# Mathematical Analysis of Two Waves of COVID-19 Disease with Impact of Vaccination as Optimal Control

Authors: P. K. Santra; D. Ghosh; G. S. Mahapatra; Ebenezer Bonyah
Journal: Computational and Mathematical Methods in Medicine
 (2022)

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## Abstract

This paper is devoted to answering some questions using a mathematical model by analyzing India’s first and second phases of the COVID-19 pandemic. A new mathematical model is introduced with a nonmonotonic incidence rate to incorporate the psychological effect of COVID-19 in society. The paper also discusses the local stability and global stability of an endemic equilibrium and a disease-free equilibrium. The basic reproduction number is evaluated using the proposed COVID-19 model for disease spread in India based on the actual data sets. The study of nonperiodic solutions at a positive equilibrium point is also analyzed. The model is rigorously studied using MATLAB to alert the decision-making bodies to hinder the emergence of any other pandemic outbreaks or the arrival of subsequent pandemic waves. This paper shows the excellent prediction of the first wave and very commanding for the second wave. The exciting results of the paper are as follows: (i) psychological effect on the human population has an impact on propagation; (ii) lockdown is a suitable technique mathematically to control the COVID spread; (iii) different variants produce different waves; (iv) the peak value always crosses its past value.


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## Body

## 1. Introduction

The novel coronavirus (COVID-19) has spread in almost all parts of the world at the pandemic level. Many researchers [1, 2] have developed different models incorporating the hazards of COVID-19 pandemic. Atangana and Araz [3] describe the model and forecast the spread of COVID-19 in Africa and Europe. Wu et al. [4] form a COVID-19 model on the use of social distancing personal protection in Ontario, Canada. Aldila et al. [5] used the community awareness as a control scheme to minimize the transmission of COVID-19 outbreak. Sen and Ibeas [6] used vaccination [7] and antiviral to control the pandemic of COVID-19. Some researchers [8, 9, 30, 32, 33, 36, 39] have discussed the effect of COVID-19 in society and optimal policy in different countries.After the initial outbreak, COVID-19 continued to spread to all provinces in India. India has controlled [10] the rate of spread of COVID-19 after the first phase of the outbreak. However, due to the negligence of people, it spread quickly more than the first variant in the second wave [11, 12]. Mathematical modelling is used to predict the number of active cases, disease spread, and duration of this pandemic and estimate the impact of measures during disease outbreaks. Ghosh et al. [13] described the transmission of COVID-19 outbreak in India based on the 1st wave. The second wave of the pandemic has come at the end of January 2021 in different countries including India; in this respect, Ershkov and Rachinskaya [14] and Glass [15] both perform a model to describe the second wave of COVID-19.This article presents a mathematical model that describes the evolution of the COVID-19 in India using the actual data in two phases,1st phase from March 23 to December 31, 2020, and 2nd phase of daily update confirmed cases, recovered, and deaths in India, in order to estimate the parameters of the model and then predict the severity of the possible infection in the coming months. Using this method, we can estimate the size of the population at risk in India and justify the growing number of new confirmed cases. With the aim to reduce the population at risk in India, we investigate an optimal control strategy by adopting vaccine which makes it very optimal, and this study may be more practical to use in developing countries. In this study, we are trying to answer some questions. How many waves come? What will be the peak value of the consequence wave? What is the effect of lockdown on COVID control? Is there any effect of fear in propagation? We are given some answers to these questions using a mathematical model.The contents of this study are organised as follows. The first section has laid the context of the work. The second section discusses the preparation of the model and its basic properties. Section3 finds the equilibrium points and checks the stability like local and global. The fourth section discusses the nonexistence of a periodic solution. The fifth section forms a COVID-19 model with the concept of optimal control. The sixth section presents the results for different waves in respect to India. Finally, section seven concludes this study and presents the precaution and future directions for this research work.


## 2. Novel Coronavirus Model with Basic Properties

Already in the literature, there are some papers to understand the dynamics of novel coronavirus spread [13, 16–18, 31, 34, 35, 37, 38, 40, 41]. This coronavirus model proposes to fill the inadequacy of previous studies for analyzing the spread dynamics incorporating the effect on human consciousness of the novel COVID-19. Based on the medical practitioners’ instructions, regular hand wash, nose and mouth cover, safe distancing, etc. affect the transmission rate. We consider α as a representative of hand wash, nasal and oral cover, and social distancing in the proposed coronavirus model, and hence, increasing disease transmission means that such instructions are not followed properly. This model also considers the consciousness of the disease as a parameter, i.e., the parameter δ measures the psychological or inhibitory effect. Furthermore, this model considers a saturated incidence rate gIuS for COVID-19 pandemic model, when Iu gets larger, i.e., gIu=αIu/1+δIu tends to be overloaded, where infection force of the disease is calculated by αIu, and 1/1+δIu measures the reticence effect from the observable change of the susceptible individuals when their number increases or from the crammed effect of the infected individuals. For COVID-19, the proposed rate of incidence [19–21] seems more justifiable compared to the other incidence rate, because it includes the detectable change and cramming effect of the infected individuals and prevents the unboundedness of the association rate by choosing apt and relevant parameters.At timet, let St,Iut,Ikt, and Rt be the densities of susceptible population S, unrevealed infected population Iu which spread the disease, known infected population Ik in isolated ward for treatment not spreading the disease, and recover population R, respectively. Our important conjecture for this model is that the disease spread by unrevealed infected populations and COVID has a psychological effect on the human population.The mathematical form of the novel coronavirus transmission as discussed above takes the following form:(1)dSdt=Λ−αSIu1+δIu−d1S,dIudt=αSIu1+δIu−βIu−d1Iu,dIkdt=βIu−γIk−d2Ik,dRdt=γIk−d1R.Here,Nt=St+Iut+Ikt+Rt stands for the total number of human community in the system at time t. The proposed COVID-19 pandemic model will analyse with the following initial densities:
(2)S0=S0,R0=R0>0,Iu0=Iu0,Ik0=Ik0≥0.The flow diagram of the proposed COVID-19 pandemic model is presented in Figure1. The model parameters with their assumed and estimated values from the real data of India during the time span of 1st February to 6th June 2021 are described in Table 1.Figure 1
Transfer diagram of the proposed COVID-19 system.Table 1
Explanation of parameters with their real field value.
ParametersInterpretationValue of the 1st wave/dayValue of the 2nd wave/dayReferenceΛRecruitment rate of new individuals1×1032×103FittedαTransmission coefficient from SP toIuP55×10−1055×10−10FittedδMeasures of the psychological or inhibitory effect33×10−8135×10−9FittedβTransmission coefficient fromIuP to IkP0.210.21FittedγTransmission coefficient fromIkP to RP0.10.1Fittedd1Natural death rate4×10−54×10−5Fittedd2Death rate due to COVID-19 plusd11×10−31×10−3FittedSince in this novel coronavirus model the variableRt has no effect in dynamics of the system, we eliminate the last equation from the pandemic model (1) for the dynamical analysis. Hence, the dynamical study of COVID-19 system is considered from India’s perspective using the following mathematical model:
(3)dSdt=Λ−αSIu1+δIu−d1S,(4)dIudt=αSIu1+δIu−βIu−d1Iu,(5)dIkdt=βIu−γIk−d2Ik.


