# Lyapunov Functionals for Delay Differential Equations Model of Viral Infections.

**ABSTRACT** We study global properties of a class of delay differential equations model for virus infections with nonlinear transmissions. Compared with the typical virus infection dynamical model, this model has two important and novel features. To give a more complex and general infection process, a general nonlinear contact rate between target cells and viruses and the removal rate of infected cells are considered, and two constant delays are incorporated into the model, which describe (i) the time needed for a newly infected cell to start producing viruses and (ii) the time needed for a newly produced virus to become infectious (mature), respectively. By the Lyapunov direct method and using the technology of constructing Lyapunov functionals, we establish global asymptotic stability of the infection-free equilibrium and the infected equilibrium. We also discuss the effects of two delays on global dynamical properties by comparing the results with the stability conditions for the model without delays. Further, we generalize this type of Lyapunov functional to the model described by n-dimensional delay differential equations.

**1**Bookmark

**·**

**194**Views

- [Show abstract] [Hide abstract]

**ABSTRACT:**We consider global stability for a heroin model with two distributed delays. The basic reproduction number of the heroin spread is obtained, which completely determines the stability of the equilibria. Using the direct Lyapunov method with Volterra type Lyapunov functions, we show that the drug use-free equilibrium is globally asymptotically stable if the basic reproduction number is less than one, and the unique drug spread equilibrium is globally asymptotically stable if the basic reproduction number is greater than one.Discrete and Continuous Dynamical Systems - Series B 05/2014; 3(3). · 0.63 Impact Factor - International Journal of Biomathematics 09/2014; 07(05):1450055. · 0.65 Impact Factor
- SourceAvailable from: B.G. Sampath Aruna Pradeep[Show abstract] [Hide abstract]

**ABSTRACT:**This article presents a new eco-epidemiological deterministic delay differential equation model considering a biological controlling approach on mosquitoes, for endemic dengue disease with variable host (human) and variable vector (Aedes aegypti) populations, and stage structure for mosquitoes. In this model, predator-prey interaction is considered by using larvae as prey and mosquito-fish as predator. We give a complete classification of equilibria of the model, and sufficient conditions for global stability/global attractivity of some equilibria are given by constructing suitable Lyapunov functionals and using Lyapunov-LaSalle invariance principle. Also, numerical simulations are presented to show the validity of our results.Electronic Journal of Differential Equations 01/2015; 2015 (2015)(10):1. · 0.42 Impact Factor

Page 1

SIAM J. APPL. MATH.

Vol. 70, No. 7, pp. 2693–2708

c ? 2010 Society for Industrial and Applied Mathematics

LYAPUNOV FUNCTIONALS FOR DELAY DIFFERENTIAL

EQUATIONS MODEL OF VIRAL INFECTIONS∗

GANG HUANG†, YASUHIRO TAKEUCHI‡, AND WANBIAO MA§

Abstract. We study global properties of a class of delay differential equations model for virus

infections with nonlinear transmissions. Compared with the typical virus infection dynamical model,

this model has two important and novel features. To give a more complex and general infection

process, a general nonlinear contact rate between target cells and viruses and the removal rate

of infected cells are considered, and two constant delays are incorporated into the model, which

describe (i) the time needed for a newly infected cell to start producing viruses and (ii) the time

needed for a newly produced virus to become infectious (mature), respectively. By the Lyapunov

direct method and using the technology of constructing Lyapunov functionals, we establish global

asymptotic stability of the infection-free equilibrium and the infected equilibrium. We also discuss

the effects of two delays on global dynamical properties by comparing the results with the stability

conditions for the model without delays. Further, we generalize this type of Lyapunov functional to

the model described by n-dimensional delay differential equations.

Key words. delay differential equations, Lyapunov functional, viral infection, global stability

AMS subject classifications. 92D25, 34A34, 34D20, 34D23

DOI. 10.1137/090780821

1. Introduction. Mathematical modeling and model analysis are very much

needed for a deeper understanding of and as a theoretical laboratory for devising

control and management strategies. The interactions between viruses and cells in an

infection process can be seen as an ecological system within the infected host. In

past decades, many simple mathematical approaches have been developed to explore

the relation between target cells, infected cells producing viruses, and virus load.

A typical model was originally proposed by Anderson and May [1] to describe the

dynamics of microparasites (virus, bacteria, or fungi) with a free-living infective stage

in a population of invertebrate hosts; then Nowak and Bangham [28] developed it to

the basic model of virus dynamics which has become quite popular among theorists

and experimentalists (see, e.g., Nowak and May [29], Perelson and Nelson [31], and

De Leenheer and Smith [4]).

It is well known that viruses are intracellular parasites that depend on the host

cells to survive and duplicate. The host cells can be damaged by antibodies, cy-

tokines, natural killer cells, and T cells. Denoting by x(t), y(t), and v(t) the numbers

of target cells, infected cells, and free viruses, the primary virus dynamical model

assumes that the incidence rate of infection is given by the product of the numbers of

target cells and free viruses, βx(t)v(t) with a constant β > 0. However, as a nonlinear

relationship between parasite dose and infection rate has been frequently observed

∗Received by the editors December 21, 2009; accepted for publication (in revised form) May 27,

2010; published electronically August 10, 2010. This research was partially supported by the Grant-

in-Aid for Scientific Research (C) 22540122, Japan Society for the Promotion of Science.

http://www.siam.org/journals/siap/70-7/78082.html

†Graduate School of Science and Technology, Shizuoka University, Hamamatsu, 4328561, Japan

(f5845034@ipc.shizuoka.ac.jp).

‡Corresponding author. Graduate School of Science and Technology, Shizuoka University, Hama-

matsu, 4328561, Japan (takeuchi@sys.eng.shizuoka.ac.jp).

§Department of Mathematics and Mechanics, School of Applied Science, University of Science

and Technology Beijing, Beijing, 100083, People’s Republic of China (wanbiao ma@sas.ustb.edu.cn).

2693

Page 2

2694

GANG HUANG, YASUHIRO TAKEUCHI, AND WANBIAO MA

in experiments in [5, 24], many researchers suggested that the bilinear incidence rate

associated with the mass action principle is insufficient to describe the infection pro-

cess in detail, and some nonlinear transmissions were proposed. For instance, Briggs

and Godfray [2] considered a nonlinear transmission of the form kx(t)ln(1+γv(t)/k).

