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- Open Access
Bifurcations of an SIRS model with generalized non-monotone incidence rate
- Jinhui Li^{1} and
- Zhidong Teng^{2}Email author
https://doi.org/10.1186/s13662-018-1675-y
© The Author(s) 2018
- Received: 30 March 2018
- Accepted: 11 June 2018
- Published: 22 June 2018
Abstract
We consider an SIRS epidemic model with a more generalized non-monotone incidence: \(\chi(I)=\frac{\kappa I^{p}}{1+I^{q}}\) with \(0< p< q\), describing the psychological effect of some serious diseases when the number of infective individuals is getting larger. By analyzing the existence and stability of disease-free and endemic equilibrium, we show that the dynamical behaviors of \(p<1\), \(p=1\) and \(p>1\) distinctly vary. On one hand, the number and stability of disease-free and endemic equilibrium are different. On the other hand, when \(p\leq1\), there do not exist any closed orbits and when \(p>1\), by qualitative and bifurcation analyses, we show that the model undergoes a saddle-node bifurcation, a Hopf bifurcation and a Bogdanov–Takens bifurcation of codimension 2. Besides, for \(p=2\), \(q=3\), we prove that the maximal multiplicity of weak focus is at least 2, which means at least 2 limit cycles can arise from this weak focus. And numerical examples of 1 limit cycle, 2 limit cycles and homoclinic loops are also given.
Keywords
- Epidemic model
- Non-monotone incidence
- Hopf bifurcation
- Bogdanov–Takens bifurcation
1 Introduction
When it comes to modeling of infectious diseases, such as measles, encephalitis, influenza, mumps et al., there are many factors that affect the dynamical behaviors of epidemic models greatly. Recently, many investigations have demonstrated that the incidence rate is a primary factor in generating the abundant dynamical behaviors (such as bistability and periodicity phenomena, which are very important dynamical features) of epidemic models [1–8].
In classical epidemic models [9], the bilinear incidence rate describing the mass-action form i.e. \(\beta SI\), where β is the probability of transmission per contact and S and I are the number of susceptible and infected individuals, respectively, is often used. Epidemic models with such bilinear incidence usually show a relatively simple dynamical behavior, that is to say, these models usually have at most one endemic equilibrium, do not have periodicity and whether the disease will die out or not is often determined by the basic reproduction number being less than zero or not [9, 10]. However, in practical applications, it is necessary to introduce the nonlinear contact rates, though the corresponding dynamics will become much more complex [11].
Actually, there are many reasons to introduce a nonlinear incidence rate into epidemic models. In [12], Yorke and London showed that the model with nonlinear incidence rate \(\beta(1-c I)IS\) with positive c and time-dependent β accorded with the results of the simulations for measles outbreak. Moreover, in order to incorporate the effect of behavioral changes, nonlinear incidence function of the form \(H_{1}(S,I)=\lambda S^{p}I^{q} \) and a more general form \(H_{2}(S,I)=\frac{\lambda S^{p}I^{q}}{1+\nu I^{p-1}} \) were proposed and investigated by Liu, Levin, and Iwasa in [13, 14]. They found that the behaviors of epidemic model with the nonlinear incidence rate \(H_{1}(S,I)\) were determined mainly by p and λ, and secondarily by q. Besides, they also explained how such a nonlinearity arise.
Remark 1.1
Actually, we will study model (1.1) in a different way and get better results compared with [21], such as the existence of equilibria, the order of Lyapunov value and Bogdanov–Takens bifurcation. In [21], the author only got the first order Lyapunov value and we get second, and they added two conditions in Theorem 4.2 to make sure the existence of Bogdanov–Takens bifurcation but we will prove these conditions are unnecessary.
The organization of this paper is as follows. In Sect. 2, we analyze the existence and stability of disease-free and endemic equilibria and show that the behavior of \(p<1\), \(p=1\) and \(p>1\) are distinctly different. When \(p<1\), there always exist an unstable disease-free equilibrium and a globally stable endemic equilibrium. When \(p=1\), there exists a unique endemic globally stable equilibrium under certain conditions. And when \(p>1\), there exist at most two endemic equilibria for some parameter values. Then we prove that the model exhibits a Hopf bifurcation when \(p>1\) and that the maximal multiplicity of the weak focus is at least 2 if we take \(p=2\), \(q=3\). Also, numerical examples of 1 limit cycle, 2 limit cycles, and a homoclinic loop are given. In Sect. 4, we show that the system will possess a Bogdanov–Takens bifurcation of codimension 2 under some conditions. Finally, we will give a brief discussion.
