Evolutionary Game of Vaccination Considering Both Epidemic and Economic Factors by Infectious Network of Complex Nodes

Abstract

Vaccines are recognized as an effective way to control the spread of epidemics. It should be noted that the vaccination of a population is influenced not only by the infectiousness of a disease but also the vaccination strategy, such as the cost of vaccination. An accurate prediction model is helpful in forecasting the most likely trend to support smart decisions. In order to solve this problem, a model of epidemic spread dynamics is proposed, which is called the Susceptible–Infected–Vaccinated with vaccine A–Vaccinated with vaccine B–Recovered ($SI{V}_A{V}_{B}R$) model. This model assesses the competition between two vaccines in terms of economic cost and protection effectiveness in an open-market economy. The optimization process of individual vaccination decision-making was studied in an evolutionary game. In addition, a novel network containing environmental nodes and individual nodes was used to simulate the increase in infection probability caused by aggregation. Using the mean-field approach, the existence and stability of the disease-free equilibrium point and the endemic equilibrium point were demonstrated. Numerous simulations were further carried out to examine the relationship between the basic reproduction number and epidemic dynamics. The results reveal that immunization hesitation reduces the immunity level of the entire population. It is important to improve vaccine efficiency and affordability for manufacturers to become more competitive. Establishing the core individuals in the network is also a means of quickly occupying the market.

1. Introduction

The emergence and global prevalence of various epidemics pose a great threat to human life and social stability. Epidemic spread dynamics can provide theoretical support and foresight for epidemic prevention and control [1]. Additionally, the use of differential equations is an important method because they reflect the transmission trends of epidemics and theoretically reveal the epidemiological patterns within populations in order to achieve optimal prevention [2,3]. However, the novel situation means that the current research does not provide sufficient support for final decisions. First, vaccines are recognized as an effective means to eliminate epidemics. However, vaccination behavior is usually voluntary. Individuals make decisions based on the evaluation of potential payoffs, which is accomplished through gaming with others. Second, the market economy has weakened the restrictions of the vaccine industry and the huge market demand has led to technological and cost competition among manufacturers. Competition will influence individuals’ decisions and epidemics’ prevalence. Finally, in addition to direct transmission causing infection, aerosol environmental transmission is another common transmission form, which is rarely discussed. The environment formed by the aggregation of individuals increases the probability of infection, which can be considered as nodes with high connectivity in complex networks in this work.

Mathematical models based on ordinary differential equations provide theoretical analysis of epidemic spread dynamics, depicting the population changes of various states in the mean-field approach [4]. The most common epidemiological models include the SI (Susceptible–Infected) model [5,6], the SIS (Susceptible–Infected–Susceptible) model [7,8,9], and the SIR (Susceptible–Infected–Recovered) model [10,11]. Subsequent research works have employed more statuses to create epidemiological models which are closer to the realistic characteristics. For example, the SIER (Susceptible–Exposed–Infected–Recovered) model considers exposed individuals who are infected but not yet infectious [12,13]. This kind of work focuses on the simulation of disease dynamics according to the pathophysiology, highlighting the changes in the whole population caused by the disease. Meanwhile, a methodology using the complex network theory emphasizes each individual of the population. Here, nodes represent individuals, whose contacts are shown by edges [14,15]. The information transmission among the nodes is represented by the weights of the links; thus, the behavior of the individuals can be measured by this type of method [16,17].

Vaccination is a cost-effective strategy to prevent epidemics such as smallpox, polio, measles, etc. [18,19]. Based on the SIR model, the SIVR (Susceptible–Infected–Vaccinated–Recovered) model is defined by the vaccinated individuals [20]. This model has been widely used by scholars because the combination of the pandemic process with effective prevention decisions can be evaluated. Many studies focus on the role of vaccination in the transmission process of epidemics [21,22,23,24,25]. Some others analyze the impact of different factors of vaccination on epidemics, such as vaccination cost [26], age [27], information spread [28], and aspiration [29]. Using complex networks, uniform immunization was the earliest studied immunization strategy, meaning that a certain percentage of the population is randomly selected for vaccination [23]. Later, targeted [16,30] and acquaintance [18] immunization strategies were proposed in order to achieve more efficient immunization based on network topologies. However, the impact of network topological characteristics on epidemic transmission and vaccination has seldomly been examined.

However, the conditions for individuals to receive a vaccine are not considered by all the above immunization strategies. In fact, vaccine hesitancy was listed by the World Health Organization as one of the top ten global health threats in 2019, in which an individual refuses to be vaccinated, potentially leading to outbreaks of vaccine-preventable diseases [31]. The lack of confidence in vaccines, cognitive biases, religious beliefs, and other reasons lead to public panic and doubt, and eventually lead to vaccine hesitancy [32,33,34]. However, it must be noted that vaccination of some people will help others indirectly gain immunity. Thus, individual decisions to vaccinate or not are influenced by the severity of epidemics and the level of immunity within the population.

According to the above analysis, it can be found that the dynamic process of vaccination decisions can be explained by game theory [35,36]. Combining complex networks with evolutionary game theory, the vaccination strategies, as external factors, have been introduced into classical epidemiological models to research the variation tendency of epidemic spread with the intervention of individual decisions [37]. Within the scope of our knowledge, the coverage of certain vaccines attracts a large amount of research, and the competition of multiple vaccines is common in the market. In fact, the presence of multiple vaccines for the same infectious disease can have an impact on the outcome of final decisions in the light of game theory.

In order to solve this problem, a novel model named Susceptible–Infected–Vaccinated with vaccine A–Vaccinated with vaccine B–Recovered ($SI{V}_A{V}_{B}R$) is proposed. In this model, two kinds of nodes, namely individuals and the environment, are defined to simulate the effectiveness provided by the clustering degree during an epidemic. Then, a complex network is constructed with different transmission channels, namely direct droplet transmission and aerosol environmental transmission. Based on the basic definition of the network, the competition between two vaccines is added to finish the construction of $SI{V}_A{V}_{B}R$. Then, game theory is employed to assess the competition of two vaccines under different immunization strategies. Based on these contents, this work has the following advantages:

(1)

A novel kind of complex network, including the environment node and aerosol environmental transmission, is presented to consider clustering’s effectiveness during the epidemic process.

(2)

Based on the mean-field approximation theory, the endemic equilibrium point and its asymptotic stability in the proposed model under random immunization are demonstrated by making use of a Jacobian matrix and the Lyapunov function to theoretically illustrate the impact of epidemiological characteristics and vaccine features on transmission trends.

(3)

The model also combines the topology of the complex network with factors affecting the immunization decision to calculate the number of vaccinated individuals in order to assess the competition of two vaccines under evolutionary game theory, which includes the epidemic risk, information transmission in the neighborhood of the network, and economic factors.

(4)

Numerical simulations provide further confirmation of the theoretical results. In the context of vaccines’ market competition, compared with a random immunization strategy, the focus is on how the cost and efficiency of vaccines affect the game between individuals, and thereby affect the prevalence of epidemics.

