
Citation: Giorgos Minas, David A Rand. Parameter sensitivity analysis for biochemical reaction networks[J]. Mathematical Biosciences and Engineering, 2019, 16(5): 3965-3987. doi: 10.3934/mbe.2019196
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Infectious diseases are known to have caused huge devastation and loss of human life since ancient times, and they have been posing a great threat, especially to developing countries. Over the last two decades, human beings have witnessed numbers of major outbreaks of infectious diseases such as SARS (2003), H7N9 (2013) in China, and the Ebola virus (2014) in West Africa [1,2]. These outbreaks have huge negative impacts on people's health and social stability. Hence, how to prevent the spread of diseases effectively becomes an important and hot issue. Meanwhile, more and more researchers have been focusing on studying the transmission mechanism of epidemic diseases in recent years; see [3,4,5,6,7].
It has been widely agreed that the mathematical modeling of infectious diseases is an effective tool and plays a crucial role in investigating the transmission mechanism and dynamics behaviors of various communicable diseases [8]. One of the pioneer works in this field was done by Kermack and McKendrick [9,10,11], where the authors established two classic compartmental epidemic models called susceptible-infected-recovered (SIR) and susceptible-infected-susceptible (SIS) models. From then on, many mathematical models about the prevalence of diseases which descend from the two fundamental epidemic models have been widely used in investigating the spreading mechanics of epidemic diseases [3,5,6,12]. In SIR models, one usually supposes that recovered individuals could not be infected again because infected individuals become permanently immune once recovered. However, for some diseases (influenza, malaria), the recovered individuals may lose immunity after a while and they will be susceptible again. By taking this factor into account, SIRS models are proposed and investigated [3,5]. Moreover, in some bacterial agent diseases such as plague and venereal diseases, the recovered individuals cannot keep immunity for a long time. Therefore, it is more reasonable and appropriate to incorporate the case that a part of infected individuals may go back directly to the susceptible class after some treatments into epidemiology modeling [5,6,12]. In [6], the authors established an SIRS model with nonlinear incidence rate as well as transfer from infectious to susceptible and investigated the global dynamics of the model. In these compartmental epidemic models, there is always an underlying assumption that the population is sufficiently large and all individuals are mixed uniformly. It means that every individual has the same possibility to contact others, whereas it cannot reflect the realistic feature of the epidemic transmission mechanics completely since different individuals may have different numbers of acquaintances, and the contact behaviors of individuals exhibit heterogeneity [13,14]. Thus, it is essential to take the effect of contact heterogeneity into consideration in investigating the transmission mechanics of epidemic diseases.
It is well known that complex networks provide a powerful framework for describing and quantifying the topological structures of various social, economic and biological systems [15]. In particular, the heterogeneity of contact patterns can be well characterized by degree fluctuation of heterogeneous networks [16]. In view of this, the structures of complex network are embedded into the traditional epidemic models, and the epidemic dynamics in complex networks have attracted increasing attention and been studied extensively in recent years [17,18,19,20,21,22,23,24,25,26,27]. In 2001, Pastor-Satorras and Vespignani [20] proposed and studied an SIS epidemic model in highly heterogeneous networks (i.e., scale-free networks) and showed that a highly heterogeneous contact pattern can result in the absence of any epidemic threshold for the first time, i, e., the threshold approaches zero in the limit of a large number of edges and nodes, and even quite a small infectious rate can lead to a major epidemic outbreaks. This is a new epidemic spreading phenomenon which is completely different from previous works. And this result was proved rigorously by Wang and Dai in [21]. The dynamics of SIR model in heterogeneous networks was also investigated in [23], and the authors found that the epidemic threshold will increase with network size on finite networks. In [27], Huo et al. proposed a fractional SIR model with birth and death rates on heterogeneous complex networks and obtained that the threshold value R0 determines the dynamics of the model. In order to investigate the heterogeneities of disease transmission about SIRS models, Li et al. [25] proposed a new SIRS model with nonlinear infectivity as well as birth and death rates on complex networks and obtained that the disease-free equilibrium is globally asymptotically stable when the basic reproduction number R0<1; when R0>1, then there exists a unique endemic equilibrium which is globally asymptotically stable. In [24] and [18], the authors proposed and investigated an SIRS epidemic model with infective individuals entering into the removed or susceptible on complex network respectively. But both of them only focused on the threshold of the epidemic models and the dynamics of equilibria were not discussed.
To the best of our knowledge, the factor of population turnover is not considered in most of aforementioned models, and there are few studies [25,27,28,29,30] about incorporating demographics into network disease models. Inspired by the works of [18,24,25] and [28], in this paper, we shall establish an improved SIRS model on complex heterogeneous networks by taking both demographics and transfer from infectious to susceptible into consideration. Our main goal is to discuss the global epidemic dynamics of the model and provide some strategies to prevent the epidemic outbreaks. The remainder of this article is organized as follows. A network-based SIRS model with demographics and transfer from infectious to susceptible is proposed in Section 2. The basic reproduction number of the model and the stability of equilibria are investigated in Section 3 and Section 4. In Section 5, the effects of three major immunization strategies are investigated and compared. In Section 6, some numerical simulations are given to support the theoretical analysis. Finally, conclusions are drawn in Section 7.
In this paper, we define the whole population as a finite size network. Every node of the network represents an individual and the edges are the interactions among individuals. The degree k of a node denotes the number of edges connected to the node. Let Sk(t), Ik(t) and Rk(t) be the number of the susceptible, infected and recovered nodes with degree k at time t, respectively, for k=1,2,...,n, where n is the maximal degree of the nodes among the network. The transmission sketch is shown in Figure 1.
Based on the mean-field theory, the dynamics of the network-based model can be described by the following differential equations
{dSk(t)dt=b−λ(k)Sk(t)Θ(t)−dSk(t)+γIk(t)+δRk(t),dIk(t)dt=λ(k)Sk(t)Θ(t)−(γ+d+α)Ik(t),dRk(t)dt=αIk(t)−(d+δ)Rk(t), | (2.1) |
where the parameter b denotes the number of newly born susceptible nodes with degree k per unit time; λ(k) (degree-dependent) is the infected rate of a susceptible individual; d represents the death rate; γ is the transfer rate from the infected class to the susceptible class; δ denotes the immunity loss rate; α represents the recovery rate from infected individuals. The parameters b,d,λ(k),α,γandδ are all nonnegative by their epidemiological meaning. Θ(t) represents the probability that a randomly chosen edge of a node degree k points to an infected node at time t and it satisfies the following equation.
