Abstract
We focus on the node-based epidemic modeling for networks, introduce the propagation medium, and propose a node-based Susceptible-Infected-Recovered-Susceptible (SIRS) epidemic model with infective media. Theoretical investigations show that the endemic equilibrium is globally asymptotically stable. Numerical examples of three typical network structures also verify the theoretical results. Furthermore, comparison between network node degree and its infected percents implies that there is a strong positive correlation between both; namely, the node with bigger degree is infected with more percents. Finally, we discuss the impact of the epidemic spreading rate of media as well as the effective recovered rate on the network average infected state. Theoretical and numerical results show that (1) network average infected percents go up (down) with the increase of the infected rate of media (the effective recovered rate); (2) the infected rate of media has almost no influence on network average infected percents for the fully connected network and NW small-world network; (3) network average infected percents decrease exponentially with the increase of the effective recovered rate, implying that the percents can be controlled at low level by an appropriate large effective recovered rate.
1. Introduction
With the development of network science, the mathematical modeling of epidemic spreading has involved in a research area across many disciplines including mathematical biology, physics, social science, and computer and information science. On the basis of classical epidemic spreading models, such as Susceptible-Infected-Susceptible (SIS) model, Susceptible-Infected-Recovered (SIR) model, and Susceptible-Infected-Recovered-Susceptible (SIRS) model, a variety of epidemic spreading models [1–13] in networks were developed. Investigations on these models have important significance in public-health domain, especially in infectious disease epidemiology, by providing a number of interesting and unexpected behaviors.
The theoretical studies of epidemic spreading models in complex networks rely mostly on the mean-field theory approaches, especially on degree-based mean-field (DBMF) theory which was the first theoretical approach presented for the analysis of general dynamical processes on complex networks [9]. This approach assumes that all nodes of degree are statistically equivalent, and any given vertex of degree is connected with the same probability to any node of degree . Therefore, the epidemic spreading model based on DBMF theory depends in general on the statistical topological properties of the underlying networks instead of the whole network structure, resulting into the loss of detailed features of network topologies such that it is difficult to deeply understand the effect of network structures on the disease (or information) propagation. To the best of our knowledge, in 2009, Mieghem et al. [10] firstly proposed the continuous-time node-based SIS epidemic spreading model for understanding the influence of network characteristics on epidemic spreading. Youssef and Scoglio [11] established a new individual-based SIR model with the whole description of network structures. Very recently, Yang et al. [12] suggested a node-based Susceptible-Latent-Exploding-Susceptible (SLBS) model, and in the same year they presented a heterogeneous node-based SIRS model where each node has the different infected and recovered rates [13]. The above models assume that disease transmission takes place between individuals in networks.
However, diseases are propagated not only by the contact between individuals in the same population, but also by the contact between individuals and infective media. For instance, many human diseases, such as dengue fever, malaria, and Chagas disease, can be transmitted by the infective mosquito. For this case, Shi et al. [14] established a new SIS epidemic model with an infective medium, which describes epidemics transmitted by infective media on various complex networks. By differentiating the infective medium from individuals, Yang et al. [15] proposed a modified SIS model. Wang et al. [16] presented a modified SIS with an infective vector by incorporating some infectious diseases. It is noteworthy that these existing models with infective media are degree-based instead of node-based.
The motivation of this paper is to build a node-based SIRS epidemic model with infective media on various complex networks by integrating the node-based approach and the infective medium and investigate the stability of the equilibrium as well as the influence of network structures, the infective medium, and the effective recovered rate on the network infected steady state.
The rest of this paper is organized as follows. Some definitions and Lemmas are introduced in Section 2. In Section 3, a node-based SIRS epidemic network model with infective media is built and then its equilibrium is given. The global asymptotical stability analysis with respect to the equilibrium is performed in Section 4. In Section 5, numerical simulations of three typical network topologies are provided for further verifying the theoretical results. The correlation between the infected percents of nodes and its degree, as well as the impact of some critical parameters on network average infected percents, is studied theoretically and numerically. Finally, some conclusions and discussions are given in Section 6.