The rest of this paper is laid out as follows. Section 2 introduces the background of complex networks and evolutionary game theory. In Section 3, an epidemiological model is constructed based on a novel complex network, and the theoretical results are demonstrated. In Section 4, extensive simulations and results are presented. Finally, the conclusion is presented in Section 5.

2. Background

Establishing a mathematical model which can simulate the spread process of epidemics is an effective way to support the indexes that measure the characteristics of the epidemics. By applying the mean-field approximation theory to solve the dynamic equation of the mathematical model, the development trend of epidemics can be estimated. Then, a prevention plan may be formulated according to the analysis of mathematical modeling. This work also aims to investigate the discrimination of different vaccinations, building the model based on an epidemiological model, complex networks, and game theory. In order to help audiences to understand this work, the basic theories about the proposed model are introduced in this section.

2.1. Epidemiological Models on Complex Networks

After reviewing the related literature, SIS and SIR were found to be the most classical epidemiological models, compared to other extensive versions [7,8,9,10,11]. Firstly, the pathological basics of these models fit the rules of most epidemics, and thus the models can yield precise simulated results. Second, they balance computational complexity and accuracy. In cases under time pressure, the demand for cutting computational complexity is very high. Both models divide individuals into different states, such as susceptibility, infection, immunity, etc. The SIS model is often used to study cases where recovered people cannot acquire immunity and have the possibility of reinfection, such as influenza [7,8,9]. The SIR model is applicable to epidemics in which infected people have immunity after recovery, such as measles, smallpox, etc. [10,11]. The basic reproduction number is a crucial threshold quantity in the study of epidemiological models, usually expressed by $R_0$, representing the possibility of susceptible individuals changing their state, in this case becoming infected. When $R_0$ is less than 1, the infected individuals cannot transmit the disease to others, and then the disease will eventually die out. If $R_0$ is greater than 1, the disease will continue to spread within the population [15]. Therefore, in order to eradicate an epidemic, it is necessary to keep the basic reproduction number below 1.

Epidemics spread through contact between susceptible individuals and infected individuals. When simulating epidemiological models, individuals are abstracted as nodes in the network, and the connections in the network indicate direct contacts through which epidemics propagate between individuals. A social network can be formed accordingly. In reality, the contact between individuals in the social network is gradually constructed, and different individuals have different amounts of contact with other individuals, leading to a differing influence of individuals in the network.

The environment can be considered as the intermediary between the susceptible and the infected because air is the carrier of viruses. Even if there is no direct contact in time and space among them, the susceptible can be indirectly infected through inhaling the pathogens in the environment. With the increase in infection probability, gathering places for the population, such as railway stations, affect the trend of the epidemic. Therefore, public places, as collections of individuals, can be regarded as big nodes in the network, and individuals entering these places can be considered as connected to these big nodes.

The networks of environment nodes and individual nodes are similar to scale-free networks, with node degrees satisfying a power-law distribution. The growth of scale-free networks reflects the fact that the scale of the real network is increasing. Additionally, the priority connection characteristic reflects the “Matthew effect”, which means that environment nodes with more connections in the network can obtain more connections with new nodes. Figure 1 presents the structure of a scale-free network.

A few environment nodes own most of the connections (such as nodes A and B in Figure 1), while individual nodes have a small number of connections (such as node C and node D in Figure 1). The edges between nodes (such as $l$ in Figure 1) represent contacts between nodes, making it possible for epidemics to spread. The probability distribution function of the node degree in scale-free networks conforms to the power-law function, with a power exponent of −3, and can be written as:

$$p(k)=\frac{2m(m+1)}{k(k+1)(k+2)}\approx 2m^2{k}^{-3}.$$

Scale-free networks are based on the following evolution rules:

(1)

The initial network contains $m_0$ isolated nodes. Each time a new node is added into the network, it is connected to m existing nodes ($m\le m_0$).

(2)

The probability of a connection between the new node and the node for which the degree value is $k_i$ can be described as:

$${\pi }_i=\frac{k_i}{{\displaystyle \sum _j{k}_j}},$$

where ${{\displaystyle \sum }}_j{k}_j$ is the total degree of the existing nodes in the scale-free network.

The construction process of a scale-free network shows that the scale of this scale-free network is gradually expanding, and its evolution direction depends on the distribution of node degrees in the network. The topological characteristics of a network affect the study on epidemiological models. Considering the changes in network topology is helpful to understand the spread of epidemics in different societies and formulate corresponding control measures. In graph theory, the average degree, average shortest path length, average clustering coefficient, and the diameter have been verified as being effective in reflecting the topological characteristics of a network.

(1)

Average degree

The average degree of a network refers to the meaning value of all nodes, which is calculated as:

$$<k>=\frac{1}{N}{\displaystyle \sum _{i=1}^N{k}_i},$$

where $N=m_0+t$.

It is clear that the average degree indicates the number of connections among individuals in the group, and thus, the addition of the average degree increases the complexity of the relationship among the nodes.

(2)

Average shortest path length

Any two nodes in a network may have different paths connected to them, among which the shortest path has the smallest number of edges, expressed as the shortest path length. The average shortest path length of a network is the average distance between any two nodes in the network. The average shortest path length is calculated as:

$$L=\frac{2}{N(N-1)}{\displaystyle \sum _{i>j}{d}_{ij}}.$$

(3)

Average clustering coefficient

The clustering coefficient of a node describes the linked proportion of its neighbor nodes, which can reflect the tightness of the connection between nodes and their neighbors. The clustering coefficient of a node whose degree is $k_i$ can be described as:

$$C_i=\frac{2E_i}{k_i(k_i-1)},$$

where $E_i$ indicates the actual number of edges among the neighbor nodes of node i, and $k_i\left(k_i-1\right)/2$ is the maximum number of edges where any neighbor node is fully linked to the other. The average clustering coefficient of a node is defined in a similar way to the above items, as follows:

$$C=\frac{1}{N}{\displaystyle {\sum }_{i=1}^N{C}_i}.$$

(4)

Diameter

The network diameter refers to the maximum distance between all nodes in the network:

$$D=Max(d_{ij}),$$

where $d_{ij}$ is the distance between node i and node j.

2.2. Game Theory in Infectious Networks

Vaccine immunization is an effective intervention to suppress the spread of epidemics [1]. An ideal vaccination strategy needs to reduce the cost of immunization while effectively mitigating epidemics. Based on models researching epidemics, random immunity and targeted immunity are the most widely studied strategies compared to other vaccination strategies. Random immunity randomly selects nodes to complete their immunization. Target immunity refers to the selective immunization of the nodes with high degree values to achieve better immune effects.

The above vaccination strategies do not consider the conditions for complete immunization. However, these strategies are limited in their application because of factors which are not included in mathematic models, such as the economic burden [33,34]. Meanwhile, individuals will voluntarily choose whether to be vaccinated. The improvement of the immune level within the whole population will reduce the risk of infection for individuals. This means that individuals can be protected even if they are not vaccinated. Therefore, under voluntary vaccination, nodes may benefit from the vaccination of others. With this pre-consideration, nodes tend to gain protection through group immunity without increasing their own costs and avoiding the possible side effects of vaccines. Nodes maximize their own benefits by checking factors including infection, economics, and so on. As most individuals lack expert judgement about infectious diseases, this may be dangerous; thus, it is possible to set non-infection as a priority decision. The conflict of benefits between nodes and the whole population in the model reduces the vaccine coverage. It is possible that this conflict may eventually lead to the deterioration of the epidemic situation.

