Computer Systems Science & Engineering DOI:10.32604/csse.2022.020869
Article

An Approximate Numerical Methods for Mathematical and Physical Studies for Covid-19 Models

Hammad Alotaibi, Khaled A. Gepreel, Mohamed S. Mohamed and Amr M. S. Mahdy*

Department of Mathematics, College of Science, Taif University, Taif, 21944, Saudi Arabia
*Corresponding Author: Amr M. S. Mahdy. Email: amattaya@tu.edu.sa
Received: 12 June 2021; Accepted: 22 September 2021

Abstract: The advancement in numerical models of serious resistant illnesses is a key research territory in different fields including the nature and the study of disease transmission. One of the aims of these models is to comprehend the elements of conduction of these infections. For the new strain of Covid-19 (Coronavirus), there has been no immunization to protect individuals from the virus and to forestall its spread so far. All things being equal, control procedures related to medical services, for example, social distancing or separation, isolation, and travel limitations can be adjusted to control this pandemic. This article reveals some insights into the dynamic practices of nonlinear Coronavirus models dependent on the homotopy annoyance strategy (HPM). We summon a novel sign stream chart that is utilized to depict the Coronavirus model. Through the numerical investigations, it is uncovered that social separation of the possibly tainted people who might be conveying the infection and the healthy virus-free people can diminish or interrupt the spread of the infection. The mathematical simulation results are highly concurrent with the statistical forecasts. The free balance and dependability focus for the Coronavirus model is discussed and the presence of a consistently steady arrangement is demonstrated.

Keywords: Covid-19 model; optimal control; existence of uniformly stable; signal stream chart; homotopy perturbation technique

1  Introduction

Over the most recent couple of years, various numerical models have been created to give smart subtleties into numerous issues of interest including the transmission and control of irresistible illnesses [1–3]. The description of the ongoing pandemic of Coronavirus, as viral pneumonia in late 2019 in Wuhan-China which has spread worldwide across 210 countries is “SARS-CoV-2” [4–6]. It is seething around the world with a tremendous cost as far as human, financial, and social effects [7–9]. Within a limited ability to focus, this brings a caution up in each country everywhere in the world like a pandemic sickness, which encourages every country to estimate the beneficiary preparatory activities to control and contain the wild spread of the infection as the seriousness of the illness will hurt human existence severely [10–12]. Since the novel Covid is new to the world to figure some effect of the pandemic circumstance and to fabricate an alleviation plan, the similitude impacts of Severe Acute Respiratory Syndrome (SARS) and (Center East Respiratory Condition) scourges in 2003 and 2009 were utilized for study and investigation [13–15]. From the investigation of the underlying spread of Coronavirus, a considerable lot of numerical models were utilized in the demonstration from benefactors across the world to decide the seriousness of the spread [16–18]. At whatever point an infectious sickness expands its feeder, it follows certain examples of spread, which broadly assist us with distinguishing and screening the elements of the illness flare-ups [19–21]. The strategy we used to appraise the spread of the sickness is a factor that drives us to finish the actions to dispose of irresistible infections [22–24]. The episode of the infection inside the country or state for time is normally nonlinear, which moves us to plan the framework where we can contemplate those dynamic nonlinear wonders [25–27]. By this framework, we can be ready to characterize the transmission of such a viral infection, which assists us to interpret the medicinal measures to stop or contain the spread of this infectious illness [28–30].

Recently, the quantity of passings throughout the planet has expanded significantly because of the spread of the new infection known as Covid-19 (Coronavirus). The quick heightening of cases in practically all nations has created a genuine test for the whole world particularly when the World Health Organization proclaimed that this infection has turned into a pandemic since its outbreak and quickly spread from China to the rest of the world. Most nations throughout the whole planet have implemented the recommended protocols to contain the development of the virus and to limit the spread of its conceivably lethal infection in all countries. Regardless of the adverse consequences on local economies and financial well-being, limiting the virus development is viewed as quite possibly the best approach to slow the infection transmission both locally and across the world. The number of infected cases globally has increased reaching 29 million so far. Consequently, the spread of Covid-19 is perceived as being quite possibly the worst disease outbreak over the last forty years. At this stage, there is no antibody against Covid-19 and most people have not yet acquired the resistance that can protect them against such a disease. As such, it is vital to address the current conditions of Covid-19 in order to forestall the disease and to take the necessary steps to contain any additional spread of its infection. In view of the reports of immunologists and clinical experts, the virus mainly spreads through respiratory droplets when an infected person coughs, sneezes, or speaks in close proximity to healthy people. Accordingly, scientists have been strongly advocating social separation between possibly contaminated people and healthy individuals to mitigate or reduce the transmission rate among local communities across countries. The test of Coronavirus is presently attracting specialists from medicine and atomic science to apply math in numerical displaying that can assume a huge part in anticipating, evaluating, and controlling the likely situations. For more extra solving the modelling mathematics [31–50].