Regoes, Ebert, and Bonhoeffer [34] considered a virus dynamics model with the in-

fection rate x(t)(v(t)/b)k/(1+ v(t)/b)k. Li and Ma [21] considered the viral infection

model with Holling type II functional response, and Huang, Ma, and Takeuchi [13]

considered a virus infection model with an infection rate βxv/(1 +ax +bv), which is

the Beddington–DeAngelis functional response. Georgescu and Hsieh in [6] considered

the nonlinear incidence rate c(x)f(v), where c(x) denotes the contact rate function at

concentration of the susceptible cells x and f(v) denotes the force of infection by virus

at concentration v. Recently Korobeinikov [16, 17] assumed that the incidence rate

is given by a general nonlinear function of target cells and free virus F(x,v), which

satisfies some conditions.

Furthermore, many processes of infected cells destructed after the replication of

viruses and producing free viruses are also unknown in detail. The removal rate

of infected cells may be different under different conditions and different virulence.

Nelson, Murray, and Perelson [26] pointed out that the death rate of virus producing

cells plays an important factor in plasma virus concentration under drug treatment.

The typical assumption is that infected cells die at a linear rate δy and viruses are

produced at a rate Nδy, where N represents the number of viruses produced by

one dead infected cell. Considering infected cells with saturated loss rate, Song and

Cha [35] introduced a term py/(1 + ay) describing the removal rate of infected cells.

Without loss of generality, in [6] Georgescu and Hsieh denoted the removal of infected

cells by c3p(y), which includes the mortality of infected cells, and the function p

is nonlinear. Here we also assume that infected cells die with a general nonlinear

function rate aG(y). Obviously, the linear term δy is a special case of aG(y).

On the other hand, for the healthy cell infected with a virus and the virus re-

production cell, there is an intracellular time delay between infection of a cell and

production of new virus particles called the latent period (see, e.g., Herz et al. [12],

Mittler et al. [25], and Nelson, Murray, and Perelson [26]). The delay describes the

finite time interval from the time when the infectious virus binds to the receptor of

a target cell to the time when the first virion is produced from the same target cell.

As in HIV, in vitro experiments, and simulations [33], the intracellular delay is esti-

mated to last between a few hours and two days. In [32], Perelson et al. estimated

the average life-span of a productively infected cell is 2.2 days. Moreover, as reported

by Gross et al. [9] and Zhou et al. [38], initially, the newly released virions are imma-

ture; subsequently, they undergo a proteolytic maturation step to become infectious

(mature). An immature virus is not infectious and may fail to enter into target cells,

and only a mature virus may contact the cells. As pointed out in [32], “This addi-

tional delay is consistent with the mechanism of action of protease inhibitors, which

render newly produced virions noninfectious but do not inhibit either the production

of virions from already infected cells or the infection of new cells by previously pro-

duced infectious virions.” Since the maturation time of a virus (0.3 days for HIV) is

much shorter than the life-span of infected cell [12, 32], usually researchers ignored

or missed the virus maturation time in viral infection dynamics. In [30], Ouifki and

Witten considered an HIV infection model including three intracellular delays which

describe different biological meanings, one of which is defined as the maturation time

of the newly produced viruses. In this paper, we will incorporate into virus infection

models two time delays which describe the latent period in infected cells and the time

Page 3

DELAY EQUATIONS MODEL OF VIRAL INFECTIONS

2695

necessary for newly produced virions to become mature and then infectious.

Due to the complexity of delay differential equations, many scientists do not

include delays in their models. However, many biological processes have inherent

delays, and including them may lead to additional insights in the study of complicated

biological processes. The first study of HIV-1 patient data using models with time

delays was done in 1996 by Herz et al. [12]. Thereafter, various models using discrete or

distributed delays to model the intracellular phase were developed (see, e.g., [3, 8, 20,

25, 27, 37, 39]). Attention was drawn to the shoulder phase by Nelson et al. [26, 27]

which used a basic model to estimate parameters from the experimental data on

HIV infected patients who are given antiretroviral therapy. The analysis in most of

them focused on the drug treatment and mathematically involved local stability of

the models. Delay is difficult to deal with mathematically, because straightforward

incorporation of it into a mathematical model generally leads to delay differential

equations which are difficult to handle mathematically.

determine whether or not bifurcations occur for various lengths of the delay.

particular, some delays are not necessarily destabilizing. The Lyapunov functional

and the LaSalle-type theorem in [11, 19] provide a direct and effective method to

establish global dynamical properties for the system of delay differential equations.

Recently, McCluskey [22, 23] and Huang et al. [14] developed a class of Lyapunov

functionals for delay epidemiological dynamical models. Li and Shu [20] established

global stability for a bilinear virus model with one intracellular delay. Generally, it

is not easy to investigate delay differential equations with multiple time delays and

nonlinearity. In this paper, we study a class of virus infection model with a nonlinear

incidence rate, a removal rate of infected cells, and two time delays. By constructing

suitable Lyapunov functionals, the global stability conditions, which depend only on

the functional properties and the basic reproductive number are established. Further,

the effects of delays are also discussed.

The paper is organized as follows. In section 2, we introduce a model described

by delay differential equations with two delays and two general nonlinear terms and

prove the existence of a positive equilibrium. In section 3, following the technique of

constructing Lyapunov functionals, we show that the global asymptotic stability of

the model depends only on the basic reproductive number under some hypotheses.

A detailed comparison between our results and the stability condition for the model

without delays and the generalization of Lyapunov functionals to n-dimensional sys-

tems are given in section 4.

It is often of interest to

In

2. Delay differential equations model. We first introduce the delay differ-

ential equations model,

x?(t) = s − dx(t) − F(x(t),v(t)),

y?(t) = e−μ1τ1F(x(t − τ1),v(t − τ1)) − aG(y(t)),

v?(t) = e−μ2τ2k G(y(t − τ2)) − uv(t),

where x(t), y(t), and v(t), defined as earlier, also represent the population of target

cells, infected cells, and free viruses, respectively. The parameters in the equations are

explained as below. The positive s is the rate at which new target cells are generated;

d is their specific death rate. The function F(x(t),v(t)) represents the rate for the

target cells to be infected by the mature viruses. Once the virus contacts the target

cell, the cell may survive the entire latent period τ1. The survival probability of

infected cells is given by e−μ1τ1. Here 1/μ1is the average lifetime of the cell during

(2.1)

Page 4

2696

GANG HUANG, YASUHIRO TAKEUCHI, AND WANBIAO MA

the latent period under the assumption that the per capita death rate of cells in the

latent class is constant as a function of duration in the latent class. It is assumed that

the death rate of the infected cells depends on the population of themselves, that is,

on the function aG(y), where a is a positive constant. Note that for G(y) = y, 1/a

represents the average lifetime of an infected cell, and free viruses are produced from

the infected cells. For this case on average each productively infected cell produces

k/a virions during its lifetime. The delay τ2 represents the time necessary for the

newly produced viruses to be infectious. The probability of survival of an immature

virus is given by e−μ2τ2, and the average lifetime of an immature virus is given by

1/μ2under the same assumption for the latent class. Finally, free viruses are cleared

from the system at a rate u.

