34
The Pharmacodynamics of Antibiotic Treatment Mudassar Imran and Hal Smith * Department of Mathematics and Statistics Arizona State University Tempe, AZ 85287 November 14, 2006 Abstract We derive models of the effects of periodic, discrete dosing or constant dosing of antibiotics on a bacterial population whose growth is checked by nutrient-limitation and possibly by host defenses. Mathematically rigorous results providing sufficient conditions for treatment success, i.e., the elimination of the bacteria, as well as for treatment failure, are obtained. Our models can exhibit bi-stability where the infection- free state and an infection-state are locally stable when antibiotic dosing is marginal. In this case, treatment success may occur only for sub-threshold level infections. Keywords: pharmacokinetics, pharmacodynamics, antibiotic, infection, plasmid. 1 Introduction The pharmacology of antibiotics (or any drug) can be divided into pharmacokinetics and pharmacodynamics. Pharmacokinetics describes the movement of antibiotic into, through and out of the body whereas pharmacodynamics describes the relationship between the con- centration of antibiotic, its effect on target bacteria (growth or decay) and factors influencing this relationship [7]. The elimination of drug either by metabolism or excretion is very im- portant since it determines the dosing frequency [1]. One of the important objectives of pharmacokinetics is to decide the optimal dosing frequency of an antibiotic for successful treatment. On the other hand pharmacodynamics describes in detail the relationship be- tween drug concentration and its effects on the bacterial population in order to achieve the maximum removal of bacteria from the host. Mathematical modeling of the effects of drug treatment has long been used side-by-side with experimental studies [1, 3, 4, 5, 26, 28, 40] However, most mathematical models of the effects of antimicrobial agents on bacterial populations assume that bacteria grow at an exponential rate in the absence of the antimicrobial agent. In that case, one need only * Supported by NSF Grant DMS 0107160 1

The Pharmacodynamics of Antibiotic Treatmenthalsmith/antibiotic-treatment-new.pdf · The Pharmacodynamics of Antibiotic Treatment ... in which antibiotic levels have attained their

Embed Size (px)

Citation preview

Page 1: The Pharmacodynamics of Antibiotic Treatmenthalsmith/antibiotic-treatment-new.pdf · The Pharmacodynamics of Antibiotic Treatment ... in which antibiotic levels have attained their

The Pharmacodynamics of Antibiotic Treatment

Mudassar Imran and Hal Smith∗

Department of Mathematics and StatisticsArizona State University

Tempe, AZ 85287

November 14, 2006

Abstract

We derive models of the effects of periodic, discrete dosing or constant dosing ofantibiotics on a bacterial population whose growth is checked by nutrient-limitationand possibly by host defenses. Mathematically rigorous results providing sufficientconditions for treatment success, i.e., the elimination of the bacteria, as well as fortreatment failure, are obtained. Our models can exhibit bi-stability where the infection-free state and an infection-state are locally stable when antibiotic dosing is marginal.In this case, treatment success may occur only for sub-threshold level infections.

Keywords: pharmacokinetics, pharmacodynamics, antibiotic, infection, plasmid.

1 Introduction

The pharmacology of antibiotics (or any drug) can be divided into pharmacokinetics andpharmacodynamics. Pharmacokinetics describes the movement of antibiotic into, throughand out of the body whereas pharmacodynamics describes the relationship between the con-centration of antibiotic, its effect on target bacteria (growth or decay) and factors influencingthis relationship [7]. The elimination of drug either by metabolism or excretion is very im-portant since it determines the dosing frequency [1]. One of the important objectives ofpharmacokinetics is to decide the optimal dosing frequency of an antibiotic for successfultreatment. On the other hand pharmacodynamics describes in detail the relationship be-tween drug concentration and its effects on the bacterial population in order to achieve themaximum removal of bacteria from the host.

Mathematical modeling of the effects of drug treatment has long been used side-by-sidewith experimental studies [1, 3, 4, 5, 26, 28, 40] However, most mathematical models ofthe effects of antimicrobial agents on bacterial populations assume that bacteria grow atan exponential rate in the absence of the antimicrobial agent. In that case, one need only

∗Supported by NSF Grant DMS 0107160

1

Page 2: The Pharmacodynamics of Antibiotic Treatmenthalsmith/antibiotic-treatment-new.pdf · The Pharmacodynamics of Antibiotic Treatment ... in which antibiotic levels have attained their

determine the pharmacodynamic function for the agent, that is, a mathematical expressionfor the decline in growth rate resulting from a given concentration of the antimicrobialagent. Once this has been achieved, then the Minimum Inhibitory Concentration (MIC)for the agent is determined as that concentration of the agent at which the growth rate,ψ = ψ(A) vanishes, where A denotes the antibiotic concentration. One can then explore theresponse of bacteria to various dosing strategies. The expected outcome of this approach isthat a dosing strategy that is insufficient to control the population results in its exponentialgrowth while an adequate dosing drives the population to extinction.

It is often found that fast growing bacteria are more susceptible to antibiotic treatmentas compared to slow growing ones [3, 6, 22, 27, 29, 38, 39]. A slow growth rate due tothe restricted availability of nutrients could be a major contributor toward insensitivity ofbacteria to antibiotic. Therefore it is plausible that in some cases the pharmacodynamicfunction should depend on both the antimicrobial agent and limiting nutrient levels, i.e.,ψ = ψ(S, A) where S denotes limiting resource level. It has been noted that bacteria multiplymore slowly in an experimental animal than in vitro [4, 6] suggesting nutrient limitation invivo. Corpet et al [4] introduce pharmacodynamic functions which depend on both limitingnutrient and antimicrobial agent. Cogan [3] does as well in his considerations of persistercells. Cozens et al [6] note that the restricted availability of iron and other nutrients appearto be typical of infection states. Roberts and Stewart [29] construct a mathematical modelto explore the possibility that the observed antibiotic tolerance of biofilms is due in partto nutrient limitation reducing bacterial growth and hence killing rates. Even if resourcesupply rates are relatively constant, one expects significant depletion in local resource levelsas a bacterial infection progresses and we expect these changes to play a role in treatmentby antimicrobial agents.

We derive simple models of the effects of periodically-administered discrete dosing orconstant antimicrobial dosing strategies on a bacterial population whose growth is checkedby nutrient-limitation and possibly by host defenses if not by the antimicrobial agent itself.Distinguishing features of our model are the inclusion of nutrient limitation of microbialgrowth and accounting for removal of antimicrobial agent by association with bacteria. Ear-lier models such as Austin et al [1] mention nutrient limitation but employ a logistic bacterialgrowth rate rather than explicitly treating nutrient limitation. Like the model of Austin etal, ours follows the pre-treatment infection dynamics as well as the post-treatment course ofbacterial levels.

Most models of antibiotic treatment, e.g. [1], assume that the pathogen has no effect onantimicrobial concentration; it’s concentration at the site of infection is simply an input tothe model. In our model, the antibiotic concentration at the site of infection is a dynamicvariable with the periodic dosing as input. Standard pharmacokinetics are used but weinclude a bacteria-dependent drug removal rate, attributable to the association of the drugwith bacterial cells, which may be important in some cases.

Mathematically rigorous results are obtained for our models that provide sufficient con-ditions for treatment success-the elimination of the bacteria, as well as for treatment failure.In the case that antibiotic concentration is assumed to be oscillatory, the MIC, which maydepend on resource levels, is no longer the critical parameter. Rather the key parameteris an “invasion eigenvalue” which determines whether bacteria can invade an environmentin which antibiotic levels have attained their asymptotic periodic pharmacokinetic regime

2

Page 3: The Pharmacodynamics of Antibiotic Treatmenthalsmith/antibiotic-treatment-new.pdf · The Pharmacodynamics of Antibiotic Treatment ... in which antibiotic levels have attained their

A∗(t): typically, a repeating cycle of exponential decay and recovery following a discretedose. See Figure 1. We refer to this regime as the APPR for simplicity. The “invasioneigenvalue”, in fact a Floquet exponent, is the average growth rate over a dose-cycle T :

λ =1

T

∫ T

0

ψ(S0, A∗(t))dt

where S0 is the resource level in the absence of bacteria and A∗(t) = A∗(t+T ) is the APPR.Although it is often remarked that the amount of time during which A∗(t) exceeds the MICis critical, λ is a much more subtle parameter. The generally nonlinear dependence of ψ onA, means that λ can differ substantially from the growth rate evaluated at the average doselevel

λ 6= ψ(S0, [A∗]m), [A∗]m =1

T

∫ T

0

A∗(t)dt

In vivo, bacteria usually become established before intervention in the form of antibiotictreatment is administered rather than the other way around where a small inoculum ofbacteria is introduced to a pre-established APPR. Therefore, one may be sceptical thatthe the invasion eigenvalue λ has relevance for treatment. However, in order to eliminatebacteria one must drive down the bacterial population to near extinction where the premiseof the invasion eigenvalue approach is satisfied because the state vector approximates theAPPR. This suggests, and we prove, that when λ > 0, then bacteria persist and treatmentfails because the bacteria-free state is unstable. This is expected since on averaging over adosing cycle bacteria introduced into the APPR grow when rare. However, λ < 0 impliesonly local stability of the bacteria-free state. This means that an indeterminately smallbacterial population introduced into the APPR is eliminated but one cannot conclude thata larger inoculum is necessarily eliminated. While there are important cases when λ < 0implies treatment success regardless of initial conditions, in general it does not. In particular,the bacteria-free state can be locally attracting, guaranteeing successful treatment for verysmall initial bacterial populations, and yet a larger initial population may lead to a stableendemic infection state. We find such bistable dynamics in our model when the antimicrobialdosing is marginal. This dynamic feature is possible when the removal rate of antibioticdepends significantly on bacterial density. In this case, provided sufficient nutrient is presentand antibiotic dosing is marginal, high bacterial density can significantly reduce antibioticconcentration thereby reducing its effect. The potential for bistability in antibiotic treatmentappears not to have been observed in previous mathematical modeling. It suggests theobvious-early treatment of infection by antibiotics, before bacterial populations have reachedhigh levels, is optimal. The larger message is that antibiotic dosing levels may need to takeaccount of nutrient availability.

Because of the failure of the inequality λ < 0 to guarantee treatment success, we aremotivated to find stronger threshold inequalities of the form λ∗ < 0 where λ < λ∗ whichdo guarantee treatment success. We are successful in identifying thresholds λ∗ expressibleentirely in terms of the mean dosing strength, the pharmacodynamic function, and removalrates of antibiotic, resource and bacteria.

The choice of pharmacodynamic function ψ is largely an educated guess even in the casewhen it can be expected to depend only on antimicrobial agent concentration (ψ = ψ(A)) [1].

3

Page 4: The Pharmacodynamics of Antibiotic Treatmenthalsmith/antibiotic-treatment-new.pdf · The Pharmacodynamics of Antibiotic Treatment ... in which antibiotic levels have attained their

The so-called Emax model, based on drug-receptor interaction, is rather standard [25, 15].In general, a pharmacodynamic function will depend on the particular antimicrobial agentas well as the organism. There appears to be no well-developed pharmacodynamic the-ory which includes both nutrient and antibiotic. Undoubtedly, there are many possibilitiesdepending on whether or not the two act independently or not. Cogan [3] introduces a phar-macodynamic function depending on the product of a function of nutrient and a function ofantibiotic. Corpet et al [4] modify the classical Monod growth rate by assuming either themaximum growth rate or the half-saturation constant is influenced by antibiotic concentra-tion. As these functions are based on educated hunches, our modeling study can only bequalitative in nature exploring possible behavior given only qualitative information aboutthe pharmacodynamic function.

