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Evolutionary Dynamics of Complex

Networks of HIV Drug-Resistant Strains:

The Case of San Francisco

Robert J. Smith?,1*† Justin T. Okano,1† James S. Kahn,2Erin N. Bodine,1‡ Sally Blower1§

Over the past two decades, HIV resistance to antiretroviral drugs (ARVs) has risen to high levels in the

wealthier countries of the world, which are able to afford widespread treatment. We have gained

insights into the evolution and transmission dynamics of ARV resistance by designing a biologically

complex multistrain network model. With this model, we traced the evolutionary history of ARV

resistance in San Francisco and predict its future dynamics. By using classification and regression trees,

we identified the key immunologic, virologic, and treatment factors that increase ARV resistance.

Our modeling shows that 60% of the currently circulating ARV-resistant strains in San Francisco are

capable of causing self-sustaining epidemics, because each individual infected with one of these strains

can cause, on average, more than one new resistant infection. It is possible that a new wave of

ARV-resistant strains that pose a substantial threat to global public health is emerging.

H

States and Europe. HIV strains began to acquire

resistance in 1987 when ARVs were introduced

as therapies for HIV-infected individuals (1).

Since then, a multitude of drug-resistant strains

have evolved that differ considerably in their

susceptibility to three major classes of ARVs:

nucleoside reverse-transcriptase inhibitors (NRTIs),

non-nucleoside reverse-transcriptase inhibitors

(NNRTIs), and protease inhibitors (PIs). These

drug-resistant strains are now being transmitted

to individuals who have never received ARVs;

that is, transmitted drug resistance (TDR) has

arisen. TDR is reported to range between 8 and

22% in many HIV-infected communities in

resource-rich countries, and if it continues to

increase, the effectiveness of therapeutic regi-

mens, as well as efforts to control the HIV

pandemic, will be compromised. We have

developed a theoretical model (the amplification

cascade model) to help understand and predict

the evolutionary dynamics of complex transmis-

sion networks composed of multiple ARV-resistant

strains. We calibrated and parameterized the

model to represent the HIV epidemic in San

Francisco in the community of men who have

sex with men (MSM), where TDR is already

high (~13%) (2). The model was able to

reproduce the observed dynamics and evolution

IVresistance to antiretroviral drugs(ARVs)

is causing serious clinical and public

health problems throughout the United

of transmitted resistance in this city over the

past 20 years. We used the model first to predict

the future evolutionary dynamics of TDR. Next,

we determined whether any of the currently

circulating ARV-resistant strains are capable of

generating self-sustaining epidemics. Third, we

identified the key drivers that generate high

levels of TDR. We also discuss here the im-

plications of our results for resource-constrained

countries where ARV treatment programs are

being rolled out.

All of the published HIV transmission

models of ARV resistance are based on simple

biological assumptions and can track only one

resistant strain (3–8). Our amplification cascade

model captures biological complexity by gen-

erating a dynamic network composed of multiple

ARV-resistant strains. We modeled the multistrain

network in San Francisco by classifying ARV-

resistant strains into seven categories; each

category was defined based on the specific class

of drugs to which the strain was resistant

(NRTIs, NNRTIs, or PIs) and the level of

resistance (single-, dual-, or triple-class) (Fig. 1A

and fig. S1). Single-class resistance was to

NRTIs, NNRTIs, or PIs. Dual-class resistance

was to NRTIs and NNRTIs, NRTIs and PIs, or

NRTIs and PIs. Triple-class resistance was to all

three. Each class of ARVs contains several drugs

(table S1) (9). In our modeling framework, if a

strain is classified as resistant to a certain class of

ARVs, then the strain is resistant to at least one

drug in that class.

We modeled treatment effects by specifying

treatment regimens and then assessing the ef-

fects of these regimens on infectivity and the

probability of developing resistance. In the model,

treated individuals receive a regimen to which

their virus is sensitive; hence, we assume that

treated individuals achieve either complete or

partial viral suppression. We consider patients

who achieve complete viral suppression to be

noninfectious and incapable of developing re-

sistance. Patients who achieve only partial viral

suppression retain some degree of infectivity and

are capable of developing resistant strains. When

individuals experience treatment failure (which

is usually determined by viral rebound), they can

beswitchedtonewdrugseitherinthesame class

or in a new class. For example, if a patient (in

themodel)isonaregimencontainingzidovudine

(NRTI), lamivudine (NRTI), and nelfinavir (PI)

and develops resistance to nelfinavir, he could be

switched to another PI (for example, indinavir).

Themodelincludesamatrixthatspecifiestherates

at which strains develop resistance; therefore,

strains are directly linked through the acquisition

and amplification of resistance.

In the model, resistant and wild-type strains

are assumed to compete to transmit HIV to un-

infected at-risk MSM. These competitive inter-

actions are mediated through strain-specific

infectivity: The greater the infectivity, the higher

the probability that the strain will be transmitted.

We ascribe a competitive advantage to wild-type

strains by assuming that they are always more

infectious than the resistant strains. Furthermore,

based on available competitive-fitness assays,

replication-capacity assays, and patterns of de-

veloped resistance, we assume that the NNRTI-

resistant strains are more transmissible than the

NRTI-resistant strains, which, in turn, are more

transmissiblethanthePI-resistantstrains(10,11).

In addition, we assume, based on the available

data, that the transmissibility of virus strains de-

creases as the number of classes of resistance in-

creases (12). Once an individual becomes infected

with a wild-type or resistant strain, the model

tracks viral dynamics, and consequently infectiv-

ity,throughfour stagesofdiseaseprogression:(i)

primary infection; (ii) not yet eligible for ARVs

(that is, CD4 count > 350 cells/ml); (iii) eligible

for ARVs (CD4 ≤ 350 cells/ml) but not currently

undergoing ARV treatment; and (iv) ARV treat-

ment.The33equationsthatspecifythemodel,as

well as a more detailed description of the struc-

ture, are given in (9). Parameter estimates are

discussedinsection2of(9),tablesS2toS11,and

fig. S2. The model can be extended to include

any number of additional drug classes, such as

integrase inhibitors, co-receptor blockers, and fu-

sion inhibitors, as they are introduced into new

therapeutic regimens.

Beforemaking predictions,weused the mod-

el (coupled with an uncertainty analysis) to re-

construct the evolution and transmission dynamics

of the network of ARV-resistant strains (13). We

calibrated the model, using Monte Carlo filtering

techniques, to match the epidemiological condi-

tions in San Francisco in 1987 when ARVs were

first introduced [section 3 of (9)] (table S12). By

the late 1980s, almost half of the MSM com-

munity was infected with HIV (14, 15). After

calibration, we used the model to simulate the

evolutionary dynamics, from 1987 to 2008, of a

network of ~4000 resistant strains, where each

strain differed in drug susceptibility and infectiv-

ity. The history of ARV therapy in San Francisco

canbedividedintofourerasspanningtwodecades

1Center for Biomedical Modeling, Semel Institute of Neu-

roscience & Human Behavior, David Geffen School of

Medicine, University of California Los Angeles, Los Angeles,

CA 90024, USA.

