# A Node-Based SIRS Epidemic Model with Infective Media on Complex Networks

Authors: Leyi Zheng; Longkun Tang
Journal: Complexity
 (2019)

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## 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.


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## Body

## 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 k are statistically equivalent, and any given vertex of degree k is connected with the same probability to any node of degree k′. 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 Section2. 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 isMetzler if its all off-diagonal entries are nonnegative.Definition 2 (see [17]).
A MatrixA isHurwitz stable if there exists a positive matrix D such that ATD+DA is negative definite.Definition 3 (see [17]).
A MatrixA is diagonally stable if there exists a positive definite diagonal matrix D such that ATD+DA is negative definite.
Obviously, the diagonally stable matrix isHurwitz stable, and the opposite is also true forMetzler matrices.Lemma 4 (see [18]).
AHurwitz andMetzler matrix is diagonally stable.Lemma 5 (see [13]).
LetA be aHurwitz andMetzler matrix, D1 be a positive definite diagonal matrix, and D2 and D3 be negative definite diagonal matrices. Then, (1)AD2D1D3is diagonally stable.Lemma 6 (see [19]).
Consider a smooth dynamical systemx˙=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 point x∗ on the boundary of C.


## 3. Model Formulation

To begin with, we consider an underlying network (or a simple graph) denote as dG=(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=(aij)N×N the adjacent matrix of graph G describing network topological structures, where aij=1 if there is an edge between node i and node j, otherwise aij=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 (Sm) and infected (Im); (H2) both states S and I convert each other with certain probability, and the state Sm is infected with the probability of γm into the state Im, but not vice versa; (H3) the state I is recovered with the probability of λ1 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 βm 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 Table1, and the schematic diagram of the model is shown in Figure 1.Table 1
Description of parameters.
Parameters Description Si(t) The percents that node i is susceptible at time t. Ii(t) The percents that node i is infected at time t. Ri(t) The percents that node i is recovered at time t. Sm(t) The percents that media is susceptible at time t. Im(t) The percents that media is susceptible at time t. βm 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. λ1 The probability of infective node turns into an immunized one. λ2 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. γm The probability of a susceptible medium transforming into an infected one.Figure 1
The schematic diagram of node-based SIRS network model with media. (a) The SIRS network part and (b) the media part.
(a) (b)LetXi(t)=0, 1, and 2 represent three states of node i at time t: the susceptible (Si(t)), the infected (Ii(t)) and the recovered (Ri(t)), respectively. Xm(t)=0, 1, and 2 represent three states of media at time t: the susceptible (Sm(t)), the infected (Im(t)), and the dead, respectively. The state of individuals at time t can be expressed as by the vector (2)Xt=X1t,X2t,…,XNt.Then (3)Sit=PXit=0,Iit=PXit=1,Rit=PXit=2,Smt=PXmt=0,Imt=PXmt=1According to the assumptions, it implies the following probability of state transition:(4)PXit+Δt=1∣Xit=0=ΔtβmImt+β∑j=1NaijIjt+oΔtPXit+Δt=2∣Xit=1=λ1Δt+oΔtPXit+Δt=0∣Xit=1=λ2Δt+oΔtPXit+Δt=0∣Xit=2=αΔt+oΔtPXmt+Δt=1∣Xmt=0=γmΔt+oΔtPXmt+Δt=2∣Xmt=0=μΔt+oΔtPXmt+Δt=2∣Xmt=1=μΔt+oΔtBy using the total probability law, one can obtain(5)Iit+Δt=PXit+Δt=1=PXit=1PXit+Δt=1∣Xit=0+PXit=1PXit+Δt=1∣Xit=1+PXit=2PXit+Δt=1∣Xit=2=Sit1-ΔtβmImt+β∑j=1NaijIjt+Iitλ2Δt+RitαΔt+oΔt.LetΔt→0, we get (6)dSitdt=-βmSitImt+βSit∑j=1NaijIjt+αRit+λ2Iit.Similarly, it is easy to get the equations dominating Ii(t), Ri(t), Rm(t) and Im(t). Collecting them together, we have the following 3N+2 dimensional dynamical system:(7)dSitdt=-βmSitImt+βSit∑j=1NaijIjt+αRit+λ2Iit,dIitdt=βmSitImt+βSit∑j=1NaijIjt-λ1+λ2Iit,dRitdt=λ1Iit-αRit,dSmtdt=μ-μSmt-γmSmt,dImtdt=γmSmt-μImt,with initial condition (S1(0),…,SN(0),Ii(0),…,IN(0),Ri(0), …,RN(0), Sm(0),Im(0))T∈Ω~, where Ω~=S1t,…,SNt,I1t,…,INt,R1t,…,RNt,Sm,ImT∈R+3N+2∣Sit+Iit+Rit=1,Sm+Im=1,i=1,…,N.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 S to state I. 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.
SinceSi(t)+Ii(t)+Ri(t)≡1, Sm+Im≡1, 1≤i≤N, system (7) can be reduced into the following system:(8)dIitdt=βm1-Iit-RitImt+β1-Iit-Rit∑j=1NaijIjt-λ1+λ2Iit,dRitdt=λ1Iit-αRit,dImtdt=γm1-Imt-μImt.with initial condition (I1(0),…,IN(0),R1(0),…,RN(0), Im(0))T∈Ω, i=1,…,N, where Ω=I1t,…,INt,R1t,…,RNt,ImT∈R+2N+1∣Iit+Rit≤1,Im≤1,i=1,…,N.
Let the right-hand terms in (8) equal to zero, one gets an equilibrium E∗=(Ii∗,Ri∗,Im∗) which is only one proven in Appendix A, here (9)Im∗=γmγm+μ,Ri∗=λ1αIi∗,Ii∗=βmγm/γm+μ+β∑j=1NaijIj∗λ1+λ2+βm1+λ1/αγm/γm+μ+β1+λ1/α∑j=1NaijIj∗.From the above equality, it is easy to get that Ii∗<1/1+λ1/α, implying that the percent with the infected state for any node is less than 1/1+λ1/α.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 mediaβm=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


