Analytical and qualitative investigation of COVID‐19 mathematical model under fractional differential operator

Abstract

In the current article, we aim to study in detail a novel coronavirus (2019‐nCoV or COVID‐19) mathematical model for different aspects under Caputo fractional derivative. First, from analysis point of view, existence is necessary to be investigated for any applied problem. Therefore, we used fixed point theorem's due to Banach's and Schaefer's to establish some sufficient results regarding existence and uniqueness of the solution to the proposed model. On the other hand, stability is important in respect of approximate solution, so we have developed condition sufficient for the stability of Ulam–Hyers and their different types for the considered system. In addition, the model has also been considered for semianalytical solution via Laplace Adomian decomposition method (LADM). On Matlab, by taking some real data about Pakistan, we graph the obtained results. In the last of the manuscript, a detail discussion and brief conclusion are provided.

1. INTRODUCTION

In the last of 2019, a dangerous out break was confirmed in the big city of China called Wuhan which was due by coronavirus. There are many theories behind the virus outbreak in present‐day literature. Some researchers have studied that it originated from bats to humans owing to the complete unlawful transmission of animals on a Wuhan seafood market. The virus involved has been identified both in dogs and in pangolin. Hence, it has been considered that several infected cases claimed that they had been working in a local fish and wild animal market in Wuhan from where they got the infection of COVID‐19. Afterward, the researchers have confirmed that the disease's widespread nature is caused by direct contact between people. The mentioned disease was then named COVID‐19. Nearly 100 000 people were infected in China up to March 2020. Then, the virus was transmitted from person to person very rapidly and spread up to the end of March 2020 in the whole globe. This pandemic is ongoing, and more than 7.38 million people almost in every country of the globe have been infected. Nearly 0.41 million people died from this pandemic. WHO announced that it is a global pandemic where scientists and researchers have not found success in controlling or curing this disease. ¹ , ²

Here, we remark that one way to open up this disease is due to immigration of the infected population from one area to another which affects the healthy population and spreads this disease very rapidly. In fact, it is like the flue which has killed millions people in past. Therefore, globally, all the countries of the entire world have reduced their activities to some restricted domain. Every country of the world restricts the movement of the people from one place to other. ³ Currently, researchers work on this outbreak. ⁴ , ⁵ Various models in mathematical form have been developed for these diseases. For some study, we refer previous works ⁶ , ⁷ , ⁸ and work cited therein. All the diseases are dangerous, but infectious diseases greatly effect the health as well as economy of a state. In the past for many diseases, controlling procedure and its future predictions are made by using mathematical modeling. ⁹ Considering the threat of current pandemic, researchers have studied the COVID‐19 disease from different aspects. ¹⁰ , ¹¹ , ¹² , ¹³ , ¹⁴ Mathematical models in this regard can play a vital role to restrict the disease spreading. For this need very recently, some models have been studied for COVID‐19; see detail in other works. ¹⁵ , ¹⁶ , ¹⁷ , ¹⁸ , ¹⁹ , ²⁰ , ²¹ , ²² , ²³ , ²⁴

It is often desirable to describe the behavior of some real‐life system or phenomenon, whether physical, sociological, or even economic, in mathematical terms. The mathematical description of a system of phenomenon is called a mathematical model and is constructed with certain goals in mind. Put it simply, mathematical modeling should become part of the toolbox of public health research and decision making. One way to study problem dynamics in the real world is to use mathematical modeling. Mathematical modeling is the ability to convert real‐world problems into mathematical formulations whose theoretical and numerical analysis can provide useful insights and guidance for native applications. In the field of mathematics, many mathematical models have been built according to the assumptions of researchers. The purpose of mathematical modeling is to represent different types of real world situation in the language of mathematics. To find out the different dynamics of a disease and therefore to overcome it at an early stage, mathematical modeling plays an important role there. Thus simply, we can say that mathematical modeling has important role to study a disease and control of the disease specially when vaccination is absent. A lot of mathematical models exist in the literature on stability, existence theory, and optimization of biological models; for example, see (other works ²⁵ , ²⁶ , ²⁷ ). Currently, for COVID‐19, some models have been constructed to study its different aspects (see Lai et al. and Fanelli and Piazza ²⁸ , ²⁹ ). In this way, Lin and his coauthors in their study ¹⁵ model COVID‐19 under integer‐order derivative as

$$\left\{\begin{array}{cc}\hfill & S^{\prime }(\mathrm{t})=\frac{-b_{0}S\mathrm{G}}{N}-\frac{b(\mathrm{t})SI}{N}-uS,\hfill \\ \hfill & E^{\prime }(\mathrm{t})=\frac{b_{0}S\mathrm{G}}{N}+\frac{b(\mathrm{t})SI}{N}-(d+u)E,\hfill \\ \hfill & I^{\prime }(\mathrm{t})=dE-(l+u)I,\hfill \\ \hfill & R^{\prime }(\mathrm{t})=lI-uR,\hfill \\ \hfill & N^{\prime }(\mathrm{t})=-uN,\hfill \\ \hfill & D^{\prime }(\mathrm{t})=vlI-zD,\hfill \\ \hfill & C^{\prime }(\mathrm{t})=dE,\hfill \end{array}\right.$$ (1)

where $b(\mathrm{t})=b_0(1-a){\left(1-\frac{D}{N}\right)}^k$, where S(t) is the susceptible populations, E(t) is exposed populations, I(t) is infectious populations, R(t) is removed population (recovered and dead), N(t) is total populations, D(t) is mimicking the public perception of risk regarding the number of severe and critical cases and deaths, and C(t) is the cumulative cases (reported and not reported). The authors establish this model on Wuhan city for the month of March 2020.

Detail of various parameters of model (1) is given as follows:

G

No. of zoonotic cases

b ₀

Rate of transmission

a

Action strength of government

k

Responds to intensity

u

Rate of emigration

d ⁻¹

Latent mean period

l ⁻¹

Infectious mean period

v

Severe cases proportion

z ⁻¹

Public reaction mean duration

As we know that most of biological models are based on the classical approach to get a system of nonlinear autonomous first‐order differential equations. Therefore, there is still room to develop such a mathematical models by the tools of advanced fractional calculus. Several novel studies, including numerous biological models, have shown to be extremely helpful and more accurate than their counterparts. For example, Ullah et al ³⁰ proposed a tuberculosis (TB) infection mathematical model for the Khyber Pakhtunkhwa province of Pakistan involving Caputo fractional‐order derivative and tested it by the real data of the mentioned province and shows the advantages of fractional‐order model. Khan et al ³¹ introduced pine wilt disease model involving fractional‐order derivative of Caputo–Fabrizio type and confirmed its productiveness by establishing its unique solution. Qureshi and Atangana ³² analyzed ordinary and fractional‐order models of dengue outbreak in Cape Verde islands during 2009 and showed that fractional‐order operator in the sense of Caputo–Fabrizio has the smallest squared sum of errors. Muhammad and Atangana ³³ analyzed the mathematical model of infectious disease called Ebola via Caputo, Caputo–Fabrizio, and Atanagana–Baleanu operators. The advanced studies show that the fractal–fractional‐order operators are the best tools to analyze the mathematical models for real‐world data; for detail, see previous studies. ³⁴ , ³⁵ , ³⁶ The beauty of fractional calculus is that fractional derivatives (and integrals) are not a local (or point) property (or quantity). Thereby, this considers the history and nonlocal distributed effects. In other words, perhaps this subject translates the reality of nature better. Therefore, to make this subject available as popular subject to science and engineering community, it adds another dimension to understand or describe basic nature in a better way.

