Soft Comput DOI 10.1007/s00500-017-2903-1

FOUNDATIONS

Epidemic spreading on a complex network with partial immunization Xuewu Zhang1

· Jiaying Wu1 · Peiran Zhao1 · Xin Su1 · Dongmin Choi2

© Springer-Verlag GmbH Germany 2017

Abstract This paper proposes a new virus spreading model, susceptible–infected–susceptible–recovered–susceptible, which is based on partial immunization and immune invalidity in a complex network. On the basis of mean-field theory, the epidemic dynamics behavior of this model in a uniform network and a scale-free network is studied. After modifying the formula for the effective spread rate, we obtain the theoretical result that the coexistence of partial immunization and immune failure does not affect the network spread threshold. At the same time, the experimental results show that the existence of the above two conditions greatly increases the spread of the virus and extends the diffusion range. The study also found that in the scale-free network, peak viral infection occurs at the beginning of the spreading process, and after the immunization strategy is applied, the early infection peak is effectively curbed. Keywords Complex network · Partial immunization · Transmission dynamics · Immunization strategy

1 Introduction With the development of complex network research, complex systems in the real world are increasingly abstracted into complex networks, and how to disseminate information in Communicated by A. Di Nola.

B

Xuewu Zhang [email protected]

1

College of Internet of Things Engineering, Hohai University, Changzhou 213022, China

2

Division of Undeclared Majors, Chosun University, Gwangju 61452, South Korea

the network has become an important research direction in complex networks. Similar phenomena, such as the spread of a virus (Pastor-Satorras and Vespignani 2000), the spread of rumors (Yu et al. 2016; Sareen et al. 2017; Junho et al. 2015; Wang et al. 2014) in a social network (Nikolaos and Kun 2014) and the spread of a crisis (Hong et al. 2017; Gianni 2015), can be regarded as network diffusion processes that obey certain rules (Yang et al. 2016). The propagation dynamics of complex networks are analyzed using the basic SIS model (Kermack and Mckendrick 1927) and SIR model (Kermack and Mckendrick 1938) in different networks. Since Watts and Strogatz (1998) proposed the small-world network model and Barabsi and Albert (1999) proposed the scale-free network model, a large number of studies have found that there is an exact threshold in the network, and when the effective propagation rate of the virus is greater than the threshold, it can burst and spread in the network. In particular, in BA networks, the propagation threshold tends to zero with increasing network size; that is to say, in a large-scale BA network, as long as the virus has a positive effective transmission rate, it can be retained in the network for a long time (Pastor-Satorras and Vespignani 2001; Moreno et al. 2002; Boguna et al. 2003; Skanderova and Fabian 2015). To study the spread of a virus under different conditions, researchers have proposed a large number of derived models. Liu et al. (2014) established the SIS model with time delay on scale-free networks; Stegehuis et al. (2016) studied the catalytic or inhibitory effects of the community structure on the percolation process (information diffusion and virus transmission); Hong et al. (2016) proposed that the mobility of nodes in heterogeneous networks will affect the propagation process of epidemics; Yun-Peng et al. (2017) proposed a dynamic model of information transmission based on the elements of social influence. In the study of the immune status

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of the nodes, some researchers have found that there is a phenomenon of immune failure in the immune system and thus cases of reinfection (Wang and Jiang 2010; Gandon and Day 2008; Shi et al. 2009). In this regard, Wang and Jiang (2010) proposed a class of SIRS model with incomplete immunity and found that immune failure and immune invalidity reduce the network transmission threshold; Gandon and Day (2008) analyzed the changes in virus transmission behavior under the condition of immune failure; Shi et al. (2009) analyzed the propagation behavior of the SIS model with incomplete immunity in scale-free networks and found that the network threshold was related to the immune failure rate. The above literature summarizes the effects of immunization and immune failure on the spread of a virus, but in real life, infection immunity and the ability to obtain the successful immunity to virus reinfection often occur in stages. Whether the infected person can gain immunity and whether the immune nodes can successfully prevent reinfection are often determined simultaneously during the immunization stage. In this paper, partial immune and immune failure are, therefore, placed in the same time sequence (that is, the step that determines whether the infected nodes can enter the immune state and whether immunity is effective is confirmed while the infected subject is recovering), and we redefine the formula of the effective spreading rate of a virus to simplify the construction of the model and to draw different conclusions from the theoretical derivation. On the premise of improving the formula of the effective spreading rate, partial immunization and immune failure cannot change the network spread threshold, but significantly affect the steadystate density of infected nodes and the spread of the virus in the early stages. Based on the experimental simulation, the target immune strategy can be flexibly adjusted according to different immune resources, which can help to curb the outbreak of the virus.

