Evolutionary optimized Padé approximation scheme for analysis of covid-19 model with crowding effect

Abstract

This work presents a novel evolutionary computation-based Padé approximation (EPA) scheme for constructing a closed-form approximate solution of a nonlinear dynamical model of Covid-19 disease with a crowding effect that is a growing trend in epidemiological modeling. In the proposed framework of the EPA scheme, the crowding effect-driven system is transformed to an equivalent nonlinear global optimization problem by assimilating Padé rational functions. The initial conditions, boundedness, and positivity of the solution are dealt with as problem constraints. Keeping in view the complexity of formulated optimization problem, a hybrid of differential evolution (DE) and a convergent variant of the Nelder-Mead Simplex algorithm is also proposed to obtain a reliable, optimal solution. The comparison of the EPA scheme results reveals that optimization results of all formulated optimization problems for the Covid-19 model with crowding effect are better than those of several modern metaheuristics. EPA-based solutions of the Covid-19 model with crowding effect are in good agreement with those of a well-practiced nonstandard finite difference (NSFD) scheme. The proposed EPA scheme is less sensitive to step lengths and converges to true equilibrium points unconditionally.

1. Literature survey

Bats were the source of Covid-19, which SARS-nCoV2 caused. SARS-CoV1 and MERS-CoV1 are similar viruses. Its genetic material, RNA, is encased in a lipid-layered surface with spike proteins that resemble the shape of a crown, hence the name corona. In December 2019, it was first seen in Wuhan, China. In February 2020, China, Italy, Iran, and South Korea were present. Then, by the middle of March 2020, it had spread all over the planet. Since then, it has appeared in waves in different parts of the world due to significantly modified forms. The elderly, immune-compromised, diabetics, cancer patients, and heart patients are the most vulnerable. It takes 4–14 days to incubate. The majorities of cases are mild, with symptoms such as fever, illness, and cough. A cytokine storm can induce shortness of breath, body aches, nausea, pneumonia, respiratory failure, and death in severe circumstances. It spreads through the inhalation of deadly virus respiratory droplets produced by infected people and breathed by adjacent people. To detect suspected patients, we employ PCR tests, and to see suspected recovered patients; we use antibody tests. Distancing, masking, hand washing, lockdown, and mass vaccination are still being used to manage the disease. Since its inception, vaccination has shown promising effects. Nonetheless, it is proving to be the only reliable way out of the problem.

Daniel Bernoulli developed the first mathematical epidemic model in 1760 to assess the efficacy of the smallpox vaccine. Modeling nowadays incorporates continual advancements in computational tools and illness data. Understanding disease dynamics, evaluating viable control techniques, and predicting future outbreaks are all aided by this information. They help to flatten the curve of the Covid-19 pandemic in the country by predicting disease patterns. The mathematical modeling of biological processes allows epidemiological assumptions to be revised. As a result, we can put our grasp of disease epidemiology to the test by comparing model findings to observed trends.

In 2020, Zeb et al. proposed a SEIQR mathematical model to control Covid-19 propagation via optimal control techniques [1]. Recently, in 2021, several complex epidemiological models have been submitted. For example, Riyapan et al. presented a SEIIQRD mathematical model of Covid-19 in Bangkok, which explains the dynamical analysis of the proposed model [2]. Oud et al. suggested a SEIAQHRM mathematical model for Covid-19 to study the quarantine effect, which describes how curable the vaccination for the dreadful virus or which strategy may be adopted during the vaccination [3]. Shaikh et al. presented a SEIARQ model of the coronavirus disease to check its outbreak in India to explain the strain of the virus through mathematical modeling [4]. Ahmed et al. presented a SEQIR model of coronavirus to study the relationship between symptomatic and asymptomatic classes [5]. In 2020, Ullah et al. introduced a SEIIQIIR model to comprehend the use of different drugs as vaccinations [6]. Peter et al. (in 2021) presented a SEIQR model to estimate the damages caused by a coronavirus in Pakistan [7]. Recently, Nazir et al. studied a SEIARW model to numerically analyze the effectiveness of the Euler method [8]. Kyrychko et al. presented a SEIARD model to calculate the propagation and role of agents in spreading coronavirus in Ukraine [9]. Khoshnaw suggested that medical awareness and keeping social distances were more effective in controlling the spread of coronaviruses [10]. Sasmita et al. presented a model that considers different subpopulations interactions for control of Covid-19 [11]. In 2020, Tiwar et al. predicted the importance of lockdown for controlling the spread of Covid-19 disease in India through the SEIRD model [12]. Wang et al., in 2021, proposed a SEIRD model which incorporates the different ways for the spread of disease [13]. Baek et al. proposed the SEIR model for controlling the spread of Covid-19 via quarantine and hospitalization [14]. Ardila et al. suggested that isolation and staying away are the key factors preventing the spread of Covid-19 [15]. Peter et al. presented the susceptible (S), Exposed (E), infected (I), Quarantine (Q), carriers (C), and recovered (R) model for controlling the disease in Nigeria [16]. In 2020, Moyles et al. proposed that social distancing is highly effective in preventing the spread of infection [17]. Ko et al. suggested the correlation of SEIR-HCD and SEIR-D model for the control of disease in Moscow [18]. Recently, Harjule has indicated the importance of modeling in controlling the spread of Covid-19 [19]. In 2020, Kim et al. proposed that staying home, keeping distance and early detection are the core points in controlling the spread of Covid-19 [20]. Some well-known models related to Covid-19 with different mathematical strategies are presented in the literature referred to in this study [21], [22], [23], [24], [25], [26], [27], [28], [29].

Modeling differential equations as equivalent optimization problems [30], [31], [32], [33], [34], [35], [36], [37], [38] is one of the most modern techniques to obtain reliable solutions, especially when the traditional method lacks handling one or more aspects of the physical phenomena. The related examples of such modeling include solving nonlinear ordinary differential equations, inverse partial differential equations, and epidemiological models. Ali et al. [37, 38] proposed a novel evolutionary Padé approximation (EPA) scheme for solving epidemiological models and nonlinear partial differential equations. Padé rational approximation functions are a vital research area due to their tremendous applications in several scientific areas [39, 40]. Henri Eugene Padé [41] established suitable Padé approximations for series expansion of functions even where the Taylor series failed to converge. In past studies, Padé approximations were employed to enhance the precision of the solution only obtained by other master methods based on power series [42], differential transform [43], homotopy analysis [44], and Adomian decomposition [45].