Existence theory is one of the interesting areas of differential equations. This area has been studied very well in the last couple of decades. In the present literature, various approaches were utilized to show existence and uniqueness of solution of differential and integral equations. Krassnoselskii's fixed point theorem, Lery–Schauder fixed point theorem, Schaefer's fixed point theorem, and degree theory are commonly used to investigate the solutions for fractional‐order differential equations; see other studies. ³⁷ , ³⁸ , ³⁹ , ⁴⁰ , ⁴¹ , ⁴²

Because most of nonlinear problems cannot be solved for exact solution, we need powerful numerical or some analytical techniques. For good numerical results, one needs stable algorithms and methods. For such need, stability theory was founded. In the literature, there are different types of stability. But the most important type is Ulam–Hyers type stability which was introduced by Ulam ⁴³ in 1940 and further studied by Hyers ⁴⁴ in 1941. This stability answers the question when is it true that a function which approximately satisfies a functional equation must be close to an exact solution of this functional equation? If the problem accepts a solution, we say that the functional equation is stable. The mentioned stabilities have been studied for different physical problem; for example, see previous works. ⁴² , ⁴⁵ , ⁴⁶

Inspired of the work of Lin et al ¹⁵ and realistic nature of fractional calculus, we consider model (1) for existence, Ulam–Hyers stability, and for semianalytical solution under Caputo fractional‐order derivative as

$$\left\{\begin{array}{cc}\hfill {}^{\mathrm{c}}{\mathrm{D}}_{0^{+}}^{\alpha }S(\mathrm{t})& =\frac{-b_{0}S\mathrm{G}}{N}-\frac{b(\mathrm{t})SI}{N}-uS,\hfill \\ \hfill {}^{\mathrm{c}}{\mathrm{D}}_{0^{+}}^{\alpha }E(\mathrm{t})& =\frac{b_{0}S\mathrm{G}}{N}+\frac{b(\mathrm{t})SI}{N}-(d+u)E,\hfill \\ \hfill {}^{\mathrm{c}}{\mathrm{D}}_{0^{+}}^{\alpha }I(\mathrm{t})& =dE-(l+u)I,\hfill \\ \hfill {}^{\mathrm{c}}{\mathrm{D}}_{0^{+}}^{\alpha }R(\mathrm{t})& =lI-uR,\hfill \\ \hfill {}^{\mathrm{c}}{\mathrm{D}}_{0^{+}}^{\alpha }N(\mathrm{t})& =-uN,\hfill \\ \hfill {}^{\mathrm{c}}{\mathrm{D}}_{0^{+}}^{\alpha }D(\mathrm{t})& =vlI-zD,\hfill \\ \hfill {}^{\mathrm{c}}{\mathrm{D}}_{0^{+}}^{\alpha }C(\mathrm{t})& =dE.\hfill \end{array}\right.$$ (2)

with 0 < α≤1, $b(\mathrm{t})=b_0(1-a){\left(1-\frac{D}{N}\right)}^k$

Initial conditions assume as follows:

$$\left\{\begin{array}{cc}\hfill S(0)& =S_0,\hfill \\ \hfill E(0)& =E_0,\hfill \\ \hfill I(0)& =I_0,\hfill \\ \hfill R(0)& =R_0,\hfill \\ \hfill N(0)& =N_0,\hfill \\ \hfill D(0)& =D_0,\hfill \\ \hfill C(0)& =C_0.\hfill \end{array}\right.$$ (3)

First, we will develop qualitative theory for the above model, then we will discuss Ulam–Hyers stability and, finally, an approximate solution of the considered model (2) to be discussed. In our analysis, we used “Banach and Schaefer's fixed point theorem” to study the existence and stability of the mentioned model. At last, a general scheme is established to obtain series type solution. Also, we testify the approximate results for the model (2) on the real data of Pakistan for the last 60 days taken from a source. ⁴⁷ At the end, graphical results are given with detail discussion.

2. FUNDAMENTAL RESULTS

Some basic results are recall as follows.

Definition 2.1

Kilbas et al. ²⁷ The fractional integral of order α ∈ R ⁺, for a function W(t) ∈ L ¹ ([a, b], R) is defined as

$${\mathbf{I}}_{a^{+}}^{\alpha }\mathbf{W}(\mathrm{t})=\frac{1}{\mathrm{\Gamma }(\alpha )}{\int }_0^{\mathrm{t}}{(\mathrm{t}-s)}^{\alpha -1}\mathbf{W}(s)\text{ds},$$ (4)

where 0 < α≤1 and integral on the right is assumed to converge.

Definition 2.2

Kilbas et al. ²⁷ The Caputo fractional‐order derivative of a function W(t) ∈ C ⁿ[a, b] is defined as

$${}^{\mathrm{c}}{\mathbf{D}}_{0^{+}}^{\alpha }\mathbf{W}(\mathrm{t})=\frac{1}{\mathrm{\Gamma }(\mathrm{n}-\alpha )}{\int }_0^{\mathrm{t}}{(\mathrm{t}-s)}^{\mathrm{n}-\alpha -1}{\mathbf{W}}^{(\mathrm{n})}(s)ds;$$ (5)

here, 0 < α≤1, $\mathrm{n}=[\alpha ]+1$, and [α] shows integer part of α.

Lemma 2.3

Kilbas et al. ²⁷ Suppose U(t) ∈ C([0, τ]), then the solution of fractional‐order differential equations of the form

$$\left\{\begin{array}{cc}\hfill {}^{\mathrm{c}}{\mathbf{D}}_{0^{+}}^{\alpha }\mathbf{W}(\mathrm{t})& =\mathrm{U}(\mathrm{t}),\mathrm{t}\in [0,\tau ],\mathrm{n}-1<\alpha <\mathrm{n},\hfill \\ \hfill \mathbf{W}(0)& ={\mathrm{U}}_0\hfill \end{array}\right.$$ (6)

is given by

$$\mathbf{W}(\mathrm{t})={\displaystyle \sum _{i=0}^{\mathrm{n}-1}}{c}_i{\mathrm{t}}^i,$$ (7)

for c _i  ∈ R, $\mathrm{i}\mathrm{=}\mathrm{0}\mathrm{,}\mathrm{1}\mathrm{,}\mathrm{2}\mathrm{,}\mathrm{\dots }\mathrm{,}\mathrm{n}\mathrm{-}\mathrm{1}$.

Definition 2.4

Podlubny. ⁴⁸ The Laplace transform of Caputo fractional‐order derivative is defined as follows:

$$\begin{array}{c}\hfill ℒ[{}^{\mathrm{c}}{\mathrm{D}}_{0^{+}}^{\alpha }\mathbf{W}(\mathrm{t})]=s^{\alpha }\mathbf{W}(s)-{\displaystyle \sum _{k=0}^{\mathrm{n}-1}}{s}^{\alpha -k-1}{\mathbf{W}}^k(0),\mathrm{n}-1<\alpha <\mathrm{n}.\end{array}$$

For a Banach's space $\mathbf{X}=\mathbf{C}([0,\tau ])$ with norm

$$\Vert \mathbf{W}\Vert =\underset{\mathrm{t}\in [0,\tau ]}{\max }\left\{|\mathbf{W}|,\text{for all}\mathbf{W}\in \mathbf{X}\right\},$$

we have the following results.

Theorem 2.5

Granas and Dugundji ⁴⁹ Let X be a Banach space and T : X → X compact and continuous. If the set

$$\mathbf{E}=\{\mathbf{W}\in \mathbf{X}:\mathbf{W}=\lambda \mathbf{T}\mathbf{W},\lambda \in (\mathrm{0,1})\},$$ (8)

is bounded, then T has a fixed point.

3. EXISTENCE THEORY

First of all, we will consider model (2) for qualitative theory. For this, we put the right hand side of model (2) as follows:

$$\left\{\begin{array}{cc}\hfill {\mathbf{\text{W}}}_1(\mathrm{t},S,E,I,R,N,D,C)& =\frac{-b_{0}S\mathrm{G}}{N}-\frac{b(\mathrm{t})SI}{N}-uS,\hfill \\ \hfill {\mathbf{\text{W}}}_2(\mathrm{t},S,E,I,R,N,D,C)& =\frac{b_{0}S\mathrm{G}}{N}+\frac{b(\mathrm{t})SI}{N}-(d+u)E,\hfill \\ \hfill {\mathbf{\text{W}}}_3(\mathrm{t},S,E,I,R,N,D,C)& =dE-(l+u)I,\hfill \\ \hfill {\mathbf{\text{W}}}_4(\mathrm{t},S,E,I,R,N,D,C)& =lI-uR,\hfill \\ \hfill {\mathbf{\text{W}}}_5(\mathrm{t},S,E,I,R,N,D,C)& =-uN,\hfill \\ \hfill {\mathbf{\text{W}}}_6(\mathrm{t},S,E,I,R,N,D,C)& =vlI-zD,\hfill \\ \hfill {\mathbf{\text{W}}}_7(\mathrm{t},S,E,I,R,N,D,C)& =dE.\hfill \end{array}\right.$$ (9)

