Global stability of a distributed delayed viral model with general incidence rate

Eric Ávila-Vales; Abraham Canul-Pech; Erika Rivero-Esquivel

doi:10.1515/math-2018-0117

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Global stability of a distributed delayed viral model with general incidence rate

Eric Ávila-Vales , Abraham Canul-Pech and Erika Rivero-Esquivel

Published/Copyright: December 26, 2018

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$Open Mathematics$

From the journal Open Mathematics Volume 16 Issue 1

Abstract

In this paper, we discussed a infinitely distributed delayed viral infection model with nonlinear immune response and general incidence rate. We proved the existence and uniqueness of the equilibria. By using the Lyapunov functional and LaSalle invariance principle, we obtained the conditions of global stabilities of the infection-free equilibrium, the immune-exhausted equilibrium and the endemic equilibrium. Numerical simulations are given to verify the analytical results.

Keywords: CTL-response; Lyapunov functional; Basic reproduction number; Viral reproduction number

MSC 2010: 34K20; 34D23

1 Introduction

During recent decades there has been a lot of research regarding mathematical modelling of viruses’ dynamics via models of ordinary differential equations. The advances in immunology have lead us to better understand the interactions between populations of virus and the immune system, therefore several nonlinear sytems of ordinary differential equations have been proposed. Nowak and Bengham [1] study the model

x′(t)=s−dx(t)−βx(t)v(t),(1)

y′(t)=βx(t)v(t)−ay(t)−py(t)z(t),(2)

v′(t)=ky(t)−uv(t),(3)

z′(t)=cy(t)z(t)−bz(t).(4)

Where x(t) denotes the number of healthy cells, y(t) denotes the infected cells, v(t) denotes the number of mature viruses and z(t) denotes the number of CTL (cytotoxic T lymphocyte response) cells. Uninfected target cells are assumed to be generated at a constant rate s and die at rate d. Infection of target cells by free virus is assumed to occur at rate β. Infected cells die at rate a and are removed at rate p by the CTL immune response. New virus is produced from infected cells at rate k and dies at rate u. The average lifetime of uninfected cells, infected cells and free virus is thus given by 1/d, 1/a and 1/u, respectively. The parameter c denotes the rate at which the CTL response is produced and b denotes death rate of the CTL response. All given constants are assumed to be positive.

Generally, the type of incidence function used in a model has an important role in modeling the dynamics of viruses. The most common, is the bilinear incidence rate βxv. However, this rate is not useful all the time. For instance, bilinear incidence suggests, that the model can not describe the infection process of hepatitis B, where individuals with small liver are more resistant to infection than the ones with a bigger liver. Recently, works about models of infections by viruses have used the incidence function of type Beddington-DeAngelis and Crowley-Martin. In [2] the authors propose a model of infection by virus with Crowley-Martin functional response βx(t)v(t)(1+ax(t))(1+bv(t)). Li and Fu in [3] study the following system:

x′(t)=s−dx(t)−βx(t)v(t)(1+ax(t))(1+bv(t)),(5)

y′(t)=βx(t−τ)v(t)e−sτ(1+ax(t−τ))(1+bv(t−τ))−ay(t)−py(t)z(t),(6)

v′(t)=ky(t)−uv(t),(7)

z′(t)=cy(t)z(t)−bz(t).(8)

They construct a Lyapunov functional to establish the global dynamics of the system. More recently, in [4], the authors consider the system

x′(t)=s−dx(t)−βx(t)v(t)1+ax(t)+bv(t),(9)

y′(t)=βx(t)v(t)1+ax(t)+bv(t)−ay(t)−py(t)z(t),(10)

v′(t)=ky(t)−uv(t),(11)

z′(t)=cy(t−τ)z(t−τ)−bz(t).(12)

They also give also results of global stability as well as, Hopf bifurcation results.

Yang and Wei in [5] consider a more general incidence rate, they study the system

x′(t)=s−dx(t)−x(t)f(v(t)),(13)

y′(t)=x(t)f(v(t))−ay(t)−py(t)z(t),(14)

v′(t)=ky(t)−uv(t),(15)

z′(t)=cy(t)z(t)−bz(t),(16)

giving some results about global stability in terms of the basic reproduction number and the immune response reproduction number. Here the f(v) is assumed to be a continuous function on v that belongs to (0, ∞) and satisfies f(0) = 0, f′(v) > 0 for all v greater or equal to 0 and f″(v) < 0 for all v greater or equal to 0.

In [6], the authors consider a more general incidence rate f(x, y, v)v, where f is assumed to be continuously differentiable in the interior of R+3 and satisfies the following hypotheses

f(0, y, v) = 0, for all y ≥ 0, v ≥ 0.
∂f∂x(x, y, v) > 0 for all x > 0, y > 0, v > 0.
∂f∂y(x, y, v) ≤ 0 and ∂f∂v(x, y, v) ≤ 0 for all x ≥ 0, y ≥ 0, v ≥ 0.

In [7], authors analyze an infection model by virus, with general incidence and immune response, which generalizes the systems [2, 6, 8], namely,

x˙=s−dx−f(x,y,v)v,y˙=f(x,y,v)v−ay−pyz,v˙=ky−uv,z˙=cyz−bz,

where f(x, y, v) is a continuous and differentiable function in the interior of R+3 and satisfies the conditions (i), (ii), (iii).

