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Role of vaccine efficacy in the vaccination behavior under myopic update rule on complex networks

Highlights • We propose a new framework that considers the imperfect vaccination on the spatially structured and heterogeneous population.• We present a new vaccination update rule: myopic update rule, which is only based on one focal player’s own perception regarding the disease outbreak.• Healthy individuals are willing to inoculate the vaccine under the myopic update rule, which can stop the infectious disease from being spread.• In regular lattice, the vaccinated fraction is more sensitive to the vaccine failure rate than the relative vaccine cost.


Introduction
Over the past two decades, the outbreak of infectious diseases has been threatening the safety of human lives and properties, such as the severe acute respiratory syndrome SARS [1] , H1N1 [2] , Ebola [3] and so on. Thus, how to prevent the extensive outbreaks of epidemics has become a challenging topic in the field of public health [4][5][6] . Meanwhile, the difference of population distribution, religious belief and regional differences may greatly affect the spread of infectious diseases, for example, Refs. [7][8][9][10] explore the impact of various topological structures within the population on the infectious diseases spread, and it is convincingly found that heterogeneous networks may quicken the disease spreading within the population, even lead to the absence of epidemic threshold [11] .
Meanwhile, the individual reactions to infectious diseases may also substantially influence the diffusion processes of epidemics. One of the most striking cases regarding the outbreaks was H1N1 pandemic in 2009 [12] , which induced around 15 , 0 0 0 deaths. During the outbreaks of H1N1, the suppression of epidemic processes can not only be attributed to the public measures, but also through personal and uncoordinated responses, that is, the human behavior has noticeably interfered with the epidemic spreading. In the long run, human behavior has been intricately correlated with the contagion of infectious diseases. In medieval ages, the deadly bubonic plague rendered many people to avoid and flee away from the sick and their close contacts so that their own immunity can be secured [13] . Similarly, the villagers of Yorkshire in Eyam tried to voluntarily quarantine themselves to stop the spread of the plague from that village [14] . In 2003, during the outbreak of SARS, many citizens spontaneously wear the face masks, some schools are temporarily closed and the students are imperatively required to stay at home so as to avoid the further epidemic infection as much as possible [15] . In addition, protective behavior when confronting the epidemics has also been observed in many other contexts, such as Measles-Mumps-Rubella(MMR) [16] , tuberculosis(TB) [17] and HIV [18] etc.
While the impact of human behaviors on the epidemic spreading process has often been mentioned anecdotally, the accurate modeling or quantitative models are relatively fewer regarding their nature, property, or the effect they may have on the spread of the disease. At present, mathematical models have been put forward to study the role of human behavior in the context of social population, such as escape panic [19] , pedestrian trails [20] , but effort s to quantitatively explore the role of human behavior in the large-scale epidemics generally focus on assessing the effectiveness of various public health measures including the social distancing, school closure etc. In the recent years, there are many fields and methods to help us to study infectious disease [21][22][23][24][25][26][27][28][29][30][31][32] , however, there is an increasing attention about the effect of spontaneous individual action or response strategy on the progression of an infectious disease, in which this kind of spontaneous actions may highly restrain from the further diffusion of epidemics and even change the fate of outbreaks. Thus, it is of great significance to fully understand the interacting mechanisms between the human behavior and disease dynamics within the specified population. It is worth mentioning that Funk et al. [33] systematically summarized the related works and provided a taxonomy framework of behavior-disease models. On the one hand, they classify these models according to source and type of information that individuals base their neighbors on, in which source of information may be local or global and the type of information that individuals change their behaviors are prevalence-based or belief-based; On the other hand, they classify the previous works based on the impact of individual behavior changes on the disease dynamics, which include the following three aspects: (i) the disease state; (ii) model parameters (infection or recovering rate); and (iii) the network contact structure relevant for the spread of epidemics.
In particular, for some preventable infectious diseases with the help of vaccines, the epidemic outbreaks are intricately linked with the individual vaccination behavior since the vaccines can help the vaccinators not to be infected by a specific disease. Meanwhile, these vaccinators may indirectly protect their nearest neighbors with whom they contact, and then these neighbors may choose not to vaccinate again (that is, free-ride the vaccinators) so as to avoid the necessary vaccine fees or other potential risk and side effects. Henceforth, the vaccination behavior may dominate the evolutionary process of vaccine preventable diseases. Among them, Bauch and Earn [34] seminally utilized the game theory to model the dilemmatic situation for an individual facing with the epidemics, and they proposed a class of vaccination game to denote the individual decision making and found that, for the well-mixed population, the Nash equilibrium is never to vaccinate if the vaccination cost is higher than that of being infected; but there exists a Nash equilibrium yielding a suboptimal vaccinated fraction if the vaccine cost is lower than that of being infected.
