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Complex dynamics in an SIS epidemic model with nonlinear incidence
- Ruixia Yuan^{1},
- Zhidong Teng^{2} and
- Jinhui Li^{1}Email authorView ORCID ID profile
https://doi.org/10.1186/s13662-019-1974-y
© The Author(s) 2019
- Received: 28 July 2018
- Accepted: 15 January 2019
- Published: 29 January 2019
Abstract
We study an epidemic model with nonlinear incidence rate, describing the saturated mass action and the psychological effect of certain serious diseases on the community. Firstly, the existence and local stability of disease-free and endemic equilibria are investigated. Then we prove the occurrence of backward bifurcations, saddle-node bifurcations, Hopf bifurcations and cusp type Bogdanov–Takens bifurcations of codimension 3. Finally, numerical simulations, including one limit cycle, two limit cycles, an unstable homoclinic loop and many other phase portraits are presented. These results show that the psychological effect of diseases and the behavior change of the susceptible individuals may affect the final spread level of an epidemic.
Keywords
- Epidemic
- Nonlinear incidence rate
- Saturated mass action
- Psychological effect
- Bifurcation
1 Introduction
In the well-known SIS epidemic model, the population is always separated into two compartments, susceptible and infective individuals. In most SIS epidemic models (see Anderson and May [1]), the incidence takes the mass-action form with bilinear interactions. However, in a practical application, to describe the transmission process more realistically, it is necessary to introduce the nonlinear contact rates [2].
The organization of this paper is as follows. In Sect. 2, we analyze the existence of the equilibria and local stability of the equilibria. In Sect. 3, we study the existence of Hopf bifurcation around the positive equilibrium at the critical value under the conditions of \(R_{0}<1\) and \(R_{0}>1\). We also show that these positive equilibria can be weak focus for some parameter values and a cusp type of Bogdanov–Takens bifurcation of codimension 3. In Sect. 4, we give some brief discussions.
2 Types and stability of the equilibria
Lemma 2.1
The set \(D=\{(x,y)|x\geq0, y\geq0, x+y\leq\frac{\varLambda}{d}\}\) is an invariant manifold of system (2.1), which is attracting in the first octant of \(\mathbb{R}^{2}\).
Proof
Theorem 2.2
- (i)When \(R_{0}<1\), we have
- (a)
if \(R_{0}< R^{*}\), then system (2.1) has no positive equilibrium;
- (b)
if \(R_{0}=R^{*}\) and \(\varLambda>(1-\sigma)/a\), then system (2.1) has a unique positive equilibrium \(E_{1}(x_{1},y_{1})\), where \(x_{1}=\frac{y_{1}^{2}+c}{ay_{1}+1}\) and \(y_{1}=\frac{a\varLambda-(1-\sigma )}{d+a(1-\sigma)}\);
- (c)
if \(R_{0}>R^{*}\) and \(\varLambda>(1-\sigma)/a\), then system (2.1) has two positive equilibria \(E_{2}(x_{2},y_{2})\) and \(E_{3}(x_{3},y_{3})\), where \(x_{k}=\frac{y_{k}^{2}+c}{ay_{k}+1}\) (\(k=2,3\)) and \(y_{2}=\frac{a\varLambda-(1-\sigma)-\sqrt{\Delta}}{2(d+a(1-\sigma))}\), \(y_{3}=\frac{a\varLambda-(1-\sigma)+\sqrt{\Delta}}{2(d+a(1-\sigma))}\);
- (a)
- (ii)
When \(R_{0}=1\) and \(\varLambda>(1-\sigma)/a\), then system (2.1) has a unique positive equilibrium \(E_{4}(x_{4},y_{4})\), where \(x_{4}=\frac{y_{4}^{2}+c}{ay_{4}+1}\) and \(y_{4}=\frac{a\varLambda-(1-\sigma)}{d+a(1-\sigma)}\);
- (iii)
When \(R_{0}>1\), then system (2.1) has a unique positive equilibrium \(E_{5}(x_{5},y_{5})\), where \(x_{5}=\frac{y_{5}^{2}+c}{ay_{5}+1}\) and \(y_{5}=\frac{a\varLambda-(1-\sigma)+\sqrt{\Delta}}{2(d+a(1-\sigma))}\).
