A delayed SIR model with general nonlinear incidence rate
- Luju Liu^{1}Email author
Received: 2 February 2015
Accepted: 25 August 2015
Published: 22 October 2015
Abstract
An SIR epidemic model is investigated and analyzed based on incorporating an incubation time delay and a general nonlinear incidence rate, where the growth of susceptible individuals is governed by the logistic equation. The threshold parameter \(\sigma_{0}\) is defined to determine whether the disease dies out in the population. The model always has the trivial equilibrium and the disease-free equilibrium whereas it admits the endemic equilibrium if \(\sigma_{0}\) exceeds one. The disease-free equilibrium is globally asymptotically stable if \(\sigma_{0}\) is less than one, while it is unstable if \(\sigma_{0}\) is greater than one. By applying the time delay as a bifurcation parameter, the local stability of the endemic equilibrium is studied and the condition which is absolutely stable or conditionally stable is established. Furthermore, a Hopf bifurcation occurs under certain conditions. Numerical simulations are carried out to illustrate the main results.
Keywords
1 Introduction
\(S(t)\), \(I(t)\), and \(R(t)\) are the numbers of susceptible, infective and recovered host individuals at time t, respectively. r denotes the intrinsic birth rate. β denotes the average number of contacts per infective per unit time. τ is the incubation time. \(\mu_{1}\) and \(\mu_{2}\) represent the death rate of infective and recovered, respectively. γ is the recovered rate of infective individuals. It is reasonable to assume that all the parameters are positive constants. Wang et al. [13] presented the dynamic properties of system (1). The global stability of the disease-free equilibrium is derived when the basic reproduction number \(R_{0}\) is less than unity. The unique endemic equilibrium is absolutely stable when \(1< R_{0}<3\), and it is conditionally stable when \(R_{0}>3\). Moreover, the existence of a Hopf bifurcation is given.
Although the bilinear incidence rate was frequently used in the literature of mathematical modeling, there are plenty of reasons why this bilinear incidence rate may require modification [5, 22]. For example, the saturated incidence rate of the form \(\frac{{\beta}S(t)I(t)}{1+{\alpha}I(t)}\) or \(\frac{{\beta}S(t)I(t)}{1+{\alpha}S(t)}\) was formulated as crowding effects of infective or behavioral changes of susceptible individuals were considered [2, 10, 18, 19, 23–25]. Moreover, other forms of nonlinear incidence rates are often developed in many papers (for details one can refer to [4, 5, 14–17, 22]). Motivated by those works, in the present paper, we attempt to extend system (1) or (2) to a more general incidence rate of the form \({\beta}F(S(t))I(t-\tau)\). It is assumed that function F is continuous on \([0,\infty)\) and continuously differentiable on \((0,\infty)\), which satisfies the following hypothesis. Furthermore, it is assumed \(F(S)\) is strictly monotonically increasing on \([0,+\infty)\) with \(F(0)=0\).
It should be noted that the general nonlinear incidence rate in system (3) includes some special cases. If \(F(S)=S\), then it becomes the classical bilinear incidence rate, which has been investigated by Wang et al. [13]. If \(F(S)=S^{q}\) (\(q>0\)), then the incidence rate is used in [20]. If \(F(S)=\frac{S}{1+{\alpha}S}\), it becomes the saturated one, which has been discussed in [18, 19].
The rest of the paper is structured as follows. In Section 2, the nonnegativity and boundedness of the solutions are discussed. In Section 3, the stabilities of the trivial equilibrium and the disease-free equilibrium are described. Section 4 deals with the existence and stability of the endemic equilibrium and the existence of a Hopf bifurcation. In Section 5, the numerical simulations are performed, followed by a brief conclusion in Section 6.
2 Nonnegativity and boundedness of solutions
Theorem 2.1
System (4) has a nonnegative and bounded solution with the initial value \((\phi_{1}(\theta),\phi_{2}(\theta),\phi_{3}(\theta))\in C([-\tau ,0],\mathbb{R}^{3}_{+})\) and \(\phi_{i}(\theta)\geq0\), \(\phi_{i}(0)>0\), \(i=1,2,3\).
