
Infectious diseases generally spread along with the asymmetry of social network propagation because the asymmetry of urban development and the prevention strategies often affect the direction of the movement. But the spreading mechanism of the epidemic remains to explore in the directed network. In this paper, the main effect of the directed network and delay on the dynamic behaviors of the epidemic is investigated. The algebraic expressions of Turing instability are given to show the role of the directed network in the spread of the epidemic, which overcomes the drawback that undirected networks cannot lead to the outbreaks of infectious diseases. Then, Hopf bifurcation is analyzed to illustrate the dynamic mechanism of the periodic outbreak, which is consistent with the transmission of COVID-19. Also, the discrepancy ratio between the imported and the exported is proposed to explain the importance of quarantine policies and the spread mechanism. Finally, the theoretical results are verified by numerical simulation.
Citation: Qianqian Zheng, Jianwei Shen, Lingli Zhou, Linan Guan. Turing pattern induced by the directed ER network and delay[J]. Mathematical Biosciences and Engineering, 2022, 19(12): 11854-11867. doi: 10.3934/mbe.2022553
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Infectious diseases generally spread along with the asymmetry of social network propagation because the asymmetry of urban development and the prevention strategies often affect the direction of the movement. But the spreading mechanism of the epidemic remains to explore in the directed network. In this paper, the main effect of the directed network and delay on the dynamic behaviors of the epidemic is investigated. The algebraic expressions of Turing instability are given to show the role of the directed network in the spread of the epidemic, which overcomes the drawback that undirected networks cannot lead to the outbreaks of infectious diseases. Then, Hopf bifurcation is analyzed to illustrate the dynamic mechanism of the periodic outbreak, which is consistent with the transmission of COVID-19. Also, the discrepancy ratio between the imported and the exported is proposed to explain the importance of quarantine policies and the spread mechanism. Finally, the theoretical results are verified by numerical simulation.
Pattern formation is a kind of spatial dynamical behavior that shows species distribution and is widely used to explain some biological mechanisms [1,2,3,4,5,6]. With the development of complex networks, more and more attention is paid to pattern formation from the perspective of complex networks [7]. Turing instability was investigated to show the effect of the network on the pattern formation [8], which is different from the reaction-diffusion system. Then, the theory of pattern formation was proposed through the distribution of the real and imaginary parts of the eigenvalues on directed networks [9]. The instability mechanism on network was explored by comparing the instability region in the reaction-diffusion and network-organized system [10,11]. Meanwhile, the concept of negative wavenumber was introduced to explain the interactions between network nodes [12], which was also used to illustrate the dynamic behaviors of the epidemic [13]. Although some work about Turing instability and pattern formation has been done on directed networks [14], the function of delay in the epidemic model with the directed network remains to be explored.
The mathematical model is a vital tool to describe the spreading of infectious diseases [15,16,17,18] and is used to explain the dynamic behaviors of COVID-19 [19,20,21,22,23,24]. The SIRS model is a classic model to describe the spreading of infectious diseases and has been extended to many infectious disease models [25,26]. The bifurcation of an SIRS model with a nonlinear incidence rate was analyzed to make strategies for controlling the epidemic [27]. And then the corresponding SDE version of the SIRS model was developed to show the effect of the basic reproduction number on the dynamical behavior and the prevalence of the epidemic [28]. The SIRS model with both noise and delay [29,30] was analyzed to help the government control the spreading of infectious diseases further. Recently, the study of infectious diseases with social networks was presented to capture the periodic outbreak behaviors of infectious diseases [31,32]. But because infectious diseases generally spread along with the directed network (social network) propagation, the SIRS model with the directed network should be considered.
Although infectious diseases generally spread along with social networks, the isolation policy also leads to the direction of propagation(the asymmetry of network propagation). Meanwhile, there is also a time delay in the propagation process of the epidemic. To understand the spread mechanism and spatiotemporal dynamic behavior of the epidemic in the directed network, pattern formation in the epidemic model with the directed network is investigated to show the effect of the directed network and delay on the outbreak of the epidemic. Firstly, we analyze a general system with the directed network and obtain the conditions of Turing instability. Then, we illustrate the effect of the directed network and delay on the Hopf bifurcation. Also, the dynamical mechanism of the epidemic model with network and delay is explained. According to our theories, some epidemic prevention strategies are given. Finally, numerical simulation verifies our results.