Another viral infection model is the studied in [9] given by:

x˙=n(x)−h(x,v),y˙=∫0∞f1(τ)h(x(t−τ),v(t−τ))dτ−ag1(y)−pw(y,z),v˙=k∫0∞f2(τ)g1(y(t−τ))dτ,z˙=c∫0∞f3(τ)w(y(t−τ),z(t−τ))dτ−bg3(z).

Where x, y, v, z denotes the non infected cells, infected cells, virus and specific virus CTL at time t respectively. Conditions on functions f_i, h, w, and g_i are specified in [9]. Related works are [10, 11, 12, 13, 14].

Based on the discussion above, we will study a delayed viral infection model with general incidence rate and CTL immune response given by

x˙=n(x)−f(x(t),y(t),v(t))v(t),(17)

y˙=∫0∞f1(τ)f(x(t−τ),y(t−τ),v(t−τ))v(t−τ)e−α1τ−aφ1(y(t))−pw(y(t),z(t)),(18)

v˙=k∫0∞f2(τ)e−α2τφ1(y(t−τ))−uv(t),(19)

z˙=c∫0∞f3(τ)w(y(t−τ),z(t−τ))−bφ2(z(t)).(20)

The dynamics of uninfected cells, x, in the absence of infection is governed by x′ = n(x). n(x) is the intrinsic growth rate of uninfected cells accounting for both productions and natural mortality. This is assumed to satisfy the following:

n(x) is continuously differentiable, and exist x̄ > 0 such as that n(x̄) = 0, n(x) > 0 for x ∈ [0, x̄), and n(x) < 0 for x < x̄. Typical functions appearing in the literature are n(x) = s − dx and n(x) = s − dx + rx(1 − x/x_max).
φ_i is strictly increasing on [0, ∞); φ_i(0) = 0; φi′(0)=1; lim_y⟶∞φ_i(y) = +∞; and there exists k_i > 0 such as φ_i(y) ≥ k_iy for any y_i ≥ 0.
w(y, z) is continuously differentiable; ∂w(y,z)∂z>0 for y ∈ (0, ∞), z ∈ [0, ∞); w(y, z) > 0 for y ∈ (0, ∞) and z ∈ (0, ∞) with w(y, z) = 0 if and if only y = 0 or z = 0.

All parameters are nonnegative and the distributions f_i for i = 1, 2 are assumed to satisfy the following (see [9, 14]):

f_i(τ) ≥ 0, for τ ≥ 0.
∫0∞f_i(τ)dτ = 1 for i = 1, 2.
∫0∞f₃(τ)dτ ≤ 1 and ∫0∞f₃(τ) e^sτdτ < ∞ for some s > 0.

In addition, the uniqueness and global stability results on the positive equilibrium require the following assumption:

w(y, z) = φ₁(y)φ₂(z).

In this paper, we will study the global dynamics model of (17)-(20). Organization is as follows: in section 2, we prove the existence and uniqueness of the infection free equilibrium, the CTL-inactivated infection equilibrium and the CTL-activated infection equilibrium. In section 3, the conditions that allow the global stability of each equilibrium are determined and proved. Section 4, provide several numerical simulations that show the results obtained in section 3 and 4. Finally, in section 5 we summarize the results we have obtained, comparing them with previous models studied in literature, and setting the guidelines for possible future work.

1.1 Positivity and boundedness

For system (17) the suitable space is 𝓒⁴ = 𝓒 × 𝓒 × 𝓒 × 𝓒, where 𝓒 is the Banach space of fading memory type ([15]):

C:={ϕ∈C((−∞,0],R),ϕ(θ)eαθis uniformly continous forθ∈(−∞,0]and∥ϕ∥<∞},

where the norm of a ϕ ∈ 𝓒 is defined as ∥ϕ∥ = sup_θ≤0 ∣ϕ(θ)∣e^αθ. The nonnegative cone of 𝓒 is defined as 𝓒₊ = C((−∞, 0], ℝ₊).

Theorem 1.1

Under the initial conditions, all solutions of system(17)-(20)are positive and ultimately uniformly bounded inX.

Proof

To see that x(t) is positive, we proceed by contradiction. Let t₁ the first value of time such that x(t₁) = 0. From (17) we see that x′(t₁) = n(0) > 0 and x(t₁) = 0, therefore there exists ϵ > 0 such that x(t) < 0 for t ∈ (t₁ − ϵ, t₁), this leads to a contradiction. It follows that x(t) is always positive. With a similar argument we see that y(t), v(t) and z(t) are positive for t ≥ 0.

The hypotheses (H₁) and equation (17) imply that limsup_t⟶∞x(t) ≤ x̄.

From (17), (18) and assumption (H₂), we obtain

∫0∞f1(τ)e−α1τx′(t−τ)dτ+y′(t)=∫0∞f1(τ)e−α1τn(x(t−τ))dτ−aφ1(y)≤M1G1−ak1y,

where M₁ = sup_x∈[0,x̄]n(x) and G₁ = ∫0∞f₁(τ)e^−α₁τdτ.

