Analysis of stochastic epidemic model with awareness decay and heterogeneous individuals on multi-weighted networks | Scientific Reports

Scientific Reports volume 14, Article number: 26765 (2024) Cite this article

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In the current study, we present a stochastic SAIS (unaware susceptible-aware susceptible-infectious-unaware susceptible) epidemic dynamic model on complex networks with multi-weights. The disease dynamic is influenced by random perturbations to the force of the infection rates, as well as awareness rates. To analyze the problem of extinction, we discuss both the stochastic asymptotic stability in the large and almost surely exponential stability of the trivial solution. Then, we get some sufficient conditions, which guarantee the stochastic persistence of infectious disease. Based on the Erdös-Réyni random graph, the numerical simulations are given. These not only validate our conclusions but also obtain else significative results. Both theoretical results and numerical simulations further reflect that improvement of risk awareness and reduction of decay in awareness are highly effective in preventing disease spread. And then, environmental noises play a significant role in disease transmission.

Infectious diseases remain a major public health concern and pose a serious threat to human survival. Using the mathematical models as analytical tools, more and more researchers make scientific quantitative forecasts and assess the effectiveness of policy interventions against epidemic development.

In 1927, Kermack and McKendrick1 first proposed the most fundamental epidemic system. In the homogeneous approximation, any individual has the equal chance of being infected or spreading infection to others, which is so-called mixed populations2. However, the heterogeneity in number and strength of contacts between individuals determines that there will be significant differences in the transmission rate and paths of disease transmission3. As driven by concerns for realism, it is the typically deficiency that such models can not capture the above feature for complex practical social contacts. After 2000, as the concept of ”random network” become more popular, the graph theory associated with it, provides innovative quantitative tools and mechanisms for modeling heterogeneity of epidemic spread between individuals, as well as the media-effects for epidemiological sense4,5. Recently, epidemic dynamic models on networks have been extensively studied6,7,8,9,10.

Though the heterogeneity networks11 draw extensive concern, it is still unknown how multi-relation impacts the dynamical behaviors of epidemic spreading for the complex network structure. If two nodes are connected, we can express the weight between them as a constant. In a social network, the weight represents the rate of direct contact between individuals. In an information network, the weight represents the rate of information transmission. Reflected in the infectious disease, it is the rate of disease transmission. Indeed, the vast majority of real-world networks, such as public traffic network, biological neural networks, and communication networks12,13,14 have inherent multi-relation connections. It is a reasonable method to use different types of weights to describe each relation of the multi-relationship network15. Due to the difficulty of theoretical analysis, few studies address the issue of multi-weighted networks on disease transmission.

The detailed social contact structure among different individuals is believed to be a central importance for the realistic characterization of epidemics, as it is immediately relevant to disease transmission16. Intuitively, the behavioural responses adopting intervention measures make it possible that infectious disease dynamics could be affected. The individuals (labelled as aware or alert) who avoid physical contact with others as much as possible, especially infected, help to prevent transmission of infection and even essentially terminate this procedure17.

Explicitly considering the impact of information dissemination on the dynamics of infectious diseases, non-infectious individuals are divided into two groups-aware and unaware. Juher et al.18 proposed a deterministic SAIS system as follows:

Here, the size of population is N. For individual i, \(p_{S_{i}}(t)\), \(p_{I_{i}}(t)\), \(p_{A_{i}}(t)\) are the probabilities to be unaware susceptible, infectious, and aware susceptible, respectively; \(\beta _{ij}\), \(\beta _{ij}^{e}\) denote the infection rate of infected node j towards unaware susceptible i and aware node i, respectively; \(\delta _{ij}\) is awareness rate of infected node j towards unaware susceptible i; \(\gamma _{i}^{e}\) is recovery rate for infected node i; \(\kappa _{i}^{e}\) is rate of awareness decay for aware node i, \(i,j=1,\ldots ,N.\)

In many aspects, the model (1) can become more realistic. It should be pointed out that, the influence of random factors cannot be ignored in the spread of infectious diseases19,20,21,22,23. We rely on each individual separately that is aimed at considering the parameters (such as the infected rate, recovery rate), which can characterize the occurrence of effective diffusion among individuals. As so often is the case, values of these parameters may not be given in advance, nevertheless statistical properties of them are known20. Specifically, distributions of heterogeneous parameters could be considered as additional dynamical variables24. Without functional forms and properties of the stochastic process, it is typically the case that we can use white noises for modeling the fluctuations in time of the parameters25,26,27.

