An SIRS model with a nonlinear incidence rate
Introduction
In classical epidemiological models, the incidence rates are bilinear, i.e., proportional to the product of the number of infective individuals and the number of susceptible individuals [4], [12], [17], [25], [26], [27]. Such models always admit a globally asymptotically stable disease free equilibrium or endemic equilibrium, corresponding to the disease free steady state or endemic steady state. However, actual data and evidences observed for many diseases show that dynamics of disease transmission are not always as simple as shown in these models. Thus, in recent years, researchers [1], [5], [6], [7], [8], [10], [11], [15], [18], [20], [21], [23], [24], [28] have taken into account oscillations in incidence rates and proposed many nonlinear incidence rates. With these nonlinear incidence rates, many interesting and complicated transmission dynamics of epidemics have been shown, such as multiple equilibria, periodic orbits, Hopf and Bogdanov–Takens bifurcations, which state clearer and more reasonable qualitative description of the disease dynamics and give better suggestions for the prediction or control of diseases.
Let S(t) be the number of susceptible individuals at time t, I(t) be the number of infective individuals at time t, and R(t) be the number of recovered individuals at time t. Capasso and Serio [5] introduced a saturated incidence rate g(I)S in an epidemic model when they studied the cholera epidemic in 1973, where g(I) is decreasing when I is large enough. Liu et al. studied the codimension-1 bifurcation for SEIRS and SIRS models with the incidence rate βIpSq in [13], [14]. Lizana and Rivero [16], Glendinning and Perry [8] and Derrick and van den Driessche [7] studied saddle-node bifurcation, Hopf bifurcation and Bogdanov–Takens bifurcation of SIRS or SIR models with the incidence rate of βIpSq. Ruan and Wang [21] studied saddle-node bifurcation, Hopf bifurcation, Bogdanov–Takens bifurcation and the existence of none, one and two limit cycles of an SIRS model with an incidence rate of kI2S/(1 + αI2), which was also proposed by Liu et al. [13]. Derrick and van den Driessche [6] considered a very general nonlinear incidence rate. van den Driessche and Watmough [23], [24] studied an incidence rate of the form βI(1 + νIk−1)S, where β > 0, ν ⩾ 0 and k > 0. When ν = 0 this incidence rate is the bilinear incidence rate βIS [4]. In [23], they studied an SIS model with this incidence rate for ν > 0 and k = 2, showing the backward bifurcation, local stability and global stability of equilibria. In [24], they further introduced the same incidence rate in an SIRS model. However, although they obtained saddle-node bifurcation for the model, they only focused on numerical examples to show possibilities of Hopf bifurcation and homoclinic orbit and to analyze the attractive area of an endemic equilibrium, paying no attention to theoretical analysis. Alexander and Moghadas [1] analyzed an SIV model with a generalized nonlinear incidence rate f(I;v) and showed the existence of bistability and various Hopf bifurcation, especially for the incidence rate β I(1 + νIq)S with β > 0, ν ⩾ 0 and 0 < q ⩽ 1. However, they neglected such dynamics as Bogdanov–Takens bifurcation. In view that former works on the incidence rate βI(1 + νIk−1)S are not complete and in detail for the effect it may have on the global transmission dynamics of a disease, in this paper we would like to continue their work and give detailed theoretical analysis of the SIRS model with the same incidence rate, so as to clearly show the effect of this nonlinear incidence rate on the transmission dynamics of epidemics. In fact, by our analysis, we have not only theoretically obtained backward bifurcation, Hopf bifurcation and Bogdanov–Takens bifurcation for this SIRS model, but also shown bistable steady states and explicit conditions for asymptotic stability of equilibria, even for globally asymptotic stability, and especially found stability switches for one of the endemic equilibria.
We still consider the incidence rate βI(1 + νIk−1)S with β > 0, ν > 0 and k = 2 and investigate the following SIRS model:where B is the recruitment rate of the population, d is the death rate of the population, α is the recovery rate of infective individuals, B > 0, d > 0, and α > 0. Assume susceptible individuals will be infective after contacting with infective ones and infective ones recover from the disease with temporary immunity and then return to the susceptible class when losing immunity. γ is the rate that recovered individuals lose immunity and return to the susceptible class.
