Original articles
Transmission dynamics, global stability and control strategies of a modified SIS epidemic model on complex networks with an infective medium☆

https://doi.org/10.1016/j.matcom.2021.03.029Get rights and content

Abstract

In this paper, a new SIS (susceptible–infected–susceptible) epidemic model on complex networks with an infective medium is proposed. The infection rate is modified by applying the theory of probability. By using mean-field approximation, iterative analysis method and mathematical analysis, the epidemic threshold, stabilities of the disease-free equilibrium and the endemic equilibrium are studied. Moreover, some disease control strategies are proposed. The results show that the epidemic threshold plays an important role in the spread of disease. Finally, the theoretical results are confirmed by some simple numerical examples.

Introduction

As is well known, epidemic diseases always bring great harm to human beings. Therefore, it is necessary to study the transmission mechanism and dynamic behavior of infectious diseases. As an important tool to study the mechanism of disease transmission, mathematical models play an important role in the research of infectious diseases [2]. The classic compartment epidemic model operates under the assumption of a homogeneous and random mixed population, in which each person is treated similarly to others and has equal opportunities to contact with anyone else [1]. However, everyone has their own social network, where the epidemic is spreading and the number of contacts is far smaller than the size of the population. Thus, the classic compartment epidemic model cannot describe disease transmission well. For this reason, the epidemic model on complex networks has been widely concerned [6], [11], [12], [28].

Since Pastor-Satorras and Vespignani built the first epidemic model on complex networks [13], more and more interesting results about the spread of diseases on complex networks have been obtained [9], [10], [15], [16], [18], [20], [22], [26], [29], [30]. Today, studying the epidemic model on complex networks is still a hot issue. However, Pastor-Satorras’ model was built by ignoring higher order corrections. When it comes to study the transmission dynamics (the epidemic threshold or basic reproduction number), there is no problem with this model, since higher order corrections effect little of it. But for the stability analysis, especially for the stability analysis of endemic equilibrium point, this point could not be ignored.

In the real world, many infectious diseases, such as rabies, swine flu, malaria, yellow fever and dengue fever which spread globally, are spread not only by infective individuals, but also by media, such as mosquitoes, birds and swines. Thus, it is necessary to study infectious diseases with infective media. In recent years, many interesting results about the spread of infectious diseases with infective media are obtained [4], [14], [23], [25]. In Ref. [8] considered both sexual transmission and the transmission by vectors of ZIKV based on complex network and the basic reproduction number and the global stability of the disease-free equilibrium are studied. In Ref. [5], [21], two different kinds of SIS models with an infective vector on complex networks were studied and the epidemic threshold and the stability of the equilibrium are studied.

In this paper, a new SIS epidemic model on complex networks with an infective medium is studied. Firstly, some conditions for disease prevalence are given. In addition, positive invariant set of the new proposed model is considered. Moreover, global stability analysis of the equilibrium points and some control strategies are studied. Finally, some numerical simulations are given to support the theorem results.

The rest of this paper is as follows: Section 2 focuses on the establishment of model and the epidemic threshold. In Section 3, the global stability of the disease-free equilibrium and endemic equilibrium are discussed. Section 4 introduced some disease control strategies. In Section 5, some numerical simulations are given by several examples. Conclusions are drawn in Section 6.

Section snippets

SIS model with an infective medium on complex networks

One considers that diseases spread not only by contacts between nodes, but also by infective medium. One treats the node contacts as scale-free, but infective medium contacts without any selectivity (like individuals and mosquitoes). From the above-mentioned considering and combining with the modeling idea of Ref. [24], the following SIS epidemic model is considered, and the transmission sketch is shown in Fig. 1. dSk(t)dt=−[1−(1−λΘ(I(t)))k(1−β1Im(t))]Sk(t)+γ1Ik(t),dIk(t)dt=[1−(1−λΘ(I(t)))k(1−β1

Global stability of disease-free equilibrium and endemic equilibrium

Theorem 2

The solutions of system (2) has a positive invariant set.

