A caputo fractional order epidemic model for evaluating the effectiveness of high-risk quarantine and vaccination strategies on the spread of COVID-19

Abstract

The recent global Coronavirus disease (COVID-19) threat to the human race requires research on preventing its reemergence without affecting socio-economic factors. This study proposes a fractional-order mathematical model to analyze the impact of high-risk quarantine and vaccination on COVID-19 transmission. The proposed model is used to analyze real-life COVID-19 data to develop and analyze the solutions and their feasibilities. Numerical simulations study the high-risk quarantine and vaccination strategies and show that both strategies effectively reduce the virus prevalence, but their combined application is more effective. We also demonstrate that their effectiveness varies with the volatile rate of change in the system’s distribution. The results are analyzed using Caputo fractional order and presented graphically and extensively analyzed to highlight potent ways of curbing the virus.

1. Introduction

In accordance with a report on global health crises by the World Health Organization (WHO) [1], COVID-19 is caused by the SARS-CoV-2 virus and has had a substantial impact on public health, healthcare systems, and the global economy [2]. The virus is highly infectious and is mainly transmitted through respiratory droplets, resulting in a pandemic that has affected millions of individuals worldwide [2]. COVID-19 can manifest a variety of symptoms, ranging from mild flu-like symptoms to severe respiratory illness and even death [3]. Preventive measures such as wearing of masks, practicing physical distancing, and frequently washing of hands [4] are critical to enhance prevention of the virus’s spread. Moreover, individuals with underlying health conditions such as diabetes, hypertension, and obesity are at greater risk of severe illness and death from COVID-19 [5]. It is, therefore, essential to safeguard vulnerable populations and provide appropriate medical care. It has also been observed that people who initially have mild symptoms of COVID-19 can as well develop long-term health complications, which include chronic fatigue, respiratory issues, and neurological problems. Consequently, ongoing medical care and monitoring of individuals who have recovered from COVID-19 are necessary.

Currently, COVID-19 vaccines are crucial tools in the fight against the ongoing pandemic caused by the SARS-CoV-2 virus [6]. Since the first COVID-19 case was reported in late 2019 [7], researchers and scientists worldwide have been working tirelessly to develop and distribute effective vaccines [8]. As at March 2023, several authorized COVID-19 vaccines have been made available, these include Pfizer-BioNTech, Moderna, Johnson & Johnson, AstraZeneca, Sinovac, and Sinopharm. These vaccines work by activating the body’s immune system to recognize and fight the SARS-CoV-2 virus if exposed [8]. Most COVID-19 vaccines require two doses, spaced several weeks apart, to achieve maximum effectiveness, while some vaccines, such as those from Johnson & Johnson, only require a single dose [9].

Clinical trials have demonstrated that COVID-19 vaccines are highly effective in preventing severe illness, hospitalization, and death from COVID-19. For instance, the Pfizer-BioNTech vaccine has shown over 90% effectiveness in preventing COVID-19 infection, while the Moderna vaccine has shown over 95% effectiveness. Johnson & Johnson vaccine has also been shown to be highly effective in preventing severe illness, hospitalization, and death [9]. Despite the high efficacy rate of COVID-19 vaccines, concerns and misconceptions regarding their safety and effectiveness still exist. Some individuals are hesitant to receive vaccinations because of concerns about side effects, the pace of development, or the vaccines’ unknown long-term effects. However, it is important to note that COVID-19 vaccines have undergone rigorous safety testing and extensive clinical trials before receiving authorization for emergency use [10]. The pace of development was a result of unprecedented global collaboration and funding, not because of shortcuts in safety testing. Adverse effects are generally mild and short-lived, such as soreness at the injection site, fever, and fatigue, with severe allergic reactions observed to be rare. Getting vaccinated against COVID-19 is a vital step in protecting oneself, loved ones, and the community from the virus. Even after vaccination, it is essential to continue practicing preventive measures such as wearing of masks, practicing physical distancing, and washing hands frequently. Vaccination is not only a personal decision but also a community responsibility in order to achieve herd immunity and end the pandemic [11].

2. Literature review

Researchers and mathematicians utilize mathematical modeling to study the transmission dynamics of infectious diseases [12]. These models can be stochastic or deterministic, and various types, including compartmental, individual-based, and network models, have been explored [13]. The benefits and drawbacks of each model type, such as the difficulty of fitting models to data and interpreting model results, have been discussed [14]. Furthermore, researchers have studied the transmission and control of COVID-19 using these mathematical models, considering within-host dynamics and between-host transmission [15]. For example, a conceptual study of the concurrent application of therapy, vaccine, and human compliance to physical limitation using a five-compartment model of coronavirus disease was conducted [16]. Additionally, an analysis of the effect of vaccination in controlling the spread of COVID-19 was studied in [17], where vaccination was shown to be an effective criterion to enable the disease to cease in the system. Recently, research developments in epidemiology have witnessed an array of applications of fractional derivatives, which generalize the standard concepts of classical derivatives and have helped researchers understand related volatile changes [18], [19], [20]. In [21], it was discussed that the effectiveness of vaccination depends not only on the vaccine itself but also on the distribution of the vaccine and the population’s response to it. Thus, several researchers often extend the use of fractional derivatives such as [22], [23], [24], [25], [26] to model the dynamics of vaccination, taking into account factors such as the rate of vaccine distribution and the rate of vaccine uptake. For example, the Caputo derivative was applied in a mathematical analysis of a coronavirus disease model in [27] to examine the dynamic reaction of the population group to vaccine uptake and distribution. The fractional-order derivative proved vital as the dynamics of each class of the population were effectively modeled using some fitted data. Apart from vaccination, high-risk immunity is another factor often considered during the eradication of diseases and refers to the quarantine of individuals who are at higher risk of contracting the disease, such as susceptible people living in infected zones, the elderly, and individuals with pre-existing medical conditions [28]. Several studies have been published on predicting COVID-19’s trajectory in the presence of high-risk immunity. A study of the effect of quarantine on the transmission of COVID-19 through a systematic review and meta-analysis was conducted in [29]. Their study, from an epidemiological point of view, analyzed quarantine strategies applicable to both travelers and contacts based on a test-and-release protocol. It was shown that through high-risk quarantine, fewer people in the population are liable to get infected by COVID-19.

Since the dynamics, behaviors, and trends of diseases in physical systems can be gained through numerical simulation, the use of computational algorithms to approximate solutions to mathematical models that describe a wide range of phenomena, such as the homotopy perturbation method, has proved powerful because it often yields good convergent simulation results [30], [31], [32], [33], [34]. For example, an analysis of the effect of two-stage vaccination was studied in [18], and the simulation process was carried out using the precise approximate results generated by the homotopy perturbation method. Their results successfully revealed the effectiveness of the method. Additionally, numerical methods such as the Laplace-Adomian decomposition method have been widely used by researchers to solve nonlinear problems involving complex geometries and initial conditions. Some studies on this can be found in [35], [36], [37], [38], [39], [40], and as evidenced in [41], where the dynamics of Lassa fever transmission were studied using the Laplace-Adomian decomposition method, the method equally produced effective simulation results that shed light on ways of curbing the virus.

The significance of vaccines and high-risk immunity in limiting the spread of COVID-19 has been emphasized by several literature sources. However, the combined effect of these two factors in the event of a re-emergence has not been examined by researchers. This study aims to fill this gap by evaluating the effectiveness of vaccine uptake, vaccine distribution, and high-risk quarantine strategies in preventing the spread of the virus. Our motivation for providing an alternative strategy for curbing the spread of COVID-19 virus in Nigeria stems from the tension and damages caused by the last pandemic on the country’s healthcare system. To achieve this goal, a mathematical model of COVID-19 is proposed, taking into account high-risk quarantine, vaccination parameters, and Caputo derivatives. The Laplace-Adomian decomposition method is employed to simulate the results of this model numerically using reported data from the Nigeria Centre for Disease Control. Therefore, the study seeks to identify effective methods of reducing virus transmission and protecting high-risk individuals to achieve herd immunity in the system.

In summary, the main objective of this study is to offer new perspectives on the effectiveness of combining vaccines and high-risk immunity to mitigate the spread of COVID-19 in Nigeria, making it a distinct research endeavor. The study plans to accomplish this by utilizing a mathematical model and numerical simulations to fill the existing research gap and provide valuable information to inform public health strategies for managing the virus in Nigeria.

Preliminaries

We discuss some essential ideas of fractional calculus applicable in this study here.

Definition 1 [42] —

A real function $\psi \left(x\right)$, for $x>0$, exists in the space ${\delta }_{\nu },\nu \in R$ if a real number $k>\nu$ exists such that $\psi \left(x\right)=x^k{\psi }_1\left(x\right).$  Where ${\psi }_1\left(x\right)\in \delta (0,\infty )$, and $\psi \left(x\right)$ is said to exists in space ${\delta }_v^{\alpha }$ if and only if ${\psi }^{\left(\alpha \right)}\in {\delta }_{\nu },\alpha \in N$.

Definition 2 [42] —

The Riemann–Liouville fractional integration of order $\gamma \ge 0$ for a real positive function $\psi \left(x\right)\in {\delta }_{\nu },\nu \ge -1x>0$ is defined as:

$$I^{\gamma }\psi \left(x\right)=\frac{1}{\Gamma \left(\gamma \right)}{\int }_0^x{(t-x)}^{\gamma -1}\psi \left(t\right)dt.$$

The Riemann–Liouville fractional integral operator $I^{\gamma }$ for $\psi \left(x\right)\in {\delta }_{\nu },\nu \ge -1\gamma ,\alpha \ge 0$ and $\beta \ge -1$ satisfies the following properties:

$$\begin{array}{c}1.I^{\gamma }{I}^{\alpha }\psi \left(x\right)=I^{\gamma +\alpha }\psi \left(x\right),\hfill \\ 2.I^{\gamma }{I}^{\alpha }\psi \left(x\right)=I^{\alpha }{I}^{\gamma }\psi \left(x\right),\hfill \\ 3.I^{\gamma }{t}^{\beta }=\frac{\Gamma (\beta +1)}{\Gamma (\gamma +\beta +1)}{t}^{\gamma +\beta }.\hfill \end{array}$$

Definition 3 [42] —

For a positively defined real function $\psi \left(x\right)\in {\delta }_{\nu }$, the Caputo fractional derivative can mathematically be expressed as;

$$D^{\gamma }\psi \left(x\right)=\frac{1}{\Gamma (\alpha -\gamma )}{\int }_0^x{(x-t)}^{\alpha -\gamma -1}{\psi }^{\left(\alpha \right)}\left(t\right)dt,\alpha -1<\gamma \le \alpha ,\alpha \in N.$$