## 4. Stability Analysis with respect to Equilibria

### 4.1. Local Stability

To analyze the local stability of system (8) at the equilibrium E∗, we start with its Jacobian (10)JE∗=BC0-γm-μ,where C=βm(S1∗,…,SN∗,0,…,0)T, B=D-βmIm∗EN-βdiag⁡AI∗λ1EN-αEN, I∗=(I1∗,I2∗,…,IN∗)T, EN is the identity matrix of order N, diag⁡(·) represents the diagonal matrix, and(11)D=βdiag⁡S1∗,S2∗,…,SN∗A-λ1+λ2diag⁡S1∗+I1∗S1∗,S2∗+I2∗S2∗,…,SN∗+IN∗SN∗.Theorem 10.
System (8) is asymptotically stable at the equilibrium E∗Proof.
Obviously,-γm-μ is a negative eigenvalue of JE∗, and other eigenvalues are determined by matrix B. Next, we show that all the eigenvalues of B have negative real part.
For convenience, define three matrices as follows:(12)K1=A-λ1+λ2βdiag⁡1S1∗,…,1SN∗,(13)K2=K1+λ1+λ2βmaxi1Si∗EN,(14)K3=A-λ1+λ2βdiag⁡1S1∗,…,1SN∗diag⁡S1∗+I1∗S1∗,…,SN∗+IN∗SN∗.Here we consider an undirected and connected graph, so the adjacent matrix A is an irreducible one, indicating that K2 is also an irreducible matrix. According to Perron-Frobenius theorem [21], K2 has a positive eigenvector v corresponding to the largest eigenvalue λmax(K2), i.e., K2v=λmax(K2)v. Then (15)K1v=λmaxK2-λ1+λ2βmaxi1Si∗v,and(16)vTK1I∗=λmaxK2-λ1+λ2βmaxi1Si∗vTI∗.
On the other hand, it is easy from the second equality of system (7) to get that (17)βmIm∗S∗+βdiag⁡S∗AI∗-λ1+λ2I∗=0,where S∗=(S1∗,…,SN∗)T. Thus, (βdiag⁡S∗)K1I∗=-βmIm∗S∗≤0 implies K1I∗≤0 when βS1∗S2∗⋯SN∗≠0.
Sincev and I∗ are positive vectors, it holds that (18)λmaxK2-λ1+λ2βmaxi1Si∗≤0,and (19)λmaxK1=λmaxK2-λ1+λ2βmaxi1Si∗≤0.
From (14) and (16), it follows that (20)λmaxK3<λmaxA-λ1+λ2βdiag⁡1S1∗,…,1SN∗=λmaxK1≤0.Thus, K3 is a negative definite matrix, and then D=βdiag⁡S∗K3 is also a negative definite one. That is to say, D is aHurwitz andMetzler matrix. According to Lemma 5, matrix B is diagonally stable. Therefore, the equilibrium E∗ 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 thatΩ=I1t,…,INt,R1t,…,RNt,ImT∣Iit+Rit≤1,Im≤1,i=1,…,N is an invariant set.Theorem 11.
Ω is a positively invariant set for system (8).Proof.
Denote∂Ω the boundary of Ω, and then it consists of the following 3N+2 hyperplanes: (21)Γi=I1t,…,INt,R1t,…,RNt,ImtT∈Ω∣Iit=0,i=1,…,N,ΓN+i=I1t,…,INt,R1t,…,RNt,ImtT∈Ω∣Rit=0,i=1,…,N,Γ2N+i=I1t,…,INt,R1t,…,RNt,ImtT∈Ω∣Iit+Rit=1,i=1,…,N,Γ3N+1=I1t,…,INt,R1t,…,RNt,ImtT∈Ω∣Imt=0,Γ3N+2=I1t,…,INt,R1t,…,RNt,ImtT∈Ω∣Imt=1,
For simplicity and convenience, system (8) is rewritten as(22)dztdt=gztwith initial condition z(0)∈∂Ω.
Take the outer normal vectors corresponding to3N+2 hyperplanes as follows: (23)Pi=0,…,-1︸i,…,0,0,…,0T,PN+i=0,…,0,0,…,-1︸N+i,…,0T,P2N+i=0,…,1︸i,…,0,0,…,1︸N+i,…,0T,P3N+1=0,…,0,0,…,0,-1︸2N+1T,P3N+2=0,…,0,0,…,0,1︸2N+1T,and let z∗=(I1∗,…,IN∗,R1∗,…,RN∗,Im∗)T be a smooth point of ∂Ω. On the basis of these hyperplanes, five different cases of z∗ are discussed, respectively. (i)
