Stability Analysis of an In-Host Viral Model with Cure of Infected Cells and Humoral Immunity

Abstract

An in-host viral model with cure of infected cells and humoral immunity is studied. We prove that the stability is completely determined by the basic reproductive number 𝑅₀ and show that the infection-free equilibrium 𝐸₀ is globally asymptotically stable if and only if 𝑅₀≤1. Moreover, if 𝑅₀>1, the infection equilibrium is locally asymptotically stable when the time delay 𝜏 is small and it loses stability as the length of the time delay increases past a critical value 𝜏₀. Finally, we confirm our analysis by providing several numerical examples.

1. Introduction

The humoral immunity is a kind of immunologic mechanism which uses B lymphocytes to produce antigen to prevent virus and there are evidences to prove that the humoral immunity is more effective than the cell-mediated immune in some infections such as malaria infection [1–3]. Many authors present and develop mathematical systems for the humoral immunity [4–7]. And the cure of virus is also important especially in HBV models [8, 9]. In the present paper, we analyze an in-host viral model with humoral immunity and intracellular delay, and we incorporate a “cure” of infected cells into it. We propose the following system: 𝑇^′(𝑡)(𝑡)𝑉(𝑡)(𝑡)(𝑡),𝐼=Λ−𝛽𝑇−𝑑𝑇+𝑏𝐼_′(𝑡)(𝑡)𝑉(𝑡)−()𝐼(𝑡),𝑉=𝛽𝑇𝑎+𝑏_′(𝑡)(𝑡)(𝑡)(𝑡)𝑉(𝑡),𝐵=𝑘𝐼−𝑢𝑉−𝑞𝐵_′(𝑡)=𝑔𝑉(𝑡−𝜏)−𝑐𝐵(𝑡), (1) where 𝑇, 𝐼, 𝑉, and 𝐵 represent the uninfected cells, the infected cells, the virus, and the B cells, respectively. Λ and 𝑑 are assumed as the birth rate and death rate of uninfected cells. 𝛽 is the infection rate and 𝑘𝐼(𝑡) represents the number of free virus which is produced during the average infected cell life span. 𝑎 is the death rate of infected cells and 𝑢 represents the death rate of virus. 𝑔 and 𝑐 represent the birth rate and death rate of B cells. The B cells neutralization rate is represented by 𝑞.

The following form is taken as the initial conditions: 𝑇(𝜃)=𝜑₁(𝜃),𝐼(𝜃)=𝜑₂(𝜃),𝑉(𝜃)=𝜑₃(𝜃),𝐵(𝜃)=𝜑₄(𝜃)𝜑_𝑖(𝜃)≥0,𝜃∈(−𝜏,0),𝜑_𝑖(0)>0(𝑖=1,2,3,4), (2) where (𝜑₁(𝜃),𝜑₂(𝜃),𝜑₃(𝜃),𝜑₄(𝜃))∈ℂ([−𝜏,0],ℝ⁴₊₀), the space of continuous functions mapping the interval (−∞,0] into ℝ⁴₊₀, and ℝ⁴₊₀=𝑥₁,𝑥₂,𝑥₃,𝑥₄∣𝑥_𝑖≥0,𝑖=1,2,3,4. (3)

The organization of this paper is as follows. In the next section, we will find threshold parameters 𝑅₀ of system (1) and it determines the existence of the equilibriums. In Section 3, by structuring suitable Lyapunov functionals and using LaSalle’s invariance principle we attain the global stability of the uninfected equilibrium if 𝑅₀≤1. In Section 4, we consider the stability of the infected equilibrium and the occurrence of local Hopf bifurcation. In Section 5, we present the numerical simulations to illustrate our results. Finally, we offer concluding remarks in the last section.

2. Existence of Equilibrium

We can easily find that system (1) always has an uninfected equilibrium 𝐸₀=(𝑇₀,0,0,0)=((Λ/𝑑),0,0,0).

