Pathogen Mutation Modeled by Competition Between Site and Bond Percolation

Laurent Hebert-Dufresne,1 Oscar Patterson-Lomba,2 Georg M. Goerg,3 and Benjamin M. Althouse4

1Departement de Physique, de Genie Physique, et d’Optique, Universite Laval, Quebec, Quebec, Canada G1V 0A62Mathematical, Computational, and Modeling Sciences Center, School of Human Evolution and Social Change,

Arizona State University, Tempe, Arizona 8528, USA3Department of Statistics, Carnegie Mellon University, Pittsburgh, Pennsylvania 15213, USA

4Department of Epidemiology, Johns Hopkins Bloomberg School of Public Health, Baltimore, Maryland 21205, USA(Received 19 October 2012; published 5 March 2013)

While disease propagation is a main focus of network science, its coevolution with treatment has yet to

be studied in this framework. We present a mean-field and stochastic analysis of an epidemic model with

antiviral administration and resistance development. We show how this model maps to a coevolutive

competition between site and bond percolation featuring hysteresis and both second- and first-order phase

transitions. The latter, whose existence on networks is a long-standing question, imply that a microscopic

change in infection rate can lead to macroscopic jumps in expected epidemic size.

DOI: 10.1103/PhysRevLett.110.108103 PACS numbers: 87.23.Cc, 64.60.ah, 64.60.aq, 87.23.Ge

With the recent focus of public health policies on plan-ning the control of the next influenza pandemic [1], morecomplex models have been introduced in epidemiology[2,3]. We expand one of these studies [2] where treatmentof influenza, as a selection pressure, favors the emergenceand spread of pathogen strains with a drug-resistant phe-notype. However, very similar adaptation dynamics couldalso be considered in the interactions of pathogens throughecological mechanisms [4], or of adaptive computerviruses [5,6], and for behavioral changes in a population[7,8] or ecosystem [9]. While we study mutation dynamics,the terms adaptation and coevolution are not used as bio-logical concepts, but simply in reference to dynamicswhere two variables influence one another.

Our model consists of a contact network where eachindividual can be in one of five states: susceptible (S),infectious and untreated (Iu), infectious and treated (It),infectious with a resistant strain (Ir), or recovered (R).The dynamics then obey the following rules: (i) A linkfrom Ix to S leads to an infection at a rate !x (x 2 fu; t; rg).(ii) A wild strain infection (through Iu or It) is untreated(S ! Iu) with a probability 1! ", or treated with a proba-bility ". (iii) Treatment is effective (S ! It) with a proba-bility 1! c, or leads to mutation (S ! Ir) with aprobability c. (iv) A resistant strain infection (through Ir)can only transmit this strain (S ! Ir). (v) Infectious indi-viduals of type Ix recover at a rate #x. Once all infectiousindividuals have recovered, the final epidemic size iscalculated.

Mean-field analysis.—One of the benefits of networkmodeling resides in the possibility to account for hetero-geneity in the contact structure of a population. Hence, weconsider both delta and fat-tailed distributions of links pernode (or degree distribution) to create homogeneous andheterogeneous networks. The distributions are detailed inthe Supplemental Material [10]. However, to accurately

follow such heterogeneity in a mean-field analysis, onemust distinguish nodes not only by their states, butalso by their degree [6]. For instance, the mean fractionof susceptible nodes of degree k at time t, Sk"t# can bewritten as

_S k $ !k"!uhIui% !thIti% !rhIri#Sk; (1)

where hIxi is the probability that a randomly chosen linkof a susceptible node leads to an infectious individual oftype x. Note that all time dependencies are implicit.Similarly for other node states, we can deduce

_I u;k $ k"!uhIui% !thIti#"1! "#Sk ! #uIu;k; (2)

_I t;k $ k"!uhIui% !thIti#""1! c#Sk ! #tIt;k; (3)

_I r;k$k"!uhIui%!thIti#"cSk%k!rhIriSk!#rIr;k: (4)

_R $X

k

#uIu;k % #tIt;k % #rIr;k: (5)

We must be careful in evaluating the mean-field quantitieshIxi as a susceptible node is less likely to be connected toan infectious node than, for example, a recently infectednode. To account for such correlations [11], we follow thedensity of each possible link attached to at least onesusceptible node (denoted [SX]):

& _SS' $ !2"!uhIui% !thIti% !rhIri#hk0si&SS'; (6)

