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Open Mathematics

formerly Central European Journal of Mathematics

Editor-in-Chief: Gianazza, Ugo / Vespri, Vincenzo


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Volume 16, Issue 1

Issues

Volume 13 (2015)

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

Eric Ávila-Vales
  • Corresponding author
  • Universidad Autónoma de Yucatán, Facultad de Matemáticas, Anillo Periférico Norte, Tablaje 13615, C.P. 97119, Mérida, Yucatán, México
  • Email
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
/ Abraham Canul-Pech
  • Universidad Autónoma de Yucatán, Facultad de Matemáticas, Anillo Periférico Norte, Tablaje 13615, C.P. 97119, Mérida, Yucatán, México
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
/ Erika Rivero-Esquivel
  • Universidad Autónoma de Yucatán, Facultad de Matemáticas, Anillo Periférico Norte, Tablaje 13615, C.P. 97119, Mérida, Yucatán, México
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
Published Online: 2018-12-26 | DOI: https://doi.org/10.1515/math-2018-0117

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)=sdx(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)=sdx(t)βx(t)v(t)(1+ax(t))(1+bv(t)),(5)

y(t)=βx(tτ)v(t)esτ(1+ax(tτ))(1+bv(tτ))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)=sdx(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(tτ)z(tτ)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)=sdx(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

  1. f(0, y, v) = 0, for all y ≥ 0, v ≥ 0.

  2. fx(x, y, v) > 0 for all x > 0, y > 0, v > 0.

  3. fy(x, y, v) ≤ 0 and fv(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˙=sdxf(x,y,v)v,y˙=f(x,y,v)vaypyz,v˙=kyuv,z˙=cyzbz,

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˙=0f1(τ)h(x(tτ),v(tτ))dτag1(y)pw(y,z),v˙=k0f2(τ)g1(y(tτ))dτ,z˙=c0f3(τ)w(y(tτ),z(tτ))dτbg3(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 fi, h, w, and gi 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˙=0f1(τ)f(x(tτ),y(tτ),v(tτ))v(tτ)eα1τaφ1(y(t))pw(y(t),z(t)),(18)

v˙=k0f2(τ)eα2τφ1(y(tτ))uv(t),(19)

z˙=c0f3(τ)w(y(tτ),z(tτ))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:

  • H1)

    n(x) is continuously differentiable, and exist > 0 such as that n() = 0, n(x) > 0 for x ∈ [0, ), and n(x) < 0 for x < . Typical functions appearing in the literature are n(x) = sdx and n(x) = sdx + rx(1 − x/xmax).

  • H2)

    φi is strictly increasing on [0, ∞); φi(0) = 0; φi(0)=1; limy⟶∞φi(y) = +∞; and there exists ki > 0 such as φi(y) ≥ kiy for any yi ≥ 0.

  • H3)

    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 fi for i = 1, 2 are assumed to satisfy the following (see [9, 14]):

  • fi(τ) ≥ 0, for τ ≥ 0.

  • 0 fi(τ) = 1 for i = 1, 2.

  • 0 f3(τ) ≤ 1 and 0 f3(τ) e < ∞ for some s > 0.

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

  • H4)

    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:={ϕC((,0],R),ϕ(θ)eαθis uniformly continous forθ(,0]andϕ<},

where the norm of a ϕ ∈ 𝓒 is defined as ∥ϕ∥ = supθ≤0ϕ(θ)∣eαθ. 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 in X.

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 ∈ (t1ϵ, 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 limsupt⟶∞ x(t) ≤ .

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

0f1(τ)eα1τx(tτ)dτ+y(t)=0f1(τ)eα1τn(x(tτ))dτaφ1(y)M1G1ak1y,

where M1 = supx∈[0,] n(x) and G1 = 0 f1(τ)eα1τ .

Let e(t) = 0 f1(τ)eα1τx(tτ), we have that e(t) ≤ G1 for t > 0. Then

(e(t)+y(t))M1G1ak1y=M1G1+M1G1M1G1ak1y2M1G1M1x¯e(t)ak1y2M1G1μ¯(e(t)+y(t)) where μ¯=minM1x¯,ak1,

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

v˙=k0f2(τ)eα2τφ1(y(tτ))dτuv(t)kM2G2uv(t),

where M2=supy0,2M1G1μ¯φ1(y) and G2 = 0 f2(τ)eα2τ , and thus limsupt⟶∞ v(t) kM2G2u.