Θ(t)=n∑i=1P(i∣k)φ(i)iIi(t)Ni(t), |
where P(i∣k) represents the probability that a node with degree k is connected to a node with degree i. In this paper, we investigate epidemic transmission on uncorrelated networks, then the probability is considered independent of the connectivity of the node from which the link is emanating[20]. Hence P(i∣k) = iP(i)/⟨k⟩, where P(k) represents the probability of a randomly chosen node with degree k and ∑nk=1P(k)=1. ⟨k⟩=∑nk=1kP(k) is the average degree of the network. φ(i) represents the infectivity of a node with degree i, that is, the mean number of edges from which a node with degree i can transmit diseases [19]. Nk(t)=Sk(t)+Ik(t)+Rk(t) denotes the number of nodes with degree k at time t.
Adding the three equations of model (2.1), we have dNk(t)dt=b−dNk(t). Then Nk(t)=Nk(0)e−dt+bd(1−e−dt). For the purpose of having a population of constant size, throughout this paper we always assume that
(H1)Sk(0)+Ik(0)+Rk(0)=Nk(0)=bd.
Hence, we have Sk(t)+Ik(t)+Rk(t)=Nk(t)=bd for all t≥0. Thus
Θ(t)=1⟨k⟩dbn∑k=1φ(k)P(k)Ik(t). |
For the practical consideration, the initial conditions of system (2.1) take the form
Sk(0)>0,Ik(0)≥0,Rk(0)≥0,Θ(0)>0,k=1,2,...,n. | (2.2) |
In this paper, beside the hypothesis (H1), we also need some hypotheses shown as below:
(H2) the infection rate is bounded, i.e., there exists two constants μ,ν>0, such that 0<μ≤λ(k)≤ν;
(H3) the network should be static, that is, the degree of each node is time invariant.
Now, we reveal some properties of solutions of system (2.1).
Lemma 2.1. Suppose (S1(t),I1(t),R1(t),⋯,Sn(t),In(t),Rn(t)) is a solution of system (2.1) with the initial conditions (2.2). Then we have the following conclusion
Sk(t)>0,Ik(t)≥0,Rk(t)≥0,foranyk=1,2,...,nandallt>0. |
Proof. Firstly, we prove Θ(t)>0 for all t>0. By the second equation of system (2.1), we have
dΘ(t)dt=1⟨k⟩dbn∑k=1φ(k)P(k)[λ(k)Sk(t)Θ(t)−(γ+d+α)Ik(t)]=1⟨k⟩dbn∑k=1λ(k)φ(k)P(k)Sk(t)Θ(t)−1⟨k⟩dbn∑k=1(γ+d+α)φ(k)P(k)Ik(t)=Θ(t)[1⟨k⟩dbn∑k=1λ(k)φ(k)P(k)Sk(t)−(γ+d+α)]. | (2.3) |
Obviously, we have
Θ(t)=Θ(0)exp[−(γ+d+α)t+1⟨k⟩db∫t0n∑k=1λ(k)φ(k)P(k)Sk(s)ds]. |
Since Θ(0)>0, one can obtain that Θ(t)>0 for all t>0.
Note that Sk(0)>0. Based on the fact of the first equation of system (2.1) and the continuity of Sk(t), we could find a sufficiently small t0>0 such that Sk(t)>0 for t∈(0,t0). We now prove that Sk(t)>0 for all t>0. Assume Sk(t)>0 is not always valid, then there exists i∈{1,2,...n} and the first time t1≥t0>0 such that Si(t1)=0 and Si(t)>0 for t∈(0,t1). From the second equation of system (2.1), we have dIi(t)dt>−(γ+d+α)Ii(t) for t∈(0,t1). Thus, Ii(t)>Ii(0)e−(γ+d+α)t≥0 for t∈(0,t1).
Combining Ii(t)≥0 for t∈(0,t1) with the third equation of system (2.1), we have dRi(t)dt>−(d+δ)Ri(t) for t∈(0,t1). Hence, one can obtain that Ri(t)>Ri(0)e−(d+δ)t for t∈(0,t1). By the continuity of Ii(t) and Ri(t), we know Ii(t1)≥0 and Ri(t)≥0. Then the first equation of system (2.1) shows
dSi(t)dt∣t=t1=b+γIi(t1)+δRi(t1)≥b>0. |
So, one can find that Si(t)<Si(t1) for t∈(0,t1). Due to Si(t1)=0, it contradicts to Sk(t)>0 obviously. Therefore, Sk(t)>0 for all t>0 and all k=1,2,...,n. By similar discussions, we also have Ik(t)≥0 and Rk(t)≥0 for all t>0. Therefore, this completes the proof.
Hence, we only need to focus on the dynamics of solutions of system (2.1) in the following feasible bounded region
Γ={(S1,I1,R1,⋯,Sn,In,Rn):Sk>0,Ik≥0,Rk≥0,Sk+Ik+Rk=bd,k=1,2,⋯,n}. |
Remark 2.1. It's remarkable that system (2.1) includes some special cases, if b=d=α=δ=0, system (2.1) turns into an SIS model [20]; if b=d=γ=0, model (2.1) is an SIR model [23]; if γ=0 and b=d, system (2.1) is an SIRS model [25].
Remark 2.2. In recent years, many different types of λ(k) and φ(k) have been studied. The most common types for λ(k) is λ(k)=λk (see, [30,21]). For φ(k), one can see φ(k)=k is choosen in [30,18]. Furthermore, in [23], the authors adopt a general form of φ(k)=akα1+vkα, where 0≤α≤1,a>0 and v≥0. The types φ(k)=C (C is a constant) [22] and φ(k)=km (0<m≤1) [17] are discussed as well.
In epidemiological investigations, the basic reproduction number R0 is widely used to measure the epidemic potential.The term basic reproduction number represents the mean number of secondary cases produced by a typical infectious individual during its entire infectious period in a completely susceptible population [31,32]. By the methods in [31], the transmission matrix F for the rate of appearance of new infections and the transition matrix V for the transfer rate of individuals among compartments around the disease-free equilibrium E0 are denoted as follows, respectively
F=(λ(1)φ(1)P(1)⟨k⟩λ(1)φ(2)P(2)⟨k⟩⋯λ(1)φ(n)P(n)⟨k⟩λ(2)φ(1)P(1)⟨k⟩λ(2)φ(2)P(2)⟨k⟩⋯λ(2)φ(n)P(n)⟨k⟩⋮⋮⋮⋯λ(n)φ(1)P(1)⟨k⟩λ(n)φ(2)P(2)⟨k⟩⋯λ(n)φ(n)P(n)⟨k⟩), |
and
V=(γ+d+α0⋯00γ+d+α⋯0⋮⋮⋮⋯00⋯γ+d+α). |
Hence, the basic reproduction number R0=ρ(FV−1) is the spectral radius of the next generation matrix FV−1. By direct calculations, we have
R0=ρ(FV−1)=⟨λ(k)φ(k)⟩(γ+d+α)⟨k⟩, |
where ⟨λ(k)φ(k)⟩=∑nk=1λ(k)φ(k)P(k).