& _SIu'$!&"!uhIui%!thIti%!rhIri#hk0si%!u%#u'&SIu'%2"!uhIui%!thIti#hk0si"1!"#&SS'; (7)

& _SIt' $ !&"!uhIui% !thIti% !rhIri#hk0si% !t % #t'&SIt'% 2"!uhIui% !thIti#hk0si""1! c#&SS'; (8)

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& _SIr' $!&"!uhIui%!thIti%!rhIri#hk0si%!r %#r'&SIr'% 2"!uhIui%!thIti#hk0si"c&SS'% 2!rhIrihk0si&SS';

(9)

& _SR' $ !"!uhIui% !thIti% !rhIri#hk0si&SR'% #u&SIu' % #t&SIt' % #r&SIr'; (10)

where hk0si is the average excess degree of susceptiblenodes. Equations (1)–(10) represent the minimal set ofequations required to describe the system at all times, inthe sense that they are sufficient to calculate all the mean-field quantities on which they depend. A simple averagingprocedure yields

hk0si $P

k k"k! 1#SkPk kSk

; (11)

hIxi $&SIx'

2&SS' % &SIu' % &SIt' % &SIr' % &SR' : (12)

Integrating this system of equations provides mean-fieldpredictions (i.e., in the infinite limit) for the final size ofepidemics.

Mapping to percolation.—Most SIR models feature anirreversible time line. For our model, there are only fourpossible scenarios for each node: S ! Iu ! R, S ! It !R, S ! Ir ! R, or S for all times, and thus none of thesescenarios can be traveled in reverse. This implies that theconsidered continuous time model can be mapped to apercolation process [12–14], or more precisely, a coevolu-tive competition arises between the site and bond percola-tion. The bond percolation represents the propagation ofthe disease under certain assumptions [15,16] (see theSupplemental Material [10] for details) while the sitepercolation represents both treatment and mutation (seeFig. 1). As we will see, these dynamics are both coevolu-tive (the disease mutates to adapt and resist treatment) andcompetitive (treatment aims to stop bond percolation, andthe two strains can hinder each other’s propagation). Thedetails of this particular process are illustrated in Fig. 1.

The different percolation probabilities involved can beeasily evaluated. In fact, treatment andmutation are alreadymodeled as site percolation in the original dynamics.Infection events on an [SIx] link are equivalent to abond percolation process where infection occurs with agiven probability Tx, i.e., that the infection event precedesthe recovery event (see the proof in the SupplementalMaterial [10]),

Tx $!x

!x % #x: (13)

Also note that while the infections map to classicBoolean bond percolation, the treatment process maps tosite percolation with three possible states (if 0< "< 1)akin to the three-state Potts model [17].

Finally, while resistance will always emerge under themean-field assumption, one can account for this by approx-imating the probability of the emergence of the resistantstrain through the probability of treatment causing at leastone mutation. The expected number of infections causedby a single infectious individual from a disease under itsepidemic threshold hni is a well-known result of networkepidemiology [12] and can be used to calculate the proba-bility P of the emergence of resistance. In hni infections,resistance develops only if at least one leads to a failedtreatment (probability "c):

P $ 1! "1! "c#hni $ 1! "1! "c#Thki="1!Thk0i#; (14)

where hki and hk0i are the mean degree and excess degreeof the network, respectively, and T is the effective trans-missibility of the treated wild strain. A more completeanalysis is given in the Supplemental Material [10].Phase transition.—Our model can lead to four possible

final states: a disease-free state, and epidemics caused byeither the wild strain (c $ 0), the resistant strain (c > 0),or a combination of both (if above their respectivethresholds). In standard epidemic and percolation models,the transition from the disease-free equilibrium to an epi-demic is observed by keeping all parameters constant andprogressively raising the transmissibility. Once the epi-demic threshold is achieved, the disease is able to spreadto an increasingly larger macroscopic fraction of the net-work [12].To highlight certain features, we consider the case

!r > !u—corresponding biologically to the developmentof compensatory mutations in the pathogen in response tothe fitness cost typically associated with treatment resist-ance [18,19]—and set #u $ #t $ #r $ # for simplicity.We note that while compensatory mutations are rare, theselective pressures exerted by treatment can still give a

FIG. 1 (color online). Competitive coevolution between siteand bond percolation. The percolative process of an [SIu] link isdesigned to be equivalent to the continuous time dynamics: 1 in-dicates the initial state, 2 indicates bond percolation, formation oflinks (infection) with probability Tu, 3 indicates three-state sitepercolation for treatment (to the untreated, treated, or mutatedstate). Events involving It nodes use bond percolation with trans-missibility Tt followed by site percolation as illustrated here,whereas events involving Ir use solely bond percolation withprobability Tr (no possible treatment, hence no site percolation).