The fractional order integral operator of Caputo derivative for $\alpha -1<\gamma \le \alpha ,$   $\alpha \in N,\psi \in {\delta }_{-1}^{\alpha },$   $\nu \ge -1$ is :

$$I^{\gamma }{D}^{\gamma }\psi \left(x\right)=\psi \left(x\right)-\sum _{m=0}^{\alpha -1}{\psi }^{\left(m\right)}\left(0\right)\frac{t^m}{m!}.$$

Definition 4 [41] —

Let $\phi \left(t\right)$ be a function defined for all positive real number $t\ge 0$

• iThe Laplace transform of $\phi \left(t\right)$ is the function $\phi \left(s\right):$ $\phi \left(s\right)={\int }_0^{\infty }{e}^{-st}\phi \left(t\right)dt$
• ii
  The Laplace transform of function $\phi \left(t\right)$ with order $\eta$ is defined as

  $$L\left[{\phi }^{\eta }\left(t\right)\right]={\alpha }^{\eta }L\left[\phi \left(t\right)\right]-{\alpha }^{\eta -1}\phi \left(0\right)-{\alpha }^{\eta -2}{\phi }^{\prime }\left(0\right)-{\alpha }^{\eta -3}{\phi }^{\prime \prime }\left(0\right)\cdots$$
• iiiThe inverse Laplace transform of $\frac{\phi \left(s\right)}{s}$ is $L^{-1}\frac{\phi \left(s\right)}{s}={\int }_0^t\phi \left(t\right)dt$
• iv
  The Laplace transform of $f\left(t\right)=t^{\lambda }$ is $L\left(t^{\lambda }\right)=\frac{\Gamma (\lambda +1)}{s^{\lambda +1}}$, and the inverse transform is

  $$L^{-1}\left(\frac{1}{s^{\lambda +1}}\right)=\frac{t^{\lambda }}{\Gamma (\lambda +1)}$$

Definition 5 [41] —

The Adomian polynomials denoted by $A_0,A_{1,\dots .}{A}_n$, consists in the decomposition of the unknown function $y\left(t\right)$ whose series can be expressed as $y\left(t\right)=y_0+y_1+y_2+\cdots y_n$ is given as:

$$A_n=\frac{1}{n}\frac{d^n}{d{\lambda }^n}{\left[G\left(t\right)\sum _{j=0}^n{y}_j{\lambda }^j\right]}_{\lambda =0}$$

3. Methods

This section outlines the methodology flowchart and algorithm utilized for solving the proposed COVID-19 epidemic model. The two methods employed for this purpose are the Laplace Adomian decomposition method and the homotopy perturbation method (see Fig. 1).

Fig. 1.

Flowchart of the methodology.

3.1. Laplace-Adomian decomposition method

Consider the following nonlinear coupled fractional order differential equation [41]

$${}_t^c{D}^{\alpha }{f}_k\left(t\right)=L_k(f_1,f_2,f_3,\dots ,f_k)+N_k(f_1,f_2,f_3,\dots ,f_k),$$ (1)

subject to

$$y_k^{i_k}\left(0\right)=c_i^k,\text{ for }k=1,2,3\dots m,\text{ and }{n}_{k-1}\le \alpha \le n_k.$$

where ${}_t^c{D}^{\alpha }{f}_k\left(t\right)$ is the Caputo-derivative of $k$ numbers of some unknown functions $f\left(t\right)$, $L_k$ and $N_k$ respectively represents the linear and nonlinear operators.

In view of applying the Laplace-Adomian decomposition method to obtain the solution of system (1), we initiate the algorithm by first taking the Laplace transform of (1):

$$L\left[{}_t^c{D}^{\alpha }{f}_k\left(t\right)\right]=L\left[L_k(f_1,f_2,f_3,\dots ,f_k)+N_k(f_1,f_2,f_3,\dots ,f_k)\right].$$ (2)

By Definition 4, Eq. (2) yields:

$$S^{{\alpha }_k}L\left[f_k\left(t\right)\right]-\sum _{i=0}^{m-1}{S}^{{\alpha }_k-i-1}{c}_i^k=L\left[L_k(f_1,f_2,f_3,\dots ,f_k)+N_k(f_1,f_2,f_3,\dots ,f_k)\right].$$ (3)

Applying the Adomian decomposition method, the unknown functions $f_k\left(t\right)$ is decomposed as:

$$f_k\left(t\right)=\sum _{j=0}^{\infty }{f}_{kj}\left(t\right),k=1,2,3\dots ,m,$$ (4)

and the nonlinear terms

$$N_k(f_1,\dots ,f_m)=\sum _{j=0}^{\infty }{A}_{kj}\left(t\right),k=1,2,\dots ,m.$$ (5)

where $A_{kj}$ is the Adomian polynomial defined in Definition 5. Thus evaluating (3) with (4), (5) yields:

$$L\left[\sum _{j=0}^{\infty }{f}_{kj}\left(t\right)\right]=\frac{1}{S^{{\alpha }_k}}\sum _{i=0}^{m-1}{S}^{{\alpha }_k-i-1}{c}_i^k+\frac{1}{S^{{\alpha }_k}}L\left[L_k\left(\sum _{j=0}^{\infty }{f}_{1j}\left(t\right),\dots ,\sum _{j=0}^{\infty }{f}_{mj}\left(t\right)\right)\right]+\frac{1}{S^{{\alpha }_k}}L\left[\sum _{j=0}^{\infty }{A}_{kj}\left(t\right)\right].$$ (6)

Next, we utilize the linearity property of the Laplace transform and establish a recursive formula as follows:

$$L\left[f_{k0}\right]=\frac{1}{s^{{\alpha }_k}}\sum _{j=0}^{m-1}{S}^{{\alpha }_k-i-1}{c}_i^k,k=1,2,3\dots ,m,$$

and

$$L\left[f_{k(j+1)}\left(t\right)\right]=\frac{1}{s^{{\alpha }_k}}L\left[L_k\left(\sum _{j=0}^{\infty }{f}_{1j}\left(t\right),\dots ,\sum _{j=0}^{\infty }{f}_{mj}\left(t\right)\right)\right]+\frac{1}{S^{{\alpha }_k}}L\left(\sum _{j=0}^{\infty }{A}_{kj}\left(t\right)\right).$$ (7)

Taking the inverse Laplace transform of both sides yields $f_{1j},f_{2j},\dots ,f_{n,j},j\ge 0$ such that

$$f_{k0}\left(t\right)=L^{-1}\left(\frac{1}{s^{{\alpha }_k}}\sum _{j=0}^{m-1}{S}^{{\alpha }_k-i-1}{c}_i^k\right),k=1,2,3\dots ,m,$$ (8)

and

$$f_{k(j+1)}\left(t\right)=L^{-1}\left(\frac{1}{s^{{\alpha }_k}}L\left[L_k\left(\sum _{j=0}^{\infty }{f}_{1j}\left(t\right),\dots ,\sum _{j=0}^{\infty }{f}_{mj}\left(t\right)\right)\right]\right)\left(+\frac{1}{S^{{\alpha }_k}}L\left[\sum _{j=0}^{\infty }{A}_{kj}\left(t\right)\right]\right).$$ (9)

3.2. Homotopy perturbation method

To extend the theory of HPM to (17), consider a general coupled differential equation of Caputo fractional order $\alpha$ defined as:

$$\frac{{}^{c}d{S}_i{}^{{\alpha }_1}\left(t\right)}{d{t}^{{\alpha }_1}}=f(t,S_1,S_2,S_3,\dots ,S_n)i\in N$$ (10)

A homotopy can be constructed for (10), such that

$$\left(1-p\right)\frac{{}^{c}d{S}_i{}^{{\alpha }_1}\left(t\right)}{d{t}^{{\alpha }_1}}-p\left(\frac{{}^{c}d{S}_i{}^{{\alpha }_1}\left(t\right)}{d{t}^{{\alpha }_1}}-f(t,S_1,S_2,S_3,\dots ,S_n)\right)=0,i\in \mathrm{N}.$$ (11)

Eq. (11) can be simplified such that

$$\frac{{}^{c}d{S}_i{}^{{\alpha }_1}\left(t\right)}{d{t}^{{\alpha }_1}}=p\left(f(t,S_1,S_2,S_3,\dots ,S_n)\right),$$ (12)

where $p\in (0,1)$ is an embedding parameter. At $p=0$, Eq. (12) becomes linear such that the following equation is obtained:

$$\frac{{}^{c}d{S}_i{}^{{\alpha }_1}\left(t\right)}{d{t}^{{\alpha }_1}}=0.$$ (13)

At $p=1$, the original equation in (12) is obtained. Let assume a solution series embedding the parameter $p$ for (10) such that

$$S_i\left(t\right)=s_{i0}+p{s}_{i1}+p^2{s}_{i2}+p^3{s}_{i3},\dots ,p^n{s}_{in},$$ (14)

substituting (14) into (12) and comparing the coefficients of equal powers of $p$, the following series of equations is obtained;

$$\begin{array}{c}{p}^{0:}:\frac{{}^{c}d{S}_{i0}{}^{{\alpha }_1}\left(t\right)}{d{t}^{{\alpha }_1}}=0\hfill \\ p^1:\frac{{}^{c}d{S}_{i2}{}^{{\alpha }_1}\left(t\right)}{d{t}^{{\alpha }_1}}=f_{i1}(t,s_{10},s_{20},s_{30},\dots ,s_{n0}),\hfill \\ p^2:\frac{{}^{c}d{S}_{i3}{}^{{\alpha }_1}\left(t\right)}{d{t}^{{\alpha }_1}}=f_{i2}(t,s_{10},s_{20},s_{30},\dots ,s_{n0},s_{11},s_{21},s_{31},\dots ,s_{n1}),\hfill \end{array}$$ (15)

and so on. In turn, these systems of equation in (15) can be solved by applying the Riemann–Liouville fractional integral operator $I^{{\alpha }_1}$ to obtain the values of $s_{i1}\left(t\right),s_{i2}\left(t\right),s_{i3}\left(t\right)\dots$ Thus, the solution of (10) is obtained as:

$$S_{iN}\left(t\right)=\sum _{k=0}^{N-1}{s}_{ik}\left(t\right).$$ (16)

4. Model formulation, description and analysis

The proposed mathematical model represents the transmission dynamics of COVID-19 in a physical system, the classes of the model includes $S\left(t\right)$ which are the susceptible individuals, exposed individuals $E\left(t\right)$, infected people of the population $I\left(t\right)$, and the recovered population $R\left(t\right)$. We introduced parameter $c$ which measures the vaccination rate of susceptible individuals living in infected zone given by $c=\frac{\tau S}{N}$, where $\tau$ is the vaccination rate, $S$ is the susceptible population and $N$ is the total population. The immunization of vulnerable persons is represented by parameter $\rho$, and conceptually we agree with [6] by assuming that all susceptible people including the current and newly recruited members of the class are administered vaccine in order to fully examine its influence on the disease prevalence. The described dynamics is represented by the following flow diagram (see Fig. 2):

Fig. 2.