The goal of this paper is to study the following network-organized system
dSidt=f(Si,Ii,Si(t−τ),Ii(t−τ))+d1n∑k=1A(1)ikh1(Sk,Si),dIidt=g(Si,Ii,Si(t−τ),Ii(t−τ))+d2n∑k=1A(2)ikh2(Ik,Ii), | (2.1) |
where f(S,I,S(t−τ),I(t−τ)),g(S,I,S(t−τ),I(t−τ)) are the interactions between species, A(1),A(2) are the adjacent matrix of the directed networks, h1(Sk,Si),h2(Ik,Ii) are the interaction functions through network, and the specific example can be found in Results and discussion. Also, we assume system (2.1) is stable when d1=d2=0.
The linear network-organized system without delay can be expressed as
dSidt=a11Si+a12Ii+d1n∑k=1L(1)ikSk,dIidt=a21Si+a22Ii+d2n∑k=1L(2)ikIk, | (2.2) |
where a11,a12,a21,a22 are the linear parts of f(Si,Ii,Si(t−τ),Ii(t−τ)),g(Si,Ii,Si(t−τ),Ii(t−τ)) at equilibrium point when τ=0, L(1)ikSk,L(2)ikIk are the linear parts of A(1)ikh1(Sk,Si),A(2)ikh2h(Sk,Si). In general, L(1),L(2) are the Laplacian matrices and generally have the same eigenvectors.
The eigenvalues and the eigenvectors of matrices L(1),L(2) can be defined [9]
n∑k=1L(1)ikvmk=(θm+Θmj)vmi,n∑k=1L(2)ikvmk=(ϕm+Φmj)vmi, |
where same eigenvector space is true for L(1),L(2).
The general solution of the linear network-organized system can be expanded as
Si=n∑k=1ckeλktvki,Ii=n∑k=1bkeλktvki, | (2.3) |
where λ represents λk for convenience in the following.
Substituting system (2.3) into system (2.2), one has the Jacobian matrix
J0=(λ−a11−d1(θi+Θij)−a12−a21λ−a22−d2(ϕi+Φij)), |
where θi,ϕi and Θi,Φi are the real parts and the imaginary parts of Λ1i,Λ2i, separately. j(j2=−1) is the imaginary part unit.
According to the Jacobian matrix, we have the characteristic equation
λ2+(c1+c2j)λ+c3+c4j=0, | (2.4) |
where
c1=−a11−d1θi−a22−d2ϕi,c2=−d2Φi−d1Θi,c3=−d1Θid2Φi−a12a21+a11a22+a11d2ϕi+d1θia22+d1θid2ϕi,c4=a11d2Φi+d1Θid2ϕi+d1Θia22+d1θid2Φi. |
The roots of system (2.4) are
λ1,2=−c1−jc2±√(c1+jc2)2−4c3−4jc42. |
Before we consider the sign of the λ1,2, a complex number z=a+bj can be defined as
√z=±(√a+|z|2+sgn(b)√−a+|z|2j), |
where sgn(.) is the standard sign function, a=c12−c22−4c3, b=2c1c2−4c4, and
λ1,2=−c1−jc2±(√a+|z|2+sgn(b)√−a+|z|2j)2. |
To investigate the stability of system (2.1), the sign of the maximum real part of λ1,2 is
s1=−c1+√a+|z|2=−c1+√a+√a2+b22>0. | (2.5) |
Namely, system (2.1) is unstable when H1 holds (s1>0), where
H1:c1=−a11−d1θi−a22−d2ϕi≤0. |
If H1 is not true,
c1=−a11−d1θi−a22−d2ϕi>0, |
and system (2.5) can be rewritten as
√a+√a2+b22>c1. |
Then, one has the following inequality
s2=−c12−c22−4c3+√c14+2c12c22−8c12c3+c24+8c22c3+16c32−16c1c2c4+16c42>0, | (2.6) |
It is found that system (2.1) is unstable (s2>0) when H2 holds, where
H2:−c12−c22−4c3>0. |
If H2 is not true,
−c12−c22−4c3<0, |
system (2.6) can be expressed as
√c14+2c12c22−8c12c3+c24+8c22c3+16c32−16c1c2c4+16c42>c12+c22+4c3 |
and it is equivalent to
H3:s3=−c12c3−c1c2c4+c42>0. |
If H3 holds, Turing instability and Hopf bifurcation may occur. Assume λ=jω, system (2.4) can be written as
−ω2+(c1j−c2)ω+c3+c4j=0. | (2.7) |
If a positive real root ω0 of system (2.7) and the transversality condition Re(dλdμ)ω=ω0,μ=μc≠0(μ is a parameter and its critical value μc) hold [33], Hopf bifurcation occurs.