Next, we will investigate the existence of equilibria of system (2.1).
Theorem 3.1. Consider system (2.1), then we have following results:
(i) There always exists a disease-free equilibrium E0=(bd,0,0,⋯,bd,0,0).
(ii) There is a unique endemic equilibrium E∗=(S∗1,I∗1,R∗1,⋯,S∗n,I∗n,R∗n) when R0>1, where
S∗k=(γ+d+α)λ(k)Θ∗I∗k,R∗k=αd+δI∗k, |
I∗k=b(d+δ)λ(k)Θ∗(d+δ)d+αdλ(k)Θ∗+d(d+δ)(γ+d+α) |
with Θ∗=1⟨k⟩db∑nk=1φ(k)P(k)I∗k,k=1,2,⋯,n.
Proof. Clearly, E0 is always a disease-free equilibrium of system (2.1).
Now, let us consider the existence of E∗. Note that the equilibrium E∗ satisfies the following equalities
{b−λ(k)S∗kΘ∗−dS∗k+γI∗k+δR∗k=0,λ(k)S∗kΘ∗−(γ+d+α)I∗k=0,αI∗k−(d+δ)R∗k=0. | (3.1) |
From the second and the third equations of (3.1), we have
S∗k=(γ+d+α)λ(k)Θ∗I∗k, | (3.2) |
R∗k=αd+δI∗k. | (3.3) |
Substituting (3.2) and (3.3) into the first equation of (3.1), one can obtain
I∗k=b(d+δ)λ(k)Θ∗d(d+δ+α)λ(k)Θ∗+d(d+δ)(γ+d+α). | (3.4) |
Substituting (3.4) into the expression of Θ∗, we obtain a self-consistency equation about Θ∗.
Θ∗=1⟨k⟩dbn∑k=1φ(k)P(k)b(d+δ)λ(k)Θ∗d(d+δ+α)λ(k)Θ∗+d(d+δ)(γ+d+α):=f(Θ∗). | (3.5) |
Obviously, Θ∗=0 is a solution of (3.5). Note that
{f(1)<1⟨k⟩dbn∑k=1φ(k)P(k)bd≤1,df(Θ∗)dΘ∗=1⟨k⟩dbn∑k=1φ(k)P(k)b(d+δ)2λ(k)d(γ+d+α)[d(d+δ+α)λ(k)Θ∗+d(d+δ)(γ+d+α)]2>0,df2(Θ∗)dΘ∗2=−21⟨k⟩dbn∑k=1φ(k)P(k)b(d+δ)2λ2(k)d(γ+d+α)[d(d+δ+α)][d(d+δ+α)λ(k)Θ∗+d(d+δ)(γ+d+α)]3<0. |
If R0>1, then we have df(Θ∗)dΘ∗|Θ∗=0>1. Consequently, the equation (3.5) has a unique nontrivial solution Θ∗(Θ∗∈(0,1)). It means that there exists one and only one endemic equilibrium when R0>1. This finishes the proof.
Remark 3.1. From Theorem 3.1, we know that the existence of the endemic equilibrium depends on R0. Obviously, R0 is in proportion to the heterogeneous parameter ⟨λ(k)φ(k)⟩⟨k⟩ which is related to the topology of the networks closely. We also find that R0 has nothing to do with the immunity loss rate δ. It seems that the transfer rate γ and the recovery rate α have the same effects on R0, i.e., increasing the transfer rate γ and the recovery rate α will decrease R0.
In this section, we will consider the global stability of equilibria E0 and E∗. To begin with we will discuss the local asymptotical stability of the disease-free equilibrium E0.
Theorem 4.1. The disease-free equilibrium E0 of system (2.1) is locally asymptotically stable when R0<1 and is unstable when R0>1.
Proof. As shown in Section 2, for the purpose of having a population of constant size, we assume that the initial values always satisfy the hypothesis (H1). Then we can obtain Sk(t)+Ik(t)+Rk(t)=Nk(t)=bd. Hence, system (2.1) can be reduced in the following form:
{dIk(t)dt=λ(k)[bd−Ik(t)−Rk(t)]Θ(t)−(γ+d+α)Ik(t),dRk(t)dt=αIk(t)−(d+δ)Rk(t). | (4.1) |
The Jacobian matrix of the disease-free equilibrium E0 of system (4.1) is given by
J=(A11A12A13⋯A1nA21A22A23⋯A2n⋮⋮⋮⋮⋯An1An2An3⋯Ann), |
where
Aii=(λ(i)φ(i)P(i)⟨k⟩−(γ+d+α)0α−(d+δ)),Aij=(λ(i)φ(j)P(j)⟨k⟩000). |
By induction, the characteristic equation can be shown as follows
(x+γ+d+α)n−1(x+d+δ)n[(x+γ+d+α)−1⟨k⟩n∑k=1λ(k)φ(k)P(k)]=0. | (4.2) |
It's obvious that J has n−1 negative eigenvalues that equal to −(γ+d+α) and n negative eigenvalues that equal to −(d+δ). The 2n-th eigenvalue completely depends on
(x+γ+d+α)−1⟨k⟩n∑k=1λ(k)φ(k)P(k)=0,i.e.,x=(γ+d+α)(R0−1). |
When R0<1, then x<0; when R0>1, then x>0. Consequently, E0 is locally asymptotically stable if R0<1 and is unstable if R0>1. The proof is complete.
Next, we proceed to study the global attractivity of E0. For this purpose, we need the following results.
Lemma 4.1. [33] If a>0,b>0, and dx(t)dt≤b−ax, when t≥0 and x(0)≥0, then lim supt→∞x(t)≤ba.
We are now in a position to establish the following results about the global asymptotical stability of the disease-free equilibrium E0 by employing above lemma.
Theorem 4.2. The disease-free equilibrium E0 of system (2.1) is globally asymptotically stable when R0<1.
Proof. Equation (2.3) implies that
dΘ(t)dt=Θ(t)[1⟨k⟩dbn∑k=1λ(k)φ(k)P(k)(bd−Ik(t)−Rk(t))−(γ+d+α)]≤Θ(t)[⟨λ(k)φ(k)⟩⟨k⟩−(γ+d+α)−1⟨k⟩dbn∑k=1λ(k)φ(k)P(k)Ik(t)]≤Θ(t)[(γ+d+α)(R0−1)−μΘ(t)]=Θ(t)(r−μΘ(t)), | (4.3) |
where r=(γ+d+α)(R0−1), it follows from R0≤1 that r≤0.
