Adv Syst Sci Appl 2021; 04:65–86 Published online at https://ijassa.ipu.ru. Dynamics of COVID-19 Outbreak and Optimal Control Strategies: A Model-Based Analysis Naba Kumar Goswami1*, B. Shanmukha2 1Department of Mathematics, PET Research Foundation, Mysuru, India 2Department of Mathematics, PES College of Engineering, Mandya, India Abstract: COVID-19 is an infectious disease caused by the SARS-CoV-2 virus, which spreads so fast in the inhabitants. The virus is transmitted through direct contact with respiratory droplets of an infected individuals through coughing and sneezing or indirect contact through contaminated objects or surface. In this article, a non-linear mathematical model is proposed and analyzed to manifest the impact of transmission dynamics of the COVID-19 pandemic based on Indian condition by considering asymptomatic and symptomatic infections. It is assumed that the transmission rates due to asymptomatic and symptomatic individuals are different. The basic reproduction number of the model is computed and studied the stability of different equilibria of the model in detail. The sensitivity analysis is presented to identify the key parameters that influence the basic reproduction number, which can be regulated to control the transmission dynamics of the disease. Also, this model is extended to the optimal control model and is analyzed by using the Pontryagin’s Maximum Principal and solved numerically. It has been observed that the optimal control model gives better result as compacted to the model without optimal control model as it reduces the number of infectives significantly in a desired interval of time. Keywords: COVID-19, basic reproduction number, stability analysis, sensitivity analysis, optimal control 1. INTRODUCTION The ongoing COVID-19 outbreak has put mathematical models in the limelight. In 1960, the human coronavirus was first identified and In 21st century, a large number of people in this world have been affected by the three outbreaks such as SARS, MERS, and 2019-nCoV. In 2003, the Severe Acute Respiratory Syndrome(SARS) outbreaks, especially in the Chinese mainland, Hong Kong, Taiwan, and Canada of the World [1] and in 2012 and 2015, the Middle East Respiratory Syndrome(MERS) outbreak in Saudi Arabin [2] and South Korea [3] respectively. Coronaviruses belonging to the family of Coronaviridae and order of Nidovirales, enveloped, non-segmented, single-stranded positive-sense RNA viruses [4]. All coronavirus are zoonotic. They start in animals and can then, following mutation, recombination, and adaptation, be passed on to humans. In the human-to-human transmission of COVID-19 can occurs via respiratory droplets directly (through droplets from coughing or sneezing) or indirectly (touching surfaces or objects contaminated with virus and touching their mouth, nose, eyes) [5]. The incubation period for the new coronavirus from 2 to 14 days in human to human transmission [6]. The common symptoms of COVID-19 are fever, cough, difficulty of breathing, and fatigue. At present, there are no specific drugs for the disease to protect the people, only hygiene measures can reduce the ∗Corresponding author: nabakrgoswami@gmail.com 66 N.K. GOSWAMI, B. SHANMUKHA rate of transmission. According CDC COVID-19 vaccination is a safer way to help build protection measure. Covid vaccines can stop or reduce most of the people from getting sick, but not everyone. Despite taking all recommended doses of vaccines and waits a few weeks for immunity to build up, still there is a chance that people can get infected. So far there is no evidence, however, that human coronaviruses can be transmitted by animals. In December 2019, a new outbreak of pneumonia of unknown cause has been identified in Wuhan city, the capital of China’s Hubei province [9]. This outbreak has some other potential causes like influenza, avian influenza, adenovirus, and SARS, but there are no symptoms like MERS [7]. Later on 7th January, 2020, the causative pathogen was identified as a Novel Coronavirus (2019-nCoV) [8]. As the virus is very closely related to SARS and MERS, so the name 2019-nCoV can distinguish the virus from both [9]. On 30th January 2020, the World Health Organization (WHO) declared that the outbreak is a Public Health Emergency of International Concern(PHEIC) [10] as it is Worldwide spread. As the number infective is increasing rapidly and the epidemiological evidence of human-to-human (doorplates, direct contact, etc.) transmission suggests that 2019-nCoV is more contagious than both SARS and MERS [11–13]. On 11th February 2020, the World Health Organization announced a new name of coronavirus disease as COVID-19 [14]. On 11th March 2020, the World Health Organization officially declares the COVID-19 outbreak as a Pandemic as it spread worldwide. The most affected countries in this world are the USA, India, Brazil, Italy, Spain, France, UK, Germany, Iran, China, Belgium, Netherlands, etc. In this ongoing outbreak, in the USA more than 43,107,628 people are infected and more than 694,619 people have died. Presently more than 229,382,253 infected cases have been reported across 223 territories, where more than 206,052,168 infected cases are recovered and more than 4,707,336 infected cases have been died till 20th Sept 2021. In India, the first Novel coronavirus case was reported on 30th January 2020 [15] in the state of Kerala. Due to the crises of coronavirus pandemic, Prime Minister of India declared 21 + 19 = 40 days nationwide lockdown from 25th of March to 3rd of May 2020 as a preventive measure for the COVID-19 [16]. The Ministry of Health and Family Welfare of India has suggested various precautionary measures to prevent the spread of viruses such as washing hands frequently, physical/social distancing, wearing mask, avoiding touching face, nose, and eyes [17, 18]. Apart from lockdown the government of India performing many awareness programs about preventative measures through media and social networks (TV, radio, newspaper, Facebook, Twitter, etc.). The effect of lockdown and social distancing play an important role to reduce the coronavirus infection. In 2020, total number of infected cases are recorded in India are 10,266,674 and January to September 2021, number infected cases are 23,211,745. In 2021, also statewide or locality wise lockdown declared according to basis of active cases in that particular state or region. On 20th September, 2021, more than 33,478,419 infected COVID-19 cases have been reported in 32 states in India, while 32,715,105 cases are recovered and 445,165 many cases have been died. Most affected states in India are Maharashtra, Karnataka, Kerla, Delhi, Gujarat, Rajasthan, Madhya Pradesh, Uttar Pradesh, Tamil Nadu, Andhra Pradesh, Telangana, West Bengal, Jammu and Kashmir. Among the cities in India, Mumbai, Bangalore and Delhi are the badly effected by COVID-19 outbreak. The primary aimed to study this model to study Indian conditions and will forecast future pandemic by using information available. In [1] authors presented a deterministic model and simplified from the SEIJR model, which is adapted to analyze the important parameters of the model of SARS epidemic; In [19] authors constructed a SEQIJR model of epidemic disease transmission which includes immunization and varying population size is studied; In [20] authors studied a mathematical about early transmission dynamics of the infection and evaluating the effectiveness of control measures; In [21] authours proposed and studied a Copyright © 2021 ASSA. Adv Syst Sci Appl (2021) DYNAMICS OF COVID-19 OUTBREAK AND OPTIMAL CONTROL STRATEGIES: A MODEL-BASED... 67 model on the various impact of the intervention on the spread of COVID-19 in India; In [22] authors discussed a data-driven analysis in the early phase of the outbreak. They estimated the basic reproduction number of novel coronavirus (2019-nCoV) in China; In [23] authors develop a mathematical model for the spread of the coronavirus disease 2019 (COVID-19); In [24] authors developed a Bats-Hosts-Reservoir-People transmission network model for simulating the potential transmission from the infection source to the human infection. In [31] authors proposed a mathematical model and they focuses on the impacts of face mask, hospitalization of symptomatic individuals and quarantine of asymptomatic individuals on the transmission dynamics of COVID-19 pandemic in India. This paper is organized as follows: In section 2 presents the model; In section 3 discussed existence of equilibria and basic reproduction number; In section 4 presents the stability analysis of the model; In section 5 deal with sensitivity analysis of basic reproduction number; In Section 6 illustrate the effects of parameters on disease outbreak ; In Section 7 we extend the model to optimal control model and analysis it. Demonstrates the numerical simulation results of the optimal control model; Finally, Section 7 we conclude the paper. 2. THE MODEL In this section, a dynamic model for COVID-19 pandemic is presented and discussed based on India condition. The model divides the total population N(t) = S + E + Ia + Is +H +R into six different compartments according to the nature of the disease such as Susceptible individuals (S), Exposed individuals (E), Asymptomatic infective individuals (Ia), Symptomatic infective individuals (Is), Hospitalized individuals (H) and Recovered individuals (R). It is assumed that the total population is varying and homogeneously mixed i.e., all people are equally likely to be infected by the infectious individuals if they come into contact. The individuals are employed in the province at a constant rate Λ and join the susceptible class. The Natural birth and deaths in the population are also considered in the model. It is assumed that susceptible individuals after being exposed to the COVID- 19 infection can progress to asymptotic infective and symptomatic infective at the rates β. Assume that an asymptomatic individual joins the symptomatic populations class at the rate ρ. Further, both asymptomatic and symptomatic infectious individuals will progress to the hospitalized or quarantine compartment with clinical symptoms of COVID-19 at the rate δ1 and δ2 respectively. Also, some asymptotic and symptomatic individuals may recover without hospitalized or quarantine at the rates γ1 and γ2 respectively. Hospitalized individuals may recover and after recovery, it progresses to recovered class at the rate γ3. However, the rates of recovery may vary from one compartment to another. The natural mortality rate of each Individuals class is µ. Due to critical illness of COVID-19 disease some of the symptomatic and hospitalized individuals class have additional mortality rates µ1 and µ2, respectively. The flow diagram of the model is given in Figure 1, and biological interpretations of parameters are shown in Table 1. Keeping the above facts/assumptions in mind, a mathematical model COVID-19 is proposed as follows: dS dt = Λ− βaSIa − βsSIs − µS dE dt = βaSIa + βsSIs − (κ+ µ)E dIa dt = ξκE − (ρ+ γ1 + µ+ δ1)Ia dIs dt = (1− ξ)κE + ρIa − (γ2 + µ1 + µ+ δ2)Is (1) Copyright © 2021 ASSA. Adv Syst Sci Appl (2021) 68 N.K. GOSWAMI, B. SHANMUKHA dH dt = δ1Ia + δ2Is − (γ3 + µ2 + µ)H dR dt = γ1Ia + γ2Is + γ3H − µR Fig. 1. Flow diagram of the model. Table 1. Biological interpretations of parameters Parameter Biological interpretations Λ : Rate of recruitment in the susceptible class, βa : Rate of infection of susceptible with asymptomatic individuals βs : Rate of infection of susceptible with symptomatic individuals κ : Rate of incubation ρ : Rate of progression from asymptomatic to symptomatic ξ : Fraction of exposed individuals not showing symptoms δ1 : Hospitalized/Quarantine rate of asymptomatic individuals δ2 : Rate of hospitalized symptomatic individuals γ1 : Recovery rate of asymptomatic individuals γ2 : Recovery rate of symptomatic individuals γ3 : Recovery rate of hospitalized individuals µ : Natural mortality rate of human µ1 : Disease related mortality rate for symptomatic individuals µ2 : Disease related mortality rate for hospitalized individuals Copyright © 2021 ASSA. Adv Syst Sci Appl (2021) DYNAMICS OF COVID-19 OUTBREAK AND OPTIMAL CONTROL STRATEGIES: A MODEL-BASED... 