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MODELING AND ANALYSIS OF A MARINE BACTERIOPHAGE

INFECTION WITH LATENCY PERIOD

EDOARDO BERETTA�

ISTITUTO DI BIOMATEMATICA

UNIVERSITA DI URBINO

I-61029 URBINO, ITALY

YANG KUANGy

DEPARTMENT OF MATHEMATICS

ARIZONA STATE UNIVERSITY

TEMPE, AZ 85287{1804, USA

E-MAIL: [email protected]

PHONE: 480-965-6915; FAX: 480-965-8119

Abstract. Amathematical model for the marine bacteriophage infection with

explicit latency period is proposed as a system of discrete time delay di�erential

equations and its important mathematical features are analyzed. Let �i rep-

resent the death constant rate in the class of infected bacteria during latency

period \T ." Only the fraction \exp(��iT )" of the bacteria infected at \t�T"

will release by lysis at time \t" \b" new phages, where b 2 (1;+1) is called

\virus replication factor." Hence, if �i > 0, the main parameter on which the

dynamics of the model depends on is \b exp(��iT )" and when the \basic repro-

duction number R0" of epidemic theory is such that R0 = b exp(��iT )=b� > 1,

b��= (�p=KC) + 1, the endemic equilibrium E+ becomes feasible. We also

allow for a possible constant supply � of free viruses from the environment.

This supply however destroys the threshold behavior based on R0. In all these

cases we prove boundedness of solutions and we study the existence of the

equilibria, performing their local stability analysis and providing global sta-

bility results. The permanence properties of the solutions are also studied.

A discussion section with extensive biological implications and simulations is

included.

Date: August 4, 2000.

Key words and phrases. marine bacteriophage infection, time delay, Liapunov functional,

global stability, persistence.� Research supported by \Gruppo Nazionale per la Fisica Matematica", C.N.R., Italy. This

author has presented this paper in the frame of the research Project Co�n 99 "Analysis of complex

systems in population biology".y Research partially supported by NSF Grant DMS-9306239. Correspondence should be di-

rected to this author.

Running title: Marine Bacteriophage Infection Analysis with Latency

Accepted by Nonlinear Analysis, B on Aug. 3, 1999.

1

2 EDOARDO BERETTA AND YANG KUANG

1. Introduction

In a previous paper [4] the authors proposed a simple model to describe the

epidemics induced by bacteriophages in marine bacteria populations like cyanobac-

teria and heterotrophic bacteria where the environment is the thermoclinic layer of

the sea within which bacteriophages and bacteria are assumed to be homogeneously

distributed. The main model simpli�cation was in modeling the latent period of

infected bacteria in order to describe the model with three nonlinear ordinary dif-

ferential equations.

However, modeling of the latent period by suitable delay terms looks to be bi-

ologically reasonable and mathematically challenging, the �ndings of which can

be interesting to compare with the outcomes of our previous model [4]. Hence,

in the following we �rst recall the biological justi�cation for the model and then

we introduce the model itself comparing it with other models on the same topic.

The experimental evidence of the bacteriophage infection of marine bacteria can be

found, for example, in the papers by Sieburth [14], Moebus [11], Bergh, Borsheim,

Bratbak, Heldal [3]; Proctor and Fuhrman [12]. It is reported (see [12]) that from

5-6% up to 70% of bacteria population is infected by bacteriophages. Thus, bacte-

riophage infection is proposed to be the main cause of bacteria mortality against

the assumption that the main mortality cause is the protozoan grazing, thus im-

plying the existence of a signi�cant new pathway of carbon and nitrogen cycling in

marine food webs. The mechanism of bacteriophage infection is assumed to be the

following. We have two populations: the bacteria whose total population density

is denoted by N ([N ] = number of bacteria/liter); the viruses or bacteriophages

whose population density is denoted by P ([P ] = number of viruses/liter).

A.1. We assume that in the absence of viruses the bacteria population density

grows according to a logistic equation with carrying capacity C ([C] = number of

bacteria/liter) and \intrinsic growth rate constant �" ([�] = day�1):

dN(t)

dt= �N(t)

�1�

N(t)

C

�:(1.1)

Here, \�" combines the growth rate constant by cellular division and bacteria

mortality rate (like protozoan grazing) excluding that resulting from viral infection.

In the presence of viruses, we divide the total bacteria population into two sub-

classes: the susceptible bacteria S(t) and the virus infected bacteria I(t) ([S] =

[I ] = [N ]),i.e.,

N(t) = S(t) + I(t):(1.2)

A susceptible bacterium S becomes infected I under the attack of many virus par-

ticles on the cellular membrane (see Bergh et al.,[3]) in a number ranging from one

up to 5 phage/cell (see, e.g., Proctor, Okubo, Fuhrman [13]), but only one virus en-

ters its head through the bacterial membrane and then starts its replication inside

MARINE BACTERIOPHAGE INFECTION ANALYSIS WITH LATENCY 3

the bacterium (now infected) and inhibiting the further attack of other viruses on

the bacterial membrane. The viruses already on the membrane return to the solu-

tion. Thus, the infecting process in the homogeneous solution of sea thermoclinic

layer seems to be \one infecting phage P infects one susceptible bacterium S" and

according to the law of mass action we assume:

A.2. the rate of infection is

KP (t)S(t)(1.3)

which is the number of new infected bacteria I per unit time and \K" is the

\e�ective per bacteria phage absorption constant rate" ([K]-ml day�1).

The viral nucleic acid inside the infected bacterium takes control of the bacterial

metabolism, inhibiting its replication by division, but directing the bacterium in the

synthesis of more viral nucleic acid and other materials needed for making copies

of complete virus. Hence the present model cannot account for lysogenic bacteria

carrying non-replicating phages, but assumes that all phages inside bacteria are

\virulent."

A.3. Accordingly, we assume that only susceptible bacteria S are capable of

reproducing by cellular division according to the logistic growth (1.1), whereas the

infected bacteria, under the genetic control of virulent phages, replicate phages

inside themselves up to the death by lysis after a \latency time" T . However the

infected bacteria I still compete with susceptible bacteria S for common resources.

According to these remarks,

dS(t)

dt= �S(t)

�1�

N(t)

C

��KS(t)P (t)(1.4)

is the balance equation for susceptible bacteria.

The time elapsing from the instant of infection, i.e., when the virus injects the

contents of the virus head inside the bacterium, to the instant of the bacterium cell

wall-lysis, at which \b" copies of assembled phages are released in solution, is called

\latent period" or \incubation time of phages inside bacteria" and is denoted by

\T" ([T ] = days).

The lysis of one infected bacterium, on the average, produces \b" copies of the

virus particles. We denote by \b" the \virus replication factor."

A.4. For a given population of bacteria we assume that latency period T , T 2

R+0 and virus replication factor \b," b 2 (1;+1) are constant and the same for

the whole population.

A.5. The infected bacteria may have mortality terms di�erent from that by viral

lysis like \protozoan grazing." We account for these terms by the \death constant

rate �i" ([�i] = day�1).

Let us construct the balance equation for infected bacteria I(t). At any time \t"

the density of infected bacteria I(t) is obtained by summation on all the rates of

infection at previous times \KS(t��)P (t��)," � � 0, multiplied by the probability


that infected bacteria have to survive from time \t��" up to time \t" with the given

mortality \�i," i.e., \e��i� ." The summation in the past cannot extend beyond

\�T" since the bacteria infected at t� � � t� T at time \t" have already left the

I class by lysis. Accordingly, at any time \t" we have:

I(t) =

Z T

0

KS(t� �)P (t � �)e��i�d�:(1.5)

By the variable change � = t� � we obtain:

I(t) =

Z t

t�T

e��i(t��)KS(�)P (�)d�:(1.6)

In the following we will use its di�erential form:

dI(t)

dt= ��iI(t) +KS(t)P (t)�KS(t� T )P (t� T )e��iT :(1.7)

Now we consider the balance equation for viruses.

A.6. We account for all kinds of possible mortality of viruses (enzymatic attack,

pH-dependence, U.V. radiation, photo-oxidation, etc.) by the death constant rate

\�p" ([�p] =day�1).

The lysis rate of infected bacteria at time \t" is \KS(t � T )P (t � T )e��iT ."

Since each bacterium delivers \b" copies of the phage in solution, the input rate for

phages at time \t" is \be��iTKS(t � T )P (t � T )." The phages leave the class of

free viruses by infecting the bacteria at the rate (1.3) or by death. Hence,

dP (t)

dt= ��pP (t)�KS(t)P (t) + be��iTKS(t� T )P (t� T );(1.8)

is the balance equation for viruses or phages.

If the virus infection to the other bacteria population is successful, an endemic

equilibrium exists, say

P (x; t) = P �(x) for all t;(1.9)

where x = (x; y; z) are the coordinates in the thermoclinic layer of the sea, and

P �(x) is the equilibrium distribution of bacteriophages. This could provide a con-

stant in ow of phages from surrounding regions accounted by the parameter

�(x) = �Dpr2P �(x)(1.10)

if di�usivity Dp of phages is assumed to be independent of coordinates.

A.7. For simplicity, we assume a constant rate of in ow, say � (� � 0, [�] =ml�1

day�1), of phages from surrounding regions. � is assumed to be independent of


coordinates. With this last assumption the model equations are:

dS(t)

dt= �S(t)

�1�

S(t) + I(t)

C

��KS(t)P (t)

dI(t)

dt= ��iI(t) +KS(t)P (t)� e��iTKS(t� T )P (t� T )(1.11)

dP (t)

dt= � � �pP (t)�KS(t)P (t) + be��iTKS(t� T )P (t� T )

where b 2 (1;+1), T 2 R+ = (0;+1), �i 2 R+0 = [0;+1):

In this paper we will study the mathematical properties of the solutions of (1.11)

under two limiting cases: � > 0, i.e., the phages have a constant input from the sur-

rounding environment; � = 0, i.e., the phages are produced by the epidemic itself.

In both cases we consider the solutions of (1.11) depending upon the parameters

(b; T ) 2 (1;+1)�R+.

Estimates for � from the doubling time of bacteria, for the latency time T and

for the virus replication factor b can be found in the paper by Proctor, Okubo and

Fuhrman [13]. Estimates forK can be obtained, for example, in the paper by Bergh,

Borsheim, Bratbak and Heldal [3]or by kinetic data from the previous referred paper

by Proctor, Okubo, Fuhrman [13]. However, for the computer simulations in this

paper we use the parameter estimates (unless di�erently speci�ed) suggested by

Prof. A. Okubo (see also Beretta and Kuang [4]), i.e.,

(C = 2� 106 ml�1; � = 1:34 day�1; T = 7 hours

K = 6:7� 10�8 ml/day; �p = 2 day�1(1.12)

with �i = 0:1�p. \b" is used as a varying parameter. In Appendix A the above

parameters are reported in dimensionless form.

It may be interesting to compare the model equations (1.11) with models on the

same topic. We start with a model by A. Campbell [6] with the following equations:

dS(t)

dt= �S(t)

�1�

S(t)

C

��KS(t)P (t)(1.13)

dP (t)

dt= bKS(t� T )P (t� T )� �pP (t)�KS(t)P (t)

where

I(t) =

Z t

t�T

KS(�)P (�)d�:(1.14)

We remark that in (1.13) the competition for common resources and additional

mortality rate endured by infected bacteria is neglected. The equations (1.13),

(1.14) can be obtained from (1.11) when � = 0, �i = 0.