The model in the current work incorporates several novel features:

Utilizing multi-weighted networks, we successfully characterize individual heterogeneity and integrated the dual processes of disease and information transmission into the model.

Taking into account the impact of environmental factors, white noise is incorporated to characterize the dynamic nature of the disease transmission rate and awareness rate.

Based on refs.17,18, this paper will fully consider the following aspects: When the infected people return to health, they may be either unaware susceptible or aware ones, And then, aware susceptible individuals become unaware class with decreased vigilance levels. In addition, when unaware susceptible individuals contact with aware ones, they may also be the aware, by capturing information and perceiving risk of contagion.

In previous research, the aforementioned factors have not been considered in the modeling process. Due to the complexity of multi-weight networks, the direct analysis of stochastic models is far from trivial and little progress has been made28,29,30. The theoretical results contained in this paper not only refer to the infectious disease dynamics of SAIS with stochastic disturbance and awareness decay, but also provide some new insight into intervention and containment of pandemics.

The rest of the paper is organized as follows. In Section “Model description” we describe our model and present the existence of the global solution. In Section “Extinction“ we concentrate on the extinction of the epidemic both in probability one and almost surely exponentially. In Section “Stochastic permanence” we study persistence of the epidemic. Numerical investigations are then given in Section “Numerical simulations”. The last part is a conclusion.

For fixed population scale N, individual i exists in one of three states: unaware susceptible (\(S_{i}\)), infective (\(I_{i}\)), or aware susceptible (\(A_{i}\)). \(1-x_{i}(t)-y_{i}(t)\), \(x_{i}(t)\), \(y_{i}(t)\) respectively denote the probabilities of them at time t. \(q_{i}\gamma _{i}\) denotes recovery rate which infected individual i become unaware susceptible; \((1-q_{i})\gamma _{i}\) denotes recovery rate which infected individual i become aware susceptible; \(\kappa _{i}\) is rate of awareness decay which aware individual i become unaware one; \(\beta ^{a}_{ij}\), \(\delta ^{a}_{ij}\) are the infection rate, the awareness rate of infected node j towards unaware susceptible node i, respectively; \(\beta ^{b}_{ij}\), \(\delta ^{b}_{ij}\) are the infection rate, the awareness rate of infected node j towards aware susceptible node i, respectively. In reality, aware susceptibles have an even lower risk of infection than unaware ones. In order to reflect this fact, we assume \(\beta ^{a}_{ij}\ge \beta ^{b}_{ij}\), \(i,j=1,\ldots ,N\). Hence, the propagating process of epidemic diseases with awareness decay is illustrated by the diagram in Fig. 1.

Graphical depictions of SAIS epidemic dynamic model on complex networks with multi-weights.

And then, taking the effect of randomly fluctuating environment into consideration, the resulting model involves that the transmission coefficients \(\beta ^{a}_{ij},\ \beta ^{b}_{ij}\) and awareness rate \(\delta ^{a}_{ij},\delta ^{b}_{ij}\) always fluctuate around some average value. In this sense, for \(i,j=1,\ldots ,N\),

Here \(\dot{B}^{\nu }_{ij}\) is white noise, and the intensity \(\sigma _{ij}^{\nu }:[0,1]^{4}\rightarrow [0,\infty )\) is locally Lipschitz continuous and satisfies

where \(m_{ij}^{\nu }\) is positive constant, \(\nu =a,b,c,d.\)

Our paper focuses on strongly connected adjacency matrix \(A=(a_{ij})_{N\times N}\). So, \(P_{a}=(\beta ^{a}_{ij})_{N\times N}\), \(P_{b}=(\beta ^{b}_{ij})_{N\times N}\), \(Q_{a}=(\delta ^{a}_{ij})_{N\times N}\), \(Q_{b}=(\delta ^{b}_{ij})_{N\times N}\) are four weighted matrixes. Obviously, we can stipulate \(\beta ^{a}_{ij}=\beta ^{a}_{ij}a_{ij}, \beta ^{b}_{ij}=\beta ^{b}_{ij}a_{ij}, \delta ^{a}_{ij}=\delta ^{a}_{ij}a_{ij}, \delta ^{b}_{ij}=\delta ^{b}_{ij}a_{ij}.\) By a mean field approximation, we give a stochastic SAIS system as follows

for \(i=1,\ldots ,N\).