The Padé approximation-based formulated optimization problem from differential equations may involve complex non-convexity, multimodality, and non-differentiability. Traditional mathematical programming approaches are not capable of solving such complex problems. On the other hand, evolutionary algorithms are capable of solving almost all kinds of optimization problems. They belong to a general class of optimization techniques known as metaheuristics. Metaheuristics are designed by imitating some natural procedures like flashing behavior of fireflies [46], student-teacher learning procedure [47], swarming [48], evolution [49, 50], water dynamics [51], [52], [53], sport strategies [54], food foraging behavior [55], animals’ hunting [56], etc. For more detailed studies, one may consult the survey article by Alexandros and Georgios [57].

Unfortunately, the theoretical authentication of metaheuristics is still deficient. Past studies [46, 58] have highlighted some severe challenges in the applications of metaheuristics. Despite their successful applications to very diverse problems, there are No Free Lunch (NFL) theorems [58] which reveal that one cannot guarantee that a method effective for one set of trials will be equally efficient for the other collection of problems. Keeping the complexity of the underlying problem in view, hybridization is an effective technique to improve the performance of the selected global search optimizer. As far as the present study is concerned, we have chosen a widely used classic version [59] of the worldwide search Differential Evolution (DE) algorithm [49] in combination with a convergent variant [60] of the Nelder-Mead Simplex (NMS) method [61] to obtain better results.

This work presents the first EPA scheme-based formulation and solution of the Covid-19 disease dynamical model with crowding effect as an optimization problem to the best of our knowledge. In the original proposal [37, 38] of the EPA scheme, the focus on optimization was minimal, and the underlying model was comparatively less complex. But the present study adds some new aspects to the theory of the EPA scheme for solving epidemiological models with crowding effect. The architecture of the EPA scheme consists of the following novel features that were lacking in the previous studies.

• (i)The impact of variation in the order of Padé approximants for state variables on resulting cost function has been studied.
• (ii)A comparative study of the performance of several modern metaheuristics on formulated equivalent unconstrained optimization problems has been conducted, and the essence of the proposed hybrid evolutionary optimizer has been elaborated.
• (iii)The strength of mathematical programming and metaheuristics for solving complex epidemiological models has been highlighted.
• (iv)The EPA scheme preserves all vital properties like positivity, boundedness, feasibility, convergence to equilibrium points, etc., by enforcing the solution to satisfy related problem constraints.
• (v)The EPA scheme's convergence speed is high compared to a well-practiced NonstandardNonstandard Finite Difference (NSFD) [62] structure-preserving method.

The remainder of the manuscript is organized as follows: Section 2 presents the formulation of the Covid-19 model with crowing effect along with stability analysis of equilibrium points. Section 3 consists of a detailed procedure of all components of the EPA scheme for solving the Covid-19 model with the crowing effect. Section 4 presents a rigorous analysis of the results of the EPA scheme in many aspects of reliability, efficiency, and structure-preserving ability. The concluding remarks are also presented in the end.

2. Model formulation

With the help of the theory of population dynamics, we consider the whole population is denoted by$N$, and subpopulations are defined as Susceptible humans $S\left(t\right)$, Infected humans $I\left(t\right),$ and Recovered humans $R\left(t\right).$ Furthermore, the fixed values of the model are defined as: $\mu :($recruitment rate /mortality rate of humans), $\beta :$ (the force of infection), $\frac{1}{1+\alpha I}$:(represents the crowding effect in a population), and $\gamma :$ (the rate of recovery due to immunity or vaccination). The dreadful situation will occur when the "α" increases continuously. Eventually, the rate of active cases is elevated. Hence, there is a direct relationship between the crowding factor and active patients of coronavirus. Here arises the inevitability of controlling the parameter α so that the dynamics of coronavirus disease may be governed. From the mathematical modeling point of view, we achieve the following coupled nonlinear dynamical system:

$$\frac{dS}{dt}=\mu N-\frac{\beta SI}{1+\alpha I}-\mu S$$ (1)

$$\frac{dI}{dt}=\frac{\beta SI}{1+\alpha I}--\left(\gamma +\mu \right)I$$ (2)

$$\frac{dR}{dt}=\gamma I--\mu R$$ (3)

2.1. Analysis of model

2.1.1. Positivity and boundedness

The feasible region of the system (1–3) as follows:

$$H=\left\{\left(S,I,R\right)ϵ{\mathbb{R}}_{+}^3:\mathrm{S}+\mathrm{I}+\mathrm{R}=N,S\ge 0,\mathrm{I}\ge 0,R\ge 0\right\}$$

For this purpose, we will investigate the following results.

Theorem 1

The solutions$\left(S,I,R\right)ϵ{\mathbb{R}}_{+}^3$of the system (1–3) are positive at any time t ≥ 0, with given non-negative initial conditions.

Proof: It is clear from the system (1–3) as follows:

$$\frac{dS}{dt}{|}_{S=0}=\mu N\ge 0,\frac{dI}{\text{dt}}{|}_{I=0}=0\ge 0,\frac{dR}{\text{dt}}{|}_{R=0}=\gamma I\ge 0.$$

Theorem 2

The solutions$\left(S,I,R\right)ϵ{\mathbb{R}}_{+}^3$of the system (1–3) are bounded.

Proof: Let us consider the population function as follows:

$$N\left(\mathrm{t}\right)=S+\mathrm{I}+\mathrm{R}$$

$$\frac{dN}{dt}=\frac{dS}{dt}+\frac{dI}{dt}+\frac{dR}{dt}$$

$$\frac{dN}{dt}=\mu N-\mu \left(S+I+R\right),S+I+R=N$$

$$\frac{dN}{dt}=\mu N-\mu N$$

$$\frac{dN}{dt}=0.$$

Since the total population is fixed, the system's solution (1–3) is bounded and lies in the feasible region $\mathrm{H}$.

2.1.2. Model equilibria

The system (1–3) admits two types of equilibria as follows:

Disease free equilibrium (DFE) $E_0=\left(N,0,0\right)$.

Endemic equilibrium (EE)$E_{*}=\left(S_{*},I_{*},R_{*}\right)$.

$$S^{*}=a_1+a_2{I}_{*}{I}_{*}=\frac{-A_2+\sqrt{A_2^2-4A_1{A}_3}}{2A_1},R^{*}=\frac{\gamma I_{*}}{\mu },$$

where, $a_1=\frac{\left(\gamma +\mu \right)}{\beta }$, $a_2=\frac{\alpha \left(\gamma +\mu \right)}{\beta }$, $A_2=$+ $(\beta a_1$ + $\mu (a_1$ + $a_2)\alpha -\mu N\alpha )$, $A_1=a_2(\beta$ + $\alpha$)$,A_3=(\mu a_1$ $--\mu N)$.

Note that, the reproduction number of the model ${\mathcal{R}}_0$ = $\rho$(F$V^{-1}$) = $\frac{\beta N}{\gamma +\mu }$.