(i) if r=0, then we have dΘ(t)dt≤−μΘ2(t). Using following comparison equation
{dx(t)dt=−μx2(t),x(0)=Θ(0). |
Obviously, we can see that Θ(t)≤x(t)=Θ(0)1+μΘ(0)t, and hence limt→∞Θ(t)=0.
(ii) if r<0, consider the comparison equation
{dy(t)dt=y(t)(r−μy(t)),y(0)=Θ(0). |
By using the comparison principle, one can obtain Θ(t)≤y(t)=rcert1+cμert, where c=Θ(0)r−μΘ(0). Therefore, limt→∞Θ(t)=0.
From (i) and (ii), we have
limt→∞1⟨k⟩dbn∑k=1φ(k)P(k)Ik(t)=0. |
Since φ(k)>0 and P(k)>0 for all k, then we have limt→∞Ik(t)=0.
Next, we will prove limt→∞Rk(t)=0. Owing to limt→∞Ik(t)=0, then for any given sufficiently small constant ε>0, there exists t1>0 such that 0≤Ik(t)≤ε for t>t1. According to the third equation of system (2.1), we have
dRk(t)dt≤αε−(d+δ)Rk(t). |
By Lemma 2, we obtain that lim supt→+∞Rk(t)≤α(d+δ)ε. Setting ε→0, then lim supt→+∞Rk(t)≤0, and hence limt→∞Rk(t)=0.
Finally, due to Sk(t)=bd−Ik(t)−Rk(t) we have limt→∞Sk(t)=bd. This implies that the disease-free equilibrium E0 of system (2.1) is globally attractive when R0≤1. Note that E0 is locally asymptotically stable if R0<1 by Theorem 4.1. Therefore, we obtain that E0 is globally asymptotically stable if R0<1. This completes the proof.
In the following, we will prove the persistence of the disease by employing Theorem 4.6 in [34].
Theorem 4.3. The disease is permanent when R0>1, i.e., there exists a ζ>0, such that
lim inft→∞n∑k=1P(k)Ik(t)≥ζ>0. |
Proof. Define
˜xt=(S1(t),I1(t),R1(t),⋯,Sn(t),In(t),Rn(t)), |
Γ={xt:Sk(t),Ik(t),Rk(t)≥0,Sk(t)+Ik(t)+Rk(t)=bd,k=1,2,⋯,n}, |
X0={xt∈Γ,n∑k=1P(k)Ik(t)>0}, |
∂X0=Γ∖X0. |
It is obvious that Γ is positively invariant with respect to system (2.1). Note that Θ(0)>0, one can deduce that Ik(0)>0 for some k and ∑nk=1P(k)Ik(0)>0 for all k. Owing to
d∑nk=1P(k)Ik(t)dt=n∑k=1P(k)dIk(t)dt≥−(γ+d+α)n∑k=1P(k)Ik(t), |
then we have
n∑k=1P(k)Ik(0)e−(γ+d+α)t>0. |
Thus, X0 is positively invariant as well. Furthermore, there exists a compact set B in which all solutions of (2.1) initiated in Γ will enter and remain forever after. The compactness condition (C4.2) for the set B in [34] is verified easily. Let
M∂={˜x0:x(t,˜x0)∈∂X0,t≥0}, |
Ω=∪{ω(˜x0):˜x0∈M∂}, |
where ˜x0=(S1(0),I1(0),R1(0),⋯,Sn(0),In(0),Rn(0)),ω(˜x0) is the omega limit set of the solutions of system (2.1) starting at ˜x0∈Γ. Restricting system (2.1) on M∂, we obtain that
{dSk(t)dt=b−dSk(t)+δRk(t),dIk(t)dt=−(γ+d+α)Ik(t),dRk(t)dt=−(d+δ)Rk(t). | (4.4) |
It is clear that system (4.4) has a unique equilibrium E0 in Γ, and E0 is also the unique equilibrium of system (2.1) in M∂. Then, we can easily prove that E0 is globally asymptotically stable. Therefore Ω={E0}. E0 is a covering of Ω, which is isolated and acyclic (there exists no nontrivial solution in M∂ which links E0 to itself). Finally, to complete the proof, we only need to show that E0 is a weak repeller for X0, i.e.
lim supt→∞d(Ψ(t;˜x0),E0)>0, | (4.5) |
where Ψ(t;˜x0) is an arbitrarily solution starting in ˜x0∈X0. In order to prove (4.5), we need to verify Ws(E0)∩X0=∅ by the method in [35], where Ws(E0) is the stable manifold of E0. Assume it is not true, then there exists a solution ˜xt∈X0, such that
limt→∞Sk(t)=bd,limt→∞Ik(t)=0,limt→∞Rk(t)=0. | (4.6) |
Hence, for any ξ>0, there exists a T>0 such that for all t>T
bd−ξ2<Sk(t)<bd+ξ2,0≤Ik(t)<ξ2,0≤Rk(t)<ξ2. |
Since R0=⟨λ(k)φ(k)⟩(γ+d+α)⟨k⟩>1, namely ⟨λ(k)φ(k)⟩⟨k⟩−(γ+d+α)>0, then we can choose a sufficiently small ξ>0, such that db⟨λ(k)φ(k)⟩⟨k⟩(bd−ξ)−(γ+d+α)>0.
Let L(t)=Θ(t), obviously, L(t) is bounded and the derivative of L along the solution is
dL(t)dt=1⟨k⟩dbn∑k=1φ(k)P(k)[λ(k)Sk(t)Θ(t)−(γ+d+α)Ik(t)]=Θ(t)[1⟨k⟩dbn∑k=1λ(k)φ(k)P(k)Sk(t)−(γ+d+α)]=Θ(t)[1⟨k⟩dbn∑k=1λ(k)φ(k)P(k)(bd−Ik(t)−Rk(t))−(γ+d+α)]>Θ(t)[1⟨k⟩dbn∑k=1λ(k)φ(k)P(k)(bd−ξ)−(γ+d+α)]. |
Due to
σ=db⟨λ(k)φ(k)⟩⟨k⟩(bd−ξ)−(γ+d+α)>0, |
then we have Θ(t)>Θ(0)eσt. Hence L(t)=Θ(t)→+∞ as t→+∞, which is inconsistent with the boundedness of L(t). This proof is complete.
Finally, we will further show that the endemic equilibrium E∗ is globally asymptotically stable by constructing a suitable Lyapunov function.
Theorem 4.4. If R0>1 and δ>γ, then the endemic equilibrium E∗ of system (2.1) is globally asymptotically stable.
