Studying the effect of lockdown using epidemiological modelling of COVID-19 and aquantum computational approach using the Ising spin interaction

Anshuman Padhi @,1, ∗ Sudev Pradhan @,2, † Pragna Paramita Sahu @,3, ‡

Kalyani S @,4, § Bikash K. Behera,5, 6, ¶ and Prasanta K. Panigrahi6, ∗∗

1School of Physical Sciences,National Institute of Science Education and Research, HBNI, Jatni 752050, Odisha, India

2Physics Department,Indian Institute of Science Education and Research, Berhampur, 760010, Odisha, India

3Maths Department,Indian Institute of Science Education and Research, Berhampur, 760010, Odisha, India

4University of Mysore, Krishnaraja Boulevard Road,K.G Koppal, Mysuru, 570006, Karnataka, India

5Bikash’s Quantum (OPC) Pvt. Ltd., Balindi, Mohanpur 741246, West Bengal, India6Department of Physical Sciences,

Indian Institute of Science Education and Research Kolkata, Mohanpur 741246, West Bengal, India

COVID-19 is a respiratory tract infection that can range from being mild to fatal. In India, thecountrywide lockdown has been imposed since 24th march, 2020, and has got multiple extensionswith different guidelines for each phase. Among various models of epidemiology, we use the SIR(D)model to analyze the extent to which this multi-phased lockdown has been active in ‘flatteningthe curve’ and lower the threat. Analyzing the effect of lockdown on the infection may give us abetter insight into the evolution of epidemic while implementing the quarantine procedures as wellas improving the healthcare facilities. For accurate modelling, incorporating various parametersalong with sophisticated computational facilities, are required. Parallel to SIRD modelling, wetend to compare it with the Ising model and derive a quantum circuit that incorporates the rate ofinfection and rate of recovery, etc as its parameters. The probabilistic plots obtained from the circuitqualitatively resemble the shape of the curve for the spread of Coronavirus. We also demonstratehow the curve flattens when the lockdown is imposed. This kind of quantum computational approachcan be useful in reducing space and time complexities of a huge amount of information related tothe epidemic.

Keywords: SIRD model of epidemiology, Ising model, Quantum circuits, IBM Quantum Experience

I. INTRODUCTION

COVID-19 (Coronavirus Disease 2019) is a diseasecaused by the virus strain known as SARS-CoV-2 (Se-vere Acute Respiratory Syndrome Coronavirus II). It haswidespread implications on the human body in the formof respiratory issues, septic shock, co-morbidity arisingfrom multiple-organ failure, and even death [1]. Routingback to its emergence in mainland China around the endof 2019, till May 2020, it has spread to over 210 countriesresulting in a total of around 40 lakh cases with almost 3lakh deaths due to the same. The World Health Organi-sation (WHO) declared it as a global pandemic on 11thMarch 2020, observing the rate it transmits. Variouscountries, including India, put forward extensive mea-

∗ anshuman.padhi@niser.ac.in; @ Those are the first authors andhave equally contributed to this work† su18@iiserbpr.ac.in; @ Those are the first authors and have

equally contributed to this work‡ pragna18@iiserbpr.ac.in; @ Those are the first authors and have

equally contributed to this work§ kalyanisuresh2608@gmail.com; @ Those are the first authors and

have equally contributed to this work¶ bikash@bikashsquantum.com∗∗ pprasanta@iiserkol.ac.in

sures to curb the viral epidemic, by extensive tracing,testing and isolating the suspected ones while improvinghealthcare systems and imposing lockdowns. Govt. ofIndia declared the countrywide lockdown on 24th March2020 to reduce the virus’s rate of transmission. To tacklethis global pandemic, the extent of spread and the timetaken by the epidemic to reach its peak and other de-tails must be well predicted so that the state can planaccordingly and fight against it.