Avoidance of Coronavirus, other than the significance of forcing general wellbeing and contamination control measures to forestall or diminish the transmission of SARS-CoV-2, the way to containing this worldwide pandemic is by inoculation to forestall SARS-CoV-2 disease in masses across the world. Exceptional endeavors in worldwide examination during this pandemic have brought about the advancement of novel immunizations against SARS-CoV-2 at a phenomenal speed to contain this viral ailment that has crushed nations worldwide and has had a descending spiraling impact on the worldwide economy.

Inoculation triggers the insusceptible framework prompting the creation of killing antibodies against SARS-CoV-2. Consequences of a progressing global, fake treatment controlled, eyewitness, dazed, urgent adequacy preliminary announced that people 16 years old or more established getting two-portion routine the preliminary immunization BNT162b2 (mRNA-based, BioNTech/Pfizer) when given 21 days separated presented 95% insurance against Coronavirus with a wellbeing profile like other viral vaccines. Results from another multicenter, Stage 3, randomized, spectator dazed, fake treatment controlled preliminary exhibited that people who were randomized to get two dosages of mRNA-1273 (mRNA based, Moderna) antibody given 28 days separated showed 94.1% viability at forestalling Coronavirus sickness, and no security concerns were noted other than mild fever and transient reactions. In light of the aftereffects of these antibody adequacy preliminaries, the FDA gave two EUAs, one on December 11, 2020, allowing the utilization of the BNT162b2 immunization, and another on December 18, 2020, conceding the utilization of the mRNA-1273 antibody for the avoidance of Coronavirus. A third immunization, Ad26.COV2.S for the avoidance of Coronavirus got EUA by the FDA on February 27, 2021, in light of a multicenter, fake treatment control, stage preliminary showed that a solitary portion of Ad26.COV2.S antibody presented 73% viability in the US in forestalling Coronavirus (information not yet distributed).

Therefore, the present study was undertaken to fill in this gap of knowledge.

S.=−u1(βC−Cq(1−β))S(I+θA)+λu2Sq,E.=βu1(1−q)S(I+θA)−σE,I.=σρE−(δI+α+γI)I,A.=σ(1−ρ)E−γAA,Sq.=u1(1−β)CqS(I+θA)−λu2Sq,Eq.=u1βCqS(I+θA)−δqEq,H.=δII+δqEq−(α+γH)H,R.=γII+γAA+γHH. (1)

Fig. 1, shows the sign stream chart G→ of the structure in which each vertex speaks with the case of the framework. There is an edge (v1,v2) if the state v1 straightforwardly influences the state v2 .

Figure 1: Suggestion signal stream chart of the model

Using a signal stream chart to act the dynamical systems is much profitable to view, for example [9,13].

A signal stream chart is a scheme agent that is used to display the interrelation among the system states and become possible to utilize scheme-theoretic stuff to find novel brow of the system.

2  Optimal Control for Covid-19 Modeling

Consider the state presented (1), in R8 , with control functions admissible [12,13]:

Ω={(u1(.),u2(.))∈(L∞(0,Tf)2)|0≤u1(.),u2(.)≤1,∀t∈[0,Tf]},

where Tf is time final, u1(.)and u2(.) are functions controls.

Defined the functional objective is (quadratic is the control variable)

J1(ζ1,ζ2)=∫0Tf[B1Sq+B2I+B3u12+B4u22]dt, (2)

Minimizes function objective at it [28–34]:

J2(ζ1,ζ2)=∫0Tfη(S,E,I,A,Sq,Eq,H,R,ζ1,ζ2,t)dt, (3)

where η(S,E,I,A,Sq,Eq,H,R,ζ1,ζ2,t)=[B1Sq+B2I+B3u12+B4u22] subjected to the constraint

DS=b1,DE=b2,DI=b3,DA=b4,DSq=b5,DEq=b6,DH=b7,DR=b8. (4)

The following initial conditions are satisfied:

S=S0,E=E0,I=I0,A=A0,Sq=Sq0,H(0)=H0,R(0)=R0. (5)

OCP is defined, consider the a modified objective function:

ϕ¯=∫0Tf[χ−∑j=18λjwj]dt, (6)

where the objective function of Hamiltonian (6) as follows:

χ=η+∑j=18λjwj, (7)