Proof. system (2.1) can be written as
{dSk(t)dt=bΛ−λ(k)Sk(t)Θ(t)−(d+γ)Sk(t)+(δ−γ)Rk(t),dIk(t)dt=λ(k)Sk(t)Θ(t)−(γ+d+α)Ik(t),dRk(t)dt=αIk(t)−(d+δ)Rk(t), | (4.7) |
where Λ=(1+γd), and the components of endemic equilibrium E∗ satisfy the following equations
{bΛ−λ(k)S∗kΘ∗−(d+γ)S∗k(t)+(δ−γ)R∗k(t)=0,λ(k)S∗kΘ∗−(γ+d+α)I∗k=0,αI∗k−(d+δ)R∗k=0. | (4.8) |
For the second equation of (4.8), multiplying by φ(k)P(k) and summing both sides with respect to k, we have
γ+d+α=1⟨k⟩dbn∑k=1λ(k)φ(k)P(k)S∗k. | (4.9) |
Submitting I∗k(t)=bd−S∗k−R∗k into the third equation of (4.8), then we obtain
αbd=αS∗k+(α+d+δ)R∗k. | (4.10) |
Now we investigate a non-negative solution (S1(t),I1(t),R1(t),⋯,Sn(t),In(t),Rn(t)) of system (2.1). Motivated by Li et al. [25], we consider the following Lyapunov function
V(t)=12n∑k=1{ρ1(k)(Sk(t)−S∗k)2+ρ2(k)(Rk(t)−R∗k)2}+(Θ(t)−Θ∗−Θ∗lnΘ(t)Θ∗), |
where ρ1(k)=dφ(k)P(k)b⟨k⟩S∗k>0 and ρ2(k)=δ−γαρ1(k)>0. Clearly, V(t)>0 except at E∗, and taking the derivative of V(t) along the positive solution of (4.7), it follows that
dV(t)dt=n∑k=1{ρ1(k)(Sk(t)−S∗k)dSk(t)dt+ρ2(k)(Rk(t)−R∗k)dRk(t)dt}+Θ(t)−Θ∗Θ(t)dΘ(t)dt=n∑k=1{ρ1(k)(Sk(t)−S∗k)[bΛ−λ(k)Sk(t)Θ(t)−(d+γ)Sk(t)+(δ−γ)Rk(t)]+ρ2(k)(Rk(t)−R∗k)[αbd−αSk(t)−(α+d+δ)Rk(t)]}+Θ(t)−Θ∗Θ(t)(1⟨k⟩dbn∑k=1φ(k)P(k)[λ(k)Sk(t)Θ(t)−(γ+d+α)Ik(t)]=n∑k=1[−ρ1(k)(λ(k)Θ+d+γ)(Sk(t)−S∗k)2−ρ1(k)λ(k)S∗k(Θ(t)−Θ∗)(Sk(t)−S∗k)+(δ−γ)ρ1(k)(Sk(t)−S∗k)(Rk(t)−R∗k)−αρ2(k)(Sk(t)−S∗k)(Rk(t)−R∗k)−ρ2(k)(α+d+δ)(Rk(t)−R∗k)2]+1⟨k⟩dbn∑k=1λ(k)φ(k)P(k)(Sk−S∗k)(Θ(t)−Θ∗)=−n∑k=1[ρ1(k)(λ(k)Θ(t)+d+γ)(Sk(t)−S∗k)2+ρ2(k)(α+d+δ)(Rk(t)−R∗k)2]≤0. |
Therefore, dV(t)dt=0 if and only if Sk=S∗k,Ik=I∗k,Rk=R∗k for k=1,2,⋯,n. It means that the largest compact invariant subset contained in the set {dV(t)dt=0} is the singleton {E∗}. From the Lasalle Invariance Principle [36], one can obtain that the endemic equilibrium E∗ is globally asymptotically stable if R0>1 and δ>γ. This completes the proof.
From Section 3 and Section 4, we know the basic reproduction number R0 is a key parameter which determines the global dynamics of the epidemic model, and if the basic reproduction number R0 can be reduced to a value less than one, the epidemic disease will die out. Moreover, immunization is one of effective ways in controlling preventable diseases [37,38,39,40] by reducing the basic reproduction number. However, it is always impossible to vaccinate all individuals in reality because of the high costs and effort. Therefore, in this section we will discuss the impacts of three immunization strategies (i.e., the random immunization, the targeted immunization and the acquaintance immunization) about system (2.1) on the heterogenous networks.
Random immunization is the simplest immunization strategy which vaccinates a fraction of the population without selectivity. Let p (0<p<1) be the immunization rate, then system (2.1) can be written as
{dSk(t)dt=b−λ(k)(1−p)Sk(t)Θ(t)−dSk(t)+γIk(t)+δRk(t),dIk(t)dt=λ(k)(1−p)Sk(t)Θ(t)−(γ+d+α)Ik(t),dRk(t)dt=αIk(t)−(d+δ)Rk(t). | (5.1) |
Through similar derivations in Section 3, we obtain the basic reproduction number Rr0 for system (5.1)
Rr0=(1−p)⟨λ(k)φ(k)⟩(γ+d+α)⟨k⟩=(1−p)R0. |
Then, similar results are obtained.
Theorem 5.1. Assume R0>1. Define pc=1−1R0, then
(i) If p>pc i.e. Rr0<1, then the disease free equilibrium of system (5.1) is globally asymptotically stable. That is to say, the disease will die out ultimately.
(ii) If 0<p<pc (1<Rr0<R0), then there always exists a unique endemic equilibrium E∗r of system (5.1) which is globally asymptotically stable when δ>γ. That indicates that the random immunization is effective, but cannot eliminate the disease in the network.
There is a fact that random immunization strategy ignores the heterogeneous connectivity of the network. In order to take this factor into account, the targeted immunization strategy was proposed [37,38]. This immunization strategy is to immunize the most highly connected nodes because they are vulnerable to selective attacks in the network. Assuming that all nodes with k≥kt will be immunized, where kt is the upper threshold. Then the immunization rate pk can be denoted as
pk={1,0,k≥kt,k<kt. |
Then, system (2.1) becomes
{dSk(t)dt=b−λ(k)(1−pk)Sk(t)Θ(t)−dSk(t)+γIk(t)+δRk(t),dIk(t)dt=λ(k)(1−pk)Sk(t)Θ(t)−(γ+d+α)Ik(t),dRk(t)dt=αIk(t)−(d+δ)Rk(t). | (5.2) |
By similar derivations in Section 3, we obtain the basic reproduction number Rt0 for system (5.2)
Rt0=1(γ+d+α)⟨λ(k)φ(k)⟩−⟨pkλ(k)φ(k)⟩⟨k⟩=R0−⟨pkλ(k)φ(k)⟩(γ+d+α)⟨k⟩. | (5.3) |
Let ˉp=∑nk=1pkP(k) be the average immunization rate, then
⟨pkλ(k)φ(k)⟩=⟨pk⟩⟨λ(k)φ(k)⟩+Cov(pk,λ(k)φ(k))=ˉp⟨λ(k)φ(k)⟩+⟨(pk−ˉp)(λ(k)φ(k)−⟨λ(k)φ(k)⟩)⟩, |
where Cov(⋅,⋅) represents the covariance of two variables. For appropriately small kt, pk−ˉp and λ(k)φ(k)−⟨λ(k)φ(k)⟩ have the same signs, except that k's where pk=ˉp or λ(k)φ(k)=⟨λ(k)φ(k)⟩, then Cov(pk,λ(k)φ(k))>0. That is to say for some kt, ⟨pkλ(k)φ(k)⟩>⟨pk⟩⟨λ(k)φ(k)⟩. Hence, we have following conclusions.