2. Preliminaries
First, some requisite definitions and lemmas are given as follows.
Definition 1 (see [17]). A Matrix is Metzler if its all off-diagonal entries are nonnegative.
Definition 2 (see [17]). A Matrix is Hurwitz stable if there exists a positive matrix such that is negative definite.
Definition 3 (see [17]). A Matrix is diagonally stable if there exists a positive definite diagonal matrix such that is negative definite.
Obviously, the diagonally stable matrix is Hurwitz stable, and the opposite is also true for Metzler matrices.
Lemma 4 (see [18]). A Hurwitz and Metzler matrix is diagonally stable.
Lemma 5 (see [13]). Let be a Hurwitz and Metzler matrix, be a positive definite diagonal matrix, and and be negative definite diagonal matrices. Then, is diagonally stable.
Lemma 6 (see [19]). Consider a smooth dynamical system defined at least in a compact set . Then, is positively invariant if is pointing into for any smooth point on the boundary of .
3. Model Formulation
To begin with, we consider an underlying network (or a simple graph) denote as d , where is the set of nodes and is the set of edges. The nodes labeled from number to number represent the individuals in propagation networks, and the edges stand for network links through which disease can propagate. In a simpler way, we denote the adjacent matrix of graph describing network topological structures, where if there is an edge between node and node , otherwise .
Assume that (H1) each node in the network has three possible states: susceptible(), infected (), and recovered (), whereas the media has two possible states: susceptible () and infected (); (H2) both states and convert each other with certain probability, and the state is infected with the probability of into the state , but not vice versa; (H3) the state is recovered with the probability of into the state , and the state is converted with the probability of into the state after the immunity is out of work; (H4) the state in the underlying network is infected with the probability of by the infective media, and the media is with the birth (death) rate of .
For simplicity, the variables and parameters in this node-based SIRS model with infective media are summarized in Table 1, and the schematic diagram of the model is shown in Figure 1.
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Let , , and represent three states of node at time : the susceptible (), the infected () and the recovered (), respectively. , , and represent three states of media at time : the susceptible (), the infected (), and the dead, respectively. The state of individuals at time t can be expressed as by the vector Then
According to the assumptions, it implies the following probability of state transition:
By using the total probability law, one can obtain
Let , we get Similarly, it is easy to get the equations dominating , , and . Collecting them together, we have the following dimensional dynamical system:with initial condition , , , where .
Remark 7. From the view point of continuous-time Markov chain [20], model (7) is an approximation one on account of the linear transition rate instead of exact one from state to state . The performance examined in Appendix C shows that model (7) is able to well forecast the epidemic dynamics of model (C.1) built by means of Markov chain technique. Furthermore, the dynamical behaviors of approximation models is more easily studied by applying the stability theory and method, and thus the similar approximation model is directly built and studied in a large number of related literatures.
Since , , , system (7) can be reduced into the following system:with initial condition , , , where .
Let the right-hand terms in (8) equal to zero, one gets an equilibrium which is only one proven in Appendix A, here From the above equality, it is easy to get that , implying that the percent with the infected state for any node is less than .
Remark 8. Obviously, the equilibrium is not virus-free, implying that the virus exists persistently in each individual. The phenomenon can be understood by the fact that the endemic disease remains safely under cover in each individual in some local areas. Although the equilibrium is given in the implicit form, it can be calculated out by the numerical iterative method.
Remark 9. When the infected rate of media , the model is reduced to the node-based SIRS epidemic model with a virus-free equilibrium, to some extent implying that our extended model is rational and practical. The infected medium terms not only increase the dimension of SIRS models but more importantly make stability analysis more complicated, especially in the part of global attractivity
4. Stability Analysis with respect to Equilibria
4.1. Local Stability
To analyze the local stability of system (8) at the equilibrium , we start with its Jacobian where , , , is the identity matrix of order , represents the diagonal matrix, and
Theorem 10. System (8) is asymptotically stable at the equilibrium
Proof. Obviously, is a negative eigenvalue of , and other eigenvalues are determined by matrix . Next, we show that all the eigenvalues of have negative real part.
