Hybrid‐quantum approach for the optimal lockdown to stop the SARS‐CoV‐2 community spread subject to maximising nation economy globally

Abstract

SARS‐CoV‐2 epidemic (severe acute respiratory corona virus 2 syndromes) has caused major impacts on a global scale. Several countries, including India, Europe, U.S.A., introduced a full state/nation lockdown to minimise the disease transmission through human interaction after the virus entered the population and to minimise the loss of human life. Millions of people have gone unemployed due to lockdown implementation, resulting in business and industry closure and leading to a national economic slowdown. Therefore, preventing the spread of the COVID‐19 virus in the world while also preserving the global economy is an essential problem requiring an effective and immediate solution. Using the compartmental epidemiology S, E, I, R or D (Susceptible, Exposed, Infectious, Recovery or Death) model extended to multiple population regions, the authors predict the evolution of the SARS‐CoV‐2 disease and construct an optimally scheduled lockdown calendar to execute lockdown over phases, using the well‐known Knapsack problem. A comparative analysis of both classical and quantum models shows that our model decreases SARS‐CoV‐2 active cases while retaining the average global economic factor, Gross Domestic Product, in contrast to the scenario with no lockdown.

INTRODUCTION

The COVID-19 affect brutally all across the world over human beings. The epidemic was spread highly across the world needed to find certain solution to stop the virus from spreading through the population. The government had to impose restrictions on people's actions in public, therefore ‘lockdown’ was implemented nationwide. The non-empirical choice of lockdown was made. The country went into months-long, as a consequence of continuous lockdown the economy and social activity of human beings both are lost, but the virus's ability to spread throughout communities was controlled to some extent. In light of this, “lockdown” emerged as a desired strategy for halting the spread, despite its drawbacks. In the subsequent events, the authorities chose to execute the lockdown on a planned basis rather than forcing it to last continuously. To some extent, this controlled the economy. As a result, difficulties were shown in balancing the “lockdown” and the “economy” while still maintaining the former to halt the infection.

Scientists from all across the world collaborated to study the virus, its effects, and potential exit strategies from this epidemic. There have been several research articles using scientific epidemiological models to forecast the development of viruses. From 1927 until the current pandemic circumstances, there have been several scientific epidemiological investigations recorded [1–7]. In 1927, Kermark et al. proposed compartmental modelling in the essay [1]. From that point forward, several mathematical models of epidemics—such as susceptible infection recovered (SIR), susceptible-infectious-recovery-death (SIRD), susceptible-exposed-infectious-recovery (SEIR), susceptible-exposed-infectious, recovery-death (SEIRD) etc.—are published in refs. [2–7]. Ivorra, Benjamin et al. published their mathematical model in ref. [8] and it includes the need for hospital beds and undiagnosed infections. However, a fixed population for a region is stated for each of these models. These models do not take into account how diseases spread or evolve in various population centres as a result of human movement or contact. Predictions from epidemiological models can assist with prior decision-making in a variety of contexts, including policy formulation and implementation timeframes for lockdowns. Again, according to all models, a lockdown is the most effective way to lessen the impact of COVID-19. However, an ongoing and protracted scenario like the one we are currently experiencing has a significant negative impact on Gross Domestic Product (GDP) growth, which is not taken into account.

In this experiment, we predicted the development of the virus using epidemiological models in a controlled setting and used the results to make the best lockdown decisions. A possible approach to meet the challenges of “lockdown” and “economy” is optimisation. Since the solution to such a problem may be found using both classical and quantum computing, both have been compared in several ways. We employed a restricted graphical representation and duration for our model, which is only a beginning point, to demonstrate its viability.

Finding a lockdown schedule that balances the “spreading of virus” and the “maintenance of economics” is the issue that our project seeks to tackle. The rate of infection within a region, which will be statistically modelled using epidemiological models, is what we mean by the phrase “spreading of the virus.” By imposing a lockdown in that region, we want to lower this incidence and prevent the infection from spreading across the neighbourhood. While “maintaining of the economy” refers to the region's resulting GDP. Maintaining value constantly is our goal in the same situation.

Instead of creating a full-stack solution to address the issue, we plan to use it as a study approach. We are eager to investigate the issue and have several potential solutions on the table. Simulated evolution predictions will be made, and the entire project's methodology will be generalised. By striving to address the issue, the initiative will assist the government in making wise policy decisions. For the framework to be applicable in a wide range of situations where the issue of optimisation between two constraints arises, we also want to construct the structure in a comparative positive manner.

Some objectives of our Research work:

• To create a mathematical model for epidemiology that accounts for the development of viruses and their spread from one place to another. Thus, a transmission function is needed to mimic the COVID-19 epidemic's current setting.

• Using strategic optimisation techniques to determine which regions should be under lockdown/open and when.

