Modeling Heterogeneous Bacterial Populations Exposed to Antibiotics: The Logistic-Dynamics Case

8/10/2019 Modeling Heterogeneous Bacterial Populations Exposed to Antibiotics: The Logistic-Dynamics Case

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Modeling Heterogeneous Bacterial Populations Exposed to

Antibiotics: The Logistic-Dynamics Case

Pratik R. Bhagunde1, Vincent H. Tam2, and Michael Nikolaou1,3

1Chemical & Biomolecular Engineering Department, University of Houston

2Department of Clinical Sciences and Administration, College of Pharmacy, University of Houston

3Author to whom all correspondence should be addressed. [email protected]

Abstract. In typical in vitrotests for clinical use or development of antibiotics, samples from abacterial population are exposed to the antibiotic at various concentrations. The resulting data

are then used to build a mathematical model suitable for dosing regimen design or for further

development. For bacterial populations that include resistant subpopulations an issue that

has reached alarming proportions building such a model is challenging. In prior work we

developed a related modeling framework for such heterogeneous bacterial populations

following linear dynamics when exposed to an antibiotic. We extend this framework to the

case of logistic dynamics, common among strongly resistant bacterial strains. Explicit formulas

are developed that can be easily used in parameter estimation and subsequent dosing regimen

design under realistic pharmacokinetic conditions. A case study using experimental data fromthe effect of an antibiotic on a gram-negative bacterial population exemplifies the usefulness of

the proposed approach.

KEYWORDS: Antibiotic resistance, nonlinear dynamics, logistic growth, bacterial population,

modeling, cumulant
mailto:[email protected]:[email protected]:[email protected]


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Bhagunde et al. Nonlinear modeling of bacterial population dynamics 2

1 Introduction

The emergence of bacterial strains that resist attacks by antibiotics was observed as early as a

decade after the initial widespread use of penicillin1. By now, bacterial resistance to antibiotics

has reached such alarming proportions2-8

that a task-force was formed in 2014 by US

Government executive order to coordinate efforts for addressing this critical public-health

issue9.

Effective clinical use or development of antibiotics typically requires a series of tests,

one of which involves in vitroexposure of samples from a bacterial population to an antibiotic

at various concentrations (time-kill test). From the outcomes of such tests conclusions are

drawn on how to design effective antibiotic dosing regimens to accommodate the antibiotics

pharmacokinetics in vivo. A mathematical model is often employed to assist the process by

capturing the dynamics of the antibiotics bactericidal activity.

Because a bacterial population may contain bacteria that cannot be killed by anantibiotic, it is common to model the population as two subpopulations, one resistant and one

susceptible to the antibiotic.10,11

Experimental evidence for the presence of resistant bacteria

in a population is easily provided in a time-kill test when, for some antibiotic concentrations,

the bacterial population initially declines, due to killing of susceptible bacteria, and

subsequently starts growing, owing to resistant bacteria taking over.

While this resistant/susceptible modeling dichotomy is conceptually valuable, it may

yield models that provide misleading quantitative information regarding what antibiotic

concentrations would eradicate an entire bacterial population (by preventing eventual growth

of its most resistant subpopulation).12-17 In previous work17 we ascribed that modelingdeficiency to the fact that the bacterial rate of killing by the antibiotic (i.e. susceptibility or

resistance) can take a multitude of values spread over the bacterial population, rather than two

distinct values corresponding to resistant or susceptible subpopulations. Correspondingly, we

developed a mathematical modeling approach that accounts for this fact. In particular, we

showed that the proposed approach can make useful long-term predictions in cases where the

two-subpopulation approach fails.

The modeling approach in our previous work relied on the assumption of linear

dynamics for physiological growth of a bacterial population. As we explain in section 2, this

assumption is reasonable for a bacterial population in decline caused by exposure to an

antibiotic. However, the assumption is not accurate when a population is not in decline,

particularly not far from the populations saturation point. Such a situation may easily arise in

practice, when an antibiotic shows weak bactericidal activity against strongly resistant strains.

In such a situation, full nonlinear description of the bacterial population logisticdynamics not

far from population saturation would offer better accuracy. The objective of this paper is to


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develop the corresponding modeling approach. Specifically, the main idea is to develop for the

logistic-dynamics case the counterpart of results developed earlier for the linear-dynamics case.

While pursuing the main idea, we present newly discovered results that bear significance for

the linear case as well.