As a further step, complex networks, beyond the well-mixed topology, provide a unified platform to characterize the topology of real-world populations, where the nodes represent individuals and links mimic the contacts among them [35] . Thus, under framework of game theory, many works are devoted to exploring the interplay between contact patterns, behavioral responses and disease dynamics. As an example, Fu et al. [36] found that heterogeneous networks, such as scale-free ones, can induce a broad range of vaccinating actions of many individuals since highdegree hubs with many neighbors become voluntary vaccinators more probably in order to reduce the risk of being infected. After that, Zhang et al. [37] demonstrated that the hubs may largely inhibit the outbreaks of infectious diseases under the voluntary vaccination policy. In the meantime, various subsidy policies on controlling the epidemic spreading have been determined from the socioeconomic perspectives within the well-mixed and networked population [38][39][40][41][42][43] . Furthermore, most previous works [44][45][46][47][48][49][50][51][52] often assume that the vaccine has a perfect efficacy, which will endow the complete immunity for the inoculated individuals, but an interesting topic is on how the epidemic spreads when the vaccine is not fully effective for the disease (i.e., 100% efficacy)? Besides, in the vaccination game, whether an individual decides to vaccinate the vaccine or not will be often determined by the estimated payoff, and the vaccinating decision may be transferred from one player to another one inside the population according to a specific role model during the outbreaks. Nevertheless, under some real-world scenarios, some individuals are not willing to imitate the behaviors of others as a result of special belief, religion, opinion and even awareness of a disease.
According to the above description, in the real world, when a new epidemic starts to outbreak, people want to take some measures to protect themselves immediately. Generally, inoculating the vaccine is considered as an effective measure, but the vaccine may fail as result of invalid storage, transportation and usage, and then rendered that the vaccinated individuals may confront the risk of being infected. Henceforth, some individuals make a decision to vaccinate just judge by themselves and don't consider their neighbors. Thus, in order to deeply analyze the role of vaccine efficacy and spontaneous individual decision mechanism in the outbreak of vaccine preventable diseases, we propose a vaccination game model to explore the impact of imperfect vaccine efficacy and myopic update rule in the spatially structure and heterogeneous populations. The rest of this paper is structured as follows. Firstly, we depict the new vaccination game model in Section 2 in detail. Then, Section 3 provides extensive numerical simulation results, which are obtained in the regular lattice and heterogeneous scalefree networks, respectively. Lastly, in Section 4 , we end this paper with some conclusions and point out the potential works in the future.

The model
As mentioned above, we consider the vaccination game for a class of emerging epidemics within the structured population, where the vaccine can be obtained after the epidemic spreads. Thus, in the current model, all individuals have no chances to vaccinate at the initial time step ( t = 0 ). After that, each time step ( t ≥ 1) is divided into two elementary sub-steps: one step is for the decision of inoculating the vaccine; the other one is used to model the process of epidemic spreading. Among them, for the epidemic sub-steps including t = 0 , we leverage the frequently used susceptible-infective-recovery (SIR) compartment model to characterize the evolution of epidemics, where each individual may lie in the susceptible (S), infective (I) or recovery (R) state. During the vaccination decision sub-steps, each susceptible individual needs to assess the risk of being infected, and then make the decision whether he will vaccinate or not.
Regarding the SIR model, any susceptible individual may be infected through the contact with infective neighbors and the transmission rate along each infective link is assumed to be β.
Meanwhile, the infected individual can be cured with the recovering rate μ, and the recovered one will not be infected again or infect any other healthy ones. Hence, the probability that the susceptible player i without inoculating the vaccine will be infected by all possible infective neighbors can be written as follows, where k i in f denotes the total number of infected neighbors of the focal player i . The epidemic continues until there are no more newly infected individuals.