Another modeling of the host-phage system is by H.J. Bremermann [5] who

proposed the following simple system:

dS(t)

dt= �S(t)

�1�

S(t)

C

��KS(t)P (t)

dI(t)

dt= KS(t)P (t)� �I(t)(1.15)

dP (t)

dt= b�I(t)� �P (t):

In both models the competition of infected bacteria is neglected in the logistic

equation. If we do the same in (1.11), in the case with � = 0, the number of model

equations reduces to two:

dS(t)

dt= �S(t)

�1�

S(t)

c

��KS(t)P (t)(1.16)

dP (t)

dt= ��pP (t)�KS(t)P (t) + be��iTKS(t� T )P (t� T )

since

I(t) =

Z t

t�T

e��i(t��)KS(�)P (�)d�:(1.17)

In the following we will refer to (1.16), (1.17) as a Campbell-like model. Another

model for host-phage system in a chemostat is introduced by Lenski and Levin [10]:

dR(t)

dt= D(R0 �R(t))� �(R)�S(t)

dS(t)

dt= �(R)S(t)�DS(t)�KS(t)P (t)(1.18)

dI(t)

dt= KS(t)P (t)�DI(t)� e�DTKS(t� T )P (t� T )

dP (t)

dt= �DP (t)�KS(t)P (t) + be�DTKS(t� T )P (t� T )

where R stands for \resource concentration" and S; I; P have the usual meaning

of previous models. In (1.18) �(R) is the uninfected multiplication rate via binary

�ssion of susceptible bacteria and � is the amount of resources for a new bacterium.

Finally, D is the wash-out rate constant of the chemostat. Of course there is

clear di�erence with (1.11), due to the chemostat structure. However, the rate of

infection is the same; and assuming that �p = �i = D the last two equations are the

same as in (1.11) (� = 0). To the best of our knowledge, all the models mentioned

above (i.e., (1.15), (1.16), (1.18)) have NOT been systematically studied from a

mathematical point of view. Hence the study of the mathematical properties of the

solutions of (1.11) may provide novel mathematical theory for these models.

The structure of the paper is as follows: Section 2 is devoted to the main math-

ematical properties of the solutions of the model equations (1.11) after having

supplemented it by the appropriate initial conditions. Section 3 is devoted to lo-

cal stability analysis by characteristic equations of the equilibria. An analysis of


the endemic equilibrium of the Campbell-like model is also performed. Section 4

deals with the global stability properties of the equilibria. Then in Section 5 we

consider properties of permanence of the solutions. Section 6 ends the paper with

a discussion about the qualitative features of the model and a comparison (in case

of � = 0) with the a previous model by the authors [4].

2. Basic Properties of the Model

In this section we will present some important properties of the solutions of

(1.11), i.e., the solutions of:8>>>>>>><>>>>>>>:

dS(t)

dt= �S(t)

�1�

S(t) + I(t)

C

��KS(t)P (t)

dI(t)

dt= ��iI(t) +KS(t)P (t)� e��iTKS(t� T )P (t� T )

dP (t)

dt= � � �pP (t)�KS(t)P (t) + be��iTKS(t� T )P (t� T )

(2.1)

where the parameters �;C;K; �i; �p 2 R+; � 2 R+0, b 2 (1;+1) and T 2 R+.

The initial conditions for (2.1) at t = 0 are:8>>><>>>:

S(�) = '1(�); P (�) = '3(�); � 2 [�T; 0]

I(0) = K

Z 0

�T

e�i�'1(�)'3(�)d�;

'i(�) 2 C([�T; 0]) : 'i(�) � 0; 'i(0) > 0; i = 1; 3:

(2.2)

In the following we de�ne R3+0 = f(S; I; P ) 2 R3

j S � 0; I � 0; P � 0g and

R3+ = f(S; I; P ) 2 R3

j S > 0; I > 0; P > 0g.

2.1. Positive invariance. Note that the plane S = 0 of R3 is invariant for (2.1).

Now we consider the variable P in [0; T ] with initial conditions (2.2). Then, for

t 2 [0; T ] the third of equations (2.1) gives

dP (t)

dt= � � (�p +KS(t))P (t) + be��iTK'1(�(T � t))'3(�(T � t))

� �(�p +KS(t))P (t); 8t 2 [0; T ]:(2.3)

By direct integration of (2.3) we obtain

P (t) � P (0) exp

��

Z t

0

(�p +KS(�))d�

�� 0(2.4)

as t 2 [0; T ] and as long asR t0S(�)d� < +1. By repeating this argument, we see

that the nonnegativity of S and P in [0; T ] can be used to infer nonnegativity of

P (t). From (2.2), I(t) can be written as

I(t) = K

Z t

t�T

e��i(t��)S(�)P (�)d�;


the nonnegativity of S(t), P (t) on [�T;+1) implies that of I(t) if t � 0. This

shows that for initial conditions (2.2) the corresponding solution of (2.1) is such

that minfS(t); I(t); P (t)g � 0 in its time interval of existence.

2.2. Boundedness of solutions. Wewill need the following lemmas due to Barb�alat

(see [2]).

Lemma 2.1. Let g be a real valued di�erentiable function de�ned on some half line

[0;+1), a 2 (�1;+1). If (i) limt!+1 g(t) = �, j�j < +1, (ii) _g(t) is uniformly

continuous for t > a, then limt!+1 _g(t) = 0.

and its integral version:

Lemma 2.2. Let f be a nonnegative function de�ned on [0;+1) such that f is

integrable on [0;+1) and uniformly continuous on [0;1). Then limt!+1 f(t) = 0.

Lemma 2.3. Assume � > 0 and the initial conditions (2.2) satisfying S(0)+I(0) <

C. Then S(t) + I(t) < C for all t � 0. On the contrary, if S(0) + I(0) � C a time

t1 > 0 exists such that S(t) + I(t) < C for all t > t1.

Proof From the �rst two equations in (2.1), we obtain

d

dt(S + I) = �S

�1�

S + I

C

�� iI � e��iTKS(t� T )P (t� T )

we see that:

d

dt[S(t) + I(t)� C] � �

�

CS(t)[S(t) + I(t)� C]

which implies:

S(t) + I(t) � C + [S(0) + I(0)� C] exp

��

�

C

Z t

0

S(�)d�

�:(2.5)

Hence if S(0)+ I(0) = C, then S(t)+ I(t) � C for all t � 0, and if S(0)+ I(0) < C,

then S(t) + I(t) < C for all t � 0. The case of S(0) + I(0) = C can give rise to

the following cases: (i) either a positive time t� exists such that S(t) + I(t) < C

for all t > t�, or (ii) S(t) + I(t) = C for all t > 0. The second case implies

(d=dt)(S(t) + I(t)) = 0 for all t > t�, which gives rise to a contradiction since if

S(t) + I(t) = C, then

d

dt(S(t) + I(t)) = ��iI � e��iTKS(t� T )P (t� T ) < 0(2.6)

for any t > 0.

Assume now that S(0)+ I(0) > C. We need only to exclude the possibility that

if S(0) + I(0) > C then S(t) + I(t) � C for all t > 0. If so, then on [0;+1),

�

d

dt(S(t) + I(t)� C) �

�

CS(t)(S(t) + I(t)� C) � 0:


Hence on [0;+1) we can de�ne the nonnegative function

f(t) := �

d

dt(S(t) + I(t)� C);(2.7)

for which S(0)+I(0) �R t0f(u)du = S(0)+I(0)�(S(t)+I(t)) � 0 for all t 2 [0;+1).

Then Lemma 2.2 implies that limt!+1 f(t) = 0, i.e.,

limt!+1

d

dt(S(t) + I(t)) = 0:(2.8)

However, from the �rst two equations (2.1) we get

limt!+1

d

dt(S(t) + I(t))(2.9)

= limt!+1

��S

�1�

S + I

C

�� iI � e��iTKS(t� T )P (t� T )

�;

If this limit does exist, unless trivial cases which give a negative limit, only two cases

are possible: (i) I(t)+S(t)! C; S(t)! 0 as t!1; (ii) I(t) +S(t)! C; I(t)! 0

as t!1: The case (i) is trivial, since it gives limt!+1ddt(S(t)+I(t)) = ��iC < 0.

The case (ii) gives

limt!+1

d

dt(S(t) + I(t)) = �e��iTKC lim

t!+1P (t� T )(2.10)

which requires a further analysis on P (t) behavior. From the �rst equation in (2.1)

we see that

lim supt!+1

S(t) � C:(2.11)

Of course there will be su�ciently large times T0 such that for t > T0, S(t) � 2C.

Then, from the third equation in (2.1) we have

dP (t)

dt� � � �pP (t)�KS(t)P (t) � � � (�p + 2KC)P (t)(2.12)

which shows that, for large t, say t > T1 > T0, then

P (t) ��

2(�p + 2KC)� � > 0:(2.13)

Therefore, (2.10) and (2.11) imply

limt!+1

d

dt(S(t) + I(t)) � ��KCe��iT < 0;(2.14)

a contradiction to (2.8).

In conclusion, let t1 := maxft�; T1g > 0. Hence, whenever S(0) + I(0) � C then

S(t) + I(t) < C for all t > t1. This proves the lemma.

The case � = 0 gives rise to a di�erent result with respect to Lemma 2.3, but

the procedure to prove it is however similar. Furthermore, the same kind of results

are presented in [1]. In the following by Ef = (C; 0; 0) we denote the free disease

equilibrium of (2.1) which is feasible only if � = 0. We omit the proof of the

following lemma to avoid repetition.


Lemma 2.4. Assume � = 0 in (2.1). If we assume that at t = 0 the initial

conditions of (2.1) satisfy that S(0) + I(0) � C, then either (i) S(t) + I(t) � C for

all t > 0 and therefore (S(t); I(t); P (t)) ! Ef = (C; 0; 0) as t ! +1, or (ii) there

exists a time, say t1 > 0, such that S(t) + I(t) < C for all t > t1. Finally, (iii) if

S(0) + I(0) < C then S(t) + I(t) < C for all t > 0.

We can further prove the following result regarding boundedness of I; P variables.

Lemma 2.5. Assume � > 0. De�ne the function:

W (t) = bI(t) + P (t); t 2 [0;+1):(2.15)

Assume further that

b < b��= 1 +

�p

KC(2.16)

and L is any positive constant such that

L >�

�m � (b� 1)KC

�= L1;(2.17)

where �m = minf�i + (b � 1)KC;�pg. Then there is a t1 = t1(L) > 0, such that,

for all t > t1, W (t) < L.

Proof From the second and third parts of equation (2.1), it follows that

dW

dt= � � �ibI � �pP + (b� 1)KSP:(2.18)

From Lemma 2.3 we know that a time, say t0, t0 � 0, exists such that S(t) < C

for all t > t0. Hence, from (2.18) we obtain that, for all t > t0,

dW

dt< � � �ibI � �pP + (b� 1)KC(W � bI)

= � � [�i + (b� 1)KC]bI � �pP + (b� 1)KCW:(2.19)

Thus we obtain:

dW

dt< � � [�m � (b� 1)KC]W:(2.20)

Observe that if �m = �p, then (2.16) indicates that �m = �p > (b � 1)KC. If

�m = �i + (b� 1)KC, then clearly (b� 1)KC < �m. Hence (2.20) implies that

lim supt!+1

W (t) ��

�m � (b� 1)KC< L:(2.21)

This proves the lemma.