For convenience, denote \(\Gamma =(0,1)^{N}\), \(\mathbb {R_{+}}=(0,\infty )\), \(\check{\varphi }=\max _{i,j\in \{1,\ldots ,N\}} \varphi _{ij}\), \(\hat{\varphi }=\min _{i,j\in \{1,\ldots ,N\}}\varphi _{ij}\), \(\check{\phi }=\max _{i\in \{1,\ldots ,N\}} \phi _{i}\), \(\hat{\phi }=\min _{i\in \{1,\ldots ,N\}}\phi _{i}\). \(X(t)=(x_{1}(t),\ldots ,x_{N}(t))^{T}\), \(Y(t)=(y_{1}(t),\ldots ,y_{N}(t))^{T}\), \(W(t)=(X(t)^{T},Y(t)^{T})^{T}\), abbreviated as \(X=X(t)\), \(Y=Y(t)\), \(W=W(t)\), unless otherwise specified. In addition, denote \(\lambda (A)\) as the maximum eigenvalue of symmetric matrix A. Let \(\Delta =\left\{ (i,j)|a_{ij}\ne 0,i,j=1,\ldots ,N\right\}\). Define the following vector notation

where \(i=1,\ldots ,N,\ \nu =a,b,c,d\).

We first rewrite (2) in terms of matrix notation as follows

Here, the operator \(F:\Gamma ^{2}\rightarrow \mathbb {R}^{2N}\) is defined as

the operator \(G:\Gamma ^{2}\rightarrow \mathbb {R}^{(2N)\times (4N^{2})}\) is defined as

where \(T_{i}(W)=\left( T_{i}^{a}(W), T_{i}^{b}(W), \textbf{0}, \textbf{0} \right) ,\) and \(D_{i}(W)=\left( \textbf{0}, D_{i}^{b}(W), D_{i}^{c}(W), D_{i}^{d}(W) \right) ,\ i=1,\ldots ,N.\) Furthermore,

where \(\textrm{d}B_{i}(t)=\left( (\textrm{d}B_{i}^{a}(t))^{T},(\textrm{d}B_{i}^{b}(t))^{T}, (\textrm{d}B_{i}^{c}(t))^{T},(\textrm{d}B_{i}^{d}(t))^{T}\right) ^{T}, i=1,\ldots ,N.\)

Given any initial values \(W(0)\in \Gamma ^{2}\), system (2) has a unique global solution W and for any \(t\in [0,\infty )\), \(W(t)\in \Gamma ^{2}\) a.s.

We omit the proof since it is analogous to that of Theorem 2.1 in31 by making use of the It\(\hat{o}\) formula in25 to \(\overline{V}(X,Y)=\sum _{i=1}^{N}\left( \frac{1}{x_{i}}+\frac{1}{1-x_{i}}\right) +\sum _{i=1}^{N}\left( \frac{1}{y_{i}}+\frac{1}{1-y_{i}}\right) .\)

In this section, using the stochastic stability theory, we discuss extinction of system (2).

Note \(F(\textbf{0})\equiv \textbf{0},\ G(\textbf{0})\equiv \textbf{0}\). So, (2) admits a trivial solution. Denote \(M_{a}=((m^{a}_{ij}a_{ij})^{2})_{N\times N}\), \(M_{b}=((m^{b}_{ij}a_{ij})^{2})_{N\times N}\), \(M_{c}=((m^{c}_{ij}a_{ij})^{2})_{N\times N}\), \(M_{d}=((m^{d}_{ij}a_{ij})^{2})_{N\times N}\) and

then the trivial solution of system (2) is stochastically asymptotically stable in the large.

Define \(V(W)=|X|^{2}+|Y|^{2}.\) Clearly, V is positive-definite, decrescent and radially unbounded.

By the It\(\hat{o}\) formula, we have

The Courant-Fischer theorem in32, together with the elementary inequality \(uv\le \frac{1}{2}(u^{2}+v^{2}),\ u,v\in \mathbb {R}_{+}\), and the fact \(z(1-z)\le \frac{1}{4},\ z\in \mathbb {R}\), yields,

Substituting the above inequalities into (5), we have

By the condition (4), LV is negative-definite. By Theorem 2.4 in25, p. 114 , Theorem 2 holds. \(\square\)

In Theorem 2, all the solutions of (2) will tend to zero almost surely. Nonetheless, It is not yet known how fast infectious disease become extinct. To improve this situation, we study exponentially extinct.

