epidemic model of covid-19 outbreak by inducing

Nonlinear Dyn (2020) 102:455–487https://doi.org/10.1007/s11071-020-05896-w

ORIGINAL PAPER

Epidemic model of COVID-19 outbreak by inducingbehavioural response in population

Sangeeta Saha · G. P. Samanta · Juan J. Nieto

Received: 27 May 2020 / Accepted: 12 August 2020 / Published online: 26 August 2020© Springer Nature B.V. 2020

Abstract COVID-19 has spread around the worldsince December 2019, creating one of the greatestpandemics ever witnessed. According to the currentreports, this is a situation when people need to be morecareful and take the precaution measures more seri-ously, unless the condition may become even worse.Maintaining social distances and proper hygiene, stay-ing at isolation or adopting the self-quarantine methodare some of the common practices that people shoulduse to avoid the infection. And the growing informationregarding COVID-19 and its symptoms help the peo-ple to take proper precautions. In this present study, weconsider an SEIRS epidemiological model on COVID-19 transmissionwhich accounts for the effect of an indi-vidual’s behavioural response due to the informationregarding proper precautions. Our results indicate thatif people respond to the growing information regardingawareness at a higher rate and start to take the protectivemeasures, then the infected population decreases sig-nificantly. The disease fatality can be controlled only

S. Saha · G. P. Samanta (B)Department of Mathematics, Indian Institute ofEngineering Science and Technology, Shibpur, Howrah711103, Indiae-mail: [email protected]

S. Sahae-mail: [email protected]

J. J. NietoInstituto de Matematicas, Universidade de Santiago deCompostela, 15782 Santiago de Compostela, Spaine-mail: [email protected]

if a large proportion of individuals become immune,either by natural immunity or by a proper vaccine. Inorder to apply the latter option, we need to wait until asafe and proper vaccine is developed and it is a time-taking process. Hence, in the latter part of the work,an optimal control problem is considered by imple-menting control strategies to reduce the disease bur-den. Numerical figures show that the control denot-ing behavioural response works with higher intensityimmediately after implementation and then graduallydecreases with time. Further, the control policy denot-ing hospitalisation of infected individuals works withits maximum intensity for quite a long time period fol-lowing a sudden decrease. As, the implementation ofthe control strategies reduce the infected populationand increase the recovered population, so, it may helpto reduce the disease transmission at this current epi-demic situation.

Keywords COVID-19 · Epidemic model ·Behavioural response · Optimal control

1 Introduction

The first outbreak caused by a novel betacoronaviruswas reported in Wuhan, capital of Hubei Chineseprovince in December 2019 [8,9,28]. Initially, mostof the cases were found around the wholesale Hua-nan seafood market, Wuhan where live animals arealso traded [24]. But within one and a half month

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456 S. Saha et al.

COVID-19 spread to all Chinese province and to therest of the world. World Health Organisation (WHO)officially declared COVID-19 as pandemic instead ofan epidemic on 17 March 2020. It is an RNA viruswhich belongs to the Coronaviridae family and oforder Nidovirales, known as SARS-CoV-2 [11,33] andit is reported that the main symptoms of the diseaseinclude viral pneumonia, fever, dry cough, tiredness,aches and pains, nasal congestion, breathing problemsor even a variety of unspecific symptoms [5,8,18,43].The severity of the infection is high enough with anestimated case fatality ratio of order 1% [13,14,40,41]and hence, the virus has made a public health priorityissue given the anticipated size of the pandemic dueto the absence of pre-existing immunity. According tothe reports of the dashboard provided by CSSE of JohnHopkinsUniversity, at April 26, the number of reportedinfected cases, the number of documented death andthe number of documented recovery reached almost2,912,421; 203,432 and825,886, respectively, through-out the world [6]. Most of the countries take the epi-demic outbreak seriously from the first day by imple-menting proper public health measures including non-pharmaceutical interventions also. Among 185 coun-tries, United States (939,249 cases), Spain (223,759cases), Italy (195,351 cases), France (161,644 cases),Germany (156,513 cases), United Kingdom (149,569cases) are facing worse epidemic situations as the acti-vated infected cases exceed 100,000 there. In particu-lar, the number of infected cases in the United Stateshas grown very fast, the number of reported infectedcases increases from15 to 939,249 till April, 26, thoughthe death case is highest in Italy with number 26,384.China was the first country where quarantine strat-egy was implemented in Wuhan on 23 January 2020.In the current situation, China has 83,909 confirmedcases with the number of documented death (in Hubei)and recovery are 4642 and 77,346, respectively. Othercountries also apply the same strategy, i.e. implement-ing national lockdown to reduce the infection transmis-sion as France has done on March 17 or United King-dom has done on March 23 or even India has done onMarch 25. Comparing the data and strategieswith othercountries, it looks like there is a large number of casesof Covid-19 in India which is not registered as Indiastill does not has the sufficient number of test kits. Theundocumented infected individuals obviously facilitatethe rapid spread of COVID-19 [28]. It is the third timewhen zoonotic human coronavirus has spread in this

century. Before this, in 2002, severe acute respiratorysyndrome coronavirus (SARS-CoV) spread among 37countries and also in 2012, Middle East respiratorysyndrome coronavirus (MERS-CoV) spread among 27countries.

The first case of COVID-19was confirmed at Keralain India on 30 January 2020. According to NIC, India,there are total 20,177 confirmed cases, 5914 recoveriesand 826 deaths are reported in the country till 26 April2020 [22,29]. The Indian government has announcedto maintain social distance or to adopt self-quarantinestrategy as precaution measures to avoid large-scaledisease transmission among the population. In fact,on 22 March 2020, central government implementeda 14-hours long “Janata curfew” when the number ofaffected cases crossed 500. Moreover, the Governmentof India also announced for a 21-days national lock-down from March 25 in order to reduce the spreadof COVID-19. But later, realising the importance ofthe current problem, the government has increased thelockdown period up to 3 May 2020. So far no vac-cine has been discovered for novel coronavirus and so,maintaining social distances or applying self-isolationare taken as common ways for prevention of diseasetransmission [15]. The lockdown includes the ban onpeople from stepping out of their homes, closure of allshops except medical stores, hospitals, banks, groceryshops, etc., suspension of all educational institutionsand offices (only work-from-home is allowed), sus-pension of all public and private transport and also theprohibition of all political, cultural, sports, entertain-ment, religious activities. It has no doubt that this cur-rent outbreak has seriously affected life both econom-ically and healthwise. According to the World Bankand RBI, after 1991, this will be the first time whenthe economic growth rate in India will be decreased by1.5–2.8% due to novel coronavirus outbreak. So, it hasbecome a matter of concern for all of us how long oursocial life will go through this calamity.

Till now there are some studies revealing someinteresting statistical results about COVID-19 outbreak[12,19,30,32,35,37,38]. Based on the data from 31December 2019 to 28 January 2020, Wu et al. pro-posed an SEIR model to analyse the disease transmis-sion dynamics on national and global basis [42]. Tanget al. proposed a compartmental model for COVID-19where clinical development of the disease, current sta-tus of infected patients and control measures are com-bined. The results reveal that the control reproduction

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Epidemic model of COVID-19 outbreak 457

number may reach up to 6.47, and the control policiesincluding social distancing, quarantine and isolationcan minimise overall COVID cases [38]. According toa report submitted by Cambridge University, India’sstrategy of announcing 21-days lockdown may not besufficient enough to prevent the large-scale outbreak ofcoronavirus epidemic as it can bounce back in monthsand cause infection at a higher rate [36]. They suggestedfor an extension of two or three lockdowns with 5-daysbreaks in between or a single 49-days lockdown.

In this manuscript, we have proposed an SEIRSepidemic model to analyse coronavirus transmissionwhere it is considered that the susceptible populationcan protect themselves from getting infected by takingproper precautions by inducing behavioural changes.India has a population of almost 135 crores and so, itis not possible to lockdown or apply home quarantineon all susceptible population. A proportion of the sus-ceptible population may take the precaution measuressuccessfully but the rest of the people become infec-tive (asymptomatically or symptomatically). Again therecovered people may become susceptible later if theydo not maintain the precaution carefully. Rest of thepaper is categorised as follows: Sect. 2 contains theproposed epidemiological model which accounts forthe information induced behavioural response of sus-ceptible individuals. Section 3 proves that the modelis well-posed while in Sect. 4, equilibrium points areobtained with basic reproduction number (R0). Sen-sitivity analysis for different parameters is performedin Sect. 5. Local and global stability conditions of theequilibria are found in Sect. 6. The consequent sectionshows that the system undergoes a forward bifurcationat R0 = 1. Section 8 contains the pictorial scenariosof the system dynamics without applying any controlinterventions. Later, a corresponding optimal controlproblem is formulated. Section 10 contains the numer-ical simulations by implementing the control strategiesand the last section includes a brief conclusion.

2 Mathematical model: basic equations

We have elaborated a compartmental epidemic modelhere to analyse how the outbreak of COVID-19 affectthe population worldwide. The total population N (t)at time t , in this work, is divided into six subclassessuch as susceptible (S), exposed (E), asymptomati-cally infected (A), symptomatically infected (I ), hos-

pitalised and under treatment (H) and recovered class(R). The susceptible population become exposed whenthey come in contact with asymptomatically or symp-tomatically infected people or even with hospitalisedindividuals through the term (β1A + β2 I + β3H)Swhere β1, β2, β3 are the rate of disease transmissionper contact by an asymptomatic infected, symptomaticinfected and hospitalised people, respectively. The con-stant recruitment rate is denoted as � which is intro-duced in susceptible class. The term d denotes the nat-ural death rate in all population, whereas δ1, δ2, δ3are disease-related death rates in asymptomaticallyinfected, symptomatically infected and hospitalisedindividuals, respectively. The people in the exposedclass can move into either asymptomatic or symp-tomatic stage with probabilities φ and (1− φ), respec-tively, depending on whether any physical symptomof the disease has been observed in the infected peo-ple [7]. The terms η, ω and γ represent the progressionrates from asymptomatic to symptomatic, symptomaticto hospitalised and hospitalised to recovered stages,respectively. Also, recovery from the disease does notguarantee permanent recovery and hence some of therecovered peoplemove back to susceptible class furtherwith rate constant ξ [34]. The degradation rate of infor-mation with time, caused by natural fading of memoryabout the consequences, is denoted by a0.Now, when a disease outbreak at a higher rate withina short time period, various media platforms like TV,newspaper try to spread awareness among individu-als. The Government also chooses social and educa-tional campaigns to demonstrate the precaution mea-sures. Due to the awareness programs, people becomecautious at a higher rate and the disease transmis-sion rate becomes lower and people move to recov-ered class directly. Now, this density of informationand awareness is directly proportional to symptomati-cally infected individuals and it depends on how manypeople become infected rapidly in a short time interval.Let, Z(t) denotes the density of awareness due to infor-mation in susceptible population so that Z(t) = 0 inabsence of symptomatic infected people. This aware-ness leads to the behavioural changes among the sus-ceptible population to protect themselves from infec-tion. The importance of information in spreading ofEbola in Senegal has been described in some studies[20]. Though the government tries to spread necessaryinformation. Everyone does not become careful enoughall the time: insufficient resources, poor financial condi-

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458 S. Saha et al.

tion and heedless nature are some of the reasons in thiscase. So, it is considered that a proportion of susceptibleadopts the changes in their behaviour by responding tothe awareness programs andmove into recovered class.Now the changes depend not only on susceptible but oninformation too and so, is a function of S and Z . More-over, the rate at which information spread depends onlyon symptomatically infected population.

Moreover, u1k is the rate of behavioural changesof the susceptible population by taking proper protec-tive measures such as maintaining hygiene, social dis-tances, self-isolation, etc., to avoid the disease preva-lence. Here, u1 and k, respectively, denote the responserate and the information interaction rate through whichindividuals adopt new behaviour and change their oldhabits. It is obvious that this type of response is notfully effective because of financial problems, heedlessnature, etc., and hence, we have considered 0 ≤ u1 ≤1. Further, the information grows according to a satu-

ration rate functionpI

1 + q Iwhere p and q represent

the ‘growth rate of information’ and the ‘level for unre-sponsiveness towards information’, respectively [3]. Itdepends only on the symptomatic infective population.At the time of an epidemic outbreak, government, dif-ferent health agencies and media platforms becomemore active to spread awareness among people regard-ing the protective measures to avoid the disease preva-lence. It is assumed that at early stages, the growth ofinformation increases with the increase in symptomaticinfective population density but it ultimately comes tosaturation with time.The proposed SEIRS model on COVID-19 mainlyanalyses the dynamics when the susceptible are pro-vided with necessary information regarding the diseaseand its precautionary measures. The overall infectedclass is divided into asymptomatic and symptomaticcompartments. There are some reports which revealthat a person may become COVID-19 positive withoutshowing any symptoms. Also, in some cases, the symp-toms occur at a later stage and so, incorporation of theasymptomatically infected compartment is justifiable.Moreover, we have considered that the virus can betransmitted to the susceptible population when they getin touch with asymptomatically infected, symptomat-ically infected and also even hospitalised individuals.The system emphasises how the information regardingthe disease, its precautions and also people’s awareness

affect the disease propagation in this current pandemicsituation.So, the proposedmodel with positive parametric valuestakes the following form:

dS

dt= � − (β1A + β2 I + β3H)S − dS

+ ξ R − u1kSZ , S(0) > 0,

dE

dt= (β1A + β2 I + β3H)S − (κ + d)E, E(0) ≥ 0,

dA

dt= κφE − ηA − (d + δ1)A, A(0) ≥ 0,

dI

dt= κ(1 − φ)E + ηA − (d + δ2)I − ωI, I (0) ≥ 0,

dH

dt= ωI − γ H − (d + δ3)H, H(0) ≥ 0,

dR

dt= γ H − dR − ξ R + u1kSZ , R(0) ≥ 0,

dZ

dt= pI

1 + q I− a0Z , Z(0) ≥ 0,

(1)

A schematic diagram is provided in Fig. 1 to get a betterinsight of the proposed system.