In the rest of the paper, background and motivation for this work is presented in moredetail in section2. The main theoretical results are presented in section3. A case study that

illustrates the usefulness of the formulas developed based on laboratory experimental data is

presented in section4,followed by discussion in section5. All proofs and experimental details

are presented in appendices.

2 Background and Motivation

As mentioned above, standard in vitro time-kill experiments expose samples of a bacterial

population to an antibiotic at various concentrations (usually twofold or fourfold dilutions) that

remain constant over time. Starting with a population of 0N similar bacterial cells in an

environment of an antibiotic at a time-invariant concentration C, a standard population

balance assuming logisticphysiological growth yields

max kill ratedue to antibiotic

logistic physiological growth rate

( )( ) 1 ( ) ( )

=

((

((((((g

dN N t K N t r C N t

dt N, 0(0)N N= (1)

where the bacterial population size ( )N t is treated as a continuous variable; gK is the specific

growth rate of the bacterial population; maxN is the saturation size of a bacterial population, at

which further growth stops; and ( )r C is the specific bacterial kill rate due to an antibiotic at

concentration C, assumed to be the same for all bacterial cells. A typical form of ( )r C is the

Hill expression10,18,19

50

( )H

k

H H

K Cr C

C C=

+ (2)

where kK is the asymptotic maximum value of the specific kill rate as C ; 50C is the

antibiotic concentration at which the kill rate reaches 50% of its asymptotic maximum value;

and H is the Hill exponent20

, which determines how sigmoidal (inflected) the shape of ( )r C is.

It is clear that for a population size far from its saturation point (i.e. max( )N t N


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populations are not homogeneous, namely they consist of subpopulations that experience

different killing rate constants ( )r C . (The growth rate constant gK is assumed to be the same

for all subpopulations, due to common physiology among bacteria of the same species, barring

small fluctuations due to bio-fitness cost incurred by resistant strains.) Consequently, eqn. (1)

is no longer valid and must somehow be modified, to account for the distribution of ( )r C overthe bacterial population.

Starting with eqn. (1) for max( )N t N


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3 Results

3.1 General case

Consider a heterogeneous bacterial population for which the bacterial kill rate induced by an

antibiotic at time-invariant concentration Cvaries among bacteria, with more resistant (less

susceptible) bacteria corresponding to lower values of the kill rate constant ( )r C . This setting

is plausible from a physiological viewpoint, given that variability is expected within any given

bacterial species, whether such variability is mild or pronounced. Therefore, the entire

bacterial population can be split into a number of subpopulations indexed by k , each

subpopulation having size ( )kN t , with

( ) ( )kk

N t N t= . (3)

Each subpopulation of size ( )kN t satisfies the counterpart of eqn. (1), namely

( )max

( )1 ( )k g k k

dN N tK r C N t

dt N

=

, 1,2,3,...k= . (4)

Note that the term max1 ( )N t N is common for all subpopulations, because it refers to

competition among bacteria for common resources. Note also that it is important to

In contrast to eqn. (1), of Bernoulli type with the standard closed-form solution

( )0

( )0 0

max max

( )1

( ) ( )e 1g

g

r C K t

g

r C

K

N t N N Nr C

N K N

= + +

, (5)

the set of equations in eqn. (4) does not admit a standard solution for ( )N t . Furthermore,

measuring each subpopulation size ( )kN t is practically infeasible; rather, only measurements

of ( )N t can reasonably be obtained. Therefore, a formula is needed for ( )N t as a function of

time for model parameter estimation. In addition, it is of paramount importance to estimate

the kill rate ( )kr C of the most resistant subpopulation, to ensure that an antibiotic

concentration Cis identified that guarantees eradication of the entire population, including itsmost resistant subpopulation under in vitro or in vivo pharmacokinetic conditions. It should

also be noted that the exact distribution of ( )kr C over the population is of secondary

importance.

Explicit formulas for ( )N t and estimate of ( )kr C for the most resistant subpopulation

are provided next. Note that index keventually drops out from all results.