University of California San Francisco, San Francisco, CA 94110,

USA.

*Present address: Department of Mathematics and Faculty of

Medicine, University of Ottawa, Ottawa, K1N 6N5, Ontario,

Canada.

†These authors contributed equally to this work.

‡Present address: Department of Mathematics, University

of Tennessee, Knoxville, TN 37996, USA.

§To whom correspondence should be addressed. E-mail:

sblower@mednet.ucla.edu

2Department of Medicine AIDS Division,

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(9, 16) (fig. S3 and table S1). Different regimens

were used in each era. We modeled the specific

regimens that were available in each era by using

data on the proportion of patients achieving viral

suppression (tables S6 to S9), degree of reduction

in viral load in partially virally suppressed pa-

tients (tables S2 and S5), rate of development of

resistance in treated patients (tables S6 to S9),

and treatment-induced increase in survival time

(table S10) (9). Because usage of ARVs has in-

creased over the past two decades, we modeled

era-specific treatment rates (table S4) (9).

The model reproduced and explained the ob-

served evolutionary dynamics of the network of

ARV-resistant strains over the four treatment eras

(Fig. 1B). The first era began in 1987 when AZT

(azidothymidine, an NRTI) was introduced as a

monotherapy. AZTwas used by a high proportion

(36 to 68%) of MSM in San Francisco (17–19).

Single-class resistance to NRTIs arose quickly

(1), because they were ineffective at suppressing

viral loads (19). In 1992, the second era began

when dual therapies (based on two NRTIs) were

introduced. These therapies were substantially

more effective than monotherapies and achieved

30to60%viralsuppression(20,21).Single-class

resistance to NRTIs decreased, but dual-class re-

sistance quickly developed, because many indi-

viduals had previously developed resistance to

AZT. In 1996, the third era [early highly active

antiretroviral therapy (HAART)] began when

NNRTIs and PIs were used in triple-therapy

regimens. Resistance to PIs was slow to emerge

andhasonlyrisentolowlevels,becausemultiple

mutations are necessary to develop resistance to

most drugs in this class (22). By 2001, more

effective triple therapies (characterized by dual

PIs combined with NRTIs) were developed,

marking the beginning of the fourth era (modern

HAART).Duringthisrecentera,theoveralllevel

of TDR appears to have stabilized (2); the model-

generatednetworkalsoexhibitsthisbehavior(Fig.

1B). Recent empirical data from San Francisco

indicate that transmission of single-class resistance

is high, that of dual-class is moderate, and that of

triple-class is low. In addition, studies indicate that

transmission of NNRTI resistance is greater than

that of NRTI resistance, which is greater than

transmission of PI resistance. The model-generated

transmission network shows these same patterns

(Fig. 1, C and D). Our modeling estimates the

overall level of TDR in 2008to be 14% [median:

interquartile range (IQR) 11.4 to 16.5%] (Fig.

1C), which is in extremely close agreement with

empirically derived estimates of 13 to 16% (2).

After reconstructing the historical epidemiol-

ogy up to 2008, we simulated the amplification

cascade model for 5 more years to predict the

levels of TDR in 2013. Our simulations revealed

that resistance to single-class NRTIs and PIs will

remainatcurrentlevels,butNNRTIresistancewill

increase(Fig.2A).Regressionanalysisdetermined

that the degree of increase in NNRTI resistance

will depend (P < 0.05) on the proportion of pa-

tients who are infected with wild-type strains and

are being treated with a regimen of two NRTIs

and one NNRTI and who achieve viral suppres-

sion (Fig. 2B). This proportion depends on the

efficacy of the regimen and adherence to it; thus,

if only 70% are virally suppressed, NNRTI re-

sistance could increase by more than 30% (Fig.

2B). This increase is predicted to be mainly due

totransmissionfromuntreatedindividualsinfected

with NNRTI-resistant strains who are in either the

acute or chronic stage of infection.

The value of a strain’s control reproduction

number Rcspecifies the average number, based

on the probability that the individual is treated,

of secondary HIV infections that an individual

generates during their entire infectious period.

Rcis a measure of a strain’s transmission po-

tential. A strain is capable of generating a self-

sustaining epidemic if Rc> 1. The Rcs of the

currently circulating ARV-resistant strains in

San Francisco vary considerably (Fig. 3A).

However, strains fall into three mutually exclu-

sive groups (Fig. 3B) [section 4 of (9)]. Almost

a quarter (24%) of the strains (Fig. 3B) cause

less than one new infection (Rc< 1) and will

eventually be eliminated (blue). Although other

strains (Fig. 3B) also cause, on average, less

than one new infection (Rc < 1), they will

continue to be transmitted, because they evolve

greater levels of resistance (green). We esti-

mated that 60% of resistant strains have an Rc>

1 (Fig. 3B; red). Approximately 75% of these

resistant strains have single-class resistance to

NNRTIs, and 20% have dual-class resistance to

NNRTIs and NRTIs. Although all have the

potential to cause self-sustaining epidemics of

resistance, they are all less infectious than the

wild-type strains in San Francisco (Fig. 3C).

Similar trends for TDR to those observed in

San Francisco and those predicted by our model

have been documented in other cities in the

Fig. 1. (A) Schematic diagram of the multiple pathways in the amplification cascade model by which

strains can acquire resistance. Strains may develop single-class resistance to NRTIs (blue), single-class

resistance to NNRTIs (red), single-class resistance to PIs (purple), dual-class resistance to NRTIs and

NNRTIs (green), dual-class resistance to NRTIs and PIs (orange), dual-class resistance to NNRTIs and PIs

(yellow), or triple-class resistance (brown). Wild-type strains are shown in gray. There are six possible

paths by which strains can develop triple-class resistance. (B) Representative simulation generated by

the amplification cascade model to show the evolution of ARV resistance in the MSM community in

San Francisco. Color coding same as in (A). (C) Estimated levels of TDR in 2008 using Monte Carlo

simulations from the uncertainty analysis of the amplification cascade model. Single-class resistance is

8.5% (median: IQR 6.8 to 9.8%) (red), dual-class resistance is 4.5% (median: IQR 3.5 to 5.8%)

(green), and triple-class resistance is 1.0% (median: IQR 0.7 to 1.3%) (blue); overall levels of TDR are

in black. (D) Box plots of estimated levels of TDR in 2008 based on Monte Carlo simulations from the

uncertainty analysis of the amplification cascade model. Color coding same as in (A). Horizontal black

lines represent medians; boxes show IQR.

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United States and Europe that have analogous

histories of ARV therapy. Potentially NNRTI-

resistant strains similar to those we have identified

in San Francisco may be increasing elsewhere.

AlthoughtheNNRTI-resistantstrainsthatwehave

identified are causing the rising wave of NNRTI

resistance, they are unlikely to lead to self-

sustaining epidemics in San Francisco or other

communities in resource-rich countries, because

new drugs will continue to become available.