3 Positivity and boundedness

For system (1): the following two theorems prove thatthe system variables are positive and bounded for alltime.

Theorem 3.1 All solutions of system (1) starting fromR7+ remain positive for all the time.

Proof Continuity and locally Lipschitzian functions ofright-hand side of model (1) on C result in occurrenceof a unique solution (S(t), E(t), A(t), I (t), H(t), R(t),Z(t)) on [0, τ ), where 0 < τ ≤ +∞ [21]. First weneed to show that, S(t) > 0, ∀ t ∈ [0, τ ). If it does nothold, then ∃ t1 ∈ (0, τ ) such that S(t1) = 0, S(t1) ≤ 0and S(t) > 0, ∀ t ∈ [0, t1). Hence we must haveE(t) ≥ 0, ∀ t ∈ [0, t1). If it is not true, then ∃ t2 ∈(0, t1) such that E(t2) = 0, E(t2) < 0 and E(t) ≥0, ∀ t ∈ [0, t2).Next we claim A(t) ≥ 0, ∀ t ∈ [0, t2).Suppose it is not true. Then ∃ t3 ∈ (0, t2) such thatA(t3) = 0, A(t3) < 0 and A(t) ≥ 0, ∀ t ∈ [0, t3).From third equation of (1), we have

dA

dt

∣∣∣∣t=t3

= κφE(t3) − (η + d + δ1)A(t3)

= κφE(t3) ≥ 0,

which is a contradiction to A(t3) < 0. So, A(t) ≥0, ∀ t ∈ [0, t2).

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Fig. 1 Schematic diagram of system (1)

Our next claim is I (t) ≥ 0, ∀ t ∈ [0, t2). If it is nottrue, then ∃ t4 ∈ (0, t2) such that I (t4) = 0, I (t4) <

0 and I (t) ≥ 0, ∀ t ∈ [0, t4). Now from the fourthequation of (1):

dI

dt

∣∣∣∣t=t4

= κ(1 − φ)I (t4) + ηI (t4)

−(ω + d + δ2)I (t4)

= κ(1 − φ)I (t4) + ηI (t4) ≥ 0,

which is a contradiction to I (t4) < 0. Hence, I (t) ≥0, ∀ t ∈ [0, t2).

Next claim is H(t) ≥ 0, ∀ t ∈ [0, t2). Let thestatement is not true. Then ∃ t5 ∈ (0, t2) such thatH(t5) = 0, H(t5) < 0 and H(t) ≥ 0, ∀ t ∈ [0, t5).

From the fifth equation of (1), we have

dH

dt

∣∣∣∣t=t5

= ωI (t5) − (γ + d + δ3)H(t5)

= ωI (t5) ≥ 0,

which is a contradiction to H(t5) < 0. So, H(t) ≥0, ∀ t ∈ [0, t2). From the second equation of (1):

dE

dt

∣∣∣∣t=t2

= (β1A(t2) + β2 I (t2) + β3H(t2))S(t2)

≥ 0 [∵ E(t2) = 0].It is a contradiction to E(t2) < 0. So, E(t) ≥ 0, ∀ t ∈[0, t1). Hence A(t) ≥ 0, I (t) ≥ 0, H(t) ≥ 0, ∀ t ∈[0, t1).

Proceeding as before, it can be proved that Z(t) ≥0, R(t) ≥ 0, ∀ t ∈ [0, t1).

From the first equation of (1), we have

dS

dt

∣∣∣∣t=t1

= � + ξ R(t1) > 0 [∵ S(t1) = 0],

which is a contradiction to S(t1) ≤ 0. So, S(t) >

0, ∀ t ∈ [0, τ ), where 0 < τ ≤ +∞. Also, by theprevious steps we have E(t) ≥ 0, A(t) ≥ 0, I (t) ≥0, H(t) ≥ 0, R(t) ≥ 0 and Z(t) ≥ 0, ∀ t ∈ [0, τ ),

where 0 < τ ≤ +∞. Hence proved. �Theorem 3.2 Solutions of system (1) which start fromR7+ remain uniformly bounded for all the time.

Proof

Let, N (t) = S(t)+E(t) + A(t)+I (t) + H(t) + R(t)

∴ dN

dt= � − dN

⇒ 0 < N (t) ≤ N (0)e−dt + �

d

(

1 − e−dt)

where N (0) = S(0) + E(0) + A(0) + I (0) + H(0) +R(0).

As t → ∞, 0 < N (t) ≤ �

d. From the last equation

of system (1), we have

dZ

dt= pI

1 + q I− a0Z ≤ pI − a0Z

⇒ dZ

dt+ a0Z ≤ p�

d(for large time t).

⇒ 0 < Z(t) ≤ Z(0)e−a0t + p�

a0d

(

1 − e−a0t)

As t → ∞, 0 < Z(t) ≤ p�

a0d.

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460 S. Saha et al.

Hence, all solutions of system (1) enter in the region:

� ={

(S, E, A, I, H, R, Z) ∈ R7+ : 0 ≤ N (t) ≤ �

d;

0 < Z(t) ≤ p�

a0d

}

.

�

4 Equilibrium analysis

Here, we obtain the equilibrium points of system (1)by solving the isoclines and determine the basic repro-duction number R0 to determine existence of endemicequilibrium point. System (1) has following equilib-rium points:

(1) Disease-free equilibrium point (DFE):

E0(S0, 0, 0, 0, 0, 0, 0) =(

�

d, 0, 0, 0, 0, 0, 0

)

.

(2) Endemic equilibrium point:E∗(S∗, E∗, A∗, I ∗, H∗, R∗, Z∗).

4.1 Basic reproduction number (R0)

Basic reproduction number R0 basically is a thresh-old value which denotes the average number of newlyinfected individuals after coming in contact with asingle infected individual in a susceptible population.A method was developed by van den Driessche andWatmough [39] which is used here to determine thebasic reproduction number. In system (1), people moveinto exposed class when the disease is introduced andalso the infected classes are asymptomatic (A), symp-tomatic stages (I ) and hospitalised stage (H). Let ustake x ≡ (E, A, I, H). Second, third, fourth and fifthequations of model (1) is written as:dx

dt= F(x) − ν(x),

F(x) =

⎛

⎜⎜⎝

(β1A + β2 I + β3H)S000

⎞

⎟⎟⎠

,

ν(x) =

⎛

⎜⎜⎝

(κ + d)E−κφE + (η + d + δ1)A

−κ(1 − φ)E − ηA + (ω + d + δ2)I−ωI + (γ + d + δ3)H

⎞

⎟⎟⎠

,

where F(x) contains only that compartment where newinfection term is introduced and ν(x) contains rest

of the terms. So, corresponding linearised matricesof F(x) and ν(x) at disease-free equilibrium E0 =(S0, 0, 0, 0, 0, 0, 0) are, respectively

F = (DF(x)) (E0) =

⎛

⎜⎜⎝

0 β1S0 β2S0 β3S00 0 0 00 0 0 00 0 0 0

⎞

⎟⎟⎠

,

V = (Dν(x)) (E0)

=

⎛

⎜⎜⎝

κ + d 0 0 0−κφ η + d + δ1 0 0

−κ(1 − φ) −η ω + d + δ2 00 0 −ω γ + d + δ3

⎞

⎟⎟⎠

Let, α0 = κ + d, α1 = η + d + δ1, α2 = ω + d +δ2, α3 = γ + d + δ3.

Here, |V | = det(V ) = (κ +d)(η+d+δ1)(ω+d+δ2)(γ + d + δ3) = α0α1α2α3.

As the spectral radius of the next generation matrixFV−1 is R0, then the basic reproduction number ofsystem (1) is given by [39]:

R0 = κS0|V |

[β1φα2α3 + (β2α3 + β3ω){φη + α1(1 − φ)}]= κ�

dα0α1α2α3

[β1φα2α3 + (β2α3 + β3ω){φη + α1(1 − φ)}](2)

Existence of endemic equilibrium point E∗(S∗, E∗,A∗, I ∗, H∗, R∗, Z∗)From system (1):

� − (β1A∗ + β2 I

∗ + β3H∗)S∗ − dS∗

+ ξ R∗ − u1kS∗Z∗ = 0,

(β1A∗ + β2 I

∗ + β3H∗)S − (κ + d)E∗ = 0,

κφE∗ − (η + d + δ1)A∗ = 0,

κ(1 − φ)E∗ + ηA∗ − (d + δ2 + ω)I ∗ = 0,

ωI ∗ − (γ + d + δ3)H∗ = 0,

γ H∗ − (d + ξ)R∗ + u1kS∗Z∗ = 0,

pI ∗

1 + q I ∗ − a0Z∗ = 0.

(3)

Solving these equations, we get

S∗ = �

dR0, Z∗ = pI ∗

a0(1 + q I ∗),

H∗ = ωI ∗

α3, E∗ = α1α2 I ∗

κ{φη + α1(1 − φ)} ,

A∗ = φα2 I ∗

{φη + α1(1 − φ)} ,

123


R∗ = [γωda0R0(1 + q I ∗) + u1kp�α3]I ∗

a0dR0α3(d + ξ)(1 + q I ∗)and I ∗ is a positive root of the equation:

P1 I2 + P2 I + P3 = 0,

where

P1 = q

[γ ξω

α3(d + ξ)− α0α1α2

κ{φη + α1(1 − φ)}]

= −q[α1{dα0α1α2+ξdα2α3+κξα3(d+δ2)+ξκω(d+δ3)}+φγ ξωκ(d+δ1)]κα3(d+ξ){φη+α1(1−φ)} < 0

P2 = q�

(

1 − 1

R0

)

+[

γ ξω

α3(d + ξ)− α0α1α2

κ{φη + α1(1 − φ)}]

− u1kp�

a0R0(d + ξ),

P3 = �

(

1 − 1

R0

)

Here, P3 > 0 when R0 > 1. As P1 is alwaysnegative, so, for R0 > 1, we get a unique endemicequilibrium point. But when R0 < 1, both of P2 andP3 become negative resulting in non-occurrence of anendemic equilibrium.Hencewe have the following the-orem.

Theorem 4.1 System (1) has a disease-free equilib-rium point E0 (S0, 0, 0, 0, 0, 0, 0) for any parametricvalues. And for R0 > 1, the system possesses a uniqueendemic equilibrium E∗(S∗, E∗, A∗, I ∗, H∗, R∗, Z∗).

5 Sensitivity analysis

For system (1), R0 depends on some of the vitalparameters such as recruitment rate (�), transmis-sion rates (β1, β2, β3), disease related death rates(δ1, δ2, δ3), natural death rate (d), progression rate ofexposed people into infected classes (κ), probability atwhich exposed people move into infected classes (φ),progression rate of asymptomatic infected individualinto symptomatic class (η), progression rate of symp-tomatic infected population into treatment class (ω),progression rate of hospitalised population into recov-ered class (γ ). But among all these parameters, wecannot control only β1, β2, β3, κ, ω.