Remark 1. Actually, model (2.1) is mathematically equivalent to a general delay

predator-prey model. For example, in [7], Georgescu and Hsieh introduced a general

predator-prey model with a stage structure for the predator as

x?= n(x) − f(x)g(y2),

y?

1= kf(x)g(y2) − c1h(y1),

y?

2= c2h(y1) − c3r(y2),

(2.2)

where x, y1, and y2 represent the densities of prey, respectively, of immature and

mature predators at time t. The functions n,f,g,h,r are allowed to be nonlinear.

If we assume n(x) = s − dx, r(y2) = y2 and incorporate two discrete delays (τ1

and τ2) into (2.2), while τ1 represents the gestation of the mature predator and τ2

due to predator maturation, we would establish a delay prey-predator model with a

stage structure, which is more appropriate to describe the real ecosystem, and that is

analogous to (2.1).

In this paper, we assume that the functions F(x,v) and G(y) in (2.1) are always

positive, differentiable, and monotonically increasing for all x > 0, y > 0, and v > 0

and that F(x,v) is concave with respect to v; that is, it satisfies the following:

(H1) F(x,v), F?

v > 0. Furthermore, F(0,v) = F(x,0) = 0, F?

(H2) G(0) = 0, G?(y) > 0 for y > 0.

The dynamics of virus infections crucially depend on the basic reproductive num-

ber R0, which in this case is the average number of new free infectious viruses derived

from a single infectious virus introduced into an entirely susceptible population of tar-

get cells. Following the definition of the basic reproductive number given by van den

Driessche and Watmough [36] for ODE systems, the number given by Korobeinikov

[16, 17] for virus dynamics models, and the number given by Li and Shu [20] and Zhu

and Zuo [39] for delay models for HIV-1 infection, the basic reproductive number for

(2.1) is presented as

x(x,v), F?

v(x,v), and −F??

vv(x,v) are positive for any x > 0 and

v(x,0) > 0 for x > 0 and v > 0.

(2.3)R0=

k

aue−μ2τ2·∂F(x0,0)

∂v

e−μ1τ1,

where x0= s/d. In the above expression, the first term k/au is the average number

of virus particles emerging from each virus-producing target cell, a fraction of e−μ2τ2

expresses the survival probability of immature virus, the term ∂F(x0,0)/∂v is the

maximal average number of cells that each virion infects, and e−μ1τ1is the survival

probability of the infected cell in the latent period. Specially, note that the basic

reproductive number (2.3) does not depend on G(y). The model with the bilinear

Page 5

DELAY EQUATIONS MODEL OF VIRAL INFECTIONS

2697

incidence rate βxv, a linear term ay, and one delay τ is considered in Li and Shu [20].

The basic reproductive number for the model is simplified to

System (2.1) always has an infection-free equilibrium E0 = (x0,0,0). Usually,

when R0 > 1, in addition, it has an infected equilibrium E∗= (x∗,y∗,v∗), which

satisfies

sβk

daue−μτ.

0 = s − dx∗− F(x∗,v∗),

0 = e−μ1τ1F(x∗,v∗) − aG(y∗),

0 = e−μ2τ2kG(y∗) − uv∗.

In the following, we give a lemma which gives the existence condition of a positive

equilibrium. First, we give a hypothesis for the function G(y):

(H3) limy→+∞G(y) = p ≤ +∞ and p ≥s

Lemma 1. Suppose that the functions F(x,v) and G(y) satisfy the conditions

(H1), (H2), and (H3). If R0> 1, then there exists a positive equilibrium E∗(x∗,y∗,v∗).

Proof. Let the right-hand sides of the three equations in system (2.1) equal zero,

and we have that

s − dx = F(x,v) = eμ1τ1aG(y) =auv

ae−μ1τ1.

(2.4)

k

eμ1τ1+μ2τ2.

After substituting the expression of x by v, we obtain the following equation for v:

?sk − auveμ1τ1+μ2τ2

It is obvious that H(0) = 0, and when v = v0= e−(μ1τ1+μ2τ2)sk/au,

H(v) = F

dk

, v

?

−auv

k

eμ1τ1+μ2τ2= 0.

H(v0) = F(0,v0) − s = −s < 0.

Since H(v) is continuous for v ≥ 0, we have that

H?(0) = lim

v→0+

=∂F(x0,0)

H(v) − H(0)

v

−au

keμ1τ1+μ2τ2(R0− 1).

∂v keμ1τ1+μ2τ2−aueμ1τ1+μ2τ2

dk

F?

x(x0,0)

=au

Thus, R0 > 1 ensures that H?(0) > 0, and there exists some v∗∈ (0,v0) such

that H(v∗) = 0. Knowing the value of v∗, from the monotonicity of the function

G(y) and (2.4), the values of x∗and y∗can be computed. It is easy to check that

g(x) = s − dx − F(x,v∗) has a positive solution x∗since g(0) > 0 and g(∞) = −∞.

Further, note that f(y) = G(y)−uv∗eμ2τ2/k = 0 has a positive solution since f(0) < 0

and

kv0eμ2τ2= p −s

lim

y→+∞f(y) >lim

y→+∞G(y) −u

ae−μ1τ1≥ 0.

Therefore, we have proved the existence of the infected equilibrium E∗(x∗,y∗,v∗) for

system (2.1) under condition R0> 1.

Let C = C([−τ,0];R3) = {φ = (ϕ1(θ),ϕ2(θ),ϕ3(θ))T| ϕi(θ) are continuous on

−τ ≤ θ ≤ 0, i = 1,2,3} be the Banach space of continuous functions from [−τ,0] to

R3equipped with the sup-norm.

