Bogdanov-Takens bifurcation in a SIRS epidemic model with a generalized nonmonotone incidence rate

In this paper, we study a SIRS epidemic model with a generalized nonmonotone incidence rate. It is shown that the model undergoes two different topological types of Bogdanov-Takens bifurcations, i.e., repelling and attracting Bogdanov-Takens bifurcations, for general parameter conditions. The approximate expressions for saddle-node, Homoclinic and Hopf bifurcation curves are calculated up to second order. Furthermore, some numerical simulations, including bifurcations diagrams and corresponding phase portraits, are given to illustrate the theoretical results.


Introduction. In 1927, Kermack and McKendrick
proposed a classic infectious disease compartmental model, where the population is divided into three classes labeled S(t), I(t) and R(t), which denote the number of susceptible individuals, the number of infected individuals, the number of individuals who have been infected and then recovered or removed at time t, respectively. Assuming the recovered individuals have temporary immunity, the classical Kermack-McKendrick model has the following form: where b is the recruitment rate of the population, d is the natural death rate of the population, µ is the natural recovery rate of the infective individuals, δ is the rate at which recovered individuals lose immunity and return to the susceptible class, g(I)S is called the incidence rate, and g(I) is a function to measure the infection force of a disease. In [11], Kemack and McKendrick assumed that infection force g(I) is a linear function of I (see Figure 1(a)), i.e., the incidence rate g(I)S is bilinear, which may be inconsistent with the reality when I(t) is getting larger.
In order to study the cholera epidemic spread in Bari, Italy, in 1973, Capasso et al. [6] and Capasso and Serio [5] proposed a saturated incidence rate where the infection force g(I) = kI 1+αI eventually tends to a saturation level k α when I is getting larger (see Figure 1(b) ).
To model the effects of psychological factor when the infection number is getting larger, Xiao and Ruan [22] proposed a specific incidence rate where g(I) is nonmonotone and eventually tends to zero (see Figure 1(c)). They showed that the disease-free equilibrium of system (1) with incidence rate (5) is globally asymptotically stable if the basic reproduction number R 0 ≤ 1, and the unique positive equilibrium is globally asymptotically stable if R 0 > 1. Then system (1) with nonmonotone incidence rate (5) has no complex dynamics and bifurcation phenomena. Xiao and Zhou [23] considered a generalized form of the nonmonotone incidence rate (5) as follows: where β is a parameter satisfied β > −2 √ α such that 1 + βI + αI 2 > 0 for all I ≥ 0. They presented qualitative analysis for model (1) with generalized nonmonotone incidence rate (6) and showed the existence of a cusp of codimension 2, bistable phenomenon and periodic oscillation. Later, Zhou et al. [24] further studied the existence of different kinds of bifurcations, such as Bogdanov-Takens bifurcation and Hopf bifurcation, by choosing several set of specific parameter values. Their results showed that model (1) with generalized nonmonotone incidence rate (6) can exhibit complex dynamics and bifurcation phenomena. However, the set of parameter values they chosen to unfold the Bogdanov-Takens bifurcation is biologically meaningless. In this paper, we continue to consider model (1) with generalized nonmonotone incidence rate (6) as follows: where b, d, δ, µ, k, α are all positive parameters and β > −2 √ α. We can see that g(I) = kI 1+βI+αI 2 increases when I is small and decreases when I is large, then tends to zero as I tends to infinite (see Figure 2). When α = 9 2 and k = 1, the difference of the curves of g(I) when β ≥ 0 and −2 √ α < β < 0 can be seen from Figure 2(a) and (b), where g(I) has sigmoid shape when −2 √ α < β < 0 and I is small. We will not choose specific parameter values, and show that model (7) undergoes two different topological types of Bogdanov-Takens bifurcations, i.e., repelling and attracting Bogdanov-Takens bifurcations, for general parameter conditions. The approximate expressions for saddle-node, Homoclinic and Hopf bifurcation curves are calculated up to second order. Furthermore, some numerical simulations, including repelling and attracting Bogdanov-Takens bifurcations diagrams and corresponding phase portraits, are given to illustrate the theoretical results.