For convenience, define three matrices as follows:Here we consider an undirected and connected graph, so the adjacent matrix is an irreducible one, indicating that is also an irreducible matrix. According to Perron-Frobenius theorem [21], has a positive eigenvector corresponding to the largest eigenvalue , i.e., . Then andOn the other hand, it is easy from the second equality of system (7) to get that where . Thus, implies when .
Since and are positive vectors, it holds that and From (14) and (16), it follows that Thus, is a negative definite matrix, and then is also a negative definite one. That is to say, is a Hurwitz and Metzler matrix. According to Lemma 5, matrix is diagonally stable. Therefore, the equilibrium of system (8) is asymptotically stable.
4.2. Global Attractivity
To proof the global attractivity, it needs to determine the positively invariant set (In brief, once a trajectory of the system enters the set, it will never leave it again). Next, it is not difficult to proof that is an invariant set.
Theorem 11. is a positively invariant set for system (8).
Proof. Denote the boundary of , and then it consists of the following hyperplanes: For simplicity and convenience, system (8) is rewritten aswith initial condition .
Take the outer normal vectors corresponding to hyperplanes as follows: and let be a smooth point of . On the basis of these hyperplanes, five different cases of are discussed, respectively. (i)Case 1: For , (ii)Case 2: For , (iii)Case 3: For , (iv)Case 4: For , (v)Case 5: For , Therefore, is pointing to , and is positively invariant according to Lemma 6.
Theorem 12. The equilibrium of system (8) is globally attractive on .
Proof. Denote where and then the virus equilibrium ; here .
To explore the asymptotic behavior of solutions of (8), we define two functions as follows: Both functions are continuous and exist right-hand derivatives along solutions of (8). Let is the solution of (8), and suppose that , for some and sufficiently small . Then we have where .
Next, we proof that the derivative of at is nonnegative. According to the definition of , it follows that For , three cases as below are discussed (here is ignored for conciseness). (i)Case 1: (ii)Case 2: . .(iii)Case 3: . Since and , one gets , implying . Similarly, if , if , and if .
Denote Obviously, and are nonnegative and continuous in , and and . Letting and , then we have and .
It follows from the LaSalle Invariance Principle that any solution of system (8) starting in approaches . Therefore, any solution with initial value satisfies ; i.e., is globally attractive in .
Remark 13. Together with local asymptotical stability, it is easily obtained that the equilibrium in the SIRS model with media is globally asymptotically stable. By the way, without the medium propagation, i.e., , system (8) has a virus-free equilibrium which is globally asymptotically stable if , otherwise unstable. Here is the largest eigenvalue of the topological matrix , and represents the actual effective recovered rate.
5. Numerical Simulations
5.1. Three Typical Network Models
In order to verify the above theoretical results, we choose three typical network structures (fully connected network, small-world network and scale-free network), and solve numerically system (8) with , , , , , , and . Here, the fully connected network means that all nodes are connected with each other, the small-world network is generated from nearest neighbor network with the probability of random adding edges (called NW small-world network), and the scale-free network is a BA scale-free one with and . Each type of network is with 100 nodes, respectively.
Figures 2–4 show the evolution of , , and over time for the fully connected network, NW small-world network, and BA scale-free network, respectively. Obviously, the infected () and susceptible () states of each node in each network tend to their equilibrium states over time, namely, and (), implying that it further verifies our theoretical results, and the infected percents of node is very low for sparsely connected networks, such as NW and BA networks.
Interestingly, for the fully connected network, all nodes tend to the same equilibrium state as time goes, namely, and (). But for the small-world network and scale-free network, all nodes approach their equilibrium but nonidentical states as time goes, namely, and (). In fact, the steady state of each node is closely related to its degree according to the formula of equilibria. For the fully connected network, each node has the same degree, resulting into that each node tends to the same equilibrium state, and the equilibrium states satisfies and thus for enough large size networks. Please refer to Appendix B for the detailed derivation.