• Analyse the spread of the infection into various regions (cities) using both tabular and graphical methods.

• To comprehend and research quantum computing for such difficult challenges, and to evaluate how well a specific model fits the issue at hand.

• To conduct a comparison of the specified problem using classical and quantum approaches in various contexts.

EPIDEMIOLOGY

The technique of epidemiology is used to identify the root causes of illnesses and other health issues in communities. In epidemiology, the population as a whole is seen as the patient. Epidemiology is defined as the scientific, methodical, and data-driven study of the distribution (frequency, pattern), determinants (causes, risk factors), and occurrences associated with health in certain populations (neighbourhood, school, city, state, country, global). Additionally, it is the application of this knowledge to the management of health issues. The SIR model is the most straightforward epidemiological model [1, 2].

The population is split into three categories: susceptible (S), infected (I), and Recovered (R) as depicted in Figure 1. Susceptible people are people who have not yet contracted the disease but who may. When someone is described as infected or removed, it refers to someone who spreads an infection. Following recovery, they are no longer affected and do not propagate the disease. According to this model, contact between infected and susceptible individuals causes new infections to arise at a rate of ß where ß is the disease's transmission rate. The population is categorised using the SIRD model as SIR with a separate section for deceased (D). N = S + I + R + D always remains constant. β is the transmission rate, γ is the recovery rate, and δ is the death rate. Where S(t), I(t), R(t), and D(t) are the number of susceptible, infected, recovered, and deceased, respectively at time t. In the SEIR model [9], the population is categorised the same as SIR with an additional compartment exposed (E). Here, ß is the transmission rate, δ is the rate of infection and γ is the rate of recovery.

[IMAGE OMITTED. SEE PDF]

SEIRD model

Forecasting the spread of diseases is crucial for reducing the effects of epidemics. Such modelling and prediction employ epidemiological modelling. To do this, mathematical methods have been used. Age-based modelling was suggested for disease transmission by Davies et al. in ref. [10]. The data flows into and out of compartments like susceptible, infected, recovered etc. are described by a mathematical epidemiological model. Sir Ronald Ross examined the efficacy of several malaria therapeutic techniques using mathematical modelling. Kermack and McKendrick used compartmental modelling to examine disease transmission in ref. [1]. I. Korolev describes the SEIRD model in detail [6]. The compartments represent:

• S: The number of susceptible people who are most likely to get the infection. After coming into touch with the sick person, the vulnerable person contracts the illness and moves to the exposed compartment.

• E: The total number of exposed people. The individuals become sick but are not yet contagious throughout a significant period of incubation. The individual is now in an exposed compartment.

• I: The total number of sick people after the incubation period, exposed individuals continue to be infectious and can infect those who are vulnerable.

• R: The number of infected people who entered the recovered compartment, survived the illness, and developed immunity to it. They are unable to infect these vulnerable people again.

• D: The total number of fatalities brought on by the illness.

However, the majority of the models are constrained by the virus's ability to evolve only inside a particular region. However, in the most recent pandemic, we saw that the virus spread quickly as a result of people moving from one location to another. When developing any epidemiological model, it is important to take viral transmission from one location to another into account. The model that we utilised for this research features a feature called “Transfer Function” that regulates the propagation of viruses.

Transfer function

The new aspect of this study is Transfer Function. It permits the spread of viruses from one state to another while assuming that moving individuals from one state to another will not affect that state's overall population.

Communication between infected and vulnerable people is how the virus spreads. The intercity contact ratio, which relies on the distance between two cities and the individual moving from one city to another, is defined by Tr, whereas the intra-city contact ratio is defined by β. Let, I _{(A, B)} and I _{(B, A)} be the number of infected populations travelling from city A to B and vice versa respectively as depicted in Figure 2. Assuming that the total population of a city remains constant over time that is, I_{(A, B)} ≈ I_{(B, A)}, which in turn does not affect the property of the SEIRD model, that is, N = S(t) + E(t) + I(t) + R(t) + D(t), which states that the total population of a city remains constant, but only transmits the virus into the city affecting the β. The distance matrix represents the distance between two cities, where the linear term is 0 and the quadratic term, I _{(A, B)} ≠ I _{(B, A)} is the distance. If the distance increases, the spread of the disease decreases, but the spread depends on several factors. Hence, we introduced a calibration constant k. Thus, the Tr function is defined as in Equation (1). 1 $$\mathrm{T}\mathrm{r}=\left(\frac{1}{\frac{{\mathrm{d}}_{\mathrm{A},\mathrm{B}}}{\mathrm{k}}}+1\right)$$

[IMAGE OMITTED. SEE PDF]

We presume that the impact of an infected city's population on nearby cities decreases with increasing distance. It is also possible to utilise and calibrate other exponential decay functions here. The objective is to simulate how an infected city's population would affect another city based on how people will migrate between the two. This may be accomplished by using several intercity transit options. By changing the calibration constant k, the whole function may be calibrated using actual data. The effect diminishes quickly with distance as ‘k’ is decreased as depicted in Figure 3 and the transfer function is explored in Algorithm 1.