However our results may have important impli-

cations for HIV treatment programs in resource-

constrained countries, where second-line regimens

arenotgenerallyavailable.NNRTI-resistantstrains

are already evolving in many of these countries,

becausetheirfirst-lineregimensarebasedontwo

NRTIs plus one NNRTI. Our current predictions

have been obtained by modeling the evolution of

resistance in individuals infected with subtype B

strains. Subtype B accounts for ~12% of world-

wide infections (and persons with subtype B are

the most ARV-experienced), but 50% of preva-

lent HIV infections and 47% of all new HIV

infections worldwide are caused by subtype C

(23). Although information is limited, prelimi-

nary data suggest that treatment response and

resistance patterns for subtype C are similar to

those of subtype B (24). These data suggest that

our results are likely to be generalizable to an

epidemic of HIV-1 resistance among individuals

infected with HIV-1 subtype C, and NNRTI-

resistant strains with Rc> 1 could emerge in

resource-constrained countries. If the Rcof the

wild-type strains is reduced below one, as could

occur by using a universal testing and treatment

strategy (25), self-sustaining epidemics of NNRTI-

resistant strains could arise (Fig. 3B and fig. S5)

[section 5 of (9)].

Current levels of TDR, as well as the bio-

logical composition of the complex multistrain

network, have emerged from two decades of treat-

ment. To identify the key drivers of ARV resist-

ance, we constructed classification and regression

trees(CART)(26)usingthe20-yeardataset(1987

to 2008) that was generated during the uncertainty

analysis of the amplification cascade model. To

build trees, we used the model's estimated level of

TDR for 2008 as the response variable and the

model’s 50 parameters as predictor variables [sec-

tion 6of (9)].The optimaltree revealedthe hidden

hierarchical structure of the data (Fig. 4). Key

drivers of TDR are the predictor variables with

Fig. 2. (A) Predictions

showing that transmis-

sion of strains that are

resistant to NNRTIs will

increase in San Fran-

cisco over the next 5

years. Predictions were

made using Monte Carlo

simulations from the un-

certainty analysis of the

amplification cascade

model. Red lines show

no increase in NNRTIs or

NRTIs over the next 5

years. Blue line indicates

an equal increase of

NNRTIs and NRTIs over

thenext5years.(B)Pre-

dictedincreaseinthelev-

el of transmitted NNRTI

resistanceinSanFrancisco

over the next 5 years as a function of the proportion of patients (who are infected with wild-type strains and are being treated with a regimen of two NRTIs and one

NNRTI) who achieve viral suppression. Predictions were made using Monte Carlo simulations from the uncertainty analysis of the amplification cascade model.

Fig. 3. (A) Box plots of the control reproduction

numbers (Rc) for all seven categories of ARV-

resistant strains in the amplification cascade model.

Color coding is the same as in Fig. 1A. Horizontal

black lines represent medians; boxes show IQR. (B)

Classification of ARV-resistant strains into three

mutually exclusive groups based on their transmis-

sion potential: resistant strains that cause less than

one new infection (Rc< 1) and will eventually be

eliminated (blue), resistant strains that cause less

than one new infection (Rc< 1) but will continue to

be transmitted (green), and resistant strains capable

of causing self-sustaining epidemics (Rc> 1) (red).

(C) Density functions showing the likelihood of

different values of TDR occurring. Dotted curves

show density functions for the relative transmissibil-

ity for all of the strains with single-class resistance to NNRTIs (red: median 86%, IQR 81 to 89%) and

dual-class resistance to NRTIs and NNRTIs (green: median 69%, IQR 60 to 76%) that are circulating in

the current network in San Francisco. Solid curves show density functions for the relative transmissibility

of NNRTI-resistant strains with Rc< 1: single-class NNRTIs (red: median 87%, IQR 83 to 90%) and

dual-class NRTIs and NNRTIs (green: median 82%, IQR 78 to 86%). Transmissibility is defined relative

to the wild type.

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thehighestimportancescores(IS)(tableS13)(9).

The most important driver (IS = 100) is the aver-

agetime(atthepopulationlevel)ittakesforCD4

cell counts in infected individuals to fall below

350 cells/ml (v–1) (Fig. 4). TDR was significantly

higher(>15%)whenCD4countsfelltothisthresh-

old within ~6 years than when counts fell more

gradually(Fig.4andfig.S6A)(9).Thisoccurred

becausefasterimmunologicaldeteriorationledto

increased treatment rates and accelerated the ac-

quisition of resistance; hence, TDR increased as

v–1decreased.

A high proportion of the transmission of

wild-type strains over the past 20 years has

occurred from asymptomatic individuals with a

CD4 count > 350 cells/ml (fig. S6B) (9).

Consequently, aH

in asymptomatic individuals, has been the

second key driver of TDR (IS = 73) (Fig. 4);

infectivity is defined in terms of the probability

of transmitting HIV per sex act. These results

can be understood in terms of classical compe-

tition theory (27): The most infectious wild-type

strains had the greatest advantage over resistant

strains andhence causedthe lowestlevels of TDR.

A recent review of empirical estimates of the

transmission probability per sex act indicates that

1, the infectiousness of strains

aH

tree (Fig. 4) reveals that if wild-type strains had

been less infectious (specifically, aH

would have been very likely (probability 0.71)

that TDR in San Francisco would be even higher

than the current level (fig. S6C) (9).

We found aT

under treatment pressure, to be the third key

driver of TDR (IS = 60) (Fig. 4). This driver

represents the probability that an individual who

is receiving current ARV regimens transmits

HIV during one sex act. In contrast to our

previous finding for aH

higher (>15%) when wild-type strains were

more infectious (aT

were less infectious (aT

paradoxical result cannot be understood in terms

of classical competition theory (27). It occurred

because the effect of evolution on network

dynamics was greater than that of competition.

Under treatment pressure, the most infectious

wild-type strains (aT

into the most infectious resistant strains;aT

had a minor effect on competition, because

treated individuals were relatively unimportant

in transmitting wild-type strains (fig. S6D) (9).

The value of aT

1is likely to be greater than 0.0024 (28). The

1≤ 0:0024), it

1, the infectiousness of strains

1, TDR was significantly

1> 0:0015) than when they

1≤ 0:0015) (Fig. 4). This

1> 0:0015) tended to evolve

1only

1can be translated into viral load

(fig. S2) [section 2 of (9)]; a value of 0.0015

corresponds to a viral load of 20,000 copies/ml.

Effective therapies used in recent years have re-

duced viral loads in patients infected with wild-

type strains to well below 20,000 copies/ml (29),

indicating that aT

than 0.0015. Given these effective treatments,

our tree shows it is highly unlikely (probability

0.22)thatTDRinSanFranciscocouldhaverisen

to more than 15% by 2008 (Fig. 4).

Our CARTanalysis also identified four other

parameters that are important drivers of TDR,

including the relative transmissibility of strains with

single-class resistance to NRTIs (l2) (IS = 51), the

degree of viral suppression in patients who are

infected with wild-type strains and not complete-

ly virologically suppressed (g1) (IS = 45), the

relative transmissibility of strains with dual-class

resistance to NRTIs and NNRTIs (l5) (IS = 40),

and finally the degree of viral suppression in

patients who are infected with strains that have

single-class resistance to NNRTIs and are not

completely virologically suppressed (g3) (IS = 39).