2.2. Local stability analysis of the model

Theorem 3

The system (1–3) is locally stable related to the virus-free equilibrium point$E_0$, if${\mathcal{R}}_0<1$and unstable if${\mathcal{R}}_0$$>$1.

Proof: For local stability, the Jacobian of the system (1–3) is as follows:

$$j=\left(\begin{array}{ccc}-\mu -\frac{\beta SI}{1+\alpha I}& -\frac{\beta S}{{\left(1+\alpha I\right)}^2}& 0\\ \frac{\beta SI}{1+\alpha I}& \frac{\beta S}{{\left(1+\alpha I\right)}^2}-\left(\mu +\gamma \right)& 0\\ 0& \gamma & -\mu \end{array}\right).$$

At $E_0,theJacobianbecomes$

$$j\left(E_0\right)=\left(\begin{array}{ccc}-\mu & -\beta N& 0\\ 0& \beta N-\left(\gamma +\mu \right)& 0\\ 0& \gamma & -\mu \end{array}\right).$$

These are the following eigen values as

$${\lambda }_1=-\mu <0,{\lambda }_2=\beta N-\left(\gamma +\mu \right)<0,\text{if}{\mathcal{R}}_0<1,\text{if}{\lambda }_3=-\mu <0.$$

Thus, point $E_0$ is locally asymptotically stable.

Theorem 4

For${\mathcal{R}}_0>$1, the system (1–3) at the positive endemic equilibrium$E_{*}$of the system (1–3) is locally stable.

Proof: The Jacobean matrix of system (1–3) is as follows:

$$j=\left(\begin{array}{ccc}-\mu -\frac{\beta I}{1+\alpha I}& -\frac{\beta S}{{\left(1+\alpha I\right)}^2}& 0\\ \frac{\beta I}{1+\alpha I}& \frac{\beta S}{{\left(1+\alpha I\right)}^2}-\left(\mu +\gamma \right)& 0\\ 0& \gamma & -\mu \end{array}\right).$$

At $E_{*},thejacobianbecomes$

$$j\left(E_{*}\right)=\left(\begin{array}{ccc}-\mu -\frac{\beta I_{*}}{1+\alpha I_{*}}& -\frac{\beta S_{*}}{{\left(1+\alpha I_{*}\right)}^2}& 0\\ \frac{\beta I_{*}}{1+\alpha I_{*}}& \frac{\beta S_{*}}{{\left(1+\alpha I_{*}\right)}^2}-\left(\mu +\gamma \right)& 0\\ 0& \gamma & -\mu \end{array}\right).$$

Which yields one eigenvalue $\lambda$ = $-\mu <0$ and the characteristics equation

$${\lambda }^2+\left(\mu +\frac{\beta I_{*}}{1+\alpha I_{*}}--\frac{\beta S_{*}}{{\left(1+\alpha I_{*}\right)}^2}+\left(\mu +\gamma \right)\right)\lambda +\left(\mu +\frac{\beta I_{*}}{1+\alpha I_{*}}\right)\left(\frac{\beta S_{*}}{{\left(1+\alpha I_{*}\right)}^2}+\left(\mu +\gamma \right)\right)+\left(\frac{\beta S_{*}}{{\left(1+\alpha I_{*}\right)}^2}\right)\left(\frac{\beta I_{*}}{1+\alpha I_{*}}\right)=0.$$

It is clear, for ${\mathcal{R}}_0$ $>$ 1, and by using Routh Hurwitz criteria for 2nd order

$(\mu +\frac{\beta I_{*}}{1+\alpha I_{*}}$$-\frac{\beta S_{*}}{{\left(1+\alpha I_{*}\right)}^2}$+ $\left(\mu +\gamma \right)$) = ($\mu +\frac{\beta I_{*}}{1+\alpha I_{*}}-\frac{\left(\mu +\gamma \right)}{{\left(1+\alpha I_{*}\right)}^2}$ + $\left(\mu +\gamma \right))>0$.

And

$$\mu +\frac{\beta I_{*}}{1+\alpha I_{*}})\left(\frac{\beta S_{*}}{{\left(1+\alpha I_{*}\right)}^2}+\left(\mu +\gamma \right)\right)+\left(\frac{\beta S_{*}}{{\left(1+\alpha I_{*}\right)}^2}\right)\left(\frac{\beta I_{*}}{1+\alpha I_{*}}\right)>0.$$

Hence, the system (1–3) is locally stable at $E_{*}$for${\mathcal{R}}_0$ $>$ 1. The proof is complete.

3. Evolutionary Padé approximation (EPA) scheme for covid-19 model with crowding effect

This section is dedicated to the architecture of the EPA scheme for solving the approximate model. The first component of the proposed scheme constructs an equivalent optimization problem by modeling the coupled system of governing equations described in (1), (2), (3) using the Padé approximation. The second component executes the optimization task by implementing a classical DE algorithm [59]. To obtain global optimum solutions of the formulated problem, DE is further hybridized with a non-stagnated Nelder-Mead Simplex (NS-NMS) algorithm [60] that is a convergent variant of a simplex search based Nelder-Mead method [61]. The unconstrained minimization objective function $\psi \left(\mathit{x}\right)$ is highly nonlinear and may contain discontinuities and several local optima.

Fig. 1.

Fig. 1.

Flow diagram of Covid-19 model with crowding effect [29].

Moreover, as the orders $\left(M_v,N_v\right)$ of Padé approximations increase, the curse of dimensionality involves. In the presence of such challenges, the use of an efficient and reliable optimization algorithm is complementary. We use the DE algorithm, which has earned very high ranks in the optimization community due to its tremendous success in almost all optimization problems.

3.1. Formulation of equivalent optimization problem

Formulation of equivalent optimization problem consists of four key factors that are (i) approximating the state variables by Padé approximations (ii) conversion of each governing equation to a residual function (iii) handling the initial conditions as problem constraints (iv) use of penalty functions for obtaining unconstrained global optimization problem. The global optimum solution of the resulting unconstrained optimization problem is analogous to the solution of the Covid-19 model with a crowding effect.