Using a similar argument, let G3 = 0 f3(τ)d τ, 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)vacG3pφ1(y)cG3φ1(y)φ2(z)+cG3φ1(y)φ2(z)bφ2(z)cG3G1pM3acG3pk1ybk2zcG3G1pM3μ~cG3py+z=cG3G1pM3μ~L(t),

therefore limsupt⟶∞ L(t) cG1G3M3pμ~. Since cG3py(t)0, we know that limsupt⟶∞ 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)vaG1φ1(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 = (, 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(, 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^=φ11bcG3,v^=kφ1(y^)G2u.

Define also H(x) = n(x) − f(x, ŷ, ), we have H(0) = n(0) > 0 and H() = − f(, ŷ, ), with f(, ŷ, ) > f(0, ŷ, ) = 0 by (ii), so there exists a ∈ (0, ) such that H() = 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(, 0, 0) > f(, y, v), for all y, v > 0, moreover for all x ∈ [0, ) we have f(x, y, v) < f(, 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))(xξ)<0 for xξ,x[0,x¯]},Xf(y,v)={ξ[0,x¯]:(f(x,y,v)f(ξ,y,v))(xξ)<0 for xξ,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.

  • (A1)

    The system (17)-(20) has an equilibrium E1 = (x1, y1, v1, 0) satisfying x1X.

  • (A2)

    The system (17) has an equilibrium E2 = (x2, y2, v2, z2) satisfying XnXf(y2, v2).

Theorem 2.1

Assume that i) − iii) and H1H4 are satisfied.

  1. If R0 ≤ 1, then E0 = (x0, 0, 0) is the unique equilibrium of the system (17)-(20).

  2. if R1 ≤ 1 < R0 then in addition to E0, system (17)-(20) has a CTL inactivated infection equilibrium E1.

  3. If R0 > R1 > 1 then, in addition to E0 and E1 system (17) has a CTL activated infection equilibrium.

When x = 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 φ11 exists. Solving n(x)=aφ1(y)G1 for y gives us that

y(x)=ϕ(x)=φ11n(x)G1a,

with ϕ() = 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, ).

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

G(x)=f(x,y(x),v(x))n(x)v(x)=fx,ϕ(x),n(x)kG1G2auaukG1G2,

we have G(0)=aukG1G2<0,G(x¯)=f(x¯,0,0)aukG1G2=aukG1G2R01. Therefore, if R0 > 1, then G(x) has a root x* ∈ (0, ) 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, ), 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 x1Xn implies that n(x1)>n(x1). By virtue of x1Xh, 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=φ11bcG3.(30)

Note that the values , ŷ, are used to define R1, so they clearly satisfy the equilibrium equations. Solving the equation (23) for z and using (H2) yields z^=ϕ21kG1G2f(x^,y^,v^)aupu=ϕ21aR11)p. Using the fact that φ21 is defined on [0, ∞) we conclude that the CTL-AE E2 = (, ŷ, , ) 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 x2Xn 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 x2Xf(y2, v2) and hence E2 is unique.

3 Global stability

Let R0 and R1 be defined as in previous section.

Theorem 3.1

If R0 < 1, then infection free-Equilibrium E0 of model (17)-(20) is globally asymptotically stable.

Proof

Define a Lyapunov functional

V1=xx0+x0xf(x0,0,0)f(s,0,0)ds+1G1y+akG1G2v+pcG1G3z+1G10f1(τ)eα1τtτtf(x(s),y(s),v(s))v(s)dsdτ+aG1G20f2(τ)eα2τtτtφ1(y(s))dsdτ+pG1G30f3(τ)tτtφ1(y(s))φ2(z(s))dsdτ,

where Gi = 0 fi(τ),i = 1, 2,3.