2 Existence and types of equilibria
2.1 Existence of equilibria
Then we will discuss the existence of positive real solution of Eq. (2.3) in three cases.
Case I. \(p<1\).
Case II. \(p=1\).
Case III. \(p>1\).
When \(p>1\), then \(h(0)=-A(p-1)<0\), \(h(+\infty)=+\infty\) and \(h'(y)=\alpha d q (q-p+1)y^{q-p}+p(1+\gamma)>0\). Thus function \(h(y)\) always has a positive real solution \(y_{m}\), which is the maximal value point of \(f(y)\) for \(y>0\) (see Fig. 2(c)).
Summarizing discussions above, the following theorem can be obtained.
Theorem 2.1
- (a)
When \(0< p<1\), then system (2.2) has a unique endemic equilibrium \(\hat{E}(\hat{x},\hat{y})\).
- (b)
- (c)When \(p>1\), we have:
- (1)
if \(d< d_{m}\), then system (2.2) has two endemic equilibria \(E_{1}=(x_{1},y_{1})\) and \(E_{2}=(x_{2},y_{2})\), with \(y_{1}< y_{2}\) and \(x_{i}=\frac {1+\alpha y_{i}^{q}}{y_{i}^{p-1}}\) (\(i=1,2\));
- (2)
if \(d=d_{m}\), then system (2.2) has a unique endemic equilibrium \(E_{*}(x_{*},y_{*})\), where \(x_{*}=\frac{1+\alpha y_{*}^{q}}{y_{*}^{p-1}}\);
- (3)
if \(d>d_{m}\), then system (2.2) has no endemic equilibrium.
- (1)
2.2 Stability of equilibria
Firstly, we can obtain the nonexistence of periodic orbits in system (2.2) when \(p\leq1\).
Theorem 2.3
For \(p\leq1\), system (2.2) does not have endemic periodic orbits.
Proof
In the following, we will also discuss the stability of equilibria in three cases.
Case I. \(p<1\).
Recall that the ω-limit set of a bounded planar flow can consist only (i) equilibria, (ii) periodic orbits, (iii) orbits connecting equilibria (heteroclinic or homoclinic orbits) (see [22]). Because there are neither limit cycles nor heterclinic or homoclinic orbits, the local asymptotic stability of the endemic equilibrium guarantees the global stability. Thus, we can obtain the global stability of Ê.
Case II. \(p=1\).
When \(p=1\), the eigenvalues of \(E_{0}\) of the Jacobian matrix are \(\frac {A}{d}-1\) and −d. Besides, when \(p=1\) and \(d< A\), then \(f'(\bar {y})<0\) and \(\rho(\bar{y})>0\), thus, Ē is an attracting node. Similarly, the locally asymptotically stable of the endemic equilibrium guarantees the global stability. Thus, we get the following theorem.
Theorem 2.4
- (I)Assume \(p<1\). Then we have:
- (a)
the disease-free equilibrium \(E_{0}\) of system (2.2) is unstable;
- (b)
the unique endemic equilibrium Ê is globally asymptotically stable.
- (a)
- (II)Assume \(p=1\), we have:
- (a)
the disease-free equilibrium \(E_{0}\) of system (2.2) is a stable hyperbolic focus if \(d>A\); a hyperbolic saddle if \(d< A\); a saddle-node if \(d=A\);
- (b)
if \(d\geq A\), then the disease-free equilibrium \(E_{0}\) is globally asymptotically stable;
- (c)
if \(d< A\), then the unique endemic equilibrium Ē is globally asymptotically stable.
- (a)
Remark 2.5
Actually, when \(p=1\), we can define the reproduction number \(R_{0}=\frac {A}{d}\). According to Theorem 2.4, we see that when \(R_{0}\leq 1\), then there is no endemic equilibrium and the disease-free equilibrium is globally stable and that when \(R_{0}>1\), then there is a unique endemic equilibrium which is globally stable. Particularly, when \(p=1\), \(q=2\), which has been studied in [18]. They defined the basic reproduction number \(R_{0}=\frac{\kappa A}{d(d+\mu)}\) for model (1.1) (\(p=1\), \(q=2\)) and got the same results.