The decision-making process of individuals can be explained by game theory. Classical game theory studies the strategic choice of completely rational individuals, who always make decisions that are beneficial to their own interests. Evolutionary game theory breaks the restrictions regarding the complete rationality of game subjects in classical game theory [38]. Game subjects keep learning and constantly find better strategies through the process of trial and error. In addition, evolutionary game theory emphasizes the dynamics of the game and the process of the population reaching equilibrium. Thus, evolutionary game theory is widely used in the study of human socioeconomic behavior. Considering that individuals are incompletely rational in reality, the optimal strategy can only be obtained through multiple games, where voluntary vaccination can be seen as an evolutionary game in a social network.

In complex networks, nodes are regarded as game players, while edges represent the game relations of players [35]. Game players compare their current payoffs with those of neighbor individuals. If their own losses are less than those of the other sides, individuals will tend to maintain their existing decisions in the next time step. On the contrary, when their own losses are greater than the losses of the other sides, they are likely to learn the strategies of their neighbors.

Figure 2 shows the change in node strategies in the first four rounds of the scale-free network presented in Figure 1, where the color of the nodes indicates their current strategy. There are more black nodes than white nodes. It can be seen that in the first round, node A chose node C in the neighborhood to play the game, and the profit of node C in this round was greater. Thus, node A would learn the strategy of node C at a greater probability in the next round. In each iteration, every node will play a game to adjust its strategy. The increasing number of black nodes illustrates that the nodes are pursuing the maximization of their own interests by continuing the game. After a period of experience accumulation, individuals will form stable strategies, and then groups will achieve the optimal allocation of social resources. Evolutionary game theory is suitable to study the preference selection of individuals and the balance of groups.

3. Epidemic $\mathit{S}\mathit{I}{\mathit{V}}_{\mathit{A}}{\mathit{V}}_{\mathit{B}}\mathit{R}$ Model

In this section, a complex network of individual nodes and environment nodes consisting of multiple individuals is constructed to simulate droplet and aerosol transmission in the spread of epidemics. On this network, an epidemic $SI{V}_A{V}_{B}R$ (Susceptible–Infected–Vaccinated with vaccine A–Vaccinated with vaccine B–Recovered) model considering the vaccination invalidity rate is established. The density evolution of the state of each node is employed to study the random immunization strategy. Then, the disease-free equilibrium point and the endemic equilibrium point of the model are calculated based on the mean-field approximation theory. After mining the rule of the proposed model, the vaccination evolutionary game discusses the competition of vaccines to measure the impact of individual voluntary immunity in the $SI{V}_A{V}_{B}R$ model.

3.1. Epidemic Transmission Network

Some common epidemics, such as measles, SARS, and COVID-19, are transmitted through the respiratory tract. The main modes of transmission include droplet transmission and aerosol transmission [39,40]. Foaming transmission refers to the direct contact between the susceptible and the infected individuals. Aerosols, existing for a long time, are mixed in the form of droplets in the air. Without closed intersections, an epidemic can spread between healthy people and infected people through aerosol transmission. The infected individuals release pathogens into the environment when coughing or sneezing. Although the susceptible may not have direct contact with the infected, inhaling the virus aerosols discharged by the infected, or touching their mouth and nose after coming into contact with the surface of a contaminated object, can cause infection. Aerosol transmission often occurs in public places, such as shopping malls, restaurants, and public transportation networks, which contain a large number of individuals.

3.2. $SI{V}_A{V}_{B}R$ Model under Random Immunization Strategy

A complex network of environments and individuals, together serving as nodes, was thus formed, as shown in Figure 3. The connections between individual nodes represent their direct contacts, providing the possibility of the direct transmission of epidemics. Importantly, the environment nodes are integrated with multiple individuals and the connections among them. As shown in Figure 3, the environment nodes (A, B, and C), consisting of multiple individuals, act as large nodes and increase the infection possibility of the individual nodes. Infected individuals are present in the environment nodes A and C, so individual nodes that enter these places may be indirectly infected. At the same time, the connections among the susceptible and the infected represent direct droplet transmission.

Figure 4 shows the transition between the various states of nodes in the $SI{V}_A{V}_{B}R$ model based on a scale-free network under a random immunization strategy. All nodes were grouped into five states: susceptible (S), infected (I), vaccinated with vaccine A ($V_A$), vaccinated with vaccine B ($V_B$), and recovered (R). $S_k, I_k, R_k, V_k^A, \mathrm{and}{V}_k^B$ represent the percentage of susceptible, infected, recovered, vaccinated with vaccine A, and vaccinated with vaccine B nodes with degree k in the total population at time t, respectively. Some of the nodes were initially set as having the infectious state, and the others were set as susceptible. Under the iteration of every time step, susceptible nodes will be infected with probability λ during contact with infected nodes. By receiving treatment or improving immunity, the probability that infected nodes can recover is μ. The recovered nodes have immunity, but the immunity will be lost with probability β. The nodes losing immunity will convert to a susceptible state. Within each time step, susceptible nodes were randomly selected to be vaccinated with vaccine A (at probability $e_A$) or vaccine B (at probability $e_B$). The vaccinated nodes gain immunity, but they also will lose immunity with probability ${\alpha }_A$ or ${\alpha }_B$ according to the kind of vaccine, transforming their status into the susceptible state.

Based on the mean-field approximation theory, the changes in the proportions of different groups at each time step can be expressed by the following equation:

$$\left\{\begin{array}{l}\frac{d{S}_k}{dt}=\beta R_k-\lambda k(1-e_A-e_B)S_k{\theta }_k-e_A{S}_k-e_B{S}_k+{\alpha }_A{V}_k^A+{\alpha }_B{V}_k^B\hfill \\ \begin{array}{l}\frac{d{I}_k}{dt}=\lambda k(1-e_A-e_B)S_k{\theta }_k-\mu I_{{}_k}\\ \frac{d{R}_k}{dt}=\mu I_k-\beta R_k\end{array}\hfill \\ \frac{d{V}_k^A}{dt}=e_A{S}_k-{\alpha }_A{V}_k^A\hfill \\ \frac{d{V}_k^B}{dt}=e_B{S}_k-{\alpha }_B{V}_k^B\hfill \end{array}\right.,$$

(1)

where ${\theta }_k$ denotes the probability of a susceptible node of degree k coming into contact with an infected node of degree k′ in its neighborhood at time t. It can be written as:

$${\theta }_k=\frac{{\displaystyle {\sum }_{k\prime }k\prime P(k\prime )I_k}}{<k>}.$$

(2)

In Equation (2), P(k) is the degree distribution of the nodes in the network and $<k>={\displaystyle {\sum }_{k}kP(k)}$ presents the average degree of the network. Changes in the number of nodes due to births and deaths are not considered in this work; that is to say:

$$S_k+I_k+R_k+V_k^A+V_k^B=1.$$

(3)

Theorem 1.

