + (1 - \tau )({a^2} + {b^2}) + 2(1 + \tau )c} = a + b. | (2.11) |
By squaring the two sides of (2.11), we have 2(1 + \tau )ab = 2(1 + \tau )c, which together with \tau \in [0, 1] . yields ab = c. By substituing ab = c into (2.2), we have \sqrt {{a^2} + {b^2} + 2ab} = a + b \geqslant 0. Since c = ab \geqslant 0, it follows that a \geqslant 0, b \geqslant 0. On the other hand, we suppose that a \geqslant 0, b \geqslant 0, ab = c, then a + b \geqslant 0 and
\sqrt {\tau {{(a - b)}^2} + (1 - \tau )({a^2} + {b^2}) + 2(1 + \tau )c} = \sqrt {{a^2} + {b^2} + 2ab} = \left| {a + b} \right| = a + b. | (2.12) |
Which implies that \phi _\tau ^c(a, b) = 0.
Lemma 2.1. Let \phi _{\tau , q}^c be defined by (2.8) with \tau \in [0, 1] and q > 1 being a positive odd interger. Let
h_\tau ^c(a, b) = \sqrt {\tau {{(a - b)}^2} + (1 - \tau )({a^2} + {b^2}) + 2(1 + \tau )c} . | (2.13) |
Then
(ⅰ)When q > 1 , \phi _{\tau , q}^c is continuously differentiable at any \left( {a, b} \right) \in {R^2} with
\nabla \phi _{\tau , q}^c = \left[ {\begin{array}{*{20}{c}} {\frac{{\partial \phi _{\tau , q}^c}}{{\partial a}}} \\ {\frac{{\partial \phi _{\tau , q}^c}}{{\partial b}}} \end{array}} \right], | (2.14) |
where
\begin{array}{l} \frac{{\partial \phi _{\tau , q}^c}}{{\partial a}} = q[{(a + b)^{q - 1}} - h_\tau ^c{(a, b)^{q - 2}}(a - \tau b)], \\ \frac{{\partial \phi _{\tau , q}^c}}{{\partial b}} = q[{(a + b)^{q - 1}} - h_\tau ^c{(a, b)^{q - 2}}(b - \tau a)]. \end{array} |
(ⅱ)When q > 3 , \phi _{\tau , q}^c is twice continuously differentiable at any \left( {a, b} \right) \in {R^2} with
{\nabla ^2}\phi _{\tau , q}^c(a, b) = \left[ {\begin{array}{*{20}{c}} {\frac{{{\partial ^2}\phi _{\tau , q}^c}}{{\partial {a^2}}}}&{\frac{{{\partial ^2}\phi _{\tau , q}^c}}{{\partial a\partial b}}} \\ {\frac{{{\partial ^2}\phi _{\tau , q}^c}}{{\partial b\partial a}}}&{\frac{{{\partial ^2}\phi _{\tau , q}^c}}{{\partial {b^2}}}} \end{array}} \right], | (2.15) |
where
\begin{array}{l} \frac{{{\partial ^2}\phi _\tau ^q}}{{\partial {a^2}}} = q\left\{ {(q - 1){{(a + b)}^{q - 2}} - {h_\tau }{{(a, b, c)}^{q - 4}}[(q - 2){{(a - \tau b)}^2} + {h_\tau }{{(a, b, c)}^2}]} \right\}, \\ \frac{{{\partial ^2}\phi _{\tau , q}^c}}{{\partial {b^2}}} = q\left\{ {(q - 1){{(a + b)}^{q - 2}} - h_\tau ^c{{(a, b)}^{q - 4}}[(q - 2){{(b - \tau a)}^2} + h_\tau ^c{{(a, b)}^2}]} \right\}, \end{array} |
\frac{{{\partial ^2}\phi _{\tau , q}^c}}{{\partial a\partial b}} = \frac{{{\partial ^2}\phi _{\tau , q}^c}}{{\partial b\partial a}} = q\left\{ {(q - 1){{(a + b)}^{q - 2}} - h_\tau ^c{{(a, b)}^{q - 4}}[(q - 2)(a - \tau b)(b - \tau a) - \tau h_\tau ^c{{(a, b)}^2}]} \right\}. |
Lemma 2.2. Let \phi _{\tau , q}^c be defined by (2.8) with \tau \in [0, 1] and q > 1 being a positive odd interger. Defining the closed and convex set \Omega \left( u \right): = \left\{ {u \in {R^2}\left| {\left\| u \right\|} \right. \leqslant \theta } \right\} , where \theta is a positive constant. Then:
(ⅰ)When q > 1 , \phi _{\tau , q}^c is Lipschitz continuous on \Omega \left( u \right) for any \theta > 0 .
(ⅱ)When q > 3 , \nabla \phi _{\tau , q}^c is Lipschitz continuous on \Omega \left( u \right) for any \theta > 0 .
Since \phi _{\tau , q}^c and \nabla \phi _{\tau , q}^c are bounded on the set \Omega \left( u \right) , therefore the conclusion (ⅰ) and (ⅱ) can be obtained from the Mean-Value-Theorem.
Given weight vector w \in R_ + ^n , let z: = (x, s, y) \in {R^{2n + m}} and
H(z) = H(x, s, y): = \left( {\begin{array}{*{20}{c}} {F(x, s, y)} \\ {\Phi _{\tau , q}^w(x, s)} \end{array}} \right), | (2.16) |
where
\Phi _{\tau , q}^w(x, s) = \left( {\begin{array}{*{20}{c}} {\begin{array}{*{20}{c}} {\phi _{\tau , q}^{{w_1}}({x_1}, {s_1})} \\ \vdots \end{array}} \\ {\phi _{\tau , q}^{{w_n}}({x_n}, {s_n})} \end{array}} \right). | (2.17) |
Then the solution of LWCP (1.3) is equivalent to the approximate solution of the system of equations H(z) = 0 .
Lemma2.3. Let H\left( z \right):{R^{2n + m}} \to {R^{2n + m}}, \Phi _{\tau , q}^w:{R^{2n}} \to {R^n} be defined by (2.16) and (2.17), respectively. Then:
(ⅰ) \Phi _{\tau , q}^w\left( {x, s} \right) is continuously differentiable at any z = \left( {x, s, y} \right) \in {R^{2n + m}} .
(ⅱ) H\left( z \right) is continuously differentiable at any z = \left( {x, s, y} \right) \in {R^{2n + m}} with its Jacobian
H'\left( z \right) = \left( {\begin{array}{*{20}{c}} {\begin{array}{*{20}{c}} {{{F'}_x}}&{{{F'}_s}}&{{{F'}_y}} \end{array}} \\ {\begin{array}{*{20}{c}} {{D_1}}&{{D_2}}&0 \end{array}} \end{array}} \right), | (2.18) |
where
\begin{array}{l} {D_1} = diag\left\{ {q\left[ {{{\left( {{x_i} + {s_i}} \right)}^{q - 1}} - h_\tau ^{{w_i}}{{\left( {{x_i}, {s_i}} \right)}^{q - 2}}\left( {{x_i} - \tau {s_i}} \right)} \right]} \right\}, i = 1, 2, \cdots , n. \\ {D_2} = diag\left\{ {q\left[ {{{\left( {{x_i} + {s_i}} \right)}^{q - 1}} - h_\tau ^{{w_i}}{{\left( {{x_i}, {s_i}} \right)}^{q - 2}}\left( {{s_i} - \tau {x_i}} \right)} \right]} \right\}, i = 1, 2, \cdots , n. \\ h_\tau ^{{w_i}}({x_i}, {s_i}) = \sqrt {\tau {{({x_i} - {s_i})}^2} + (1 - \tau )({x_i}^2 + {s_i}^2) + 2(1 + \tau ){w_i}} , i = 1, 2, \cdots , n. \end{array} |
Let H(z) be defined by (2.16), then its value function M:{R^{2n + m}} \to {R_ + } can be defined as:
M(z): = \frac{1}{2}{\Vert H\left(z\right)\Vert }^{2}. | (2.19) |
Obviously, the solution of LWCP (1.3) is also equivalent to the approximate solution of the system of equations M(z) = 0. In addition, the following conclusion can be obtained from the Lemma 2.3.