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Theorem 1 Aggregate dynamics of heterogeneous bacterial population with logistic growth

Assume that a heterogeneous population of 0N bacteria is exposed to an antibiotic at time-

invariant concentration C. The population has a number of subpopulations, each of which

satisfies eqn. (4). Then the aggregate growth or decline of the entire bacterial population is

characterized by the following equations:

max

( )1 ( ) ( )g

dN N t K t N t

dt Nm

=

(6)

2( )d

tdt

m= (7)

2

3( )d

tdt

= (8)

1( )n

nd t

dt += , 3n (9)

where

( )( ) ( )

( )

kk

k

N tt r C

N tm = (10)

and

[ ]22 ( )( ) ( ) ( )

( )

kk

k

N tt r C t

N t m= (11)

are the average and variance, respectively, of the kill rate constant over the entire population

at time t; and ( )n t are the cumulants22, p. 928

of the kill rate constant distribution function

( ) ( )

( ), ( )

kk

N tf r C t

N t= (12)

at time t, defined through the moment- and cumulant-generating functions, ( , )M s t and

( , )s t , respectively, as follows.

( )( )( , ) ( ), ksr C kk

M s t e f r C t= . (13)

[ ] 1!1

( , ) ln ( , ) ( ) n

nn

n

s t M s t t s

=

= = . (14)

0

( , )( )

n

n n

s

s tt

s

=

=

, 1,2,...n= (15)


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with 1( ) ( )t t m= and2

2 ( ) ( )t t = .

Proof: SeeAppendix B.

Remark 1 Comparison of linear and logistic growth dynamics

Interestingly enough, the equations resulting from Theorem 1, proved for logistic-growth

dynamics, are remarkably similar to these in Theorem 1 of Nikolaou & Tam17

. In fact, eqns. (7)-

(9) turn out to be similar, whereas eqn. (6) is of the same nature, i.e. it is the logistic-growth

dynamics counterpart of the linear-growth dynamics equation

( )( ) ( )gdN

K t N t dt

m= (16)

Consequently, the following remarks can be made, which are the counterparts of the linear-

growth dynamics case.

- The first four cumulants ( )n t , 1,...,4n= , are directly related to the average, ,m

variance, 2 , skewness, 33m

, and kurtosis excess, 44 3

m

, of the kill rate constant

distribution, namely

1 m= ,2

2 = , 3 3 m= , and4

4 4 3 m = (17)

where ( )tm

are the central moments defined as

( ) ( )( ) ( ) ( ),k kk

t r t f r C t m m=

, 0 (18)

for the kill rate constant distribution function f .

- Because 2( ) 0t , eqn. (7) implies that the average kill rate constant ( )tm (a measure

of susceptibility or inverse measure of resistance) of a heterogeneous bacterial

population will decrease monotonically with time. Furthermore, because ( ) 0tm ,

( )tm will converge to a non-negative value. This is because more susceptible bacteria

will be preferentially eradicated by the antibiotic over more resistant bacteria, until only

the most resistant bacteria, if any, remain in the population.

Theorem 1 can be used to derive closed-form expressions for the time-dependence of the

bacterial population size as well as of the average and variance of the kill rate constant

distribution, as indicated next.


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Theorem 2 Closed-form expressions for time-dependence of population characteristics

Under the assumptions ofTheorem 1,the total bacterial population, kill rate constant average,

and kill rate constant variance depend explicitly on the moment- and cumulant-generating

functions of the initial distribution of the kill rate constant ( )r C as

0

00 max

( )ln ( , 0) ln 1 e ( , 0)g

t K

g g

NN tK t t K M d

N N

tt t

= + +

, (19)

2 2 31 13 42! 3!

( ,0)( ) (0) (0) (0) (0) ...

s t

st t t t

sm m

=

= = + +

, (20)

22 2 2 31 1

3 4 52! 3!2

( ,0)( ) (0) (0) (0) (0) ...

s t

st t t t

s

=

= = + +

, (21)

respectively. Moreover, the higher-order cumulants satisfy

2 31 11 2 32! 3!

( ,0)( ) (0) (0) (0) (0) ...

n

n n n n nn

s t

st t t t

s + + +

=

= = + +

, 3n (22)

Proof: SeeAppendix C.

Remark 2 Implications ofTheorem 2

- For a population size far from its saturation point, maxN , corresponding to

approximately linear-growth kinetics, eqn. (19) generalizes the result obtained by

Nikolaou & Tam17

as

0

( )ln ( , 0)g

N tK t t

N

= +

(23)

- The terms ( , 0)t and ( , 0)M t of eqn. (19), defined in eqns. (13) and (14),

respectively, refer to the cumulant-generating function and to the moment-generating

function of the initial bacterial population (they depend on the initial values of the

cumulants (0)n ).