Case 1: ForIi∗=0, (dz/dt|z∗∈Γi,Pi)=-βm(1-Ri∗)Im∗-β(1-Ri∗)∑j=1NaijIj∗<0.(ii)
Case 2: ForRi∗=0, (dz/dt|z∗∈ΓN+i,PN+i)=-λ1Ii∗<0.(iii)
Case 3: ForIi∗+Ri∗=1, (dz/dt|z∗∈Γ2N+i,P2N+i)=-λ2Ii∗-αRi∗<0.(iv)
Case 4: ForIm∗=0, (dz/dt|z∗∈Γ3N+1,P3N+1)=-γm<0.(v)
Case 5: ForIm∗=1, (dz/dt|z∗∈Γ3N+2,P3N+2)=-μ<0.
Therefore,g(z∗) is pointing to Ω, and Ω is positively invariant according to Lemma 6.Theorem 12.
The equilibriumE∗ of system (8) is globally attractive on Ω-0.Proof.
Denotey(t)=(y1(t),…,y2N(t),y2N+1(t))T where yi(t)=Ii(t),yN+i=Ri(t),y2N+1=Im(t)(i=1,2,…,N) and then the virus equilibrium y∗=(y1∗,…,y2N+1∗); here yi∗≠0(i=1,…,N).
To explore the asymptotic behavior of solutions of (8), we define two functions as follows: (24)Fyt=maxiyityi∗:Ω→R,fyt=miniyityi∗:Ω→R.Both functions are continuous and exist right-hand derivatives along solutions of (8). Let y(t) is the solution of (8), and suppose that F(y(t))=yi0(t)/yi0∗,t∈[t0,t0+ε], for some t0 and sufficiently small ε>0. Then we have (25)F′yt02=yi0′t0yi0∗,where F′(y(t))|2≜limh→0+⁡sup⁡F(y(t+h))-F(y(t))/h.
Next, we proof that the derivative ofF(y(t)) at t0 is nonnegative. According to the definition of F(y(t)), it follows that (26)yi0t0yi0∗≥yit0yi∗,i=1,2,…,2N+1.For F(y(t))>1, three cases as below are discussed (here t0 is ignored for conciseness). (i)
Case 1:1≤i0≤N.(27)yi0∗yi0′t0yi0t0=yi0∗yi0βm1-yi0-yN+i0y2N+1+β1-yi0-yN+i0∑j=1Naijyj-λ1+λ2yi0<βm1-yi0∗-yN+i0∗y2N+1yi0∗yi0+β1-yi0∗-yN+i0∗yi0∗yi0∑j=1Naijyj-λ1+λ2yi0∗<βm1-yi0∗-yN+i0∗y2N+1∗+β1-yi0∗-yN+i0∗∑j=1Naijyj∗-λ1+λ2yi0∗=0.(ii)
Case 2:N+1≤i0≤2N. yi0∗yi0′(t0)/yi0(t0)=λ1yi0-N-αyi0yi0∗/yi0<λ1yi0-N∗-αyi0∗=0.(iii)
Case 3:i0=2N+1. yi0∗yi0′(t0)/yi0(t0)=λ1yi0-N-αyi0yi0∗/yi0<λ1yi0-N∗-αyi0∗=0.
Sinceyi0∗>0 and yt0(t0)>0, one gets yi0′(t0)<0, implying F′(y(t0))<0. Similarly, F′|(2)(y(t0))≤0 if F(y(t0))=1, f′|(2)(y(t0))>0 if f(y(t0))<1, and f′|(2)(y(t0)≥0 if f(y(t0))=1.
Denote(28)Uy=max⁡Fy-1,0,y∈Ω,Vy=min⁡1-fy,0,y∈Ω.Obviously, U(y) and V(y) are nonnegative and continuous in Ω, and U′(y)|(5)≤0 and V′(y)|(5)≤0. Letting HU=y∈Ω∣U′2yt=0 and HV=y∈Ω∣V′2yt=0, then we have HU=y:0≤yjt≤yj∗∪0 and HV=y:yj∗≤yjt≤1∪0.
It follows from the LaSalle Invariance Principle that any solution of system (8) starting in Ω approaches HU∩HV=y∗∪0. Therefore, any solution y(t) with initial value y(0)∈Ω satisfies limt→∞y(t)=y∗; i.e., y∗ is globally attractive in Ω.Remark 13.
Together with local asymptotical stability, it is easily obtained that the equilibriumE∗ in the SIRS model with media is globally asymptotically stable. By the way, without the medium propagation, i.e., βm=0, system (8) has a virus-free equilibrium E0 which is globally asymptotically stable if σmax(A)<(λ1+λ2)/β, otherwise unstable. Here σmax(A) is the largest eigenvalue of the topological matrix A, and (λ1+λ2)/β represents the actual effective recovered rate.