Now, R0 = κ�

dα0α1α2α3[β1φα2α3 + (β2α3 +

β3ω){φη + α1(1 − φ)}], where α0 = κ + d, α1 =η + d + δ1, α2 = ω + d + δ2 and α3 = γ + d + δ3.From the expression of R0:

∂R0

∂β1= �κφ

dα0α1> 0

∂R0

∂β2= �κ{φη + α1(1 − φ)}

dα0α1α2> 0

∂R0

∂β3= �κω{φη + α1(1 − φ)}

dα0α1α2α3> 0

∂R0

∂κ= �[β1φα2α3+(β2α3+β3ω){φη+α1(1−φ)}]

α20α1α2α3

> 0

∂R0

∂ω= �κ{φη+α1(1−φ)}

dα0α1α22α3

[β3(d + δ2) − β2(γ + d + δ3)]

So,∂R0

∂ωis monotonic decreasing when β3(d + δ2) <

β2(γ + d + δ3) holds. Next we compute normalisedforward sensitivity index with respect to each of theparameters β1, β2, β3, κ and ω to analyse the sensi-tivity of R0 (to each of the parameters) by the methodof Arriola and Hyman [1]:

�β1 =∣∣∣∣∣

∂R0R0∂β1β1

∣∣∣∣∣=

∣∣∣∣

β1

R0

∂R0

∂β1

∣∣∣∣

=∣∣∣∣

β1φα2α3

[β1φα2α3 + (β2α3 + β3ω){φη + α1(1 − φ)}]∣∣∣∣< 1

�β2 =∣∣∣∣∣

∂R0R0∂β2β2

∣∣∣∣∣=

∣∣∣∣

β2

R0

∂R0

∂β2

∣∣∣∣

=∣∣∣∣

β2α3{φη + α1(1 − φ)}[β1φα2α3 + (β2α3 + β3ω){φη + α1(1 − φ)}]

∣∣∣∣< 1

�β3 =∣∣∣∣∣

∂R0R0∂β3β3

∣∣∣∣∣=

∣∣∣∣

β3

R0

∂R0

∂β3

∣∣∣∣

=∣∣∣∣

β3ω{φη + α1(1 − φ)}[β1φα2α3 + (β2α3 + β3ω){φη + α1(1 − φ)}]

∣∣∣∣< 1

�κ =∣∣∣∣∣

∂R0R0∂κκ

∣∣∣∣∣=

∣∣∣∣

d

κ + d

∣∣∣∣< 1

123

462 S. Saha et al.

�ω =∣∣∣∣∣

∂R0R0∂ωω

∣∣∣∣∣=

∣∣∣∣

ω

R0

∂R0

∂ω

∣∣∣∣

=∣∣∣∣

ω{φη + α1(1 − φ)}{β3(d + δ2) − β2(γ + d + δ3)}α2[β1φα2α3 + (β2α3 + β3ω){φη + α1(1 − φ)}]

∣∣∣∣

From the calculation, it is observed that the diseasetransmission rates from asymptomatically infected(β1), symptomatically infected (β2) and hospitalisedindividuals (β3) maintain direct proportional relationwith basic reproduction number which is biologicallyacceptable too. Increasing disease transmission ratescan stimulate the probability of occurrence of an epi-demic outbreak. Moreover, if more people from theexposed class move into infected classes whether inasymptomatic stage or symptomatic stage, then thechance of getting an infected system increase. Thecalculation also shows that R0 increases for increas-ing value of κ . On the other hand, ω denotes therate at which the symptomatically infected individu-als are admitted to hospitals for treatment. So, it isevident that increasing ω helps to reduce the diseaseprevalence and hence ω maintains an inversely propor-tional relation with R0, i.e. increment in this parameter

leads to a decrease in R0. Here,∂R0

∂ω< 0 only when

β3(d+δ2) < β2(γ +d+δ3) holds. So,maintaining thisinequality can reduce the disease outbreak for increas-ing hospitalisation rate. The calculations and numericalsimulations reveal that R0 is more sensitive to changesin βi for i = 1, 2, 3 than κ andω. So, if we try to reducethe transmission rates by maintaining social distancesand taking proper precautions, then this epidemic situ-ation may be handled.

6 Stability analysis

We discuss the local and global stability conditions forthe disease-free equilibrium point as well as endemicequilibriumpoint in this section. Let,α0 = d+κ, α1 =η + d + δ1, α2 = ω + d + δ2 and α3 = γ + d + δ3.The Jacobian matrix of system (1) is given as:

J

=

⎛

⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

a11 0 −β1S −β2S −β3S ξ −u1kS

β1A + β2 I + β3H −α0 β1S β2S β3S 0 0

0 κφ −α1 0 0 0 0

0 κ(1 − φ) η −α2 0 0 0

0 0 0 ω −α3 0 0

u1kZ 0 0 0 γ −(d + ξ) u1kS

0 0 0 p(1+q I )2

0 0 −a0

⎞

⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(4)

where a11 = −(β1A + β2 I + β3H) − d − u1kZ .

6.1 Local stability of E0

Jacobian matrix at the disease-free equilibrium point

E0=(

�

d, 0, 0, 0, 0, 0, 0

)

is given as follows

(

S0=�

d

)

:

J |E0 =

⎛

⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

−d 0 −β1S0 −β2S0 −β3S0 ξ −u1kS0

0 −α0 β1S0 β2S0 β3S0 0 0

0 κφ −α1 0 0 0 0

0 κ(1 − φ) η −α2 0 0 0

0 0 0 ω −α3 0 0

0 0 0 0 γ −(d + ξ) u1kS0

0 0 0 p 0 0 −a0

⎞

⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

Some of the eigenvalues of the Jacobian matrix J |E0

are λ1 = −d, λ2 = −(d + ξ), λ3 = −a0 and otherfour eigenvalues are obtained from the roots of the fol-lowing equation: λ4+Q1λ

3+Q2λ2+Q3λ+Q4 = 0,

where, Q1 = α0 + α1 + α2 + α3, Q2 = α0α1 +α2α3 + (α0 + α1)(α2 + α3) − κS0{β1φ + β2(1− φ)},Q3 = α0α1(α2+α3)+α2α3(α0+α1)−S0κ[β1φ(α2+α3) + β2{φη + α1(1 − φ)} + (β2α3 + β3ω)(1 − φ)]and Q4 = α0α1α2α3(1− R0). So, Q4 > 0 for R0 < 1.So for R0 < 1, the characteristic equation has rootswith negative real parts only when Q2, Q3 > 0 andwe have the following theorem

Theorem 6.1 The DFE (E0) is locally asymptomat-ically stable (LAS) for R0 < 1 when Q2 > 0 andQ3 > 0 hold.

6.2 Global stability of E0

Theorem 6.2 DFE E0 of system (1) is globally asymp-totically stable (GAS) when p < d and u1k� < a0dhold.

Proof Let us consider the Lyapunov function as

V1 =(

S − S0 − S0ln

(S

S0

))

+E + A + I + H + R + Z

123


Here, V1 is a positive definite function for all (S, E, A,

I, H, R, Z) other than the DFE. Time derivative of V1computed along the solutions of system (1) is given as:

dV1dt

=(

1 − S0S

)

[�−(β1A+β2 I+β3H)S−dS+ξ R−u1kSZ ]+ (β1A + β2 I + β3H)S

− d(E + A + I + H + R) − δ1A − δ2 I

− δ3H − ξ R + u1kSZ + pI

1 + q I− a0Z

= � − d(E + A + I + H + R) − δ1A − δ2 I

− δ3H − �S0S

+ dS0 − ξ S0R

S

+ u1kS0Z + pI

1 + q I− a0Z

≤(

�

S− d

)

(S − S0) + (p − d)I

+ 1

d(u1k� − a0d)Z

= − d

S(S − S0)

2 + (p − d)I

+ 1

d(u1k� − a0d)Z

Hence,dV1dt

< 0 when p < d and u1k� < a0d hold.

Also, dV1dt = 0 when S = S0 and E = A = I = H =R = Z = 0.By LaSalle’s invariance principle [27], E0

is globally asymptotically stable when R0 < 1 whenthe mentioned parametric restrictions are fulfilled. �

6.3 Local stability of E∗

Theorem 6.3 The endemic equilibrium point E∗ isLAS for R0 > 1 when the conditions if Ri > 0 fori = 1, 7 and �i > 0 for i = 2, 3, 4, 5, 6 hold.

Proof Proof is given in the “Appendix”. �

6.4 Global stability of E∗

Theorem 6.4 The endemic equilibrium point E∗ ofsystem (1) is globally asymptomatically stable (GAS)in

� ={

(S, E, A, I, H, R, A) ∈ R7+ :

(m2β1S)2 <4

9m2α0

(δ1 + m3α1), (m2β3S − δ3)2

<1

6m2δ3α3,

(m2β2S + m4κφ)2

<m2α0

3(δ2 + m4α2),

with (i) − (i i i) as stated at the end in the proof} ,

where αi for i = 0, 1, 2, 3 and m j for j = 1, 2, 3, 4are mentioned in the proof.

Proof Consider a Lyapunov function V2 as:

V2(t) = 1

2[(S − S∗) + (E − E∗) + (A − A∗)

+ (I − I ∗) + (H − H∗) + (R − R∗)]2+m1

2(S − S∗)2 + m2

2(E − E∗)2

+ m3

2(A − A∗)2 + m4

2(I − I ∗)2

+ 1

2(Z − Z∗)2

Time derivative of V2 along the solutions of system (1)is given as:

dV2dt

= [(S − S∗) + (E − E∗) + (A − A∗)

+ (I − I ∗) + (H − H∗) + (R − R∗)]d

dt(S + E + A + I + H + R) + m1(S − S∗)dS

dt

+ m2(E − E∗)dEdt

+ m3(A − A∗)dAdt

+ m4(I − I ∗)dIdt

+ (Z − Z∗)dZdt

Now steady state of system (1) at E∗ gives

� − (β1A∗ + β2 I

∗ + β3H∗)S∗ − dS∗

+ξ R∗ − u1kS∗Z∗ = 0,

(β1A∗ + β2 I

∗ + β3H∗)S − (κ + d)E∗ = 0,

κφE∗ − (η + d + δ1)A∗ = 0,

κ(1 − φ)E∗ + ηA∗ − (d + δ2 + ω)I ∗ = 0,

ωI ∗ − (γ + d + δ3)H∗ = 0,

γ H∗ − (d + ξ)R∗ + u1kS∗Z∗ = 0,

123

464 S. Saha et al.

pI ∗

1 + q I ∗ − a0Z∗ = 0,

� − d(S∗ + E∗ + A∗ + I ∗ + H∗ + R∗)−δ1A

∗ − δ2 I∗ − δ3H

∗ = 0

Consider α0 = κ + d, α1 = η + d + δ1, α2 = d +δ2 + ω, α3 = γ + d + δ3. So, we have

dV2dt

= [(S − S∗) + (E − E∗) + (A − A∗)

+ (I − I∗) + (H − H∗) + (R − R∗)][� − d(S + E + A + I + H + R) − δ1A

− δ2 I − δ3H ]+ m1(S − S∗)[� − dS + ξ R

− (β1A + β2 I + β3H)S − u1kSZ ]+ m2(E − E∗)[(β1A + β2 I + β3H)S − α0E]+ m3(A − A∗)[κφE − α1A]+ m4(I − I∗)[κ(1 − φ)E + ηA − α2 I ]

+ (Z − Z∗)

(pI

1 + q I− a0Z

)

= [(S − S∗) + (E − E∗) + (A − A∗) + (I − I∗)

+ (H − H∗) + (R − R∗)][−d(S − S∗) − d(E − E∗) − (d + δ1)(A − A∗)

− (d + δ2)(I − I∗) − (d + δ3)

(H − H∗) − d(R − R∗)]+ m1(S − S∗)

[� − (β1A + β2 I + β3H)S − dS + ξ R − u1kSZ ]+ m2(E − E∗)

[(β1A + β2 I + β3H)S − α0E]+ m3(A − A∗)[κφE − α1A] + m4(I − I∗)

[κ(1 − φ)E + ηA − α2 I ] + (Z − Z∗)

(pI

1 + q I− a0Z

)

dV2dt

= [(S − S∗) + (E − E∗) + (A − A∗)

+ (I − I∗) + (H − H∗) + (R − R∗)][−d(S − S∗) − d(E − E∗) − (d + δ1)(A − A∗)

− (d + δ2)(I − I∗)

− (d + δ3)(H − H∗) − d(R − R∗)]+ m1(S − S∗)

[−β1(AS − A∗S∗) − β2(I S − I∗S∗)

− β3(HS − H∗S∗) − d(S − S∗)

+ ξ(R − R∗) − u1k(SZ − S∗Z∗)]+ m2(E − E∗)

[β1(AS − A∗S∗) + β2(I S − I∗S∗)

+ β3(HS − H∗S∗) − α0(E − E∗)]+ m3(A − A∗)

[κφ(E − E∗) − α1(A − A∗)]+ m4(I − I∗)

[κ(1 − φ)(E − E∗ + η(A − A∗) − α2(I − I∗)]+ (Z − Z∗)[

p(I − I∗)

(1 + q I )(1 + q I∗)− a0(Z − Z∗)

]

= −d(S − S∗)2

− d[(E − E∗) + (A − A∗) + (I − I∗) + (H − H∗)]2

− m2α0(E − E∗)2

− (δ1 + m3α1)(A − A∗)2

− (δ2 + m4α2)(I − I∗)2

− δ3(H − H∗)2 − d(R − R∗)2

+ [−2d + m2(β1A∗ + β2 I

∗ + β3H∗)]

(S − S∗)(E − E∗)

− [2d + δ1 + m1β1S∗](S − S∗)(A − A∗)

− [2d + δ2 + m1β2S∗](S − S∗)(I − I∗)

− [2d + δ3 + m1β3S∗](S − S∗)(H − H∗)

+ (−2d + m1ξ)(S − S∗)(R − R∗)

+ [−δ1 + m2β1S + m3κφ](E − E∗)(A − A∗)

+ [−δ2 + m2β2S + m4κφ](E − E∗)(I − I∗)

+ [−δ3 + m2β3S](E − E∗)(H − H∗)

− 2d(E − E∗)(R − R∗)

+ [−(δ1 + δ2) + m4η](A − A∗)(I − I∗)

− (δ1 + δ3)(A − A∗)(H − H∗)

− δ1(A − A∗)(R − R∗)

− (δ2 + δ3)(I − I∗)(H − H∗)

− δ2(I − I∗)(R − R∗)

− δ3(H − H∗)(R − R∗)

+ p(I − I∗)(Z − Z∗)

(1 + q I )(1 + q I∗)− a0(Z − Z∗)2

Let, m1 = 2d

ξ, m2 = 2d

β1A∗ + β2 I ∗ + β3H∗ , m3 =δ1

κφand m4 = δ1 + δ2

η. Then we have

123


dV2dt

= −d(S − S∗)2

− d[(E − E∗) + (A − A∗) + (I − I∗) + (H − H∗)]2

− m2α0(E − E∗)2

− (δ1 + m3α1)(A − A∗)2

− (δ2 + m4α2)(I − I∗)2

− δ3(H − H∗)2 − d(R − R∗)2

− [2d + δ1 + m1β1S∗](S − S∗)(A − A∗)

− [2d + δ2 + m1β2S∗](S − S∗)(I − I∗)

− [2d + δ3 + m1β3S∗](S − S∗)(H − H∗)

+ m2β1S(E − E∗)(A − A∗)

+ [−δ2 + m2β2S + m4κφ](E − E∗)(I − I∗)

+ [−δ3 + m2β3S](E − E∗)(H − H∗)

− 2d(E − E∗)(R − R∗)

− (δ1 + δ3)(A − A∗)(H − H∗)

− δ1(A − A∗)(R − R∗)

− (δ2 + δ3)(I − I∗)(H − H∗)

− δ2(I − I∗)(R − R∗) − δ3(H − H∗)(R − R∗)

+ p(I − I∗)(Z − Z∗)

(1 + q I )(1 + q I∗)− a0(Z − Z∗)2

So,dV2dt

< 0 in�when the following conditions hold:

(i) (δ1+m3α1)>max

{

3

2d(2d+δ1+m1β1S

∗)2,9δ214d

}

,

(ii) (δ2 + m4α2) > max

{4

d(2d + δ2 + m1β2S

∗)2,

12δ22d

,8(δ2 + δ3)

2

δ3,

2p4d4

a0(d + q�)4

}

,

(iii)48

d(2d + δ3 + m1β3S

∗)2 < 12δ3 < d <m2α0

18.