Page 6

2698

GANG HUANG, YASUHIRO TAKEUCHI, AND WANBIAO MA

The initial condition of (2.1) is given as

(2.5)x(θ) = ϕ1(θ), y(θ) = ϕ2(θ),v(θ) = ϕ3(θ),

where ϕ = (ϕ1,ϕ2,ϕ3)T∈ C, such that ϕi(θ) ≥ 0 (θ ∈ [−max{τ1,τ2},0], i = 1,2,3).

Further, we assume the following:

(H4) x − x0−?x

x∗

(H6) y − y∗−?y

The following theorem establishes the nonnegativity of solutions of (2.1).

Theorem 1. Let (x(t),y(t),v(t))Tbe the solution of system (2.1) with the initial

conditions (2.5). Then (x(t),y(t),v(t))Tis existent and nonnegative on [0,+∞) when

(H1)–(H6) are satisfied.

The proof is given in the appendix.

Hence we discuss system (2.1) in the closed set

Ω =?(ϕ1,ϕ2,ϕ3)T|

It is easy to show that Ω is positively invariant with respect to system (2.1).

x0limv→0+F(x0,v)

F(x∗,v∗)

F(σ,v∗)dσ → +∞ as x → 0+or x → +∞;

G(y∗)

G(σ)dσ → +∞ as y → 0+or y → +∞.

Note that all functions cited in the introduction satisfy conditions (H4)–(H6).

F(σ,v)dσ → +∞ as x → 0+;

(H5) x − x∗−?x

y∗

? ϕ1?≤ x0, ϕi≥ 0, i = 1,2,3?.

3. Global asymptotic stability. In this section, we shall consider the global

stability of the infection-free equilibrium and the infected equilibrium of system (2.1)

by the Lyapunov direct method, respectively. First, we assume the following:

(H7)F?

From (H7), it is easy to see that the following inequalities hold:

v(x,0) is increasing with respect to x > 0.

(3.1)

F?

F?

v(x0,0)

v(x,0)

> 1 for x ∈ (0,x0) and

F?

F?

v(x0,0)

v(x,0)

< 1 for x > x0.

Theorem 2. Suppose that conditions (H1)–(H7) are satisfied.

(i) If R0≤ 1, then the infection-free equilibrium E0(x0,0,0) is globally asymptot-

ically stable.

(ii) If R0> 1, then the positive equilibrium E∗(x∗,y∗,v∗) is globally asymptotically

stable.

Proof. (i) Define a Lyapunov functional

(3.2)U1= V1(xt,yt,vt) + U++ a eμ1τ1U−,

where

V1= x(t) − x0−

?τ1

U−=

0

?x(t)

x0

lim

v→0+

F(x0,v)

F(σ,v)dσ + eμ1τ1y(t) +aeμ1τ1eμ2τ2

k

v(t),(3.3)

U+=

0

F(x(t − θ),v(t − θ))dθ,(3.4)

?τ2

G(y(t − η))dη.(3.5)

By (H1)–(H4), it is obvious that U1is defined and continuous for all x(t),y(t),v(t) > 0,

and U1= 0 at (x0,0,0). The time derivative of V1along the solution of (2.1) is given

Page 7

DELAY EQUATIONS MODEL OF VIRAL INFECTIONS

2699

by

dV1

dt

=

?

− eμ1τ1aG(y(t))+ eμ1τ1aG(y(t − τ2)) −au

1 − lim

v→0+

F(x0,v)

F(x,v)

?

(dx0− dx − F(x,v)) + F(x(t − τ1),v(t − τ1))

keμ1τ1eμ2τ2v.

Further, we have

dU+

dt

=d

dt

?τ1

?τ1

dtF(x(t − θ),v(t − θ))dθ

?τ1

= −F(x(t − τ1),v(t − τ1)) + F(x,v)

0

F(x(t − θ),v(t − θ))dθ

=

0

d

= −

0

d

dθF(x(t − θ),v(t − θ))dθ

and

dU−

dt

=

?τ2

0

d

dtG(y(t − η))dη = −

?τ2

0

d

dηG(y(t − η))dη = −G(y(t − τ2)) + G(y).

Then we have

dU1

dt

=dV1

dt

+dU+

?x0

+au

keμ1τ1eμ2τ2v

dt

+ aeμ1τ1dU−

??

?k

dt

= dx

x− 1

1 − lim

v→0+

F(x0,v)

F(x,v)

F(x,v)

v

?

au

e−μ1τ1e−μ2τ2lim

v→0+

F(x0,v)

F(x,v)− 1

?

.

By (3.1), we have

(3.6)

?x0

v(x0,0)/F?

x− 1

??

1 − lim

v→0+

F(x0,v)

F(x,v)

?

=

?x0

x− 1

??

1 −F?

v(x0,0)

F?

v(x,0)

?

≤ 0.

Note that F?

holds only if x = x0because of the inequality F?

the concavity of F(x,v) with respect to v implies that

v(x,0) ?= 1 for x ?= x0, x > 0, and v > 0, and the equality in (3.6)

v(x,0) > 0 and (H7). Furthermore,

k

au

F(x,v)

v

e−μ1τ1e−μ2τ2lim

v→0+

∂F(x0,0)

∂v

∂F(x,0)

∂v

F(x0,v)

F(x,v)

=

k

au

F(x,v)

v

e−μ1τ1e−μ2τ2

≤

= R0.

k

au

∂F(x0,0)

∂v

e−(μ1τ1+μ2τ2)

Hence, for R0 ≤ 1, E0 is globally asymptotically stable. In fact, from (H4) and

dU1/dt ≤ 0, any solution is also bounded on [0,+∞). If R0< 1, from Corollary 5.2

of Kuang [19], E0 is globally asymptotically stable. Also, for R0 = 1, dU1/dt = 0

Page 8

2700

GANG HUANG, YASUHIRO TAKEUCHI, AND WANBIAO MA

implies that x(t) = x0. It is easy to show that E0(x0,0,0) is the largest invariant

set in {(xt,yt,vt) |˙U1= 0}. By the classical Lyapunov–LaSalle invariance principle

(Theorem 5.3 of Kuang [19]), E0is globally asymptotically stable.

(ii) From Lemma 1, there exists at least one positive equilibrium E∗when R0> 1.

Define a Lyapunov functional for E∗,

(3.7)U2= V2+ F(x∗,v∗)(U++ U−),

where

V2= x − x∗−

?x

y − y∗−

x∗

F(x∗,v∗)

F(σ,v∗)dσ

?y

v − v∗− v∗lnv

+ eμ1τ1

keμ1τ1+μ2τ2?