Let e(t) = ∫0∞f₁(τ)e^−α₁τx(t − τ)dτ, we have that e(t) ≤ x̄G₁ for t > 0. Then

(e(t)+y(t))′≤M1G1−ak1y=M1G1+M1G1−M1G1−ak1y≤2M1G1−M1x¯e(t)−ak1y≤2M1G1−μ¯(e(t)+y(t)) where μ¯=minM1x¯,ak1,

and thus limsup_t⟶∞ (e(t) + y(t)) ≤2M1G1μ¯. Since e(t) ≥ 0, we know that limsup_t⟶∞y(t) ≤2M1G1μ¯. From (19), we have that,

v˙=k∫0∞f2(τ)e−α2τφ1(y(t−τ))dτ−uv(t)≤kM2G2−uv(t),

where M2=supy∈0,2M1G1μ¯φ1(y) and G₂ = ∫0∞f₂(τ)e^−α₂τdτ, and thus limsup_t⟶∞v(t) ≤kM2G2u.

Using a similar argument, let G₃ = ∫0∞f₃(τ)dτ, L(t)=cG3py(t) + z(t), μ̃ = min{ak₁, bk₂}

and M3=sup(x,y,v)∈[0,x¯]×[0,2M1G1μ¯]×[0,kM2G2u]f(x,y,v)v, then

L′(t)=cG3pG1f(x,y,v)v−acG3pφ1(y)−cG3φ1(y)φ2(z)+cG3φ1(y)φ2(z)−bφ2(z)≤cG3G1pM3−acG3pk1y−bk2z≤cG3G1pM3−μ~cG3py+z=cG3G1pM3−μ~L(t),

therefore limsup_t⟶∞L(t) ≤cG1G3M3pμ~. Since cG3py(t)≥0, we know that limsup_t⟶∞z(t) ≤cG1G3M3pμ~.

Therefore, x(t), y(t), v(t) and z(t) are ultimately uniformly bounded in 𝓒⁴. □

Previous theorem implies that the omega limit set of system (17)-(20) is contained in the following bounded feasible region:

Γ=(x,y,v,z)∈C+4:∥x∥≤x¯,∥y∥≤2M1G1μ¯,∥v∥≤kM2G2u,∥z∥≤cG1G3M3pμ~.

It can be verified that the region Γ is positively invariant with respect to model (17)-(20) and that the model is well posed.

2 Existence and uniqueness of equilibria of system

At any equilibrium we have

n(x)−f(x,y,v)v=0,(22)

f(x,y,v)v−aG1φ1(y)−pG1φ1(y)φ2(z)=0,(23)

kφ1(y)−uG2v=0,(24)

cφ1(y)φ2(z)−bG3φ2(z)=0.(25)

System (17)-(20) always has an infection free equilibrium E₀ = (x̄, 0, 0, 0). In addition to E₀, the system could have two types of chronic infection equilibria E₁ = (x₁, y₁, v₁, 0) and E₂ = (x₂, y₂, v₂, z₂) in Γ, where the entries of E₁ and E₂ are strictly positive. The equilibria E₁ and E₂ are called CTL-inactivated infection equilibrium (CTL-IE) and CTL-activated infection equilibrium (CTL-AE), respectively.

We define the general reproduction number as

R(x,y,v)=kG1G2f(x,y,v)au,

which is the ratio of the per capita production and decay rates of mature viruses at an equilibrium (x, y, v, z) with z = 0. In particular, at the infection free equilibrium, E₀, we denote R(x̄, 0, 0) by R₀, representing the basic production number for viral infection:

R0=R(x¯,0,0)=kG1G2f(x¯,0,0)au.(26)

From assumption (H2) we have that φ₁ is invertible, so we can define from equation (25):

y^=φ1−1bcG3,v^=kφ1(y^)G2u.

Define also H(x) = n(x) − f(x, ŷ, v̂)v̂, we have H(0) = n(0) > 0 and H(x̄) = − f(x̄, ŷ, v̂)v̂, with f(x̄, ŷ, v̂) > f(0, ŷ, v̂) = 0 by (ii), so there exists a x̂ ∈ (0, x̄) such that H(x̂) = 0. We denote:

R1=R(x^,y^,v^),(27)

and refer it as the viral reproduction number. From assumptions on f it is easy to see that f(x̄, 0, 0) > f(x̄, y, v), for all y, v > 0, moreover for all x ∈ [0, x̄) we have f(x, y, v) < f(x̄, y, v), so:

R0=R(x¯,0,0)>R(x¯,y,v)>R(x,y,v),∀x∈[0,x¯),y,v>0.(28)

Particularly, R₀ > R₁. The basic reproduction number for the CTL response is given by:

RCTL=cG3φ1(y1)b.

In order to prove of the existence and uniqueness of equilibria, we require two additional assumptions. First, we define the following sets:

Xn={ξ∈[0,x¯]:(n(x)−n(ξ))(x−ξ)<0 for x≠ξ,x∈[0,x¯]},Xf(y,v)={ξ∈[0,x¯]:(f(x,y,v)−f(ξ,y,v))(x−ξ)<0 for x≠ξ,x∈[0,x¯]},X=∩y,v∈(0,y¯)×(0,v¯)Xf(y,v)∩Xn.

The following conditions are used to guarantee the uniqueness of the equilibria.

The system (17)-(20) has an equilibrium E₁ = (x₁, y₁, v₁, 0) satisfying x₁ ∈ X.
The system (17) has an equilibrium E₂ = (x₂, y₂, v₂, z₂) satisfying X_n ∩ X_f(y₂, v₂).

Theorem 2.1

Assume that i) − iii) andH₁ − H₄are satisfied.