Mathematical modeling can come handy in these pro-cesses, as they can predict how the epidemics evolve whileanalyzing the current set of available data. Any such pre-diction system requires a set of assumptions and consider-ations, which helps to formulate the necessary equationsthat can later project the regular convolutions. Preciseconsideration of networks within a population while themodel is being formulated yields an accurate prediction.Later, the prediction can be informed to the healthcaresector and the stakeholders for necessary implementa-tions.

Here, we use a time-dependent SIRD (Susceptible-Infected-Recovered-Deceased) model to predict the evo-lution of this epidemic in India. The SIRD model is oneamong several compartmental studies in epidemiology [2]such as SEIR, where E stands for exposed and rest stay-ing the same; SEIRD, SIR, SIS model, etc. They have its

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origin from the Kermack-McKendrick theory of infectionspread, a very rigorous statistical analysis performed in1927. Here, the population is divided into various com-partments, and their interactions are studied further. Itis a simple yet an instrumental model of epidemiologysince it takes into account various factors such as the rateat which infection spreads, the rate at which the activecases recover, etc. In this current study, we use the datapresent at the crowd-sourced COVID-19 tracker [32]. Weuse classical computation to demonstrate how a changein spread rates might stabilize figures related to infection,recovery, and death.

Several studies have been done with the available setof data of COVID-19 spread in India and other countries.Some of the notable works have used several other sta-tistical approaches while some have used compartmentalstudies to model and predict the viral spread [4–12].

In our study, we focus on breaking the time scale intosmaller intervals (a period of ten days) and analyzingthem individually to determine the value of the param-eters specified in these periods. We demonstrate howstep-wise constraints on such systems have effectively re-duced infection spread over time. Here, we introduce twopossibilities concerning a set of parameters, one whichdepicts a situation with no lockdowns and another withcontrolled movement of people. We discuss and analyzethe curves obtained from the simulation in terms of ratesof infection spread, as they are mostly dependent on theinteraction network of subjects in a concerned systemexplicitly. As we study this parameter, we aim to estab-lish a pattern in their evolution based on data obtainedand extrapolate them to obtain a specific value in thetimeline towards the end of the multi-phased lockdown(supposedly on 17th May). Further, this set of predictedparameters is used to deduce the progress of the systemin the near future with a fixed population. These deduc-tions with current constraints on the system are put upwith a system with no such constraints to demonstratethe efficacy of multi-phased lockdown in widening anddelaying the peak of infection reporting.

Quantum computation can be useful in the assessmentof such systems since an accurate prediction of a viralspread needs to encompass various factors that mightpose complexity challenges in classical computation. Fac-tors like quarantine measures, social distancing, popula-tion networking, self-protection actions, etc, can give riseto a complex set of problems, challenging for a classicalcomputer to solve. Assuming such factors could be eas-ily fitted into quantum computation facilities, given itsintrinsic ability to hold substantial information and par-allelly process them, underlies this project’s hypothesis.

Parallel to the SIRD modelling in the study, we form ananalogy to the Ising model of magnetic lattice to form aHamiltonian. Then we build a quantum circuit to demon-strate how they are efficient enough to qualitatively showthe nature of the epidemic through the obtained graphsas outputs. We also demonstrate how the considered pa-rameters must be varied to reduce the number of infec-

tions when at its peak and also to delay the time by whichviral spread peaks in the country. This delay of the peakwith lowering of its height is given a term called ‘flatten-ing the curve’ and this becomes crucial, as the affectedpopulation is more spread out for a given time interval.Hence, this might not flood the country’s healthcare fa-cilities, unlike otherwise. The reduction of the number ofcases at its peak also ensures that the current healthcarefacility faces a shortage of resources while treating thepatients.

To build the quantum circuit, we have used the IBMQuantum Experience platform. Various prototypes ofquantum operators have been designed and have beenmade available through IBM quantum experience, a free-web based platform. Researchers have used it to theirstrength to experiment with circuits and also to simulateresults which have furthered their research [16–31].