Lemma 2.4. Let M:{R^{2n + m}} \to {R_ + } be defined by (2.19), then M(z) is continuously differentiable at any z \in {R^{2n + m}} , and \nabla M(z) = H'{\left( z \right)^T}H\left( z \right).
In this section, based on the WCP function in Section 2, we will give the smooth L-M type algorithm and its convergence.
Algorithm3.1 (A smooth L-M method)
Step 0: Choose \theta , \sigma , \gamma , \delta \in \left( {0, 1} \right) and {z^0}: = \left( {{x^0}, {s^0}, {y^0}} \right) \in {R^{2n + m}} , let 0 \leqslant \varepsilon \leqslant 1 , and {C_0} = M\left( {{z^0}} \right) . Choose a sequence \left\{ {{\eta _k}\left| {\forall k \geqslant 0, } \right.{\eta _k} \in \left( {0, 1} \right)} \right\} , set k: = 0.
Step 1: Compute H({z^k}) . If \left\| {H({z^k})} \right\| \leqslant \varepsilon then stop.
Step 2: Let {\mu _k}: = \theta {\left\| {H\left( {{z^k}} \right)} \right\|^2} . Compute the search direction {d_k} \in {R^{2n + m}} by
\nabla M({z^k}) + \left( {{H^{'}}{{({z^k})}^T}{H^{'}}({z^k}) + {\mu _k}I} \right){d_k} = 0. | (3.1) |
Step 3: If {d_k} satisfies
\left\| {H({z^k} + {d_k})} \right\| \leqslant \sigma \left\| {H({z^k})} \right\|. | (3.2) |
Then let {\alpha _k}: = 1 , and go to step 5. Otherwise, go to step 4.
Step 4: Set {j_k} be the smallest nonnegative integer j satisfying
M({z^k} + {\delta ^j}{d_k}) \leqslant {C_k} - \gamma {\left\| {{\delta ^j}{d_k}} \right\|^2}. | (3.3) |
let {\alpha _k}: = {\delta ^{{j_k}}} , and go to step 5.
Step 5: Set {z^{k + 1}}: = {z^k} + {\alpha _k}{d_k} and
{Q_{k + 1}}: = {\eta _k}{Q_k}, {C_{k + 1}}: = \frac{{{\eta _k}{Q_k}{C_k} + M\left( {{z^{k + 1}}} \right)}}{{{Q_{k + 1}}}}. | (3.4) |
Step 6: Let k: = k + 1 , and go to step 1.
Existing L-M type methods [16,17,18] are usually designed based on the Armijo line search. While algorithm 3.1 adopts a nonmonotone derivate free line search. The choice of {\eta _k} controls the degree of nonmonotoicity. If {\eta _k} \equiv 0 , then the line search is monotone.
Theorem3.1. Let \left\{ {{z^k}} \right\} be the sequence generated by Algorithm 3.1. Then, \left\{ {{z^k}} \right\} satisfying M({z^k}) \leqslant {C_k} for all k \geqslant 0 .
Proof. By Algorithm 3.1 {C_0} = M\left( {{z^0}} \right). We first assume that M\left( {{z^k}} \right) \leqslant {C_k} . If \nabla M\left( {{z^k}} \right) = 0, then Algorithm 3.1 terminates. Otherwise \nabla M\left( {{z^k}} \right) \ne 0 which implies that H\left( {{z^k}} \right) \ne 0 , hence {\mu _k} = \theta {\left\| {H\left( {{z^k}} \right)} \right\|^2} > 0 . So the matrix H'{\left( {{z^k}} \right)^T}H'\left( {{z^k}} \right) + {\mu _k}I is positive definite. Thus the search direction {d_k} in step 3 is well-defined and {d_k} \ne 0 . Since \nabla M\left( {{z^k}} \right) \ne 0 , we have
\nabla M{\left( {{z^k}} \right)^T}{d_k} = - {d_k}^T\left( {H'{{\left( {{z^k}} \right)}^T}H'\left( {{z^k}} \right) + {\mu _k}I} \right){d_k} < 0. | (3.5) |
This implies that {d_k} is a descent direction of M\left( {{z^k}} \right) at the point {z^k} . Next we will prove that at least one step size is obtained by step 4. Inversely, we assume that for any j , M\left( {{z^k} + {\delta ^j}{d_k}} \right) > {C_k} - \gamma {\left\| {{\delta ^j}{d_k}} \right\|^2} , then
M\left( {{z^k} + {\delta ^j}{d_k}} \right) > {C_k} - \gamma {\left\| {{\delta ^j}{d_k}} \right\|^2} \geqslant M\left( {{z^k}} \right) - \gamma {\left\| {{\delta ^j}{d_k}} \right\|^2}, | (3.6) |
thereby
\frac{{M\left( {{z^k} + {\delta ^j}{d_k}} \right) - M\left( {{z^k}} \right) + \gamma {{\left\| {{\delta ^j}{d_k}} \right\|}^2}}}{{{\delta ^j}}} > 0. | (3.7) |
By letting j \to \infty in (3.7), we have \nabla M{\left( {{z^k}} \right)^T}{d_k} \geqslant 0 , which contradicts (3.5). Therefore, we can always get {z^{k + 1}} by Step 3 or Step 4. If {z^{k + 1}} is generated by step 3, i.e., \left\| {H\left( {{z^k} + {d_k}} \right)} \right\| \leqslant \sigma \left\| {H\left( {{z^k}} \right)} \right\| , then \frac{1}{2}{\left\| {H\left( {{z^k} + {d_k}} \right)} \right\|^2} \leqslant \frac{1}{2}{\sigma ^2}{\left\| {H\left( {{z^k}} \right)} \right\|^2} , so M\left( {{z^{k + 1}}} \right) \leqslant {\sigma ^2}M\left( {{z^k}} \right) . And because, \sigma \in \left( {0, 1} \right) , therefore, we have M\left( {{z^{k + 1}}} \right) \leqslant {\sigma ^2}M\left( {{z^k}} \right) < M\left( {{z^k}} \right) \leqslant {C_k} . If {z^{k + 1}} is generated by step 4, we can get M\left( {{z^{k + 1}}} \right) \leqslant {C_k} directly. So, from(3.4), we can get that {C_k} \geqslant \frac{{{\eta _k}{Q_k}M\left( {{z^{k + 1}}} \right) + M\left( {{z^{k + 1}}} \right)}}{{{Q_{k + 1}}}} = M\left( {{z^{k + 1}}} \right) . Hence, we conclude that M({z^k}) \leqslant {C_k} for all k \geqslant 0 .
Next, we first suppose that \nabla M\left( {{z^k}} \right) \ne 0 for all k \geqslant 0 . In order to discuss the convergence of algorithm 3.1, we need the following lemma.