69 3. ANALYSIS OF THE MODEL SYSTEM From the system (1), we have dS dt ∣∣∣ S=0 = Λ > 0, dE dt ∣∣∣ E=0 = βaSIa + βsSIs ≥ 0, dIa dt ∣∣∣ Ia=0 = ξκE ≥ 0 dIs dt ∣∣∣ Is=0 = (1− ξ)κE + ρIa ≥ 0, dH dt ∣∣∣ H=0 = δ1Ia + δ2Is ≥ 0, dR dt ∣∣∣ R=0 = γ1Ia + γ2Is + γ3H ≥ 0 Here, all the rates are non-negative on the bounding planes. So, if we start in the interior of the 6-dimensional closed hyperoctant R6 +, we will always remain there, in view of the fact that the direction of the vector field is inward on all the bounding planes. Thus, non-negativity of all the solutions of the model system (1) is guaranteed. Further, from the model system (1), we note that the total human population N = S1 + S2 + I +H +Q+R satisfies, dN dt = Λ− µN − µ1Is − µ2H This gives lim sup t→∞ N ≤ Λ µ Therefore, all the solutions S(t), E(t), Ia(t), Is(t), H(t), R(t) are bounded by Λ µ . Hence, the biologically feasible region for the system (1) is given by the following positively invariant set: Ω = {(S,E, Ia, Is, H,R) ∈ R6 + : 0 ≤ S + E + Ia + Is +H +R ≤ Λ µ } 3.1. Basic Reproduction Number The disease-free equilibrium for the model (1) as E0 = (S0, E0, I0 a , I 0 s , H 0, R0) = ( Λ µ , 0, 0, 0, 0, 0). We find the basic reproduction number R0 by using the next generation matrix method [26]. The new infection terms of the matrix F and the transition terms of the matrix V of the system (1) are respectively, as follows: F =  βaSIa + βsSIs 0 0 0  , V =  (κ+ µ)E −ξκE + (ρ+ γ1 + δ1 + µ)Ia −(1− ξ)κE − ρIa + (γ2 + δ2 + µ1 + µ)Is −δ1Ia − δ2Is + (γ3 + µ2 + µ)H  Now, we find the matrices F (of new infection terms) and V (of the transition terms) as Copyright © 2021 ASSA. Adv Syst Sci Appl (2021) 70 N.K. GOSWAMI, B. SHANMUKHA F =  0 βaS βsS 0 0 0 0 0 0 0 0 0 0 0 0 0  , V =  κ+ µ 0 0 0 −ξκ ρ+ γ1 + δ1 + µ 0 0 −(1− ξ)κ −ρ γ2 + δ2 + µ1 + µ 0 0 −δ1 −δ2 γ3 + µ2 + µ  It follows that FV −1 =  m11 m12 m13 0 0 0 0 0 0 0 0 0 0 0 0 0  where, m11 = κξβaΛ µ(κ+ µ)(ρ+ γ1 + δ1 + µ) + βsΛκ(ξρ+ (1− ξ)(ρ+ γ1 + δ1 + µ)) µ(κ+ µ)(ρ+ γ1 + δ1 + µ)(γ2 + δ2 + µ1 + µ) m12 = βaΛ µ(ρ+ γ1 + δ1 + µ) − ρβsΛ µ(ρ+ γ1 + δ1 + µ)(γ2 + δ2 + µ1 + µ) , m13 = βsΛ µ(γ2 + δ2 + µ1 + µ) The basic reproduction number is same as the spectral radius of the next-generation matrix FV −1. Thus, from above, we obtain the expression for R0 as R0 = κΛ µ(κ+ µ)(ρ+ γ1 + δa + µ) [ ξβa + βs{ξρ+ (1− ξ)(ρ+ γ1 + δ1 + µ)} γ2 + δ2 + µ1 + µ ] The quantity R0 is known as basic reproduction number, the expected number of secondary cases produced in completely susceptible population, by a typical infective individual for the system (1). 3.2. Existence of Endemic Equilibrium Point The endemic equilibrium of the system (1) satisfies the following algebraic equations such as dS dt = 0, dE dt = 0, dIa dt = 0, dIs dt = 0, dH dt = 0, dR dt = 0 The system (1) realize a unique positive solution E1 = (S∗, E∗, I∗a , I ∗ s , H ∗, R∗) S∗ = κ+ µ (βad1 + βsd2) , E∗ = Λ(βad1 + βsd2)− µ(κ+ µ) (βad1 + βsd2)(κ+ µ) , I∗a = ξκE∗ ρ+ γ1 + δ1 + µ = d1E ∗, I∗s = (1− ξ)κE∗ + ρd1E ∗ γ2 + δ2 + µ1 + µ = d2E ∗, H∗ = (δ1d1 + δ2d2)E∗ γ3 + µ2 + µ = d3E ∗, R∗ = (γ1d1 + γ2d2 + γ3d3)E∗ µ where, d1 = κξ ρ+ γ1 + δ1 + µ , d2 = (1− ξ)κ+ ρd1 γ2 + δ2 + µ1 + µ , d3 = (δ1d1 + δ2d2) γ3 + µ2 + µ Copyright © 2021 ASSA. Adv Syst Sci Appl (2021) DYNAMICS OF COVID-19 OUTBREAK AND OPTIMAL CONTROL STRATEGIES: A MODEL-BASED... 71 4. STABILITY ANALYSIS OF THE MODEL Theorem 4.1: For model system (1), the disease-free equilibrium E0 is locally asymptotically stable if R0 < 1 and unstable if R0 > 1. 4.1. Global stability of disease-free equilibrium To prove the global stability of disease-free equilibrium, we are using the theorem by Castillo- chavez et al. [25] Theorem 4.2: If the given mathematical model can be written in the form: dX dt = F (X, Y ), and dY dt = G(X, Y ), G(X, 0) = 0 (2) where X = S , Y = (E, Ia, Is, H)T , denoting the number of uninfected and denoting the number of Covid-19 infected people respectively. Then the disease-free equilibrium is represented here by E0 = (X0, 0) = ( Λ µ , 0) For the global asymptotically stable, the condition (H1) and (H2) given below must be satisfied. H1 : for dX dt = F (X0, 0), H2 : G(X, Y ) = AY − Ĝ(X, Y ), Ĝ(X, Y ) ≥ 0, Here A = DYG(X0, 0) is M-matrix (In M-matrix, all the off diagonal element of matrix are non-negative) . If the given system of differential equation in mathematical model satisfies the given condition in (2) then the point E0 = (X0, 0) is a global asymptotically stable equilibrium of given mathematical model provided R0 < 1. And for the given mathematical model, the result is shown in the next theorem, as given below. Theorem 4.3: The point E0 = (X0, 0) of the system (1) is global asymptotically stable (G.A.S.), provided R0 < 1. and the condition given in (2) are satisfied. Proof By using theorem (2.1) to our model system (1), we get F (X0, 0) = Λ− µS, G(X, Y ) = AY − Ĝ(X, Y ) where, A = F − V =  −(κ+ µ) βaS βsS 0 ξκ −(ρ+ γ1 + δ1 + µ) 0 0 (1− ξ)κ ρ −(γ2 + δ2 + µ1 + µ) 0 0 δ1 δ2 −(γ3 + µ2 + µ)  Copyright © 2021 ASSA. Adv Syst Sci Appl (2021) 72 N.K. GOSWAMI, B. SHANMUKHA then G(X, Y ) = AY − Ĝ(X, Y ) = AY −  Ĝ1(X, Y ) Ĝ2(X, Y ) Ĝ3(X, Y ) Ĝ4(X, Y )  = (S0 − S)(βaIa + βsIs) 0 0 0  where AY =  −(κ+ µ)E + S0(βaIa + βsIs) ξκE − (ρ+ γ1 + δ1 + µ)Ia (1− ξ)κE + ρIa + (γ2 + δ2 + µ1 + µ)Is δ1Ia + δ2Is + (γ3 + µ2 + µ)H  and Ĝ(X, Y ) =  (κ+ µ)E + S(βaIa + βsIs) ξκE − (ρ+ γ1 + δ1 + µ)Ia (1− ξ)κE + ρIa + (γ2 + δ2 + µ1 + µ)Is δ1Ia + δ2Is + (γ3 + µ2 + µ)H  Here, we can easily see S0 ≥ S, hence G(X, Y ) ≥ 0 for all (X, Y ). Also by the defination of M matrix we can say that the matrix A is M matrix. Hence, disease-free equilibrium (E0) is global asymptotically stable. 