Quantum complex network

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As part of network science, the study of quantum complex networks aims to explore the impact of complexity science and network architectures in quantum systems. According to quantum information theory, it is possible to improve communication security and data transfer rates by taking advantage of quantum mechanics. In this context, the study of quantum complex networks is motivated by the possibility of quantum communications being used on an enterprise scale in the future. In this case, it is likely that quantum communication networks will acquire nontrivial features, as is common in existing communication networks today.

Motivation

In theory, it is possible to take advantage of quantum mechanics to create secure and faster communications. For example, quantum key distribution is an application of quantum cryptography that enables secure communications, and quantum teleportation can be used to transfer data at a higher rate than one can using only classical channels.

Successful quantum teleportation experiments in 1998, followed by the development of the first quantum communication networks in 2004, open the possibility of quantum communication being used on a large scale in the future. According to findings in network science, network topology is extremely important in many cases, and existing large scale communication networks tend to have non-trivial topologies and characteristics, such as small world effect, community structure, or being a scale-free network. The study of networks with quantum properties and complex network topologies can help us to not only better understand such networks but also use the network topology to improve the efficiency of communication networks in the future.

Important concepts

Qubits

In quantum information theory, qubits are analogous to bits in classical systems. A qubit is a quantum object that, when measured, can be found to be in one of only two states, and that is used to transmit information. Photon polarization or nuclear spin are examples of two-state systems that can be used as qubits.

Entanglement

Quantum entanglement is a physical phenomenon characterized by correlation between the quantum states of two or more particles. While entangled particles do not interact in the classical sense, the quantum state of such particles cannot be described independently due to their entanglement. Particles can be entangled in many ways. The maximally entangled states are the ones that maximize the entropy of entanglement. In the context of quantum communication, entangled qubits are used as a quantum channel, which is capable of transmitting information when combined with a classical channel.

Bell measurement

Bell measurement is a kind of joint quantum-mechanical measurement of two qubits such that, after the measurement, the two qubits will be maximally entangled.

Entanglement swapping

Entanglement swapping is a frequently used strategy used in the study of quantum networks that allows connections in the network to change. For example, suppose that we have 4 qubits, A, B, C and D, such that qubits C and D belong to the same station, while A and C belong to two different stations, and where qubit A is entangled with qubit C and qubit B is entangled with qubit D. By performing a Bell measurement for qubits A and B, not only will qubits A and B be entangled, but it is also possible to entangle qubits C and D, despite the fact that these two qubits never interacted directly with each other. Following this process, the entanglement between qubits A and C, and qubits B and D will be lost. This strategy can be used to shape connections on the network.

Network structure

While not all models for quantum complex networks follow exactly the same structure, usually, in such models, a node represents a set of qubits in the same station (where operations like Bell measurements and entanglement swapping can be applied) and an edge between node ${\displaystyle i}$ and ${\displaystyle j}$ means that a qubit in node ${\displaystyle i}$ is entangled to a qubit in node ${\displaystyle j}$, although those two qubits are in different places and so cannot physically interact. Quantum networks where the links are interaction terms instead of entanglement may also be considered but for very different purposes.

Notation

Each node in the network contains a set of qubits that can be in different states. To represent the quantum state of these qubits, it is convenient to use Dirac notation and represent the two possible states of each qubit as ${\displaystyle |0\rangle }$ and ${\displaystyle |1\rangle }$. In this notation, two particles are entangled if the joint wave function, ${\displaystyle |\psi _{ij}\rangle }$, cannot be decomposed as

${\displaystyle |\psi _{ij}\rangle =|\phi \rangle _{i}\otimes |\phi \rangle _{j},}$

where ${\displaystyle |\phi \rangle _{i}}$ represents the quantum state of the qubit at node i and ${\displaystyle |\phi \rangle _{j}}$ represents the quantum state of the qubit at node j.

