
The increasing focus on environmentally sustainable technologies has catalyzed significant advancements in electric vehicles (EVs), marking a pivotal shift in the transportation industry. Among these developments, wireless charging technology for EVs has emerged as a transformative innovation, attracting substantial interest in research and development. These advancements aim to reduce greenhouse gas emissions, establish EVs as a primary mode of transportation, and contribute to a more sustainable future. Electric vehicles are generally categorized into hybrid and pure electric types. Hybrid EVs combine internal combustion engines with onboard energy storage systems for propulsion and charging, serving as an intermediate step toward full electrification. In contrast, pure EVs rely entirely on external energy sources, necessitating the use of dedicated charging infrastructure. Despite the widespread adoption of conductive charging methods, they face challenges such as reliance on physical connections, which can be cumbersome and inefficient. Wireless charging, particularly through contactless power transfer (CPT), has demonstrated significant potential to overcome these challenges. This technology offers key benefits, including higher efficiency, tolerance to misalignment, and reduced charging times, positioning it as a viable alternative to conventional charging methods. Recent advancements, driven by extensive research, have further enhanced the practicality and efficiency of wireless charging systems. This study investigates dynamic wireless charging systems for EVs, addressing the critical need to reduce environmental impact and enhance charging convenience. It utilizes simulations to analyze the performance of dynamic charging systems employing resonant inductive power transfer technology under real-world conditions. Variations in parameters such as alignment and gap are examined to provide insights into system performance and optimization. The findings of this research aim to advance the development of wireless charging technology, emphasizing its potential to revolutionize the transportation sector. By addressing existing challenges and offering innovative solutions, this study contributes to the pursuit of sustainable and efficient mobility systems.
Citation: Nikolaos S. Korakianitis, Nikolaou Nikolaos, Georgios A. Vokas, George Ch. Ioannidis, Stavros D. Kaminaris. Simulation-Based evaluation of dynamic wireless charging systems for electric vehicles: Efficiency, limitations, and future directions[J]. AIMS Energy, 2025, 13(3): 590-631. doi: 10.3934/energy.2025023
[1] | Hongfan Lu, Yuting Ding, Silin Gong, Shishi Wang . Mathematical modeling and dynamic analysis of SIQR model with delay for pandemic COVID-19. Mathematical Biosciences and Engineering, 2021, 18(4): 3197-3214. doi: 10.3934/mbe.2021159 |
[2] | Sarita Bugalia, Jai Prakash Tripathi, Hao Wang . Mathematical modeling of intervention and low medical resource availability with delays: Applications to COVID-19 outbreaks in Spain and Italy. Mathematical Biosciences and Engineering, 2021, 18(5): 5865-5920. doi: 10.3934/mbe.2021295 |
[3] | A. Q. Khan, M. Tasneem, M. B. Almatrafi . Discrete-time COVID-19 epidemic model with bifurcation and control. Mathematical Biosciences and Engineering, 2022, 19(2): 1944-1969. doi: 10.3934/mbe.2022092 |
[4] | Xinyu Liu, Zimeng Lv, Yuting Ding . Mathematical modeling and stability analysis of the time-delayed SAIM model for COVID-19 vaccination and media coverage. Mathematical Biosciences and Engineering, 2022, 19(6): 6296-6316. doi: 10.3934/mbe.2022294 |
[5] | Yuting Ding, Gaoyang Liu, Yong An . Stability and bifurcation analysis of a tumor-immune system with two delays and diffusion. Mathematical Biosciences and Engineering, 2022, 19(2): 1154-1173. doi: 10.3934/mbe.2022053 |
[6] | Qingwen Hu . A model of regulatory dynamics with threshold-type state-dependent delay. Mathematical Biosciences and Engineering, 2018, 15(4): 863-882. doi: 10.3934/mbe.2018039 |
[7] | Jinhu Xu, Yicang Zhou . Bifurcation analysis of HIV-1 infection model with cell-to-cell transmission and immune response delay. Mathematical Biosciences and Engineering, 2016, 13(2): 343-367. doi: 10.3934/mbe.2015006 |
[8] | Tao Chen, Zhiming Li, Ge Zhang . Analysis of a COVID-19 model with media coverage and limited resources. Mathematical Biosciences and Engineering, 2024, 21(4): 5283-5307. doi: 10.3934/mbe.2024233 |
[9] | Honghua Bin, Daifeng Duan, Junjie Wei . Bifurcation analysis of a reaction-diffusion-advection predator-prey system with delay. Mathematical Biosciences and Engineering, 2023, 20(7): 12194-12210. doi: 10.3934/mbe.2023543 |
[10] | Huan Dai, Yuying Liu, Junjie Wei . Stability analysis and Hopf bifurcation in a diffusive epidemic model with two delays. Mathematical Biosciences and Engineering, 2020, 17(4): 4127-4146. doi: 10.3934/mbe.2020229 |
The increasing focus on environmentally sustainable technologies has catalyzed significant advancements in electric vehicles (EVs), marking a pivotal shift in the transportation industry. Among these developments, wireless charging technology for EVs has emerged as a transformative innovation, attracting substantial interest in research and development. These advancements aim to reduce greenhouse gas emissions, establish EVs as a primary mode of transportation, and contribute to a more sustainable future. Electric vehicles are generally categorized into hybrid and pure electric types. Hybrid EVs combine internal combustion engines with onboard energy storage systems for propulsion and charging, serving as an intermediate step toward full electrification. In contrast, pure EVs rely entirely on external energy sources, necessitating the use of dedicated charging infrastructure. Despite the widespread adoption of conductive charging methods, they face challenges such as reliance on physical connections, which can be cumbersome and inefficient. Wireless charging, particularly through contactless power transfer (CPT), has demonstrated significant potential to overcome these challenges. This technology offers key benefits, including higher efficiency, tolerance to misalignment, and reduced charging times, positioning it as a viable alternative to conventional charging methods. Recent advancements, driven by extensive research, have further enhanced the practicality and efficiency of wireless charging systems. This study investigates dynamic wireless charging systems for EVs, addressing the critical need to reduce environmental impact and enhance charging convenience. It utilizes simulations to analyze the performance of dynamic charging systems employing resonant inductive power transfer technology under real-world conditions. Variations in parameters such as alignment and gap are examined to provide insights into system performance and optimization. The findings of this research aim to advance the development of wireless charging technology, emphasizing its potential to revolutionize the transportation sector. By addressing existing challenges and offering innovative solutions, this study contributes to the pursuit of sustainable and efficient mobility systems.
COVID-19 is a respiratory infectious disease and was first reported in Wuhan, China. It is caused by a novel coronavirus named SARS-CoV-2. SARS-COV-2 appeared after severe acute respiratory syndrome coronavirus (SARS) and Middle East respiratory syndrome coronavirus (MERS) [1]. Coronaviruses are enveloped nonsegmented positive-sense RNA viruses and are broadly distributed in mammals, including humans. They can destroy the human respiratory systems and cause severe acute respiratory syndrome [2]. Infected individuals usually cough, have a fever, and have difficulty breathing. Some of them even die of COVID-19. Moreover, there are many ways to spread COVID-19, such as direct transmission, aerosol transmission and contact transmission. This caused COVID-19 to spread rapidly worldwide. As a global infectious disease, COVID-19 has caused more than one million deaths. We attempt to establish a model to analyze the spread of COVID-19 and provide some advice to make COVID-19 controllable.
In recent investigations, many researchers have studied different epidemic models applied to the spread of COVID-19. In ref. [3] and ref. [4], Varotsos et al. established an SPRD model that included people that were susceptible, infected, recovered and dead. The model was based on COVID-19 data and was established to predict COVID-19 spread. We think there is no need to include people who died as a main part of the model because the proportion of people who died is very small. In ref. [5], León et al. developed an SEIARD mathematical model to investigate the COVID-19 outbreak in Mexico. The SEIARD model included infection with symptoms, infection without symptoms (asymptomatic), recovery from symptomatic infection and recovery from asymptomatic infection. If we consider only the infectivity of COVID-19, both infection with symptoms and infection without symptoms (asymptomatic) can infect the suspected cases. There is no need to divide them into infected with symptoms and infected without symptoms (asymptomatic). Both the recovery from symptomatic infection and the recovery from asymptomatic infection have very small possibilities for reinfection. There is no need to divide them into recovered from symptomatic infection and recovered from asymptomatic infection. In ref. [6], an SEIR epidemic model was developed for COVID-19. In ref. [7] and ref. [8], two SIR epidemic models were developed for COVID-19. For COVID-19, many countries have taken quarantine measures, which play an important part in the spread of COVID-19. These authors did not consider the influence of these quarantine measures. The quarantine measures taken by many countries make the proportion of quarantined people large. These quarantine measures can prevent infected individuals from infecting suspected individuals. In addition, exposure to air pollution can damage the heart and lungs and increase vulnerability to more serious coronavirus effects [9]. These quarantine measures can reduce the influence of air pollution. We think it is necessary to consider quarantined people because of the importance of these quarantine measures and the number of quarantined people. We attempt to construct an SIQR epidemic model including quarantined people to describe the spread of COVID-19.