Using Equation (9), model (2) can be expressed as follows:

$$\left\{\begin{array}{cc}\hfill {}^{\mathrm{c}}{\mathbf{D}}_{0^{+}}^{\alpha }\mathbf{W}(\mathrm{t})& =\mathbf{Q}(\mathrm{t},\mathbf{W}(\mathrm{t})),\mathrm{t}\in [0,\tau ],0<\alpha \le 1,\hfill \\ \hfill \mathbf{W}(0)& ={\mathbf{W}}_0,\hfill \end{array}\right.$$ (10)

where

$$\mathbf{W}(\mathrm{t})=\left\{\begin{array}{cc}\hfill & S(\mathrm{t}),\hfill \\ \hfill & E(\mathrm{t}),\hfill \\ \hfill & I(\mathrm{t}),\hfill \\ \hfill & R(\mathrm{t}),\hfill \\ \hfill & N(\mathrm{t}),\hfill \\ \hfill & D(\mathrm{t}),\hfill \\ \hfill & C(\mathrm{t}).\hfill \end{array}\right.,{\mathbf{W}}_0(\mathrm{t})=\left\{\begin{array}{cc}\hfill & S_0,\hfill \\ \hfill & E_0,\hfill \\ \hfill & I_0,\hfill \\ \hfill & R_0,\hfill \\ \hfill & N_0,\hfill \\ \hfill & D_0,\hfill \\ \hfill & C_0,\hfill \end{array}\right.,\mathbf{Q}(\mathrm{t},\mathbf{W}(\mathrm{t}))=\left\{\begin{array}{cc}\hfill & {\mathbf{\text{W}}}_1(\mathrm{t},S,E,I,R,N,D,C),\hfill \\ \hfill & {\mathbf{\text{W}}}_2(\mathrm{t},S,E,I,R,N,D,C),\hfill \\ \hfill & {\mathbf{\text{W}}}_3(\mathrm{t},S,E,I,R,N,D,C),\hfill \\ \hfill & {\mathbf{\text{W}}}_4(\mathrm{t},S,E,I,R,N,D,C),\hfill \\ \hfill & {\mathbf{\text{W}}}_5(\mathrm{t},S,E,I,R,N,D,C),\hfill \\ \hfill & {\mathbf{\text{W}}}_6(\mathrm{t},S,E,I,R,N,D,C),\hfill \\ \hfill & {\mathbf{\text{W}}}_7(\mathrm{t},S,E,I,R,N,D,C).\hfill \end{array}\right.$$ (11)

On Lemma 2.3, Equation (10) can be converted to an equivalent integral form as follows:

$$\mathbf{W}(\mathrm{t})={\mathbf{W}}_0(\mathrm{t})+\frac{1}{\mathrm{\Gamma }(\alpha )}{\int }_0^{\mathrm{t}}{(\mathrm{t}-s)}^{\alpha -1}\mathbf{Q}(s,\mathbf{W}(s))ds.$$ (12)

The following assumptions are key to our analysis:

(H1)

If there exist constants K _Q , M _Q , and q ∈ [0, 1), such that

$$|\mathbf{Q}(\mathrm{t},\mathbf{W}(\mathrm{t}))|\le K_{\mathbf{Q}}|\mathbf{W}{|}^q+M_{\mathbf{Q}}.$$

(H2)

If there exist constants L _Q  > 0, and for each $\mathbf{W},\overline{\mathbf{W}}\in \mathbf{X}$, such that

$$|\mathbf{Q}(\mathrm{t},\mathbf{W})-\mathbf{Q}(\mathrm{t},\overline{\mathbf{W}})|\le L_{\mathbf{Q}}[|\mathbf{W}-\overline{\mathbf{W}}|].$$

Define the map T : X → X as follows:

$$\mathbf{T}\mathbf{W}(\mathrm{t})={\mathbf{W}}_0(\mathrm{t})+\frac{1}{\mathrm{\Gamma }(\alpha )}{\int }_0^{\mathrm{t}}{(\mathrm{t}-s)}^{\alpha -1}\mathbf{Q}(s,\mathbf{W}(s))ds.$$ (13)

Theorem 3.1

If assumptions H ₁ and H ₂ hold, then problem (10) possesses at least one solution. Consequently, our considered model (2) has at least one solution.

With the help of Schaefer's fixed point theorem, the proof of the results will be present in four steps.

Step 1: First, we have to show T is continuous. Assume W _i is continuous for $i=\mathrm{1,2},3,..,7$. Thus, Q(t, W(t)) is continuous. Let W _j, W ∈ X such that W _j → W, and we must have TW _j → TW.

Consider

$$\begin{array}{cc}\hfill \Vert \mathbf{T}{\mathbf{W}}_{\mathrm{j}}-\mathbf{T}\mathbf{W}\Vert & =\underset{\mathrm{t}\in [0,\tau ]}{\max }\left|\frac{1}{\mathrm{\Gamma }(\alpha )}{\int }_0^{\mathrm{t}}{(\mathrm{t}-s)}^{\alpha -1}{\mathbf{Q}}_{\mathrm{j}}(s,{\mathbf{W}}_{\mathrm{j}}(s))ds-\frac{1}{\mathrm{\Gamma }(\alpha )}{\int }_0^{\mathrm{t}}{(\mathrm{t}-s)}^{\alpha -1}\mathbf{Q}(s,\mathbf{W}(s))ds\right|\hfill \\ \hfill & \le \underset{\mathrm{t}\in [0,\tau ]}{\max }{\int }_0^{\mathrm{t}}\left|\frac{{(\mathrm{t}-s)}^{\alpha -1}}{\mathrm{\Gamma }(\alpha )}\right|\left|{\mathbf{Q}}_{\mathrm{j}}(s,{\mathbf{W}}_{\mathrm{j}}(s))-\mathbf{Q}(s,\mathbf{W}(s))\right|ds\hfill \\ \hfill & \le \frac{{\tau }^{\alpha }{L}_{\mathbf{Q}}}{\mathrm{\Gamma }(\alpha +1)}\Vert {\mathbf{W}}_{\mathrm{j}}-\mathbf{W}\Vert \to 0\text{as}\mathrm{j}\to \infty .\hfill \end{array}$$

Since Q is continuous, thus TW _j → TW; as a result ,T is continuous.

Step 2: The operator T is bounded. For any W ∈ X, the function T enjoys the growth condition as follows:

$$\begin{array}{cc}\hfill \Vert \mathbf{T}\mathbf{W}\Vert & =\underset{\mathrm{t}\in [0,\tau ]}{\max }\left|{\mathbf{W}}_0(\mathrm{t})+\frac{1}{\mathrm{\Gamma }(\alpha )}{\int }_0^{\mathrm{t}}{(\mathrm{t}-s)}^{\alpha -1}\mathbf{Q}(s,\mathbf{W}(s))ds\right|\hfill \\ \hfill & \le |{\mathbf{W}}_0|+\underset{\mathrm{t}\in [0,\tau ]}{\max }\frac{1}{\mathrm{\Gamma }(\alpha )}{\int }_0^{\mathrm{t}}\left|{(\mathrm{t}-s)}^{\alpha -1}\right|\left|\mathbf{Q}(s,\mathbf{W}(s))\right|ds\hfill \\ \hfill & \le |{\mathbf{W}}_0|+\frac{{\tau }^{\alpha }}{\mathrm{\Gamma }(\alpha +1)}\left[K_{\mathbf{Q}}\Vert \mathbf{W}{\Vert }^q+M_{\mathbf{Q}}\right].\hfill \end{array}$$

Let S be any bounded subset of X. We need to show T(S) is bounded. For any W ∈ S, since S is bounded, therefore, there exists K≥0, such that

$$\Vert \mathbf{W}\Vert \le K,\forall \mathbf{W}\in \mathbf{S}.$$ (14)

Hence, for any W ∈ S using above growth conditions, one has

$$\Vert \mathbf{T}\mathbf{W}\Vert \le |{\mathbf{W}}_0|+\frac{{\tau }^{\alpha }}{\mathrm{\Gamma }(\alpha +1)}\left[K_{\mathbf{Q}}\Vert \mathbf{W}{\Vert }^q+M_{\mathbf{Q}}\right]\le |{\mathbf{W}}_0|+\frac{{\tau }^{\alpha }}{\mathrm{\Gamma }(\alpha +1)}\left[K_{\mathbf{Q}}{K}^q+M_{\mathbf{Q}}\right].$$

Hence, T(S) is bounded.

Step 3: For equi‐continuity, let t₁, t₂ ∈ [0, τ], such that t₁≥t₂, then

$$\begin{array}{cc}\hfill |\mathbf{T}\mathbf{W}({\mathrm{t}}_1)-\mathbf{T}\mathbf{W}({\mathrm{t}}_2)|& =\left|\frac{1}{\mathrm{\Gamma }(\alpha )}{\int }_0^{{\mathrm{t}}_1}{({\mathrm{t}}_1-s)}^{\alpha -1}\mathbf{Q}(s,\mathbf{W}(s))ds-\frac{1}{\mathrm{\Gamma }(\alpha )}{\int }_0^{{\mathrm{t}}_2}{({\mathrm{t}}_2-s)}^{\alpha -1}\mathbf{Q}(s,\mathbf{W}(s))ds\right|\hfill \\ \hfill & \le \left|\frac{1}{\mathrm{\Gamma }(\alpha )}{\int }_0^{{\mathrm{t}}_1}{({\mathrm{t}}_1-s)}^{\alpha -1}-\frac{1}{\mathrm{\Gamma }(\alpha )}{\int }_0^{{\mathrm{t}}_2}{({\mathrm{t}}_2-s)}^{\alpha -1}\right|\left|\mathbf{Q}(s,\mathbf{W}(s))\right|ds\hfill \\ \hfill & \le \frac{{\tau }^{\alpha }}{\mathrm{\Gamma }(\alpha +1)}\left[K_{\mathbf{Q}}\Vert \mathbf{W}{\Vert }^q+M_{\mathbf{Q}}\right][{\mathrm{t}}_1^{\alpha }-{\mathrm{t}}_2^{\alpha }]\to 0\text{as}{\mathrm{t}}_1\to {\mathrm{t}}_2.\hfill \end{array}$$

Hence, by Arzelá–Ascoli theorem, T(S) is relatively compact.