We present global qualitative and bifurcation analysis for the model and obtain rich dynamics. A disease free equilibrium exists for all parameters and is asymptotically stable when the basic reproduction number R0 < 1 and unstable when R0 > 1. When R0 > 1 there is a unique endemic equilibrium, but when R0 < 1 there may be none, one or two endemic equilibria and a backward bifurcation may emerge. As for the two endemic equilibria, one is always a saddle; the other may be stable or unstable. When R0 increases, the stability of this latter equilibrium may change from stable to unstable then back to stable, which indicates the existence of stability switches. The system may admit bistable steady states: a stable disease free equilibrium and a stable endemic equilibrium or a stable disease free equilibrium and a stable limit cycle. In this case, the initial condition is very important for the eventual steady state the system settles into. When all equilibria are unstable, at least one stable limit cycle emerges and the disease is destined to break out periodically. The disease free equilibrium or the endemic equilibrium may be globally asymptotically stable for suitable parameters. The system may also undergo Hopf bifurcation or Bogdanov–Takens bifurcation, which are important for strategies for control of a disease. The criteria for Hopf bifurcation and appropriate curves of Bogdanov–Takens bifurcation have been obtained.
To simplify the model, adding all equations of (1.1) and denoting the number of the total population by N(t) (N = S + I + R), we obtainThen for any initial condition, N(t) will tend to a constant N0 ≡ B/d when t tends to infinity. In this paper, we assume the population is in equilibrium and investigate the behavior of the system on the plane S + I + R = N0 > 0. So (1.1) becomesRescaling (1.2) by I1 = βI/(d + γ), R1 = βR/(d + γ), t1 = (d + γ)t and still writing (I1, R1, t1) as (I, R, t), we obtainwhere p = ν(d + γ)/β, A = βN0/(d + γ), R0 = βN0/(d + α), q = α/(d + γ), p > 0, A > 0, R0 > 0, q > 0.
The organization of this paper is as follows. In the next section, we present a qualitative analysis of (1.3). We analyze the existence and stability of equilibria, find stability switches for an endemic equilibrium and the existence of a stable limit cycle and bistable states, and analyze the nonexistence of limit cycles and globally asymptotic stability of equilibria. In Section 3, we show that the system admits Hopf bifurcation. In Section 4, we show that the system undergoes a Bogdanov–Takens bifurcation at the degenerate equilibrium.
Section snippets
Disease free equilibrium
Eq. (1.3) admits a unique disease free equilibrium (0, 0). The Jacobian matrix of (1.3) at (0, 0) isIt is easy to obtain the following result: Theorem 2.1 (0, 0) is asymptotically stable if R0 < 1, and unstable if R0 > 1.
Endemic equilibria
To obtain endemic equilibria of (1.3), it suffices to consider the following equations:which yieldThis equation may admit positive solutions
Hopf bifurcation
In this section, we study the Hopf bifurcation of (1.3). Since R axis is invariant and E1 must be a saddle, there is no closed orbit surrounding (0, 0) or E1. Thus Hopf bifurcations only emerge at E2. For (1.3), let x = I − I2, y = R − R2 to translate E2 to (0, 0). Then (1.3) becomeswhereTo obtain the Hopf bifurcation, we fix parameters such that tr(M2) = 0,
Bogdanov–Takens bifurcation
The Bogdanov–Takens bifurcation (for short, BT bifurcation) is a type of codimension-2 bifurcation that emerges when (1.3) admits a unique degenerate equilibrium. Assume the following two assumptions hold.
- (S1)
Ap − q − 1 > 0.
- (S2)
.
Let x = I − I∗, y = R − R∗ to translate E∗ to (0, 0). (1.3) becomeswhere
Discussion
In this paper we study an SIRS epidemic model with a nonlinear incidence rate βI(1 + νIk−1)S with β > 0, k = 2, and ν > 0, which was introduced in epidemic models in [23], [24]. Previous studies of analogous model with (1.1) in [24] mainly focused on simulations and obtained such dynamics as saddle-node, Hopf and Bogdanov–Takens bifurcations, and the endemic basin. In contrast, we analyze the model theoretically and found richer dynamics of the model, from which the effect of the nonlinear incidence
Acknowledgement
Research is supported by the National Science Fund of PR China: 10571143 and the Science Fund of Southwest China Normal University.
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