None of the 43 other predictor variables was

found to be important (IS < 30). The tree shows

that TDR has remained below 15% because of

specific immunologic, virologic, and treatment

factors operating in San Francisco (Fig. 4).

The amplification cascade model can be

recalibrated and reparameterized to assess the

dynamics of networks of ARV-resistant strains

of HIVin any setting where ARVs are available.

We have applied it to San Francisco. We have

shown that a complex network of HIV strains

has arisen in this city due to two decades of

sequential selection for resistance; first with

single agents, then dual agents, and, more re-

cently, a combination of multiple-class agents.

By designing a biologically complex multistrain

network model, we have obtained important

insights into the otherwise hidden dynamics of

drug-resistant strains of HIV. We have identified

the key immunological, virological, and treat-

ment variables, as well as the hierarchical

interactions among these variables, which have

had a key role in driving resistance. Our results

have shown that effective treatments have

prevented TDR from increasing to greater than

15% in San Francisco. However, our modeling

shows that the network is continuing to evolve.

We found that the majority of the resistant

strains currently being transmitted in this city

are capable of causing self-sustaining epidemics,

and we have estimated that an individual with

an NNRTI-resistant strain can cause, on average,

more than one new infection. We predict that a

wave of NNRTI-resistant strains will emerge

over the next 5 years in San Francisco due to

transmission from untreated individuals. Our

results also have implications for resource-

constrained countries where first-line regimens

are based on NNRTIs. If the resistant strains we

have identified in our analyses evolve in these

countries, they could substantially compromise

HIV treatment programs. Consequently, current-

1is (and was) significantly less

Fig. 4. A pruned version of the optimal tree. The root node contains data from the 3827 filtered Monte

Carlo simulations that were generated by the amplification cascade model; filtered simulations are after

model calibration [section 3 of (9)]. Inside each node is the total number of simulations it contains (N),

as well as the distribution of the response variable TDR. Low levels of TDR (<15%) are blue, whereas

high levels of TDR (>15%) are red. The most important variable (IS = 100) is v–1, the average time (at

the population level) it takes for CD4 cell counts in infected individuals to fall below 350 cells/ml. The

variable aH

infection, where infectivity is specified as the probability of transmitting HIV during one sex act. The

variable aT

during one sex act. The remaining variables are as follows: the transmissibility of strains (relative to the

wild type) with single-class resistance to NRTIs (l2), the degree of viral suppression in patients who are

infected with wild-type strain and are not completely virologically suppressed (g1), and the degree of

viral suppression in patients who are infected with strains that have single-class resistance to NNRTIs

and are not completely virologically suppressed (g3). Because the pruned tree is a subtree of the

optimal tree, not every variable deemed important appears in it. The optimal tree has 84% predictive

power in correctly identifying which simulations will generate high levels of TDR and 82% predictive

power in correctly identifying which simulations will generate low levels of TDR.

1reflects the degree of infectivity of wild-type strains during the asymptomatic stage of

1represents the probability that an individual receiving a current ARV regimen transmits HIV

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ly circulating NNRTI-resistant strains in San

Francisco pose a great and immediate threat to

global public health.

References and Notes

1. B. A. Larder, G. Darby, D. D. Richman, Science 243, 1731

(1989).

2. H. M. Truong et al., AIDS 20, 2193 (2006).

3. S. M. Blower, H. B. Gershengorn, R. M. Grant, Science

287, 650 (2000).

4. J. Goudsmit et al., AIDS 15, 2293 (2001).

5. A. Phillips, Nat. Med. 7, 993 (2001).

6. E. Tchetgen, E. H. Kaplan, G. H. Friedland, J. Acquir.

Immune Defic. Syndr. 26, 118 (2001).

7. R. Vardavas, S. Blower, J. J. Miranda, PLoS ONE 2, e152

(2007).

8. G. S. Zaric, M. L. Brandeau, A. M. Bayoumi, D. K. Owens,

Simulation 71, 262 (1998).

9. Materials and methods are available as supporting

material on Science Online.

10. C. L. Booth, A. M. Geretti, J. Antimicrob. Chemother. 59,

1047 (2007).

11. J. Martinez-Picado, M. A. Martínez, Virus Res. 134, 104

(2008).

12. L. Ross, N. Parkin, R. Lanier, AIDS Res. Hum. Retroviruses

24, 617 (2008).

13. S. Blower, H. Dowlatabadi, Int. Stat. Rev. 62, 229 (1994).

14. W. Lang et al., JAMA 257, 326 (1987).

15. W. Winkelstein Jr. et al., JAMA 257, 321 (1987).

16. J. K. Louie, L. C. Hsu, D. H. Osmond, M. H. Katz,

S. K. Schwarcz, J. Infect. Dis. 186, 1023 (2002).

17. W. Lang, D. Osmond, K. Page-Bodkin, A. Moss,

W. Winkelstein Jr., J. Acquir. Immune Defic. Syndr. 6, 191

(1993).

18. W.Langetal.,J.Acquir.ImmuneDefic.Syndr.4,713(1991).

19. J. C. Schmit et al., AIDS 12, 2007 (1998).

20. J. J. Eron et al., N. Engl. J. Med. 333, 1662 (1995).

21. C. Katlama et al., JAMA 276, 118 (1996).

22. D. R. Kuritzkes, AIDS Patient Care STDS 18, 259 (2004).

23. B. S. Taylor, M. E. Sobieszczyk, F. E. McCutchan,

S. M. Hammer, N. Engl. J. Med. 358, 1590 (2008).

24. A. J. Kandathil et al., Indian J. Med. Microbiol. 27, 231

(2009).

25. R. M. Granich, C. F. Gilks, C. Dye, K. M. De Cock,

B. G. Williams, Lancet 373, 48 (2009).

26. L. Breiman, J. H. Freidman, R. A. Olshen, C. J. Stone,

Classification and Regression Trees (Chapman & Hall,

Boca Raton, FL, 1984).

27. C. Darwin, The Origin of Species (Signet, London,

new ed. 1, 2003).

28. K. A. Powers, C. Poole, A. E. Pettifor, M. S. Cohen, Lancet

Infect. Dis. 8, 553 (2008).

29. S. M. Hammer et al., JAMA 300, 555 (2008).

30. R.J.S., J.T.O., E.N.B., and S.B. acknowledge the financial

support of the National Institute of Allergy and Infectious

Diseases (NIAID) (RO1 AI041935). R.J.S. is supported

by a Natural Sciences and Engineering Research Council

of Canada discovery grant, an Early Researcher award,

and funding from Mathematics of Information

Technology and Complex Systems. In addition, S.B.

acknowledges the John Simon Guggenheim Foundation,

the National Academies Keck Foundation, and the Semel

Institute for Neuroscience & Human Behavior. J.S.K.

acknowledges NIAID grants P30-AI27763, NCRR

K24RR024369, and AHRQ R18-HS017784. We thank

R. Breban, D. Freimer, J. Freimer, N. Freimer, N. Jewell,

E. Kajita, T. Pylko, V. Supervie, and R. Vardavas for useful

discussions throughout the course of this research.