We organize this paper as follows. Section II discussesthe theoretical backgrounds of the SIRD compartmentalmodel and the quantum computational approach. Sec-tion III proposes how both of them have been imple-mented and what results were obtained from them. Sec-tion IV is dedicated to a general discussion on the ob-tained results, including the drawbacks. Furthermore,at last, we conclude this article by citing the future im-plications of the proposal in Section V. For additionalreferences, an appendix has been attached at VI.

II. THEORETICAL BACKGROUND

A. SIRD Model

As already mentioned, the SIRD model is one of thecompartmental models used in epidemiology. It dividesthe whole population into categories where S stands forthe part of the population which is susceptible to beinginfected by the virus. I is the population that has beeninfected and has the potential to spread the infection.R is the group which has successfully recovered from thedisease. D stands for the portion which has been deceasedafter getting infected due to it. N, the country’s totalpopulation assumed to be time-independent, is the sumof the susceptible, infected, recovered and the deceased.

To develop the operating mathematical model equa-tions, some assumptions have been made here to keepthe model computationally simple. They are

• The average birth rate and mortality rate of Indiahave not been considered.

• The mode of transmission has been considered tobe from person to person.

• Once a person has recovered from the disease,he/she has attained the immunity for infection,hence does not fall back to the category of suscep-tibles.

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Here α is the rate at which the infection is transmit-ted to the susceptibles due to possible contact betweeninfected and susceptible ones. β indicates the rate atwhich the infected individuals recover, which is the re-ciprocal of the number of days in the treatment period.γ describes the fatality of the virus as the rate at whichinfected individuals lose their life due to the virus. Inthis article, we have assumed that the incubation periodof the virus is 14 days for patients in India. From thechart presented above, we describe the SIRD modelingof the virus by the time rate of change of the differentcompartments of the population using coupled ordinarydifferential equations.

dS

dt= −αSI

NdI

dt=αSI

N− βI − γI

dR

dt= βI

dD

dt= γI (1)

N = S + I +R+D (2)

The much talked about R0 value of a viral epidemic isthe number of individuals to whom one infected personcan transmit the virus in a day. It can be calculated byusing the expression R0 = α x Incubation period (Incuba-tion Period= 1

β , if appropriate unit is used). If R0 values

become less than 1, then we can think that the situationwill be under control as the disease now will eventuallydie down. It is so because the number of people infectedper day would become less than the number of peoplerecovering during that period.

We present the plot of the cumulative data of I, R, andD, obtain the parameters, α, β, and γ by the curves forfixed time intervals with multiple iterations. Later, wesolve the mentioned differential equations for the multi-ple sets of parameters, and their corresponding time in-tervals, which nearly fits the data. Once the phase-wiseplotting is done, we note down two sets of parameters,one corresponding to a no-lockdown/constraint systemand another with some constraints. It must also be notedthat this analysis assumes the occurrence of both extrem-ities, and the period of the analysis is from fourth march2020 to 12th May 2020.

B. Ising model

On the other hand, we use the Ising model [3] to builda quantum circuit to demonstrate the effect of ‘curveflattening’. The Ising model of atomic spin discusses thespin interaction of an individual atom in the lattice withits neighboring lattice points and how the spins behavein the presence of a magnetic field. Each can have an

atomic spin of +12 or − 1

2 . The model describes thespin-spin coupling and the exchange interaction betweenthe lattice points and the associated energy value tothem. Interaction matrix depicts the fashion in whichtwo lattice points interact with each other.

Here, Squillante et al. [3], attempted to study theCOVID-19 spread by comparing it with the Ising model,with the analogy of each atom as an individual. The spinof each lattice point (individual) describes whether theperson is infected with the virus or not. A spin of +1

2(p=probability of getting infected [success]) depicts thatthe person is infected with the virus and the spin of − 1

2(q=1-p, probability of not being infected [failure]) indi-cates otherwise, while considering the total population =N. Hence, just like the Ising model, here also the spins(infected or susceptible individuals) interact with eachother. The effect of magnetic field has not been consid-ered since it doesn’t carry in relevance in the analogy tothe viral spread.