Moreover,dV2dt

∣∣∣∣E∗

= 0. So, by Lyapunov LaSalle’s

theorem [27], E∗ is GAS in the interior of � when thementioned restrictions are fulfilled. �

7 Forward bifurcation at R0 = 1

A unique endemic equilibrium point E∗ of system (1)exists when basic reproduction number (R0) exceeds1. Also, for R0 less than unity, there is no endemicequilibrium point. Hence, there exists of a transcriticalbifurcation around the disease-free equilibrium pointE0 when R0 = 1.

Theorem 7.1 The system undergoes a transcriticalbifurcationwith respect to the bifurcation parameterβ1

around E0(S0, 0, 0, 0, 0, 0, 0) ≡(

�

d, 0, 0, 0, 0, 0, 0

)

for R0 = 1.

Proof Consider the RHS of system (1) as: g =(g1, g2, g3, g4, g5, g6, g7)T where

g1 = � − (β1A + β2 I + β3H)S − dS + ξ R − u1kSZ;g2 = (β1A + β2 I + β3H)S − (κ + d)E;g3 = κφE − (η + d + δ1)A;g4 = κ(1 − φ)E + ηA − (d + δ2 + ω)I ;g5 = ωI − (γ + d + δ3)H ;g6 = γ H − (d + ξ)R + u1kSZ;g7 = pI

1 + q I− a0Z .

For system (1), R0 = κ�

dα0α1α2α3[β1φα2α3+(β2α3+

β3ω){φη + α1(1 − φ)}], where α0 = d + κ, α1 =η + d + δ1, α2 = ω + d + δ2, α3 = γ + d + δ3.

J |E0 =

⎛

⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

−d 0 −β1S0 −β2S0 −β3S0 ξ −u1kS00 −α0 β1S0 β2S0 β3S0 0 00 κφ −α1 0 0 0 00 κ(1 − φ) η −α2 0 0 00 0 0 ω −α3 0 00 0 0 0 γ −(d + ξ) u1kS00 0 0 p 0 0 −a0

⎞

⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

The eigenvalues are λ1 = −d, λ2 = −(d + ξ), λ3 =−a0 and other four eigenvalues are roots of the equa-tion: λ4 + Q1λ

3 + Q2λ2 + Q3λ + Q4 = 0,

where, Q1 = α0 +α1+α2 +α3, Q2 = α0α1+α2α3+(α0 + α1)(α2 + α3) − κS0{β1φ + β2(1 − φ)}, Q3 =α0α1(α2+α3)+α2α3(α0+α1)− S0κ[β1φ(α2+α3)+β2{φη+α1(1−φ)}+ (β2α3+β3ω)(1−φ)] and Q4 =α0α1α2α3(1 − R0).

Let β1[TC] = 1

φα2α3

[dα0α1α2α3

�κ− (β2α3 + β3ω)

{φη + α1(1 − φ)}] be the value of β1 such that R0 = 1which implies J |E0 has a simple zero eigenvalue atβ1 = β1[TC]. Also, V = (v1, v2, v3, v4, v5, v6, v7)

T

and W = (w1, w2, w3, w4, w5, w6, w7)T , respec-

tively, be the eigenvectors of J |E0 and J |TE0correspond-

ing to the zero eigenvalue.Calculations give v1 = κ(ξγωa0−u1k�pα3){φη+

α1(1−φ)}−a0α0α1α2α3(d+ξ), v2 = da0α0α1α2α3(d+ξ), v3 = dφa0κα2α3(d + ξ), v4 = da0α3κ(d +ξ){φη+α1(1−φ)}, v5 = ωda0κ(d+ξ){φη+α1(1−

123

466 S. Saha et al.

φ)}, v6 = κ{φη + α1(1 − φ)}(γωa0d + u1k�pα3)

and v7 = pdα3κ(d + ξ){φη + α1(1 − φ)}. Also,w1 = w6 = w7 = 0, w2 = dα1α2α3, w3 =�{β1α2α3+η(β2α3+β3ω)}, w4 = �α1(β2α3+β3ω)

and w5 = �α1α2β3. Therefore,

�1 = WT .gβ1(E0, β1[TC])= w2(AS)

∣∣E0

= 0,

�2 = WT [

Dgβ1(E0, β1[TC])V]

= S0v3w2 �= 0,

�3 = WT[

D2g(E0, β1[TC])(V, V )]

= 2(β1v3 + β2v4 + β3v5)v1w2 �= 0.

Thus, in viewof Sotomayor’s Theorem, system (1) pos-sesses a transcritical bifurcation around E0 at R0 = 1taking β1 as bifurcating parameter. �

8 Numerical simulation without any control policy

Pictorial scenarios helpus to understand systemdynam-ics more clearly. The human population in India inApril 2020 is about 136 million, the annual birth rateis 18.7 births/1000 people and the annual death rateis 7.3 deaths/1000 people. So, we are taking S(0) =1.36 × 109 and � = 7 × 104 by applying unit con-version from year to day. And the death rate per daywe get is near about 0.00002. For the sake of calcula-tion, we are taking d = 0.000055. From the data pro-vided in the dashboard by the centre for system scienceand engineering (CSSE) at John Hopkins Universityon 13th April 2020, India has 9352 corona activatedcases [6]. And till the date, total death cases are 324and recovered cases are 980. Hence, unit conversionto day gives δ3 as 8 × 10−4 and ξ as 3 × 10−3. Astotal active cases till 13th is 8048 among 9352 cases;so, we get ω as 0.02 approximately [29]. Accordingto the current epidemic situation of coronavirus, thenew human cases infected per unit time is denoted byβSI ≡ (β1A + β2 I + β3H)S. Human cases infectedby COVID-19 in March was (I ) is about 1393, thepopulation in India (S) in April is approximately by1.35×109, the new human cases (β2SI ) till 13th Aprilis about 9352 [29], hence we have β2 ≈ 1.5×10−10 bydoing the unit conversion frommonth to day. As per thedata ofMarch provided byMinistry ofHealth and Fam-ily Welfare, Government of India, the infected casesby COVID-19 is about 1393, so, for sake of simplic-

0 50 1000

5

10

15 x 105

t

S

0 50 1000

500

1000

t

E

0 50 1000

20

40

t

A

0 50 1000

200

400

t

I

0 50 100

5101520

t

H

0 50 1000

20406080

t

R

0 20 40 60 80 100 1200

0.050.1

0.150.2

0.250.3

0.350.4

0.450.5

t

Z

(a)

(b)

Fig. 2 Stability of the populations around disease-free equilib-rium E0

ity I (0) is taken as 40. Now, all the assumed and esti-mated parameters are listed in Table 1. ByCDC reports,mostly 25% of infected may not show any symptoms,i.e. remain as asymptomatic [7]. So, we have assumedA(0) = 10. Also, E(0) = 103, H(0) = 15, R(0) = 5and Z(0) = 0.5 have been assumed.

Figure 2 shows that for the parametric values inTable 1 and d = 0.6, the trajectory starting frommentioned initial point ultimately converges to DFEE0(1.273×109, 0, 0, 0, 0, 0, 0) and as we get the basicreproduction number R0 as 0.000013 here which liesbelow unity, so, the disease cannot invade in the systemin this case.

Now if we start to decrease the value of d, thenfor d = 0.000055 along with parametric values inTable 1, the trajectory starting from mentioned initialpoint approaches to unique endemic equilibrium pointE∗(1.3788 × 108, 2.2419 × 106, 895850.6, 6.5435 ×107, 2.6129 × 106, 4.2916 × 108, 0.1667) with time(see Fig. 3). For these parametric values we get R0 =

123


Table 1 Parameter values used for numerical simulation of system (1)

Parametric values

� 7 × 104 β1 0.37 × 10−10

β2 1.48 × 10−10 β3 0.15 × 10−10

ξ 3 × 10−10 u1 0.1

k 0.002 κ 0.6

φ 0.1 η 0.15

δ1 10−4 δ2 5 × 10−4

δ3 8 × 10−4 ω 0.02

γ 0.5 d 0.000055

p 0.01 q 1

a0 0.06

0 500 10000

1

2 x 109

t

S

0 1000 20000

5

10 x 107

t

E

0 1000 2000 30000

2

4 x 107

t

A

0 2 4x 104

0

5

10 x 108

t

I

0 2 4 6

x 104

0

2

4 x 107

t

H

0 2 4 6

x 104

0

1

2 x 109

t

R

0 500 1000 1500

0.2

0.25

0.3

0.35

0.4

0.45

0.5

t

Z

(a)

(b)

Fig. 3 Stability of the populations around endemic equilibriumE∗

9.2305 > 1 indicating the presence of infection in thesystem.

Now E0 changes its stability when d comes belowof a threshold value d[TC] and becomes stable for

1 2 3 4 5 6 7 8 9x 10−4

0

1

2

3

4

5

6

7

8 x 108

d

Susc

eptib

le P

opul

atio

n (S

)

Unstable Branch

Stable Branch

Bifurcation Threshold

Fig. 4 Trancritical bifurcation around E0 taking d as bifurcationparameter

d > d[TC]. So, the system undergoes a transcriticalbifurcation at d = d[TC] = 0.0004965 around DFE(E0) (see Fig. 4). Besides of the mortality rate, dis-ease transmission rate from symptomatically infectedindividuals to susceptible (β2) and hospitalisation rateof symptomatically infected (ω) also play key roles tocontrol the system dynamics. Figure 5a, b show thesystem possesses transcritical bifurcations around E0

at β2 = β2[TC] = 1.50478 × 10−11 and ω = ω[TC] =0.2018, respectively.

Figure 6 demonstrates the sensibility of some ofthe vital parameters on disease transmission. Figure 6ashows that β2 is most sensitive to control the transmis-sion of the disease than β1, β3 and κ . A small increasein β2 can increase the value of R0 significantly. On theother hand, ω is inversely proportional with R0, i.e. if

123

468 S. Saha et al.

0 1 2 3 4 5 6 7 8

x 10−11

1.2727

1.2727

1.2727

1.2727

1.2727

1.2727x 109

β2

Susc

eptib

le P

opul

atio

n (S

)

Stable Branch

Unstable Branch

Bifurcation threshold

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.51.2727

1.2727

1.2727

1.2727

1.2727

1.2727x 109

ω

Susc

eptib

le P

opul

atio

n (S

)

Bifurcation ThresholdStable Branch

Unstable Branch

(a) (b)

Fig. 5 Trancritical bifurcation around E0 taking a β2 and b ω as bifurcation parameters

more people enter the observation centre or hospital,then the disease fatality starts to decrease with time.

Here, k denotes the information interaction ratewhich brings behavioural changes in susceptible indi-viduals. People becomemore cautiouswhen thediseasestarts to outbreak at higher rate. At early stage, changein information density does not make any significantimpact but later, increasing value of k decreases thenumber of symptomatically infected individuals and itis observed in Fig. 7.