Given any initial values \(W(0)\in \Gamma ^{2}\), if (4) holds, then \(\limsup _{t\rightarrow \infty }\frac{1}{t}\ln |W(t)|\le \frac{C}{2},\) where \(C=\max \{C_{1},C_{2}\}\).

One can apply It\(\hat{o}\)’s formula and (6) to show that,

In addition, \(\textrm{trace} \left( G^{T}(W)V_{W}^{T}(W)V_{W}(W)G(W) \right) = V_{W}(W)G(W)G^{T}(W)V_{W}^{T}(W).\) So,

where \(H(t)=\int _{0}^{t}\frac{1}{V(W(s))}V_{W}(W(s))G(W(s))\textrm{d}B(s)\) is local martingale with \(H(0)=0\).

Let \(n=1,2,\ldots\). According to the exponential martingale inequality in25, for arbitrary \(\varepsilon >0\), we have

Since \(\sum _{n=1}^{\infty }\frac{1}{n^{2}}<\infty\), and Borel-Cantelli’s lemma in25, we see that there exist \(\Omega _{0}\in \mathcal {F}\) with \(\mathbb {P}(\Omega _{0})=1\) and an integer \(n_{0}=n_{0}(\omega )\), such that if \(n\ge n_{0}\),

holds for all \(0\le t\le n\) and almost all \(\omega \in \Omega _{0}\).

Substituting the above inequality into (7) and letting \(\varepsilon =1\), for any \(\omega \in \Omega _{0},\ n\ge n_{0},\ 0\le t\le n\), we have

which yields

for all \(\omega \in \Omega _{0},\ 0\le n-1\le t\le n,\ n\ge n_{0}\). Letting \(n\rightarrow \infty\), we obtain

This completes the proof. \(\square\)

Even though both theorems have the same conditions, Theorem 3 states the solutions of (2) will tend to zero almost surely exponentially fast, which can’t be known from Theorem 2.

By Perron-Frobenius theorem in32, there is a unique real vector \(u=(u_{1},u_{2},\ldots ,u_{N})^{T}>0\) such that \(P_{a}u=\rho (P_{a})u\) and \(\sum _{i=1}^{N}u_{i}=1\), where \(\rho (P_{a})\) is the spectral radius of \(P_{a}\).

Given any initial values \(W(0)\in \Gamma ^{2}\), if \(\eta >0\) and

then there exists \(H>0\), such that solution of system (2) satisfies

Define \(R(X)=\frac{1}{\sum _{i=1}^{N}u_{i}x_{i}}\) and \(V_{4}(X,t)=e^{k t}\big (1+R(X)\big )^{\eta }\), where k is positive constant and satisfies

Thanks to the It\(\hat{o}\) formula, we derive

Furthermore,

Hence,

where

Then,

Notice that \(R(X(s))\ge \frac{1}{N}\), for \(0\le s\le t\). And then, there exists a constant \(\bar{K}>0\), such that \((1+R(X(s)))^{\eta -2}[C_{3} + C_{4}R(X(s)) +C_{5}R^{2}(X(s))]\le \bar{K}\), \(0\le s\le t\). Obviously,

By the fact that \(R^{-1}(X(t))\le |u||X(t)|\), we know \(|X(t)|^{-\eta }\le R^{\eta }(X(t))\). Hence, (12) yields

This proof is completed. \(\square\)

then system (2) is stochastically permanent.

we can choose a constant \(\eta >0\) to make the condition of Theorem 5 hold. By Chebyshev’s inequality and Theorem 5, for any \(\varepsilon >0\), there is \(\xi =(\frac{\varepsilon }{H})^{\frac{1}{\eta }}\), such that

This proof is completed. \(\square\)

