A node-based SIRS epidemic model with infective media on complex networks

In this paper, 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.


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, computer and information science, and so on. On the basis of classical epidemic spreading models, such as, Susceptible-Infected-Susceptible (SIS) 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 Sec. 2. In Sec. 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 Sec. 4. In Sec. 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, are studied theoretically and numerically. Finally, some conclusions and discussions are given in Sec. 6.

Preliminaries
First, some requisite definitions and lemmas are given as follows.
Definition 1 [17]: A Matrix is Metzler if its all off-diagonal entries are non-negative.
Definition 2 [17]: A Matrix A is Hurwitz stable if there exists a positive matrix D such that A T D + DA is negative definite.
Definition 3 [17]: A Matrix A is diagonally stable if there exists a positive definite diagonal matrix D such that A T D + DA is negative definite.
Obviously, the diagonally stable matrix is Hurwitz stable, and the opposite is also true for Metzler matrices.
Lemma 2 [13]: Let A be a Hurwitz and Metzler matrix, D 1 be a positive definite diagonal matrix, D 2 and D 3 be negative definite diagonal matrices. Then, is diagonally stable.
Lemma 3 [19]: Consider a smooth dynamical systemẋ = g(x) defined at least in a compact set C. Then, C is positively invariant if g(x * ) is pointing into C for any smooth The percents that node i is susceptible at time t.
The percents that node i is infected at time t.
The percents that node i is recovered at time t. S m (t) The percents that media is susceptible at time t. I m (t) The percents that media is susceptible at time t. β m The probability that a susceptible node is infected by an infective media. β The probability that a susceptible node is infected by an infected neighbor. λ 1 The probability of infective node turns into an immunized one. λ 2 The probability of infective node turns into a susceptible one. α The probability that an immunized node loses immunity into a susceptible one. µ The birth (death) rate of the medium. γ m The probability of a susceptible medium transforming into an infected one. point x * on the boundary of C.

Model formulation
To begin with, we consider an underlying network( or a simple graph) denoted G = (V, E) where V is the set of nodes and E is the set of edges. The nodes labeled from number 1 to number N represent the individuals in propagation networks, and the edges stand for network links through which disease can propagate. In a simpler way, we denote A = (a ij ) N ×N the adjacent matrix of graph G describing network topological structures, where a ij = 1 if there is an edge between node i and node j, otherwise a ij = 0.
Assume that (H1) each node in the network has three possible states: susceptible(S), infected (I), and recovered (R), whereas the media has two possible states: susceptible (S m ) and infected (I m ); (H2) both states S and I convert each other with certain probability, and the state S m is infected with the probability of γ m into the state I m , but not vice versa; (H3) the state I is recovered with the probability of λ 1 into the state R , and the state R is converted with the probability of α into the state S after the immunity is out of work; (H4) the state S in the underlying network is infected with the probability of β m by the infective media, and the media is with the birth (death) rate of µ.
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 Fig.1.
Let X i (t) = 0, 1, and 2 represent three states of node i at time t: the susceptible (S i (t)), the infected (I i (t)) and the recovered (R i (t)), respectively. X m (t) = 0, 1, and 2 represent three states of media at time t: the susceptible (S m (t)), the infected (I m (t)) 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 ∆t → 0, we get Similarly, it is easy to get the equations dominating I i (t), R i (t), R m (t) and I m (t). Collecting them together, we have the following 3N + 2 dimensional dynamical system: with initial condition (S 1 (0), · · · , S N (0), I i (0), · · · , I N (0), R i (0), · · · , R N (0), S m (0), I m (0)) T ∈ Ω, where Ω = {(S 1 (t), · · · , S N (t), I 1 (t), · · · , I N (t), R 1 (t), · · · , R N (t), S m , I m ) T ∈ R 3N +2 Since S i (t) + I i (t) + R i (t) ≡ 1, S m + I m ≡ 1, 1 ≤ i ≤ N , system (1) can be reduced into the following system: with initial condition (I 1 (0), · · · , I N (0), R 1 (0), · · · , R N (0), I m (0)) T ∈ Ω, i = 1, · · · , N , Let the right-hand terms in (2) equal to zero, one gets an equilibrium E * = (I * i , R * i , I m * ) which is only one proven in Appendix A, here From the above equality, it is easy to get that I * i < 1 1+λ 1 /α , implying that the percent with the infected state for any node is less than 1 1+λ 1 /α . Remark 2: 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 3: When the infected rate of media β m = 0, 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.

Stability analysis with respect to equilibria 4.1 Local stability
To analyze the local stability of system (2) at the equilibrium E * , we start with its Jacobian represents the diagonal matrix, and For convenience, define three matrices as follows Here we consider an undirected and connected graph, so the adjacent matrix A is an irreducible one, indicating that K 2 is also an irreducible matrix. According to Perron-Frobenius theorem [27], K 2 has a positive eigenvector v corresponding to the largest On the other hand, it is easy from the second equality of system (1) to get that Since v and I * are positive vectors, it holds that From (6) and (7), it follows that Thus, K 3 is a negative definite matrix, and then D = βdiag(S * )K 3 is also a negative definite one. That is to say, D is a Hurwitz and Metzler matrix. According to Lemma 2, matrix B is diagonally stable. Therefore, the equilibrium E * of system (2) is asymptotically stable.