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

V˙1=1f(x0,0,0)f(x,0,0)x˙+1G1y˙+akG1G2v˙+pcG1G3z˙+f(x,y,v)v1G10f1(τ)eα1τf(x(tτ),y(tτ),v(tτ))dτ+aG1φ1(y)aG1G20f2(τ)eα2τφ1(y(tτ))dτ+pG1φ1(y)φ2(z)pG1G30f3(τ)φ1(y(tτ))φ2(z(tτ))dτ=1f(x0,0,0)f(x,0,0)n(x)f(x,y,v)v+1G10f1(τ)eα1τf(x(tτ),y(tτ),v(tτ))v(tτ)aφ1(y)pφ1(y)φ2(z)+akG1G2k0f2(τ)eα2τφ1(y(tτ))uv+pcG1G3c0f3(τ)φ1(y(tτ)φ2(z(tτ))bφ2(z)+f(x,y,v)v1G10f1(τ)eα1τf(x(tτ),y(tτ),v(tτ))dτ+aG1φ1(y)aG1G20f2(τ)eα2τφ1(y(tτ))dτ+pG1φ1(y)φ2(z)pG1G30f3(τ)φ1(y(tτ))φ2(z(tτ))dτ.

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

V˙1=n(x)n(x0)1f(x0,0,0)f(x,0,0)+aukG1G2vf(x,y,v)f(x,0,0)f(x0,0,0)kG1G2au1pbcG1G3φ2(z)=n(x)n(x0)1f(x0,0,0)f(x,0,0)+aukG1G2vf(x,y,v)f(x,0,0)R01pbcG1G3φ2(z)n(x)n(x0)1f(x0,0,0)f(x,0,0)+aukG1G2vR01pbcG1G3φ2(z).

Using the following inequalities:

n(x)n(x0)<0,1f(x0,0,0)f(x,0,0)0forxx0,n(x)n(x0)>0,1f(x0,0,0)f(x,0,0)0forx<x0.

We have that

n(x)n(x0)1f(x0,0,0)f(x,0,0)0.

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

Theorem 3.2

If R0 > 1 and R1 < 1, then CTL-IE E1, of model (17) is globally asymptotically stable.

Proof

Consider the following Lyapunov functional:

V1=L^(t)

where

L^=xx1x1xf(x1,y1,v1)f(s,y1,v1)ds+1G1y1y1φ1(y1)φ1(σ)dσ+akG1G2v1v1v1σdσ+pcG1G3z+f(x1,y1,v1)v1G10f1(τ)eα1τ0τHf(x(tω),y(tω),v(tω))v(tω)f(x1,y1,v1)v1dωdτ+aφ1(y1)G1G20f2(τ)eα2τ0τHφ1(tω)φ1(y1)dωdτ+pG1G30f3(τ)0τφ1(y(tω))φ2(z(tω))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 along the positive solutions of (17), we get

V˙1=1f(x1,y1,v1)f(x,y1,v1)x˙+1G11φ1(y1)φ1(y)y˙+akG1G21v1vv˙+pcG1G2z˙+f(x1,y1,v1)v1G10f1(τ)eα1τHf(x,y,v)vf(x1,y1,v1)v1Hf(x(tτ),y(tτ),v(tτ))v(tτ)f(x1,y1,v1)v1dτ+aφ1(y1)G1G20f2(τ)eα2τHφ1(y)φ1(y1)Hφ1(y(tτ))φ1(y1)dτ+pG1φ1(y)φ2(z)pG1G30f3(τ)φ1(y(tτ))φ2(z(tτ))dτ=1f(x1,y1,v1)f(x,y1,v1)n(x)f(x,y,v)v+1G11φ1(y1)φ1(y)0f1(τ)eα1τf(x(tτ),y(tτ),v(tτ))v(tτ)aφ1(y)pφ1(y)φ2(z)+akG1G21v1vk0f2(τ)eα2τφ1(yτ)uv+pcG1G2c0f3(τ)φ1(y(tτ))φ2(z(tτ))bφ2(z)+f(x1,y1,v1)v1G10f1(τ)eα1τHf(x,y,v)vf(x1,y1,v1)v1Hf(x(tτ),y(tτ),v(tτ))v(tτ)f(x1,y1,v1)v1dτ+aφ1(y1)G1G20f2(τ)eα2τHφ1(y)φ1(y1)Hφ1(y(tτ))φ1(y1)dτ+pG1φ1(y)φ2(z)pG1G30f3(τ)φ1(y(tτ))φ2(z(tτ))dτ.