It is well-known that bacterial populations are heterogeneous in their susceptibility toantimicrobial agents [3, 15, 26]. Furthermore, many human pathogens have acquired resis-tance to the antibiotic of choice which has had an important impact on public health. Wealso formulate a model where the bacterial population is heterogeneous in its response tothe antimicrobial agent either due to mutation or the presence or absence of a transmissi-ble plasmid encoding resistance to the antimicrobial agent. Successful treatment requireseradication of both resistant and susceptible populations. This model has potentially twotypes of disease states: one with only the susceptible population present and one with bothsusceptible and resistant populations. We are able to give sufficient conditions for successfultreatment and sufficient conditions for treatment failure-i.e., the persistence of a bacterialpopulation. In this case, there are two distinct values of λ, λs for susceptible and λr for theresistant population. It is important to understand the conditions favoring the establishmentof the mixed population of resistant and susceptible types.

Our work appears to be the first to attempt to obtain mathematically rigorous, explicitsufficient conditions for treatment success and for treatment failure in simple, but hopefullynot too simple, mathematical models of periodic drug dosing in vitro and in vivo. Ourmodels combine both the pharmacokinetics of the antibiotic and the pharmacodynamics ofits effect on a bacterial population in order to study antibiotic treatment. The mathematicaltheory of persistence plays a major role here as it is used to characterize treatment failure.

In the next section we formulate and analyze models of antibiotic treatment of a bacterialpopulation which is homogeneous in its response to the antimicrobial agent. In the followingsection, models containing a susceptible and resistant population are developed. Proofs ofour results are relegated to an appendix.

2 Model of Antibiotic Treatment

We first assume an in vitro environment that can be modeled as a chemostat with the addedfeature that an antibiotic is also input from the external source. Later we will introduce amore general model suitable for in vivo environments. Let S denotes the concentration ofnutrient sustaining microbial growth; u denote the density of bacteria; and A denotes theconcentration of an antibiotic. If we view the antibiotic as bactericidal, that is as killingbacteria, then the equations take the form

4

Page 5: The Pharmacodynamics of Antibiotic Treatmenthalsmith/antibiotic-treatment-new.pdf · The Pharmacodynamics of Antibiotic Treatment ... in which antibiotic levels have attained their

S ′ = D(S0 − S)− γ−1f(S)u

A′ = D(A0(t)− A)− up(A)

u′ = [f(S)−D − g(S,A)]u (1)

We assume fresh nutrient at constant concentration S0 is input and A0(t) is the con-centration of the antibiotic at time t in the input. Parameter γ is the yield constant andf(S) is the growth rate of bacteria at nutrient concentration S. Typically, we take f to beMonod type but our results hold more generally. We require only that the growth(or uptake)function f(S) be monotone increasing in S:

f(0) = 0, f ′(S) ≥ 0

D is the dilution rate.Typically, antibiotics are administered in either a constant dose A0(t) ≡ A0 = constant

or periodically A0(t) = A0(t + T ) ≥ 0 with dosing period T . While our model allows ageneral non-negative periodic dosing function A0(t), in practice it is typically a sequence ofdiscrete doses which might be approximated by:

A0(t) =∑

i

d δ(t− iT )

Parameter d measures the dose, and δ is the Dirac impulse function.Function g = g(S,A) is the so-called pharmacodynamic function which describes the kill

rate or alternatively, the reduction in the growth rate, induced by the antibiotic agent perunit of bacteria. In general, the killing rate depends on the bacteria and the antibiotic usedas well as the nutrient levels. Borrowing from Emax theory [1], one choice is

g(S,A) = f(S)AH

aH + AH

where H > 0 is the Hill exponent, a the half-saturation concentration and f is specificgrowth rate. In this case the net growth rate is

f(S)− g(S, A) = f(S)aH

aH + AH

Cogan takes

g(S,A) = kS + α

a + SA

where α = 0 for antibiotic that kills in proportion to growth rate (if f(S) = mS/a + S) andα > 0 for antibiotic which is partially effective in killing non-growing bacteria. The simplestterm commonly used is independent of nutrient levels, namely

g(S, A) = kA

which is usually interpreted as kill rate.

5

Page 6: The Pharmacodynamics of Antibiotic Treatmenthalsmith/antibiotic-treatment-new.pdf · The Pharmacodynamics of Antibiotic Treatment ... in which antibiotic levels have attained their

Commonly used functions appearing in the literature are described in the following table.

Table1. Pharmacodynamic Functions

Pharmacodynamic function Reference Descriptiong(A) = kA [14] Linear in Ag(A) = mlog(A) + I [10] Log formg(S,A) = k S

a+SA [3] Linear in A

g(A) = k AH

LH+AH [28, 26] Emax model

g(A) = k An

(L+A)n [2] Binomial form

g(A) = k AL0+L1A+L2A2+...+LnAn [2] Cooperativity form

We make only qualitative assumptions regarding the pharmacodynamic function g(S, A).It should vanish in the absence of drug and increase with antibiotic concentration:

g(S, 0) = 0,∂g

∂A≥ 0

Furthermore, adding nutrient should not decrease the net bacterial growth rate:

S → f(S)− g(S,A) is nondecreasing for 0 ≤ S ≤ S0

Finally, (1) includes a removal rate of antibiotic due to its association with bacteria,modeled by the term −p(A)u. p might be taken to be Michaelis-Menten or more generally ofHill type, or simply p(A) = cA. As mentioned in the introduction, most pharmacodynamicmodels do not include such a term. In some cases, it may be reasonable to neglect thisterm entirely, taking p = 0, if drug removal rate is relatively independent of u. This caseis mathematically attractive since it decouples the pharmacokinetics from the rest of themodel. We assume that p vanishes with A and is nondecreasing in A:

p(0) = 0, p′(A) ≥ 0

The chemostat-based model requires considerable modification if we wish to view it as anin vivo model of the application of antibiotic to a tissue in an organism hosting an infectiousbacteria. In this setting, the circulating blood may deliver the antibiotic to the infection siteand the nutrient may be supplied by the infected tissue or by the blood circulation. For thein vivo interpretation, the assumption of an identical removal rate D for all components isunreasonable. More generally, we consider

S ′ = dSS0 − dSS − γ−1f(S)u

A′ = dAA0(t)− dAA− up(A)

u′ = [f(S)− du − g(S,A)]u (2)

v′ = g(S, A)u− dvv

6

Page 7: The Pharmacodynamics of Antibiotic Treatmenthalsmith/antibiotic-treatment-new.pdf · The Pharmacodynamics of Antibiotic Treatment ... in which antibiotic levels have attained their

where F = dSS0 is a flux of nutrient into the infected region, possibly supplied by surroundingtissues or the blood, dS is the removal rate of nutrient which might also include uptake byhost cells. dAA0(t) is now interpreted as flux of antibiotic into the infected region and dA itsremoval rate, perhaps due to metabolism in the liver and excretion by the kidney. As for (1),an additional antibiotic removal rate −up(A) due to association of drug with bacterial cellsis included. In some cases, this term may be neglected if it is small in relation to removalby metabolism and excretion. The removal rate du for viable bacteria u may include effectsof specific or non-specific immune response [1]. We have included an equation for v, thedensity of nonviable cells as the antibiotic may be bacteriostatic rather than bacteriocidal.However, as this equation is decoupled from the others, we ignore it hereafter.

Prior to antimicrobial treatment A0(t) = 0 and A(t) = 0 so the model reduces to slightmodification of the classical chemostat model of microbial growth on a single substrate [32].

S ′ = dS(S0 − S)− γ−1f(S)u

u′ = [f(S)− du]u (3)

Assuming that f(S0) > du (otherwise, there is no need for treatment), the untreated infectionsteady state is given by

S = S, u = u = γ(S0 − S), f(S) = du

This can be viewed as the pre-treatment state since this state is most likely approximatedjust prior to treatment.

If the effect of the antibiotic is to inhibit growth and uptake of nutrient then the followingequations may be more natural for in vitro environments

S ′ = D(S0 − S)− γ−1f(S, A)u

A′ = D(A0(t)− A)− up(A) (4)

u′ = (f(S, A)−D) u.

where we always assume that the pharmacodynamic function f(S,A) ≥ 0 is monotonicallyincreasing in S and monotonically decreasing in A. The function f(S,A) could be chosenfrom among the following:

Table2. Pharmacodynamic Functions

Pharmacodynamics Reference Descriptionf(S, A) = m S

a+Sexp(−ξA) [20, 14] Reduce maximum growth rate

f(S, A) = m S(a+α(A))+S

[4] Reduce ability of bacteria to use substrate

f(S, A) = (m− β(A)) Sa+S

[4] Reduce maximum growth rate

where α(A) and β(A) are in the units of concentration and time respectively. m denotesthe maximum growth rate and a is the Michaelis-Menton(or half-saturation) constant.

7

Page 8: The Pharmacodynamics of Antibiotic Treatmenthalsmith/antibiotic-treatment-new.pdf · The Pharmacodynamics of Antibiotic Treatment ... in which antibiotic levels have attained their

The following equations are more appropriate for an in vivo environment.

S ′ = dSS0 − dSS − γ−1f(S, A)u

A′ = dAA0(t)− dAA− up(A) (5)

u′ = (f(S, A)− du) u.

In our models (2) and (5), we assume the simplest possible pharmacokinetics, introducingonly a single drug compartment A, representing the drug at the site of infection. As drug isusually administered orally or intravenously and then must pass through a number of organsand tissues before reaching the site of infection, more realistic models typically includeadditional compartments for drug levels in these tissues. Such extensions of our modelscould be included.

Let T denote the length of the dosing period. If g is a continuous T -periodic functionthen [g]m will denote the mean value of g:

[g]m =1

T

∫ T

0

g(s)ds

Hereafter, all periodic functions are to be understood to be T -periodic unless otherwisenoted.

It is natural to expect the existence of periodic responses to periodic antibiotic dosing; theconstant dosing case is included as a special case. We expect two different types. Naturally,there is the “sterile state” or “infection-free state”

E0(t) = (S0, A∗(t), 0)

with no bacteria present and nutrient levels matching feed level. A∗(t) is the unique periodicsolution of

A′ = dAA0(t)− dAA

A∗(t) may be called the asymptotic pharmacokinetics since every solution of this simpledifferential equation is asymptotic to it as t becomes large. It has the following properties

[A∗]m = [A0]m, mint

A0 ≤ mint

A∗ ≤ maxt

A∗ ≤ maxt

A0

The infection-free state may be viewed as the desired target state in the sense that successfultreatment must drive the system state to it.

In addition, there may or may not be one or more “disease states” or “infection states”of the form

Eu(t) = (S(t), A(t), u(t))

where u(t) > 0 and where all components are positive periodic functions. Such statescorrespond to treatment failure. Our results will focus on the existence, uniqueness andstability of these solutions.

