Fractional Order Modeling of Predicting COVID-19 with Isolation and Vaccination Strategies in Morocco

1  Introduction

COVID-19 is an infectious viral disease brought about by a virus known as severe acute syndrome Coronavirus [1]. This virus was first reported at the end of 2019 in Wuhan, China. Then, it was reported in other places in China surprisingly [2]. In three months, the disease was transmitted in many countries of the world. Therefore, in April 2020, the World Health Organization (WHO) announced it as an outbreak. Its symptoms involve breathing trouble, exhaustion, fever, dry hack, sleepiness, conjunctivitis, chest torment, loss of discourse, loose bowels, and painful throat. Furthermore, in serious cases, pneumonia, multiorgan letdown, intense respiratory pain condition, septic shock, abnormal heart rhythm, myocarditis, blood clumps, cardiovascular breakdown, encephalitis, stroke, and Guillain Barré disorder, reciprocal lung entrance have been reported [3,4]. Additionally, a few patients might experience the ill effects of looseness of the bowels, loss of craving, taste, or smell, with no indications of breathing problem [5–7].

Investigating infectious diseases through various procedures is an attractive area of research in recent times. One of the important areas in this regard is devoted to the mathematical modeling of infectious diseases. Mathematical modeling is a powerful tool to study infectious diseases for analysis of the dynamic of diseases and backing control systems [8]. Plenty of research work has been published by researchers. The mentioned work is devoted to investigating disease transmission dynamics and their control. In the same fashion to understand the dynamics of COVID-19 and the tracking down—of the reasonable result of the outbreak is helpful for public health initiatives and consciousness programs, for which significant work has been published, and we refer to a few such as [9–14].

Fractional calculus can more precisely describe natural phenomena as compared to ordinary calculus, where derivatives and integrations have integer-orders. Numerous scientists have given more attention to investigating the dynamics of fractional-order models regarding the COVID-19 pandemic [15–38]. There are many papers showing the effectiveness of fractional calculus (see [39–45]). Further, the dynamics of a stochastic epidemic model has been studied, for which we refer [46,47] and mathematical modeling of COVID-19 pandemic using the Caputo-Fabrizio fractional derivative (see [48]). Also, a model of COVID-19 disease has been investigated under the stochastic concept in [49].

In this research work, an SEIR-type model that concentrates on the transmission dynamics of the COVID-19 outbreak are considered. Here, with a special spotlight on the transmissibility of people with mild, severe, and without side effects like the existence of people who tested positive for sharp, mild, or asymptomatic manifestations, and separating infectious compartment into two fundamental compartments of hospitalized people with mild symptoms and those in dense care units are involved in our proposed model.

This work is organized as: Part 2 is devoted to the formulation of the proposed model for COVID-19. In Parts 3 and 4, we analyze the proposed model with fractional order. Some qualitative analysis, computation of ℛ0, and local stability for the disease-free equilibrium (DFE) in terms of ℛ0, of the proposed model are obtained in Part 5. The stability analysis is studied in Part 6. Part 7 is dedicated to data fit and explanation of our model via numerical simulation and comparison with real data for Morocco. The conclusion is given in the last section.

2  The Proposed Epidemic Model

Here in this section, the proposed model is formulated. The flowchart of the model is given in Fig. 1 to understand the evolution of the mentioned disease. Let the total human population at time ϑ be denoted by N(ϑ). The proposed model divides the human population at time ϑ into eleven compartments which are described in Table 1. We generalize the SEIR model of the spread of COVID-19 with a specific focus on the transmissibility of people who are Individuals who follow precautionary health measures Sa and Individuals who do not follow precautionary health measures Su. We separate hospitalized cases into severe Hs and mild cases Hm and incorporate awareness programs. Moreover, individuals with severe Iss, mild symptoms Ims and asymptomatically infected Ia are also considered. Furthermore, the immunization of a sample of aware susceptible individuals in the proposed model to forecast the effect of vaccination V is also considered.

Figure 1: Flowchart of model (1)

Thus, we have

N(ϑ)=Sa(ϑ)+Su(ϑ)+E(ϑ)+Iss(ϑ)+Ims(ϑ)+Ia(ϑ)+Hm(ϑ)+Hs(ϑ)+Icu(ϑ)+R(ϑ)+D(ϑ).

