Hopf Bifurcation of an Epidemic Model with Delay

A spatiotemporal epidemic model with nonlinear incidence rate and Neumann boundary conditions is investigated. On the basis of the analysis of eigenvalues of the eigenpolynomial, we derive the conditions of the existence of Hopf bifurcation in one dimension space. By utilizing the normal form theory and the center manifold theorem of partial functional differential equations (PFDs), the properties of bifurcating periodic solutions are analyzed. Moreover, according to numerical simulations, it is found that the periodic solutions can emerge in delayed epidemic model with spatial diffusion, which is consistent with our theoretical results. The obtained results may provide a new viewpoint for the recurrent outbreak of disease.


Introduction
Currently, new infectious diseases continuously emerge, and existing diseases recurrently outbreak [1][2][3][4][5][6][7][8][9]. Ebola virus was firstly discovered in 1976, which began to outbreak in Guinea in February 2014, then spread to West Africa. It caused serious death and social panic. After the outbreak of 2014, Ebola once again emerged in Guinea in March 2016 [10][11][12]. These diseases have brought a great threat to the public health. In order to provide some suggestions for the prevention and control of the disease, it is necessary to establish rational mathematics model based on infectious mechanism of disease, the route of transmission, and the symptoms of the infected individuals. In particular, the incidence rate describes the number of new infections per unit time, which largely reflects the transmission mechanism of the disease [13][14][15][16][17]. For example, Capasso et al. proposed saturated incidence rate βSI/(1 + kI) to model the cholera epidemics in Bari in 1973, which reflects the psychological effect or the inhibition effect [18]. By taking appropriate preventive measures, May and Anderson gave nonlinear incidence rate β(SI/(1 + αS)) [19]. Therefore, some reasonable suggestions can be provided for the prevention and effective control of infectious diseases.
It takes an individual a period of time to show the corresponding symptoms based on the infectious mechanism of disease, after an individual is infected disease, such as, dengue, rabies, cholera and so on [20][21][22][23][24][25][26][27]. Therefore, time delay describing the incubation period of disease is a significant quantity. In fact, these potentially asymptomatic individuals (incubation individuals) may promote the wide spread of disease [28,29]. Thus, it is necessary for us to introduce time delay in the epidemic models.
Because of all the species living in the space environment, and they could diffuse the surrounding area. The individual diffusion in space has an effect on the disease contagion. For example, Zhang et al. indicated that dog movement led to the traveling wave of dog and human rabies and had a large influence on the minimal wave speed [30]. However, previous works on epidemic models did not account for the spatial diffusion factors. McCluskey proved that the endemic equilibrium was globally asymptotically stable whenever it existed for an SIR epidemic model with delay and nonlinear incidence rate [31]. A delayed predator-prey system with disease in the prey was investigated by Han et al., they considered the existence of Hopf bifurcation with time delay in terms of degree 2 [32]. Hence, it is more suitable for us to consider time delay and spatial factor in epidemic model. This paper is organized as below. In Section II, the eigenpolynomials of spatiotemporal epidemic model with nonlinear incidence rate are given, we further analyze the existence of Hopf bifurcation for two cases. In Section III, by using the normal form theory and the center manifold theorem, some properties of Hopf bifurcation are showed. In Section IV, on the basis of numerical simulations, we show that the epidemics will display recurrent behavior if time delay exceeds a critical point. Finally, some conclusions are obtained.