The local stability of the sterile state can be determined by the Floquet exponents of thevariational equation about E0(t). It turns out that two of these are negative; the third, the“invasion exponent” is given by

λ = f(S0)− du − [g(S0, A∗(t))]m

8

Page 9: The Pharmacodynamics of Antibiotic Treatmenthalsmith/antibiotic-treatment-new.pdf · The Pharmacodynamics of Antibiotic Treatment ... in which antibiotic levels have attained their

for system (2) andλ = [f(S0, A∗(t))]m − du

for system (5). In either case, λ depends on the net, time-averaged bacterial growth rate inthe asymptotic pharmacokinetic state where S = S0 and A = A∗(t), involving the growthdynamics, the pharmacodynamic function and the bacterial removal rate.

The following theorem states our main result for both (2) and (5). It includes theautonomous case, A0 = constant, and the in vitro equations (1) and (4) as a special cases.

Theorem 2.1 For systems (2) and (5), the sterile state E0(t) is locally asymptotically stableif λ < 0 and unstable if λ > 0. Moreover:

(a) If λ < 0, E0(t) is globally attracting in the following cases:

(i) for (5) when p = 0 and for (4) if no Eu(t) exists.

(ii) for (2) when p = 0.

(b) If λ > 0, then bacteria persist. More precisely there exists ε > 0, independent ofinitial data (S(0), A(0), u(0)) provided u(0) > 0, and there exists t0 > 0, dependingon initial data, such that

u(t) > ε, t > t0.

Furthermore, at least one infection state Eu(t) exists. Assuming λ > 0, the followingalso hold:

(iii) for (4), every solution with u(0) > 0 converges to one of the periodic states Eu(t);if Eu(t) is unique, in particular when p = 0, it attracts all solutions with u(0) > 0.

(iv) If A0 is constant and p = 0 holds, then Eu is unique and globally asymptoticallystable for (2) and for (5).

In summary, Theorem 2.1 establishes that the invasion exponent λ, whose sign char-acterizes the local stability of the sterile state, can provide useful information concerningthe global dynamics of the models in certain cases. Treatment failure is guaranteed whenλ > 0 in all cases since bacteria can grow when rare. Furthermore, there exists a least one(periodic) infection state Eu(t); in general, it may be non-unique and we do not know itsstability properties. An important exception is the case of constant dosing and p = 0 whereit is globally attracting.

Only in special cases have we shown that the reverse inequality λ < 0 guarantees treat-ment success, in the sense of bacterial eradication, regardless of initial conditions. Both suchcases require that the removal rate of antibiotic be independent of bacterial density (p = 0).

We note that the system (4) is much more mathematically tractable due to a conservationprinciple common to chemostat models [32]; the corresponding reduced system has specialmonotonicity properties.

The left side of Figure 1 depicts periodic dosing of an antibiotic, given every four hours,with a mean value of 0.2. The right side describes the asymptotic pharmacokinetics A∗(t)after initial transients die out. Parameters were chosen as by [28].

9

Page 10: The Pharmacodynamics of Antibiotic Treatmenthalsmith/antibiotic-treatment-new.pdf · The Pharmacodynamics of Antibiotic Treatment ... in which antibiotic levels have attained their

0 4 8 12 16 20 240

0.5

1

1.5

2

t(hours)

A0(t

)

0 4 8 12 16 20 240

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

t(hours)

Ant

ibio

tic C

once

ntra

tion

AMIC

Figure 1: The left figure depicts periodic discrete dosing of an antibiotic. The resultingpharmacokinetics A∗(t) and the MIC level of A at which λ = 0 appears on the right.

0 4 8 12 16 20 240

0.2

0.4

0.6

0.8

1

1.2

1.4

t(hours)

Bac

teria

l pop

ulat

ion

u

0 12 24 36 48 69 720

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

t(hours)

Bac

teria

l Pop

ulat

ion

u

Figure 2: The left figure shows treatment success when λ = −0.2804 < 0. The right depictstreatment failure when λ = 0.1555 > 0. Same dosing as Figure 1.

10

Page 11: The Pharmacodynamics of Antibiotic Treatmenthalsmith/antibiotic-treatment-new.pdf · The Pharmacodynamics of Antibiotic Treatment ... in which antibiotic levels have attained their

0 12 24 36 48 60 72 84 960

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

t(hours)

Bac

teria

l Pop

ulat

ion

u

Figure 3: An illustration of pre-treatment infection for 0 < t < 60h followed by treatmentstarting at t = 60h resulting in bacterial eradication. Same dosing as Figure 1.

0 12 24 36 48 60 720

0.5

1

1.5

2

2.5

3

t(hours)

Bac

teria

l Pop

ulat

ion

uu

Figure 4: Initial-condition-dependent outcomes are possible when λ < 0: successful andunsuccessful treatment result from the same system but different initial data. Same dosingas Figure 1.

11

Page 12: The Pharmacodynamics of Antibiotic Treatmenthalsmith/antibiotic-treatment-new.pdf · The Pharmacodynamics of Antibiotic Treatment ... in which antibiotic levels have attained their

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

A0

u(B

acte

rial P

opul

atio

n)

LP

BP

Eu Stable Branch

UnstableBranch

E0 StableE

0 Unstable

0 0.5 1 1.50

1

2

3

4

5

6

A0

S0

No Eu

2 Eu

1 Eu

λ=0

Figure 5: Bifurcation diagram for constant dosing on left shows the existence of a large-amplitude, stable branch of Eu while E0 is stable; A0 is the bifurcation parameter. Theright figure shows the MIC(λ = 0) curve and regions corresponding to different numbers ofdisease steady states in the (A0, S

0)-plane for constant dosing.

Our simulations use only (2) with f(S) = mSa+S

, p(A) = νAL1+A

and g(S, A) = k Sa+S

AL+A

.The left side of Figure 2 shows successful treatment and the right figure depicts unsuccessfultreatment. All the parameters and functions are the same in both the figures except themaximum killing rate and it is chosen in such a way that λ < 0 in left figure and λ > 0 in theright. Output has been scaled by S/a, u/(aγ), A/L. Time t is scaled by 1/dS, S0 is scaledby 1/(a), A0 is scaled by 1/L, and dA, du, m and k are all scaled by 1/dS. Parameter valuesare chosen as in [3, 29, 17]. In particular: yield coefficient γ = 0.8, maximum specific growthrates m = 0.417 , ν = 0.28, dilution rate dS = 0.23, half saturation constants a = 0.1,L = 0.1 and L1 = 0.1, maximum disinfection rate k = 0.96 (left) and k = 0.13 (right),concentration of the substrate in the feed S0 = 0.2 and Hill exponent H = 1.

Figure 4 illustrates that λ < 0 only guarantees successful treatment of small bacterialpopulations except in certain special cases. The bacterial population versus time is plottedfor a small and for a large initial inoculum of bacteria (u(0) = 0.25 and u(0) = 0.45)at start of a treatment-all other parameters and initial data are the same. The solutioncorresponding to the small bacterial inoculum shows successful treatment while the solutioncorresponding to a large initial bacterial level exhibits treatment failure. Both E0(t) andEu(t) are simultaneously locally stable. All parameters are the same as used in the previousfigure except k = 0.96 and S0 is increased to S0 = 0.5. The phenomena of bistability mayhave important implications for antibiotic treatment and appears not to have been observedin earlier modeling work. It suggests that early treatment, before bacterial populationsbecome large, is best.

Figures 5,6 consider the constant dosing case for (2). Figure 5 (left) plots the scaledvalue of u at steady state Eu versus the constant dosing concentration of antibiotic A0.Parameter values are the same used in previous figures except for S0 and k, which areS = 0.5 and k = 0.45. The bifurcating branch Eu is stable from (u,A0) = (3.8, 0) to LP

12

Page 13: The Pharmacodynamics of Antibiotic Treatmenthalsmith/antibiotic-treatment-new.pdf · The Pharmacodynamics of Antibiotic Treatment ... in which antibiotic levels have attained their

Figure 6: The dependence of the infection steady state (Eu) value of u on parameters (S0, A0)for constant dosing, illuminating Figure 5. The surface intersects the u = 0 plane along theλ = 0 curve.

and is unstable from LP to (0, 0.44).. This figure shows that although the concentration ofantibiotic exceeds the MIC, treatment is unsuccessful. If we increase the initial concentrationof antibiotic more than 0.67 then the bacteria population will be cleared from the host. Thenumber of infection steady states as a function of input parameters S0 and A0 is shown inthe right side of Figure 5. The MIC curve λ = 0 is also plotted, showing that there is onlyone disease state when concentration of antibiotic is less than the MIC value and zero or twodisease states when concentration is more than MIC.

Figure 6 shows the surface u = u(S0, A0), giving the bacterial density at the infectionstate Eu. A fold in this surface illuminates the information contained in the right side ofFigure 5.

Theorem 2.1 does not provide an adequate set of sufficient conditions for treatmentsuccess in the general case p 6= 0. Our next result applies much more generally, providing athreshold condition for treatment success in terms of parameters and functions appearing inthe model. It is a special case of a more general result proved in the appendix.

Theorem 2.2 Let

B0 =γdSS0

mindS, duand suppose that

p(A) ≤ cA, 0 ≤ A ≤ maxt

A0(t) (6)

If, in addition, A → g(S0, A) is concave on 0 ≤ A ≤ maxt

A0(t) (e.g., ∂2g∂A2 ≤ 0) and

f(S0)− du − g(S0,dA

dA + B0c[A0]m) < 0 (7)

then u(t) → 0 as t →∞ for every solution of (2).

13

Page 14: The Pharmacodynamics of Antibiotic Treatmenthalsmith/antibiotic-treatment-new.pdf · The Pharmacodynamics of Antibiotic Treatment ... in which antibiotic levels have attained their

The concavity assumption on g is not necessary but it simplifies the statement of theresult. It is satisfied for many of the functions commonly used including the Emax modelwith exponent one. Analogous results can be proved for (5). Assumption (6) is a mild oneas c can be taken to be supp(A)/A : 0 < A ≤ max

tA0(t) so long as p is differentiable at

A = 0. Note that the expression on the left in (7), which exceeds λ, depends only on thesupplied antibiotic level.

3 Treatment of Susceptible and Resistant Bacteria

Bacteria may acquire resistance to an antibiotic via spontaneous mutation or the acquisitionof new genetic material by transduction, transformation or conjugation. Transduction isthe process by which bacterial DNA is moved from one bacterium to another by a virus.Transformation is a process where pieces of DNA in the exterior environment are taken upby the bacteria. Conjugation is the transfer of genetic material between bacteria throughcell-to-cell contact and it is thought to play a significant role in the proliferation of antibioticresistant pathogens [30, 36]. We consider a bacterial population consisting of two types ofcells: a plasmid-free cell which is susceptible to an antibiotic and a plasmid-bearing cellwhich is partially or totally resistant to the antibiotic due to a resistance gene encoded bythe plasmid. Hereafter we refer to the cell types as resistant and susceptible. We assumethat the resistant, plasmid-bearing cell may transfer a copy of the plasmid to a susceptible,plasmid-free cell via conjugation. Furthermore, a plasmid-bearing cell may miss-segregateplasmid at cell division with probability q resulting in one daughter cell receiving no plasmidand hence becoming susceptible. Our model takes the form:

S ′ = dS(S0 − S)− γ−1[f(S)u + f+(S)u+]

A′ = dA(A0(t)− A)− p(A)[u + u+]

u′ = [f(S)− du − g(S, A)]u + qf+(S)u+ − µuu+ (8)

u′+ = [f+(S)(1− q)− du − g+(S,A)]u+ + µuu+

For simplicity, we assume both cells types have the same growth yield and removal rate andremove antibiotic similarly. We expect that resistant cells grow no faster than susceptibleones and, of course, are at least partially resistant to antibiotic:

f+(S) ≤ f(S) and g+(S, A) < g(S, A)

Such a cost of plasmid carriage is well documented in the literature, see Lenski [21] althoughit may decline over many generations due to the effects of natural selection as observed byDahlberg and Chao [8].

