# Qualitative Analysis of a Generalized Virus Dynamics Model with Both Modes of Transmission and Distributed Delays

Authors: Khalid Hattaf; Noura Yousfi
Journal: International Journal of Differential Equations
 (2018)

---

## Abstract

We propose a generalized virus dynamics model with distributed delays and both modes of transmission, one by virus-to-cell infection and the other by cell-to-cell transfer. In the proposed model, the distributed delays describe (i) the time needed for infected cells to produce new virions and (ii) the time necessary for the newly produced virions to become mature and infectious. In addition, the infection transmission process is modeled by general incidence functions for both modes. Furthermore, the qualitative analysis of the model is rigorously established and many known viral infection models with discrete and distributed delays are extended and improved.


---

## Body

## 1. Introduction

Viruses are microscopic organisms that need to penetrate into a cell of their host to duplicate and multiply. Many human infections and diseases are caused by viruses such as the human immunodeficiency virus (HIV) that is responsible for acquired immunodeficiency syndrome (AIDS), Ebola that can cause an often fatal illness called Ebola hemorrhagic fever, and the hepatitis B virus (HBV) that can lead to chronic infection, cirrhosis, or liver cancer.In viral dynamics, infection processes and virus production are not instantaneous. In reality, there are two kinds of delays: one in cell infection and the other in virus production. In the literature, these delays are modeled by discrete time delays [1–6], by finite distributed delays [7–9], and by infinite distributed delays [9–12]. The delay in cell infection can be modeled by an explicit class of latently infected cells (see, e.g., [13–15]).On the other hand, viruses can spread by two fundamental modes, one by virus-to-cell infection through the extracellular space and the other by cell-to-cell transfer involving direct cell-to-cell contact [16–19]. For these reasons, we propose the following generalized virus dynamics model with both modes of transmission and distributed delays: (1)x˙t=λ-dxt-fxt,yt,vtvt-gxt,ytyt,y˙t=∫0∞h1τe-α1τfxt-τ,yt-τ,vt-τvt-τ+gxt-τ,yt-τyt-τdτ-ayt,v˙t=k∫0∞h2τe-α2τyt-τdτ-μvt,where x(t), y(t), and v(t) are the concentrations of uninfected cells, infected cells, and free virus particles at time t, respectively. The uninfected cells are produced at rate λ, die at rate dx, and become infected either by free virus at rate f(x,y,v)v or by direct contact with an infected cell at rate g(x,y)y. Hence, the term f(x,y,v)v+g(x,y)y denotes the total infection rate of uninfected cells. The parameters a and μ are, respectively, the death rates of infected cells and free virus. k is the production rate of free virus by an infected cell. In this proposed model, we assume that the virus or infected cell contacts an uninfected target cell at time t-τ, and the cell becomes infected at time t, where τ is a random variable taken from a probability distribution h1(τ). The term e-α1τ represents the probability of surviving from time t-τ to time t, where α1 is the death rate for infected but not yet virus-producing cells. Similarly, we assume that the time necessary for the newly produced virions to become mature and infectious is a random variable with a probability distribution h2(τ). The term e-α2τ denotes the probability of surviving the immature virions during the delay period, where 1/α2 is the average life time of an immature virus. Therefore, the integral ∫0∞h2(τ)e-α2τy(t-τ)dτ describes the mature viral particles produced at time t.As in [20], the incidence functions f(x,y,v) and g(x,y) for the two modes are continuously differentiable and satisfy the following hypotheses:(H0)
g(0,y)=0, for all y≥0; ∂g/∂x(x,y)≥0 (or g(x,y) is a strictly monotone increasing function with respect to x when f≡0) and ∂g/∂y(x,y)≤0, for all x≥0 and y≥0.(H1)
f(0,y,v)=0, for all y≥0 and v≥0.(H2)
f(x,y,v) is a strictly monotone increasing function with respect to x (or ∂f/∂x(x,v,y)≥0 when g(x,y) is a strictly monotone increasing function with respect to x), for any fixed y≥0 and v≥0.(H3)
f(x,y,v) is a monotone decreasing function with respect to y and v.Biologically, the four hypotheses are reasonable and consistent with the reality. For more details on the biological significance of these four hypotheses, we refer the reader to the works [20–22]. Further, the general incidence functions f(x,y,v) and g(x,y) include various types of incidence rates existing in the literature.The probability distribution functionsh1(τ) and h2(τ) are assumed to satisfy hi(τ)≥0 and ∫0∞hi(τ)dτ=1 for i=1,2. When h1(τ)=δ(τ-τ1) and h2(τ)=δ(τ-τ2), where δ· is the Dirac delta function, system (1) becomes a model with two discrete time delays τ1 and τ2 which is the generalization of the models presented in [2–6]. When f(x,y,v)=β1x and g(x,y)=0, where β1 is the virus-to-cell infection rate, we get the HIV infection model with distributed intracellular delays investigated by Xu [11]. On the other hand, the model proposed by Lai and Zou [23] is a special case of our model (1) when f(x,y,v)=β1x, g(x,y)=β2x, and h2(τ)=δ(τ), where β2 is the cell-to-cell transmission rate. It is important to note that the model studied by Nelson and Perelson in [10] is a particular case of [23].The main objective of this work is to investigate the dynamical behavior of system (1). For this end, we start with the existence, the positivity, and boundedness of solutions, which implies that our model is well posed. After that, we determine the basic reproduction number and steady states of the model. The global stability of the disease-free equilibrium and the chronic infection equilibrium is established in Sections 3 and 4 by constructing appropriate Lyapunov functionals. An application of our results is presented in Section 5. Finally, the conclusion is summarized in Section 6.