Since, in the case of viral epidemics, the infection isspread through ‘contact’ between individuals [13, 14], weassume in a population of N, out of which ‘S’ are sus-ceptible ‘r’ people are infected. Here, we consider whentwo infected people interact, the net interaction is 0 orno effect. So a general probability distribution can beobtained from the Bernoulli’s equation as follows-

P (X = r) = CSr ∗ pr ∗ qS−r (3)

From the previous SIRD model, we get a reproductionrate (Ro = α

β ) which also tells about the average number

of secondary cases arising from primary cases in an en-tirely susceptible population, so we can consider p = Ro

Sor pS = Ro. The resultant equation turns out to be

limS−>∞P (r) = limS−>∞[Rror! (1− Ro

S )S

(1− RoS )−r S!

(S)r(S−r)! ]

(4)

On using Stirling approximation ?? formula and fur-ther solving (4) we get a Poisson distribution (for furtherdetails, refer Appendix VI),

P (r) =Rroe

−Ro

r!(5)

As analysed, new cases per day rise nearly exponen-tially to maximum value and then decrease, which showsthe trend of a peak function, from which the equationcould be derived (17). We can derive its time evolu-tion factor with the probability of cases by implement-ing Hamiltonian operator of its function dependent ontime. The spin systems interact with each other by theexchange interaction. It is defined by a Hamiltonian op-erator as-

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Hr =N∑n=1

An(r)σxnσxn +

N∑n=1

Bn(r)σynσyn

+N∑n=1

Cn(r)σznσzn (6)

where, An, Bn and Cn are the probability functions.The strength of the exchange interaction decreases asthe distance between these particles increases (whichcan also deduce that if people maintain distance, thespread of infection may decrease). In the neighboringregion, we can almost assume that the interaction of oneparticle is almost similar to all of the neighbors, and theinteraction is followed in all three directions, and thusthe exchange interaction takes place; the stronger theexchange interaction more will be its infection rate.

Hence to find its co-relation with the time evolutionoperator and deriving it into quantum circuits here, forn=1, we get a more straightforward form i.e.

Hr = An(r)σxnσxn +Bn(r)σynσ

yn + Cn(r)σznσ

zn (7)

We apply the time evolution unitary operator on ourHamiltonian operator

U(t) = e−iHt(2π)

h (8)

by taking h2π = 1 and putting the Hamiltonian of (7) in

unitary operator of (8) we deduce.

U(t) = e−it(An(r)σxnσ

xn+Bn(r)σ

ynσ

yn+Cn(r)σ

znσ

zn) (9)

= e−it(An(r)σxnσ

xn) ∗ e−it(Bn(r)σ

ynσ

yn) ∗ e−it(Cn(r)σ

znσ

zn)

where, σx=

[0 11 0

], σy=

[0 −ii 0

]and σz=

[1 00 −i

]Dividing the equation (10) into 3 parts as -

U1(t) = e−it(An(r)σxnσ

xn) (10)

U2(t) = e−it(Bn(r)σynσ

yn) (11)

U3(t) = e−it(Cn(r)σznσ

zn) (12)

On solving the equation (10)

U1(t) = e(−itAn(r))(σxnσ

xn)

(13)

which follows the condition of

e−iθA = cos(θ)I − isin(θ)A

(14)

where I is the identity matrix and A is of same order ofthe identity matrix.Here, A=σx ⊗ σx, which follow the identity rule and hasa order 4 and we break down it to

U1(t) = cos(An(r)t)I − isin(An(r)t)σx ⊗ σx

[cos(An(r)t) 0 0 −isin(An(r)t)

0 cos(An(r)t) −isin(An(r)t) 00 −isin(An(r)t) cos(An(r)t) 0

−isin(An(r)t) 0 0 cos(An(r)t)

]Hence, solving the matrix and comparing with the U3

matrix we derive a following circuit 1:

FIG. 1: The derived circuit has a combination ofa CNOT gate, U3 gate and a CNOT gate, where

θ = 2An(r)t, φ = −π/2 and λ = π/2.

