In this section, we constructed the Quantum Healthy-Infected Model (QHIM) based on the concept of quantum superposition in quantum mechanics. Quantum superposition is a basic concept in quantum mechanics that describes the phenomenon where a quantum system can exist in multiple different states simultaneously21. Unlike the deterministic states in classical physics, the state of a quantum system is described probabilistically and can exist as a superposition of several possible states. The meaning of quantum superposition is that a quantum system can simultaneously be in a ‘superposition’ of multiple possible states, and it only ‘collapses’ into a definite state upon measurement.
In quantum mechanics, the state of a system is typically represented by a state vector, denoted as \( |\left. \psi \right\rangle\). Quantum superposition refers to a linear combination of several basis states. Suppose there is a quantum system that can exist in two states, \(|\left. 0 \right\rangle\) and \(|\left. 1 \right\rangle\). These two states are orthogonal to each other, meaning they are independent. If the quantum system is in a superposition state, its state can be represented as:
$$ \begin{array}{*{20}c} {\left| {\left. \psi \right\rangle } \right. = \alpha \left| {\left. 0 \right\rangle } \right. + \beta \left. {\left| 1 \right.} \right\rangle } \\ \end{array}. $$
(1)
where \( |\left. 0 \right\rangle\) and \( |\left. 1 \right\rangle\) represent the basis states of the system.\( \alpha\) and \(\beta \) are complex coefficients that represent the probability amplitudes for the different states. The quantities \( \left| \alpha \right|^{2}\) and \( \left| \beta \right|^{2} \) represent the probabilities of the system being in states \( |\left. 0 \right\rangle\) and \( |\left. 1 \right\rangle\), respectively. These coefficients satisfy the normalization condition \( \left| \alpha \right|^{2} + \left| \beta \right|^{2} = 1\).
In classical physics, a system can only be in one specific state, such as a coin being either heads or tails. However, in quantum mechanics, before measurement, a quantum system can exist in multiple states simultaneously. Although the quantum system is not in a definite state while in superposition, when a measurement is made, the system collapses to a definite state.
In the QHIM, we assume that the individual’s ground state consists of the infected ground state and the healthy ground state:
$$ \begin{array}{*{20}c} {\left| {\left. \psi \right\rangle } \right. = a\left| {\left. 0 \right\rangle } \right. + b\left| {\left. 1 \right\rangle } \right..} \\ \end{array} $$
(2)
In the model, \(|\left. 0 \right\rangle\) represents the healthy ground state of an individual, while \(|\left. 1 \right\rangle\) denotes the infected ground state. The individual’s state during the propagation phase is represented as a linear combination of these two ground states. Intuitively, an individual’s state exists in a superposition of healthy and infected states, and only upon observation does the individual’s state become definite. The coefficients \( a\) and \(b\) are the amplitudes corresponding to the healthy and infected states, respectively, and they also satisfy the normalization condition \(\left| a \right|^{2} + \left| b \right|^{2} = 1\). The values \(\left| a \right|^{2}\) and \(\left| b \right|^{2}\) not only indicate the probability of an individual being in each of the ground states but also reflect their health and infection levels. Figure 1 illustrates the network constructed using QHIM, where the degree of infection of a node varies continuously with the amplitude due to the linear superposition of ground states.
An example of a network constructed using QHIM.
Infection processes
We use quantum gate operations to control the state of an individual’s quantum superposition in order to simulate dynamic processes such as infection and recovery. Quantum gates are fundamental operations in quantum computing, which modify quantum states by rotating, superposing, or entangling23,24. By adjusting the parameters of these quantum gates, we can control the intensity and speed of infection spread, thereby accurately simulating the dynamic changes in disease transmission.