4.2. Global stability of endemic equilibrium Theorem 4.4: The endemic equilibrium E1 = (S∗, E∗, I∗a , I ∗ s , H ∗, R∗) of the given mathematical model is globally asymptotically stable. Proof For the global stability result, we will use the method discussed in Korobeinikov and Wake [27], Li and Muldowney [28, 29]. Here we consider the following Lyapunov function: L = C1 ( S − S∗ − S∗ln S S∗ ) + C2 ( E − E∗ − E∗ln E E∗ ) + C3 ( Ia − I∗a − I∗a ln Ia I∗a ) +C4 ( Is − I∗s − I∗s ln Is I∗s ) Then the time derivative of L is given by dL dt = C1 ( 1− S∗ S ) dS dt + C2 ( 1− E∗ E ) dE dt + C3 ( 1− I∗1 I1 ) dI1 dt + C4 ( 1− I∗2 I2 ) dI2 dt Now from the mathematical model we put the expressions for dS dt , dE dt , dIa dt , dIs dt , in the above equation, which gives dL dt = C1 ( 1− S∗ S ) {Λ− βaIaS − βsIsS − µS} + C2 ( 1− E∗ E ) {βaIaS + βsIsS − (κ+ µ)E} + C3 ( 1− I∗a Ia ) {ξκE − ρIa − (γ1 + µ)Ia − δ1Ia} Copyright © 2021 ASSA. Adv Syst Sci Appl (2021) DYNAMICS OF COVID-19 OUTBREAK AND OPTIMAL CONTROL STRATEGIES: A MODEL-BASED... 73 + C4 ( 1− I∗s Is ) {(1− ξ)κE + ρIa − (γ2 + µ1 + µ)Is − δ2Is} (3) The mathematical model system satisfies the following relation at the equilibrium point. Λ = βaI ∗ aS ∗ + βsI ∗ sS ∗ + µS∗, κ+ µ = βaI ∗ aS ∗ + βsI ∗ sS ∗ E∗ , ρ+ γ1 + δ1 + µ = ξκE∗ I∗a , γ2 + δ2 + µ1 + µ = (1− ξ)κE∗ + ρI∗a I∗s Putting all the above expressions in (3) we get, dL dt = C1 ( 1− S∗ S ) [βaI ∗ aS ∗ + βsI ∗ sS ∗ + µS∗ − βaIaS − βsIsS − µS] + C2 ( 1− E∗ E )[ βaIaS + βsIsS − ( βaI ∗ aS ∗ + βsI ∗ sS ∗ E∗ ) E ] + C3 ( 1− I∗a Ia )[ ξκE − ( ξκE∗ I∗a ) Ia ] + C4 ( 1− I∗s Is )[ (1− ξ)κE + ρIa − ( (1− ξ)κE∗ + ρI∗a I∗s ) Is ] Then, dL dt = −C1 ( (S∗ − S)2 S ) µ+ C1 [( 1− S∗ S ) {βaI∗aS∗ + βsI ∗ sS ∗ − β1IaS − β2IsS} ] + C2 ( 1− E∗ E )[ βaIaS + βsIsS − ( βaI ∗ aS ∗ + βsI ∗ sS ∗ E∗ ) E ] + C3 ( 1− I∗a Ia )[ ξκE − ( ξκE∗ I∗a ) Ia ] + C4 ( 1− I∗s Is )[ (1− ξ)κE + ρIa − ( (1− ξ)κE∗ + ρI∗a I∗s ) Is ] dL dt = −C1 ( (S∗ − S)2 S ) µ+ g(x1, x2, x3, x4) where, S S∗ = x1, E E∗ = x2, Ia I∗a = x3, Is I∗s = x4, βaS ∗I∗a = p, βsS ∗I∗s = q, ξκE∗ = r, (1− ξ)κE∗ = s Now, g(x1, x2, x3, x4) = C1(p+ q − px1x3 − qx1x4)− C1p 1 x1 − C1q 1 x1 + C1px3 + C1qx4 + C2(px1x2 + qx1x4 − ax2 − qx2)− C2p ( x1x3 x2 ) − C2q ( x1x4 x2 ) Copyright © 2021 ASSA. Adv Syst Sci Appl (2021) 74 N.K. GOSWAMI, B. SHANMUKHA + C2(p+ q) + C3(rx2 − rx3 − r x2 x3 + r) + C4(sx2 − sx4 − s x2 x4 + s) = x2(−C2p+ C3r + C4s) + x3(C1p− C3r) + x4(C1q − C4s) + x1x3(−C1p+ C2p) + x1x4(C1q + C2q) + C1(2p+ q) + C2(p+ q) + C3r − C3r ( x2 x3 ) + C4s− C4s ( x2 x4 ) − C1(p+ q) ( 1 x1 ) − C2(px1x3 + qx1x4) ( 1 x2 ) To get the values of C1, C2, C3, C4 we take the coefficients of x1x3, x1x4, x2, x3, x4 equal to zero and solve the algebraic equations in C1, C2, C3, C4. This gives C1 = C2; C3 = C1p r ; C4 = C1q s Choosing C1 = C2 = 1 and n = 1, we get g(x1, x2, x3, x4) = p ( 3− 1 x1 − x1x3 x2 − x2 x3 ) + q ( 3− 1 x1 − x1x4 x2 − x2 x4 ) Since the arithmetic mean is greater than or equal to geometric mean, we have 1 x1 + x1x3 x2 + x2 x3 ≥ 3 and 1 x1 + x1x4 x2 + x2 x4 ≥ 3 Hence, dL dt = − ( (S∗ − S)2 S ) µ+ p ( 3− 1 x1 − x1x3 x2 − x2 x3 ) + q ( 3− 1 x1 − x1x4 x2 − x2 x4 ) Thus it is easy to observe that dL dt ≤ 0 and the equality dL dt = 0 hold on;y for x1 = x2 = x3 = x4 = 1 for which S = S∗, E = E∗, Ia = I∗a , Is = I∗s . From the LaSalle’s invariance principle [30], the equilibrium E1 of the given system is globally asymptotically stable for R0 > 1. 5. SENSITIVITY ANALYSIS OF REPRODUCTION NUMBER In this section, we also perform sensitivity analysis for the parameters involved in reproduction number R0, which reflects that increase or decrease in these parameter causes increase or decrease in R0. The sensitivity of R0 to different parameters is shown in Figure 3. It is used to discover the parameters that have a high impact on R0 and should be targeted by intervention strategies. Sensitivity indices allows to measure the relative change in a variable when parameter changes. For that we use the forward sensitivity index of a variable, with Copyright © 2021 ASSA. Adv Syst Sci Appl (2021) DYNAMICS OF COVID-19 OUTBREAK AND OPTIMAL CONTROL STRATEGIES: A MODEL-BASED... 75 respect to a given parameter, which is defined as the ratio of the relative change in the variable to the relative change in the parameter. If such variable is differentiable with respect to the parameter, then the sensitivity index is defined using partial derivatives. The normalized forward sensitivity index of R0, which is differentiable with respect to a given parameter α, is defined by Y R0 α = α R0 ∂R0 ∂α The above formula can be used to compute the analytical expression for the sensitivity of R0 to each parameter that it includes. Accordingly, the sensitivity indexes of the model (1) are illustrate in Figure 2. Consequently, the value of R0 increases with increase in the values of all positive indices parameters Λ, βa, βs, κ, and ξ with R0. Also, the parameters γ1, γ2, δ1, δ2, ρ, µ, and µ1 have negative index with R0. It is clearly observed that the effect of the parameter Λ is the maximum and hence it is the most sensitive parameter of R0. It means that small change (increase or decrease) in the parameters (Λ) will significant change in the value of R0 by 100%. It is obvious that phenomenon of a lower value of R0 will boost to prevent the disease prevalence. Thus, to control the disease from the population, we have to control the increase of parameters having positive indices with R0, whereas parameters with negative indices should be maintained. 