Dynamical analysis of an SIRS network model with direct immunization and infective vector
 Rongzhong Yu^{1}Email author,
 Kezan Li^{2},
 Baidi Chen^{1} and
 Dingqin Shi^{1}
https://doi.org/10.1186/s1366201504364
© Yu et al.; licensee Springer. 2015
Received: 10 October 2014
Accepted: 3 March 2015
Published: 9 April 2015
Abstract
With the awareness of risk in infective disease spreading, healthy individuals (the susceptible ones) will take some measures to acquire temporary immunity. This paper addresses an SIRS model with direct immunization and an infective vector in complex networks and performs the dynamical analysis for this model. By theoretical analysis, we obtain the epidemic threshold \(\lambda_{c}\) and prove that if infection rate \(\lambda<\lambda_{c}\), the diseasefree equilibrium is globally asymptotically stable; if \(\lambda> \lambda_{c}\), there exists a unique endemic equilibrium, and it is globally attractive. These theoretical results are confirmed by numerical simulations.
Keywords
1 Introduction
In recent years, many epidemic models on complex networks, such as SIS (susceptibleinfectedsusceptible) [1–8] and SIR (susceptibleinfectedremoved) [9–16] and so on, have been widely studied by researchers from different subjects. Classical studies have revealed that there is an epidemic threshold \(\lambda_{c}\) for an epidemic model on homogeneous networks, below which the disease will die out; otherwise there will exist a persistence. However, PastorSatorras and Vespignani further showed a striking result that the epidemic threshold \(\lambda_{c}\) will vanish for a heterogenous network with sufficiently large sizes [1, 2, 4].
In fact, apart from the human behavior [17, 18] and the external environment [19], the infection vector (e.g., mosquitoes) may also play an important role in epidemic transmission [3, 20, 21]. The infective vector generally acts as a carrier of an infective disease and can transmit it to a human. By considering the disease spreading on a human network caused by an infection vector, Cooke and Busenberg [22, 23] have addressed some epidemic compartment models. As we known, some diseases spread not only by contacts between people and infected vectors but also by blood contacts within human. By noting this fact, Shi et al. [21] proposed a new SIS model with an infective medium on complex networks, which models the spread of a class of infectious diseases. Then a modified SIS model is proposed in [3] by assuming that the human contacts can be considered as a scalefree network, but the infective media may contact a person without any selectivity. The epidemic threshold and the stability of endemic equilibrium are investigated theoretically. A more general modified SIS model with an infective medium on complex networks was introduced in [20], and the authors investigated the global attraction of endemic equilibrium by the basic reproduction number \(R_{0}\). However, the direct relation between the epidemic threshold \(\lambda_{c}\) in [3] and the basic reproduction number \(R_{0}\) in [20] has not been revealed.
In the real world, with the awareness of an infectious disease spreading, some healthy individuals will usually take some protective measures (e.g., vaccine inoculation) to acquire temporary immunity. By investigating an SIRS epidemic model [24] with direct immunization on complex networks, the result shows that the direct immunization can increase the epidemic threshold and reduce the prevalence of infectious disease.
In this paper, we propose a new SIRS model with direct immunization and an infective vector on complex networks. We get the epidemic threshold \(\lambda_{c}\), below which the diseasefree equilibrium is globally stable; otherwise, the diseasefree equilibrium is unstable and a unique endemic equilibrium exists, and it is also globally attractive. More importantly, according to our method, one can directly determine the relation between the epidemic threshold and the basic reproduction number.
The rest of the paper is organized as follows. In Section 2, we propose a new SIRS model with direct immunization and an infective vector on complex networks. Section 3 analyzes the dynamics of the model and shows some theoretical results. Some numerical simulations are performed to confirm our theoretical predictions in Section 4.
2 The model

Two infection mechanisms: the disease spreads not only by contacts between individuals, but also by contacts between individuals and infective vectors.

Two removed ways: (i) with awareness of risk infective disease spreading, some health individuals may make some protective measures (vaccination) to acquire temporary immunity; (ii) an infected individual becoming a removed individual after cure may acquire temporary immunity.