Another important concept is maximally entangled states. The four states (the Bell states) that maximize the entropy of entanglement between two qubits can be written as follows:

${\displaystyle |\Phi _{ij}^{+}\rangle ={\frac {1}{\sqrt {2}}}(|0\rangle _{i}\otimes |0\rangle _{j}+|1\rangle _{i}\otimes |1\rangle _{j}),}$

${\displaystyle |\Phi _{ij}^{-}\rangle ={\frac {1}{\sqrt {2}}}(|0\rangle _{i}\otimes |0\rangle _{j}-|1\rangle _{i}\otimes |1\rangle _{j}),}$

${\displaystyle |\Psi _{ij}^{+}\rangle ={\frac {1}{\sqrt {2}}}(|0\rangle _{i}\otimes |1\rangle _{j}+|1\rangle _{i}\otimes |0\rangle _{j}),}$

${\displaystyle |\Psi _{ij}^{-}\rangle ={\frac {1}{\sqrt {2}}}(|0\rangle _{i}\otimes |1\rangle _{j}-|1\rangle _{i}\otimes |0\rangle _{j}).}$

Models

Quantum random networks

The quantum random network model proposed by Perseguers et al. (2009) can be thought of as a quantum version of the Erdős–Rényi model. In this model, each node contains ${\displaystyle N-1}$ qubits, one for each other node. The degree of entanglement between a pair of nodes, represented by ${\displaystyle p}$, plays a similar role to the parameter ${\displaystyle p}$ in the Erdős–Rényi model: in the Erdős–Rényi model, two nodes form a connection with probability ${\displaystyle p}$, whereas in the context of quantum random networks, ${\displaystyle p}$ refers to the probability of an entangled pair of qubits being successfully converted to a maximally entangled state using only local operations and classical communication.

Using the Dirac notation introduced previously, we can represent a pair of entangled qubits connecting the nodes ${\displaystyle i}$ and ${\displaystyle j}$ as

${\displaystyle |\psi _{ij}\rangle ={\sqrt {1-p/2}}|0\rangle _{i}\otimes |0\rangle _{j}+{\sqrt {p/2}}|1\rangle _{i}\otimes |1\rangle _{j},}$

For ${\displaystyle p=0}$, the two qubits are not entangled:

${\displaystyle |\psi _{ij}\rangle =|0\rangle _{i}\otimes |0\rangle _{j},}$

and for ${\displaystyle p=1}$, we obtain the maximally entangled state:

${\displaystyle |\psi _{ij}\rangle ={\sqrt {1/2}}(|0\rangle _{i}\otimes |0\rangle _{j}+|1\rangle _{i}\otimes |1\rangle _{j})}$.

For intermediate values of ${\displaystyle p}$, ${\displaystyle 0<p<1}$, any entangled state is, with probability ${\displaystyle p}$, successfully converted to the maximally entangled state using LOCC operations.

One feature that distinguishes this model from its classical analogue is the fact that, in quantum random networks, links are only truly established after measurements in the networks being made, and it is possible to take advantage of this fact to shape the final state of the network. For an initial quantum complex network with an infinite number of nodes, Perseguers et al. showed that, by doing the right measurements and entanglement swapping, it is possible to collapse the initial network to a network containing any finite subgraph, provided that ${\displaystyle p}$ scales with ${\displaystyle N}$ as ${\textstyle p\sim N^{Z}}$, where ${\displaystyle Z\geq -2}$. This result is contrary to what we find in classical graph theory, where the type of subgraphs contained in a network is bounded by the value of ${\displaystyle z}$.

Entanglement percolation

The goal of entanglement percolation models is to determine if a quantum network is capable of establishing a connection between two arbitrary nodes through entanglement, and to find the best strategies to create such connections.

In a model proposed by Cirac et al. (2007) and applied to complex networks by Cuquet et al. (2009), nodes are distributed in a lattice or in a complex network, and each pair of neighbors share two pairs of entangled qubits that can be converted to a maximally entangled qubit pair with probability ${\displaystyle p}$. We can think of maximally entangled qubits as the true links between nodes. In classical percolation theory, with a probability ${\displaystyle p}$ of two nodes being connected, there is a critical value of ${\displaystyle p}$ (denoted by ${\displaystyle p_{c}}$), so that if ${\displaystyle p>p_{c}}$ there is a finite probability of a path between two randomly selected nodes existing, and for ${\displaystyle p<p_{c}}$ the probability of such a path existing is asymptotically zero. ${\displaystyle p_{c}}$ depends only on the topology of the network.