The basic reproduction number$R_0=\frac{<k^2>}{<k>}\frac{{\alpha }_A{\alpha }_B\lambda (1-e)}{\mu ({\alpha }_A{\alpha }_B+e_A{\alpha }_B+e_B{\alpha }_A)}$. When $R_0<1$, the epidemic has a disease-free equilibrium point. When $R_0>1$, the epidemic has an endemic equilibrium point.

Proof of Theorem 1.

When the evolution of the epidemic system is stable, the number of individuals in each state is constant. It is reasonable to set $\frac{d{S}_k}{dt}=\frac{d{I}_k}{dt}=\frac{d{R}_k}{dt}=\frac{d{V}_k^A}{dt}=\frac{d{V}_k^B}{dt}=0$, and obtain the disease-free equilibrium point $E_1$ and the endemic equilibrium point $E_2$:

$$E_1=(S_1,I_1,V_1^A,V_1^B,R_1)=(\frac{1}{e}(\frac{{\alpha }_A{e}_A}{{\alpha }_A+e_A}+\frac{{\alpha }_B{e}_B}{{\alpha }_B+e_B}),0,\frac{e_A}{{\alpha }_A+e_A},\frac{e_B}{{\alpha }_B+e_B},0),$$

(4)

$$E_2=(S_2,I_2,V_2^A,V_2^B,R_2),$$

(5)

where $e=e_A+e_B$ and

$$\left\{\begin{array}{l}{S}_2=\frac{\mu }{\lambda k(1-e){\theta }_k}{I}_2\hfill \\ V_2^A=\frac{e_A\mu }{{\alpha }_A\lambda k(1-e){\theta }_k}{I}_2\hfill \\ V_2^B=\frac{e_B\mu }{{\alpha }_B\lambda k(1-e){\theta }_k}{I}_2\hfill \\ R_2=\frac{\mu }{\beta }{I}_2\hfill \end{array}\right..$$

(6)

Combining Equations (3) and (6), we obtain $I_2=\frac{\beta }{\beta +\mu }-\frac{\beta \mu ({\alpha }_A{\alpha }_B+e_A{\alpha }_B+e_B{\alpha }_A)}{{\alpha }_A{\alpha }_B\lambda k_0(1-e)(\beta +\mu )}$. Substituting this into Equation (2), the expression of ${\theta }_k$ is:

$${\theta }_k=\frac{1}{<k>}{\displaystyle \sum _{k\prime }kP(k)\frac{\beta {\alpha }_A{\alpha }_B\lambda k(1-e){\theta }_k}{\beta ({\alpha }_A{\alpha }_B+\mu e_A{\alpha }_B+\mu e_B{\alpha }_A)+(\beta +\mu )(1-e){\alpha }_A{\alpha }_B\lambda k{\theta }_k}}.$$

(7)

Based on Equation (7), we set the expression as $F({\theta }_k)={\theta }_k$. It can be observed that ${\theta }_k=0$ is always the root of this function and $F^{″}({\theta }_k)<0$. It can be calculated that:

$$\begin{array}{ll}F(1)& =\frac{1}{<k>}{\displaystyle \sum _{k}kP(k)\frac{\beta {\alpha }_A{\alpha }_B\lambda k(1-e)}{\beta [{\alpha }_A{\alpha }_B\mu +e_A{\alpha }_B\mu +e_B{\alpha }_A\mu +{\alpha }_A{\alpha }_B\lambda k(1-e)]+\mu {\alpha }_A{\alpha }_B\lambda k(1-e)}}\\ & <\frac{1}{<k>}{\displaystyle \sum _k{k}^{2}P(k)\frac{\beta {\alpha }_A{\alpha }_B\lambda (1-e)}{\beta {\alpha }_A{\alpha }_B\lambda (1-e)+\mu {\alpha }_A{\alpha }_B\lambda k(1-e)}}=\frac{\beta }{\mu +\beta }<1.\end{array}$$

(8)

Thus, if $F({\theta }_k)={\theta }_k$ has a solution in the interval (0,1], it needs to satisfy $F\prime (0)>1$:

$$\frac{<k^2>}{<k>}\frac{{\alpha }_A{\alpha }_B\lambda (1-e)}{{\alpha }_A{\alpha }_B\mu +e_A\mu {\alpha }_B+e_B\mu {\alpha }_A}>1.$$

(9)

Equation (9) shows that there exists a basic reproduction number $R_0=\frac{<k^2>}{<k>}\frac{{\alpha }_A{\alpha }_B\lambda (1-e)}{\mu ({\alpha }_A{\alpha }_B+e_A{\alpha }_B+e_B{\alpha }_A)}$. When $R_0<1$, the epidemic system has a disease-free equilibrium point, which means that the epidemic can eventually be eliminated at the point $E_1$. When $R_0>1$, there exists an endemic equilibrium point $E_2$ for the epidemic. The epidemic does not die out, but the scale of infection stabilizes at $I_2$. □

Theorem 2.

When $R_0<1$, the disease-free equilibrium point is globally asymptotically stable.

Proof of Theorem 2.

Firstly, the local stability of the disease-free equilibrium point is provided. Combining Equations (1) and (3), we can obtain:

$$\left\{\begin{array}{l}\frac{d{S}_k}{dt}=\beta (1-S_k-I_k-V_k^A-V_k^B)-\lambda k(1-e)S_k{\theta }_k-e{S}_{{}_k}+{\alpha }_A{V}_k^A+{\alpha }_B{V}_k^B\hfill \\ \frac{d{I}_k}{dt}=\lambda k(1-e)S_k{\theta }_k-\mu I_{{}_k}\hfill \\ \frac{d{V}_k^A}{dt}=e_A{S}_{{}_k}-{\alpha }_A{V}_k^A\hfill \\ \frac{d{V}_k^B}{dt}=e_B{S}_{{}_k}-{\alpha }_B{V}_k^B\hfill \end{array}\right.,$$

(10)

when $R_0<1$, the Jacobian matrix of Equation (10) can be written as:

$$J_1=\left[\begin{array}{cccc}-e-\beta & -\beta -(1-e)\lambda k_0{S}_1& {\alpha }_A-\beta & {\alpha }_B-\beta \\ 0& (1-e)\lambda k_0{S}_1-\mu & 0& 0\\ e_A& 0& -{\alpha }_A& 0\\ e_B& 0& 0& -{\alpha }_B\end{array}\right].$$

(11)

We use ${\lambda }^{\prime }$ to represent the eigenvalue of the matrix $J_1$, so the characteristic equation of the matrix is:

$$[(1-e)\lambda k_0{S}_1-\mu -\lambda \prime ][\lambda {\prime }^3+P\lambda {\prime }^2+Q\lambda \prime +T]=0,$$

(12)

where $P={\alpha }_A+{\alpha }_B+\beta +e$, $Q={\alpha }_B{e}_B+e{\alpha }_B+{\alpha }_A{\alpha }_B+e_B{\alpha }_A+\beta {\alpha }_A+\beta e_A$, and $T=2e_B{\alpha }_A{\alpha }_B+\beta {\alpha }_A{\alpha }_B+\beta e_A{\alpha }_B-\beta e_B{\alpha }_A$.