Schematic diagram of the proposed model.

The system of non-linear, fractional-order ordinary differential equations that follows is the proposed model’s dynamics (see Table 2).

$$\begin{array}{c}\frac{{}^c{d}^{{\alpha }_1}S\left(t\right)}{d{t}^{{\alpha }_1}}=\Lambda -\beta (1-c)S\left(t\right)I\left(t\right)-(\mu +\rho )S\left(t\right)\hfill \\ \frac{{}^c{d}^{{\alpha }_2}E\left(t\right)}{d{t}^{{\alpha }_2}}=\beta (1-c)S\left(t\right)I\left(t\right)-(\mu +\varepsilon )E\left(t\right)\hfill \\ \frac{{}^c{d}^{{\alpha }_3}I\left(t\right)}{d{t}^{{\alpha }_3}}=\left(\varepsilon E\left(t\right)-(\mu +\gamma +d)\right)I\left(t\right)\hfill \\ \frac{{}^c{d}^{{\alpha }_3}R\left(t\right)}{d{t}^{{\alpha }_3}}=\gamma I\left(t\right)-\mu R\left(t\right)+\rho S\left(t\right)\hfill \end{array}$$ (17)

Table 2.

Values of the model’s parameters and references.

Parameters	Value	References
$\Lambda$	$750{\mathrm{day}}^{-1}$	[43]
$\beta$	$0.0000124{\mathrm{day}}^{-1}$	[43]
$\sigma$	$0.010939586{\mathrm{day}}^{-1}$	[43]
$\gamma$	$4.013000000\times 10^{-8}{\mathrm{day}}^{-1}$	[43]
$\theta$	$0.0766169{\mathrm{day}}^{-1}$	[43]
$\mu$	$0.001466848{\mathrm{day}}^{-1}$	[43]
$c$	0.05	–
$\rho$	0.05	–

4.1. Model analysis

In this section, we shall conduct the qualitative analysis of the mathematical model, and we shall do this in an integer order $\alpha =1$ to fully study the properties of the mathematical model and show its potential for real-life applications.

4.1.1. Existence and uniqueness of solution

Theorem 1

Let

$$\begin{array}{c}{m}_1^{\prime }=f_1(s_1,s_2,s_3,\dots ,s_n,t),s_1\left(t_0\right)=s_{10},\hfill \\ m^{\prime }{}_2=f_2(s_1,s_2,s_3,\dots ,s_n,t),s_2\left(t_0\right)=s_{20},\hfill \\ m^{\prime }{}_3=f_3(s_1,s_2,s_3,\dots ,s_n,t),s_3\left(t_0\right)=s_{30},\hfill \\ ⋮\hfill \\ .m^{\prime }{}_k=f_n(s_1,s_2,s_3,\dots ,s_n,t),s_k\left(t_0\right)=s_{k0}.\hfill \end{array}$$

Suppose D exist in domain of $(k+1)$ dimensional space for time $t$ and space $x$ and the partial derivative $\frac{\partial f}{\partial s_i}$ where $i=1,2,3,\dots ....n$ are continuous in $D=\{(s,t):\setminus t-t_0\setminus \le a,\setminus s-s_0\setminus \le b\}$ then there exist a constant $\varepsilon \ge 0$ such that a unique solution exists for continuous vector. $m=[s_1\left(t\right),s_2\left(t\right),s_3\left(t\right)\cdots s_k\left(t\right)]$ in the interval $\setminus t-t_0\setminus \le \varepsilon$

Proof

A Lipchitz criterion will be employed to ensure that the solution exists and is unique. Thus from Eq. (17), let:

$$\begin{array}{c}{m}_1=\Lambda -\left(\mu +\rho \right)S-\beta \left(1-c\right)SI,\hfill \\ m_2=\left(1-c\right)\beta SI-(\mu +\varepsilon )E,\hfill \\ m_3=\varepsilon E-\left(\gamma +\mu +d\right)I,\hfill \\ m_4=\gamma I-\mu R+\rho S.\hfill \end{array}$$ (18)

Following the criterion, we obtain the system’s partial derivatives.

The partial derivatives of $m_1=\Lambda -\left(\mu +\rho \right)S-\beta \left(1-c\right)SI$ with respect to the classes yields:

$$\left|m_{1S}\right|=|-\left(\mu +\rho \right)|<\infty ,\left|m_{1E}\right|=\left|0\right|<\infty ,\left|m_{1I}\right|=|-\beta \left(1-c\right)S|<\infty ,\left|m_{1R}\right|=\left|0\right|<\infty .$$

Similarly for $m_2=\left(1-c\right)\beta SI-(\mu +\varepsilon )E$ we obtain:

$$\left|m_{2S}\right|=\left|\left(1-c\right)\beta I\right|<\infty ,\left|m_{2E}\right|=\left|(\varepsilon +\mu )\right|<\infty ,\left|m_{2I}\right|=\left|\left(1-c\right)\beta S\right|<\infty ,\left|m_{2R}\right|=\left|0\right|<\infty .$$

For $m_3=\varepsilon E-\left(\gamma +\mu +d\right)I$,

$$\left|m_{3S}\right|=\left|0\right|<\infty ,\left|m_{3E}\right|=\left|\varepsilon \right|<\infty ,\left|m_{3I}\right|=|-\left(\mu +\gamma +d\right)|<\infty ,\left|m_{3R}\right|=\left|0\right|<\infty .$$

For $m_{4S}=\gamma I-\mu R+\rho S$,

$$\left|m_{4S}\right|=\left|\rho \right|<\infty ,\left|m_{4E}\right|=\left|0\right|<\infty \left|m_{4I}\right|=\left|\gamma \right|<\infty ,\left|m_{4R}\right|=|-\mu |<\infty .$$

The partial derivatives of these functions exist and are continuous and bounded; therefore, system (18) exists and has a unique solution in ${\Re }^5$.

4.1.2. Positivity of invariant region

The feasible domain for the stability of solution of an epidemiological model is the invariant region. At an increasing time $t\ge 0$, we prove that the region covered by the model solution remain positively invariant. Thus, let the total human population be:

$N\left(t\right)=S\left(t\right)+E\left(t\right)+I\left(t\right)+R\left(t\right)$. Since the human population varies throughout time, therefore:

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

Which yields:

$$\frac{dN}{dt}=\Lambda -\mu (S+E+I+R)$$ (19)

Theorem 2

The resulting solutions provided analytically for Eq. (17) are feasible in $\Pi$ for $t\ge 0$ .

Proof

Let $D=\left\{S,E,I,R\right\}\in {\Re }^4$ contains the solution of (17) for $\left\{S\left(t\right),E\left(t\right),I\left(t\right),R\left(t\right)\ge 0\right\}$, assume that the population is devoid of infection, then E, I, and Q are set to zero:

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

and

$$\frac{dN}{dt}\le \Lambda -\mu N.$$ (20)

Separating the variables and integrating both sides of Eq. (20) yields:

$$-\frac{1}{\mu }lin(\Lambda -\mu N)\le t.$$ (21)

Such that

$$lin(\Lambda -\mu N)\le -\mu t.$$

Thus, solving for the total human population $N$ in (21),

$$\Lambda =\mu N+e^{-\mu t}.$$

$$\text{As }t\to \infty \text{ we have }N\le \frac{\Lambda }{\mu }.$$ (22)

This implies that the suggested model in (17) may be investigated in the viable zone

$$\Pi =\left\{\left(S,E,I,R\in {\Re }^4:N\le \frac{\Lambda }{\mu }\right)\right\}.$$

4.1.3. Non-negativity of solution

Theorem 3

Given $S>0,E>0,I>0,R>0$ , then the solutions $\left\{\left(S,E,I,R\in {\Re }^4:N\le \frac{\Lambda }{\mu }\right)\right\}$ are positively invariant for $t\ge 0$ .

Proof

From Eq. (17),

$$\frac{dS\left(t\right)}{dt}\ge \left(\mu +\rho +\beta (1-c)I\right)S\left(t\right).$$

Separating the variables,

$$\frac{dS\left(t\right)}{S\left(t\right)}\ge (\mu +\rho +\beta (1-c)I)dt.$$

Integrating both sides and applying the initial conditions,

$S\left(t\right)\ge s_0{e}^{-\left(\mu +\rho +\beta (1-c)I\right)t}\ge 0$. This indicates that $S\left(t\right)>0$ for all $t\ge 0$.

Following the same procedure, we demonstrate the positivity of the other classes.

$$E\left(t\right)\ge e_0{e}^{-(\mu +\varepsilon )t}\ge 0,I\left(t\right)\ge i_0{e}^{-\left(\mu +d+\gamma \right)t}\ge 0.,R\left(t\right)\ge R_0{e}^{-\mu t}\ge 0.$$

Thus $\left\{\left(S\left(t\right),E\left(t\right),I\left(t\right),R\left(t\right)>0,\forall t\ge 0inS\left(t\right),E\left(t\right),I\left(t\right),R\left(t\right)\in {\Re }^4;and\right)\right)$ $\left(\left(N\le \frac{\Lambda }{\mu }\right)\right\}$.

The solutions are positive and this completes the proof.

After satisfying all of the fundamental requirements for an epidemiology model, we conclude that the suggested model is appropriate for studying the dynamics of COVID-19 in the general population.

4.2. Equilibrium analysis

4.2.1. Disease free equilibrium

At disease free equilibrium, there is no infection in the system. Thus, $E=I=0$. Thus, concurrently solving for $S,E,I,R$ at these points, the disease free threshold of the model class are.

$$S=\frac{\Lambda }{\mu +\rho },E=0,I=0,R=\frac{\Lambda \rho }{(\mu +\rho )\mu }$$

4.2.2. Endemic equilibrium points

At these point, there is transmission of infection in the system. Hence, $E\ne I\ne 0$ and the following thresholds are obtained.

$$\begin{array}{c}S=\frac{(\mu +\varepsilon )(\gamma +\mu +d)}{\beta (1-c)\varepsilon },\hfill \\ E=\frac{\beta (1-c)\varepsilon \Lambda -(\mu +\rho )(\mu +\varepsilon )(\gamma +\mu +d)}{\beta (1-c)\varepsilon (\mu +\varepsilon )\varepsilon },\hfill \\ I=\frac{\beta (1-c)\varepsilon \Lambda -(\mu +\rho )(\mu +\varepsilon )(\gamma +\mu +d)}{\beta (1-c)(\mu +\varepsilon )(\mu +\gamma +d)},\hfill \\ R=\frac{\beta (1-c){\varepsilon }^2\Lambda \gamma -(\mu +\rho )(\mu +\varepsilon )(\gamma +\mu +d)\varepsilon \gamma +{(\mu +\varepsilon )}^2{(\gamma +\mu +d)}^2\rho }{\beta (1-c)(\mu +\varepsilon )(\gamma +\mu +d)\varepsilon \mu }.\hfill \end{array}$$ (23)

4.3. Basic reproduction number

The reproduction ratio of the SEIR model is associated with the reproductive power of the disease and is defined by $R_0=\rho \left(G\right)$ where $\rho$ is the spectral radius of the next generation matrix $G=F{V}^{-1}$.