In Fig. 8, the impact of the growth of informationand the ‘level for unresponsiveness towards informa-tion’ on the infected population have been observed.Here, p and q represent the ‘growth rate of informa-tion’ and the ‘level for unresponsiveness towards infor-mation’, respectively. Figure 8a depicts that increase ininformation can lower the infectedpopulationwith timeas people can successfully save themselves from get-ting infected by induced behavioural changes. On theother hand, decreasing the level of people unrespon-siveness towards information (smaller value of q) ulti-mately decreases the symptomatically infected individ-uals with time (see Fig. 8b). It is known that increasinginformation reduce the infected population but if theresponse rate (u1) also starts to increase along withinformation, then the rate of decrease is higher. Fig-ure 9 depicts that symptomatically infected population

1 1.5 2

x 10−11

02468

10

β1

R0

1.6 1.8 2

x 10−11

0.91

1.11.21.31.4

β2

R0

1 1.5 2

x 10−11

02468

10

β3

R0

0.2 0.4 0.6 0.802468

10

κ

R0

R 0 = 1 R 0 = 1

R 0 = 1 R 0 = 1

0.02 0.04 0.06 0.08 0.1 0.12 0.140

2

4

6

8

10

12

14

16

18

ω

R0

R 0 = 1

Fig. 6 Relationship between basic reproduction number R0 withβ1, β2, β3, κ and ω

123


0 20 40 60 80 100 1200

1

2

3

4

5

6

7

8

9 x 108

t

I

k = 0.4

k = 0.04

k = 0.004

Fig. 7 Trajectory profiles of symptomatically infected popula-tion (I ) for different values of k

decrease significantly for increasing value of u1 forincreasing p.

9 Optimal control problem

We formulate the corresponding optimal control prob-lem here to observe how suitable control interven-tions reduce the disease burden on the population. (a)The awareness programs among susceptible and symp-tomatically infected people regarding the informationabout COVID-19 and its symptoms so that the sus-ceptible can take precautions and infected can admitto hospitals without neglecting the symptoms and (b)better treatment policies on hospitalised people havebeen considered as the control policies. We have anal-ysed analytically and also numerically how these con-trol policies make their impact on disease transmissionand try to optimise the cost burden for their implemen-tations.

(i) Increase the awareness among susceptible indi-viduals and symptomatically infected individualsthrough information: Susceptible individuals start tobecome aware of a disease and its preventionwhen theyare provided with enough information and this resultsin behavioural changes in population. The awarenessprograms to spread the information regarding COVID-19 outbreak has been considered as a possible tool toactivate the sensibility of those individuals who livein a susceptible environment. These days Governmentand media sources have spread the news about this dis-ease fatality regularly. And due to the regular broad-cast, the people have started to take protective mea-

sures at a higher rate by maintaining social distancesand proper hygiene, staying at isolation and even adopt-ing the self-quarantine strategy. In system (1), u1 rep-resents the intensity of response through informationwith the restriction 0 ≤ u1 ≤ 1. Here, 0 and 1, respec-tively, denotes no response and the full response of theinformed population. As a consequence, u1(t) changesaccording to the individual’s behavioural response andwe have taken this response intensity as one of the con-trol variables. Incurred cost is involved as a nonlinearfunction of u1 to stimulate the response of individualsand their behavioural changes. If the information startsto spread at a higher rate, then it may help to find theoptimal response of susceptible individuals. Moreover,these awareness programs are conducted to aware notonly the susceptible population but infected people too.It is a person’s responsibility to consult a medical per-son or to admit to a hospital if the slightest symptom isshown in his body.Anyone should not ignore by assum-ing it as a simple cough and cold case. By admittingto the hospital at an early stage can also decrease thedisease burden and so, a saturated hospitalisation rate

functionε1u2 I

1 + ζ1 Iis incorporated in the system where

ε1 denotes the rate at which symptomatic infected indi-viduals move to hospitals without ignoring the symp-toms with intensity u2 and saturation constant is ζ−1

1 .The costs incurred in hospitalisation, medicines, etc.,during the time a person admitted to a hospital or isola-tion centre is taken into consideration. The awarenessintensity u2 is taken as another control variable withrestriction 0 ≤ u2 ≤ 1 where 1 denotes when a personadmits into a hospital without ignoring the symptomswhenever feel sick. And 0 denotes the case when a per-son becomes ignorant about his sickness and does notconsult a doctor.(ii) Better treatment policy for hospitalised individ-ual: Providing proper and better antidote of a virus tothe hospitalised people at an early stage of infection canlower the disease fatality. It affects disease progressiontoo. It is considered that the treatmentwhich is availableand provided to the individuals admitted in the hospi-tals is of limited quantity. The resource availabilitiesdepend on medical diagnosis, financial stability, treat-ment, etc., and all these things are limited. Considering

this fact, a saturated treatment rate functionε2u3H

1 + ζ2His incorporated in the system where ε2 denotes treat-ment rate with intensity u3 and saturation constant is

123

470 S. Saha et al.

0 20 40 60 80 100 1200

1

2

3

4

5

6

7

8

9x 108

t

I

p = 0.001

p = 1

0 20 40 60 80 100 1200

1

2

3

4

5

6

7

8

9x 108

tI

q = 0.01

q = 10

(a) (b)

Fig. 8 Trajectory profiles of symptomatically infected population (I ) for different values of a p and b q

0 0.2 0.4 0.6 0.8 10

5

10

15 x 107

p

I

u 1 = 0.8

u 1 = 0.4

u 1 = 0.1

Fig. 9 Variation of symptomatically infected population (I ) dueto change in growth of information, p for different values of u1

ζ−12 . The cost incurred in vaccination, medicines, diag-nosis, health care, hospitalisation, etc., at the time oftreatment period is taken into consideration. The treat-ment intensity u3 is taken as another control variablewith restriction 0 ≤ u3 ≤ 1 where 0 and 1 denote theno response and full response to the given treatment,respectively.

The main work is to determine optimal responseintensity and optimal treatment with minimum cost bythe help of provided information. So, the region for the

control interventions u1(t), u2(t) and u3(t) is givenas:

� = {(u1(t), u2(t), u3(t)) | (u1(t), u2(t), u3(t))∈ [0, 1] × [0, 1] × [0, 1], t ∈ [0, T f ]

}

,

where T f is the final time up to which the control poli-cies are executed, and also ui (t) for i = 1, 2, 3 aremeasurable and bounded functions.

9.1 Determination of total cost

We determine the incurred cost which needs to be min-imised in order to apply control interventions.(i) Cost involved to spread awareness among suscep-tible and symptomatic infected people:The total costincurred during awareness spreading programs amongpeople is given as:∫ T f

0

[

w1 I (t) + w3u21(t) + w4u

22(t)

]

dt

The cost for spreading awareness among susceptibleregarding the disease, necessary precautions and pre-vention by maintaining social distance and hygiene viasocial campaigns, newspapers, television, social net-works, etc., is denoted by w3u21(t). The term considersthe cost of associated efforts to realise the individual

123


about the importance of maintaining social distancesand it is observed that this cost is high enough. More-over, the cost incurred when symptomatically infectedindividual consults a medical person and admits toa hospital without ignoring the symptoms is repre-sented by the termw1 I (t)+w4u22(t). This cost includesthe expenditure of hospitalisation, medicines, etc., andw4u22(t) considers the productivity loss due to illness.There are some studies already exist which reveal theimpact of the cost associated with awareness programs,screening and self-protective measures and nonlinear-ity up to order two have been taken [2,4,25]. We, inthis work, now emphasise on the fact how the aware-ness programs and practical cautions reduce the diseaseburden at the time of this epidemic outbreak. (ii) Costinvolved during treatment at hospitals: Total costassociated with treatment for symptomatic infectedindividual is given as:

∫ T f

0

[

w2H(t) + w5u23(t)

]

dt,

where w2H(t) and w5u23(t), respectively, denote thecost associated with hospitalised population for losingman power [17,23,25] and the cost at the time of treat-ment due to diagnosis charges, the expenditure of hos-pitalisation, etc. The later term considers the opportu-nity loss including productivity loss due to admittanceto the hospital. So, the nonlinearity of u3(t) is taken upto order two for treatment policy [17,23,25].

The following control problem is considered basedon previous discussions along with the cost functionalJ to be minimised:

J [u1(t), u2(t), u3(t)]=

∫ T f

0

[

w1 I (t) + w2H(t) + w3u21(t)

+w4u22(t) + w5u

23(t)

]

dt (5)

subject to the model system:

dS

dt= � − (β1A + β2 I + β3H)S − dS

+ξ R − u1(t)kSZ ,

dE

dt= (β1A + β2 I + β3H)S − (κ + d)E,

dA

dt= κφE − ηA − (d + δ1)A,

dI

dt= κ(1 − φ)E + ηA − (ω + d + δ2)I

−ε1u2(t)I

1 + ζ1 I,

dH

dt= ωI − γ H − (d + δ3)H + ε1u2(t)I

1 + ζ1 I

−ε2u3(t)H

1 + ζ2H,

dR

dt= γ H − dR − ξ R + u1(t)kSZ

+ε2u3(t)H

1 + ζ2H,

dZ

dt= pI

1 + q I− a0Z , (6)

with initial conditions S(0) > 0, E(0) ≥ 0, A(0) ≥0, I (0) ≥ 0, H(0) ≥ 0, R(0) ≥ 0 and Z(0) ≥ 0.Here, the functional J denotes the total incurred costas stated and the integrand:

L(S, E, A, I, H, R, Z , u1(t), u2(t), u3(t))

= w1 I (t) + w2H(t) + w3u21(t)

+w4u22(t) + w5u

23(t)

denotes the cost at time t . Positive parameters w1, w2,

w3, w4 andw5 are weight constants balancing the unitsof the integrand [17,25]. The optimal control interven-tions u∗

1, u∗2 and u∗

3, exist in �, mainly minimise thecost functional J .

Theorem 9.1 Theoptimal control interventions u∗1, u

∗2

and u∗3 in� of the control system (5)–(6) exist such that

J (u∗1, u

∗2, u

∗3) = min[J (u1, u2, u3)].

Proof Proof is done in “Appendix”. �

Pontryagin’s Maximum Principle helps to obtainoptimal controls u∗

1, u∗2 and u∗

3 of system (5)–(6).

Theorem 9.2 If u∗1, u∗

2 and u∗3 are the optimal control

variables and S∗, E∗, A∗, I ∗, H∗, R∗, Z∗ are corre-sponding optimal state variables of the control sys-tem (5)–(6), then there exist adjoint variables λ =(λ1, λ2, . . . , λ7) ∈ R

7 satisfying the canonical equa-tions:

123

472 S. Saha et al.

dλ1dt

= λ1[β1A + β2 I + β3H + d + u1kZ ]− λ2(β1A + β2 I + β3H) − λ6(u1kZ)

dλ2dt

= λ2(d + κ) − λ3(κφ) − λ4{κ(1 − φ)}dλ3dt

= λ1(β1S) − λ2(β1S)

+ λ3(η + d + δ1) − λ4(η)

dλ4dt

= −w1 + λ1(β2S) − λ2(β2S)

+ λ4

{

ω + d + δ2 + ε1u2(1 + ζ1 I )2

}

− λ5

{

ω + ε1u2(1 + ζ1 I )2

}

− λ7

{p

(1 + q I )2

}

dλ5dt

= −w2 + λ1(β3S) − λ2(β3S)

+ λ5

{

γ + d + δ3 + ε2u3(1 + ζ2H)2

}

− λ6

{

γ + ε2u3(1 + ζ2H)2

}

dλ6dt

= −λ1(ξ) + λ6(d + ξ)

dλ7dt

= λ1(u1kS) − λ6(u1kS) + λ7(a0)

(7)

with transversality conditions λi (T f ) = 0 for i =1, 2, . . . , 7 and corresponding optimal controls u∗

1, u∗2

and u∗3 are given as:

u∗1 = min

{

max

{

0,

(kS∗Z∗2w3

(λ1 − λ6)

)}

, 1

}

,

u∗2 = min

{

max

{

0,

(ε1 I

∗2w4 (1 + ζ1 I∗)

(λ4 − λ5)

)}

, 1

}

,

and u∗3 = min

{

max

{

0,

(ε2H

∗2w5 (1 + ζ2H∗)

(λ5 − λ6)

)}

, 1

}

.

(8)

Proof Proof is given in “Appendix”. �

10 Numerical results with control policies

In system (6), different control strategies have beenapplied to reduce the disease burden and to minimisethe total cost by finding the optimal control paths. Thegrowth of information varies with time as it dependson disease fatality and behavioural response. So, u1

is taken as a control variable. Also, there exist twoother control variables u2 and u3 representing hospital-isation of symptomatically infected without neglectingsymptoms and better treatment of hospitalised people,respectively. The positive weights are taken as w1 =1, w2 = 1, w3 = 2500, w4 = 10 and w5 = 100[17,25]. In order to draw the numerical figures, weslightly adjustβi for i = 1, 2, 3andφ and all the param-eters are listed inTable 2. The effects of implementationof one or all control strategies to find the minimal costhave been analysed one by one here. Correspondingcontrol system in Eqs. (5)–(6) is solved here with theinitial population size: S(0) = 135 × 104, E(0) =500, A(0) = 10, I (0) = 40, H(0) = 15, R(0) = 5and Z(0) = 1. Four different cases are considered: (i)whenonlyu1 is applied, (ii)whenu1 andu2 are applied,(iii) when u1 and u3 are applied and (iv) when all con-trol policies are applied. The numerical simulation forall cases is obtained by MATLAB. The optimal con-trol variables are found by Forward-backward sweepmethod where the optimal state system and the adjointstate system are solved by forward and backward intime, respectively. In the next step, the steepest descentmethod is used to update the optimal controls byHamil-tonian for the optimality of the system [26] and theprocess continues until the convergence. It is assumedthat the control policies are applied for approximatelyT f = 90 days. In India, the first case of COVID-19was registered on 30 January 2020 and still now thenumber of affected cases are increasing day by day.The first lockdown is announced in March and observ-ing the severity, the Government has extended its dura-tion. The people are strictly advised to maintain socialdistancing and proper hygiene in order to keep them-selves safe. As there is no vaccine discovered still now,so, natural immunity and physical distancing are theonly way-outs to avoid from being infected. We havetaken (i) awareness among susceptible individuals (u1),(ii) awareness among infected individuals (u2) and (iii)better treatment (u3) as our control policies and lookingat the current situation, it is observed that people needto maintain these precautions for quite a long time.