In the spreading process of COVID-19, there is a time delay from infection to isolation. There are some studies about other epidemics with time delays. Liu et al. [10] developed an SEIRU epidemic model with a time delay before an infected person can transmit the infection to another person. They evaluated the effect of the latency period on the dynamics of COVID-19. Xu [11] and Yang et al. [12] also developed epidemic models with time delays before an infected person can transmit the infection to another person. Although this kind of time delay exists, human beings have difficulty changing this kind of time delay. In addition, Wang et al. [13] and Liu et al. [14] made time delays in their models the sojourn times in an infective state. Lu et al. [15] presented an SIQR model with a time delay from infection to recovery. The influence of the time delay from infection to recovery was discussed in detail. According to their conclusions, the smaller the time delay from infection to isolation was, the better COVID-19 was controlled. We decide to focus on the time delay from infection to isolation and discuss the effect of the time delay from infection to isolation on the spread of COVID-19, which is the main difference between our work and the works of others.
Stability and bifurcation have great significance to epidemic models, and some studies have analyzed stability and bifurcation for some epidemic models. In ref. [16], Greenhalgh investigated the stability of some SEIRS epidemiological models with vaccination and temporary immunity. Greenhalgh proposed that there was a threshold parameter R0, and the disease could persist if and only if R0 exceeded one. In addition, disease-free equilibrium always existed and was locally stable if R0<1 and unstable if R0>1. However, for R0>1, the endemic equilibrium was unique and locally asymptotically stable. Based on this conclusion, we think it is necessary for us to calculate the threshold parameter R0 in our model. In ref. [17], Xie et al. investigated the global stability of endemic equilibrium of an SIS epidemic model in complex networks. They proved that the endemic equilibrium was globally asymptotically stable by using a combination of the Lyapunov function method and a monotone iterative technique. However, the Hopf bifurcation analysis was not mentioned in this paper. In refs. [18,19], two SIS models were proposed. In ref. [20], an SIQRS model was proposed. The stability of the models was analyzed, but bifurcation analysis was not mentioned. In refs. [21,22,23,24], the stability and bifurcation of some SIR models were analyzed by different authors. In refs. [25,26,27,28], different SIR models were developed. The stability and bifurcation of them were analyzed. In ref. [29], an SIR epidemic model with time delay was proposed. In ref. [30], an SIR model with information-dependent vaccination was proposed. We compare the methods they used to analyze the stability and bifurcation of these SIR models. Then, we use the multiple time scales method, which was also used in ref. [21]. The multiple time scales method is systematic. It is a standard method for calculating the normal form of Hopf bifurcation. It can be directly applied to the original nonlinear dynamical system, which is described by ordinary differential equations and delayed differential equations, without application of the center manifold theory [31].
It is obvious that the isolation measures taken by many countries play an important role in controlling the spread of COVID-19. The time delay also has a considerable influence on the spread. Thus, we attempt to develop one mathematical model with a time delay to investigate the spread of COVID-19 and analyze the stability of this model. In some studies such as refs. [3,4,5], the authors forecasted how the epidemic situation changed with detailed data. However, we decide to focus on the stability of equilibria in our model, calculate the critical time delay from infection to isolation and analyze the dynamical property of our model to discuss the tendency of the spread of COVID-19.
The main objective of this paper includes constructing the DDE model for COVID-19 and analyzing the stability of the model, the existence of Hopf bifurcation and the stability of the bifurcating periodic solution. The rest of the paper is organized as follows. In Section 2, we construct the DDE model according to the characteristics of the spread of COVID-19. In Section 3, we analyze the existence and stability of the equilibria and the existence of Hopf bifurcation associated with the COVID-19 model with a time delay. We calculate the critical time delay from infection to isolation and analyze the dynamic properties of our model. In Section 4, we derive the normal form of the Hopf bifurcation for the above model and analyze the stability of the bifurcating periodic solution. In Section 5, we present the results of simulations to verify the correctness of our analysis. Finally, the conclusion is drawn in Section 6.
Suspected individuals can be infected with COVID-19 by infected individuals. After individuals are infected with COVID-19, some of them will be quarantined, and some of them will die of COVID-19. The quarantined individuals will not infect others. For COVID-19, we ignore the effect of reinfection because the probability of reinfection is very small. In addition, we ignore the probability of transforming the suspected individuals into recovered individuals directly since the probability is also small. We divide people into four kinds (suspected, infected, quarantined and recovered) based on their infectivity. The specific conversion between the four kinds of people is given in Figure 1.
The variables and parameters in Figure 1 are given in Table 1.
Symbol | Definition |
S | The number of suspected people |
I | The number of infected people |
Q | The number of quarantined people |
R | The number of recovered people |
Λ | The population growth |
μ | The natural mortality |
α1 | The rate of COVID-19 death of infected people |
α2 | The rate of COVID-19 death of quarantined people |
β | The transmission rate from S to I |
ε | The transmission rate from I to Q |
γ1 | The transmission rate from I to R |
γ2 | The transmission rate from Q to R |
For COVID-19, there is a latent period after the suspected individuals are infected. In the latent period, the infected individuals have no symptoms. However, these asymptomatic infected individuals can also infect suspected individuals. If we consider only their infectivity, there is little difference between asymptomatic infected individuals and symptomatic infected individuals. Thus, both of them can be divided into infected. Additionally, some suspected individuals may be quarantined, and they will not be infected. This means that the number of suspected persons who can be infected will decrease. Considering the influence of the suspected individuals who are quarantined, we can reduce the transmission rate from S to I, and there is no need to include the suspected individuals who are quarantined in the model.
Based on Figure 1, we construct the DDE model (2.1):
{S′(t)=Λ−μS(t)−βS(t)I(t),I′(t)=βS(t)I(t)−εI(t−τ)−μI(t)−α1I(t)−γ1I(t),Q′(t)=εI(t−τ)−γ2Q(t)−α2Q(t)−μQ(t),R′(t)=γ1I(t)+γ2Q(t)−μR(t), | (2.1) |
where the definitions of parameters and variables are presented in Figure 1 and Table 1, and τ>0 denotes the time delay from infection to isolation.
The initial condition of system (2.1) is ϕS(θ)≥0,ϕI(θ)≥0,ϕQ(θ)≥0,ϕR(θ)≥0, where Φ=(ϕS(θ),ϕI(θ),ϕQ(θ),ϕR(θ))∈C([−τ,0],R4+0) for θ∈[−τ,0], and C([−τ,0],R4+0) is the Banach space of continuous functions mapping interval [−τ,0] into R4+0. For the system (2.1), R4+0={(S,I,Q,R)|S≥0,I≥0,Q≥0,R≥0}.
According to the initial condition of system (2.1), we present a theorem addressing the nonnegativity and the boundedness of the solution to system(2.1).
Theorem 2.1. If S(0)≥0,I(0)≥0,Q(0)≥0,R(0)≥0, the solution S∗(t), I∗(t), Q∗(t), R∗(t) to system (2.1) with τ=0 is nonnegative and bounded when t>0.
Proof. First, we prove S∗(t)≥0 when t≥0 under the initial condition of system (2.1).
We assume that S∗(t) is not always nonnegative for t≥0 and make t1 the first time that S∗(t1)=0, S′(t1)<0. According to the first equation of system (2.1), we can obtain S′(t1)=Λ>0. The two conclusions we obtain are contradictory.
Therefore, S∗(t)≥0 when t>0. In the same way, I∗(t)≥0, Q∗(t)≥0, R∗(t)≥0 when t>0. The solution to system (2.1) is positive when t>0.
Then, let N(t)=S(t)+I(t)+Q(t)+R(t), and N(t) represent the total number of people at time t.
Add four equations of Eq. (2.1), and we obtain N′(t)=Λ−μ(S(t)+I(t)+Q(t)+R(t)). Then, we can obtain limt→∞supN(t)=Λμ. Thus, the solution S∗(t), I∗(t), Q∗(t), R∗(t) to system (2.1) is bounded when t>0.
Remark 1: We prove if S(0)≥0,I(0)≥0,Q(0)≥0,R(0)≥0, the solution S∗(t), I∗(t), Q∗(t), R∗(t) to system (2.1) with τ=0 is positive. It is not easy for us to prove that the solution to system (2.1) is positive when τ>0. However, according to our numerical simulation, we can find that when system (2.1) is stable, the solution to system (2.1) is always positive, which is not contradictory to the positivity of the solution to system (2.1).
Next, we consider the dynamic phenomena of system (2.1).
In this section, the system (2.1) is considered, and we first determine the equilibria of the above system. Obviously, the system (2.1) has two equilibria:
E1=(S∗1,I∗1,Q∗1,R∗1),E2=(S∗2,I∗2,Q∗2,R∗2), | (3.1) |
with
S∗1=Λμ,I∗1=0,Q∗1=0,R∗1=0,S∗2=ε+μ+α1+γ1β,I∗2=Λε+μ+α1+γ1−μβ,Q∗2=εα2+γ2+μ(Λε+μ+α1+γ1−μβ),R∗2=1μ(γ1+εγ2α2+γ2+μ)(Λε+μ+α1+γ1−μβ). | (3.2) |
Transferring the equilibria Ek(k=1,2) to the origin, we make w=S+S∗k, x=I+I∗k, y=Q+Q∗k, z=R+R∗k. We can obtain S=w−S∗k, I=x−I∗k, Q=y−Q∗k and R=z−R∗k. Substituting these equations into the model (2.1), we use S, I, Q and R to represent w, x, y and z. We obtain the following model:
{S′(t)=−μS(t)−βS(t)I(t)−βS∗kI(t)−βS(t)I∗k,I′(t)=βS(t)I(t)+βS∗kI(t)+βS(t)I∗k−εI(t−τ)−μI(t)−α1I(t)−γ1I(t),Q′(t)=εI(t−τ)−γ2Q(t)−α2Q(t)−μQ(t),R′(t)=γ1I(t)+γ2Q(t)−μR(t),k=1,2. | (3.3) |
In this subsection, we first demonstrate the stability of the equilibrium E1=(S∗1,I∗1,Q∗1,R∗1)=(Λμ,0,0,0). We obtain the characteristics equation of the linearized system of Eq. (3.3) at the equilibrium E1 as follows:
(λ+μ)2(λ+μ+α2+γ2)(λ+μ+α1+γ1−βS∗1+εe−λτ)=0. | (3.4) |
When τ=0, Eq. (3.4) becomes:
(λ+μ)2(λ+μ+α2+γ2)(λ+μ+α1+γ1−βS∗1+ε)=0. | (3.5) |
We show the following assumption:
(H1) βS∗1−(ε+μ+α1+γ1)<0.