Based on33, we show numerical simulations by Milstein method. Here, we consider an Erdös-Réyni random graph with the number of nodes \(N=30\), introducing a edge between each pair of nodes with probability \(p=0.35\). Take \(\sigma _{ij}^{a}(x_{i},y_{i},x_{j},y_{j})= m_{ij}^{a}x_{i}(x_{j}+y_{j}), \sigma _{ij}^{b}(x_{i},y_{i},x_{j},y_{j})= m_{ij}^{b}x_{i}y_{j}, \sigma _{ij}^{c}(x_{i},y_{i},x_{j},y_{j})= m_{ij}^{c}y_{i}(x_{j}+y_{j}), \sigma _{ij}^{d}(x_{i},y_{i},x_{j},y_{j})= m_{ij}^{d}y_{i}(x_{j}+y_{j}).\) where \(m_{ij}^{\nu }\) is positive constant in some interval, \(\nu =a,b,c,d.\) Furthermore, we discuss the model (2) with the same initial data \(x_{i}(0)=0.5,\ y_{i}(0)=0.25,\ i=1,2,\cdots ,30.\)

In Fig. 2, the parameters of system (2) take random numbers on Data class 1 of Table 1. By simulation, we get the corresponding adjacency matrix \(A=(a_{ij})_{30\times 30}\) and \(\lambda (A)=\rho (A)=11.5097\), \(\lambda \left( \frac{1}{2}(P_{a}+P_{a}^{T})\right) =11.9866\), \(\lambda \left( \frac{1}{2}(Q_{b}+Q_{b}^{T})\right) =0.6346\), \(\lambda \left( \frac{1}{2}(M_{a}+M_{a}^{T})\right) =0.0349\), \(\lambda \left( \frac{1}{2}(M_{b}+M_{b}^{T})\right) =0.3132\), \(\lambda \left( \frac{1}{2}(M_{d}+M_{d}^{T})\right) =0.4406\), \(C_{1}=-1.7197<0\), \(C_{2}=-0.2097<0\). That is, conditions of Theorems 2 and 3 have been checked. Thus, from Theorem 3, epidemic become extinct. Fig. 2 clearly supports Theorems 2 and 3.

Dynamics of infection probability and awareness probability for nodes 2, 12, 22 and 30, with parameters taking random numbers on Date Class 1 of Table 1.

In Fig. 3, the parameters of system (2) take random numbers on Data class 2 of Table 1. Hence, choose \(\eta =1\). By simulation, we get \(\hat{u}=0.0149\), \(\check{u}=0.0435\), \(\lambda (A)=11.5097\), \(\rho (P_{a})=23.0823\), \(\lambda \left( \frac{1}{2}(P_{a}+P_{a}^{T})\right) =23.0823\), \(\lambda \left( \frac{1}{2}(M_{a}+M_{a}^{T})\right) =0.0059\), \(\lambda \left( \frac{1}{2}(M_{b}+M_{b}^{T})\right) =0.006\), \(\min _{(i,j)\in \Delta }\left\{ \beta _{ij}^{b}/\beta _{ij}^{a}\right\} =2.2016\), and \(-\min _{(i,j)\in \Delta }\left\{ \beta _{ij}^{b}/\beta _{ij}^{a}\right\} \rho (P_{a}) +\check{\gamma } +\frac{\eta +1}{8}\frac{\check{u}^{2}}{\hat{u}^{2}} \bigg ( \lambda \big (\frac{1}{2}(M_{a}+M_{a}^{T})\big ) + \frac{1}{4} \lambda \big (\frac{1}{2}(M_{b}+M_{b}^{T})\big ) \bigg ) =-43.0354<0.\) Therefore conditions of Theorems 5 and 6 have been checked, respectively. There is a good agreement between Theorem 6 and Fig. 3.

Dynamics of infection probability for nodes 2, 12, 22 and 30, with parameters taking random numbers on Data Class 2 of Table 1.

Average number of individuals with the parameters taking random numbers on Data Class 3 of Table 1.

In Fig. 4, except parameters \(\beta _{ij}^{a}\) and \(\beta _{ij}^{b}\), everything else is the same taking on Data Class 3 of Table 1. The solid lines represent the situation without considering risk awareness, while the dashed lines represent the situation with risk awareness. In case (a), they take random numbers from value range (7.5,7.55) when \(\beta _{ij}^{a}=\beta _{ij}^{b}\). \(\beta _{ij}^{a}\) takes from (7.5,7.55), \(\beta _{ij}^{b}\) takes from (0.13,0.18) when \(\beta _{ij}^{a}>\beta _{ij}^{b}\). In case (b), they take random numbers from (0.65,0.7) when \(\beta _{ij}^{a}=\beta _{ij}^{b}\). \(\beta _{ij}^{a}\) takes from (0.65,0.7), \(\beta _{ij}^{b}\) takes from (0.13,0.18) when \(\beta _{ij}^{a}>\beta _{ij}^{b}\). From case (a), although the system (2) is stochastically permanent, the average number of infectious individuals taking awareness into consideration, is far lower than that without regard to influence factor of awareness. From case (b), the risk awareness of susceptible people can lead to faster disease extinction. These also reflect from the side that risk awareness of disease is highly effective in preventing disease spread corresponding to dashed lines.