### 2.1. Nonnegativity of Solution of COVID-19 Model

Theorem 1.
Every solution of COVID-19 system (1) with initial conditions (2) exists in the interval 0,∞ and St>0,Iut≥0,Ikt≥0, and Rt>0 for all t≥0.Proof.
Since the right hand side of COVID-19 model (1) is continuous and locally Lipschitzian, then the solution St,Iut,Ikt,Rt of (1) with respect to the initial conditions is unique on 0,ξ, where 0<ξ<+∞.
From the model (1) and using the initial conditions, we have
(6)dSdt=Λ−d1+αIut1+δIutSt.
We thus have(7)ddtStexpd1t+∫0tαIus1+δIusds=Λexpd1t+∫0tαIus1+δIusds.
Hence,(8)Stexpd1t+∫0tαIus1+δIusds−S0=∫0tΛexpd1t+∫0tαIuω1+δIuωdωdt,
so that(9)St=S0exp−d1t+∫0tαIus1+δIusds+exp−d1t+∫0tαIus1+δIusds×∫0tΛexpd1t+∫0tαIuω1+δIuωdωdt>0.
The second equation of the model (1) yields dIu/dt≥−β+d1Iut which gives Iut≥Iu0exp−β+d1t≥0.
From the third equation of system (1), we get dIk/dt≥−γ+d2Ik which gives Ikt≥Ik0exp−γ+d2t≥0.
The last equation of the system (1) yields dR/dt≥−d1Rt which gives Rt≥R0exp−d1t>0.
Therefore, we can see thatSt,Rt>0, and Iut,Ikt≥0,∀t≥0.
This completes the proof.


### 2.2. Invariant Region of Solutions of COVID-19 Model

Theorem 2.
All solutions of COVID-19 system (1) in ℝ+4 are bounded and lie in the region Ω defined by Ω=S,Iu,Ik,R∈ℝ+4:0<Nt≤Λ/μ as t⟶∞, where μ=mind1,d2.Proof.
AssumeSt,Iut,Ikt,Rt be any solution of system (1). Now, we consider a function like Nt=St+Iut+Ikt+Rt. Differentiating both sides with respect to t, we have
(10)dNtdt=dStdt+dIutdt+dIktdt+dRtdt,dNtdt=Λ−d1S−d1Iu−d2Ik−d1R,dNtdt+μNt=Λ−d1−μS−d1−μIu−d2−μIk−d1−μR.
dNt/dt+μNt≤Λ, assuming μ=mind1,d2.
Then, by comparison theorem, we obtain0<Nt≤N0e−μt+Λ/μ and for t⟶∞,0<Nt≤Λ/μ. Therefore, all solutions of coronavirus system (1) enter into the region Ω=S,Iu,Ik,R∈ℝ+4:0<Nt≤Λ/μ.


### 2.3. The Basic Reproduction Number

Definition 3 (basic reproduction number (BRN)).
“The BRN is defined as the number of newly infected individuals produced by a single infected individual during his or her effective infectious period when it is introduced into the susceptible population.”Here, the BRN (R0) for the proposed COVID-19 model (1) is given by
(11)R0=αΛβ+d1d1.Impact of transmission coefficientα from S to Iu is measured qualitatively on the coronavirus disease transmission dynamics.Since∂R0/∂α=Λ/β+d1d1>0, it is obvious that if α decreases, then BRN R0 also decreases and therefore reduces the disease burden. On the other side, if α increase, then R0 would rise leading to the rise of the infection burden, and therefore, the scenario changes to be a very harmful one.From Figure2, we observe that if the values of α increases then the value of R0 also increases (both cases first wave and second wave), and after certain value of α, R0 becomes greater than 1. From these two figures, we also observed that the second wave in India is more dangerous than the 1st wave. Our study finds that the mathematical model of this type does not predict the wave. So different waves come due to different strains of the COVID-19 virus.Figure 2
Change ofR0 with respect to α in two different waves using data in Table 1.


## 2.1. Nonnegativity of Solution of COVID-19 Model

Theorem 1.
Every solution of COVID-19 system (1) with initial conditions (2) exists in the interval 0,∞ and St>0,Iut≥0,Ikt≥0, and Rt>0 for all t≥0.Proof.
Since the right hand side of COVID-19 model (1) is continuous and locally Lipschitzian, then the solution St,Iut,Ikt,Rt of (1) with respect to the initial conditions is unique on 0,ξ, where 0<ξ<+∞.
From the model (1) and using the initial conditions, we have
(6)dSdt=Λ−d1+αIut1+δIutSt.
We thus have(7)ddtStexpd1t+∫0tαIus1+δIusds=Λexpd1t+∫0tαIus1+δIusds.
Hence,(8)Stexpd1t+∫0tαIus1+δIusds−S0=∫0tΛexpd1t+∫0tαIuω1+δIuωdωdt,
so that(9)St=S0exp−d1t+∫0tαIus1+δIusds+exp−d1t+∫0tαIus1+δIusds×∫0tΛexpd1t+∫0tαIuω1+δIuωdωdt>0.
The second equation of the model (1) yields dIu/dt≥−β+d1Iut which gives Iut≥Iu0exp−β+d1t≥0.
From the third equation of system (1), we get dIk/dt≥−γ+d2Ik which gives Ikt≥Ik0exp−γ+d2t≥0.
The last equation of the system (1) yields dR/dt≥−d1Rt which gives Rt≥R0exp−d1t>0.
Therefore, we can see thatSt,Rt>0, and Iut,Ikt≥0,∀t≥0.
This completes the proof.