Under (H1), the four roots of Eq. (3.5) have negative real parts due to μ>0,α1>0,γ1>0, and the equilibrium E1 is locally asymptotically stable when τ=0.
Similarly, for the stability of the other equilibrium E2=(S∗2,I∗2,Q∗2,R∗2), we obtain the characteristic equation of the linearized system of Eq. (3.3) at the equilibrium E2 as follows:
(λ+μ)(λ+μ+α2+γ2)[(λ−ε+εe−λτ)(λ+μ+βI∗2)+β2S∗2I∗2]=0. | (3.6) |
When τ=0, Eq. (3.6) becomes:
(λ+μ)(λ+μ+α2+γ2)[λ2+λ(μ+βI∗2)+β2S∗2I∗2]=0. | (3.7) |
If (H1) is not satisfied, the four roots of Eq. (3.7) have negative real parts due to μ>0,α2>0,γ2>0, and the equilibrium E2 of system (2.1) is locally asymptotically stable when τ=0.
We can find that E1 is the disease-free equilibrium, and E2 is the endemic equilibrium. We calculate the basic reproduction number R0, which is the number of suspected individuals who are infected by the same infectious individual and can estimate the infectiousness of an infectious disease. According to the system (2.1), we can obtain the new infections matrix F and the transition matrix V.
F=[0βS(t)I(t)00],V=[−Λ+μS(t)+βS(t)I(t)εI(t)+μI(t)+α1I(t)+γ1I(t)−εI(t)+γ2Q(t)+α2Q(t)+μQ(t)−γ1I(t)−γ2Q(t)+μR(t)]. |
Then, we make F0 represent the derivative of F at E1 and V0 represent the derivative of V at E1:
F0=[00000βS∗10000000000],V0=[μβS∗1000ε+μ+α1+γ1000−εγ2+α2+μ00−γ1−γ2 μ]. |
The inverse of V0 is:
V−10=[1μ−βS∗1μ(ε+μ+α1+γ1)0001ε+μ+α1+γ1000ε(ε+μ+α1+γ1)(γ2+α2+μ)1γ2+α2+μ001μ[γ1ε+μ+α1+γ1+εγ2(ε+μ+α1+γ1)(γ2+α2+μ)]γ2μ(γ2+α2+μ)1μ.]. |
And we can obtain:
F0V−10=[00000βS∗1ε+μ+α1+γ10000000000]. |
The maximum eigenvalue of F0V−10 is R0:
R0=ρ(F0V−10)=βS∗1ε+μ+α1+γ1. |
We compare R0 with (H1), and we find that if R0<1, the inequality of (H1) exists constantly, and if R0>1, (H1) is not satisfied. This represents if R0<1, the disease-free equilibrium E1 is locally asymptotically stable when τ=0 and if R0>1, the endemic equilibrium E2 is locally asymptotically stable when τ=0.
For equilibrium E1, when τ>0, let λ=iω0(ω0>0) be a root of the following equation:
λ+μ+α1+γ1−βS∗1+εe−λτ=0. | (3.8) |
Substituting λ=iω0 into Eq. (3.8) and separating the real and imaginary parts, we have
{ω0=εsin(ω0τ),βS∗1−(μ+α1+γ1)=εcos(ω0τ). | (3.9) |
Thus, we obtain:
{Q1≜sin(ω0τ)=ω0ε,P1≜cos(ω0τ)=βS∗1−(μ+α1+γ1)ε. | (3.10) |
Adding the square of two equations (3.9), we can obtain
ω20=ε2−[βS∗1−(μ+α1+γ1)]2. | (3.11) |
Thus, ω0=√ε2−[βS∗1−(μ+α1+γ1)]2 when ε2−[βS∗1−(μ+α1+γ1)]2≥0, and substituting it into Eq. (3.10). Since ω0>0 and ε≥0, we can obtain Q1>0 and obtain:
τ(j)1=1ω0[arccos(P1)+2jπ],Q1≥0, | (3.12) |
where Q1,P1 are given in Eq. (3.10), j=0,1,2⋯.
Let λ(τ)=α(τ)+iω(τ) be the root of Eq. (3.8) satisfying α(τ(j)1)=0,ω(τ(j)1)=ω0,j=0,1,2⋯. Then, we have the transversality conditions
Re(dλdτ)−1|λ=iω0,τ=τ(j)1=1ε2>0,j=0,1,2⋯. |
Thus, the system (2.1) undergoes a Hopf bifurcation at equilibrium E1 of system (2.1) when τ=τ(j)1,j=0,1,2,⋯.
For the other equilibrium E2, when τ>0, let λ=iω(ω>0) be the root of the following equation:
(λ−ε+εe−λτ)(λ+μ+βI∗2)+β2S∗2I∗2=0. | (3.13) |
Substituting it into the equation and separating the real and imaginary parts, we have
{ω2+με+εβI∗2−β2S∗2I∗2=(ωε)sin(ωτ)+(με+βεI∗2)cos(ωτ),μω−εω+βωI∗2=(−ωε)cos(ωτ)+(με+βεI∗2)sin(ωτ). | (3.14) |
Equation (3.14) can be presented as:
{Q2≜sin(ωτ)=(ω2+ε(βI∗2+μ)−β2S∗2I∗2)ω−(ε−(βI∗2+μ))(βI∗2+μ)ωε[ω2+(βI∗2+μ)2],P2≜cos(ωτ)=(ω2+ε(βI∗2+μ)−β2S∗2I∗2)(βI∗2+μ)+(ε−(βI∗2+μ))ω2ε[ω2+(βI∗2+μ)2]. | (3.15) |
Adding the square of two equations Eq. (3.14), let z=ω2, and
h(z)=z2+c1z+c0=0, | (3.16) |
where c1=(βI∗2+μ)2−2β2S∗2I∗2,c0=(β2S∗2I∗2)2−2β2S∗2I∗2(βI∗2+μ)ε.
Therefore, we have the following assumption:
(H2) c21−4c0>0,c1<0,c0>0.
(H3) c0<0.
If (H2) holds, then Eq. (3.16) has two positive roots, z1 and z2 (z1<z2). If (H3) holds, then Eq. (3.16) has only one positive root z3. Without loss of generality, substituting ωk=√zk(k=1,2,3) into Eq. (3.15), we obtain:
τ(j)2,k={1ωk[arccos(P2,k)+2jπ],Q2,k≥0,1ωk[2π−arccos(P2,k)+2jπ],Q2,k<0,j=0,1,2⋯, | (3.17) |
where Q2,P2 are given in Eq. (3.15) and Q2,k=Q2|ω=ωk,τ=τ(j)2,k,P2,k=P2|ω=ωk,τ=τ(j)2,k,k=1,2,3.
Note that since Q1=ω0ε is always nonnegative, we can obtain τ(j)1 directly. However, whether Q2 is positive depends on the parameters of Eq. (3.15), and we need to obtain τ(j)2,k, (j=1,2,3) according to the value of Q2.
Furthermore, let λ(τ)=α(τ)+iωk(τ) be the root of Eq. (3.17) satisfying α(τ(j)2,k)=0, ω(τ(j)2,k)=ωk(k=1,2,3;j=0,1,2…).
If (H2) or (H3) holds and zk=ω2k(k=1,2,3), then we can deduce Re(dτdλ)|τ=τ(j)2,k:
Re(dτdλ)|τ=τ(j)2,k=ωk2[(βI∗2+μ)2−2β2S∗2I∗2]ε2[ωk2+(βI∗2+μ)2]=ωk2h′(zk)ε2[ωk2+(βI∗2+μ)2]≠0,k=1,2,3. |
There is a Hopf bifurcation at equilibrium E2 of system (2.1) when τ=τ(j)2,k,j=0,1,2⋯.
Then, we present a stability theorem of system (2.1)'s equilibria E1, E2 and the existence of Hopf bifurcation.
Theorem 3.1. We consider the model (2.1):
(1) If (H1) holds, the equilibrium E1 of the model (2.2) undergoes a Hopf bifurcation when τ=τ(j)1,j=0,1,2⋯, and we obtain the following. When τ∈[0,τ(0)1), the equilibrium E1 is locally asymptotically stable. When τ∈[τ(0)1,+∞), the equilibrium E1 is locally asymptotically unstable.
(2) Under the condition that (H1) is not satisfied, and if (H2) or (H3) holds, the equilibrium E2 of the model (2.1) undergoes a Hopf bifurcation when τ=τ(j)2,k(k=1,2,3), and we obtain the following.