Denote 𝑅₀=Λ𝑘𝛽().𝑢𝑑𝑎+𝑏 (4) We call 𝑅₀ the basic reproductive number. It is easy to prove that if 𝑅₀>1, model (1) exist an infected equilibrium 𝐸^∗(𝑇^∗,𝐼^∗,𝑉^∗,𝐵^∗), where 𝑉^∗=𝛽𝑐𝑢𝑎−(𝑎+𝑏)𝑑𝑔𝑞²+4Λ𝑔𝑞𝑎𝑐𝑘𝛽²()×−𝛽𝑐𝑢𝑎−𝑎+𝑏𝑑𝑔𝑞2𝑎𝛽𝑔𝑞⁻¹,𝑇^∗=𝑔𝑞(𝑎+𝑏)𝑉𝑘𝛽𝑐^∗+𝑢(𝑎+𝑏),𝐵𝑘𝛽_∗=𝑔𝑐𝑉^∗,𝐼^∗=𝑔𝑞𝑉𝑘𝑐^∗2+𝑢𝑘𝑉^∗. (5)

3. Global Attraction of Infection-Free Equilibrium

Theorem 1. The infection-free equilibrium 𝐸₀ of system (1) is globally attractive if 𝑅₀≤1.

Proof. Let (𝑇(𝑡),𝐼(𝑡),𝑉(𝑡),𝐵(𝑡)) be any positive solution of system (1) with initial conditions (2).
Define 𝐿(𝑡)=𝑔𝑘()⋅2𝑞𝑎+𝑏𝑇−𝑇₀²𝑇₀+𝑏𝑔𝑘()⋅2𝑞𝑎+𝑏)(𝑎+𝑑𝑇−𝑇₀+𝐼²𝑇₀+𝑔𝑘𝑞()𝑔𝑎+𝑏𝐼+𝑞1𝑉+2𝐵²+𝑔𝐵^𝑡_𝑡−𝜏𝑉(𝜉)𝑑𝜉. (6) Calculating the derivative of 𝐿(𝑡) along positive solutions of system (1) and noting Λ=𝑑𝑇₀ and 𝑇𝑉(𝑇−𝑇₀)=𝑉(𝑇−𝑇₀)²+𝑇₀𝑉(𝑇−𝑇₀), it follows that 𝐿̇=−𝑔𝑘𝛽𝑞()()𝑎+𝑏𝑉+𝑑𝑔𝑘𝑎+𝑑+𝑏𝑑𝑔𝑘𝑞()×𝑎+𝑏)(𝑎+𝑑𝑇−𝑇₀²𝑇₀−𝑎𝑏𝑔𝑘𝑞()𝐼𝑎+𝑏)(𝑎+𝑑²−𝑐𝐵²+𝑔𝑢𝑞𝑅₀−1𝑉. (7) If 𝑅₀≤1, it follows from (7) that 𝐿̇≤0. The solutions are limited to the largest invariant subset 𝑀 of {𝐿̇(𝑡)=0}. It is clear that 𝐿̇(𝑡)=0 if and only if (𝑇,𝐼,𝑉,𝐵)=(𝑇₀,𝐼₀,𝑉₀,𝐵₀). So we can obtain the global attraction of 𝐸₀ by using LaSalle’s invariance principle. This completes the proof.

4. Permanence of the System for 𝑅₀>1

Theorem 2. If 𝑅₀>1, the infected equilibrium 𝐸^∗ of system (1) is locally asymptotically stable when 𝜏=0.