Case III. \(p>1\).
When \(p>1\) and \(d< d_{m}\), it can be seen from Fig. 2(b) that \(f'(y_{1})>0\), \(f'(y_{2})<0\), so \(E_{1}\) is a hyperbolic saddle and \(E_{2}\) is an anti-saddle. Besides, \(E_{2}\) is an attracting node or focus if \(\rho(y_{2})>0\); \(E_{2}\) is a repelling node or focus if \(\rho (y_{2})<0\); \(E_{2}\) is a weak focus or center if \(\operatorname{tr} J(E_{2})=0\).
Obviously, when \(p>1\), then the eigenvalues of \(E_{0}\) are −1 and −d. Thus, equilibrium \(E_{0}\) is always locally asymptotically stable for all parameters allowable when \(p>1\).
Theorem 2.6
When \(p>1\), \(E_{0}\) is always locally asymptotically stable.
Theorem 2.7
- (1)
equilibrium \(E_{2}\) is attracting if \(\bar{d}< d< d_{m}\);
- (2)
equilibrium \(E_{2}\) is repelling if \(d<\min\{d_{m},\bar{d}\}\);
- (3)
equilibrium \(E_{2}\) is a weak focus or center if \(d=\bar{d}< d_{m}\).
According to Theorem 2.7, the following corollary is obtained.
Corollary 2.8
When \(d>p-1\), then \(E_{2}\) is always an attracting node.
When \(d=d_{m}\), the equilibria \(E_{1}\) and \(E_{2}\) coalesce at \(E_{*}\), which is degenerate because the Jacobian matrix of the linearized system of (2.2) at \(E_{*}\) has determinant 0. Then we get the following result.
Theorem 2.9
When \(p>1\) and \(d=d_{m}\), \(E_{*}\) is a saddle-node if \(d\neq d^{*}\).
Proof
When \(d=d_{m}\), system (2.2) has only one endemic equilibrium \(E_{*}\). If \(d\neq d^{*}\), we have \(\operatorname{tr} J(E_{*})\neq0\).
Number and stability of endemic equilibria
p | Condition | Number | Stability |
---|---|---|---|
p<1 | d>0 | 1 | globally stable |
p = 1 | d<A | 1 | globally stable |
d ≥ A | 0 | ||
p>1 | \(d< d_{m}\) | 2 | a saddle and an anti-saddle |
\(d=d_{m}\) | 1 | degenerate sigularity | |
\(d>d_{m}\) | 0 |
Remark 2.10
Summarizing the three cases discussed above, one can easily observe that the dynamical behaviors of system (2.2) are completely different for these three cases.
3 Hopf bifurcation
Theorem 3.1
- (1)
if \(\gamma\neq\gamma^{*}\) or \((y_{2},\alpha)\notin\Omega\), then \(E_{2}\) is a multiple focus of multiplicity 1;
- (2)
if \(\gamma=\gamma^{*}\) for \((y_{2},\alpha)\in\Omega\), then \(E_{2}\) is a multiple focus of multiplicity at least 2.
Proof
Remark 3.2
As shown above, the Lyapunov value of order 2 is very small, thus, there may exist other parameter values that make \(L_{5}\) equal to zero, which means the equilibrium \(E_{2}\) is a multiple focus of multiplicity at least 3.
Remark 3.3
4 Bogdanov–Takens bifurcation
The purpose of this section is to study the Bogdanov–Takens bifurcation of system (2.2), when there is a unique degenerate endemic equilibrium. Since when \(p<1\), the equilibrium Ê of system (2.2) is globally stable for any allowable parameter values, when \(p=1\) and \(d< A\), the equilibrium Ē of system (2.2) is globally stable for any allowable parameter values and when \(p>1\), we get system (2.2) has a unique endemic equilibrium \(E_{*}(x_{*},y_{*})\) of multiplicity 2 if \(d=d_{m}\), according to Theorem 2.1. Thus, for system (2.2), when \(p>1\) and \(d=d_{m}\), there may exist a Bogdanov–Takens singularity.
Lemma 4.1 is from Perko [25], it will be used in the proof of Theorem 4.2.