Supporting Online Material

www.sciencemag.org/cgi/content/full/science.1180556/DC1

Materials and Methods

Figs. S1 to S6

Tables S1 to S13

References

13 August 2009; accepted 16 December 2009

Published online 14 January 2010;

10.1126/science.1180556

Include this information when citing this paper.

Optimal Localization by

Pointing Off Axis

Yossi Yovel,1Ben Falk,2Cynthia F. Moss,2Nachum Ulanovsky1*

Is centering a stimulus in the field of view an optimal strategy to localize and track it? We

demonstrated, through experimental and computational studies, that the answer is no. We trained

echolocating Egyptian fruit bats to localize a target in complete darkness, and we measured the

directional aim of their sonar clicks. The bats did not center the sonar beam on the target, but instead

pointed it off axis, accurately directing the maximum slope (“edge”) of the beam onto the target.

Information-theoretic calculations showed that using the maximum slope is optimal for localizing the

target, at the cost of detection. We propose that the tradeoff between detection (optimized at

stimulus peak) and localization (optimized at maximum slope) is fundamental to spatial localization and

tracking accomplished through hearing, olfaction, and vision.

M

is this more evident than in echolocating bats

(4, 7–10), which control many aspects of their

sonar signal design (4, 7, 9, 11–16) and use

returning echoes to orient and forage in the dark

(4, 7–16). We trained Egyptian fruit bats to fly

in a large flight room and land on a spherical

target while relying exclusively on sonar (17).

The bats’ three-dimensional (3D) position was

measured with two infrared cameras, and the

shape and direction of their sonar beam pattern

were measured with a 20-microphone array (17)

(Fig. 1, A to D, and movie S1).

At the beginning of each trial, the target was

randomly repositioned. Subsequently, the bat

ost sensory systems allow some active

control over the information acquired

from the environment (1–6). Nowhere

searched for the target, approached it, and landed

on it, either by a straight flight or a curved

trajectory (Fig. 1C and fig. S1). Unlike micro-

bats (microchiropteran bats), which emit laryn-

geal tonal calls, Egyptian fruit bats are megabats

(megachiropteran bats) that produce very short

(50- to 100-ms) impulse-like tongue clicks, with

frequencies centered at 30 to 35 kHz (fig. S2).

While flying, bats typically emitted pairs of

clicks, with an ~20-ms interval within the click

pair and an ~100-ms interval between the

pairs (Fig. 1A and fig. S3) (18, 19). The bats

pointed their sonar beam toward the left or

the right, in an alternating manner as follows:

left→right→100-ms interval→right→left (Fig. 1D

and movie S1).

We observed two different phases of behav-

ior. During the first stage, the bats did not

necessarily lock their click pairs onto the target,

and the directions of clicks were widely dis-

tributed (the “unlocked” phase). At the final

stage, the bats directed their sonar clicks so that

the vector average of the pair of clicks pointed

toward the target with accuracy better than 30°

(17). We refer to this as the “locked” phase

(Figs. 1E, arrows, and 2A, top, and fig. S1C).

During this phase, 0.5 s before landing, 80% of

the click pairs were locked with accuracy better

than 15° (Fig. 2A, bottom, gray lines). In 10%

of the trials, the bats locked onto the target with

average accuracy better than 5°. The left-right

orientation of the clicks in the locked phase

implies that the bats did not direct the maximum

intensity of the click toward the target, contra-

dicting the common notion that bats steer their

sonar beam in order to maximize the signal-to-

noise ratio (SNR) of the echoes (13, 20).

Another possible strategy would be for the

bats to direct the maximal slope of the beam’s

emission curve toward the target, because this

would maximize changes in reflected echo

energy that result from changes in the relative

position of the bat and the target. Plotting the

directional span of the beams between the right

and left maximum slope (green lines in Fig. 1, E

and F, and fig. S1, C and D) showed that the

bats consistently placed the maximum slope of

their beams onto the target (Fig. 1F and fig.

S1D; the top and bottom of the green lines are

close to direction 0°). Next, we examined the

population distribution of the directions of the

beams’ maximum intensity and maximum slope

(Fig. 2, B and C, top two rows). Before locking,

the bats directed their sonar beams over a wide

range of angles, spanning >100° around the tar-

get (Fig. 2B, top). After locking, however, they

clearly directed their beam so that the maximum

slope of the intensity curve of the beam, and not

its peak, was on the target (Fig. 2C, middle row).

All six bats exhibited this behavior (fig. S4).

When the maximum slope of the beam is di-

rected toward an object, any motion of the object

relativetothebatwillresultinthelargestpossible

change in echo intensity. The sign of the energy

change (positive or negative) corresponds to the

1Department of Neurobiology, Weizmann Institute of Science,

Rehovot 76100, Israel.2Department of Psychology, Institute

for Systems Research and Neuroscience and Cognitive Science

Program,Universityof Maryland,College Park, MD 20742,USA.

*To whom correspondence should be addressed. E-mail:

nachum.ulanovsky@weizmann.ac.il

www.sciencemag.org

SCIENCE

VOL 327 5 FEBRUARY 2010

701

REPORTS

on February 10, 2010

www.sciencemag.org

Downloaded from

Page 6

www.sciencemag.org/cgi/content/full/science.1180556/DC1

Supporting Online Material for

Evolutionary Dynamics of Complex Networks of HIV Drug-Resistant

Strains: The Case of San Francisco

Robert J. Smith?, Justin T. Okano, James S. Kahn, Erin N. Bodine, Sally Blower*

*To whom correspondence should be addressed. E-mail: sblower@mednet.ucla.edu

Published 14 January 2010 on Science Express

DOI: 10.1126/science.1180556

This PDF file includes:

Materials and Methods

Figs. S1 to S6

Tables S1 to S13

References

Page 7

2

Table of Contents

Page

Materials and Methods

1. Amplification Cascade Model: Structure & Equations

2. Parameter Estimates for Monte Carlo Simulations

3. Model Calibration using Monte Carlo filtering

4. A classification system for resistant strains

5. Evaluating the impact of NNRTI-resistant strains in resource-constrained countries

6. CART Methods and Results

Tables

Figures

References

3

5

8

10

13

13

15

30

39

Page 8

Section 1: Amplification Cascade Model: Structure & Equations

All published HIV transmission models of ARV-resistance are based on simple biological assumptions

and can only track one resistant strain (S1-S8). These models were useful predictive tools for ARV

resistance in the early years of treatment, but are now inadequate for predicting the complex dynamics

of the multitude of ARV-resistant strains circulating in resource-rich countries. Therefore a new

generation of predictive models is needed. We have designed the first of this generation of

mathematical models: the Amplification Model. Our innovative model captures a high degree of

biological complexity by generating by generating a dynamic network composed of multiple strains of

HIV, both wild type and ARV resistant.

In the model, susceptible/uninfected individuals ( )

ARV-resistant strains in the network. Once an individual becomes infected, they progress through four

stages: (i) primary infection (

, (ii) not yet eligible for ARVs (i.e., CD4 count > 350 cells/microL)

)

P

(iii) eligible for ARVs (i.e., CD4 ≤ 350 cells/microL) but not on treatment ( )

ARV treatment drugs are documented in Table S1. Viral load is modeled (see Section 2 and Table S2)

such that P has the highest viral load, H a low viral load, Y a higher viral load than the previous state

and T the lowest viral load. N represents the total number of individuals in the sexually active

community.