Assume system (2.1)is stable when d1=d2=0. Then, the network-organized system with delay is considered, and the linear parts of system (2.1) can be expressed as
dSidt=b11Si+b12Ii+b13Si(t−τ)+b14Ii(t−τ)+d1n∑k=1LikSk,dIidt=b21Si+b22Ii+b23Si(t−τ)+b24Ii(t−τ)+d2n∑k=1LikIk, | (2.8) |
where b11,b12,b13,b14,b21,b22,b23,b24 are the linear parts of f(Si,Ii,Si(t−τ),Ii(t−τ)),g(Si,Ii,Si(t−τ),Ii(t−τ)) at equilibrium point.
Substituting system (2.3) into system (2.8), one has the Jacobian matrix
J=(λ−b11−b13e−λτ−d1(θi+Θij)−b12−b14e−λτ−b21−b23e−λτλ−b22−b24e−λτ−d2(ϕi+Φij)) |
and the characteristic equation is
λ2+(c11e−λτ+c12+c13j)λ+(c14+c15j)e−λτ+c16+c17j=0, | (2.9) |
where
c11=−b13−b24,c12=−d2ϕi−θid1−b11−b22,c13=−d2Φi−Θid1,c14=b13d2ϕi+b24θid1−b23b12+b11b24+b13b22−b14b21,c15=b13d2Φi+b24Θid1,c16=−d2ΦiΘid1+d2ϕiθid1+b11d2ϕi+b22θid1−b21b12+b11b22,c17=d2Φiθid1+d2Θiϕid1+b11d2Φi+b22Θid1. |
To investigate the Hopf bifurcation of system (2.9), we substitute λ=jω into system (2.9), and have
−ω2+ωc11sin(ωτ)−ωc13+cos(ωτ)c14+sin(ωτ)c15+c16+j(ωc11cos(ωτ)+ωc12−sin(ωτ)c14+cos(ωτ)c15+c17)=0. | (2.10) |
Separating the real and imaginary parts of system (2.10), one has
−ω2+ωc11sin(ωτ)−ωc13+cos(ωτ)c14+sin(ωτ)c15+c16=0,ωc11cos(ωτ)+ωc12−sin(ωτ)c14+cos(ωτ)c15+c17=0. | (2.11) |
and the solutions are
cos(ωτ)=−ω2c11c12−ω2c14+ωc11c17+ωc12c15−ωc13c14+c14c16+c15c17ω2c112+2ωc11c15+c142+c152,sin(ωτ)=ω3c11+ω2c11c13+ω2c15−ωc11c16+ωc12c14+ωc13c15+c14c17−c15c16ω2c112+2ωc11c15+c142+c152. | (2.12) |
According to cos2(ωτ)+sin2(ωτ)=1, we have
p0ω6+p1ω5+p2ω4+p3ω3+p4ω2+p5ω+p6=0, | (2.13) |
where
p0=c112,p1=2c112c13+2c11c15,p2=−c114+c112c122+c112c132−2c112c16+4c11c13c15+c142+c152,p3=−4c113c15+2c112c12c17−2c112c13c16+2c11c122c15+2c11c132c15−4c11c15c16+2c13c142+2c13c152,p4=−2c112c142−6c112c152+c112c162+c112c172+4c12c15c17c11−4c11c13c15c16+c122c142+c122c152+c132c142+c132c152−2c142c16−2c152c16,p5=−4c11c142c15−4c11c153+2c11c15c162+2c11c15c172+2c12c142c17+2c12c152c17−2c13c142c16−2c13c152c16,p6=−c144−2c142c152+c142c162+c142c172−c154+c152c162+c152c172. |
Based on Hurwitz criterion[33], if H4 holds, system (2.2) is always stable, otherwise Turing instability induced by network occurs.