From Fig. 1, we formulate our proposed model as

{Sa′(ϑ)=Λ−βSa(ϑ)N(Iss(ϑ)+Ims(ϑ)+Ia(ϑ))+vSu(ϑ),Su′(ϑ)=−βSu(ϑ)N(Iss(ϑ)+Ims(ϑ)+Ia(ϑ))−vSu(ϑ),E′(ϑ)=βSa(ϑ)+Su(ϑ)N(Iss(ϑ)+Ims(ϑ)+Ia(ϑ))−θE(ϑ),Iss′(ϑ)=θp1E(ϑ)+φ1Ims(ϑ)−(ψ1+h)Iss(ϑ),Ims′(ϑ)=θp2E(ϑ)+ψ1Iss(ϑ)−(ηi+φ1)Ims(ϑ),Ia′(ϑ)=θ(1−p1−p2)E(ϑ)−ηaIa(ϑ),Hm′(ϑ)=hτIss(ϑ)+ψ2Hs(ϑ)−(φ2+ηHm)Hm(ϑ),Hs′(ϑ)=hρIss(ϑ)+φ2Hm(ϑ)−(ψ2+ηHs+δHs)Hs(ϑ),Icu′(ϑ)=h(1−τ−ρ)Iss(ϑ)−Icu(ϑ),R′(ϑ)=ηiIms(ϑ)+ηaIa(ϑ)+ηHsHs(ϑ)+ηHmHm(ϑ)+(1−δi)Icu(ϑ),D′(ϑ)=δHsHs(ϑ)+δiIcu(ϑ),(1)

with initial conditions as

{Sa(0)=Sa,0,Su(0)=Su,0,E(0)=E0,Iss(0)=Iss,0,Ims(0)=Ims,0,Ia(0)=Ia,0,Hm(0)=Hm,0,Hs(0)=Hs,0,Icu(0)=Icu,0,D(0)=D0,R(0)=R0.

3  Fractional Order Model

In this part, we recall some definitions from fractional calculus which are needed throughout the paper.

Definition 3.1. [50] The Caputo fractional derivative of f of order μ is described as

f(μ)(ϑ)=1Γ(n−μ)∫0ϑf(n)(s)(ϑ−s)μ−n+1ds,(2)

where n−1<μ≤n and Γ(x)=∫0∞e−ϑϑx−1dϑ, which is called the Euler’s Gamma function.

The fractional integral [51] having of order μ>0 is defined by

Iμ(f(ϑ))=1Γ(μ)∫0ϑf(s)(ϑ−s)1−μds,

where ϑ>0, satisfies:

{(Iμf(ϑ))(μ)=f(ϑ),Iμ(f(ϑ)(μ))=f(ϑ)−f(0).(3)

Now, let us present the fractional order model of the COVID-19 by means of the Caputo derivative as follows:

{Sa(μ)(ϑ)=Λ−βSa(ϑ)N(Iss(ϑ)+Ims(ϑ)+Ia(ϑ))+vSu(ϑ),Su(μ)(ϑ)=−βSu(ϑ)N(Iss(ϑ)+Ims(ϑ)+Ia(ϑ))−vSu(ϑ),E(μ)(ϑ)=βSa(ϑ)+Su(ϑ)N(Iss(ϑ)+Ims(ϑ)+Ia(ϑ))−θE(ϑ),Iss(μ)(ϑ)=θp1E(ϑ)+φ1Ims(ϑ)−(ψ1+h)Iss(ϑ),Ims(μ)(ϑ)=θp2E(ϑ)+ψ1Iss(ϑ)−(ηi+φ1)Ims(ϑ),Ia(μ)(ϑ)=θ(1−p1−p2)E(ϑ)−ηaIa(ϑ),Hm(μ)(ϑ)=hτIss(ϑ)+ψ2Hs(ϑ)−(φ2+ηHm)Hm(ϑ),Hs(μ)(ϑ)=hρIss(ϑ)+φ2Hm(ϑ)−(ψ2+ηHs+δHs)Hs(ϑ),Icu(μ)(ϑ)=h(1−τ−ρ)Iss(ϑ)−Icu(ϑ),R(μ)(ϑ)=ηiIms(ϑ)+ηaIa(ϑ)+ηHsHs(ϑ)+ηHmHm(ϑ)+(1−δi)Icu(ϑ),D(μ)(ϑ)=δHsHs(ϑ)+δiIcu(ϑ).(4)

4  Existence Theory

In this part, we study the existence of a solution for the fractional order model Eq. (4). The existence theory is an important consequence of applied analysis and has many applications. It provides us with whether the model we study exists or not and if it exists whether it has a solution or not. If a problem has a solution, whether it will be unique or multiple. It also helps us in investigating the qualitative behavior of the solution to the problem.