IfR₀ ≤ 1, thenE₀ = (x₀, 0, 0) is the unique equilibrium of the system(17)-(20).
ifR₁ ≤ 1 < R₀then in addition toE₀, system(17)-(20)has a CTL inactivated infection equilibriumE₁.
IfR₀ > R₁ > 1 then, in addition toE₀andE₁system(17)has a CTL activated infection equilibrium.

When x = x̄ and y = v = z = 0 the equations (22)-(25)are satisfied, therefore E₀ = (x₀, 0, 0, 0) is a steady state called the infection free equilibrium. To prove that it is unique when R₀ < 1, we look for the existence of a positive equilibrium.

To find a positive equilibrium we proceed as follows: from equation (25)φ₂(z) = 0 or φ₁(y) = bcG1. If φ₂(z) = 0 then from assumption (H₂) we get, z = 0 and using (22)-(24)

n(x)=f(x,y,v)v=aφ1(y)G1=auvkG1G2.(29)

By (H₂), we know that φ1−1 exists. Solving n(x)=aφ1(y)G1 for y gives us that

y(x)=ϕ(x)=φ1−1n(x)G1a,

with ϕ(x̄) = 0 and ϕ(0) = y⁰ being a unique root of equations n(0)=aφ1(y0)G1. Solving (24) for v we obtain:

v(x)=kn(x)G1G2au.

Note that, from (ii) we have f(x, y, v) > f(0, y, v) = 0, ∀x > 0. So if E^* = (x^*, y^*, v^*, z^*) is a positive equilibrium, then n(x^*) = f(x^*, y^*, v^*)v^* > 0, so x^* ∈ (0, x̄).

Now, using (29) define on the interval [0, x̄) the function G,

G(x)=f(x,y(x),v(x))−n(x)v(x)=fx,ϕ(x),n(x)kG1G2au−aukG1G2,

we have G(0)=−aukG1G2<0,G(x¯)=f(x¯,0,0)−aukG1G2=aukG1G2R0−1. Therefore, if R₀ > 1, then G(x) has a root x^* ∈ (0, x̄) such that

f(x∗,y(x∗),v(x∗))−n(x∗)v(x∗)=0,

or equivalently

n(x∗)−f(x∗,y(x∗),v(x∗))v(x∗)=0.

We conclude that, for R₀ > 1, there exists another equilibrium E₁ = (x₁, y₁, v₁, 0) with x₁ = x^* ∈ (0, x̄), y₁ = ϕ(x₁) and v1=kG2ϕ(x1)u. Moreover, using the fact that R₀ > R(x, y, v), for all x, y, v > 0, when R₀ < 1 we have R(x, y, v) < 1. Using (29) we arrive to

n(x)−f(x,y,v)v>0,∀x,y,v>0,

so equation (22) never holds. Therefore, E₁ exists iff R₀ > 1.

Next we show that E₁ = (x₁, y₁, v₁, 0) is unique. Suppose, to the contrary, that there exists another CTL-IE equilibrium E1∗=(x1∗,y1∗,v1∗,0). Without loss of generality, we assume that x1∗<x1. Then x₁ ∈ X_n implies that n(x1∗)>n(x1). By virtue of x₁ ∈ X_h, we get f(x1∗,y1∗,v1∗) < f(x₁, y₁, v₁). On the other hand, it follows from (29) that f(x1∗,y1∗,v1∗) = f(x₁, y₁, v₁) = au/kG₁G₂. This is a contradiction, and thus E₁ is the unique CTL-IE.

Note that the CTL-AE E₂ = (x₂, y₂, v₂, z₂) exists if (x₂, y₂, v₂, z₂) ∈ R+4 satisfies the equilibrium equations (22)-(25) and φ₁(y) = b/cG₃. Therefore according to system (17)-(20) and (H₂) we have

y2=φ1−1bcG3.(30)

Note that the values x̂, ŷ, v̂ are used to define R₁, so they clearly satisfy the equilibrium equations. Solving the equation (23) for z and using (H2) yields z^=ϕ2−1kG1G2f(x^,y^,v^)−aupu=ϕ2−1aR1−1)p. Using the fact that φ2−1 is defined on [0, ∞) we conclude that the CTL-AE E₂ = (x̂, ŷ, v̂, ẑ) exists if and only if R₁ > 1.

Now we will prove that E₂ is unique. Suppose that there exists another CTL-AE, E2∗=(x2∗,y2∗,v2∗,z2∗). Then y2=y2∗ and v2=v2∗. Without loss of generality, we assume that x2∗<x2. Then x₂ ∈ X_n implies n(x2∗)>n(x2). Note that n(x2∗)=f(x2∗,y2,v2) and n(x₂) = f(x₂, y₂, v₂), implies that f(x2∗,y2,v2) > f(x₂, y₂, v₂). This contradicts that x₂ ∈ X_f(y₂, v₂) and hence E₂ is unique.

3 Global stability

Let R₀ and R₁ be defined as in previous section.

Theorem 3.1

IfR₀ < 1, then infection free-EquilibriumE₀of model(17)-(20)is globally asymptotically stable.

Proof

Define a Lyapunov functional

V1=x−x0+∫x0xf(x0,0,0)f(s,0,0)ds+1G1y+akG1G2v+pcG1G3z+1G1∫0∞f1(τ)e−α1τ∫t−τtf(x(s),y(s),v(s))v(s)dsdτ+aG1G2∫0∞f2(τ)e−α2τ∫t−τtφ1(y(s))dsdτ+pG1G3∫0∞f3(τ)∫t−τtφ1(y(s))φ2(z(s))dsdτ,

where G_i = ∫0∞f_i(τ),i = 1, 2,3.