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large advantage to uncompensated resistant strains. In fact,our results are qualitatively similar with or without thesemutations as long as the resistant-strain epidemic under-goes a phase transition before the treated wild-type strain.

The phase transition from the disease-free state to theepidemic state, dominated mostly by the resistant strain, isdemonstrated in Fig. 2. The main feature is the explosivetransition, where the observed maximal epidemic sizejumps suddenly from zero to almost 10%. While the tran-sition is technically second order (i.e., continuous), theprobability of resistance emergence falls to zero when !u

diminishes such that some epidemic sizes are practicallyimpossible to observe. In fact, in the case of extremely raremutations (i.e., c ! 0), even the infinite system will fea-ture a discontinuous jump as the probability of a mutationbecomes a step function (see Fig. 3). The system thusfeatures a first-order phase transition in the limit of raremutations.

This is of great interest for research in percolationprocesses as discontinuous phase transitions in percolationmodels on networks have been claimed before [20,21], butdisproven [22]. We show here for the first time that thesetransitions can actually occur on a general network struc-ture, as opposed to fractal networks [23]. While the mecha-nisms potentially leading to such transitions in percolationon networks are generally not well understood [21], thediscontinuity in our biologically inspired model can beexplained by a classic phase transition concept. In short,a first-order (or explosive) phase transition is achieved

because the resistant strain must wait for the wild strainto spread and then for treatment to allow resistance tospread throughout the system. While this is the most likelyoutcome above the threshold of the wild strain, this sce-nario is almost impossible for smaller infection rates. Ifmore transmissible than the treated wild strain, the resistant

FIG. 2 (color online). Explosive phase transition. Emergenceof the giant component (i.e., of epidemics) as infection ratesincrease. The results for over 106 simulations of the percolationprocess on fat-tailed networks with 2:5( 105 nodes are plotted(points). Point color represents the proportion of cases landingon either branch (lighter $ less likely, darker $ more likely).This coloring highlights the second transition, or invasive thresh-old, which marks the end of bistability. Analytical curves areobtained by integrating our equations with (c > 0) and without(c $ 0) mutation for upper and lower branches, respectively. Thecolor code of the upper branch corresponds to logP (log of theprobability of resistance emergence) and encodes the probabilityof reaching that branch: likely in color, unlikely in white.

FIG. 3 (color online). From continuous to discontinuousepidemics. (a) Equation (14) for the probability of resistanceemergence for various mutation probabilities c. As c goesto zero, the transition at threshold converges toward a stepfunction leading to a discontinuity in possible epidemic size.(b) Probability of reaching a resistant strain epidemic as afunction of infection rate !u for various mutation probabilitiesc in simulations. Notice how Eq. (14) correctly predicts thatprobabilities below the treated-disease threshold (dotted line) aremore affected by variations in c than for above the threshold.(c) Effect of population size N on the probability of reaching aresistant strain epidemic. Scenarios above threshold featuremacroscopic epidemics (fractions of N) of the wild strain andare thus significantly more affected by population size than thosebelow thresholds where epidemics are microscopic (independentof N), also as predicted by Eq. (14). (d) Combining thesebehaviors for c ! 0 and N ! 1, we can expect that a veryeffective treatment in a very large population will feature adiscontinuity in the observed or expected total epidemic size(as shown by the solid red line). Other simulation parameters areset to the values of Fig. 2.

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strain then contains an ‘‘infection potential,’’ conceptuallyequivalent to latent heat in classical phase transition theory,resulting in a discontinuity at the transition.

Bistable and competitive regimes.—For c > 0, thereexists a regime of bistability where a given disease caneither stay in the disease-free state or reach the epidemicstable branch (hysteresis). Interestingly, when integratingour mean-field analysis with a finite precision, there existsa critical manifold (roughly P) precision) marking a limitabove which the initial conditions escape the disease-freestate towards the epidemic state. The analytical observa-tion of the bistability was thus achieved by using differentinitial conditions (all< 10!5). Though our finite simula-tions start with a single infectious individual, they canstochastically tunnel through this manifold and reach theepidemic state. Figures 2 and 4 color the simulation resultsto illustrate the likelihood of such events for each trans-missibility (points). As transmission rates increase, thesystem features a second phase transition correspondingto the epidemic threshold of the case without mutation(c $ 0), after which all epidemics reach the highest branch(see Fig. 3).