Step 4: Finally, to show that the set

$$\mathbf{E}=\left\{\mathbf{W}\in \mathbf{X}:\mathbf{W}=\lambda \mathbf{T}\mathbf{W},\lambda \in (\mathrm{0,1})\right\},$$ (15)

is bounded, let W ∈ E, then for each t ∈ [0, τ], we have

$$\Vert \mathbf{W}\Vert =\lambda \Vert \mathbf{T}\mathbf{W}\Vert \le \lambda \left[|{\mathbf{W}}_0|+\frac{{\tau }^{\alpha }}{\mathrm{\Gamma }(\alpha +1)}\left[K_{\mathbf{Q}}\Vert \mathbf{W}{\Vert }^q+M_{\mathbf{Q}}\right]\right]$$

which shows E is bounded. Hence, by Schaefer's theorem, T possesses a fixed point; consequently, our considered problem (10) has at least one solution.

Remark 3.2

If the assumption (H ₁ ) is constructed for $q=1$, then Theorem 3.1 still holds if $\frac{{\tau }^{\alpha }{K}_{\mathbf{Q}}}{\mathrm{\Gamma }(\alpha +1)}<1$.

Theorem 3.3

Our proposed model (10) has unique solution if $\frac{{\tau }^{\alpha }{L}_{\mathbf{Q}}}{\mathrm{\Gamma }(\alpha +1)}<1$ hold.

Thanks to “Banach's contraction theorem,” let $\mathbf{W},\overline{\mathbf{W}}\in \mathbf{X}$, then

$$\begin{array}{cc}\hfill \Vert \mathbf{T}\mathbf{W}-\mathbf{T}\overline{\mathbf{W}}\Vert \le & \underset{\mathrm{t}\in [0,\tau ]}{\max }\frac{1}{\mathrm{\Gamma }(\alpha )}{\int }_0^{\mathrm{t}}\left|{(\mathrm{t}-s)}^{\alpha -1}\right|\left|\mathbf{Q}(s,\mathbf{W}(s))-\mathbf{Q}(s,\overline{\mathbf{W}}(s))\right|ds\hfill \\ \hfill \le & \frac{{\tau }^{\alpha }{L}_{\mathbf{Q}}}{\mathrm{\Gamma }(\alpha +1)}\Vert \mathbf{W}-\overline{\mathbf{W}}\Vert .\hfill \end{array}$$

Hence, T has unique fixed point. Therefore, our problem (10) has unique solution.

4. ULAM–HYERS STABILITY

Now in this part of the manuscript, we will discuss the Ulam–Hyers type stability which was introduced by Ulam, ⁴³ in 1940 and studied further by Hyers. ⁴⁴ The stated stability generalized further by Rassiass, ⁵⁰ to Ulam–Hyer–Rassiass stability. For the last two decades this stability was studied very well. ⁴⁵ , ⁴⁶ , ⁵⁰ , ⁵¹ , ⁵² First, we will present different definition of Ulam–Hyers type stability which can be found in Ali et al. ⁵²

Let H : X → X be an operator satisfying

$$\mathbf{H}\mathbf{W}=\mathbf{W},\text{for}\mathbf{W}\in \mathrm{X}.$$ (16)

Definition 4.1

Problem (16) is Ulam–Hyers type stable for ϵ > 0 and assume W ∈ X represents any solution of

$$\Vert \mathbf{W}-\mathbf{H}\mathbf{W}\Vert \le ϵ,\text{for}\mathrm{t}\in [0,\tau ].$$ (17)

There exists unique solution $\overline{\mathbf{W}}$ of problem (16) such that C _q  > 0 satisfying

$$\Vert \overline{\mathbf{W}}-\mathbf{W}\Vert \le {\mathbf{C}}_qϵ,\mathrm{t}\in [0,\tau ].$$ (18)

Definition 4.2

For Υ ∈ C(R, R) with $\mathrm{{\rm Y}}(0)=0$, for any solution W of (17), and let $\overline{\mathbf{W}}$ be at most one solution of (16) such that

$$\Vert \overline{\mathbf{W}}-\mathbf{W}\Vert \le \mathrm{{\rm Y}}(ϵ),$$ (19)

then problem (16) is generalized Ulam–Hyers type stable.

Remark 4.3

For χ(t) ∈ C([0, τ]; R), then $\overline{\mathbf{W}}\in \mathbf{X}$ satisfies (17) if

• (i)|χ(t)| ≤ ϵ,  ∀ t ∈ [0, τ]
• (ii) $\mathbf{H}\overline{\mathbf{W}}(\mathrm{t})=\overline{\mathbf{W}}+\chi (\mathrm{t}),\forall \mathrm{t}\in [0,\tau ]$.

The following relation is needed in future work. The perturb problem corresponding to Equation (10) is given by

$$\left\{\begin{array}{cc}\hfill {}^{\mathbf{c}}{\mathbf{D}}_{+0}^{\alpha }\mathbf{W}(\mathrm{t})& =\mathbf{Q}(\mathrm{t},\mathbf{W}(\mathrm{t}))+\chi (\mathrm{t}),\hfill \\ \hfill \mathbf{W}(0)& ={\mathbf{W}}_0.\hfill \end{array}\right.$$ (20)

Lemma 4.4

Perturb problem (20) satisfies the relation given by

$$\begin{array}{c}\hfill |\mathbf{W}(\mathrm{t})-\mathbf{T}\mathbf{W}(\mathrm{t})|\le aϵ,\text{where}a=\frac{{\tau }^{\alpha }}{\mathrm{\Gamma }(\alpha +1)}.\end{array}$$ (21)

By Remark 4.3 and Lemma 2.3, one may obtain the required result.

Theorem 4.5

Solution to problem (10) is Ulam–Hyers type and generalized‐Ulam–Hyers type stable on Lemma 4.4 if $\frac{{\tau }^{\alpha }{L}_{\mathbf{Q}}}{\mathrm{\Gamma }(\alpha +1)}<1$.

For any solution W ∈ X of problem (20) and $\overline{\mathbf{W}}\in \mathbf{X}$ represents unique solution to problem (10), then

$$\begin{array}{cc}\hfill |\mathbf{W}(\mathrm{t})-\overline{\mathbf{W}}(\mathrm{t})|=|\mathbf{W}(\mathrm{t})-\mathbf{T}\overline{\mathbf{W}}(\mathrm{t})|& \le |\mathbf{W}(\mathrm{t})-\mathbf{T}\mathbf{W}(\mathrm{t})|+|\mathbf{T}\mathbf{W}(\mathrm{t})-\mathbf{T}\overline{\mathbf{W}}(\mathrm{t})|\hfill \\ \hfill & \le aϵ+\frac{{\tau }^{\alpha }{L}_{\mathbf{Q}}}{\mathrm{\Gamma }(\alpha +1)}|\mathbf{W}(\mathrm{t})-\overline{\mathbf{W}}(\mathrm{t})|\hfill \\ \hfill & \le \frac{aϵ}{1-\frac{{\tau }^{\alpha }{L}_{\mathbf{Q}}}{\mathrm{\Gamma }(\alpha +1)}}.\hfill \end{array}$$ (22)

Thus, problem (10) is Ulam–Hyers and generalized Ulam–Hyers type stable.

Definition 4.6

Problem (16) is Ulam–Hyers–Rassias type stable with Ω ∈ C[[0, τ], R], if for ϵ > 0 and for any solution W ∈ X of

$$\Vert \mathbf{W}-\mathbf{H}\mathbf{W}\Vert \le \mathrm{\Omega }(\mathrm{t})ϵ,\text{for}\mathrm{t}\in [0,\tau ],$$ (23)

there exists unique solution $\overline{\mathbf{W}}$ to problem (16) with C _q  > 0 satisfying

$$\Vert \overline{\mathbf{W}}-\mathbf{W}\Vert \le {\mathbf{C}}_q\mathrm{\Omega }(\mathrm{t})ϵ,\forall \mathrm{t}\in [0,\tau ].$$ (24)

Definition 4.7

For Ω ∈ C[[0, τ], R] if there exist C _q, Ω and for ϵ > 0, let W represent any solution to problem (23) and $\overline{\mathbf{W}}$ represent unique solution to problem (16) such that

$$\Vert \overline{\mathbf{W}}-\mathbf{W}\Vert \le {\mathbf{C}}_{q,\mathrm{\Omega }}\mathrm{\Omega }(\mathrm{t}),\forall \mathrm{t}\in [0,\tau ],$$ (25)

then problem (16) is generalized Ulam–Hyers–Rassias type stable.