Similarly, working for the equation 11 we get a reducedmatrix format

U2(t) = cos(Bn(r)t)I − isin(Bn(r)t)σy ⊗ σy

[cos(Bn(r)t) 0 0 −isin(Bn(r)t)

0 cos(Bn(r)t) −isin(Bn(r)t) 00 −isin(Bn(r)t) cos(Bn(r)t) 0

isin(Bn(r)t) 0 0 cos(Bn(r)t)

]Where the equivalent matrix is reduced in the form of

circuit as, 2

FIG. 2: The derived circuit has a combination ofa CNOT gate, control U3 gate, Anti-control U3gate and a CNOT gate, where θ1 = 2Bn(r)t, φ1 =−π/2 and λ1 = π/2 and θ2 = 2Bn(r)t, φ2 = π/2 andλ2 = −π/2.

Similarly, working for the equation 12 we get a reducedmatrix format

U3(t) = cos(Cn(r)t)I − isin(Cn(r)t)σz ⊗ σz

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e−itCn(r) 0 0 0

0 eitCn(r) 0 00 0 eitCn(r)) 00 0 0 e−itCn(r)

Where the equivalent matrix is reduced in the form of

circuit as, 3

FIG. 3: The derived circuit has a combination ofa CNOT gate, U1 gate and a CNOT gate, whereθ = 2Cn(r)t.

Combining these 3 circuits, we get an equivalent circuitas in Ref. 4,

FIG. 4: The equivalent circuit describes theHamiltonian operator 7 in a N=2 state.

Finding the average value helps us find the Energyeigenstate of the operator, which we can relate to thenumber of infected people at a particular time.

En = An(r)t < σxσx > +Bn(r)t < σyσy >

+ Cn(r)t < σzσz >

(15)

where, <> = shows average value and< σxσx >= P00 − P01 − P10 − P11

which is determined by putting 2 Hadamard gates at theend of the equivalent circuit, which is useful for movinginformation between the x and z bases, which is shownin the following diagram (5).

FIG. 5: Measurement bases for the calculation ofaverage values of < σxσx > for N=2 in circuit.

Similarly, < σyσy > can be represented by putting ainverse S gate followed by a H gate in each qubit as itmoves information from y to z bases, which is shown inthe following diagram (6).

FIG. 6: Measurement bases for the calculation ofaverage values of < σyσy > for N=2 in circuit.

And for < σzσz >, we only measure the qubits in zzbasis, which is shown in the following diagram (7).

FIG. 7: Measurement bases for the calculation ofaverage values of < σzσz > for N=2 in circuit.

An(r), Bn(r) and Cn(r), all three functions are takento be P(r) which we have derived in the above. Hence theHamiltonian, whose time evolution we study for COVID-19 spread now turns-

Hr =Rroe

−Ro

r![σxnσ

xn + σynσ

yn + σznσ

zn] (16)

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III. IMPLEMENTATION AND RESULTS

A. SIRD model

(a)

(b)

FIG. 8: Simulating the prediction curves by usingthe best fit over the entire range of dataFig. (a) depicts the data points and respective curveswith α = 0.144, β = 0.021 and γ=0.0041 (b) representsthe same curve with the time axis extended upto 1000days, thus giving a trend of the infection.

The first Covid-19 case was reported in India on 30thJanuary 2020. Considering it as Day 1, on 4th March(the 33rd day), the number of cases jumped suddenlyfrom 5 to 28. So, in this study, we start analyzing thedata from March 4th onwards. For solving the differ-ential equations, we use the Odeint module from scipy,which uses the LSODA algorithm to solve the systemof coupled ODEs. As Rajesh et al. [10] pointed out in

their SIRD model prediction of Covid-19 in India, thatthere is no reported error in the database, so we cannotuse reduced chi-sq fitting for the above data. So, weemployed the approach mentioned to carry out our anal-ysis i.e., the method of eye-approximation for the best fit.