- While the initial values (0)n of the cumulants may be practically impossible to

estimate from experimental data alone, it is demonstrated in the following section that

approximations of the initial distribution of the kill rate constant, ( )r C , over the

bacterial population results in convenient expressions that explicitly contain only a few

parameters, which can be estimated from standard experimental data, namely from the

size of the bacterial population at a few distinct points in time, measured during time-

kill experiments.


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3.2 Explicit formulas for specific distributions of the kill rate constant

It is interesting to examine the following two special cases of the above results, namely when

the initial ( )r C approximately follows the normal or Poisson distribution, because closed-form

expressions can be derived.

3.2.1 Normally distributed initial kill rate constant

Strictly speaking, it is impossible to have a distribution of the initial kill rate constant ( )r C that

is exactly normal, because that would entail negative values for ( )r C . Nevertheless, it is

interesting to consider an approximatelynormally distributed ( )r C , because, excluding the first

two cumulants (average and variance), all higher-order cumulants of the normal distribution

are identically22

(0) 0n = , 3n . (24)

In that case, the following explicit formulas can be developed with applicability over limited

time t.

Theorem 3 Explicit formulas for normally distributed initial kill rate constant

For initially normally distributed ( )r C with average m and variance 2 , it follows that

( ) ( )2 2

0

2 2 2 2max

2 212

0

( ) (0)

2 2 2 2

( )ln ( )

ln 1 exp erfi ,g g g

g

K K K t N

g N

N tK t t

N

K m m m p

m

+

= +

+

(25)

where erfi( ) erf( )z iz i= ,2

2

0erf( )

ztz e dt

p

= ,

2( )t tm m = (26)

and

2 2( )t = (27)

for a time period not exceeding

max 2t

m

= . (28)

In addition, for the same time period,

( ) 0n t = , 3n (29)

i.e. the distribution of ( )r C will remain normal for the same time period.

Proof: SeeAppendix D.


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The above normality assumption yields simple explicit formulas and provides reasonable

predictions for relatively short time periods, as has been demonstrated for linear growth

dynamics by Nikolaou & Tam17

. However, the normal distribution assumption is problematic

when predictions over longer periods of time are sought, e.g. because ( )tm , eqn. (26), becomes

negative. A better alternative is presented next, using a Poisson distribution.

3.2.2 Poisson-distributed initial kill rate constant

Assuming an underlying Poisson distribution for the kill rate constant ( )r C over the initial

bacterial population yields explicit formulas that are valid over arbitrarily long time periods and,

because they are explicit, they are convenient for parameter identification from total bacterial

population time-kill data alone. Specifically, the initial kill rate distribution function is assumed

to be

( ) min( )

( ),0 [ ]

!

k

k

r C r ef r C P k

a kX

= = =

((

, 0,1,2,...k= (30)

where the average and variance, 0> , of the Poisson-distributed variable X are related to

the average, m, and variance, 2 , of the kill rate constant, min( ) r C aX r = + , as

min 0a rm = + . (31)

2 2

min( ) 0a a r m= = . (32)

The parameter 0> roughly determines the shape of the discrete unimodal

distribution f , as can be visualized inFigure 2.

Figure 2. Family of the discrete distributions min !( ) [ ] ke

kf r P r r ak

= = + = , 0k , for the kill

rate constant ( )r C parametrized in terms of and minr . All distributions correspond to the

same average 1m= and 1a= . Note that ( ) 0f r = for 0r< .


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Given 0> , the parameters 2 , eqn. (32) and m , eqn. (31), determine the spread and

location of f , respectively. Equivalently, the parameters min 0r and 0a> in eqn. (30) are the

corresponding parameters for translation and dilation of ( )r C , respectively.

Within the above framework, the following explicit formulas can be developed.

Theorem 4 Explicit formulas for Poisson-distributed initial kill rate constant

Given eqn. (30), eqns. (19), (20), and (21) yield

( ) ( )

( ) ( )

( ) ( )

( )min

min

min

0

0min

max 0

min

0

max

( )ln 1

ln 1 exp 1

1

ln 1 , ,g

g

at

g

t

a

g g

at

g

K rr Kg ata

a

N tK r t e

N

NK K r e d

N

K r t e

K Ne e

a N

t

t t

= +

+ +

= +

+

(33)

( ) minmin min

min

( ) exp

at

rt r r t

r ae

mm m

= +

= +

(34)

and

( )2

min2 min

2

( ) exp

at

r rt t

a e

m m

=

=

(35)

respectively, where the parameters min ,r ,a ,m andminr

a

m

= depend on C , and the

incomplete gamma function is defined as

( ) 1

0

1

0 1, , z

c z

zc z z z e dz

= . (36)

Proof: SeeAppendix E.