Theorem 2.3
- (i)
an attracting node if \(R_{0}<1\);
- (ii)
a hyperbolic saddle if \(R_{0}>1\);
- (iii)
a saddle-node of codimension 1 if \(R_{0}=1\) and \(2a-\frac{1-\sigma }{dc}\neq0\); a repelling node if \(R_{0}=1\) and \(2a-\frac{1-\sigma}{dc}=0\).
Proof
Obviously, at equilibrium \(E_{0}\), we have \(\det(J_{0})=d(1-R_{0})\) and \(\operatorname{Tr} (J_{0})=-d-(1-R_{0})\). Therefore, \(E_{0}\) is a stable node if \(R_{0}<1\) and a hyperbolic saddle if \(R_{0}>1\), and degenerate if \(R_{0}=1\).
From the expression of \(\psi_{1}\) and \(\phi_{1}\), we can see that one of the eigenvalues of the characteristic matrix of \(E_{1}\) is zero and the other is nonzero if \(\psi_{1}\neq0\). The type of \(E_{1}\) can be directed proved by checking the conditions in Zhang et al. ([14], Theorems 7.1–7.3). So, we have the following results.
Theorem 2.4
- (a)
if \(\psi_{1}\neq0\), then \(E_{1}\) is a saddle-node;
- (b)
if \(\psi_{1}=0\), then \(E_{1}\) is a cusp.
Theorem 2.5
- (i)
\(E_{3}\) is a stable focus or node if \(\psi_{3}<0\);
- (ii)
\(E_{3}\) is a weak focus or center if \(\psi_{3}=0\);
- (iii)
\(E_{3}\) is an unstable focus or node if \(\psi_{3}>0\).
Proof
Note that \(\phi_{2}\) is less than zero since \(\Delta<(a\varLambda+1-\sigma)^{2}\), then \(E_{2}\) is a hyperbolic saddle for any choices of the parameters. And at \(E_{3}\), we have \(\phi_{3}>0\). Thus, the stability of the equilibrium \(E_{3}\) depends on the sign of \(\psi_{3}\). □
Theorem 2.6
When \(R_{0}=1\) and \(c>\frac{1-\sigma}{a}\), then system (2.1) has a unique positive equilibrium \(E_{4}(x_{4},y_{4})\), and the equilibrium \(E_{4}\) is stable if \(\psi_{4}<0\).
Proof
In fact, when \(R_{0}=1\) and \(c>\frac{1-\sigma}{a}\), then \(\phi_{4}>0\). Thus, the stability of \(E_{4}\) is determined by the sign of \(\psi_{4}\). □
Theorem 2.7
- (i)
\(E_{5}\) is stable if \(\psi_{5}<0\);
- (ii)
\(E_{5}\) is a weak focus or center if \(\psi_{5}=0\);
- (iii)
\(E_{5}\) is unstable if \(\psi_{5}>0\).
Proof
Obviously, when \(R_{0}>1\), then \(\phi_{5}>0\), and then \(E_{5}\) is stable if \(\psi_{5}<0\). □
Lemma 2.8
From the expression of \(\psi_{k}\) (\(k=1,3,5\)), we can see that \(E_{k}\) (\(k=1,3,5\)) is always stable if \(d\geq1-\frac{1}{ac}\).
Remark 2.9
In fact, Fig. 2(a) shows the occurrence of bi-stability, in which solution may converge to one of the two equilibria, depending on the initial conditions. And in practical cases, this interesting phenomenon implies that initial states determine whether the disease dies out or not.
Remark 2.10
From Fig. 2(a), we can see that there exist two separatrices. All solutions tend to the disease-free equilibrium \(E_{0}\) except the two green lines tend to equilibrium \(E_{2}\).