A similar phenomenon was found in the model proposed by Cirac et al. (2007), where the probability of forming a maximally entangled state between two randomly selected nodes is zero if ${\displaystyle p<p_{c}}$ and finite if ${\displaystyle p>p_{c}}$. The main difference between classical and entangled percolation is that, in quantum networks, it is possible to change the links in the network, in a way changing the effective topology of the network. As a result, ${\displaystyle p_{c}}$ depends on the strategy one uses to convert partially entangled qubits to maximally connected qubits. With a naïve approach, ${\displaystyle p_{c}}$ for a quantum network is equal to ${\displaystyle p_{c}}$ for a classic network with the same topology. Nevertheless, it was shown that is possible to take advantage of quantum swapping to lower ${\displaystyle p_{c}}$ both in regular lattices and complex networks.

Mathematical modelling of infectious disease

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https://en.wikipedia.org/wiki/Mathematical_modelling_of_infectious_disease 

Mathematical models can project how infectious diseases progress to show the likely outcome of an epidemic (including in plants) and help inform public health and plant health interventions. Models use basic assumptions or collected statistics along with mathematics to find parameters for various infectious diseases and use those parameters to calculate the effects of different interventions, like mass vaccination programs. The modelling can help decide which intervention(s) to avoid and which to trial, or can predict future growth patterns, etc.

History

The modeling of infectious diseases is a tool that has been used to study the mechanisms by which diseases spread, to predict the future course of an outbreak and to evaluate strategies to control an epidemic.

The first scientist who systematically tried to quantify causes of death was John Graunt in his book Natural and Political Observations made upon the Bills of Mortality, in 1662. The bills he studied were listings of numbers and causes of deaths published weekly. Graunt's analysis of causes of death is considered the beginning of the "theory of competing risks" which according to Daley and Gani  is "a theory that is now well established among modern epidemiologists".

The earliest account of mathematical modelling of spread of disease was carried out in 1760 by Daniel Bernoulli. Trained as a physician, Bernoulli created a mathematical model to defend the practice of inoculating against smallpox. The calculations from this model showed that universal inoculation against smallpox would increase the life expectancy from 26 years 7 months to 29 years 9 months. Daniel Bernoulli's work preceded the modern understanding of germ theory.

In the early 20th century, William Hamer and Ronald Ross applied the law of mass action to explain epidemic behaviour.

The 1920s saw the emergence of compartmental models. The Kermack–McKendrick epidemic model (1927) and the Reed–Frost epidemic model (1928) both describe the relationship between susceptible, infected and immune individuals in a population. The Kermack–McKendrick epidemic model was successful in predicting the behavior of outbreaks very similar to that observed in many recorded epidemics.

Recently, agent-based models (ABMs) have been used in exchange for simpler compartmental models. For example, epidemiological ABMs have been used to inform public health (nonpharmaceutical) interventions against the spread of SARS-CoV-2. Epidemiological ABMs, in spite of their complexity and requiring high computational power, have been criticized for simplifying and unrealistic assumptions. Still, they can be useful in informing decisions regarding mitigation and suppression measures in cases when ABMs are accurately calibrated.

Assumptions

Models are only as good as the assumptions on which they are based. If a model makes predictions that are out of line with observed results and the mathematics is correct, the initial assumptions must change to make the model useful.

• Rectangular and stationary age distribution, i.e., everybody in the population lives to age L and then dies, and for each age (up to L) there is the same number of people in the population. This is often well-justified for developed countries where there is a low infant mortality and much of the population lives to the life expectancy.
• Homogeneous mixing of the population, i.e., individuals of the population under scrutiny assort and make contact at random and do not mix mostly in a smaller subgroup. This assumption is rarely justified because social structure is widespread. For example, most people in London only make contact with other Londoners. Further, within London then there are smaller subgroups, such as the Turkish community or teenagers (just to give two examples), who mix with each other more than people outside their group. However, homogeneous mixing is a standard assumption to make the mathematics tractable.