Calculating the derivative of V₁ along the positive solutions of system (17)-(20), it follows that

V˙1=1−f(x0,0,0)f(x,0,0)x˙+1G1y˙+akG1G2v˙+pcG1G3z˙+f(x,y,v)v−1G1∫0∞f1(τ)e−α1τf(x(t−τ),y(t−τ),v(t−τ))dτ+aG1φ1(y)−aG1G2∫0∞f2(τ)e−α2τφ1(y(t−τ))dτ+pG1φ1(y)φ2(z)−pG1G3∫0∞f3(τ)φ1(y(t−τ))φ2(z(t−τ))dτ=1−f(x0,0,0)f(x,0,0)n(x)−f(x,y,v)v+1G1∫0∞f1(τ)e−α1τf(x(t−τ),y(t−τ),v(t−τ))v(t−τ)−aφ1(y)−pφ1(y)φ2(z)+akG1G2k∫0∞f2(τ)e−α2τφ1(y(t−τ))−uv+pcG1G3c∫0∞f3(τ)φ1(y(t−τ)φ2(z(t−τ))−bφ2(z)+f(x,y,v)v−1G1∫0∞f1(τ)e−α1τf(x(t−τ),y(t−τ),v(t−τ))dτ+aG1φ1(y)−aG1G2∫0∞f2(τ)e−α2τφ1(y(t−τ))dτ+pG1φ1(y)φ2(z)−pG1G3∫0∞f3(τ)φ1(y(t−τ))φ2(z(t−τ))dτ.

Using n(x₀) = 0 and simplifying, we get

V˙1=n(x)−n(x0)1−f(x0,0,0)f(x,0,0)+aukG1G2vf(x,y,v)f(x,0,0)f(x0,0,0)kG1G2au−1−pbcG1G3φ2(z)=n(x)−n(x0)1−f(x0,0,0)f(x,0,0)+aukG1G2vf(x,y,v)f(x,0,0)R0−1−pbcG1G3φ2(z)≤n(x)−n(x0)1−f(x0,0,0)f(x,0,0)+aukG1G2vR0−1−pbcG1G3φ2(z).

Using the following inequalities:

n(x)−n(x0)<0,1−f(x0,0,0)f(x,0,0)≥0forx≥x0,n(x)−n(x0)>0,1−f(x0,0,0)f(x,0,0)≤0forx<x0.

We have that

n(x)−n(x0)1−f(x0,0,0)f(x,0,0)≤0.

Since R₀ ≤ 1, we have V̇₁ ≤ 0. Therefore the disease free E₁ is stable, V̇₁ = 0 if and only if x = x₀, y = 0, v = 0, z = 0. So, the largest compact invariant set in {(x, y, v, z):V̇₁ = 0} is just the singleton E₁. From LaSalle invariance principle, we conclude that E₁ is globally asymptotically stable. □

Theorem 3.2

IfR₀ > 1 andR₁ < 1, then CTL-IEE₁, of model (17) is globally asymptotically stable.

Proof

Consider the following Lyapunov functional:

V1=L^(t)

where

L^=x−x1−∫x1xf(x1,y1,v1)f(s,y1,v1)ds+1G1∫y1y1−φ1(y1)φ1(σ)dσ+akG1G2∫v1v1−v1σdσ+pcG1G3z+f(x1,y1,v1)v1G1∫0∞f1(τ)e−α1τ∫0τHf(x(t−ω),y(t−ω),v(t−ω))v(t−ω)f(x1,y1,v1)v1dωdτ+aφ1(y1)G1G2∫0∞f2(τ)e−α2τ∫0τHφ1(t−ω)φ1(y1)dωdτ+pG1G3∫0∞f3(τ)∫0τφ1(y(t−ω))φ2(z(t−ω))dωdτ.

At infected equilibrium

n(x1)−f(x1,y1,v1)v1=0,(31)

f(x1,y1,v1)v1=aφ1(y1)G1,(32)

ukG2=φ1(y1)v1.(33)

Calculating the derivative of L̂ along the positive solutions of (17), we get

V˙1=1−f(x1,y1,v1)f(x,y1,v1)x˙+1G11−φ1(y1)φ1(y)y˙+akG1G21−v1vv˙+pcG1G2z˙+f(x1,y1,v1)v1G1∫0∞f1(τ)e−α1τHf(x,y,v)vf(x1,y1,v1)v1−Hf(x(t−τ),y(t−τ),v(t−τ))v(t−τ)f(x1,y1,v1)v1dτ+aφ1(y1)G1G2∫0∞f2(τ)e−α2τHφ1(y)φ1(y1)−Hφ1(y(t−τ))φ1(y1)dτ+pG1φ1(y)φ2(z)−pG1G3∫0∞f3(τ)φ1(y(t−τ))φ2(z(t−τ))dτ=1−f(x1,y1,v1)f(x,y1,v1)n(x)−f(x,y,v)v+1G11−φ1(y1)φ1(y)∫0∞f1(τ)e−α1τf(x(t−τ),y(t−τ),v(t−τ))v(t−τ)−aφ1(y)−pφ1(y)φ2(z)+akG1G21−v1vk∫0∞f2(τ)e−α2τφ1(y−τ)−uv+pcG1G2c∫0∞f3(τ)φ1(y(t−τ))φ2(z(t−τ))−bφ2(z)+f(x1,y1,v1)v1G1∫0∞f1(τ)e−α1τHf(x,y,v)vf(x1,y1,v1)v1−Hf(x(t−τ),y(t−τ),v(t−τ))v(t−τ)f(x1,y1,v1)v1dτ+aφ1(y1)G1G2∫0∞f2(τ)e−α2τHφ1(y)φ1(y1)−Hφ1(y(t−τ))φ1(y1)dτ+pG1φ1(y)φ2(z)−pG1G3∫0∞f3(τ)φ1(y(t−τ))φ2(z(t−τ))dτ.