The organization of this paper is as follows. In section 2, we show that model (7) undergoes repelling and attracting Bogdanov-Takens bifurcations of codimension 2 for general parameter conditions. In section 3, some numerical simulations are given to illustrate the theoretical results. A brief discussion is given in the last section.
2. Bogdanov-Takens bifurcation. As is shown in Xiao and Zhou [23] and Zhou et al. [24], the limit set of system (7) is on the plane S + I + R = b d . Thus, we focus on the reduced system For simplicity, we rescale system (8) by then system (8) becomes (we still denote τ by t) where and q < p ≤ q + 1, m > −2 √ n, A, p, q, n > 0.
We first define The following lemma is from Lemma 2.7 in Xiao and Zhou [23].

Note that
since p, q > 0, m < − 1+q p < 0 and m = m * . Thus the parameter transformation (18) is a homeomorphism in a small neighborhood of the origin, and µ 1 and µ 2 are independent parameters.

Remark 1.
Zhou et al. [24] showed that system (9) exhibits an attracting Bogdanov-Takens bifurcation of codimension 2 by choosing a set of specific parameter values, which is biologically meaningless because it does not satisfy p > q. We have shown that system (9) undergoes two different topological types of Bogdanov-Takens bifurcation, i.e., repelling and attracting Bogdanov-Takens bifurcations, for general parameter conditions. 3. Numerical simulations. Next, we illustrate our theoretical conclusions by numerical simulations. Firstly, we fixed p = 3, q = 2, and m = −3 < m * = −2, then get A = 9 4 and n = 4 from A = A * and n = n * , respectively. This set of specific parameter values satisfy the conditions in the case (i) of Theorem 2.2. Then system (17) undergoes repelling Bogdanov-Takens bifurcation when (λ 1 ,λ 2 ) vary in a small neighbor of (0, 0) with this set of specific parameter values. From (18), we can get The repelling Bogdanov-Takens bifurcation diagram of codimension 2 and corresponding phase portraits of system (11) with p = 3, q = 2, m = −3, A = 9 4 , n = 4 are given in Figure 4. These bifurcation curves H, HL and SN divide the small neighborhood of the origin in the parameter (λ 1 , λ 2 )-plane into four regions (see Figure 4(a)).
(a) When (λ 1 , λ 2 ) = (0, 0), the unique positive equilibrium is a cusp of codimension 2 (see Figure 3(a)). (b) There are no positive equilibria when the parameters lie in region I (see Figure  4(b)), we can see that the disease will die out for all positive initial populations. (c) When the parameters lie on the curve SN , there is a unique positive equilibrium, which is a saddle-node.  (d) Two positive equilibria, one is an unstable focus and the other is a saddle, will occur through the saddle node bifurcation when the parameters cross SN into region II (see Figure 4(c)). (e) An unstable limit cycle will appear through the subcritical Hopf bifurcation when the parameters cross H into region III (see Figure 4(d)), where the focus is stable, whereas the focus is an unstable one with multiplicity one when the parameters lie on the curve H. (f) An unstable homoclinic cycle will occur through the homoclinic bifurcation when the parameters pass region III and lie on the curve HL (see Figure 4(e)). (g) The relative location of one stable and one unstable manifold of the saddle will be reverse when the parameters cross III into region IV (compare Figure  4(c) and Figure 4(f)). Secondly, we fixed p = 3, q = 2, and m = − 3 2 > m * = −2, then get A = 36 13 and n = 13 16 from A = A * and n = n * , respectively. This set of specific parameter values satisfy the conditions in the case (ii) of Theorem 2.2. Then system (17) undergoes attracting Bogdanov-Takens bifurcation when (λ 1 ,λ 2 ) vary in a small neighbor of (0, 0) with this set of specific parameter values. From (18), we can get , and the approximate expressions for saddle-node, Homoclinic and Hopf bifurcation curves up to second order are given as follows: (i) The saddle-node bifurcation curve SN = The attracting Bogdanov-Takens bifurcation diagram of codimension 2 and corresponding phase portraits of system (11) with p = 3, q = 2, m = − 3 2 , A = 36 13 , n = 13 16 are given in Figure 5. These bifurcation curves H, HL and SN divide the small neighborhood of the origin in the parameter (λ 1 , λ 2 )-plane into four regions (see Figure 5(a)).