2 Theoretical model construction In this class of SIRS model, each node can be in 1 of 3 states: susceptible, infected and recovered. The infection process is the same as in the traditional SIRS model. In each time series, a node susceptible to contact with infected persons is infected with probability β. When the infected nodes recover, they gain immunity with probability δ or return to the susceptible state with probability μ without obtaining immunity. For the node to obtain immunity, immune failure must occur with probability α, similar to that of susceptible individuals. Therefore, we treat them as susceptible individuals in the actual processing. For those who have access to the immune state, recovered individuals enter the immune invalidity process with lower probability, η, of loss of immunity into the susceptible state.

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Fig. 1 Diagrammatic representation of the SISRS model in terms of reaction–diffusion processes

In summary, the essence of this paper’s model is to integrate the immune failure and partial immunity in the same time series. On the one hand, it fits the reason that immune failure occurs in the real world. On the other hand, it is possible to simplify the model by addressing immune and infection separately, and the theoretical results are different from those of previous studies. Propagation reaction equation is given by: β

→ 2I S+I − μ

I − →S

η

R− →S (1−α)δ

I −−−−→ R

αδ

I −→ S.

As shown in Fig. 1, the SIS, SIR and SIRS models are included in the partial loop of this model (hereinafter referred to as SISRS). Hence, the SISRS model can be regarded as a unified model; that is, SIS, SIR and SIRS belong to the three special parameter points on the SISRS model parameters. Therefore, the dynamic properties of the SISRS model derived from theoretical analysis should satisfy the characteristics of the other three models. 2.1 SISRS model in a uniform network First, the assumption of increasing uniformity is given: It is assumed that the degree distribution of nodes has a peak at k and that the two sides are steep. According to this assumption, the node degree in the network is approximately k (k is the first-order average degree of nodes). s(t), i(t) and r(t) are defined as the node density of the S nodes, I nodes and R nodes in time t. Based on mean-field theory, the time response equations of s(t), i(t) and r(t) can be obtained: ⎧ ds(t) ⎪ ⎪ = −β · k · i(t) · s(t) + μ · i(t) + η · r (t) + α · δ · i(t) ⎪ ⎪ dt ⎪ ⎪ ⎨ di(t) (1) = β · k · i(t) · s(t) − μ · i(t) − δ · i(t) ⎪ dt ⎪ ⎪ ⎪ ⎪ dr (t) ⎪ ⎩ = δ · i(t) − η · r (t) − α · δ · i(t). dt

In Eq. (1.1), the first term on the right-hand side is a newly infected node. The second means that the infected nodes do not inherit immunity to become susceptible. The third term indicates that the immune node loses immunity and becomes

Epidemic spreading on a complex network with partial immunization

infected. The last term is an immune node that is infected and recovers, thereby gaining immunity. After a period of time (t → ∞), the propagation process reaches a stable state, making ddi(t) = 0. Then t β · k · i(t) · s(t) − (μ + δ) · i(t) = 0. According to the normalization condition s(t) + i(t) + r (t) = 1 and in order to simplify the formula, we define 

μ+δ β·k

=A δ − α · δ = B.

(2)

r  (t) + (η + B) · r (t) = B − B · A. Solving the differential equation, we have

=



B − B · A (η+B)·t ·e +c η+B



B−B·A + c · e−(η+B)·t . η+B

(3)

In order to determine the constant C, the initial conditions of disease propagation are required: Assuming that at the initial time, the virus begins to spread, the infected node density is small and randomly distributed in the entire network. Then s(0) = 1; i(0) = 0; r (0) = 0. By combining the initial conditions, constant C is obtained c=−

B−B·A . η+B

So r (t) = −

B−B·A B − B · A −(η+B)·t . − ·e η+B η+B

Substituting into i(t) = 1 i(t) = · B



B2 − B2 · A η+B

1 B

· (r  (t) + η · r (t)) gives

 ·e

−(η+B)·t

B−B·A . +η· η+B

Because i(t) represents the density of infected nodes, it must be nonnegative λc ≥