• (I)Approximation by Padé rational functions

First, we approximate the state variables $S\left(t\right)$, $I\left(t\right)$, and $R\left(t\right)$ by Padé rational functions of appropriate orders. For any $v\in \left\{S,I,\mathrm{R}\right\}$, let $A_v\left(t\right)$ and $B_v\left(t\right)$ denote polynomial functions of variable $t$ having degrees $M_v$ and $N_v$, respectively. Then Padé rational approximation functions of orders $\left(M_v,N_v\right)\in {\mathbb{W}}^2$ [37, 38] for state variables are:

$$S\left(t\right)\approx p_S\left(t\right)=\frac{A_S\left(t\right)}{B_S\left(t\right)},I\left(t\right)\approx p_I\left(t\right)=\frac{A_I\left(t\right)}{B_I\left(t\right)},R\left(t\right)\approx p_R\left(t\right)=\frac{A_R\left(t\right)}{B_R\left(t\right)}.$$

Here $A_v\left(t\right)$ and $B_v\left(t\right)$are polynomial functions with real coefficients $a_{vj}$, $b_{vj}$ for each $j\in \mathbb{W}$ defined by:

$$A_v\left(t\right)=\sum _{j=0}^{M_v}{a}_{vj}{t}^j;B_v\left(t\right)=\sum _{j=0}^{N_v}{b}_{vj}{t}^j.$$

The derivative of $p_v\left(t\right)$ with respect to $t$ can be expressed as under:

$$p{\prime }_v\left(t\right)=\frac{{A\prime }_v\left(t\right)-p_v\left(t\right){B\prime }_v\left(t\right)}{B_v\left(t\right)}\forall v\in \left\{S,I,R\right\}$$

• (I)Formation of residual functions

The domain of $t$ for the Covid-19 model is $\left[0,\infty \right)$. For some positive integer $K$, let us consider a monotone strictly increasing sequence $\left\{t_0,t_1,t_2,t_3,\dots ,t_K\right\}$ as a subset of the domain $\left[0,\infty \right)$. The proposed scheme does not depend on step length to create such a sequence, but any appropriate step length can be used for ease of implementation. Let us present the unknown coefficients of Padé approximation functions for each variable $v\in \left\{S,I,R\right\}$ by a point ${\mathit{x}}_v=\left(a_{v0},a_{v1},\dots ,a_{v{M}_v},b_{v0},b_{v1},b_{v2},\dots ,b_{v{M}_v}\right)\in {\mathbb{R}}^{M_v+N_v+2}$, then there is a total number of $n=M_S+N_S+M_I+N_I+M_R+N_R+6$ of unknown coefficients. Representing the corresponding approximate solutions and their derivatives by $p_{vr},p{\prime }_{vr}$ at $t_r\in \left\{t_0,t_1,t_2,t_3,\dots ,t_K\right\}$ and then substituting in (1), (2), (3) we get a system of $3\times \left(K+1\right)$ nonlinear algebraic equations. Let us define real-valued residual functions ${\epsilon }_S\left(\mathit{x},t\right),{\epsilon }_I\left(\mathit{x},t\right)$ and${\epsilon }_R\left(\mathit{x},t\right)$ of vector $\mathit{x}=\left({\mathit{x}}_S,{\mathit{x}}_I,{\mathit{x}}_R\right)\in {\mathbb{R}}^n$ of unknown Padé approximation coefficients at any time $t\in \left[0,\infty \right)$. From (1–3), at each $t_r\in \left\{t_0,t_1,t_2,t_3,\dots ,t_K\right\}$ we get the following residuals from:

$${\epsilon }_S\left(\mathit{x},t_r\right)=\frac{{A\prime }_{Sr}-p_{Sr}{B\prime }_{Sr}}{B_{Sr}}-\mu N+\frac{\beta A_{Sr}{A}_{Ir}}{B_{Sr}\left(B_{Ir}+\alpha A_{Ir}\right)}+\mu \frac{A_{Sr}}{B_{Sr}}$$ (4)

$${\epsilon }_I\left(\mathit{x},t_r\right)=\frac{{A\prime }_{Ir}-p_{Ir}{B\prime }_{Ir}}{B_{Ir}}-\frac{\beta A_{Sr}{A}_{Ir}}{B_{Sr}\left(B_{Ir}+\alpha A_{Ir}\right)}+\left(\gamma +\mu \right)\frac{A_{Ir}}{B_{Ir}}$$ (5)

$${\epsilon }_R\left(\mathit{x},t_r\right)=\frac{{A\prime }_{Rr}-p_{Rr}{B\prime }_{Rr}}{B_{Rr}}-\gamma \frac{A_{Ir}}{B_{Ir}}+\mu \frac{A_{Sr}}{B_{Sr}}$$ (6)

• (I)Handling initial conditions as problem constraints

The initial conditions and the physical restrictions give rise to the following equality and non-negativity constraints functions.

$$H_S\left(\mathit{x}\right)=p_S\left(t_0\right)-S_0=0$$ (7)

$$H_I\left(\mathit{x}\right)=p_I\left(t_0\right)-I_0=0$$ (8)

$$H_R\left(\mathit{x}\right)=p_R\left(t_0\right)-R_0=0$$ (9)

$$g_{vr}\left(\mathit{x}\right)=p_v\left(t_r\right)\ge 0,\forall v\in \left\{S,I,R\right\}\text{and}{t}_r\in \left\{t_0,t_1,t_2,t_3,\dots ,t_K\right\}.$$ (10)

• (I)Formulation of unconstrained global optimization problem

The Padé approximation-based mathematical modeling of the underlying problem aims to find optimized decision vector $\mathit{x}=\left({\mathit{x}}_S,{\mathit{x}}_I,{\mathit{x}}_R\right)\in {\mathbb{R}}^n$ of unknown coefficients so that the residuals at each discrete time step are minimized by meeting all the problem constraints. It can be observed from (7), (8), (9), (10) that the residual functions are highly nonlinear and hence lead to a highly nonlinear optimization problem. Due to the high complexity of the resulting optimization problem, an efficient optimizer is needed to be used. Evolutionary algorithms are modern methods that are believed to be less sensitive to the nature of the underlying problem, but they require the problem to be unconstrained. For this purpose, we use the penalty function approach to embed problem constraints into the objective function. Let $L$ be a large positive real number and $P\left(\mathit{x}\right)$ be the function of the following relation.

$$P\left(\mathit{x}\right)=\max \left\{0,\left|H_S\left(\mathit{x}\right)\right|,\left|H_I\left(\mathit{x}\right)\right|,\left|H_R\left(\mathit{x}\right)\right|,-g_S\left(\mathit{x}\right),-g_I\left(\mathit{x}\right),-g_R\left(\mathit{x}\right)\right\}.$$

Then the following unconstrained minimization problem is obtained

$$Minimize\psi \left(\mathit{x}\right)=\frac{1}{3\left(K+1\right)}\sum _{r=0}^K\left(\sum _{v\in \left\{S,I,R\right\}}{\left[{\epsilon }_v\left(\mathit{x},t_r\right)\right]}^2\right)+L\times P\left(\mathit{x}\right).$$ (11)

3.2. Differential evolution (DE) global search optimizer

DE algorithm [49] is a population-based random search algorithm that starts with a randomly generated set of several solutions known as population. DE algorithm tries to improve the current population by developing new solutions through mutation and crossover evolutionary operations. The individual of the population with the smallest objective function value is termed as the best solution. The iteration process of the DE algorithm continues until a preset criterion is satisfied. The standard DE algorithm uses only three parameters to work that are (i) the population size $\kappa \in \mathbb{N}$ (ii) crossover rate $c_r\in \left[0,1\right]$ and (iii) differential weight $\zeta >0$. Geometric view of DE operations is shown in Fig. 2 (a). Several studies describe the classic DE algorithm, also referred to as DE/rand/1/bin, in precise and smarter forms. We have implemented the traditional DE algorithm as Price, Storn and Lampinen explained in a detailed study [59] of the DE algorithm. To minimize a function of the form $\psi :{\mathbb{R}}^n\to \mathbb{R}$, the sequence of operations of classic DE algorithm [59] is given below:

Fig. 2.