Proof. The characteristic equation associated with system (1) at 𝐸^∗ is 𝜆⁴+𝑚₃𝜆³+𝑚₂𝜆²+𝑚₁𝜆+𝑚₀+𝑒^−𝜆𝜏𝑛₂𝜆²+𝑛₁𝜆+𝑛₀=0. (8) Noting that 𝑇^∗=((𝑔𝑞(𝑎+𝑏))/(𝑘𝛽𝑐))𝑉^∗+((𝑢(𝑎+𝑏))/(𝑘𝛽)), 𝐵^∗=(𝑔/𝑐)𝑉^∗, we can obtain 𝑚₃=𝑎+𝑏+𝑐+𝑑+𝑢+𝑞𝐵^∗+𝛽𝑉^∗,𝑚₂=𝑏𝑑+𝑎𝑑+𝛽𝑉^∗+𝑢+𝑞𝐵^∗𝑑+𝛽𝑉^∗+𝑐𝑎+𝑏+𝑑+𝑢+𝑞𝐵^∗+𝛽𝑉^∗,𝑚₁=(𝑎+𝑐)𝑔𝑞𝑐𝛽𝑉^∗2+(𝑎𝑢+𝑐𝑢+𝑎𝑐)𝛽𝑉^∗(),𝑚+𝑐𝑑𝑎+𝑢+𝑏₀=𝑐𝑎𝑢𝛽𝑉^∗+𝑎𝑞𝛽𝑉^∗𝐵^∗,𝑛₂=𝑔𝑞𝑉^∗,𝑛₁=𝑔𝑞𝑉^∗𝑎+𝑏+𝑑+𝛽𝑉^∗,𝑛₀=𝑑𝑔𝑞(𝑎+𝑏)𝑉^∗+𝑎𝑔𝑞𝛽𝑉^∗2. (9) If 𝜏=0, (8) becomes 𝜆⁴+𝑚₃𝜆³+𝑚₂+𝑛₂𝜆²+𝑚₁+𝑛₁𝜆+𝑚₀+𝑛₀=0. (10) Then we can obtain 𝑚₀+𝑛₀>0, 𝑚₁+𝑛₁>0, 𝑚₂+𝑛₂>0, and 𝑚₃>0. It follows from Routh-Hurwitz criterion that all roots of (10) have negative real parts when 𝜏=0. This competes the proof.

Let 𝜆=𝑖𝑣 and then substitute it into (8). Separating the real and imaginary parts, we will gain 𝑣⁴−𝑚₂𝑣²+𝑚₀=𝑛₂𝑣²−𝑛₀cos𝑣𝜏−𝑛₁𝑚𝑣sin𝑣𝜏,₃𝑣³−𝑚₁𝑣=𝑛₂𝑣²−𝑛₀sin𝑣𝜏+𝑛₁𝑣cos𝑣𝜏. (11)

Squaring and adding (11) yield 𝑣⁸+𝑚²₃−2𝑚₂𝑣⁶+𝑚²₂−2𝑚₁𝑚₃+2𝑚₀−𝑛²₂𝑣⁴+𝑚²₁−2𝑚₀𝑚₂−𝑛²₁+2𝑛₀𝑛₂𝑣²+𝑚²₀−𝑛²₀=0. (12) Denote 𝐺(𝑥)=𝑥⁴+𝑚²₃−2𝑚₂𝑥³+𝑚²₂−2𝑚₁𝑚₃+2𝑚₀−𝑛²₂𝑥²+𝑚²₁−2𝑚₀𝑚₂−𝑛²₁+2𝑛₀𝑛₂𝑥+𝑚²₀−𝑛²₀. (13) We have that 𝐺^′(𝑥)=4𝑥³+3𝑚²₃−2𝑚₂𝑥²+2𝑚²₂−2𝑚₁𝑚₃+2𝑚₀−𝑛²₂𝑥+𝑚²₁−2𝑚₀𝑚₂−𝑛²₁+2𝑛₀𝑛₂. (14) Hence if 𝐺(𝑥)=0 has a positive root 𝑥=𝑣², then 𝜆=±𝑖𝑣 are a couple of purely imaginary characteristic roots. Moreover, for a real practice model with fully known coefficients, the exactly four roots of 𝐺(𝑥)=0 can be numerically calculated with the help of computational software such as Matlab. Let 𝑥_𝑖(1≤𝑖≤𝑘,1≤𝑘≤4) be the positive roots of 𝐺(𝑥)=0 and 𝑣_𝑖=√𝑥_𝑖.

Solving (11) with respect to 𝜏, we can obtain 𝜏_𝑖^(𝑗)=1𝑣_𝑖𝑛arccos²𝑣⁶_𝑖+−𝑛₀+𝑛₁𝑚₃−𝑛₂𝑚₂𝑣⁴_𝑖𝑛²₁𝑣²_𝑖+𝑛₂𝑣²_𝑖−𝑛₀²+𝑛₀𝑚₂−𝑛₁𝑚₁+𝑛₂𝑚₀𝑣²_𝑖−𝑛₀𝑚₀𝑛²₁𝑣²_𝑖+𝑛₂𝑣²_𝑖−𝑛₀²+1𝑣_𝑖⋅2𝑗𝜋, (15) where 1≤𝑖≤𝑘;1≤𝑘≤4;𝑗=0,1,2,…. Let 𝜏₀=min𝜏_𝑖^(𝑗)∣1≤𝑖≤𝑘;𝑗=0,1,2,…. (16) This means that 𝜏₀ is the least value of 𝜏 which can be used to make the characteristic equation have purely imaginary roots.