Remark 3 Implications ofTheorem 4

- The entire bacterial population will be eradicated if and only if min gr K> .


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- The entire population will eventually be dominated by the most resistant subpopulation,

as eqn. (34) implies that minlim ( )t

t rm

= .

- The logarithmic part on the right-hand side of eqn. (33) is negligible when the

population size ( )N t is far from maxN , but becomes significant when maxN is close to

maxN (Figure 3).

Figure 3. Typical shape of ( )0

log NN

for a heterogeneous bacterial population withmin gr K<

(left) andmin gr K> (right) when a saturation bound is absent max( )N = or present max( )N < .

- Following the convention that a declining bacterial population will be completely

eradicated if there exists a time 0t> such that ( ) 1N t = , eqn. (33) implies that such

time will satisfy the equation

( ) ( )0 minln 1 0atgN K r t e + + = (37)

whose solution is

0

min

( ln )

0

min min

expln 1ProductLog

g

a N

K r

g g

aN

K r a K r

(38)

where the function ProductLog( )z is defined23

as the principal solution, w , of the

equationw

z we= .

- Estimating the dependence of minr as a function of the antibiotic concentration C(e.g.

eqn. (2)) is crucial for determining whether a bacterial population will be eventually

eradicated or grow, even if there is an initial decline due to elimination of the more

susceptible subpopulations.


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Figure 5. Data fit for 0.5 MICC= through 128 MICC= (along with 0C= ) using eqn. (33)

Table 1. Estimates of the parameters min ,r , a, for 0.5 MICC= through 128 MICC=

( s denotes standard error)

C

( MIC) minr s

(1/h)

(1/h)

a

(1/h)

0.5 0.58 0.21 0.38 0.11

2 0.89 0.04 1.3 4.1

8 0.97 0.04 11 6.4

32 1.02 0.03 12 4.9

128 1.04 0.02 12 5.4

The estimates of minr for 0.5 MICC= through 128 MICC= can be fitted by eqn. (2)

as shown inFigure 6,with corresponding parameter estimates shown inTable 2.


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Figure 6. Fit (95% confidence zone) of the kill-rate constant minr as a function of antibiotic

concentration C.

Table 2. Estimates of the parametersk

K , 50C , H for the most resistant bacterial subpopulation

Parameter Estimate standard error

kK (1/h) 1.03 0.02

50C ( MIC) 0.39 0.04

H 1.06 0.15

An important outcome of the above results is that an estimate of the antibioticconcentration Ccan be easily calculated that ensures eradication of the entire bacterial

population, including its most resistant subpopulation. Indeed, eradication is guaranteed if

min ( ) gr C K= , which, by eqn. (2), implies

1/

50 2.2

H

k g

g

K KC C

K

=

= ( MIC) (39)

as can be visualized inFigure 6. Thus, for this bacterial population, the antibiotic concentration

that would ensure complete eradication is more than double the conventional MIC, because ofthe subpopulations comprising resistant strains.

5 Discussion

A mathematical modeling framework was developed to describe the effect of antibiotics on

heterogeneous bacterial populations (entailing subpopulations of non-uniform susceptibility or


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resistance to antibiotics). The essence of the framework is captured by Theorem 2, which

generalizes and expands prior work developed for antibiotic/bacteria systems following linear

dynamics17

to the case of logistic dynamics. From a practical viewpoint, Theorem 4 offers

convenient expressions that can be easily used for parameter estimation based on

measurements of an entire bacterial population in time-kill experiments.Because of its generality,Theorem 2 makes it possible to study the outcomes of various

cases, including bimodal distributions.

Acknowledgement. The study was supported in part by the National Science Foundation (CBET-

0730454), National Institutes of Health (1R56AI111793-01) and an unrestricted grant from

AstraZeneca.