For the NW small-world network, the node with large (small) degree has large (small) infected equilibrium state, as shown in Figure 5(a). However, the correlation is a little weaker for the BA scale-free network which is a heterogeneous one; please see Figure 5(b). In a word, the infected (susceptible) equilibrium state () is positively (negatively) correlated with the degree of nodes, and the node with higher degree is easier to be infected, further verifying the result obtained by the article [22].
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5.2. Impact of System Parameters
To learn more about this proposed model, we now analyze the influence of several important parameters, namely, , , , and , on network average infected state.
Let us look back at these parameters: means the infected rate resulted from the medium, represents the infected rate from the node’s neighbour, and can be considered as the potential infected rate due to the fact that the state of nodes never transforms back once it changes to the susceptible state from the recovered state, and the susceptible state is the potential part transferred into the infected state. represents the recovered rate from the infected state to the recovered state. Here we call and the actual and potential effective recovered rate, respectively.
We firstly analyze the influence of , , , and on network average infected state for the general topologies. According to the implicit differentiation theorem, it is easy to obtain the following theoretical results through the formula of equilibria.
Theorem 14. Under the assumptions (H1)-(H4), it follows that (a) , (b) , (c) , and (d) , where called network average infected state (please see Appendix D for the proof).
The above theorem implies that the network average infected state rises with the increase of and decreases with the increase of or .
Furthermore, we numerically verify the theoretical implications for three typical network topologies, respectively. Figure 6(a) shows that, with fixed , as increases, the network average infected state rises gradually for BA scale-free network but almost is unchanged for the fully connected network and NW small-world network, implying that the heterogenous network is sensitive to the infected rate of the medium, but the homogeneous network is the opposite.
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It is shown from Figures 6(c), 6(b), and 6(d) that the average infected state of three typical networks decreases exponentially as the potential recovered rate increases with fixed , and it can be located at low level when is large enough. It implies that the epidemic spreading on networks can be significantly suppressed by the even small increase of the potential effective recovered rate .
Similarly, the average infected state goes down with the increase of , and the virus decreases exponentially for NW small-world network, faster than those for the fully connected network and BA scale-free network, as shown in Figure 7. A possible reason is that the small-world network has the properties of the short average path length and small average degree.
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6. Conclusions and Discussions
In summary, this paper has presented a node-based SIRS epidemic model with media for understanding the disease spreading of networks with media propagation, where there is only an equilibrium yet not virus-free one that is always globally asymptotically stable through the stability analysis. Without the medium propagation, the model has a virus-free equilibrium which is globally asymptotically stable when the maximum eigenvalue of topological matrices is less than the effective recovered rate .
Three typical networks, i.e., the fully connected, small-world, and scale-free networks, are applied to numerical investigations for further verifying the theoretical results. Numerical simulations also show that the sparse network has less infected percents. In addition, it shows that the infected percents of network nodes have the positive correlation with the degree of the node, in particular for the homogenous network, such as the fully connected network and small-world network.
Finally, theoretical and numerical studies on the influence of the effective recovered rate and medium propagation rate on network average infected percents imply that network average infected percents go up (down) with the increase of the medium propagation rate (the effective recovered rate). Numerical investigations further show that the medium propagation rate does nothing with network average infected percents for homogenous networks, and the infected percents decease exponentially with the increase of the effective recovered rate. Moreover, the percents can be controlled at low level only if the effective recovered rate is enough large, in other words, only if the effective infected rate is small enough.
Appendix
A. Proof of Uniqueness of Equilibria
Here we now prove the equilibrium is one and only equilibrium fixed point. First of all, define a continuous mapping as below: We can assert that the equilibrium is one and only if is monotonic and exist a unique fixed point.
Claim 1. is monotonic.
Proof. Let , , (). Then, which implies ; the proof of Claim 1 is completed.
Claim 2. admits a unique fixed point in .