Theorem 5.2. Consider system (5.2), then we have following results:
(i) If Rt0<1<R0, then the disease free equilibrium of system (5.2) is globally asymptotically stable, i.e. the disease can be controlled by the targeted immunization strategy.
(ii) If 1<Rt0<R0, there will always exists a unique endemic equilibrium E∗t of system (5.2), which is globally asymptotically stable when δ>γ.
(iii) For some kt, ⟨pkλ(k)φ(k)⟩>⟨pk⟩⟨λ(k)φ(k)⟩, then Rt0<Rr0. This shows that the targeted immunization is more effective than random immunization.
Since the targeted immunization requires us to know the global connectivity information of each node in networks, and it is always much difficult to carry out in reality. In order to deal with this dilemma, R. Cohen, S. Havlin and D. Ben-Avraham [37] proposed the acquaintance immunization scheme for the immunization of random acquaintances of random nodes. Specifically, choose a fraction q of N nodes randomly, the probability that a particular node with degree k is selected for immunization is kP(k)/(N⟨k⟩) [37,41]. Hence, substituting pk=pN⋅kP(k)/(N⟨k⟩)=kP(k)(p/⟨k⟩) to (5.3), then the basic reproduction number for acquaintance immunization is
Ra0=1(γ+d+α)⟨λ(k)φ(k)⟩−⟨kP(k)(p/⟨k⟩)λ(k)φ(k)⟩⟨k⟩=R0−p⟨kP(k)λ(k)φ(k)⟩(γ+d+α)⟨k⟩2. |
This implies that acquaintance immunization scheme is effective as well.
Theorem 5.3. For system (5.2), we have following results:
(i) If Ra0<1<R0, then the disease can be controlled by this immunization scheme, i.e., the disease will be eradicated from the network ultimately.
(ii) If 1<Ra0<R0, then the disease is uniformly persistent in the network and there always exists a unique endemic equilibrium E∗a.
In this section, we present several numerical simulations to demonstrate the correctness and effectiveness of the main results. These simulations are based on scale-free networks. We first start with a graph with 5 vertexes, a node with one edge is added into the graph in each time step, and then a BA scale-free network G with 100 nodes is obtained by using preferential attachment mechanism in [42]. There will give several numerical examples with different infectivities φ(k) to illustrate the theoretical analysis.
In Figures 2, 3 and 4, we adopt the infected rate λ(k)=βk and give infectivity φ(k) three types such as linear, constant and nonlinear infectivity to verify the stability of the disease-free equilibrium.
Case ⅰ. Let n=11,φ(k)=k, b=2,γ=0.04,β=0.01,d=0.025,α=0.04,δ=0.03 and the initial values: Sk(0)=40,Ik(0)=30,Rk(0)=10. Then we have R0=0.8884<1. From our results, the disease will die out if R0=0.8884<1, and the time evolution of the densities of each state for the initial value is drawn in Figure 2. It agrees with Theorem 4.2.
Case ⅱ. Choose n=13,φ(k)=4 and other parameters are the same as in Case ⅰ. Then one can obtain R0=0.2857<1. It follows from Theorem 4.2, disease-free equilibrium E0 is globally asymptotically stable. The Figure 3 shows the time evolution of the number of each state for the initial value.
Case ⅲ. We choose n=13, the nonlinear infectivity φ(k)=k0.5 and other parameters are the same as in Case ⅰ as well. Then R0=0.2369<1, and by Theorem 4.2 the disease-free equilibrium E0 is globally asymptotically stable. The time evolution of the densities of each state for the initial value is shown in Figure 3.
Next, in the same network architecture in Subsection 6.1, we will verify the global asymptotical stability of the endemic equilibrium E∗. Similarly, we choose λ(k)=βk and give three types of the infectivity φ(k).
Case ⅳ. We make n=13,φ(k)=k, b=2,γ=0.02,β=0.06,d=0.025,α=0.04,δ=0.03 and the initial value: Sk(0)=40,Ik(0)=30,Rk(0)=10. We obtain R0=5.9765>1, which implies that E∗ is global asymptotical stable from Theorem 4.4. The time evolution of the densities of each state for the initial value is depicted in Figure 5.
Case ⅴ. Setting n=15,φ(k)=4 and other parameters are the same as in Case ⅳ. Then one can verify R0=2.1176>1. Hence it follows from Theorem 4.4 that the endemic equilibrium E∗ is globally asymptotically stable. The Figure 6 shows the time evolution of the densities of each state for the initial values.
Case ⅵ. We choose n=13, the nonlinear infectivity φ(k)=k0.5 and other parameters are the same as in Case ⅳ as well. Then we obtain R0=1.7420>1, and the time evolution of the densities of each state for the initial value is presented in Figure 7. It agrees with Theorem 4.4.
As shown in the Figures 5, 6 and 7, it follows from Theorem 4.4 that the endemic equilibrium E∗ is globally asymptotically stable when R0>1 and δ>γ. In addition, if we choose n=14,λ(k)=βk, φ(k)=k, b=2,γ=0.04,β=0.06,d=0.025,α=0.04 and δ=0.03, then we can obtain that R0=4.1789>1 and δ≤γ hold. The time evolution of the densities of each state is shown in Figure 8. It seems that the endemic equilibrium E∗ is also globally attractive. However, we could not give a rigorous proof for the global attractivity of E∗ when R0>1 and δ≤γ, which we leave as our future work.
By incorporating demographics and transfer from infectious to susceptible individuals into network disease models, we propose an improved SIRS model on complex networks in this paper. We derive the basic reproduction number R0 by using the next generation matrix method and obtain that the basic reproduction number R0 determines not only the existence of the endemic equilibrium E∗ but also the global stability of equilibria E0 and E∗, i.e., the disease-free equilibrium E0 is globally asymptotically stable when R0<1; when R0>1, there exists a unique endemic equilibrium E∗ and the disease is permanent on the network, which indicates that reducing the basic reproduction number R0 below one could eradicate the disease. Meanwhile, by employing the method of Lyapunov function, we prove that if R0>1 and δ>γ, the endemic equilibrium E∗ is globally asymptotically stable. Compared with the model in literature [27], our model is more general by considering the degree-related infection rate λ(k) and the general form of infectivity φ(k). At the same time, the global dynamics of the disease-free equilibrium and the endemic equilibrium under different forms of infectivity are discussed by some numerical simulations. It also confirms the correctness of the theoretical analysis. In addition, from our numerical simulations, the endemic equilibrium E∗ is globally asymptotically stable when R0>1 with δ<γ which is not proved in the present paper and will be investigated in further exploration.