The basic reproductive ratio of the model equation (1) at disease free equilibrium point is obtained as:

$$R_0=\frac{\beta \Lambda (1-c)\varepsilon }{(\mu +\rho )(\mu +\varepsilon )(\mu +\gamma +d)}$$

Remark

If $R_0<1$ the disease will cease to exist in the system and if $R_0>1$, epidemic will occur.

Without the implementation of the control parameter, the ratio is evaluated as $R_0=35.74811197$, which is large and can cause rapid progression of the disease in the system; thus, we investigate the effect of raising the levels of the two introduced mitigating factors on $R_0$ (see Table 3).

Table 3.

Response of $R_0$ to control parameters.

$c,\rho$	$R_0$
0	35.74811197
0.03	1.616378873
0.06	0.80188346895
0.09	0.5216772740

4.4. Stability analysis

We shall study the stability of the model equilibrium points by finding the eigenvalues of the Jacobian matrix of system (1)

Definition 6

The equilibrium point $\overline{v}$ is stable if the reproductive ratio $R_0$ is positive and the following are true $\forall$ $R<R_0$ $\exists$ an $r$ on the interval $0<r<R$ such that $v\left(0\right)$ is inside $S(\overline{v},R)$, whenever $\overline{v}\left(t\right)$ is inside $S(\overline{v},R)$ $\forall$ $t>0$

Definition 7

An equilibrium point $\overline{v}$ is unstable if it is not stable. Thus $\overline{v}$ is unstable $R_0>0$ for which the following is true. For every $R<R_0$ there exist $r$ on $0<r<R$, such that $v\left(0\right)$ is inside $S(\overline{v},R)$ when $\overline{v}\left(t\right)$ is inside $S(\overline{v},R)$ $\forall$ $t>0$.

Lemma 1

The model’s disease-free equilibrium is locally asymptotically stable if $R_0<1$ and unstable if $R_0>1$ .

Proof

For brevity, let $\beta (1-c)=A,(\mu +\rho )=D,(\mu +\varepsilon )=F,(\mu +\gamma +d)=G$ in Eq. (1). Thus the Jacobian of Eq. (1) after evaluation at the disease free equilibrium points yields:

$$J_1=\left[\begin{array}{cccc}\hfill -D\hfill & \hfill 0\hfill & \hfill -\frac{\Lambda A}{D}\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill -F\hfill & \hfill \Lambda A\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill \varepsilon \hfill & \hfill -G\hfill & \hfill 0\hfill \\ \hfill \rho \hfill & \hfill 0\hfill & \hfill \gamma \hfill & \hfill -\mu \hfill \end{array}\right]$$ (24)

and the eigenvalues of the matrices are obtained as:

$${\lambda }_1=-D,$$

$${\lambda }_2=-\mu$$

$${\lambda }_3=-\frac{1}{2}\left[\frac{DF+DG+\sqrt{4AD\Lambda \varepsilon +D^2{F}^2-2D^{2}FG+D^2{G}^2}}{D}\right]$$

$${\lambda }_4=-\frac{1}{2}\left[\frac{DF+DG-\sqrt{4AD\Lambda \varepsilon +D^2{F}^2-2D^{2}FG+D^2{G}^2}}{D}\right]$$

Remark

The disease-free equilibrium is locally asymptotically stable since all eigenvalues are negative.

5. Numerical solution

By means of qualitative analysis, we have verified the uniqueness and existence of the solution to the model and its suitability for addressing physical problems. Thus, we proceed with the application of the methodology outlined in Section 4 to numerically compute the solution of the Caputo derivative mathematical model through the utilization of the Laplace-Adomian decomposition method and the homotopy perturbation method.

5.1. The homotopy perturbation method

Following the iterative scheme of the homotopy perturbation method described in the methodology section, we can construct a homotopy for (1):

$$\begin{array}{c}\frac{{}^c{d}^{{\alpha }_1}S\left(t\right)}{d{t}^{{\alpha }_1}}=p(\Lambda -\beta (1-c)S\left(t\right)I\left(t\right)-(\mu +\rho )S\left(t\right)),\hfill \\ \frac{{}^c{d}^{{\alpha }_2}E\left(t\right)}{d{t}^{{\alpha }_2}}=p\left(\beta (1-c)S\left(t\right)I\left(t\right)-(\mu +\varepsilon )E\left(t\right)\right),\hfill \\ \frac{{}^c{d}^{{\alpha }_3}I\left(t\right)}{d{t}^{{\alpha }_3}}=p\left(\varepsilon E\left(t\right)-(\mu +\gamma +d)I\left(t\right)\right),\hfill \\ \frac{{}^c{d}^{{\alpha }_3}R\left(t\right)}{d{t}^{{\alpha }_3}}=p\left(\gamma I\left(t\right)-\mu R\left(t\right)+\rho S\left(t\right)\right).\hfill \end{array}$$ (25)

We assume the following series results for the system state variables:

$$\begin{array}{c}S\left(t\right)=s_0\left(t\right)+p{s}_1\left(t\right)+p^2{s}_2\left(t\right)+\cdots p^n{s}_n\left(t\right),\hfill \\ E\left(t\right)=e_0\left(t\right)+p{e}_1\left(t\right)+p^2{e}_2\left(t\right)+\cdots p^n{e}_n\left(t\right),\hfill \\ I\left(t\right)=i_0\left(t\right)+p{i}_1\left(t\right)+p^2{i}_2\left(t\right)+\cdots p^n{i}_n\left(t\right),\hfill \\ R\left(t\right)=r_0\left(t\right)+p{r}_1\left(t\right)+p^2{r}_2\left(t\right)+\cdots p^n{r}_n\left(t\right).\hfill \end{array}$$ (26)

Eq. (17) is evaluated with (18), and the coefficients of $p^n$ are compared such that for

$$p^0:\frac{{}^c{d}^{{\alpha }_1}{s}_0\left(t\right)}{d{t}^{{\alpha }_1}}=0,\frac{{}^c{d}^{{\alpha }_1}{e}_0\left(t\right)}{d{t}^{{\alpha }_1}}=0,\frac{{}^c{d}^{{\alpha }_1}{i}_0\left(t\right)}{d{t}^{{\alpha }_1}}=0,\frac{{}^c{d}^{{\alpha }_1}{r}_0\left(t\right)}{d{t}^{{\alpha }_1}}=0.$$

Applying operator $I^{{\alpha }_1}$, the following initial approximations are obtained:

$$s\left(t\right)=s_0,e\left(t\right)=e_0,i\left(t\right)=i_0,r\left(t\right)=r_0.$$

Also, the coefficient of $p^1$

$$\begin{array}{c}\frac{{}^c{d}^{{\alpha }_1}{s}_1\left(t\right)}{d{t}^{{\alpha }_1}}=p(\Lambda -\beta (1-c)s_0\left(t\right)i_0\left(t\right)-(\mu +\rho )s_0\left(t\right))\hfill \\ \frac{{}^c{d}^{{\alpha }_2}{e}_1\left(t\right)}{d{t}^{{\alpha }_2}}=p\left(\beta (1-c)s_0\left(t\right)i_0\left(t\right)-(\mu +\varepsilon )e_0\left(t\right)\right)\hfill \\ \frac{{}^c{d}^{{\alpha }_3}{i}_1\left(t\right)}{d{t}^{{\alpha }_3}}=p\left(\varepsilon s_0\left(t\right)-(\mu +\gamma +d)i_0\left(t\right)\right)\hfill \\ \frac{{}^c{d}^{{\alpha }_3}{r}_1\left(t\right)}{d{t}^{{\alpha }_3}}=p\left(\gamma i_0\left(t\right)-\mu r_0\left(t\right)+\rho s_0\left(t\right)\right)\hfill \end{array}$$ (27)

The Riemann–Liouville integral operator $I^{{\alpha }_1}$ is equally applied on (19) to give the first approximation results which yields:

$$\begin{array}{c}{s}_1\left(t\right)=(\Lambda -\beta (1-c)s_0\left(t\right)i_0\left(t\right)-(\mu +\rho )s_0\left(t\right))\frac{t^{{\alpha }_1}}{\Gamma \left({\alpha }_1+1\right)},\hfill \\ e_1\left(t\right)=\left(\beta (1-c)s_0\left(t\right)i_0\left(t\right)-(\mu +\varepsilon )e_0\left(t\right)\right)\frac{t^{{\alpha }_2}}{\Gamma \left({\alpha }_2+1\right)},\hfill \\ i_1\left(t\right)=\left(\varepsilon s_0\left(t\right)-(\mu +\gamma +d)i_0\left(t\right)\right)\frac{t^{{\alpha }_3}}{\Gamma \left({\alpha }_3+1\right)},\hfill \\ r_1\left(t\right)=\left(\gamma i_0\left(t\right)-\mu r_0\left(t\right)+\rho s_0\left(t\right)\right)\frac{t^{{\alpha }_4}}{\Gamma \left({\alpha }_4+1\right)}.\hfill \end{array}$$ (28)