1 . k

(s(∞), i(∞)) = (1, 0) or (s(∞), i(∞)) =

By inserting Eq. (2) into Eq. (1.3), we obtain

r (t) = e−(η+B)·t ·

Here, in contrast to the previous literature, the formula of β . the effective spread rate of the virus is redefined as λ = μ+δ Based on the above results, we can see that the SISRS model 1 for uniform network propagation has a threshold value of k under the new virus effective spread rate. At the same time, when the network is in a stable state, ds(t) = 0 and ds(t) = 0, and we can find the steady-state dt dt solution of s(t) and i(t), (s(∞), i(∞)), as

(4)



 μ + δ ηβk − η · (μ + δ) . , β · k βk · (η + δ − α · δ)

(5)

Thus, the local stability of the solution of the equation shows that when the effective spread rate λ is greater than or equal to λc , the virus will spread in the uniform network and eventually

converge to the local equilibrium point μ+δ ηβk−η·(μ+δ) β·k , βk·(η+δ−α·δ) . When λ is less than λc , the virus will quickly die out at the point of no disease (1, 0). Similar to the SIS and SIRS models, the uniform network spread threshold is uniform in the effective spread rate and the spread threshold is the same. The SISRS model proposed in this paper can be regarded as an intermediate model of SIS and SIRS in a uniform network. 2.2 SISRS model in scale-free networks As discussed for the model in the network with no uniformity assumption, because the degree distribution of scale-free networks is represented by a power-law function approximation, the definition of Θ(t) states that any given edge with an infected node exists with a distinct probability. For unrelated scale-free networks (i.e., there is no correlation between different node degrees), the probability of any node having the specified edge orientation of node S is s P(s) k , where P(s) is the degree distribution of scale-free networks. The expression of Θ(t) is  k · P(k) · i k (t) Θ(t) = k s · P(s) (6)  s k k · P(k) · i k (t) . = k Define sk (t), i k (t) and rk (t) as the density of nodes S, I and R with node degree K at moment t. Using mean-field theory ⎧ dsk (t) ⎪ ⎪ = −β · k · Θ(t) · i k (t) · sk (t) + μ · i k (t) + η · rk (t) + α · δ · i k (t) ⎪ ⎪ dt ⎪ ⎪ ⎨ di k (t) (7) = β · k · Θ(t) · i k (t) · sk (t) − μ · i k (t) − δ · i k (t) ⎪ dt ⎪ ⎪ ⎪ ⎪ dr (t) ⎪ ⎩ k = δ · i k (t) − η · rk (t) − α · δ · i k (t). dt

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After a period of time (t → ∞), the spread of the virus reaches a stable period; that is, ⎧ ds (t) k ⎪ =0 ⎨ dt ⎪ ⎩ di k (t) = 0. dt Combining the normalization formula sk (t) + i k (t) + rk (t) = 1, the steady-state solution i k (∞) of i k (t) is obtained as i k (∞) =

ηβkΘ(∞) . η(μ + δ) + βk(η + δ − αδ) · Θ(∞)

(8)

Simultaneously Eqs. (6) and (8) can obtain a self-consistent equation of θ (∞) 

k · P(k) · i k (∞) k 1 ηβkΘ(∞) = k P(k) · k η(μ + δ) + βk(δ + η − α · δ)Θ(∞) k

Θ(t) =

where the effective propagation rate of the scale-free netβ and the spread threshold λc of work is defined as λ = μ+δ

the SISRS model in the scale-free network is kk2  . When the effective transmission rate is less than the threshold value, the virus disappears in the network and the network tends to disease-free equilibrium. When the effective spreading rate is greater than or equal to the threshold, the virus is spread to the local equilibrium point and the equilibrium point i(∞) is Eq. (8). In particular, for scale-free networks, when the network size N → 0, λc will be close to zero. This reflects the vulnerability of scale-free networks to the SISRS model. As long as there is a positive rate of spread, the virus will spread and ultimately become stable in a state of equilibrium. To compare the propagation characteristics of SIS and SIR in the scale-free network, based on the conclusion in Sect. (2.1), when improving the effective spreading rate definition, partial immunization and immune failure do not affect the transmission threshold of the network, but play a role in the steady-state density of infected nodes.

k

= f (Θ(∞)).