$\left(1-e\right)\lambda k_0{S}_1-\mu$ is one of the eigenvalues, which is negative when $R_0<1$. We set $g\left({\lambda }^{\prime }\right)={{\lambda }^{\prime }}^3+P{{\lambda }^{\prime }}^2+Q{\lambda }^{\prime }+T$, and thus the other eigenvalues are determined by $g\left({\lambda }^{\prime }\right)=0$. Obviously, when $R_0<1$, $g\left(0\right)>0$, the value of $g^{\prime }\left({\lambda }^{\prime }\right)$ remains positive in $\left(0,+\infty \right)$. The above analysis illustrates that the roots of $g\left({\lambda }^{\prime }\right)=0$ are negative, which means the eigenvalues of $J_1$ are all negative. In conclusion, the point $E_1$ is locally asymptotically stable.

Furthermore, the Lyapunov function is constructed to prove the globally asymptotic stability of the disease-free equilibrium point. Letting $L=I_k$, it is obtained that:

$$L\prime =\frac{d{I}_k}{dt}=\lambda k_0(1-e_A-e_B)S_k{I}_k-\mu I_k=[\lambda k_0\frac{1-e}{e}(\frac{{\alpha }_A{e}_A}{{\alpha }_A+e_A}+\frac{{\alpha }_B{e}_B}{{\alpha }_B+e_B})-\mu ]I_k.$$

(13)

When $R_0<1$, $\lambda k_0\frac{1-e}{e}(\frac{{\alpha }_A{e}_A}{{\alpha }_A+e_A}+\frac{{\alpha }_B{e}_B}{{\alpha }_B+e_B})-\mu <0$ results in $L^{\prime }\le 0$. Therefore, there is a unique positive solution $E_1$ for $\frac{d{I}_k}{dt}=0$, and the disease-free equilibrium point is globally asymptotically stable. □

Theorem 3.

When $R_0>1$, the endemic equilibrium point is locally asymptotically stable.

Proof of Theorem 3.

The Jacobian matrix of Equation (8) can be written as Equation (14) when $R_0>1$:

$$J_2=\left[\begin{array}{cccc}{\pi }_1& -\beta -\mu & {\alpha }_A-\beta & {\alpha }_B-\beta \\ {\pi }_2& 0& 0& 0\\ e_A& 0& -{\alpha }_A& 0\\ e_B& 0& 0& -{\alpha }_B\end{array}\right].$$

(14)

In Equation (14), ${\pi }_1=-\lambda k_0\left(1-e\right)I_2-e-\beta$, ${\pi }_2=\lambda k_0\left(1-e\right)I_2$. It can be seen that:

$${\pi }_1<0,$$

(15)

$$\left|\begin{array}{cc}{\pi }_1& -\beta -\mu \\ {\pi }_2& 0\end{array}\right|={\pi }_2(\beta +\mu )>0,$$

(16)

$$\left|\begin{array}{ccc}{\pi }_1& -\beta -\mu & {\alpha }_A-\beta \\ {\pi }_2& 0& 0\\ e_A& 0& -{\alpha }_A\end{array}\right|=(\beta +\mu )\left|\begin{array}{cc}{\pi }_2& 0\\ e_A& -{\alpha }_A\end{array}\right|=-{\alpha }_A{\pi }_2(\beta +\mu )<0,$$

(17)

$$\left|\begin{array}{cccc}{\pi }_1& -\beta -\mu & {\alpha }_A-\beta & {\alpha }_B-\beta \\ {\pi }_2& 0& 0& 0\\ e_A& 0& -{\alpha }_A& 0\\ e_B& 0& 0& -{\alpha }_B\end{array}\right|=(\beta +\mu )\left|\begin{array}{ccc}{\pi }_2& 0& 0\\ e_A& -{\alpha }_A& 0\\ e_B& 0& -{\alpha }_B\end{array}\right|={\pi }_2(\beta +\mu ){\alpha }_A{\alpha }_B>0.$$

(18)

As Equations (15)–(18) show, when $R_0>1$, all odd-order principal minors of matrix $J_2$ are smaller than zero, and all even-order principal minors are greater than zero. As a result, $J_2$ is a negative definite matrix. The four characteristic roots all have negative real parts. Therefore, when $R_0>1$, the endemic equilibrium point $E_2$ is locally asymptotically stable. □

3.3. Vaccination Evolutionary Game Model

The above analysis of $SI{V}_A{V}_{B}R$ shows that vaccination plays an important role in controlling the progression of epidemics, especially for preventable epidemics such as smallpox and avian influenza. Universal vaccination can effectively inhibit the spread of epidemics in the population. For class II vaccines, which are not free, individuals can choose whether to be vaccinated. Their decision depends on multiple factors, including the cost of vaccination, the protection rate of the vaccine, the severity of the epidemic, etc. In recent years, vaccines for epidemics, such as influenza vaccines, have been commercialized, pushing vaccine manufacturers to continually update their products to increase their market shares. The main methods of achieving this include offering differentiated products and reducing vaccine prices.

Instead of a random immunization strategy, a vaccination evolutionary game model is established in the network to discuss the change, caused by the competition of vaccines, in the proposed model. Two vaccines against the same disease compete with each other in the market. Each node represents an individual who voluntarily chooses whether and which vaccine to receive by comparing the cost with neighbor nodes. The epidemic model is based on the $SI{V}_A{V}_{B}R$ model described before.

It is assumed that the cost of vaccination is calculated in both direct and implicit ways. The direct cost is the expense of the vaccine measured in monetary terms. This is a fixed value during a short economic cycle. The indirect mode refers to the implied cost, such as the transportation fee, the possible psychological impact, and the loss of time. Nodes will evaluate the implicit cost based on the available number of vaccinations and the infection risk. When a node has many neighbor nodes that are vaccinated, the node may be reluctant to get vaccinated due to immunization fluke in order to save the cost of the vaccine. However, this means that that the infectious factor in the implied cost dynamically increases to become the predominate value. On the other hand, when the infected population grows, increasing the infectious risk, the nodes will tend to be vaccinated, cutting the implied cost.