(i) When (λ 1 , λ 2 ) = (0, 0), the unique positive equilibrium is a cusp of codimension 2 (see Figure 3(b)). (ii) There are no positive equilibria when the parameters lie in region I (see Figure  5(b)), we can see that the disease will die out for all positive initial populations. (iii) When the parameters lie on the curve SN , there is a unique positive equilibrium, which is a saddle-node. (iv) Two positive equilibria, one is a stable focus and the other is a saddle, will occur through the saddle node bifurcation when the parameters cross SN into region II (see Figure 5(c)). (v) A stable limit cycle will appear through the subcritical Hopf bifurcation when the parameters cross H into region III (see Figure 5(d)), where the focus is unstable, whereas the focus is a stable one with multiplicity one when the parameters lie on the curve H. (vi) A stable homoclinic cycle will occur through the homoclinic bifurcation when the parameters pass region III and lie on the curve HL (see Figure 5(e)). (vii) The relative location of one stable and one unstable manifold of the saddle will be reverse when the parameters cross III into region IV (compare Figure  5(c) and Figure 5(f)).
4. Discussion. In some epidemic diseases, when the infectious number is getting larger, the psychological factor can play an important influence on the disease, since people and government can take a series of protection measures and intervention policies to control the disease ( [22]). The psychological effect can be modelled by a nonmonotone incidence rate, i.e., the infection force g(I) increases firstly when the infection number is small, and then decreases when the infection number is getting larger. In [23], Xiao and Zhou proposed a generalized nonmonotone incidence rate (6) to model the psychological effect. For model (1) with generalized nonmonotone incidence rate (6) (i.e., model (9)), they showed the existence of a cusp of codimension 2, bistable phenomenon and periodic oscillation. Later, Zhou et al. [24] further studied the existence of Bogdanov-Takens bifurcation by choosing a set of specific parameter values. However, the set of parameter values they chosen to unfold the Bogdanov-Takens bifurcation is biologically meaningless because it does not satisfy p > q. In this paper, we continue to consider model (1) with generalized nonmonotone incidence rate (6), for general parameter conditions, we showed that model (9) undergoes two different topological types of Bogdanov-Takens bifurcations, i.e., repelling and attracting Bogdanov-Takens bifurcations. We calculated the approximate expressions for saddle-node, Homoclinic and Hopf bifurcation curves up to second order. Furthermore, some numerical simulations, including bifurcations diagrams and corresponding phase portraits, are also given to illustrate the theoretical results. Our results can be seen as a complement to the work in [23] and [24]. The stable limit cycle arising from the supercritical Hopf bifurcation implies the existence of sustained periodic oscillation for the disease, and the disease tends to periodic outbreak when the initial population lies in the attracting region of the stable limit cycle. It is very important to understand the underlying mechanics of periodic oscillations, which have been observed in the real world [1,7,8,16,13,17,18].
From Lemma 2.1, if (A, m, n, p, q) ∈ Γ, n = n * , A = A * and m = m * , then the unique positive equilibrium E * (x * , y * ) of system (9) is a cusp of codimension 2. Moreover, we find that E * (x * , y * ) is a cusp of codimension at least 3 when (A, m, n, p, q) ∈ Γ, n = n * , A = A * and m = m * , then system (9) may undergo Bogdanov-Takens bifurcation of codimension more than 2. We will consider these problems in the future.