## 4.1. Local Stability

To analyze the local stability of system (8) at the equilibrium E∗, we start with its Jacobian (10)JE∗=BC0-γm-μ,where C=βm(S1∗,…,SN∗,0,…,0)T, B=D-βmIm∗EN-βdiag⁡AI∗λ1EN-αEN, I∗=(I1∗,I2∗,…,IN∗)T, EN is the identity matrix of order N, diag⁡(·) represents the diagonal matrix, and(11)D=βdiag⁡S1∗,S2∗,…,SN∗A-λ1+λ2diag⁡S1∗+I1∗S1∗,S2∗+I2∗S2∗,…,SN∗+IN∗SN∗.Theorem 10.
System (8) is asymptotically stable at the equilibrium E∗Proof.
Obviously,-γm-μ is a negative eigenvalue of JE∗, and other eigenvalues are determined by matrix B. Next, we show that all the eigenvalues of B have negative real part.
For convenience, define three matrices as follows:(12)K1=A-λ1+λ2βdiag⁡1S1∗,…,1SN∗,(13)K2=K1+λ1+λ2βmaxi1Si∗EN,(14)K3=A-λ1+λ2βdiag⁡1S1∗,…,1SN∗diag⁡S1∗+I1∗S1∗,…,SN∗+IN∗SN∗.Here we consider an undirected and connected graph, so the adjacent matrix A is an irreducible one, indicating that K2 is also an irreducible matrix. According to Perron-Frobenius theorem [21], K2 has a positive eigenvector v corresponding to the largest eigenvalue λmax(K2), i.e., K2v=λmax(K2)v. Then (15)K1v=λmaxK2-λ1+λ2βmaxi1Si∗v,and(16)vTK1I∗=λmaxK2-λ1+λ2βmaxi1Si∗vTI∗.
On the other hand, it is easy from the second equality of system (7) to get that (17)βmIm∗S∗+βdiag⁡S∗AI∗-λ1+λ2I∗=0,where S∗=(S1∗,…,SN∗)T. Thus, (βdiag⁡S∗)K1I∗=-βmIm∗S∗≤0 implies K1I∗≤0 when βS1∗S2∗⋯SN∗≠0.
Sincev and I∗ are positive vectors, it holds that (18)λmaxK2-λ1+λ2βmaxi1Si∗≤0,and (19)λmaxK1=λmaxK2-λ1+λ2βmaxi1Si∗≤0.
From (14) and (16), it follows that (20)λmaxK3<λmaxA-λ1+λ2βdiag⁡1S1∗,…,1SN∗=λmaxK1≤0.Thus, K3 is a negative definite matrix, and then D=βdiag⁡S∗K3 is also a negative definite one. That is to say, D is aHurwitz andMetzler matrix. According to Lemma 5, matrix B is diagonally stable. Therefore, the equilibrium E∗ 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 thatΩ=I1t,…,INt,R1t,…,RNt,ImT∣Iit+Rit≤1,Im≤1,i=1,…,N is an invariant set.Theorem 11.
Ω is a positively invariant set for system (8).Proof.
Denote∂Ω the boundary of Ω, and then it consists of the following 3N+2 hyperplanes: (21)Γi=I1t,…,INt,R1t,…,RNt,ImtT∈Ω∣Iit=0,i=1,…,N,ΓN+i=I1t,…,INt,R1t,…,RNt,ImtT∈Ω∣Rit=0,i=1,…,N,Γ2N+i=I1t,…,INt,R1t,…,RNt,ImtT∈Ω∣Iit+Rit=1,i=1,…,N,Γ3N+1=I1t,…,INt,R1t,…,RNt,ImtT∈Ω∣Imt=0,Γ3N+2=I1t,…,INt,R1t,…,RNt,ImtT∈Ω∣Imt=1,
For simplicity and convenience, system (8) is rewritten as(22)dztdt=gztwith initial condition z(0)∈∂Ω.