## 2.2. Invariant Region of Solutions of COVID-19 Model

Theorem 2.
All solutions of COVID-19 system (1) in ℝ+4 are bounded and lie in the region Ω defined by Ω=S,Iu,Ik,R∈ℝ+4:0<Nt≤Λ/μ as t⟶∞, where μ=mind1,d2.Proof.
AssumeSt,Iut,Ikt,Rt be any solution of system (1). Now, we consider a function like Nt=St+Iut+Ikt+Rt. Differentiating both sides with respect to t, we have
(10)dNtdt=dStdt+dIutdt+dIktdt+dRtdt,dNtdt=Λ−d1S−d1Iu−d2Ik−d1R,dNtdt+μNt=Λ−d1−μS−d1−μIu−d2−μIk−d1−μR.
dNt/dt+μNt≤Λ, assuming μ=mind1,d2.
Then, by comparison theorem, we obtain0<Nt≤N0e−μt+Λ/μ and for t⟶∞,0<Nt≤Λ/μ. Therefore, all solutions of coronavirus system (1) enter into the region Ω=S,Iu,Ik,R∈ℝ+4:0<Nt≤Λ/μ.


## 2.3. The Basic Reproduction Number

Definition 3 (basic reproduction number (BRN)).
“The BRN is defined as the number of newly infected individuals produced by a single infected individual during his or her effective infectious period when it is introduced into the susceptible population.”Here, the BRN (R0) for the proposed COVID-19 model (1) is given by
(11)R0=αΛβ+d1d1.Impact of transmission coefficientα from S to Iu is measured qualitatively on the coronavirus disease transmission dynamics.Since∂R0/∂α=Λ/β+d1d1>0, it is obvious that if α decreases, then BRN R0 also decreases and therefore reduces the disease burden. On the other side, if α increase, then R0 would rise leading to the rise of the infection burden, and therefore, the scenario changes to be a very harmful one.From Figure2, we observe that if the values of α increases then the value of R0 also increases (both cases first wave and second wave), and after certain value of α, R0 becomes greater than 1. From these two figures, we also observed that the second wave in India is more dangerous than the 1st wave. Our study finds that the mathematical model of this type does not predict the wave. So different waves come due to different strains of the COVID-19 virus.Figure 2
Change ofR0 with respect to α in two different waves using data in Table 1.


## 3. Existence of Equilibrium Points and Stability

The equilibrium points of the proposed COVID-19 system (3) are (i) disease-free equilibrium point E1Λ/d1,0,0 and (ii) endemic equilibrium E∗S∗,Iu∗,Ik∗, where S∗=1+δIu∗β+d1/α,Iu∗=αΛ−d1β+d1/β+d1δd1+α, and Ik∗=β/γ+d2Iu∗. The endemic equilibrium point exists when R0>1.


### 3.1. Local Stability Analysis

The local stability analysis of the coronavirus model (3) is presented at the equilibrium points.


#### 3.1.1. Disease-Free Equilibrium

Theorem 4.
Disease-free equilibrium pointE1Λ/d1,0,0 is locally asymptotically stable if R0<1, marginally stable if R0=1, and unstable if R0>1.The proof of this theorem is in the Appendix.


#### 3.1.2. Endemic Equilibrium

Theorem 5.
Endemic equilibrium pointE∗S∗,Iu∗,Ik∗ is locally asymptotically stability if R0>1.The proof of this theorem is in the Appendix.Therefore, up to a certain value ofα disease-free equilibrium point is stable (Theorem 4) and beyond that value of α endemic equilibrium point is stable (Theorem 5).


### 3.2. Global Stability Analysis

The global stability analysis of the proposed COVID-19 model (3) is presented here.


#### 3.2.1. Disease-Free Equilibrium

Theorem 6.
IfR0<1, the disease-free equilibrium E1 is globally asymptotically stable.The proof of this theorem is in the Appendix.


#### 3.2.2. Global Stability of Endemic Equilibrium: Geometric Approach

The global stability of the endemic equilibriumE∗ will be discussed when R0>1 using the geometric approach for global dynamics [22]. For some preliminary discussion on the geometric approach, consider the autonomous dynamical system:
(12)x.=fx,where f:D⟶ℝn,D∈ℝn open set and simply connected and f∈C1D.LetAx be an n2×n2 matrix value function that is C1 on D and consider Q=AfA−1+AJ2A−1, where the matrix Af is qijxf=∂qijx/∂xT. fx=∇qij.fx, and here, J2 represents the second additive compound matrix of Jx=Dx. Let the Lozinskii measure [23] μ of Q concerning a vector norm . in ℝn2 be μQ=limh⟶0+I+hQ−1/h. Define a quantity q¯2 as q¯2=limsupt⟶∞supx0⟶K1/t∫0tμQxs,x0ds. We will apply the following theorem.Theorem 7.
LetD be simply connected, and [23] (H1) “there exists a compact absorbing set K⊂D,” (H2) “the system (12) has a unique equilibrium x~ in D,” then x~ of (12) is globally asymptotically stable in D if q¯2<0.Theorem 8.
The COVID-19 model (3) admits an unique endemic equilibrium which globally asymptotically stable if R0>1.The proof is in the Appendix.Since disease-free equilibrium pointE1Λ/d1,0,0 is globally stable for R0<1 and an unique endemic equilibrium point E∗S∗,Iu∗,Ik∗ is globally stable for R0>1, so there exists no limit cycle for this model. Therefore, we include Nonexistence Periodic Solution part for only the mathematical purpose for the proof in a different way.


## 3.1. Local Stability Analysis

The local stability analysis of the coronavirus model (3) is presented at the equilibrium points.


### 3.1.1. Disease-Free Equilibrium

Theorem 4.
Disease-free equilibrium pointE1Λ/d1,0,0 is locally asymptotically stable if R0<1, marginally stable if R0=1, and unstable if R0>1.The proof of this theorem is in the Appendix.


### 3.1.2. Endemic Equilibrium

Theorem 5.
Endemic equilibrium pointE∗S∗,Iu∗,Ik∗ is locally asymptotically stability if R0>1.The proof of this theorem is in the Appendix.Therefore, up to a certain value ofα disease-free equilibrium point is stable (Theorem 4) and beyond that value of α endemic equilibrium point is stable (Theorem 5).


## 3.1.1. Disease-Free Equilibrium

Theorem 4.
Disease-free equilibrium pointE1Λ/d1,0,0 is locally asymptotically stable if R0<1, marginally stable if R0=1, and unstable if R0>1.The proof of this theorem is in the Appendix.


## 3.1.2. Endemic Equilibrium

Theorem 5.
Endemic equilibrium pointE∗S∗,Iu∗,Ik∗ is locally asymptotically stability if R0>1.The proof of this theorem is in the Appendix.Therefore, up to a certain value ofα disease-free equilibrium point is stable (Theorem 4) and beyond that value of α endemic equilibrium point is stable (Theorem 5).