S can become infected with any of the wild-type or

( )

H ,

Y , and (iv) treated ( )

T . The

Resistant strains in the network are grouped into seven categories which are defined based upon the

specific class of drugs they are resistant to (Nucleoside Reverse Transcriptase Inhibitors (NRTIs), non-

Nucleoside Reverse Transcriptase Inhibitors (NNRTIs) and Protease Inhibitors (PIs)) and their level of

resistance (single, dual or triple class). Categories are denoted by subscript : wild-type strains

single-class resistance to NRTIs (

, single-class resistance to NNRTIs (

resistance to PIs (

, dual-class resistance to NRTIs and NNRTIs ()4

i =

NRTIs and PIs , dual-class resistance to NNRTIs and PIs ((6)

=

i

NRTIs, NNRTIs and PIs .

(8

i = )

i

i

, dual-class resistance to

, and triple-class resistance to

(1)

i =

,

, single-class

2

i = ))3

=

5)

i =

7)

i =

We model treatment effects by specifying treatment regimens and then assessing the effects of these

regimens on both infectivity and the probability of developing resistance. In our modeling, treated

individuals generally receive a regimen to which their virus is sensitive; hence, we assume treated

individuals either achieve complete or partial viral suppression. In our model, patients who achieve

complete viral suppression become uninfectious and are incapable of developing resistance. We

represent the percentage of patients who achieve viral suppression by (1 - γi). Patients who only

achieve partial viral suppression retain some degree of infectivity (which is based on the degree of

treatment-induced reduction in viral load) and are capable of developing resistant strains.

The amplification cascade model can be extended to include any number of additional drug classes,

such as integrase inhibitors, co-receptor blockers and fusion inhibitors which are gradually being

introduced into new therapeutic regimens. These classes are currently used almost exclusively among

patients who have failed other classes of ARVs.

A flow diagram of the complete model is shown in Fig. S1. The model consists of 33 ordinary

differential equations, but using matrix notation (matrices in bold), the model can be written as:

3

Page 9

()

()

8

0

1

0

0

P

i

H

i

Y

i

T

iiiiii

i

PHYT

Y

T

cS

N

S

&

SPHY

cS

N

θP

μβββγ

μ

μ

−

−

=

= Ω−−+++

=+++−−

=

=

=

−−

−

−

+

−

∑

P

&

β P β H β YΓβ TP θP

H

&

&

&

HνH

YνH ρY μ Y ωT

TρY ωT μ T KT

T

β

1

2

3

4

5

6

7

8

T

T

T

T

T

T

T

T

(1)

where .

∑

=

i

++++=

8

1

)(

iiii

TYHPSN

The variable state vectors are

P

P

P

P

P

P

P

P

⎢⎥⎢

⎣⎦⎣

The parameter matrices are

(

i

diag

β

111

222

333

444

555

666

777

888

, , , ,

H

H

H

H

H

H

H

H

Y

Y

Y

Y

Y

Y

Y

Y

⎡

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎤

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎡

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎤

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎦

⎡

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎣

⎤

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎦

⎡

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎣

⎤

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎦

====

PHYT

)()()()

( )

γ

, , , ,

PHYT

PHYT

iii

diagdiagdiag diag

i

ββββ=====

βββΓ

( )

θ

( )

ν

()()

()()

, , , , ,

YT

YT

iiiii

diagdiag

}

.

diag diagdiagdiag

ρωμ=

{

=====

θνρωμμ

,

iμ

where

1,2,3,4,5,6,7,8

i∈

Here,

is the average time period for acquiring new sex partners; c is the average number of new sex partners

per individual per year;is the per-partnership probability of an individual in the primary stage of

is the rate at which susceptible/uninfected individuals join the sexually active community; 1

Ω

μ0

infection transmitting strains in category i; is the per-partnership probability of an infected

individual with a CD4 count greater than 350 cells/microL transmitting strains in category ;

per-partnership probability of an infected untreated individual with a CD4 count less than or equal to

350 cells/microL transmitting strains in category ;

i

1

is the

is the per-partnership probability of an individual

on treatment transmitting strains in category ;

P

i β

H

i β

i

Y

i β

i

T

i β

is the average duration of primary infection;

iθ

i ν

1

is

4

Page 10

the average time that an infected individual's CD4 counts remain above 350 cells/microL; 1

iρ is the

average time an individual spends in the treatment-eligible stage;

an individual (infected with strains in category i) who has a CD4 count less than or equal to 350

cells/microL and is not on treatment; ωi is the rate at which individuals go off treatment; and

the average survival time of an individual (infected with strains in category ) on treatment.

Y

i μ

1

is the average survival time of

T

i μ

1

is

i

The K matrix specifies the degree of acquisition and amplification of resistance from one state to the

next, accounting for the fact that the order in which resistance develops is important.

1 12

k

γ

⎡

⎢

⎢

⎢

⎢

⎢

=⎢

⎢

⎢

⎢

⎢

⎢⎣

1 13

k

k

k

k

γ

−

1 14

k

γ

1 12

γ

γ

−

22 26

k

γ

1 133 35

k

γ

3 37

k

γ

1 14

0

0

0

0

4 46

k

γ

4 47

k

γ

2 25

k

k

γ

−

0

0

3 35

0

k

γ

−

5 58

0

0

k

γ

−

2 264 46

k

k

γ

−

0

6 68

0

k

γ

−

3 37

0

4 477 78

k

k

γ

−

5 58 6 687 78

0

+

0

0

+

0

0

0

0

k

0

0

0

0

0

0

0

0

0

0

+

0

000

0

0

0

k

k

k

γ

−γ

γγγ

γγ

γ

++

⎤

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

−−

−

⎦

K

25

00

0

0

0

0

0

0

0

Here, is the per-capita rate at which individuals in treatment class

, ' {1

i i

∈

,2,3

−

develop resistance and move

to treatment class , where . An equivalent, but more physically

meaningful measure is 1

develop resistance per year.

, which represents the proportion of individuals in treatment class who

'

iik

iT

' iT

,4,5,7,8| '}

ii

' i

6,

<

ik

e

iT

The parameter

and who are not virally suppressed.

i

iγ represents the proportion of treated individuals who are resistant to strains in category

Section 2: Parameter Estimates for Monte Carlo Simulations

We estimated all model parameters from virologic, epidemiological or clinical data (Tables S2-S11).

We calculated per-act and per-partner transmission probabilities based on viral load data which was a

function of the stage of HIV infection. We used a relationship between viral load and the per-act

transmission probability

references (S10) and (S11).

j

i α (Equation 1) that Smith and Blower (S9) derived based on data from

5

Page 11

()

10

log/

( )

( )

w

2.45

j

v w

i

j

i

v

α

α

=

(2)

This relationship, depicted in Fig. S2, enables us to calculate the probability of transmission for an

individual in disease stage

{ ,, , }

jP H Y T

∈

and resistance category i. Equation 2 compares the viral

load of interest,v, with a baseline viral load, . We used data from Gray et al., which compares viral

load to per-act transmission probability (S11), to calculate a baseline. Since a range of viral loads were

given, we used the largest viral load from each range for 25-29 year olds. To test the baseline, we set

, and . Applying Equation 2 gives

close to the actual per-act transmission probability of 0.0026 observed in the Gray et al. study. This

suggests that these data can be used as a reasonable baseline for Equation 2.