After computing the second approximation, the following solutions are equally obtained:

$$s_2\left(t\right)=\left(\begin{array}{c}{\beta }^2{c}^2{i}_0^2-2{\beta }^{2}c{i}_0^2{s}_0+{\beta }^2{i}_0^2{s}_0\hfill \\ -2\beta c\mu i_0{s}_0-2\beta c\rho i_0{s}_0+\Lambda \beta c{i}_0\hfill \\ +2\beta \rho i_0{s}_0-\Lambda \beta i_0+{\mu }^2{s}_0+2\mu \rho s_0\hfill \\ +{\rho }^2{s}_0-\Lambda \mu -\Lambda \rho \hfill \end{array}\right)\frac{t^{2{\alpha }_1}}{\Gamma (2{\alpha }_1+1)}-\left(\begin{array}{c}\beta s_0(cd{i}_0-c\varepsilon e_0\hfill \\ +c\gamma i_0+c\mu i_0-\hfill \\ d{i}_0+\varepsilon e_0-\gamma i_0-\mu i_0)\hfill \end{array}\right)\frac{t^{{\alpha }_1+{\alpha }_3}}{\Gamma ({\alpha }_1+{\alpha }_3+1)},$$

$$e_2\left(t\right)=-\left(\begin{array}{c}{\beta }^2{c}^2{i}_0^2-2{\beta }^{2}c{i}_0^2{s}_0\hfill \\ +{\beta }^2{i}_0^2{s}_0-\beta c\mu i_0{s}_0\hfill \\ -\beta c\rho i{s}_0+\beta \Lambda c\hfill \\ +\beta \rho s_0-\Lambda \beta +\mu \beta s_0\hfill \end{array}\right)\frac{t^{{\alpha }_1+{\alpha }_2}}{\Gamma ({\alpha }_1+{\alpha }_2+1)}-\left(\begin{array}{c}\beta c{s}_0\varepsilon i_0+{\mu }^2{e}_0\hfill \\ +\beta c{\mu }_0{i}_0{s}_0\hfill \\ -\beta c{i}_0\varepsilon e_0{e}_0{\varepsilon }^2\hfill \\ +2\varepsilon \mu e_0\hfill \end{array}\right)\times \frac{t^{2{\alpha }_{13}}}{\Gamma (2{\alpha }_1+1)}+\left(\begin{array}{c}\beta s_{0}cd{i}_0-\beta s_{0}c\varepsilon e_0\hfill \\ +\beta s_{0}c\gamma i_0+\beta s_{0}c\mu i_0\hfill \\ -\beta s_{0}d{i}_0+\beta s_0{e}_0\varepsilon \hfill \\ +\beta s_0\gamma i_0-\beta s_0\mu i_0\hfill \end{array}\right)\frac{t^{{\alpha }_2+{\alpha }_3}}{\Gamma ({\alpha }_2+{\alpha }_3+1)},$$

$$i_2\left(t\right)=\left(-\varepsilon \beta c{i}_0{s}_0+\varepsilon \beta i_0{s}_0-{\varepsilon }^2{e}_0-\varepsilon \mu e_0\right)\frac{t^{{\alpha }_2+{\alpha }_3}}{\Gamma ({\alpha }_2+{\alpha }_3+1)}-\left(\begin{array}{c}{d}^2{i}_0-d\varepsilon e_0+2d\gamma i_0+2d\mu i_0-\varepsilon e_0\gamma \hfill \\ +\varepsilon e_0\mu +{\gamma }^2{i}_0+2\gamma \mu i_0+{\mu }^2{i}_0\hfill \end{array}\right)\frac{t^{2{\alpha }_3}}{\Gamma (2{\alpha }_3+1)},$$

$$r_2\left(t\right)=\left(\begin{array}{c}-\gamma d{i}_0+\gamma \varepsilon e_0\hfill \\ -{\gamma }^2{i}_0-\mu \gamma i_0\hfill \end{array}\right)\frac{t^{{\alpha }_4+{\alpha }_3}}{\Gamma ({\alpha }_4+{\alpha }_3+1)}-\left(\begin{array}{c}\mu \gamma i_0\hfill \\ -\mu r_0\hfill \\ +\rho \mu s_0\hfill \end{array}\right)\frac{t^{2{\alpha }_4}}{\Gamma (2{\alpha }_4+1)}+\left(\begin{array}{c}\rho \beta c{i}_0{s}_0-\rho \beta i_0{s}_0\hfill \\ -\rho \mu s_0-{\rho }^3{s}_0+\Lambda \rho \hfill \end{array}\right)\frac{t^{{\alpha }_4+{\alpha }_4}}{\Gamma ({\alpha }_4+{\alpha }_1+1)}.$$

The third iteration was equally computed using MATHEMATICA 12 software package, and the approximate solution of each class is obtained by adding the iterative solutions:

$$S\left(t\right)=\sum _{n=0}^3{s}_n\left(t\right),C\left(t\right)=\sum _{n=0}^3{c}_n\left(t\right).I\left(t\right)=\sum _{n=0}^3{i}_n\left(t\right),$$

$$R\left(t\right)=\sum _{n=0}^3{r}_n\left(t\right),Q\left(t\right)=\sum _{n=0}^3{q}_n\left(t\right)$$

5.2. The Laplace Adomian decomposition method

Again, the iterative scheme of the Laplace-Adomian decomposition method will be applied to obtain the model’s solution. Thus, following the algorithm of the method as described in the methodology,

$$\begin{array}{c}{}^{c}D{}^{{\alpha }_1}S\left(t\right)=\Lambda -\mu S\left(t\right)-\beta S\left(t\right)I\left(t\right)(1-c)-\rho S\left(t\right),\hfill \\ {}^{c}D{}^{{\alpha }_2}E\left(t\right)=\beta S\left(t\right)I\left(t\right)(1-c)-(\mu +\varepsilon )E\left(t\right),\hfill \\ {}^{c}D{}^{{\alpha }_3}I\left(t\right)=\varepsilon E\left(t\right)-(\gamma +\mu +d)I\left(t\right),\hfill \\ {}^{c}D{}^{{\alpha }_4}R\left(t\right)=\gamma I\left(t\right)-\mu R\left(t\right)+\rho S\left(t\right).\hfill \end{array}$$ (29)

Subject to initial conditions $e_0\left(t\right)=s_0,e_0\left(t\right)=e_0,i_0\left(t\right)=i_0,r_0\left(t\right)=r_0$.

We initiate the process by taking the Laplace transforms of class of the system such that

$$\begin{array}{c}L\left\{{}^{c}D{}^{{\alpha }_1}S\left(t\right)\right\}=L\left\{\Lambda \right\}-L\{\mu S\left(t\right)-\beta S\left(t\right)I\left(t\right)(1-c)-\rho S\left(t\right)\},\hfill \\ L\left\{{}^{c}D{}^{{\alpha }_2}E\left(t\right)\right\}=L\{\beta S\left(t\right)I\left(t\right)(1-c)-(\mu +\varepsilon )E\left(t\right)\},\hfill \\ L\left\{{}^{c}D{}^{{\alpha }_3}I\left(t\right)\right\}=L\{\varepsilon E\left(t\right)-(\gamma +\mu +d)I\left(t\right)\},\hfill \\ L\left\{{}^{c}D{}^{{\alpha }_5}R\left(t\right)\right\}=L\{\gamma I\left(t\right)-\mu R\left(t\right)+\rho S\left(t\right)\}.\hfill \end{array}$$ (30)

Applying Definition (4) on (30) yields

$$\begin{array}{c}{\mathrm{\ell }}^{{\alpha }_1}L\left\{S\left(t\right)\right\}-{\mathrm{\ell }}^{{\alpha }_1-1}S\left(0\right)=L\left\{\Lambda \right\}-L\{\mu S\left(t\right)-\beta S\left(t\right)I\left(t\right)(1-c)-\rho S\left(t\right)\},\hfill \\ {\mathrm{\ell }}^{{\alpha }_2}L\left\{E\left(t\right)\right\}-{\mathrm{\ell }}^{{\alpha }_2-1}E\left(0\right)=L\{\beta S\left(t\right)I\left(t\right)(1-c)-(\mu +\varepsilon )E\left(t\right)\},\hfill \\ {\mathrm{\ell }}^{{\alpha }_3}L\left\{I\left(t\right)\right\}-{\mathrm{\ell }}^{{\alpha }_3-1}I\left(0\right)=L\{\varepsilon E\left(t\right)-(\gamma +\mu +d)I\left(t\right)\},\hfill \\ {\mathrm{\ell }}^{{\alpha }_4}L\left\{R\left(t\right)\right\}-{\mathrm{\ell }}^{{\alpha }_4-1}R\left(0\right)=L\{\gamma I\left(t\right)-\mu R\left(t\right)+\rho S\left(t\right)\}.\hfill \end{array}$$ (31)

Such that we have

$$\begin{array}{c}L\left\{S\left(t\right)\right\}=\frac{s_0}{\mathrm{\ell }}+\frac{1}{{\mathrm{\ell }}^{{\alpha }_1}}L\left\{\Lambda \right\}-L\{\mu S\left(t\right)-\beta S\left(t\right)I\left(t\right)(1-c)-\rho S\left(t\right)\},\hfill \\ L\left\{E\left(t\right)\right\}=\frac{e_0}{\mathrm{\ell }}+\frac{1}{{\mathrm{\ell }}^{{\alpha }_2}}L\{\beta S\left(t\right)I\left(t\right)(1-c)-(\mu +\varepsilon )E\left(t\right)\},\hfill \\ L\left\{I\left(t\right)\right\}=\frac{i_0}{\mathrm{\ell }}+\frac{1}{{\mathrm{\ell }}^{{\alpha }_3}}L\{\varepsilon E\left(t\right)-(\gamma +\mu +d)I\left(t\right)\},\hfill \\ L\left\{R\left(t\right)\right\}=\frac{r_0}{\mathrm{\ell }}+\frac{1}{{\mathrm{\ell }}^{{\alpha }_4}}L\{\gamma I\left(t\right)-\mu R\left(t\right)+\rho S\left(t\right)\}.\hfill \end{array}$$ (32)

The solution of each class can be represented by an infinite series given by:

$$S\left(t\right)=\sum _{n=0}^{\propto }{s}_n\left(t\right),E\left(t\right)=\sum _{n=0}^{\propto }{e}_n\left(t\right),I\left(t\right)=\sum _{n=0}^{\propto }{i}_n\left(t\right),.R\left(t\right)=\sum _{n=0}^{\propto }{r}_n\left(t\right),$$ (33)

the infinite series of the nonlinear term $I\left(t\right)S\left(t\right)$ is represented with:

$$I\left(t\right)S\left(t\right)=\sum _{n=0}^{\propto }{X}_n$$ (34)

where $X_n\left(t\right)$ is the Adomian polynomial of the nonlinear term given by

$$X_n=\frac{1}{\Gamma (n+1)}\frac{d^n}{dt}\left[\sum _{c=0}^n{\lambda }^c{I}_c\sum _{c=0}^n{\lambda }^c{S}_c\right].$$ (35)