Types of epidemic models

Stochastic

"Stochastic" means being or having a random variable. A stochastic model is a tool for estimating probability distributions of potential outcomes by allowing for random variation in one or more inputs over time. Stochastic models depend on the chance variations in risk of exposure, disease and other illness dynamics. Statistical agent-level disease dissemination in small or large populations can be determined by stochastic methods.

Deterministic

When dealing with large populations, as in the case of tuberculosis, deterministic or compartmental mathematical models are often used. In a deterministic model, individuals in the population are assigned to different subgroups or compartments, each representing a specific stage of the epidemic.

The transition rates from one class to another are mathematically expressed as derivatives, hence the model is formulated using differential equations. While building such models, it must be assumed that the population size in a compartment is differentiable with respect to time and that the epidemic process is deterministic. In other words, the changes in population of a compartment can be calculated using only the history that was used to develop the model.

Sub-exponential growth

A common explanation for the growth of epidemics holds that 1 person infects 2, those 2 infect 4 and so on and so on with the number of infected doubling every generation. It is analogous to a game of tag where 1 person tags 2, those 2 tag 4 others who've never been tagged and so on. As this game progresses it becomes increasing frenetic as the tagged run past the previously tagged to hunt down those who have never been tagged. Thus this model of an epidemic leads to a curve that grows exponentially until it crashes to zero as all the population have been infected. i.e. no herd immunity and no peak and gradual decline as seen in reality.

A better analogy is a game of roulette. Consider a wheel made of only red and black pockets, spin the wheel and roll a ball. The rules of the game are that if a ball lands in a red or black pocket then the pocket turns green and, on the next spin, an extra ball is rolled. If, instead, a ball lands in a green pocket then it is removed from play. The game is played until there are no balls left to roll at which point there might have been exponential growth observed but it is highly improbable. Taking the average of a number of games and it is expected that the initial growth is sub-exponential and that, at the end of the game, not all of the pockets will be green which corresponds to herd immunity.

Reproduction number

Main article: Basic reproduction number

The basic reproduction number (denoted by R₀) is a measure of how transferable a disease is. It is the average number of people that a single infectious person will infect over the course of their infection. This quantity determines whether the infection will increase sub-exponentially, die out, or remain constant: if R₀ > 1, then each person on average infects more than one other person so the disease will spread; if R₀ < 1, then each person infects fewer than one person on average so the disease will die out; and if R₀ = 1, then each person will infect on average exactly one other person, so the disease will become endemic: it will move throughout the population but not increase or decrease.

Endemic steady state

An infectious disease is said to be endemic when it can be sustained in a population without the need for external inputs. This means that, on average, each infected person is infecting exactly one other person (any more and the number of people infected will grow sub-exponentially and there will be an epidemic, any less and the disease will die out). In mathematical terms, that is:

${\displaystyle \ R_{0}S\ =1.}$

The basic reproduction number (R₀) of the disease, assuming everyone is susceptible, multiplied by the proportion of the population that is actually susceptible (S) must be one (since those who are not susceptible do not feature in our calculations as they cannot contract the disease). Notice that this relation means that for a disease to be in the endemic steady state, the higher the basic reproduction number, the lower the proportion of the population susceptible must be, and vice versa. This expression has limitations concerning the susceptibility proportion, e.g. the R₀ equals 0.5 implicates S has to be 2, however this proportion exceeds the population size.

Assume the rectangular stationary age distribution and let also the ages of infection have the same distribution for each birth year. Let the average age of infection be A, for instance when individuals younger than A are susceptible and those older than A are immune (or infectious). Then it can be shown by an easy argument that the proportion of the population that is susceptible is given by:

${\displaystyle S={\frac {A}{L}}.}$

We reiterate that L is the age at which in this model every individual is assumed to die. But the mathematical definition of the endemic steady state can be rearranged to give:

${\displaystyle S={\frac {1}{R_{0}}}.}$

Therefore, due to the transitive property:

${\displaystyle {\frac {1}{R_{0}}}={\frac {A}{L}}\Rightarrow R_{0}={\frac {L}{A}}.}$

This provides a simple way to estimate the parameter R₀ using easily available data.