Theorem 3. 𝐺^′(𝑣²_𝑖) and Re[𝑑𝜆/𝑑𝜏]⁻¹_{𝜏=𝜏𝑖(𝑗)} have the same sign.

Proof. Let the characteristic equation be in the following form: 𝑓(𝜆)+𝑔(𝜆)𝑒^−𝜆𝜏=0. (17) We can obviously know from (8) that 𝑓(𝜆)=𝜆⁴+𝑚₃𝜆³+𝑚₂𝜆²+𝑚₁𝜆+𝑚₀,𝑔(𝜆)=𝑛₂𝜆²+𝑛₁𝜆+𝑛₀. (18) Calculating the derivative of (17) with respect to 𝜏, we can obtain that 𝑑𝜆𝑓𝑑𝜏^′(𝜆)+𝑔^′(𝜆)𝑒^−𝜆𝜏−𝑔(𝜆)𝑒^−𝜆𝜏(𝜆)𝑒⋅𝜏=𝑔^−𝜆𝜏⋅𝜆,𝑑𝜆𝑑𝜏⁻¹_{𝜏=𝜏𝑖}^(𝑗)=𝑓^′(𝜆)+𝑔^′(𝜆)𝑒^−𝜆𝜏𝑔(𝜆)e^−𝜆𝜏−𝜏⋅𝜆𝜆=𝑓^′𝑖𝑣_𝑖+𝑔^′𝑖𝑣_𝑖𝑒^{−𝑖𝑣𝑖𝜏}𝑔𝑖𝑣_𝑖𝑒^{−𝑖𝑣𝑖𝜏}⋅𝑖𝑣_𝑖−𝜏𝑖𝑣_𝑖. (19) Therefore, Re𝑑𝜆𝑑𝜏⁻¹_{𝜏=𝜏𝑖}^(𝑗)𝑓=Re^′𝑖𝑣_𝑖+𝑔^′𝑖𝑣_𝑖𝑒^{−𝑖𝑣𝑖𝜏}𝑔𝑖𝑣_𝑖𝑒^{−𝑖𝑣𝑖𝜏}⋅𝑖𝑣_𝑖𝑓=Re^′𝑖𝑣_𝑖𝑖𝑣_𝑖⋅−𝑓𝑖𝑣_𝑖+𝑔^′𝑖𝑣_𝑖𝑖𝑣_𝑖⋅𝑔𝑖𝑣_𝑖1=Re𝑖𝑣_𝑖⋅−𝑓^′𝑖𝑣_𝑖⋅𝑓𝑖𝑣_𝑖𝑓𝑖𝑣_𝑖⋅𝑓𝑖𝑣_𝑖+𝑔^′𝑖𝑣_𝑖⋅𝑔𝑖𝑣_𝑖𝑔𝑖𝑣_𝑖⋅𝑔𝑖𝑣_𝑖1=Re𝑖𝑣_𝑖⋅𝑔^′𝑖𝑣_𝑖⋅𝑔𝑖𝑣_𝑖−𝑓^′𝑖𝑣_𝑖⋅𝑓𝑖𝑣_𝑖𝑓𝑖𝑣_𝑖². (20) Obviously, signRe𝑑𝜆𝑑𝜏⁻¹_{𝜏=𝜏𝑖}^(𝑗)=signIm𝑔^′𝑖𝑣_𝑖⋅𝑔𝑖𝑣_𝑖−𝑓^′𝑖𝑣_𝑖⋅𝑓𝑖𝑣_𝑖. (21) And after some calculations, we can get Im𝑔^′𝑖𝑣_𝑖⋅𝑔𝑖𝑣_𝑖−𝑓^′𝑖𝑣_𝑖⋅𝑓𝑖𝑣_𝑖=4𝑣⁷_𝑖+3−2𝑚₂+𝑚²₃𝑣⁵_𝑖+2⋅𝑚²₂−2𝑚₁𝑚₃+2𝑚₀−𝑛²₂𝑣³_𝑖+𝑚²₁−2𝑚₀𝑚₂−𝑛²₁+2𝑛₀𝑛₂𝑣_𝑖. (22) Therefore, 𝐺^′(𝑣²_𝑖) and Re[𝑑𝜆/𝑑𝜏]⁻¹_{𝜏=𝜏𝑖(𝑗)} have the same sign. This completes the proof.