As for the individual vaccination decision, each susceptible one will evaluate the risk of being infected and compare the difference between the vaccine cost ( C V ) and the potential expenses once he has been infected. Without loss of generality, we fix the infection cost C I = 1 , while the vaccine cost is usually lower than C I and then its relative cost can be re-scaled as 0 < c = C V /C I < 1 . In order to quantitatively perform the decision, by borrowing from the terms in game theory, we assume that the decision process is based on the comparison between the perceived payoffs of vacci-nation i V and the perceived payoffs of being infected i NV if he is not vaccinated, which can be expressed as follows, respectively, where λ i denotes the potential infection probability that can be calculated according to Eq. (1) . Then, the susceptible agent i will independently decide to inoculate the vaccine with the following Fermi-like probability, where K represents the impact of the noise or its reverse 1/ K means the strength of strategy selection, which reflects the uncertainty of vaccination strategy adoption. Here, we term this vaccination decision as the myopic update rule just based on one player's own perception, which is different from imitating the vaccination strategy of others in many previous works [37,38,[43][44][45][46][47] . When K → 0, agent i is totally rational and whether he will vaccinate or not is fully determined by comparing i V and i NV , On the contrary, when K → ∞ , agent i will randomly perform the vaccination choice.
In addition, we consider the imperfect vaccination program, that is, the vaccine efficacy is not 100% and the vaccination may fail as a result of incorrect transportation, storage, and usage of vaccine. Thus, we introduce an independent parameter θ to characterize the vaccine failure rate, which implies that the susceptible individual to choose the vaccination is still kept in the susceptible state with the probability θ , while the probability changing from S into V ( S → V ) state is set to be ( 1 − θ ). Once the healthy individual decides to vaccinate, he will have a chance to enter into the vaccinated ( V ) state, the SIR epidemic dynamics will evolve into the SIRV model as illustrated in Fig. 1 , in which the successfully vaccinated individual will be equivalent to the R-type one at the next time step, that is, the successfully vaccinated (V-type) agent does not get the infection or infect others, either. We do not know when does vaccine fails, so that at each propagating time step, inoculation individuals are judged whether the vaccine is fails or not. To be noted, the vaccinated healthy individuals will not be vaccinated again during this epidemics even if the vaccine lost its effect.
In order to further explore the impact of network topology on the evolutionary process under the imperfect vaccination, we simulate the current mechanism on L × L regular lattices and scale-free networks with N = L 2 nodes, respectively. Initially, we stochastically choose I 0 = 10 individuals as the infective seeds within the population, and other ( N − I 0 ) individuals are kept in the susceptible state, and thus there have no vaccinators. At t = 0 step, the system starts the evolution according to the SIR model. After that, from t = 1 , each susceptible agent has the opportunity to receive the vaccination and then the system carries out the evolution of epidemics based on the above-mentioned two elementary steps. The system continuously evolves until there are no infective individuals. In addition, the current results are averaged over 10 0 0 independent runs so that the large fluctuations can be removed. In all the numerical simulations, the population size is fixed to be N = 10 0 0 0 . In the homogenous topology, we use the regular lattice satisfying the periodic boundary with the size L = 100 as the underlying networks, and each individual has 4 nearest neighbors (that is, the von-Neumann neighborhood). As for the heterogeneous topology, we generate the scale-free network by using the configuration model, in which the average degree is fixed to be < k > = 4 and the power exponent is set to be γ = 3 .

Results and analyses
In this section, we conduct extensive numerical simulations to demonstrate the vaccination behavior on the regular lattices and scale-free networks, respectively. Among them, we mainly discuss the influence of vaccine cost c and failure rate θ on the collective vaccination level within the population. Without loss of generality, we set the value of parameters in the SIR model as β = 0 . 46 and γ = 1 / 3 , which are identical with those in Ref. [36] .

Regular lattices
First, we investigate the equilibrium fraction of both vaccinated ( ρ V ) and recovered ( ρ R ) state individual size for different values of relative cost of vaccination c and vaccine failure rate θ . Fig. 2 plots 0.06 (yellow triangle), respectively. On the one hand, for a specific vaccine failure rate θ , ρ V declines with the increase of the relative vaccination cost c , while ρ R increases as c augments, which means that the vaccination cost will markedly affect the willingness of individuals to inoculate the vaccine. As an example, when c ≤ 0.5, ρ V and ρ R can almost keep the similar vaccination level as c increases; However, c > 0.5 leads to the substantial reduction of ρ V and the continuous rising of ρ R since the vaccination cost is comparable to the infection cost; In particular, ρ V will be dramatically reduced when c is up to 0.95, even tends to zero as the vaccination cost is too high, especially for c = 0 . 95 . On the other hand, under the same vaccination cost c, ρ V decreases as the vaccine failure rate becomes higher, for instance, the fraction of adopting the vaccination strategy under c = 0 . 06 is much less than that with c = 0 . 05 , which implies that the vaccinated fraction within the whole population will be a little more sensitive to the vaccine failure rate.