This final regime also differs from regular percolation, asboth strains end up competing for the potential infections.That is, if the wild strain spreads to high degree nodes earlyon, the system is less easily invaded by the resistant strain.To illustrate this competition for high degree nodes, con-sider the narrower spread of results on the uniform networkof Fig. 4 as opposed to the large competitive regimeobserved on the heterogeneous network of Fig. 2. Within

this competitive regime, the dynamics become highlysensitive to the initial conditions. Although the differentstrains compete for the highest degree node even in thelimit of an infinite population (i.e., the analytical system),this competition and sensitivity is always stronger in afinite system. Inthe limit of rare mutations and largeinfection rates, our model is akin to previous models ofcompeting epidemics [24,25].Finally, note that these results are valid as long as the

resistant strain propagates faster than the treated wildstrain, i.e., !r > "1! "#!u % ""1! c#!t which is likelyin practice according to realistic estimates [2]. Otherwise,the dynamics still feature competition, but lacks bothbistability and the explosive phase transition as the diseasenever accumulates infection potential.Discussion.—In light of recent studies on first-order

transitions in percolation on networks, our simple biolog-ically inspired model of coevolutive competition providesdeep insight into how discontinuous transitions can emergein such systems due to the build up of potential connectiv-ity (latent heat) from coevolution. Similar results hadpreviously been observed on adaptive networks whosestructure changes through time [26] and in jamming tran-sitions for a network with traffic awareness where routingprotocols depend on the network’s state [27]. This arguablyhints at a new universality class corresponding to coevol-utive dynamics on networks.Our results also have important implications for the

control of epidemics in finite structured populations.Because of the presence of bistability and hysteresis, treat-ment effectiveness depends highly on the initial conditions[28]. This is especially important given the relative ease ofmany pathogens to evolve resistance to treatment [29–33]and the potential morbidity and mortality associated withtreatment failure (for example, the neuraminidase inhibitorsoseltamivir and zanamivir for the treatment of influenza)[2,34–36]. From that point of view, future work will studythe implications of resistance development for the optimaltargeting, timing, and scale of treatment strategies. Finallyand most importantly, the first-order phase transition indi-cates that a microscopic change in transmission rate canlead to a severe macroscopic jump in the expected epidemicsize. It is thus primordial that future efforts focus not only onreducing mutation probability in treatment, but also ondetecting and controlling the emergence of resistance.L. H.-D. is grateful to the Natural Sciences and

Engineering Research Council of Canada and to CalculQuebec for computing facilities. O. P.-L. is supported bythe WAESOBD LSAMPBD NSF Cooperative AgreementHRD-1025879. G.M.G. was supported by an INET grant(Grant No. IN01100005). B.M.A. holds an NSF GraduateResearch Fellowship (Grant No. DGE-0707427). Theauthors also wish to thank the Santa Fe Institute and theirComplex Systems Summer School at which this work wasperformed.

FIG. 4 (color online). Importance of state correlations andheterogeneity. Using the uniform network (i.e., pk $ $k;4), wesee a very similar phase transition. For comparison, the dottedline corresponds to the prediction of classic epidemiologicalmodels, neglecting state correlations, as used in the originalstudy of the present model [2]. Results of over 2( 106 simula-tions on networks with 2:5( 105 nodes are plotted, and bothpoints and lines use the same color scheme as Fig. 2. The lowerbranch of our model, corresponding to epidemics of the treatedwild strain, is barely visited above its threshold (at !u * #u) asthe upper branch is by far the most likely outcome at this point.

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[10] See Supplemental Material at http://link.aps.org/supplemental/10.1103/PhysRevLett.110.108103 for thefull mathematical model, simulation details and additionalresults.

[11] J. C. Miller, A. C. Slim, and E.M. Volz, J. R. Soc. Interface9, 890 (2012).