Proof
First we show that \(S(t)\) is nonnegative for all \(t\geq0\). On the contrary, it is assumed that there exists \(t_{1}>0\) such that \(S(t_{1})=0\) and \(S'(t_{1})<0\). Then the first equation of system (4) implies \(S'(t_{1})=0\), which is a contradiction. Therefore, it follows that \(S(t) \geq 0\) for all \(t \geq 0\).
3 Stabilities of the trivial equilibrium and the disease-free equilibrium
Before the main results are established, the following lemma will be given first.
Lemma 3.1
(see [27])
- (i)
if \(a< b\), then \(\lim_{t\rightarrow\infty}u(t)=0\);
- (ii)
if \(a>b\), then \(\lim_{t\rightarrow\infty}u(t)=+\infty\).
Theorem 3.1
The trivial equilibrium \(E_{0}\) of system (4) is always unstable.
Proof
Theorem 3.2
If \(\sigma_{0}<1\), the disease-free equilibrium \(E_{1}\) for system (4) is globally asymptotically stable; and if \(\sigma_{0}>1\), the disease-free equilibrium \(E_{1}\) for system (4) is unstable.
Proof
If \(\sigma_{0}>1\), then \(G(0)=(\mu_{1}+\gamma)(1-\sigma_{0})<0\). When \(\lambda\rightarrow{+\infty}\), \(G(\lambda)\rightarrow{+\infty}\). Then \(G(\lambda)=0\) has at least one positive root. Therefore \(E_{1}\) is unstable. □
4 The stability of endemic equilibrium and Hopf bifurcation
In this section, we pay attention to the stability of the endemic equilibrium and Hopf bifurcation when \(\sigma_{0}>1\).
Theorem 4.1
Proof
At the endemic equilibrium \(E_{*}\), it follows from the second equation of system (4) that \(F(S^{*})=\mu_{1}+\gamma\). Let \(H(S)=F(S)-(\mu_{1}+\gamma)\). It is obvious that \(H(0)=F(0)-(\mu_{1}+\gamma)=-(\mu_{1}+\gamma)<0\). For all \(S \geq 1\), \(H(S) \geq H(1)=F(1)-(\mu_{1}+\gamma)=(\mu_{1}+\gamma)(\sigma_{0}-1)>0\) because \(H(S)\) is monotonically increasing on the interval \([0,+\infty)\) and \(\sigma_{0}>1\). Therefore \(H(S)=0\) has exactly one root \(S^{*}\in(0,1)\). It is not difficult to compute the expressions \(I^{*}\) and \(R^{*}\) from system (4) at the endemic equilibrium \(E_{*}\). □
Proposition 4.1
Assume \(\sigma_{0}>1\) and \(I^{*}F'(S^{*})>r(1-2S^{*})\), then all the roots of (15) have a negative real part for \(\tau=0\).
Proof
Proposition 4.2
- (i)
If \(I^{*}F'(S^{*})\geq2r(1-2S^{*})\), then all the roots of (19) have a negative real part for \(\tau>0\).
- (ii)
If \(I^{*}F'(S^{*})<2r(1-2S^{*})\), then there exists a monotone increasing sequence \(\{\tau_{n}\}^{\infty}_{n=0}\) with \(\tau_{0}>0\) such that (15) has a pair of imaginary roots for \(\tau=\tau_{n}\) (\(n=0,1,2,\ldots\)).
Proof
Firstly assume that \(I^{*}F'(S^{*})\geq2r(1-2S^{*})\). Then we arrive at \(a^{2}-2b-c^{2}>0\) and \(b+d\geq0\). That is to say, (19) has no positive real root ω, which is a contradiction. Therefore, all the roots of (15) have negative real part for \(\tau>0\). The first part of the proof is completed.
We give the following proposition without any proof, since the proof is similar to that of [6].
Proposition 4.3
Summarizing the above propositions, we obtain the following theorem.
Theorem 4.2
- (i)
If \(I^{*}F'(S^{*})\geq2r(1-2S^{*})\), then the endemic equilibrium of system (4) is locally asymptotically stable for \(\tau\geq0\).