χ=B1Sq+B2I+B3u12++B4u22++B5RC+λ1(−u1(βC−Cq(1−β))S(I+θA)+λu2Sq)+λ2(βu1(1−q)S(I+θA)−σE)+λ3(σρE−(δI+α+γI)I)+λ4(σ(1−ρ)E−γAA)+λ5(u1(1−β)CqS(I+θA)−λu2Sq)+λ6(u1βCqS(I+θA)−δqEq)+λ7(δII+δqEq−(α+γH)H)+λ8(γII+γAA+γHH). (8)

Applying Pontryagin's maximum principal, from (6) and (8), conditions necessary and sufficient OPC is

Dλ1=∂χ∂S, Dλ2=∂χ∂E, Dλ3=∂χ∂I, Dλ4=∂χ∂A, Dλ5=∂χ∂Sq,Dλ6=∂χ∂Eq, Dλ7=∂χ∂H, Dλ8=∂χ∂R. (9)

0=∂χ∂ξ1,0=∂χ∂ξ2. (10)

DS=∂χ∂λ1,DE=∂χ∂λ2,DI=∂χ∂λ3,DA=∂χ∂λ4,DSq=∂χ∂λ5,DEq=∂χ∂λ6,DH=∂χ∂λ7,DR=∂χ∂λ8 (11)

λj,(Tj)=0,j=1,...,8. (12)

where λj are multipliers Lagrange. Eqs. (10), (11) exemplify the conditions necessary of Hamiltonian for the OPC.

We construct the theorem at similar [12,13,30,31]:

Theorem 1.

If u1 and u2 are controls optimal with state corresponding variables, consequently they work out adjoint variables λj∗,j=1,...,8, accepts:

(i) Co-state equation

Dλ1=λ1*(−u1(βC+Cq(1−β))S(I+θA))+λ2*(βu1(1−q)S(I+θA))+λ5*(u1(1−β)CqS(I+θA)) +λ6*(u1βCqS(I+θA)),

Dλ2=λ2∗(−δ)+λ3∗(δρ)+λ4∗(δ(1−ρ)),

Dλ3=B2+λ1∗(−u1(βC+Cq(1−β))S)+λ2∗(βu1(1−q)S)+λ3∗(−(δI+α+γI))+λ5∗(u1(1−β)CqS)+λ6∗(u1βCqS)+λ7∗(δI)+λ8∗(γI),

−u1(βC+Cq(1−β))Sθ 

Dλ5=B1+λ1∗(−u2λ)+λ2∗(−λu2),Dλ6=λ7∗(δq)+λ6∗(−δq),Dλ7=λ7∗(−(α+γH))+λ8∗(γH),Dλ8=B5. (13)

(ii) With condition transversality:

λj∗(Tf)=0. (14)

(iii) Optimality conditions

χ=min0≤uV∗,uR∗≤1χ, (15)

So, the functions control u1∗,u2∗ .

For a lot of optimal control and model differentials, equations for solving models see [1,29].

3  Existence of Uniformly Stable Solution

This subsection explores the existence of uniformly stable solution. Let us define

g1=−u1(βC−Cq(1−β))S(I+θA)+λu2Sq,g2=βu1(1−q)S(I+θA)−σE,g3=σρE−(δI+α+γI)I,g4=σ(1−ρ)E−γAA,g5=u1(1−β)CqS(I+θA)−λu2Sqg6=u1βCqS(I+θA)−δqEqg7=δII+δqEq−(α+γH)Hg8=γII+γAA+γHH. (16)

Let Δ={S,E,I,A,Sq,Eq,H,R∈ℜ:|S,E,I,A,Sq,Eq,H,R|≤c,t∈[0,T]} .

This means such all of the 8 functions f1,f2 ∂g∂S, ∂g∂E,∂g∂I, ∂g∂A, ∂g∂Sq, ∂g∂Eq, ∂g∂H, and ∂g∂R coincide with the Lipschitz z case with regard to the 8 arguments, thereafter all of the 8 functions f1,f2 have absolutely continuous with regard to the 8 arguments [12,13]. For extra of fractional models [32–50].

4  Invariance and Symmetry of the Proposed Model

For system (1), the transformation

(S,E,I,A,Sq,Eq,H,R)→(S,−E,−I,−A,−Sq,−Eq,−H,−R) implies that the system is invariant.

Then if (S,E,I,A,Sq,Eq,H,R) is a solution of model (1), then (S,−E,−I,−A,−Sq,−Eq,−H,−R) is a solution of the same model too.