Lemma 3.1. Let \left\{ {{z^k}} \right\} be the sequence generated by Algorithm 3.1, then there exists a nonnegative constant {C^ * } such that
\mathop {\lim }\limits_{k \to \infty } M\left( {{z^k}} \right) = \mathop {\lim }\limits_{k \to \infty } {C_k} = {C^ * }. | (3.8) |
Proof. By Theorem3.1, we can get 0 \leqslant M\left( {{z^k}} \right) \leqslant {C_k} for all k \geqslant 0 and {C_{k + 1}} \leqslant \frac{{{\eta _k}{Q_k}{C_k} + {C_k}}}{{{Q_{k + 1}}}} = {C_k}. Hence, by The Monotone Bounded Theorem, there exists a nonnegative constant {C^ * } such that \mathop {\lim }\limits_{k \to \infty } {C_k} = {C^ * } . By the definition of {Q_k} , we have
{Q_{k + 1}} = 1 + \mathop \Sigma \limits_{i = 0}^k \mathop \Pi \limits_{j = 0}^i {\eta _{k - j}} \leqslant 1 + \mathop \Sigma \limits_{i = 0}^k \eta _{\max }^{i + 1} \leqslant \mathop \Sigma \limits_{i = 0}^\infty \eta _{\max }^i = \frac{1}{{1 - {\eta _{\max }}}}. | (3.9) |
Hence, we conclude that {\eta _k}{Q_k} \leqslant \frac{{{\eta _{\max }}}}{{1 - {\eta _{\max }}}} is bounded, which together with \mathop {\lim }\limits_{k \to \infty } {C_k} = {C^ * } yields \mathop {\lim }\limits_{k \to \infty } {\eta _{k - 1}}{Q_{k - 1}}\left( {{C_k} - {C_{k - 1}}} \right) = 0. So, it follows from (3.4) that
\begin{array}{l} M\left( {{z^{k + 1}}} \right) = {Q_{k + 1}}{C_{k + 1}} - {\eta _k}{Q_k}{C_k} = \left( {{\eta _k}{Q_k} + 1} \right){C_{k + 1}} - {\eta _k}{Q_k}{C_k} \\ \;\;\;\;\;\;\;\;\;\;\;\;\;= {\eta _k}{Q_k}\left( {{C_{k + 1}} - {C_k}} \right) + {C_{k + 1}}. \end{array} | (3.10) |
Hence
\mathop {\lim }\limits_{k \to \infty } M\left( {{z^k}} \right) = \mathop {\lim }\limits_{k \to \infty } \left[ {{\eta _{k - 1}}{Q_{k - 1}}\left( {{C_k} - {C_{k - 1}}} \right) + {C_k}} \right] = {C^ * }. | (3.11) |
We complete the proof.
Theorem3.2. Let \left\{ {{z^k}} \right\} be the sequence generated by Algorithm 3.1. Then any accumulation point {z^ * } of \left\{ {{z^k}} \right\} is a stationary point of M\left( z \right) .
Proof. By Lemma 3.1, we have \mathop {\lim }\limits_{k \to \infty } M\left( {{z^k}} \right) = \mathop {\lim }\limits_{k \to \infty } {C_k} = {C^ * }, {C^ * } \geqslant 0 . If {C^ * } = 0 , then \mathop {\lim }\limits_{k \to \infty } H\left( {{z^k}} \right) = 0 which together with Lemma 2.4 yields \nabla M\left( {{z^ * }} \right) = 0 . In the following, we discuss the case of {C^ * } > 0 . Set N: = \left\{ {k\left| {\left\| {H\left( {{z^k} + {d_k}} \right)} \right\| \leqslant \sigma \left\| {H\left( {{z^k}} \right)} \right\|} \right.} \right\} . Then N must be a finite set, otherwise M\left( {{z^{k + 1}}} \right) \leqslant {\sigma ^2}M\left( {{z^k}} \right) holds for infinitely many k . By letting k \to \infty with k \in N , we can have {C^ * } \leqslant {\sigma ^2}{C^ * } and 1 \leqslant {\sigma ^2} which contradicts \sigma \in \left( {0, 1} \right) . Therefore, we can suppose that there exists an index \bar k > 0 such that \left\| {H\left( {{z^k} + {d_k}} \right)} \right\| > \sigma \left\| {H\left( {{z^k}} \right)} \right\| for all k \geqslant \bar k . Thereby, there exists a {j_k} such that M\left( {{z^{k + 1}}} \right) \leqslant {C_k} - \gamma {\left\| {{\delta ^{{j_k}}}{d_k}} \right\|^2} , i.e.,
\gamma {\left\| {{\delta ^{{j_k}}}{d_k}} \right\|^2} \leqslant {C_k} - M\left( {{z^{k + 1}}} \right). | (3.12) |
Next, we suppose that {z^ * } is the limit of the subsequence {\left\{ {{z^k}} \right\}_{k \in K}} \subset \left\{ {{z^k}} \right\} where K \in \left\{ {0, 1, 2, \cdots } \right\} , i.e., \mathop {\lim }\limits_{k\left( { \in K} \right) \to \infty } {z^k} = {z^ * } . Hence, by the continuity, we have {C^ * } = M\left( {{z^ * }} \right) = \frac{1}{2}{\left\| {H\left( {{z^ * }} \right)} \right\|^2} . By \mathop {\lim }\limits_{k \to \infty } {\mu _k} = \mathop {\lim }\limits_{k \to \infty } \theta {\left\| {H\left( {{z^k}} \right)} \right\|^2} = \mathop {\lim }\limits_{k \to \infty } 2\theta M\left( {{z^k}} \right) = 2\theta {C^ * } , we can get that
\mathop {\lim }\limits_{k\left( { \in K} \right) \to \infty } \left[ {H'{{\left( {{z^k}} \right)}^T}H'\left( {{z^k}} \right) + {\mu _k}I} \right] = H'{\left( {{z^ * }} \right)^T}H'\left( {{z^ * }} \right) + 2\theta {C^ * }I. | (3.13) |
According to the proof process of theorem 3.1, the matrix H'{\left( {{z^k}} \right)^T}H'\left( {{z^k}} \right) + {\mu _k}I is a symmetric positive definite matrix. In addition, because of {C^ * } > 0 , the matrix H'{\left( {{z^ * }} \right)^T}H'\left( {{z^ * }} \right) + 2\theta {C^ * }I is also symmetric positive definite matrix. Hence, we have
\mathop {\lim }\limits_{k\left( { \in K} \right) \to \infty } {\left[ {H'{{\left( {{z^k}} \right)}^T}H'\left( {{z^k}} \right) + {\mu _k}I} \right]^{ - 1}} = {\left[ {H'{{\left( {{z^ * }} \right)}^T}H'\left( {{z^ * }} \right) + 2\theta {C^ * }I} \right]^{ - 1}}. | (3.14) |
and
\mathop {\lim }\limits_{k\left( { \in K} \right) \to \infty } {d_k} = {d^ * } = - {\left[ {H'{{\left( {{z^ * }} \right)}^T}H'\left( {{z^ * }} \right) + 2\theta {C^ * }I} \right]^{ - 1}}\nabla M\left( {{z^ * }} \right). | (3.15) |