H4:pi>0(i=0,...,6),Dn>0(n=1,...,6), |
where
Dn=|p1p00000p3p2p1000p5p4p3p2p1p00p6p5p4p3p2000p6p5p400000p6| |
If s(1≤s≤6) positive real roots ωi(1≤i≤s) exist in system (2.13), Hopf bifurcation occurs. And the critical value τc can be obtained from system (2.11),
τc=min1≤i≤s{1ωiarccos(−Q1Q2)}, | (2.14) |
where Q1=ωi2c11c12−ωi2c14+ωic11c17+ωic12c15−ωic13c14+c14c16+c15c17, Q2=ωi2c112+2ωic11c15+c142+c152, and τ0 is the critical value of τc without network, τ1 is the critical value of τc with network. Also, the transversality condition Re(dλdτ)≠0 [34,35].
Theorem 1 In the network-organized system, If one of H1,H2,H3 and the transversality condition Re(dλdτ)>0 hold, Turing instability induced by network occurs; if H4 does not hold, Turing instability induced by network may occur when τ0>τ1 and τ0>τ>τ1.
Proof: It is well known that system (2.1) without network is stable when τ<τ0. And the network could induce the decrease of τc to τ1. Namely, system (2.1) is unstable when τ>τ1. As a result, τ0>τ>τ1 means Turing instability occurs.
In this section, we take an SIRS model as an example to illustrate our theoretical results through numerical simulation. A general SIRS model can be written as
dSdt=α−βS(t−τ)I(t−τ)−dS+δR,dIdt=βS(t−τ)I(t−τ)−γI−dI,dRdt=γI−δR−dR. |
Assume dRdt=0, one has the following system,
dSdt=α−βS(t−τ)I(t−τ)−dS+δγd+δI,dIdt=βS(t−τ)I(t−τ)−dI−γI, |
where the equilibrium point (S∗,I∗) is E1=(αd,0), E2=(d+γβ,−−αdβ−αδβ+d3+d2δ+d2γ+dδγβd(γ+d+δ)).
In this paper, we consider the network-organized system
dSidt=α−βSi(t−τ)Ii(t−τ)−dSi+δγd+δIi+d1n∑k=1Aikf(Sk,Si),dIidt=βSi(t−τ)Ii(t−τ)−dIi−γIi+d2n∑k=1Aikg(Ik,Li), | (3.1) |
Suppose f(Sk,Si)=Sj−qSi,g(Ik,Li)=Ij−qIi, we obtain
dSidt=α−βSi(t−τ)Ii(t−τ)−dSi+δγd+δIi+d1n∑k=1Aik(Sj−qSi),dIidt=βSi(t−τ)Ii(t−τ)−dIi−γIi+d2n∑k=1Aik(Ij−qIi), |
where α=0.2,β=1/14,γ=0.01,δ=0.1,d=1/60; q is the discrepancy ratio between the imported and the exported, q=1 means the imported equal to the exported, q>1 means the imported is smaller than the exported, q<1 means the imported is larger than the exported.