(a) Geometry of DE algorithm in ${\mathbb{R}}^2$ (b) Schematic flowchart of NMS method.

1. Initialization

• •The parameters $\kappa$, $c_r$ and $\zeta$ are set. A positive integer $T_{max}$ as a maximum number of iterations is also fixed. The iteration counter $\tau$ is initiated as $\tau =0$.
• •A population of $\kappa$ solutions $\left\{{\mathit{x}}_j=\left(x_j^1,x_j^2,x_j^3,\dots ,x_j^n\right)\in {\mathbb{R}}^n;1\le j\le \kappa \right\}$ is generated randomly.
• •The $\kappa$solutions are evaluated i. e. ${\psi }_j=\psi \left({\mathit{x}}_j\right)\forall j=1,2,3,\dots ,\kappa$. The best solution ${\mathit{x}}^{best}=arg\left(\min \left\{{\psi }_j:1\le j\le \kappa \right\}\right)$ is preserved.

2. Iterative process

• •Set $\tau =\tau +1$.
• •For each target solution ${\mathit{x}}_j$, $j\in \left\{1,2,3,\dots ,\kappa \right\}$, choose mutually distinct indices$j_1,j_2,j_3\in \left\{1,2,3,\dots ,\kappa \right\}\setminus \left\{j\right\}$.
• •Construct mutant vector $\mathit{u}$ using differential mutation according as $\mathit{u}={\mathit{x}}_{j_1}+\zeta \times \left({\mathit{x}}_{j_2}-{\mathit{x}}_{j_3}\right)$.
• •Apply uniform/discrete crossover as:

For each dimension $i=1,2,3,\dots ,n$, choose a random integer $j_{rand}\in \left\{1,2,3,\dots ,n\right\}$ and generate a uniformly distributed random number $i_{rand}\in \left(0,1\right)$ to find a trial solution $\mathit{y}=\left(y^i,y^2,y^3,\dots ,y^n\right)\in {\mathbb{R}}^n$ with coordinates:

$$y^i=\{\begin{array}{c}{u}^i,if{i}_{rand}\le c_{r}ori=j_{rand}\\ x_j^i,otherwise.\end{array}$$

If $\psi \left(\mathit{y}\right)>{\psi }_j$ discard $\mathit{y}$ otherwise${\mathit{x}}_j\leftarrow \mathit{y}$ and ${\psi }_j\leftarrow \psi \left(\mathit{y}\right)$.

• •Update the best solution as: If ${\psi }_j\le \psi \left({\mathit{x}}^{best}\right)$ then ${\mathit{x}}^{best}\leftarrow {\mathit{x}}_j$.

3. Termination criterion

If $\tau <T_{max},$then repeat step 2; otherwise, display the best solution and stop.

4. Optimum results

Display ${\mathit{x}}^{best}$ and $\psi \left({\mathit{x}}^{best}\right)$ as the optimum solution to the problem.

3.3. Nelder-Mead simplex (NMS) local optimizer

Nelder-Mead Simplex (NMS) algorithm [61] is a well-practiced gradient-free optimization method. For an $n$-dimensional problem, the NMS method starts with an initial $n$-simplex having non-zero volume. An $n$-simplex is a convex hull of points ${\mathit{x}}_1,{\mathit{x}}_2,\dots ,{\mathit{x}}_{n+1}$ in ${\mathbb{R}}^n$. The vertices are ordered from the best to the worst according to as $\psi \left({\mathit{x}}_1\right)\le \psi \left({\mathit{x}}_2\right)\le \psi \left({\mathit{x}}_2\right)\le \dots \le \psi \left({\mathit{x}}_{n+1}\right)$. At each iteration, the centroid $\mathit{G}=\frac{1}{n}\sum _{j=1}^n{\mathit{x}}_j$ of all the non-worst vertices is computed. Then the operations of reflection, expansion, and contractions about $\mathit{G}$ are used to replace the worst vertex. In case of failure, the simplex is squeezed towards the best vertex using shrink operation. The equations of four NMS operations are given below.

• (i)Reflection: $\mathit{y}={\mathit{x}}_{n+1}+\alpha \times \left(\mathit{G}-{\mathit{x}}_{n+1}\right)$.
• (ii)Expansion: $\mathit{y}={\mathit{x}}_{n+1}+\beta \times \left(\mathit{G}-{\mathit{x}}_{n+1}\right)$.
• (iii)Contraction outside: $\mathit{y}={\mathit{x}}_{n+1}+\gamma \times \left(\mathit{G}-{\mathit{x}}_{n+1}\right)$.

Contraction inside: $\mathit{y}={\mathit{x}}_{n+1}-\gamma \times \left(\mathit{G}-{\mathit{x}}_{n+1}\right)$.

• (i)Shrink: ${\mathit{x}}_j={\mathit{x}}_j+\delta \times \left({\mathit{x}}_1-{\mathit{x}}_j\right)\forall j=2,3,\dots ,n+1$.

The typical values of these parameters in the original study [61] are $\alpha =2,\beta =3,\gamma =\delta =0.5$. We use a non-stagnated convergent variant of the NMS method proposed by Ali et al. [60]. The order of operations of the NMS method is outlined as a flowchart presented in Fig. 2(b). The complete schematic framework of the proposed EPA scheme for the underlying model is presented in Fig. 3 .

Fig. 3.

Schematic flowchart of the proposed EPA scheme.

4. Results and discussion

This section consists of essential components of the current work. The principal objective of the present study is to obtain the semi-analytical solution of the considered model with high accuracy and preserve vital properties of the dynamics of the model. It is imperative to investigate the EPA scheme's accuracy, efficiency, and reliability when applied to the considered model. Therefore, in the following subsections, we conduct a twofold study of the solution obtained by the EPA scheme. Firstly, we analyze the optimization results of our proposed EPA scheme. Secondly, we compare the solutions obtained by the EPA scheme with a well-practiced nonstandard finite difference (NSFD) scheme to highlight the prominent aspects of the proposed method.

