1 Introduction

Infectious diseases can be transmitted between people, animals and goods which always threaten human survival and take great challenges to whole world [1–3]. In recent decades, a large number of mathematical models for infectious diseases have been built and studied widely to realize the infectious disease. The basic mathematical model representing the dynamical behavior of the three main populations which include the susceptible(S(t)), the infected(I(t)) and the recovered(R(t)), was firstly proposed in 1927 by Kermack and McKendricks [4] called SIR model. Two metapopulation SIR models from individual and population perspectives were proposed in reference [5], which studied the significant influence of contact-dependent infection and migration on epidemic propagation. Khyar et al. [6] considered the multi strain SEIR epidemic model with general incidence rate and gave the equilibrium point stability theorem of different strains. Reference [7] analyzed modified SLIR model with nonlinear incidence and equilibriums of the proposed model are both globally asymptotically stable. Gumel et al. [8] proposed extended models of COVID-19 in which the stability of the equilibrium point and parameter estimation was studied.

As we all know, the accurate modeling can more effectively explore the mechanism of the infectious disease. Therefore, the main factors reflected the practical infectious diseases must be considered, such as delay-time, vaccination and random disturbance. Vaccination [9–12] has always been one of the effective measures to control infectious diseases. Xing et al. [13] studied a recurrent nonautonomous SVIR epidemic model with vaccination, who proved the existence and uniqueness of globally attractive near periodic solutions for the model. A deterministic SVIRS epidemic model with Holling type II incidence rate and vaccination was investigated in reference [14], which explicitly discussed the local stability of the disease equilibrium and the existence of Hopf bifurcation. In order to enhance the immune effect and increase the probability of antibody production, most vaccines adopt a vaccination program of two or more doses. Gabrick et al. [15] propose a SEIR model with two doses of vaccine administration, and analyze that administering two doses of vaccine can significantly reduce the number of infections. Omar et al. [16] generated fractional order model based on the secondary vaccination and analyzed various vaccination strategies. Reference [17] established the SIRS model by introducing vaccination passes and made predictions based on real-world parameter values.

In real life, infectious diseases are inevitably influenced by various random factors during the transmission process, so considering the influence of random factors in infectious disease models will be more practical. In addition, many studies shown that environmental fluctuations also have a huge impact on the development of epidemics with vaccination. Therefore, stochastic differential equation model [18–20] became a more appropriate method for modeling epidemic diseases. A stochastic cholera model with saturation recovery rate is discussed in the reference [21], then the optimal control is added and studied to provide a theoretical basis for the prevention and control of cholera. Zhang et al. [22] established a stochastic SVIR model with general incidence rate, who obtained the sufficient conditions of the extinction and persistence for the model affected by white noise. Reference [23] built a SIVS epidemic model with white noise and gave sufficient conditions of the existence for the periodic solutions. It can be found in reference [24] that the random threshold of the outcome for the stochastic SIS model with vaccination can be determined in case the white noises are small. The SVIR model with white noise was proposed in reference [25], which showed that that environmental white noise is helpful for controlling the disease. Reference [26] further shown that the disease gradually disappeared due to the influence of environmental white noise in the stochastic COVID-19 model. Wang et al. [27] studied the influence of vaccination rates, vaccine effectiveness and immune loss rates on infectious disease in a stochastic mathematical model with vaccination. We can conclude from the above references that appropriate white noise intensity can accelerate the extinction of diseases under some conditions.