By (3.5), we can get
\nabla M{\left( {{z^ * }} \right)^T}{d^ * } = \mathop {\lim }\limits_{k\left( { \in K} \right) \to \infty } \nabla M{\left( {{z^k}} \right)^T}{d^k} \leqslant 0. | (3.16) |
By letting k \to \infty with k \in N in (3.12), we have \mathop {\lim }\limits_{k\left( { \in K} \right) \to \infty } \left\| {{\delta ^{{j_k}}}{d_k}} \right\| = 0 . If {\delta ^{{j_k}}} > 0 , then \mathop {\lim }\limits_{k\left( { \in K} \right) \to \infty } {d_k} = {d^ * } = 0 which together with (3.15) yields \nabla M\left( {{z^ * }} \right) = 0 . Otherwise, \mathop {\lim }\limits_{k\left( { \in K} \right) \to \infty } {\delta ^{{j_k}}} = 0 . From step 4 and Theorem 3.1
M({z^k} + {\delta ^{{j_k} - 1}}{d_k}) > {C_k} - \gamma {\left\| {{\delta ^{{j_k} - 1}}{d_k}} \right\|^2} \geqslant M({z^k}) - \gamma {\left\| {{\delta ^{{j_k} - 1}}{d_k}} \right\|^2}, | (3.17) |
i.e.,
\frac{{M({z^k} + {\delta ^{{j_k} - 1}}{d_k}) - M({z^k})}}{{{\delta ^{{j_k} - 1}}}} + \gamma {\left\| {{\delta ^{{j_k} - 1}}{d_k}} \right\|^2} > 0. | (3.18) |
Now that M\left( z \right) is continuously differentiable at {z^ * } , so we have
\nabla M{\left( {{z^ * }} \right)^T}{d^ * } \geqslant 0. | (3.19) |
Then, from (3.16), we can get \nabla M{\left( {{z^ * }} \right)^T}{d^ * } = 0 and
\begin{array}{l} {\left( {{d^ * }} \right)^T}\left( {H'{{\left( {{z^ * }} \right)}^T}H'\left( {{z^ * }} \right) + 2\theta {C^ * }I} \right){d^ * } = - \nabla M{\left( {{z^ * }} \right)^T}{\left[ {H'{{\left( {{z^ * }} \right)}^T}H'\left( {{z^ * }} \right) + 2\theta {C^ * }I} \right]^{ - 1}}\left( {H'{{\left( {{z^ * }} \right)}^T}H'\left( {{z^ * }} \right) + 2\theta {C^ * }I} \right){d^ * } \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\; \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;= - \nabla M{\left( {{z^ * }} \right)^T}{d^ * } = 0. \end{array} |
Since the matrix H'{\left( {{z^ * }} \right)^T}H'\left( {{z^ * }} \right) + 2\theta {C^ * }I is a positive matrix, so we have
{d^ * } = - {\left[ {H'{{\left( {{z^ * }} \right)}^T}H'\left( {{z^ * }} \right) + 2\theta {C^ * }I} \right]^{ - 1}}\nabla M{\left( {{z^ * }} \right)^T} = 0. | (3.20) |
Now that the matrix {\left[ {H'{{\left( {{z^ * }} \right)}^T}H'\left( {{z^ * }} \right) + 2\theta {C^ * }I} \right]^{ - 1}} is also positive matrix, we can get \nabla M\left( {{z^ * }} \right) = 0.
In this section, we carry out some numerical experiments on the LWCP by Algorithm 3.1. All experiments were conducted on a ThinkPad480 with a 1.8GHz CPU and 8.0GB RAM. The codes are run in MATLAB R2018b under Win10.
We first generate the matrices P, Q, R and vector a by following way:
P = \left( {\begin{array}{*{20}{c}} A \\ M \end{array}} \right), Q = \left( {\begin{array}{*{20}{c}} 0 \\ I \end{array}} \right), P = \left( {\begin{array}{*{20}{c}} 0 \\ { - {A^T}} \end{array}} \right), a = \left( {\begin{array}{*{20}{c}} b \\ { - f} \end{array}} \right), | (4.1) |
where A \in {R^{m \times n}} is a full row rank matrix with m < n , the matrix M is an n \times n symmetric semidefinite matrix, b \in {R^m}, f \in {R^n}. In our algorithm we set: \gamma = 0.01, \sigma = 0.5, \delta = 0.8, \theta = {10^{ - 4}}. The initial points are choosing as : {x^0} = \left( {1, \cdots , 1} \right), {s^0} = \left( {1, \cdots , 1} \right), {y^0} = \left( {0, \cdots , 0} \right).
In the course of experiments, we generate LWCP (1.3) by the following two ways.
(ⅰ) We take A = randn\left( {m, n} \right) with rank\left( A \right) = m , and M = \frac{{B{B^T}}}{{\left\| {B{B^T}} \right\|}} with B = rand\left( {n, n} \right) . we first generate \hat x = rand\left( {n, 1} \right), f = rand\left( {n, 1} \right) , then we set \hat b: = A\hat x, \hat s = M\hat x + f, w = \hat x\hat s .
(ⅱ) We choose a = \left( {\begin{array}{*{20}{c}} b \\ { - f} \end{array}} \right) - \xi where \xi \in {R^{n + m}} is a noise. We choose M = diag(v) with v = rand\left( {n, 1} \right) . The matrix A and vectors b, f are generated in the same way as (ⅰ). In the course of experiments, we take \xi = {10^{ - 4}}rand(1, 1)p with p: = {\left( {1, 1, 0, \cdots , 0} \right)^T} \in {R^{n + m}} .
First, in order to observe the local convergence of algorithm 3.1, we conducted two sets of random test experiments on LWCP (ⅰ) with n = 1000, m = 500 . Figure 1 gives the convergence curve of \left\| {H\left( {{z^k}} \right)} \right\| at the k -th iteration. We can clearly see that algorithm 3.1 is locally fast, or at least locally superlinear.
Next, we conducted comparative experiments with [13]. In the course of experiments, the parameters in the WCP functions \phi _{\tau , q}^w are respectively taken as \tau = 0.5, q = 3, \tau = 1, q = 3 and \tau = 0.3, q = 3, \tau = 0.8, q = 3 . The numerical results are presented in Tables 1, 2, Figures 2 and 3 respectively. Where AIT, ACPU, ANH are respectively the average number of iterations, the average CPU time (unit seconds), and the average number \left\| {H\left( {{z^k}} \right)} \right\| of iterations at the end of 10 random experiments. LM represents our experimental result, TLM is the experimental result of [13].