Theorem 2.11
Suppose that \(R_{0}=1\) and \(\varLambda>\frac{1-\sigma}{a}\), then system (2.1) has a unique positive equilibrium \(E_{4}\). If \(\psi_{4}>0\), then there exists at least one stable limit cycle in the interior of the first quadrant.
Proof
Similarly, when \(\psi_{5}>0\), we have the following result.
Theorem 2.12
Suppose that \(R_{0}>1\). If \(\psi_{5}>0\), then there exists at least one stable limit cycle in the interior of the first quadrant.
Proof
Indeed, the Jacobian \(J_{5}\) has eigenvalues \(\lambda_{1}=-d\) and \(\lambda _{2}=R_{0}-1>0\), when \(R_{0}>1\). Thus, we find that there exists a \(E_{5}\), which is the unique repelling equilibrium in the region \(D_{2}\) shown in (b) of Fig. 5. Consequently, by the Poincaré–Bendixson theorem, at least one stable limit cycle appears in the interior of the first quadrant. □
3 Bifurcations
3.1 Backward bifurcation
Theorem 3.1
When \(R_{0}=1\) and \(\varLambda>\frac{1-\sigma}{a}\), model (2.1) exhibits a backward bifurcation at equilibrium \(E_{0}\).
3.2 Hopf bifurcation
In this subsection, we will study the Hopf bifurcation of system (2.1) for (i) \(R^{*}< R_{0}<1\) and \(\varLambda>(1-\sigma)/a\); (ii) \(R_{0}>1\). From the discussion in Sect. 2, it can be seen that Hopf bifurcation may occur at \(E_{3}\), \(E_{5}\). The expressions of the equilibria \(E_{3}\) and \(E_{5}\) are the same, not considering the values of every parameters. Based on Theorem 2.5, Theorem 2.6 and Theorem 2.7, we know that the stability of \(E_{3}\) and that of \(E_{5}\) are similar and when \(\psi_{k}=0\), \(E_{k}\) (\(k=3,5\)) is a weak focus or center. Thus, we show the existence of a Hopf bifurcation around \(E_{k}\) (\(k=3,5\)).
Theorem 3.4
- (a)
if \(\eta<0\), there is a family of stable periodic orbits of model (2.1) as \(\psi_{k}\) decreases from 0;
- (b)
if \(\eta=0\), there are at least two limit cycles in (2.1), where η will be defined below;
- (c)
if \(\eta>0\), there is a family of unstable periodic orbits of (2.1) as \(\psi_{k}\) increases from 0.
Proof
Remark 3.5
What we need to note here is that the expression \(b_{12}\) in Theorem 3.4 is nonzero. Otherwise, we have \(\det J_{k}=b_{11}b_{22}<0\), since \(b_{11}+b_{22}=0\), which is a contradiction.
Next, we present examples to show that equilibrium \(E_{k}\) can be a stable weak focus of multiplicity two, and under a small perturbation, system (2.1) undergoes a degenerate Hopf bifurcation and produces two limit cycles.
Remark 3.6
As a matter of fact, the reproduction number is equal to zero in [13, 18], which simplifies the condition that a Hopf bifurcation occur. In our model, we also comprehensively discuss the existence of a Hopf bifurcation when \(R_{0}<1\), \(R_{0}=1\) and \(R_{0}>1\). Besides, the authors in [13] did not show the appearance of a homoclinic loop, which is an interesting bifurcation phenomenon given in Fig. 7.
3.3 Bogdanov–Takens bifurcation
In this subsection, we investigate the Bogdanov–Takens bifurcation in system (2.1). Lemma 3.7 is from Perko [19], and Lemma 3.8 is Proposition 5.3 in Lamontage et al. [20].
Lemma 3.7
Lemma 3.8
Theorem 3.9
- (a)
if \(f(y_{1})g(y_{1}) \neq0\), then \(E_{1}\) is a cusp of codimension 2;
- (b)
if \(f(y_{1})g(y_{1}) =0\), then \(E_{1}\) is a cusp of codimension greater than or equal to 3.