Also, we proved the divergence of the proposed paradigm (1) as follows

∇.Z=∂ S⋅∂S+∂ E⋅∂E+∂ I⋅∂I+∂ A⋅∂A+∂Sq⋅∂Sq+∂Eq⋅∂Eq+∂ H⋅∂H+∂ R⋅∂R=−(βC−Cq(1−β))S(I+θA)−σ−(δI+α+γI)−γA−λ−Sq−(α+γH).

Then proposed model paradigm (1) is dissipative such as

∇.Z=−(βC−Cq(1−β))S(I+θA)−σ−(δI+α+γI)−γA−λ−Sq−(α+γH)<0.

5  HPM Approximates the Solution for Nonlinear COVID-19 Model

Here, we present the scientific approximate solution to the COVID-19 (1). By utilizing MHPM procedure [22,23], we develop a homotopy Hi(t,p):R+×[0,1]→R+ , which satisfies:

H1=S′(t)+p[(βc+cq(1−β))S(t)(I(t)+θA(t))−λSq(t)]=0, (17)

H2=E′(t)+σE(t)]−p[βc(1−q)S(t)(I(t)+θA(t))]=0, (18)

H3=I′(t)+(δi+α+γi)I(t)−pσρE(t)=0, (19)

H4=A′(t)+γAA(t)−pσ(1−ρ)E(t)=0, (20)

H5=Sq′(t)+λSq(t)−p(1−β)cqS(t)(I(t)+θA(t))=0, (21)

H6=Eq′(t)+δqEq(t)−pβcqS(t)(I[t]+θA(t))=0, (22)

H7=H′(t)+(α+γH)H(t)−p(δII(t)+δqEq(t)=0, (23)

H8=R′(t)−p(γiI(t)+γAA(t)+γHH(t))=0. (24)

As per the HPM method, we guess the arrangements of conditions (17) and (24) as a force arrangement in p , where p the installing little boundary:

S(t)=S0(t)+pS1(t)+p2S2(t)+p3S3(t)+… (25)

E(t)=E0(t)+pE1(t)+p2E2(t)+p3E3(t)+… (26)

I(t)=I0(t)+pI1(t)+p2I2(t)+p3I3(t)+… (27)

A(t)=A0(t)+pA1(t)+p2A2(t)+p3A3(t)+… (28)

Sq(t)=Sq0(t)+pSq1(t)+p2Sq2(t)+p3Sq3(t)+… (29)

Eq(t)=Eq0(t)+pEq1(t)+p2Eq2(t)+p3Eq3(t)+… (30)

H(t)=h0(t)+ph1(t)+p2h2(t)+p3h3(t)+… (31)

R(t)=R0(t)+pR1(t)+p2R2(t)+p3R3(t)+… (32)

Inserting Eqs. (25)–(32) into Eqs. (17)–(24) and setting the coefficient of p to be zero, we deduce the following system of ODE’s as follows:

S0′(t)=0, (33)

−λSq0(t)+cqθA0(t)S0(t)+cβθA0(t)S0(t)−cqβθA0(t)S0(t)+cqI0(t)S0(t)

+cβI0(t)S0(t)]−cqβI0(t)S0(t)]+S1′(t)=0, (34)

−λSq1(t)+cqθA1(t)S0(t)+cβθA1(t)S0(t)−cqβθA1(t)S0(t)+cqI1(t)S0(t)

+cβI1(t)S0(t)−cqβI1(t)S0(t)+cqθA0(t)S1(t)+cβθA0(t)S1(t)−cqβθA0(t)S1(t)

+cqI0(t)S1(t)+cβI0(t)S1(t)−cqβI0(t)S1(t)+S2′(t)=0,…, (35)

σE0(t)+E0′(t)=0, (36)

E1′(t)+σE1(t)−cβθA0(t)S0(t)+cqβθA0(t)S0(t)−cβI0(t)S0(t)+cqβI0(t)S0(t)=0, (37)

E2′+σE2(t)−cβθA1(t)S0(t)+cqβθA1(t)S0(t)−cβI1(t)S0(t)+cqβI1(t)S0(t) −cβθA0(t)S1(t)+cqβθA0(t)S1(t)−cβI0(t)S1(t)+cqβI0(t)S1(t)=0,... (38)

I0′(t)+(α+γi+δi)I0(t)=0, (39)

I1′(t)+(α+γi+δi)I1(t)−ρσE0(t)=0, (40)

I2′(t)+(α+γi+δi)I2(t)−ρσE1(t)=0,…, (41)

A0′(t)+γAA0(t)=0, (42)

A1′(t)+γAA1(t)−σE0(t)+ρσeE0(t)=0, (43)