m | n | LM | TLM | ||||
AIT | ACPU | ANH | AIT | ACPU | ANH | ||
200 | 500 | 7.9 | 0.6960 | 5.0131×10-12 | 8.0 | 0.7015 | 6.0903×10-13 |
7.8 | 0.6974 | 5.5794×10-12 | 7.7 | 0.6906 | 1.2630×10-11 | ||
7.6 | 0.6703 | 8.5289×10-12 | 7.9 | 0.7025 | 3.5098×10-13 | ||
400 | 800 | 8.1 | 2.4705 | 5.5548×10-13 | 8.8 | 3.1241 | 5.2707×10-13 |
8.2 | 2.5097 | 7.6171×10-13 | 8.9 | 2.6100 | 2.2961×10-13 | ||
8.2 | 2.6300 | 2.4813×10-13 | 8.1 | 2.4039 | 3.6750×10-12 | ||
500 | 1000 | 8.1 | 4.4569 | 1.2136×10-12 | 8.1 | 4.3590 | 2.2894×10-12 |
8.4 | 4.7697 | 3.1192×10-13 | 8.4 | 4.4993 | 4.5153×10-12 | ||
8.2 | 4.8820 | 2.7039×10-12 | 8.4 | 4.4767 | 9.3738×10-13 | ||
600 | 1500 | 7.9 | 11.2160 | 9.7961×10-12 | 8.0 | 11.6639 | 1.0240×10-12 |
8.0 | 11.4230 | 1.0008×10-13 | 8.0 | 11.6522 | 9.3154×10-13 | ||
8.0 | 11.5575 | 1.0238×10-12 | 7.9 | 11.4497 | 1.0559×10-11 | ||
1000 | 1500 | 9.6 | 18.4934 | 5.6351×10-12 | 9.5 | 18.6699 | 1.6880×10-11 |
9.9 | 19.0396 | 5.2759×10-12 | 11.1 | 21.6384 | 5.9206×10-11 | ||
8.4 | 16.3751 | 1.2735×10-11 | 10.9 | 21.3177 | 7.6313×10-12 |
m | n | LM | TLM | ||||
AIT | ACPU | ANH | AIT | ACPU | ANH | ||
200 | 500 | 7.5 | 0.6642 | 1.5973×10-11 | 8.0 | 0.7155 | 9.7275×10-12 |
7.4 | 0.6675 | 8.8485×10-12 | 8.4 | 0.7429 | 8.7482×10-13 | ||
7.6 | 0.6661 | 2.5321×10-12 | 8.1 | 0.7167 | 6.7422×10-13 | ||
400 | 800 | 8.0 | 2.3642 | 2.4919×10-13 | 8.8 | 2.6212 | 4.6000×10-12 |
8.0 | 2.3791 | 4.5892×10-13 | 8.2 | 2.5740 | 2.4604×10-13 | ||
8.2 | 2.4293 | 9.2368×10-13 | 9.0 | 2.6885 | 1.3216×10-12 | ||
500 | 1000 | 8.0 | 4.3592 | 6.2691×10-13 | 8.3 | 4.5328 | 3.5736×10-12 |
8.1 | 4.3174 | 3.1221×10-13 | 8.2 | 4.3540 | 3.2290×10-13 | ||
7.9 | 4.2089 | 9.7440×10-12 | 9.9 | 5.3469 | 6.5691×10-12 | ||
600 | 1500 | 7.9 | 11.3807 | 9.2057×10-12 | 8.9 | 12.9567 | 9.7825×10-13 |
7.8 | 11.2766 | 1.3435×10-11 | 8.0 | 11.6116 | 9.9437×10-13 | ||
8.0 | 11.5494 | 9.8875×10-13 | 9.2 | 13.3792 | 1.0247×10-12 | ||
1000 | 1500 | 9.3 | 17.6422 | 7.8120×10-12 | 8.9 | 17.2609 | 3.4824×10-12 |
8.7 | 16.3247 | 4.9019×10-12 | 8.8 | 17.3407 | 4.7999×10-11 | ||
9.3 | 18.1968 | 7.8112×10-12 | 9.4 | 18.4024 | 1.3738×10-11 |
Tables 1 and 2 show the numerical results for LWCP (ⅰ). Where, the parameters are taken as \tau = 0.5, q = 3;\tau = 1, q = 3 respectively. It can be seen from the table that no matter what value \tau takes, our algorithm 3.1 has less iteration time or higher accuracy than algorithm 1 in [13].
Figures 2 and 3 show the numerical results for solving LWCP (ⅱ). Where, the parameters are respectively taken as \tau = 0.3, q = 3, m = \frac{n}{2};\tau = 0.8, q = 3, m = \frac{n}{2} . It can be seen from the figure that with the increase of dimension, the AIT of algorithm 3.1 fluctuates slightly, but it is always smaller than the AIT in [13]. The ACPU increases steadily and always smaller than the ACPU in [13].
When \tau = 0.6, q = 3, m = \frac{n}{2} , Figure 4 shows the ACPU and AIT comparison line graphs for LWCP (ⅰ) and LWCP (ⅱ) solved by algorithms 3.1 and [13] respectively. It can be seen from the figure that after adding noise to LWCP (ⅰ), the solution speed of both algorithms decreases, but our algorithm still has certain advantages.
In general, the problems generated by numerical experiments converge in a few iterations. The number of iterations varies slightly with the dimension of the problem. Our algorithm is effective for the linear weighted complementarity problem LWCP (1.3), because each problem can be successfully solved in a very short time with a small number of iterations. Numerical results show the feasibility and effectiveness of the algorithm 3.1.
Based on the idea of L-M method, with the help of a new class of WCP functions {\varphi }_{\tau , q}^{c}(a, b), we give the algorithm 3.1 for solving the LWCP (1.3). Under certain conditions, our algorithm can obtain the approximate solution of LWCP (1.3). Numerical experiments show the feasibility and effectiveness of the algorithm 3.1.
The authors declare no conflicts of interest.
[1] | World Health Organization (WHO), Naming the coronavirus disease (COVID-19) and the virus that causes it, 2020. |
[2] |
D. S. Hui, E. I Azhar, T. A. Madani, F. Ntoumi, R. Kock, O. Dar, et al., The continuing 2019-nCoV epidemic threat of novel coronaviruses to global health-The latest 2019 novel coronavirus outbreak in Wuhan, China, Int. J. Infect. Dis., 91 (2020) 264–266. https://doi.org/10.1016/j.ijid.2020.01.009 doi: 10.1016/j.ijid.2020.01.009
![]() |
[3] |
S. Zhao, Q. Lin, J. Ran, S. S. Musa, G. Yang, W. Wang, et. al, Preliminary estimation of the basic reproduction number of novel coronavirus (2019-nCoV) in China, Int. J. Infect. Dis., 92 (2020), 214–217. https://doi.org/10.1016/j.ijid.2020.01.050 doi: 10.1016/j.ijid.2020.01.050
![]() |
[4] |
S. Zhao, S. S. Musa, Q. Lin, J. Ran, G. Yang, W. Wang, et al., Estimating the unreported number of novel coronavirus (2019-nCoV) cases in China in the first half of January 2020: A data-driven modelling analysis of the early outbreak, J. Clin. Med., 9 (2020), 388. https://doi.org/10.3390/jcm9020388 doi: 10.3390/jcm9020388
![]() |
[5] |
A. Parasher, COVID-19: Current understanding of its pathophysiology, clinical presentation and treatment, Postgrad. Med. J., 97 (2021), 312–320. https://doi.org10.1136/postgradmedj-2020-138577 doi: 10.1136/postgradmedj-2020-138577
![]() |
[6] |
W. M. El-Sadr, A. Vasan, A. El-Mohandes, Facing the new Covid-19 reality, N. Engl. J. Med., 388 (2023), 385–387. https://doi.org/10.1056/NEJMp2213920 doi: 10.1056/NEJMp2213920
![]() |
[7] | I. Nesteruk, Statistics based predictions of coronavirus 2019-nCoV spreading in mainland China, 2020. MedRxiv. |
[8] |