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Appendix A Details on experiments performed for time-kill data collection

Time-kill studies were performed with a wild-type Escherichia coli strain (MG 1655) with an

inoculum of approximately 1108CFU/ml at baseline. Apart from the placebo control ( 0)C= ,

five concentrations of moxifloxacin in Mueller-Hinton broth were used with fourfold increase

between subsequent concentrations. All concentrations were normalized to fraction/multiples

of the minimum inhibitory concentration (MIC), equal to 0.0625 g/ml , from 0.5MIC to

128MIC. The experiments were performed in a shaker water bath set at 35C. Serial samples(0.5 ml) were obtained in duplicate over 24 hours at baseline, 2, 4, 8, 12 and 24 hours; viable

bacterial burden was determined by quantitative culture. Before being cultured quantitatively,

the bacterial samples were centrifuged (10000G for 15 minutes at 4C) and reconstituted withsterile normal saline to minimize the drug carryover effect. Total bacterial populations were

quantified by spiral plating 10serial dilutions of the samples (50 ml) onto Mueller-Hinton agar

plates. The media plates were incubated at 35C for up to 24 hours in a humidified incubator,then bacterial density from each sample was enumerated visually.


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Appendix B Proof ofTheorem 1

Summation of eqn. (4) over all i yields

( )max

max

( )1 ( )

( )1 ( ) ( )

g k k

k

g

dN N t K r C N t

dt N

N tK t N t

Nm

=

=

. (40)

which is eqn. (6).

Taking derivatives of eqn. (12) with respect to time and combining it with eqns. (4) and

(6) yields

( )

( )

( ) ( )

2

max

2

max

( )1( ),

( ) ( )

( ) 11 ( )

( )

( ) ( )1 ( ) ( )

( )

( ) ( ) ( ),

k kk

g k k

kg

k k

dN N t d dNf r C t

dt dt N t N t dt

N tK r C N t

N N t

N t N tK t N t

N t N

t r C f r C t

m

m

=

=

=

(41)

Taking derivatives of the moment-generating function, eqn. (13), with respect to time,

and combining the resulting equation with eqn. (41) yields

( ) ( ) ( )

( )

( ) ( )

( )

( ),( , )( ) ( ) ( ),

( ) ( , ) ( ) ( ),

k k

k

sr C sr C k

k k

k k

sr C

k k

k

df r C t M s te e t r C f r C t

t dt

t M s t e r C f r C t

m

m

= =

=

. (42)

Taking derivatives of the cumulant-generating function, eqn. (14), with respect to time,

and combining the resulting equation with eqn. (41) yields


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( )

[ ]

( )

( , ) 1 ( , )

( , )

1( ) ( , ) ( ) ( ),

( , )

1 ( , )( )( , )

ln ( , )( )

( , )( )

ksr C

k k

k

s t M s t

t M s t t

t M s t e r C f r C t M s t

M s ttM s t s

M s tt

s

s tt

s

m

m

m

m

=

=

=

=

=

(43)

Expanding ( , )s t around 0s= using Taylor series as

1 2 3

/ / / 2 / / / 3

0 ( ) ( ) ( )

2 3

1 2 3

1 1( , ) (0, ) (0, ) (0, ) (0, ) ...

2! 3!1 1

( ) ( ) ( ) ...2! 3!

t t t

s t t t s t s t s

t s t s t s

= + + + +

= + + +

(( (( (( (44)

and using the definition of cumulants in eqn. (15) yields

2 331 2( , ) 1 1 ...2! 3!

dd ds ts s s

t dt dt dt

= + + +

(45)

and

2 3

1 2 3 4

( , ) 1 1

( ) ( ) ( ) ( ) ...2! 3!

s t

t t s t s t ss

= + + + (46)

Substituting the expansions of eqns. (45) and (46) into eqn. (43) implies

2 331 2

2 3

1 2 3 4

0

1 1...

2! 3!

1 1( ) ( ) ( ) ( ) ( ) ...

2! 3!

dd ds s s

dt dt dt

t t t s t s t s

m

=

+ + + =

((

(47)

Equating like powers of s in the above equation yields

1( )n

nd t

dt += , 1n (48)

which is eqn. (7) for 1n= , eqn. (8) for 2n= , and eqn. (9) for 3n .