First, we consider the case when people only adoptbehavioural changes in a susceptible environment. Fig-ure 10 depicts the population profiles when u1 = u∗

1and u2 = u3 = 0. At T f = 90, the popula-tion become (7,647,536.63, 6.2058, 6.0621, 203.3930,13.79, 561.30, 0.169704). The susceptible populationincreases with time. Both asymptomatically and symp-

123


Table 2 Parametric values used in model system (6)

Parametric values

� 7 × 104 β1 0.5 × 10−10

β2 2 × 10−9 β3 1.2 × 10−10

ξ 3 × 10−10 u1 0.1

k 0.002 κ 0.6

φ 0.3 η 0.15

δ1 10−4 δ2 5 × 10−4

δ3 8 × 10−4 ω 0.02

γ 0.5 d 0.000055

p 0.01 q 1

a0 0.06 ε1 0.1

ε2 0.9 ζ1 0.05

ζ2 0.01 w1 1

w2 1 w3 2500

w4 10 w5 100

0 20 40 60 800

5

10 x 106

t

S

0 20 40 60 800

200400

t

E

0 20 40 60 800

50

100

t

A

0 20 40 60 800

500

t

I

0 20 40 60 8010

20

30

t

H

0 20 40 60 800

500

1000

t

R

0 20 40 60 800

0.5

1

t

Z

Fig. 10 Profiles of populations with applied optimal control u∗1

only and u2 = u3 = 0

tomatically infected population increase steeply within7–10days and reach their maximum values but afterthat, the slope of the trajectories start to decrease.It is noted that the rate of declination is higher forasymptomatically infected people. Also, the numberof asymptomatically infected people is lower than thenumber of symptomatically infected people. Hospi-talised people also decreases with time almost after thefirst 2weeks as recovered people increases. The cor-responding optimal control path is given in Fig. 11.The intensity of the control variable works with higher

0 10 20 30 40 50 60 70 80 900

0.2

0.4

0.6

0.8

1

1.2

1.4 x 10−3

t

u 1 u 1*

Fig. 11 Profiles of populations with applied optimal control u∗1

only and u2 = u3 = 0

0 20 40 60 8005

10 x 106

tS

0 20 40 60 800

200400

t

E

0 20 40 60 800

50100

t

A

0 20 40 60 800

500

t

I0 20 40 60 80

0

2040

t

H

0 20 40 60 800

5001000

tR

0 20 40 60 800

0.51

t

Z

Fig. 12 Profiles of populations with applied optimal controls u∗1

and u∗2 only and u3 = 0

intensity at earlier days but later it decreases with time.The declination of the graph may be caused because ofpeople ignorance, etc.

Next, we consider the case when susceptible peo-ple take precautions to avoid being infected (u1) andalso symptomatically infected individuals enter intohospitals even when they are shown slightest symp-toms (u2). Figure 12 shows the population trajecto-ries when u1 = u∗

1 and u2 = u∗2 but u3 = 0.

At T f = 90, the population becomes (7,647,620.98,3.2498, 3.3487, 106.0726, 8.60, 587.64, 0.169182). It isobserved that the number of symptomatically infectedpeople decreases significantly in this case. Also, the

123

474 S. Saha et al.

0 20 40 60 80 900

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6x 10−3

t

u 1

0 20 40 60 80 900

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

tu 2

u 1* when u3 =0

u 2* when u3 =0

Fig. 13 Optimal controls u∗1 and u∗

2 when u3 = 0

recovered population increases here. Both the infectedpopulation first increase and then decrease after almost1week. Figure 13 shows the optimal control paths ofu1 and u2. From this figure, it is observed that u1 workswith lesser intensity than u2. Moreover, the intensity ofu1 gradually decreases with time. But u2 works withits maximum intensity for quite a long time and thendecreases suddenly.

Now we consider the case when people inducebehavioural changes to protect themselves in suscepti-ble stage (u1) and also hospitalised people are providedwith better treatment (u3). Figure 14 depicts the pop-ulation profiles for u1 = u∗

1 and u3 = u∗3 but u2 = 0.

At T f = 90, the population becomes (7,647,547.32,6.1961, 6.0521, 203.2271, 11.7627, 553.23, 0.169703).Implementation of u3 increases susceptible popula-tion as more people are recovered in this case and somoved to susceptible class further. Corresponding opti-mal control paths are depicted in Fig. 15. The inten-sity of the control variable representing behaviouralresponse decreases continuously with time and theintensity itself is not much higher. On the other hand,the intensity of u3 first increases for almost 3weeksand then decreases with a slower rate for next 2monthsand then suddenly decreases. It represents that the treat-ment works with higher intensity at the earlier state and

0 20 40 60 800

5

10 x 106

t

S

0 20 40 60 800

200400

t

E0 20 40 60 80

0

50

100

t

A

0 20 40 60 800

500

tI

0 20 40 60 8010

15

20

t

H

0 20 40 60 800

500

1000

t

R

0 20 40 60 800

0.5

1

t

Z

Fig. 14 Profiles of populations with applied optimal controls u∗1

and u∗3 only and u2 = 0

with time the intensity decreases when people becomeaware of the disease and its precautions.

It is obvious that implementing all the control poli-cies is beneficial for the proposed system. So, weconsider the combination of all three control poli-cies in the system, i.e. a system where people inducebehavioural changes with time to protect themselvesfrom infection, symptomatically infected people move

123


0 20 40 60 80 900

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6x 10−3

t

u 1

0 20 40 60 80 900

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

tu 3

u 1* when u2 =0

u 3* when u2 =0

Fig. 15 Optimal controls u∗1 and u∗

3 when u2 = 0

to a hospital without neglecting symptoms and bet-ter treatment is applied to hospitalised people. Fig-ure 16 depicts the population trajectories in the pres-ence of all control policies. At T f = 90, popula-tion becomes (7,647,630.40, 3.2391, 3.3381, 105.8244,7.2543, 580.2986, 0.169180). The susceptible popula-tion increases at the highest level in this case.Also, boththe infected population decreases at its lowest popula-tion here. Moreover, the recovered population is higherthan the case when only u1 is applied. Figure 17 showsthe paths of optimal control strategies. The control pol-icy on behavioural response works with higher inten-sity for few days but after that, it starts to decrease. It isjustifiable as people become curious when epidemic orpandemic first outbreaks and they try to adopt changesin behaviour based on the information received but laterit turns into their usual habit. Moreover, the infectedpeople, if take the slightest symptoms seriously andconsult medical person immediately, then it can reducethe disease also. Again, if the hospitalised people areprovided with a proper antidote and better treatment, ithelps to increase the number of recovered people. Here,u2 actswithmaximum intensity for a long time and thenit decreases. u3 also acts with its higher intensity foralmost one and a half month following a declination ata very slow rate.

0 20 40 60 800

5

10 x 106

t

S

0 20 40 60 800

200400

t

E0 20 40 60 80

0

50

100

t

A

0 20 40 60 800

500

tI

0 20 40 60 800

20

40

t

H

0 20 40 60 800

500

1000

t

R

0 20 40 60 800

0.5

1

t

Z

Fig. 16 Profiles of populations with both optimal control poli-cies u∗

1, u∗2 and u∗

3

In Fig. 18, cost design analysis has been performedin the absence and presence of u2 and u3 but we haveconsidered u1 in both the cases. There is one casewhere all three control policies are applied and the nextcase is when only the behavioural response is consid-ered. Optimal cost profiles are shown in Fig. 18a forthe cases and trajectory profiles for symptomaticallyinfected individuals are depicted in Fig. 18b. In the

123

476 S. Saha et al.

0 10 20 30 40 50 60 70 80 900

0.2

0.4

0.6

0.8

1

1.2

1.4x 10−3

t

u 1* u1*

0 10 20 30 40 50 60 70 80 900

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

t

u 2* , u3*

u3*

u2*

Fig. 17 Profiles of optimal controls u∗1, u∗

2 and u∗3

0 10 20 30 40 50 60 70 80 9050

100

150

200

250

300

350

400

450

500

t

J

Optimal Cost for u1* , u2

* , u3*

Productivity Loss for u1=u1* , u2, u3 = 0

0 10 20 30 40 50 60 70 80 900

50

100

150

200

250

300

350

400

450

t

I

I when u1=u1* , u2=u2

* , u3=u3*

I when u1 = u1* , u2=0, u3=0

(a) (b)

Fig. 18 a Cost distribution in presence and absence of control policies. b Profiles of symptomatic infected population under differentcontrol policies

absence of u2 and u3, cost occurs due to productivityloss by infected only. So, the opportunity loss is higherdue to an epidemic outbreak and overall infected pop-ulation increases in this case. On the other hand, theoptimal cost is lower when all control interventions areapplied. And as the infected population is lower in thiscase, it reduces the cost incurred because of opportunityloss.

10.1 Effect of hospitalisation rate and saturationconstant on optimal control policies

If more symptomatically infected individuals admitinto a hospital, then we observe how the hospitalisa-tion rate and saturation constant have effects on thedisease dynamics in the presence of the control policieswith optimal intensities. Saturation rate (ζ−1

1 ) and thehospitalisation rate (ε1) have been varied in the follow-

123


0 10 20 30 40 50 60 70 80 900

50

100

150

200

250

300

350

400

450

t

I ζ1 = 0.05

ζ1 = 0.5

0 10 20 30 40 50 60 70 80 9050

100

150

200

250

300

350

400

450

500

t

J

ζ1 = 0.5

ζ1 = 0.05

(a) (b)

Fig. 19 a Profiles of symptomatic infective population for various ζ1 with u∗1, u∗

2 and u∗3. b Profiles of cost for various ζ1 with u∗

1, u∗2

and u∗3

ing figures. Figure 19 shows the graphs of symptomaticinfective population (I ) and corresponding cost for dif-ferent values of ζ1. For smaller saturation rate (largervalues of ζ1), infective individuals and associated costincrease due to increase in productivity loss. Corre-sponding optimal control paths are drawn in Fig. 20from which it is observed that for increasing value ofζ1, the control policy denoting behavioural responsein susceptible individuals acts with a higher intensitywhich means smaller saturation rate increases the timeperiod during which the control policy is implementedwith higher intensity successfully. On the other hand,decreasing value of ζ1 increases the intensity level ofthe other two control interventions. Thus, if the satu-ration constant is high enough, then it is economicallyviable and hence, requires comparatively lesser effortswhile implementing the control policies.

Further with the increase in treatment rate (ε1) from0.01 to 0.1, the both symptomatic infective populationand associated cost decrease with time (Fig. 21). FromFig. 22, it is observed that for increasing value of ε1,the control policy denotingbehavioural response in sus-ceptible people works for a smaller time period. Also,increasing value of ε1 increases the intensity level of theother two control variables. In Fig. 22b, c it is observedthat the higher hospitalisation rate increases the lengthsof maximum intensities of these two optimal controlimplementation periods. Itmeans lesser efforts on these

two applied controls are sufficient to reduce the overallinfective population.

11 Conclusion

Coronavirus or Covid-19 first appeared in China inDecember 2019, but today it has spread all over theworld in the form of a pandemic. Reports of the cur-rent situation reveal that almost 30 lakhs people areinfected with the virus worldwide. Though the Gov-ernments and medical persons of each and every coun-try are trying to provide protective measures to peo-ple, the infection rate is still high enough as the properantidote for this virus is still unknown. According tothe data till 26th April 2020, provided by the dash-board of CSSE at John Hopkins University, US hasthe highest confirmed cases with the number 939, 249[6]. Though according to the official reports, almost26, 384 people have died in Italy which is the highestin number among 185 countries or religions. If we con-sider the current situation in India, then from the reportsof NIC, India, there are 20, 177 confirmed cases, 826death cases and 5, 914 recovered cases are reportedtill 26th April [29]. Keeping this pandemic situation inmind, in this work, we have formulated a compartmen-tal SEIRS model of Covid-19 where a separate equa-tion is incorporated to reflect the information which

123

478 S. Saha et al.

0 10 20 30 40 50 60 70 80 900

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6x 10−3

t

u* 1

ζ1 = 0.05

ζ1 = 0.5

0 10 20 30 40 50 60 70 80 900

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

t

u* 2

ζ1 = 0.05

ζ1 = 0.5

0 10 20 30 40 50 60 70 80 900

0.05

0.1

0.15

0.2

0.25

t

u* 3

ζ1 = 0.05

ζ1 = 0.5

(a)

(c)

(b)

Fig. 20 a Plots of control u∗1 for various ζ1. b Plots of control u∗

2 for various ζ1. c Plots of control u∗3 for various ζ1

induces behavioural changes. The growth rate of infor-mation is based on symptomatically infected individu-als, awareness programs, social activities, etc. Positiv-ity andboundedness of systemvariables guarantees thatthe proposed system is well-defined. Feasibility condi-tions of equilibrium points show that DFE exists for allparametric values where the unique endemic equilib-rium point exists only when basic reproduction numberR0 exceeds unity. Both local and global stability condi-tions have been derived in Sect. 6. As we have only oneendemic equilibrium point for R0 > 1 and there doesnot exist any endemic point for R0 < 1, hence, it isconcluded that the system undergoes a forward (tran-

scritical) bifurcation around the disease-free equilib-rium. If the people start to take the information regard-ing the virus seriously and take its precautions, thenit is obvious that overall infected population is lesserand it is shown in Fig. 7. Moreover, increasing val-ues of the growth of information and decreasing ‘levelfor unresponsiveness towards information’ also helpto reduce the infected population. Also, symptomat-ically infected population with respect to the growthrate of information decreases for increasing value ofu1, i.e. if people respond to the growing informationregarding the awareness at a higher rate, then infectedpopulation decreases significantly. As the behavioural

123


0 10 20 30 40 50 60 70 80 900

50

100

150

200

250

300

350

400

450

t

I

ε1 = 0.01

ε1 = 0.1

0 10 20 30 40 50 60 70 80 9050

100

150

200

250

300

350

400

450

500

tJ

ε1 = 0.01

ε1 = 0.1

(a) (b)

Fig. 21 a Profiles of infective population for various ε1 with u∗1, u∗

2 and u∗3. b Profiles of cost for various ε1 with u∗

1, u∗2 and u∗

3

changes include keeping social distances, maintainingproper hygiene, staying at isolation and adopting theself-quarantinemethod, so, these precautions can reallybe useful to prevent the disease transmission at a higherrate.