Remark 4.8

If there exist χ(t) ∈ C([0, τ]; R), then $\overline{\mathbf{W}}\in \mathbf{X}$ satisfies (17) if

• (i)|χ(t)| ≤ ϵΩ(t),  ∀ t ∈ [0, τ]
• (ii) $\mathbf{H}\overline{\mathbf{W}}(\mathrm{t})=\overline{\mathbf{W}}+\chi (\mathrm{t}),\forall \mathrm{t}\in [0,\tau ]$.

Lemma 4.9

For perturb problem (20), the following hold.

$$\begin{array}{c}\hfill |\mathbf{W}(\mathrm{t})-\mathbf{T}\mathbf{W}(\mathrm{t})|\le a\mathrm{\Omega }(\mathrm{t})ϵ,\text{where}a=\frac{{\tau }^{\alpha }}{\mathrm{\Gamma }(\alpha +1)}.\end{array}$$ (26)

Using Remark (4.8) and Lemma 2.3, one may get the required result.

Theorem 4.10

Solution to problem (10) is Ulam–Hyers–Rassiass and generalized‐Ulam–Hyers–Rassiass type stable on Lemma 4.9 if $\frac{{\tau }^{\alpha }{L}_{\mathbf{Q}}}{\mathrm{\Gamma }(\alpha +1)}<1$.

Let W ∈ X be any solution and $\overline{\mathbf{W}}\in \mathbf{X}$ represent unique solution to problem (10), then

$$\begin{array}{cc}\hfill |\mathbf{W}(\mathrm{t})-\overline{\mathbf{W}}(\mathrm{t})|=|\mathbf{W}(\mathrm{t})-\mathbf{T}\overline{\mathbf{W}}(\mathrm{t})|& \le |\mathbf{W}(\mathrm{t})-\mathbf{T}\mathbf{W}(\mathrm{t})|+|\mathbf{T}\mathbf{W}(\mathrm{t})-\mathbf{T}\overline{\mathbf{W}}(\mathrm{t})|\hfill \\ \hfill & \le a\mathrm{\Omega }(\mathrm{t})ϵ+\frac{{\tau }^{\alpha }{L}_{\mathbf{Q}}}{\mathrm{\Gamma }(\alpha +1)}|\mathbf{W}(\mathrm{t})-\overline{\mathbf{W}}(\mathrm{t})|\hfill \\ \hfill & \le \frac{a\mathrm{\Omega }(\mathrm{t})ϵ}{1-\frac{{\tau }^{\alpha }{L}_{\mathbf{Q}}}{\mathrm{\Gamma }(\alpha +1)}}.\hfill \end{array}$$ (27)

Thus, problem (10) is Ulam–Hyers–Rassias stable and generalized Ulam–Hyers–Rassias type stable.

5. SERIES SOLUTION TO PROPOSED MODEL

In this part of the article, a general algorithm for series solution to model (2) is developed. Using Laplace transform, we have

$$\left\{\begin{array}{cc}\hfill ℒ[S(\mathrm{t})]& =\frac{S_0}{s}+\frac{1}{s^{\alpha }}ℒ\left[\frac{-b_{0}S\mathrm{G}}{N}-\frac{b(\mathrm{t})SI}{N}-uS\right],\hfill \\ \hfill ℒ[E(\mathrm{t})]& =\frac{E_0}{s}+\frac{1}{s^{\alpha }}ℒ\left[\frac{b_{0}S\mathrm{G}}{N}+\frac{b(\mathrm{t})SI}{N}-(d+u)E\right],\hfill \\ \hfill ℒ[I(\mathrm{t})]& =\frac{I_0}{s}+\frac{1}{s^{\alpha }}ℒ\left[dE-(l+u)I\right]\hfill \\ \hfill ℒ[R(\mathrm{t})]& =\frac{R_0}{s}+\frac{1}{s^{\alpha }}ℒ\left[lI-uR\right]\hfill \\ \hfill ℒ[N(\mathrm{t})]& =\frac{N_0}{s}+\frac{1}{s^{\alpha }}ℒ\left[-uN\right]\hfill \\ \hfill ℒ[D(\mathrm{t})]& =\frac{D_0}{s}+\frac{1}{s^{\alpha }}ℒ\left[vlI-zD\right]\hfill \\ \hfill ℒ[C(\mathrm{t})]& =\frac{C_0}{s}+\frac{1}{s^{\alpha }}ℒ[dE].\hfill \end{array}\right.$$ (28)

Assuming solution in the form of series as

$$\left\{\begin{array}{cc}\hfill S(\mathrm{t})& ={\displaystyle \sum _{\mathrm{j}=0}^{\infty }}{S}_{\mathrm{j}}(\mathrm{t}),\hfill \\ \hfill E(\mathrm{t})& ={\displaystyle \sum _{\mathrm{j}=0}^{\infty }}{E}_{\mathrm{j}}(\mathrm{t}),\hfill \\ \hfill I(\mathrm{t})& ={\displaystyle \sum _{\mathrm{j}=0}^{\infty }}{I}_{\mathrm{j}}(\mathrm{t}),\hfill \\ \hfill R(\mathrm{t})& ={\displaystyle \sum _{\mathrm{j}=0}^{\infty }}{R}_{\mathrm{j}}(\mathrm{t}),\hfill \\ \hfill N(\mathrm{t})& ={\displaystyle \sum _{\mathrm{j}=0}^{\infty }}{N}_{\mathrm{j}}(\mathrm{t}),\hfill \\ \hfill D(\mathrm{t})& ={\displaystyle \sum _{\mathrm{j}=0}^{\infty }}{D}_{\mathrm{j}}(\mathrm{t}),\hfill \\ \hfill C(\mathrm{t})& ={\displaystyle \sum _{\mathrm{j}=0}^{\infty }}{C}_{\mathrm{j}}(\mathrm{t}).\hfill \end{array}\right.$$ (29)

Decomposing the nonlinear terms S(t)I(t) by Adomian polynomials as

$$S(\mathrm{t})I(\mathrm{t})={\displaystyle \sum _{\mathrm{j}=0}^{\infty }}{A}_{\mathrm{j}}(\mathrm{t}),\text{where}{A}_{\mathrm{j}}(\mathrm{t})=\frac{1}{\mathrm{j}!}\frac{d^{\mathrm{j}}}{d{z}^{\mathrm{j}}}\left[{\displaystyle \sum _{k=0}^{\mathrm{j}}}{z}^k{S}_k(\mathrm{t}){\displaystyle \sum _{k=0}^{\mathrm{j}}}{z}^k{I}_k(\mathrm{t})\right]{|}_{z=0}.$$ (30)