(a) If (H2) holds, h(z) has two positive roots, z1 and z2; then we assume z1<z2, and we obtain h′(z1)<0,h′(z2)>0. Then, ∃m∈N, which can make 0<τ(0)2,2<τ(0)2,1<τ(1)2,2<τ(1)2,1<⋯<τ(m−1)2,1<τ(m)2,2<τ(m+1)2,2. When τ∈[0,τ(0)2,2)∪m⋃l=1(τ(l−1)2,1,τ(l)2,2), the equilibrium of the model is locally asymptotically stable. When τ∈m−1⋃l=0(τ(l)2,2,τ(l)2,1)∪(τ(m)2,2,+∞), the equilibrium is locally asymptotically unstable.
(b) If (H3) holds, h(z) has only one positive root z3, when τ∈[0,τ(0)2,3), the equilibrium E2 is locally asymptotically stable. When τ∈(τ(0)2,3,+∞), the equilibrium E2 is unstable.
Without loss of generality, we denote the critical value τ=τ∗ when the characteristic Eq. (3.4) and Eq. (3.6) have eigenvalue λ=iω∗k,k=1,2, at which system (2.1) undergoes a Hopf bifurcation at equilibrium Ek=(S∗k,I∗k,Q∗k,R∗k),k=1,2.
First, we transform the equilibrium to the origin; then, we use the multiple time scales method, and system (2.1) can be written as
˙X(t)=AX(t)+BX(t−τ)+F(X(t),X(t−τ)), | (4.1) |
where X(t)=(^Sk(t),^Ik(t),^Qk(t),^Rk(t))T, X(t−τ)=(^Sk(t−τ),^Ik(t−τ),^Qk(t−τ),^Rk(t−τ))T,
A=[−μ−βI∗k−βS∗k00βI∗kβS∗k−(μ+α1+γ1)0000−(γ2+α2+μ)00γ1γ2−μ],B=[00000−ε000ε000000],F(X(t),X(t−τ))=[FSFIFQFR]=[Λ−μS∗k−βSkIk−βS∗kI∗kβSkIk+βS∗kI∗k−(ε+μ+α1+γ1)I∗kεI∗k−(μ+α2+γ2)Q∗kγ1I∗k+γ2Q∗k−μR∗k]. |
We make ˜˙X(t)=AX(t)+BX(t−τ) the linear system of system (4.1). We assume that hk(k=1,2) is the eigenvector of the eigenvalue λ=iω∗k(k=1,2) of the linear system and that h∗k is the eigenvector of the eigenvalue λ=−iω∗k of the linear system. We make ˜I the unit matrix, and hk satisfies (λ˜I−A−Be−λτ)hk=0, where λ=iω∗k, and h∗k satisfies (λ˜I−A−Be−λτ)Thk∗=0, where λ=−iω∗k. Additionally, ⟨h∗k,hk⟩=¯h∗kThk=1, and we can obtain
hk=(hk1,hk2,hk3,hk4)T=(−βS∗kλ+μ+βI∗k,1,εe−λτλ+μ+α2+γ2,γ1λ+μ+γ2εe−λτ(λ+μ)(λ+μ+α2+γ2))T,hk∗=(h∗k1,h∗k2,h∗k3,h∗k4)T=dk(βI∗kλ+μ+βI∗k,1,0,0)T, | (4.2) |
where dk=(−β2S∗kI∗k(μ+βI∗k)2+ω∗k2+1)−1,k=1,2.
We use the multiple time scale method to deduce the normal form of the Hopf bifurcation and assume the solution to Eq. (4.1) is
X(t)=X(T0,T1,T2,⋯)=∞∑k=1ϵkXk(T0,T1,T2,⋯), | (4.3) |
where X(T0,T1,T2,⋯)=[S(T0,T1,T2,⋯),I(T0,T1,T2,⋯),Q(T0,T1,T2,⋯),R(T0,T1,T2,⋯)]T, Xk(T0,T1,T2,⋯)=[Sk(T0,T1,T2,⋯),Ik(T0,T1,T2,⋯),Qk(T0,T1,T2,⋯),Rk(T0,T1,T2,⋯)]T, k=1,2, Ti=ϵit,i=0,1,2,⋯ and Ti is the scaling transform in the time direction.
The derivative with respect to t is
ddt=∂∂T0+ϵ∂∂T1+ϵ2∂∂T2+⋯=D0+ϵD1+ϵ2D2+⋯, |
where Di=∂∂Ti,i=0,1,2,⋯.
Note that Xj=(Sj,Ij,Qj,Rj)T=Xj(T0,T1,T2,⋯),Xj,τc=(Sj,τc,Ij,τc,Qj,τc,Rj,τc)T=Xj(T0−τc,T1,T2,⋯),j=1,2,⋯.
Then, we can obtain
⋅X(t)=ϵD0X1+ϵ2D1X1+ϵ3D2X1+ϵ2D0X2+ϵ3D1X2+ϵ3D0X3+⋯. | (4.4) |
We consider that τ is the bifurcation parameter, and we set τ=τc+ϵτε, where τc=τ(j)k(k=1,2) is the critical value of the Hopf bifurcation, τε is the perturbation parameter, and ϵ is the dimensionless parameter.
Using Taylor series expansion of X(t−τ), we obtain that
X(t−τ)=ϵX1,τc+ϵ2X2,τc+ϵ3X3,τc−ϵ2τεD0X1,τc−ϵ3τεD0X2,τc−ϵ2τcD1X1,τc−ϵ3τεD1X1,τc−ϵ3τcD2X1,τc−ϵ3τcD1X2,τc+⋯, | (4.5) |
where Xj,τc=Xj(T0−τc,T1,T2,⋯),j=1,2,⋯.
Then, substituting the solution with the multiple scales Eqs. (4.3)-(4.5) into Eq. (4.1) and balancing the coefficients of ϵn(n=1,2,3), we obtain a set-ordered linear differential equation:
{D0Sk1+(μ+βI∗k)Sk1+βS∗kIk1=0,D0Ik1−βI∗kSk1+(μ+α1+γ1)Ik1−βS∗kIk1+εIk1,τc=0,D0Qk1+(μ+α2+γ2)Qk1−εQk1,τc=0,D0Rk1−γ1Ik1−γ2Qk1+μRk1=0,k=1,2. | (4.6) |
Since ±iω∗k(k=1,2) are the eigenvalues of the characteristic Eq. (4.1), the solution to Eq. (4.6) can be expressed in the form of Eq. (4.7):
X1(T1,T2,T3,⋯)=G1(T1,T2,T3,⋯)eiω∗kT0hk+ˉG1(T1,T2,T3,⋯)e−iω∗kT0ˉhk,k=1,2. | (4.7) |
where G1 and ˉG1 represent the coordinates on the center manifold, and the linear part of the center manifold can be presented as equation (4.7).
Next, for the ϵ2-order terms, we can obtain the following equations:
{D0Sk2+(μ+βI∗k)Sk2+βS∗kIk2=−D1Sk1−βSk1Ik1,D0Ik2−βI∗kSk2+(μ+α1+γ1)Ik2−βS∗kIk2+εIk2,τc=ετεD0Ik1,τc+ετcD1Ik1,τc−D1Ik1+βSk1Ik1,D0Qk2+(μ+α2+γ2)Qk2−εQk2,τc=−ετεD0Qk1,τc−ετcD1Qk1,τc−D1Qk1,D0Rk2−γ1Ik2−γ2Qk2+μRk2=−D1Rk1,k=1,2. | (4.8) |
We substitute Eq. (4.7) into the right expression of Eq. (4.8), and the coefficients before eiω∗kT0 are denoted by the vector m1. According to the solvability condition ⟨h∗k,m1⟩=0, we can solve ∂G∂T1 as follows:
∂G1∂T1=MkτεG1, | (4.9) |
where Mk=εe−iω∗kτc[βI∗k(iω∗k+μ+βI∗k)(μ+βI∗k)2+ω∗k2βS∗kiω∗k+μ+βI∗k+1−ετce−iω∗kτc]−1,k=1,2.