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(2007).[16] J. C. Miller, Phys. Rev. E 76, 010101(R) (2007).[17] F.-Y. Wu, Rev. Mod. Phys. 54, 235 (1982).[18] B. R. Levin, V. Perrot, and N. Walker, Genetics 154, 985

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787 (2012).[24] V. Marceau, P.-A. Noel, L. Hebert-Dufresne, A. Allard,

and L. J. Dube, Phys. Rev. E 84, 026105 (2011).[25] B. Karrer and M. E. J. Newman, Phys. Rev. E 84, 036106

(2011).[26] T. Gross, C. J. Dommar D’Lima, and B. Blasius, Phys.

Rev. Lett. 96, 208701 (2006).[27] P. Echenique, J. Gomez-Gardenes, and Y. Moreno,

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L. Hebert-Dufresne, PLoS Comput. Biol. 9, e1002912(2013).

[29] D.M. Weinstock and G. Zuccotti, JAMA, J. Am. Med.Assoc. 301, 1066 (2009).

[30] M. Lipsitch, J. B. Plotkin, L. Simonsen, and B. Bloom,Science 336, 1529 (2012).

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Pathogen mutation modeled by competition

between site and bond percolation

Supplemental Material

Laurent Hebert-DufresneDepartement de Physique, de Genie Physique, et d’Optique, Universite Laval, Quebec, Canada G1V 0A6

Oscar Patterson-LombaMathematical, Computational, and Modeling Sciences Center, School of Human Evolution and Social Change,

Arizona State University, Tempe, AZ, 85287

Georg M. GoergDepartment of Statistics, Carnegie Mellon University, Pittsburgh, PA, 15213

Benjamin M. AlthouseDepartment of Epidemiology, Johns Hopkins Bloomberg School of Public Health, Baltimore, MD, 21205

Pathogen mutation model

The studied network dynamics is based on a recent epidemic model of influenza which con-siders five possibles states for individuals: susceptible (S), infectious and untreated (I

u

),infectious and treated (I

t

), infectious with a resistant strain (Ir

), or recovered (R) [1]. Themodel obeys the following rules:

• a link from I

x

to S leads to an infection at rate �

x

;

• a wild strain infection (through I

u

or I

t

) is untreated with probability 1� ⇢, S ! I

u

,or treated with probability ⇢;

• treatment is e↵ective with probability 1� c, S ! I

t

, or leads to mutation with proba-bility c, S ! I

r

;

• an infection caused by the resistant strain (through I

r

) can only transmit this strain,S ! I

r

;

• infectious individuals of type I

x

recover at rate �

x

.

The rules are iterated until no infectious individuals remain, and the final epidemic size iscalculated.

Network model

One of the main advantages of network modeling resides in the possibility to account forheterogeneity in the contact structure of a population. To take full advantage of this fact,the main text considers both a uniform and a heterogeneous fat-tailed degree distributions(distribution of links per node) both shown on Fig. 1.

HEBERT-DUFRESNE et al. (2012) Pathogen Mutation: SOM

10

-5

10

-4

10

-3

10

-2

10

-1

10

0

1 4 10 25 50 75

k

%

uniform

heavy-tail

Figure 1: Degree distributions used in the main text. The uniform distribution correspondsto a delta distribution, i.e. p

k

= �k,4, while the heavy-tail corresponds to an initial binomial regime

leading into a power-law tail with exponential cuto↵ (for mean degree hki ⇠ 4.1 and mean excessdegree hk0i = 16.6).

From the degree distribution, networks are created with the so-called configuration model[2]. A number N of nodes are created with a degree k

i

randomly taken from the degreedistribution {p

k

} under the unique constraint thatP

N

i=1 k

i

is even. Degree (or stubs) arethen randomly matched without restrictions, to create a random network of size N with thecorrect degree distribution. One unique network is created for every single simulation of thedynamics and the mean-field analysis is then expected to reproduce the average behavior ofthis network ensemble.

Complete mean field analysis

To accurately follow the consequences of heterogeneity in the chosen contact structure, onemust distinguish nodes not only by their states, but also by their degree [3]. For instance, themean fraction of susceptible nodes of degree k at time t, S

k

(t) can be written as (droppingexplicit time dependencies):

S

k

= �k

��

u

hIu

i+ �

t

hIt

i+ �

r

hIr

i�S

k

(1)

where hIx

i is the probability that a randomly chosen links of a susceptible node leads to aninfectious individual of type x. Similarly for other node states, we can deduce:

I

u,k

= k

��

u

hIu

i+ �

t

hIt

i�(1� ⇢)S

k

� �

u

I

u,k

(2)