4.1. Analysis of optimization results

We observe that the global minimum value of the objective function $\psi \left(\mathit{x}\right)$ defined by Eq. (11) is zero, which corresponds to a global optimal point ${\mathit{x}}^{*}$. Keeping in mind the positivity of residual functions ${\epsilon }_S\left(\mathit{x},t\right),{\epsilon }_I\left(\mathit{x},t\right),{\epsilon }_R\left(\mathit{x},t\right)$ and the penalty function $P\left(\mathit{x}\right)$ for all $t$ and $\mathit{x}$, we have that:

$$\psi \left({\mathit{x}}^{*}\right)=0\iff {\epsilon }_v\left({\mathit{x}}^{*},t\right)=0\forall v\in \left\{S,I,R\right\}\text{and}P\left({\mathit{x}}^{*}\right)=0.$$

The equation ${\epsilon }_v\left({\mathit{x}}^{*},t\right)=0$ implies that ${\mathit{x}}^{*}$ is the exact solution of the governing equation of $v\in \left\{S,I,R\right\}$, whereas $P\left({\mathit{x}}^{*}\right)=0$ guarantees that all initial conditions and the mentioned structure-preserving properties of the model are accurately satisfied.

The proposed EPA scheme tries to find the approximate solution, say ${\mathit{x}}^{**}$, of the model with acceptable accuracy. Hence the objective of the optimization process is to minimize $\psi \left(\mathit{x}\right)$ as much as possible by ensuring that all problems constraints are satisfied. To optimize the formulated problem by NMS-DE, the parameters used in the proposed EPA scheme have been presented in Table 1 .

Table 1.

EPA parameters.

Parameter	Description	Value	Reference
$\kappa$	Population size for DE algorithm	50	[49, 59]
$c_r$	The crossover acceptance rate of DE algorithm	0.9	[49, 59]
$\zeta$	Differential weight of DE algorithm	0.5	[49, 59]
$T_{max}$	Maximum number of iterations for DE algorithm to produce the optimum solution	2000	Assumed
$\left(M_v,N_v\right)$	Order of Padé approximation for state variable $v$	(4,4), (5,5), (6,6)
$n$	Problem dimension denoting the number of unknown Padé approximation coefficients	$\sum _{v\in \left\{S,I,R\right\}}\left(M_v+N_v\right)$
$L$	The penalty factor of penalty function	106
$\mu$	Death rate	0.1	[29]
$\beta$	Bilinear incidence rate	0.2464 (EE) 0.1464 (DFE)	[29]
$\gamma$	Recovered rate	0.07	[29]
$\alpha$	Infection rate	0.01	[29]
$V_{threshold}$	The threshold value for convergence speed	10−04	Assumed

The optimal values of the Padé approximation coefficients have been found under various options of orders Padé approximations at endemic and disease-free equilibrium states. To validate the performance of the proposed hybrid optimizer reasonably and explore the interpolating strength of Padé approximants, we have considered Padé rational functions of orders (4, 4), (5, 5) and (6, 6) to approximate each state variable for endemic as well as disease-free equilibrium. Table 2 presents the statistical measures (best, mean, and standard deviation (SD)) of optimization results of $\psi \left(\mathit{x}\right)$ produced by the proposed hybrid optimizer in 20 independent runs. Mean convergence curves of 2000 iterations of optimization progress using (4, 4), (5, 5), and (6, 6) order approximants have been exhibited in Fig. 4 .

Table 2.

Optimization results for two equilibrium states.

Order	DFE			EE
	Best	Mean	SD	Best	Mean	SD
(4,4)	2.5e-07	8.1e-06	4.9e-04	8.9e-08	8.9e-06	3.1e-05
(5,5)	1.7e-11	3.6e-05	1.1e-05	4.2e-09	1.2e-05	1.6e-05
(6,6)	8.4e-07	9.4e-06	5.3e-06	3.7e-08	5.7e-05	7.1e-06

Fig. 4.

Mean convergence curves of DE-NMS at (a) DFE (b) EE state.

Tables 3 and 4 represent the optimum values of unknown coefficients, which belong to the smallest optimal values of the objective function over 20 runs.

Table 3.

Optimal values of coefficients at DFE state.

Order	$i$	$a_{Si}$	$b_{Si}$	$a_{Ii}$	$b_{Ii}$	$a_{Ri}$	$b_{Ri}$
(4,4)	0	0.5	1	0.2	1	0.3	1
	1	11.62412	23.19136	−0.4410504	−1.99006	4.518759	15.1704
	2	17.96686	27.99012	21.62841	129.7301	41.81043	151.2964
	3	−4.589881	−4.792515	−0.5847947	−40.31095	−0.7980348	−13.7038
	4	3.214352	3.215415	0.0045273	71.09928	0.0038315	14.43962
(5,5)	0	0.5	1	0.2	1	0.3	1
	1	1.3390699	2.6037853	0.7072810	3.7697691	2.1501625	7.2888593
	2	788.49054	1570.245	−0.8496248	−5.7637786	1.4227432	4.0616136
	3	10.919599	36.620152	9.3823315	56.037332	7.9462428	30.799049
	4	4.7504681	3.9790506	−0.1915813	−7.5008662	−0.152888	−2.5337034
	5	4.7330172	4.7324437	0.0001154	5.4266051	−0.0001095	0.7767076
(6,6)	0	0.5	1	0.2	1	0.3	1
	1	1.08109	2.0329	1.2794	6.50122	0.452094	1.55232
	2	3.2641	8.91808	8.57519	47.9268	1.48791	5.34671
	3	10.8111	10.5012	6.6515	31.3825	6.66342	21.5409
	4	8.40516	26.7405	0.304318	20.9876	5.69837	24.053
	5	2.10008	1.75478	−0.0081925	−1.98912	−0.119462	−1.33771
	6	1.41142	1.41119	9.945e-05	1.2269	6.887e-05	0.603988

Table 4.

Optimal values of coefficients at EE state.