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

V˙1=n(x)n(x1)1f(x1,y1,v1)f(x,y1,v1)+aφ1(y1)G11f(x1,y1,v1)f(x,y1,v1)+f(x,y,v)vf(x,y1,v1)v1+aφ1(y1)G111G10f1(τ)eα1τφ1(y1)φ1(y)v(tτ)v1f(x(tτ,y(tτ),v(tτ)))f(x1,y1,v1)dτ+aφ1(y1)G11G10f1(τ)eα1τlnf(x(tτ),y(tτ),v(tτ))v(tτ)f(x1,y1,v1)v1dτ+aφ1(y1)G11G20f2(τ)eα2τlnφ1(y(tτ))φ1(y1)dτlnf(x,y,v)vφ1(y)f(x1,y1,v1)v1φ1(y1)+pbφ1(z)cG1G3cG3φ1(y1)b1.

Therefore

V˙1=n(x)n(x1)1f(x1,y1,v1)f(x,y1,v1)+aφ1(y1)G11vv1+f(x,y1,v1)f(x,y,v)+f(x,y,v)vf(x,y1,v1)v1aφ1(y1)G1Hf(x1,y1,v1)f(x,y1,v1)+Hf(x,y1,v1)f(x,y,v)aφ1(y1)G11G10f1(τ)eα1τHφ1(y1)φ1(y)v(tτ)v1f(x(tτ),y(tτ),z(tτ))f(x1,y1,v1)dτaφ1(y1)G11G20f2(τ)eα2τHφ1(y(tτ))v1φ1(y1)vdτ+pbφ1(z)cG1G3R11.

Using the inequalities:

n(x)n(x1)<0,1f(x1,y1,v1)f(x,y1,v1)0forxx1,n(x)n(x1)>0,1f(x1,y1,v1)f(x,y1,v1)0forx<x1.

We have that

1xx11f(x1,y1,v1)f(x,y1,v1)0,1vv1+f(x,y1,v1)f(x,y,v)+vv1f(x,y,v)f(x,y1,v1)=1f(x,y,v)f(x,y1,v1)f(x,y1,v1)f(x,y,v)vv10.

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

Theorem 3.3

Assume that (i)−(iii), H1H4 hold and f3(τ) = δ (τ). If R1 > 1, then the CTL-AE, E2 is Globally asymptotically stable.

Proof

Define a Lyapunov functional for E2.

V2=xx2x2xf(x2,y2,v2)f(s,y2,v2)ds+1G1y2y1φ1(y2)φ1(σ)dσ+a+pφ2(z2)kG1G2v2v1v2σdσ+pcG1z2z1φ2(z2)φ2(σ)dσ+1G1f(x2,y2,v2)v20f1(τ)eα1τtτtHf(x(s),y(s),v(s))v(s)f(x2,y2,v2)v2dsdτ+a+pφ2(z2)kG1G2φ1(y2)0f2(τ)eα2τtτtHφ1(y(s))φ1(y2)dsdτ.

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

V2˙=1f(x2,y2,v2)f(x,y2,v2)x˙+1G11φ1(y2)φ1(y)y˙+a+pφ2(z2)kG1G21v2vv˙+pcG1z2z1φ2(z2)φ2(z)z˙+1G1f(x2,y2,v2)v20f1(τ)eα1τHf(x,y,v)vf(x2,y2,v2)v2Hf(x(tτ),y(tτ),v(tτ))v(tτ)f(x2,y2,v2)v2dτ+a+pφ2(z2)kG1G2φ1(y2)0f2(τ)eα2τHφ1(y)φ1(y2)Hφ1(y(tτ))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)1f(x2,y2,v2)f(x,y2,v2)+aφ1(y2)+pφ1(y2)φ2(z2)G11f(x2,y2,v2)f(x,y2,v2)+f(x,y,v)vf(x,y2,v2)v2+aφ1(y2)+pφ1(y2)φ2(z2)G111G10f1(τ)eα1τφ1(y2)φ1(y)v(tτ)v2f(x(tτ,y(tτ),v(tτ)))f(x1,y2,v2)dτ+aφ1(y2)+pφ1(y2)φ2(z2)G11G10f1(τ)eα1τlnf(x(tτ),y(tτ),v(tτ))v(tτ)f(x2,y2,v2)v2dτ+aφ1(y2)+pφ1(y2)φ2(z2)G11G20f2(τ)eα2τlnφ1(y(tτ))φ1(y2)dτlnf(x,y,v)vφ1(y)f(x2,y2,v2)v2φ1(y2).