6. EFFECTS OF PARAMETERS ON DISEASE OUTBREAK For the Numerical simulation of the model, we consider all the parameters are in per day basis. First we consider the following set of parameters which corresponds to disease-free equilibrium. Λ = 2; βa = 0.00085; βs = 0.00029; γ1 = 1/14; γ2 = 0.00245; γ3 = 0.002;κ = 1.19; µ1 = 0.1;µ2 = 0.015;µ = 0.00425; ρ = 0.001; ξ = 0.0015; δ1 = 0.16; δ2 = 0.01; For the above set of parameters we get R0 = 0.0932 < 1 and the disease-free equilibrium point E0(462.52, 0, 0, 0, 0, 0) is stable. This fact is demonstrated in Figure 4 (a). Later, we did few changes for endemic equilibrium as follows: Λ = 40; βa = 0.00085; βs = 0.00029; γ1 = 1/14; γ2 = 0.000245; γ3 = 0.002; κ = 0.89; µ1 = 1.05; µ2 = 0.015; µ = 0.0125; ρ = 0.69; ξ = 0.95; δ1 = 0.012; δ2 = 0.019; From the above set of parameter we get R0 = 2.0075 > 1, and the endemic equilibrium E1(801.56, 26.82, 25.04, 24, 43, 38.75, 208.6, ) is stable. The stability of the equilibrium point E1 is shown in Figure 4(b). The effect of different values of recovery rate from asymptomatic (γ1), symptomatic (γ2) and hospitalized (γ3), which corresponds to infective human is demonstrated in Figure 4(c), 4(d), and 4(e) respectively. It is clear that the parameter of recovery rate γ1, γ2, and γ3 increases, simultaneously the infected population decreases. The effect of S, H, and R also shown in the figure 4(f) with respect to endemic equilibrium point. Copyright © 2021 ASSA. Adv Syst Sci Appl (2021) 76 N.K. GOSWAMI, B. SHANMUKHA a s 1 2 1 1 2 Parameters -1.5 -1 -0.5 0 0.5 1 F or w ar d S en si tiv ity o f R 0 Fig. 2. Forward sensitivity of R0 7. THE OPTIMAL CONTROL MODEL In this section, the model (1) is extended to formulate optimal control problem by incorporating two time-dependent optimal control parameters, namely u1(t), and u2(t). If u1, and u2 equal zero, no effort is placed in these controls at time t, and if they equal one, maximum effort is applied. Thus, optimal control variables are given, as follows: The control variable u1(t) represents the reduction in the transmission between human- to-human via using surgical face masks, social distancing, self-isolation, sensitization and awareness of transmission of the disease. The control variable u2(t) represents the increase in the testing facility and treatment, which can lead to fast detection of Covid infected cases and recovery and add additional time-dependent parameter ηu2(t) in the rate of direction δ2. Keeping in view of the above assumptions, the optimal control model is formulated as follows: dS dt = Λ− (1− u1)βaIaS − (1− u1)βsIsS − µS dE dt = (1− u1)βaIaS + (1− u1)βsIsS − (κ+ µ)E dIa dt = ξκE − (ρ+ γ1 + µ+ δ1)Ia dIs dt = (1− ξ)κE + ρIa − (γ2 + µ1 + µ)Is − (δ2 + ηu2)Is (4) dH dt = δ1Ia + (δ2 + ηu2)Is − (γ3 + µ2 + µ)H dR dt = γ1Ia + γ2Is + γ3H − µR Copyright © 2021 ASSA. Adv Syst Sci Appl (2021) DYNAMICS OF COVID-19 OUTBREAK AND OPTIMAL CONTROL STRATEGIES: A MODEL-BASED... 77 0 1 1 1 R 0 0.8 s 0.5 0.6 a 2 0.4 0.20 0 0 0.5 1 1.5 (a) 0 1 1 1 R 0 0.80.5 0.6 a 2 0.4 0.20 0 0.5 1 1.5 (b) 0 1 1 1 R 0 0.80.5 0.6 s 2 0.4 0.20 0 0 2 4 6 8 10 12 (c) 0 1 1 1 R 0 0.80.5 0.6 a 2 0.4 0.20 0 0.5 1 1.5 (d) 0 1 1 1 R 0 0.80.5 0.6 2 0.4 0.20 0 0 0.5 1 1.5 (e) 0 1 1 1 R 0 0.80.5 0.6 3 2 0.4 0.20 0 0 0.5 1 1.5 2 (f) Fig. 3. (a) Influence of βa and βs on R0, (b) Influence of βa and ρ on R0, (c) Influence of βs and ξ on R0 (d) Influence of βa and ξ on R0, (e) Influence of Λ and ξ on R0, (f) Influence of γ3 and κ on R0 Copyright © 2021 ASSA. Adv Syst Sci Appl (2021) 78 N.K. GOSWAMI, B. SHANMUKHA 0 200 400 600 800 1000 Time(in Days) 0 200 400 600 800 1000 P op ul at io ns S E I a I s H R Fig. 4. Variation of S, E, Ia, Is, H , R performing the stability of disease-free equilibrium point with R0 = 0.0932 < 1. 0 200 400 600 800 1000 Time(in Days) 0 200 400 600 800 1000 1200 P op ul at io ns S E I a I s H R Fig. 5. Variation of S, E, Ia, Is, H , R performing the stability of endemic equilibrium point with R0 = 2.0075 > 1. 0 50 100 150 200 250 300 350 400 450 500 Time(in Days) 0 50 100 150 In fe ct ed P op ul at io ns 1 =0.526 1 =0.0526 1 =0.00526 Fig. 6. Variation of Ia with time performing different values of γ1 Copyright © 2021 ASSA. Adv Syst Sci Appl (2021) DYNAMICS OF COVID-19 OUTBREAK AND OPTIMAL CONTROL STRATEGIES: A MODEL-BASED... 79 0 50 100 150 200 250 300 350 400 450 500 Time(in Days) 0 50 100 150 In fe ct ed P op ul at io ns 2 =1.5 2 =0.5 2 =.005 Fig. 7. Variation of Is with time performing different values of γ2 0 50 100 150 200 250 300 350 400 450 500 Time(in Days) 0 50 100 150 In fe ct ed P op ul at io ns 3 =0.2 3 =0.02 3 =0.0002 Fig. 8. Variation of H with time performing different values of γ3 7.1. The Optimal Control Problem In this section, we study the behavior of the proposed model by using optimal control theory. The objective functional for fixed time Tf if given by J(u1, u2) = ∫ Tf 0 [ A1Ia + A2Is + 1 2 (B1u 2 1 +B2u 2 2) ] dt (5) subject to the model system (4). The parameter A1 ≥ 0, A2 ≥ 0, B1 ≥ 0, B2 ≥ 0 are the weight and balancing constants, which measure the respective cost involvement over the interval [0, Tf ]. In order to find an optimal control, u1 ∗, and u2 ∗ such that J(u1 ∗, u2 ∗) = min (u1,u2)∈Ω J(u1, u2), (6) where Ω is the control set and is defined as Ω = {(u1, u2) : 0 ≤ u1, u2 ≤ 1, t ∈ [0, Tf ]} Here, all the controls are bounded and measurable. 7.1.1. Existence and characterization of optimal controls Here, we shall first establish the existence of such control functions that minimizes the cost functional J . The Lagrangian L Copyright © 2021 ASSA. Adv Syst Sci Appl (2021) 80 N.K. GOSWAMI, B. SHANMUKHA of this problem is defined as: L(Ia, Is, u1, u2) = A1Ia + A2Is + 1 2 B1u1 2 + 1 2 B2u 2 2 Now, we shall use Pontryagin’s maximum principle [32, 33] for necessary conditions for optimal controls system (4). For that by choosing X = (S,E, Ia, Is, H,R) ,Ω = (u1, u2) and λ = (λ1, λ2, λ3, λ4, λ5, λ6), the associated HamiltonianH can be written as H(X,Ω, λ) = L(Ia, Is, u1, u2) + λ1 dS dt + λ2 dE dt + λ3 dIa dt + λ4 dIs dt + λ5 dH dt + λ6 dR dt (7) Since u∗1, and u∗2 are solutions to the control problem (4), there exists the adjoint variables λ1, λ2, λ3, λ4, λ5, λ6 satisfying the following conditions. dx dt = ∂H(t, x, u∗1, u ∗ 2, λ1, λ2, λ3.λ4, λ5, λ6) ∂λ 0 = ∂H(t, x, u∗1, u ∗ 2, λ1, λ2, λ3.λ4, λ5, λ6) ∂u dλ dt = −∂H(t, x, u∗1, u ∗ 2, λ1, λ2, λ3.