(9)

3 Simulation results and discussion It is obvious that Θ(∞) = 0 is a nontrivial solution of the equation. When Θ(∞) = 0, i k (∞) = 0; this shows that when the network is in a stable state, there are no infected nodes. To make the virus spread in the network, it is necessary to have a nontrivial solution of f (Θ(∞)) = Θ(∞). The observation equation is knowable, if f (Θ(∞)) is continuous and differentiable, and f (Θ(∞)) on Θ(∞) is strictly monotonically increasing. If the equation has a nontrivial solution, it must satisfy 0 < Θ(∞) ≤ 1. d f (Θ(∞)) |Θ(∞) = 0 ≥ 1. dΘ

(10)

From Eqs. (9) and (10), we have 1≤ 1≤

d f (Θ(∞)) |Θ(∞) = 0 dΘ

k P(k) k

k

ηβkΘ(∞) η(μ + δ) + βk(δ + η − α · δ)Θ(∞) k P(k) βk 1≤ · k μ+δ ·



|Θ(∞) = 0

k

β k 2  · k μ + δ k λc ≥ 2 , k  1≤

123

(11)

With respect to the network model, this paper chooses the WS network and BA network as the uniform network and scale-free network models to conduct the simulations. As a type of SIR model, the SISRS model has a priori knowledge about the direction of the propagation threshold. Therefore, the experimental contents mainly focus on two aspects: the spread of the virus and the immune strategy. 3.1 WS small-world networks Because there is no strict uniform network in the real world, the WS small-world network generated by Watts and Strogatz (1998) is used to simulate the propagation characteristics of the virus in a uniform network. The simulation parameters are as follows: The number of nodes in the initial nearestneighbor coupled network is 16383, the random reconnection probability p = 0.7, and the propagation time period is 300. The virus parameters are set as follows: infection probability β = 0.05, immune probability δ = 0.15, nonimmune probability μ = 0.15, immune failure probability α = 0.02 and immune invalidity probability η = 0.2. In the initial time of virus transmission, 4 random nodes are selected to be in the infectious state with infection density ρi = 0.024%. Figure 2 shows the image of the WS network with the above parameters obtained from the statistical average of 20 data points. According to the formula of the local equilibrium point density of the uniform network in Sect. (2.1), the theoretical value should be (23.0%, 16.7%) and the actual

Epidemic spreading on a complex network with partial immunization

⎧ ⎨1 f (x) ∼ 4 ⎩ (log2x)/4x

Fig. 2 Density of infected and recovered individuals versus time in a WS network

Fig. 3 Density of infected individuals versus reconnection probability p based on the same nearest-neighbor coupled network

simulation value is (21.1%, 16.2%). There are some errors, but the theoretical values are basically consistent with the simulation values. Figure 3 shows the spread of the virus in the WS smallworld network with different reconnection probabilities, p. Three types of WS networks are derived from the same nearest-neighbor coupled network, and an increase in the reconnection probability p will accelerate and expand the spread of the virus. According to Barrat and Weigt (2000), the clustering coefficient of the WS network is C( p) =

3(K − 2) (1 − p)3 . 4(K − 1)

Newman and Watts (1999), and Newman et al. (2000) give the approximate expression of the average path length of the WS network L( p) =

2N f (N K p/2) K

for x 1 for x  1.

Therefore, the relationships between P and the average path length and clustering coefficient of the WS network are given in Table 1. As shown in Table 1, the influence of the reconnection probability P on the spread of the virus is due to the fact that the side reconnection of the nearest-neighbor coupled network changes the nature of the network. The data in the table show that the average path length and clustering coefficient of the WS network are reduced, which can accelerate the spread and increase the density of the virus. Theoretically, the decrease in the average path length represents a more uniform mixing of nodes. In the real world, population mobility exacerbates the spread of the virus. There are few nodes far from the center of the network, and the virus can spread to the whole network after a certain point. As for the clustering coefficient, a decrease in node clustering reduces the probability of community appearance and weakens the closeness of the community to the host network, which is beneficial to the infection of the community with the virus. Researchers have previously studied the propagation characteristics of viruses in different networks, but have rarely addressed how specific network parameters impact the spread of the virus. For example, the clustering characteristics of the network inhibit or catalyze the virus (Baagyere et al. 2015). However, because of the correlations and interactions between the network characteristics, it is very difficult for researchers to perform an experiment for a single parameter. This is also one of the reasons why academic circles seldom perform such studies. 3.2 BA scale-free network BA network generation methods refer to (Gianni 2015). The network settings are as follows: number of initial nodes M0 = 3, number of edges for new nodes M = 3 and total number of nodes N = 16383. The virus model parameters are set as follows: infection probability β = 0.05, immune probability δ = 0.15, nonimmune probability μ = 0.15, immune failure probability α = 0.02 and immune invalidity probability η = 0.2. At the initial time of virus transmission, 4 random nodes are selected to be in the infectious state with infection density ρi = 0.024%. Figure 4 is the BA network with the above parameters used for the virus transmission experiment of the infected nodes and immune nodes. The results of the BA network are different from those of the WS network: In the early stage of the BA network, there is an outbreak peak of infected nodes. In a short period of time, the peak rapidly disappears and the density of the infection returns to the normal level.