Proof
Therefore, \(E_{1}\) is a cusp of codimension 2 if \(f(y_{1})g(y_{1}) \neq0\), by the results in Perko [19], or else, \(E_{1}\) is a cusp of codimension at least 3. □
Remark 3.10
- (1)
If \(f(y_{1})\neq0\) and \(g(y_{1})=0\), \(E_{1}\) is a cusp point;
- (2)
If \(f(y_{1})=0\) and \(g(y_{1})\neq0\), \(E_{1}\) is nilpotent focus/elliptic point;
- (3)
If \(f(y_{1})=g(y_{1})=0\), \(E_{1}\) is a nilpotent focus.
Unfortunately, due to the complexity of \(f(y_{1})\) and \(g(y_{1})\), we cannot determine which of these three situations occurs theoretically. But we will show for some parameter values that \(f(y_{1})\neq0\) and \(g(y_{1})=0\), i.e. \(E_{1}\) is a cusp point.
In the following, we will give an example to show that Theorem 3.9(b) occurs.
Theorem 3.11
Proof
Remark 3.12
The authors in [13, 18] proved their epidemic model with saturated incidence rate undergoes a Bogdanov–Takens bifurcation of codimension 2. When we consider the incidence of a combination of the saturated incidence rate and a non-monotonic incidence, the codimension of Bogdanov–Takens bifurcation can grow up to 3.
Remark 3.13
Xiao and Ruan (see [11]) showed that either the number of infective individuals tends to zero as time evolves or the disease persists. The authors in [13, 18] proved that their epidemic model with saturated incidence rate undergoes a Bogdanov–Takens bifurcation of codimension 2. When we consider the incidence of a combination of the saturated incidence rate and a non-monotonic incidence, the codimension of Bogdanov–Takens bifurcation can grow up to 3.
4 Conclusions
In this paper, by combining qualitative and bifurcation analyses we study an SIS epidemic model with the incidence rate \(\frac {aI^{2}}{c+I^{2}}+\frac{bI}{c+I^{2}}\), which is a combination of the saturated incidence rate studied in [13, 18], describing the inhibition effect from the behavioral change and the non-monotonic incidence studied by Ruan in [11], interpreting the “psychological” effect. In Sect. 2, we give a full-scale analysis for the types and stability of the equilibria \(E_{i}\) (\(i=0,1,2,3,4,5\)). We prove that for system (2.1) there can occur backward bifurcation and the backward bifurcation will disappear if \(a=0\). At equilibrium \(E_{i}\) (\(i=3, 5\)), a degenerate Hopf bifurcation arises under certain conditions. When the critical condition \(\varPsi_{i}\) (\(i=3, 5\)) satisfied, we calculate the Liapunov value of the weak focus and obtain the maximal multiplicity of the weak focus is two, indicating that there exist at most two limit cycles around \(E_{i}\) (\(i=3, 5\)). In Fig. 6 and Fig. 8, we give the phase portraits corresponding to equilibrium \(E_{3}\) and \(E_{5}\) exhibiting a unique limit cycle and adding a new limit cycle after a small perturbation of the parameters Λ and c. In Sect. 3.3, we proved that the model exhibits Bogdanov–Takens bifurcation of codimension 2 and codimension 3, under certain conditions. If the parameter \(a=0\), the model can just have a Bogdanov–Takens bifurcation of codimension 2, shown in [13].
In reality, we show that the model exhibits multi-stable states. This interesting phenomenon indicates that the initial states of an epidemic can determine the final states of an epidemic to go extinct or not. Moreover, the periodical oscillations signify that the trend of the disease may be affected by the behavior of the susceptible and the effect of psychology of the disease.
Declarations
Acknowledgements
The authors are grateful to both reviewers for their helpful suggestions and comments.
Availability of data and materials
Not applicable.
Funding
This work was supported by the National Natural Science Foundation of China [11771373, 11001235].
Authors’ contributions
The authors have made 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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