Lemma 4.1
Theorem 4.2
Assume \(p>1\). Suppose that \(d=d_{m}=d^{*}\), then the only interior equilibrium \(E_{*}\) of system (2.2) is a cusp of codimension 2.
Proof
- (i)The saddle-node bifurcation curve:$$\mathit{SN}=\bigl\{ (\varepsilon_{1},\varepsilon_{2})| \mu_{1}(\varepsilon_{1},\varepsilon _{2})=0\bigr\} . $$
- (ii)The Hopf bifurcation curve:$$H=\bigl\{ (\varepsilon_{1},\varepsilon_{2})| \mu_{1}(\varepsilon_{1},\varepsilon _{2})=- \mu_{2}^{2}(\varepsilon_{1},\varepsilon_{2}), \mu_{2}>0\bigr\} . $$
- (iii)The homoclinic bifurcation curve:$$H=\biggl\{ (\varepsilon_{1},\varepsilon_{2})\Big| \mu_{1}(\varepsilon_{1},\varepsilon _{2})=- \frac{49}{25}\mu_{2}^{2}(\varepsilon_{1}, \varepsilon_{2})+O\bigl(\mu _{2}^{\frac{5}{2}}\bigr), \mu_{2}>0\biggr\} . $$
On the basis of the bifurcation curves, the dynamics of system (4.3) in a small neighborhood of \(E_{*}\) as parameters \((A,d)\) vary in a small neighborhood of \((A_{0},d_{0})\) can be concluded as the following theorem.
Theorem 4.3
- (i)
system (4.3) has a unique positive equilibrium if \((\varepsilon_{1},\varepsilon_{2})\) are on the SN curve;
- (ii)
system (4.3) has two positive equilibria (a saddle and a weak focus) if parameters \((\varepsilon_{1},\varepsilon_{2})\) are on the H curve;
- (iii)
system (4.3) has two positive equilibria (a saddle and a hyperbolic focus) and a homoclinic loop if the parameters \((\varepsilon_{1},\varepsilon_{2})\) are on the HL curve;
- (iv)
system (4.3) has two positive equilibria (a saddle and a hyperbolic focus) and a limit cycle if parameters \((\varepsilon _{1},\varepsilon_{2})\) are in the region between the H curve and the HL curve.
Remark 4.4
The existence of a Hopf bifurcation and a Bogdanov–Takens bifurcation when \(p>1\) further shows that the dynamical behaviors tend to be more complex with the increasing of p.
5 Conclusions
In this paper, we study an SIRS epidemic model with a more generalized non-monotone incidence rate \(\kappa SI^{p}/(1+\alpha I^{q})\) with \(0< p< q\), which describes the psychological effect when there are a large number of infective individuals. We prove that the behavior of the model can be classified into three various cases: \(p<1\), \(p=1\) and \(p=1\). When \(p<1\), there is a unique globally asymptotically stable endemic equilibrium and the disease-free equilibrium is unstable; when \(p=1\), there is a unique globally asymptotically stable endemic equilibrium provided by \(d< A\) and no endemic equilibrium when \(d\geq A\); and when \(p>1\), there exist two endemic equilibria if \(d< d_{m}\), a unique equilibrium if \(d=d_{m}\) and no endemic equilibrium if \(d>d_{m}\). By qualitative and bifurcation analysis, we prove that a saddle-node bifurcation, a Hopf bifurcation, and a Bogdanov–Takens bifurcation can happen for the system when \(p>1\). Moreover, for \(p=2\), \(q=3\), we calculate the first and second order Lyapunov values and prove that the maximal multiplicity of weak focus \(E_{2}\) is at least 2, which implies that at least 2 limit cycles can appear from the weak focus with suitable parameters. And we present numerical examples about 1 limit cycle, 2 limit cycles and a homoclinic loop for \(p=2\), \(q=3\).
In fact, we show that the model exhibits multi-stable states. This interesting phenomenon indicates that the beginning states of an epidemic can determine the final states of an epidemic to go extinct or not. Moreover, the periodical oscillation signifies that the trend of the disease may be affected by the behavior of susceptible population.
Declarations
Acknowledgements
The authors are grateful to both reviewers for their helpful suggestions and comments.
Funding
This work was supported by the National Natural Science Foundation of China [11771373, 11001235].
Authors’ contributions
The authors have achieved equal contributions. All authors read and approved the manuscript.
Competing interests
The authors declare that they have no competing interests.
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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