## 3.2. Global Stability Analysis

The global stability analysis of the proposed COVID-19 model (3) is presented here.


### 3.2.1. Disease-Free Equilibrium

Theorem 6.
IfR0<1, the disease-free equilibrium E1 is globally asymptotically stable.The proof of this theorem is in the Appendix.


### 3.2.2. Global Stability of Endemic Equilibrium: Geometric Approach

The global stability of the endemic equilibriumE∗ will be discussed when R0>1 using the geometric approach for global dynamics [22]. For some preliminary discussion on the geometric approach, consider the autonomous dynamical system:
(12)x.=fx,where f:D⟶ℝn,D∈ℝn open set and simply connected and f∈C1D.LetAx be an n2×n2 matrix value function that is C1 on D and consider Q=AfA−1+AJ2A−1, where the matrix Af is qijxf=∂qijx/∂xT. fx=∇qij.fx, and here, J2 represents the second additive compound matrix of Jx=Dx. Let the Lozinskii measure [23] μ of Q concerning a vector norm . in ℝn2 be μQ=limh⟶0+I+hQ−1/h. Define a quantity q¯2 as q¯2=limsupt⟶∞supx0⟶K1/t∫0tμQxs,x0ds. We will apply the following theorem.Theorem 7.
LetD be simply connected, and [23] (H1) “there exists a compact absorbing set K⊂D,” (H2) “the system (12) has a unique equilibrium x~ in D,” then x~ of (12) is globally asymptotically stable in D if q¯2<0.Theorem 8.
The COVID-19 model (3) admits an unique endemic equilibrium which globally asymptotically stable if R0>1.The proof is in the Appendix.Since disease-free equilibrium pointE1Λ/d1,0,0 is globally stable for R0<1 and an unique endemic equilibrium point E∗S∗,Iu∗,Ik∗ is globally stable for R0>1, so there exists no limit cycle for this model. Therefore, we include Nonexistence Periodic Solution part for only the mathematical purpose for the proof in a different way.


## 3.2.1. Disease-Free Equilibrium

Theorem 6.
IfR0<1, the disease-free equilibrium E1 is globally asymptotically stable.The proof of this theorem is in the Appendix.


## 3.2.2. Global Stability of Endemic Equilibrium: Geometric Approach

The global stability of the endemic equilibriumE∗ will be discussed when R0>1 using the geometric approach for global dynamics [22]. For some preliminary discussion on the geometric approach, consider the autonomous dynamical system:
(12)x.=fx,where f:D⟶ℝn,D∈ℝn open set and simply connected and f∈C1D.LetAx be an n2×n2 matrix value function that is C1 on D and consider Q=AfA−1+AJ2A−1, where the matrix Af is qijxf=∂qijx/∂xT. fx=∇qij.fx, and here, J2 represents the second additive compound matrix of Jx=Dx. Let the Lozinskii measure [23] μ of Q concerning a vector norm . in ℝn2 be μQ=limh⟶0+I+hQ−1/h. Define a quantity q¯2 as q¯2=limsupt⟶∞supx0⟶K1/t∫0tμQxs,x0ds. We will apply the following theorem.Theorem 7.
LetD be simply connected, and [23] (H1) “there exists a compact absorbing set K⊂D,” (H2) “the system (12) has a unique equilibrium x~ in D,” then x~ of (12) is globally asymptotically stable in D if q¯2<0.Theorem 8.
The COVID-19 model (3) admits an unique endemic equilibrium which globally asymptotically stable if R0>1.The proof is in the Appendix.Since disease-free equilibrium pointE1Λ/d1,0,0 is globally stable for R0<1 and an unique endemic equilibrium point E∗S∗,Iu∗,Ik∗ is globally stable for R0>1, so there exists no limit cycle for this model. Therefore, we include Nonexistence Periodic Solution part for only the mathematical purpose for the proof in a different way.


## 4. Nonexistence Periodic Solution

This section presents suitable conditions for the COVID system (3) for nonperiodic solutions around the positive equilibria E∗ based on the criterion of [23]; let an autonomous ordinary differential equation as follows:
(13)dxdt=fx,wheref is a function in C1 in open subset of ℝN. Let J=df/dx be the Jacobian matrix of system (13), and J2 be the matrix of N2×N2 which represents as the second additive compound matrix [23] associated with the Jacobian matrix J. Let the matrix J=aijn×n for i=1,2,3,⋯N2, let i=i1,i2 be the ith member in the lexicographic ordering of integer pairs i1,i2 for 1≤i1≤i2≤n. Then, the i×jth element of J2 is
(14)ai1i1+ai2i2,ifi=j,−1r+sairjs,ifexactlyoneentryirofidoesnotoccurinjandjsdoesnotoccurinj0,ifneitherentryfromioccursinj.For a general3×3 matrix
(15)J=a11a12a13a21a22a23a31a32a33,its second additive compound matrixJ2 is
(16)J2=a11+a22a23−a13a32a11+a33a12−a31a21a22+a33.In this case,1=1,2,2=1,3,3=2,3.Theorem 9.
Bendixson’s criterion [23]: a simple closed rectifiable function cannot exist which is invariant under the system (13) for x∈ℝn if any one of the following conditions is satisfied:
(i)
sup∂fr/∂xr+∂fs/∂xs+Σq≠r,s∂fq/∂xr+∂fq/∂xs:1≤r<s≤n<0(ii)
sup∂fr/∂xr+∂fs/∂xs+Σq≠r,s∂fr/∂xq+∂fs/∂xq:1≤r<s≤n<0(iii)
λ1+λ2<0(iv)
inf∂fr/∂xr+∂fs/∂xs−Σq≠r,s∂fq/∂xr+∂fq/∂xs:1≤r<s≤n>0(v)
inf∂fr/∂xr+∂fs/∂xs−Σq≠r,s∂fr/∂xq+∂fs/∂xq:1≤r<s≤n>0(vi)
λn−1+λn>0where λ1≥λ2≥λ3≥...≥λn are the eigenvalues of 1/2∂f/∂x∗+∂f/∂x where ∗ denotes the transposition, and ∂f/∂x is the Jacobian matrix of f.The corresponding logarithmic norm ofJ2 is denoted by μ∞J2 and provided by the vector norm x=supixi as follows:
(17)μ∞J2=sup∂fr∂xr+∂fs∂xs+Σq≠r,s∂fq∂xr+∂fq∂xs:1≤r<s≤n,whereμ∞J2<0 implies the diagonal dominance by row matrix J2. Then, the following result holds.Theorem 10.
A simple closed rectifiable curve that is invariant under system (3) cannot exist [22] if μ∞J2<0.The nonexistence of periodic solutions of system (3) will be discussed by applying Li–Muldowney’s criterion. The logarithm norm μ∞ of the second additive compound matrix J2, for the Jacobian J, is negative if the following conditions satisfy:
(18)2αS1+δIu2−β+γ+d1+d2<0,(19)α1+δIu2S−Iu−δIu2−2d1<0,(20)−γ+d1+d2<0,Now the left hand side of inequality (18)
(21)2αS1+δIu2−β+γ+d1+d2≤β+d1−γ−d2.Thus, inequality (18) will follow if β+d1−γ−d2<0. So the inequalities (18), (19), and (20) can be easily demonstrated if (i), (ii), and (iii) hold, respectively, where iβ+d1−γ−d2<0,iid1>β, and (iii) γ+d1+d2>0.