λ4, λ5, λ6) ∂x (8) Theorem 7.1: For the objective functional (5) and the control set (8) subject to control system (4) there exists an optimal control u∗ = (u1 ∗, u2 ∗) ∈ Ω such that J(u1 ∗, u2 ∗) = min Ω J(u1, u2). Proof To establish this result, we follow the Theorem 4.1 mentioned in [40] for the existence of optimal controls. As, we have discussed above that all the state variables (population) are bounded for each bounded controls coming from the control set Ω. Furthermore, Lipschitz condition with respect to state variables is satisfied by the right hand part of the model system (4). The control variable set Ω is also convex and closed by the definition and the model system (4) is linear in control variables. Thus, all the conditions for the existence of controls are fulfilled (for more details one can follow [28, 29]). Hence the result. Theorem 7.2: For optimal controls measures u∗1, u ∗ 2 and the state solutions S∗, E∗, I∗a , I ∗ s , H ∗, R∗ of the state system (4), there exists adjoint variables λ = (λi) Tf ∈ R6, i = 1, 2, 3, 4, 5, 6 such that dλ1 dt = (1− u1)βaIa(λ1 − λ2) + (1− u1)βsIs(λ1 − λ2) + µλ1 dλ2 dt = (κ+ µ)λ2 + ξκ(λ4 − λ3)− κλ4 dλ3 dt = −A1 + (1− u1)βaS(λ1 − λ2) + ρ(λ3 − λ4) + γ1(λ3 − λ6) + (µ+ δ1)λ3 dλ4 dt = −A2 + (1− u1)βsS(λ1 − λ2) + γ2(λ4 − λ6) + (µ1 + µ)λ4 + (δ2 + ηu2)(λ4 − λ5) dλ5 dt = γ3(λ5 − λ6) + (µ2 + µ)λ5 (9) dλ6 dt = µλ6 Copyright © 2021 ASSA. Adv Syst Sci Appl (2021) DYNAMICS OF COVID-19 OUTBREAK AND OPTIMAL CONTROL STRATEGIES: A MODEL-BASED... 81 with transversality conditions λ1(Tf ) = λ2(Tf ) = λ3(Tf ) = λ4(Tf ) = λ5(Tf ) = λ6(Tf ) = 0 (10) Proof Let u∗1, u ∗ 2 be the optimal control functions and S∗, E∗, I∗a , I ∗ s , H ∗, R∗ are the corresponding state variables. Then, Pontryagin’s Maximum Principle ensures the existence of the following adjoint variable λi(i = 1, 2, 3, 4, 5, 6) ∈ R6, which satisfies the following canonical equations: dλ1 dt = −∂H ∂S , dλ2 dt = −∂H ∂E , dλ3 dt = −∂H ∂Ia , dλ4 dt = −∂H ∂Is , dλ5 dt = −∂H ∂H , dλ6 dt = −∂H ∂R with transversality conditions (10) and the Hamiltonian (7). The adjoint system (9) can be obtained. In the following result, we shall state the analytical forms of the optimal controls. Theorem 7.3: The optimal controls (u∗1, u ∗ 2) which minimizes J over the region Ω are given by u∗1 = min{1, max(0, ũ1)} u∗2 = min{1, max(0, ũ2)} where ũ1 = (β1Ia + β2Is)S(λ2 − λ1) B1 , ũ2 = ηIs(λ4 − λ5) B2 , Proof Using optimally condition, we have ∂H ∂u1 = 0, ∂H ∂u2 = 0 We have ∂H ∂u1 = B1u1 + (βaIa + βsIs)S(λ1 − λ2) = 0 This gives u1 = ((βaIa + βsIs)S)(λ2 − λ1) B1 := ũ1 Similarly, ∂H ∂u2 = B2u2 + ηIs(λ5 − λ4) = 0 This implies u2 = ηIs(λ4 − λ5) B2 := ũ2 Moreover, lower and upper bounds of these control are 0 and 1 respectively. Thus, if ũ1 > 1, ũ2 > 1, then u1 = u2 = 1. Copyright © 2021 ASSA. Adv Syst Sci Appl (2021) 82 N.K. GOSWAMI, B. SHANMUKHA Also, ũ1 < 0, ũ2 < 0 then u1 = u2 = 0. Otherwise, we have u1 = ũ1, and u2 = ũ2 Hence, for these controls u∗1, u ∗ 2 we get optimum value of the function J . 8. SIMULATION OF OPTIMAL CONTROL PROBLEM In this section, we simulate our optimal control model using MATLAB. The parameter values are keeping same the parameters corresponding to stability of endemic equilibrium point E1 of the model (1). The weight constants for the optimal control problem are taken as A1 = 1, A2 = 1, B1 = 45, B2 = 65. We solve the optimality system by iterative method with the help of forward and backward difference approximations [32, 34–36]. We consider the time interval as [0,180]. First we solve the state equations by the forward difference approximation method then we use the backward difference approximation method to solve the adjoint equations. We consider different types of strategies to see the impact of optimal control in the total number of human infectives. 8.1. Strategy A: Employing hygiene promotion, social distancing and self-isolation (u1), only. Here, only control measure u1(t) is used to optimize the objective function J , while control intervention u2(t) = 0, were not employed. The influence of u1(t) is demonstrated in Figure 5(a), to minimize the objective function, the optimal control u1(t) is maintained at the maximum level. A single preventive measure can influence the spread of the Covid-19 in the population. Maintaining social distancing and self-isolation leads to control significant number reduction in asymptomatic and symptomatic cases in the population. From the figures, it is clear that the optimal control u1(t) is a little more effective compared to other types of controls but we need to maintain it to one for a longer period which is not easy to achieve. This control strategy is for using surgical face masks, social distancing, self- isolation, awareness of the transmission of disease, and sensitization. 8.2. Strategy B: Increase testing facility and treatment of the symptomatic individuals (u2) only. Here, only control measure u2(t) is used to optimize the objective function J , while control intervention u1(t) = 0, were not employed. In Figure 5(b), we present the plots of population and the effects of the increase in testing facility and treatment are demonstrated to minimizing the cost and reducing the number of coronavirus infections in the population. 8.3. Strategy C: Employing both the control interventions (u1, u2). Here both the control interventions (u1(t), u2(t)) are used to optimize the objective function J. From Figure 5(c), it is easy to say that by combining both optimal controls u1(t) and u2(t), the total number of infectious individuals decreases significantly. The simulation result indicates the effectiveness of optimal control strategies in reducing the number of infectives. It is observed that combined controls are more useful in reducing the number of infected cases significantly. Finally, it observed that from Figure 5(d), the optimal control model gives a better result as compacted to the model without the optimal control model as it reduces the number of infectives significantly in a desired interval of time. Copyright © 2021 ASSA. Adv Syst Sci Appl (2021) DYNAMICS OF COVID-19 OUTBREAK AND OPTIMAL CONTROL STRATEGIES: A MODEL-BASED... 