The following is an important result in the study of global stability:

Theorem 2.1. Assume that � = 0. Then for all b 2 (1; b� = 1 + (�p=KC)) the

free disease equilibrium Ef = (S� = C; I� = 0; P � = 0) is globally asymptotically

stable in R3+.


Proof We assume � = 0. According to Lemma 2.4 we have the following. Assume

that S(0)+I(0) > C. Then either (i) (S(t); I(t); P (t)) ! Ef = (C; 0; 0) as t! +1

or (ii) there exists a time, say t1 > 0, such that S(t)+I(t) < C for all t > t1. Finally

(iii) if S(0) + I(0) < C, then S(t) + I(t) < C for all t > 0. Thus, in case (i) the

theorem is proven, whereas for cases (ii), (iii) we can assume that a time, say

t1 � 0, exists such that S(t) < C for all t > t1. De�ne W (t) = bI(t) + P (t) for all

t 2 [0;+1). Then from the second and third equations of (2.1) we obtain (note

� = 0)

dW (t)

dt= ��ibI � �pP + (b� 1)KSP:(2.22)

This implies (see (2.19)):

dW (t)

dt� �[�m � (b� 1)KC]W (t)(2.23)

for all t > t1 and where �m = minf�i + (b� 1)KC;�pg. Furthermore if b 2 (1; b�),

b� = 1 + (�p=KC), then �m � (b � 1)KC > 0. Hence (2.23) implies that W (t) =

bI(t) + P (t) ! 0 as t ! +1, i.e., I(t) ! 0 and P (t) ! 0 as t ! +1. Since S(t)

is bounded, then for any 1 > " > 0 there is a T" > t1 such that for t > T",

�S(t)

�1� "�

S(t)

C

�<dS(t)

dt< �S(t)

�1 + "�

S(t)

C

�:(2.24)

Hence (1 � ")C � lim inf t!+1 S(t) � lim supt!+1 S(t) � (1 + ")C. Hence

limt!+1 jS(t)� Csj < "C and the conclusion follows by letting "! 0.

Note that for � = 0, a necessary condition for the existence of the positive

(endemic) equilibrium E+ is b > b�. Thus, Theorem 2.1 holds true when E+ is not

feasible.

Our next theorem shows that there is an L2 > 0 such that, regardless of the

value of b and �, and independent of initial conditions, lim supt!+1 P (t) � L2.

Observe �rst that, for large t, say t � t1 > T , S(t) < C + 1. Hence for t � t1,

P 0(t) � �[�p +K(C + 1)]P (t)(2.25)

which implies that

P (t) � P (t� T )e�[�p+K(C+1)]T(2.26)

thus (C1 = e[�p+K(C+1)]T )

P (t� T ) � P (t)e[�p+K(C+1)]T�= P (t)C1:(2.27)

Hence, for t � t1,

P 0(t) � � +Kbe��iTC1(C + 1)P (t):(2.28)

Let C2�= bKe��iTC1(C + 1), then for t > t0 > t1,

P (t) � ��C�12 + (P (t0) + �C�12 )eC2(t�t0):(2.29)


It is also easy to observe that if, for all large t,

P (t) ��

K+ 1

then limt!+1 S(t) = 0 and limt!+1 P (t) = Cp, in which case L2 = Cp. We let

` = maxf�K�1; �C�12 g+ 1;(2.30)

T1 = T +1

K`� �ln

�2Kb(C + 1)

�pe�iT

��= T + ;(2.31)

L2 = ��C�12 + (`+ �C�12 )eC2T1 + 2�

�p:(2.32)

Clearly L2 > ` and L2 > 2�=�p = 2Cp. We have, regardless of the value of � (� 0)

and b:

Theorem 2.2. With L2 de�ned above, we have

lim supt!+1

P (t) � L2:(2.33)

Proof Recall that it is impossible for P (t) � ` for all large t. Assume the theorem

is not true; then there is a t�> t1 + T1, such that

P (t�) = L2; P 0(t

�) � 0;(2.34)

and for all t < t�, P (t) � P (t

�). In addition, there is a t0, t1 < t0 < t

�, such that

P (t0) = `; P 0(t0) � 0(2.35)

and ` � P (t) � L2 for t 2 [t0; t�]. By (2.29), we see that (letting t = t�)

t�� t0 �

1

C 2ln

�L2 + �C�12

`+ �c�12

�> T1:(2.36)

For t 2 [t0; t�], we have

S0(t) � �S(t)�KP (t)S(t) < (K`� �)S(t):(2.37)

Hence

S(t�� T ) � S(t0)e

�(K`��)(t�� T � t0) < (C + 1)e�(K`��)(T1�T ):(2.38)

This implies that

P 0(t�) < � � �pP (t�) + be��iTKS(t

�� T )P (t

�� T )

� � � L2[�p � bK(C + 1)e��iT e�(K`��) ]

= � � L2

��p �

�p

2

�

= � ��p

2L2 < 0;

a contradiction to the second part of (2.34). This proves the theorem.


In the rest of this paper, we de�ne

L = maxfL1; L2g;(2.39)

2.3. Equilibria. A) Case � > 0.

In the following we denote by Cp = �=�p the \carrying capacity" of bacterio-

phages. The equilibria of (2.1) are solutions of:8><>:

S[�(C � (S + I))�KCP ] = 0

��iI +KSP�(T ) = 0

�p(Cp � P ) +KSP�(T ) = 0

(2.40)

where, for the sake of simplicity, we set

�(T ) = be��iT � 1; �(T ) = 1� e��iT(2.41)

with �(T ) 2 (�1; b� 1], �(T ) 2 (0; 1) as T 2 [0;+1). Furthermore, �(T ) = 0 at

T = T �, where

T � =1

�ilog b:(2.42)

Straightforward computation yields the following.

Proposition 2.1. Assume T = T � (see (2.41)), i.e., �(T ) = 0. Then (2.1) admits

the following two equilibria: Ef = (0; 0; Cp) which is always feasible and the positive

equilibrium E+ = (C(��KCp)

�(1+�Cp);�CpC(��KCp)

�(1+�Cp); Cp), provided that Cp < �=K. When

Cp = �=K, E+ coincides with Ep. The constant � is de�ned in (2.43) below.

Standard but tedious analysis yields the following.

Proposition 2.2. Assume �(T ) > 0, i.e., T < T �. Then a unique positive equi-

librium E+ exists provided that Cp < �=K. When Cp = �=K, E+ coincides with

Ep = (0; 0; Cp).

Now let us consider the case (ii) �(T ) < 0, i.e., T > T �. Let

S1(P )�=�p(Cp � P )

Kj�(T )jP; S2(P )

�=C(� �KP )

�(1 + �P ); �

�=K�(T )

�i:(2.43)

S1(P ) is positive on (0; Cp), vanishes as P = Cp and has limP!0+ S1(P ) = +1.

Furthermore, S1(P ) is monotone decreasing as P increases in (0; Cp) with positive

concavity.

Let us de�ne the function (P ) = S1(P ) � S2(P ) on (0; Cp). Hence 2

C((0; Cp]) is such that (P )! +1 as P ! 0+ and (Cp) < 0. Hence (P ) = 0 has

at least one root P � 2 (0; Cp) and in any case the number of roots of (P ) = 0 on

(0; Cp) must be odd. Since (P ) = 0 gives a second order of algebraic equation, the

number of the roots of (P ) = 0 on (0; Cp) must be one. This proves the existence

and uniqueness of P � 2 (0; Cp) such that S1(P�) = S2(P

�) and, correspondingly, of

the positive equilibrium E+ whose other components are S� := S1(P�) = S2(P

�),


S� 2 (0; C) and I� = �S�P �. Of course, if Cp = �=k the positive equilibrium

becomes Ep = (0; 0; Cp). Hence the following is proven:

Proposition 2.3. Assume �(T ) < 0, i.e., T > T � and Cp < �=K. Then a unique

positive equilibrium E+ exists with P � 2 (0; Cp) and S�

2 (0; C). When Cp = �=k,

E+ coincides with Ep = (0; 0; Cp).

As a summary we may say that if � > 0 then the positive equilibrium E+ is

feasible provided that Cp < �=k. If Cp = �=k, then E+ becomes Ep. Finally, if

�(T ) > 0, i.e., T < T � then P � 2 (Cp; (�=k)), whereas if �(T ) < 0, i.e., T > T �,

then P � 2 (0; Cp).

B) Case � = 0.

The equilibria are solutions of8><>:

S[�(C � (S + I))�KCP ] = 0

I = �SP

P [��p +KS�(T )] = 0;

(2.44)

where � = (K�(T )=�i) and �(T );�(T ) have been de�ned in (2.41). Assume �rst

�(T ) = 0, i.e., T = T �. Then (2.44) shows that P � = 0, I� = 0 and S� = C, in

agreement with Proposition 2.1 in which we set Cp = (�=�p) = 0.

Now let us consider the case �(T ) 6= 0.

It is easy to check that boundary equilibria

E0 = (S� = 0; I� = 0; P � = 0); Ef = (S� = C; I� = 0; P � = 0);(2.45)

are both feasible for all parameter values. For the positive equilibria, if they exist,

their components are:

E+ =

�S� =

�p

K�(T ); I� = �S�P �; P � =

�(C � S�)

KC + ��S�

�:(2.46)

Clearly, if �(T ) < 0, then S� < 0 and we cannot have positive equilibria. Hence

we need only to consider the case �(T ) > 0, i.e., T < T �.

If, in addition, S� = �p=(k�(T )) < C, the unique positive equilibrium is feasible

and it is given by (2.46). When �(T ) is such that S� = C then the positive

equilibrium becomes Ef = (C; 0; 0), and E+ is not feasible if S� > C. If T = 0,

then S� = �p=(K(b � 1)), whereas for T increasing in (0; T �), T � = (1=�i) log b,

S� = �p=(K�(T )) is a monotonically increasing function of T and that S� % +1

as T % T � from the left.

Let \Tc" be the incubation time at which �(T ) = �p=KC, i.e., S� = C:

Tc�=

1

�ilog

b

b�(2.47)

where, Tc < T � and b� = 1+�p=KC. Some comments on (2.46) are in order. If the

virus's replication factor \b" is such that b � b� = 1+�p=KC, then Tc � 0 and the

endemic equilibrium E+ cannot be feasible. Hence, the feasibility of the endemic


equilibrium requires as a necessary condition that b > b�. This implies Tc > 0,

and the endemic equilibrium is feasible only if the time T taken in replicating the

phages is not too high, i.e., T < Tc. We summarize the above results in the:

Proposition 2.4. Assume � = 0. Then the boundary equilibria E0 = (0; 0; 0)

and Ef = (C; 0; 0) are feasible for all parameter values. If it exists, the endemic

equilibrium E+ is unique and given by (2.46). A necessary and su�cient condition

for its existence is that T 2 (0; Tc), where Tc = (1=�i) log(b=b�) and b� = 1 +

�p=KC.

We would like to point out that the presence of � > 0, i.e., of a resource of

bacteriophages from the surrounding environment, ensures the existence of a unique

endemic equilibrium for any latency period (or incubation time) T . While for � = 0,

it is necessary that b > b� and T 2 (0; Tc = (1=�i) log(b=b�)) in order to have E+

feasible.