Proof. (1) Existence. Since is monotonic and continue for , it follows that . On the other hand, and ,
implying and , such that , where and . We conclude that , so the restriction on the compact convex set maps into . It follows from Brouwer Fixed point Theorem [23] that H exists in a fixed point .
(2) Uniqueness. Suppose exists the other fixed point . Let and , Without loss of generality, we may assume , and it follows that which contradicts the assumption that . Hence, the fixed point is unique. This completes proof.
B. Computation of the Equilibrium
In this Appendix, we give the computation process of the equilibrium for the fully connected network with nodes. Assuming that , then Denote , , , and . Then, the above equality can be rewritten as
As , it follows from the Hurwitz criterion [24] that (B.2) has two opposite sign roots. It is easy to verify that the positive root satisfies , so is the equilibrium due to the uniqueness of solutions. Since , it follows that Therefore, the equilibrium for the large size fully connected network.
C. Comparison with Exact Markov Models
For the purpose of showing the performance of our model, the following exact Markov model is established by means of continuous-time Markov chain technique [20]:where represents the probability of node being state and node being state .
As matter of fact, model (C.1) turns into model (7) when will be replaced with , namely, the transition rate from state to state is linear and equal to , as shown in Figure 3. For the relation between exact Markov model and approximation model please refer to [25].
Next, we select three typical networks with 100 nodes and use the Gillespite algorithm [26] to simulate the solution of the Markov model (C.1) where model parameters and initial conditions are the same as those in Section 5.1. In the experiment, we select randomly initial nodes including susceptible, infected and recovered nodes, and make 200 realizations for fully connected networks and 2000 realizations for NW small-world and BA scale-free networks.
Figure 8 shows the comparison of network average states and between the Markov model and the approximation model. As time goes, both states and of approximation models are able to describe those of Markov models, although there is a little overestimation. Furthermore, the estimation for small-world and scale-free networks is better than that for the fully connected networks. On the whole, the performance of this new model is good for describing the real Markov model.
D. Proof of Theorem 14
Proof of Theorem 14: Denote where . Then, it follows from the formula of equilibria in model (8) that
Taking the partial derivatives of with respect to and , respectively, one gets Here , , , and the same is below.
According to the implicit differentiation formula, it follows that Obviously, , namely, . Next, we prove that is invertible and all the elements of are negative.
It is easy to obtain that due to Denote , and . Obviously, is nonnegative matrix, and it is irreducible due to the fact that is irreducible on account of the connectedness of the graph .
According to the Perron-Frobenius Theorem [21], has a simple positive eigenvalue and a positive eigenvector , such that . So , implying that is also eigenvector of . On the other hand, it follows from (D.2) that , indicating that is a positive eigenvector of belonging to eigenvalue . As , combining the simplicity of resulted from the simplicity of , one gets , indicating that has a zero eigenvalue and the other eigenvalues are negative.
On the other hand, . It follows that all the eigenvalues of matrix are negative and is Metzler and irreducible. According to the result in [27], all the elements of matrix are negative. Thus and consequently .
Similarly, we get where , , and . Therefore, , , and . The proof of Theorem 14 is completed.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest regarding the publication of this paper.
Acknowledgments
This work is supported in part by the National Natural Science Foundation of China (Grants no. 61573004, 11871231, and 11501221), in part by the Promotion Program for Young and Middle-Aged Teacher in Science and Technology Research of Huaqiao University (Grants no. ZQN-YX301 and ZQN-PY401), in part by the Program for New Century Excellent Talents in Fujian Province University in 2016, in part by the Program for Innovative Research Team in Science and Technology in Fujian Province University, in part by Quanzhou High-Level Talents Support Plan under Grant 2017ZT012, and in part by the Project of Education and Scientific Research for Middle and Young Teachers in Fujian Province (Grant no. JAT170027 and JAT170028). The authors also thank the Biomathematics Society of Fujian Province for providing a chance to report the main results of this paper at 2018 Annual Meeting of the Biomathematics Society of Fujian Province.