Based on the above assumptions, the cost of vaccination for vaccines A and B can be described by the following equation:

$$\left\{\begin{array}{c}{C}_v^A=[a+f(x,I)]C_I\\ C_v^B=[b+f(x,I)]C_I\end{array}\right.,$$

(19)

where $a$ and $b$, respectively, represent the direct vaccination cost of vaccine A and vaccine B, $f(x,I)$ represents the implicit cost of vaccination, x is the number of vaccinated nodes, and I is the number of infected nodes. Therefore, $f(x,I)$ reflects the immune fluke mentality of nodes, which is directly proportional to x and inversely proportional to I. Here, an assumption is set as: $f(x,I)=-I/x$. $C_I$ represents the loss suffered by infected nodes. In order to simplify parameters without losing generality, $C_I=1$. Then, the income of node i can be denoted as $P_i$, which is shown in Equation (20):

$$P_i=\left\{\begin{array}{l}-f(x,I)-a,\mathrm{if}\mathrm{i}\mathrm{is}\mathrm{vaccinated}\mathrm{with}\mathrm{vaccine}\mathrm{A}\hfill \\ -f(x,I)-b,\mathrm{if}\mathrm{i}\mathrm{is}\mathrm{vaccinated}\mathrm{with}\mathrm{vaccine}\mathrm{B}\hfill \\ -1,\mathrm{if}\mathrm{i}\mathrm{is}\mathrm{not}\mathrm{vaccinated}\mathrm{and}\mathrm{is}\mathrm{infected}\hfill \\ 0,\mathrm{if}\mathrm{i}\mathrm{is}\mathrm{not}\mathrm{vaccinated}\mathrm{and}\mathrm{is}\mathrm{susceptible}\hfill \end{array}\right..$$

(20)

At each time step, susceptible nodes will compare their own incomes with those of their neighbors and make a decision on whether to be vaccinated. Since nodes are not completely rational in reality, it is assumed that susceptible nodes learn strategies according to the Fermi function. The node x receives the income information from a node (expressed as y) picked from the neighborhood of x with the same chance, and then the probability of the node x learning from y is:

$$W_{y\to x}=\frac{1}{1+\exp [-\rho (P_y-P_x)]},$$

(21)

where $P_x$ and $P_{\mathrm{y}}$ denote the income of nodes 𝑥 and 𝑦 at the previous time step, respectively, and ρ evaluates the irrational characteristic. The addition of ρ enhances the rational choice. Therefore, the rate of change in the number of vaccinated nodes within each time step can be rewritten as:

$$\begin{array}{l}\frac{d{V}_k^A}{dt}=\frac{1}{1+\exp \{-\rho [-f(x,I)-a-0]\}}\frac{k{\displaystyle {\sum }_{k\prime }k\prime P(k\prime )V_k^A}}{<k>}{S}_k-{\alpha }_A{V}_k^A,\\ \frac{d{V}_k^B}{dt}=\frac{1}{1+\exp \{-\rho [-f(x,I)-b-0]\}}\frac{k{\displaystyle {\sum }_{k\prime }k\prime P(k\prime )V_k^B}}{<k>}{S}_k-{\alpha }_B{V}_k^B.\end{array}$$

(22)

Based on Equation (22), it is clear that the number of vaccinated nodes is determined by the number of existing infections, the number of existing vaccinations, the cost of vaccination, and the vaccination invalidity rate.

4. Simulations and Discussion

Based on the above epidemic $SI{V}_A{V}_{B}R$ model, this section simulates the spread process of an epidemic under different parameter settings. The experiments compared the competition of vaccines and a vaccination evolutionary game model with random immunization. To explore the prevalence of epidemics in the network containing large environmental nodes and small individual nodes, scale-free networks and small-world networks were used to simulate the interaction of nodes. Considering the randomness of the experiment, every simulation was repeated 100 times. The figures depict the average of the results of all simulations.

4.1. Simulations of Random Immunization Strategy

Firstly, in order to demonstrate the rationality of the above theoretical analysis, the impact of different basic reproduction numbers and parameters on the epidemic were subsequently explored. In a network whose size was 3000, 100 nodes were randomly selected as the initial infected nodes, and 50 nodes were randomly selected as the initial immune nodes of vaccines A and B, separately.

Figure 5 shows the evolution of the number of various nodes with time when the basic regeneration number was greater than 1 or smaller than 1. In Figure 5a–d, the basic reproduction numbers were as follows: 0.342, 2.463, 0.185, and 1.333, respectively. In Figure 5a,c, the infected nodes continuously decreased and eventually stabilized at zero, representing the disappearance of the epidemic and the existence of a disease-free equilibrium point with global asymptotic stability when the basic reproduction number was smaller than 1.

On the contrary, in Figure 5b,d, the number of infected nodes increased, but the growth rate was slowing down. After a period of time, the number of infected nodes became stable, representing the final influence of the epidemic. The spread of the epidemic reached an endemic equilibrium point which was locally asymptotically stable. Under this circumstance, the epidemic will become an endemic, which means it neither spreads nor dissipates. This illustrates that epidemic transmission varies with the setting of parameters and networks and tests the applicability of this model in different kinds of networks.

To explore the impact of vaccines on the epidemic terminal condition and investigate effective methods to inhibit the spread of the epidemic, the vaccination invalidity rates of vaccines A and B were assumed to be the same here. Figure 6 shows the change in the basic reproduction number with respect to the vaccination invalidity rate and the vaccination rate. It can be seen that the basic reproduction number increased with the increase in the vaccine invalidity rate but decreased with the increase in the random vaccination rate. This proved that improving the protection capability of vaccines and promoting vaccination can help to cause epidemics to reach the disease-free equilibrium point.

Figure 7 further describes the relationship between the vaccination invalidity rate and the random vaccination rate, resulting in the basic reproduction number becoming 1. Meanwhile, the infected rate and the recovery rate were changed. The values of the other parameters remained constant. The points on the curve represent $R_0=1$, the points on the right side of the curve represent $R_0<1$, and the points on the left side of the curve represent $R_0>1$. Figure 7 reveals that $R_0$ had positive correlations with the infection rate and the vaccination invalidity rate. However, the correlations between $R_0$ and the vaccination rate and the recovery rate were negative. Therefore, by reducing the vaccination invalidity rate and improving the vaccination rate, the system can reach the disease-free balance point.

In addition, for any set of parameters, there is always a threshold of the vaccination rate. When the vaccination rate was greater than the threshold, $R_0$ was smaller than 1, regardless of the value of the vaccination invalidity rate. This indicates that the impact of vaccination was positive. The active participation in vaccination of individuals can control the spread of epidemics and cause epidemics to eventually die out.

Furthermore, the effects of different vaccination rates and vaccination invalidity rates on epidemic spread under the random immunization strategy were studied by setting different $e$($e=e_A+e_B)$, ${\alpha }_A$, and ${\alpha }_B$ parameters. The vaccination invalidity rates of vaccines A and B were kept the same (${\alpha }_A={\alpha }_B)$. Figure 8 and Figure 9, respectively, show the time evolution curves of the number of infected and vaccinated nodes with different vaccination rates (controlled by $e$) or different invalidity rates (controlled by $\alpha$) in scale-free networks and small-world networks. Although the epidemic will always eventually die out, the decrease in the vaccination rate and the increase in the vaccination invalidity rate will delay the dissipation of the epidemic, which is unfavorable to the control work. In addition, when the system is stable, the number of vaccinated nodes becomes greater with the addition of the vaccination rate and the decline in the vaccination invalidity rate, which illustrates the raised market share of the vaccine.

It can be seen that the above phenomenon was more weakly reflected in the infected nodes of a small-world network. Thus, increasing the efficiency of vaccines is more effective in epidemic prevention for the scale-free network containing large environmental nodes and individual nodes assumed in this paper. Based on the above analyses, improving the vaccination rate and reducing the vaccination invalidity rate are effective measures to inhibit epidemic transmission.