3. Local Stability of Boundary Equilibria

For convenience, we denote by

x(t) = col(S(t)� S�; I(t)� I�; P (t)� P �); x 2 R3; t > 0:(3.1)

Then (2.1) can be written as

d

dtx(t) = F (x(t);x(t � T ))(3.2)

where F : C([�T; 0];R3) ! R3 is a continuously di�erentiable vector function.

Hence, de�ne the matrices A;B 2 R3�3

A =

�@F

@x(t)

�x=0

; B =

�@F

@x(t� T )

�x=0

;(3.3)

the equations (2.1), linearized around 0, takes the form:

dx(t)

dt= Ax(t) +Bx(t � T )(3.4)

and the corresponding characteristic equation is

det [A+Be��T � �I ] = 0(3.5)

where � are the corresponding characteristic roots. It is easy to check that the

characteristic equation takes the form of (3.6) below:

��

�1�

2S� + I�

C

��KP � � � �

�

CS� �KS�

KP �(1� e�(�i+�)T ) �(�i +�) KS�(1� e�(�i+�)T )

�KP �(1� be�(�i+�)T ) 0 �(�p +�)�KS�(1� be�(�i+�)T )

��= 0:

(3.6)

In the following we consider two cases:

Case 1. � > 0.


From (3.6) it is easy to check that the following proposition holds true:

Proposition 3.1. Assume � > 0. The equilibrium Ep = (0; 0; Cp) is an asymptot-

ically stable node if �=K < Cp, i.e., E+ is not feasible; if �=K > Cp it becomes an

unstable saddle point and as �=K = Cp, Ep becomes critically stable (if �=K > Cp,

E+ is feasible).

Case 2. � = 0.

In this case we have two boundary equilibria, i.e., E0 = (0; 0; 0) and Ef =

(C; 0; 0) which are feasible for all parameter values. If Tc > 0, i.e., b > b�, the

endemic equilibrium E+ is feasible provided that T 2 (0; Tc), and it is not feasible

if T > Tc. E+ coincides with Ef if T = Tc. We can prove:

Proposition 3.2. Assume � = 0. The equilibrium E0 = (0; 0; 0) is always an un-

stable saddle point. As far as Ef = (C; 0; 0) is concerned, it is locally asymptotically

stable whenever T > Tc and unstable if T < Tc (Tc = (1=�i) log(b=b�)). If T = Tc,

then Ef becomes critically stable.

The proof of Proposition 3.2 is trivial where it concerns the equilibrium E0, as it

can be easily checked from (3.6). The proof of the remaining part of Proposition 3.2

is not trivial but still it is quite standard and therefore can be found in Appendix B.

The analysis of local stability properties of the endemic equilibrium for both cases

� > 0 and � = 0 are very di�cult. However, for the case � = 0, we may have

an insight about the possible stability behavior of the endemic equilibrium E+

considering equations very close to our model equations (1.11) but which enable a

remarkable simpli�cation of the characteristic equation: the Campbell-like model

(1.16), (1.17) with �i > 0 and the Campbell model (1.13), (1.14) where �i = 0. For

both models we just have to consider two variables: S and P .

Before considering the characteristic equation for the Campbell-like model (Camp-

bell model is just a particular case with �i = 0) we recall that the equilibria have the

same structure as in (1.11). The equilibria E0 = (0; 0) and Ef = (c; 0) are feasible

for all parameter values. The endemic equilibrium E+ = (S�; P �) has components:

S� =�p

K(be��iT � 1); P � =

�

K

�1�

S�

C

�(3.7)

and it is feasible provided that b > b� = 1 + �p=(KC) and

T < Tc =1

�ilog

b

b�:(3.8)

The same holds for the Campbell model, where �i = 0; however, this implies that

the endemic equilibrium

E+ =

�S� =

�p

K(b� 1); P � =

�

K

�1�

S�

C

��(3.9)


is now independent from the latency time T , and it is feasible provided that b > b�.

For both cases �i � 0 in the following we set

� =S�

C=

b� � 1

be��iT � 1; p� =

P �

C=

�

KC(1� �)(3.10)

with � 2 (0; 1). Remark that if �i > 0 then � = b��1be��iT�1

is a monotone increasing

function of T as T is increasing in [0; Tc] and � = 1 at T = Tc. In the case of �i = 0,

� is a monotone decreasing function of b in (b�;+1) and � = 1 at b = b�.

In both cases (�i � 0) the characteristic equation at the endemic equilibrium

E+ is: �� (�� +�) �KC�

�p�

�(KC� � �(�)e��T ) ��(�)(1� e��T )� �

�� = 0(3.11)

where � are the characteristic roots and �(�) = �p +KC�. We obtain:

�2 + a�+ b�e��T + c+ de��T = 0:(3.12)

where

a = �� + �(�) b = ��(�) c = �d+ ��p(1� �) d = �(1� 2�)�(�):

Note that c + d > 0, for � 2 (0; 1) and at T = 0 the characteristic roots have

negative real parts.

When increasing T; a stability shift may occur only with a pair of purely imag-

inary characteristic roots, say � = �i!, ! > 0, crossing the imaginary axis from

left to right. Hence, we look for characteristic roots � = �i!, ! > 0 of (3.12) We

obtain that ! > 0 must be a solution of

!2 =1

2f(b2 + 2c� a2)� [(b2 + 2c� a2)2 � 4(c2 � d2)]1=2g(3.13)

where it is easy to check that

b2 + 2c� a2 < 0; � 2 (0; 1):(3.14)

Of course, if c2 � d2 there are no characteristic roots � = �i!, ! > 0.

If d2 > c2 there is one pair, say � = �i!+, !+ > 0, such that

dRe(�)

d�

��=�i!+

> 0:(3.15)

Hence, by a simple analysis of the function

�(�)�= d2(�)� c2(�) = (1� �)[��2�2p(1� �) + 2�2�p(1� 2�)�(�)];

as � 2 [0; 1] it is simple to prove:

Lemma 3.1. There exists a unique value, say �1; �1 2 (0; 12) such that d2(�) >

c2(�) for all � 2 (0; �1) (and d2(�) < c2(�) in (�1; 1)).

Then we can conclude:


Theorem 3.1. Let us denote by

bc�= (b� � (1� �1))=�1:(3.16)

(i) If �i > 0 then for b 2 (b�; bc) no stability shifts can occur and the positive

equilibrium remains (locally) asymptotically stable for all T 2 [0; Tc].

(ii) Assume �i = 0. If b 2 (b�; bc] no stability shifts can occur and the positive

equilibrium remains (locally) asymptotically stable for all T > 0.

If b > bc there exists a T0 > 0,

T0 =�1

!+; 0 � �1 � 2�(3.17)

where

sin �1 =da!+ � b!+(c� !2+)

b2!2+ + d2cos �1 = �

ab!2+ + (c� !2+)d

b2!2+ + d2

such that the positive equilibrium is (locally) asymptotically stable if T 2

(0; T0), unstable if T > T0.

The proof of Theorem 3.1 is in Appendix B. Comments on the interpretation of

the results in the theorem are in the discussion section.

4. Global Stability Analysis

As in the previous sections we distinguish between the two cases A) � = 0 and

B) � > 0. Let us consider �rst

Case A) � = 0.

We have two results concerning the global asymptotic stability of Ef . The �rst

is concerned with the case b � b�, i.e., Tc � 0, in which the endemic equilibrium

E+ is not feasible. A second result regards the case b > b� where Tc > 0. The �rst

result slightly extends the one already proven in Theorem 2.1 for the case b = b�.

Theorem 4.1. Assume � = 0. Then if b � b� the free disease equilibrium Ef =

(C; 0; 0) is globally asymptotically stable in R3+.

Proof Recall �rst that Lemma 2.4 provides that, if S(0) + I(0) � C then either

S(t) + I(t) � C for all t > 0 in which case (S(t); I(t); P (t)) ! Ef as t ! +1,

or a positive time, say t1 > 0, exists such that S(t) + I(t) < C for all t > t1.

Furthermore, if S(0) + I(0) < C then S(t) + I(t) < C for all t > 0. In conclusion,

either (S(t); I(t); P (t)) ! Ef as t! +1 or a nonnegative time exists, say t0 � 0,

such that S(t) + I(t) < C for all t � t0. Hence we study below the global stability

of Ef assuming this last case. Let us consider the Liapunov function U : R3

+0 ! R,

where R3

+0 := f(S; I; P ) 2 R3+0 j S > 0g, de�ned by

U = (S � C logS) + w(bI + P ); w 2 R+(4.1)


which is lower bounded and di�erentiable on R3

+0. We obtain:

U 0j(2.1) = �

�

C(C � S(t))[C � (S(t) + I(t))]� (w�p �KC)P (t)

�wb�iI(t)�K(1� w(b� 1))S(t)P (t):(4.2)

Since S(t) + I(t) � C for all t � t0, (4.2) shows that for all t � t0 U0

j(2.1) � 0 and

U 0j(2.1) = 0 if and only if (S; I; P ) coincides with Ef provided that the arbitrary

positive constant w in (4.1) can be chosen in such a way that:

KC

�p< w �

1

b� 1:(4.3)

Hence (4.3) becomes a su�cient condition for the global asymptotic stability of Ef

in R3

+. Since (4.3) reduces to b < b� = �p=(KC) + 1, this proves the theorem if

b < b�. If b = b� then the unique choice for w is: w�p �KC = w(b � 1)� 1 = 0.

Hence, from (4.2) we obtain:

U 0j(2.1) = �

�

C(C � S(t))[C � (S(t) + I(t))] � wb�iI(t):(4.4)

In R3

+0 let us consider the set E = f(S; I; P ) : U 0 = 0g, i.e., E = f(S; I; P ) 2 R3

+0 :

I = 0; S = Cg. Say M is the largest invariant subset of E. Assume S(t) = C for

all t. Then dS=dt = 0 thus implying P (t) = 0 for all t. Since in E, I(t) = 0 for all

t, then M = fEfg. Hence, by the Liapunov-LaSalle theorem, (e.g., Kuang (1993),

[9]), the global asymptotic stability of Ef in R3

+0 follows.

We now present a global asymptotic stability result about Ef in the case in

which b > b�.

Theorem 4.2. Assume � = 0 and b > b�. Then, the free disease equilibrium

Ef = (C; 0; 0) is globally asymptotically stable in R3+ provided that

T > T1�=

1

�ilog

�b� 1

b� � 1

�:(4.5)

Proof Recall that either (S(t); I(t); P (t)) ! Ef as t ! +1 or there is a t0 � 0

such that for t � t0, S(t) < C. In the following we assume t � t0 + T . Then, from

the last two equations of (2.1) we get:

d

dt(I(t) + P (t)) = ��iI(t)� �pP (t) + (b� 1)e��iTKS(t� T )P (t� T )

� ��iI(t)� �pP (t) + (b� 1)e��iTKCP (t� T ):(4.6)

Let us consider the functional:

U(t) = I(t) + P (t) + (b� 1)e��iTKC

Z t

t�T

P (�)d�:(4.7)

Then, from (4.6), (4.7), we get

U 0(t) � ��iI(t)� [�p � (b� 1)e��iTKC]P (t):(4.8)


Let us de�ne �= �p � (b� 1)e��iTKC and assume it is positive, i.e., > 0. Then

T must satisfy that

T >1

�ilog

�b� 1

b� � 1

��= T1:

Hence if (4.5) holds true, then � = minf�i; g > 0, and we get

U 0(t) � ��iI(t)� P (t) � ��(I(t) + P (t)):(4.9)

So U(t) is a Liapunov functional for global asymptotic stability of the equilibrium

(I� = 0; P � = 0) of the last two equations in (2.1), i.e., (I(t); P (t)) ! (0; 0) as

t! +1. This in turn implies S(t)! C as t! +1.