A2′(t)+γAA2(t)−σE1(t)+ρσeE1(t)=0,… (44)

Sq0′(t)+λSq0(t)=0 (45)

Sq1′(t)+λSq1(t)−cqθA0(t)S0(t)+cqβθA0(t)S0(t)−cqI0(t)S0(t)+cqβI0(t)S0(t)=0 (46)

Sq2′(t)+λSq2(t)−cqθA1(t)S0(t)+cqβθA1(t)S0(t)−cqI1(t)S0(t)+cqβI1(t)S0(t)−cqθA0(t)S1(t)+cqβθA0(t)S1(t)−cqI0(t)S1(t)+ cqβI0(t)S1(t)=0 (47)

Eq0′(t)+δqEq0(t)=0 (48)

Eq1′(t)+δqEq1(t)−cqβθA0(t)S0(t)−cqβI0(t)S0(t)=0 (49)

Eq2′(t)+δqEq2(t)−cqβθA1(t)S0(t)−cqβI1(t)S0(t)−cqβθA0(t)S1(t)

−cqβI0(t)S1(t)=0,… (50)

αh0(t)+h0(t)γH+h0′(t)=0 (51)

αh1(t)+h1(t)γH−I0(t)δi−Eq0(t)δq+h1′(t)=0 (52)

αh2(t)+h2(t)γH−I1(t)δi−Eq1(t)δq+h2′(t)=0,… (53)

R0′(t)=0, (54)

−A0(t)γA−h0(t)γH−I0(t)γi+R1′(t)=0, (55)

−A1(t)γA−h1(t)γH−I1(t)γi+R2′(t)=0,… (56)

With the initial conditions:

S0(0)=S0,S1(0)=0,S2(0)=0,S3(0)=0,…. (57)

E0(0)=E0,E1(0)=0,E2(0)=0,E3(0)=0,…. (58)

I0(0)=I0,I1(0)=0,L2(0)=0,I3(0)=0,…. (59)

A0(0)=A0,A1(0)=0,A2(0)=0,A3(0)=0,…. (60)

Sq0(0)=Sq0,Sq1(0)=0,Sq2(0)=0,Sq3(0)=0,…. (61)

Eq0(0)=Eq0,Eq1(0)=0,Eq2(0)=0,Eq3(0)=0,…. (62)

h0(0)=h0,h1(0)=0,h2(0)=0,h3(0)=0,…. (63)

R0(0)=R0,R1(0)=0,R2(0)=0,R3(0)=0,…. (64)

Now, we solve the over system of ordinary differential Eqs. (33)–(56) with the initial conditions (57)–(64) to get the results:

The results of the first iteration are given by:

S0(t)=S0,

E0(t)= e−tσE0,

I0(t)=et(−α−γi−δi)I0,

A0(t)=e−tγAA0,

Sq0(t)=e−tλSq0,

Eq0(t)=e−tδqEq0,

h0(t)=et(−α−γH)h0,

R0(t)=R0. (65)

The results of the second iteration are given by

S1(t)=1γA(α+γi+δi)e−t(α+λ+γi+δi)(c(et(α+λ+γi+δi)−et(α+λ−γA+γi+δi)) (q(−1+β)−β)θA0S0(α+γi+δi)−γA(c(etλ−et(α+λ+γi+δi)) (q(−1+β)−β)I0S0−et(α+γi+δi)(−1+etλ)Sq0(α+γi+δi))),

E1(t)=1(σ−γA)(α−σ+γi+δi)ce−t(α+γi+δi)(−1+q)βS0(−(−1+et(α−σ+γi+δi))I0(σ−γA)+(et(α−σ+γi+δi)−et(α−γA+γi+δi))θA0(α−σ+γi+δi)),

I1(t)=et(−α−γi−δi)(−1+et(α−σ+γi+δi))ρσE0α−σ+γi+δi,

A1(t)=e−t(σ−γA)−tγA(−1+et(σ−γA))(−1+ρ)σE0−σ+γA,

Sq1(t)=1(λ−γA)(α−λ+γi+δi)ce−t(α+γi+δi)q(−1+β)S0(−(−1+et(α−λ+γi+δi))I0(λ−γA)+(et(α−λ+γi+δi)−et(α−γA+γi+δi))θA0(α−λ+γi+δi)),

Eq1(t)=1(α+γi+δi−δq)(−γA+δq){ce−t(α+γi+δi)qβS0((et(α−γA+γi+δi)−et(α+γi+δi−δq))θA0(α+γi+δi−δq)+(−1+et(α+γi+δi−δq))I0(−γA+δq)),