In the later part, a corresponding optimal con-trol problem is considered. Implementation of controlinterventions helps to reduce the disease burden. Thebehavioural changes in susceptible population changeswith time and so, it is considered as one of the controlpolicy. Further, the symptomatically infected peoplecan also become cautious by the current disease fatal-ity and may consult doctors or admit to hospitals ifslightest symptoms are shown. Again, during the treat-ment period, better and proper medicines or diagno-siscan be provided to a hospitalised person. So, all these

things can be considered as control strategies. Numeri-cal figures show that the control presenting behaviouralresponse (u1)works with higher intensity immediatelyafter implementation but gradually it decreases withtime. On the other hand, the control policy denotinghospitalisation of infected individuals (u2) works withits maximum intensity for quite a long time period andthen it decreases. Again, the control presenting bet-ter treatment of hospitalised people works with higherintensity for almost 2months following a declinationat a later stage, though this intensity is lower than theintensity of u2. Implementation of a single strategy isuseful but all the three control policies together canreduce the infected population and increase the recov-ered population at a higher rate. Hence, applying allthe control policies together may help to reduce dis-ease transmission at this current epidemic situation.

123

480 S. Saha et al.

0 10 20 30 40 50 60 70 80 900

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6x 10−3

t

u* 1

ε1 = 0.01

ε1 = 0.1

0 10 20 30 40 50 60 70 80 900

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

t

u* 2

ε1 = 0.1

ε1 = 0.01

0 10 20 30 40 50 60 70 80 900

0.05

0.1

0.15

0.2

0.25

t

u* 3

ε1 = 0.01

ε1 = 0.1

(a) (b)

(c)

Fig. 22 a Plots of control u∗1 for various ε1. b Plots of control u∗

2 for various ε1. c Plots of control u∗3 for various ε1

Acknowledgements The authors are grateful to the anony-mous referees, Dr. Jun Ma, Associate Editor, for their carefulreading, valuable comments and helpful suggestions, which havehelped them to improve the presentation of this work signifi-cantly. The first author (Sangeeta Saha) is thankful to the Univer-sity Grants Commission, India for providing SRF. The researchof J.J. Nieto has been partially supported by the Agencia Estatalde Investigacion (AEI) of Spain, cofinanced by the EuropeanFund for Regional Development (FEDER) corresponding to the

2014–2020 multiyear financial framework, Project MTM2016-75140-P; and byXunta deGalicia underGrant ED431C 2019/02.

Compliance with ethical standards

Conflict of interest The authors declare that they have no con-flict of interest.

123


Appendix

11.1 Local stability of the endemic equilibrium point

Proof of Theorem 6.3 The Jacobian matrix at endemicequilibrium point E∗ is given as

J =

⎛

⎜⎜⎜⎜⎜⎜⎜⎜⎝

a11 0 a13 a14 a15 a16 a17a21 a22 a23 a24 a25 0 00 a32 a33 0 0 0 00 a42 a43 a44 0 0 00 0 0 a54 a55 0 0a61 0 0 0 a65 a66 a670 0 0 a74 0 0 a77

⎞

⎟⎟⎟⎟⎟⎟⎟⎟⎠

where a11 = −(β1A∗ + β2 I ∗ + β3H∗) − d −u1kZ∗, a13 = −β1S∗, a14 = −β2S∗, a15 =−β3S∗, a16 = ξ, a17 = u1kS∗, a21 = β1A∗ +β2 I ∗ + β3H∗, a22 = −α0, a23 = β1S∗, a24 =β2S∗, a25 = β3S∗, a32 = κφ, a33 = −α1, a42 =κ(1 − φ), a43 = η, a44 = −α2, a54 = ω, a55 =−α3, a61 = u1kZ∗, a65 = γ, a66 = −(d+ξ), a67 =u1kS∗, a74 = p

(1+q I ∗)2 and a77 = −a0.

Characteristic equation of J |E∗ is λ7+ R1λ6+ R2λ

5+R3λ

4 + R4λ3 + R5λ

2 + R6λ + R7 = 0,

R1 = −(a11 + a22 + a33 + a44 + a55 + a66 + a77),

R2 = a11(a22 + a33 + a44 + a55 + a66 + a77)

+ a22(a33 + a44 + a55 + a66 + a77)

+ a33(a44 + a55 + a66 + a77)

+ a44(a55 + a66 + a77) + a55(a66 + a77) + a66a77

− a16a61 − a23a32 − a24a42

R3 = −[a11a22(a33 + a44 + a55 + a66 + a77)

− a11a23a32 − a11a24a42 + a11a55(a66 + a77)

+ a11a33(a44 + a55 + a66 + a77)

+ a11a44(a55 + a66 + a77)

+ a11a66a77 + a13a21a32 + a14a42a21

− a16a61(a22 + a33 + a44 + a55 + a77)

+ a22a33(a44 + a55 + a66 + a77)

+ a22a44(a55 + a66 + a77)

+ a22a55(a66 + a77) + a22a66a77

− a23a32(a44 + a55 + a66 + a77) + a43a32a24

− a24a42(a33 + a55 + a66 + a77) + a54a42a25

+ a33a44(a55 + a66 + a77) + a33a55(a66 + a77)

+ a33a66a77 + a44a55(a66 + a77)

+ a66a77(a44 + a55)]

R4 = −[−a11a22a33(a44 + a55 + a66 + a77)

− a11a22a44(a55 + a66 + a77)

− a11a22a55(a66 + a77) − a11a22a66a77

+ a11a23a32(a44 + a55 + a66 + a77)

− a11a24a32a43 + a11a24a42

(a33 + a55 + a66 + a77) − a11a25a42a54

− a11a33a44(a55 + a66 + a77)

− a11a33a55(a66 + a77) − a11a33a66a77

− a11a44a55(a66 + a77)

− a11a66a77(a44 + a55)

− a13a32a21(a44 + a55 + a66 + a77)

+ a14a43a32a21

− a14a42a21(a33 + a55 + a66 + a77)

+ a15a54a42a21

+ a16a22a61(a33 + a44 + a55 + a77)

− a16a61(a23a32 + a24a42)

+ a16a33a61(a44 + a55 + a77)

+ a16a44a61(a55 + a77)

+ a16a55a61a77

+ a74a42a21a17

− a22a33a44(a55 + a66 + a77)

− a22a33a55(a66 + a77) − a22a33a66a77

− a22a44a55(a66 + a77)

− a22a66a77(a44 + a55)

+ a23a32a44(a55 + a66 + a77)

+ a23a32a55(a66 + a77) + a22a32a66a77

− a24a32a43(a55 + a66 + a77)

+ a24a33a42(a55 + a66 + a77)

+ a24a42a55(a66 + a77) + a24a42a66a77

+ a25a32a43a54 − a24a42a54(a33 + a66 + a77)

− a33a44a55(a66 + a77) − a33a66

a77(a44 + a55) − a44a55a66a77]

R5 = −[a11a22a33a44(a55 + a66 + a77)

+ a11a22a33a55(a66 + a77)

+ a11a32a33a66a77

123

482 S. Saha et al.

+ a11a22a44a55(a66 + a77)

+ a11a22a66a77(a44 + a55)

− a11a23a32a44(a55 + a66 + a77)

− a11a23a32a55(a66 + a77) − a11a23a32a66a77

+ a11a24a32a43(a55 + a66 + a77)

− a11a24a33a42(a55 + a66 + a77)

− a11a24a42a55(a66 + a77)

− a11a24a42a66a77 − a11a25a32

a43a54 + a11a25a33a42a54

+ a11a25a42a54(a66 + a77)

+ a11a33a44a55(a66 + a77)

+ a11a33a66a77(a44 + a55)

+ a11a44a55a66a77

+ a13a21a32a44(a55 + a66 + a77)

+ a13a21a32a55(a66 + a77)

+ a13a21a32a66a77

− a14a21a32a43(a55 + a66 + a77)

+ a14a21a33a42(a55 + a66 + a77)

+ a44a21a42a55(a66 + a77)

+ a14a21a42a66a77

+ a15a21a32a43a54

− a15a21a42a54(a33 + a66 + a77)

+ a65a54a42a21a16 + a16a67a74a42a21

− a16a22a33a61(a44 + a55 + a77)

− a16a22a44a61(a55 + a77)

− a16a22a55a61a77

+ a16a23a32a61(a44 + a55 + a77)

− a16a24a32a43a61

+ a16a24a42a61(a33 + a55 + a77)

− a16a25a42a54a61

− a16a33a44a61(a55 + a77)

− a16a33a55a61a77

− a16a44a55a61a77 + a17a21a32a43a74

− a74a42a21a17(a33a55 + a66)

+ a22a33a44a55(a66 + a77)

+ a22a33(a44 + a55)a66a77

+ a22a44a55a66a77

− a23a32a44a55(a66 + a77)

− a23a32(a44 + a55)a66a77

+ a24a32a43a55(a66 + a77)

+ a24a32a43a66a77

− a24a33a42a55(a66 + a77)

− a24a42a66a77(a33 + a55)

− a25a32a43a54(a66 + a77)

+ a25a33a42a54(a66 + a77)

+ a25a42a54a66a77 + a33a44a55a66a77]

R6 = −[−a11a22a33a44a55(a66 + a77)

− a11a22a33(a44 + a55)a66a77

− a11a22a44a55a66a77

+ a11a23a32a44a55(a66 + a77)

+ a11a23a32(a44 + a55)a66

− a11a24a32a43a55(a66 + a77)

− a11a24a32a43a66a77

+ a11a24a33a42a55(a66 + a77)

+ a11a24a42(a55 + a33)a66a77

+ a11a25a32a43a54(a66 + a77)

− a11a25a33a42a54(a66 + a77)

− a11a25a42a54a66a77

− a11a33a44a55a66a77

− a13a21a32a44a55(a66 + a77)

− a13a21a32(a44 + a55)a66a77

+ a14a21a32a43a55(a66 + a77)

+ a14a21a32a43a66a77

− a14a21a33a42a55(a66 + a77)

− a14a21(a33 + a55)a42a66a77

− a15a21a32a43a54(a66 + a77)

+ a15a21a33a42a54(a66 + a77)

+ a15a21a42a54a66a77

+ a16a21a32a43(a54a65 + a67a74)

− a16a21a33a42(a54a65 + a67a74)

− a16a21a42a54a65a77

− a16a21a42a55a67a74

+ a16a22a33a44a61(a55 + a77)

+ a16a22a77a55a61(a33 + a44)

− a16a23a32a44a61(a55 + a77)

− a16a23a32a55a61a77

+ a16a24a32a43a61(a55 + a77)

123


− a16a24a33a42a61(a55 + a77)

+ a16a24a42a55a61a77

− a16a25a32a43a54a61

+ a16a25a42a54a61(a33 + a77)

+ a16a33a44a55a61a77

− a17a21a32a43a74(a55 + a66)

+ a17a21a33a42a74(a55 + a66)

+ a17a21a42a55a66a74

− a22a33a44a55a66a77

+ a23a32a44a55a66a77

− a24a43a55a66a77

+ a24a33a42a55a66a77

+ a25a32a43a54a66a77

− a25a33a42a54a66a77]

R7 = a11a23a32a44a55a66a77

− a11a24a32a43a55a66a77

+ a11a24a33a42a55a66a77

+ a11a25a32a43a54a66a77

− a11a25a33a42a54a66a77

− a13a21a32a44a55a66a77

+ a14a21a32a43a55a66a77

− a14a21a33a42a55a66a77

− a15a21a32a43a54a66a77

+ a15a21a33a42a54a66a77

+ a16a21a32a43a54a65a77

+ a16a21a32a43a55a67a74

− a16a21a33a42a54a65a77

− a16a21a33a42a55a67a74

+ a16a22a33a44a55a61a77

− a16a23a32a44a55a61a77

+ a16a24(a32a43 − a33a42)a55a61a77

+ a16a25(a33a42 − a32a43)a54a61a77

+ a17a21(a33a42 − a32a43)a55a66a74

− a11a22a33a44a55a66a77

Let us consider

�1 = R1, �2 =∣∣∣∣

R1 1R3 R2

∣∣∣∣, �3 =

∣∣∣∣∣∣

R1 1 0R3 R2 R1

R5 R4 R3

∣∣∣∣∣∣

,

�4 =

∣∣∣∣∣∣∣∣

R1 1 0 0R3 R2 R1 1R5 R4 R3 R2

R7 R6 R5 R4

∣∣∣∣∣∣∣∣

,

�5 =

∣∣∣∣∣∣∣∣∣∣

R1 1 0 0 0R3 R2 R1 1 0R5 R4 R3 R2 R1

R7 R6 R5 R4 R3

0 0 R7 R6 R5

∣∣∣∣∣∣∣∣∣∣

,

�6 =

∣∣∣∣∣∣∣∣∣∣∣∣

R1 1 0 0 0 0R3 R2 R1 1 0 0R5 R4 R3 R2 R1 1R7 R6 R5 R4 R3 R2

0 0 R7 R6 R5 R4

0 0 0 0 R7 R6

∣∣∣∣∣∣∣∣∣∣∣∣

,

�7 = Det (J |E∗).