Using (31)-(33), we, get

V˙1=n(x)−n(x1)1−f(x1,y1,v1)f(x,y1,v1)+aφ1(y1)G11−f(x1,y1,v1)f(x,y1,v1)+f(x,y,v)vf(x,y1,v1)v1+aφ1(y1)G11−1G1∫0∞f1(τ)e−α1τφ1(y1)φ1(y)v(t−τ)v1f(x(t−τ,y(t−τ),v(t−τ)))f(x1,y1,v1)dτ+aφ1(y1)G11G1∫0∞f1(τ)e−α1τlnf(x(t−τ),y(t−τ),v(t−τ))v(t−τ)f(x1,y1,v1)v1dτ+aφ1(y1)G11G2∫0∞f2(τ)e−α2τlnφ1(y(t−τ))φ1(y1)dτ−lnf(x,y,v)vφ1(y)f(x1,y1,v1)v1φ1(y1)+pbφ1(z)cG1G3cG3φ1(y1)b−1.

Therefore

V˙1=n(x)−n(x1)1−f(x1,y1,v1)f(x,y1,v1)+aφ1(y1)G11−vv1+f(x,y1,v1)f(x,y,v)+f(x,y,v)vf(x,y1,v1)v1−aφ1(y1)G1Hf(x1,y1,v1)f(x,y1,v1)+Hf(x,y1,v1)f(x,y,v)−aφ1(y1)G11G1∫0∞f1(τ)e−α1τHφ1(y1)φ1(y)v(t−τ)v1f(x(t−τ),y(t−τ),z(t−τ))f(x1,y1,v1)dτ−aφ1(y1)G11G2∫0∞f2(τ)e−α2τHφ1(y(t−τ))v1φ1(y1)vdτ+pbφ1(z)cG1G3R1−1.

Using the inequalities:

n(x)−n(x1)<0,1−f(x1,y1,v1)f(x,y1,v1)≥0forx≥x1,n(x)−n(x1)>0,1−f(x1,y1,v1)f(x,y1,v1)≤0forx<x1.

We have that

1−xx11−f(x1,y1,v1)f(x,y1,v1)≤0,−1−vv1+f(x,y1,v1)f(x,y,v)+vv1f(x,y,v)f(x,y1,v1)=1−f(x,y,v)f(x,y1,v1)f(x,y1,v1)f(x,y,v)−vv1≤0.

Since R₁ ≤ 1, we have V̇₁ ≤ 0, thus E₁ is stable. V̇₁ = 0 if and only if x = x₁, y = y₁, v =₁, z = 0. So, the largest compact invariant set in {(x, y, v, z) : v̇₁ = 0} is the singleton E₁. From LaSalle invariance principle, we conclude that E₁ is globally asymptotically stable. □

Theorem 3.3

Assume that (i)−(iii), H₁ − H₄hold andf₃(τ) = δ (τ). IfR₁ > 1, then the CTL-AE, E₂is Globally asymptotically stable.

Proof

Define a Lyapunov functional for E₂.

V2=x−x2−∫x2xf(x2,y2,v2)f(s,y2,v2)ds+1G1∫y2y1−φ1(y2)φ1(σ)dσ+a+pφ2(z2)kG1G2∫v2v1−v2σdσ+pcG1∫z2z1−φ2(z2)φ2(σ)dσ+1G1f(x2,y2,v2)v2∫0∞f1(τ)e−α1τ∫t−τtHf(x(s),y(s),v(s))v(s)f(x2,y2,v2)v2dsdτ+a+pφ2(z2)kG1G2φ1(y2)∫0∞f2(τ)e−α2τ∫t−τtHφ1(y(s))φ1(y2)dsdτ.

The derivative of V₂ along with the solutions of system (17) is

V2˙=1−f(x2,y2,v2)f(x,y2,v2)x˙+1G11−φ1(y2)φ1(y)y˙+a+pφ2(z2)kG1G21−v2vv˙+pcG1∫z2z1−φ2(z2)φ2(z)z˙+1G1f(x2,y2,v2)v2∫0∞f1(τ)e−α1τHf(x,y,v)vf(x2,y2,v2)v2−Hf(x(t−τ),y(t−τ),v(t−τ))v(t−τ)f(x2,y2,v2)v2dτ+a+pφ2(z2)kG1G2φ1(y2)∫0∞f2(τ)e−α2τHφ1(y)φ1(y2)−Hφ1(y(t−τ))φ1(y2)dτ.