Proof

Let ik=Ik for k=1,2,…,n, in+1=I′ and ∂Ω be the boundary of Ω. Define ∂Ωn+11={i∈Ωn+1|ij=1forsomej},∂Ωn+12={i∈Ωn+1|ij=0forsomej},where Ωn+1=∏i=1n+1[0,1] and j=1,2,…,n+1. Let υj1 be (0,…,1︷j,…,0) and υj2=−υj1 as the outer normal of Ωn+11 and Ωn+12. Through Nagumo’s result [27], it is easy to see that didt|ij=1⋅υj1=−γ1≤0,j=1,2,…,n,didt|ij=1⋅υj1=−γ2≤0,j=n+1,didt|ij=0⋅υj2=−[1−(1−λΘ(I(t)))k(1−β1I′(t))]≤0,j=1,2,…,n,didt|ij=0⋅υj2=−β2∑k=1nIk(t)≤0,j=

Uniform immunization control

Uniform immunization control is to immunize the susceptible nodes at a ratio of 0<μ<1. Therefore, system (2) becomes dIk(t)dt=[1−(1−λΘ(I(t)))k(1−β1Im(t))](1−μ)(1−Ik(t))−γ1Ik(t),k=1,2,…,ndIm(t)dt=β2∑k=1nP(k)Ik(t)(1−Im(t))−γ2Im(t).With the same method in Section 2, one can get that if γ1γ2>(1−μ)β1β2. λ>〈k〉γ1(γ1γ2−(1−μ)β1β2)(1−μ)[(γ1γ2−(1−μ)β1β2)〈k2〉+(1−μ)β1β2〈k〉2].The epidemic threshold is the minimum value of λ satisfying the above inequality. That is λˆc=〈k〉γ1(γ1γ2−(1−μ)β1β2)(1−μ)[(γ1γ2−(1−μ)β1β

Numerical simulation

In this section, some simple examples will be used to verify the correctness of the results. Let P(5)=P(10)=P(15)=13 and P(k≠5,10,15)=0, then 〈k〉=10 and 〈k2〉=116.6667.

Example 1

Let λ=0.01, β1=0.3, β2=0.5, γ1=0.4 and γ2=0.8. In this case, γ1γ2>β1β2 and λc=0.0195. Choose the initial value as I5=0.1k,I10=1−0.2k,I15=0,im=0 for k=1,2,3,4 to get Fig. 2. From Fig. 2, one can get that when λ<λc, the disease-free equilibrium is globally asymptotically stable. The result of Fig. 2 is consistent with Theorem 3.

Example 2

Let λ

Conclusions

In this paper, a modified SIS epidemic model on complex networks with an infective medium is proposed. The results show that the recovery and infection rates play an important role in the global stability of the equilibrium points, moreover the epidemic threshold has the same effect. The results show that if γ1γ2>β1β2 and λ<λc, then the disease-free equilibrium is globally asymptotically stable. Otherwise, there exists a positive equilibrium, which is globally asymptotically stable. One also

References (30)

  • XieY. et al.

    Stability analysis and control strategies for a new SIS epidemic model in heterogeneous networks

    Appl. Math. Comput.

    (2020)
  • YangM. et al.

    A modified SIS model with an infective medium on complex networks and its global stability

    Physica A

    (2011)
  • YangH. et al.

    Suppression of epidemic spreading in time-varying multiplex networks

    Appl. Math. Model.

    (2019)
  • ZhangX. et al.

    Complex dynamics of epidemic models on adaptive networks

    J. Differ. Equ.

    (2019)
  • ZhuG. et al.

    Spreading dynamics and global stability of a generalized epidemic model on complex heterogeneous networks

    Appl. Math. Model.

    (2012)

    • Cited by (0)

    ☆

    This work was supported by the National Natural Science Foundation of China (Nos. 61573008, 61973199, 51778351), Taishan Scholar Project of Shandong Province of China and Shandong University of Science and Technology, China Research Fund (2018 TDJH101).

    View full text
    © 2021 International Association for Mathematics and Computers in Simulation (IMACS). Published by Elsevier B.V. All rights reserved.