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Appendix C Proof ofTheorem 2

Eqn. (48) implies

1 1

2 2

3 3

( )

/ ( )0 1 0

/ ( )0 1

0/ ( )

d dt t

d dt t

d dt t

d dt t

=

J

((((((((

. (49)

The corresponding solution of the above system of eqns. (49), which is in Jordan form, is

[ ] 1!0

( ) exp (0) (0) k k

k

k

t t t

=

= = J J , (50)

from which

2 31 11 1 2 3 42! 3!( ) (0) (0) (0) (0) ...t t t t = + + (51)

or

2 2 31 13 42! 3!

( ,0)( ) (0) (0) (0) (0) ...

s t

st t t t

sm m

=

= + + =

(52)

which is eqn. (20). Similarly,

2 31 12 2 3 4 52! 3!( ) (0) (0) (0) (0) ...t t t t = + + (53)

or

22 2 2 31 1

3 4 52! 3! 2

( ,0)( ) (0) (0) (0) (0) ...

s t

st t t t

s

=

= + + =

(54)

which is eqn. (21), and

2 31 11 2 32! 3!

( ,0)( ) (0) (0) (0) (0) ...

n

n n n n n n

s t

st t t t

s + + +

=

= + + =

, 3n (55)

which is eqn. (22).

Next, integration of the Bernoulli differential equation in eqn. (6) implies

[ ]00

0 max

( )ln ( ) ln 1 exp ( )

t

g

NN tG t K G d

N Nt t

= +

(56)

where


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0( ) ( )

t

gG t K d m = . (57)

which, by eqn. (20), becomes

( ) ( ,0)gG t K t t = + . (58)

Therefore, substituting ( )G t from the above eqn. (58) into eqn. (56) results in

0

00 max

0

0max

( )ln ( ,0) ln 1 exp ( ,0)

( ,0) ln 1 e ( ,0)g

t

g g g

t K

g g

NN tK t t K K d

N N

NK t t K M d

N

t

t t t

t t

= + + +

= + +

, (59)

which is eqn. (19).


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Appendix D Proof ofTheorem 3

Equations (27) and (26) follow immediately from eqns. (20) and (21), along with eqn. (24).

Combining eqns. (19) and (44) with eqn. (24) yields

2 212!

0

2 20 12!0

max

2 212

2 20 120

max

( )ln ( ) ( )

ln 1 exp ( ) ( )

( )

ln 1 exp ( )

g

t

g

g

t

g g

N t K t t t N

NK d

N

K t t

NK K d

N

m

m t t t

m

m t t t

= + +

+ +

= +

+ +

(60)

which immediately yields eqn. (25).

Equation (29) follows immediately from eqns. (24) and (22).

Finally, eqn. (28) follows immediately from eqn. (26) and the fact that ( ) 0tm .


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Appendix E Proof ofTheorem 4

The proof follows straightforward facts on moment- and cumulant-generating functions for

Poisson distributions and is provided for completeness.

Given eqn. (30), eqn. (13) implies

( )

( )

min

( )

0

( )

0

min

minmin

( , 0) ( ), 0

!

exp 1

exp exp 1

ktr C

k

k

kt ak r

k

at

M t e f r C

ee

k

r t e

rr t t

m

+

=

=

= +

= +

(61)

which, combined with eqn. (14), implies that the cumulant-generating function is

[ ]

( )

minmin

min

( , 0) ln ( , 0)

exp 1

1at

t M t

rr t t

r t e

m

=

= +

= +

(62)

Substituting eqns. (61) and (62) into eqn. (19) immediately yields eqn. (33).

To get a closed-form expression in terms of standard functions for the integral in the

nonlinear term of eqn. (33), consider the transformation

a

x e t= (63)

to get

( ) ( )

( )

( )

min

min min

min

min

min0

min

1

11

1

exp 1

exp ln 1

, ,

at

g

at

g g

at

g

g

ta

g

eg

r K

xe aa

e

K r r K

ze a aa

e

K rr K ate a

a a

K r e d

K r dxx x

a ax

x e dx

z e dz

e

t

t t

+

= +

=

=

=

(64)

leading to


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( ) ( )

( )min

min

0

00 max

min

0

max

( )ln ( ,0) ln 1 e ( ,0)

1

ln 1 , ,

g

g

g

t K

g g

at

g

K rr Kg

ataa

NN tK t t K M d

N N

K r t e

K Ne e

a N

t

t t

= + +

= +

+

. (65)

Modeling Heterogeneous Bacterial Populations Exposed to Antibiotics: The Logistic-Dynamics Case

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