2

HEBERT-DUFRESNE et al. (2012) Pathogen Mutation: SOM

I

t,k

= k

��

u

hIu

i + �

t

hIt

i�⇢(1 � c)S

k

� �

t

I

t,k

(3)

I

r,k

= k

��

u

hIu

i + �

t

hIt

i�⇢cS

k

+ k�

r

hIr

iSk

� �

r

I

r,k

(4)

R =X

k

�

u

I

u,k

+ �

t

I

t,k

+ �

r

I

r,k

. (5)

We must be careful in evaluating the mean-field quantities (hIx

i) as a susceptible may beless likely to be connected to an infectious node than, for example, a recently infected node.To account for such correlations, we follow the density of each possible link attached to atleast one susceptible node (denoted as [SX]):

˙[SS] = �2��

u

hIu

i + �

t

hIt

i + �

r

hIr

i�hk0

s

i[SS] (6)

˙[SI

u

] = �⇥�

�

u

hIu

i+�

t

hIt

i+�

r

hIr

i�hk0

s

i+�

u

+�

u

⇤[SI

u

]

+ 2��

u

hIu

i+�

t

hIt

i�hk0

s

i(1 � ⇢)[SS] (7)

˙[SI

t

] = �⇥�

�

u

hIu

i+�

t

hIt

i+�

r

hIr

i�hk0

s

i+�

t

+�

t

⇤[SI

t

]

+ 2��

u

hIu

i+�

t

hIt

i�hk0

s

i⇢(1 � c)[SS] (8)

˙[SI

r

] = �⇥�

�

u

hIu

i+�

t

hIt

i+�

r

hIr

i�hk0

s

i+�

r

+�

r

⇤[SI

r

]

+ 2��

u

hIu

i+�

t

hIt

i�hk0

s

i⇢c[SS] + 2�r

hIr

ihk0s

i[SS] (9)

˙[SR] = ���

u

hIu

i + �

t

hIt

i + �

r

hIr

i�hk0

s

i[SR]

+ �

u

[SI

u

] + �

t

[SI

t

] + �

r

[SI

r

] (10)

where hk0s

i is the average excess degree of susceptible nodes. Equations (1-10) represent theminimal set of equations required to describe the system with desired dynamics, in the sensethat they are su�cient to calculate all the mean field quantities on which they depend. Asimple averaging procedure yields:

hk0s

i =

Pk

k(k � 1)SkP

k

kS

k

(11)

hIx

i =[SI

x

]

2[SS] + [SI

u

] + [SI

t

] + [SI

r

] + [SR]. (12)

3

HEBERT-DUFRESNE et al. (2012) Pathogen Mutation: SOM

In order to follow the full state of the system, one can also rely on additional equations:

˙[Iu

I

u

] = �2�u

[Iu

I

u

]

+⇥�

�

u

hIu

i+�

t

hIt

i�hk0

s

i + �

u

⇤(1 � ⇢)[SI

u

] (13)

˙[Iu

I

t

] = ���

u

+ �

t

�[I

u

I

t

]

+⇥�

�

u

hIu

i+�

t

hIt

i�hk0

s

i + �

t

⇤(1 � ⇢)[SI

t

]

+⇥�

�

u

hIu

i+�

t

hIt

i�hk0

s

i + �

u

⇤⇢(1 � c)[SI

u

] (14)

˙[Iu

I

r

] = ���

u

+ �

r

�[I

u

I

r

] + hIr

ihk0s

i�r

[SI

u

]

+��

u

hIu

i+�

t

hIt

i�hk0

s

i(1 � ⇢)[SI

r

]

+⇥�

�

u

hIu

i+�

t

hIt

i�hk0

s

i + �

u

⇤⇢c[SI

u

] (15)

˙[Iu

R] = ��

u

[Iu

R] + 2�u

[Iu

I

u

] + �

t

[Iu

I

t

] + �

r

[Iu

I

r

]

+��

u

hIu

i+�

t

hIt

i�hk0

s

i(1 � ⇢)[SR] (16)

˙[It

I

t

] = �2�t

[It

I

t

]

+⇥�

�

u

hIu

i+�

t

hIt

i�hk0

s

i + �

t

⇤⇢(1 � c)[SI

t

] (17)

˙[It

I

r

] = ���

t

+ �

r

�[I

t

I

r

] + hIr

ihk0s

i�r

[SI

t

]

+��

u

hIu

i+�

t

hIt

i�hk0

s

i⇢(1 � c)[SI

r

]