?

y∗

G(y∗)

G(σ)dσ

?

(3.8)

+a

v∗

?

and

U+=

?τ1

?τ2

0

?F(x(t − θ),v(t − θ))

?G(y(t − θ))

F(x∗,v∗)

− 1 − lnF(x(t − θ),v(t − θ))

F(x∗,v∗)

− 1 − lnG(y(t − θ))

G(y∗)

?

dθ,(3.9)

U−=

0

G(y∗)

?

dθ.(3.10)

The time derivative of V2along the solution of (2.1) is given by

?

+

G(y)

?

It is easy to see that

?τ1

=

0

dtF(x∗,v∗)

?τ1

= −

F(x∗,v∗)

= −F(x(t − τ1),v(t − τ1))

F(x∗,v∗)

dV2

dt

=1 −F(x∗,v∗)

F(x,v∗)

?

1 −v∗

?

(s − dx − F(x,v))

1 −G(y∗)

?

eμ1τ1aG(y(t − τ2)) − eμ1τ1eμ2τ2au

(F(x(t − τ1),v(t − τ1)) − eμ1τ1aG(y))

+

v

??

kv

?

.

dU+

dt

=

d

dt

?τ1

0

?F(x(t − θ),v(t − θ))

?F(x(t − θ),v(t − θ))

d

dθF(x∗,v∗)

?F(x(t − θ),v(t − θ))

F(x∗,v∗)

− 1 − lnF(x(t − θ),v(t − θ))

F(x∗,v∗)

− 1 − lnF(x(t − θ),v(t − θ))

F(x∗,v∗)

− 1 − lnF(x(t − θ),v(t − θ))

− 1 − lnF(x(t − θ),v(t − θ))

F(x∗,v∗)

+F(x(t),v(t))

F(x∗,v∗)

?

?

dθ

d

dθ

?

= −

0

?F(x(t − θ),v(t − θ))

F(x∗,v∗)

dθ

?????

τ1

θ=0

+ lnF(x(t − τ1),v(t − τ1))

F(x(t),v(t))

.

Similarly, we obtain

dU−

dt

= −G(y(t − τ2))

G(y∗)

+G(y(t))

G(y∗)

+ lnG(y(t − τ2))

G(y(t))

.

Page 9

DELAY EQUATIONS MODEL OF VIRAL INFECTIONS

2701

Since

dU2

dt

=dV2

dt

+ F(x∗,v∗)dU+

dt

+ F(x∗,v∗)dU−

dt

,

and by using F(x∗,v∗) = eμ1τ1aG(y∗) = eμ1τ1+μ2τ2auv∗/k, we obtain

??

+ F(x∗,v∗)

F(x,v)

?

+ F(x∗,v∗)

v∗+

F(x,v∗)

dU2

dt

= dx∗?

1 −x

x∗

1 −F(x∗,v∗)

F(x,v∗)

lnF(x(t − τ1),v(t − τ1))

?

?

+ lnG(y(t − τ2))

G(y)

?

+ F(x∗,v∗)3 −F(x∗,v∗)

?

F(x,v∗)−G(y∗)

F(x,v)

G(y)

?

F(x(t − τ1),v(t − τ1))

F(x∗,v∗)

−G(y(t − τ2))

G(y∗)

v∗

v

?

−v

.

By the properties of a logarithm function, we have that

lnF(x(t − τ1),v(t − τ1))

F(x,v)

= lnF(x∗,v∗)

+ lnG(y(t − τ2))

G(y)

F(x,v∗)+ lnG(y∗)

+ lnG(y(t − τ2))

G(y∗)

G(y)

v∗

v

F(x(t − τ1),v(t − τ1))

F(x∗,v∗)

+ lnv

v∗

F(x,v∗)

F(x,v),

and we obtain

= dx∗?

dU2

dt

1 −x

x∗

??

?

?

?

?

?

1 −F(x∗,v∗)

F(x,v∗)

1 −F(x∗,v∗)

1 −G(y∗)

G(y)

1 −G(y(t − τ2))

G(y∗)

1 −v

v∗

F(x,v)

−1 −v

?

+ F(x∗,v∗)

F(x,v∗)+ lnF(x∗,v∗)

F(x(t − τ1),v(t − τ1))

F(x∗,v∗)

v∗

v

F(x,v∗)

+ lnv

F(x,v∗)

?

+ F(x∗,v∗)

+ lnG(y∗)

G(y)

v∗

v

F(x(t − τ1),v(t − τ1))

F(x∗,v∗)

?

+ F(x∗,v∗)

+ lnG(y(t − τ2))

G(y∗)

F(x,v∗)

F(x,v)

F(x,v∗)

F(x,v)+F(x,v)

?

+ F(x∗,v∗)

v∗

?

+ F(x∗,v∗)

v∗+

v

v∗

F(x,v∗)

?

.

By factoring the last term, we have

dU2

dt

= dx∗?

1 −x

?

?

x∗

??

F(x,v∗)+ lnF(x∗,v∗)

1 −G(y∗)

G(y)

1 −F(x∗,v∗)

F(x,v∗)

?

(3.11)

+ F(x∗,v∗)1 −F(x∗,v∗)

F(x,v∗)

?

(3.12)

+ F(x∗,v∗)

F(x(t − τ),v(t − τ))

F(x∗,v∗)

+ lnG(y∗)

G(y)

F(x(t − τ),v(t − τ))

F(x∗,v∗)

?

(3.13)

Page 10

2702

GANG HUANG, YASUHIRO TAKEUCHI, AND WANBIAO MA

+ F(x∗,v∗)

?

?

?v

1 −G(y(t − τ2))

G(y∗)

1 −v

v∗

v∗−F(x,v)

F(x,v∗)

v∗

v

+ lnG(y(t − τ2))

G(y∗)

F(x,v∗)

F(x,v)

??F(x,v∗)

v∗

v

?

(3.14)

+ F(x∗,v∗)

F(x,v∗)

F(x,v)+ lnv

v∗

?

.

(3.15)

+ F(x∗,v∗)

F(x,v)

− 1

?

(3.16)

From the monotonicity of the function F(x,v) on x, the following inequality holds:

?

Furthermore, from the concavity and monotonicity of the function F(x,v) on v, the

inequalities

?

hold, which implies that

?v

Note that the function

1 −x

x∗

??

1 −F(x∗,v∗)

F(x,v∗)

?