However, some sudden environmental impacts such as earthquakes, floods, large scale human activities, etc. also affect the infectious diseases in reality which cannot be described by white noise. Therefore, researchers began to use non Gaussian Lévy noise to depict these sudden environmental phenomena in nature [28, 29]. Reference [30] presents a class of the stochastic time-delayed SEIRS epidemic model incorporating both public health education driven by Lévy noise and how to prevent and control diseases. Sabbar et al. [31] established a stochastic COVID model under the influence of quadratic Lévy noise and quadratic jump-diffusion has no impact on the threshold value, but it remarkably influences the dynamics of the infection and may worsen the pandemic situation. The stochastic SIRS epidemic model with white noise and Lévy noise was discussed by Mu et al. [32], who verified that Lévy noise can further inhibit the outbreak of disease. Fan [33] proposed a stochastic SIV epidemic model with non-linear incidence rate, who found that the Lévy noise intensity and the vaccination have a great impact on the transmission dynamics of the disease. Reference [34] introduced white noise together with Lévy noise into the stochastic SIVS model and proposed random threshold affected by noise intensity which completely determined the development of epidemic disease. In addition, a hybrid switching SVIR epidemic model with Lévy noise was studied in reference [35] and the sufficient conditions for the existence of positive recursion for the solutions were obtained. Jaouad et al. [36] established a stochastic infectious disease model with isolation strategy for COVID-19 and studied the stochastic dynamical properties of the random solutions around the equilibrium point. Based on the above references, although various mathematical models have been extensively studied and applied, it should be noted that the complex modeling of secondary vaccination for infectious diseases is still not enough and the propagation mechanism of the corresponding complex models is not clear. In addition, the research on discontinuous random disturbances in such vaccination models should be further explored. Therefore, building upon the foundation established by previous works, this study seeks to explore the dynamical properties of the stochastic model with secondary vaccination, which is mainly driven by discontinuous noise, namely Lévy noise. In particular, we will discuss the effects of discontinuous stochastic interference and its intensity on the average extinction of diseases.

The present work will be organized as follows. In the second section, we establish the disease model with secondary vaccination and multisource noises. The existence and uniqueness of positive solutions proposing model is proved in the third section. In the fourth section, asymptotic behavior at the disease-free equilibrium point is studied and analyzed. We give the random threshold of disease extinction for the proposed model in the fifth section. In the last section, the theoretical results are verified by numerical simulation.

2 Stochastic model

In this paper, in order to be more consistent with the early development of infectious disease and obtain the required results, we present the following assumptions:

1. (1)The recovered population has immunity and will not be infected again.
2. (2)Most people who receive the second vaccination will have immunity and will not be infected again.
3. (3)Vaccinated people will transfer to exposure after being infected.
4. (4)Assuming that the interval between two vaccinations is short, the first vaccinated person will not be infected temporarily.

We divide the population into seven compartments based on the above assumptions as shown in Fig 1. That is susceptible person S(t), exposed person E(t), first vaccinator V₁(t), second vaccinator V₂(t), asymptomatic infected person A(t), symptomatic infected person I(t) and recovering person R(t). By introducing white noise, the model is built to reflect the more real state of disease transmission. Among them, white noise represents small disturbances in the environment, such as temperature changes, climate and other impacts. The established stochastic model is as follows [37]: (1) where Λ is the constant migration rate of the susceptible population, β₁ is the transmission rate of asymptomatic infected persons, β₂ is the transmission rate of symptomatic infected persons, ρ₁ is the first vaccination rate, ρ₂ is the second vaccination rate, σ is the infection rate of the exposed to the infected, ω is the proportion of infections in symptomatic patients, α is the ratio from asymptomatic infected people to symptomatic infected people, α₂(0 < α₂ < 1) is the infection rate of secondary vaccinators, so 1 − α₂ is the effectiveness of the vaccine. γ₁ and γ₂ respectively represent the recovery rate of asymptomatic and symptomatic infected persons. μ₁ and μ₂ represent the natural mortality rate and disease-related mortality rate respectively. All parameters are positive. B_i(t), i = 1, 2, 3, ⋯, 7 denotes the independent standard Brownian motion defined on a complete probability space (Ω, F, P) with filtering {F_t}_t>0, σ_i ≥ 0 is the intensity of B_i(t), i = 1, 2, 3, ⋯, 7.