K. Shah, R. U. Din, W. Deebani, P. Kumam, Z. Shah, On nonlinear classical and fractional order dynamical system addressing COVID-19, Results Phys., 24 (2021), 104069. https://doi.org/10.1016/j.rinp.2021.104069 doi: 10.1016/j.rinp.2021.104069
![]() |
[9] |
A. J. Lotka, Contribution to the theory of periodic reactions, J. Phys. Chem., 14 (2002), 271–274. https://doi.org/10.1021/j150111a004 doi: 10.1021/j150111a004
![]() |
[10] |
N. S. Goel, S. C. Maitra, E. W. Montroll, On the Volterra and other nonlinear models of interacting populations, Rev. Mod. phys., 43 (1971), 231–276. https://doi.org/10.1103/RevModPhys.43.231 doi: 10.1103/RevModPhys.43.231
![]() |
[11] |
M. M. Khalsaraei, An improvement on the positivity results for 2-stage explicit Runge-Kutta methods, J. Comput. Appl. Math., 235 (2010), 137–143. https://doi.org/10.1016/j.cam.2010.05.020 doi: 10.1016/j.cam.2010.05.020
![]() |
[12] |
P. Zhou, X. L. Yang, X. G. Wang, B. Hu, L. Zhang, W. Zhang, et al., A pneumonia outbreak associated with a new coronavirus of probable bat origin, Nature, 579 (2020), 270–273. https://doi.org/10.1038/s41586-020-2012-7 doi: 10.1038/s41586-020-2012-7
![]() |
[13] |
Q. Li, X. Guan, P. Wu, X. Wang, L. Zhou, Y. Tong, et al., Early transmission dynamics in Wuhan, China, of novel coronavirus infected pneumonia, N. Engl. J. Med., 382 (2020), 1199–1207. https://doi.org/10.1056/NEJMoa2001316 doi: 10.1056/NEJMoa2001316
![]() |
[14] |
I. I. Bogoch, A. Watts, A. Thomas-Bachli, C. Huber, M. U. G. Kraemer, K. Khan, Pneumonia of unknown aetiology in Wuhan, China: Potential for international spread via commercial air travel, J. Travel Med., 27 (2020), taaa008. https://doi.org/10.1093/jtm/taaa008 doi: 10.1093/jtm/taaa008
![]() |
[15] |
A. B. Gumel, S. Ruan, T. Day, J. Watmough, F. Brauer, P van den Driessche, et al., Modelling strategies for controlling SARS out breaks, Proc. Biol. Sci., 271 (2004), 2223–2232. https://doi.org/10.1098/rspb.2004.2800 doi: 10.1098/rspb.2004.2800
![]() |
[16] |
R. Kahn, I. Holmdahl, S. Reddy, J. Jernigan, M. J. Mina, R. B. Slayton, Mathematical modeling to inform vaccination strategies and testing approaches for coronavirus disease 2019 (COVID-19) in nursing homes, Clin. Infect. Dis., 74 (2022), 597–603. https://doi.org/10.1093/cid/ciab517 doi: 10.1093/cid/ciab517
![]() |
[17] |
J. Mondal, S. Khajanchi, Mathematical modeling and optimal intervention strategies of the COVID-19 outbreak, Nonlinear Dyn., 2022 (2022), 177–202. https://doi.org/10.1007/s11071-022-07235-7 doi: 10.1007/s11071-022-07235-7
![]() |
[18] |
A. I. Abioye, O. J. Peter, H. A. Ogunseye, F. A. Oguntolu, T. A. Ayoola, A. O. Oladapo, A fractional-order mathematical model for malaria and COVID-19 co-infection dynamics. Healthc. Anal., 4 (2023), 100210. https://doi.org/10.1016/j.health.2023.100210 doi: 10.1016/j.health.2023.100210
![]() |
[19] |
J. T. Wu, K. Leung, G. M. Leung, Nowcasting and forecasting the potential domestic and international spread of the 2019-nCoV outbreak originating in Wuhan, China: A modelling study, Lancet, 395 (2020), 689–697. https://doi.org/10.1016/S0140-6736(20)30260-9 doi: 10.1016/S0140-6736(20)30260-9
![]() |
[20] |
J. T. Machado, V. Kiryakova, F. Mainardi, Recent history of fractional calculus, Commun. Nonlinear Sci. Numer. Simul., 16 (2011), 1140–1153. https://doi.org/10.1016/j.cnsns.2010.05.027 doi: 10.1016/j.cnsns.2010.05.027
![]() |
[21] |
F. C. Meral, T. J. Royston, R. Magin, Fractional calculus in viscoelasticity: An experimental study, Commun. Nonlinear Sci. Numer. Simul., 15 (2010), 939–945. https://doi.org/10.1016/j.cnsns.2009.05.004 doi: 10.1016/j.cnsns.2009.05.004
![]() |
[22] |
R. L. Magin, Fractional calculus in bioengineering, Crit. Rev. Biomed. Eng., 32 (2004), 1–104. https://doi.org/10.1615/critrevbiomedeng.v32.i1.10 doi: 10.1615/critrevbiomedeng.v32.i1.10
![]() |
[23] | M. Dalir, M. Bashour, Applications of fractional calculus, Appl. Math. Sci., 4 (2010), 1021–1032. |
[24] | R. L. Magin, Fractional calculus in bioengineering, Redding: Begell House, 2006. |
[25] |
Y. A. Rossikhin, M. V. Shitikova, Applications of fractional calculus to dynamic problems of linear and nonlinear hereditary mechanics of solids, Appl. Mech. Rev., 50 (1997), 15–67. https://doi.org/10.1115/1.3101682 doi: 10.1115/1.3101682
![]() |
[26] | R. Gorenflo, F. Mainardi, Fractional calculus. In: Fractals and fractional calculus in continuum mechanics, Vienna: Springer, 1997. https://doi.org/10.1007/978-3-7091-2664-6 |
[27] |
E. Addai, A. Adeniji, O. J. Peter, J. O Agbaje, K. Oshinubi, Dynamics of age-structure smoking models with government intervention coverage under fractal-fractional order derivatives, Fractal Fract., 7 (2023), 370. https://doi.org/10.3390/fractalfract7050370 doi: 10.3390/fractalfract7050370
![]() |
[28] |
M. Shimizu, W. Zhang, Fractional calculus approach to dynamic problems of viscoelastic materials. JSME Int. J. Ser. C Mech. Syst. Mach. Elem. Manuf., 42 (1999), 825–837. https://doi.org/10.1299/jsmec.42.825 doi: 10.1299/jsmec.42.825
![]() |
[29] |
F. Mainardi, An historical perspective on fractional calculus in linear viscoelasticity, Fract. Calc. Appl. Anal., 15 (2012), 712–717. https://doi.org/10.2478/s13540-012-0048-6 doi: 10.2478/s13540-012-0048-6
![]() |
[30] |
Z. Dai, Y. Peng, H. A. Mansy, R. H. Sandler, T. J Royston, A model of lung parenchyma stress relaxation using fractional viscoelasticity, Med. Eng. Phys., 37 (2015), 752–758. https://doi.org/10.1016/j.medengphy.2015.05.003 doi: 10.1016/j.medengphy.2015.05.003
![]() |
[31] |
M. M. Amirian, Y. Jamali, The concepts and applications of fractional order differential calculus in modeling of viscoelastic systems: A primer, Crit. Rev. Biomed. Eng., 47 (2019), 249–276. https://doi.org/10.1615/CritRevBiomedEng.2018028368 doi: 10.1615/CritRevBiomedEng.2018028368