Applying Theorems 2 and 3 and the Hopf bifurcation theorem for functional differential equation [10], we derive the existence of a Hopf bifurcation as stated in Theorem 4.

Theorem 4. Suppose that 𝐺(𝑥)=0 has at least one simple positive root and 𝜏₀ is defined in (16). Then there is a Hopf bifurcation for the system (1) as s passes upwards through 𝜏₀ leading to a periodic solution that bifurcates from 𝐸^∗, where 𝜏_𝑖^(𝑗)=1𝑣_𝑖𝑛arccos²𝑣⁶_𝑖+−𝑛₀+𝑛₁𝑚₃−𝑛₂𝑚₂𝑣⁴_𝑖𝑛²₁𝑣²_𝑖+𝑛₂𝑣²_𝑖−𝑛₀²+𝑛₀𝑚₂−𝑛₁𝑚₁+𝑛₂𝑚₀𝑣²_𝑖−𝑛₀𝑚₀𝑛²₁𝑣²_𝑖+𝑛₂𝑣²_𝑖−𝑛₀²+1𝑣_𝑖⋅2𝑗𝜋,1≤𝑖≤𝑘;𝑗=0,1,2,…. (23)

5. Numerical Examples

Example 1. For system (1), consider all parameters as follows: 𝑎=0.1, 𝑏=0.1, 𝛽=0.1, 𝑐=0.001, 𝑑=0.2, 𝑔=0.1, 𝑘=0.1, 𝜆=1, 𝑞=0.001, 𝑢=10, and 𝜏=10. In this case, we can obtain that 𝑅₀=0.375<1; the global attraction of the infection-free equilibrium 𝐸₀ is illustrated by Figure 1.

Figure 1 

Equilibrium 𝐸0 is globally asymptotically stable if 𝑅0<1.

Example 2. Consider all parameters of system (1) as follows: 𝑎=2.41979, 𝑏=2.80472, 𝛽=2.34292, 𝑐=2.2057, 𝑑=0.9129, 𝑔=4.2074, 𝑘=2.6415, 𝜆=4.60059, 𝑞=3.738, and 𝑢=1.3978. In this case, we can obtain that 𝑅₀=4.2709>1.(1)When 𝜏=0.75, 𝐸^∗ is locally asymptotically stable and the stability is illustrated by Figure 2.(2)When 𝜏=0.9, the delay can destabilize 𝐸^∗ and lead to Hopf bifurcation. We will show that in Figure 3.

Figure 2 

Equilibrium 𝐸∗ is locally asymptotically stable when 𝜏 is very small.

Figure 3 

Delay can destabilize 𝐸∗ and lead to Hopf bifurcation when 𝜏 increases past a critical value 𝜏0.

6. Discussions

In this paper, we consider an in-host viral model humoral immunity. In addition, a “cure” of infected cells is incorporated. Firstly, it has been shown that if the basic reproductive number is less than unity, the infection-free equilibrium 𝐸₀ is globally attractive, and the time delay has no effect on the dynamics of the system. Then, a detailed analysis on the local asymptotic stability of the infection equilibrium 𝐸^∗ of the model is carried out. If 𝐸^∗ is feasible, the basic reproduction number of the virus is greater than unity; 𝐸^∗ is globally attractive for any time delay under some parameter conditions. By taking the discrete time delay as a bifurcation parameter, it is shown that this system undergoes a sequence of Hopf bifurcations and stability switches are observed by using simulations.

Acknowledgments

This work was supported by National Natural Science Foundation of China (61174209 and 11071013), the Basic Scientific Research Foundation of Central University (FRF-BR-11-048B and FRF-BR-12-004), and the Basic Theory Research Foundation for Engineering Research Institute of USTB (YJ2012-001).