The properties of Hopf bifurcating period solutions
The above section gives the conditions of the existence of Hopf bifurcation for two cases. In this section, we investigate properties of these bifurcating periodic solutions from the positive constant steady state E Ã (S Ã , I Ã ) of system (1) by employing the normal form theory and the center manifold theorem of partial functional differential equations (PFDEs) [39][40][41][42], these properties include the direction, stability and period. It's simple for mathematical calculation to mark t c ¼ t j k ðk ¼ 0; k 0 ; j ¼ 0; 1; 2; :::Þ. LetŜðt; xÞ ¼ Sðtt; xÞ,Î ðt; xÞ ¼ I ðtt; xÞ, then system (3) can be expressed as In Next, the linear part of the system (19) is given by From the conclusions of section II, an equilibrium of the system (20) is the origin, the corresponding characteristic equation of the system (20) at origin has two pairs of purely imaginary eigenvalues ±iw 0 τ c , AEiŵ 0 t c for k = 0, and only a pair of purely imaginary eigenvalues ±iw k τ c for k 2 N. We account for purely imaginary eigenvalues ±iw 0 τ c for the case k = 0, and set Λ 0 = {iw k τ c , −iw k τ c }, (k = 0, k 0 ).
By some computations, the following Lemma is directly given: Lemma 2.3 A basis of P with Λ 0 is q 1 ðyÞ ¼ e iw k t c y ð1; xÞ T ; q 2 ðyÞ ¼ q 1 ðyÞ; À 1 y 0; and a basis of P Ã with Λ 0 is F = (F 1 , F 2 ) and F Ã ¼ ðF Ã 1 ; F Ã 2 Þ T are obtained by separating the real and imaginary parts of q 1 (θ) and q Ã 1 ðsÞ, respectively. Obviously, F is the basis of P, F Ã is the basis of P Ã , and According to the bilinear pairing form Eq (26), we can compute: þt c a 23 w k ðd 1 k 2 À a 11 Þ a 13 a 21 þ 1 2 t c w k 1 þ sin2w k t c 2w k t c þt c a 13 a 21 ðd 1 k 2 À a 11 Þ þ a 23 ½w 2 k þ ðd 1 k 2 À a 11 Þ 2 a 13 a 21 þt c a 23 w k ðd 1 k 2 À a 11 Þ a 13 a 21 À 1 2 t c w k 1 À sin2w k t c 2w k t c þt c a 13 a 21 ðd 1 k 2 À a 11 Þ À a 23 ½w 2 k þ ðd 1 k 2 À a 11 Þ 2 a 13 a 21 t c a 13 a 21 ðd 1 k 2 À a 11 Þ À a 23 ðd 1 k 2 À a 11 Þ 2 a 13 a 21 1 À sin2w k t c 2w k t c : Next, we construct a new basis C for P Ã , where C = (C 1 , On the basis of the theory of decomposition of the phase space, we have } = P CN } + P s }, where P CN } is the center subspace of linear Eq (20), and P s } is the complement subspace of P CN }.

Numerical results
Compared with the theoretical analyses, we perform a series of extensive numerical simulations of the spatiotemporal epidemic model with nonlinear incidence rate in one-dimensional space, and investigate the incubation period how to affect the spread of epidemics. We solve the numerical solutions of system (1) by using Matlab. The reaction-diffusion system is solved in a discrete domain with N x × N y lattice sites. The Laplacian describing diffusion is approximated by using finite differences, and we also discretize the time evolution.

Discussion
In this study, the characteristic equation at the positive constant steady state E Ã (S Ã , I Ã ) is derived. In order to study the influence of incubation period on epidemic transmission, we choose time delay τ as a bifurcation parameter. Moreover, we get the two classes conditions of the existence of Hopf bifurcation: one is the absence of diffusion k = 0, the other is the presence of diffusion k = k 0 2 N. With increasing of parameter τ, the stability of positive constant steady state E Ã (S Ã , I Ã ) will change, and Hopf bifurcation will concurrently occur in system (1) at the critical point τ c (t j 0 or t j k 0 ). In the following, we obtain the properties of bifurcating period solutions including direction, stability and period by utilizing the normal formal theory and the center manifold theorem of partial functional differential equations (PFDs).
It should be noted that spatial pattern may be found in epidemic model (1). Based on pattern dynamics of model (1), one can obtain the pattern structures in different parameters space [44,45]. In this case, we can reveal the distributions of disease with high density or low density and thus provide useful control measures to eliminate the disease.

Conclusion
The numerical results validate our theoretical findings, which show that the length of the incubation period have significant impacts on epidemic transmission. The biennial outbreaks of measles is the signature of an endemic infectious disease, which becomes non-endemic if there were a minor increase in infectivity or a decrease in the length of the incubation period [15]. Based on this paper, we provide a possible mechanism to explain the recurrent outbreak of disease.