## 5. COVID-19 Model with Control

In this section, we extend the basic model (1) by including a particular control measure aimed at controlling the spread of the COVID-19 infection and formulate the optimal control problem by proposing the control objectives. The aim of the control measures is to reduce the infection in the population, and thus, there is the need to formulate the optimal control problem to achieve this goal. The control function σt is applied as a vaccine for the susceptible, which reduces the number of infected people which spread the disease per unit of time. Under these control measure, the proposed model (1) is modified as
(22)dSdt=Λ−αStIut1+δIut−σtSt−d1St,dIudt=αStIut1+δIut−βIut−d1Iut,dIkdt=βIut−γIkt−d2Ikt,dRdt=σtSt+γIkt−d1Rt,with nonnegative initial conditions
(23)S0=S¯0,R0=R¯0>0,Iu0=I¯u0,Ik0=I¯k0≥0.The flow diagram of the proposed COVID-19 pandemic model with control is presented in Figure3. The control is completely effective when σt=1, and the control is not effective when σt=0, i.e., 0≤σt≤1. Our focus is to minimize the number of exposed individuals under the cost of applying control measures, which can be done by considering the following fractional optimal control problem to minimize the objective functional given by
(24)Jσt=∫0τQ1S+12Q2σ2,subjected to the state system given in (22) along nonnegative initial conditions (23). In Equation (24), Q1 and Q2 represent the positive constants to keep a balance in the size of the terms. The square of the control variable reflects the severity of the side-effects of the vaccine. Our objective is to minimize the cost function Jσt given in (22) so that the spread of infection rate can be minimized. So, we seek an optimal control σ∗ such that
(25)Jσ∗=minYσ:σ∈U,subjected to the state system given in (22), where the control set is defined as
(26)U=σσtisLebesguemeasurableon0,1.Figure 3
Transfer diagram of COVID-19 model with control.


### 5.1. Existence of an Optimal Control

Lemma 11.
Every solution of system (22) with initial conditions (23) exists in the interval 0,∞ and St>0,Iut≥0,Ikt≥0, and Rt>0 for all t≥0.Proof.
Since the right hand side of COVID-19 model (22) is continuous and locally Lipschitzian, then the solution St,Iut,Ikt,Rt of (22) using the initial conditions is unique on 0,ξ, where 0<ξ<+∞.
From the model (22) and using the initial conditions, we have
(27)dSdt=Λ−σ+d1+αIut1+δIutSt.
We thus have(28)ddtStexpσ+d1t+∫0tαIus1+δIusds=Λexpσ+d1t+∫0tαIus1+δIusds.
Hence,(29)Stexpσ+d1t+∫0tαIus1+δIusds−S0=∫0tΛexpσ+d1t+∫0tαIuω1+δIuωdωdt,
so that(30)St=S0exp−σ+d1t+∫0tαIus1+δIusds+exp−σ+d1t+∫0tαIus1+δIusds×∫0tΛexpσ+d1t+∫0tαIuω1+δIuωdωdt>0.
The second equation of the model (22) yields
(31)dIudt≥−β+d1Iut,
which providesIut≥Iu0exp−β+d1t≥0.
From the third equation of system (1), we get
(32)dIkdt≥−γ+d2Ik,
which givesIkt≥Ik0exp−γ+d2t≥0.
Finally, the last equation of the system (22) yields
(33)dRdt≥−d1Rt,
which providesRt≥R0exp−d1t>0.
Therefore, we can see thatSt,Rt>0 and Iut,Ikt≥0,∀t≥0. This completes the proof.Lemma 12.
All solutions of COVID-19 system (22) in ℝ+4 are bounded and lie in the region Ω defined by Ω=S,Iu,Ik,R∈ℝ+4:0<Nt≤Λ/μ as t⟶∞, where μ=mind1,d2.Proof.
AssumeSt,Iut,Ikt,Rt be any solution of system (22). Now, we consider a function like Nt=St+Iut+Ikt+Rt. Differentiating both sides with respect to t, we have
(34)dNtdt=dStdt+dIutdt+dIktdt+dRtdt,dNtdt=Λ−d1S−d1Iu−d2Ik−d1R,dNtdt+μNt=Λ−d1−μS−d1−μIu−d2−μIk−d1−μR,
dNt/dt+μNt≤Λ, assuming μ=mind1,d2.
Then by comparison theorem, we obtain0<Nt≤N0e−μt+Λ/μ and for t⟶∞,0<Nt≤Λ/μ. Therefore, all solutions of coronavirus system (22) enter into the region Ω=S,Iu,Ik,R∈ℝ+4:0<Nt≤Λ/μ.Theorem 13.
Given the objective functional(35)Jσt=∫0τQ1S+12Q2σ2dt,where U=σσtisLebesguemeasurableon0,1 subject to the system [24] with [25], then there exists an optimal control σ∗ such that Jσ∗=minYσ:σ∈U, if the following conditions are satisfied:
(1)
The class of all initial conditions with a controlσt in the admissible control set along with each state equation being satisfied is not empty(2)
The admissible control setU is closed and convex(3)
Each right hand side of the state system (22) is continuous and is bounded above by a sum of the bounded control and the state and can be written as a linear function of σ with coefficients depending on time and the state(4)
The integrand ofJσ is convex on U and is bounded below by p1σ2−p2 with p1,p2>0The proof is in the Appendix.