Note that T1 = (1=�i) log[(b � 1)=(b� � 1)] > Tc = (1=�i) log(b=b�) whenever

Tc > 0. Therefore it is yet to be investigated what happens when T 2 (Tc; T1]. At

the moment, this problem is open.

In the case of � > 0, the following results hold:

Lemma 4.1. Assume � > 0 and initial conditions (2.2) such that S(0) > 0, P (0) >

�=K. Furthermore, if

Cp >�

K

�1 +

S(0)K

�p

�(4.10)

then the corresponding solutions are such that

(S(t); I(t); P (t)) ! Ep = (0; 0; Cp) as t! +1:(4.11)

Proof Consider the �rst and third of equations (2.1):8<: S0(t) = �K

�P (t)�

�

K

�S(t)�

�

CS(t)(S(t) + I(t))

P 0(t) = ��p(P (t)� Cp)�KS(t)P (t) + be��iTKS(t� T )P (t� T ):

(4.12)

Let us �rst assume that P (t) � �=K for all t � 0. Thus, we can de�ne the

nonnegative function f(t) = �S0(t) on [0;+1) since the �rst of (4.12) shows that

S0(t) < 0 for all t � 0. Furthermore, remark thatR t0f(�)d� exists for all t 2 [0;+1)

since negativity of S0(t) implies

S(0) �

Z t

0

f(�)d� = S(0)� S(t) � 0 for all t � 0:(4.13)

Therefore, Barb�alat's Lemma 2.2 implies limt!+1 f(t) = 0. In other words,

limt!+1 S0(t) = 0, and nonnegativity of P (t)��=K requires that limt!+1 S(t) =

0. Since P (t) is bounded, this implies that

limt!+1

[P 0(t) + �p(P (t)� Cp)] = 0:(4.14)

From (4.14) we can say that for all " > 0, T" > 0 exists such that

jP 0(t) + �p(P (t)� Cp)j < �p"(4.15)


for all t > T", or equivalently, lim supt!+1 jP (t) � Cpj � ". Letting " ! 0 we

obtain:

limt!+1

P (t) = Cp:(4.16)

Finally, recalling that

I(t) = K

Z t

t�T

e��i(t��)S(�)P (�)d�(4.17)

we get limt!1 I(t) = 0. In conclusion, every solution of (2.1) satisfying that

P (t) � �=K for all t � 0 is such that (S(t); I(t); P (t)) ! Ep = (0; 0; Cp) as

t! +1. Since P (0) > �=K we denote by t�, t� > 0, the �rst time at which P (t)

assumes the value �=K. At t = t�, the second of (4.12) gives:

P 0(t�) = �p

�Cp �

�

K

�� S(t�) + be��iTKS(t� � T )P (t� � T ):

Since S0(t) < 0 in (0; t�], S(t�) < S(0) and therefore

P 0(t�) > �p

�Cp �

�

K

�� S(0) + be��iTKS(t� � T )P (t� � T ):(4.18)

Thus (4.10), (4.18) imply

P 0(t�) > be��iTKS(t� � T )P (t� � T ) � 0;(4.19)

i.e., P (t) cannot cross the �=k value at any t� > 0. Hence P (t) � �=k for all t � 0

and the �rst part of the proof implies the theorem.

Theorem 4.3. Assume � > 0. If

�K

�(�p +KC)> 1;(4.20)

then Ep = (0; 0; Cp) is globally asymptotically stable. If (�K=�(�p + KC)) < 1,

then

Cp >�

K

�1 +

KC

�p

�1�

�K

�(�p +KC)

��(4.21)

implies the global asymptotic stability of Ep = (0; 0; Cp).

Proof In Lemma 2.3, we stated that a time t0, t0 � 0, exists such that S(t) < C

for all t > t0. This remark is used in the third of equations (2.1) to imply

dP (t)

dt> � � (�p +KC)P (t)(4.22)

for all t � t0. Hence for all " > 0, a time exists, say t1("), t1(") > t0, such that

inft>t1(")

P (t) ��

�p +KC� ":(4.23)


From the �rst of equations (2.1) we obtain:

S0(t) = �KS(t)

�P (t)�

�

K

��

�

CS(t)(I(t) + S(t))

� �KS(t)

��

K

��K

�(�p +KC)� 1

�� "

�;(4.24)

for all t > t1("). Since (4.20) holds true, we chose ":

0 < " <�

K

��K

�(�p +KC)� 1

�(4.25)

thus obtaining

S0(t) < 0 for all t > t1("):(4.26)

By Barb�alat's Lemma 2.2, (4.26) implies limt!+1 S(t) = 0, which in turn implies

limt!+1 P (t) = Cp and limt!+1 I(t) = 0. (For details, see Lemma 4.1.)

Assume now �K=(�(�p + KC)) < 1. Hence (4.23) implies that for all " > 0,

there exists t1("):

S0(t) � �S(t)

�1�

K

�P (t)�

S(t)

C

�

� �S(t)

�1�

K

�

��

�p +KC� "

��

S(t)

C

�(4.27)

holds true for all t > t1("). Hence, for all " > 0,

lim supt!+1

S(t) � C

�1 + "

K

��

�K

�(�p +KC)

�:(4.28)

Hence, (4.28) and the third part of (2.1) imply

P 0(t) > � �

��p +KC

��1 + "

K

�

��

�K

�(�p +KC)

��P (t)

for all t > t1("). Again, for all "0 > 0 there exists t2("0), t2("

0) > t1(") and for

t > t2("0)

inft>t2("0)

P (t) �Cp

1 + KC�p

h�1 + "K

�

��

�K�(�p+KC)

i � "0:(4.29)

From the �rst equation of (2.1), we obtain:

S0(t) � �KS(t)

��Cp

1 + KC�p

h�1 + "K

�

��

�K�(�p+KC)

i � �

K

�� "0

�

= �KS(t)

��Cp � �K

h1 + KC

�p

�1� �K

�(�p+KC)

�i�

KC�p"

1 + KC�p

h�1 + "K

�

��

�K�(�p+KC)

i �� "0

�(4.30)

for all t > t2("0).


Since inequality (4.21) holds true, then we can choose ":

0 < " <

�Cp �

�

K

�1 +

KC

�p

�1�

�K

�(�p +KC)

��p

KC;(4.31)

for which the choice of "0:

0 < "0 <Cp �

�K

h1 + KC

�p

�1� �K

�(�p+KC)

�i�

KC�p"

1 + KC�p

h�1 + "K

�

��

�K�(�p+KC)

i(4.32)

implies that (see (4.30)) S0(t) < 0 for all t > t2("0).

Again the Barb�alat Lemma 2.2 implies limt!+1 S(t) = 0 and the global asymp-

totic stability of Ep follows.

Let us remark that �K=�(�p +KC) > 1 is the same as Cp > (�=K)(�p +KC).

Here we address our attention to some particular cases which are not included

among the results of Lemma 4.1 and Theorem 4.3.

Proposition 4.1. Assume initial conditions (2.2) such that:

'1(�) � S(0) > 0; '3(�) ��

K; P (0) >

�

K; � 2 [�T; 0]:(4.33)

Denote by t� > 0 the �rst time at which (eventually) P (t�) = �=K. Then, if

�p

�Cp �

�

K

�+ �S(t�)�(T ) > 0;(4.34)

where �(T ) = be��iT � 1, all the solutions of (2.1) with initial conditions (4.33)

satisfy that (S(t); I(t); P (t)) ! Ep = (0; 0; Cp) as t! +1.

Proof In Lemma 4.1 we proved that if P (t) > �=K for all t � 0 then (S(t); I(t); P (t)) !

Ep as t ! +1. Hence, it is su�cient to prove P 0(t�) > 0 if t� is the �rst time at

which P (t) assumes the �=K value. Assume t� > T . Then the second equation

(4.12) gives

P 0(t�) = �p

�Cp �

�

K

�� S(t�) + be��iTKS(t� � T )P (t� � T ):(4.35)

Since S0(t) < 0 in (0; t�], then S(0) > S(t��T ) > S(t�). Furthermore P (t��T ) >

P (t�) = �=K. Hence, from (4.35) we have

P 0(t�) > �p

�Cp �

�

K

�+ �S(t�)(be��iT � 1):(4.36)

Assume now t� 2 (0; T ]. Equation (4.12) gives:

P 0(t�) = �p

�Cp �

�

K

�� S(t�) + be��iTK'1(t

�

� T )'3(t�

� T ):

Due to the initial condition (4.33) we obtain

P 0(t�) � �p

�Cp �

�

K

�� S(t�) + be��iTK � S(0) �

�

K:


Since S0(t) < 0 in (0; t�], then S(0) > S(t�) and again we obtain (4.36). Hence, if

(4.34) holds true P (t) cannot cross the �=K value for all t � 0 and the proposition

follows.

Recall that �(T ) = be��iT �1 > 0 if T 2 [0; T �), �(T ) < 0 if T 2 (T �;+1) and

�nally �(T ) = 0 if T = T � = ��1i log b. In the following we consider some cases in

which Proposition 4.1 applies:

i) Cp � �=K and �(T ) > 0, i.e., T 2 [0; T �). Remark that as Cp = �=K the

endemic equilibrium collapses into Ep.

ii) Cp < �=K and T < T � such that

�S(t�)be��iT > �S(t�) + �p

��

K� Cp

�;

from which we have

T < ~T�=

1

�ilog

�b�S(t�)

�S(t�) + �p((�=K)� Cp)

�:(4.37)

In (4.37), of course, ~T < T � and ~T > 0 provided that S(t�) is such that (b�S(t�)=

(�S(t�) + �p((�=K)� Cp))) > 1. Recall that in this case the endemic equilibrium

E+ is feasible.

iii) Cp > �=K and �(T ) < 0. Then (4.34) becomes

�p

�Cp �

�

K

�� S(t�)j�(T )j > �p

�Cp �

�

K

�� S(0)j�(T )j;

since S(t�) < S(0). Hence if

�p

�Cp �

�

K

�� S(0)j�(T )j > 0;(4.38)

Proposition 4.1 holds true. It is easy to check that (4.38) is satis�ed if

T � < T < ~T1�=

1

�ilog

�b�S(0)

�S(0)� �p(Cp � (�=K))

�;(4.39)

and �S(0) > �p(Cp � (�=K)). Thus, we have proved:

Corollary 4.1. Assume Cp > �=K but that inequality (4.10) is reversed, i.e.

�S(0) > �p

�Cp �

�

K

�:(4.40)

Then if the latency time T satis�es (4.39), all the solutions of (2.1) with initial

conditions (4.33) converge to Ep = (0; 0; Cp) as t! +1.