Horizontal transmission of the plasmid between the two sub-populations is assumed tooccur at a rate proportional to the product of their densities: µuu+.

Our model equations (8) are based on the plasmid model of Stephanopoulus and Lapidus[35]. See also Hsu et. al. [11, 12, 13, 14] for a similar modeling of segregative loss in theirmodel of competition between plasmid-bearing and plasmid-free organisms in selective mediaand [17, 18] for plasmid transfer in biofilms.

14

Page 15: The Pharmacodynamics of Antibiotic Treatmenthalsmith/antibiotic-treatment-new.pdf · The Pharmacodynamics of Antibiotic Treatment ... in which antibiotic levels have attained their

Removing the antibiotic from the model results in the “pre-treatment model”

S ′ = dS(S0 − S)− γ−1[f(S)u + f+(S)u+]

u′ = [f(S)− du)]u + qf+(S)u+ − µuu+ (9)

u′+ = [f+(S)(1− q)− du]u+ + µuu+.

which is studied in detail in [18] in the case dS = du. If u+ = 0, it reduces to (3) and iff(S0) > du it has the pretreatment steady state (S, u, 0) with no resistant bacteria which islocally stable for (9) if f+(S)(1−q)−du+µu < 0 and unstable if the reverse inequality holds.If f+(S)(1−q)−du +µu > 0, then (9) has another pretreatment state (S, u, u+) consisting ofboth susceptible and resistant bacteria. Appropriate initial conditions for the post-treatmentsystem (8) are expected to be near the stable equilibrium of the pre-treatment system (9).

If the antibiotic inhibits the growth and uptake of the nutrient then the following equa-tions may be more appropriate

S ′ = dS(S0 − S)− γ−1[f(S, A)u + f+(S,A)u+]

A′ = dA(A0(t)− A)− p(A)[u + u+]

u′ = [f(S,A)− du]u + qf+(S, A)u+ − µuu+ (10)

u′+ = [f+(S, A)(1− q)− du]u+ + µuu+.

The periodic solutions of (8) and (10) include the ”sterile state”

E0(t) = (S0, A∗(t), 0, 0),

”infection states” with only susceptible cells of the form

Eu(t) = (S(t), A(t), u(t), 0)

and infection states with susceptible and resistant organisms of the form

Eu∗(t) = (S∗(t), A∗(t), u∗(t), u∗+(t))

where all components are positive periodic functions. Such infection states need not beunique. Note that if u+(0) = 0 then u+(t) = 0 for all t and this case is discussed in theprevious section. Therefore we assume that u(0) > 0 and u+(0) > 0.

As in the previous section, the local stability of the sterile state can be determined byFloquet exponents of the variational equation associated with the periodic solution E0(t).Two of these, −dS, −dA are negative; the third and the fourth are the invasion exponentsfor susceptible and resistant organisms:

λ = f(S0)− du − [g(S0, A∗(t))]m, λ+ = f+(S0)(1− q)− du − [g+(S0, A∗(t))]m

for system (8) and

λ = [f(S0, A∗(t))]m − du, λ+ = [f+(S0, A∗(t))]m(1− q)− du

for system (10).The counterpart to Theorem 2.1 for treatment of susceptible and resistant populations

is the following result.

15

Page 16: The Pharmacodynamics of Antibiotic Treatmenthalsmith/antibiotic-treatment-new.pdf · The Pharmacodynamics of Antibiotic Treatment ... in which antibiotic levels have attained their

Theorem 3.1 For systems (8) and (10), the sterile state E0(t) is locally asymptoticallystable if both λ < 0 and λ+ < 0 and unstable if either λ > 0 or λ+ > 0. Moreover:

(a) If λ < 0, A0 is constant, and p = 0, then E0 is globally attracting for (8) whenf+(S0)− du − g+(S0, A0) < 0 and for (10) when f+(S0, A0)− du < 0.

(b) If λ > 0 or λ+ > 0, then treatment fails. In particular, there exists ε > 0, independentof initial data, and t0 > 0 depending on initial data, such that

u(t) + u+(t) > ε, t > t0 (11)

At least one infection state Eu(t) or Eu∗(t) exists. If resistant population survive thenso does susceptible in the sense if (11) hold with only u+ then a similar statement holdsfor u. Assuming λ < 0, λ+ > 0 and p = 0, then both u and u+ are uniformly persistentfor (10) and (8) and at least one infection state Eu∗ exists.

(c) If A0 is constant, λ > 0, and p = 0 then Eu = (S, A, u, 0) exists, is unique, and islocally asymptotically stable for (8) if λ+ = f+(S)(1− q)− du− g+(S, A) + µu < 0 andfor (10) if λ+ = f+(S, A)(1− q)− du + µu < 0. Eu is unstable if λ+ > 0, in this caseu and u+ uniformly persist, and Eu∗ exists for (8) and (10).

Theorem 3.1 describes the extent to which we have been able to link the local stabilityproperties of the sterile state, as determined by the sign of the invasion exponents λ andλ+, to the global dynamics of the models. In the case of non-constant, periodic dosing, thedisease-free state is only locally stable when both λ < 0 and λ+ < 0. This means treatmentsuccess is only guaranteed for indeterminately small infections. On the other hand, treatmentfailure is guaranteed when either λ > 0 or λ+ > 0, in which case there exists at least oneperiodic disease state Eu(t) or Eu∗(t). Special attention is focused on the case p = 0 withλ < 0 and λ+ > 0 where the resistant, but not the susceptible, strain can invade the sterilestate. Of course, treatment failure occurs, both resistant and susceptible organisms persist,and there is a periodic disease state Eu∗ with both resistant and susceptible populations.

More information is available for the constant dosing case. In particular, global stabilityof the disease-free state holds under certain conditions when λ < 0 and p = 0. If λ > 0 andp = 0, then Eu exists and is unique and its stability is determined by whether or not u+ cansuccessfully invade it, as determined by the sign of λ+. If the latter is positive then bothresistant and susceptible bacteria persist and there is a corresponding steady state with bothpresent.

Our simulation in this section involve only (8) with f(S), p(A), g(S, A) as defined in theprevious section and f+(S) = m+S

a+S, g+(S, A) = k+

Sa+S

AL+A

. Figure 7 depicts a successfultreatment when both λ and λ+ are negative. The output has been scaled by S/a, A/L,u/(aγ) and u+/(aγ). Time t is scaled by 1/dS, ν is measured in units of (aγ)/(dSL), µ ismeasured in units of aγ, dA, du, m, m+, k and k+ are all scaled by 1/dS, S0 is scaled by1/(aγ) and A0 is scaled by 1/L. Parameter values are: yield coefficient γ = 0.8, maximumspecific growth rates m = 0.417, m+ = 0.416 and ν = 0.345, dilution rate dS = 0.23, halfsaturation constants a = 0.1 and L = 0.1, Hill exponent H = 1, maximum disinfection ratek = 0.96 and k+ = 0.87, concentration of the substrate in the feed S0 = 0.2, segregation lossq = 0.01 and conjugation µ = 0.0000001.

16

Page 17: The Pharmacodynamics of Antibiotic Treatmenthalsmith/antibiotic-treatment-new.pdf · The Pharmacodynamics of Antibiotic Treatment ... in which antibiotic levels have attained their

0 4 8 12 16 20 240

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

t(hours)

Bac

teria

l Pop

ulat

ion

uu

+

0 12 24 36 48 60 72 840

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

t(hours)

Bac

teria

l Pop

ulat

ion

uu

+

Figure 7: The left figure shows treatment success when λ = −0.2804 and λ+ = −0.2511.The right figure is an illustration of pre-treatment infection for 0 < t < 60h followed bytreatment starting at t = 60h resulting in bacterial eradication. The value of λ and λ+ areas in left figure during the treatment. Same dosing as Figure 1.

0 24 48 72 96 1200

0.1

0.2

0.3

0.4

0.5

0.6

0.7

t(hours)

Bac

teria

l Pop

ulat

ion

uu

+

0 48 96 144 192 2400

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

t(hours)

Bac

teria

l Pop

ulat

ion

uu

+

Figure 8: Simulation of the effect of unsuccessful treatment on the bacterial population.(Left) λ = −0.3654 is negative and λ+ = 0.0558 is positive. In this case both populationspersists and the treatment is ineffective. (Right) Both λ = 0.0598, λ+ = 0.1316 are positiveand the treatment is ineffective. Same dosing as Figure 1.

17

Page 18: The Pharmacodynamics of Antibiotic Treatmenthalsmith/antibiotic-treatment-new.pdf · The Pharmacodynamics of Antibiotic Treatment ... in which antibiotic levels have attained their

0 12 24 36 48 60 72 84 960

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

t(hours)

Bac

teria

l Pop

ulat

ion

uu

+

0 24 48 72 96 1200

0.5

1

1.5

2

2.5

3

3.5

4

t(hours)

Bac

teria

l Pop

ulat

ion

uu

+

Figure 9: Simulation of the effect of treatment on the bacterial population. In both thefigures λ and λ+ are negative. The right figure shows a successful treatment and the leftfigure shows an unsuccessful treatment. Same dosing as Figure 1.

Figure 8 shows that the treatment could fail either by λ+ > 0 and λ < 0 or λ > 0 andλ+ > 0. Initially the culture contains the mixture with a minority of resistant bacteria anda majority of sensitive bacteria. In the first case (left figure), the susceptible populationdeclines for first 24 hours and then increases. In the second case the susceptible populationgrows for first 24 hours and then start decreasing before reaching a periodic solution andthe resistant population grows. In both cases the resistance population becomes dominantand overcomes the susceptible population because the killing rate of resistant population ismuch smaller than for the susceptible one. All the parameters are the same as used in theprevious figure except k and k+ are now 0.96 and 0.25 for figure (a) and 0.29 and 0.125 forfigure (b).

Figure 9 shows that λ < 0 and λ+ < 0 only guarantees successful treatment of a smallbacterial populations except in certain special cases. Both resistant and susceptible pop-ulations versus time are plotted for a small and for a large initial inoculum of bacteria(u(0) = 0.5, u+(0) = 0.01 and u(0) = 0.7, u+(0) = 0.01). All other parameters and initialdata are the same as in previous figures. Solutions corresponding to small initial data showssuccessful treatment while solutions corresponding to a large initial bacterial populationsexhibits treatment failure. In the failure case the susceptible population grows and takesover the resistant population. All parameters are the same as used in left Figure 7 exceptS0 is increased to S0 = 0.5.

The following analog of Theorem 2.2 gives simple sufficient conditions guaranteeing treat-ment success. It is a special case of a more general result proved in an appendix.

Theorem 3.2 Let B0 and c be as in Theorem 2.2. Suppose that f+ ≤ f , g+ ≤ g, and in

addition, A → g+(S0, A) is concave on [0, maxt

A0(t)] (e.g., ∂2g+

∂A2 ≤ 0). Finally, assume that

λ∗ = f(S0)− du − g+(S0,dA

dA + B0c[A0]m) < 0 (12)

18

Page 19: The Pharmacodynamics of Antibiotic Treatmenthalsmith/antibiotic-treatment-new.pdf · The Pharmacodynamics of Antibiotic Treatment ... in which antibiotic levels have attained their

Then u(t) + u+(t) → 0 as t →∞ for every solution of (8).