Assuming that the infection process exists between a pair of individuals \(u\) and \(v\), the model describes the process by constructing a controlled revolving gate \(CRY\left( \theta \right)\) by combining the revolving gate \(RY\left( \theta \right)\) in the quantum gate operation. The \(RY\left( \theta \right)\) gate is a basic quantum gate operation that changes the state of the superposition by a rotation angle \(\theta\):
$$ \begin{array}{*{20}c} {RY\left( \theta \right) = \left[ {\begin{array}{*{20}c} {\cos \left( {\frac{\theta }{2}} \right)} & { – \sin \left( {\frac{\theta }{2}} \right)} \\ {\sin \left( {\frac{\theta }{2}} \right)} & {\cos \left( {\frac{\theta }{2}} \right)} \\ \end{array} } \right],} \\ \end{array} $$
(3)
The controlled revolving gate \(CRY\left( \theta \right)\), constructed using \(RY\left( \theta \right)\), is represented as:
$$ \begin{array}{*{20}c} {CRY\left( \theta \right) = \left| {\left. 0 \right\rangle \left\langle 0 \right.\left| { \otimes E + } \right|\left. 1 \right\rangle \left\langle 1 \right.} \right| \otimes RY\left( \theta \right),} \\ \end{array} $$
(4)
where \(E\) represents the unit matrix. \(CRY\left( \theta \right)\) indicates that when \(u\) is in a perfectly healthy base state, \(v\) will be unaffected. When \(u\) is in a superimposed state of infection and health, the state of \(v\) will be altered by the action of \(RY\left( \theta \right)\). Since \( \theta\) controls the strength of the revolving gate action, this parameter is used in this study to represent the strength of infection when individuals interact with each other. After applying \(CRY\left( \theta \right)\), the joint state of \(u\) and \(v\) can be represented as the tensor product of these two states:
$$ \begin{aligned} |\left. {{\uppsi }_{{{\text{uv}}}} } \right\rangle & = \left( {a_{u} \left| 0 \right. + b_{u} |\left. 1 \right\rangle } \right) \otimes \left( {a_{v} \left| {\left. 0 \right\rangle } \right. + b_{v} |\left. 1 \right\rangle } \right) \\ & = (a_{u} a_{v} \left| {\left. {00} \right\rangle } \right. + a_{u} b_{v} \left| {\left. {01} \right\rangle } \right. + b_{u} a_{v} \left| {\left. {10} \right\rangle } \right. + b_{u} b_{v} \left| {\left. {11} \right\rangle } \right.. \\ \end{aligned} $$
(5)
The joint state after applying \(CRY\left( \theta \right)\) is
$$ \begin{array}{*{20}c} {\left| {\left. {{\uppsi }_{{{\text{uv}}}} } \right\rangle } \right. = a_{u} a_{v} \left| {\left. {00} \right\rangle } \right. + a_{u} b_{v} \left| {\left. {01} \right\rangle } \right. + \left( {b_{u} a_{v} + b_{u} \sin \left( {\frac{\theta }{2}} \right)b_{v} } \right)\left| {\left. {10} \right\rangle } \right. + b_{u} \cos \left( {\frac{\theta }{2}} \right)b_{v} \left| {\left. {11} \right\rangle } \right..} \\ \end{array} $$
(6)
We can extract the final state of \(v\) from the joint state:
$$ \begin{array}{*{20}c} {\left| {\left. {{\uppsi }_{{\text{v}}} } \right\rangle } \right. = \left( {a_{u} a_{v} + b_{u} a_{v} + b_{u} \sin \left( {\frac{\theta }{2}} \right)b_{v} } \right)\left| {\left. 0 \right\rangle } \right. + \left( {a_{u} b_{v} + b_{u} \cos \left( {\frac{\theta }{2}} \right)b_{v} } \right)\left| {\left. 1 \right\rangle } \right..} \\ \end{array} $$
(7)
This implies that the state change of healthy individuals upon exposure to infected individuals is influenced by both the state of the infected person and the intensity of their interaction.
Self-evolutionary process
It is hypothesized that after infection, an individual undergoes a period of self-evolution, which consists of three stages: Stage 1 is a phase of increasing symptoms \(\left( {0 \le t \le T_{1} } \right)\). Stage 2 represents a gradual recovery period \(\left( {T_{1} \le t \le T_{2} } \right)\). Finally, Stage 3 is the immunization phase \(\left( {t \ge T_{2} } \right)\). This process can be described by the piecewise function \(w\left( t \right)\):
$$ w\left( t \right) = \left\{ {\begin{array}{*{20}l} {\pi \left( {1 – \frac{1}{{1 + e^{{\lambda_{1} \left( {t – \frac{{T_{1} }}{2}} \right)}} }}} \right),{ }0 \le t \le T_{1} } \hfill \\ {\pi e^{{ – \lambda_{2} \left( {t – T_{1} } \right)}} ,{ }T_{1} \le t \le T_{2} } \hfill \\ {0, { }T_{2} \le t} \hfill \\ \end{array} .} \right. $$
(8)