### 5.2. Characterization of the Optimal Control Pair

The LagrangianL and Hamiltonian H for the fractional optimal problem Equations (22)–(26) are as follows:
(36)LS,σ=Q1S+Q22σ2,H≪LS,σ+λSdSdt+λIudIudt+λIkdIkdt+λRdRdt.This further implies(37)H≪Q1S+Q22σ2+λSΛ−αStIut1+δIut−σtSt−d1St+λIuαStIut1+δIut−βIut−d1Iut+λIkβIut−γIkt−d2Ikt+λRσtSt+γIkt−d1Rt,whereλS, λIu, λIk, and λR are the adjoint variables to be determined suitably.The forms of the adjoint equations and transversality conditions are standard results from Pontryagin’s maximum principle. The adjoint system can be obtained as follows:(38)λS′=−∂H∂S=λS−λIuαIu1+δIu+d1λS+λS−λRσ−Q1,λIu′=−∂H∂Iu=λS−λIuαS1+δIu2+β+d1λIu−βλIk,λIk′=−∂H∂Ik=γ+d2λIk−γλR,λR′=−∂H∂R=d1λR,with transversality conditions or boundary conditionsλSτ=0,λIuτ=0,λIkτ=0 and λRτ=0.By the optimality condition, we have(39)∂H∂σ=Q2σ∗−λS−λRS¯∗=0atσ=σ∗.By using the bounds for the controlσt, we get
(40)σ∗=λS−λRS¯∗Q2,if0≤λS−λRS¯∗Q2≤1,0,ifλS−λRS¯∗Q2≤0,1,ifλS−λRS¯∗Q2≥1.In compact notation:(41)σ∗=minmax0,λS−λRS¯∗Q2,1.Using (22), we obtain the following optimality system:
(42)dSdt=Λ−αStIut1+δIut−minmax0,λS−λRStQ2,1St−d1St,dIudt=αStIut1+δIut−βIut−d1Iut,dIkdt=βIut−γIkt−d2Ikt,dRdt=minmax0,λS−λRStQ2,1St+γIkt−d1Rt,dλSdt=λS−λIuαIu1+δIu+d1λS−Q1+λS−λRminmax0,λS−λRStQ2,1,dλIudt=λS−λIuαS1+δIu2+β+d1λIu−βλIk,dλIkdt=γ+d2λIk−γλR,dλRdt=d1λR,with nonnegative initial conditions
(43)S0>0,Iu0≥0,R0>0,Ik0≥0,λSτ=0,λIuτ=0,λIkτ=0,λRτ=0.The previous analysis can be summarized in the following theorem.Theorem 14.
LetS¯∗,I¯u∗,I¯k∗, and R¯∗ be optimal state solutions with associated optimal control variable σ∗ for the optimal control problems (22) and (23). Then there exist adjoint variables λS, λIu, λIk, and λR satisfying
(44)λS′=−∂H∂S=λS−λIuαIu1+δIu+d1λS+λS−λRσ−Q1,λIu′=−∂H∂Iu=λS−λIuαS1+δIu2+β+d1λIu−βλIk,λIk′=−∂H∂Ik=γ+d2λIk−γλR,λR′=−∂H∂R=d1λR,
with transversality conditions or boundary conditionsλSτ=0,λIuτ=0,λIkτ=0 and λRτ=0.Furthermore, the control functionsσ∗ is given by
(45)σ∗=minmax0,λS−λRS¯∗Q2,1.Proof.
The adjoint system (42), i.e., λS′,λIu′,λIk′, and λR′, is obtained from the Hamiltonian H as
(46)−dλSdt=∂H∂S,−dλIudt=∂H∂Iu,−dλIkdt=∂H∂Ik,−dλRdt=∂H∂R,
with zero final time conditions (transversality), conditionsλSτ=0,λIuτ=0,λIkτ=0 and λRτ=0, and the characterization of the fractional optimal control given by (45) is obtained by solving the equation ∂H/∂σ=0 on the interior of the control set and using the property of the control space U.Hence, that is the theorem.


## 5.1. Existence of an Optimal Control

Lemma 11.
Every solution of system (22) with initial conditions (23) exists in the interval 0,∞ and St>0,Iut≥0,Ikt≥0, and Rt>0 for all t≥0.Proof.
Since the right hand side of COVID-19 model (22) is continuous and locally Lipschitzian, then the solution St,Iut,Ikt,Rt of (22) using the initial conditions is unique on 0,ξ, where 0<ξ<+∞.
From the model (22) and using the initial conditions, we have
(27)dSdt=Λ−σ+d1+αIut1+δIutSt.
We thus have(28)ddtStexpσ+d1t+∫0tαIus1+δIusds=Λexpσ+d1t+∫0tαIus1+δIusds.
Hence,(29)Stexpσ+d1t+∫0tαIus1+δIusds−S0=∫0tΛexpσ+d1t+∫0tαIuω1+δIuωdωdt,
so that(30)St=S0exp−σ+d1t+∫0tαIus1+δIusds+exp−σ+d1t+∫0tαIus1+δIusds×∫0tΛexpσ+d1t+∫0tαIuω1+δIuωdωdt>0.
The second equation of the model (22) yields
(31)dIudt≥−β+d1Iut,
which providesIut≥Iu0exp−β+d1t≥0.
From the third equation of system (1), we get
(32)dIkdt≥−γ+d2Ik,
which givesIkt≥Ik0exp−γ+d2t≥0.
Finally, the last equation of the system (22) yields
(33)dRdt≥−d1Rt,
which providesRt≥R0exp−d1t>0.
Therefore, we can see thatSt,Rt>0 and Iut,Ikt≥0,∀t≥0. This completes the proof.Lemma 12.
All solutions of COVID-19 system (22) in ℝ+4 are bounded and lie in the region Ω defined by Ω=S,Iu,Ik,R∈ℝ+4:0<Nt≤Λ/μ as t⟶∞, where μ=mind1,d2.Proof.
AssumeSt,Iut,Ikt,Rt be any solution of system (22). Now, we consider a function like Nt=St+Iut+Ikt+Rt. Differentiating both sides with respect to t, we have
(34)dNtdt=dStdt+dIutdt+dIktdt+dRtdt,dNtdt=Λ−d1S−d1Iu−d2Ik−d1R,dNtdt+μNt=Λ−d1−μS−d1−μIu−d2−μIk−d1−μR,
dNt/dt+μNt≤Λ, assuming μ=mind1,d2.
Then by comparison theorem, we obtain0<Nt≤N0e−μt+Λ/μ and for t⟶∞,0<Nt≤Λ/μ. Therefore, all solutions of coronavirus system (22) enter into the region Ω=S,Iu,Ik,R∈ℝ+4:0<Nt≤Λ/μ.Theorem 13.
Given the objective functional(35)Jσt=∫0τQ1S+12Q2σ2dt,where U=σσtisLebesguemeasurableon0,1 subject to the system [24] with [25], then there exists an optimal control σ∗ such that Jσ∗=minYσ:σ∈U, if the following conditions are satisfied:
(1)
The class of all initial conditions with a controlσt in the admissible control set along with each state equation being satisfied is not empty(2)
The admissible control setU is closed and convex(3)
Each right hand side of the state system (22) is continuous and is bounded above by a sum of the bounded control and the state and can be written as a linear function of σ with coefficients depending on time and the state(4)
The integrand ofJσ is convex on U and is bounded below by p1σ2−p2 with p1,p2>0The proof is in the Appendix.