+⇥�

�

u

hIu

i+�

t

hIt

i�hk0

s

i + �

u

⇤⇢c[SI

t

] (18)

˙[It

R] = ��

t

[It

R] + 2�t

[It

I

t

] + �

u

[Iu

I

t

] + �

r

[It

I

r

]

+��

u

hIu

i+�

t

hIt

i�hk0

s

i⇢(1 � c)[SR] (19)

˙[Ir

I

r

] = �2�r

[Ir

I

r

] +��

u

hIu

i+�

t

hIt

i�hk0

s

i⇢c[SI

r

]

+�hI

t

ihk0s

i + 1��

r

[SI

r

] (20)

˙[Ir

R] = ��

r

[Ir

R] + 2�r

[Ir

I

r

] + �

u

[Iu

I

r

] + �

t

[It

I

r

]

+��

u

hIu

i+�

t

hIt

i�hk0

s

i⇢c[SR] + �

r

hIr

ihk0s

i[SR] (21)

˙[RR] = �

u

[Iu

R] + �

t

[It

R] + �

r

[Ir

R] . (22)

It is easily shown that the complete system conserves the density of both node and linkstates, i.e.

R +X

k

hS

k

+ I

u,k

+ I

t,k

+ I

r,k

i= 0 (23)

and X

{X,Y }

˙[XY ] = 0 . (24)

4

HEBERT-DUFRESNE et al. (2012) Pathogen Mutation: SOM

Mapping to percolation

In this model, there are only four possible scenarios for each node: S ! I

u

! R, S ! I

t

! R,S ! I

r

! R, or S for all t. None of these scenarios can be traveled in reverse. This impliesthat the considered continuous time model can be mapped unto a percolation process, ormore precisely, a coevolutive competition between site and bond percolation. The bondpercolation represents the propagation of the disease, while the site percolation representsboth treatment and mutation.

The di↵erent percolation probabilities involved can be easily evaluated. In fact, treatmentand mutation are already modeled as site percolation in the original dynamics. One thenonly needs to evaluate the total probability of infection T

x

through an [SI

x

] link during thetime ⌧

x

spent in a given infectious state I

x

:

T

x

(⌧x

) = 1� lim�t!0

(1� �

x

�t)⌧

x

/�t = 1� e

��

x

⌧

x

. (25)

To evaluate the probability of a given ⌧

x

, first consider its cumulative distribution

F (⌧x

) = 1� lim�t!0

(1� �

x

�t)⌧

x

/�t = 1� e

��

x

⌧

x

. (26)

from which the distribution of infectious period f(⌧x

) is straightforwardly obtained,

f(⌧x

) =dF (⌧

x

)

d⌧

x

= �

x

e

��

x

⌧

x

. (27)

The total probability of transmission thus becomes

T

x

=

Z 1

0

T

x

(⌧x

)f(⌧x

)d⌧

x

=�

x

�

x

+ �

x

. (28)

The disease spread dynamics thus map to a bond percolation process using this proba-bility of occupation for links between infectious and susceptible individuals. However, map

might not be the most appropriate expression here, as one distinction exist between thetwo processes: if an individual stays infectious for a short/long period of time, all of hislinks will have a lower/higher e↵ective T

x

. Bond percolation does not consider these cor-relations between links sharing the same infectious nodes. Hence, the SIR epidemic modelis not isomorphic to the bond percolation model. We can however convince ourselves thatthese correlations do not have any significant impact on our results. Consider Fig. 2 whichcompares results obtained with our mean-field formalism to simulations of both the bondpercolation process and the continuous time SIR dynamics.

Network model

Mutation around the epidemic threshold

The classic one-strain bond percolation process has been studied at great length [2, 4, 5] andwe here rely on previous results to estimate the probability of resistance emergence.

5

HEBERT-DUFRESNE et al. (2012) Pathogen Mutation: SOM

Figure 2: Comparison between bond percolation and continuous SIR dynamics. Thecurve is obtained by integrating the mean-field model (with initial conditions as small as possible(in this case

Pk

It,k

(0) = 10�8) and its color is given by log P with blue for low probabilities andgreen for high probabilities. Notice how the results of the continuous SIR dynamics are qualitativelysimilar to those of the bond percolation process. Both use roughly 106 simulations.