Applying n(x₂) = f(x₂, y₂, v₂)v₂, f(x₂, y₂, v₂)v2=1G1 (aφ₁(y₁) + pφ₁(y₁)φ₂(z₂)), φ1(y1)=bc,ukG2=φ1(y2)v2, we obtain

V2˙=n(x)−n(x1)1−f(x2,y2,v2)f(x,y2,v2)+aφ1(y2)+pφ1(y2)φ2(z2)G11−f(x2,y2,v2)f(x,y2,v2)+f(x,y,v)vf(x,y2,v2)v2+aφ1(y2)+pφ1(y2)φ2(z2)G11−1G1∫0∞f1(τ)e−α1τφ1(y2)φ1(y)v(t−τ)v2f(x(t−τ,y(t−τ),v(t−τ)))f(x1,y2,v2)dτ+aφ1(y2)+pφ1(y2)φ2(z2)G11G1∫0∞f1(τ)e−α1τlnf(x(t−τ),y(t−τ),v(t−τ))v(t−τ)f(x2,y2,v2)v2dτ+aφ1(y2)+pφ1(y2)φ2(z2)G11G2∫0∞f2(τ)e−α2τlnφ1(y(t−τ))φ1(y2)dτ−lnf(x,y,v)vφ1(y)f(x2,y2,v2)v2φ1(y2).

Therefore

V2˙=n(x)−n(x1)1−f(x2,y2,v2)f(x,y2,v2)+aφ1(y2)+pφ1(y2)φ2(z2)G11−vv2+f(x,y2,v2)f(x,y,v)+f(x,y,v)vf(x,y2,v2)v2−aφ1(y2)+pφ1(y2)φ2(z2)G1Hf(x2,y2,v2)f(x,y2,v2)+Hf(x,y2,v2)f(x,y,v)−aφ1(y2)+pφ1(y2)φ2(z2)G11G1∫0∞f1(τ)e−α1τHφ1(y2)φ1(y)v(t−τ)v2f(x(t−τ),y(t−τ),z(t−τ))f(x2,y2,v2)dτ−aφ1(y2)+pφ1(y2)φ2(z2)G11G2∫0∞f2(τ)e−α2τHφ1(y(t−τ))v2φ1(y2)vdτ.

Therefore V̇₂ ≤ 0, thus E₂ is stable. V̇₂ = 0 if and only if x = x₂, y = y₂, v = v₂, z = z₃. So, the largest compact invariant set in {(x, y, v, z) : V̇₂ = 0} is the singleton E₂. From LaSalle invariance principle, we conclude that E₂ is globally asymptotically stable. □

4 Numerical simulations

In this section we present some numerical simulations to illustrate the results of stability, obtained in our theorems from previous sections.

Example 4.1

Consider the functionsn(x)=λ−dx+rx1−xK,ϕ₁(y) = y, ϕ₂(z) = z, w(y, z) = yzandf(x,y,v)=βxαy+γx.Letτ₁, τ₂ ∈ [0, ∞) two fixed delays, and setf₁(τ) = δ(τ−τ₁), f₂(τ) = δ(τ−τ₂), f₃(τ) = δ(τ), whereδis the Dirac delta function defined as

∫0∞δ(τ−τi)F(τ)dτ=F(τi).

Then, the model takes the form:

x˙=λ−dx+rx1−xK−βxvαy+γx,(34)

y˙=βx(t−τ1)v(t−τ1)αy(t−τ1)+γx(t−τ1)e−μτ1−ay−pyz,(35)

v˙=ke−α2τ2y(t−τ2)−uv,(36)

z˙=cyz−bz,(37)

Fix the parameters as λ = 200, d = 0.1, r = 0.6, K = 500, p = 1, k = 0.8, α₂ = 0.05, u = 3.5, c = 0.03, b = 0.75, τ₁ = 5, τ₂ = 10, μ = 0.1, α = γ = 0.001. The trivial equilibrium point is given byE₀ = (666.6666, 0, 0, 0), so the basic reproduction number 𝓡₀is obtained by:

R0=kG1G2f(x0,0,0)au=ke−μτ1e−α2τ2βauγ=105.10841176326923474β.

𝓡₀ ≤ 1 iffβ ≤ 0.009513986. We setβ = 0.003 . By Theorem 2.1 i), we have the single equilibriumE₀ = (666.6666, 0, 0, 0) which is globally asymptotically stable by Theorem 3.1. Using a constant history functionS = (25, 50, 10, 5) fort ∈ (0, 10) and the tool DDE 23 from Matlab, we can compute numerically the solution of system(34)-(37)fort ∈ (10, 200). The results are shown in Figure 1, where we can see that the solution goes to the equilibria pointE₀.

$Fig. 1 Global stability of infection free equilibrium for 𝓡0 < 1.$

Fig. 1

Global stability of infection free equilibrium for 𝓡₀ < 1.

Example 4.2

Now, setβ = 0.0096, soR₀ > 1. ComputingR₁from its definition we haveR₁ = ke−μτ1e−α2τ2auf(x^,y^,v^),where

y^=bc,v^=y^ke−α2τ2u,n(x^)−f(x^,y^,v^)v^=0,

thereforeR₁ = 0.97091 < 1 and we have then, a second equilibrium

E1=(659.461141,5.962025,0.826548,0).

Using the history function from previous example we can plot the solution with Matlab. By Theorem 3.2, E₁is globally asymptotically stable as we can see in Figure 2.