We assume:
Sk2=fk1eiω∗kT0G1+ˉfk1e−iω∗kT0ˉG1+gk1e2iω∗kT0G12+ˉgk1e−2iω∗kT0ˉG12+lk1G1ˉG1,Ik2=fk2eiω∗kT0G1+ˉfk2e−iω∗kT0ˉG1+gk2e2iω∗kT0G12+ˉgk2e−2iω∗kT0ˉG12+lk2G1ˉG1,Qk2=fk3eiω∗kT0G1+ˉfk3e−iω∗kT0ˉG1+gk3e2iω∗kT0G12+ˉgk3e−2iω∗kT0ˉG12+lk3G1ˉG1,Rk2=fk2eiω∗kT0G1+ˉfk4e−iω∗kT0ˉG1+gk4e2iω∗kT0G12+ˉgk4e−2iω∗kT0ˉG12+lk4G1ˉG1. | (4.10) |
Substituting Eq. (4.10) into Eq. (4.8), we obtain:
fk1=U−1τε(iω∗k+μ+α2+γ2−εe−iω∗kτc)[(−Mkhk1)(iω∗k+μ+α1+γ1−βS∗k+εe−iω∗kτc)−βS∗k(iω∗kεe−iω∗kτchk2+ετcMkhk2−Mkhk2)](iω∗k+μ),fk2=U−1τε(iω∗k+μ+α2+γ2−εe−iω∗kτc)[(iω∗k+μ+βI∗k)(iω∗kεe−iω∗kτchk2+ετcMkhk2−Mkhk2)+βI∗k(−Mkhk1)](iω∗k+μ),fk3=τε(iω∗k+μ)[(iω∗k+μ+βI∗k)(iω∗k+μ+α1+γ1−βS∗k+εe−iω∗kτc)(iω∗k+μ+α2−εe−iω∗kτc+γ2)+β2S∗kI∗k(iω∗k+μ+α2+γ2−εe−iω∗kτc)]U−1,fk4=τε{(iω∗k+μ+βI∗k)[(−Mkhk4)(iω∗k+μ+α1+γ1−βS∗k+εe−iω∗kτc)(iω∗k+μ−εe−iω∗kτc+α2+γ2)+γ1(iω∗kεe−iω∗kτchk2+ετcMkhk2−Mkhk2)(iω∗k+μ+α2+γ2−εe−iω∗kτc)+γ2(iω∗k+μ+α1+γ1−βS∗k+εe−iω∗kτc)(−Mkhk3−ετεMke−iω∗kτchk3−iω∗kεe−iω∗kτchk3)]+βI∗k[βS∗k(iω∗k+μ+α2+γ2−εe−iω∗kτc)(−Mkhk4)+γ1(iω∗kεe−iω∗kτchk2+ετcMkhk2−Mkhk2)(iω∗k+μ+α2+γ2−εe−iω∗kτc)+γ2βS∗k(−Mkhk3−iω∗kεMke−iω∗kτchk3−εe−iω∗kτchk3)]}U−1,gk1=(2iω∗k+μ)(2iω∗k+μ+α2+γ2−εe−2iω∗kτc)[(−βhk1hk2)(2iω∗k+μ+α1+γ1+εe−2iω∗kτc−βS∗k)+β2S∗khk1hk2]V−1,gk2=(2iω∗k+μ)(2iω∗k+μ+α2+γ2−εe−2iω∗kτc)[βhk1hk2(2iω∗k+μ+βI∗k)−β2I∗khk1hk2]V−1,gk3=0,gk4=(2iω∗k+μ+α2+γ2−εe−2iω∗kτc)[(2iω∗k+μ+βI∗k)(βhk1ˉhk2+βhk2ˉhk1)+βI∗k(−βhk1ˉhk2−βhk2ˉhk1)]γ1V−1,lk1=μ(μ+α2+γ2−ε)[(−βhk1ˉhk2−βhk2ˉhk1)(μ+α1+γ1−βS∗k+ε)+βS∗k(βhk1ˉhk2+βhk2ˉhk1)]W−1,lk2=W−1μ(μ+α2+γ2−ε)[(βhk1ˉhk2+βhk2ˉhk1)(μ+βI∗k)+βI∗k(−βhk1ˉhk2−βhk2ˉhk1)],lk3=0,lk4=(μ+α2+γ2−ε)γ1[(βhk1ˉhk2+βhk2ˉhk1)(μ+βI∗k)+βI∗k(−βhk1ˉhk2−βhk2ˉhk1)]W−1,U=(iω∗k+μ)(iω∗k+μ+α2+γ2−εe−iω∗kτc)[(iω∗k+μ+βI∗k)(iω∗k+μ+α1+γ1−βS∗k+εe−iω∗kτc)+β2S∗kI∗k],V=(2iω∗k+μ)(2iω∗k+μ+α2+γ2−εe−2iω∗kτc)[(2iω∗k+μ+βI∗k)(2iω∗k+μ+α1+γ1−βS∗k+εe−2iω∗kτc)+β2S∗kI∗k],W=μ(μ+α2+γ2−ε)[(μ+βI∗k)(μ+α1+γ1−βS∗k+ε)+β2S∗kI∗k],k=1,2. | (4.11) |
Next, balancing the ϵ3-order terms, we can obtain equations as follows:
{D0S3+(μ+βI∗k)S3+βS∗kI3=−D2S1−D1S2−βS2I1−βS1I2,D0I3−βI∗kS3+(μ+α1+γ1)I3−βS∗kI3+εI3,τc=−D2I1−D1I2+βS2I1+βS1I2+ε[τε(D0I2,τc−D1I1,τc)−τc(D2I1,τc+D1I2,τc)],D0Q3+(μ+α2+γ2)Q3−εQ3,τc=−ε(τεD0Q2,τc−τεD1Q1,τc−τcD2Q1,τc−τcD1Q2,τc)−D2Q1−D1Q2,D0R3−γ1I3−γ2Q3+μR3=−D2R1−D1R2,k=1,2. | (4.12) |
Substituting Eq. (4.7) and Eq. (4.10) into the right expression of Eq. (4.12), and the coefficients before \rm{e}^{i \omega_k T_0} are denoted by the vector m_2 . According to the solvability condition \langle {{h}_1^{*}}, m_2 \rangle = 0 and note that \tau_\varepsilon^2 is small enough for small unfolding parameter \tau_\varepsilon , we ignore the term \tau_\varepsilon^{2}G_1 . Then, we have:
\begin{equation} \frac{\partial G_1}{\partial {{T}_{2}}}\rm{ = }H_k\chi_k {{G}_1^{2}}{\bar G_1}, \end{equation} | (4.13) |
where
\begin{equation*} \begin{split} \begin{aligned} & {{H}_{k}} = \frac{1}{\frac{\beta I_{k}^{*}(\mu +\beta I_{k}^{*}-i\omega_k^{*} )}{{{(\mu +\beta I_{k}^{*})}^{2}}+{{\omega_k^{*} }^{2}}}{{h}_{k1}}+{{h}_{k2}}+\varepsilon {{\tau }_{c}}{\rm{e}^{-i\omega_k^{*} {{\tau }_{c}}}}{{h}_{k2}}}, \\ & {{\chi }_{k}} = -\frac{\beta I_{k}^{*}(\mu +\beta I_{k}^{*}-i\omega_k^{*} )}{{{(\mu +\beta I_{k}^{*})}^{2}}+{{\omega_k^{*} }^{2}}}\beta {{g}_{k1}}{\bar {h}_{k2}}-\frac{\beta I_{k}^{*}(\mu +\beta I_{k}^{*}-i\omega_k^{*} )}{{{(\mu +\beta I_{k}^{*})}^{2}}+{{\omega_k^{*} }^{2}}}\beta {{g}_{k2}}{\bar {h}_{k2}}+\beta { {g}_{k1}}{\bar {h}_{k2}}+\beta {{g}_{k2}}{\bar {h}_{k2}}, k = 1, 2, \\ \end{aligned} \end{split} \end{equation*} |
where g_k are given in Eq. (4.11) and h_k are given in Eq. (4.2).
Let G_1 \mapsto (G_1/\varepsilon) , and we can obtain the normal form of Hopf bifurcation of system (2.1) as:
\begin{equation} \overset{\centerdot }{\mathop{G_1}}\, = M_k\tau_{\varepsilon} G_1+H_k\chi_k {{G}^{2}}\overset{-}{\mathop{G_1}}, \end{equation} | (4.14) |
where M_k is given in Eq. (4.9), and H_k, \chi_k are given in Eq. (4.13).
Let G = r{\rm{e}^{i\theta_k^{*} }}(k = 1, 2) and substitute it into Eq. (4.14), and we can obtain the normal form of Hopf bifurcation in polar coordinates:
\begin{equation} \left\{ \begin{matrix} \overset{\centerdot }{\mathop{r}}\, = \operatorname{Re}(M_k){{\tau }_{\varepsilon }}r+\operatorname{Re}(H_k\chi_k ){{r}^{3}}, \\ \overset{\centerdot }{\mathop{\theta_k^{*} }}\, = \operatorname{Im}(M_k){{\tau }_{\varepsilon }}+\operatorname{Im}(H_k\chi_k ){{r}^{2}}, \\ \end{matrix} \right. \end{equation} | (4.15) |
where M_k is given in Eq. (4.9), and H_k, \chi_k are given in Eq. (4.13) (k = 1, 2) .
According to the normal form of bifurcation in polar coordinates, there is a theorem as follows:
Theorem 4.1. For system (4.15), if \frac{\operatorname{Re}(M_k){{\tau }_{\varepsilon }}}{\operatorname{Re}(H_k\chi_k)} < 0 (k = 1, 2) holds, then system (2.1) exists periodic solutions near equilibrium E_k :
(1) If \operatorname{Re}(M_k){{\tau }_{\varepsilon }} < 0 , the bifurcating periodic solutions are unstable.
(2) If \operatorname{Re}(M_k){{\tau }_{\varepsilon }} > 0 , the bifurcating periodic solutions are locally asymptotically stable.
In this section, according to the natural mortality in China and the data presented in ref. [32], we choose \Lambda = 100, \mu = 0.00713, \varepsilon = 0.8, \gamma_1 = 0.1, \gamma_2 = 0.1, \alpha_1 = 0.025, \alpha_2 = 0.025, \beta = 0.00001 . According to Eq. (3.2), we obtain S_{1}^{*} = \frac{\Lambda }{\mu } = 14025, I_{1}^{*} = 0, Q_{1}^{*} = 0, R_{1}^{*} = 0, S_2^{*} = 93213, I_2^{*} = -605.719, Q_2^{*} = -3667.4, R_2^{*} = -59932. Obviously, assumption (H1) holds and equilibrium E_1 is locally asymptotically stable when \tau = 0 . However, the equilibrium E_2 is unstable for I_2^{*} < 0 .
Remark 2: If I_2^{*} < 0 , the number of infected individuals is less than zero, which is inconsistent with the facts. Additionally, according to our theoretical analysis, when I_2^{*} < 0 , the equilibrium E_2 is unstable, which is consistent with the facts.
Substituting these parameter values into Eqs. (3.10)-(3.12), by using MATLAB, we can obtain \omega_0 = 0.8000, Q_1 = 0.0102, P_1 = 0.9999, \tau_1^{(0)} = 1.9509 .