Consider the following subsystem:

{Su(μ)(ϑ)=−βSu(ϑ)N(Iss(ϑ)+Ims(ϑ)+Ia(ϑ))−vSu(ϑ),E(μ)(ϑ)=βSa(ϑ)+Su(ϑ)N(Iss(ϑ)+Ims(ϑ)+Ia(ϑ))−θE(ϑ),Iss(μ)(ϑ)=θp1E(ϑ)+φ1Ims(ϑ)−(ψ1+h)Iss(ϑ),Ims(μ)(ϑ)=θp2E(ϑ)+ψ1Iss(ϑ)−(ηi+φ1)Ims(ϑ),Ia(μ)(ϑ)=θ(1−p1−p2)E(ϑ)−ηaIa(ϑ),Hm(μ)(ϑ)=hτIss(ϑ)+ψ2Hs(ϑ)−(φ2+ηHm)Hm(ϑ),Hs(μ)(ϑ)=hρIss(ϑ)+φ2Hm(ϑ)−(ψ2+ηHs+δHs)Hs(ϑ),Icu(μ)(ϑ)=h(1−τ−ρ)Iss(ϑ)−Icu(ϑ).(5)

Since

{Sa(μ)(ϑ)=Λ−βSa(ϑ)N(Iss(ϑ)+Ims(ϑ)+Ia(ϑ))+vSu(ϑ),R(μ)(ϑ)=ηiIms(ϑ)+ηaIa(ϑ)+ηHsHs(ϑ)+ηHmHm(ϑ)+(1−δi)Icu(ϑ),D(μ)(ϑ)=δHsHs(ϑ)+δiIcu(ϑ).(6)

Therefore, to study the existence and uniqueness of solutions of the system Eq. (4), we focus to study Eq. (5).

Assume that

{K1(ϑ,Su)=−βSuN(Iss(ϑ)+Ims(ϑ)+Ia(ϑ))−vSu,K2(ϑ,E)=βSa(ϑ)+Su(ϑ)N(Iss(ϑ)+Ims(ϑ)+Ia(ϑ))−θE,K3(ϑ,Iss)=θp1E(ϑ)+φ1Ims(ϑ)−(ψ1+h)Iss,K4(ϑ,Ims)=θp2E(ϑ)+ψ1Iss(ϑ)−(ηi+φ1)Ims,K5(ϑ,Ia)=θ(1−p1−p2)E(ϑ)−ηaIa,K6(ϑ,Hm)=hτIss(ϑ)+ψ2Hs(ϑ)−(φ2+ηHm)Hm,K7(ϑ,Hs)=hρIss(ϑ)+φ2Hm(ϑ)−(ψ2+ηHs+δHs)HsK8(ϑ,Icu)=h(1−τ−ρ)Iss−Icu.(7)

Applying the fractional integral to Eq. (5), it follows from initial conditions that

{Su(ϑ)=Su,0+1Γ(μ)∫0ϑ(ϑ−s)μ−1K1(s,Su(s))ds,E(ϑ)=E0+1Γ(μ)∫0ϑ(ϑ−s)μ−1K2(s,E(s))ds,Iss(ϑ)=Iss,0+1Γ(μ)∫0ϑ(ϑ−s)μ−1K3(s,Iss(s))ds,Ims(ϑ)=Ims,0+1Γ(μ)∫0ϑ(ϑ−s)μ−1K4(s,Ims(s))ds,Ia(ϑ)=Ia,0+1Γ(μ)∫0ϑ(ϑ−s)μ−1K5(s,Ia(s))ds,Hm(ϑ)=Hm,0+1Γ(μ)∫0ϑ(ϑ−s)μ−1K6(s,Hm(s))ds,Hs(ϑ)=Hs,0+1Γ(μ∫0ϑ(ϑ−s)μ−1K7(s,Hs(s))ds,Icu(ϑ)=Icu,0+1Γ(μ)∫0ϑ(ϑ−s)μ−1K8(s,Icu(s))ds.(8)

Assume that ‖E(ϑ)‖≤c1, ‖Iss(ϑ‖≤c2, ‖Ims(ϑ)‖≤c3, where ci,i=1,…,3, are some positive constants. Denote L1=βN(c1+c2+c3)+v,L2=θ,L3=ψ1+h,L4=φ1+ηi,L5=ηa,L6=φ1+ηHm,L7=ψ2+ηHs+δHs.