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Fig 1. The relationship diagram between different compartments in the infectious disease model.

https://doi.org/10.1371/journal.pone.0296183.g001

When sudden phenomena occur in nature, the impact of measures taken by people during the epidemic and government intervention on infectious diseases is described by Lévy noise in this article. Therefore, a class of improved models driven by both white noise and Lévy noise are as follows: (2) where is Poisson compensation measure, N(dt, du) is Poisson random measure, λ(du)dt is smooth compensation. λ is a feature measure defined on a measurable subset of Y ⊂ [0, ∞) and satisfies λ(Y) < ∞. In addition, N(dt, du) and B_i(t) are independent, D_i(u) > −1(i = 1, 2, 3, ⋯, 7) represents the jump diffusion coefficient.

For σ_i(i = 1, 2, 3, ⋯, 7) and D_i(u)(i = 1, 2, 3, ⋯, 7) of Eq (2), we make the following assumptions:

1. (A) D_i(u) is bounded and , where u ∈ Y.
2. (B) Suppose ϵ > 1 exists make where
3. (C) There is a normal number D that satisfies

It should be pointed out that Eq (2) is an idealized model. From a real perspective, Eq (2) can approximately describe the spread of diseases in areas with sudden random population changes. In the following content, we mainly focus on how the discontinuous environmental shocks described by Lévy jumps affect the dynamic behavior of Eq (2), especially the threshold for disease extinction.

3 Existence and uniqueness of global positive solution

In order to study the dynamics of the infectious disease model, we first concern whether the solution of Eq (2) is global and unique. Here, we give the following theorem. Since the first six formulas of Eq (2) which do not contain R(t), this section only considers the subsystem composed of the first six formulas of Eq (2), i.e Eq (3): (3)

Theorem 3.1 Let Assumption (A) hold, for given initial value (S(0), E(0), V₁(0), V₂(0), A(0), I(0)), Eq (3) has a unique positive solution (S(t), E(t), V₁(t), V₂(t), A(t), I(t)) at t ≥ 0, and this solution will stay in with probability 1. So for all t ≥ 0, solution a.s..

Proof: Since the coefficients of the equation are locally Lipschitz continuous, for given initial value (S(0), E(0), , there is a unique local solution (S(t), E(t), V₁(t), V₂(t), A(t), I(t)), t ∈ [0, τ_e), where τ_e stands for explosion time. To prove that the local solution is the global solution, we need to prove τ_e = ∞ a.s.. To do that, we have to make k₀ > 0 and sufficiently large, so that S(t), E(t), V₁(t), V₂(t), A(t), I(t) is in the interval . For each integer k ≥ k₀, we define the stopping time In this section, we set inf ∅ = ∞ (∅ denotes the empty set). It is easy to get τ_k is increasing as k → ∞. Set τ_∞ = lim_k→∞ τ_k which implies τ_∞ < τ_e a.s.. If the hypothesis τ_∞ = ∞ is true, then τ_e = ∞ a.s.. For all t ≥ 0. This means Use proof by contradiction, assume τ_e = ∞ a.s.. And then there are constants T > 0 and ε ∈ (0, 1) which make Hence there is an integer k₁ > k₀ such that (4) Define a C²–function , as follows (5) the non-negativity of Eq (5) can be obtained from Using Itô formula, we can get (6) where is defined by we let there are where Using assumption (A) and Taylor’s formula where θ_i(i = 1, ⋯, 6) is any value in (0, 1).

So we can get where K(K ∈ N⁺) is a positive constant, which isn’t rely on S, E, V₁, V₂, A, I and t.

So there is (7)

Integrating both sides of Eq (7) from 0 to τ_k ∧ T = min{τ_k, T}, then take the expectation, we can get (8) Set Ω_k = {τ_k ≤ T} when k ≥ k₁, we obtain P{Ω_k} ≥ ε by Eq (4). Now notice that for every ω ∈ Ω_k, there is at least one of S(τ_k, ω), E(τ_k, ω), V₁(τ_k, ω), V₂(τ_k, ω), A(τ_k, ω), I(τ_k, ω) equals to k or .