≤ 0.

(3.17)

1 ≥ F(x,v)/F(x,v∗) ≥ v/v∗

1 ≤ F(x,v)/F(x,v∗) ≤ v/v∗

for 0 < v ≤ v∗,

for v ≥ v∗

v∗−F(x,v)

F(x,v∗)

??F(x,v∗)

F(x,v)− 1

?

≤ 0.

I(t) = 1 − i(t) + lni(t)

is always nonpositive for any function i(t) > 0, and I(t) = 0 if and only if i(t) = 1.

Therefore, the terms (3.12)–(3.15) are always nonpositive.

The above shows that all sufficient conditions given in Corollary 5.2 of Kuang [19]

are satisfied under the conditions R0 > 1 and (H1)–(H7). This implies the global

asymptotic stability of E∗.

Remark 2. The sufficient conditions for the global asymptotic stability of two

equilibria depend on the basic reproductive number R0 and conditions (H1)–(H7).

Actually, as pointed out by Korobeinikov in [15, 16, 17, 18], the concavity of the

function F(x,v) for v is a sufficient but not a necessary condition for the global

stability of system (2.1) without time delay. It is obvious that a weaker condition

(3.17) suffices for the global stability. From the proof of Theorem 2, (3.17) is also

sufficient for the global stability of the system with time delays.

Remark 3.Recently, two terms βxv and μy, instead of the functional rates

F(x,v) and G(y) in (2.1), have been considered in Zhu and Zou [39]. But in [39],

the two delays represent a latent period between the time when the target cells are

contacted by the virus particles and the time when the virions enter the cells, and a

virus production period for new virions to be produced within and released from the

infected cells. Actually, those two delays in [39] could be combined into one delay τ1

of our model (2.1); from a biological viewpoint, they both belong to a latent period

of infected cells.

In particular, for model (2.1), if we let Y (t) = y(t − τ2) or Y (t) = y(t + τ1), we

can simplify (2.1) to the following models, respectively:

x?(t) = s − dx(t) − F(x(t),v(t)),

Y?(t) = e−μ1τ1F(x(t − τ1− τ2),v(t − τ1− τ2)) − aG(Y (t)),

v?(t) = e−μ2τ2kG(Y (t)) − uv(t)

(3.18)

Page 11

DELAY EQUATIONS MODEL OF VIRAL INFECTIONS

2703

or

x?(t) = s − dx(t) − F(x(t),v(t)),

Y?(t) = e−μ1τ1F(x(t),v(t)) − aG(Y (t)),

v?(t) = e−μ2τ2kG(Y (t − τ1− τ2)) − uv(t).

(3.19)

When τ = τ1+ τ2, it is clear that (3.18) and (3.19) include only one time delay τ.

Mathematically, they are equivalent to (2.1) and have the same dynamical properties,

but the structures of the former are much simpler. The global stability of (3.18) and

(3.19) could still be proved by using Lyapunov functionals which are parts of (3.2)

and (3.7). That is to say, (3.2) without the term U− and (3.7) without the term

U−are Lyapunov functionals for model (3.18). Similarly, (3.2) without the term U+

and (3.7) without the term U+are Lyapunov functionals for model (3.19). Note that

both y(t) and Y (t) express the number of productivity infected cells, but y(t) is the

number at time t and Y (t) is the number at time t − τ2in (3.18) or at time t + τ1in

(3.19). When we experiment with viral infection, model (2.1) is easier to carry out

since it needs to count the numbers of the populations of target cells, infected cells,

and free viruses simultaneously.

4. Discussions and conclusions. In section 3, we analyzed the global asymp-

totic properties of two equilibrium states. From Theorem 2, it is obvious that the

global stability of the equilibria depends only on the basic reproductive number when

the conditions (H1)–(H7) are satisfied. Generally, the nonlinear incidence rate F(x,v)

and virus-producing rate G(y) do not destabilize two equilibria. As a special instance,

Li and Ma [21] considered an infection rate with a Holling type II functional response

βxv/(1 + av) and only one delay τ1. Zhu and Zou [39] considered a model with two

delays (τ1 and τ2) but with bilinear incidence rates βxv and py. They both leave

the global stability of the positive equilibrium as an open problem. By using our

Lyapunov functional and Theorem 2 in section 3, we have solved these problems.

Similarly, the new results for the bilinear delay differential equations model with one

delay in [20] can also be obtained from Theorem 2. In particular, the basic reproduc-

tive number R0is a decreasing function on time delays τ1and τ2. To study the effect

of time delays, one needs to compare the delay model with the model without time

delay.

4.1. Comparison with the model without time delays. If we consider

system (2.1) without any delay, that is, τ1= τ2= 0, system (2.1) is simply reduced

to an ordinary differential equation (ODE) model as follows:

x?(t) = s − dx(t) − F(x(t),v(t)),

y?(t) = F(x(t),v(t)) − aG(y(t)),

v?(t) = kG(y(t)) − uv(t).

(4.1)

Now the basic reproductive number for system (4.1) is rewritten as

¯R0=

k

au

∂F(x0,0)

∂v

.

The system (4.1) has the two similar equilibria¯E0and¯E∗(when¯R0> 1) as system

(2.1). In fact, the functions V1 and V2 in section 3 given as the parts of Lyapunov

Page 12

2704

GANG HUANG, YASUHIRO TAKEUCHI, AND WANBIAO MA

functionals for delay differential equations model (2.1) become Lyapunov functions

for the ODE model (4.1) as follows:

V1= x − x0−

?x

?x

v − v∗− v∗lnv

x0

lim

v→0+

F(x∗,v∗)

F(σ,v∗)dσ +

F(x0,v)

F(σ,v)dσ + y +a

?

?

kv,

(4.2)

V2= x − x∗−

x∗

y − y∗−

?y

y∗

G(y∗)

G(σ)dσ

?

(4.3)

+a

k

?

v∗

.

We can obtain the global properties of (4.1) as follows.

Corollary 1. Suppose that conditions (H1)–(H7) are satisfied.

(i) If¯R0≤ 1, then the infection-free equilibrium¯E0of (4.1) is globally asymptot-

ically stable.

(ii) If¯R0> 1, then the positive equilibrium¯E∗of (4.1) is globally asymptotically

stable.