Interval/ Date Day ofLeg no. spread

1 4 Mar-13 Mar 33-422 14 Mar-23 Mar 43-523 24 Mar-2 Apr 53-624 3 Apr-12 Apr 63-725 13 Apr-22 Apr 73-826 23 Apr- 2 May 83-927 3 May-12 May 93-102

TABLE I: Distribution into various legs

As per this fitting, in Fig. 8, the peak of the infec-tion curve suggests that a maximum of 109 people willget infected for a particular set of parameters. However,data plotting suggests a change in the concavity of thecurve it follows periodically, hence in its slope. Thus tak-ing a single fitting for the whole period might not be awell scientific method. Hence, we tried to break the timeinterval into smaller legs (here ten days) as per Table Iand record the values of the parameters (Table II) in therespective leg. The simulated curves are thus made tonearly fit the data points in Fig. 9.

Interval/ α βLeg no.

1 0.127 0.0022 0.167 0.0103 0.165 0.0144 0.146 0.0145 0.101 0.0176 0.0825 0.0207 0.091 0.024

TABLE II: Values of α(t) and β(t) for various intervalsfor making the curve fit with the respective data sets.

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FIG. 9: Simulating the prediction curves by usingthe best fit for every single interval obtained bydividing the time axis by the dashed cyan lines.

By obtaining values of parameters, we proceed to makethe curves for infection, recovery, and deaths that ap-proximately match the data points available. Further-more, thus we obtain alpha and beta as for each periodi.e., α(t) and β(t). Here, our only concern remains α(t),and how it evolves in time as it is dependent on interac-tions between components of the population that’s beingcurbed and regulated during a lockdown [4]. β(t) and, onthe other hand, is dependent on various other factors likethe efficacy of treatment, the capacity of the healthcaresystem, and demographics of the individual population,which is not the point of discussion in this paper, hencenot given much importance. Also, changes in their val-ues are in the order of 10−3 if any were made during thefitting and duly presented.

In India, many cases start exhibiting the symptoms af-ter completing their 14-day quarantine. Hence, for safetypurposes, many state governments have extended theirquarantine period to 28 days. So it might be unsafeto assume any incubation period of fewer than 14 days.Hence we consider it to be exactly 14 days. Moreover,that makes the R0 value for the 3rd leg (March 24-April2) 2.31, which falls to 1.274 by the end of the 7th leg.Hence, the lockdown has been able to contain the spreadto some extent, but it should continue until it reaches avalue closer to 1.

If we go on plotting the evolution of α(t) from theabove data, we can see that a lockdown can successfullyreduce the infection rate. To know the future of α underthe lockdown, we consider the rate of infections of theseveral smaller periods under lockdown while excludingthe first 2 data points for day 33-42 and 43-52 (since thelockdown nearly began on the day 53 of the arrival ofthe virus in India) and do an exponential decay fit. Welater go on extrapolating the curve to know the estimatedalpha values in the few next legs. As shown in figure 10,we later use a straight-line fitting to extrapolate beta forsubsequent legs. Here, we can interpret that the recovery

rates have increased in a pretty uniform manner. It canbe attributed to various aspects, such as improvementin the healthcare scenario, change in the demography ofthe infected individuals, or a boost in the quality andquantity of available treatment methods. Further studiesand accurate models might be able to justify this trend(Fig. 10).

(a)

(b)

FIG. 10: Curve fitting for studying the evolutionof α and β over timeFig (a) α(t) being fit to an exponential decay functiony = 0.063 + 0.376 · e−x/t(b) β(t) being fit yo a straight line y = 0.00256x +0.00508.

We use the obtained values of α and β for the nextlegs and simulate the SIRD curve and present it cumula-tively. We have tried to analyze how the future numbersmight be if the lockdown continues as it is until the dis-ease is entirely null. After 3-4 legs further, the change inα would have started becoming negligible (and R0 value

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slowly tending to 1). The recovery rate also would havebecome nearly constant, we have taken the values of theparameters at the end of the leg of period 132-142 to beconstant throughout, for the time to come. We extrapo-lated the curve to obtain a trend (Fig. 11).