Therefore is no less than thus (9)

Substituting Eq (8) into Eq (9), we have (10) where denotes the indicator function of Ω_k. Letting k → ∞, then we have (11) Eq (11) is a contradiction. Therefore we have τ_∞ = ∞, i.e the proof is completed.

From Theorem 3.1, it can be seen that regardless of the noise intensity D_i and σ_i, the stochastic model almost always has a unique global positive solution for any given initial value.

4 Asymptotic behavior of disease-free equilibrium point

The disease-free equilibrium point of the deterministic form of Eq (1) can be obtained where

Based on the disease-free equilibrium point and its expression obtained from the above equation, we can then discuss the asymptotic behavior of the system’s solution in the disease-free equilibrium point of the corresponding deterministic system, which reflects whether the disease is extinct to a certain extent.

Theorem 4.1 For any given initial value (S(0), E(0), V₁(0), V₂(0), A(0), I(0), R(0)) , then the solution of the Eq (2) satisfies the following property where Proof: Let Eq (2) is rewritten as Define a C²–function where c_i(i = 1, ⋯, 5) are the normal numbers to be determined.

Using Itô formula, we have (12) where

At this time, we make where c_i > 0, i = 1, ⋯, 5, then we use 2ab ≤ a² + b² and (a + b + c)² ≤ 3a² + 3b² + 3c² to get therefore (13) where Let integrate both sides of Eq (12) from 0 to τ_k ∧ T = min{τ_k, T}, then take the expectation, we can obtain (14) Substituting Eq (13) into Eq (14) to get (15) We conclude that where hence, this theorem has proven that the disease-free equilibrium point is globally asymptotically stable.

According to the above theorem, if P₀ is globally asymptotically stable, the disease will disappear for a period of time.

5 Extinction of disease

In this section, we shall discuss the condition of extinction of the diseases. To this end, we first provide the following relevant lemma and its proof.

Lemma 5.1 For any initial value Eq (2) always has a unique global positive solution it has the following properties (16) and (17) as well as (18) Proof: Define X(t) = S(t) + E(t) + V₁(t) + V₂(t) + A(t) + I() + R(t), Let Y(X) = X^ϵ. Applying Itô formula, we can get where (19) where constants σ² and d are defined in assumption (B). Choose ϵ > 1 such that So (20) For 0 < k < bϵ, we have (21) Integrating both sides of Eq (21) from 0 to t yields (22) Taking expectation on both sides of Eq (22) yields (23) Combining Eq (19), we get where Thus, it follows from Eq (23) that that is, we obtain For convenience, we denote Q = X^ϵ(0) + ϵH, then we have (24) Integrating both sides of Eq (20) from 0 to t, for sufficiently small δ > 0, k = 1, 2, ⋯, we obtain where and where we have used the Burkholder-Davis-Gundy inequality in the aboved, l₁ = bϵ and is positive constant. Therefore In particular, choose δ > 0 such that then combine Eq (24), we get Let ϵ_X > 0 be arbitrary. Applying Chebyshev’s inequality, we obtain According to Borel-Cantelli lemma, we obtain that for almost all ω ∈ Ω (25) holds for all but finitely many k. Then, there exists a k₀(ω), for almost all ω ∈ Ω, for which Eq (25) holds whenever k ≥ k₀. Consequently, for almost all ω ∈ Ω, if k ≥ k₀ and kδ < t < (k + 1)δ, Hence Let ϵ_X → 0, we obtain For , we get , and so Namely, for any small , there exists a constant T = T(ω) such that for t ≥ T, So which together with the positivity of the solution implies Then we have By the same way, our subsequent proof is similar to the proof of [38], therefore the lemma has been proven.