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X. Zhang et al. Table 1 Influence of reconnection probability p on network parameters and virus spreading Reconnection probability

Average shortest paths

Clustering coefficient

Speed of the virus spread

Diffusion ranges %

p = 0.3

3.600

0.2287

Slowest

19.30

p = 0.5

2.263

0.0833

Faster

20.42

p = 0.7

1.664

0.018

Fastest

20.80

Fig. 4 Density of individuals in each state in the BA network

Fig. 5 Influence of h on the density of infected individuals in the BA network

However, the immune nodes in the network do not have a corresponding immune peak but remain stable. To explain this phenomenon, we conduct an experiment for η. Figure 5 shows that a change in the immune invalidity probability η has a great impact on the density of infected nodes. Specifically, with respect to the spread of the virus, the smaller the value η, the more prominent the peak. When η is greater than 0.3, the impact on the density of infected nodes gradually decreases. The reaction in the real world is that the disease gradually recedes and goes into a stable state after the initial eruption. Considering that the value η represents the immune extinction rate, from the expectation of the

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immunization period, the value η means that the duration of immunity can be preserved after the infected node is recovered and it can acquire immunity. The smaller the value of η, the longer the retention period. The information in Fig. 5 and Table 2 shows that a change in the value of η has a strong influence on the steady-state density of the infected nodes. During the early outbreak of the virus, it can produce a certain control effect, but the effect is far less significant than at the later stage. Additionally, it has little effect on the spread of the virus. Based on the existing data, the following conclusion can be drawn: The value η affects the survival rate of immune nodes in the network. The smaller the η, the longer the retention period and the more obvious the immune lag. During the stable period of virus infection, the immune nodes, which are transformed from the initially infected nodes, spread over the network. These immune nodes play an irreplaceable role in containing the spread of the virus and form a dynamic balance, but in the early stages of virus transmission, because of the lag of immunity, the growth rate of immune individuals is much lower than the growth rate of infected individuals. Therefore, this period becomes an immune vacuum in the overall spread process. It is difficult to reflect the effect of low-density immunity on virus transmission, so in the early stages of transmission, there is often a peak in infection. With the infection peak, the number of immune individuals increases synchronously and gradually begins to function. The peak of infection enters a rapid decline, and the network enters a stable period. However, when η is large, the immune hysteresis decreases and the infection peak coincides with the steady-state density.

3.3 Immune strategies for BA networks Figure 6 shows the spread of the virus in the BA network, which is used in Fig. 4 (parameter settings are as follows: β = 0.05, δ = 0.15, μ = 0.15, α = 0.02), when disposable immunization is applied to nodes with a degree of more than 20. The disposable immunization strategy is used to impose a disposable immune resource on the target. Set initial infection density, ρi (0), is 0.024%. When the target node is first infected, the initial infection is successfully immunized. Then, the target node is transferred to the general immune state and enters the immune invalidity process. Therefore, we

Epidemic spreading on a complex network with partial immunization Table 2 Influence of η on the infection peak and the steady-state density of infected individuals