Thus, the equilibrium E_1 is locally asymptotically stable when \tau\in[0, \tau_1^{(0)}) , and Hopf bifurcation occurs near the equilibrium E_1 when \tau = \tau_1^{(0)} . According to Eq. (4.9) and Eq. (4.13), we obtain \rm{Re}(M_1) < 0 , \rm{Re}({{H}_{1}}{{\chi }_{1}}) > 0 . The bifurcating periodic solution is unstable due to Theorem 4.1. We find that if the time delay from infection to isolation is over the critical value, the epidemic will not be controlled under a stable situation, and the epidemic situation will be more severe.
When \tau = 0 , we choose the initial value (1500, 1000,100,200) , and the solution corresponding to a locally asymptotically stable equilibrium is shown in Figure 2. Thus, when \tau = 0 , once the individuals are infected, they will be quarantined and they will not infect others, and the epidemic situation can be controlled into a stable situation.
When \tau = 1.5\in (0, \tau_1^{(0)}) , we choose initial values (1,500, 1,000,100,200) , and the equilibrium E_1 of system (2.1) is locally asymptotically stable. See Figure 3.
When \tau = 2\in (\tau_1 ^{(0)}, +\infty) , we choose the initial values (1,500, 1,000,100,200) , and the equilibrium E_1 is unstable, as shown in Figure 4. This means that if \tau is larger than the critical value \tau_1^{(0)} , the epidemic situation cannot be controlled into a stable state.
According to Figures 2-4, we find that the time delay from infection to isolation has a great influence on the epidemic situation in E_1 , and with the increasing time delay, the epidemic situation will become increasingly severe.
According to the natural mortality in China and the data presented in ref. [32], we choose another group of parameters, namely, \Lambda = 200, \mu = 0.00713, \varepsilon = 0.8, \gamma_1 = 0.05, \gamma_2 = 0.05, \alpha_1 = 0.025, \alpha_2 = 0.025, \beta = 0.0001 . According to Eq. (3.2), we obtain S_1^{*} = 28,050, I_1^{*} = 0, Q_1^{*} = 0, R_1^{*} = 0, S_{2}^{*} = 8,821.300, I_{2}^{*} = 155.424, Q_{2}^{*} = 1,513.9, R_{2}^{*} = 11,703. We can find that assumption (H1) does not hold and the equilibrium E_1 is unstable. However, the equilibrium E_2 is locally asymptotically stable when \tau = 0 .
Substituting these parameter values into Eq. (3.16), we can obtain c_1 = -0.0269, c_2 = -0.00030938 , and Eq. (3.16) has only one positive root, z_1 = 0.0356 . According to Eq. (3.15)and Eq. (3.17), we can obtain \omega_2 = 0.1887, Q_2 = 0.9892, P_1 = 0.1463, \tau_{2, 3}^{(0)} = 0.7782 by using MATLAB.
Thus, the equilibrium E_2 is locally asymptotically stable for \tau\in[0, \tau_{2, 3}^{(0)}) , and Hopf bifurcation occurs near the equilibrium E_2 when \tau = \tau_{2, 3}^{(0)} . According to Eq. (4.9) and Eq. (4.13), we obtain \rm{Re}(M_1) > 0, \rm{Re}({{H}_{1}}{{\chi }_{1}}) > 0 . Thus, \rm{Re}(M_1)\tau_{\varepsilon } < 0 , the bifurcation solution is unstable. In E_2 , the epidemic situation cannot be controlled under a stable situation.
When \tau = 0 , we choose the initial values (1,000, 50, 2,000, 1,000) , and the solution corresponding to a locally asymptotically stable equilibrium is shown in Figure 5.
When \tau = 0.5\in (0, \tau_{2, 3}^{(0)}) , we choose the initial value (1,000, 50, 2,000, 1,000) , and the equilibrium E_2 of system (2.1) is locally asymptotically stable. See Figure 6.
When \tau\in (\tau_{2, 3}^{(0)}, +\infty), \tau_{2, 3}^{(0)} = 0.7782 , the equilibrium E_2 is unstable. We choose \tau = 0.8 > \tau_{2, 3}^{(0)} = 0.7782 and the initial value (1,000, 50, 2,000, 1,000) . See Figure 7.
According to numerical simulations, we can find that the smaller \tau is, the better the control effect is. The results of the numerical simulation are consistent with this fact. Then, we provide some explanations for the above results.
Remark 3: For the two groups of parameter values that are \Lambda = 100, \mu = 0.00713, \varepsilon = 0.8, \gamma_1 = 0.1, \gamma_2 = 0.1, \alpha_1 = 0.025, \alpha_2 = 0.025, \beta = 0.00001 and \Lambda = 200, \mu = 0.00713, \varepsilon = 0.8, \gamma_1 = 0.05, \gamma_2 = 0.05, \alpha_1 = 0.025, \alpha_2 = 0.025, \beta = 0.0001 , we have the following conclusions.
For the first group of parameter values, we can control \tau < 1.9509 , and the epidemic situation can gradually become stable. However, if \tau > 1.9509 , the epidemic cannot be controlled into a stable situation. This means that if we can make the time delay from infection to isolation within 1.9509, we can make the number of infected individuals gradually tend to zero, and the epidemic situation will be controllable. However, if we cannot make the time delay from infection to isolation within 1.9509, the epidemic situation will be uncontrollable.
For the second group of parameter values, we can control \tau < 0.7782 , and the epidemic situation can be gradually stable. If \tau > 0.7782 , the epidemic cannot be controlled into a stable situation. This means that if we can make the time delay from infection to isolation within 0.7782, we can make the number of infected individuals gradually tend to stabilize, and the epidemic situation will be controllable. However, if we cannot make the time delay from infection to isolation within 0.7782, the epidemic situation will be uncontrollable.
Remark 4: According to our theoretical analysis, with the increase in \tau , the epidemic situation will be increasingly severe, and COVID-19 will further threaten the health of humans. To control the epidemic situation, we need to take some measures to shorten the time from infection to isolation. In addition, for epidemic situations in different areas, we can deduce the corresponding parameter values and use the model we constructed to obtain the stability determination and the strategies to control epidemic situations.
Our work is reproducible. For different regions, we can calculate the different critical time delays by changing variables such as the transmission rate from S to I , the transmission rate from I to Q and the transmission rate from I to R . We can obtain the critical time delay based on Eqs. (3.10)-(3.12) and Eqs. (3.15)-(3.17). Then, according to Theorem 3.1, we know that the equilibria are locally asymptotically stable if the time delay is within the critical time delay. Finally, we can obtain the normal form of Hopf bifurcation by using the multiple time scales method and determine whether the epidemic situation can be controlled into a stable situation based on Theorem 4.1. The time delay from infection to isolation is the most important variable in our model. We calculate the critical time delay based on the parameter values. We choose parameter values according to the natural mortality in China and the data presented in ref. [32]. Then, we change the value of the time delay from infection to isolation, and we know that the equilibria are locally asymptotically stable if the time delay is within the critical time delay through numerical simulations.
In this paper, we constructed a DDE according to the characteristics of the COVID-19 epidemic phenomenon based on the SIQR model. We considered the existence and stability of equilibria in the above delayed SIQR model. We also analyzed the existence and dynamic properties of Hopf bifurcation associated with both equilibria. We chose two groups of parameter values according to the natural mortality in China and the data presented in ref. [32]. Numerical simulations were carried out to verify the analytical results.
According to the results of numerical simulations, the epidemic solution would be more severe with increasing the time delay from infection to isolation. In addition, we predicted the stability of epidemic solutions in different areas and provided effective strategies to control the epidemic.
This study was funded by Fundamental Research Funds for the Central Universities of China. (Grant No. 2572019BC14), the Heilongjiang Provincial Natural Science Foundation of China (Grant No. LH2019A001) and College Students Innovations Special Project funded by Northeast Forestry University of China (No. 202010225035).
All authors declare no conflicts of interest in this paper.