## 2. Well-Posedness and Equilibria

For biological reasons, we suppose that the initial conditions of system (1) satisfy (2)xθ=ϕ1θ≥0,yθ=ϕ2θ≥0,vθ=ϕ3θ≥0,θ∈-∞,0.Define the Banach space for fading memory type as follows: (3)Cα=φ∈C-∞,0,R+3:φθeαθisuniformlycontinuouson-∞,0,φ=supθ≤0⁡φθeαθ<∞,where α is a positive constant and R+3=x1,x2,x3:xi≥0,i=1,2,3.Theorem 1.
For any initial conditionϕ=(ϕ1,ϕ2,ϕ3)∈Cα satisfying (2), system (1) has a unique solution on 0,+∞. Furthermore, this solution is nonnegative and bounded for all t≥0.Proof.
By the fundamental theory of functional differential equations [24–26], system (1) with initial condition ϕ∈Cα has a unique local solution on (0,Tmax), where Tmax is the maximal existence time for solution of system (1).
First, we prove thatx(t)>0 for all t∈(0,Tmax). In fact, supposing the contrary, let t1>0 be the first time such that x(t1)=0 and x˙(t1)≤0. From the first equation of system (1), we have x˙(t1)=λ>0 which is a contradiction. Then x(t)>0 for all t∈(0,Tmax).
From the second and third equations of system (1), we get (4)yt=ϕ20e-at+∫0te-at-s∫0∞h1τe-α1τfxs-τ,ys-τ,vs-τvs-τ+gxs-τ,ys-τys-τdτds,vt=ϕ30e-μt+k∫0te-μt-s∫0∞h2τe-α2τys-τdτds,which implies that y(t) and v(t) are nonnegative for all t∈(0,Tmax).
Now, we prove the boundedness of the solutions. From the first equation of  (1), we have x˙(t)≤λ-dx(t) which implies that (5)limsupt→+∞⁡xt≤λd.Then x(t) is bounded. Let (6)Tt=yt+∫0∞h1τe-α1τxt-τdτ.Since x(t) is bounded and ∫0∞h1(τ)dτ=1, the integral in T(t) is well defined and differentiable with respect to t. Hence, (7)dTtdt=λ∫0∞h1τe-α1τdτ-d∫0∞h1τe-α1τxt-τdτ-ayt≤λη1-δTt,where δ=min⁡a,d and (8)ηi=∫0∞hiτe-αiτdτ,i=1,2.Thus, T(t)≤M≔max⁡T0,λη1/δ, which implies that y(t) is bounded.
It remains to prove thatv(t) is bounded. By third equation of system (1) and the boundedness of y(t), we deduce that (9)v˙t≤kMη2-μvt.Then v(t)≤max⁡v0,kMη2/μ. Therefore, v(t) is also bounded. We have proved that all variables of system (1) are bounded which implies that Tmax=+∞ and the solution exists globally.If in addition to (2) we assume that ϕi(0)>0 for all i=1,2,3, we easily obtain the following result.Remark 2.
Whenϕ=(ϕ1,ϕ2,ϕ3)∈Cα satisfying (2) with ϕi(0)>0 (i=1,2,3), all solution of (1) with initial condition ϕ is positive for all t≥0.