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 Theorem 2: Ω is a positively invariant set for system (2).
Proof: Denote ∂Ω the boundary of Ω, and then it consists of the following 3N + 2 hyperplanes: For simplicity and convenience, system (2) is rewritten as: with initial condition z(0) ∈ ∂Ω.
Take the outer normal vectors corresponding to 3N + 2 hyperplanes as follows: On the basis of these hyperplanes, five different cases of z * are discussed respectively.
Therefore, g(z * ) is pointing to Ω, and Ω is positively invariant according to Lemma 3.
To explore the asymptotic behavior of solutions of Eq. (2), we define two functions as follows: Both functions are continuous and exist right-hand derivatives along solutions of Eq.(2).
Let y(t) is the solution of Eq.(2), and suppose that F (y(t)) = some t 0 and sufficiently small ε > 0. Then we have Next, we proof that the derivative of F (y(t)) at t 0 is non-negative. According to the definition of F (y(t)), it follows that For F (y(t)) > 1, three cases as below are discussed (here t 0 is ignored for conciseness).
Remark 4: Together with local asymptotical stability, it is easily obtained that the equilibrium E * in the SIRS model with media is globally asymptotically stable. By the way, without the medium propagation, i.e., β m = 0, system (2) has a virus-free equilibrium E 0 which is globally asymptotically stable if σ max (A) < (λ 1 + λ 2 )/β, otherwise unstable.
Here σ max (A) is the largest eigenvalue of the topological matrix A, and (λ 1 + λ 2 )/β represents the actual effective recovered rate. Interestingly, for the fully connected network, all nodes tend to the same equilibrium state as time goes, namely, I i (t) → I * and S i (t) → S * (t → ∞). But for the small-world network and scale-free network, all nodes approach their equilibrium but nonidentical states as time goes, namely, I i (t) → I * i and S i (t) → S * i (t → ∞). In fact, the steady state of each node is closely related with 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 I * satisfies , and thus I * ≈ 1 1+λ 1 /α 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 Fig. 5 (a). However, the correlation is a little weaker for the BA scale-free network which is a heterogeneous one, please see Fig. 5 (b).
In a word, the infected (susceptible) equilibrium state I * i (S * i ) 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].

Impact of system parameters
To learn more about this proposed model, we now analyze the influence of several important parameters, namely, β m , β, λ 1 , and α, on network average infected state.
Let us look back these parameters, β m means the infected rate resulted from the medium, β represents the infected rate from the node's neighbour, and α 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. λ 1 represents the recovered rate from the infected state to the recovered state. Here we call λ 1 /β and λ 1 /α the actual and potential effective recovered rate, respectively.
We firstly analyze the influence of β m , α, β and λ 1 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.
The above theorem implies that the network average infected state rises with the increase of β m , and decreases with the increase of λ 1 /α or λ 1 /β. Furthermore, we numerically verify the theoretical implications for three typical network topologies, respectively. Fig. 6(a) shows that with fixed λ 1 = 0.3, λ 2 = 0.3, β = 0.06, α = 0.49, γ m = 0.15, µ = 0.5, as β m increases, the network average infected state rises gradually for BA scale-free network, but almost 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.
It is shown from Fig. 6(c)(b)(d) that the average infected state of three typical networks decreases exponentially as the potential recovered rate λ 1 /α increases with fixed α = 0.3, and it can be located at low level when λ 1 /α is enough large. It implies that the epidemic spreading on networks can be significantly suppressed by the even small increase of the potential effective recovered rate λ 1 /α.
Similarly, the average infected state goes down with the increase of λ 1 /β, and the virus decreases exponentially for NW small-world network, faster than those for the fullyconnected network and BA scale-free network, as shown in Fig. 7. A possible reason is that the small-world network has the properties of the short average path length and small average degree.

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 virusfree equilibrium which is globally asymptotically stable when the maximum eigenvalue of topological matrices is less than the effective recovered rate (λ 1 + λ 2 )/β. 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.

Conflicts of Interest
The authors declare that they have no conflicts of interest regarding the publication of this paper.

Appendix A: Proof of uniqueness of equilibria
Here we now prove the equilibrium E * is one and only equilibrium fixed point. First of all, Define a continuous mapping H = (h 1 , · · · , h N ) : (0, ∞) N → (0, 1) N as below: We can assert that the equilibrium is one and only if H is monotonic and exist a unique fixed point.
which implies H(X) ≤ H(Z), the proof of Claim 1 is completed.
(2) Uniqueness. Suppose H exists the other fixed point , Without loss of generality, we may assume τ > 1, it follows that which contradicts the assumption that U * i 0 = τ V * i 0 . Hence, the fixed point is unique. This completes proof.
where P {X i (t) = 0, X j (t) = 1} represents the probability of node i being state R and node j being state I.
As matter of fact, model (9) turns into model (1) when P {X i (t) = 0, X j (t) = 1} will replaced with I j (t), namely, the transition rate from state R to state I is linear and equals to β m I m (t) + N i=1 a ij I j (t), as shown in Fig.3. The relation between exact Markov model and approximation model please refers to the reference [23].
Next, we select three typical networks with 100 nodes, and use the Gillespite algorithm [24] to simulate the solution of the Markov model (9) where model parameters and initial conditions are the same as those in Sec. 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.
It is easy to obtain that (1 + λ 1 α )I * i − 1 < 0 (i = 1, ..., N ) due to φ i = (λ 1 + λ 2 )I * i + [(1 + Obviously, M is non-negative matrix, and it is irreducible due to the fact that A is irreducible on account of the connectedness of the graph G.