Observe that λ∗ appearing in (12) depends on the (larger) specific growth rate of suscep-tible bacteria and the (smaller) pharmacodynamic function of resistant bacteria.

4 Appendices

Proofs of our results are provided in these appendices. The proof of Theorem (2.1) is given inAppendix 1, Theorem 3.1 in Appendix 2, and Theorem (2.2) and Theorem (3.2) are provedin Appendix 3.

4.1 Appendix 1

We prove results for (2); similar arguments apply to (4). Some of part (b) of theorem (2.1)holds for (4) only and so we prove that part only for (4).

Before proving the main result, we prove the following lemmas for (2). The first isstandard so we omit its proof.

Lemma 4.1 The nonnegative cone R3+ is positively invariant for (2).

Lemma 4.2 All nonnegative solutions of (2) are ultimately uniformly bounded in forwardtime, and thus they exists for all positive time. Moreover, the system (2) is dissipative in R3

+

Proof: Multiplying the first equation of (2) by γ and adding to third equation, using anew variable z = γS + u, we arrive at the differential inequality:

z′ = (dSγS0 − dSγS − duu)− g(S, A)u

≤ dSγS0 − d(S + u) where d = mindS, du≤ dSγS0 − dz.

This implies

z(t) ≤(

z(0)− γS0dS

d

)e−dt + γ

S0dS

d.

Thus for any initial condition in R3+

lim supt→∞

z(t) ≤ lim supt→∞

((z(0)− γ

S0dS

d)e−dt + γ

S0dS

d

)≤ γ

S0dS

d(13)

Now, from the second equation we have

A′ ≤ dA(A0(t)− A)

which implies that A is ultimately bounded by M1 = maxt

A0(t).

These inequalities prove that (2) is dissipative.

19

Page 20: The Pharmacodynamics of Antibiotic Treatmenthalsmith/antibiotic-treatment-new.pdf · The Pharmacodynamics of Antibiotic Treatment ... in which antibiotic levels have attained their

For a real-valued function p on [0,∞) we define

p∞ = lim inft→∞

p(t), p∞ = lim supt→∞

p(t).

Lemma 4.3 There exist a constant η > 0, independent of initial data, such that S∞ ≥ η forall solutions S(t), where η is a unique, nonzero, root of the equation f(η)S0 + d(η−S0) = 0.

Proof: The equation G(v) = f(v)S0 + d(v − S0) is an increasing function of v satisfyingG(0) < 0 and G(S0) > 0. So the intermediate value theorem gives the unique value of

η ∈ (0, S0) such that G(η) = 0. Now, suppose that S∞ < η. Since u∞ ≤ γ S0dS

d, applying

corollary 2.4(a) of [37] to the S equation of (2)

0 ≥ dS(S0 − S∞) + γ−1f(S∞) lim inft→∞

[−u(t)]

≥ dS(S0 − S∞)− γ−1f(S∞) lim supt→∞

[u(t)]

≥ dS(S0 − S∞)− f(S∞)S0dS

d> d(S0 − η)− f(η)S0

and so, f(η)S0 + dη > dS0 which gives a contradiction due to the choice of η.

Note that for f(S) = m Sa+S

, the value of η is, η =−(da+mS0−dS0)+

√(da+mS0−dS0)2+4d2S0a)

2a.

Lemma 4.4 There is a constant θ > 0, independent of initial data, such that A∞ ≥ θ for allsolutions A(t), where θ is a unique, nonzero, root of the equation dSγS0p(θ)+ddAθ−ddAa1 =0 where min

tA0(t) = a1.

Proof: For any fixed a1, the equation F (w) = ddAw + dSγS0p(w) − ddAa1 is increasingfunction of w, satisfying F (0) < 0 and F (a1) > 0. So by the intermediate value theorem there

exists a unique root θ ∈ (0, a1) of F (w). Now, suppose that A∞(t) < θ. Since u∞ ≤ γ S0dS

d,

applying corollary 2.4(a) of [37] to the A equation of (2)

0 ≥ dA(a1 − A∞) + p(A∞) lim inft→∞

[−u(t)]

≥ dA(a1 − A∞)− p(A∞) lim supt→∞

[u(t)]

≥ ddA(a1 − A∞)− γS0p(A∞)dS

> ddA(a1 − θ)− γS0p(θ)dS

this implies that ddAθ + γS0p(θ)dS > ddAa1 which gives a contradiction due to the choice ofθ.

Proof of Theorem (2.1): (a) The local stability of E0(t) can be determined by theFloquet exponents of the variational equation. The variational equation corresponding toE0(t) of (2) is:

Z ′ =

−dS 0 −γ−1f(S0)

0 −dA −p(A∗(t))0 0 f(S0)− du − g (S0, A∗(t))

Z.

20

Page 21: The Pharmacodynamics of Antibiotic Treatmenthalsmith/antibiotic-treatment-new.pdf · The Pharmacodynamics of Antibiotic Treatment ... in which antibiotic levels have attained their

A computation yields the fundamental matrix Φ(t)

Φ(t) =

e−dSt 0 .0 e−dAt .

0 0 eR t0(f(S0)−du−g(S0,A∗(s)))ds

.

Note that the entries denoting ”.” play no role in the stability of E0(t). Evaluating Φ(t) at

t = T , we obtain the multipliers e−dST , e−dAT and eT(f(S0)−du−[g(S0,A∗(t))]m). Then it followsimmediately that λ1 = −dS, λ2 = −dA and λ = f(S0)−du− [g (S0, A∗(t))]m are the Floquetexponents. Two of the exponents, λ1, λ2 are automatically negative and the third, λ isnegative if f(S0) − du − [g (S0, A∗(t))]m < 0. Thus E0(t) is asymptotically stable if λ < 0and is unstable if λ > 0.

In order to prove global stability we first consider the case when S(t) ≥ S0 for all t ∈ R+.The S equation implies that

S ′ ≤ 0,

i.e., S(t) is non-increasing for all t ≥ 0. Since S is bounded from below, so it approaches toa limit, denote lim

t→∞S(t) = L ≥ S0, then lim

t→∞S ′(t) = 0 because the limt→∞ S(t) exists. Then

limt→∞ u(t) exists and limt→∞

u(t) =γdS(S0 − L)

f(L). The positivity of u implies that L ≤ S0. So

the limt→∞

S(t) = S0 and limt→∞

u(t) = 0.

Now, consider S(t) ≤ S0 for some t. By positively invariance of [0, S0] under S ′ =dS(S0 − S), we have that S(t) ≤ S0 for all alrge t. Since λ < 0, we can choose ε > 0small enough such that λ0 = f(S0)− du − [g (S0, A∗(t) + ε)]m < 0. Then from the equationA′ = dA(A0(t)−A), we conclude that there exists a t0 such that A(t) ≤ A∗(t) + ε for t > t0and where ε > 0 is chosen above. From the u equation of (2) we have

u((n + 1)T ) ≤ u(nT )eR (n+1)T

nT (f(S(s))−du−g(S(s),A∗(s)+ε))ds

= u(nT )λn

where λn = eR (n+1)T

nT (f(S(s))−du−g(S(s),A∗(s)+ε))ds. This implies that

λn ≤ eR (n+1)T

nT (f(S0)−du−g(S0,A∗(s)+ε))ds

= e(f(S0)−du−[g(S0,A∗(t)+ε)]m)T

= eλ0T

and this implies that limn→∞

u(nT ) = 0.

In order to show that limt→∞

S(t) = S0 we use Corollary 2.4(a) of [37]. From the inequality

S ′ ≤ dS(S0 − S) we conclude that S∞ ≤ S0. Let ε > 0 and take t large so that u∞ < εdSγf(S0)

(because limt→∞

u(t) = 0). Applying Corollary 2.4(a) of [37] to the S equation in (2)

0 ≥ dSS0 − dSS∞ + γ−1 lim inft→∞

[−u(t)]f(S∞)

≥ dSS0 − dSS∞ − γ−1 lim supt→∞

[u(t)]f(S0)

≥ dSS0 − dSS∞ − εdS

21

Page 22: The Pharmacodynamics of Antibiotic Treatmenthalsmith/antibiotic-treatment-new.pdf · The Pharmacodynamics of Antibiotic Treatment ... in which antibiotic levels have attained their

and hence we have S0 ≥ S∞ ≥ S∞ ≥ S0 − ε, which is true for any ε > 0. This implies thatfor t large enough we have lim

t→∞S(t) = S0. This implies that E0(t) is globally asymptotically

stable for (2).Part (b) We apply theorem (4.1) of [16]. Using the notation, we set X = (S, u,A) ∈

R3+ : γS + u + A ≤ γ S0dS

d+ M1, where M1 = max

0≤t≤T|A0(t)| and d = mindS, du, X1 =

(S, u,A) ∈ X : u 6= 0 and X2 = (S, u,A) ∈ X : u = 0. Define a map P such asP (S(0), A(0), u(0)) = (S(T ), A(T ), u(T )). We want to show that there exists ε > 0 suchthat

lim infn→∞

d(P n(X), X2) > ε.

Given that

(i) X is compact metric space.

(ii) P : X → X is continuous map.

(iii) P (X1) ⊂ X1

(iv) M is the maximal compact invariant set in X2.

In our case M = E0(0), since the omega limit set of solutions starting in X2 is E0(0) whereE0(0) = (S0, A∗(0), 0). We want to show that

(i) M is isolated in X

(ii) W s(M) ⊂ X2

In order to show that M is isolated in X we will apply the following theorem 2.3 of [24]. LetV be the neighborhood given by this theorem. Assume that there exists an invariant set Ksuch that

M ⊂ K ⊆ V ∩X.

Since K is positively invariant, all solutions that begin in K stay in K and so in V forpositive time. Thus K ⊂ W s(E1).Since K is negatively invariant, all solutions that begin in K stay in K and so in V fornegative time. Thus K ⊂ W u(E1).

W s(E1) ∩W u(E1) = E1

thus K = M = E1. Therefore K is an isolated compact invariant set in X.It is clear that in this case W s(M) = X2.

For the existence of a positive periodic solution, we assume that X1 is a convex andrelatively open subset in X. The map P satisfies the following conditions

(i) P : X → X is point dissipative (since all the positive trajectories eventually lie in abounded set);

(ii) P is compact (since P is continuous in R+3);

22

Page 23: The Pharmacodynamics of Antibiotic Treatmenthalsmith/antibiotic-treatment-new.pdf · The Pharmacodynamics of Antibiotic Treatment ... in which antibiotic levels have attained their

(iii) P is uniformly persistent with respect to (X1, X2).

The existence of a positive T periodic solution follows directly by theorem 1.3.6 of [41].For (4), it follows from theorem 4.2 of [32], every solution with u(0) > 0 converges to one

of these states.If p(A) = 0 and dS = dA = d, then the limiting system

u′ = (f(S,A∗)− d)u = g1(t, u) (14)

S =S0

d− γ−1u.

has concave nonlinearities. By theorem 3.1 of [33] and lemma 2.7 of [11], if λ > 0, everysolution with u(0) > 0 converges to Eu(t).