[IMAGE OMITTED. SEE PDF]

The algorithm for the transfer function is given below.

1

ALGORITHM

Transfer Function TF

Therefore, the SEIRD Model along with the transfer function can work in one or more cities, hence a Multi-City Epidemiology Model. The model is depicted in Figure 4 and the proposed formulations of susceptible rate, exposed rate, and infected rate are expressed in Equations (2), (3), and (4) respectively. Whereas, the rate of recovery and death rates is illustrated in Equations (5) and (6) respectively. 2 $$\frac{dS}{dt}=-\beta .\frac{S\,.\,(I+Tr)}{N}$$ 3 $$\frac{dE}{dt}=\beta .\frac{S\,.(I+Tr)}{N}-\delta .E$$ 4 $$\frac{dI}{dt}=\delta .E-(1-\alpha ).\gamma .I-\alpha .\rho .I$$ 5 $$\frac{dR}{dt}=(1-\alpha ).\gamma .I$$ 6 $$\frac{dD}{dt}=\alpha .\rho .I$$

[IMAGE OMITTED. SEE PDF]

QUANTUM COMPUTER

The origins of quantum computation (QC) lie in the ideas of Benioff relating to quantum mechanical versions of the Turing machine [11], and of Feynman about the need for a simple quantum computational system that can efficiently simulate more complicated ones [12]. In the years that have ensued since these ideas were put forward, the notion of a quantum computer processing quantum information has been scientifically well established, with several equivalent models of quantum computing available.

The two most prominent ones are the quantum circuit model [13], which is popular due to its similarity to the electrical circuit model that prevails in engineering disciplines, and the adiabatic quantum computation model [14], which is based on the adiabatic theorem of quantum mechanics in which a quantum system's lowest energy state stays invariant under a “slow” enough evolution of the system's Hamiltonian. Either model has advantages depending on the nature of the analysis one is interested in performing. The two models can be transformed into each other in polynomial time.

Analogous to “bit” in a classical computer, which has two states, either 0 or 1, synonyms to OFF and ON respectively, Quantum Computer has quantum bits or “qubits”. Qubits are fundamentally different from those classical bits. The concept and property of qubit form the basic building block of the quantum computer, which exploits the property of quantum mechanics to do computation. To formally define, a qubit can exist in the state |0> or the state |1>, but it can also exist in what we call a superposition state. This state is a linear combination of the states |0> and |1>. If we label this state |φ>, a superposition state is written as $$\vert \varphi > =\,\alpha \vert 0 > +\,\beta \vert 1 > $$ here α, and β are the complex numbers.

Operators are employed in the quantum circuit model to manipulate qubits in the form of circuits, exactly like a logic circuit. All of the operators, however, are unitary and are shown as matrices. The platform that enables us to work with the quantum circuit model is called “IBM Qiskit”, along with some other emerging platforms like Rigetti, IonQ etc. The quantum circuit model is more adapted to tackle issues with sorting and searching. Due to technical limitations and quantum noise, IBM Qiskit's most recent state-of-the-art technology cannot handle a large number of Qubits. Quantum Circuit Model cannot effectively calculate such a difficult issue since optimisation problems demand a large number of qubits, but quantum annealer can.

Simulated annealing was introduced by S. Kirkpatrick et al. to solve real-world optimisation problems [15]. They used thermal fluctuation property to find a global minimum value by escaping local minimums that a system can get stuck in. Geman and Geman proved that if the thermal fluctuation is decreased at the rate of T = c/ln t or slower, then it will reach the global minimum value (where the constant c depends on the system's size) [6]. Otherwise, it will stay stuck at some local minimum value. T Sallis et al. proposed another probabilistic method based on conventional Boltzmann-type probability distributions to find out better convergence to solutions than the thermal fluctuation methodology [16]. In 1998, Kadowaki et al. proposed quantum annealing [17] and in 2001, Farhi et al. proposed quantum adiabatic evolution algorithm [18] as a way to solve combinatorial optimisation problems. Compared to the thermal fluctuation methodology, quantum annealers is a more efficient metaheuristic for detecting global minimum.