Order	$i$	$a_{Si}$	$b_{Si}$	$a_{Ii}$	$b_{Ii}$	$a_{Ri}$	$b_{Ri}$
(4,4)	0	0.5	1	0.2	1	0.3	1
	1	1.962976	3.873777	10.50587	52.57363	4.722648	115.74965
	2	5.110822	9.836035	16.82871	96.37293	32.34391	111.6259
	3	−0.2354903	−0.3935344	−14.07242	−77.26163	−0.5267137	−1.305307
	4	0.561096	0.8118427	3.553253	19.55911	.082824	8.514212
(5,5)	0	0.5	1	0.2	1	0.3	1
	1	−0.2527975	−0.5398225	4.208253	21.04604	2.752185	9.216065
	2	21.55926	43.33501	1.465227	11.21728	17.88936	64.84665
	3	71.79745	139.7148	−7.792861	−44.28272	16.88882	54.20429
	4	3.595083	4.53876	2.867689	15.78287	1.212053	11.0835
	5	8.729702	12.6298	1.85344	10.20357	0.6390833	5.025223
(6,6)	0	0.5	1	0.2	1	0.3	1
	1	−2.70156	−5.4178	4.88544	24.4964	7.69363	25.8147
	2	−6.08778	−12.5332	0.197156	8.6703	11.8406	39.6912
	3	−4.29643	−4.72722	9.74047	28.7846	9.08754	50.9515
	4	−8.35969	−17.1982	8.09084	66.408	7.75748	15.1099
	5	3.24166	4.79335	−0.899575	−5.46805	−0.0815287	0.269303
	6	−2.99372	−4.33166	1.66903	9.18781	1.07708	8.47145

In the following subsections, the optimization results are presented and analyzed as:

• (i)The accuracy of the solution is assessed by the fact that how small $\psi \left({\mathit{x}}^{**}\right)$ is found.
• (ii)Consistency of NMS-DE algorithm demonstrated by computing best, mean and standard deviations of final optimal values over several optimizer attempts.
• (iii)The convergence speed of NMS-DE is assessed by many iterations used to achieve a threshold value denoted by ${}_{theVthreshold}$.
• (iv)The effectiveness of the proposed method is justified by comparing the obtained solutions with multiple modern metaheuristics.

4.1.1. Accuracy

From Table 2, we observe that the best minimum values of $\psi \left({\mathit{x}}^{**}\right)$ ranges from 2.5e-07 to 1.7e-11 at DFE point and lie in the range [3.7e-08, 4.2e-09] at EE point for various orders of Padé approximants. Fig. 5, Fig. 6, Fig. 7 present the squared residuals of governing equations of state variables over a time of up to 150 days. One can observe from figures that the squared residuals can be minimized up to sufficiently small values 10e-15. Moreover, Fig. 5 shows that the solutions of governing equation of susceptible population achieve accuracies below 10e-6. In contrast, Fig. 6 shows that the accuracy of an infected population is below 10e-10 at both the equilibrium points and for all orders of the Padé approximants. Similarly, the accuracy of solutions of governing equation of susceptible population falls around 10e-10 and 10e-06 at DFE and EE points, respectively. These results ensure that the solutions found by the EPA scheme can achieve approximate solutions of the model with good accuracies.

Fig. 5.

Squared residuals of governing equation of class $S$ at (a) EE (b) DFE state.

Fig. 6.

Squared residuals of governing equation of class $I$ at (a) EE (b) DFE state.

Fig. 7.

Squared residuals of governing equation of class $R$ at (a) EE (b) DFE state.

4.1.2. Consistency and reliability

To assess the consistency of NMS-DE, we do not rely on the solution achieved by a single run of the optimizer. To investigate the reliability of the results, we perform a statistical analysis of marks obtained in multiple runs of the NMSE-DE algorithm. The results presented in Table 2 describe that the mean of the optimal value of $\psi \left({\mathit{x}}^{**}\right)$ remains within the range [8.1e-06, 5.7e-05] for all considered orders and at both equilibrium points with corresponding range [5.3e-06, 4.9e-04] for standard deviations. These facts and figures demonstrate that the performance of NMS-DE in optimization tasks is consistent, and results are reliable with good accuracy.

4.1.3. Convergence speed

Fig. 4 presents the mean convergence curves of the optimization process at two equilibrium points for all orders of Padé approximants. We can observe from Fig. 4(a) that, at DFE point, the threshold value $V_{threshold}$ is achieved in less than 500, 600, and 900 iterations on average for (4, 4), (5, 5), and (6,6) order solutions respectively. Similarly, $V_{threshold}$ is achieved in around 400, 500, and 750 iterations (please see Fig. 4(b)) for (4, 4) in corresponding solutions at EE point. These observations highlight that the order of Padé approximants is directly proportional to computational cost and speed of convergence.

4.1.4. Performance comparison with other metaheuristics

In this subsection, we present a comparison of the optimization results of our proposed method with original DE and NMS algorithms and with a few modern metaheuristics under the same settings of population size, the maximum number of iterations, and the threshold value. The other built-in parameters of all competing algorithms are kept the same as recommended in their original propositions. The metaheuristics used for comparison are Teaching-Learning Based Optimization (TLBO) [47], Particle Swarm Optimization (PSO) [48], DE [49], Controlled Showering Optimization (CSO) [52], Water Cycle Algorithm (WCA) [53], Artificial Bee Colony (ABC) [55] and grey Wolf Optimizer (GWO) [56]. Tables 5 and 6 exhibit the best minimum value, percentage ($P\%$) of successful runs to achieve $V_{threshold}$, and an average number of iterations (T_a) in successful runs to achieve $V_{threshold}$ over 20 independent runs of each algorithm that are used as performance measures to compare the results of all competing metaheuristics.

Table 5.

Optimization results of competing metaheuristics at DFE point.

Algorithm	(4, 4) order solution			(5, 5) order solution			(6, 6) order solution
	Best	$P\%$	Ta	Best	$P\%$	Ta	Best	$P\%$	Ta
TLBO	1.4e-04	58	1078	5.3e-05	74	1142	1.5e-06	68	1212
PSO	3.8e-06	64	1685	1.0e-06	31	1715	7.1e-05	48	1729
DE	3.3e-06	71	1011	9.2e-07	70	1501	8.7e-06	81	1811
NMS	7.0e+06	0	–	3.5e+06	0	–	8.1e+06	0	–
CSO	7.1e-03	53	1623	4.6e-06	63	1495	5.6e-05	57	1677
WCA	5.1e-04	0	–	8.0e-04	0	–	1.4e-04	0	–
ABC	3.3e-01	0	–	2.2e00	0	–	9.1e-01	0	–
GWO	6.2e-02	0	–	2.8e-04	0	–	7.9e-03	0	–
NMS-DE	2.5e-07	100	413	1.7e-11	100	668	8.4e-07	100	891

Table 6.

Optimization results of competing metaheuristics at EE point.