Parallelly, we have studied the effect of the currentlockdown being lifted. If the lockdown is lifted on May17, we had assumed the other extreme that the infectionrate might well reach the value it had before the impo-sition, but certainly not anything greater than that andplotted the curves.

FIG. 11: Analysis of number of infections uponvarious scenarios of lockdown. The time axis is seg-mented by using green lines, which signify the start and(Probable) end of lockdown. It can be observed that thenumber of cases at the peak almost halves with a distinctshift in the abscissa of peak, when the lockdown contin-ues. When lockdown is removed all of a sudden (andassumed that the α= 0.165, just like the value beforelockdown), there is less decline in the number of cases atthe peak with a subtle shift in the abscissa of peak.

B. Ising model

From Equation (16) from the previous section, we havea Hamiltonian for the COVID-19 spread when comparedto the Ising model. We were able to find a function which,when fed into the quantum circuit, qualitatively depictsthe shape of the curve for the infected individuals, as wealso obtained in the SIRD modeling.

While trying out various functions, we finally foundthat a Poisson probability distribution function gives aconsiderable resemblance to an actual graph and also thetype obtained from SIRD modelling (Fig. 12). In thefunction, the parameter R0 signifies the ratio of α to thatof β. We can observe that when R0 is decreased whilekeeping other variables and parameters constant, we can

observe that the peak of the curve shifts forward in thetime axis with a reduction in its height and making itbroader.

From these, we can easily derive that a decline in therate of infection (α) will infect lesser people than before,thus flattening the curve. Also, an increase in the rateof recovery (β) will imply an improvement in healthcarefacilities, changing demographics of the infected patients,etc. and show a change in the height and position of thepeak. Using proper parameters and constants, this canbe a novel way to simulate approximate curves for givenvalues. An analysis of epidemic spreads can be carriedout with the inclusion of more and more factors thatotherwise gets difficult for classical computers.

IV. INTERPRETATION AND DRAWBACKS

The three graphs, with I, R, D, show two trends postlockdown- one set of parameters without any distancing(constraints), and their respective carrier limits, and timeis taken to attain them. The latter case depicts anotherset of parameters that have a few constraints imposed onthem, as in the last phase of lockdown. So latter caseparameters postponed the attainment of carrier limitsand are not as rapidly growing as the former case. Also,the no lockdown case is started from the day of datacollection, along with lockdown trends and post lockdowntrends growing by two parameters to give an effectivepicture of what would have been the picture without alockdown and how it would evolve once a lockdown islifted off considering the extreme case scenario too. Itis evident that the lockdown shouldn’t be lifted all of asudden, as it will lead to a massive upsurge in the numberof cases (nearly same as that of no-lockdown case) witha very narrow delaying of the peak.

The SIRD model discussed here has its own set oflimitations pertaining to the assumptions made and themethods adopted while formulating it. The first limita-tion of this analysis is the use of the slope-estimation ofdata plotted curves to find the simulated curve in thelockdown period while obtaining a periodic split up ofparameters. This might account for errors in the anal-ysis. The assumptions like not considering birth andmortality rates, the permanent immunization of a recov-ered patient, unavoidable migration and interactions, etc.make the model lose its precision. Here, the susceptibleare considered to be the entire population except the in-fected, recovered, and deceased individuals, initially at atime t. No considerations have been made to distinguishthe exposed individuals out of the susceptible and modelthem accordingly. This might be a significant drawbackof the model. The trend in recovery and its rate also is in-comprehensible due to the lack of data on the demograph-ics of the recovered patients in terms of their age, sex,etc. In this model, the number of tests performed andtheir rate has not been considered, but they might playa significant role in reporting the cases, hence the α(t).

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9

(a)

(b)

FIG. 12: Used Hamiltonian being simulated using the IBM Quantum Experience, demonstrating thephenomena of curve flattening. Fig (a) A 2-D graph depicting the shift of the peak along X-axis when R0 is

decreased (b) A 3-D curve showing the decline in the height of the peak when R0 is decreased.