### 2.1. Equilibria

Obviously, system (1) has always one disease-free equilibrium of the form Efλ/d,0,0. Therefore, the basic reproduction R0 of system (1) can be defined by (10)R0=kη1η2fλ/d,0,0+μη1gλ/d,0aμ.As in [20], R0 can be rewritten as R0=R01+R02, where R01=kη1η2fλ/d,0,0/aμ is the basic reproduction number corresponding to virus-to-cell infection mode and R02=η1gλ/d,0/a is the basic reproduction number corresponding to cell-to-cell transmission mode.Theorem 3.
(i)
IfR0≤1, then system (1) has a unique disease-free equilibrium of the form Ef(λ/d,0,0).
(ii)
IfR0>1, the disease-free equilibrium is still present and system (1) has a unique chronic infection equilibrium of the form E∗(x∗,y∗,v∗) with x∗∈(0,λ/d), y∗>0, and v∗>0.Proof.
It is clear thatEfλ/d,0,0 is the unique steady state of system (1) when R0≤1. To find the other equilibria, we resolve the following system: (11)λ-dx-fx,y,vv-gx,yy=0,η1fx,y,vv+gx,yy-ay=0,kη2y-μv=0.From (11), we obtain the equation (12)kη1η2fx,η1λ-dxa,kη1η2λ-dxaμ+μη1gx,η1λ-dxa=aμ.We have y=η1λ-dx/a≥0, which implies that x≤λ/d. Thus, there is no biological equilibrium when x>λ/d.
Define the functionψ1 on the interval 0,λ/d by (13)ψ1x=kη1η2fx,η1λ-dxa,kη1η2λ-dxaμ+μη1gx,η1λ-dxa-aμ.Clearly, ψ1(0)=-aμ<0, ψ1λ/d=aμ(R0-1), and (14)ψ1′x=kη1η2∂f∂x-dη1a∂f∂y-kdη1η2aμ∂f∂v+μη1∂g∂x-dη1a∂g∂y>0.Hence, if R0>1, there exists another biologically meaningful equilibrium E∗(x∗,y∗,v∗) with x∗∈(0,λ/d), y∗>0, and v∗>0. This completes the proof.