In order to get the conditions of diseases extinction, we have

Theorem 5.1 If assumption (A),(B),(C) hold, let (S(t), E(t), V₁(t), V₂(t), A(t), I(t), be a positive solution for Eq (2), the initial solution of Eq (2) is (S(0), E(0), V₁(0), V₂(0), A(0), I(0), R(0)).

If random threshold then the solution of the system has the following property, i.e (26)

Proof: First, integrating Eq (2) to obtain (27)

Integrating both ends of Eq (27) from 0 to t divides by t, we can obtain (28) at this time, we have (29) where

Using Itô formula and Eq (2), we can get (30)

Integrating the Eq (30) from 0 to t and divide by t, so there is (31) where Using assumption (C), we can obtain According to the strong number theorem, we have (32)

It can be get by combining Eqs (29), (31) and (32) (33) where the random threshold is denoted as If R₁ < 1, There are Therefore, this conclusion is proved.

According to the random threshold R₁ in the above theorem, if R₁ < 1, the disease will disappear for a period of time, otherwise the disease will continue and further develop into endemic diseases.

6 Numerical simulations

In this section, numerical simulations will be conducted to verify the proposed theoretical results. To this end, we will apply the Euler numerical approximation method [39, 40] to calculate Eq (2). By assumption (A) and the constraints on c_i > 0(i = 1, ⋯, 5) in Theorem 4.1, we provide the following parameters for numerical simulation: at this time, the disease-free equilibrium is P₀ = (0.0103, 0, 0.01, 0.0127, 0, 0, 0.1592).

Based on the constraints proposed in the theorem and the conditions in the hypothesis, we choose In this case, the random threshold R₁ = 0.0855 < 1.

The initial value of Eq (2) is set as (1, 1, 0, 0, 2, 1, 0). Using the data given above, Fig 2 shows the asymptotic behavior of the stochastic model with different noise at the disease-free equilibrium point. From Fig 2, it can be clearly observed that the two curves for A(t) shown in Fig 2(a) and I(t) shown in Fig 2(b) with different noise gradually approach zero, which means that the infected persons of the disease gradually approach extinction. That is consistent with the theory we found about the disappearance of infected people. It also can be seen from the Fig 2 that the curve with Lévy noise fluctuates greatly, which indicates that Lévy noise has an prominent effect on both asymptomatic and infected persons. Obviously it has a greater impact on asymptomatic infected people shown in Fig 2(a) for Lévy noise, which means that asymptomatic infected people are not easy to find under common conditions. Therefore the control measures for asymptomatic infected people should be scientifically deployed.

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Fig 2. The trend of infected persons in stochastic model (2) with white noise and Lévy noise.

https://doi.org/10.1371/journal.pone.0296183.g002

Based on the above parameters, we let ρ₂ is equal to 0.65, 0.83 and 0.96 respectively, the extinction trend of asymptomatic infected person A(t) and symptomatic infected person I(t) is shown in Fig 3(a) and 3(b) respectively. It can be clearly seen from Fig 3 that with the increase of secondary vaccination rate, the two types of infected populations gradually become extinct at a faster rate, which means that the extinction rate of the disease is faster.

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Fig 3. The trend of infected persons in stochastic model (2) with different secondary vaccination rate ρ₂, where ρ₂ = 0.65, 0.83, 0.96, respectively.

https://doi.org/10.1371/journal.pone.0296183.g003

Next, in order to further investigate the impact of vaccination on diseases, the extinction trends of infected individuals are plotted under different vaccination rates in Fig 4. It is observed from Fig 4 that the trend of disease extinction is faster when both vaccination rates are high. However for lower value of vaccination rates, for example ρ₁ = 0.32, ρ₂ = 0.42, the longer it takes to achieve disease extinction. This indicates that increasing the vaccination rate has a significant impact on diseases, and when the vaccination rate is low, prevention and control work will take longer.