Take the outer normal vectors corresponding to3N+2 hyperplanes as follows: (23)Pi=0,…,-1︸i,…,0,0,…,0T,PN+i=0,…,0,0,…,-1︸N+i,…,0T,P2N+i=0,…,1︸i,…,0,0,…,1︸N+i,…,0T,P3N+1=0,…,0,0,…,0,-1︸2N+1T,P3N+2=0,…,0,0,…,0,1︸2N+1T,and let z∗=(I1∗,…,IN∗,R1∗,…,RN∗,Im∗)T be a smooth point of ∂Ω. On the basis of these hyperplanes, five different cases of z∗ are discussed, respectively. (i)
Case 1: ForIi∗=0, (dz/dt|z∗∈Γi,Pi)=-βm(1-Ri∗)Im∗-β(1-Ri∗)∑j=1NaijIj∗<0.(ii)
Case 2: ForRi∗=0, (dz/dt|z∗∈ΓN+i,PN+i)=-λ1Ii∗<0.(iii)
Case 3: ForIi∗+Ri∗=1, (dz/dt|z∗∈Γ2N+i,P2N+i)=-λ2Ii∗-αRi∗<0.(iv)
Case 4: ForIm∗=0, (dz/dt|z∗∈Γ3N+1,P3N+1)=-γm<0.(v)
Case 5: ForIm∗=1, (dz/dt|z∗∈Γ3N+2,P3N+2)=-μ<0.
Therefore,g(z∗) is pointing to Ω, and Ω is positively invariant according to Lemma 6.Theorem 12.
The equilibriumE∗ of system (8) is globally attractive on Ω-0.Proof.
Denotey(t)=(y1(t),…,y2N(t),y2N+1(t))T where yi(t)=Ii(t),yN+i=Ri(t),y2N+1=Im(t)(i=1,2,…,N) and then the virus equilibrium y∗=(y1∗,…,y2N+1∗); here yi∗≠0(i=1,…,N).
To explore the asymptotic behavior of solutions of (8), we define two functions as follows: (24)Fyt=maxiyityi∗:Ω→R,fyt=miniyityi∗:Ω→R.Both functions are continuous and exist right-hand derivatives along solutions of (8). Let y(t) is the solution of (8), and suppose that F(y(t))=yi0(t)/yi0∗,t∈[t0,t0+ε], for some t0 and sufficiently small ε>0. Then we have (25)F′yt02=yi0′t0yi0∗,where F′(y(t))|2≜limh→0+⁡sup⁡F(y(t+h))-F(y(t))/h.
Next, we proof that the derivative ofF(y(t)) at t0 is nonnegative. According to the definition of F(y(t)), it follows that (26)yi0t0yi0∗≥yit0yi∗,i=1,2,…,2N+1.For F(y(t))>1, three cases as below are discussed (here t0 is ignored for conciseness). (i)
Case 1:1≤i0≤N.(27)yi0∗yi0′t0yi0t0=yi0∗yi0βm1-yi0-yN+i0y2N+1+β1-yi0-yN+i0∑j=1Naijyj-λ1+λ2yi0<βm1-yi0∗-yN+i0∗y2N+1yi0∗yi0+β1-yi0∗-yN+i0∗yi0∗yi0∑j=1Naijyj-λ1+λ2yi0∗<βm1-yi0∗-yN+i0∗y2N+1∗+β1-yi0∗-yN+i0∗∑j=1Naijyj∗-λ1+λ2yi0∗=0.(ii)
Case 2:N+1≤i0≤2N. yi0∗yi0′(t0)/yi0(t0)=λ1yi0-N-αyi0yi0∗/yi0<λ1yi0-N∗-αyi0∗=0.(iii)
Case 3:i0=2N+1. yi0∗yi0′(t0)/yi0(t0)=λ1yi0-N-αyi0yi0∗/yi0<λ1yi0-N∗-αyi0∗=0.
Sinceyi0∗>0 and yt0(t0)>0, one gets yi0′(t0)<0, implying F′(y(t0))<0. Similarly, F′|(2)(y(t0))≤0 if F(y(t0))=1, f′|(2)(y(t0))>0 if f(y(t0))<1, and f′|(2)(y(t0)≥0 if f(y(t0))=1.
Denote(28)Uy=max⁡Fy-1,0,y∈Ω,Vy=min⁡1-fy,0,y∈Ω.Obviously, U(y) and V(y) are nonnegative and continuous in Ω, and U′(y)|(5)≤0 and V′(y)|(5)≤0. Letting HU=y∈Ω∣U′2yt=0 and HV=y∈Ω∣V′2yt=0, then we have HU=y:0≤yjt≤yj∗∪0 and HV=y:yj∗≤yjt≤1∪0.