If A0 is constant, p = 0 and λ > 0. Then, a unique Eu state can be obtained by settingthe right side of (2) equal to zero and then solving for a none-zero S, A, and u. Then F (S) =f(S) − du − g(S, A0), by intermediate value theorem, gives a unique S ∈ (0, S0), becauseF (0) < 0 and F (S0) > 0 such that F (S) = 0. Also G(u) = (dSS0 − dSS) − uγ−1f(S) = 0gives a unique u such that G(u) = 0. Thus in this case Eu is unique. If λ < 0 then there doesnot exist a S such that F (S) = 0. The local stability of Eu is proved in the next theorem.For the global stability of Eu, we consider the limiting system

S ′ = dS(S0 − S)− γ−1f(S)u

u′ = (f(S)− du − g(S, A∗)) u. (15)

We apply the Dulac criterion with the auxiliary function g(S, u) = 1u

to (15) and find that

∂S[g(S, u)S ′)] +

∂u[g(S, u)u′)] = −dS

u− γ−1f ′(S)

< 0.

Hence the above system (15) does not have any periodic solution. This together withPoincare-Bendixson theoem implies the global stability of (15). For the global stabilityof (2) we use the theorem (F.1) of [32]. Since all the hypotheses of the theorem (F.1) aresatisfied, see [18], so we conclude that Eu is globally asymptotically stable.

4.2 Appendix 2

It is easy to see that all the lemmas of appendix 1 also hold for (8) and (10). The upperbound of the solutions for (8) as given in lemma (4.2) is

lim supt→∞

(γS(t) + u(t) + u+(t) + A(t)) ≤ γS0dS

d+ M1,

where M1 = maxt|A0(t)|, d = mindS, du.

All the results that are proved only for (8) can also be proved for (10) in exactly thesame way. So we proved them only for (8).

23

Page 24: The Pharmacodynamics of Antibiotic Treatmenthalsmith/antibiotic-treatment-new.pdf · The Pharmacodynamics of Antibiotic Treatment ... in which antibiotic levels have attained their

Proof of Theorem 3.1: (a) The local stability of E0(t) can be determined by theFloquet exponents of the variational equation. The variational equation corresponding toE0(t) of (8) is:

Z ′ =

−dS 0 −γ−1f(S0) −γ−1f(S0)0 −dA −g1(A

∗(t)) 00 0 f(S0)− du − g (S0, A∗(t)) qf+(S0)0 0 0 z44

Z

where z44 = f+(S0)(1 − q) − du − g+ (S0, A∗(t)). Note thatthe entries denoting ”.” play norole in the stability of E0(t). A computation yields the fundamental matrix Φ(t)

Φ(t) =

e−dSt 0 . .0 e−dAt . 0

0 0 eR t0(f(S0)−du−g(S0,A∗(s)))ds .

0 0 0 eR t0(f+(S0)(1−q)−du−g+(S0,A∗(s)))ds

Evaluating Φ(t) at t = T , we obtain the multipliers e−dST , e−dAT , eT(f(S0)−du−[g(S0,A∗(t))]m)

and eT(f+(S0)(1−q)−du−[g+(S0,A∗(t))]m). Then it follows immediately that λ1 = −dS, λ2 = −dA,λ = f(S0)−du− [g (S0, A∗(t))]m and λ+ = f+(S0)(1− q)−du− [g+ (S0, A∗(t))]m are Floquetexponents. Two of the exponents, λ1, λ2 are automatically negative and λ and λ+ arenegative if f(S0)− du − [g (S0, A∗(t))]m < 0 and f+(S0)(1− q)− du − [g+ (S0, A∗(t))]m < 0.Thus E0(t) is asymptotically stable if λ, λ+ both are negative and is unstable if either λ > 0or λ+ > 0.

For the global stability of E0 in the constant case, since f(S0) − du − g(S0, A0) < 0and f+(S0) − du − g+(S0, A0) < 0, we choose ε > 0 small enough so that f(S0 + ε) − du −g(S0 + ε, A0 + ε) < 0 and f+(S0 + ε) − du − g+(S0 + ε, A0 + ε) < 0. From the inequalitiesS ′ ≤ dS(S0 − S) and A′ ≤ dA(A0 − A), we conclude that for all large t > 0, S(t) ≤ S0 + εand A(t) ≤ A0 + ε, where ε > 0 is chosen above.

From the u and u+ equations of the system (8)

u′ + u′+ = (f(S)− du − g(S, A)) u + (f+(S)− du − g+(S, A)) u+

≤ (f(S0 + ε)− du − g

(S0 + ε, A0 + ε

))u

+(f+(S0 + ε)− du − g+(S0 + ε, A0 + ε)

)u+.

Take m = minf+(S0 + ε)− du − g+(S0 + ε, A0 + ε), f(S0 + ε)− du − g(S0 + ε, A0 + ε) < 0.Then

u′ + u′+ ≤ m(u + u+)

and this implies that lim supt→∞

(u(t)+u+(t)) ≤ 0. From the positivity of u and u+, we conclude

that limt→∞

u(t) = 0 and limt→∞

u+(t) = 0

Part (b) First we will consider the case when λ+ > 0. We can choose ε > 0 suchthat f+(S0 − ε)(1 − q) − du − [g+(S0 + ε, A∗(t) + ε)]m = B1 > 0. From the inequalityA′(t) ≤ dA(A0(t)− A), there exists a t0 > 0 such that A(t) ≤ A∗(t) + ε, t > t0.

24

Page 25: The Pharmacodynamics of Antibiotic Treatmenthalsmith/antibiotic-treatment-new.pdf · The Pharmacodynamics of Antibiotic Treatment ... in which antibiotic levels have attained their

Pick a solution (S(t), A(t), u(t), u+(t)) with u+(0) > 0. We suppose that u and u+ do notuniformly weak persist and derive a contradiction. We can assume that

u∞ ≤ εγdS

4nf(S0)≤ ε

u∞+ ≤ εγdS

4nf+(S0)≤ ε (16)

where n = max1, γ and ε > 0 is chosen above. From the inequality S ′ ≤ dS(S0 − S) weconclude that S∞ ≤ S0. We apply Corollary 2.4(a) of [37] to the S equation of (2)

0 ≥ dSS0 − dSS∞ + γ−1 lim inft→∞

[− (u(t)f(S∞) + u+(t)f+(S∞))]

≥ dSS0 − dSS∞ − γ−1 lim supt→∞

[u(t)f(S0) + u+(t)f+(S0)]

≥ dSS0 − dSS∞ − εdS

4n− εdS

4n

≥ S0 − S∞ − ε

4n− ε

4n

≥ −S∞ + S0 − ε

2n

from this, we conclude that

S0 ≥ S∞ ≥ S∞ ≥ S0 − ε

2.

This implies that there exists a t1 > 0 such that S(t) ∈ [S0 − ε, S0 + ε] for t > t1. Pick asolution (S(t), A(t), u(t), u+(t)). For a large t > maxt0, t1 (say t1), we have from the u+

equation of (8):

u′+ ≥ (f+(S0 − ε)(1− q)− du − g+(S0 + ε, A∗(t) + ε) + µu

)u+

≥ (f+(S0 − ε)(1− q)− du − g+(S0 + ε, A∗(t) + ε)

)u+ (17)

By integrating (17) from NT ≥ t1 to nT , with n > N , we have

u+(nT ) ≥ u+(NT )eB1(n−N)T , for n > N

which gives a contradiction to (16) for a large enough t.Now we will consider the case when λ > 0. We can pick ε > 0 (it could be different then

above ) such that f(S0−ε)−du−[g(S0+ε, A∗(t)+ε)]m−µε = B2 > 0. Also, from the inequalityA′ ≤ dA(A0(t) − A) we conclude that A(t) ≤ A∗(t) + δ for some 0 < δ ≤ ε and for t > t2large enough. Assume that u∞ ≤ εγdS

4nf(S0)≤ ε and u∞+ ≤ εγdS

4nf(S0)≤ ε where n = max1, γ. As

shown above, in this case there exists a t1 > 0 such that S(t) ∈ [S0− ε, S0 + ε] where ε > 0 ischosen above. Pick a solution (S(t), A(t), u(t), u+(t)) with u(0) > 0. For t > t′ = maxt1, t2,we have from the u equation of (10):

u′ = [f(S)− du − g (S,A(t))] u + qf+(S)u+ − µuu+

≥ [f(S0 − ε)− du − g(S0 + ε, A∗(t) + ε)− µε

]u. (18)

25

Page 26: The Pharmacodynamics of Antibiotic Treatmenthalsmith/antibiotic-treatment-new.pdf · The Pharmacodynamics of Antibiotic Treatment ... in which antibiotic levels have attained their

Integrate (18) from NT ≥ t′ to nT we have:

u(nT ) ≥ u(NT )eB2(n−N)T , for n > N

which gives a contradiction to u∞ < ε for large enough n. This shows that, if λ > 0 orλ+ > 0 then, there exists a constant ε > 0 independent of initial data such that

lim supt→∞

(u(t) + u+(t)) > ε

for all solutions.Now, we will prove the uniformly strong persistence of either u or u+. Using the nota-

tion, we set X = (S, A, u, u+) ∈ R4+ : 0 ≤ γS + u + u+A ≤ γ S0dS

d+ M1, where M1 =

max0≤t≤T

|A0(t)|, d = mindS, du, X2 = (S, A, u, u+) ∈ X : u + u+ = 0 and X1 =

(S, A, u, u+) ∈ X : u 6= 0 or u+ 6= 0. Define a map P such that

P (S(0), A(0), u(0), u+(0)) = (S(T ), A(T ), u(T ), u+(T )) .

This map P satisfies the following

(i) P : X → X is continuous map;

(ii) P (X1) ⊂ X1;

(iii) P has a global attractor.

(iii) holds for P because the map P is compact and point dissipative. By theorem 1.3.3. of[41], there exists an ε > 0 independent of initial data such that lim inf

n→∞(u(nT )+u+(nT )) > ε.

From this, we conclude thatlim inf

t→∞(u(t) + u+(t)) > ε

for all solutions with u(0) > 0, u+(0) > 0.Now, we will show that if u+∞ > ε for some ε > 0, then there exists some ε1 > 0 such

that u∞ > ε1. Since A∞ ≤ M1 = max0≤t≤T

A∗(t), we apply Corollary 2.4(a) of [37] to the u

equation in (8)

0 ≥ lim inft→∞

[(f (S(t))− du − g(S(t), A(t))) u∞ + qf+ (S(t)) u+(t)− µu∞u+(t)]

≥ lim inft→∞

[(f (S(t))− du − g(S(t), A(t)))] u∞ + lim inft→∞

(qf+(S(t))u+(t))

−µu∞ lim supt→∞

u+(t)

≥ (f(η)− du − g(S0,M1)

)u∞ + qf+(η)u+∞ − µu∞γ

S0dS

d.

Here we have used that lim inft→∞

S(t) ≥ η, by the lemma (4.3). Solving for u∞ we obtain

u∞ > qf+(η)ε„−f(η)+du+g(S0,M1)+µγ

S0dSd

« = ε1, which is positive by lemma (4.3), independent of the

initial data. This implies that if u+ persists then so does u.

26

Page 27: The Pharmacodynamics of Antibiotic Treatmenthalsmith/antibiotic-treatment-new.pdf · The Pharmacodynamics of Antibiotic Treatment ... in which antibiotic levels have attained their

The existence of at least one non-zero, positive, periodic solution of form Eu(t) or Eu∗(t)follows by theorem 1.3.6 of [41] as shown in the previous theorem.