Average number of individuals with parameters take random numbers on Data Class 4 of Table 1 in case (a) and parameters take random numbers on Data Class 5 of Table 1 in case (b).

In Fig. 5, except parameters \(q_{i}\), the parameters of system (2) take random numbers on Data classes 4 and 5 of Table 1,respectively. In case (a), the solid line is the average number corresponding to \(q_{i}\in (0.93,0.98)\); the deshed line is the average number corresponding to \(q_{i}\in (0.02,0.03)\). In case (b), the solid line is the average number corresponding to \(q_{i}\in (0.9,0.95)\); the dashed line is the average number corresponding to \(q_{i}\in (0.05,0.06)\), for any \(i=1,\ldots ,N\). The red curve represents the average number of infected individuals; the red solid lines are above the red dashed lines. This means that the higher the proportion of infected individuals recovering and becoming aware susceptible individuals, the fewer the average number of infected individuals. This also reflects from the side that the greater the proportion \((1-q_i)\), which infected individual become aware susceptible, the better the prevention of infectious diseases.

Average number of individuals with parameters taking random numbers on Data Class 6 of Table 1.

In Fig. 6, except parameters \(\kappa _{i}\), the parameters of system (2) take random numbers on Data class 6 of Table 1. In numerical simulation, the solid linea are the average numbers corresponding to \(\kappa _{i}\in (160,161)\), and the dashed lines are the average numbers corresponding to \(\kappa _{i}\in (0.3,0.41)\), for any \(i=1,\ldots ,N\). Through observation, infectious disease becomes extinct when decay rate \(\kappa _{i}\in (0.3,0.41)\), and it becomes persistent when decay rate \(\kappa _{i}\in (160,161)\). That is, A lower rate of awareness decay corresponds to a reduced likelihood of disease transmission. We are not difficult to notice that the increase of decay rate leads to qualitative changes in the long-term asymptotic behavior of the disease.

Numerical simulations with different noise intensities.

In cases (a)–(c) of Fig. 7, \(\ m_{ij}^{b},\ m_{ij}^{c},\ m_{ij}^{d}\) take random value on (0, 0.1) and the other take from Date class 1 of Table 2; In cases (e)–(g) of Fig. 7, \(\ m_{ij}^{a},\ m_{ij}^{c},\ m_{ij}^{d}\) take random value on (0, 0.1) and the other take from Date class 1 of Table 2; In cases (i)–(k) of Fig. 7, \(\ m_{ij}^{a},\ m_{ij}^{b},\ m_{ij}^{d}\) take random value on (0, 0.1) and the other take from Date class 2 of Table 2; In cases (m)–(o) of Fig. 7, \(\ m_{ij}^{a},\ m_{ij}^{b},\ m_{ij}^{c}\) take random value on (0, 0.1) and the other take from Date class 2 of Table 2; Obviously, solutions of stochastic system fluctuate around positive equilibrium of the corresponding deterministic system. Hence, no matter what kind of noise intensities increase, the fluctuations become stronger.

In practical terms, stochastic disturbance and heterogeneity are essentially inevitable in the spread of infectious diseases. This paper focuses on SAIS system on networks with heterogeneous individuals and awareness decay, considering the fluctuation in time both of infectious disease transmission and information spreading. By using stochastic stability theory, we discuss the stochastic asymptotic stability in the large and almost surely exponential stability of the trivial solution. That is to say, the sufficient conditions for disease extinction are given. And then, according to the Perron-Frobenius theorem, we get some sufficient conditions, which guarantee the stochastic persistence of infectious disease. Based on the Erd\(\mathrm {\ddot{o}}\)s-R\(\mathrm {\acute{e}}\)yni random graph, the numerical simulations validate our conclusions.