## 2.1. Equilibria

Obviously, system (1) has always one disease-free equilibrium of the form Efλ/d,0,0. Therefore, the basic reproduction R0 of system (1) can be defined by (10)R0=kη1η2fλ/d,0,0+μη1gλ/d,0aμ.As in [20], R0 can be rewritten as R0=R01+R02, where R01=kη1η2fλ/d,0,0/aμ is the basic reproduction number corresponding to virus-to-cell infection mode and R02=η1gλ/d,0/a is the basic reproduction number corresponding to cell-to-cell transmission mode.Theorem 3.
(i)
IfR0≤1, then system (1) has a unique disease-free equilibrium of the form Ef(λ/d,0,0).
(ii)
IfR0>1, the disease-free equilibrium is still present and system (1) has a unique chronic infection equilibrium of the form E∗(x∗,y∗,v∗) with x∗∈(0,λ/d), y∗>0, and v∗>0.Proof.
It is clear thatEfλ/d,0,0 is the unique steady state of system (1) when R0≤1. To find the other equilibria, we resolve the following system: (11)λ-dx-fx,y,vv-gx,yy=0,η1fx,y,vv+gx,yy-ay=0,kη2y-μv=0.From (11), we obtain the equation (12)kη1η2fx,η1λ-dxa,kη1η2λ-dxaμ+μη1gx,η1λ-dxa=aμ.We have y=η1λ-dx/a≥0, which implies that x≤λ/d. Thus, there is no biological equilibrium when x>λ/d.
Define the functionψ1 on the interval 0,λ/d by (13)ψ1x=kη1η2fx,η1λ-dxa,kη1η2λ-dxaμ+μη1gx,η1λ-dxa-aμ.Clearly, ψ1(0)=-aμ<0, ψ1λ/d=aμ(R0-1), and (14)ψ1′x=kη1η2∂f∂x-dη1a∂f∂y-kdη1η2aμ∂f∂v+μη1∂g∂x-dη1a∂g∂y>0.Hence, if R0>1, there exists another biologically meaningful equilibrium E∗(x∗,y∗,v∗) with x∗∈(0,λ/d), y∗>0, and v∗>0. This completes the proof.


## 3. Stability of the Disease-Free Equilibrium

In this section, we establish the stability of the disease-free equilibrium.Theorem 4.
The disease-free equilibriumEf is globally asymptotically stable when R0≤1 and becomes unstable when R0>1.Proof.
To study the global stability ofEf, we consider the following Lyapunov functional: (15)Vt=1η1yt+fλ/d,0,0μvt+1η1∫0∞h1τe-α1τ∫t-τtfxs,ys,vsvs+gxs,ysysdsdτ+kfλ/d,0,0μ∫0∞h2τe-α2τ∫t-τtysdsdτ.Calculating the time derivative of V along the positive solution of system (1), we get (16)V˙t1=fx,y,v-fλd,0,0v+aη1ykη1η2fλ/d,0,0+μη1gx,yaμ-1.We have lim supt→∞⁡x(t)≤λ/d, which implies that all omega limit points satisfy x(t)≤λ/d. Thus, it is sufficient to consider solutions for which x(t)≤λ/d. By (10) and (H1)–(H3), we obtain (17)V˙t1≤fx,0,0-fλd,0,0v+aη1R0-1y≤aη1R0-1y.Therefore, V˙1≤0 when R0≤1. In addition, it is not hard to verify that the largest compact invariant set in x,y,v∣V˙=0 is the singleton {Ef}. From LaSalle invariance principle [27], we deduce that the disease-free equilibrium Ef is globally asymptotically stable when R0≤1.
On the other hand, the characteristic equation atEf is given by (18)ξ+dξ+μξ+a-η-1ξgλd,0-η-1ξη-2ξkfλa,0,0=0,where η-i(ξ)=∫0∞hi(τ)e-(ξ+αi)τdτ. Define a function ψ2 on 0,+∞ by (19)ψ2ξ=ξ+μξ+a-η-1ξgλd,0-η-2ξη-2ξkfλd,0,0.We have ψ2(0)=aμ(1-R0)<0 and limξ→+∞ψ2(ξ)=+∞, which implies that ψ2 has a positive real root. Consequently, Ef is unstable for R0>1.