We firstly consider the system (2.1) without delay. Based on Theorem 1, Turing instability never occur when q=1 (the imported equals to the exported) (Figure 1(a)). Namely, the diffusion (network) of the infected(the susceptible) does not work in the outbreak, which does not match the reality[13,31]. And H1,H2,H3 hold if q=0.99, the directed network leads to Turing instability (Figure 1(b)). Therefore, the discrepancy ratio is vital for the spread of the epidemic (Figure 2). That's why the importation of cases is strictly controlled in some countries. From Figure 2, one of H1,H2,H3 may hold if a θi exists (Figure 2(a)). Meanwhile, the outbreak intensity and maximum θmax(θmax is the maximum real part of θi) decreases with the increase of q (Figure 2(b), Figure 3). Namely, q is a key factor in instability (Figures 2 and 3). Namely, controlling the imported cases is a good way to reduce the outbreaks of infectious diseases. Also, the directed network is more suitable than the undirected network to explain the transmission mechanism of infectious diseases because the undirected network doesn't work in some models[13,31].
Then, we consider the system (2.1). Supposing the incubation period is a time delay, the delay could affect the stability of system (2.1) and make the periodic oscillation state occur when τ=2.18>τ0=2.175 (Figure 4). Also, the outbreak intensity is in direct proportion to τ (Figure 5(a)) and τ0 is inversely proportional to θi,Θi[Figure 5(b)]. From Figure 5(b), Θi and θi could lead to the decrease of τc and Turing instability in the directed network-organized system. From Figure 4, system (2.1) without network is stable when τ=2.17, but Turing instability occurs in system (2.1) when τ=2.17 because the critical τc decreases to τ1 and τ0>τ>τ1(Figure 6). Furthermore, the diffusion(on network) of the susceptible (Figure 6(a))(or the infected[Figure 6(b)]) alone leads to instability. Of course, the diffusion of the infected bring about the larger outbreak (Figure 6(a), (b)). It is found that the diffusion of the susceptible and infected could make outbreaks of the epidemic more synchronous (Figure 6(c)). So is Figure 6(c) when τ is larger than τ0. Finally, we investigate the effect of the discrepancy ratio q on the pattern formation of system (2.1) (Figures 7 and 8). If p=0.01 is small, system (2.1) is stable (Figure 7(a)), and Turing instability occurs when q=0.99,p=0.01 (Figure 7(a)). Also, the discrepancy ratio q could make the regional differences even greater (Figure 7). Although the maximum Imax is reduced, the minimum Imin is increasing, it ultimately becomes more uniform (Figure 7). Finally, the numerical results qualitatively agree with the periodic outbreaks and distribution of COVID-19 (Figure 9).
Through the above analysis, it is found that the discrepancy ratio q and time delay τ play a vital role in the outbreak of the epidemic. And controlling the imported cases is a good way to reduce the outbreaks of infectious diseases, which could increase the discrepancy ratio. Because the incubation period τ0 is the inherent nature of infectious diseases, it is impossible to change. But we can control the region of τ1 through the directed network, which also provides a novel to prevent the epidemic.
In this paper, the effect of the directed network and delay on the spread of the epidemic is investigated, which overcome the shortage of the undirected network (The undirected network can't lead to the outbreak of infectious diseases in some model [13,31]). The conditions of Turing instability are given in a general system with the directed network, which is an important indicator to determine whether an outbreak occurs. Then, Hopf bifurcation is analyzed to illustrate the role of the delay and directed network in Turing instability, which can be controlled through the directed network. Also, the proposed discrepancy ratio could make the regional differences even more significant[Figure.9(a)], which is an essential indicator in assessing quarantine policies. Finally, although the combination of the directed network and delay may be a novel way to investigate the pattern dynamics, the general interaction function through the network is difficult to express in the stability of a network-organized system, which will be further studied.
This work is supported by National Natural Science Foundation of China (12002297), Basic research Project of Universities in Henan Province (21zx009), Program for Science & Technology Innovation Talents in Universities of Henan Province (22HASTIT018), The Scientific Research Innovation Team of Xuchang University (2022CXTD002), Outstanding Young Backbone Teacher of Xuchang University (2022), Key scientific research projects of Henan Institutions of Higher learning in 2021 (21B130004).
The authors declare there is no conflict of interest.
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