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Fig 4. The trend of infected persons in stochastic model (2) with vaccination rate ρ₁ and ρ₂, where (1)ρ₁ = 0.32, ρ₂ = 0.42, (2)ρ₁ = 0.92, ρ₂ = 0.42, (3) ρ₁ = 0.32, ρ₂ = 0.96, (4)ρ₁ = 0.92, ρ₂ = 0.96.

https://doi.org/10.1371/journal.pone.0296183.g004

In addition, the curve fluctuation of asymptomatic infected persons fluctuates significantly shown in Figs 3(a) and 4(a) when the vaccination rates is smaller, which means that in real life, the population’s immunity decreases after reducing vaccination. Due to the difficulty in detecting asymptomatic infected individuals, there will be significant changes in the number of infected individuals in such situations. Therefore, we should improve the secondary vaccination rate and increase the body immunity rate to effectively control the epidemic in real life.

According to Theorem 5.1, the random threshold R₁ will affect the extinction of the disease, therefore it is necessary to analyze the influence of R₁ on its extinction. Assume that the given system parameters and partial noise intensity remain unchanged, we make that σ₆ = 0.62 and σ₅ is 0.11, 0.37 and 0.62 respectively, then the random threshold value R₁ is equal to 0.5375, 0.5348 and 0.5299 respectively. From this, we know that the influence of white noise intensity σ₅ on random threshold is insensitive. The trend of asymptomatic infected person A(t) and symptomatic infected person I(t) with different white noise intensity σ₅ is shown in Fig 5(a) and 5(b) respectively. It can be intuitively concluded that when the white noise disturbance intensities increase, the fluctuation amplitude of the curve is relatively low, and with the increase of noise disturbance intensity, the extinction rate of the disease will be faster. As can be seen from Fig 5 that the change of asymptomatic infected person more fluctuated significantly shown in Fig 5(a) with the increase of σ₅.

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Fig 5. The trend of infected persons in stochastic model (2) with different white noise intensity σ₅, where σ₅ = 0.11, 0.37, 0.62, respectively and σ₆ = 0.62.

https://doi.org/10.1371/journal.pone.0296183.g005

In addition, We set σ₅ = 0.65 and σ₆ is equal to 0.05, 0.35 and 0.65 respectively, the other parameters remain unchanged, then the random threshold R₁ is equal to 0.5375, 0.5351 and 0.5296 respectively. It is also found that the the influence of white noise intensity σ₆ on random threshold is insensitive. The trend of asymptomatic infected person A(t) and symptomatic infected person I(t) with different white noise intensity σ₆ is shown in Fig 6(a) and 6(b) respectively. It can be seen from Fig 6 that the change of symptomatic infected people fluctuates more obviously shown in Fig 6(b) as σ₆ is increased.

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Fig 6. The trend of infected persons in stochastic model (2) with different white noise intensity σ₆, where σ₆ = 0.05, 0.35, 0.65, respectively and σ₅ = 0.65.

https://doi.org/10.1371/journal.pone.0296183.g006

Figs 5 and 6 indicate that the minor events disturbances have insensitive impact on the extinction of disease. However the intensity σ₅ of the white noise B₅(t) caused by the environment changes for asymptomatic infected people just affected themselves obviously. The same as the σ₆ to symptomatic infected person. Therefore epidemic prevention and control should be treated in different categories for different kind of infected people.

Let the proposed parameters remain unchanged, D₆ = 0.45 and D₅ is chosen as 0.32, 0.36 and 0.39 respectively, the random threshold R₁ is equal to 0.1043, 0.6043 and 0.9793 respectively. It is obvious that the small change of D₅ can make the lager change of R₁. The trend of asymptomatic infected person A(t) and symptomatic infected person I(t) in stochastic model (2) with different diffusion coefficient D₅ is shown in Fig 7(a) and 7(b) respectively. It can be seen from Fig 7 that the later to disappear of asymptomatic infected people shown in Fig 7(a) as D₅ is increased, which means that the sudden fluctuation encountered by asymptomatic infected people have a stronger impact on the extinction of disease.