Now suppose that λ < 0, λ+ > 0 and p = 0. It is easy to see that if S(t) ≥ S0 for allt ≥ 0, then in this case u+ uniformly persists.

Consider S(t) ≤ S0 for large t. Since λ+ > 0, there exists some ε0 > 0 such thateither u∞ > ε0 or u∞+ > ε0. Since λ < 0, we can pick some δ > 0 small enough so thatf(S0)−du−[g (S0, A∗(t)− δ)]m < 0. From the fact that lim

t→∞A(t) = A∗(t), we have A∗(t)−δ ≤

A(t) ≤ A∗(t) + δ where δ > 0 is chosen above for t large enough. As f(S)− du − g(S,A∗) isnondecreasing in S ∈ [0, S0] for each fixed t ∈ [0, T ], the u equation of (8) can be written as

u′(t) ≤ [f(S0)− du − g(S0, A)

]u + qf+(S0)u+ − µuu+

u′(t) ≤ [f(S0)− du − g

(S0, A∗(t)− δ

)]u + qf+(S0)u+.

This implies, by using the comparison theorem and the well know result from [9], that thereexists a constant e > 0 such that u∞ ≤ e lim sup

t→∞|qf+(S0)u+(t)|. Next, we suppose that the

u+ is not uniformly persistant and derive a contradiction. Choose some 0 < ε < ε0 such thatu∞+ < minε, ε

eqf+(S0) where e is defined above. Then, from u∞ ≤ e lim sup

t→∞

(qf+(S0)u+(t)

),

we have u∞ ≤ ε < ε0, in contradiction to the fact that u∞ > ε. Thus u∞+ > ε for some ε > 0.Now we will prove that u+∞ > ε for some ε > 0. Using the notation, we set X =

(S, A, u, u+) ∈ R4+ : 0 ≤ γS + u + u+A ≤ γ S0dS

d+ M1, where M1 = max

0≤t≤T|A0(t)|, d =

mindS, du, X2 = (S,A, u, u+) ∈ X : u+ = 0 and X1 = (S, A, u, u+) ∈ X : u+ 6=0. By theorem 1.3.3. of [41], there exists an ε > 0 independent of initial data such thatlim infn→∞

u+(nT ) > ε and we conclude that lim inft→∞

u+(t) > ε with ε > 0 not depending on the

initial data.The existence of one periodic solution Eu∗(t) follows from theorem 1.3.6 of [41].Part (c) The local stability of Eu = (S, A, u, 0) is determined by the eigenvalues of the

matrix

C =

−dS − γ−1uf(S) 0 −γ−1f(S) 00 −dA − up′(A) −p(A) 0

u(f ′(S)− gS(S, A)

) −ugA(S, A) 0 qf+(S)− µu0 0 0 c44

where c44 = f+(S)(1− q)− du − g+(S, A)− µu, c33 = f(S)− du − g(S, A) = 0, gS(S, A)and gA(S, A) are the derivatives of g(S,A) with respect to S and A at (S, A) respectively.Here we have used that c33 = 0 at Eu steady state. Note that in this case A = A0. Theeigenvalues of the matrix C are the eigenvalue

λ+ = f+(S)(1− q)− du − g+(S, A) + µu

and the eigenvalues of the matrix

C1 =

−dS − γ−1uf ′(S) 0 −γ−1f(S)0 −dA − up′(A) −p(A)(

f ′(S)− gS(S, A))u −ugA(S, A) 0

27

Page 28: The Pharmacodynamics of Antibiotic Treatmenthalsmith/antibiotic-treatment-new.pdf · The Pharmacodynamics of Antibiotic Treatment ... in which antibiotic levels have attained their

We compute the coefficients of the characteristic polynomial P of the matrix of C1. LetP (λ) = λ3 + a1λ

2 + a2λ + a3. Then

a1 = −Tr(C1) = dS + γ−1uf(S) + dA + up′(A) > 0,

and

a3 = −Det(C1)

= −(dS + γ−1uf(S))p(A)ugA(S, A) + (dA + up′(A))(f ′(S)− gS(S, A)

)γ−1uf(S)

and

a2 =(dS + γ−1uf(S)

) (dA + up′(A)

)+

(f ′(S)− gS(S, A)

)γ−1uf(S)− p(A)ugA(S, A).

Now, if p(A) = 0, then a3 = dA

(f ′(S)− gS(S, A)

)γ−1uf(S) > 0 and

s = a1a2 − a3

=(dS + dA + γ−1uf(S)

)[(dS + γ−1uf(S)

)dA +

(f ′(S)− gS(S, A)

)γ−1uf(S)]

−[dA

(f ′(S)− gS(S, A)

)γ−1uf(S)]

=(dS + γ−1uf(S)

)[dS + γ−1uf(S)dA +

(f ′(S)− gS(S, A)

)γ−1uf(S)]

+[(dS + γ−1uf(S)

)dA]

= (dS + γ−1uf(S))[dA

(dS + γ−1uf(S)

)+ (dA)2 +

(f ′(S)− gS(S, A)

)uγ−1f(S)] > 0.

By the Routh-Hurwitz criterion all three eigenvalues have negative real parts. Thus, E1 isasymptotically stable if f+(S)(1− q)− du − g+(S, A) + µu < 0 and is unstable if f+(S)(1−q)− du − g+(S, A) + µu > 0.

Now we will prove the persistence result for the autonomous case. We follow a similarargument used in Theorem 5.3 of [34], applying Theorem 4.6 in [37]. Using the notation of

that result, we set X = (S,A, u, u+) ∈ R4+ : 0 ≤ γS + A + u + u+ ≤ γ S0dS

d+ A0, where d =

mindS, du, X2 = (S,A, u, u+) ∈ X : u+ = 0, and X1 = X \ X2. We want to showthat solutions which start in X1 are eventually bounded away from X2. Using the notationx(t) = (S(t), A(t), u(t), u+(t)) for a solution of (8), define

Y2 = x(0) ∈ X2 : x(t) ∈ X2, t ≥ 0 = x(0) ∈ X : u+(0) = 0and Ω2, the union of omega limit sets of solutions starting in X2, is, by theorem (2.1),the set E0, Eu where E0 := (S0, A0, 0, 0) and Eu := (S, A, u, 0). We will show that ifM0 = E0 and M1 = Eu, then M0,M1 is an isolated acyclic covering of Ω2 in Y2 andeach Mi is a weak repeller. All solutions starting in Y2 but not on the SA-plane convergeto Eu while those on the SA-plane converge to E0. Eu, being locally asymptotically stablerelative to Y2, cannot belong to the alpha limit set of any full orbit in X2 different fromEu itself. Similar arguments apply to E0; the only solutions converging to it lie on the SA-plane and these are either unbounded or leave X in backward time. Thus M0,M1 is anacyclic covering of Ω2. If M1 were not a weak repeller for X1, there would exist an x(0) ∈ X1

such that x(t) → Eu as t →∞. Let V (t) = (u(t), u+(t))t and define the matrix P (S, A, u) by

(f(S)− du − g(S, A) qf+(S)− µu

0 f+(S)(1− q)− du − g+(S, A) + µu

)

28

Page 29: The Pharmacodynamics of Antibiotic Treatmenthalsmith/antibiotic-treatment-new.pdf · The Pharmacodynamics of Antibiotic Treatment ... in which antibiotic levels have attained their

Then C = P (S, A, u) and we may write the equation satisfied by V (t) as

V = P (S, A, u)V + [P (S,A, u)− P (S, A, u)]V

If P (S, A, u)tW = qW where q = s(P (S, A, u)) = s(C) > 0 and W = (m,n)t with m,n > 0is the Perron-Frobenius eigenvector, then on taking the scalar product of both sides of thedifferential equation by W and using that S(t) → S, A(t) → A and u(t) → u, we have

d

dt(mu + nu+) ≥ q/2(mu + nu+)

for all large t. But this leads to the contradiction to x(t) → Eu, namely that mu(t) +nu+(t) → ∞ as t → ∞. Thus M1 is a weak repeller. The argument above together withthe fact that Eu is locally asymptotically stable relative to the subspace u+ = 0 implies thatit is an isolated compact invariant set in X. Similar arguments show that M0 is a weakrepeller and an isolated compact invariant set in X. Therefore, Theorem 4.6 in [37] impliesour result: there exists ε > 0 such that lim inft→∞ d(x(t), X2) > ε for all x(0) ∈ X1, whered(x,X2) is the distance from x to X2.

4.3 Appendix 3

Let

B0 =γdSS0

mindS, duand let (s, U) be the unique solution of

B0 = u + γS

0 = dS(S0 − S)− γ−1f(S)u (19)

with s, U > 0. (s, U) relates to the pretreatment equilibrium (S, u) as follows:

s < S, u < U < B0

If f(S) = mSa+S

then U is the positive root of

(m− dS)U2 + γ[dSa + 2dSxS0 − dSS0 −mxS0]U + dSγ2S0(a + xS0)(1− x)

and x = γdS

mindS ,du . Due to the ugliness of this expression, the reader may wish to substituteB0 for U in the formulae to follow; the results continue to hold in this case.

Consider the T -periodic scalar equation

A′ = dA(A0(t)− A)− Up(A)

in which u has been set to the constant value u = U . It has a unique T -periodic solutionA∗(t) = A∗(t + T ) and it satisfies

0 < A∗(t) ≤ A∗(t)

29

Page 30: The Pharmacodynamics of Antibiotic Treatmenthalsmith/antibiotic-treatment-new.pdf · The Pharmacodynamics of Antibiotic Treatment ... in which antibiotic levels have attained their

Also, observe thatp = 0 ⇒ A∗(t) = A∗(t).

Now defineλ∗ = λ∗(S0) = f(S0)− du − [g(S0, A∗)]m

Observe thatλ∗ ≥ λ

andp = 0 ⇒ λ∗ = λ.

First of all we will prove that if λ∗ < 0 then u(t) → 0 as t →∞ for every solution of (2).

The following generalizes Theorem 2.1 (a):

Theorem 4.1 If λ∗ < 0 then u(t) → 0 as t →∞ for every solution of (2).

Proof: From the lemma (4.2) we have

lim supt→∞

u ≤ B0.

The lemma (4.3) giveslim inf

t→∞S ≥ η

It follows that

lim supt→∞

u ≤ lim supt→∞

(γS + u)− γ lim inft→∞

S ≤ B0 − γη(B0/γ) = B1

Applying the lemma (4.3) again, we obtain a better (bigger) lower bound for the limit inferiorof S:

lim inft→∞

S ≥ η(B1)

where η = η(B1) is the unique root S = η of

0 = dS(S0 − S)− γ−1f(S)B1.

If F (b) = B0−γη(b/γ), then F (0) = B0 and F ′(b) < 0 because η(b) is an increasing functionof b. Moreover, B0 > U = F (U) > F (B0) = B1, from which we conclude that

lim supt→∞

u ≤ U.