In this context, \(\lambda_{1} \) and \(\lambda_{2}\) are parameters that control the rates of symptom progression and the immunization process, respectively. Combined with the rotation gate \(RY\left( {w\left( t \right)} \right)\), which describes the individual’s self-evolution, the state of the individual at time \(t\) can be represented as:
$$ \begin{array}{*{20}c} {\left| {\left. {\psi \left( t \right)} \right\rangle } \right. = \left( {a\cos \left( {\frac{w\left( t \right)}{2}} \right) + ibsin\left( {\frac{w\left( t \right)}{2}} \right)} \right)\left| {\left. 0 \right\rangle } \right. + \left( {bcos\left( {\frac{w\left( t \right)}{2}} \right) – iasin\left( {\frac{w\left( t \right)}{2}{ }} \right)} \right)\left. {\left| 1 \right.} \right\rangle .} \\ \end{array} $$
(9)
In a network with \(N\) nodes, the number of healthy and infected nodes observed at time \(t\) is:
$$ \begin{array}{*{20}c} {N_{0} \left( t \right) = \mathop \sum \limits_{u = 1}^{N} \left| {a_{u} \left( t \right)} \right|^{2} ,N_{1} \left( t \right) = \mathop \sum \limits_{u = 1}^{N} \left| {b_{u} \left( t \right)} \right|^{2} .} \\ \end{array} $$
(10)
Disease-free equilibrium points of the model
Near the disease-free equilibrium of the model, the states of all individuals are assumed to be very close to the healthy base state. From this assumption, the Jacobian matrix \(J\) can be constructed, with its elements given by:
$$ \begin{array}{*{20}c} {J_{ij} = \frac{{\partial \left( {\frac{{d\left| {b_{u} } \right|^{2} }}{dt}} \right)}}{{\partial \left| {b_{v} } \right|^{2} }} = \sin^{2} \left( {\frac{{\theta_{uv} }}{2}} \right) – \gamma \left( t \right)\delta_{uv} , \exists u,v \in N.} \\ \end{array} $$
(11)
$$ \begin{array}{*{20}c} {\gamma \left( t \right) = \frac{{d\left( {\cos^{2} \left( {\frac{w\left( t \right)}{2}} \right.} \right)}}{dt}.} \\ \end{array} $$
(12)
\(\gamma \left( t \right)\) represents the rate of change of the amplitude of the healthy base state, and \(\delta_{uv}\) denotes the Kronecker delta function. The stability of the disease-free equilibrium is determined by the eigenvalues of the Jacobian matrix \(J\). Let \(\lambda_{max}\) be the maximum eigenvalue. According to Gershgorin’s disk theorem, the upper bound of \(\lambda_{max}\) is determined by the sum of the diagonal and off-diagonal elements of each row of the matrix, given by:
$$ \begin{array}{*{20}c} {\lambda_{max} \le max_{u} \left( {\left| {J_{uu} } \right| + \mathop \sum \limits_{v \ne u} \left| {J_{uv} } \right|} \right)} \\ \end{array} $$
(13)
Substituting \(J_{uu}\) and \(J_{uv}\), the upper limit of \(\lambda_{max}\) can be expressed as:
$$ \begin{array}{*{20}c} {\lambda_{max} \le {\text{max}}_{{\text{u}}} \left( {\mathop \sum \limits_{u} \sin^{2} \left( {\frac{{\theta_{uv} }}{2}} \right) – \gamma \left( t \right)} \right)} \\ \end{array} $$
(14)
It can be deduced that a sufficient condition for the existence of disease-free equilibrium points of the model is:
$$ \begin{array}{*{20}c} {\mathop \sum \limits_{u} \mathop \sum \limits_{v} \sin^{2} \left( {\frac{{\theta_{uv} }}{2}} \right) < \gamma \left( t \right), \exists u,v \in N.} \\ \end{array} $$
(15)
The basic reproduction number of the model, \(R_{0}\), is defined as the ratio of the total infectious intensity to the recovery intensity. It can be derived based on condition for the existence of a disease-free equilibrium:
$$ \begin{array}{*{20}c} {R_{0} = \frac{1}{N\gamma \left( t \right)}\mathop \sum \limits_{u} \mathop \sum \limits_{v} \sin^{2} \left( {\frac{{\theta_{uv} }}{2}} \right).} \\ \end{array} $$
(16)
\(R_{0}\) depends on the parameters of both the infection process and the evolution process, exhibiting dynamic changes over time. When \(R_{0} > 1\), the virus will continue to spread until it potentially infects the entire population. Conversely, when \(R_{0} < 1\), the spread of the virus will diminish over time, eventually leading to its extinction25,26. The dynamic \(R_{0}\) can help develop more refined intervention strategies, adjusting measures flexibly according to the changes in the epidemic. By tracking the real-time variations of \(R_{0}\), the acceleration of the epidemic can be promptly detected, allowing for stronger intervention measures to be implemented. When \(R_{0}\) decreases, social restrictions can be gradually eased or adjusted, reducing economic losses and social impact.
Based on the above, the description of the parameters of the QHIM are summarized in Table 1.