It follows from the LaSalle Invariance Principle that any solution of system (8) starting in Ω approaches HU∩HV=y∗∪0. Therefore, any solution y(t) with initial value y(0)∈Ω satisfies limt→∞y(t)=y∗; i.e., y∗ is globally attractive in Ω.Remark 13.
Together with local asymptotical stability, it is easily obtained that the equilibriumE∗ in the SIRS model with media is globally asymptotically stable. By the way, without the medium propagation, i.e., βm=0, system (8) has a virus-free equilibrium E0 which is globally asymptotically stable if σmax(A)<(λ1+λ2)/β, otherwise unstable. Here σmax(A) is the largest eigenvalue of the topological matrix A, and (λ1+λ2)/β 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 βm=0.02, β=0.06, λ1=0.3, λ2=0.3, α=0.49, μ=0.5, and γm=0.15. 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 p=0.05 of random adding edges (called NW small-world network), and the scale-free network is a BA scale-free one with m0=10 and m=3. Each type of network is with 100 nodes, respectively.Figures2–4 show the evolution of Ii(t), Ri(t), and Im(t) over time for the fully connected network, NW small-world network, and BA scale-free network, respectively. Obviously, the infected (I) and susceptible (S) states of each node in each network tend to their equilibrium states over time, namely, Ii(t)→Ii∗ and Si(t)→Si∗(t→∞), 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.Figure 2
The statesIi(t), Ri(t), and Im(t) of time for fully connected networks and random initial conditions.Figure 3
The statesIi(t), Ri(t), and Im(t) of time for NW small-world networks and random initial conditions.Figure 4
The statesIi(t), Ri(t), and Im(t) of time for BA scale-free networks and random initial conditions.Interestingly, for the fully connected network, all nodes tend to the same equilibrium state as time goes, namely,Ii(t)→I∗ and Si(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, Ii(t)→Ii∗ and Si(t)→Si∗(t→∞). 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 I∗ satisfies (29)-λ1+λ2β1+λ1/αN-1<I∗-11+λ1/α<-λ1+λ22β1+λ1/αN-1,and thus I∗≈1/1+λ1/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 Figure5(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 Ii∗ (Si∗) 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].Figure 5
The correlation between the node degree (red solid line) and its infected equilibrium state (blue solid line) for NW small-world (a) and BA scale-free networks (b).
(a) (b)