Now suppose that λ∗ < 0. This implies that there exists ε > 0 such that

µ = f(S0)− du − [g(S0, A∗ − ε)]m < 0

Let (S, A, u) be a solution of (2). Given δ > 0, there exists τ > 0 such that

u(t) ≤ U + δ, t ≥ τ

30

Page 31: The Pharmacodynamics of Antibiotic Treatmenthalsmith/antibiotic-treatment-new.pdf · The Pharmacodynamics of Antibiotic Treatment ... in which antibiotic levels have attained their

ThenA′ ≥ dA(A0(t)− A)− (U + δ)p(A), t ≥ τ

so A(t) ≥ Aδ(t), t ≥ τ where Aδ(t) is the solution of

A′ = dA(A0(t)− A)− (U + δ)p(A)

satisfying Aδ(τ) = A(τ). But Aδ(t) → A∗δ(t) where the latter is the unique T -periodic

solution of this differential equation. It is not difficult to verify that A∗δ(t) → A∗(t) as δ → 0

uniformly on compact sets so by choosing δ sufficiently small, there exists τ > τ so that

A(t) ≥ Aδ(t) ≥ A∗(t)− ε, t ≥ τ

This implies that

f(S(t))− du − g(S(t), A(t)) ≤ f(S0)− du − g(S0, A∗(t)− ε), t ≥ τ

Sinceu′ ≤ u[f(S0)− du − g(S0, A∗(t)− ε)], t ≥ τ

we have thatu(t + T ) ≤ u(t) exp(µT ), t ≥ τ

implying that if λ∗ < 0 then u(t) → 0 as t →∞ for every solution of (2).

Proof of Theorem (2.2) and Theorem (3.2): Now, if

p(A) ≤ cA, 0 ≤ A ≤ maxt

A0(t) (20)

then A∗(t) ≥ A(t) where A(t) is the unique T -periodic solution of the linear equation

A′ = dA(A0(t)− A)−B0cA.

Observe that

[A]m =dA

dA + B0c[A0]m

By monotonicity of gλ∗ ≤ f(S0)− du − [g(S0, A)]m

Consequently, if f(S0)− du − [g(S0, A)]m < 0 then λ∗ < 0. If g is concave in A, then −g isconvex so by Jensen’s inequality, we have

f(S0)− du− [g(S0, A)]m ≤ f(S0)− du− g(S0, [A)]m) = f(S0)− du− [g(S0,dA

dA + B0c[A0]m).

We note that

A(t) = dA

∫ T

0

G(t, s)A0(s)ds

where Greens function G(t, s) is given by

G(t, s) =

exp(−µ(t−s)) exp(µT )

exp(µT )−1, 0 ≤ s < t ≤ T

exp(−µ(t−s))exp(µT )−1

, 0 ≤ t ≤ s ≤ T

and µ = dA + cU .Theorem (3.2) can be proved as above by adding the equations for u and u+, defining

v = u + u+, and noticing that f(S)− du− g(S, A) ≤ f(S)− du− g+(S, A) and f+(S)− du−g+(S, A) ≤ f(S)− du − g+(S, A).

31

Page 32: The Pharmacodynamics of Antibiotic Treatmenthalsmith/antibiotic-treatment-new.pdf · The Pharmacodynamics of Antibiotic Treatment ... in which antibiotic levels have attained their

References

[1] D.J. Austin, N.J. White, and R.M. Anderson., The dynamics of drug action onthe within-host population growth of infectious agents: melding pharmacokineticswith pathogen population dynamics, J Theor Biol., 194(3): 313-39 (Oct 1998).

[2] M. J. Chappell, Structure identifiability and indistinguishability of certain two-compartment models incorporating nonlinear efflux from the peripheral compart-ment, Mathematical Biosciences 125: 61-81 (1995).

[3] N.G. Cogan, Persisters, tolerance and dosing, Journal of Theoretical Biology238(3): 694-703 (2006).

[4] D. E. Corpet, S. Lumeau, and F. Corpet Minimum antibiotics levels for selectinga resistance plasmid in a gnotobiotic animal model, Antimicrobial Agents andChemotherapy 33(4): 535-540, (Apr 1989).

[5] S. Corvaisier, P. Maire, M. Bouvier D’Yvoire, X. Barbaut, N. Bleyzac, and R. Jel-liffe, Comparisons between antimicrobial pharmacodynamic indices and bacterialkilling as described by Using the Zhi model, Antimicrob Agents Chemother 42(7):1731-1737(July 1998).

[6] R. M. Cozens, E. Tuomanen, W. Tosch, O. Zak, J. Suter, and A. Tomasz, Eval-uation of the bactericidal activity of beta-lactam antibiotics on slowly growingbacteria cultured in the chemostat, Antimicrob Agents Chemother 29(5): (May1986).

[7] W. A. Craig, Pharmacokinetics/Pharmacodynamic parameters: rationale for an-tibacterial dosing of mice and men, Clinical Infectious Diseases 26: 1-12 (1998).

[8] C. Dahlberg, and L. Chao, Amelioration of the cost of conjugative plasmid carriagein Eschericha coli K12, Genetics 165: 1641-1649 (Dec. 2003).

[9] J. K. Hale., Ordinary Differential Equations, R. E. Krieger Publishing Company,INC., second edition (1980).

[10] H. Nicholas. G. Holford, and L. B. Sheiner, Kinetics of Pharmacologic Response,Pharmac. Ther. 16: 143-166 (1982).

[11] G. J. Butler, S. B. Hsu, and P. Waltman, A mathematical model of the chemostatwith periodic washout Rate , SIAM Journal on Applied Mathematics 45(3): 435-449 (June 1985) .

[12] S. B. Hsu, P. Waltman, and G. Wolkowicz, Global analysis of a model of plasmid-bearing, plasmid-free competition in the chemostat, J. Math. Biol. (32): 731-742(1994).

[13] S. B. Hsu, and P. Waltman, Competition between plasmid-bearing and plasmid-freeorganisms in selective media, Chem. Engng. Sci. 52: 23-35 (1997).

32

Page 33: The Pharmacodynamics of Antibiotic Treatmenthalsmith/antibiotic-treatment-new.pdf · The Pharmacodynamics of Antibiotic Treatment ... in which antibiotic levels have attained their

[14] S. B. Hsu, and P. Waltman, A survey of mathematical models of competition withan inhibitor, Mathematical Biosciences 187: 53-91 (2004) .

[15] M. Lipsitch, and B.R. Levin., The population dynamics of antimicrobialchemotherapy , Antimicrob Agents Chemother. 41(2): 363-73 (Feb 1997)..

[16] J. Hofbaer, and J. W. -H. So, Proceedings of the american mathematical society,(American Mathematical Society 107(4): (Dec 1989).

[17] M. Imran, D. Jones, and H. L. Smith, Biofilms and the plasmid maintenancequestion, Mathematical Biosciences 193: 183-204 (2005).

[18] M. Imran, and H. L. Smith, A mathematical model of gene transfer in a biofilm,Springer book vol.1 ”Mathematics for Ecology and Environmental Sciences”, (ac-cepted).

[19] M. Imran Mathematical Models in Biofilms and Antibiotic Treatment, Ph.D. the-sis, Arizona State University, Tempe, AZ, (Dec 2006).

[20] R. Lenski and S. Hattingh, Coexistence of two competitors on one resource andone inhibitor , J. Theor. Biol. 122: 83-92 (1986).

[21] R. Lenski, Bacterial evolution and the cost of antibiotic resistance, Internatl. Mi-crobiol. 1: 265-270 (1998).

[22] K. Lewis, Riddle of biofilm resistence, Antimicrobial Agents and Chemotherapy45(4): 999-1007 (Apr 2001) .

[23] C. T. Bergstrom, M. Lipsitch, B. R. Levin, Natural selection, infectious transferand existence conditions for bacterial plasmids, Genetics 155: 1505-1519 (2000) .

[24] Y. A. Kuznetsov, Elements of Applied Bifurcation Theory, Springer-Verlag NewYork (1995).

[25] P. Macheras, and A. Iliadis Modeling in Biopharmaceutics, Pharmacokinetics, andPharmacodynamics Homogeneous and Heterogeneous Approaches, Springer 2006,New York.

[26] M. Nikolaou, and V. H. Tam A new modeling approach to the effect of antimi-crobial agents on heterogeneous microbial population, J. Math. Biol. 52: 154-182(2006).

[27] I. Odenholt , E. Lowdin, and Cars O., Studies of the killing kinetics of benzylpeni-cillin, cefuroxime, azithromycin, and sparfloxacin on bacteria in the postantibioticphase, Antimicrob Agents Chemother. 41(11): 2522-6 (Nov 1997).

[28] R. R. Regoes, C. Wiuff, R. M. Zappala, K. N. Garner, F. Baquero, and B. R.Levin, Pharmacodynamic functions: a multiparameter approach to the design ofantibiotic treatment regimens, Antimicrobial Agents and Chemotherapy 48(10):3670-3676 (Oct 2004).

33

Page 34: The Pharmacodynamics of Antibiotic Treatmenthalsmith/antibiotic-treatment-new.pdf · The Pharmacodynamics of Antibiotic Treatment ... in which antibiotic levels have attained their

[29] M. E. Roberts, and P. S. Stewart, Modeling antibiotic tolerance in biofilms byaccounting for nutrient limitation, Antimicrobial Agents and Chemotherapy 48(1):48-52 (Jan 2004).

[30] R. S. Schwalbe, C. W. Hoge, J. G. Morris, P. N. O’Hanlon, R. A. Crawford, andP. H. Gilligan., In vivo selection for transmissible drug resistance in Salmonellatyphi during antimicrobial therapy, Antimicrob Agents Chemother. 34 (1): 161-163, (Jan. 1990).

[31] L. Simonsen, The existence conditions for bacterial plasmids: theory and reality,Microbial Ecology 22: 187-205 (1991).

[32] H.L. Smith, and P. Waltman, The Theory of the Chemostat , Cambridge Univer-sity Press, New York (1995).

[33] H.L. Smith Cooperative systems of differential equations with concave nonlin-earties, Nonlinear Anal., Theory, Methods & Appli. 10: 1037-1052 (1986).

[34] E. Stemmons, and H.L. Smith, Competition in a chemostat with wall attachment,SIAM J. Appl.Math. 61: 567-595 (2000).

[35] G. Stephanopoulus, and G. Lapidus, Chemostat dynamics of plasmid-bearingplasmid-free mixed recombinant cultures, Chem. Engng. Sci. 43: 49-57 (1988).

[36] D. Summers, The Biology of Plasmids, Blackwell Science 1996, London (1996).

[37] H.R. Thieme, Persistence under relaxed point-dissipativity (with application to anepidemic model), SIAM J. Math. Anal. 24: 407-435 (1993).

[38] E. Tuomanen., Phenotypic tolerance: the search for beta-lactam antibiotics thatkill nongrowing bacteria., Rev Infect Dis. 8(3): 279-91 (Jul 1986).

[39] E. Tuomanen, R. Cozens, W. Tosch, O. Zak, and A. Tomasz. The rate of killingof Escherichia coli by beta-lactam antibiotics is strictly proportional to the rate ofbacterial growth, J Gen Microbiol. 132(5): 1297-304 (May 1986).

[40] C. Wiuff, R. M. Zappala, R. R. Regoes, K.N Garner, F. Baquero, and B. R.Levin.,Phenotypic tolerance: antibiotic enrichment of noninherited resistance inbacterial populations, Antimicrob Agents Chemother. 49(4):1483-94 (Apr 2005).

[41] X.-Q. Zhao, Dynamical Systems in Population Biology, CMS Books in Mathemat-ics, (Springer, New York, 2003).

34