As far as the global asymptotic stability of the endemic equilibrium is concerned,

we consider the standard Liapunov functional approach on the equations centered

on E+. However, to avoid many (simple but) tedious algebraic computations, we

decided to work on the dimensionless form of the equations as they appear in

Appendix A (see (A.1)). We can prove the following:


Theorem 4.4. Assume that the parameters of equations (A.1) satisfy that

a > 2=3; mi > 1=2; p� < (2a� 1)m=a:(4.41)

Then, for any incubation time T� satisfying that

T� > max

�1

mi

ln

�p�(w�2 + bw3)

(2a� 1)� p�w3;1

mi

ln

�p� + 1

2mi � 1

�;

1

mi

ln

�w�2 + bw3(p

� + 2)

w3(2m� p�)� 1

��(4.42)

the endemic equilibrium E+, if feasible, is globally asymptotically stable in R3+.

Proof We start by centering the model equations (A.1) at the endemic equilibrium

E+ = (s�; i�; p�):

u1 = s� s�; u2 = i� i�; u3 = p� p�:

8><>:

u01(�) = s(�)f�au1 � au2 � u3g

u02(�) = p�u1 �miu2 + s(�)u3 � p�e�miT�u1(� � T� )� e�miT� s(� � T� )u3(� � T� )

u03(�) = �p�u1 � (m+ s(�))u3 + bp�e�miT�u1(� � T� ) + bs(� � T� )e�miT�u3(� � T� ):

(4.43)

where E+ has been transformed into (0,0,0). Let us consider

R3�

�= fu 2 R3

j u1 > �s�; u2 � �i

�; u3 � �p�

g

on which we can de�ne V : R3�! R+0, V 2 C1(R3

�) such that:

V (u) = u1 � s� log

�u1 + s�

s�

�+

1

2(w2u

22 + w3u

23)

where wi 2 R+, i = 2; 3, are at the moment arbitrary real constants. We know

that (see Lemma 2.3) there exists a time, say �1 > 0, such that s(�) + i(�) < 1 for

all � > �1. Hence, for all � > �1 + T� we have

s(�) < 1 and s(� � T� ) < 1:

V 0(u)j(4:43) = �au21 � au1u2 � u1u3 + w2p�u1u2 � w2miu

22 + w2su2u3

�w2p�e�miT�u2u1(� � T� )� w2e

�miT� s(� � T� )u2u3(� � T� )

�w3p�u1u3 � w3(m+ s)u23 + w3bp

�e�miT�u3u1(� � T� )

+w3be�miT� s(� � T� )u3(� � T� )u3

� �au21 + [�a+ w2p�]u1u2 � [1 + p�w3]u1u3 � w2miu

22

+1

2w2su

22 +

1

2w2su

23 � w3mu

23 � w3su

23

�w2p�e�miT�u2u1(� � T� )� w2e

�miT� s(� � T� )u2u3(� � T� )

+w3bp�e�miT�u3u1(� � T� ) + w3be

�miT� s(� � T� )u3u3(� � T� ):


Choosing w2 = w�2 =ap�, we obtain that

V 0(u)j(4:43) � �

�a�

1

2�

1

2p�w3

�u21 � w�2

�mi �

1

2

�u22 �

�w3m�

1

2�

1

2p�w3

�u23

�s

�w3 �

1

2w�2

�u23

+1

2w�2p

�e�miT�u22 +1

2w�2p

�e�miT�u21(� � T� )

+1

2w�2e

�miT� s(� � T� )u22 +

1

2w�2e

�miT� s(� � T� )u23(� � T� ) (

+1

2w3bp

�e�miT�u23 +1

2w3bp

�e�miT�u21(� � T� )

+1

2w3be

�miT� s(� � T� )u23 +

1

2w3be

�miT� s(� � T� )u23(� � T� ):

Assume now that a > 12, mi >

12, m > 1

2p�. We choose

max

�a

2p�;

1

2m� p�

�< w3 <

2a� 1

p�

i.e. we must assume

max

�a

2p�;

1

2m� p�

�<

2a� 1

p�:

This is possible provided that:

(H): a > 23; mi >

12; p� < (2a� 1)m=a:

For su�ciently large time, e.g., � > �1 + T� we have:

V 0(u)j(4:43) � �

��a�

1

2

��

1

2p�w3

�u21 �

a

p�

�mi �

1

2

�u22 �

�w3

�2m� p�

2

��

1

2

�u23

+1

2(p� + 1)w�2e

�miT�u22 +1

2(p� + 1)w3be

�miT�u23 (4:48)

+1

2p�(w�2 + bw3)e

�miT�u21(� � T� )

+1

2(w�2 + bw3)e

�miT�u23(� � T� )

Choose the Liapunov functional

U(ut) = V (u)+1

2p�(w�2+bw3)e

�miT�

Z �

��T�

u21(�)d�+1

2(w�2+bw3)e

�miT�

Z �

��T�

u23(�)d�:)

U 0(ut)��(4:43)

= V 0(u)��(1)

+1

2p�(w�2 + bw3)e

�miT�u21(�)

�

1

2p�(w�2 + bw3)e

�miT�u21(� � T� ) +1

2(w�2 + bw3)e

�miT�u23(�)

�

1

2(w�2 + bw3)e

�miT�u23(� � T� ):


Hence, we have

U 0(ut)��(4:43)

� �

��a�

1

2

��

1

2p�w3

�u21 �

a

p�

�mi �

1

2

�u22 �

1

2[w3(2m� p�)� 1]u23

+1

2p�(w�2 + bw3)e

�miT�u21(�) +1

2(p� + 1)w�2e

�miT�u22(�)

+1

2(w�2 + bw3(p

� + 2))e�miT�u23(�)

= �

1

2f[(2a� 1)� p�w3]� p�(w�2 + bw3)e

�miT�gu21(�)

�

1

2

a

p�f(2mi � 1)� (p� + 1)e�miT�

gu22(�)

�

1

2f[w3(2m� p�)� 1]� (w�2 + bw3(p

� + 2))e�miT�gu23(�)

which is negative de�nite provided that:

T� > max

�1

mi

log

�p�(w�2 + bw3)

(2a� 1)� p�w3

�;1

mi

log

�p� + 1

2mi � 1

�;

1

mi

log

�w�2 + bw3(p

� + 2)

w3(2m� p�)� 1

��This completes the proof.

5. Permanence Results

We say system (2.1) with initial condition (2.2) is permanent (or uniformly per-

sistent) if there exist positive constants, independent of initial conditions, m;M ,

m <M , such that for solutions of (2.1){(2.2), we have

minflim inft!+1

S(t); lim inft!+1

I(t); lim inft!+1

P (t)g � m(5.1)

and

maxflim supt!+1

S(t); lim supt!+1

I(t); lim supt!+1

P (t)g �M:(5.2)

In view of the fact that S(t) and P (t) are eventually bounded, so must be I(t)

(since (1.5)). Also, we see that if there is an m1 > 0 such that

minflim inft!+1

S(t); lim inft!+1

P (t)g > m1;(5.3)

then, by (1.5), we have

lim inft!+1

I(t) � Km21(1� e��iT )=�i

�= m2:(5.4)

If �i = 0, then m2 = KTm21. Hence (5.1) holds for m = minfm1;m2g. Therefore,

to establish permanence for (2.1){(2.2), we need only to �nd m1 > 0, such that

(5.3) holds.

We consider �rst the case when � > 0. From (4.22), we see that

lim inft!+1

P (t) � �=(�p +KC):(5.5)


Thus we need only to show that there is an m > 0, such that

lim inft!+1

S(t) > m:(5.6)

We shall prove the following:

Theorem 5.1. If � > 0 and Cp < �=K, then (2.1) with (2.2) is permanent.

To prove the above, we need a few preparations. Lemma 2.3, Lemma 2.5 and

Theorem 2.2 imply that for any L > L1, there is a t1 > 0 such that, for t � t1,

S(t) < C and P (t) < L:(5.7)

Let l 2 (0; �=(�p+KC)). Due to (5.5), and for convenience, we assume below that

for t � t2 > t1 + T ,

P (t) � l:(5.8)

Lemma 5.1. Assume t4 � T > t3 > t2 and for t 2 [t3; t4], S(t) < �1. Then for

t 2 [t3 + T; t4],

P (t) �

�L�

� + be��iTL�1

�p

�e��p(t�t3�T ) +

� + be��iTL�1

�p

�= f1(t; �1)(5.9)

and

P (t) �

�l�

�

�p +K�1

�e�(�p+K�1)(t�t3�T ) +

�

�p +K�1

�= f2(t; �1):(5.10)

Proof Equation (5.9) follows directly from the fact that for t 2 [t3 + T; t4 + T ],

P (t) < L, and

P 0(t) � � + be��iTKL�1 � �pP

and (5.10) follows directly from the fact that for t 2 [t3; t4], P (t) > l and

P 0(t) � � � �pP �K�1P:

It is clear that

lim�1!0

t!+1

f1(t; �1) = lim�1!0

t!+1

f2(t; �1) =�

�p= Cp;

uniformly. Assume Cp � �=K and let

" =1

4

��

K� Cp

�:(5.11)

We de�ne

g(�i) =

(T �i = 0

��1i (1� e��iT ) �i > 0:

Let �1 = �1(") > 0 such that �1 < 1 and

�1[1 +KLg(�i)] � "KC��1(5.12)


max

��Cp � � + be��iTL�1

�p

��;��Cp � �

�p +K�1

��< "(5.13)

and T1 = T1(") > 0, such that

max

��L� � + be��iTL�1

�p

��e��pT1 ;��l� �

�p +K�1

��e�(�p+K�1)T1

�< ":(5.14)

Hence, if t4 > t3 + T + T1, then for t 2 [t3 + T + T1; t4],

maxfjf1(t; �1)� Cpj; jf2(t; �1)� Cpjg < 2":(5.15)

In which case, this together with (5.12), implying that for t 2 [t3 + T + T1; t4],

maxfjf1(t; �1)j; jf2(t; �1)jg <�

K� 2":(5.16)

Now we are ready to present the proof of Theorem 5.1.

Proof of Theorem 5.1. Let " be de�ned by (5.12) and �2 = �1e�KL(2T+T1). We

claim that

lim inft!+1

S(t) � �2:(5.17)

If not, there exists a t�> T + T1(") + t2, such that

S(t�) < �2 and S0(t

�) � 0:(5.18)

Note that for t > t2,

S0(t) � �KSP � �KLS

which implies that for t > t0 > t2

S(t) � S(t0)e�KL(t�t0)

and hence

S(t0) � S(t)eKL(t�t0):(5.19)

Therefore for t 2 [t�� T � T1; t�],

S(t) � �2eKL(T+T1) = �1e

�KLT < �1:(5.20)

Applying Lemma 5.1 with t4 = t�, t3 = t

�� T � T1, then we have for

P (t�) � f1(t�; �1) < Cp + 2":(5.21)

However, S0(t�) � 0 implies that

0 � S(t�)

�K

��

K� P (t

�)

��

�[S(t�) + I(t

�))

C

�� S(t

�)[2K"� �C�1(�1 + I(t

�))]:(5.22)

Note that

I(t�) � K

Z t�

t��T

e��i(t��)�1Ld�:


For �i = 0, we have I(t�) � KTL�1 and for �i > 0, we have I(t

�) � ��1i KL(1�

e��iT )�1. Hence we have

I(t�) � KLg(�i)�1:

Therefore (5.22) implies that

0 � S(t�)[2K"� �C�1(1 +KLg(�i))�1]

� S(t�)[2K"�K"] = K"S(t

�);

a contradiction. Let m = �2. Then (5.6) holds and hence the theorem.