In this paper, we use D-Wave quantum computer to find an optimal lockdown implementation in the context of the COVID-19 pandemic. We first construct the binary variable model or quadratic unconstrained binary optimization (QUBO) model and then convert it into the equivalent Ising or bipolar model. Basic QUBO formulation is expressed as 7 $$\mathrm{Q}\mathrm{U}\mathrm{B}\mathrm{O}:\mathrm{Y}={\mathrm{x}}^{\mathrm{T}}\mathrm{Q}\mathrm{x}\,\mathrm{o}\mathrm{r}\,\mathrm{f}(\mathrm{x})=\sum\limits _{i}{Q}_{i,i}{x}_{i}+\sum\limits _{i< j}{Q}_{i,j}{x}_{i}{x}_{j}$$ where x is a vector of binary variables and Q is N × N upper bound triangular matrix. Q_{i, i} are the linear diagonal coefficient, and Q_{i, j}, represents off-diagonal non-zero coefficients. In this manuscript upper bound triangular structure of the Q, the matrix is used to define our constraints in the QUBO model.

D-Wave Quantum Computer

D-wave Systems is the first commercial vendor of a quantum computer based on quantum annealing to solve combinatorial optimisation problems. D-wave quantum processing unit (QPU) can be accessed from the D-Wave Quantum Leap that has three aspects—Build, test, and run. Users can test using a simulator of the Quantum program by underlying software. Users can run the program to D-wave QPU and other QPU.

D-Wave used Quantum annealer which is specialised in optimisation problems. Using quantum tunning and entanglement, it finds the lowest energy of a problem space in efficient time; however, the absolute advantage of quantum annealer against classical computation is unknown and still under active research.

The logical Topology of the qubits in the D-Wave Quantum Computer Chip is arranged in 8 qubits in a bipartite graph as directed in Figure 5. A large ensemble of pairs behaves as a single quantum state. The D-Wave 2000Q QPU supports a C16 Chimera graph: its more than 2000 qubits are logically mapped into a 16 × 16 matrix of unit cells of 8 qubits. In Figure 6, qubits are represented as horizontal and vertical loops. This graphic shows three rows of 12 vertical qubits and three columns of 12 horizontal qubits for a total of 72 qubits, 36 vertical and 36 horizontal. In Figure 7, the physical chip is shown, which must be kept in extremely cold condition, much below 9000 mK. D-Wave Quantum Computer runs at 10.45 mK, 0.01 C above absolute zero.

[IMAGE OMITTED. SEE PDF]

[IMAGE OMITTED. SEE PDF]

[IMAGE OMITTED. SEE PDF]

OPTIMISATION

An optimisation problem is a computational issue where the goal is to find the best available solution. Finding the highest or lowest value that a function may have been, in mathematical language, the goal of optimisation problems. Discrete mathematics, computer science, and operational research all heavily rely on combinatorial optimisation. The goal of this discipline is to resolve several challenging combinatorial optimisation issues. Finding the highest GDP value while lowering the infection rate is the goal of our project activity. An optimisation program will calculate the best lockdown approach for a city based on the number of infected, bed capacity, and total GDP. In layman's terms, a city is supposed to go into lockdown mode when the number of infected there is greater than or equal to the number of beds available in the hospital. This straightforward lockdown strategy is covered in Chapter 3 of Practical & Visualisation, but it won't effectively solve our complex problem because it does not take GDP into account.

It is still unclear how to find the model that fits an issue the best. The Knapsack Model is appropriate for choosing a city from a list of cities to retain open while lowering the threshold of infection (described below) across the country. It is crucial to develop an objective function that strikes a compromise between the competing dynamics of the need to keep a city open to sustain an economy and, ultimately, GDP, versus the necessity to close a city to limit the spread. We consider the objective function at a high level to be the statement below.

Objective = Maximise (GDP) + Minimise (Illness).

Classical knapsack

Definition: 0–1 Knapsack problem: Given a set of items of different values and volumes, find the most valuable set of items that fit the knapsack of fixed volume. In mathematical terms there is a knapsack of capacity c > 0 and N items. Each item has a value v_i > 0 and weight w_i > 0. Find the selection of items (δ_i = 1 if selected, 0 if not) that fit, $$\sum\limits _{i=1}{1}^{N}\,{\delta }_{i}{w}_{i}\le c$$, and the total value, $$\sum\limits _{i=1}{1}^{N}{\delta }_{i}{v}_{i}$$, is maximised.

The list of items to include in the knapsack can be found using one of the two methods: brute-force, which thoroughly searches all potential scenarios to find the best fitting answer; or dynamic, which adopts an optimisation strategy to break a large issue down into a series of smaller problems. When developing a dynamic-programming algorithm, we follow a sequence of four steps:

• (i)

  Characterise the structure of the optimal solution.

• (ii)

  Recursively define the value of an optimal solution.

• (iii)

  Compute the value of the optimal solution, typically in a bottom-up fashion.

• (iv)

  Construct an optimal solution from computed information

Hence, the dynamic 0–1 knapsack problem algorithm (Algorithm 2) is given below from the perspective of our project problem formulation.