Theorem 4.1. The kernels Ki(i=1,…,8), satisfy the Lipschitz and contractive conditions if

0≤Li<1(9)

hold.

Proof. Let S∗ and S∗∗ be two functions, then

‖K1(ϑ,S∗)−K1(ϑ,S∗∗)‖=‖(−βN(Iss(ϑ)+Ims(ϑ)+Ia(ϑ))−v)(S∗∗−S∗)‖≤[βN(‖Iss‖+‖Ims‖+‖Ia‖)+v]‖S∗(ϑ)−S∗∗(ϑ)‖≤L1‖S∗(ϑ)−S∗∗(ϑ)‖.

Similarly

‖K2(ϑ,E∗)−K2(ϑ,E∗∗)‖≤L2‖E∗(ϑ)−E∗∗(ϑ)‖,

‖K3(ϑ,Iss∗)−K3(ϑ,Iss∗∗)‖≤L3‖Iss∗(ϑ)−Iss∗∗(ϑ)‖,

‖K4(ϑ,Ims∗)−K4(ϑ,Ims∗∗)‖≤L4‖Ims∗(ϑ)−Ims∗∗(ϑ)‖,

‖K5(ϑ,Ia∗)−K5(ϑ,Ia∗∗)‖≤L5‖Ia∗(ϑ)−Ia∗∗(ϑ)‖,

‖K6(ϑ,Hm∗)−K6(ϑ,Hm∗∗)‖≤L6‖Hm∗(ϑ)−Hm∗∗(ϑ)‖,

‖K7(ϑ,Hs∗)−K7(ϑ,Hs∗∗)‖≤L7‖Hs∗(ϑ)−Hs∗∗(ϑ)‖,

‖K8(ϑ,Icu∗)−K8(ϑ,Icu∗∗)‖≤L8‖Icu∗(ϑ)−Icu∗∗(ϑ)‖.

Thus, Ki satisfy Lipschitz condition, for i=1,….,8. From Eq. (9), Ki (i=1,…,8) are contraction.

Theorem 4.2. The model Eq. (4) has a unique solution provided

1μΓ(μ)rμLi<1, for i=1,2,…,8,

when r∈[0,ϑ].

Proof. The proof for the above result is the same as the proof of Theorem 1 in [52].

5  Reproductive Number and Stability Analysis

5.1 Reproductive Number

We follow the references [53,54] to calculate ℛ0. In order to calculate ℛ0, we consider the subsystem Eq. (5).

The DFE point is given by ℰ0=(N,0,0,0,0,0,0,0,0,0,0). From the system (4), we can write

F=(0βSa+SuN(Iss+Ims+Ia)000000),

and V associated with the net rate outside of the corresponding compartments

v=(−βSuN(Iss+Ims+Ia)−vSu−θEθp1E+φ1Ims−(ψ1+h)Issθp2E+ψ1Iss−(ηi+φ1)Imsθ(1−p1−p2)E−ηaIahτIss+ψ2Hs−(φ2+ηHm)HmhρIss+φ2Hm−(ψ2+ηHs+δHs)Hsh(1−τ−ρ)Iss−Icu)

It follows that the Jacobian matrices of F and V at ℰ0 respectively are

ℱ=(0000000000βββ000000000000000000000000000000000000000000000000000),

and 𝒱=(−v00000000−θ0000000θp1−(h+ψ1)φ100000θp2ψ1−(ηi+φ1)00000θ(1−p1−p2)00−ηa00000hτ00−(φ2+ηHm)ψ2000hρ00φ2−(ψ2+ηHm+δHs)000h(1−τ−ρ)0000−1).

Thus, the ℛ0 is obtained as the spectral radius of −ℱ𝒱−1 as

ℛ0=maxλ∈Sp(−ℱ⋅𝒱−1)|λ|=Sp(−ℱ⋅𝒱−1)=β(1−p1−p2)ηa−β(p2(h+ψ1)+p1ψ1+p1(ηi+φ1)+p2φ1)hηi+hφ1+ψ1ηi.

5.2 Stability Analysis

The reported result below is given in Theorem 2 of [54].

Theorem 5.1. The DFE ℰ0 of model Eq. (4), is locally asymptotically stable if ℛ0<1 and unstable if ℛ0>1.