By system (29) and (30), system (28) gives

$$\left\{\begin{array}{cc}\hfill ℒ\left[{\displaystyle \sum _{\mathrm{j}=0}^{\infty }}{S}_{\mathrm{j}}(\mathrm{t})\right]& =\frac{S_0}{s}+\frac{1}{s^{\alpha }}ℒ\left[\frac{-b_0{\displaystyle {\sum }_{\mathrm{j}=0}^{\infty }}{S}_{\mathrm{j}}(\mathrm{t})\mathrm{G}}{{\displaystyle {\sum }_{\mathrm{j}=0}^{\infty }}{N}_{\mathrm{j}}(\mathrm{t})}-\frac{b(\mathrm{t}){\displaystyle {\sum }_{\mathrm{j}=0}^{\infty }}{A}_{\mathrm{j}}(\mathrm{t})}{{\displaystyle {\sum }_{\mathrm{j}=0}^{\infty }}{N}_{\mathrm{j}}(\mathrm{t})}-u{\displaystyle \sum _{\mathrm{j}=0}^{\infty }}{S}_{\mathrm{j}}(\mathrm{t})\right],\hfill \\ \hfill ℒ\left[{\displaystyle \sum _{\mathrm{j}=0}^{\infty }}{S}_{\mathrm{j}}(\mathrm{t})\right]& =\frac{E_0}{s}+\frac{1}{s^{\alpha }}ℒ\left[\frac{b_0{\displaystyle {\sum }_{\mathrm{j}=0}^{\infty }}{S}_{\mathrm{j}}(\mathrm{t})\mathrm{G}}{{\displaystyle {\sum }_{\mathrm{j}=0}^{\infty }}{N}_{\mathrm{j}}(\mathrm{t})}+\frac{b(\mathrm{t})S{\displaystyle {\sum }_{\mathrm{j}=0}^{\infty }}{A}_{\mathrm{j}}(\mathrm{t})}{{\displaystyle {\sum }_{\mathrm{j}=0}^{\infty }}{N}_{\mathrm{j}}(\mathrm{t})}-(d+u){\displaystyle \sum _{\mathrm{j}=0}^{\infty }}{E}_{\mathrm{j}}(\mathrm{t})\right],\hfill \\ \hfill ℒ\left[{\displaystyle \sum _{\mathrm{j}=0}^{\infty }}{I}_{\mathrm{j}}(\mathrm{t})\right]& =\frac{I_0}{s}+\frac{1}{s^{\alpha }}ℒ\left[d{\displaystyle \sum _{\mathrm{j}=0}^{\infty }}{E}_{\mathrm{j}}(\mathrm{t})-(l+u){\displaystyle \sum _{\mathrm{j}=0}^{\infty }}{I}_{\mathrm{j}}(\mathrm{t})\right],\hfill \\ \hfill ℒ\left[{\displaystyle \sum _{\mathrm{j}=0}^{\infty }}{R}_{\mathrm{j}}(\mathrm{t})\right]& =\frac{R_0}{s}+\frac{1}{s^{\alpha }}ℒ\left[l{\displaystyle \sum _{\mathrm{j}=0}^{\infty }}{I}_{\mathrm{j}}(\mathrm{t})-u{\displaystyle \sum _{\mathrm{j}=0}^{\infty }}{R}_{\mathrm{j}}(\mathrm{t})\right],\hfill \\ \hfill ℒ\left[{\displaystyle \sum _{\mathrm{j}=0}^{\infty }}{N}_{\mathrm{j}}(\mathrm{t})\right]& =\frac{N_0}{s}+\frac{1}{s^{\alpha }}ℒ\left[-u{\displaystyle \sum _{\mathrm{j}=0}^{\infty }}{N}_{\mathrm{j}}(\mathrm{t})\right],\hfill \\ \hfill ℒ\left[{\displaystyle \sum _{\mathrm{j}=0}^{\infty }}{D}_{\mathrm{j}}(\mathrm{t})\right]& =\frac{D_0}{s}+\frac{1}{s^{\alpha }}ℒ\left[vl{\displaystyle \sum _{\mathrm{j}=0}^{\infty }}{I}_{\mathrm{j}}(\mathrm{t})-z{\displaystyle \sum _{\mathrm{j}=0}^{\infty }}{D}_{\mathrm{j}}(\mathrm{t})\right],\hfill \\ \hfill ℒ\left[{\displaystyle \sum _{\mathrm{j}=0}^{\infty }}{C}_{\mathrm{j}}(\mathrm{t})\right]& =\frac{C_0}{s}+\frac{1}{s^{\alpha }}ℒ\left[d{\displaystyle \sum _{\mathrm{j}=0}^{\infty }}{E}_{\mathrm{j}}(\mathrm{t})\right].\hfill \end{array}\right.$$ (31)

Compare like terms on each sides of (31), we get

$$\left\{\begin{array}{cc}\hfill ℒ[S_0(\mathrm{t})]& =\frac{S_0}{s},\hfill \\ \hfill ℒ[E_0(\mathrm{t})]& =\frac{E_0}{s},\hfill \\ \hfill ℒ[I_0(\mathrm{t})]& =\frac{I_0}{s},\hfill \\ \hfill ℒ[R_0(\mathrm{t})]& =\frac{R_0}{s},\hfill \\ \hfill ℒ[N_0(\mathrm{t})]& =\frac{N_0}{s},\hfill \\ \hfill ℒ[D_0(\mathrm{t})]& =\frac{D_0}{s},\hfill \\ \hfill ℒ[C_0(\mathrm{t})]& =\frac{C_0}{s}.\hfill \end{array}\right.$$ (32)

$$\left\{\begin{array}{cc}\hfill ℒ[S_1(\mathrm{t})]& =\frac{1}{s^{\alpha }}ℒ\left[\frac{-b_0{S}_0(\mathrm{t})\mathrm{G}}{N_{\mathrm{j}}(\mathrm{t})}-\frac{b(\mathrm{t})A_0(\mathrm{t})}{N_0(\mathrm{t})}-u{S}_0(\mathrm{t})\right],\hfill \\ \hfill ℒ[E_1(\mathrm{t})]& =\frac{1}{s^{\alpha }}ℒ\left[\frac{b_0{S}_0(\mathrm{t})\mathrm{G}}{N_0(\mathrm{t})}+\frac{b(\mathrm{t})S{A}_0(\mathrm{t})}{N_0(\mathrm{t})}-(d+u)E_0(\mathrm{t})\right],\hfill \\ \hfill ℒ[I_1(\mathrm{t})]& =\frac{1}{s^{\alpha }}ℒ\left[d{E}_0(\mathrm{t})-(l+u)I_0(\mathrm{t})\right],\hfill \\ \hfill ℒ[R_1(\mathrm{t})]& =\frac{1}{s^{\alpha }}ℒ\left[l{I}_0(\mathrm{t})-u{R}_0(\mathrm{t})\right],\hfill \\ \hfill ℒ[N_1(\mathrm{t})]& =\frac{1}{s^{\alpha }}ℒ\left[-u{N}_0(\mathrm{t})\right],\hfill \\ \hfill ℒ[D_1(\mathrm{t})]& =\frac{1}{s^{\alpha }}ℒ\left[vl{I}_0(\mathrm{t})-z{D}_0(\mathrm{t})\right],\hfill \\ \hfill ℒ[C_1(\mathrm{t})]& =\frac{1}{s^{\alpha }}ℒ\left[d{E}_0(\mathrm{t})\right].\hfill \end{array}\right.$$ (33)

$$\left\{\begin{array}{cc}\hfill ℒ[S_2(\mathrm{t})]& =\frac{1}{s^{\alpha }}ℒ\left[\frac{-b_0{S}_1(\mathrm{t})\mathrm{G}}{N_1(\mathrm{t})}-\frac{b(\mathrm{t})A_1(\mathrm{t})}{N_1(\mathrm{t})}-u{S}_1(\mathrm{t})\right],\hfill \\ \hfill ℒ[E_2(\mathrm{t})]& =\frac{1}{s^{\alpha }}ℒ\left[\frac{b_0{S}_1(\mathrm{t})\mathrm{G}}{N_0(\mathrm{t})}+\frac{b(\mathrm{t})A_1(\mathrm{t})}{N_1(\mathrm{t})}-(d+u)E_1(\mathrm{t})\right],\hfill \\ \hfill ℒ[I_2(\mathrm{t})]& =\frac{1}{s^{\alpha }}ℒ\left[d{E}_1(\mathrm{t})-(l+u)I_1(\mathrm{t})\right],\hfill \\ \hfill ℒ[R_2(\mathrm{t})]& =\frac{1}{s^{\alpha }}ℒ\left[l{I}_1(\mathrm{t})-u{R}_1(\mathrm{t})\right],\hfill \\ \hfill ℒ[N_2]& =\frac{1}{s^{\alpha }}ℒ\left[-u{N}_1(\mathrm{t})\right],\hfill \\ \hfill ℒ[D_2(\mathrm{t})]& =\frac{1}{s^{\alpha }}ℒ\left[vl{I}_1(\mathrm{t})-z{D}_1(\mathrm{t})\right],\hfill \\ \hfill ℒ[C_2(\mathrm{t})]& =\frac{1}{s^{\alpha }}ℒ\left[d{E}_1(\mathrm{t})\right].\hfill \end{array}\right.$$ (34)