For a population with an exponential age distribution,

${\displaystyle R_{0}=1+{\frac {L}{A}}.}$

This allows for the basic reproduction number of a disease given A and L in either type of population distribution.

Compartmental models in epidemiology

Main article: Compartmental models in epidemiology

Compartmental models are formulated as Markov chains. A classic compartmental model in epidemiology is the SIR model, which may be used as a simple model for modeling epidemics. Multiple other types of compartmental models are also employed.

The SIR model

Diagram of the SIR model with initial values ${\textstyle S(0)=997,I(0)=3,R(0)=0}$, and rates for infection ${\textstyle \beta =0.4}$ and for recovery ${\textstyle \gamma =0.04}$

Animation of the SIR model with initial values ${\textstyle S(0)=997,I(0)=3,R(0)=0}$, and rate of recovery ${\textstyle \gamma =0.04}$. The animation shows the effect of reducing the rate of infection from ${\textstyle \beta =0.5}$ to ${\textstyle \beta =0.12}$. If there is no medicine or vaccination available, it is only possible to reduce the infection rate (often referred to as "flattening the curve") by appropriate measures such as social distancing.

In 1927, W. O. Kermack and A. G. McKendrick created a model in which they considered a fixed population with only three compartments: susceptible, ${\displaystyle S(t)}$; infected, ${\displaystyle I(t)}$; and recovered, ${\displaystyle R(t)}$. The compartments used for this model consist of three classes:

• ${\displaystyle S(t)}$ is used to represent the individuals not yet infected with the disease at time t, or those susceptible to the disease of the population.
• ${\displaystyle I(t)}$ denotes the individuals of the population who have been infected with the disease and are capable of spreading the disease to those in the susceptible category.
• ${\displaystyle R(t)}$ is the compartment used for the individuals of the population who have been infected and then removed from the disease, either due to immunization or due to death. Those in this category are not able to be infected again or to transmit the infection to others.

Other compartmental models

There are many modifications of the SIR model, including those that include births and deaths, where upon recovery there is no immunity (SIS model), where immunity lasts only for a short period of time (SIRS), where there is a latent period of the disease where the person is not infectious (SEIS and SEIR), and where infants can be born with immunity (MSIR).

Infectious disease dynamics

Mathematical models need to integrate the increasing volume of data being generated on host-pathogen interactions. Many theoretical studies of the population dynamics, structure and evolution of infectious diseases of plants and animals, including humans, are concerned with this problem. Research topics include:

• antigenic shift
• epidemiological networks
• evolution and spread of resistance
• immuno-epidemiology
• intra-host dynamics
• Pandemic
• pathogen population genetics
• persistence of pathogens within hosts
• phylodynamics
• role and identification of infection reservoirs
• role of host genetic factors
• spatial epidemiology
• statistical and mathematical tools and innovations
• Strain (biology) structure and interactions
• transmission, spread and control of infection
• virulence

Mathematics of mass vaccination

If the proportion of the population that is immune exceeds the herd immunity level for the disease, then the disease can no longer persist in the population and its transmission dies out. Thus, a disease can be eliminated from a population if enough individuals are immune due to either vaccination or recovery from prior exposure to disease. For example, smallpox eradication, with the last wild case in 1977, and certification of the eradication of indigenous transmission of 2 of the 3 types of wild poliovirus (type 2 in 2015, after the last reported case in 1999, and type 3 in 2019, after the last reported case in 2012).