## 4. Stability of the Chronic Infection Equilibrium

In this section, we investigate the global stability of the chronic infection equilibriumE∗ by assuming that R0>1 and the functions f and g satisfy, for all x,y,v>0, the following hypothesis: H41-fx,y,vfx,y∗,v∗fx,y∗,v∗fx,y,v-vv∗≤0,1-fx∗,y∗,v∗gx,yfx,y∗,v∗gx∗,y∗fx,y∗,v∗gx∗,y∗fx∗,y∗,v∗gx,y-yy∗≤0.Therefore, we get the following result.Theorem 5.
Assume thatH4 holds. If R0>1, then the chronic infection equilibrium E∗ is globally asymptotically stable.Proof.
We define a Lyapunov functional as follows:(20)Wt=xt-x∗-∫x∗xtfx∗,y∗,v∗fs,y∗,v∗ds+1η1y∗Hyty∗+fx∗,y∗,v∗v∗kη2y∗v∗Hvtv∗+1η1fx∗,y∗,v∗v∗∫0∞h1τe-α1τ∫t-τtHfxs,ys,vsvsfx∗,y∗,v∗v∗dsdτ+1η1gx∗,y∗y∗∫0∞h1τe-α1τ∫t-τtHgxs,ysysgx∗,y∗y∗dsdτ+1η2fx∗,y∗,v∗v∗∫0∞h2τe-α2τ∫t-τtHysy∗dsdτ,where H(x)=x-1-ln⁡x, x>0. Clearly, H:(0,+∞)→[0,+∞) attains its strict global minimum at x=1 and H(1)=0. Hence, H(x)≥0. Further, the functional W is nonnegative.
In order to simplify the presentation, we will use the following notations:z=z(t) and zτ=z(t-τ) for any z∈{x,y,v}. The time derivative of W along the positive solution of system (1) is given by (21)W˙t1=1-fx∗,y∗,v∗fx,y∗,v∗x˙+1η11-y∗yy˙+fx∗,y∗,v∗v∗kη2y∗1-v∗vv˙+1η1fx∗,y∗,v∗v∗∫0∞h1τe-α1τHfx,y,vvfx∗,y∗,v∗v∗-Hfxτ,yτ,vτvτfx∗,y∗,v∗v∗dτ+1η1gx∗,y∗y∗∫0∞h1τe-α1τHgx,yygx∗,y∗y∗-Hgxτ,yτyτgx∗,y∗y∗dτ+1η2fx∗,y∗,v∗v∗∫0∞h2τe-α2τHyy∗-Hyτy∗dτ.Applying λ=dx∗+f(x∗,y∗,v∗)v∗+g(x∗,y∗)y∗=dx∗+a/η1y∗ and kη2y∗=μv∗, we obtain (22)W˙t1=dx∗1-xx∗1-fx∗,y∗,v∗fx,y∗,v∗+1η1fx∗,y∗,v∗v∗∫0∞h1τe-α1τ3-fx∗,y∗,v∗fx,y∗,v∗+fx,y,vvfx,y∗,v∗v∗-vv∗-fxτ,yτ,vτvτy∗fx∗,y∗,v∗v∗y+ln⁡fxτ,yτ,vτvτfx,y,vvdτ+1η1gx∗,y∗y∗∫0∞h1τe-α1τ2-fx∗,y∗,v∗fx,y∗,v∗+fx∗,y∗,v∗gx,yyfx,y∗,v∗gx∗,y∗y∗-yy∗+gxτ,yτyτgx∗,y∗y+ln⁡gxτ,yτyτgx,yydτ-1η2fx∗,y∗,v∗v∗∫0∞h2τe-α2τv∗yτvy∗-ln⁡yτydτ.Hence, (23)W˙t1=dx∗1-xx∗1-fx∗,y∗,v∗fx,y∗,v∗+fx∗,y∗,v∗v∗-1-vv∗+fx,y∗,v∗fx,y,v+fx,y,vvfx,y∗,v∗v∗+gx∗,y∗y∗-1-yy∗+fx,y∗,v∗gx∗,y∗fx∗,y∗,v∗gx,y+fx∗,y∗,v∗gx,yyfx,y∗,v∗gx∗,y∗y∗-1η1fx∗,y∗,v∗v∗∫0∞h1τe-α1τHfx∗,y∗,v∗fx,y∗,v∗+Hfxτ,yτ,vτvτy∗fx∗,y∗,v∗v∗y+Hfx,y∗,v∗fx,y,vdτ-1η1gx∗,y∗y∗∫0∞h1τe-α1τHfx∗,y∗,v∗fx,y∗,v∗+Hgxτ,yτyτgx∗,y∗y+Hfx,y∗,v∗gx∗,y∗fx∗,y∗,v∗gx,ydτ-1η2fx∗,y∗,v∗v∗∫0∞h2τe-α2τHv∗yτvy∗dτ.Since the function f(x,y,v) is strictly monotonically increasing with respect to x, we have (24)1-xx∗1-fx∗,y∗,v∗fx,y∗,v∗≤0.From H4, we have (25)-1-vv∗+fx,y∗,v∗fx,y,v+vv∗fx,y,vfx,y∗,v∗=1-fx,y,vfx,y∗,v∗fx,y∗,v∗fx,y,v-vv∗≤0,-1-yy∗+fx,y∗,v∗gx∗,y∗fx∗,y∗,v∗gx,y+fx∗,y∗,v∗gx,yyfx,y∗,v∗gx∗,y∗y∗=1-fx∗,y∗,v∗gx,yfx,y∗,v∗gx∗,y∗fx,y∗,v∗gx∗,y∗fx∗,y∗,v∗gx,y-yy∗≤0.Since H(x)≥0 for x>0, we have W˙1≤0 with equality if and only if x=x∗, y=y∗, and v=v∗. It follows from LaSalle invariance principle that E∗ is globally asymptotically stable.