In addition, we further suppose that the individuals’ contacts can be treated as heterogeneous, but the contacts between individuals and vectors can be considered as homogeneous. This assumption results from the selective contacts in a human network and nonselective contacts between people and vectors [3, 20].
Let \(S_{k}(t)\), \(I_{k}(t)\) and \(R_{k}(t)\) be the densities of susceptible, infected and removed nodes with degree k at time t respectively, and let \(V(t)\) be the density of the infective medium at time t. Let \(\rho(t) = \sum p(k)I_{k}(t)\) denote the density of infected individuals on the network, and Θ represents the probability that a randomly chosen link emanating from a node of degree k leads to infected nodes. In this paper, we consider the situation of uncorrelated networks, then Θ can be written as \(\Theta= \frac{1}{\langle k\rangle}\sum_{k'}k'p(k')I_{k'}(t)\) [1, 2], where \(p(k)\) denotes the degree distribution of the network, \(\langle k\rangle=\sum kp(k)\) is its average degree.
We assume that the susceptible nodes become the removed nodes with rate α for acquiring temporary immunity. At the same time, each susceptible (health) node is infected with rates λ and \(\gamma_{1}\) if it is contacted to infected nodes and infective vectors, respectively. Infected nodes are cured with rate β and removed nodes again become susceptible with rate δ for immunizationlost. Health vectors are infected with rate \(\gamma_{2}\) if they are contacted to infected individuals and infected vectors recover with rate ξ.
3 Epidemic threshold and global analysis
3.1 Epidemic threshold
Theorem 1
Let \(\lambda_{c} = \frac{[\beta(\alpha+\delta)\delta\gamma_{1}\gamma_{2}]\beta(\alpha+\delta )\langle k\rangle}{\delta^{2}\gamma_{1}\gamma_{2} (\langle k\rangle^{2}\langle k^{2}\rangle)+\delta\beta(\delta+\alpha)\langle k^{2}\rangle}\) and \(\beta \frac{\delta}{\alpha+\delta}\gamma_{1}\gamma_{2} > 0\), if \(\lambda> \lambda_{c}\), then one and only one endemic equilibrium solution of system (2) exists, i.e., the epidemic propagation may outbreak on complex networks.
Proof
The reader should find that the model parameters are general in this model. If \(\gamma_{1}\) or \(\gamma_{2}\) vanishes, the proposed model may become an SIRS model with direct immunization via one infection mechanism (contacts between individuals), and the epidemic \(\lambda_{c} = \frac{\beta\langle k \rangle}{\tau\langle k^{2} \rangle}\) which agrees with the one of paper [24]. In addition, the proposed model may become one SIRS model with an infection vector via two infection mechanisms when \(\alpha= 0\). And the epidemic threshold \(\lambda_{c}= \frac{[\beta\gamma_{1}\gamma_{2}]\beta\langle k\rangle}{\gamma_{1}\gamma_{2} (\langle k\rangle^{2}\langle k^{2}\rangle)+\beta\langle k^{2}\rangle}\), especially, \(\lambda_{c}= \frac{[1\gamma_{1}\gamma_{2}]\langle k\rangle}{\gamma_{1}\gamma_{2} (\langle k\rangle^{2}\langle k^{2}\rangle)+\langle k^{2}\rangle}\) if \(\beta= 1\), which is in accordance with the one of paper [3].
3.2 Global stability of diseasefree equilibrium
Theorem 2
For system (1), let \(\lambda_{c}\) be the epidemic threshold defined as (18). If \(\lambda< \lambda_{c}\), then the diseasefree equilibrium is globally asymptotically stable. Otherwise, there exists a unique endemic equilibrium.