6

HEBERT-DUFRESNE et al. (2012) Pathogen Mutation: SOM

Probability of mutation

It is well known that the bond percolation process of the treated wild strain will undergoa phase transition for a given value of �

u

[4]. In the thermodynamic limit, macroscopicepidemics can occur only above this threshold. Assuming that the treated disease roughlybehaves as if it propagated with an e↵ective rate �e↵ = (1 � ⇢)�

u

+ ⇢(1 � c)�t

(or totaltransmissiblity Te↵ ⌘ T = �e↵/ (� + �e↵)), the epidemic threshold (critical point) is given by[4]:

T

⇤e↵ =

1

hk0i (29)

or�

⇤u

=�✓

hk0i � 1

◆✓1� ⇢ + ⇢�

t

/�

u

◆ (30)

where hk0i is the mean excess degree and where we once again assumed that both the ratio�

t

/�

u

and the recovery rates �

t

= �

u

= � are kept fixed at all times.The interesting feature here is the probability to get resistance emergence even if the

treated disease is under its threshold (the bistability regime of the main text). In thisregime, the size s

n

of the microsopic epidemics (i.e. epidemics of finite size which correspondto 0% of the infinite population) follow a known distribution [5]. However, assuming thata number I0 & 1 of individuals are infectious when treatment begins, we can neglect thefull distribution and simply expect a number I0hsi � I0 of new infections by considering theaverage epidemic size [4]

hsi = 1 +T hki

1� T hk0i . (31)

From this result, we can write the probability P

T<T

c

of getting at least one mutation inthe I0(hsi � 1) new infections as

P

T<T

c

= 1� (1� ⇢c)I0

Te↵hki1�Te↵hk0i

, (32)

which is only valid under the epidemic threshold (29) and independent of system size N .From the pidemic threshold, the disease will invade a progressively bigger fraction S of theinfinite system (starting with S = 0 at threshold) which is easily calculated from the degreedistribution (see Ref. [2]). The probability P

T�T

c

of getting at least one mutation can thenbe written as

P

T�T

c

= 1� (1� ⇢c)SN

. (33)

In the limit of very rare mutations (i.e. c ! 0), it is easily shown that the probability ofgetting at least one mutation also undergoes a (continuous/second-order) phase transitionat the epidemic threshold since

limc!0

P

T<T

c

P

T�T

c

⇠ 0

N

= 0 (34)

and

limc!0

P

T�T

c

P

T<T

c

⇠ N

0= 1 , (35)

7

HEBERT-DUFRESNE et al. (2012) Pathogen Mutation: SOM

easily evaluated from l’Hospital’s rule as the order of P

T<T

c

is finite (no matter how close �

u

is to �

⇤u

) and the order of P

T�T

c

is infinite due to the phase transition of the classic epidemicdynamics.

In short, for a given c � 0, one can calculate the (non-zero) probability of getting atleast one mutation for any given �

u

(32). However, in the limit of very rare mutations, thenon-zero probability of resistance emergence exists only above the epidemic threshold.

Implication for possible states

For a given set of �

u

, �

t

, and �

r

, there are always two branches of possible states: onecorresponding to the wild strain and the second to the resistant strain. In the case of�

r

> �e↵, the latter is systematically higher than the former. The dynamics then featurea regime of bistability where the epidemics can reach either branch; and the probability ofobserving the higher resistance-dominated state is given by Eqs. (32) or (33).

The results of the previous section indicate that in the limit c! 0, the higher branch isa possible final state of the system only above the epidemic threshold of the wild strain; theprobability of reaching this state being zero below the threshold. Hence, there is a disconti-nuity at this critical point resulting in a first-order phase transition of the full dynamics.

References

[1] M. Lipsitch, T. Cohen, M. Murray, and B. R. Levin, “Antiviral resistance and the controlof pandemic influenza,” PLoS Med, vol. 4, p. e15, Jan 2007.

[2] M. E. J. Newman, S. H. Strogatz, and D. J. Watts, “Random graphs with arbitrarydegree distributions and their applications,” Phys. Rev. E, vol. 64, p. 026118, 2001.

[3] R. Pastor-Satorras and A. Vespignani, “Epidemic spreading in scale-free networks,” Phys.

Rev. Lett., vol. 86, pp. 3200–3203, 2001.

[4] M. E. J. Newman, “Spread of epidemic disease on networks,” Phys Rev E, vol. 66,p. 016128, Jul 2002.

[5] M. E. J. Newman, “Component sizes in networks with arbitrary degree distributions,”Phys. Rev. E, vol. 76, p. 045101, 2007.

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