Evaluating Eq. (32) with expressions in (33), (34) and taking the inverse Laplace transform of both sides yields the recurrence relations:

$$\begin{array}{c}\sum _{n=0}^{\infty }{S}_{n+1}\left(t\right)=s_0+\frac{\Lambda t^{{\alpha }_1}}{\Gamma ({\alpha }_1+1)}+L^{-1}\left\{\frac{1}{{\mathrm{\ell }}^{{\alpha }_1}}L\left(-\mu S_n-\beta X_n(1-c)-\rho S_n\right)\right\},\hfill \\ \sum _{n=0}^{\infty }{E}_{n+1}\left(t\right)=e_0+L^{-1}\left\{\frac{1}{{\mathrm{\ell }}^{{\alpha }_2}}L\left(X_n(1-c)-(\mu +\varepsilon )E_n\right)\right\},\hfill \\ \sum _{n=0}^{\infty }{I}_{n+1}\left(t\right)=i_0+L^{-1}\left\{\frac{1}{{\mathrm{\ell }}^{{\alpha }_3}}L\left(\varepsilon E_n-(\gamma +\mu +\alpha )I_n\right)\right\},\hfill \\ \sum _{n=0}^{\infty }{R}_{n+1}\left(t\right)=r_0+L^{-1}\left\{\frac{1}{{\mathrm{\ell }}^{{\alpha }_4}}L\left(\gamma I_n-\mu R_n+\rho S_n\right)\right\}.\hfill \end{array}$$ (36)

From (36), the initial approximations are:

$$s_0\left(t\right)=s_0+\frac{\Lambda t^{{\alpha }_1}}{\Gamma ({\alpha }_1+1)},e_0\left(t\right)=e_0,i_0\left(t\right)=i_0,r_0\left(t\right)=r_0$$

Matching both sides of (36) to obtain subsequent approximations terms of $n\ge 0$, at $n=0$, we have

$$\begin{array}{c}{s}_1\left(t\right)=L^{-1}\left\{\frac{1}{{\mathrm{\ell }}^{{\alpha }_1}}L\left(-\mu S_0-\beta X_0(1-c)-\rho S_0\right)\right\},\hfill \\ e_1\left(t\right)=L^{-1}\left\{\frac{1}{{\mathrm{\ell }}^{{\alpha }_2}}L\left(X_0(1-c)-(\mu +\varepsilon )E_0\right)\right\},\hfill \\ i_1\left(t\right)=L^{-1}\left\{\frac{1}{{\mathrm{\ell }}^{{\alpha }_3}}L\left(\varepsilon E_0-(\gamma +\mu +\alpha )I_0\right)\right\},\hfill \\ r_1\left(t\right)=L^{-1}\left\{\frac{1}{{\mathrm{\ell }}^{{\alpha }_4}}L\left(\gamma I_0-\mu R_0+\rho S_0\right)\right\}.\hfill \end{array}$$ (37)

Which leads to the following first approximate results obtained as:

$$\begin{array}{c}{s}_1\left(t\right)=(\Lambda -\mu s_0-\beta s_0{i}_0(1-c)-\rho s_0)\frac{t^{{\alpha }_1}}{\Gamma ({\alpha }_1+1)},\hfill \\ e_1\left(t\right)=\left(\beta (1-c)s_0\left(t\right)i_0\left(t\right)-(\mu +\varepsilon )e_0\left(t\right)\right)\frac{t^{{\alpha }_2}}{\Gamma \left({\alpha }_2+1\right)},\hfill \\ i_1\left(t\right)=\left(\varepsilon s_0\left(t\right)-(\mu +\gamma +d)i_0\left(t\right)\right)\frac{t^{{\alpha }_3}}{\Gamma \left({\alpha }_3+1\right)},\hfill \\ r_1\left(t\right)=\left(\gamma i_0\left(t\right)-\mu r_0\left(t\right)+\rho s_0\left(t\right)\right)\frac{t^{{\alpha }_4}}{\Gamma \left({\alpha }_4+1\right)}.\hfill \end{array}$$ (38)

At $n=1$, we have

$$\begin{array}{c}{s}_2\left(t\right)=L^{-1}\left\{\frac{1}{{\mathrm{\ell }}^{{\alpha }_1}}L\left(-\mu s_1-\beta X_1(1-c)-\rho s_1\right)\right\},\hfill \\ e_2\left(t\right)=L^{-1}\left\{\frac{1}{{\mathrm{\ell }}^{{\alpha }_2}}L\left(X_1(1-c)-(\mu +\varepsilon )e_1\right)\right\},\hfill \\ i_2\left(t\right)=L^{-1}\left\{\frac{1}{{\mathrm{\ell }}^{{\alpha }_3}}L\left(\varepsilon e_1-(\gamma +\mu +\alpha )i_1\right)\right\},\hfill \\ r_2\left(t\right)=L^{-1}\left\{\frac{1}{{\mathrm{\ell }}^{{\alpha }_4}}L\left(\gamma i_1-\mu r_1+\rho s_1\right)\right\}.\hfill \end{array}$$ (39)

Such that the following solutions are obtained:

$$s_2\left(t\right)=\left(\begin{array}{c}-(\mu +\rho )\beta (1-c)\hfill \\ (\Lambda -\mu s_0-\beta s_0{i}_0\hfill \\ (1-c)-\rho s_0)\hfill \end{array}\right)\frac{t^{2{\alpha }_1}}{\Gamma (2{\alpha }_1+1)}-\left(\begin{array}{c}\beta (1-c)s_0(\varepsilon e_0\hfill \\ -(\gamma +\mu +d)i_0)\hfill \end{array}\right)\times \frac{t^{{\alpha }_1+{\alpha }_3}}{\Gamma ({\alpha }_1+{\alpha }_3+1)}+\left(\begin{array}{c}{i}_0(\Lambda -\mu j_1\hfill \\ -\beta s_0{s}_0-\rho s_0\hfill \\ (1-c))\hfill \end{array}\right)\frac{t^{2{\alpha }_1}}{\Gamma (2{\alpha }_1+1)}$$

$$e_2\left(t\right)=\left(\begin{array}{c}\beta (1-c)s_0\varepsilon e_0-\beta \hfill \\ (1-c)(\gamma +\mu +d)i_0\hfill \end{array}\right)\frac{t^{{\alpha }_2+{\alpha }_3}}{\Gamma ({\alpha }_2+{\alpha }_3+1)}+i_0\left(\begin{array}{c}\Lambda -\mu s_0-\rho s_0\hfill \\ -\beta s_0{i}_0(1-c)\hfill \end{array}\right)\times \frac{t^{{\alpha }_2+{\alpha }_1}}{\Gamma ({\alpha }_2+{\alpha }_1+1)}-\left(\begin{array}{c}\beta {\alpha }_1{\alpha }_3(1-c)+\varepsilon \hfill \\ (\mu +\varepsilon ){\alpha }_2\mu \hfill \end{array}\right)\frac{t^{2{\alpha }_2}}{\Gamma (2{\alpha }_2+1)}$$

$$i_2\left(t\right)=\varepsilon \beta s_{01}{i}_0(1-c)-(\mu +\varepsilon )e_0\frac{t^{{\alpha }_3+{\alpha }_2}}{\Gamma ({\alpha }_2+{\alpha }_3+1)}-(\gamma +\mu +\alpha )i_0(\varepsilon e_0-(\gamma +\mu +\alpha )i_0)\frac{t^{2{\alpha }_{31}}}{\Gamma (2{\alpha }_3+1)}$$

$$r_2\left(t\right)=\left(\gamma \varepsilon e_0-\gamma (\gamma +\mu +\alpha )i_0\right)\frac{t^{{\alpha }_4+{\alpha }_3}}{\Gamma ({\alpha }_4+{\alpha }_3+1)}-\mu (\gamma i_0-\mu r_0+\rho s_0)\times \frac{t^{{\alpha }_4}}{\Gamma ({\alpha }_4+1)}+\rho \left(\begin{array}{c}\Lambda -\mu s_0-\beta s_0{i}_0\hfill \\ (1-c)-\rho s_0\hfill \end{array}\right)\frac{t^{{\alpha }_4+{\alpha }_1}}{\Gamma ({\alpha }_4+{\alpha }_1+1)}$$

We also computed the third-order approximation of each class using the MATHEMATICA 12 software package, and the approximate results of the classes are obtained as follows: i.e., 

$$\text{i.e., }S\left(t\right)=\sum _{n=0}^3{s}_n\left(t\right),C\left(t\right)=\sum _{n=0}^3{c}_n\left(t\right).I\left(t\right)=\sum _{n=0}^3{i}_n\left(t\right),$$

$$R\left(t\right)=\sum _{n=0}^3{r}_n\left(t\right),Q\left(t\right)=\sum _{n=0}^3{q}_n\left(t\right)$$

6. Results

The model’s raw series results, produced via the homotopy perturbation method and Laplace Adomian decomposition methods in Sections 5.1, 5.2, are examined with live COVID-19 data from Nigeria’s Center for Disease Control [44]. The outcomes, presented in the subsequent section, offer a third-order polynomial function with two key control parameters introduced in the model.

6.1. Numerical results of the homotopy perturbation method

$$S\left(t\right)≔470200+\left(\begin{array}{c}-4.70200\cdot 10^5\rho \hfill \\ +2425.479680\mathrm{c}\hfill \\ -2365.169040\hfill \end{array}\right)\mathrm{t}+\left(\begin{array}{c}-2425.479680\mathrm{c}\rho \hfill \\ +9.565559022\hfill \\ +2740.169040\rho \hfill \\ +2.35100\cdot 10^5{\rho }^2\hfill \\ +6.255797190{\mathrm{c}}^2\hfill \\ -15.86558804\mathrm{c}\hfill \end{array}\right)\frac{{\mathrm{t}}^2}{2}-\left(\begin{array}{c}17.08728806\mathrm{c}\rho -6.255797191\rho {\mathrm{c}}^2\hfill \\ +1212.739840\mathrm{c}{\rho }^2-0.05987847224\hfill \\ +0.1283929616\mathrm{c}-1432.584520{\rho }^2\hfill \\ -10.97060904\rho -78366.66667{\rho }^3\hfill \\ -0.07924949775{\mathrm{c}}^2+0.01075663474{\mathrm{c}}^3\hfill \end{array}\right)\frac{{\mathrm{t}}^3}{6}+\cdots$$

$$E\left(t\right)≔2003+\left(\begin{array}{c}-2425.479680\mathrm{c}\hfill \\ +2422.516842\hfill \end{array}\right)\mathrm{t}+\left(\begin{array}{c}1212.739840\rho \mathrm{c}\hfill \\ -9.622637507\hfill \\ -6.255797191{\mathrm{c}}^2\hfill \\ +15.88062601\mathrm{c}\hfill \\ -1212.739840\rho \hfill \end{array}\right)\frac{{\mathrm{t}}^2}{2}-\left(\begin{array}{c}-0.01075663474{\mathrm{c}}^3\hfill \\ -12.94152250\mathrm{c}\rho \hfill \\ +4.170531461\rho c^2\hfill \\ -404.2466133\mathrm{c}{\rho }^2\hfill \\ +0.06865180673\hfill \\ -0.1460992507\mathrm{c}\hfill \\ +404.2466133{\rho }^2\hfill \\ +8.770991040\rho \hfill \\ +0.08820299825{\mathrm{c}}^2\hfill \end{array}\right)\frac{{\mathrm{t}}^3}{6}+\cdots$$