In theory, when all individuals are vaccinated, the immune effect among the population is the best and the epidemics can be quickly controlled. However, in reality, it is extremely difficult to vaccinate all individuals due to limited vaccine resources or the voluntary behavior of vaccination. It is more feasible to achieve the coverage of vaccines with a basic reproduction number of 1. Meanwhile, the vaccination rate and the vaccination invalidity rate can reduce the basic number of vaccinated people and resource input, making this an economical and effective method of epidemic prevention. Following the selection and game of vaccines, individuals having an immune fluke mentality will be simulated.

4.2. Simulations on Vaccination Evolutionary Game Model

In this section, based on vaccines’ competition and a vaccination evolutionary game model, the evolutionary process of individual voluntary vaccination behavior and the epidemic spread process are simulated. In this paper, $\rho =1$. The simulations were developed on scale-free networks, which can reflect the connection characteristics of the networks of environment nodes and individual nodes.

Firstly, the cost of vaccination was assumed to be the same for both vaccines (a = b = 0.5) to compare the random immunization strategy and the evolutionary game model by setting the same vaccination invalidity rate. Meanwhile, based on vaccines’ competition and the vaccination evolutionary game model, the effect of the vaccination invalidity rate on epidemic spread was explored. One hundred nodes were randomly selected as the initially infected nodes. Fifty nodes each were randomly selected as the initial vaccinated nodes for vaccine A and vaccine B.

When ${\alpha }_A={\alpha }_B=0.2$, the spread of the epidemic when considering the vaccination evolutionary game model and a random immunization strategy were compared. As shown in Figure 10, it can be seen that the number of infected nodes gradually approached zero under different strategies. That is to say that the system can reach a disease-free equilibrium point which is globally asymptotically stable. This is consistent with our theoretical conclusion. However, the game among nodes slowed down the demise of the epidemic and reduced the coverage of vaccines. It was found that when nodes decided whether to vaccinate voluntarily, it took longer to control the epidemic, and fewer nodes chose to vaccinate. This means that individuals make decisions to maximize their own benefits, whereas the decisions that are beneficial to individuals are not conducive to the immunity of the whole community.

For an endemic disease, based on vaccines’ competition and the vaccination evolutionary game model, Figure 11 shows the relationship between the vaccination invalidity rate and the number of nodes in each state. The number of individuals in each state eventually became stable, but the epidemic cannot be eliminated. This proves that when $R_0>1$, the endemic equilibrium is locally asymptotically stable. In addition, it can be observed that when ${\alpha }_A={\alpha }_B=0.5$, the variation of the number of nodes in each state almost coincided with the values when ${\alpha }_A={\alpha }_B=0.8$. Meanwhile, after the evolution became stable, the number of vaccinated nodes was zero. However, when ${\alpha }_A={\alpha }_B=0.2$, there were still nodes choosing to vaccinate. This suggests that when the invalidity rate of vaccination is too high, the value of the vaccine is less obvious. Additionally, the number of vaccinated nodes was negatively correlated with the vaccination invalidity rate. In conclusion, when the game model is considered, the higher the vaccination invalidity rate is, the fewer nodes that choose to be vaccinated. Additionally, a too high vaccination invalidity rate will lead to no nodes choosing to vaccinate after the evolution becomes stable.

Next, the spread of an epidemic under different costs of vaccination was simulated. As shown in Figure 12, the direct costs of vaccine A and vaccine B were assumed to be the same, the cost was changed to 0.2, 0.5, and 0.8, respectively, and $\beta =0.2$,$\lambda =0.1$, and $\mu =0.9$ were employed. It is shown that the higher the cost of vaccination, the longer the system takes to reach a steady state, and the fewer individuals that choose to be vaccinated. Figure 10, Figure 11 and Figure 12 illustrate that improving the efficiency of the vaccines and reducing the cost of vaccination can encourage individuals to get vaccinated and accelerate the demise of the epidemic.

When $\beta =0.2$, $\lambda =0.1$, and $\mu =0.9$, the impact of the vaccination cost on the vaccination scale was investigated. The vaccination invalidity rate of both vaccines A and B was 0.2. Figure 13 depicts the time evolution curves of the number of susceptible, infected, recovered, and vaccinated nodes based on vaccines’ competition and the evolutionary game model under different costs of vaccine A and vaccine B. At the beginning of the input of the vaccine, the growth rate of the number of nodes vaccinated with the lower-cost vaccine was greater than that of the number of nodes vaccinated with the higher-cost vaccine. The number of nodes which chose to get vaccinated with the lower-cost vaccine kept increasing, but the growth rate gradually slowed down until it became stable. Those who chose to receive the higher-cost vaccine increased by a small amount at the beginning. Then, this value slowly declined to zero. After the system reached the equilibrium state, the total numbers of nodes vaccinated with the two kinds of vaccines were constant. In addition, the smaller the cost gap between the two vaccines, the longer they will compete in the market; that is, the longer the lower-cost vaccine will circulate in the population.

To explore the effect of vaccination invalidity rates on vaccines’ competition, the vaccination costs of vaccine A and vaccine B were kept equal (a = b = 0.5), and we set $\beta =0.2$, $\lambda =0.1$, $\mu =0.9$. We fixed ${\alpha }_B=0.5$ and modified ${\alpha }_A$ to 0.1, 0.3, 0.5, 0.7, and 0.9 in turn. Figure 14 shows the time evolution curves of the number of nodes vaccinated with vaccine A or vaccine B. By observing Figure 14, it can be concluded that without considering the cost, the coverage of the vaccines depends on the efficiency of the vaccines. The main reason for this is that the invalidity rate of the vaccine affects the opportunity for nodes to learn from their neighbors. When the invalidity rates of vaccine A and vaccine B were the same, there was no difference between vaccines A and B for nodes. Therefore, the numbers of nodes covered by vaccine A and vaccine B were the same in the steady state. When the invalidity rate of vaccine B was greater than that of vaccine A, although the number of nodes vaccinated with vaccine B initially increased, it soon began to decrease to zero. It may be because in the early stage of the outbreak of the epidemic, susceptible nodes chose to get vaccinated with vaccine B after the game between them and their neighbors. However, as the invalidity rate of vaccine B was greater than that of vaccine A, individuals who were vaccinated with vaccine B had a lower probability of being learned from. After a period of time, no node will choose to get vaccinated with vaccine B.

Figure 15 shows the influence of the initial vaccinated nodes on the vaccination competition. In the simulation, the 50 nodes with the greatest centrality were selected as the nodes for the initial vaccination of vaccine B, and 50 nodes were randomly selected to be vaccinated with vaccine A. The nodes were still vaccinated according to the rules of the evolutionary game model. As shown in Figure 15, the vaccination costs and invalidity rates of the two vaccines were the same, but the number of people covered by vaccine B was far greater than those covered by vaccine A. This indicates that the node with greater centrality has a stronger influence and can drive the neighbor node to get vaccinated with the same vaccine. In addition, compared with the simulation results in Figure 8, selecting individuals with a high degree as the initial vaccinated individuals shortened the transmission time of the epidemic. The degree of nodes in a network refers to the number of edges connected to the node. On the basis of the generating rule of a scale-free network, a node with a large degree is more likely to be infected by other nodes connected to this node. Then, the susceptible nodes linked to the node with a large degree value increase their infection possibility because of the indirect effectiveness. The nodes with a high degree are the key to inhibiting the spread of epidemics.