By Routh–Hurwitz criterion, E∗ is locally asymp-tomatically stable (LAS) if and only if �i > 0 fori = 1, 2, 3, 4, 5, 6, 7, i.e. equivalently

(i) Ri > 0 for i = 1, 7;(ii) �i > 0 for i = 2, 3, 4, 5, 6.

�

11.2 Existence of optimal control functions

Now we derive the conditions for existence of opti-mal control interventions which also minimise the costfunction J in a finite time period.

Proof of Theorem 9.1 The optimal control variables,when exist, satisfy the following conditions:

(i) Solutions of system (6) with control variablesu1, u2 and u3 in � �= φ.

(ii) The mentioned set � is closed, convex and thestate system is represented with linear functionof control variables where coefficients depend ontime and also on state variables.

(iii) Integrand of (5): L is convex on � and L(S, E,

A, I, H, R, u1, u2, u3) ≥ f (u1, u2, u3) wheref (u1, u2, u3) is continuous and ||(u1, u2, u3)||−1

f (u1, u2, u3) → ∞when ||(u1, u2, u3)|| → ∞;||.|| represents the L3(0, T f ) norm.

123

484 S. Saha et al.

From (6), the total population N = S + E + A + I +H + R.

So,dN

dt= � − dN − δ1A − δ2 I − δ3H ≤ � − dN

⇒ 0 < N (t) ≤ N (0)e−dt + �

d

(

1 − e−dt)

,

where N (0) = S(0) + E(0) + A(0) + I (0) + H(0) +R(0).

As t → ∞, 0 < N (t) ≤ �

d.

Also,dZ

dt= pI

1 + q I− a0Z ≤ pI − a0Z

⇒ dZ

dt+ a0Z ≤ p�

d(for large time t)

⇒0 < Z(t) ≤ Z(0)e−a0t + p�

da0

(

1 − e−a0t)

As t → ∞, 0 < Z(t) ≤ p�

a0d.

For each of the control variable in �, solution of (6)is bounded and right-hand side functions are locallyLipschitzian too. Picard − Lindelo f theorem showsthat condition (i) is satisfied [10].

The control set� is closed and convex by definition.Again all the equations of system (6) are written as lin-ear equations in u1, u2 and u3 where state variablesdepend on coefficients and hence condition (ii) is satis-fied also. Moreover, the quadratic nature of all controlvariables guarantee the convex property of integrandL(S, E, A, I, H, R, Z , u1, u2, u3).

Also, L(S, E, A, I, H, R, Z , u1, u2, u3)

= w1 I + w2H + w3u21 + w4u

22 + w5u

23

≥ w3u21 + w4u

22 + w5u

23

Let, u = min(w3, w4, w5) > 0 and f (u1, u2, u3) =u(u21 + u22 + u23).Then L(S, E, A, I, H, R, Z , u1, u2, u3)≥ f (u1, u2, u3).

Here, f is continuous and ||(u1, u2, u3)||−1 f (u1, u2,u3) → ∞ whenever ||(u1, u2, u3)|| → ∞. Hence,condition (iii) is also satisfied. So, it is concluded thatthere exist control variables u∗

1, u∗2 and u∗

3 with thecondition J [u∗

1, u∗2, u

∗3] = min[J [u1, u2, u3]] [16,17].

�

11.3 Characterisation of optimal control functions

By Pontryagin’s Maximum Principle, we have derivedhere the necessary conditions for optimal control func-

tions for system (5)–(6) [16,31]. Let us define theHamiltonian function as:

H (S, E, A, I, H, R, Z , u1, u2, u3, λ)

= L(S, E, A, I, H, R, Z , u1, u2, u3)

+ λ1dS

dt+ λ2

dE

dt+ λ3

dA

dt

+ λ4dI

dt+ λ5

dH

dt+ λ6

dR

dt+ λ7

dZ

dt

So, H = w1 I + w2H + w3u21 + w4u

22 + w5u

23

+ λ1[� − (β1A + β2 I + β3H)S

− dS + ξ R − u1(t)kSZ ]+ λ2[(β1A + β2 I + β3H)S − (κ + d)E]+ λ3[κφE − ηA − (d + δ1)A]+ λ4 [κ(1 − φ)E + ηA − (ω + d + δ2)I

−ε1u2(t)I

1 + ζ1 I

]

+ λ5

[

ωI − γ H − (d + δ3)H + ε1u2(t)I

1 + ζ1 I

−ε2u3(t)H

1 + ζ2H

]

+ λ6[

γ H − dR − ξ R + u1(t)kSZ

+ε2u3(t)H

1 + ζ2H

]

+ λ7

[pI

1 + q I− a0Z

]

(9)

Here, λ = (λ1, λ2, λ3, λ4, λ5, λ6, λ7) are the adjointvariables. We get minimised Hamiltonian by Pontrya-gin’s Maximum Principle to minimise the cost func-tional. Pontryagin’s Maximum Principle mainly adjointhe cost functional with the state equations by intro-ducing adjoint variables.

Proof of Theorem 9.2 Let u∗1, u∗

2 and u∗3 be optimal

control variables and S∗, E∗, A∗, I ∗, H∗, R∗, Z∗ arecorresponding optimal state variables of the controlsystem (6) which minimise the cost functional (5). So,byPontryagin’sMaximumPrinciple, there exist adjointvariables λ1, λ2, λ3, λ4, λ5, λ6, λ7 which satisfy thefollowing canonical equations:

123


dλ1dt

= −∂H

∂S,

dλ2dt

= −∂H

∂E,

dλ3dt

= −∂H

∂A,

dλ4dt

= −∂H

∂ I,

dλ5dt

= −∂H

∂H,

dλ6dt

= −∂H

∂R,

dλ7dt

= −∂H

∂Z.

So, we have

dλ1dt

= λ1[β1A + β2 I + β3H + d + u1kZ ]− λ2(β1A + β2 I + β3H) − λ6(u1kZ)

dλ2dt

= λ2(d + κ) − λ3(κφ)

− λ4{κ(1 − φ)}dλ3dt

= λ1(β1S) − λ2(β1S)

+ λ3(η + d + δ1) − λ4(η)

dλ4dt

= −w1 + λ1(β2S) − λ2(β2S)

+ λ4

{

ω + d + δ2 + ε1u2(1 + ζ1 I )2

}

− λ5

{

ω + ε1u2(1 + ζ1 I )2

}

− λ7

{p

(1 + q I )2

}

dλ5dt

= −w2 + λ1(β3S) − λ2(β3S)

+ λ5

{

γ + d + δ3 + ε2u3(1 + ζ2H)2

}

− λ6

{

γ + ε2u3(1 + ζ2H)2

}

dλ6dt

= −λ1(ξ) + λ6(d + ξ)

dλ7dt

= λ1(u1kS) − λ6(u1kS) + λ7(a0)

(10)

with the transversality conditions λi (T f ) = 0, for i =1, 2, 3, 4, 5, 6, 7.

From optimality conditions :∂H∂u1

∣∣∣∣u1=u∗

1

= 0, ∂H∂u2

∣∣∣∣u2=u∗

2

= 0 and ∂H∂u3

∣∣∣∣u3=u∗

3

= 0.

So,u∗1 = kS∗Z∗

2w3(λ1 − λ6),u∗

2 = ε1 I ∗2w4(1+ζ1 I ∗) (λ4 − λ5)

and u∗3 = ε2H∗

2w5 (1 + ζ2H∗)(λ5 − λ6) .Now from these

findings along with the characteristics of control set�,

we have

u∗1 =

⎧

⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

0, if kS∗Z∗2w3

(λ1 − λ6) < 0

kS∗Z∗2w3

(λ1 − λ6) , if 0 ≤ kS∗Z∗2w3

(λ1 − λ6) ≤ 1

1, if kS∗Z∗2w3

(λ1 − λ6) > 1

u∗2 =

⎧

⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

0, if ε1 I∗

2w4(1+ζ1 I∗)(λ4 − λ5) < 0

ε1 I∗

2w4(1+ζ1 I∗)(λ4 − λ5) , if 0 ≤ ε1 I

∗2w4(1+ζ1 I∗)

(λ4 − λ5) ≤ 1

1, if ε1 I∗

2w4(1+ζ1 I∗)(λ4 − λ5) > 1

u∗3 =

⎧

⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

0, if ε2H∗

2w5(1+ζ2H∗)(λ5 − λ6) < 0

ε2H∗

2w5(1+ζ2H∗)(λ5 − λ6) , if 0 ≤ ε2H

∗2w5(1+ζ2H∗)

(λ5 − λ6) ≤ 1

1, if ε2H∗

2w5(1+ζ2H∗)(λ5 − λ6) > 1

which is equivalent as (8). �

11.4 Optimal system

We state the optimal system with optimal control vari-ables u∗

1, u∗2 and u∗

3 below. The optimal system withminimisedHamiltonianH

∗at (S∗, E∗, A∗, I ∗, H∗, R∗,

Z∗, λ1, λ2, λ3, λ4, λ5, λ6, λ7) is as follows:

dS∗

dt= � − (β1A

∗ + β2 I∗ + β3H

∗)S∗

− dS∗ + ξ R∗ − u∗1kS

∗Z∗,dE∗

dt= (β1A

∗ + β2 I∗ + β3H

∗)S∗ − (κ + d)E∗,

dA∗

dt= κφE∗ − ηA∗ − (d + δ1)A

∗,

dI

dt= κ(1 − φ)E∗ + ηA∗ − (ω + d + δ2)I

∗

− ε1u∗2 I

∗

1 + ζ1 I ∗ ,

dH∗

dt= ωI ∗ − γ H∗ − (d + δ3)H

∗

+ ε1u∗2 I

∗

1 + ζ1 I ∗ − ε2u∗3H

∗

1 + ζ2H∗ ,

dR∗

dt= γ H∗ − dR∗ − ξ R∗ + u∗

1kS∗Z∗

+ ε2u∗3H

∗

1 + ζ2H∗ ,

123

486 S. Saha et al.

dZ∗

dt= pI ∗

1 + q I ∗ − a0Z∗, (11)

with initial conditions: S∗(0) > 0, E∗(0) ≥ 0, A∗(0) ≥0, I ∗(0) ≥ 0, H∗ ≥ 0, R∗(0) ≥ 0 and Z∗ ≥ 0. Thecorresponding adjoint system is given as:

dλ1dt

= λ1[β1A∗ + β2 I

∗ + β3H∗ + d + u∗

1kZ∗]

− λ2(β1A∗ + β2 I

∗ + β3H∗)

− λ6(u∗1kZ

∗)dλ2dt

= λ2(d + κ) − λ3(κφ)

− λ4{κ(1 − φ)}dλ3dt

= λ1(β1S∗) − λ2(β1S

∗)

+ λ3(η + d + δ1) − λ4(η)

dλ4dt

= −w1 + λ1(β2S∗) − λ2(β2S

∗)

+ λ4

{

ω + d + δ2 + ε1u∗2

(1 + ζ1 I ∗)2

}

− λ5

{

ω + ε1u∗2

(1 + ζ1 I ∗)2

}

− λ7

{p

(1 + q I ∗)2

}

dλ5dt

= −w2 + λ1(β3S∗) − λ2(β3S

∗)

+ λ5

{

γ + d + δ3 + ε2u∗3

(1 + ζ2H∗)2

}

− λ6

{

γ + ε2u∗3

(1 + ζ2H∗)2

}

dλ6dt

= −λ1(ξ) + λ6(d + ξ)

dλ7dt

= λ1(u∗1kS

∗)

− λ6(u∗1kS

∗) + λ7(a0), (12)

with transversality conditions λi (T f ) = 0, for i =1, 2, . . . , 7 and u∗

1, u∗2 and u∗

3 are same as (8).

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epidemic model of covid-19 outbreak by inducing

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