- (ii)
If \(I^{*}F'(S^{*})<2r(1-2S^{*})\), then the endemic equilibrium of system (4) is locally asymptotically stable for \(0\leq\tau<\tau_{0}\) and it is unstable for \(\tau>\tau_{0}\).
Remark 4.1
If both \(\sigma_{0}>1\) and \(I^{*}F'(S^{*})<2r(1-2S^{*})\) hold true, system (4) undergoes a Hopf bifurcation at the endemic equilibrium \(E_{*}\) when τ crosses \(\tau_{n}\) (\(n=0,1,\ldots\)).
5 Numerical results
6 Conclusion
In this paper, a delayed SIR vector disease model with incubation time delay is established, in which the growth of susceptible individuals follows the logistic function in the absence of disease and the more general form of the nonlinear incidence rate is considered. The stability of the equilibria has been discussed by analyzing the roots of characteristic equations and applying the theory of asymptotic autonomous systems. It is shown that the trivial equilibrium is always unstable. The stability of the disease-free equilibrium is completely determined by the threshold parameter \(\sigma_{0}\): the disease-free equilibrium is globally asymptotically stable if \(\sigma_{0}<1\) while it is unstable if \(\sigma_{0}>1\). Moreover, if \(\sigma_{0}>1\), there exists a unique endemic equilibrium. It is found that \(I^{*}F'(S^{*})=2r(1-2S^{*})\) is the condition which determines the absolute stability or conditional stability of the endemic equilibrium. To be specific, the endemic equilibrium is absolutely stable if \(I^{*}F'(S^{*})\geq2r(1-2S^{*})\) holds true, while it is conditionally stable if \(I^{*}F'(S^{*})<2r(1-2S^{*})\) is satisfied. Furthermore, there is a certain threshold time value \(\tau_{0}\) such that the endemic equilibrium is locally asymptotically stable when \(0<\tau<\tau_{0}\), whereas it is unstable when \(\tau>\tau_{0}\). It is worth noting that, if \(\sigma_{0}>1\) and \(I^{*}F'(S^{*})<2r(1-2S^{*})\), the system exhibits a Hopf bifurcation when the time delay τ crosses \(\tau_{n}\) (\(n=0,1,\ldots\)).
References [4, 5, 8, 10, 11] have discussed the delayed SIR vector disease models with nonlinear incidence functions. But the growth of the number of susceptible individuals is governed by a constant rate rather than the logistic function. They have proved that the endemic equilibrium is globally asymptotically stable for any delay and the model does not exhibit a Hopf bifurcation, which implies that the incubation delay does not cause any periodic oscillations. On the other hand, [6, 13, 19] have also investigated the delayed SIR vector disease models with the logistic growth of susceptible individuals. They have found that the endemic equilibrium is unstable and a Hopf bifurcation occurs under some conditions for some delays. For example, Wang et al. [13] investigated system (1) with the incidence function \(F(S)=S\). They have proved if \(R_{0}>3\), the endemic equilibrium is stable when the delay \(\tau<\tau_{0}\) is satisfied, while the endemic equilibrium is unstable and the model undergoes Hopf bifurcation when \(\tau=\tau_{n}\), \(n=0, 1, 2, \ldots \) . Therefore, the logistic growth of susceptible individuals should be more responsible for the instability of the endemic equilibrium, and Hopf bifurcation may be the result of the logistic growth of susceptible individuals.
Wang et al. [13] analyzed system (1) for the incidence function \(F(S)=S\). Zhang et al. [19] also formulated system (2) for the incidence function \(F(S)=\frac{S}{1+{\alpha}S}\). As a matter of fact, two systems in the above-mentioned papers could be studied as special cases for system (3). It should be pointed out here that the threshold parameter \(\sigma_{0}\) defined in the present paper is the same as \(R_{0}\) derived in [13] and is equivalent to \(R_{0}\) given in [19]. Furthermore, our results for the stability of equilibria extend the results in [13] and [19]. The numerical simulations performed further illustrate the theoretical results.
Declarations
Acknowledgements
This work is supported in part by the National Nature Science Foundation of China (NSFC11101127 and 11301543), the Scientific Research Foundation for Doctoral Scholars of Haust (09001535), and the Educational Commission of Henan Province of China (14B110021).
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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