We are using GDP as the cost and infection as the weight in our knapsack, and the total bed capacity of all cities is the bag's weight capacity. The sheer act of adding up the bed capacity for all cities reduces the precision of the lockdown decision. The lockdown policy is based on each city's bed capacity in real life; however, this is not the case for our knapsack method. As a result, a future goal is to strengthen the limitation and compare the infection with each city's bed capacity rather than as a whole.

2

ALGORITHM

Classical Knapsack Algorithm

Quantum knapsack

In ref. [19], Andrew Lucas et al. discussed about quantum knapsack algorithm. In term of Quantum Computer Algorithm Formulation, we have a list of N object, labelled by indices α, with the weight of each object given by w_α, and its value given by c_α, and we have a knapsack which can only carry weight W. If x_α is a binary variable, object α is contained in the knapsack, the total weight in the knapsack is 8 $$\mathcal{W}=\sum\limits _{\alpha =1}^{N}{w}_{\alpha }{x}_{\alpha }$$ and the total cost is 9 $$C=\sum\limits _{\alpha =1}^{N}{c}_{\alpha }{x}_{\alpha }$$

The NP-hard [20] knapsack problem asked us to maximise C subject to the constraint that $$\mathcal{W}\le W$$.

Using binary quadratic model (BQM) this objective is transformed into an energy function and then minimised using classical and D-Wave Quantum Annealer. The knapsack formulation is written in Equations (10–12). 10 $$H={H}_{A}+{H}_{B}$$ 11 $${H}_{A}=A{\left(1-\sum\limits _{n=1}^{W}{y}_{n}\right)}^{2}+A\,{\left(\sum\limits _{n=1}^{W}n{y}_{n}-\sum\limits _{\alpha }{w}_{\alpha }{x}_{\alpha }\right)}^{2}$$ 12 $${H}_{B}=-B\sum\limits _{\alpha }{c}_{\alpha }{x}_{\alpha }$$ where H_A expresses the inequality of minimising the weight of the knapsack, and H_B denotes maximisation of the total profit of knapsack. The cities are selected to put in the knapsack in such a way that total death should not cross the weight of the knapsack whereas total bed capacity can be maximised. The Quantum knapsack problem is mapped to solve the optimal lockdown for SARS-CoV-2 pandemic. The Quantum Knapsack BQM Algorithm is given below in Algorithm 3. Using the bqm we solve it by D-Wave sampler and get a sample set, which consist of open cities their sample sets and respective energy.

3

ALGORITHM

Quantum Knapsack Algorithm

WORKFLOW

In this part, we have spoken about how the SEIRD Model, Transfer Function, Model Evolution, Classical Knapsack, Quantum Knapsack, and Lockdown are all related to one another and work together to produce the chain of outcomes that we are after. A graphic flowchart, shown in Figure 8 is more appropriate to help grasp the project's complicated workflow. Below the project's overall work flow. Three models—No-Lockdown, Classical Lockdown, and Quantum Lockdown Model—are included in the project. For the No-Lockdown model, the evolution function evolve simulates the evolution of the S, E, I, R, and D compartments by diffusing the model equations within the derive function, which uses the transfer function Tr to determine the overall influx of people from other cities, given that the city to and from which the transfer is to take place is open. We plot the simulation results in several forms using visualisation tools.

[IMAGE OMITTED. SEE PDF]

We apply the aforementioned procedure with an additional component for lockdown choice for models dependent on lockdown. The evolution function runs the SEIRD Model's evolve function with a modified beta parameter if lockdown is to be enforced after first obtaining the list of cities that are to be placed under lockdown from the knapsack algorithm. The knapsack uses the simulated data again to make new decisions at different time points. Over all the time points with the supplied total days and interval, these procedures are repeated. The lockdown is implemented based on the value of interval; for example, if interval is 5, lockdown is to be implemented for 5 consecutive days. All of the cumulative data are cleaned up and combined when the simulation is finished in order to visualise the outcomes.

All of the models have been compared across various scopes.

PRACTICAL & VISUALISATION– DATA AND PRE-REQUISITES

Python 3.0 with Jupyter Notebook served as the experimental platform for our project. The Python module pandas dataframe has been used for data imports. The data were estimated for experimental purposes after being obtained from the World Data Forum. The original dataset is made up of the following variables: City, Population Exposed, Infected, Death, Bed Capacity, and GDP, where GDP is expressed in US dollars. We selected four cities, but only City 1 disclosed a significant amount. This is to support our hypothesis about viral spread from one city to another. We utilised the Python package matplotlib for graph and chart visualisation, the Python program D-wave for quantum simulation, and the Python tool odeint from scipy for the differential equation of the SEIRD model.

For all the models, the parameters used in the Multicity-SEIRD model are experimental and derived from various research articles [5, 6]. Contact Ratio: β = 1.25, Incubation period: δ = 0.2, Death rate: α = 0.02, proportion of infected recovering per day: γ = 0.25, total number of people an infected person infects: R⁰ = 5.0, number of days an infected person has and can spread the disease D = 4.0, days from infection until death: ρ = 0.16 [5, 6].