Proof
In a word, there exists a unique positive eigenvalue of J if and only if \(\lambda> \lambda_{c}\), below which the unique epidemic equilibrium exists. Otherwise all realvalued eigenvalues of J are negative, this implies that the diseasefree equilibrium is globally stable according to Lemma 1 in paper [14]. □
In paper [20], the authors follow the concepts of nextgeneration matrix (NGM) to give a threshold  the basic reproduction number \(R_{0}\), by which the global stability of a modified SIS model is studied. The NGM is a matrix that relates the numbers of newly infected individuals in various categories in consecutive generation, and the basic reproduction number \(R_{0}\) is the spectral radius of the NGM (refer to the papers [25, 26] for details). However, the direct relationship between the epidemic threshold and the basic reproduction number is not clearly revealed. In fact, we can reveal that \(\lambda= \lambda_{c}\) if and only if \(R_{0} = 1\) by the same way as above.
3.3 Global attraction of endemic equilibrium
In this part, we show a proposition and prove the global attraction of the endemic equilibrium by the same way as the one in [3, 8]. Inequalities (27) and (28) in Proposition 2 are helpful to prove the main result (Theorem 3).
Proposition 1
Proof
Proposition 2
Proof
Denote \(\bigtriangleup_{k} =\{(I_{k},R_{k})0 \leq I_{k}+R_{k}\leq1,I_{k}\geq 0,R_{k}\geq0\}\), \(k=1,2,\ldots,n\), and \(\bigtriangleup=\prod_{k=1}^{n}\bigtriangleup_{k} \times[0,1]\).
Theorem 3
If \(\lambda> \lambda_{c}\) and \(\alpha\geq\beta\), then system (2) has a unique endemic equilibrium \(E^{*}=\{ I_{1}^{*},R_{1}^{*},\ldots,I_{n}^{*},R_{n}^{*},V^{*}\}\) which is of global attraction in \(\bigtriangleup\{F^{*}\}\), where \(F^{*}=\{0,\frac{\alpha}{\alpha+\delta},0,\frac{\alpha}{\alpha+\delta }, \ldots,0,\frac{\alpha}{\alpha+\delta},0\}\) is the diseasefree equilibrium of system (2).
Proof
Moreover, for all \(k=1,2,\ldots,n\), we can testify the convergence of the sequences \(\{U_{k}^{(m)}\}_{m=1}^{+\infty}\) by induction. First, it is obvious that \(U_{k}^{(2)} < U_{k}^{(1)} = 1\). Secondly, if \(U_{k}^{(m+1)} \leq U_{k}^{(m)}\), then the reader can easily verify that \(U_{k}^{(m+2)} \leq U_{k}^{(m+1)}\). It implies that the sequence \(\{U_{k}^{(m)}\}\) is convergent. Denoted by \(U_{k}=\lim_{m\rightarrow\infty}U_{k}^{(m)}\), we then have \(\limsup_{t\rightarrow\infty}I_{k}(t)\leq U_{k}\), \(k=1,2,\ldots,n\).
Let \(\mathcal{H}(\Theta) = \frac{1}{\langle k\rangle}\langle k\mathcal{G}_{k}\rangle=\frac{1}{\langle k \rangle}\langle k \frac{[\lambda\frac{k}{\langle k\rangle}\langle kx_{k}\rangle+\frac{\gamma_{1}\gamma_{2}\langle x_{k}\rangle}{1+\gamma_{2}\langle x_{k}\rangle}][\frac{\delta+(\alpha\beta)x_{k}}{\alpha+\delta}]}{\beta +\lambda \frac{k}{\langle k\rangle}\langle kx_{k}\rangle+\frac{\gamma_{1}\gamma_{2}\langle x_{k}\rangle}{1+\gamma_{2}\langle x_{k}\rangle}}\rangle\), where \(\Theta= \frac{1}{\langle k \rangle}\langle kx_{k}\rangle\). One obtains that \(\mathcal{H}'(\Theta)_{\Theta= 0} > 1\) if \(\lambda >\lambda_{c}\). It implies that \(\frac{1}{\langle k \rangle}\langle k\mathcal{G}_{k}\rangle> \frac{1}{\langle k \rangle}\langle k x_{k}\rangle\) when \(x_{k} >0\) (\(k=1,2,\ldots,n\)) is small enough.