$Fig. 2 Global stability of CTL-IE E1 for 𝓡0 > 1 > R1.$

Fig. 2

Global stability of CTL-IE E₁ for 𝓡₀ > 1 > R₁.

Finally, forβ = 1, we haveR₀ > 1 andR₁ = 6.088529, so there exists an infection free equilibriumE₀and two equilibria

E1=(1.461792,152.184859,21.098236,0),

and

E2=(1.537198,25,3.465889,4.070823).

By Theorem 3.3 we have global stability of equilibriumE₂. We can see in Figure 3 that the solutions approach toE₂.

$Fig. 3 Global stability of CTL-AE E2 for R0 > 1, R1 > 1.$

Fig. 3

Global stability of CTL-AE E₂ for R₀ > 1, R₁ > 1.

Example 4.3

In order to show that our model, generalizes the previous articles, we propose the following incidence function to show our results:

f(x,y,v)=βx(1+αy)(1+γv).

This function is not included in the cases studied by [9], since the general form in that work ish(x, v). Our functionfincludes the three variablesx, y, vand satisfies our conditions (i)-(iii) to be an admisible incidence rate.

f(0, y, v) = 0, ∀y, v ≥ 0.
∂f∂x=β(1+αy)(1+γv)>0,∀x,y,v>0.
∂f∂y=−βxααy+12γv+1≤0∀x,y,v≥0.
∂f∂v=−βxγαy+1γv+12≤0,∀x,y,v≥0.

Setting the other functions and parameters, as in Example 4.1, then we obtain model:

x˙=λ−dx+rx1−xK−βxv(1+y)(1+v),y˙=βx(t−τ1)v(t−τ1)(1+y(t−τ1))(1+v(t−τ1))e−μτ1−ay−pyz,v˙=ke−α2τ2y(t−τ2)−uv,z˙=cyz−bz,

The infection free equilibrium isE₀ = (666.6666, 0, 0, 0) as in previous cases, withR₀ = 70.0722745 β, soR₀ > 1 iffβ > 0.01427097. Therefore, if we setβ = 0.1 then we haveR₀ > 1, R₁ = 4.9234560677357286281 and there exists two more equilibria points,

E1=(115.436331,183.268932,25.407594,0),E2=(481.791432,25,3.465889,3.138764)

withE₂globally asymptotically stable. Figure 4 shows how the solutions approach to the equilibriumE₂.

$Fig. 4 Global stability of CTL-AE E2 for f(x,y,v)=βx(1+αy)(1+γv),$\begin{array}{} f(x,y,v)= \frac{\beta x}{(1+\alpha y)(1+ \gamma v)}, \end{array} $ with R0 > 1, R1 > 1.$

Fig. 4

Global stability of CTL-AE E₂ for f(x,y,v)=βx(1+αy)(1+γv), with R₀ > 1, R₁ > 1.

5 Conclusions

In this paper we studied the global properties of a model of infinitely distributed delayed viral infection. That considers a nonlinear CTL immune response, given by w(y, z) = ϕ₁(y) ϕ₂(z) and a general incidence function of the form f(x, y, v)v, where w and f satisfy certain conditions derived from previous works and biological meanings. Even when there exists variety of papers that include the CTL immune response (see for example [1, 3, 4, 5, 7]) and general incidence functions of various types (see [5, 7, 9]), the model proposed in this article includes a family of the works studied by several authors, and their conclusions can be seen as a particular case of our theorems. There lies its importance and relevance.

The model always presents an infection free positive equilibrium E₀ = (x̄, 0, 0, 0), and two types of chronic infection equilibria: the CTL inactivated infection equilibrium (CTL-IE) E₁ = (x₁, y₁, v₁, 0) and the CTL activated infection equilibrium (CTL-AE) E₂ = (x₂, y₂, v₂, z₂). The coexistence of these equilibria is determined by the basic reproduction number R₀ and the viral reproduction number R₁. These were defined in section 2 and are given in terms of parameters and the functions f(x, y, v), f_i(τ), ϕ₁(y) and ϕ₂(z). The results show that R₀ > R₁ and the system admits always a positive infection free equilibrium E₀, which is the unique equilibrium when R₀ ≤ 1. If R₀ > 1 which in addition to E₀ provides only the CTL-IE (when R₁ ≤ 1), or the coexistence of the CTL-IE and CTL-AE (R₁ > 1).

We proved, by construction of a Lyapunov function, that whenever the equilibrium E₀ is unique (R₀ ≤ 1) and R₀ ≠ 1, E₀ is globally asymptotically stable. Moreover when R₀ > 1 and R₁ < 1 the CTL-IE, E₁ is globally asymptotically stable. In the case of CTL-AE, E₂ we obtained conditions for global stability only in the case f₃(τ) = δ (τ), i.e., when the equation ż does not present delay. The results indicate that in this case, the equilibrium E₂ is globally asymptotically stable when R₁ > 1 and conditions (i)−(iii), H₁ − H₄ hold. It will be of interest to find conditions that guarantee the global stability of the E₂ with a general f₃(τ), this topic can be taken as a future work.

Acknowledgement

This article was supported in part by Mexican SNI under grant 15284 and CONACYT Scholarship 295308.

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Received: 2018-02-08

Accepted: 2018-10-15

Published Online: 2018-12-26

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

Articles in the same Issue

https://doi.org/10.1515/math-2018-0117

Keywords for this article

CTL-response; Lyapunov functional; Basic reproduction number; Viral reproduction number

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