83 0 20 40 60 80 100 120 140 160 180 Time(in Days) 0 0.2 0.4 0.6 0.8 1 u 1(C o n tr o l P ro fi le ) u 1 , when u 2 = 0 Fig. 9. Influence of u1, when u2 = 0 0 20 40 60 80 100 120 140 160 180 Time(in Days) 0 0.2 0.4 0.6 0.8 1 u 2(C o n tr o l P ro fi le ) u 2 , when u 1 = 0 Fig. 10. Influence of u2, when u1 = 0 0 20 40 60 80 100 120 140 160 180 Time(in Days) 0 0.2 0.4 0.6 0.8 1 u 1(C o n tr o l P ro fi le ) u 1 u 2 Fig. 11. Influence of u1, u2 0 20 40 60 80 100 120 140 160 180 Time(in Days) 0 50 100 150 200 In fe ct ed P op ul at io ns without control with control Fig. 12. Variation of infected human populations against time with and without optimal control Copyright © 2021 ASSA. Adv Syst Sci Appl (2021) 84 N.K. GOSWAMI, B. SHANMUKHA 9. CONCLUSION Dynamics of Covid-19 pandemic and its optimal control strategies are discussed in this present study through a mathematical model. The host population of the model is subdividing into six different compartment according to nature of the disease such as susceptible, exposed, asymptomatic infective,symptomatic infective, hospitalized and recovery. This non-linear mathematical model is proposed based on Indian condition by considering asymptomatic and symptomatic infections populations. It is assumed that the transmission rates due to asymptomatic and symptomatic individuals are different. The epidemiological threshold (basic reproduction number) is computed by using next generation matrix method and discussed different equilibria of the model in details. The model is globally asymptotically stable for the global stability of disease-free equilibrium when basic reproduction number is less than unity using the Castillo-chavez theorem. Also the theoretical analysis is carried out the global stability of endemic equilibrium is globally asymptotically stable using Lyapunov method when basic reproduction number is greater than one. Sensitivity analysis of the model performed to identify the key parameter that influence the basic reproduction number, which will regulate to control transmission dynamics of the disease. It emphasized that Λ and µ are the most sensitive parameter, followed by asymptomatic transmission coefficient β1 and ξ. Furthermore, the mathematical model is extended to optimal control problem by incorporating two time-dependent optimal control parameters to reducing the burden due to Covid-19 using Pontryagin’s Maximum Principal. Introduced control parameter u1 in the model as surgical face masks, social distancing, self-isolation, sensitization and awareness of transmission of the disease in asymptomatic and symptomatic invective individuals, which effectively reduce transmission rate. Social distancing should always implement at a higher percentage than self-isolation. The optimal control drives a significant reduction in the asymptomatic and symptomatic infected populations. The control parameter u2 included reducing the cost infrastructure like testing facility, which can lead to fast detection of Covid infected cases. The optimal control suggests that unless the cost is very high, the social distancing should implement at the maximum level throughout the time examined. The optimal control model provides a more reliable result as compacted to the model without the optimal control model. This control strategy reduces the number of infectives significantly in a desired interval of time. ACKNOWLEDGEMENTS The authors would like to thank the editor and anonymous referees for their valuable comments and suggestions which led to an improvement of our original manuscript. REFERENCES 1. Guanghong, D.,Chang,L.,Jianqiu,G.,Ling, W., CHENG, K., and ZHANG, D. (2004) SARS epidemical forecast research in mathematical model,Chinese Science Bulletin, 49 (21), 2332—2338. 2. Killerby,M., H. Biggs, H., Midgleys, C., Gerber, S., Watson, J. (2020) Middle Esat Respiratory Syndrome coronavirus transmission, Emerg.Infect.Dis. 26 (4),191-198. 3. Willman, M., Kobasa, D., Kindrachuk, J. (2019) A comparative analysis of factors influencing two outbreaks of middle eastern respiratory syndrome (MERS) in Saudi Arabia and South Korea, Viruses 11(12):1119.Doi :10.3390/v11121119.PMID 31817037. 4. Kolifarhood, G., Aghaali, M., Saadati, H. M., Taherpour, N., Rahimi, S., Izadi, N.,and Nazari, S.S.H. (2020) Epidemiological and Clinical Aspects of COVID-19: A Narative review,Archives of Academic Emergency Medicine, 8(1). 5. Chan, J.F.W. ,Yuan, S., Kok , K.H., Chu,H., Yang, J. (2019) Afamilial cluster of pneumonia associated with the 2019 novel coronavirus indicating person-to-person Copyright © 2021 ASSA. Adv Syst Sci Appl (2021) DYNAMICS OF COVID-19 OUTBREAK AND OPTIMAL CONTROL STRATEGIES: A MODEL-BASED... 85 transmission: a study of a family cluster. Lancet. http://dx.doi.org/10.1016/S0140- 6736(20) 30154-9. 6. Bai, Y., Yao, L., Wei, T., Tian, F., Jin, D. Y., Chen, L. (2020) Presumed Asymptomatic Carrier Transmission of COVID-19. Jama. 2020 Feb 21. PubMed PMID: 32083643. Pubmed Central, PMCID: PMC7042844. Epub 2020/02/23. eng. 7. World Health Organization. Novel coronavirus – China. Geneva,Switzerland: World Health Organization.https://www.who.int/csr/https://www.who.int/csr/don/12-january- 2020-novel-coronavirus-china/en/ 8. Hui, D. S., Azhar, E. I. , Madani, T. A., Ntoumi, F., Kock, R., Dar , O. (2020) The continuing 2019− nCoV epidemic threat of novel coronaviruses to global health - the latest 2019 novel coronavirus outbreak in Wuhan, China. Int. J. Infect. Dis. 9. N. Zhu, D. Zhang, W. Wang,X. W. Li, B. Yang, J. D. Song , et al. A novel coronavirus from patients with pneumonia in China, 2019, N Engl JMed.http: //dx.doi.org/10.1056/NEJMoa2001017. [2020-01-24]. 10. World Health Organization. Statement on the second meeting of the International Health Regulations (2005) Emergency Committee regarding the outbreak of novel coronavirus (2019-nCoV). Geneva, Switzerland: World Health Organization.https://www.who.int/newsroom/detail/30-01-2020-statement-on-the- second-meeting-of-theinternational-health-regulations-(2005)-emergency-committee- regar ding-the-outbreak-of-novel-coronavirus-(2019-ncov). [2020-01-30]. 