It can be inferred that the spread of an epidemic in scale-free networks is heavily affected by a few nodes with strong connectivity, which are called environment nodes here. Therefore, the introduction of the targeted immunization of big environment nodes is a strategy to improve vaccine coverage.

Then, the spread of the epidemic in different scale-free networks was simulated to explore the impact of network topological properties on vaccination coverage and epidemic spread. Once a new node is added to the scale-free network, it will be connected to m existing nodes. N represents the number of nodes in the scale-free network.

Table 1. The topological properties’ values of scale-free networks, with m = {2, 3, 5, 10} and N = {300, 5000, 10,000, 20,000}.

N	Topological Property	m = 2	m = 3	m = 5	m = 10
3000	average degree	3.997	5.994	9.983	19.933
	average shortest path length	4.557	3.864	3.316	2.805
	average clustering	0.012	0.014	0.018	0.028
	diameter	8	6	5	4
5000	average degree	3.998	5.996	9.990	19.960
	average shortest path length	4.775	4.051	3.469	2.910
	average clustering	0.007	0.009	0.012	0.019
	diameter	8	7	5	4
10,000	average degree	3.999	5.998	9.995	19.980
	average shortest path length	5.047	4.307	3.670	3.062
	average clustering	0.003	0.005	0.007	0.011
	diameter	9	7	6	5
20,000	average degree	4.000	5.999	9.998	19.990
	average shortest path length	5.316	4.522	3.859	3.237
	average clustering	0.003	0.003	0.004	0.007
	diameter	9	8	6	5

shows the topological properties of a scale-free network with scales of 3000, 5000, 10,000, and 20,000 when m = {2,3,5,10}. Each network was constructed ten times and the average of the topological properties is shown.

The average degree, average shortest path length, average clustering, and the diameter can reflect the characteristics of connections among nodes in a network. The average degree represents the sparsity of the network. The average clustering reflects the aggregation of nodes. The average shortest path length measures the distance between nodes. The diameter is the maximum length of the shortest path in the network, and its value represents the size of the social network in the topic of this work. It is obvious that a larger value of the average degree and average clustering in the network means a greater risk of mutual infection of nodes. The disease easily spread in networks with small values of the average shortest path length and diameter. According to Table 1, it can be seen that with the increase in m, the connection between nodes became tighter, and the possibility of an interaction between nodes was greater.

Figure 16 and Figure 17 depict the time evolution curves of nodes of each state with different topological properties. In Figure 16, the number of nodes was kept at 3000 and the number of connected edges of the new joined nodes was changed to 2, 3, 5, and 10, in sequence. When a node joins the network, the large value of m supports more connections of nodes. Therefore, with a greater m, the connections among nodes are closer, and then it is easier for the epidemic to spread in the network. In Figure 16, with the increase in m, more nodes are infected and fewer nodes are vaccinated. When the density is high enough, the epidemic is prevalent on a large scale and no node will be vaccinated in the network. This indicates that reducing contact among individuals to make the network sparse is beneficial for the prevention of an epidemic.

In Figure 17, the number of connected edges of the newly joined nodes was 3 and the number of nodes was changed to 3000, 5000, 10,000, and 20,000, in sequence. Initially, 5% of nodes were randomly selected as infected nodes and 2.5% of nodes were randomly selected to be vaccinated with vaccine A or vaccine B, separately. In networks with different total numbers of nodes, the proportion of nodes in each state was nearly the same. This means that the population size has little impact on the spread trend of an epidemic when the other properties are the same.

5. Conclusions

A novel model named the Susceptible–Infected–Vaccinated with vaccine A–Vaccinated with vaccine B–Recovered ($SI{V}_A{V}_{B}R$) model was presented based on a complex network that considers individual nodes and environment nodes formed by individual aggregation. This novel network not only indicates direct contact transmission among individuals, but also indicates a higher risk of aerosol environmental transmission.

First, the transformation of individual states in the complex network was explained by the differential equations based on the mean-field approximation theory. By calculating the basic reproduction number, $R_0$, using a Jacobian matrix and the Lyapunov function, the conditions for the system to achieve asymptotic stability were obtained. $R_0$ is mainly determined by the properties of vaccines and epidemics. Then, the competition between two vaccines in the vaccine market in terms of efficiency and cost was expressed through the evolutionary game of individual voluntary vaccination. Differing from random immunization strategies, the evolutionary game makes vaccination relevant to vaccine costs and vaccine efficiency.

Second, numerous simulations were presented in scale-free networks. The effect of the parameter values on $R_0$ was verified in simulations. It was found that there is a threshold value for the vaccination rate. For a vaccination rate higher than this threshold, no matter how effective the vaccine is, the epidemic will eventually be eliminated. In addition, epidemics spreading in scale-free networks are extraordinarily sensitive to the vaccine invalidity rate. This means that when considering both environmental and contact infection, vaccination plays a critical role in epidemic control. However, further simulations indicate that egoism caused vaccine coverage to decline when individuals voluntarily decided whether to be vaccinated. For the overall communities, a higher connection density led to lower vaccine coverage and more infected people.

Finally, when market competition between vaccines was analyzed, both improving vaccine efficiency and reducing vaccination costs can encourage individuals to get vaccinated and accelerate the demise of epidemics. Vaccines with a high cost and high inactivation efficiency have a relatively short lifecycle, which means that they are not strong in market competition. In addition, if the manufacturer chooses highly connected individuals as the immunizers after completing the vaccine, it will gain a bigger market share than when the initial immunizers are randomly selected. Additionally, the disease will therefore reach disease-free equilibrium much faster.

This paper provided a more realistic model of epidemic spread dynamics and proposed practical suggestions for epidemic prevention and control through theoretical analysis and simulation. In general, the evolutionary game and complex networks can help to elucidate the interactions between epidemic transmission and individual immunization decisions, especially when vaccine hesitancy and vaccine competition coexist. In addition to improving the immune efficacy of vaccines, implementing price strategies and targeted immunization strategies have proven to be effective for vaccine manufacturers to be dominant the market. For the community, it helps to improve public health.

A complex network was proposed, which includes environmental nodes and aerosol environmental transmission. On this complex network, the competition between two vaccines was evaluated, and evolutionary game theory was used to explain individual choices of different vaccines. This provides a more realistic research method for studying the epidemic clustering, while expanding the traditional epidemic spread dynamic model from the perspective of vaccine market competition. This paper focused on the vaccination game of individuals and preliminarily simulated the impact of different initial vaccination individuals on vaccine coverage. In the future, based on this model, the focus of research can be shifted to the strategies of vaccine suppliers. The impact of different competitive methods on the market share can be further explored, including acquaintance immunization, changing entry times, and penetration pricing.