$$I\left(t\right)≔416+\left(-.5936732694\right)\mathrm{t}-\left(\begin{array}{c}-0.01503797402\mathrm{c}\hfill \\ +0.01546094232\hfill \end{array}\right)\frac{{\mathrm{t}}^2}{2}+\left(\begin{array}{c}-0.00002585729505{\mathrm{c}}^2\hfill \\ +0.005012658004\mathrm{c}\rho \hfill \\ +00000.7309276088\mathrm{c}\hfill \\ -0.005012658004\rho \hfill \\ -0.0004743603205\hfill \end{array}\right)\frac{{\mathrm{t}}^3}{6}+\cdots$$

$$R\left(t\right)≔115+\left(470200\rho +0.1686803306\right)\mathrm{t}-\left(\begin{array}{c}-1527.429200\rho \hfill \\ +0.0001237089633\hfill \\ +1212.739840\rho \mathrm{c}\hfill \\ -2.35100000000{\rho }^2\hfill \end{array}\right)\frac{{\mathrm{t}}^2}{2}-\left(\begin{array}{c}-2.011579657.10^{-11}\mathrm{c}+0.0004\hfill \\ 123673942+2.085265730\rho {\mathrm{c}}^2\hfill \\ -808.4932267\rho {\mathrm{c}}^2+78366.66\hfill \\ 667{\rho }^3+398.9580840\rho \mathrm{c}-77453\hfill \\ .27699{\rho }^2-505.9545470\rho \hfill \end{array}\right)\frac{{\mathrm{t}}^3}{6}+\cdots$$

6.2. Numerical results of the Laplace adomian decomposition method

$$S\left(t\right)≔470200+\left(\begin{array}{c}-2365.1690399\hfill \\ +2425.47968\mathrm{c}\hfill \\ -470200\rho \hfill \end{array}\right)\mathrm{t}+\left(\begin{array}{c}5.746065097++235100{\rho }^2\hfill \\ -5.240246921129955\mathrm{c}\hfill \\ -1212.73984\rho \mathrm{c}+1902.4292\rho \hfill \end{array}\right)\frac{{\mathrm{t}}^2}{2}-\left(\begin{array}{c}0.0036807742703296002\hfill \\ -0.0036807742703296002\mathrm{c}\hfill \end{array}\right)\frac{{\mathrm{t}}^3}{6}+\cdots$$

$$E\left(t\right)≔2003+\left(\begin{array}{c}-2425.479680\mathrm{c}\hfill \\ +2422.516842\hfill \end{array}\right)\mathrm{t}+\left(\begin{array}{c}5.253093580456996\hfill \\ 5.253093580456996\mathrm{c}\hfill \end{array}\right){\mathrm{t}}^2+\left(\begin{array}{c}-0.0036807742703296002\hfill \\ 0.0036807742703296002\mathrm{c}\hfill \end{array}\right){\mathrm{t}}^3+\cdots$$

$$I\left(t\right)≔416+\left(-0.5936732694\right)\mathrm{t}-\left(\begin{array}{c}0.015460942322563322\hfill \\ -0.015037974015999999\mathrm{c}\hfill \end{array}\right){\mathrm{t}}^2+\left(\begin{array}{c}0.000029375250495871813\hfill \\ +0.000029174684271396126\mathrm{c}\hfill \end{array}\right){\mathrm{t}}^3+\cdots$$

$$R\left(t\right)≔115+\left(\begin{array}{c}470200\rho \hfill \\ -0.1686803306\hfill \end{array}\right)\mathrm{t}-\left(\begin{array}{c}-5.240246921129955\mathrm{c}\hfill \\ +235100{\rho }^2-1212.73984\rho \mathrm{c}\hfill \\ +1902.4292\rho \hfill \end{array}\right){\mathrm{t}}^2-\left(\begin{array}{c}0.0036513990198337285\hfill \\ -0.0036515995860582043\mathrm{c}\hfill \end{array}\right){\mathrm{t}}^3+\cdots$$

6.3. Error analysis

Performing an analysis of the model solution generated by HPM and LADM is crucial to determining which method is most appropriate for numerical simulation. In this section, an error analysis of both methods is performed to observe their convergence rates. Additionally, comparison plots of the two methods are provided to evaluate their agreement, while ensuring that all parameters retain their baseline values as defined in Table 1. The following results were obtained:

$$\begin{array}{c}{S}_H\left(t\right)≔470200-2365.169040\mathrm{t}+5.565559020{\mathrm{t}}^2-0.05987847222{\mathrm{t}}^3\hfill \\ S_L\left(t\right)≔470200-2365.17\mathrm{t}+5.746060810{\mathrm{t}}^2+0.003680775000{\mathrm{t}}^3\hfill \end{array}$$

$$\begin{array}{c}{E}_H\left(t\right)≔2003+2422.516842\mathrm{t}-5.593447004{\mathrm{t}}^2-0.02495188340{\mathrm{t}}^3\hfill \\ E_L\left(t\right)≔2003+2422.52\mathrm{t}-5.253090810{\mathrm{t}}^2-0.0036807750{\mathrm{t}}^2\hfill \end{array}$$ (40)

$$\begin{array}{c}{I}_H\left(t\right)≔416-0.5936732694\mathrm{t}+0.01546094232{\mathrm{t}}^2-0.00004743603205{\mathrm{t}}^3\hfill \\ I_L\left(t\right)=416-0.593673\mathrm{t}+0.0154609{\mathrm{t}}^2-0.0000293753{\mathrm{t}}^3\hfill \end{array}$$

$$\begin{array}{c}{R}_H\left(t\right)≔115-.1686803306\mathrm{t}+0.0001237089632{\mathrm{t}}^2\hfill \\ -0.00004123673942{\mathrm{t}}^3\hfill \\ R_L\left(t\right)=115-.016868\mathrm{t}+0.000123709{\mathrm{t}}^2-0.00000006.04648{\mathrm{t}}^8\hfill \end{array}$$

Table 1.

Variables, parameters, and descriptions.

Variable	Description
$S\left(t\right)$	Time relying count of vulnerable individuals
$E\left(t\right)$	Time relying count of exposed individuals
$I\left(t\right)$	Time relying count of infected individuals
$R\left(t\right)$	Time relying count of recovered individuals
Incorporated parameters	Description
$c$	High risk immunity rate
$\rho$	Vaccination rate of susceptible Covid-19 population
Parameters	Description
$\Lambda$	Influx rate
$\beta$	Effective transmission rate
$\varepsilon$	Rate of progression from exposed to infected
$\gamma$	Progression rate from infected to recovered
$\mu$	Natural death rate
$d$	Disease induced death rate

Eq. (40) represents the evaluated results of the homotopy perturbation method and the Laplace-Adomian decomposition method, respectively, on the susceptible class at classical order $\alpha =1$.

Convergence

The convergence of the methods strictly depends on the contraction of the approximation solution to the exact solution [45].

Theorem

Let there exist a mapping $k:m\to n$ defined on two Banach spaces $m$ , $n$ for all $s,t\in m$ then ${\Vert k\left(s\right)-k\left(t\right)\Vert }_n\le \delta {\Vert s-t\Vert }_m$ , $0<\delta <1$ such that the sequence $s_{r+1}=k^n\left(s_0\right)=\tau \left(s_0\right)$ for some $s_0\in m$ which converges to a unique fixed point $k$

Proof

We consider a Picard sequence $s_{r+1}=k\left(s_r\right)\subseteq n$ to prove the theorem. It is required to show that $s_r$ is convergent in $n$ for all $r\ge \nu$ such that $\Vert s_r-s_v\Vert \le \Vert s_r-s_{r+1}\Vert +\Vert s_{r+1}-s_{r+2}\Vert ++\Vert s_{r+2}-s_{r+3}\Vert +\cdots$ $+\Vert s_{r-1}-s_v\Vert$. The proof is defined by applying mathematical induction on the contractive property of contraction c such that $\Vert s_r-s_{r+1}\Vert \le {\delta }^r\Vert s_0-s_1\Vert$. This implies, ${lim}_{v\to \infty }\Vert s_r-s_v\Vert \le \frac{{\delta }^r}{1+\delta }\Vert s_0-s_1\Vert =0asr\to \infty$.

This proves that $\left(s_r\right)$ is Convergent in $n$ and through completeness of $n$, we can find $\omega \in n$: ${lim}_{r\to \infty }\left(s_r\right)=\omega \in n$. Clearly, contraction C ensures the continuity of $k$. Thus $\omega ={lim}_{r\to \infty }{s}_{r+1}=s_v$.

Lemma

The proposed mathematical model’s convergence of $s_r$ to $s_v$ cannot be asserted since no exact solution to the model exists.

Proof

This rate can be examined using Maclaurin’s error approximation [46] hence it follows:

Definition 8

Let, $\phi \left(x\right)$ be a polynomial of order $k$ with $k+1$ derivatives on the interval $\left|x\right|\le \varepsilon$ and $\left|{\phi }^{k+1}\left(x\right)\right|\le \mathrm{B}$, The Maclaurin’s truncation error of the $k\text{th}$ order Maclaurin’s polynomial of a function has the following bound as maximum error [46]

$${\delta }_n\left(t\right)=\underset{x\in [0,\varepsilon ]}{max}\left[\frac{\mathrm{B}}{(k+1)!}\cdot {\left|x\right|}^{{}^{k+1}}\right].$$

We apply this definition to compute the $3\text{rd}$ order maximum truncation error of the model’s series results for each class on an interval $[0,1]$. Hence, we construct the following functions (see Table 4, Table 5):

$$S_3\left(t\right)=\underset{t\in [0,1]}{max}\left[\frac{d^{3}S\left(t\right)}{d{t}^3}\cdot \frac{{\left|t\right|}^3}{3!}\right]E_3\left(t\right)=\underset{t\in [0,1]}{max}\left[\frac{d^{3}E\left(t\right)}{d{t}^3}\cdot \frac{{\left|t\right|}^3}{3!}\right],$$

$$I_3\left(t\right)=\underset{t\in [0,1]}{max}\left[\frac{d^{3}I\left(t\right)}{d{t}^3}\cdot \frac{{\left|t\right|}^3}{3!}\right],$$

$$R_3\left(t\right)=\underset{t\in [0,1]}{max}\left[\frac{d^{3}R\left(t\right)}{d{t}^3}\cdot \frac{{\left|t\right|}^3}{3!}\right],$$

Table 4.

$3\text{rd}$ order maximum truncation error for $t\in [0,1]$.