![]() |
[32] |
H. Khan, J. F. Gómez-Aguilar, A. Alkhazzan, A. Khan, A fractional order HIV-TB coinfection model with nonsingular Mittag-Leffler law, Math. Method. Appl. Sci., 43 (2020), 3786–3806. https://doi.org/10.1002/mma.6155 doi: 10.1002/mma.6155
![]() |
[33] |
C. Celauro, C. Fecarotti, A. Pirrotta, A. C. Collop, Experimental validation of a fractional model for creep/recovery testing of asphalt mixtures, Constr. Build. Mater., 36 (2012), 458–466. https://doi.org/10.1016/j.conbuildmat.2012.04.028 doi: 10.1016/j.conbuildmat.2012.04.028
![]() |
[34] |
G. C. Wu, M. Luo, L. L. Huang, S. Banerjee, Short memory fractional differential equations for new memristor and neural network design, Nonlinear Dyn., 100 (2020), 3611–3623. https://doi.org/10.1007/s11071-020-05572-z doi: 10.1007/s11071-020-05572-z
![]() |
[35] | A. Atangana, D. Baleanu, New fractional derivatives with non-local and non-singular kernel, 2016. arXiv: 1602.03408. |
[36] |
E. F. D. Goufo, Application of the Caputo-Fabrizio fractional derivative without singular kernel to Korteweg-de Vries-Burgers equation, Math. Model. Anal., 21 (2016), 188–198. https://doi.org/10.3846/13926292.2016.1145607 doi: 10.3846/13926292.2016.1145607
![]() |
[37] |
E. F. D. Goufo, A biomathematical view on the fractional dynamics of cellulose degradation, Fract. Calc. Appl. Anal., 18 (2015), 554–564. https://doi.org/10.1515/fca-2015-0034 doi: 10.1515/fca-2015-0034
![]() |
[38] |
M. B. Jeelani, Stability and computational analysis of COVID-19 using a higher order galerkin time discretization scheme, Adv. Appl. Stat., 86 (2023), 167–206. https://doi.org/10.17654/0972361723022 doi: 10.17654/0972361723022
![]() |
[39] |
A. Al Elaiw, F. Hafeez, M. B. Jeelani, M. Awadalla, K. Abuasbeh, Existence and uniqueness results for mixed derivative involving fractional operators, AIMS Mathematics, 8 (2023), 7377–7393. https://doi.org/10.3934/math.2023371 doi: 10.3934/math.2023371
![]() |
[40] |
S. K. Kabunga, E. F. D. Goufo, V. H. Tuong. Analysis and simulation of a mathematical model of tuberculosis transmission in democratic Republic of the Congo, Adv. Differ. Equ., 2020 (2020), 642. https://doi.org/10.1186/s13662-020-03091-0 doi: 10.1186/s13662-020-03091-0
![]() |
[41] |
A. Atangana, S. I. Araz, Mathematical model of COVID-19 spread in Turkey and South Africa: Theory, methods and applications, Adv. Differ. Equ., 2020 (2020), 659. https://doi.org/10.1186/s13662-020-03095-w doi: 10.1186/s13662-020-03095-w
![]() |
[42] |
A. Atangana, S. I. Araz, New concept in calculus: Piecewise differential and integral operators, Chaos Soliton. Fract., 145 (2021), 110638. https://doi.org/10.1016/j.chaos.2020.110638 doi: 10.1016/j.chaos.2020.110638
![]() |
[43] |
M. A. Khan, A. Atangana, Modeling the dynamics of novel coronavirus (2019-nCov) with fractional derivative, Alex. Eng. J., 59 (2020), 2379–2389. https://doi.org/10.1016/j.aej.2020.02.033 doi: 10.1016/j.aej.2020.02.033
![]() |
[44] |
M. A. Khan, A. Atangana, E. Alzahrani, Fatmawati, The dynamics of COVID-19 with quarantined and isolation, Adv. Differ. Equ., 2020 (2020), 425. https://doi.org/10.1186/s13662-020-02882-9 doi: 10.1186/s13662-020-02882-9
![]() |
[45] |
O. Dyer, Covid-19: China stops counting cases as models predict a million or more deaths, BMJ, 380 (2023), 2. https://doi.org/10.1136/bmj.p2 doi: 10.1136/bmj.p2
![]() |
[46] |
A. Moumen, R. Shafqat, A. Alsinai, H. Boulares, M. Cancan, M. B. Jeelani, Analysis of fractional stochastic evolution equations by using Hilfer derivative of finite approximate controllability, AIMS Mathematics, 8 (2023), 16094–16114. https://doi.org/10.3934/math.2023821 doi: 10.3934/math.2023821
![]() |
[47] |
A. Zeb, A. Atangana, Z. A. Khan, S. Djillali, A robust study of a piecewise fractional order COVID-19 mathematical model, Alex. Eng. J., 61 (2022), 5649–5665. https://doi.org/10.1016/j.aej.2021.11.039 doi: 10.1016/j.aej.2021.11.039
![]() |
[48] |
C. Y. Li, J. Yin, A pedestrian-based model for simulating COVID-19 transmission on college campus, Transportmetrica A, 19 (2023), 2005182. https://doi.org/10.1080/23249935.2021.2005182 doi: 10.1080/23249935.2021.2005182
![]() |
[49] |
M. S. Arshad, D. Baleanu, M. B. Riaz, M. Abbas, A novel 2-stage fractional Runge-Kutta method for a time fractional logistic growth model, Discrete Dyn. Nat. Soc., 2020 (2020), 1020472. https://doi.org/10.1155/2020/1020472 doi: 10.1155/2020/1020472
![]() |
[50] |
F. Liu, K. Burrage, Novel techniques in parameter estimation for fractional dynamical models arising from biological systems, Comput. Math. Appl., 62 (2011), 822–833. https://doi.org/10.1016/j.camwa.2011.03.002 doi: 10.1016/j.camwa.2011.03.002
![]() |
[51] | M. T. Hoang, O. F. Egbelowo, Dynamics of a fractional-order hepatitis B epidemic model and its solutions by nonstandard numerical schemes, In: Mathematical Modelling and Analysis of Infectious Diseases, Springer, Cham, 302 (2020), 127–153. https://doi.org/10.1007/978-3-030-49896-2_5 |
[52] |
Z. J. Fu, Z. C. Tang, H. T. Zhao, P. W. Li, T. Rabczuk, Numerical solutions of the coupled unsteady nonlinear convection-diffusion equations based on generalized finite difference method, Eur. Phys. J. Plus, 134 (2019), 272. https://doi.org/10.1140/epjp/i2019-12786-7 doi: 10.1140/epjp/i2019-12786-7
![]() |
[53] |
B. Wang, L. Li, Y. Wang, An efficient nonstandard finite difference scheme for chaotic fractional-order Chen system, IEEE Access, 8 (2020), 98410–98421. https://doi.org/10.1109/ACCESS.2020.2996271 doi: 10.1109/ACCESS.2020.2996271
![]() |
[54] |