The argument above in fact shows that we can (with quite some computation)

�nd an explicit expression (in terms of the parameters of (2.1)) for m in (5.1). This

can be very useful in practice. To save space, we choose not to do it here.

When � = 0 and �i > 0, the persistence or permanence question becomes

very challenging. The di�culty is that even though we know that T < Tc (=

��1i (ln(b=b�)) implies the instability of Ef , we cannot locate all the eigenvectors

of the in�nitely many eigenvalues with negative real part (the root of g(�) = 0 of

(3.12)). Other methods (e.g., those used in [7], [8] and [15]) are equally di�cult to

implement.

6. Discussion

In this paper we proposed a delay di�erential equation model which describes

the bacteriophage infection of marine bacteria in the thermoclinic level of the sea

during the warm season; the experimental evidence has been reported by several

authors (e.g., by Proctor and Fuhrman (1990), see [12]). We have to describe

the time evolution of three population densities (assumed spatially homogeneous),

namely the susceptible bacteria S(t), the phage-infected bacteria I(t), and the

infecting agent: the phages P (t). The justi�cation for the equations is given in the

introductory Section 1. Here we recall that one goal of the authors in this paper is

that of providing a better description of the infected class of bacteria with respect

to a previous mathematical model (see [4]), in which the infection was described

by a system of three nonlinear odes. Here by T we denote the \average latency

time," In the previous model the time evolution of the density of infected bacteria

was described by the o.d.e.

dI(t)

dt= KS(t)P (t)� �I(t)(6.1)

where \�," the lysis death constant rate of infected bacteria was just assumed to

be � = 1=T . In the present model (see Section 1 for its justi�cation) we substitute

the above ode with the delay di�erential equation

dI(t)

dt= ��iI(t) +KS(t)P (t)� e��iTKS(t� T )P (t� T )(6.2)


or with its integral form (see (1.6))

I(t) = K

Z t

t�T

e��i(t��)S(�)P (�) d�:(6.3)

where \�i" is a mortality rate constant, which accounts for possible extra-mortality

besides lysis of infected bacteria. However, attention must be paid to the equiva-

lence between (6.2) and (6.3), since any function

I(t) = C +K

Z t

t�T

e��i(t��)S(�)P (�)d�(6.4)

where C 2 R is a constant, of course satis�es (6.2). However, when S(t) ! S�,

P (t) ! P � as t ! +1, then I(t) does not converge to I� = (k�(T )=�i)S�P � as

t! +1 but to the value I� + C. Since at t = 0

C = I(0)�K

Z 0

�T

e�i�'1(�)'3(�)d�;(6.5)

we see that if we apply initial conditions (2.2), then necessarily C = 0. Of course,

all the mathematical results obtained in this paper assume initial conditions (2.2)

for which C = 0 in (6.4) and (6.5). We still have to remark that the mathematical

analysis of this model has been performed with �i 2 R+, i.e., �i > 0. The case

�i = 0 is in fact to be studied with care since the equivalence between di�erential

form (6.2) for I and its integral form (6.3) is lost. Assume �i = 0. Equation (6.2)

gives

dI

dt= KS(t)P (t)�KS(t� T )P (t� T )(6.6)

whereas from (6.3) we obtain

I(t) = K

Z t

t�T

S(�)P (�)d�:(6.7)

For instance, at the equilibrium in which S(t) = S�, P (t) = P � for all t 2 R, (6.6)

gives I� = �, � an arbitrary real constant, whereas (6.7) provides I� = KTS�P �.

Accordingly, the equation for phages has been changed from

dP (t)

dt= ��pP (t)�KS(t)P (t) + b�I(t)(6.8)

of the previous model into

dP (t)

dt= � � �pP (t)�KS(t)P (t) + be��iTKS(t� T )P (t� T )(6.9)

where � is a possible constant supply for phages from the surrounding environment.

An issue of this discussion is the comparison of the outcomes of this model with

the previous one. The comparison is meaningful only if we assume in the present

model � = 0 and �i > 0 but close to zero.

The model in [4] is such that two equilibria are always feasible, but one E0 =

(0; 0; 0) is unstable and the other Ef = (1; 0; 0) (at which the infection dies out) is

globally asymptotically stable whenever the endemic equilibrium E+ is not feasible.


However, as the \phage multiplication factor" b satis�es that b � b� = m + 1 '

15:925 the endemic equilibrium E+ bifurcates from Ef (E+ � Ef as b = b�) and

becomes the one asymptotically stable as b > b�, whereas Ef = (1; 0; 0) becomes a

uniform strong repeller in a suitable subseto

ofR3+ as b > b� (for the nomenclature,

see Thieme (1993) in [15]). For a further increase of b a bifurcation value b0 ' 95

exists in [4] after which E+ is unstable and there exist orbitally asymptotically

stable periodic solutions. For the parameters of the present model, we refer to the

dimensionless equations (A.1) where dimensionless parameters have been estimated

according to [4] in (A.3), (A.4). Thus we consider the case in which Cp = 0 and

T� = 1=` ' 0:0406, whereas we choose as the mi value mi = 0:1. The endemic

equilibrium is feasible if b > b� = m + 1 = 15:925 and T� < Tc = m�1i log(b=b�).

Hence, a �rst di�erence with the model in [4] is that the endemic equilibrium

bifurcates from Ef = (1; 0; 0) not at b = b� but at a value \b�emiT� " which ensures

that condition T� < Tc is satis�ed: see Fig. 6.1 with T� = 0:0406 and increasing b.

Hence, in contrast to [4], if b 2 (b�; bc) still E+ is not feasible and Ef is the

asymptotically stable equilibrium (E0 = (0; 0; 0) is always an unstable saddle point).

We remark that the analysis by the characteristic equation shows that Ef is locally

asymptotically stable if T� > Tc, unstable if T� < Tc and critically stable at T� = Tc

(see Proposition 3.2). Furthermore, Ef is globally asymptotically stable in R3+ if

b 2 (1; b�] (see Corollary ?? for b < b� and Theorem 4.1 for b � b�), otherwise if b >

b� the global asymptotic stability of Ef follows if T� > T1�= m�1i log[(b�1)=(b��1)]

(see Theorem 4.2). We remark that T1 > Tc; thus the global behavior of Ef as

T 2 (Tc; T1] is an open problem. For b > bc = 15:989, the condition T� < Tc

is satis�ed, e.g., if b = 16 then Tc �= 0:0469 and T� = 0:0406) and the endemic

equilibrium E+ is feasible. The only result that we have (see Theorem 4.4) about

the global stability of E+ in R3+ requires that mi > 0:5 and therefore it cannot be

applied to the present case (mi = 0:1). The situation of known stability results is

reported in Fig. 6.1 below.

Fig. 6.1

However we could have an insight about the possible stability behavior of the

endemic equilibrium E+ considering equations very close to our model equations

(1.11): the Campbell-like model with mi > 0 (see (1.16), (1.17)) and the Campbell

model with mi = 0 (see (1.13), (1.14)). Let us �rst consider the Campbell-like

model. The equilibria have the same behavior of that depicted in Fig. 6.1. Partic-

ularly the endemic equilibrium E+ is feasible if b > b� and T� < Tc. Theorem 3.1


states that whenever b � bc =b��(1��1)

�1then E+ is locally asymptotically stable.

As b > bc a stability shift toward stable oscillations might arise, but still we cannot

prove it since the coe�cients of characteristic equation (3.12) are explicitly depen-

dent upon the latency time T� . In other words, for b > bc and �xed T� there exists

T0 such that for T� > T0 a stability shift has occurred. However, it is a challenging

mathematical problem to prove that T� > T0 (or T� < T0). Computer simulations

at �xed T� = 0:0406 and b = 16, 17, 20, 29 show a global asymptotic behavior of

E+ (e.g. Figure 6.3) in R3+. Furthermore, increasing \b" a bifurcation value for b

exists, say bc ' 72, such that at \bc" E+ loses its stability and as b > bc the simu-

lations show the existence of stable periodic solutions around E+ (e.g., Figure 6.5

with b = 75), whereas at b < bc the oscillations around E+ are damped and still

the solutions converge to E+ as t ! +1 (see Figure 6.4 with b = 70). Hence, at

least for small T� values (T� = 0:0406) there is a good agreement (suggested by the

computer simulations when b > bc) between the present and our previous model.

However, for increasing T� this agreement is lost. For example, assume b �xed with

b > bc and to increase T� up to Tc. Since at Tc the endemic equilibrium coin-

cides with Ef which is asymptotically stable, the stable oscillations should become

damped as T� approaches Tc (see Figure 6.6) and the solutions tend to Ef when

Tc is exceeded (Figure 6.7). This result is also in agreement with Theorem (4.4) in

the case of Cp = 0, mi > 0:5 and the critical latency time on the right of (4.42)

less than Tc. In fact the theorem implies global stability of E+. A further remark

concerns the Campbell-like model. It is not di�cult to check that all results proved

for the present model (in the case Cp = 0) still hold true for the Campbell-like

model. Hence, Fig. 6.1 depicts the stability properties of Campbell-like model's

equilibria where, whenever E+ is feasible, then it is asymptotically stable if b � bc.

What happens for E+ when it is feasible and b > bc is an open problem.

The agreement of our previous model [4] is closer with the Campbell model

(mi = 0). In fact, as in our �rst model, the endemic equilibrium E+ is feasible

whenever b > b� without any constraint on latency time T� . Furthermore, Theorem

3.1 (ii) states that a bc = (b� � (1 � �1))=�1 exists such that E+ is asymptotically

stable if b 2 (b�; bc), whereas for any b > bc there exists a critical value T0 (see

(3.17), (3.18)) such that undamped oscillations occur for any T� > T0 without any

further constraint on T� .

Another feature of the model [4] concerns the close relation

I% =i�

s� + i�=

a

l + a

b� b�

b� 1(6.10)

between the number \b" of phages inside the bacteria (just before lysis) and the

percentage of infected bacteria on the whole bacteria population. Equation (6.10)

shows that, in any case, such a percentage cannot overcome the threshold value

I1%

�=

a

l + a' 29%


(according to the estimates in (A.3)).

How large can the percentage of the infected bacteria at endemic equilibrium be

in the present model with Cp = 0 and mi > 0? An inspection of the equilibrium

components of E+ in (B.3) shows that

I% =�(T� )a

mi + �(T� )a

�be�miT�

� b�

be�miT� � 1

�(6.11)

where �(T� ) = 1�e�miT� and T� < Tc = m�1i log(b=b�). For a �xed T� assume that

b % +1. Hence Tc % +1 and limb!+1 I% = �(T� )a=(mi + �(T� )a). This is an

increasing function of �(T� ), which however cannot exceed 1. Thus, the estimates

in Appendix A give:

I1% =

a

mi + a= 99%;

assuming mi = 0:1. Therefore, depending on b;mi; T� , I% can assume almost all

the values between [0; 100%]. Let us now recall that, in model [4] the threshold

condition b > b� = m + 1, m = �p=�c for existence of the endemic equilibrium

E+ can be viewed as a threshold on the \basic reproduction number" R0 of epi-

demic theory (e.g. [1]), i.e., R0 is the \number of new infections that an infectious

individual makes, on the average, in a susceptible population." It is shown in [4]

that

R0 = bKC

�p +KC=

b

m+ 1;(6.12)

i.e., R0 = b=b�, gives the number of new phages per phage. Thus the threshold

R0 = b=b� > 1 in [4] simply states that the phage-population can only survive if

a phage, on the average, produces at least one new phage during its life when the

bacteria are kept at their carrying capacity. Let us now consider the present case

in which Cp = 0 and mi > 0. The di�erence with respect to the previous case is

that, due to mi > 0, during the latency period \T�" not all the bacteria infected

at \� � T�" arrive to the lysis at time \� ," i.e.., only the fraction \e�miT� " of the

infected bacteria is capable of arriving at the lysis. Thus the basic reproduction

number R0 which gives the number of new phages per phage has to be changed

into:

R0 =be�miT�

m+ 1=be�miT�

b�;(6.13)

and the endemic equilibrium E+ is therefore feasible if R0 > 1, or equivalently if

T� < Tc =1

mi

logb

b�(6.14)

where of course it must be b > b�. The same threshold result can be obtained by

the positivity requirement of E+ components (see Appendix B, (B.3)).