## 5.2. Characterization of the Optimal Control Pair

The LagrangianL and Hamiltonian H for the fractional optimal problem Equations (22)–(26) are as follows:
(36)LS,σ=Q1S+Q22σ2,H≪LS,σ+λSdSdt+λIudIudt+λIkdIkdt+λRdRdt.This further implies(37)H≪Q1S+Q22σ2+λSΛ−αStIut1+δIut−σtSt−d1St+λIuαStIut1+δIut−βIut−d1Iut+λIkβIut−γIkt−d2Ikt+λRσtSt+γIkt−d1Rt,whereλS, λIu, λIk, and λR are the adjoint variables to be determined suitably.The forms of the adjoint equations and transversality conditions are standard results from Pontryagin’s maximum principle. The adjoint system can be obtained as follows:(38)λS′=−∂H∂S=λS−λIuαIu1+δIu+d1λS+λS−λRσ−Q1,λIu′=−∂H∂Iu=λS−λIuαS1+δIu2+β+d1λIu−βλIk,λIk′=−∂H∂Ik=γ+d2λIk−γλR,λR′=−∂H∂R=d1λR,with transversality conditions or boundary conditionsλSτ=0,λIuτ=0,λIkτ=0 and λRτ=0.By the optimality condition, we have(39)∂H∂σ=Q2σ∗−λS−λRS¯∗=0atσ=σ∗.By using the bounds for the controlσt, we get
(40)σ∗=λS−λRS¯∗Q2,if0≤λS−λRS¯∗Q2≤1,0,ifλS−λRS¯∗Q2≤0,1,ifλS−λRS¯∗Q2≥1.In compact notation:(41)σ∗=minmax0,λS−λRS¯∗Q2,1.Using (22), we obtain the following optimality system:
(42)dSdt=Λ−αStIut1+δIut−minmax0,λS−λRStQ2,1St−d1St,dIudt=αStIut1+δIut−βIut−d1Iut,dIkdt=βIut−γIkt−d2Ikt,dRdt=minmax0,λS−λRStQ2,1St+γIkt−d1Rt,dλSdt=λS−λIuαIu1+δIu+d1λS−Q1+λS−λRminmax0,λS−λRStQ2,1,dλIudt=λS−λIuαS1+δIu2+β+d1λIu−βλIk,dλIkdt=γ+d2λIk−γλR,dλRdt=d1λR,with nonnegative initial conditions
(43)S0>0,Iu0≥0,R0>0,Ik0≥0,λSτ=0,λIuτ=0,λIkτ=0,λRτ=0.The previous analysis can be summarized in the following theorem.Theorem 14.
LetS¯∗,I¯u∗,I¯k∗, and R¯∗ be optimal state solutions with associated optimal control variable σ∗ for the optimal control problems (22) and (23). Then there exist adjoint variables λS, λIu, λIk, and λR satisfying
(44)λS′=−∂H∂S=λS−λIuαIu1+δIu+d1λS+λS−λRσ−Q1,λIu′=−∂H∂Iu=λS−λIuαS1+δIu2+β+d1λIu−βλIk,λIk′=−∂H∂Ik=γ+d2λIk−γλR,λR′=−∂H∂R=d1λR,
with transversality conditions or boundary conditionsλSτ=0,λIuτ=0,λIkτ=0 and λRτ=0.Furthermore, the control functionsσ∗ is given by
(45)σ∗=minmax0,λS−λRS¯∗Q2,1.Proof.
The adjoint system (42), i.e., λS′,λIu′,λIk′, and λR′, is obtained from the Hamiltonian H as
(46)−dλSdt=∂H∂S,−dλIudt=∂H∂Iu,−dλIkdt=∂H∂Ik,−dλRdt=∂H∂R,
with zero final time conditions (transversality), conditionsλSτ=0,λIuτ=0,λIkτ=0 and λRτ=0, and the characterization of the fractional optimal control given by (45) is obtained by solving the equation ∂H/∂σ=0 on the interior of the control set and using the property of the control space U.Hence, that is the theorem.