Korobeinikov established much research on Lyapunov functions and global stabil-

ity for ODE models of disease transmission, viral infection, and predator-pray dynam-

ics in [17, 18]. Here we omit the detailed process of the proof of Corollary 1 since it is

similar to what appears in [17, 18], and we show only the Lyapunov functions (4.2),

(4.3). From the above corollary, we can see that the intracellular delays τ1and τ2do

not affect the stability of the equilibrium if the sign of R0−1 is uncharged when the

delays are set to 0. However, when all other parameters are fixed and delays are suf-

ficiently large, R0becomes less than one, which makes the infection-free equilibrium

globally asymptotically stable. From a biological viewpoint, intracellular delays play

a positive role in the virus infection process in order to eliminate virus. Sufficiently

large intracellular delay makes the virus development slower, and the virus is con-

trolled and disappears. This gives us some suggestions on new drugs to prolong the

time of the latent period in infected cells or the time for viruses to mature (infectious).

4.2. n-dimensional dynamical model with multiple delays. The function

of the form l(x) = x − x∗− x∗ln(x/x∗) was usually used to construct Lyapunov

functions for the Lotka–Volterra system at first; then it was successfully applied for

ODEs of epidemic models by Korobenikov [15, 16, 17]. It appears to be a sound

basis for constructing Lyapunov functions for more advanced models, including n-

dimensional models. In particular, in this paper we construct the functional

?τ

where x∗is a positive equilibrium value. Obviously, here L(t) ≥ 0 for all x > 0, and

L(t) = 0 if and only if x(t−σ) = x∗. This latter form of the Lyapunov functional was

first introduced by McCluskey [22, 23] in order to investigate the global stability of

endemic equilibrium of the SEIR epidemic model with an infinite delay, and Huang

et al. [14] generalized them to delay SIR and SEIR epidemic models with nonlinear

transmissions.It would be worthwhile to apply a combination of the above two

functions l and L for a more complex delay differential equations model.

For example, we can generalize the above functions and functionals to an n-

dimensional system with multiple delays. Due to ongoing viral replication in the

HIV infection process, the time from the contact of viruses and CD4 cells to the

L(t) =

0

?x(t − σ)

x∗

− 1 − lnx(t − σ)

x∗

?

dσ,

Page 13

DELAY EQUATIONS MODEL OF VIRAL INFECTIONS

2705

death of the cells is modeled by dividing the process into several short steps. Gross-

man et al. in [10] introduced several n-dimensional ODE models of HIV infections.

And global properties of the model with nonlinear terms have also been extended by

Georgescu and Hsieh in [6]. When incorporating time delays into Grossman’s ongoing

viral replication model with nonlinearities, we would consider the following model

with n stages in the infected cells:

x?(t) = s − dx(t) − F(x(t),v(t)),

y?

1(t) = F(x(t − ω),v(t − ω)) − k1G1(y1(t)),

y?

y?

......

y?

v?(t) =˜knGn(yn(t − τn)) − uv(t).

The above system has n (n ≥ 2) stages in the infected cells before producing a virus.

Denote the basic reproductive number of system (4.4) as

⎛

j=1

kj

2(t) =˜k1G1(y1(t − τ1)) − k2G2(y2(t)),

3(t) =˜k2G2(y2(t − τ2)) − k3G3(y3(t)), (4.4)

n(t) =˜kn−1Gn−1(yn−1(t − τn−1)) − knGn(yn(t)),

ˆR0=1

u

⎝

n

?

˜kj

⎞

⎠∂F(x0,0)

∂v

,

where x0= s/d. Similarly, system (4.4) has an infection-free equilibriumˆE0(s/d,0,...,

0,0) and a positive equilibriumˆE∗(x∗,y∗

give the following Lyapunov functionals, W1forˆE0and W2forˆE∗, respectively:

0,y∗

1,...,y∗

n,v∗) whenˆR0> 1. We also can

W1= W1+ W++ W−,

W2= W2+ F(x∗,v∗) (W++ W−),

(4.5)

(4.6)

where

W1= x(t) − x0−

?x(t)

?x(t)

˜kj

kj

⎞

x0

lim

v→0+

F(x0,v)

F(σ,v)dσ +

n

?

i=1

⎛

⎝

i−1

?

j=1

˜kj

kj

⎞

⎠yi(t) +

⎛

⎝

n

?

j=1

˜kj

kj

⎞

⎠v(t),

W2= x(t) − x∗−

x∗

F(x∗,v∗)

F(σ,v∗)dσ

?

+

n

?

⎛

i=1

⎛

⎝

n

?

i−1

?

˜kj

kj

j=1

⎞

⎠

v(t) − v∗− v∗lnv

yi(t) − y∗

i−

?yi(t)

y∗

i

Gi(y∗)

Gi(σ)dσ

?

+

⎝

j=1

⎠

?

v∗

?

and

W+=

?ω

n

?

0

F(x(t − θ),v(t − θ))dθ,

⎛

j=1

kj

W−=

i=1

⎝

i−1

?

˜kj

⎞

⎠

?τi

0

Gi(yi(t − θ))dθ,

Page 14

2706

GANG HUANG, YASUHIRO TAKEUCHI, AND WANBIAO MA

W+=

?ω

n

?

0

?F(x(t − θ),v(t − θ))

⎛

j=1

kj

F(x∗,v∗)

⎞

0

− 1 − lnF(x(t − θ),v(t − θ))

F(x∗,v∗)

?Gi(yi(t − θ))

?

?

dθ,

W−=

i=1

⎝

i−1

?

˜kj

⎠

˜kj

kj= 1.

?τi

Gi(y∗

i)

− 1 − lnGi(yi(t − θ))

Gi(y∗

i)

dθ,

with the convention?0

(4.4).

Corollary 2. Suppose that conditions (H1)–(H7) are satisfied.

(i) IfˆR0 ≤ 1, then the infection-free equilibriumˆE0(s/d,0,...,0,0) is globally

asymptotically stable.

(ii) IfˆR0 > 1, then the positive equilibrium ˆE∗(x∗,y∗

asymptotically stable.

In particular, the delay in model (4.4) is found in the food chain model, and for

arbitrary large time delay (τ = Σn

i=1τi, τ → +∞), we can regard the model as one with

a continuous infinite delay on (0,∞). Recently, the viral dynamics with continuously

distributed intracellular delays were also introduced and studied in [25, 27]. The

construction of new Lyapunov functionals for the model with infinite delays would be

left to future work.

j=1

Similar to the proofs in section 3, we can obtain the following corollary for system

1,...,y∗

n,v∗) is globally

Appendix: Proof of Theorem 1. For any ? > 0, consider the following systems

with the parameter ? > 0:

x?(t) = s − dx(t) − F(x(t),v(t)),

y?(t) = e−μ1τ1F(x(t − τ1),v(t − τ1)) − aG(y(t)) + ?,

v?(t) = e−μ2τ2kG(y(t − τ2)) − uv(t) + ?.