11. Wang, C., Hornby, P. W., Hayden, F. W., Gao, G. F. (2020) A novel coronavirus outbreak of global health concern. Lancet. http://dx.doi.org/10. 1016/S0140-6736(20)30185-9.. 12. Munster, V.J., Koopmans, M., Doremalen , N. V., Riel, D. V., Wit, E. d. (2020) A novel coronavirus emerging in China – key questions for impact assessment. N Engl J Med. http://dx.doi.org/10.1056/NEJMp2000929. 13. Huang, C., Wang, Y., Li, X., Ren, L., Zhao, J., Hu, Y. (2020) Clinical features of patients infected with 2019 novel coronavirus in Wuhan, China. Lancet. http://dx.doi.org/10.1016/S0140-6736(20)30183-5. 14. WHO, (2020) Naming the coronavirus disease (COVID-19) and the virus that causes it. Available online:,https://www.who.int/emergencies/diseases/novel-coronavirus- 2019/technical-guidance/[Retrieved: 25/03/2020]. 15. WHO, (2020) Coronavirus disease 2019 (COVID-19), situation report -10. Available online:, https://www.who.int/emergencies/diseases/novel-coronavirus-2019/situation- reports/[Retrieved: 25/03/2020]. 16. Pulla, P. (2020) Covid-19: India imposes lockdown for 21 days and cases rise, BMJ 368, Doi: 10.1136/bmj.m1251. 17. Ministry of Health and Family Welfare (MOHFW), (2020) Coronavirus disease 2019 (COVID-19). Available online:, https://www.mohfw.gov.in/, [Retrieved: 25/03/2020] . 18. Khanna, R.C., Honavar, S.G., (2020) All eyes on coronavirus - what do we need to know as ophthalmologists, Indian Journal of Ophthalmology, 68(4) 549–553. 19. Siriprapaiwana, .S , Moorea, E. J., Koonpraserta, S. (2018) Generalized reproduction numbers, sensitivity analysis and critical immunity levels of an SEQIJR disease model with immunization and varying total population size, Mathematics and Computers in Simulation,146,70-89. 20. Kucharski, A.J., Russell, T. W., Diamond, C., Liu, Y., Edmunds, J., Funk, S.,Eggo, R. M. (2020) Early dynamics of transmission and control COVID-19: A mathematical modelling study, Lancet Infect. Dis. Doi:10.1016/S1473–3099(20)30144–4. 21. Senapati, A., Rana1b, S., Dasb, T., Chattopadhyaya, J. (2021)Impact of intervention on the spread of COVID-19 in India: A model based study, J. Theor Biol., doi:10.1016/j.jtbi.2021.110711, PMID:33862090. 22. Zhao, S., Lin, Q., Ran, J. (2020) Preliminary estimation of the basic reproduction number of novel coronavirus (2019-nCoV) in China, from 2019 to 2020: A data-driven analysis in the early phase of the outbreak, Int. J. Infect. Dis. 92, 214–217. Copyright © 2021 ASSA. Adv Syst Sci Appl (2021) 86 N.K. GOSWAMI, B. SHANMUKHA 23. Ivorra, B., Ferrández, M. R., Vela-Pérez, M.,and Ramos.A. M. (2020) Mathematical modeling of the spread of the coronavirus disease 2019 (COVID-19) considering its particular characteristics. The case of China. Communications in Nonlinear Science and Numerical Simulation. Accepted . Preprint version in ResearchGate , DOI link: http://www.doi.org/10.13140/RG.2.2.21543.29604. 24. Chen, T., Rui, J.,Wang, Q., Zhao1, Z., Cui, J,.and Yin, L. (2020) A mathematical model for simulating the phase-based transmissibility of a novel coronavirus, Infectious Diseases of Poverty 9:24, https://doi.org/10.1186/s40249-020-00640-3 25. Castillo-Chavez, C., Feng, Z., and Huang, W. (2002) On the computation of R0 and its role on global stability, in: Mathematical Approaches for For Emerging and Reemerging Infectious Diseases. Springer-Verlag, 229-250. 26. Driessche, P.V. and Watmough, J. (2002) Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission Mathematical Biosciences, 180, 29-48. 27. Korobeinikov, A., Wake, G. C. (2002) Lyapunov function and global stability for SIR, SIRS, and SIS epidemiological models. Appl. Math. Lett. 15:955–960. 28. Li MY, Muldowney JS (1996) A geometric approach to global stability problems. SIAM J Math Anal 27(4):1070–1083. 29. Li, M. Y., Muldowney, J. S. (1995) Global stability for the SEIR model in epidemiology. Math Biosci 125:155–164. 30. LaSalle, J.P. (1976) The stability of dynamical systems. Regional conference series in applied mathematics. SIAM, Philadelphia. 31. Srivastav, A.K., Tiwari, P.K., Srivastava, P.K., Ghosh, M., Kang, Y. (2020) A mathematical model for the impacts of face mask, hospitalization and quarantine on the dynamics of COVID-19 in India: deterministic vs. stochastic, Mathematical Biosciences and Engineering, 18 (1), 182–213. 32. Lenhart, S., Workman, J. T. (2007) Optimal control applied to biological models. CRC Press, Boca Raton. 33. Pontryagin, L.S., Boltyanskii, V. G., Gamkrelidze, R.V., Mishchenko, E. F. (1962) The mathematical theory of optimal processes, Inter science Publishers, Geneva. 34. Goswami, N. K. and Shanmukha, B. (2020)‘Modeling and Analysis of Symptomatic and Asymptomatic Infections of Zika Virus Disease with Non-Monotonic Incidence Rate’, Applied Mathematics and Information Sciences, 14(4), 655-671. 35. Goswami , N. K. (2021)’Modeling analysis of Zika virus with saturated incidence using optimal control theory’, International journal of dynamical systems and differential equations, 11 (3/4), 287–301. 36. Olaniyi, S., Obabiyi, O.S., Okosun, K.O., Adewale, A.T., Adewale, S.O. (2020)Mathe- matical Modeling and Optimal cost-effecting control of Covid-19 transmission dynamics. Eur Phys J Plus.135(11). Copyright © 2021 ASSA. Adv Syst Sci Appl (2021) Introduction The Model Analysis of the model system Basic Reproduction Number Existence of Endemic Equilibrium Point Stability analysis of the Model Global stability of disease-free equilibrium Global stability of endemic equilibrium Sensitivity Analysis of Reproduction Number Effects of Parameters on Disease Outbreak The Optimal Control Model The Optimal Control Problem Existence and characterization of optimal controls Simulation of Optimal Control Problem Strategy A: Employing hygiene promotion, social distancing and self-isolation (u1), only. Strategy B: Increase testing facility and treatment of the symptomatic individuals (u2) only. Strategy C: Employing both the control interventions (u1,u2). Conclusion