Then, we discuss the influence of the noise factor K on the vaccination behavior within the population in Fig. 3 , where we set σ = 1 K , termed as the strength of selection (0 < σ < ∞ ), as 2 and 2.5, which are slightly different from that in Fig. 2 . Likewise, it can be clearly shown that ρ V declines and ρ R increases slowly when the relative cost of vaccination c lies between 0 and 0.4. Afterwards, when the relative cost of vaccination c is more than 0.5, the greater the noise selection strength, the more rapidly the varying trend of ρ V and ρ R . In fact, as the strength of selection increases, individuals become much more rational and will not tend to take the vaccination strategy since they will take their own economic cost and the related interests, say, free-riding behavior, into account. In particular, the relative cost c is beyond 0.5, or the vaccine loss rate is higher (i.e., θ = 0 . 06 ), the un-vaccination behavior of rational individuals become much more prominent, and thus it is unable to prevent the outbreaks of epidemics, which can be observed from the larger ρ R as c > 0.5 or θ = 0 . 06 .
Next, in order to fully check the impact of relative vaccination cost c and the vaccine failure rate θ on the vaccination behavior, Fig. 4 illustrates the evolution of ρ V and ρ R within the broader ranges of c and θ . It is clearly indicated that at the lower vaccine failure rates (say, θ < 0.04), the fraction of vaccinated individuals is often more than half of the total population, even if relative cost of vaccination c is large (e.g., 0.9); Meanwhile, a plethora of vacci- Thus, creating the high quality vaccine is significant, which greatly determines the individual vaccination inclination. Furthermore, to deeply understand individual state change in the lattice as SIRV model evolves, we record the evolutionary snapshots of individual states at various time steps for θ = 0 . 03 , c = 0 . 3 and θ = 0 . 08 , c = 0 . 8 in Fig. 5 . Among them, the upper eight panels denote the snapshots under θ = 0 . 03 and c = 0 . 3 , while the lower eight panels represent the ones for θ = 0 . 08 and c = 0 . 8 . At time step t = 0 , there are no vaccinated individuals on the lattice and only I 0 = 10 randomly infected seeds, and then the epidemic starts to propagate at this time. After that ( t ≥ 1), the susceptible individuals have the opportunity to determine whether they will inoculate the vaccine or not. It is clearly observed that most individuals choose to vaccinate under these two cases when epidemic begins to spread. However, when θ = 0 . 03 is lower, there is fewer vaccinated individuals to become susceptible, and then most of vaccinated individuals are immunized, in which the epidemic is hard to spread and finally tends to be extinct. Reversely, for the higher vaccine failure rate (i.e., θ = 0 . 08 ), the vaccine is easy to be invalid, many vaccinated individuals become susceptible due to the loss of vaccine efficacy. Therefore, the epidemic can be pandemic and then most of individuals enter the recovered state in the end. All these results again demonstrate that the vaccine efficacy plays the significant role in the evolution of vaccination behavior of epidemic outbreaks within the structured population.

Scale-free networks
In the real world, many networks are often heterogeneous, and thus it is necessary to understand the mechanics of myopic update rule better on heterogeneous topology. To this end, we formulated the game of taking the vaccine on the scale-free network. Here, we generate the scale-free network with 10,0 0 0 node under the configuration model, where the average degree of the whole network is equal to 4 and the power exponent 3. After the fundamental networks are created, the system evolves according to the SIRV model, which is identical with the iteration procedure on regular lattices, and the epidemic continues until there are no more newly infected individuals.
