Bifurcation analysis for a delayed SEIR epidemic model with saturated incidence and saturated treatment function

A delayed SEIR epidemic model with saturated incidence and saturated treatment function is considered in this paper. Sufficient conditions for the existence of local Hopf bifurcation are established by regarding the possible combination of the two delays as the bifurcation parameter. General formula for the direction, period and stability of the bifurcated periodic solutions are derived by using the normal form method and the centre manifold theory. Finally, some numerical simulations are given to illustrate the obtained results.


Introduction
In recent years, epidemic models have been studied by many scholars to study the dynamics of disease transmissions. Incidence rate plays an important role in the modelling of epidemic dynamics. Many epidemic models are proposed based on the bilinear incidence rate βSI and the standard incidence rate βSI N [1,8,10,11]. Capasso and Serio [4] introduced the saturated incidence rate βSI 1+αI which includes the behavioural change and crowding effect of the infective individuals and prevents the unboundedness of the contact rate by choosing suitable parameters. This makes the saturated incidence rate βSI 1+αI seems more reasonable than the bilinear incidence rate. On the other hand, most of the present epidemic models assume that the treatment rate of the infection is proportional to the number of the infective individuals. Obviously, this assumption disregards the limitation of the medical resources. Based on this, Zhang et al. [19] proposed the following SEIR epidemic model with saturated incidence and saturated treatment function: CONTACT Juan Liu my7216@163.com where S(t), E(t), I(t), and R(t) represent the numbers of susceptible, exposed but not yet infectious, infective and recovered individuals at time t, respectively. A is the recruitment rate of the population, d is the natural death rate and γ is the death rate due to the disease. rI 1+kI is the saturated treatment rate in which r is the maximal medical resources supplied per unit time and k is the saturation factor that measures the effect of the infected being delayed for treatment. βSI 1+αI is the saturated incidence rate in which α is the saturation factor that measures the inhibitory effect and β is the transmission rate. ε is the rate of transformation from incubation period individuals to the infective ones. δ is the natural recovery rate of the infective individuals. Zhang et al. [19] studied the stability and backward bifurcation of system (1).
In fact, many diseases have different kinds of delays when they spread, such as latent period delay [5,6,12,[14][15][16]20] and immunity period delay [13,18,21]. In [12], Xue and Li analysed existence of Hopf bifurcation for a delayed SIR epidemic model with logistic growth by regarding the latent period delay of the disease as a bifurcation parameter and studied the properties of the Hopf bifurcation. In [18], Zhang et al. obtained the sufficient conditions for the linear stability and existence of Hopf bifurcation of a delayed epidemic model with a nonlinear birth in population and vertical transmission by regarding the immunity period delay of the recovered individuals as a bifurcation parameter. They also derived the explicit formulas determining the direction and stability of the Hopf bifurcation. Motivated by the work above, we consider the Hopf bifurcation of the following epidemic model with two delays: where τ 1 is the time delay due to the latent period of the disease. τ 2 is the time delay due to the period that the infected individuals use to move into the recovered class. The reminder of the paper is organized as follows. In Section 2, we focus on the local stability of the endemic equilibrium and the existence of the Hopf bifurcation of system (2). Then, we obtain explicit formulas that determine the stability and direction of the Hopf bifurcation by using the normal theory and the centre manifold theorem in Section 3. Numerical simulations are carried out in Section 4 to illustrate the main theoretical results and a brief discussion is given in the last part to conclude this work.
τ 10 is defined as in Equation (14). where Let λ = iω 2 (ω 2 > 0) be the root of Equation (15), then from which one can get with Let ω 2 2 = v 2 , then Equation (17) becomes Similar as in Case 2, we can make the following assumption.
Further we have where the sign μ 2 determines the direction of the Hopf bifurcation; the sign β 2 determines the stability of the bifurcating periodic solutions and the sign of T 2 determines the period of the bifurcating periodic solutions. According to the analysis above, we have the following. (2), if μ 2 > 0(μ 2 < 0), the Hopf bifurcation is supercritical(subcritical). If β 2 < 0(β 2 > 0), the bifurcating periodic solutions are stable (unstable). If T 2 > 0(T 2 < 0), the period of the bifurcating periodic solutions increases (decreases).
1.25 ∈ (0, τ 10 ) can be shown in Figure 4. In addition, we obtain μ 2 = 149.6526 > 0, β 2 = −96.0988 < 0 and T 2 = 264.0964 > 0. Thus, based on the results in Theorem 3.1, we can deduce that the Hopf bifurcation is supercritical, the bifurcating periodic solutions are stable and the period of the bifurcating periodic solutions increase.

Conclusions
This paper is concerned with a delayed SEIR epidemic model with saturated incidence and saturated treatment function. The effect of the two delays on the model is investigated and the main results are given in terms of local stability and local Hopf bifurcation. It has been shown that the model is stable when the value of the bifurcation parameter is below the critical value, which means that the disease can be controlled easily. However, when the value of the bifurcation parameter is above the critical value, a Hopf bifurcation will occur. In this condition, the disease is out of control. Accordingly, we should shorten the delays in the model as much as possible so that we can predict and control the disease propagation. For further investigation, the properties of the Hopf bifurcation are studied with the aim of the normal form method and centre manifold theorem. Finally, some numerical simulations are carried out to support our theoretical results.