6  Fractional Order Model with Vaccination

In this part, we discuss the fractional order model with vaccination which is overseeing and controlling contagious diseases by giving security to powerless and susceptible individuals. We suppose that a specified proportion p of people in the mindful susceptible class are vaccinated. For this situation, inoculated people are moved to another compartment V(ϑ). Due to the vaccine does not supply impunity to all vaccine recipients, vaccinated people might become infected yet a lower rate than unvaccinated. In this situation, let σv∈[0,1] such that (1−σv) be the vaccine efficacy. The flowchart of the fractional order model is as given in Fig. 2.

Figure 2: Flowchart of the fractional model (10)

We can interpret this diagram into fractional differential equations as follows:

{Sa(μ)(ϑ)=Λ−pSa(ϑ)−βSa(ϑ)N(Iss(ϑ)+Ims(ϑ)+Ia(ϑ))+vSu(ϑ),Su(μ)(ϑ)=−βSu(ϑ)N(Iss(ϑ)+Ims(ϑ)+Ia(ϑ))−vSu(ϑ),E(μ)(ϑ)=βSa(ϑ)+Su(ϑ)N(Iss(ϑ)+Ims(ϑ)+Ia(ϑ))+σVβV(ϑ)N(Iss(ϑ)+Ims(ϑ)+Ia(ϑ))−θE(ϑ),Iss(μ)(ϑ)=θp1E(ϑ)+φ1Ims(ϑ)−(ψ1+h)Iss(ϑ),Ims(μ)(ϑ)=θp2E(ϑ)+ψ1Iss(ϑ)−(ηi+φ1)Ims(ϑ),Ia(μ)(ϑ)=θ(1−p1−p2)E(ϑ)−ηaIa(ϑ),Hm(μ)(ϑ)=hτIss(ϑ)+ψ2Hs(ϑ)−(φ2+ηHm)Hm(ϑ),Hs(μ)(ϑ)=hρIss(ϑ)+φ2Hm(ϑ)−(ψ2+ηHs+δHs)Hs(ϑ),Icu(μ)(ϑ)=h(1−τ−ρ)Iss(ϑ)−Icu(ϑ),R(μ)(ϑ)=ηiIms(ϑ)+ηaIa(ϑ)+ηHsHs(ϑ)+ηHmHm(ϑ)+(1−δi)Icu(ϑ),D(μ)(ϑ)=δHsHs(ϑ)+δiIcu(ϑ),V(μ)(ϑ)=pSa(ϑ)−σVβV(ϑ)N(Iss(ϑ)+Ims(ϑ)+Ia(ϑ)),(10)

with Sa(0)=(1−p)Sa,0,Su(0)=(1−p)Su,0,E(0)=E0,Iss(0)=Iss,0,Ims(0)=Ims,0,Ia(0)=Ia,0, Hm(0)=Hm,0,Hs(0)=Hs,0,Icu(0)=Icu,0,D(0)=D0,R(0)=R0,V(0)=pN. Then, the fractional order model (10) has one equilibrium point

ℰv=((1−p)N,0,0,0,0,0,0,0,0,0,0,pN).

The components of infection in this model are, Su,E,Iss,Ims,Ia,Hm,Hs, and Icu.

Theorem 6.1. The reproduction number ℛv for the vaccinated fractional order model (10) is given by

ℛv=(1−(1−σv)p)ℛ0.

Proof. Let F and V be given by

F=(0βSa+SuN(Iss+Ims+Ia)+βσvVN(Iss+Ims+Ia)000000),

and

V=(−βSuN(Iss+Ims+Ia)−vSu−θEθp1E+φI1Ims−(ψI1+h)Issθp2E+ψI1Iss−(ηi+φI1)Imsθ(1−p1−p2)E−ηaIahτIss+ψI2Hs−(φI2+ηHm)HmhρIss+φI2Hm−(ψI2+ηHs+δHs)Hsh(1−τ−ρ)Iss−Icu).

By the differential of F and V with respect to E,Iss,Ims,Ia,Hm, and Icu and evaluating at the ℰv, respectively, we get

ℱ:=JF(ℰv)=(1−(1−σv)p)(0000000000βββ000000000000000000000000000000000000000000000000000)

and 𝒱:=JV(ℰv)=

(−v00000000−θ0000000θp1−(h+ψI1)φI100000θp2ψI1−(ηi+φI1)00000θ(1−p1−p2)00−ηa00000hτ00−(φI2+ηHm)ψI2000hρ00φI2−(ψI2+ηHm+δHs)000h(1−τ−ρ)0000−1).