The Transfer function Tr uses the distance matrix M_d, (element d_ij is the distance (in Km) between states i and j), and calibrating constant k = 2. $${M}_{d}\,=\left[\begin{array}{@{}cccc@{}}0& 1200& 1300& 2000\\ 1200& 0& 2100& 1600\\ 1300& 2100& 0& 1700\\ 2000& 1600& 1700& 0\end{array}\right]$$

The Transfer of people between the cities is given in Table 1. It depicts the influx to the cities.

TABLE 1 Influx of people into cities due to transmission of the virus.

City	Population	Influx
City_1	1,563,490	2078
City_2	1,249,034	2601
City_3	980,045	2401
City_4	1,107,086	1561

The population though a dependent factor of GDP is not a solely necessary factor. There might not be any dependence between population and GDP, and hence, implementing a lockdown for the city having the highest population might stop the community from spreading the virus, but not affect the overall GDP to a greater extent. For our project, we took the dataset keeping in mind the above issue. Hence, the dependence of Population Composition on GDP Composition is depicted in the pie chart as given in Figure 9.

[IMAGE OMITTED. SEE PDF]

EVOLUTION OF EPIDEMIOLOGICAL MODEL

Using the SEIRD model we run the program for 51 days without lockdown. For open cities, the transmission rate is considered 1.25, and for closed cities, it is much lesser than 0.1. The total time is split into smaller time frames, assuming that the state will be in lockdown or open condition for every interval period. We use the SEIRD Model to design the optimal lockdown over each interval period for every city using the classical and Quantum Knapsack algorithm.

Model 0: No lockdown at any city

In model 0, we have simulated the evolution of the Epidemiological Model without considering lockdown for any city. The odeint function of scipy.integrate differentiate the SEIRD Model equations and return the different compartments of the model after 51 days' time points. For all the cities, a loop has been set and the different compartments of the models are stored in a list dataset.

Table 2 displays the outcome dataset following the model simulation. Since only City 1 was set in the initial dataset before evolution with 100 persons exposed and all the cities' compartments are now non-zero, it may be inferred that the virus has spread to and from other cities, achieving the goal of the Transfer Function. Additionally, the Total Population is fixed after evolution, implicitly supporting the Multicity-SEIRD Model and the idea that only the spread of viruses, rather than real population movement within and between cities, is important. Because the ordinary differential equation uses floating-point numbers, there has been a rounding mistake that may be further corrected by managing data and datasets with exceptional accuracy.

TABLE 2 Dataset after no lockdown SEIRD model evolution.

City	Population	Susceptible	Exposed	Infected	Recovered	Death	Total	Death rate
City_1	1,563,490	10,464	6998	16,720	1,508,778	20,527	1,563,489	0.01312
City_2	1,249,034	7813	4182	10,192	1,210,377	16,467	1,249,033	0.01318
City_3	980,045	5951	2948	7226	950,980	12,938	980,045	0.01320
City_4	1,107,089	7353	4796	11,488	1,068,903	14,542	1,107,085	0.01313

To understand the evolution of the virus into different compartments of the model, a graphical representation of the evolution is needed (Shown in Figure 10–13). We implemented two types of visualisation namely (a) One State One graph that is, each state will have all the compartments of the model itself in a single graph (b) One compartment One Graph that is, each compartment will have all the state's respective evolution graph. Both types of graphs are given below with a brief explanation.

[IMAGE OMITTED. SEE PDF]

[IMAGE OMITTED. SEE PDF]

[IMAGE OMITTED. SEE PDF]

[IMAGE OMITTED. SEE PDF]

The above graphs in Figures 14–18 show the disease's evolution without imposing any lockdown. The graphs cover the four cities where the disease is spreading rapidly. Infection rises exponentially, and with no strict measures like lockdown, it will be a threat to the medical infrastructure. Our objective for the optimisation is to decrease death to the point that the cities can be kept functioning, maintaining an optimised lockdown schedule and sufficient bed capacity available. Table 2 above shows the result of the 51-day run using the SEIRD model without imposing lockdown conditions. The total number of deaths for no lockdown implementation is 64 thousand approximately. The bed capacity is inversely proportional to the infected population's increase with time, as shown in Figure 19. The bed capacity reduces drastically in all four cities, creating the need to implement lockdown, which we want to optimise using the knapsack formulation.