The herd immunity level will be denoted q. Recall that, for a stable state:

${\displaystyle R_{0}\cdot S=1.}$

In turn,

${\displaystyle R_{0}={\frac {N}{S}}={\frac {\mu N\operatorname {E} (T_{L})}{\mu N\operatorname {E} [\min(T_{L},T_{S})]}}={\frac {\operatorname {E} (T_{L})}{\operatorname {E} [\min(T_{L},T_{S})]}},}$

which is approximately:

${\displaystyle {\frac {\operatorname {\operatorname {E} } (T_{L})}{\operatorname {\operatorname {E} } (T_{S})}}=1+{\frac {\lambda }{\mu }}={\frac {\beta N}{v}}.}$

Graph of herd immunity threshold vs basic reproduction number with selected diseases

S will be (1 − q), since q is the proportion of the population that is immune and q + S must equal one (since in this simplified model, everyone is either susceptible or immune). Then:

${\displaystyle {\begin{aligned}&R_{0}\cdot (1-q)=1,\\[6pt]&1-q={\frac {1}{R_{0}}},\\[6pt]&q=1-{\frac {1}{R_{0}}}.\end{aligned}}}$

Remember that this is the threshold level. Die out of transmission will only occur if the proportion of immune individuals exceeds this level due to a mass vaccination programme.

We have just calculated the critical immunization threshold (denoted q_c). It is the minimum proportion of the population that must be immunized at birth (or close to birth) in order for the infection to die out in the population.

${\displaystyle q_{c}=1-{\frac {1}{R_{0}}}.}$

Because the fraction of the final size of the population p that is never infected can be defined as:

${\displaystyle \lim _{t\to \infty }S(t)=e^{-\int _{0}^{\infty }\lambda (t)\,dt}=1-p.}$

Hence,

${\displaystyle p=1-e^{-\int _{0}^{\infty }\beta I(t)\,dt}=1-e^{-R_{0}p}.}$

Solving for ${\displaystyle R_{0}}$, we obtain:

${\displaystyle R_{0}={\frac {-\ln(1-p)}{p}}.}$

When mass vaccination cannot exceed the herd immunity

If the vaccine used is insufficiently effective or the required coverage cannot be reached, the program may fail to exceed q_c. Such a program will protect vaccinated individuals from disease, but may change the dynamics of transmission.

Suppose that a proportion of the population q (where q < q_c) is immunised at birth against an infection with R₀ > 1. The vaccination programme changes R₀ to R_q where

${\displaystyle R_{q}=R_{0}(1-q)}$

This change occurs simply because there are now fewer susceptibles in the population who can be infected. R_q is simply R₀ minus those that would normally be infected but that cannot be now since they are immune.

As a consequence of this lower basic reproduction number, the average age of infection A will also change to some new value A_q in those who have been left unvaccinated.

Recall the relation that linked R₀, A and L. Assuming that life expectancy has not changed, now:

${\displaystyle R_{q}={\frac {L}{A_{q}}},}$

${\displaystyle A_{q}={\frac {L}{R_{q}}}={\frac {L}{R_{0}(1-q)}}.}$

But R₀ = L/A so:

${\displaystyle A_{q}={\frac {L}{(L/A)(1-q)}}={\frac {AL}{L(1-q)}}={\frac {A}{1-q}}.}$

Thus, the vaccination program may raise the average age of infection, and unvaccinated individuals will experience a reduced force of infection due to the presence of the vaccinated group. For a disease that leads to greater clinical severity in older populations, the unvaccinated proportion of the population may experience the disease relatively later in life than would occur in the absence of vaccine.

When mass vaccination exceeds the herd immunity

If a vaccination program causes the proportion of immune individuals in a population to exceed the critical threshold for a significant length of time, transmission of the infectious disease in that population will stop. If elimination occurs everywhere at the same time, then this can lead to eradication.

Elimination

Interruption of endemic transmission of an infectious disease, which occurs if each infected individual infects less than one other, is achieved by maintaining vaccination coverage to keep the proportion of immune individuals above the critical immunization threshold.

Eradication

Elimination everywhere at the same time such that the infectious agent dies out (for example, smallpox and rinderpest).

Reliability

Models have the advantage of examining multiple outcomes simultaneously, rather than making a single forecast. Models have shown broad degrees of reliability in past pandemics, such as SARS, SARS-CoV-2, Swine flu, MERS and Ebola.