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Fig 7. The trend of infected persons in stochastic model (2) with different diffusion coefficient D₅, where D₅ = 0.32, 0.36, 0.39, respectively and D₆ = 0.45.

https://doi.org/10.1371/journal.pone.0296183.g007

In addition, let D₅ = 0.56 and D₆ be 0.35, 0.38 and 0.41 respectively, then the random threshold R₁ at this time is 0.1601, 0.5351 and 0.9101 respectively. We can find that the small change of D₆ can make the lager change of R₁. The trend of asymptomatic infected person A(t) and symptomatic infected person I(t) with different diffusion coefficient D₆ is shown in Fig 8(a) and 8(b) respectively. It can be seen from Fig 8 that when the diffusion coefficient D₆ of Lévy noise caused by symptomatic infected persons I(t) changes, the symptomatic infected persons changed significantly shown in Fig 8(b).

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Fig 8. The trend of infected persons in stochastic model (2) with different diffusion coefficient D₆, where D₆ = 0.35, 0.38, 0.41, respectively and D₅ = 0.56.

https://doi.org/10.1371/journal.pone.0296183.g008

From the Figs 7 and 8, we know the smaller the random threshold R₁, the faster the disease reaches the disease-free equilibrium point. This means that both the diffusion coefficients have a sensitive impact on disease extinction. In real life, some emergencies can affect the spread and control of infectious disease. The drug treatment and passive physical isolation play a role in promoting the extinction of the disease. At the same time, the virus detection ability and drug treatment need to be further improved.

7 Conclusions

Nowadays, infectious disease models with vaccination have been used by more and more researchers to predict the infectious diseases. On the other hand, in real life, there are many random factors which affect the spread of the epidemic, such as large scale human activities, preventional policies and drug improvement. On this basis, the mathematical epidemic model of vaccination affected by white noise and Lévy noise is established, then the dynamics of the stochastic model was analyzed.

Firstly, the model is proved to have a globally unique positive solution by establishing Lyapunov function. Secondly, the asymptotic behavior of disease-free equilibrium is studied. Then, we obtain the theoretical results about the disease extinction for the random threshold R₁. If R₁ < 1, the disease tends to go extinct. Finally, the numerical simulation results show that the Lévy noise has a great influence on disease dynamics which verified the theoretical proof. When the random threshold of the infectious disease is less than 1, the proposed random model can become extinct in an average sense. The main reason may be random interference, which can lead to the disappearance of diseases due to large white noise interference. However, in the stochastic model driven by Lévy noise, the factors affected by white noise cannot account for the majority of the reasons, and Lévy noise is the main factor determining the development of diseases. Levy noise may also inhibit the spread of diseases or promote their spread.

This work provides a stochastic infectious disease dynamics model with secondary vaccination, which helps to understand the impact of vaccination and random noise on infectious disease prevention and control. In terms of disease prevention and control in the real world, we find that when some sudden situations occur for infected people, it may directly promote or control the spread of infectious diseases. For example, develop more effective vaccines and drugs for infected people and let more people to receive multiple vaccinations, which must provide a powerful help to control the spread of infectious disease. In addition, during the outbreak of the epidemic, susceptible populations can be encouraged to receive vaccines to improve their effectiveness, and efforts should be made to increase the immunity of the population through such measures to prevent the further spread and spread of infectious diseases.

Based on the Virus evolution and technological development, we need improve the researches for the infection disease deeply. Such as considering the effects of impulsive perturbations on system (2) or the impact of viral mutations on vaccine administration. In our future researches the machine learning method, data-driven modeling and intelligent prediction will be applied to explore the infectious disease more effectively.

Acknowledgments

The authors would like to thank the editors and reviewers for their valuable suggestions on the logic and preciseness of this article.