Some new insights in this paper may help us to understand the spreading of diseases and provide some effective intervening measures to prevent the epidemic outbreaks. As one can see, the basic reproduction number R0 is an important index in disease control, which is closely related to the topology of the networks and some model parameters. One can obtain that increasing the transfer rate γ and the recovery rate α will decrease R0 to restrict the spreading of the epidemic diseases. Moreover, it is worth noting that R0 is not dependent on the immunity loss rate δ, which indicates that strengthening the treatment is very important. According to the investigation on three immunization strategies about system (2.1) on the heterogenous networks, we can find that immunizations are also effective ways in controlling diseases transmission by decreasing R0. Furthermore, one can see that the targeted immunization is more effective than random immunization, and the acquaintance immunization is more economical and more easier to be carried out.
The authors are grateful to the anonymous referee for the valuable comments. The work is partially supported by NNSF of China (Nos. 11771059, 11401051, 71471020), Hunan Provincial Natural Science Foundation (Nos. 2015JJ3013, 2016JJ1001) and the Scientific Research Fund of Hunan Provincial Education Department of China (No. 18B152).
All authors declare no conflicts of interest in this paper.
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15. | Xin Long, Novel stability criteria on a patch structure Nicholson’s blowflies model with multiple pairs of time-varying delays, 2020, 5, 2473-6988, 7387, 10.3934/math.2020473 | |
16. | Lihong Huang, Huili Ma, Jiafu Wang, Chuangxia Huang, GLOBAL DYNAMICS OF A FILIPPOV PLANT DISEASE MODEL WITH AN ECONOMIC THRESHOLD OF INFECTED-SUSCEPTIBLE RATIO, 2020, 10, 2156-907X, 2263, 10.11948/20190409 | |
17. | Xin Yang, Shigang Wen, Xian Zhao, Chuangxia Huang, Systemic importance of financial institutions: A complex network perspective, 2020, 545, 03784371, 123448, 10.1016/j.physa.2019.123448 | |
18. | Wentao Wang, Wei Chen, Persistence and extinction of Markov switched stochastic Nicholson’s blowflies delayed differential equation, 2020, 13, 1793-5245, 2050015, 10.1142/S1793524520500151 | |
19. | Xin Yang, Shigang Wen, Zhifeng Liu, Cai Li, Chuangxia Huang, Dynamic Properties of Foreign Exchange Complex Network, 2019, 7, 2227-7390, 832, 10.3390/math7090832 | |
20. | Qian Cao, Guoqiu Wang, Dynamic analysis on a delayed nonlinear density-dependent mortality Nicholson's blowflies model, 2020, 0020-7179, 1, 10.1080/00207179.2020.1725134 | |
21. | Hong Zhang, Chaofan Qian, Convergence analysis on inertial proportional delayed neural networks, 2020, 2020, 1687-1847, 10.1186/s13662-020-02737-3 | |
22. | Ajendra singh, Jitendra Nath Rai, Stability of Fractional Order Fuzzy Cellular Neural Networks with Distributed Delays via Hybrid Feedback Controllers, 2021, 1370-4621, 10.1007/s11063-021-10460-7 | |
23. | Qian Cao, Xin Long, New convergence on inertial neural networks with time-varying delays and continuously distributed delays, 2020, 5, 2473-6988, 5955, 10.3934/math.2020381 | |
24. | Chuangxia Huang, Xiaoguang Yang, Jinde Cao, Stability analysis of Nicholson’s blowflies equation with two different delays, 2020, 171, 03784754, 201, 10.1016/j.matcom.2019.09.023 | |
25. | Jian Zhang, Chuangxia Huang, Dynamics analysis on a class of delayed neural networks involving inertial terms, 2020, 2020, 1687-1847, 10.1186/s13662-020-02566-4 | |
26. | Qian Wang, Hui Wei, Zhiwen Long, A non-reduced order approach to stability analysis of delayed inertial genetic regulatory networks, 2021, 33, 0952-813X, 227, 10.1080/0952813X.2020.1735531 | |
27. | Jiafu Wang, Su He, Lihong Huang, Limit Cycles Induced by Threshold Nonlinearity in Planar Piecewise Linear Systems of Node-Focus or Node-Center Type, 2020, 30, 0218-1274, 2050160, 10.1142/S0218127420501606 | |
28. | Gang Yang, Qian Cao, Stability for patch structure Nicholson’s blowflies systems involving distinctive maturation and feedback delays, 2020, 0952-813X, 1, 10.1080/0952813X.2020.1836032 | |
29. | Chuangxia Huang, Xin Long, Jinde Cao, Stability of antiperiodic recurrent neural networks with multiproportional delays, 2020, 43, 0170-4214, 6093, 10.1002/mma.6350 | |
30. | Qian Cao, Guoqiu Wang, New findings on global exponential stability of inertial neural networks with both time-varying and distributed delays, 2021, 0952-813X, 1, 10.1080/0952813X.2021.1883744 | |
31. | Ruihan Chen, Song Zhu, Yongqiang Qi, Yuxin Hou, Reachable set bounding for neural networks with mixed delays: Reciprocally convex approach, 2020, 125, 08936080, 165, 10.1016/j.neunet.2020.02.005 | |
32. | Shigang Wen, Yu Tan, Mengge Li, Yunke Deng, Chuangxia Huang, Analysis of Global Remittance Based on Complex Networks, 2020, 8, 2296-424X, 10.3389/fphy.2020.00085 | |
33. | Zhenhua Duan, Changjin Xu, Anti-periodic behavior for quaternion-valued delayed cellular neural networks, 2021, 2021, 1687-1847, 10.1186/s13662-021-03327-7 | |
34. | Yi Wang, Jinde Cao, Gang Huang, Further dynamic analysis for a network sexually transmitted disease model with birth and death, 2019, 363, 00963003, 124635, 10.1016/j.amc.2019.124635 | |
35. | Luogen Yao, Qian Cao, Anti-periodicity on high-order inertial Hopfield neural networks involving mixed delays, 2020, 2020, 1029-242X, 10.1186/s13660-020-02444-3 | |