A. J. Arenas, G. González-Parra, B. M. Chen-Charpentier, Construction of nonstandard finite difference schemes for the SI and SIR epidemic models of fractional order, Math. Comput. Simulat., 121 (2016), 48–63. https://doi.org/10.1016/j.matcom.2015.09.001 doi: 10.1016/j.matcom.2015.09.001
![]() |
[55] |
R. Lewandowski, Z. Pawlak, Dynamic analysis of frames with viscoelastic dampers modelled by rheological models with fractional derivatives, J. Sound Vib., 330 (2011), 923–936. https://doi.org/10.1016/j.jsv.2010.09.017 doi: 10.1016/j.jsv.2010.09.017
![]() |
[56] | Pakistan population (LIVE), Available from: https://www.worldometers.info/world-population/pakistan-population/. |
[57] | Pakistan COVID-19 corona tracker, Available from: https://www.coronatracker.com/country/pakistan/. |
[58] | Current information about COVID-19 in Pakistan, Available from: https://www.worldometers.info/. |
[59] |
K. Shah, T. Abdeljawad, R. Ud Din, To study the transmission dynamic of SARS-CoV-2 using nonlinear saturated incidence rate, Physica A, 604 (2022), 127915. https://doi.org/10.1016/j.physa.2022.127915 doi: 10.1016/j.physa.2022.127915
![]() |
[60] |
R. Ouncharoen, K. Shah, R. Ud Din, T. Abdeljawad, A. Ahmadian, S. Salahshour, et al., Study of integer and fractional order COVID-19 mathematical model, Fractals, 31 (2023), 2340046. https://doi.org/10.1142/S0218348X23400467 doi: 10.1142/S0218348X23400467
![]() |
m | n | LM | TLM | ||||
AIT | ACPU | ANH | AIT | ACPU | ANH | ||
200 | 500 | 7.9 | 0.6960 | 5.0131×10-12 | 8.0 | 0.7015 | 6.0903×10-13 |
7.8 | 0.6974 | 5.5794×10-12 | 7.7 | 0.6906 | 1.2630×10-11 | ||
7.6 | 0.6703 | 8.5289×10-12 | 7.9 | 0.7025 | 3.5098×10-13 | ||
400 | 800 | 8.1 | 2.4705 | 5.5548×10-13 | 8.8 | 3.1241 | 5.2707×10-13 |
8.2 | 2.5097 | 7.6171×10-13 | 8.9 | 2.6100 | 2.2961×10-13 | ||
8.2 | 2.6300 | 2.4813×10-13 | 8.1 | 2.4039 | 3.6750×10-12 | ||
500 | 1000 | 8.1 | 4.4569 | 1.2136×10-12 | 8.1 | 4.3590 | 2.2894×10-12 |
8.4 | 4.7697 | 3.1192×10-13 | 8.4 | 4.4993 | 4.5153×10-12 | ||
8.2 | 4.8820 | 2.7039×10-12 | 8.4 | 4.4767 | 9.3738×10-13 | ||
600 | 1500 | 7.9 | 11.2160 | 9.7961×10-12 | 8.0 | 11.6639 | 1.0240×10-12 |
8.0 | 11.4230 | 1.0008×10-13 | 8.0 | 11.6522 | 9.3154×10-13 | ||
8.0 | 11.5575 | 1.0238×10-12 | 7.9 | 11.4497 | 1.0559×10-11 | ||
1000 | 1500 | 9.6 | 18.4934 | 5.6351×10-12 | 9.5 | 18.6699 | 1.6880×10-11 |
9.9 | 19.0396 | 5.2759×10-12 | 11.1 | 21.6384 | 5.9206×10-11 | ||
8.4 | 16.3751 | 1.2735×10-11 | 10.9 | 21.3177 | 7.6313×10-12 |
m | n | LM | TLM | ||||
AIT | ACPU | ANH | AIT | ACPU | ANH | ||
200 | 500 | 7.5 | 0.6642 | 1.5973×10-11 | 8.0 | 0.7155 | 9.7275×10-12 |
7.4 | 0.6675 | 8.8485×10-12 | 8.4 | 0.7429 | 8.7482×10-13 | ||
7.6 | 0.6661 | 2.5321×10-12 | 8.1 | 0.7167 | 6.7422×10-13 | ||
400 | 800 | 8.0 | 2.3642 | 2.4919×10-13 | 8.8 | 2.6212 | 4.6000×10-12 |
8.0 | 2.3791 | 4.5892×10-13 | 8.2 | 2.5740 | 2.4604×10-13 | ||
8.2 | 2.4293 | 9.2368×10-13 | 9.0 | 2.6885 | 1.3216×10-12 | ||
500 | 1000 | 8.0 | 4.3592 | 6.2691×10-13 | 8.3 | 4.5328 | 3.5736×10-12 |
8.1 | 4.3174 | 3.1221×10-13 | 8.2 | 4.3540 | 3.2290×10-13 | ||
7.9 | 4.2089 | 9.7440×10-12 | 9.9 | 5.3469 | 6.5691×10-12 | ||
600 | 1500 | 7.9 | 11.3807 | 9.2057×10-12 | 8.9 | 12.9567 | 9.7825×10-13 |
7.8 | 11.2766 | 1.3435×10-11 | 8.0 | 11.6116 | 9.9437×10-13 | ||
8.0 | 11.5494 | 9.8875×10-13 | 9.2 | 13.3792 | 1.0247×10-12 | ||
1000 | 1500 | 9.3 | 17.6422 | 7.8120×10-12 | 8.9 | 17.2609 | 3.4824×10-12 |
8.7 | 16.3247 | 4.9019×10-12 | 8.8 | 17.3407 | 4.7999×10-11 | ||
9.3 | 18.1968 | 7.8112×10-12 | 9.4 | 18.4024 | 1.3738×10-11 |
m | n | LM | TLM | ||||
AIT | ACPU | ANH | AIT | ACPU | ANH | ||
200 | 500 | 7.9 | 0.6960 | 5.0131×10-12 | 8.0 | 0.7015 | 6.0903×10-13 |
7.8 | 0.6974 | 5.5794×10-12 | 7.7 | 0.6906 | 1.2630×10-11 | ||
7.6 | 0.6703 | 8.5289×10-12 | 7.9 | 0.7025 | 3.5098×10-13 | ||
400 | 800 | 8.1 | 2.4705 | 5.5548×10-13 | 8.8 | 3.1241 | 5.2707×10-13 |
8.2 | 2.5097 | 7.6171×10-13 | 8.9 | 2.6100 | 2.2961×10-13 | ||
8.2 | 2.6300 | 2.4813×10-13 | 8.1 | 2.4039 | 3.6750×10-12 | ||
500 | 1000 | 8.1 | 4.4569 | 1.2136×10-12 | 8.1 | 4.3590 | 2.2894×10-12 |
8.4 | 4.7697 | 3.1192×10-13 | 8.4 | 4.4993 | 4.5153×10-12 | ||
8.2 | 4.8820 | 2.7039×10-12 | 8.4 | 4.4767 | 9.3738×10-13 | ||
600 | 1500 | 7.9 | 11.2160 | 9.7961×10-12 | 8.0 | 11.6639 | 1.0240×10-12 |
8.0 | 11.4230 | 1.0008×10-13 | 8.0 | 11.6522 | 9.3154×10-13 | ||
8.0 | 11.5575 | 1.0238×10-12 | 7.9 | 11.4497 | 1.0559×10-11 | ||
1000 | 1500 | 9.6 | 18.4934 | 5.6351×10-12 | 9.5 | 18.6699 | 1.6880×10-11 |
9.9 | 19.0396 | 5.2759×10-12 | 11.1 | 21.6384 | 5.9206×10-11 | ||
8.4 | 16.3751 | 1.2735×10-11 | 10.9 | 21.3177 | 7.6313×10-12 |
m | n | LM | TLM | ||||
AIT | ACPU | ANH | AIT | ACPU | ANH | ||
200 | 500 | 7.5 | 0.6642 | 1.5973×10-11 | 8.0 | 0.7155 | 9.7275×10-12 |
7.4 | 0.6675 | 8.8485×10-12 | 8.4 | 0.7429 | 8.7482×10-13 | ||
7.6 | 0.6661 | 2.5321×10-12 | 8.1 | 0.7167 | 6.7422×10-13 | ||
400 | 800 | 8.0 | 2.3642 | 2.4919×10-13 | 8.8 | 2.6212 | 4.6000×10-12 |
8.0 | 2.3791 | 4.5892×10-13 | 8.2 | 2.5740 | 2.4604×10-13 | ||
8.2 | 2.4293 | 9.2368×10-13 | 9.0 | 2.6885 | 1.3216×10-12 | ||
500 | 1000 | 8.0 | 4.3592 | 6.2691×10-13 | 8.3 | 4.5328 | 3.5736×10-12 |
8.1 | 4.3174 | 3.1221×10-13 | 8.2 | 4.3540 | 3.2290×10-13 | ||
7.9 | 4.2089 | 9.7440×10-12 | 9.9 | 5.3469 | 6.5691×10-12 | ||
600 | 1500 | 7.9 | 11.3807 | 9.2057×10-12 | 8.9 | 12.9567 | 9.7825×10-13 |
7.8 | 11.2766 | 1.3435×10-11 | 8.0 | 11.6116 | 9.9437×10-13 | ||
8.0 | 11.5494 | 9.8875×10-13 | 9.2 | 13.3792 | 1.0247×10-12 | ||
1000 | 1500 | 9.3 | 17.6422 | 7.8120×10-12 | 8.9 | 17.2609 | 3.4824×10-12 |
8.7 | 16.3247 | 4.9019×10-12 | 8.8 | 17.3407 | 4.7999×10-11 | ||
9.3 | 18.1968 | 7.8112×10-12 | 9.4 | 18.4024 | 1.3738×10-11 |