We used a weighting factor

wild-type strains and included sampling constraints so that transmissibility decreased as the number of

classes of resistance increased (i.e.,

123

,,

αααα>

on available competitive fitness assays, replication capacity assays and patterns of developed

resistance, we assumed NNRTI-resistant strains were more transmissible than NRTI-resistant strains,

which in turn were more transmissible than PI-resistant strains (S12-S17). The values for

( )0.0018

j

i w

α=

, which is

w

38,500

v =

12

=

,500

w

( )0.0028

j

i v

α=

iλ to ensure that resistant strains were less transmissible (per act) than

4567

,,

8

PPPPPPPP

αααα>>

and similar for

H

i α

etc). Based

iλ are given in

Table S3. We then used these estimates to determine the per-partnership transmission probability

as follows:

1

i

β

= −

for an individual in disease stage

average number of sex acts per year, c (as defined previously) is the average number of new sex

partners per individual per year and is therefore the number of sex acts per partnership. For our

Monte Carlo simulations, we sampled from a triangular distribution (1

c

from a triangular distribution (, with a peak at 10).

/

n c

5/60

n c

<<

Probability density functions (pdfs) for estimates of per-act and per-partnership transmission

probabilities for individuals infected with wild-type strains are given in Table S2. To estimate

parameters, we sampled with constraints so that the per-partnership probabilities satisfied the following

relationships:

ii

ββ

<

,

ii

ββ

<

and

ii

ββ

≤

. These constraints ensured that: (i) individuals had the

highest viral load (and therefore transmissibility) during primary infection (ii) viral load (and

transmissibility) decreased after the primary infection stage, (iii) viral load (and transmissibility)

increased as the individuals CD4 counts fell below 350 cells/microL and (iv) viral load decreased under

treatment (except in the first treatment era of monotherapy).

In San Francisco, monotherapy began in 1987 with the NRTI Zidovudine (AZT). At this time, almost half

of the MSM community was estimated to be infected with HIV (S18-S20). In the late eighties ARV

resistance arose quickly (S21) because AZT was ineffective in suppressing viral loads (S22) and was

very widely used (S23-S26). In 1992, dual therapy became available, featuring treatment regimens of

two NRTIs at a time (for example, Zidovudine (then AZT) and Didanosine). Dual therapies were

substantially more effective than monotherapies and achieved 30-60% viral suppression (S27-S30).

j

iβ

/

1( )

n c

jj

i w

α

⎡

⎣

⎤

⎦

−

transmitting strains in category i. Here, is the

{ ,

P H Y T

, , }

j

∈

/

n c

n

5

c

< <

, with a peak at 1) and

HPHYT

Y

6

Page 12

In 1996, Highly Active Antiretroviral Therapy (HAART) became available with the arrival of two new

classes of drugs: NNRTIs and PIs. Due to the different efficacy of treatment regimens, we subdivided

the HAART era into two: Early HAART (1996-2000) and Modern HAART (2001-2008). The treatment

eras reflect the approval and rollout of new medications. Fig. S3 shows a detailed timeline of the

introduction of drugs used to treat HIV infection; these drugs are also categorized under each treatment

era in Table S1. The specific treatment regimens that we model are given in Tables S6 (for Era 1), S7

(for Era 2), S8 (for Era 3) and S9 (for the current era of HAART: Era 4).

For each era, we modeled the impact of the specific regimens that were available by varying: (i) the

percentage of treatment-eligible individuals who began treatment per year (10

act and per-partner transmissibility (

⋅−

), (iv) the percentage of patients who acquired resistance (

(v) the average time spent on treatment (1/

used in each era are given in Tables S2-S11. Notably, resistance rates vary considerably for different

classes of ARVs. The rate at which a strain develops resistance to a drug depends on the genetic

barrier (i.e., the number of mutations needed to acquire resistance to that specific drug). For example,

single point mutations lead to resistance for some NRTIs (e.g., lamivudine and emtricitabine); however,

multiple mutations are necessary to develop resistance to most PIs.

), (ii) the per-

0 (1

⋅

)

i

e

ρ−

−

j

i α and

j

iβ ), (iii) the percentage of patients who were virally

suppressed (100 (1)

iγ

), and

'

(1)

ii

ke

⋅−

100

T

iμ ). Pdfs for these parameter estimates for the regimens

For each era, we computed 1/

with strains in category i (Table S11). To make these computations, we first derived Equation 3.

T

iμ , the average time spent in the treatment stage for individuals infected

'

1111

1

Y

i

gT

i

Y

ii

T

iiiik

μμμ

ω

+

μρ

μωγ

+=+

⎛

⎜

⎜

⎝

⎞

⎟

⎟

⎠

+−

+

∑

(3)

Here,

Y

i μ

1

is the average survival time of an individual (infected with strains in category ) who is

untreated but has a CD4 count less than or equal to 350 cells/microL. 1/

life years gained due to treatment (

class resistance, crepresents dual-class resistance and

parameters were estimated in (S31) and appear in Table S10. It should be noted that 1/

represents the average survival time from when a treated individual's CD4 count falls below 350

cells/microL. We then estimated the average time an infected individual spent in the treatment-eligible

stage (i.e., with a CD4 count less than 350 cells/microL) before they received treatment. Finally, using

Equation 3, we calculated the average time spent on treatment (1/

The remaining parameter estimates that specify the natural history of HIV infection were obtained from

reference (S32) and are given in Table S10.

To reconstruct the evolution and transmission dynamics of the network of ARV-resistant strains from

1987 to 2008 (that we present in the main text) we analyzed the amplification cascade model using a

time-dependent uncertainty analysis, (S33, S34).

represents the additional

where represents wild-type, b represents single-

represents triple-class resistance). These

d

+ 1/

i

Y

iμ

g

μ

{ , , , }

a b c dg

∈

a

g

μ

T

iμ ) (Table S11).

7

Page 13

Section 3: Model Calibration using Monte Carlo filtering

We calibrated our model using Monte Carlo filtering techniques. We began this calibration process by

deriving an analytical expression for the Basic Reproduction Number (

secondary infections caused by a single infectious individual in a wholly susceptible population in the

absence of treatment. It is a threshold condition that determines whether the pathogen will die out or

become endemic.

0 R );

0 R is the average number of

We derived

0 R from the amplification cascade model using the following method (S35, S36):

Step 1:

Construct matrices

infections in each infected state and V is an

transported (non-new infections) in each infected state and where

infected states.

and where

FVF is an

1

1

n× column vector representing the new

n× column vector representing the

is the number of

n

Step 2:

Construct matrices F and

at the disease-free equilibrium.