## 5. Application

In this section, we consider the following HIV infection model with distributed delays:(26)x˙t=λ-dxt-β1xtvt1+γ1vt-β2xtyt1+γ2yt,y˙t=∫0∞h1τe-α1τβ1xt-τvt-τ1+γ1vt-τ+β2xt-τyt-τ1+γ2yt-τdτ-ayt,v˙t=k∫0∞h2τe-α2τyt-τdτ-μvt,where γ1 and γ2 are nonnegative constants that measure the saturation effect. The parameters β1 and β2 are the virus-to-cell infection rate and the cell-to-cell transmission rate, respectively. The other parameters have the same biological meanings as in model (1). Further, system (26) is a special case of (1) with f(x,y,v)=β1x/1+γ1v and g(x,y)=β2x/1+γ2y. Notice that the HIV infection model presented by Lai and Zou [23] is a particular case of our model (26), it suffices to take γ1=γ2=0 and h2(τ)=δ(τ). In addition, system (26) always has a disease-free equilibrium Ef(λ/d,0,0) and a unique chronic infection equilibrium E∗(x∗,y∗,v∗) when R0=β1λkη1η2/daμ+β2λη1/da>1.On the other hand, it is easy to see that the hypotheses(H0)–(H3) are satisfied. Furthermore, we have (27)1-fx,y,vfx,y∗,v∗fx,y∗,v∗fx,y,v-vv∗=-δ1v-v∗2v∗1+δ1v∗1+δ1v≤0,(28)1-fx∗,y∗,v∗gx,yfx,y∗,v∗gx∗,y∗fx,y∗,v∗gx∗,y∗fx∗,y∗,v∗gx,y-yy∗=-δ2y-y∗2y∗1+δ1y∗1+δ1y≤0.Consequently, the hypothesis H4 is satisfied. By applying Theorems 4 and 5, we get the following result.Corollary 6.
(i)
IfR0≤1, then the disease-free equilibrium Ef of system (26) is globally asymptotically stable.
(ii)
IfR0>1, then the disease-free equilibrium Ef becomes unstable and the chronic infection equilibrium E∗ of (26) is globally asymptotically stable.


## 6. Conclusion

In this work, we have proposed a mathematical model that describes the dynamics of viral infections, such as HIV and HBV, and takes into account the two modes of transmission and the two kinds of delays, one in cell infection and the other in virus production. The transmission process for both modes is modeled by two general incidence functions that include many types of incidence rates existing in the literature. Further, the two delays are modeled by infinite distributed delays. Under some assumptions on the general incidence functions, we have proved that the global stability of the proposed model is fully determined by one threshold parameter that is the basic reproduction numberR0. In addition, the viral infection models with infinite distributed delays and the corresponding results presented in several previous studies are extended and generalized.In this study, we have neglected the mobility of cells and virus. Motivated by the works in [28–32], we will consider this mobility in our future project in order to improve our present model.



---

*Source: 9818372-2018-02-01.xml*