36. | Anbalagan Pratap, Ramachandran Raja, Jinde Cao, Chuangxia Huang, Michal Niezabitowski, Ovidiu Bagdasar, Stability of discrete‐time fractional‐order time‐delayed neural networks in complex field, 2021, 44, 0170-4214, 419, 10.1002/mma.6745 | |
37. | Luogen Yao, Global exponential stability on anti-periodic solutions in proportional delayed HIHNNs, 2021, 33, 0952-813X, 47, 10.1080/0952813X.2020.1721571 | |
38. | Yanli Xu, Qian Cao, Xiaojin Guo, Stability on a patch structure Nicholson’s blowflies system involving distinctive delays, 2020, 105, 08939659, 106340, 10.1016/j.aml.2020.106340 | |
39. | Yanli Xu, Qian Cao, Global asymptotic stability for a nonlinear density-dependent mortality Nicholson’s blowflies system involving multiple pairs of time-varying delays, 2020, 2020, 1687-1847, 10.1186/s13662-020-02569-1 | |
40. | Sudesh Kumari, Renu Chugh, Jinde Cao, Chuangxia Huang, On the construction, properties and Hausdorff dimension of random Cantor one pth set, 2020, 5, 2473-6988, 3138, 10.3934/math.2020202 | |
41. | A. Pratap, R. Raja, Jinde Cao, J. Alzabut, Chuangxia Huang, Finite-time synchronization criterion of graph theory perspective fractional-order coupled discontinuous neural networks, 2020, 2020, 1687-1847, 10.1186/s13662-020-02551-x | |
42. | Qian Cao, Guoqiu Wang, Hong Zhang, Shuhua Gong, New results on global asymptotic stability for a nonlinear density-dependent mortality Nicholson’s blowflies model with multiple pairs of time-varying delays, 2020, 2020, 1029-242X, 10.1186/s13660-019-2277-2 | |
43. | Rundong Zhao, Qiming Liu, Meici Sun, Dynamical behavior of a stochastic SIQS epidemic model on scale-free networks, 2021, 1598-5865, 10.1007/s12190-021-01550-9 | |
44. | Xianhui Zhang, Convergence analysis of a patch structure Nicholson’s blowflies system involving an oscillating death rate, 2021, 0952-813X, 1, 10.1080/0952813X.2021.1908433 | |
45. | Roberto Galizia, Petri T. Piiroinen, Regions of Reduced Dynamics in Dynamic Networks, 2021, 31, 0218-1274, 2150080, 10.1142/S0218127421500802 | |
46. | Jian Zhang, Ancheng Chang, Gang Yang, Periodicity on Neutral-Type Inertial Neural Networks Incorporating Multiple Delays, 2021, 13, 2073-8994, 2231, 10.3390/sym13112231 | |
47. | Jiaying Zhou, Yi Zhao, Yong Ye, Yixin Bao, Bifurcation Analysis of a Fractional-Order Simplicial SIRS System Induced by Double Delays, 2022, 32, 0218-1274, 10.1142/S0218127422500687 | |
48. | Shuping Li, Xiaorong Zhao, Ruixia Zhang, Site-bond percolation model of epidemic spreading with vaccination in complex networks, 2022, 85, 0303-6812, 10.1007/s00285-022-01816-1 | |
49. | Lian Duan, Lihong Huang, Chuangxia Huang, Spatial dynamics of a diffusive SIRI model with distinct dispersal rates and heterogeneous environment, 2021, 20, 1534-0392, 3539, 10.3934/cpaa.2021120 | |
50. | Jie Li, Jiu Zhong, Yong-Mao Ji, Fang Yang, A new SEIAR model on small-world networks to assess the intervention measures in the COVID-19 pandemics, 2021, 25, 22113797, 104283, 10.1016/j.rinp.2021.104283 | |
51. | Chaouki Aouiti, Mahjouba Ben Rezeg, Impulsive multidirectional associative memory neural networks: New results, 2021, 14, 1793-5245, 10.1142/S1793524521500601 | |
52. | Hong Zhang, Qian Cao, Hedi Yang, Dynamics analysis of delayed Nicholson-type systems involving patch structure and asymptotically almost periodic environments, 2022, 34, 0952-813X, 725, 10.1080/0952813X.2021.1924869 | |
53. | Reinhard Schlickeiser, Martin Kröger, Determination of a Key Pandemic Parameter of the SIR-Epidemic Model from Past COVID-19 Mutant Waves and Its Variation for the Validity of the Gaussian Evolution, 2023, 5, 2624-8174, 205, 10.3390/physics5010016 | |
54. | Xiaojin Guo, Chuangxia Huang, Jinde Cao, Nonnegative periodicity on high-order proportional delayed cellular neural networks involving D operator, 2020, 6, 2473-6988, 2228, 10.3934/math.2021135 | |
55. | Qian Cao, Attractivity analysis on a neoclassical growth system incorporating patch structure and multiple pairs of time-varying delays, 2021, 14173875, 1, 10.14232/ejqtde.2021.1.76 | |
56. | Reinhard Schlickeiser, Martin Kröger, Key Epidemic Parameters of the SIRV Model Determined from Past COVID-19 Mutant Waves, 2023, 3, 2673-8112, 592, 10.3390/covid3040042 | |
57. | Shixiang Han, Guanghui Yan, Huayan Pei, Wenwen Chang, Dynamical Analysis of an Improved Bidirectional Immunization SIR Model in Complex Network, 2024, 26, 1099-4300, 227, 10.3390/e26030227 | |
58. | Guangyu Li, Haifeng Du, Jiarui Fan, Xiaochen He, Wenhua Wang, The Effect of Fangcang Shelter Hospitals under Resource Constraints on the Spread of Epidemics, 2023, 20, 1660-4601, 5802, 10.3390/ijerph20105802 | |
59. | Chuangxia Huang, Jiafu Wang, Lihong Huang, Asymptotically almost periodicity of delayed Nicholson-type system involving patch structure, 2020, 2020, 1072-6691, 61, 10.58997/ejde.2020.61 | |
60. | 德玉 郭, Construction and Dynamic Analysis of a Class of Hepatitis C Model with Population Heterogeneity, 2023, 12, 2324-7991, 4665, 10.12677/AAM.2023.1211458 | |
61. | Guojin Wang, Wei Yao, An application of small-world network on predicting the behavior of infectious disease on campus, 2024, 9, 24680427, 177, 10.1016/j.idm.2023.12.007 | |
62. | Bingjie Wu, Liang’an Huo, Studying the impact of individual emotional states on the co-evolution of information, behavior and disease in multiplex networks, 2025, 03784371, 130480, 10.1016/j.physa.2025.130480 | |
63. | Samuel Lopez, Natalia L. Komarova, An optimal network that promotes the spread of an advantageous variant in an SIR epidemic, 2025, 00225193, 112095, 10.1016/j.jtbi.2025.112095 |