[IMAGE OMITTED. SEE PDF]

[IMAGE OMITTED. SEE PDF]

[IMAGE OMITTED. SEE PDF]

[IMAGE OMITTED. SEE PDF]

[IMAGE OMITTED. SEE PDF]

[IMAGE OMITTED. SEE PDF]

LOCKDOWN OPTIMISATION

We comprehended the necessity and effects of applying lockdown to a certain metropolis in the Theory portion. Before putting the optimised strategic lockdown into practice, we investigated a straightforward lockdown strategy with an if-then structure. The methods compute by iterating over all time points. $$\frac{dE}{ds}=\frac{Exposed\,of\,a\,city\,over\,time}{Susceptible\,of\,a\,city\,over\,time}$$

We set the threshold as 0.05 of $$\frac{dE}{ds}$$, beyond that and if the Infected of the city is more or equal to the bed capacity of the same, we implement lockdown, else kept the city open. The straightforward approach, however ineffective, nonetheless communicates the fundamental goal of lockdown. The lockdown/open choice made by the aforementioned algorithm is shown in Figure 20. The algorithm closes all four cities for a lengthy period of 45 days, which is inefficient and what many nations did, leading to the decline of their economies.

[IMAGE OMITTED. SEE PDF]

Model 1: Optimal Lockdown Scheduling as an application of classical knapsack algorithm

In Table 3, the initial state is listed. We use the same settings for Model 1 that were specified for SEIRD. The complete period of 60 days has been divided with an interval of n = 5 days. The optimised lockdown schedule is determined every 5 days using the conventional knapsack technique, and it is then implemented at the start of the subsequent interval. Table 4 depicts the result of the classical knapsack. Figure 21 shows the lockdown schedule for each time interval as obtained by the classical knapsack algorithm.

TABLE 3 Initial status of our experimental model.

City	Population	Exposed	Infected	Death	Bed capacity	GDP
City_1	1,563,490	100	0	0	20,000	310.0
City_2	1,249,034	0	0	0	15,000	293.0
City_3	980,045	0	0	0	15,050	78.6
City_4	1,107,086	0	0	0	25,000	150.0

TABLE 4 Final values after 60 days of the progress of the SEIRD Model considering optimal lockdown using classical knapsack problem.

City	Population	Susceptible	Exposed	Infected	Recovered	Dead	Total
City_1	1,563,490	561,687	63,122	69,792	857,224	11,662	1,563,490
City_2	1,249,034	390,285	70,305	47,785	730,716	9941	1,249,033
City_3	980,045	389,727	23,721	31,695	527,721	7179	980,044
City_4	1,107,086	595,956	29,484	36,173	439,492	5957	1,107,085

[IMAGE OMITTED. SEE PDF]

The number of deaths by implementing an optimal lockdown schedule classically is approximately 34,736 which is higher than that of quantum knapsack-assisted lockdown.

The bed capacity over time for all four cities is shown in Figure 22, but as the number of lockdown periods increases the bed availability increases.

[IMAGE OMITTED. SEE PDF]

After applying optimal lockdown, the graph significantly differs from the No-Lockdown Model; at intervals of 5 days, this is because of the implementation of lockdown, which causes the transmission rate and transfer function to drop and the rate of characteristics in all compartments, shown in Figures 23–27.

[IMAGE OMITTED. SEE PDF]

[IMAGE OMITTED. SEE PDF]

[IMAGE OMITTED. SEE PDF]

[IMAGE OMITTED. SEE PDF]

[IMAGE OMITTED. SEE PDF]

Model 2: Optimal Lockdown Scheduling as an application of quantum knapsack algorithm

The initial circumstances are given in Table 3. For Model 2 we are using the same parameters as mentioned in SEIRD. With an interval of n = 5 days, the total time duration of 60 days has been split. After 5 days, we apply the quantum knapsack algorithm to determine the optimal lockdown schedule. After 60 days run the result is depicted in Table 5. Figure 28 shows the lockdown schedule for every 5 days as obtained by the knapsack algorithm. For knapsack formulation, the cost is bed capacity and weight is the number of infected. Select the cities to put in a knapsack (carries open cities) such that death is minimised and bed capacity increases. The other cities not in knapsack had to be closed. The trend of the result is chaotic and influenced by many variables. The Quantum Lockdown Algorithm is described above (Algorithm 3).

TABLE 5 Final values after 60 days of the progress of the SEIRD model considering optimal lockdown using quantum knapsack Problem.

City	Population	Susceptible	Exposed	Infected	Recovered	Dead	Total
City_1	1,563,490	1,458,969	26,008	22,167	55,588	756	1,563,489
City_2	1,249,034	1,127,418	46,749	18,508	55,600	756	1,249,034
City_3	980,045	99,109	84,194	84,194	727,000	9891	980,044
City_4	1,107,086	968,528	12,100	13,771	111,173	1512	1,107,086

[IMAGE OMITTED. SEE PDF]

The total infected 12,915 by implementing an optimal quantum lockdown schedule, whereas 70% of bed capacity is used after 60 days using quantum knapsack assisted lockdown.