For j≥0, we generalized the terms as

$$\left\{\begin{array}{cc}\hfill ℒ[S_{\mathrm{j}+1}(\mathrm{t})]& =\frac{1}{s^{\alpha }}ℒ\left[\frac{-b_0{S}_{\mathrm{j}}(\mathrm{t})\mathrm{G}}{N_{\mathrm{j}}(\mathrm{t})}-\frac{b(\mathrm{t})A_{\mathrm{j}}(\mathrm{t})}{N_{\mathrm{j}}(\mathrm{t})}-u{S}_{\mathrm{j}}(\mathrm{t})\right],\hfill \\ \hfill ℒ[E_{\mathrm{j}+1}(\mathrm{t})]& =\frac{1}{s^{\alpha }}ℒ\left[\frac{b_0{S}_{\mathrm{j}}(\mathrm{t})\mathrm{G}}{N}+\frac{b(\mathrm{t})S{A}_{\mathrm{j}}(\mathrm{t})}{N_{\mathrm{j}}(\mathrm{t})}-(d+u)E_{\mathrm{j}}(\mathrm{t})\right],\hfill \\ \hfill ℒ[I_{\mathrm{j}+1}(\mathrm{t})]& =\frac{1}{s^{\alpha }}ℒ\left[d{E}_{\mathrm{j}}(\mathrm{t})-(l+u)I_{\mathrm{j}}(\mathrm{t})\right],\hfill \\ \hfill ℒ[R_{\mathrm{j}+1}(\mathrm{t})]& =\frac{1}{s^{\alpha }}ℒ\left[l{I}_{\mathrm{j}}(\mathrm{t})-u{R}_{\mathrm{j}}(\mathrm{t})\right],\hfill \\ \hfill ℒ[N_{\mathrm{j}+1}(\mathrm{t})]& =\frac{1}{s^{\alpha }}ℒ\left[-u{N}_{\mathrm{j}}(\mathrm{t})\right],\hfill \\ \hfill ℒ[D_{\mathrm{j}+1}(\mathrm{t})]& =\frac{1}{s^{\alpha }}ℒ\left[vl{I}_{\mathrm{j}}(\mathrm{t})-z{D}_{\mathrm{j}}(\mathrm{t})\right],\hfill \\ \hfill ℒ[C_{\mathrm{j}+1}(\mathrm{t})]& =\frac{1}{s^{\alpha }}ℒ\left[d{E}_{\mathrm{j}}(\mathrm{t})\right].\hfill \end{array}\right.$$ (35)

By application of inverse Laplace transform to systems (32)–(35), we have

$$\left\{\begin{array}{cc}\hfill S_0(\mathrm{t})& =S_0,\hfill \\ \hfill E_0(\mathrm{t})& =E_0,\hfill \\ \hfill I_0(\mathrm{t})& =I_0,\hfill \\ \hfill R_0(\mathrm{t})& =R_0,\hfill \\ \hfill N_0(\mathrm{t})& =N_0,\hfill \\ \hfill D_0(\mathrm{t})& =D_0,\hfill \\ \hfill C_0(\mathrm{t})& =C_0.\hfill \end{array}\right.$$ (36)

$$\left\{\begin{array}{cc}\hfill S_1(\mathrm{t})& {=ℒ}^{-1}\left[\frac{1}{s^{\alpha }}ℒ\left[\frac{-b_0{S}_0(\mathrm{t})\mathrm{G}}{N_{\mathrm{j}}(\mathrm{t})}-\frac{b(\mathrm{t})A_0(\mathrm{t})}{N_0(\mathrm{t})}-u{S}_0(\mathrm{t})\right]\right],\hfill \\ \hfill E_1(\mathrm{t})& {=ℒ}^{-1}\left[\frac{1}{s^{\alpha }}ℒ\left[\frac{b_0{S}_0(\mathrm{t})\mathrm{G}}{N_0(\mathrm{t})}+\frac{b(\mathrm{t})S{A}_0(\mathrm{t})}{N_0(\mathrm{t})}-(d+u)E_0(\mathrm{t})\right]\right],\hfill \\ \hfill I_1(\mathrm{t})& {=ℒ}^{-1}\left[\frac{1}{s^{\alpha }}ℒ\left[d{E}_0(\mathrm{t})-(l+u)I_0(\mathrm{t})\right]\right],\hfill \\ \hfill R_1(\mathrm{t})& {=ℒ}^{-1}\left[\frac{1}{s^{\alpha }}ℒ\left[l{I}_0(\mathrm{t})-u{R}_0(\mathrm{t})\right]\right],\hfill \\ \hfill N_1(\mathrm{t})& {=ℒ}^{-1}\left[\frac{1}{s^{\alpha }}ℒ\left[-u{N}_0(\mathrm{t})\right]\right],\hfill \\ \hfill D_1(\mathrm{t})& {=ℒ}^{-1}\left[\frac{1}{s^{\alpha }}ℒ\left[vl{I}_0(\mathrm{t})-z{D}_0(\mathrm{t})\right]\right],\hfill \\ \hfill C_1(\mathrm{t})& {=ℒ}^{-1}\left[\frac{1}{s^{\alpha }}ℒ\left[d{E}_0(\mathrm{t})\right]\right].\hfill \end{array}\right.$$ (37)

$$\left\{\begin{array}{cc}\hfill S_2(\mathrm{t})& {=ℒ}^{-1}\left[\frac{1}{s^{\alpha }}ℒ\left[\frac{-b_0{S}_1(\mathrm{t})\mathrm{G}}{N_1(\mathrm{t})}-\frac{b(\mathrm{t})A_1(\mathrm{t})}{N_1(\mathrm{t})}-u{S}_1(\mathrm{t})\right]\right],\hfill \\ \hfill E_2(\mathrm{t})& {=ℒ}^{-1}\left[\frac{1}{s^{\alpha }}ℒ\left[\frac{b_0{S}_1(\mathrm{t})\mathrm{G}}{N_0(\mathrm{t})}+\frac{b(\mathrm{t})A_1(\mathrm{t})}{N_1(\mathrm{t})}-(d+u)E_1(\mathrm{t})\right]\right],\hfill \\ \hfill I_2(\mathrm{t})& {=ℒ}^{-1}\left[\frac{1}{s^{\alpha }}ℒ\left[d{E}_1(\mathrm{t})-(l+u)I_1(\mathrm{t})\right]\right],\hfill \\ \hfill R_2(\mathrm{t})& {=ℒ}^{-1}\left[\frac{1}{s^{\alpha }}ℒ\left[l{I}_1(\mathrm{t})-u{R}_1(\mathrm{t})\right]\right],\hfill \\ \hfill N_2(\mathrm{t})& {=ℒ}^{-1}\left[\frac{1}{s^{\alpha }}ℒ\left[-u{N}_1(\mathrm{t})\right]\right],\hfill \\ \hfill D_2(\mathrm{t})& {=ℒ}^{-1}\left[\frac{1}{s^{\alpha }}ℒ\left[vl{I}_1(\mathrm{t})-z{D}_1(\mathrm{t})\right]\right],\hfill \\ \hfill C_2(\mathrm{t})& {=ℒ}^{-1}\left[\frac{1}{s^{\alpha }}ℒ\left[d{E}_1(\mathrm{t})\right]\right].\hfill \end{array}\right.$$ (38)

Thus, for j≥0, the general term is given by

$$\left\{\begin{array}{cc}\hfill S_{\mathrm{j}+1}(\mathrm{t})& {=ℒ}^{-1}\left[\frac{1}{s^{\alpha }}ℒ\left[\frac{-b_0{S}_{\mathrm{j}}(\mathrm{t})\mathrm{G}}{N_{\mathrm{j}}(\mathrm{t})}-\frac{b(\mathrm{t})A_{\mathrm{j}}(\mathrm{t})}{N_{\mathrm{j}}(\mathrm{t})}-u{S}_{\mathrm{j}}(\mathrm{t})\right]\right],\hfill \\ \hfill E_{\mathrm{j}+1}(\mathrm{t})& {=ℒ}^{-1}\left[\frac{1}{s^{\alpha }}ℒ\left[\frac{b_0{S}_{\mathrm{j}}(\mathrm{t})\mathrm{G}}{N}+\frac{b(\mathrm{t})S{A}_{\mathrm{j}}(\mathrm{t})}{N_{\mathrm{j}}(\mathrm{t})}-(d+u)E_{\mathrm{j}}(\mathrm{t})\right]\right],\hfill \\ \hfill I_{\mathrm{j}+1}(\mathrm{t})& {=ℒ}^{-1}\left[\frac{1}{s^{\alpha }}ℒ\left[d{E}_{\mathrm{j}}(\mathrm{t})-(l+u)I_{\mathrm{j}}(\mathrm{t})\right]\right],\hfill \\ \hfill R_{\mathrm{j}+1}(\mathrm{t})& {=ℒ}^{-1}\left[\frac{1}{s^{\alpha }}ℒ\left[l{I}_{\mathrm{j}}(\mathrm{t})-u{R}_{\mathrm{j}}(\mathrm{t})\right]\right],\hfill \\ \hfill N_{\mathrm{j}+1}(\mathrm{t})& {=ℒ}^{-1}\left[\frac{1}{s^{\alpha }}ℒ\left[-u{N}_{\mathrm{j}}(\mathrm{t})\right]\right],\hfill \\ \hfill D_{\mathrm{j}+1}(\mathrm{t})& {=ℒ}^{-1}\left[\frac{1}{s^{\alpha }}ℒ\left[vl{I}_{\mathrm{j}}(\mathrm{t})-z{D}_{\mathrm{j}}(\mathrm{t})\right]\right],\hfill \\ \hfill C_{\mathrm{j}+1}(\mathrm{t})& {=ℒ}^{-1}\left[\frac{1}{s^{\alpha }}ℒ\left[d{E}_{\mathrm{j}}(\mathrm{t})\right]\right].\hfill \end{array}\right.$$ (39)