### 5.2. Impact of System Parameters

To learn more about this proposed model, we now analyze the influence of several important parameters, namely,βm, β, λ1, and α, on network average infected state.Let us look back at these parameters:βm 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. λ1 represents the recovered rate from the infected state to the recovered state. Here we call λ1/β and λ1/α the actual and potential effective recovered rate, respectively.We firstly analyze the influence ofβm, α, β, and λ1 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)∂I¯∗/∂βm>0, (b) ∂I¯∗/∂α>0, (c) ∂I¯∗/∂β>0, and (d) ∂I¯∗/∂λ1<0, where I¯∗=1/N∑i=1N‍Ii∗ 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 ofβm and decreases with the increase of λ1/α or λ1/β.Furthermore, we numerically verify the theoretical implications for three typical network topologies, respectively. Figure6(a) shows that, with fixed λ1=0.3,λ2=0.3,β=0.06,α=0.49,γm=0.15,μ=0.5, as βm 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.Figure 6
(a) The influence of the infected rateβm of media on network average infected states, and the curves from top to bottom correspond to the fully connected network, NW small-world network and BA scale-free network, respectively. The influence of the potential spreading rate λ1/α on network average virus at different values of β for the fully connect network (b), NW small-world network (c) and BA scale-free networks (d). The curves from bottom to top in each subplot correspond to the value of β=0.05,0.3,0.6,0.9, respectively.
(a) (b) (c) (d)It is shown from Figures6(c), 6(b), and 6(d) that the average infected state of three typical networks decreases exponentially as the potential recovered rate λ1/α increases with fixed α=0.3, and it can be located at low level when λ1/α 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 λ1/α.Similarly, the average infected state goes down with the increase ofλ1/β, 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.Figure 7
The influence of the actual spreading rateλ1/β on network average infected steady state at different values of α for the fully connect network (a), NW small-world network (b) and BA scale-free networks (c). The curves from bottom to top in each subplot correspond to the value of α=0.06,0.3,0.65,0.85, respectively.
(a) (b) (c)