RESULT

The solution, which incorporates the knapsack problem on the two goals is provided with sample data from four cities. The knapsack formulation is being run on the classical and quantum annealing computers D-Wave QPU DW 2000Q 6, Advantage system5.2, using sample data from 4 cities and their sample. The classical knapsack runs in Ο(n²) while the Quantum knapsack runs in Ο(n).

The GDP and mortality count under no lockdown, classical knapsack assisted lockdown, and quantum knapsack assisted lockdown are compared in the following graphs. It has been noted that lockdown reduces the mortality rate. Quantum lockdown, compared to traditional lockdown, lowers the death count over 60 days, except in city 3.

Below is a snapshot of the Quantum Knapsack's output. The output of the result is presented for each time point. The output consists of a raw data sample from a D-Wave Quantum Computer, salvaged GDP, depleted hospital capacity, lockdown choices, and many more.

The latter exhibits a more promising result when the classical and quantum optimisation strategies are compared. However, the optimisation technique, which is the foundation of the entire project, still includes heuristic properties. Figure 29 shows a comparison of the Infected compartment between the quantum and classical theories. Except for city 3, which exhibits an increasing peak after the 19th day, all other cities in the quantum lockdown optimisation strategic infection compartment have significantly fewer infections than those in the classical lockdown optimisation compartment. Furthermore, for all the cities, there are sufficient days beforehand to take the necessary steps to prevent the peak and maintain the economy.

[IMAGE OMITTED. SEE PDF]

It has been compared between classical and quantum approaches to preserve the economy and prevent the transmission of viruses inside communities, which ultimately aims to reduce mortality rates. Since the GDP of one city can affect the GDP of the entire collective, and since Quantum Computer, City 3 failed to stop the virus in comparison to a classical computer, the resultant infection of all the cities increased at a certain time point and it forced the implementation of all city lockdown, which decreased the overall GDP but kept it from falling in a zigzag pattern, causing havoc and economic instability. Thus, in the example of GDP, we can say that the lockdown decision made by the quantum computer is significantly steadier and more effective than the decision made by the classical computer. The comparison of GDP is depicted in Figure 30.

[IMAGE OMITTED. SEE PDF]

We discovered that the quantum computer's choice to apply the lockdown schedule reduced the death count when compared to the classical method in the case of the death count shown in Figure 31. Even though City 3 displays a greater lockdown, Quantum Knapsack executed Lockdown with 12,916 deaths compared to Classical Knapsack's 34,763 deaths, making Quantum Computer advantageous to Classical Computer for the challenging optimisation task.

[IMAGE OMITTED. SEE PDF]

CONCLUSION

In the past two years, COVID-19 has been responsible for a large number of mortalities. We have experienced an emergency when hospital beds were in short supply due to an outbreak of infection. Strict lockdown is unavoidable to lessen the effects of the epidemic. However, a lockdown cannot last forever since it would have a significant negative impact on the nation. There is a pressing need for an optimised lockdown schedule to reduce mortality and increase the economy. We provide an optimised lockdown calendar to enforce lockdown across short periods using the Knapsack algorithm, employing the S, E, I, R, or D model for forecasting the spread of the SARS-CoV-2 illness. The towns are chosen for the backpack such that the maximum number of beds may be accommodated while ensuring that total death does not exceed the weight of the knapsack. Since there is no lockdown condition, we have analysed the SEIRD model and discovered that imposing lockdown death is repressed. The outcome demonstrates that employing a quantum knapsack for lockdown gave better results. It is discovered that the quantum solution lowers the number of mortalities in each city.

Although quantum computing is still in its early stages, the optimisation model we have presented may be modified to solve more broad issues in the current and post-COVID situations. When the vaccine is ready, the model will be updated to incorporate a decline in the vulnerable population that is reliant on immunity. An ideal vaccination schedule may also be created using the presented methodology. Our suggested methodology might be used to open the city or town, business, school, colleges, institutions etc. after the lockdown.

We demonstrate that this model may be utilised to achieve the goal of facilitating city opening via the COVID-19 crisis with the proper medical and policy monitoring, validation, and ongoing calibration with real data. The quantum computer still has a long way to go, there is little question that in the future, real-world complicated issues that concern humanity will be solved by utilising a well-planned, broad quantum methodology.

AUTHOR CONTRIBUTION

All authors contributed in this work.

ACKNOWLEDGEMENT

No funding is available for this research work.

CONFLICT OF INTEREST STATEMENT

The authors have no conflict of interest.

DATA AVAILABILITY STATEMENT

The data that support the findings of this study are available on request from the corresponding author. The data are not publicly available due to privacy or ethical restrictions.

References

Kermack, W.O., McKendrick, A.G.: Contributions to the mathematical theory of epidemics—I. Bull. Math. Biol. 53(1‐2), 33–55 (1991). [DOI: https://dx.doi.org/10.1007/bf02464423]