Similarly, we obtain series solution as

$$\left\{\begin{array}{cc}\hfill S(\mathrm{t})& =S_0(\mathrm{t})+S_1(\mathrm{t})+S_2(\mathrm{t})+S_3(\mathrm{t})+\dots ,\hfill \\ \hfill E(\mathrm{t})& =E_0(\mathrm{t})+E_1(\mathrm{t})+E_2(\mathrm{t})+E_3(\mathrm{t})+\dots ,\hfill \\ \hfill I(\mathrm{t})& =I_0(\mathrm{t})+I_1(\mathrm{t})+I_2(\mathrm{t})+I_3(\mathrm{t})+\dots ,\hfill \\ \hfill R(\mathrm{t})& =R_0(\mathrm{t})+R_1(\mathrm{t})+R_2(\mathrm{t})+R_3(\mathrm{t})+\dots ,\hfill \\ \hfill N(\mathrm{t})& =N_0(\mathrm{t})+N_1(\mathrm{t})+N_2(\mathrm{t})+N_3(\mathrm{t})+\dots ,\hfill \\ \hfill D(\mathrm{t})& =D_0(\mathrm{t})+D_1(\mathrm{t})+D_2(\mathrm{t})+D_3(\mathrm{t})+\dots ,\hfill \\ \hfill C(\mathrm{t})& =C_0(\mathrm{t})+C_1(\mathrm{t})+C_2(\mathrm{t})+C_3(\mathrm{t})+\dots .\hfill \end{array}\right.$$ (40)

6. NUMERICAL RESULTS AND DISCUSSION

For justification of developed results, values have been taken about COVID‐19 in Pakistan and also presented in Table 1, ⁴⁷:

• Case‐I: $\mathrm{G}=0,u=0.02205/day,a=0.4239/day$,   $b_0=0.5944/day$

  In Figures 1, 2, 3, 4, 5, 6, 7, we draw the graph of the solution for the given values of parameter defined as above. Using Matlab‐16, the series solutions defined in Equation (40) have been plotted.

  From Figure 1, we see that the susceptible population was decreasing for the initial population at given time in different orders. In same line as in Figure 2, exposed class was increasing with the decrease of susceptive class, and hence, the number of infection is increasing as in Figure 3 because people are exposed to infection. This was due to the fact that people were not taking the matter seriously in Pakistan so the number of infections grew with proper speed. Initially, in first 15–20 days, the numbers of recovered class were increasing as people initially were getting ride easily due to the reason that cases were limited and treatment was proper as in Figure 4. After that, the recovered rate is becoming slow now as numbers of infection increased, and also there is no proper cure yet introduced. Also now the infection has attacked mostly on aged people whose recovery will take time. From Figure 5, we see that total population of healthy people is gradually decreasing with different rate due to fractional order derivative. Similarly, in Figure 6, the people initially not taking the matter seriously but now the people are taking it seriously and so the Mimicking class is decreasing with different rates because of fractional order. The class of cumulative cases is also increasing with different rate as in Figure 7. From Figures 1, 2, 3, 4, 5, 6, 7, we see that fractional calculus provides a global dynamics of the novel coronavirus infection model. The smaller the fractional order, the faster the concerned decay or growth and vice versa.

• Case‐II:  $\mathrm{G}=100,u=0.000002205/day,a=0.0004239/day$,   $b_0=0.00003345/day$.

TABLE 1.

Interpretation of the numerical values and parameters

G	No. of zoonotic cases	0, 100
b 0	Rate of transmission	0.5944,  0.00003345/day
a	Action strength of the Government	0.4239,  0.0004239
k	Responds intensity	1117.3
u	Rate of emigration	0.0205/day,  0.000002205/day
d −1	Latent period mean	3
l −1	Infectious period mean	5
z −1	Public reaction mean duration	11.2/day
v	Severe cases proportion	0.2
N 0	Population size initially	220 Millions
S 0	Susceptible population initially	10.25N 0
E 0	Exposed population initially	12 Millions
I 0	Infection population initially	0.01194 Millions
R 0	Recovered population initially	0.003008 Millions
D 0	Mimicking people initially	7 Millions
C 0	Cumulative cases initially (both reported and not reported)	0.054 Millions

FIGURE 1.

Dynamics of susceptible people S at various fractional order [Colour figure can be viewed at wileyonlinelibrary.com]

FIGURE 2.

Dynamics of exposed population E at various fractional order [Colour figure can be viewed at wileyonlinelibrary.com]

FIGURE 3.

Dynamics of infected population I at various fractional order [Colour figure can be viewed at wileyonlinelibrary.com]

FIGURE 4.

Dynamics of recovered population R at various fractional order [Colour figure can be viewed at wileyonlinelibrary.com]

FIGURE 5.

Dynamics of total population N at various fractional order [Colour figure can be viewed at wileyonlinelibrary.com]

FIGURE 6.

Dynamics of mimicking population D at various fractional order [Colour figure can be viewed at wileyonlinelibrary.com]

FIGURE 7.

Dynamics of cumulative population C at various fractional order [Colour figure can be viewed at wileyonlinelibrary.com]

In Figures 8, 9, 10, 11, 12, 13, 14, we draw the graph of the solution for the given values of parameter defined above. Using Matlab‐16, the series solutions defined in Equation (40) have been plotted.

FIGURE 8.

Dynamics of susceptible population S at various fractional order [Colour figure can be viewed at wileyonlinelibrary.com]

FIGURE 9.

Dynamics of exposed population E at various fractional order [Colour figure can be viewed at wileyonlinelibrary.com]

FIGURE 10.

Dynamics of infected population I at various fractional order [Colour figure can be viewed at wileyonlinelibrary.com]

FIGURE 11.

Dynamics of recovered people R at various fractional order [Colour figure can be viewed at wileyonlinelibrary.com]

FIGURE 12.

Dynamics of total population N at various fractional order [Colour figure can be viewed at wileyonlinelibrary.com]

FIGURE 13.

Dynamics of mimicking population D at various fractional order [Colour figure can be viewed at wileyonlinelibrary.com]

FIGURE 14.

Dynamics of cumulative population C at various fractional order [Colour figure can be viewed at wileyonlinelibrary.com]

From Figure 8, we see that as susceptible papulation is decreasing in the presence of zoonotic cases and immigration for the given initial papulation at given time at different order, then in the same line as in Figure 9, exposed class was also decreasing with the decrease of susceptive class, and hence, the number of infection was increasing as in Figure 10, because more people were exposed to catch infection. This is due to the reason that people first were not taking the matter serious in Pakistan so the number of infection rate raised up with proper fast speed. Initially, in first 15–20 days, the numbers of recovered class were increasing as people initially were getting ride easily due to the reason that cases were limited and treatment was proper as in Figure 11. The recovery rate is also gradually increased due to the increase of people getting ride from infection or going to die. From Figure 12, we see that total papulation of healthy people is gradually decreasing with different rate due to fractional order derivative. Similarly, in Figure 13, the people initially not taking the matter seriously but now the people are taking it seriously and so the Mimicking class is decreasing with different rates because of fractional order. The class of cumulative cases is also increasing with different rates as in Figure 14. From Figures 8, 9, 10, 11, 12, 13, 14, we see that fractional calculus provides a global dynamics of the novel coronavirus‐19 infection model. The smaller the fractional order, the faster the concerned decay or growth and vice versa.

7. CONCLUSION

We have examined a novel model of coronavirus‐19 under fractional order derivative. We first have established the existence theory of the model along with stability results of Ulam type via the use of nonlinear analysis. After that by a coupled method of Laplace transform and Adomian decomposition, we have established a general algorithm for series type solution to the considered model. By using Matlab, we have plotted the graphs against some real data of Pakistan, we observed that permitting immigration and not taking the matter seriously by the people in first sixty days the infection rate has been raised roughly in the country. But now the people are taking the matter seriously, but in the previous sixty days proper, great numbers of people have been coughing throughout the country. By fractional calculus approach, we have examined the global dynamics of the current novel disease. We see that fractional calculus approach globally describes the dynamics of all the compartments of the considered model more comprehensively.

CONFLICT OF INTEREST

There exists no competing interest regarding this research work.

Shah K, Sher M, Rabai'ah H, Ahmadian A, Salahshour S, Pansera BA. Analytical and qualitative investigation of COVID‐19 mathematical model under fractional differential operator. Math Meth Appl Sci. 2021;1‐20. 10.1002/mma.7704