$t$	$S_{L3}\left(t\right)$	$S_{H3}\left(t\right)$	$E_{L3}\left(t\right)$	$E_{H3}\left(t\right)$	$I_{L3}\left(t\right)$	$I_{H3}\left(t\right)$	$R_{L3}\left(t\right)$	$R_{H3}\left(t\right)$
0	0	0	0	0	0	0	0	0
0.2	0.00000294	0.000479	0.0000294	0.0002	0.0000000235	0.0000000379	0.000000000484	0.0000000329894
0.4	0.000236	0.003832	0.000236	0.001597	0.000000188	0.000000304	0.00000000387	0.000000263915
0.6	0.000795	0.012934	0.000795	0.00539	0.000000635	0.00000102	0.00000000131	0.000000890714
0.8	0.001885	0.030658	0.001885	0.012775	0.00000015	0.000000243	0.00000000317	0.00000211132
1.0	0.003681	0.059878	0.003681	0.024952	0.000000294	0.000000474	0.0000000605	0.00000412367

Table 5.

Error range of each class at $t\in [0,1]$.

$S_{e3}\left(t\right)$	$0\le E_L\le 10^{-3}$	$0\le E_H\le 10^{-2}$
$E_{e3}\left(t\right)$	$0\le E_L\le 10^{-3}$	$0\le E_H\le 10^{-2}$
$I_{e3}\left(t\right)$	$0\le E_L\le 10^{-7}$	$0\le E_r\le 10^{-7}$
$R_{e3}\left(t\right)$	$0\le E_L\le 10^{-8}$	$0\le E_r\le 10^{-6}$

The error analysis of the approximation series shows the model is suitable for actual data simulation with a small error from third-order approximation. LADM had a lower error compared to HPM, indicating it is the best method for numerical simulation.

Comparison plots

Fig. 3 reflects the comparison and extent of agreement of the results produced by the two methods. A good correlation of the results is measured early on the graph before the graph diverges dramatically over time. Overall, it follows that the results of the two methods for the four classes follow the same trend. The error analysis performed showed that the third-order truncation error of the LADM is of lower order when compared to that of the HPM. Hence, we proceed to conduct the numerical simulation of the mathematical model using the LADM.

Fig. 3.

Comparison plots of HPM and LADM on the model classes.

6.4. Numerical simulation

In this section, we mainly emphasize the effects of the parameters: High risk immunity $c$, vaccination control $\rho$ and the fractional-order of the system in the presence of live COVID-19 data acquired from the Nigeria Centre for Disease Control on 1st December 2020 given that $E\left(0\right)=2003$, $I\left(0\right)=416$, $Q\left(0\right)=404$, $R\left(0\right)=115$ [26]. We shall conduct a case study on the real life acquired population of Ikeja Lagos, Nigeria [47] given that $S\left(0\right)=470200$. The dynamics are studied to determine the impact of the Caputo fractional order derivative on the distribution of the model population, as well as the impact of high-risk immunity and vaccination of susceptible individuals, in order to examine the effects on methods of controlling the COVID-19 virus and preventing its epidemic outbreak or future proliferation. Two scenarios are specifically examined to determine the influence of the factors on viral propagation.

The scenarios are:

• A.effective analysis of control applications which involves:
• i.effects of varying $0\le c<1$ on the model classes to study the population response and distributions to high risk immunity;
• ii.analysis of variation of $0\le \rho <1$ on the Susceptible, Exposed, infected and recovered humans for proposed controls scheme;
• iii.analysis of fractional order effect of combinations of both controls on the model class.

• B.analysis of model population reaction in the presence of combined
• i.implemented control strategies;
• ii.unimplemented control strategies;

6.4.1. Simulation outcome of scenario A

• 1.In this analysis, the effectiveness of vaccination in controlling the spread of the disease is examined by simulating the model class population. The data presented below was obtained through numerical simulations of each model equation.

It is worth noting that increasing vaccination rates in the population may not have a drastic impact on reducing the number of infected cases. The data recorded in the table suggests that increasing the vaccination rate can potentially decrease the number of exposed and susceptible individuals, but not necessarily in a significant way (see Table 6).

Table 6.

Dynamical variation in population response to vaccination $\rho$.

Class	$\rho =0$	$\rho =0.3$	$\rho =0.9$
Susceptible	346 607.0930	254 624.9961	179 323.1486
Exposed	4091.154396	3839.110328	3659.830648
Infected	415.4067058	415.3924197	415.3781336
Recovered	12 176.72336	21 398.21368	28 945.49406

• 1.Here the impact of high-risk quarantine is analyzed on the model population to observe the population distribution towards its implementation

The findings from the simulation suggest that high-risk quarantine has an impact on the population comparable to vaccination but with a slower rate of effectiveness. This highlights the importance of vaccination in controlling the spread of the disease, particularly in protecting susceptible individuals who may be at high risk of exposure in infected areas. Even in situations where a complete curfew cannot be implemented, it is suggested that other health regulations, such as social distancing and regular hand washing should be strictly enforced in order to reduce the transmission of the disease (see Table 7).

Table 7.

Dynamical variation in population response to vaccination $\rho =0$.

Class	$c=0$	$c=0.3$	$c=0.9$
Susceptible	398 003.4643	401 075.3989	400 969.9147
Exposed	9760.772200	2953.560982	3106.563423
Infected	413.2887348	415.4126917	413.0775985
Recovered	64 843.43772	65 084.51221	65 326.99425

• 2.Population distributions with respect to combined fractional implementation of vaccination and imposition of high risk quarantine

6.4.2. Simulation outcome of scenario B:

Here, we do some mathematical analysis and present some comparison plots of the results obtained for the combined maximum applications of mitigation strategies and various responses that were received for each class at both classical order $\alpha =1$ and fractional order $\alpha =0.75$.

The result of the analysis shows that the implementation of control strategies, particularly vaccination, has a significant impact on reducing the spread of the disease in the population. In the susceptible class, a decline was observed when control measures were implemented, with a more drastic reduction in population when the control measure was implemented at a classical derivative $\alpha =1$. This takes more time than when control measures are implemented at a fractional level $\alpha =0.75$. Similarly, in the infected class, a sharp decline was observed after a month of control implementation at $\alpha =1$. The number of infected individuals (390) recorded at $\alpha =1$ is much lower (406) than when control measures are implemented at $\alpha =0.75$. The recovered class also shows the importance of the control measures studied, with a steady increase in population observed in both cases, although the progression is higher when the control measure is implemented at an integer derivative.

7. Discussion

This section provides a detailed discussion of our analysis results on the transmission control of the COVID-19 virus, which were obtained through numerical simulations using real-time data from Nigeria’s Ikeja region. The evaluation of the basic reproductive ratio in the area is relatively high, making it an ideal setting for investigating the efficacy of vaccination and high-risk quarantine measures in reducing disease prevalence and new cases. Our results reveal that the combination of both vaccination and quarantine measures produces a faster decline in disease prevalence and new cases, as depicted in Fig. 4, Fig. 5. It was found that in the initial phases of the pandemic, the effectiveness of vaccines did not have a significant influence on transmission. Nevertheless, as the pandemic entered remission, a more potent vaccination implementation was linked to a faster decline in cases. Vaccinating susceptible individuals is particularly effective in preventing disease spread and decreasing overall prevalence, as demonstrated in scenario A. In addition, we investigated the impact of a high-risk quarantine period on the transmission of COVID-19. The findings, as presented in Fig. 5, suggest that a longer period of high-risk quarantine can aid in preventing the spread of the pandemic. Although high-risk quarantine measures offer protection, their efficacy is lower than that of vaccination. Moreover, we explored the impact of fractional ordering on control measures and observed that higher orders result in a greater decrease in disease prevalence. In scenario B, we studied the effects of maximum and zero control implementation on susceptible, infected, and recovered individuals. Our findings, as observed in Fig. 6, suggest that full control implementation at classical orders lead to a slower spread of the disease than fractional orders. Fig. 7 provides evidence that the dual implementation of vaccination and high-risk quarantine measures produces a rapid and positive response in reducing the reproductive ratio. Hence, we recommend a dual implementation of vaccination and high-risk quarantine measures to significantly reduce disease transmission and spread. Moreover, it is essential to ensure that vaccines are widely distributed to the masses and that the rate of vaccine uptake among them is high.

Fig. 4.

Reaction of the four population class to vaccination at time variations.

Fig. 5.

Impact of high risk quarantine on the model classes.

Fig. 6.

Fractional order dynamics of vaccination and high risk quarantine on recovered population.

Fig. 7.

Shows the reaction of SIR population class to control strategies at a Caputo classical order $\alpha =1$ and fractional order $\alpha =0.75$.

8. Conclusion

The proposed model of COVID-19 transmission dynamics has been analyzed to investigate the combined effects of high-risk quarantine and vaccination. A separate study [48] titled “vaccination and quarantine effect on COVID-19 transmission dynamics incorporating the Chinese spring festival travel rush” suggests that a minimum of 61.38% of people must be vaccinated to achieve herd immunity. When vaccination and quarantine are implemented simultaneously, a quarantine rate of 38.74 percent is necessary to prevent further spread of the disease. These findings are consistent with our simulation results, which demonstrate the efficacy of combining these two methods. Additionally, a higher number of vaccinations and greater vaccine effectiveness lead to a more significant positive effect, while a longer immune protection period helps to help suppress the spread of the pandemic.

Theoretical and practical implications

Our analysis indicates that combining vaccination measures with high-risk quarantine could be an effective strategy for reducing the spread of the virus. These measures may involve protecting vulnerable populations and minimizing the negative impact of the pandemic, which could potentially require implementing temporary curfews. However, it is crucial to ensure the welfare of those living in affected regions to gain their cooperation with the proposed measures. Furthermore, careful planning, resource allocation, and ongoing development of vaccine distribution are essential to ensure timely access to protection against the virus and prevent prolonging any curfew measures. Implementing our research findings could lead to several positive outcomes, including lower hospitalization and mortality rates as well as economic recovery.

Recommendations for future research

Our future goal is to develop a model that integrates screening and vaccination protocols for immigrants, providing a more accurate assessment of COVID-19 transmission within a larger population that incorporates vaccines and quarantine measures. Through this research, we intend to offer valuable insights to policymakers who seek effective strategies for preventing the spread of COVID-19. Ultimately, our findings are believed to play a vital role in shaping public health policies aimed at curbing the transmission of this pandemic.

CRediT authorship contribution statement

Morufu Oyedunsi Olayiwola: Innovation, Model design, Read and approve the manuscript. Adedapo Ismaila Alaje: Model solution, Computations, Read and approve the manuscript. Akeem Yunus Olarewaju: Qualitative analysis, Editing, Read and approve the manuscript. Kamilu Adewale Adedokun: Simulations, Results discussion, Read and approve the manuscript.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Data availability

Data will be made available on request.