## 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 βm=0.02, β=0.06, λ1=0.3, λ2=0.3, α=0.49, μ=0.5, and γm=0.15. 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 p=0.05 of random adding edges (called NW small-world network), and the scale-free network is a BA scale-free one with m0=10 and m=3. Each type of network is with 100 nodes, respectively.Figures2–4 show the evolution of Ii(t), Ri(t), and Im(t) over time for the fully connected network, NW small-world network, and BA scale-free network, respectively. Obviously, the infected (I) and susceptible (S) states of each node in each network tend to their equilibrium states over time, namely, Ii(t)→Ii∗ and Si(t)→Si∗(t→∞), 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.Figure 2
The statesIi(t), Ri(t), and Im(t) of time for fully connected networks and random initial conditions.Figure 3
The statesIi(t), Ri(t), and Im(t) of time for NW small-world networks and random initial conditions.Figure 4
The statesIi(t), Ri(t), and Im(t) of time for BA scale-free networks and random initial conditions.Interestingly, for the fully connected network, all nodes tend to the same equilibrium state as time goes, namely,Ii(t)→I∗ and Si(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, Ii(t)→Ii∗ and Si(t)→Si∗(t→∞). 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 I∗ satisfies (29)-λ1+λ2β1+λ1/αN-1<I∗-11+λ1/α<-λ1+λ22β1+λ1/αN-1,and thus I∗≈1/1+λ1/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 Figure5(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 Ii∗ (Si∗) 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].Figure 5
The correlation between the node degree (red solid line) and its infected equilibrium state (blue solid line) for NW small-world (a) and BA scale-free networks (b).
(a) (b)


## 5.2. Impact of System Parameters

To learn more about this proposed model, we now analyze the influence of several important parameters, namely,βm, β, λ1, and α, on network average infected state.Let us look back at these parameters:βm 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. λ1 represents the recovered rate from the infected state to the recovered state. Here we call λ1/β and λ1/α the actual and potential effective recovered rate, respectively.We firstly analyze the influence ofβm, α, β, and λ1 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)∂I¯∗/∂βm>0, (b) ∂I¯∗/∂α>0, (c) ∂I¯∗/∂β>0, and (d) ∂I¯∗/∂λ1<0, where I¯∗=1/N∑i=1N‍Ii∗ 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 ofβm and decreases with the increase of λ1/α or λ1/β.Furthermore, we numerically verify the theoretical implications for three typical network topologies, respectively. Figure6(a) shows that, with fixed λ1=0.3,λ2=0.3,β=0.06,α=0.49,γm=0.15,μ=0.5, as βm 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.Figure 6
(a) The influence of the infected rateβm of media on network average infected states, and the curves from top to bottom correspond to the fully connected network, NW small-world network and BA scale-free network, respectively. The influence of the potential spreading rate λ1/α on network average virus at different values of β for the fully connect network (b), NW small-world network (c) and BA scale-free networks (d). The curves from bottom to top in each subplot correspond to the value of β=0.05,0.3,0.6,0.9, respectively.
(a) (b) (c) (d)It is shown from Figures6(c), 6(b), and 6(d) that the average infected state of three typical networks decreases exponentially as the potential recovered rate λ1/α increases with fixed α=0.3, and it can be located at low level when λ1/α 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 λ1/α.Similarly, the average infected state goes down with the increase ofλ1/β, 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.Figure 7
The influence of the actual spreading rateλ1/β on network average infected steady state at different values of α for the fully connect network (a), NW small-world network (b) and BA scale-free networks (c). The curves from bottom to top in each subplot correspond to the value of α=0.06,0.3,0.65,0.85, respectively.
(a) (b) (c)


## 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(λ1+λ2)/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.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.



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*Source: 2849196-2019-02-03.xml*