## 6. Numerical Demonstration

The numerical part of this paper is introduced to obtain some sound results based on some data using MATLAB. For parameter estimation, we have not used any mathematical method; we use the trial and error method to fit our model to the actual data. This work intends not to find the exact value of the parameter; we want to see the hidden fact of the outspread speed. We consider a small amount for the initial condition of the susceptible population in both waves, and our model almost represents the same result as accurately (Figures4–7). That is an exciting result that COVID is not homogeneously spread all over India and we can control it by lockdown. However, lockdown can pull down the economy, which is an extensive issue to use this method. Our study hints that another technique to control is to find unrevealed COVID patients as early as possible (Figures 8 and 9). The detected patients are not major responsible for the spread, rather the unrevealed patient mainly spreads COVID. Another question that has a great impact on society is that how many waves face India? The answer depends on the vaccination speed and the variant; however, mathematically, we did not find multiple waves from a single COVID-19 variant. If vaccines work for all variants, then this is the first and last strategy to defend COVID-19 and various waves. If we do not complete the vaccination within the ongoing wave, then we may face the next wave, and so on. Moreover, the community of poor, uneducated, insanitary, and highly dense populations would not control waves without vaccine help. Another question that can answer our model is what will be the peak value of the subsequent consequence wave? The answer is straightforward; the peak value always crosses its past value because COVID spreads from big town to small town to the village in successive waves (see Figures 10 and 11). We draw the figures from Figures 4–13 based on the parameter value and initial condition for both waves using Tables 1 and 2. For Figure 5, we take Iu0=70000 and the rest of the initial values and parameter values are the same shown in Tables 1 and 2. For our study, we take statistical data from https://www.worldometers.info/ [26].Figure 4
Time graph of the first wave of unrevealed infected populationIu and known infected population Ik for α=54×10−10, α=55×10−10, and α=56×10−10 in India.Figure 5
Second wave time graph of unrevealed infected populationIu and known infected population Ik for α=54×10−10, α=55×10−10, and α=56×10−10 in India.Figure 6
Time history of the susceptibleS and recovery population R of the first wave for α=54×10−10, α=55×10−10, and α=56×10−10 in India.Figure 7
Time graph of the second wave for susceptibleS and recovery population R for α=54×10−10, α=55×10−10, and α=56×10−10 in India.Figure 8
Time history of unrevealed infected populationIu and known infected population Ik of the first wave for β=0.20, β=0.21, and β=0.22 in India.Figure 9
Time history of unrevealed infected populationIu and known infected population of the second wave Ik for β=0.20, β=0.21, and β=0.22 in India.Figure 10
First wave time history of unrevealed infected populationIu and known infected population Ik for S0=5×107, S0=6×107, and S0=7×107 in India.Figure 11
Time history of unrevealed infected populationIu and known infected population of the second wave Ik for S0=6×107, S0=7×107, and S0=8×107 in India.Table 2
Initial population for different waves.
Initial populationInitial value for the 1st waveInitial value for the 2nd waveS05×1076×107Iu025015×104Ik0256170126Our chosen parameter set of the model by trial and error method almost satisfies the actual situation. Figure4 represents the known and unrevealed infected population in the first wave and Figure 5 for the second wave. Figure 6 is for the susceptible and recover class for the 1st wave and Figure 7 is for the 2nd wave. The 1st and 2nd wave end their journey. We studied them to know the hidden dynamics. Our numerical study satisfies our model assuming the unknown infected population is critical for spreading the disease. So controlling the 3rd wave is a big challenge for India due to its vast population.Figure8 for the 1st wave and Figure 9 for the 2nd wave show exciting findings. If we can identify unknown infected people quickly, then we can control COVID spread effectively; however, it is a difficult job for a country with a large population. Figures 10 and 11 for the 1st and 2nd wave, respectively, answer the effectiveness of lockdown. Implementing lockdown can only control the spread more effectively since the initial population for the susceptible variable will be small. Figures 12 and 13 for the 1st and 2nd wave, respectively, show the psychological effect on COVID propagation. Media-created fear on the human population has a clear impact on propagation.Figure 12
First wave time history of unrevealed infected populationIu and known infected population Ik for δ=25×10−8, δ=33×10−8, and δ=4×10−7 in India.Figure 13
Time history of unrevealed infected populationIu and known infected population of the second wave Ik for δ=1×10−7, δ=135×10−9, and δ=17×10−8 in India.


### 6.1. Optimal Control

Here, we use some numerical simulations to investigate the effect of the suggested control strategy, vaccine, on the outbreak of COVID-19. From Figure14, it is clear that when time increases, then optimal vaccine control strategies decreases time to time in a country like India. It is clear that the vaccine reduces the number of infected people. Figure 14 shows the speed of vaccination to control the disease within 30 days. This is mathematical analysis; reality is complicated. To prevent illness within 30 days, massive vaccine and huge trained human experts are required. Figures 15 plots the variation in the number of susceptible, unrevealed infected, known infected, and recovery people in the presence and the absence of the control strategy in India. Our goal was to reduce the number of infected people; the results confirm that the number of infected people decreased, and since the initial number of infected people was small, this wave ended faster, and the spread of the disease was controlled by the vaccine strategy. We use Tables 2 and 3 (2nd wave) to present Figures 14 and 15 for the model with controls (22).Figure 14
The optimal control diagrams for the vaccine control, with input values from Table3.Figure 15
Plots forS,Iu,Ik, and R in the presence and absence of control using data from Table 3.Table 3
Parameter values for the optimal control problem.
ParametersValue per dayΛ2×103α55×10−10δ135×10−9β0.21γ0.1d14×10−5d21×10−3Q140Q21×105


## 6.1. Optimal Control

Here, we use some numerical simulations to investigate the effect of the suggested control strategy, vaccine, on the outbreak of COVID-19. From Figure14, it is clear that when time increases, then optimal vaccine control strategies decreases time to time in a country like India. It is clear that the vaccine reduces the number of infected people. Figure 14 shows the speed of vaccination to control the disease within 30 days. This is mathematical analysis; reality is complicated. To prevent illness within 30 days, massive vaccine and huge trained human experts are required. Figures 15 plots the variation in the number of susceptible, unrevealed infected, known infected, and recovery people in the presence and the absence of the control strategy in India. Our goal was to reduce the number of infected people; the results confirm that the number of infected people decreased, and since the initial number of infected people was small, this wave ended faster, and the spread of the disease was controlled by the vaccine strategy. We use Tables 2 and 3 (2nd wave) to present Figures 14 and 15 for the model with controls (22).Figure 14
The optimal control diagrams for the vaccine control, with input values from Table3.Figure 15
Plots forS,Iu,Ik, and R in the presence and absence of control using data from Table 3.Table 3
Parameter values for the optimal control problem.
ParametersValue per dayΛ2×103α55×10−10δ135×10−9β0.21γ0.1d14×10−5d21×10−3Q140Q21×105


## 7. Observations and Conclusion

This paper has proposed a model for infectious novel coronavirus disease for the dynamical study. The BRNR0 is the threshold condition that determines the disease propagation dynamics. This study has shown that when R0<1, the system has only a globally stable disease-free equilibrium E1 which leads to the eventual death of the disease. The coronavirus system has a unique endemic equilibrium E∗ for R0>1, which is globally stable under the same condition. In this paper, mainly we consider data of two different waves in India and checked which of the waves is more dangerous in India. We checked the effect of different parameters through the figures; the truth tells that the second wave is more dangerous than the first wave. The next focus of this paper is to set up an optimal control problem relative to the COVID-19 epidemic model to minimize the daily infected people. We have considered the vaccine rate as a function of time by σt. σ is representing the vaccine control in this COVID model. The control function σ is designed in such a way that it minimizes the objective functional (cost function) J as given in [27]. Finally, we found some exciting results by studying the presented model numerically based on accurate data. Lockdown is the best technique mathematically to control the disease spread. However, ideal lockdown is not possible in practice, so we did not find satisfying practical results by using this method. These measures have proven unpopular due to their social and economic consequences, so it is essential to find new standards. Quick detection of undetected patents can control the spread. Vaccination can handle the situation; otherwise, we observe many waves. Mathematically, we did not find multiple waves from a single COVID-19 variant since the new variants produce distinct waves. If vaccines work for each variant, this is the most acceptable strategy to defend the various upcoming waves. We have monitored the psychological effect on infection propagation. This study has observed that media-created fear negatively affects human psychology and affects COVID dynamics positively. The peak value of subsequent consequence waves always crosses its past value because this disease spreads from the big town to the small town to the village in successive waves, and new strains are more quickly spread than the previous version.



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*Source: 2684055-2022-04-14.xml*