Immune invalidity probability η

0.1

0.15

0.2

0.25

0.3

Peak infection density

26.43%

27.76%

28.49%

30.55%

32.07%

Steady-state infection density

18.01%

22.97%

26.44%

29.17%

31.44%

Fig. 6 Density of individuals in each state by loading disposable immunization on individuals whose degree is above 20

can treat disposable immunization as a strategic short-term immunization. As shown in Fig. 6, nodes with disposable immunization decrease rapidly in the early stage of infection and disappear completely before the network reaches a steady state. Comparison of Figs. 6 and 4 shows the effect of the application of a certain density of target nodes with a disposable immunization resource: 1. The peak of infection in Fig. 4 disappears; 2. The time period of virus spreading from zero density to steady-state density almost doubles; 3. The disposable immunization strategy has little effect on the steady-state density of the virus. In the event of certain diseases or crises and with the lack of effective containment measures, targeting a specific target to impose a disposable immunization strategy helps the whole system to reach a certain number of immune individuals in the early stages of transmission, thus smoothly transitioning through the immune vacuum phase. Because target immunity is temporary, in contrast to the permanent immunization strategy, although the final steady-state infection density is not reduced, the occupation of immune resources is very small and the initial effect is excellent. Figure 7 and Table 3 show the virus spread in the BA network, respectively, for 20, 30, 40 and more than 50 nodes. In Table 3, the rate of containment of infected nodes indicates that the relative density of infected nodes is reduced

Fig. 7 Density of infected individuals by loading permanent immunization, respectively, on individuals whose degrees are above 20, 30, 40, 50

after the immunization strategy is applied. The above data show that targeted immunization to suppress viral transmission is significant. For a small number of nodes, although the immune strategy cannot prevent virus outbreak in the network, it can greatly slow the spread of the virus, control the virus’s steady-state infection density and prevent the initial outbreak peak. The effect of the target immune strategy increases with increasing number of immunizations. Compared to the disposable immunization strategy, under conditions of abundant immune resources, permanent immunity has a great competitive advantage. In particular, for scale-free networks, nodes with higher values generally have greater global influence and become the key nodes of the network, which is necessary and effective to achieve immunity. However, the real-world network system is different from the scale-free network. Although it has the same scale-free property, the parameters, such as the Kshell and betweenness centrality, are more representative and practical indexes to evaluate the influence of nodes. Therefore, the selection strategy of the target node in the actual network must be adjusted according to the situation.

4 Conclusion In this paper, based on the SIR model, a new propagation model (SISRS) is proposed to address the phenomenon of partial immunity and immune invalidity. After modify-

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X. Zhang et al. Table 3 Influence of loading permanent immunization on individuals with different degrees

Degree of nodes

30

40

50

Nonimmune

Density of infected nodes

12.38%

18.27%

20.82%

22.32%

26.44%

Density of permanent immune nodes

3.02%

1.36%

0.78%

0.49%

Nonexistent

Rate of containment of the virus

53.18%

30.90%

21.26%

15.59%

Nonexistent

ing the formula for the effective spread rate, the epidemic dynamics behavior of this model in a homogeneous network and a scale-free network is studied based on mean-field theory. After the theoretical analysis, we found that partial immunization and immune failure not only increase the steady-state density of infected nodes, but also accelerate the spread of the virus. However, the length of immune invalidity significantly affects the steady-state density of the infected nodes and has little effect on the initial outbreak. The experimental data show some problems that cannot be addressed by the theoretical derivation. First, we find that a change in the reconnection probability p can affect the spread of the virus, that is, the network parameters, such as the clustering coefficient and average path length, play roles in the process of virus spreading. However, it is difficult to set up an experiment for a single network parameter, so research on the influence of the network characteristic parameters on the spread of viruses is rare. This is an important direction for future research. Second, in the scale-free network experiment, during the spread of the virus in the immune regression process, due to the lag of acquired immunity, a virus with a low immune invalidity rate has an obvious peak and the peak outbreak rapidly declines, indicating that the increase in immune retention time has a beneficial effect on the long-term control of the virus. Finally, two methods to realize the target immune system are discussed. According to the different protection targets and the quantity of immune resources, it can be flexibly adjusted to optimize the immune effect. Acknowledgements This paper is supported by the National Natural Science Foundation of China (Grant Nos. 61671202, 61573128, 61273170), the National Key Research and Development Program of China (Grant No. 2016YFC0401606) and the Fundamental Research Funds for the Central Universities of Ministry of Education of China (Grant No. 2015B25214). This study was funded by Xuewu Zhang (Grant Nos. 61671202, 61573128, 61273170, 2016YFC0401606, 2015B25214). Compliance with ethical standards

Conflict of interest The authors declare that they have no conflicts of interest to this work. Ethical approval This article does not contain any studies with human participants or animals performed by any of the authors.

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