Therefore

V2˙=n(x)n(x1)1f(x2,y2,v2)f(x,y2,v2)+aφ1(y2)+pφ1(y2)φ2(z2)G11vv2+f(x,y2,v2)f(x,y,v)+f(x,y,v)vf(x,y2,v2)v2aφ1(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)G11G10f1(τ)eα1τHφ1(y2)φ1(y)v(tτ)v2f(x(tτ),y(tτ),z(tτ))f(x2,y2,v2)dτaφ1(y2)+pφ1(y2)φ2(z2)G11G20f2(τ)eα2τHφ1(y(tτ))v2φ1(y2)vdτ.

Therefore 2 ≤ 0, thus E2 is stable. 2 = 0 if and only if x = x2, y = y2, v = v2, z = z3. So, the largest compact invariant set in {(x, y, v, z) : 2 = 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 functions n(x)=λdx+rx1xK, ϕ1(y) = y, ϕ2(z) = z, w(y, z) = yz and f(x,y,v)=βxαy+γx. Let τ1, τ2 ∈ [0, ∞) two fixed delays, and set f1(τ) = δ(ττ1), f2(τ) = δ(ττ2), f3(τ) = δ(τ), where δ is the Dirac delta function defined as

0δ(ττi)F(τ)dτ=F(τi).

Then, the model takes the form:

x˙=λdx+rx1xKβxvαy+γx,(34)

y˙=βx(tτ1)v(tτ1)αy(tτ1)+γx(tτ1)eμτ1aypyz,(35)

v˙=keα2τ2y(tτ2)uv,(36)

z˙=cyzbz,(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 by E0 = (666.6666, 0, 0, 0), so the basic reproduction number 𝓡0 is obtained by:

R0=kG1G2f(x0,0,0)au=keμτ1eα2τ2βauγ=105.10841176326923474β.

𝓡0 ≤ 1 iff β ≤ 0.009513986. We set β = 0.003 . By Theorem 2.1 i), we have the single equilibrium E0 = (666.6666, 0, 0, 0) which is globally asymptotically stable by Theorem 3.1. Using a constant history function S = (25, 50, 10, 5) for t ∈ (0, 10) and the tool DDE 23 from Matlab, we can compute numerically the solution of system (34)-(37) for t ∈ (10, 200). The results are shown in Figure 1, where we can see that the solution goes to the equilibria point E0.

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

Global stability of infection free equilibrium for 𝓡0 < 1.

Example 4.2

Now, set β = 0.0096, so R0 > 1. Computing R1 from its definition we have R1 = keμτ1eα2τ2auf(x^,y^,v^), where

y^=bc,v^=y^keα2τ2u,n(x^)f(x^,y^,v^)v^=0,

therefore R1 = 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, E1 is globally asymptotically stable as we can see in Figure 2.

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

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

Finally, for β = 1, we have R0 > 1 and R1 = 6.088529, so there exists an infection free equilibrium E0 and 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 equilibrium E2. We can see in Figure 3 that the solutions approach to E2.

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 is h(x, v). Our function f includes the three variables x, y, v and satisfies our conditions (i)-(iii) to be an admisible incidence rate.

  • i)

    f(0, y, v) = 0,   ∀y, v ≥ 0.

  • ii)

    fx=β(1+αy)(1+γv)>0,x,y,v>0.

  • iii a)

    fy=βxααy+12γv+10x,y,v0.

  • iii b)

    fv=βxγαy+1γv+120,x,y,v0.

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

x˙=λdx+rx1xKβxv(1+y)(1+v),y˙=βx(tτ1)v(tτ1)(1+y(tτ1))(1+v(tτ1))eμτ1aypyz,v˙=keα2τ2y(tτ2)uv,z˙=cyzbz,

The infection free equilibrium is E0 = (666.6666, 0, 0, 0) as in previous cases, with R0 = 70.0722745 β, so R0 > 1 iff β > 0.01427097. Therefore, if we set β = 0.1 then we have R0 > 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)

with E2 globally asymptotically stable. Figure 4 shows how the solutions approach to the equilibrium E2.

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 = (, 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), fi(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), H1H4 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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About the article

Received: 2018-02-08

Accepted: 2018-10-15

Published Online: 2018-12-26


Citation Information: Open Mathematics, Volume 16, Issue 1, Pages 1374–1389, ISSN (Online) 2391-5455, DOI: https://doi.org/10.1515/math-2018-0117.

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© 2018 Ávila-Vales et al., published by De Gruyter. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License. BY-NC-ND 4.0

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