Denote \(L_{k}^{(m+1)}=\mathcal{G}_{k}(L_{1}^{(m)},L_{2}^{(m)},\ldots,L_{n}^{(m)})\) for each \(k = 1,2,\ldots,n\). According to Lemma 1 in paper [14], we can take \(L_{k}^{(1)}\) small enough such that ∀k, \(0 < L_{k}^{(1)} < \liminf_{t\rightarrow\infty}I_{k}(t)\) and \(L^{(2)}_{k} > L^{(1)}_{k}\). If \(L_{k}^{(m)} \geq L_{k}^{(m1)}\), it is easy to testify that \(L_{k}^{(m+1)} \geq L_{k}^{(m)}\). In result, the sequences \(\{L_{k}^{(m)}\}_{m=1}^{+\infty}\) are convergent for all k, and denote \(L_{k} = \lim_{m\rightarrow\infty} L_{k}^{(m)}\).
4 Simulations
In the section above, some theoretical results for the proposed meanfield equations are revealed. For a finite size network, there exists an epidemic threshold \(\lambda_{c}\), if the infection rate \(\lambda< \lambda_{c}\), then the diseasefree equilibrium is globally asymptotically stable. Otherwise, the diseasefree equilibrium is unstable and a unique globally attracting endemic equilibrium exists.
In this section, we will perform some numerical simulations to confirm the theoretical results over BA (BarabásiAlbert) scalefree networks which are generated by the preferential attachment algorithm [27]. All the networks used in the simulations were built using \(N=10^{4}\) nodes.

The epidemic threshold \(\lambda_{c}\) that changes as functions of different model parameters.

The final density of the infected nodes that changes as functions of different model parameters.
In stochastic simulations, the dynamics are totally evolved for \(2\text{,}000\) time steps, we set the time interval \(h=0.1\) and let \(\rho=\frac{1}{T}\sum_{t=t_{0}}^{t_{0}1+T}\rho(t)\) (here, \(T=100\), \(t_{0}=1\text{,}901\)) be the time average to reduce the fluctuation of \(\rho(t)\). At the same time, to minimize random fluctuation caused by the initial conditions, we make average of ρ over 100 realizations of different initial infectious nodes. Let λ increase systematically by Δλ beginning with \(\lambda=0\), if \(\rho > 0.0005\) as \(\lambda= \lambda^{*}\) and \(\rho< 0.0005\) as \(\lambda< \lambda^{*}\), we set \(\lambda_{c}=\lambda^{*}\Delta\lambda\).
5 Conclusions
In order to better explain the mechanism of spreading of epidemics, we have investigated a novel SIRS model with direct immunization via two infection mechanisms in this paper. The model is approximately described by the meanfield method neglecting contact duration.
The diseasefree equilibrium and endemic equilibrium and their dynamics are discussed in this paper. Our theoretic results show that for finite size networks, there exists epidemic threshold \(\lambda_{c}\), below which the diseasefree equilibrium is globally asymptotically stable; otherwise a unique endemic equilibrium exists. We prove theoretically that the endemic equilibrium is of attraction when \(\lambda> \lambda_{c}\) under the assumption that \(\alpha\geq\beta\). To go a step further, we perform some numerical simulations to test and verify our theoretical results.
From simulations as above, we can find that the discrepancies between stochastic simulations and theoretical predictions remain for the effect of a finite size network [28] and stochastic factors. Disregarding these slight errors, we think that the numerical simulations confirm to the theoretical results, and the meanfield approach is of effectiveness. Especially, one can see that the larger transmission rates from infected vectors to susceptible individuals \(\gamma_{1}\) or from infected individuals to susceptible vectors \(\gamma_{2}\), the better the simulations accord with the meanfield predictions. Just as mentioned above, this may attribute to the homogeneity of contacts between individuals and vectors.
Declarations
Acknowledgements
This research was jointly supported by NSFC Grant 11401274 and the Natural Science Foundation of Jiangxi Province (Grants 20142BDH80027 and 20132BAB201012). RZ Yu was also supported by the Foundations of Jiangxi Education Bureau (Grants GJJ12617 and GJJ13714).
Open Access This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.
Authors’ Affiliations
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