Our N (the total population) has been assumed constant,but immigration, new birth, and deaths might vary theN, thus making us compromise on its accuracy. Moreefficient models like the SEIRD (Susceptible- Exposed-Infected-Recovered-Deceased), and higher compartmen-tal models can be implemented for better prediction andanalysis.

The quantum circuit design only gives the qualitativeshape of a ‘usual’ curve of an epidemic infection spreadwith the proper use of the parameters and some con-stants, this can be used to simulate the curves for re-spective regions. Adding various complexities, out ofwhich some are discussed in the above, is a matter oftime. With better and smarter use of quantum gates,efficient circuits can be made to minimize the space andtime complexities. Here, we have assumed only two pa-rameters, the rate of infection and the rate of recovery.This can be intricately designed to accommodate vari-ous other detailed parameters that encompass factors likethe number of tests, the age group of infected individu-als, government measures in curbing the epidemic, etc.Moreover, this can give us a more accurate simulation ofthe epidemics as significant as COVID-19.

V. CONCLUSION

COVID-19 has turned out to be a global crisis, af-fecting all the countries. The current model study reporthints at a frightening upsurge of the viral epidemic in thetimes to come. Measures like quarantine and lockdown

have been successful enough to reduce their impact, yeta lot needs to be taken care of. However, the current pro-longed lockdown has started worrying national as well asglobal economies, pushing them into a tremendous cri-sis; hence the lockdown cannot be sustained forever. Butas per our analysis, the lifting of lockdown shouldn’t beall of a sudden, and be more gradual in the approach.We must start practicing the concept of social distancingand personal hygiene to keep the viral spread at bay. Indesperate times like these, researchers of all fields mustcome together and contribute towards finding more in-formation regarding the virus through experiments anddata analysis. This shall let us be more aware and helpus in tackling the risk. We sincerely hope that, just likeprevious global pandemics, we can pass through this withthe advent of science and technology.

ACKNOWLEDGMENTS

S.P., A.P., P.S. and K.S. would like to thank Bikash’sQuantum (OPC) Pvt. Ltd. for providing hospitalityduring the course of this project. A.P. would like tothank Dr. Victor Roy, SPS, NISER for clearing some ofthe doubts and sharing valuable resources in the initialphase of the project. B.K.B. acknowledges the presti-gious Prime Minister’s Research Fellowship awarded byDST, Govt. India. The authors acknowledge the supportof IBM Quantum Experience for producing the basic cir-cuits. The views expressed are those of the authors anddo not reflect the official policy of IBM or IBM Quantum

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11

VI. APPENDIX

p is the probability of getting infected.q is the probability of not getting infected,i.e.,q=1-p.

So, P (X = r) = CSr ∗ pr ∗ qS−r

P (X = r)= S!(S−r)!r!

Rr0Sr (1− R0

S )S−r

Assuming the susceptible to be infinite,R0 is a constant.

limS−>∞P (r) = limS−>∞[Rror!

(1− RoS

)S(1− RoS

)−rS!

(S)r(S − r)!]

=Rror!limS−>∞(1− Ro

S)S(1− Ro

S)−r

S!

(S)r(S − r)!]

=> limS−>∞(1− RoS

)S = e−Ro

=(Ro)

re−Ro

r!limS−>∞

S!

(S)r(S − r)!Using : Stirling′s− approximation− formula

S! =√

2ΠSe−SSS

Then, P (r) =(Ro)

re−Ro

r!limS−>∞

√2ΠSe−SSS

Sr√

2Π(S − r)e−S+r(S−r)S−r

=(Ro)

re−Ro

r!limS−>∞

e−S+S+0.5SS−r+0.5

(S − r)S−r+0.5

=(Ro)

re−Ro

r!e−rlimS−>∞(

S

S − r)S−r+0.5

=(Ro)

re−Ro

r!e−rlimS−>∞

1

(1− rS )S

=(Ro)

re−Ro

r!e−rer

=(Ro)

re−Ro

r!

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