(A?)

For any ? > 0, let (x(t,?),y(t,?),v(t,?))Tbe the solution of (A?) with the initial

conditions (2.5), and its maximal existence interval is denoted by [0,ρ?), where 0 <

ρ?≤ +∞.

For ? = 0, let us show that the solution (x(t),y(t),v(t))Tis nonnegative on [0,ρ0).

In fact, for any closed interval [0,b] ⊂ [0,ρ?), since the solution (x(t,?),y(t,?),v(t,?))T

is a continuous function of the parameter ?, for sufficiently small ? > 0 the solution

(x(t,?),y(t,?),v(t,?))Tis uniformly existent on [0,b].

Furthermore, note that, if for some t ≥ 0, x(t) = 0, or y(t) = 0, or v(t) = 0, then

we have that x?(t) = s > 0, or y?(t) ≥ ? > 0, or v?(t) ≥ ? > 0, respectively. Hence, for

any t ∈ (0,b],

x(t,?) > 0,y(t,?) > 0,v(t,?) > 0.

Letting ? → 0+gives that for any t ∈ (0,b],

x(t,0) ≥ 0,y(t,0) ≥ 0,v(t,0) ≥ 0.

Therefore, for any t ∈ [0,ρ0),

x(t) ≥ 0,y(t) ≥ 0,v(t) ≥ 0.

Next, let us further show that ρ0= +∞.

Page 15

DELAY EQUATIONS MODEL OF VIRAL INFECTIONS

2707

First, note that

x?(t) ≤ s − dx(t)

implies that x(t) is bounded on [0,ρ0).

Case (i). When R0≤ 1, from the Lyapunov functional U1for E0, we know that

dU1/dt ≤ 0 for any t ∈ [0,ρ0). Hence, y(t) and v(t) are also bounded on [0,ρ0).

Case (ii). When R0> 1, from the Lyapunov functional U2for E∗, we also have

dU2/dt ≤ 0 for any t ∈ [0,ρ0). Hence, y(t) and v(t) are also bounded on [0,ρ0).

From the proof of Theorem 2, the solution is ensured to be bounded under the

conditions (H1)–(H6); the above discussion can be extended for [0,+∞). Hence, the

solution (x(t),y(t),v(t))Tis existent and nonnegative on [0,+∞).

This completes the proof of Theorem 1.

Acknowledgment. The authors would like to express their sincere thanks to

two reviewers for the useful comments on the revision of the paper. The fact that

model (2.1) with two delays can be rewritten as one delay model (3.18) or (3.19)

especially makes the mathematical structure of the former much simpler.

REFERENCES

[1] R. M. Anderson and R. M. May, The population dynamics of microparasites and their in-

vertebrate hosts, Philos. Trans. Roy. Soc. Lond. Ser. B, 291 (1981), pp. 451–524.

[2] C. J. Briggs and H. C. J. Godfray, The dynamics of insect-pathogen interactions in stage-

structured populations, Am. Nat., 145 (1995), pp. 855–887.

[3] R. V. Culshaw and S. Ruan, A delay-differential equation model of HIV infection of CD4+

T cells, Math. Biosci., 165 (2000), pp. 27–39.

[4] P. De Leenheer and H. L. Smith, Virus dynamics: A global analysis, SIAM J. Appl. Math.,

63 (2003), pp. 1313–1327.

[5] D. Ebert, C. D. Zschokke-Rohringer, and H. J. Carius, Does effects and density-dependent

regulation of two microparasites of Daphnia magna, Oecologia, 122 (2000), pp. 200–209.

[6] P. Georgescu and Y.-H. Hsieh, Global stability for a virus dynamics model with nonlinear

incidence of infection and removal, SIAM J. Appl. Math., 67 (2006), pp. 337–353.

[7] P. Georgescu and Y.-H. Hsieh, Global dynamics of a predator-prey model with stage structure

for the predator, SIAM J. Appl. Math., 67 (2007), pp. 1379–1395.

[8] S. A. Gourley, Y. Kuang, and J. D. Nagy, Dynamics of a delay differential equation model

of hepatitis B virus infection, J. Biol. Dynam., 2 (2008), pp. 140–153.

[9] I. Gross, H. Hohenberg, T. Wilk, K. Wiegers, M. Gr¨ attinger, B. M¨ uller, S. Fuller,

and H.-G. Kr¨ ausslich, A conformational switch controlling HIV-1 morphogenesis, EMBO

J., 19 (2000), pp. 103–113.

[10] Z. Grossman, M. Polis, M. B. Feinberg, Z. Grossman, I. Levi, S. Jankelevich, R.

Yarchoan, J. Boon, F. De Wolf, J. M. A. Lange, J. Goudsmit, D. S. Dimitrov,

and W. E. Paul, Ongoing HIV dissemination during HAART, Nat. Med., 5 (1999), pp.

1099–1104.

[11] J. Hale and S. M. Verduyn Lunel, Introduction to Functional Differential Equations, Appl.

Math. Sci. 99, Springer-Verlag, New York, 1993.

[12] V. Herz, S. Bonhoeffer, R. Anderson, R. M. May, and M. A. Nowak, Viral dynamics in

vivo: Limitations on estimations on intracellular delay and virus delay, Proc. Natl. Acad.

Sci. USA, 93 (1996), pp. 7247–7251.

[13] G. Huang, W. Ma, and Y. Takeuchi, Global properties for virus dynamics model with

Beddington-DeAngelis functional response, Appl. Math. Lett., 22 (2009), pp. 1690–1693.

[14] G. Huang, Y. Takeuchi, W. Ma, and D. Wei, Global stability for delay SIR and SEIR

epidemic models with nonlinear incidence rate, Bull. Math. Biol., 72 (2010), pp. 1192–

1207.

[15] A. Korobeinikov, Global properties of basic virus dynamics models, Bull. Math. Biol., 66

(2004), pp. 879–883.

[16] A. Korobeinikov, Global properties of infectious disease models with nonlinear incidence,

Bull. Math. Biol., 69 (2007), pp. 1871–1886.