[1] | Korakianitis N, Vokas GA, Ioannides G (2019) Review of wireless power transfer (WPT) on electric vehicles (EVs) charging. AIP Conf Proc, 020072. https://doi.org/10.1063/1.5138558 |
[2] |
Panchal C, Stegen S, Lu J (2018) Review of static and dynamic wireless electric vehicle charging system. Eng Sci Technol, Int J 21: 922–937. https://doi.org/10.1016/j.jestch.2018.06.015 doi: 10.1016/j.jestch.2018.06.015
![]() |
[3] |
Zhang W, Mi CC (2015) Compensation topologies of high-power wireless power transfer systems. IEEE Trans Veh Technol 65: 4768–4778. https://doi.org/10.1109/TVT.2015.2454292 doi: 10.1109/TVT.2015.2454292
![]() |
[4] |
Wu J, Kong W, Gao P, et al. (2021) Design considerations of bidirectional wireless power transfer and full-duplex communication system via a shared inductive channel. Energies 14: 4918. https://doi.org/10.3390/en14164918 doi: 10.3390/en14164918
![]() |
[5] |
Budhia M, Covic GA, Boys JT (2011) Design and optimization of circular magnetic structures for lumped inductive power transfer systems. IEEE Trans Power Electron 26: 3096–3108. https://doi.org/10.1109/TPEL.2011.2143730 doi: 10.1109/TPEL.2011.2143730
![]() |
[6] |
Buja G, Rim CT, Mi CC (2016) Dynamic charging of electric vehicles by wireless power transfer. IEEE Trans Ind Electron 63: 6530–6532. https://doi.org/10.1109/TIE.2016.2596238 doi: 10.1109/TIE.2016.2596238
![]() |
[7] |
Laporte S, Coquery G, Deniau V, et al. (2019) Dynamic wireless power transfer charging infrastructure for future EVs: From experimental track to real circulated roads demonstrations. World Electr Veh J 10: 84. https://doi.org/10.3390/wevj10040084 doi: 10.3390/wevj10040084
![]() |
[8] |
Li W, Zhao H, Deng J, et al. (2016) Comparison study on SS and double-sided LCC compensation topologies for EV/PHEV wireless chargers. IEEE Trans Veh Technol 65: 4429–4439. https://doi.org/10.1109/TVT.2015.2479938 doi: 10.1109/TVT.2015.2479938
![]() |
[9] |
Lu J, Zhu G, Wang H, et al. (2019) Sensitivity analysis of inductive power transfer systems with voltage-fed compensation topologies. IEEE Trans Veh Technol 68: 4502–4513. https://doi.org/10.1109/TVT.2019.2903565 doi: 10.1109/TVT.2019.2903565
![]() |
[10] | Zhao J, Cai T, Duan S, et al. (2016) A general design method of primary compensation network for dynamic WPT system maintaining stable transmission power. IEEE Trans Power Electron 31: 8343–8358. |
[11] | Yang J, Zhang X, Zhang K, et al. (2020) Design of LCC-S compensation topology and optimization of misalignment tolerance for inductive power transfer. IEEE Access 8: 191309–191318. |
[12] | Qi J (2020) Analysis, design, and optimisation of an LCC/S compensated WPT system featured with wide operation range. IET Power Electron 13: 1687–1924. |
[13] | Wang Y, Wang Z, Xu Z (2024) Analysis and comparison of the compensation topologies of magnetic coupled resonant wireless charging for electric vehicles. Highlights Sci Eng Technol 119: 394–405. |
[14] |
Zhu Q, Wang L, Guo Y, et al. (2016) Applying LCC compensation network to dynamic wireless EV charging system. IEEE Trans Ind Electron 63: 6557–6567. https://doi.org/10.1109/TIE.2016.2529561 doi: 10.1109/TIE.2016.2529561
![]() |
[15] | Chen Y, Yang N, Li Q, et al. (2019) New parameter tuning method for LCC/LCC compensated IPT system with constant voltage output based on LC resonance principles. IET Power Electron 12: 2433–2712. https://doi.org /10.1049/iet-pel.2018.6129 |
[16] |
Feng H, Cai T, Duan S, et al. (2016) An LCC-compensated resonant converter optimized for robust reaction to large coupling variation in dynamic wireless power transfer. IEEE Trans Ind Electron 63: 6591–6601. https://doi.org/10.1109/TIE.2016.2589922 doi: 10.1109/TIE.2016.2589922
![]() |
[17] |
Li S, Li W, Deng J, et al. (2015) A double-sided LCC compensation network and its tuning method for wireless power transfer. IEEE Trans Veh Technol 64: 2261–2273. https://doi.org/10.1109/TVT.2014.2347006 doi: 10.1109/TVT.2014.2347006
![]() |
[18] | Korakianitis N, Theodorakis A, Vokas GA, et al. (2023) A simulation comparison and evaluation of wireless power transfer methods and topologies applicable on static and dynamic wireless electric vehicles charging systems. AIP Conf Proc, 2769. |
[19] | Kurs A, Karalis A, Moffatt R, et al. (2007) Wireless power transfer via strongly coupled magnetic resonances. Science 317: 83–86. |
[20] | Hekal S, Allam A, Abdel‐Rahman A, et al. (2019) Compact size wireless power transfer using defected ground structures. Energy Syst Electr Eng. https://doi.org/10.1007/978-981-13-8047-1 |
[21] | Ko YD, Jang YJ (2013) The optimal system design of the online electric vehicle utilizing wireless power transmission technology. IEEE Trans Intell Transp Syst 14: 1255–1265. |
[22] |
Mi CC, Buja G, Choi SY, et al. (2016) Modern advances in wireless power transfer systems for roadway powered electric vehicles. IEEE Trans Ind Electron 63: 6533–6545. https://doi.org/10.1109/TIE.2016.2574993 doi: 10.1109/TIE.2016.2574993
![]() |
[23] | KAIST (2025) KAIST OLEV (On-Line Electric Vehicle) to begin operation! Available from: https://www.kaist.ac.kr/newsen/html/news/?mode=V&mng_no=3960. |
[24] | Kavitha M, Mohan Reddy D, Kalyan Chakravarthy NS (2022) Electrical vehicles (EVs)—An application of wireless power transfer (WPT) system. AI Enabled IoT Electrification Connected Transp, 165–189. https://doi.org/10.1007/978-981-19-2184-1_8 |
[25] | Guidi G, Suul JA (2018) Transient control of dynamic inductive EV charging and impact on energy efficiency when passing a roadside coil section. IEEE PELS Workshop Emerg Technol Wireless Power Transfer. https://doi.org/10.1109/WOW.2018.8450893 |
[26] |
Fisher TM, Farley KB, Gao Y, et al. (2014) Electric vehicle wireless charging technology: A state-of-the-art review of magnetic coupling systems. Wireless Power Transfer 1: 87–96. https://doi.org/10.1017/wpt.2014.8 doi: 10.1017/wpt.2014.8
![]() |
[27] |
Sagar A, Kashyap A, Nasab MA, et al. (2023) A comprehensive review of the recent development of wireless power transfer technologies for electric vehicle charging systems. IEEE Access 11: 83703–83751. https://doi.org/10.1109/ACCESS.2023.3300475 doi: 10.1109/ACCESS.2023.3300475
![]() |
[28] | Electreon (2025) Building the infrastructure to support mass EV adoption. Available from: https://electreon.com/projects. |
[29] |
Si P, Hu AP, Malpas S, et al. (2008) A frequency control method for regulating wireless power to implantable devices. IEEE Trans Biomed Circuits Syst 2: 22–29. https://doi.org/10.1109/TBCAS.2008.918284 doi: 10.1109/TBCAS.2008.918284
![]() |
[30] |
Liu H, He L, Cheng B (2024) Powerless fractional-order tuning wireless power transfer system with zero phase angle input. IEEE Trans Circuits Syst I 72: 433–442. https://doi.org/10.1109/TCSI.2024.3404647 doi: 10.1109/TCSI.2024.3404647
![]() |
[31] |
Liu H, He L, Cheng B (2024) A powerless fractional-order WPT system with extended zero voltage switching region. IEEE Trans Ind Electron 72: 3525–3533. https://doi.org/10.1109/TIE.2024.3447733 doi: 10.1109/TIE.2024.3447733
![]() |
[32] |
He L, Huang X, Cheng B (2023) Robust class E2 wireless power transfer system based on parity-time symmetry. IEEE Trans Power Electron 38: 4279–4288. https://doi.org/10.1109/TPEL.2022.3230852 doi: 10.1109/TPEL.2022.3230852
![]() |
[33] | Assawaworrarit S (2020) Robust dynamic wireless power transfer based on parity-time symmetry. Available from: https://purl.stanford.edu/sd395gx8779. |
[34] |
Zhang H, Zhu K, Guo Z, et al. (2023) Robustness of wireless power transfer systems with parity-time symmetry and asymmetry. Energies 16: 4605. https://doi.org/10.3390/en16124605 doi: 10.3390/en16124605
![]() |
[35] | Huang X, He LZ (2021) Stable and efficient class E2 wireless power transfer system based on parity-time symmetry. IEEE PELS Workshop Emerg Technol Wireless Power Transfer. https://doi.org/10.1109/WOW51332.2021.9462866 |
[36] |
Houran MA, Yang X, Chen W (2018) Magnetically coupled resonance WPT: review of compensation topologies, resonator structures with misalignment, and EMI diagnostics. Electronics 7: 296. https://doi.org/10.3390/electronics7110296 doi: 10.3390/electronics7110296
![]() |
[37] |
Villa JL, Sallán J, Sanz Osorio JF, et al. (2012) High-misalignment tolerant compensation topology for ICPT systems. IEEE Trans Ind Electron 59: 945–951. https://doi.org/10.1109/TIE.2011.2161055 doi: 10.1109/TIE.2011.2161055
![]() |
[38] |
Zhao L, Thrimawithana DJ, Madawala UK (2017) Hybrid bidirectional wireless EV charging system tolerant to pad misalignment. IEEE Trans Ind Electron 64: 7079–7086. https://doi.org/10.1109/TIE.2017.2686301 doi: 10.1109/TIE.2017.2686301
![]() |
[39] |