Algorithm	(4, 4) order solution			(5, 5) order solution			(6, 6) order solution
	Best	$P\%$	Ta	Best	$P\%$	Ta	Best	$P\%$	Ta
TLBO	2.3e-05	60	1053	1.7e-05	70	1083	1.5e-05	71	976
PSO	4.6e-06	60	1723	2.1e-05	20	1690	1.5e-05	70	1729
DE	6.1e-07	64	1303	1.2e-06	73	1429	5.8e-06	68	1691
NMS	3.4e+08	0	–	5.9e+06	0	–	7.7e+06	0	–
CSO	5.1e-05	50	1225	3.7e-05	60	1551	5.6e-05	60	1677
WCA	2.9e-04	0	–	4.0e-04	0	–	9.1e-04	0	–
ABC	4.0e00	0	–	1.8e00	0	–	2.4e01	0	–
GWO	3.7e-02	0	–	9.1e-03	0	–	1.5e-02	0	–
NMS-DE	8.9e-08	100	973	4.2e-09	100	1023	3.7e-08	100	1264

The features of results presented in Tables 5 and 6 are comprised of six test instances of solutions of various Padé approximants’ orders at two equilibrium points. We observe that due to significant dependency on provided initial guesses and the complexity of the underlying problem, the performance of the NMS algorithm is nowhere comparable to all other global search algorithms. Hence we exclude NMS results from further comparative analysis. One can observe from these results that the metaheuristics WCA, ABC, and GWO were unsuccessful in achieving the threshold value $V_{threshold}$ in all attempts and returned poor approximations of optimum values. Their mutual comparison is based on the best objective function values only. WCA was able to minimize the objective function better than ABC and GWO. Therefore, further competition occurs between TLBO, PSO, DE, and CSO, and NMS-DE algorithms.

Comparing the results of TLBO, PSO, DE, CSO, and NMS-DE, the following vital facts can be observed:

• 1The DE algorithm computed the best minimum objective function values are better than TLBO, PSO, and CSO for all test instances. The best minimum values found by NMS-DE are far better than those of all of its competitors. The optimal values found by NMS-DE are 92.42%, 99.998%, and 90.34% better than those of the DE algorithm for respective cases of DFE point. At EE point, NMSE found 85.41%, 99.65%, and 99.36% better values than the DE algorithm for three respective cases of endemic equilibrium.
• 2NMS-DE algorithm outperformed other metaheuristics in the sense of success rates (100%) of achieving pre-set accuracy $V_{threshold}$ in all scenarios. On the other hand, none of the competing metaheuristics achieved such a consistent performance while solving the formulated optimization problem of the CovidCE model. For the 5th order solution at DFE point and 6th order solution at EE point, only TLBO showed success rates of 74% and 71%, respectively, better than those of DE, but DE outperformed TLBO, PSO, and CSO on the rest of the cases. This observation justifies the selection of DE as a global solver and the need for further hybridization to achieve a more efficient hybrid solver (NMS-DE).
• 3The final notable point is the speed of convergence of the proposed NMS-DE is faster than those of TLBO, PSO, DE, and CSO. The computational cost is measured by the number of iterations required to achieve the threshold value $V_{threshold}$. At DFE point, NMS-DE consumed around 413 (20.65%), 668 (33.4%), and 891 (44.55%) iterations for 4th, 5th, and 6th order solution, respectively. At EE point these amounts are 973 (48.65%), 1023 (51.15%) and 1274 (63.2%) respectively. Fig. 8 presents the competitive percentage measure of the faster or slower convergence speed of NMS-DE over each successful metaheuristic. The positive and negative percentages reflect that the convergence speed of NMS-DE is faster or slower than a competing algorithm. The convergence speed of NMS-DE is better than all of the competitors except for the 6th order solution at EE point, where TLBO was better than NMS-DE.

Fig. 8.

Percentage competitive measures of convergence speed of NMS-DE.

4.2. Model simulations

The graphical views of evolutions of subpopulations of the underlying model are presented in Fig. 9 –11 . One can observe that the optimized solutions found in be EPA scheme for all considered orders of Padé approximants at both equilibrium points converge exactly to steady-state points. This fact reflects that solution by EPA scheme is in good agreement with that of the NSFD method in the vicinity of equilibrium points [62]. It is worth mentioning that the convergence speed of EPA-based solutions is significantly faster than those of NSFD; we may also observe that the model's physically vital properties like positivity, boundedness, dynamical consistency, and converges to actual steady states (disease-free and endemic equilibria) are also preserved by EPA scheme. Furthermore, the proposed scheme is also independent of the choice of step lengths and unconditionally converges to both states of the model.

Fig. 10.

Dynamics of Infected population at (a) EE (b) DFE states.

Fig. 9.

Dynamics of susceptible population at (a) EE (b) DFE states.

Fig. 11.

Dynamics of the recovered population at (a) EE (b) DFE states.

5. Conclusions

The context of the current study is the evolvement of an evolutionary solver for a complex Covid-19 with a crowding effect (CovidCE) model. The main focus of this research was the formulation and solution of the underlying CovidCE model as an equivalent global optimization problem. This work is a blend of mathematical programming, epidemiology, and metaheuristics. The first contribution relates to the theory of optimization that includes two prominent innovations (i) formulation of a new general order Padé approximation based equivalent global optimization problem preserving vital properties (positivity, boundedness, and stability) of the CovidCE model. This generalization allows varying or adjusting the orders of Padé approximants for approximating the solution more accurately (ii) The formulated optimization problem's complexity barred not only the use of deterministic optimizers but also modern metaheuristics. Therefore, the second contribution of this work was to tune a new hybrid method, named NMS-DE, to solve the formulated problem. This step was accompanied by a statistical analysis of the results of various orders of solutions. The comparative analysis witnessed that the proposed hybrid method handled the optimization phase of the EPA scheme efficiently and outperformed other metaheuristics in terms of accuracy, consistency, and speed of convergence. The Final contribution relates to the characteristics of the obtained solution of the CovidCE model. To the best of our knowledge, the exact analytical solution of such epidemiological models with all physical properties has not been reported so far. The current work enables us to observe an approximate closed-form solution of the CovidCE type model. The graphics of model simulations also reveal that the solution found by the EPA scheme is in good agreement with NSFD in convergence to equilibrium points and possesses a notably faster convergence speed.

Conflict of interest and authorship conformation form

Please check the following as appropriate:

• •All authors have participated in (a) conception and design, or analysis and interpretation of the data; (b) drafting the article or revising it critically for important intellectual content; and (c) approval of the final version.
• •This manuscript has not been submitted to, nor is under review at, another journal or other publishing venue.
• •The authors have no affiliation with any organization with a direct or indirect financial interest in the subject matter discussed in the manuscript

CRediT authorship contribution statement

Javaid Ali: Conceptualization, Methodology. Ali Raza: Data curation, Writing – original draft. Nauman Ahmed: Data curation, Writing – original draft. Ali Ahmadian: Supervision, Validation, Writing – review & editing, Writing – original draft. Muhammad Rafiq: Writing – review & editing, Validation. Massimiliano Ferrara: Writing – review & editing, Validation.