First of all, we plot the time courses of fraction of susceptible, vaccinated and recovered individuals for different the relative cost of vaccination c and vaccine failure rate θ in Fig. 6 . In all panels, the red, blue and yellow lines denote the evolution of susceptible, recovered and vaccinated individuals, respectively. It can be found that the vaccinated individuals increase rapidly in a very short time, and then reach a peak. Vaccinated individuals are rarely become susceptible because of the vaccine failure θ is lower (as shown in Fig. 6 a,c) so that the number of recovered ones increases a little and arrives at the equilibrium quickly, which states clearly that the epidemic is eliminated and has not become pandemic. Due to the vaccine failure rate, the fraction of vaccinated individuals goes down and then tends to be zero after reaching the peak. However, when the value of vaccine failure rate θ is raised (as shown in Fig. 6 b,d), even though the vaccinated individuals increase rapidly, they become susceptible quickly due to the high vaccine failure rate θ , it can't prevent the epidemic spread so that the number of recovered individuals increases. Additionally, we found that for the value of vaccine failure rate θ = 0 . 03 , whatever the values of relative cost of vaccination c , the number of recovered agents is the same as that at the equilibrium. Generally, when the epidemic starts to spread, many susceptibles take the vaccine in the population at a short due to the perception of infection risk. Also, this vaccination behavior is almost widespread regardless of the values of relative cost of vaccination c and vaccine failure rate θ . At the lower vaccine failure rate, vaccinated individuals are hard to become susceptible, which leads to the disease propagates difficultly and be eliminated as soon as possible. These results are also consistent with the work of Zhang et al. [37] , since the hub nodes are often vaccinated immediately after the disease starts to spread. But for the higher vaccine failure rate, the vaccinated individuals become susceptible quickly, the disease can outbreak.
We also consider the equilibrium fraction of both vaccinated ( ρ V ) and recovered ( ρ R ) state individual size for different values of  It can be found that in the Figs. 7 and 8 , whatever the ways of vaccination, the epidemic can outbreak and vaccinated individuals are more sensitive to the vaccine failure rate. Therefore, except β = 0 . 46 , we consider the epidemic evolution of SIRV model under the lower transmission rate β = 0 . 3 . Meanwhile, we set the I 0 = 10 initial infective seeds as the top 10 largest degree nodes, which is here termed as the hub infection scheme. Correspondingly, we call the randomly selecting one for the generous case as the random infection.

Conclusions
In summary, based on the SIR epidemic dynamics, we investigate the imperfect vaccine immunity under the myopic update rule in different foundation topology including the regular lattice and scale-free networks, where the focal player makes the vaccination decision just according to his own judgement about the epidemic situation. Extensive numerical simulations show that most unvaccinated susceptible individuals are willing to inoculate the vaccine under the myopic update rule, whatever the type of network is, in particular for the lower vaccine cost ( c ≤ 0.5) and failure rate ( θ ≤ 0.06). After the epidemic starts to propagate, and most of individuals change their strategies to adopt the vaccine in a short time since the individual can estimate the infection risk at the early stage, which leads the epidemics to be hard to spread within the population. However, due to the failure of vaccine or the free-riding behavior of susceptible individuals, vaccinated individuals become susceptible again and then confront the risk of being infected, which creates the potential epidemics situation.
To be of great interest, we find that the impact of vaccine failure rate on the vaccination coverage becomes much higher, when compared to the role of the relative cost of vaccine. For example, the value of vaccine failure rate θ is usually assumed to be no more than 10%, or else most vaccinated individuals become re-susceptible again. At a fixed θ , the fraction of vaccinated individuals almost keep unchanged when the relative vaccine cost is not beyond c = 0 . 5 , but this value will become lower and lower after c is more than 0.5, which is basically consistent with the reality of vaccine usage. On the contrary, on the scale-free networks, the number of vaccinated individuals is more sensitive to the effect on vaccine failure rate θ . Hub nodes have a stronger inclination to adopt the vaccine under the myopic update rule, which can effectively prevent the diffusion of epidemics. Hence, the disease will be eliminated quickly in heterogeneous topology. However, when the values of vaccine failure rate increase a little bit, hub nodes become susceptible and cannot prevent the epidemic spread, which leads to the epidemic outbreak. Meanwhile, when the relative cost of vaccination c increases, unvaccinated susceptible individuals are not willing to choose the vaccination strategy, which can not stop the outbreak of infectious diseases. When the vaccine failure rate further increases, such as 0.03, the number of vaccinated individuals decays to zero before the epidemic is eliminated, which is almost equivalent with the classic SIR model. Anyway, current results are conducive to better understanding the individual vaccination behaviors when confronting the real epidemics.

Declaration of Competing Interest
The authors declare that they do not have any financial or nonfinancial conflict of interests.