Ramezani A, Narimani M (2020) Optimized electric vehicle wireless chargers with reduced output voltage sensitivity to misalignment. IEEE J Emerg Sel Top Power Electron 8: 3569–3581. https://doi.org/10.1109/JESTPE.2019.2958932 doi: 10.1109/JESTPE.2019.2958932
![]() |
[40] |
Dehui W, Qisheng S, Xiaohong W, et al. (2018) Analytical model of mutual coupling between rectangular spiral coils with lateral misalignment for wireless power applications. IET Power Electron 11: 781–960. https://doi.org/10.1049/iet-pel.2017.0470 doi: 10.1049/iet-pel.2017.0470
![]() |
[41] |
Chen Y, He S, Yang B, et al. (2023) Reconfigurable rectifier-based detuned series-series compensated IPT system for anti-misalignment and efficiency improvement. IEEE Trans Power Electron 38: 2720–2729. https://doi.org/10.1109/TPEL.2022.3204592 doi: 10.1109/TPEL.2022.3204592
![]() |
[42] |
Lopez-Alcolea FJ, Del Real JV, Roncero-Sanchez P, et al. (2020) Modeling of a magnetic coupler based on single- and double-layered rectangular planar coils with in-plane misalignment for wireless power transfer. IEEE Trans Power Electron 35: 5102–5121. https://doi.org/10.1109/TPEL.2019.2944194 doi: 10.1109/TPEL.2019.2944194
![]() |
[43] |
Rahulkumar J, Narayanamoorthi R, Vishnuram P, et al. (2023) An empirical survey on wireless inductive power pad and resonant magnetic field coupling for in-motion EV charging system. IEEE Access 11: 4660–4693. https://doi.org/10.1109/ACCESS.2022.3232852 doi: 10.1109/ACCESS.2022.3232852
![]() |
[44] | ICNIRP (1998) Guidelines for limiting exposure to time-varying electric, magnetic, and electromagnetic fields (up to 300 GHz). Available from: https://www.icnirp.org/cms/upload/publications/ICNIRPemfgdl.pdf. |
[45] |
Ziegelberger G (2009) ICNIRP statement on the guidelines for limiting exposure to time-varying electric, magnetic, and electromagnetic fields (up to 300 GHz). Health Phys 97: 257–258. https://doi.org/10.1097/HP.0b013e3181aff9db doi: 10.1097/HP.0b013e3181aff9db
![]() |
[46] |
ICNIRP (2010) Guidelines for limiting exposure to time-varying electric and magnetic fields (1 Hz to 100 kHz). Health Phys 99: 818–836. https://doi.org/10.1097/HP.0b013e3181f06c86 doi: 10.1097/HP.0b013e3181f06c86
![]() |
[47] | ICNIRP (2020) Guidelines for limiting exposure to electromagnetic fields (100 kHz to 300 GHz). Health Phys 118: 483–524. |
[48] | IEEE (2019) IEEE standard for safety levels with respect to human exposure to electric, magnetic, and electromagnetic fields, 0 Hz to 300 GHz. IEEE. https://doi.org/10.1109/IEEESTD.2019.8859679 |
[49] |
Xiao C, Cheng D, Wei K (2018) An LCC-C compensated wireless charging system for implantable cardiac pacemakers: theory, experiment, and safety evaluation. IEEE Trans Power Electron 33: 4894–4905. https://doi.org/10.1109/TPEL.2017.2735441 doi: 10.1109/TPEL.2017.2735441
![]() |
[50] | Jiang H, Brazis P, Tabaddor M, et al. (2012) Safety considerations of wireless charger for electric vehicles—a review paper. IEEE Symp Prod Compliance Eng Proc: 51–56. https://doi.org/10.1109/ISPCE.2012.6398288 |
[51] |
Tan L, Li G, Xie Q, et al. (2023) Study on the safety assessment and protection design of human exposure to low-frequency magnetic fields in electric vehicles. Radiat Prot Dosimetry 200: 60–74. https://doi.org/10.1093/rpd/ncad269 doi: 10.1093/rpd/ncad269
![]() |
[52] |
Zhao J, Wu ZJ, Li NL, et al. (2020) Study on safe distance between human body and wireless charging system of electric vehicles with different power and frequencies. Electr Eng 102: 2281–2293. https://doi.org/10.1007/s00202-020-01026-2 doi: 10.1007/s00202-020-01026-2
![]() |
[53] | Baikova EN, Romba L, Valtchev S, et al. (2019) Electromagnetic influence of wpt on human's health. Adv Wireless Technol Telecommun, 141–161. https://doi.org/10.4018/978-1-5225-5870-5.ch006 |
[54] |
Baikova EN, Valtchev SS, Melicio R, et al. (2016) Study of the electromagnetic interference generated by wireless power transfer systems. Int Rev Electr Eng (IREE) 11: 526. https://doi.org/10.15866/iree.v11i5.9773 doi: 10.15866/iree.v11i5.9773
![]() |
[55] | Zhang Y, Yuan J, Li L, et al. (2019) Design and application of early warning model for electric vehicle charging safety. J Phys Conf Ser 1873: 012052. |
[56] |
Li S, Mi CC (2015) Wireless power transfer for electric vehicle applications. IEEE J Emerg Sel Top Power Electron 3: 4–17. https://doi.org/10.1109/JESTPE.2014.2319453 doi: 10.1109/JESTPE.2014.2319453
![]() |
[57] | Rim CT, Mi C (2017) Wireless power transfer for electric vehicles and mobile devices. Wireless Power Transfer, John Wiley & Sons, Ltd. https://doi.org/10.1002/9781119329084 |
[58] |
Zhang W, Mi CC (2016) Compensation topologies of high-power wireless power transfer systems. IEEE Trans Veh Technol 65: 4768–4778. https://doi.org/10.1109/TVT.2015.2454292 doi: 10.1109/TVT.2015.2454292
![]() |
[59] |
Covic GA, Boys JT (2013) Modern trends in inductive power transfer for transportation applications. IEEE J Emerg Sel Top Power Electron 1: 28–41. https://doi.org/10.1109/JESTPE.2013.2264473 doi: 10.1109/JESTPE.2013.2264473
![]() |
[60] |
Hui SYR, Zhong W, Lee CK (2014) A critical review of recent progress in mid-range wireless power transfer. IEEE Trans Power Electron 29: 4500–4511. https://doi.org/10.1109/TPEL.2013.2249670 doi: 10.1109/TPEL.2013.2249670
![]() |
1. | Zimeng Lv, Jiahong Zeng, Yuting Ding, Xinyu Liu, Stability analysis of time-delayed SAIR model for duration of vaccine in the context of temporary immunity for COVID-19 situation, 2023, 31, 2688-1594, 1004, 10.3934/era.2023050 | |
2. | Ruqi Li, Yurong Song, Haiyan Wang, Guo-Ping Jiang, Min Xiao, Reactive–diffusion epidemic model on human mobility networks: Analysis and applications to COVID-19 in China, 2023, 609, 03784371, 128337, 10.1016/j.physa.2022.128337 | |
3. | S. Suganya, V. Parthiban, Optimal control analysis of fractional order delayed SIQR model for COVID-19, 2024, 1951-6355, 10.1140/epjs/s11734-024-01294-0 | |
4. | Fatima Cherkaoui, Fatima Ezzahrae Fadili, Khalid Hilal, Stability analysis of a delayed fractional-order \mathcal {SIR} epidemic model with Crowley–Martin type incidence rate and Holling type II treatment rate, 2024, 1982-6907, 10.1007/s40863-024-00470-3 | |
5. | Xi-Xi 习习 Huang 黄, Min 敏 Xiao 肖, Leszek Rutkowski, Hai-Bo 海波 Bao 包, Xia 霞 Huang 黄, Jin-De 进德 Cao 曹, Mechanism analysis of regulating Turing instability and Hopf bifurcation of malware propagation in mobile wireless sensor networks, 2024, 33, 1674-1056, 060202, 10.1088/1674-1056/ad24d5 | |
6. | Zafer ÖZTÜRK, Halis BİLGİL, Sezer SORGUN, Application of Fractional SPR Psychological Disease Model in Turkey and Stability Analysis, 2023, 6, 2636-8692, 49, 10.33187/jmsm.1196961 | |
7. | Debao Gao, Xiangfeng Yang, Dynamic Analysis of the Project Investment Model with Time Delay and Density Constraints, 2023, 2023, 2314-4785, 1, 10.1155/2023/3791676 | |
8. | Guo-Feng Feng, Jiaqi Chen, Bin Ge, A class of natural pinus koraiensis population system with time delay and diffusion term, 2024, 17, 1793-5245, 10.1142/S1793524523500195 | |
9. | Zafer Öztürk, Halis Bilgil, Sezer Sorgun, Application of Fractional SIQRV Model for SARS-CoV-2 and Stability Analysis, 2023, 15, 2073-8994, 1048, 10.3390/sym15051048 | |
10. | Youssra Hajri, Amina Allali, Saida Amine, A delayed deterministic and stochastic SIRICV model: Hopf bifurcation and stochastic analysis, 2024, 215, 03784754, 98, 10.1016/j.matcom.2023.07.027 | |
11. | Chen Wang, Ruizhi Yang, Hopf bifurcation analysis of a pine wilt disease model with both time delay and an alternative food source, 2025, 33, 2688-1594, 2815, 10.3934/era.2025124 |
Symbol | Definition |
S | The number of suspected people |
I | The number of infected people |
Q | The number of quarantined people |
R | The number of recovered people |
\Lambda | The population growth |
\mu | The natural mortality |
\alpha_1 | The rate of COVID-19 death of infected people |
\alpha_2 | The rate of COVID-19 death of quarantined people |
\beta | The transmission rate from S to I |
\varepsilon | The transmission rate from I to Q |
\gamma_1 | The transmission rate from I to R |
\gamma_2 | The transmission rate from Q to R |
Symbol | Definition |
S | The number of suspected people |
I | The number of infected people |
Q | The number of quarantined people |
R | The number of recovered people |
\Lambda | The population growth |
\mu | The natural mortality |
\alpha_1 | The rate of COVID-19 death of infected people |
\alpha_2 | The rate of COVID-19 death of quarantined people |
\beta | The transmission rate from S to I |
\varepsilon | The transmission rate from I to Q |
\gamma_1 | The transmission rate from I to R |
\gamma_2 | The transmission rate from Q to R |