, the Jacobian matrices of

V

F and V, respectively, evaluated

Step 3:

The basic reproduction number

0 R is the maximum eigenvalue of .

()

1

−

⋅ −

FV

In Step 1, we derived

()

111111

0

0

PHY

cS

N

PHY

βββ

⎡

⎢

⎢

⎤

⎥

⎥

⎥

⎥

⎥

⎥

⎦

++

=⎢

⎢

⎢

⎢

⎣

F

where the first row represents new infections in primary infection, the second row represents new

infections in individuals with a CD4 count greater than 350 cells/microL and the third row represents

new infections in individuals with a CD4 count less than or equal to 350 cells/microL. Furthermore,

μ

θ

ν

⎢

⎣

where the first row represents infections transported to primary infection, the second row represents

infections transported to individuals with a CD4 count greater than 350 cells/microL and the third row

represents infections transported to individuals with a CD4 count less than or equal to 350 cells/microL.

In Step 2, we calculated the Jacobian matrices of F and V evaluated at the disease-free equilibrium,

i.e., and found that

S,0,0,0

()

0 1

P

μ

1 1

P

ν

−

1 1

P

01

μ

11

111

Y

H

−

H

HY

θ

−−

⎡

⎢

⎢

⎤

⎥

⎥

⎥

⎦

=−

V

8

Page 14

111

0

0

0

0

0

0

PHY

ccc

βββ

⎡

⎢

⎤

⎥

⎥

⎥

⎦

=⎢

⎢

⎣

F

and

0

θ

1

101

11

00

0

μ

0

Y

μθ

μ

ν

ν

−−

⎡

⎢

⎢

⎢

⎣

⎤

⎥

⎥

⎥

⎦

=−−

−

V

.

Lastly, in Step 3, we found that

F⋅ −V

()

−1=

c β1

(

⎢

⎢

⎢

⎢

⎢

⎢

⎢

Pμ1

Yμ0+ν1

(

μ1

(

)+ β1

)μ1

0

Hθ1μ1

Yμ0+ν1

(

Y+ β1

Yθ1ν1

)

Yμ0+θ1

)

c β1

(

Hθ1μ1

μ1

Y+ β1

Yμ0+ν1

(

0

Yθ1ν1

)

)

cβ1

μ1

0

Y

Y

000

⎡

⎣

⎤

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎦

Eigenvalues of this matrix are 0,0,c β1

Pμ1

Yμ0+ν1

(

μ1

(

)+ β1

)μ1

Hθ1μ1

Yμ0+ν1

(

Y+ β1

Yθ1ν1

()

Yμ0+θ1

)

⎧

⎨

⎩ ⎪

⎪

⎫

⎬

⎭ ⎪

⎪

, the greatest of which can be

rewritten as

1111

Y

1

0

010101010

1

PHY

ccc

μ

R

β

+

β

+

θ

+

βθ

+

ν1

+

1

μθμνμθμθμν

=+⋅+⋅⋅

(4)

This expression can be understood biologically as the sum of the following three R0's:

P

P

c

R

μθ

+

0101

μνμθ

++

P

P

c

R

μθ

+

()

(

01

1/

μθ+

(i.e., the primary infection stage).

)

H

H

c

R

μνμθ

++

(

(

01

1/

)

μν+

(i.e., after primary infection but before CD4 counts have fallen

)

1

H

1

0

01

β

=

,

11

0

H

H

c

R

βθ

=⋅

and

1

Y

11

0

1010

Y

Y

c

μ

R

βθ

+

ν

+

1

μθμν

=⋅⋅

1

0

01

β

=

represents the number of infections ()

1

P

cβ

an individual causes throughout the duration

)

of (

1P

11

0

0101

βθ

=⋅

represents the number of infections ()

1

H

cβ

the individual causes throughout the

duration

)

of stage (

below 350 cells/microL) multiplied by the probability

1

01

θ

+

μθ

⎛

⎜

⎞

⎟

⎠⎝

they survive to the stage.

1

H

9

Page 15

1

Y

11

0

1010

Y

Y

c

μ

R

βθ

+

ν

+

of stage (

⎞

⎟ they survive to the

1

μθ

μ

μ

)

ν

=⋅⋅

represents the number of infections ()

1

Y

cβ

the individual causes throughout

the duration (

1

1/

Y

(i.e., when they are untreated but treatment eligible) multiplied by the

probability

)

1Y

1

01

θ

+

μθ

⎛

⎜

⎝⎠

stage and then the probability

1

H

1

01

ν

+

μν

⎛

⎜

⎝

⎞

⎟ they survive to

⎠

the stage.

1Y

We then used the expression that we had derived for R0 to calculate the values of theR0’s for wild-type

strains in San Francisco in the pre-treatment era. We made these calculations using Equation 4, Latin

Hypercube Sampling (S33) and pdfs for the model's parameter estimates given in Tables S2-S11; each

pdf was sampled 10,000 times. This procedure resulted in 10,000 parameter sets, which led to a

median estimate of R0 of 0.84 (Inter-Quartile-Range (IQR) 0.49 to 1.43).

We then filtered these parameter sets in order to calibrate the model to match the prevalence of HIV in

the MSM community in San Francisco in 1987 when treatment was first introduced. Prevalence in this

city in the late eighties has been estimated to be as high as 50% (S18-S20); this corresponds to an R0

of 2.0. We used the value of R0 as a filtering criterion and determined how many of the 10,000

parameter sets generated an R0 value between 1 and 3. This Monte Carlo filtering procedure reduced

our parameter sets from 10,000 to 3,827; after filtering, the median R0 value was 1.5 (Inter-Quartile-

Range (IQR) 1.2 to 2.0).

We used Kolmogorov-Smirnov tests to compare the distributions of the model's parameters before and

after filtering. Four of the parameter ranges were statistically different; for these, see Table S12.

The 3,827 filtered parameter sets were used to conduct the Monte Carlo simulations for our time-

dependent uncertainty analysis and historical reconstructions (S33). For our Monte Carlo simulations,

we modeled monotherapy for 5 years (1987-1991), dual therapy for 4 years (1992-1995), early HAART

for 5 years (1996-2000) and modern HAART thereafter.

Section 4: A classification system for ARV-resistant strains

We derived a classification system for the resistant strains in the network by deriving an analytical

expression for the Control Reproduction Number, Rc. The quantity Rc is a measure of the average

number of secondary HIV infections an individual generates during their entire infectious period; it is

calculated as a weighted average based on the probability the individual is treated. We used the same

methods to derive Rc from the amplification cascade model that we had used previously to deriveR0

(S35, S36). We calculated a Control Reproduction Number (Rc) for each of the seven categories of

ARV resistance included in the model, as well as for wild-type strains. Categories are denoted by

subscript i: wild-type strains , single-class resistance to NRTIs

(1)

i =

NNRTIs (

, single-class resistance to PIs (3)

i =

4

i =

, dual-class resistance to NRTIs and PIs

(5

i = )(

i =

and triple-class resistance to NRTIs, NNRTIs and PIs

(7

i = )

(2

i

)

=

, single-class resistance to

)

6

, dual-class resistance to NRTIs and NNRTIs

)

, dual-class resistance to NNRTIs and PIs

(i

8)

=

.

10