Dynamics analysis of a delayed virus model with two different transmission methods and treatments

In this paper, a delayed virus model with two different transmission methods and treatments is investigated. This model is a time-delayed version of the model in (Zhang et al. in Comput. Math. Methods Med. 2015:758362, 2015). We show that the virus-free equilibrium is locally asymptotically stable if the basic reproduction number is smaller than one, and by regarding the time delay as a bifurcation parameter, the existence of local Hopf bifurcation is investigated. The results show that time delay can change the stability of the endemic equilibrium. Finally, we give some numerical simulations to illustrate the theoretical findings.

Using mathematical models to help discover the mechanism of viral transmission to predict the development of infectious diseases has become the mainstream method for controlling and preventing infectious diseases [9][10][11][12]. Therefore, for over a century, lots of mathematical models have been established to explain the evolution of the free virus in a body, and mathematical analysis was implemented to explore the threshold associated with eradication and persistence of the virus; for example, [13][14][15][16] studied the global dynamic behavior of HIV models, [17][18][19][20][21][22][23][24] analysed the global dynamics of HBV models [25][26][27][28]. A general class of models describing the process of virus invading the target cells and release of the virus due to the infected cell apoptosis has been established and analyzed by Perelson et al. [29,30] and Nelson et al. [31] as follows: where x, y, and v represent the concentrations of uninfected target cells, infected cells, and virus, respectively. Λ and d are the generation rate and mortality rate of uninfected target cells respectively, α is the infection rate, a is the mortality rate of infected cells, k and u are the generation rate and mortality rate of free virus respectively.
However, the above model only considers that free viruses can infect uninfected cells by direct contact with them. Recent studies have shown that virus can be transmitted directly from cell to cell by virological synapses, i.e., cell-to-cell transmission [32][33][34][35][36][37][38][39]. Spouge et al. [40] built a model to characterize this phenomenon: where C, I, M represent the concentrations of uninfected cells, infected cells, and dead cells, respectively. k I is the rate constant for cell-to-cell spread, r C is the reproductive rate of uninfected cell. μ I , μ C are the rate constants at which uninfected or infected cells die respectively, and the term k I IC represents the cell-to-cell transmission. However, research [41] shows that there is a delay between the time an uninfected cell becomes infected and when it begins to infect other uninfected cells, then Culshaw et al. [42] improved the model by introducing a distributed delay into model (2) and got the following model: where F(u) is the delay kernel.
Lai et al. [43] proposed a model containing two different types of infection as follows: where T, T * , and V are the concentrations of uninfected cells, infected cells, and free virus, respectively. β 1 is the infection rate of cell-free virus transmission and β 2 is the infection rate of cell-to-cell transmission. For more details on parameters, please see [43]. The authors proved that a Hopf bifurcation can occur under certain conditions. Recently, Zhang et al. [1] proposed an ordinary differential equations virus model with both two different types of infection and cure rate as follows: where x, y, and v represent the concentrations of uninfected cells, infected cells, and free virus respectively. π is the regeneration rate of uninfected cells, d, a, u are the death rates of three kinds of cells. ρ represents the cure rate, ky is the rate at which infected cells produce free viruses. By constructing suitable Lyapunov function, the authors proved that the equilibria are globally asymptotically stable under some conditions. However, the authors did not consider the time delay in model (3). In order to understand whether the introduction of time delay or not will change the stability of the equilibria, then motivated by the works [42,43] and based on [1], we further consider model (4) by introducing a discrete delay into model (3) as follows: where τ is time delay. βxy is the term of cell-to-cell transmission. αvx represents cellfree virus transmission. For more details on parameters, please see [1]. Considering the biological meanings, we analyze model (4) The paper is organized as follows. Firstly, we summarize some basic results about model (4) in Sect. 2. The local stability of the free equilibrium and Hopf bifurcation of the system are discussed in Sect. 3. We give the properties of Hopf bifurcation in Sect. 4. Finally, we perform some numerical simulations to verify the results in Sect. 5.

Some basic results
From [1], we can conclude some basic results and summarize them in the following theorem in this section.
is the basic reproduction number.

Local stability of the free equilibrium and Hopf bifurcation
Then linearization at the original results in a characteristic equation is as follows: Clearly, it has a root λ = -d < 0. Thus we only need to analyze the distribution of roots, which determines the stability of solution of system (5) of the equation where When τ = 0, (7) reduces to Since R = (αk+βu)x 0 (a+ρ)u < 1 implies A + C > 0 and B + D > 0, we know the two roots of (8) always have a negative real part. Next, we assume that equation (7) with τ = 0 has two pure imaginary roots ±i ( > 0), which implies that the equation has at least one positive solution.
Obviously, B 2 -D 2 > 0, then we can see that where a + ρ > βx 0 is used. Then we can conclude that when R < 1, equation (9) has no positive real root, which leads to that equation (7) does not have a pure imaginary root. Thus, all the roots of (7) always have negative real parts. Therefore, E 0 is locally stable. When R > 1, it is easy to see B + D < 0, then equation (7) has at least one positive root, thus E 0 is unstable. This proof is completed.

Property of Hopf bifurcation at E *
From Theorem 3.2, we got the sufficient conditions for the Hopf bifurcation to appear. We assume that when τ = τ 0 , system (4) produces a Hopf bifurcation at E * . Next by the normal form theory and the center manifold [46], we try to establish the explicit formula determining the directions, stability, and period of periodic solutions bifurcating from E * at τ = τ 0 and ω(τ 0 ) = ω 0 .

Conclusion and numerical simulations
In this paper, we have mainly considered the effect of time delay on the dynamics of a virus model with two different transmission methods and treatments. Our results show that the introduction of time delay has a significant effect on the dynamics of the system. However, it can be seen from [1] that when there is no time delay in system (3), the positive equilibrium E * of system (3) is asymptotically stable if it exists. The appearance of the time delay causes the positive equilibrium of the model (4) to be inverted from stable to unstable, and a periodic solution of small amplitude is generated near the positive equilibrium E * . Biologically, the number of healthy, infected, and free viruses exhibits periodic changes. Next, we take some numerical simulations to validate our main results. We set the basic parameters as follows [48]: d = 0.2, β = 0.000024, α = 0.000024, ρ = 0.2, a = 0.15, k = 150, u = 0.2. Firstly, we set π = 2, direct calculations with Maple 14 show that R = 0.514971 < 1 and E 0 = (10, 0, 0). By Theorem 3.1 the virus-free equilibrium E 0 of the system is stable G z * 1 = -0.008284 < 0, then equation (12) has two positive roots z 1 = 0.0089 and z 2 = 0.3680, and equation (11) has two positive roots 1 = 0.09441603258 and 2 = 0.6066529567. Therefore we have the Hopf bifurcation value τ 0 2 = 3.828340005. If we set τ = 3.8 < 3.828340005, by Theorem 3.2, the virus equilibrium E * is asymptotically stable (see Figs. [5][6][7][8]. If we set τ = 3.9 > 3.828340005, by Theorem 3.2, the virus equilibrium E * is unstable and periodic oscillations occur (see Figs. 9-16 with different initial values).

Availability of data and materials
Data sharing not applicable to this article as all data sets are hypothetical during the current study.