Let us comment now on the case � > 0 (or Cp > 0). In this case, the phages being

supplied from the environment, the basic reproduction number R0 loses its meaning


and the phage population cannot ever go extinct. In particular, if �=�p > �=K, or

equivalently,

Cp > aC;(6.15)

(i.e., the number of new phages per unit time exceeds the maximum number of

new susceptible bacteria per unit time when the bacteria are kept at the carrying

capacity) then the endemic equilibrium E+ is not feasible and the unique equilib-

rium is the boundary one Ep = (0; 0; Cp=C) (in dimensionless units) at which both

bacteria populations are extinct. On the contrary, if inequality (6.15) is reversed,

the endemic equilibrium E+ becomes feasible for all the values of parameters b and

T� . Unless for the case T� = T ��= m�1i log b (see (2.34)), as can be argued by

the discussion in Section 2, it is hard to provide the explicit analytic form for the

equilibrium components. Depending on T� we can only say that p� 2 ((Cp=C); a)

if T� < T �, p� = (Cp=C) at T� = T � and p� 2 (0; (Cp=C)) if T� > T �. In all these

cases s� 2 (0; 1) and i� = �(T� )m�1i s�p�.

In Proposition 3.1 we have shown that Ep is locally asymptotically stable if

Cp > Ca, it is critically stable when Cp = Ca (and E+ coincides with Ep) and

becomes an unstable saddle point if Ca > Cp. In Lemma 4.1, Proposition 4.1 and

Corollary 4.1 we have proven attractivity results for the equilibrium Ep, attractivity

that might still hold even if E+ is feasible (see point (ii) after Proposition 4.1). The

global stability results of Ep in R3+ are given in Theorem 4.3. We proved that if

Cp > aCb�

b� � 1(6.16)

then Ep is globally asymptotically stable in R3+. If Cp < aCb�=(b�� 1), then Ep is

still globally asymptotically stable in R3+ if

Cp > aC

�1 +

1

b� � 1

�1�

Cp(b�

� 1)

aCb�

��:(6.17)

Finally, in Theorem 5.1 we proved that, as long as the equilibrium E+ exists (i.e.,

aC > Cp), then system (2.1) with initial conditions (2.2) is permanent (see (5.1),

(5.2)). That is, whenever the endemic equilibrium exists, both the bacteria popu-

lations cannot go to extinction. The result in Theorem 4.3 that proves the global

asymptotic stability of E+ with respect to R3+ provided that T� is larger than a

given threshold (4.43) and that parameters satisfy (4.41), seems more suitable for

the case in which Cp > 0 than for the case Cp = 0, in which the feasibility of E+

requires T� < Tc.

APPENDIX

A. Dimensionless form of the equations

We choose the dimensionless time as: � = KCt. Note that one unit of the

dimensionless time scale, i.e., � = 1 corresponds to t� = (1=KC). We also need


these dimensionless variables:

s =S

c; p =

P

C; i =

I

C:

These are the dimensionless parameters:

a =�

KC; m =

�p

KC; mi =

�i

KCand

�

KC= m

Cp

Cwhere Cp =

�

�p:

The equations (2.1) have the dimensionless form8>>>><>>>>:

ds(�)

d�= as(�) � as(�)(s(�) + i(�))� s(�)p(�)

di(�)

d�= �mii(�) + s(�)p(�) � e�miT� s(� � T� )p(� � T� )

dp(�)

d�= m

Cp

C�mp(�)� s(�)p(�) + be�miT� s(� � T� )p(� � T� )

(A.1)

We also have

i(�) =

Z �

��T�

e�mi(��)s(�)p(�)d�:(A.2)

The values for the dimensionless parameters and the dimensionless time scale are

taken from the previous model (see [4]) of the authors and the estimates are due

to Prof. A. Okubo. In that model the lysis death constant rate \�" was assumed

to be the reciprocal of the average incubation time \T" and its dimensionless form

was denoted by ` = (�=KC) = (1=KCT ). Hence T� = (1=`). The estimates in

were:

a = 10; ` = 24:628; m = 14:925;(A.3)

with t� = (1=KC) = 7:4627 days. Since b� = (�p=KC) + 1 we get

b� = m+ 1 = 15:925; T� = `�1 ' 0:0406(A.4)

corresponding to an average latency time T ' 0:303 days. We have no estimates

for mi = (�i=KC) but it seems reasonable to assume it is smaller than m (since

the main cause of mortality is the lysis of infected cells). We assume mi ' 0:1m.

B. Local stability analysis of the equilibria

Proof of Proposition 3.2. Consider the equilibrium Ef = (C; 0; 0). At Ef the

characteristic equation (3.6) becomes:��(a+�) �a �KC

0 �(�i +�) KC(1� e�(�i+�)T )

0 0 �(�p +KC �KCbe�(�i+�)T +�)

�� = 0:(B.1)

Clearly, two characteristic roots are real and negative, say

�1 = ��; �2 = ��i(B.2)

and the others are the roots of

g(�) := �p +KC � bKCe��iT e��T +� = 0:(B.3)


� = 0 is a root of (B.3) if T is such that �p +KC = bKCe��iT ; i.e., if T = Tc

where Tc = (1=�i) log(b=b�), b� = (�p=KC)+1. In this case, g(�) = a�ae��T +�

where a�= �p + KC. If � = u + iv is a root of g(�) = 0, then one must have

u � 0, since (a+u)2+ v2 = a2e�2uT : This proves that as T = Tc the characteristic

equation (B.1) has no root with positive real part. Note that g0(0) = 1 + aTc > 0.

Hence the root �3 = 0 is simple. Therefore, at T = Tc the equilibrium Ef is linearly

neutrally stable. Assume now that T > Tc, i.e.:

�p +KC > bKCe��iT :(B.4)

Then g(�) = 0 implies that

�p +KC +� = bKCe��iT e��T :(B.5)

If Re � � 0, then

j�p +KC +�j � j�p +KCj > bKCe��iT je��T j:

This shows that all roots of g(�) = 0 must have negative real parts. Therefore, if

T > Tc, then Ef is locally asymptotically stable.

Assume now that T < Tc, i.e.,

�p +KC < bKCe��iT :(B.6)

Then we see that g(0) < 0 and g(+1) = +1. Hence g(�) = 0 has at least one

positive characteristic root. Therefore, Ef is unstable when T < Tc.

Proof of Theorem 3.1.

(i) Remember that �(T ) = b��1be��iT�1

is an increasing function of T . If at T = 0,

�(0) > �1 (i.e. b 2 (b�; bc)) then �(T ) > �1 for all T 2 [0; Tc], i.e., according

to Lemma 3.1 no characteristic roots � = �i!, ! > 0 can occur.

(ii) If we assume �i = 0 the positive equilibrium components are independent

of T . Therefore, the coe�cients of characteristic equation (3.12) are also

independent of T . If b 2 (b�; bc] then � > �1, i.e., no characteristic roots

� = �i!, ! > 0 can occur. If b > bc then � < �1 and we have characteristic

roots � = �i!+, !+ > 0 at any T value given by

Tn =�1

!++

2n�

!+; n 2 N [ f0g

where �1 satis�es (3.18). The stability switch occurs at the smallest Tn value,

i.e., T0 = �1=!+.

References

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Biosc., 149:57{76 (1998).

[5] H.J. Bremermann, Parasites at the origin of life, J. Math. Biol., 16:165{180 (1983).

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delays, Canadian Appl. Math. Quart., 6:321{354 (1998).

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Figure Legends

Figure 6.1. Cp = 0. The �gure describes the dynamic behavior of the model in

the parameter space (b; T�) 2 (1;+1) � R+0, where T� is the non-dimensional

latency time. The endemic equilibrium E+ is feasible in the region between the

curve Tc and the \b" axis. In the �gure GAS means \global asymptotic stability."

In the dashed region between Tc; T1, Ef is LAS (locally asymptotically stable) (see

Proposition 3.2). E0 is always unstable.

Figure 6.2. � = 0, b = 15, s0 = 0:3, i0 = 0:0609, p0 = 5, a = 10, T = 0:0406,

m = 14:925, mi = 0:1. Top curve depicts s(t), middle one depicts p(t) and the

bottom one depicts i(t). Clearly, solution approaches the boundary steady state

(1; 0; 0).

Figure 6.3. � = 0, b = 17, s0 = 0:3, i0 = 0:0609, p0 = 5, a = 10, T = 0:0406,

m = 14:925, mi = 0:1. Again, top curve depicts s(t), middle one depicts p(t) and

the bottom one depicts i(t). Clearly, the solution approaches E+.

Figure 6.4. � = 0, b = 70, s0 = 0:3, i0 = 0:0609, p0 = 5, a = 10, T = 0:0406,

m = 14:925, mi = 0:1. Top curve is p(t), middle one is s(t) and the bottom one

depicts i(t). Clearly, the solution oscillates and approaches E+.

Figure 6.5. � = 0, b = 75, s0 = 0:3, i0 = 0:0609, p0 = 5, a = 10, T = 0:0406,


depicts i(t). Clearly, the solution approaches a positive periodic solution.

Figure 6.6. � = 0, b = 75, s0 = 0:3, i0 = 0:0609, p0 = 5, a = 10, T = 0:35,


depicts i(t). Clearly, the positive periodic solution seen from previous �gure disap-

peared and the solution approaches E+.

Figure 6.7. � = 0, b = 75, s0 = 0:3, i0 = 0:0609, p0 = 5, a = 10, T = 16 > Tc,

m = 14:925, mi = 0:1. Top curve is s(t), middle one is i(t) and the bottom one

depicts p(t). Clearly, the positive steady state E+ is no longer feasible and the

solution approaches Ef .

1 20 40 60 80 1000

5

10

15

20

25

30

.0406 T

T

b b emi T bc

b

Ef is

GAS for

b b

(Theor 4.1)

Ef is

GAS for

b b and T T1

(Theor 4.2)

T1

Tc

E is feasible

Ef is unstable

(Proposition 3.2)

00.2

0.4

0.6

0.81s,i,p 0

2040

6080

100

t

0246810s,i,p 0

510

1520

2530

3540

4550

t

024681012s,i,p 0

510

1520

2530

3540

4550

t

012345678s,i,p 0

2040

6080

100

t

00.2

0.4

0.6

0.81s,i,p 0

2040

6080

100

t

modeling - arizona state universitykuang/paper/bk99.pdf · 2003. 2. 2. · modeling and anal ysis...

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