Model (2) is more realistic than (1) on a technical level.Hence, the Fig. 7 shows that no matter what kind of noise intensities increase, the fluctuations will become stronger. It is worthy to point out that the various parameters and intensities of Brownian noise will not affect that the solution of (2) may not explode from Lemma 1. Then, the long time behavior of the epidemic is comprehensively analysed. We obtain the extinction of epidemic by proving stochastically asymptotically stable in the large of the zero solution for SAIS model. Although the epidemic eventually ends, we place more emphasis on the rate of extinction. When \(2\lambda \left( \frac{1}{2}(P_{a}+P_{a}^{T})\right) -2\hat{\gamma } +N\check{\delta ^{a}}+(1-\hat{q})\check{\gamma } +\frac{1}{4}\lambda \left( \frac{1}{2}(M_{a}+M_{a}^{T})\right) +\frac{N}{16}(\check{m^{c}})^{2}<0,\) and \(N\check{\delta ^{a}}+2\lambda \left( \frac{1}{2}(Q_{b}+Q_{b}^{T})\right) -2\hat{\kappa }+(1-\hat{q})\check{\gamma } +\frac{1}{8}\lambda \left( \frac{1}{2}(M_{b}+M_{b}^{T})\right) +\frac{1}{4}\lambda \left( \frac{1}{2}(M_{d}+M_{d}^{T})\right) <0,\) the trivial solution of (2) is not only stochastically asymptotically stable in the large, but also almost surely exponentially stable. Besides, we discuss the sufficient condition for stochastic permanence, which is \(-\min _{(i,j)\in \Delta }\left\{ \beta _{ij}^{b}/\beta _{ij}^{a}\right\} \rho (P_{a}) +\check{\gamma } +\frac{1}{8}\frac{\check{u}^{2}}{\hat{u}^{2}} \left( \lambda \left( \frac{1}{2}(M_{a}+M_{a}^{T})\right) + \frac{1}{4} \lambda \left( \frac{1}{2}(M_{b}+M_{b}^{T})\right) \right) < 0\). It needs to be emphasized that the smaller the parameter \(\beta _{ij}^{b}\) is, the stronger the awareness is, the harder the condition is to satisfy, and the more difficult the disease is to be persistent.

Hence, we discover that different values of the infected rates \(\beta _{ij}^{a}, \beta _{ij}^{b}\), decay rates \(\kappa _{i}\), recovery rates \((1-q_{i})\gamma _{i}\) from infected individuals to aware susceptible ones, could all lead to completely different trends in infectious diseases through numerical simulations. These further reflect that risk awareness of disease is highly effective in preventing disease spread.

Although degree is the characteristic of network vertices, we do not make assumptions about degree. So, the networks we are studying can be scale-free networks, can be Poisson networks, and so on. Our conclusions hold for all of the above networks.

Under the conditions of extinction and persistence, \(\hat{\gamma }>\lambda \left( \frac{1}{2}(P_{a}+P_{a}^{T})\right)\) and \(\check{\gamma }<\rho (P_{a})\) hold, respectively, given in terms of the parameters of the model. Unfortunately, the gap which is due to \(\rho (P_{a})\le \lambda \left( \frac{1}{2}(P_{a}+P_{a}^{T})\right)\) (8.3.P10 in32) is not included in current results.

All data generated or analysed during this study are included in this paper.

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This work was supported by National Natural Science Foundation of China (Nos. 12371494, 12231012, 11971279), and Shanxi Provincial Key Research and Development Project (No. 202202020101010).

School of Mathematics and Statistics, Taiyuan Normal University, Jinzhong, 030619, People’s Republic of China

Xin Yi

School of Mathematical Sciences, Shanxi University, Taiyuan, 030006, People’s Republic of China

Guirong Liu

Key Laboratory of Complex Systems and Data Science of Ministry of Education, Shanxi University, Taiyuan, 030006, People’s Republic of China

Guirong Liu

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X.Y.: Writing - original draft. G.L.: Writing - review and editing.

Correspondence to Guirong Liu.

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Yi, X., Liu, G. Analysis of stochastic epidemic model with awareness decay and heterogeneous individuals on multi-weighted networks. Sci Rep 14, 26765 (2024). https://doi.org/10.1038/s41598-024-78218-4

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Received: 26 July 2024

Accepted: 29 October 2024

Published: 05 November 2024

DOI: https://doi.org/10.1038/s41598-024-78218-4

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