Download Pathogen Mutation Modeled by Competition

PRL 110, 108103 (2013)
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PHYSICAL REVIEW LETTERS
Pathogen Mutation Modeled by Competition Between Site and Bond Percolation
Laurent Hébert-Dufresne,1 Oscar Patterson-Lomba,2 Georg M. Goerg,3 and Benjamin M. Althouse4
1
Département de Physique, de Génie Physique, et d’Optique, Université Laval, Québec, Québec, Canada G1V 0A6
2
Mathematical, Computational, and Modeling Sciences Center, School of Human Evolution and Social Change,
Arizona State University, Tempe, Arizona 8528, USA
3
Department of Statistics, Carnegie Mellon University, Pittsburgh, Pennsylvania 15213, USA
4
Department 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 planning the control of the next influenza pandemic [1], more
complex models have been introduced in epidemiology
[2,3]. We expand one of these studies [2] where treatment
of influenza, as a selection pressure, favors the emergence
and spread of pathogen strains with a drug-resistant phenotype. However, very similar adaptation dynamics could
also be considered in the interactions of pathogens through
ecological mechanisms [4], or of adaptive computer
viruses [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 biological concepts, but simply in reference to dynamics
where two variables influence one another.
Our model consists of a contact network where each
individual 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 link
from 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 probability ". (iii) Treatment is effective (S ! It ) with a probability 1 ! c, or leads to mutation (S ! Ir ) with a
probability c. (iv) A resistant strain infection (through Ir )
can only transmit this strain (S ! Ir ). (v) Infectious individuals of type Ix recover at a rate #x . Once all infectious
individuals have recovered, the final epidemic size is
calculated.
Mean-field analysis.—One of the benefits of network
modeling resides in the possibility to account for heterogeneity in the contact structure of a population. Hence, we
consider both delta and fat-tailed distributions of links per
node (or degree distribution) to create homogeneous and
heterogeneous networks. The distributions are detailed in
the Supplemental Material [10]. However, to accurately
0031-9007=13=110(10)=108103(5)
follow such heterogeneity in a mean-field analysis, one
must distinguish nodes not only by their states, but
also by their degree [6]. For instance, the mean fraction
of susceptible nodes of degree k at time t, Sk ðtÞ can be
written as
S_ k ¼ !kð!u hIu i þ !t hIt i þ !r hIr iÞSk ;
(1)
where hIx i is the probability that a randomly chosen link
of a susceptible node leads to an infectious individual of
type x. Note that all time dependencies are implicit.
Similarly for other node states, we can deduce
I_ u;k ¼ kð!u hIu i þ !t hIt iÞð1 ! "ÞSk ! #u Iu;k ;
(2)
I_ t;k ¼ kð!u hIu i þ !t hIt iÞ"ð1 ! cÞSk ! #t It;k ;
(3)
I_ r;k ¼ kð!u hIu iþ!t hIt iÞ"cSk þk!r hIr iSk !#r Ir;k :
R_ ¼
X
#u Iu;k þ #t It;k þ #r Ir;k :
(4)
(5)
k
We must be careful in evaluating the mean-field quantities
hIx i as a susceptible node is less likely to be connected to
an infectious node than, for example, a recently infected
node. To account for such correlations [11], we follow the
density of each possible link attached to at least one
susceptible node (denoted [SX]):
_ ¼ !2ð!u hIu i þ !t hIt i þ !r hIr iÞhk0s i½SS';
½SS'
(6)
_ u ' ¼ !½ð!u hIu iþ!t hIt iþ!r hIr iÞhk0s iþ!u þ#u '½SIu '
½SI
þ2ð!u hIu iþ!t hIt iÞhk0s ið1!"Þ½SS';
(7)
_ t ' ¼ !½ð!u hIu i þ !t hIt i þ !r hIr iÞhk0s i þ !t þ #t '½SIt '
½SI
108103-1
þ 2ð!u hIu i þ !t hIt iÞhk0s i"ð1 ! cÞ½SS';
(8)
! 2013 American Physical Society
PHYSICAL REVIEW LETTERS
PRL 110, 108103 (2013)
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_ r ' ¼ !½ð!u hIu i þ !t hIt i þ !r hIr iÞhk0s i þ !r þ #r '½SIr '
½SI
þ 2ð!u hIu i þ !t hIt iÞhk0s i"c½SS' þ 2!r hIr ihk0s i½SS';
(9)
_ ¼ !ð!u hIu i þ !t hIt i þ !r hIr iÞhk0s i½SR'
½SR'
þ #u ½SIu ' þ #t ½SIt ' þ #r ½SIr ';
(10)
where hk0s i is the average excess degree of susceptible
nodes. Equations (1)–(10) represent the minimal set of
equations required to describe the system at all times, in
the sense that they are sufficient to calculate all the meanfield quantities on which they depend. A simple averaging
procedure yields
P
kðk ! 1ÞSk
0
hks i ¼ k P
;
(11)
k kSk
hIx i ¼
½SIx '
:
2½SS' þ ½SIu ' þ ½SIt ' þ ½SIr ' þ ½SR'
(12)
Integrating this system of equations provides mean-field
predictions (i.e., in the infinite limit) for the final size of
epidemics.
Mapping to percolation.—Most SIR models feature an
irreversible time line. For our model, there are only four
possible scenarios for each node: S ! Iu ! R, S ! It !
R, S ! Ir ! R, or S for all times, and thus none of these
scenarios can be traveled in reverse. This implies that the
considered continuous time model can be mapped to a
percolation process [12–14], or more precisely, a coevolutive competition arises between the site and bond percolation. The bond percolation represents the propagation of
the disease under certain assumptions [15,16] (see the
Supplemental Material [10] for details) while the site
percolation represents both treatment and mutation (see
Fig. 1). As we will see, these dynamics are both coevolutive (the disease mutates to adapt and resist treatment) and
competitive (treatment aims to stop bond percolation, and
the two strains can hinder each other’s propagation). The
details of this particular process are illustrated in Fig. 1.
The different percolation probabilities involved can be
easily evaluated. In fact, treatment and mutation are already
modeled as site percolation in the original dynamics.
Infection events on an [SIx ] link are equivalent to a
bond percolation process where infection occurs with a
given probability Tx , i.e., that the infection event precedes
the recovery event (see the proof in the Supplemental
Material [10]),
Tx ¼
!x
:
!x þ #x
(13)
Also note that while the infections map to classic
Boolean bond percolation, the treatment process maps to
site percolation with three possible states (if 0 < " < 1)
akin to the three-state Potts model [17].
FIG. 1 (color online). Competitive coevolution between site
and bond percolation. The percolative process of an [SIu ] link is
designed to be equivalent to the continuous time dynamics: 1 indicates the initial state, 2 indicates bond percolation, formation of
links (infection) with probability Tu , 3 indicates three-state site
percolation for treatment (to the untreated, treated, or mutated
state). Events involving It nodes use bond percolation with transmissibility Tt followed by site percolation as illustrated here,
whereas events involving Ir use solely bond percolation with
probability Tr (no possible treatment, hence no site percolation).
Finally, while resistance will always emerge under the
mean-field assumption, one can account for this by approximating the probability of the emergence of the resistant
strain through the probability of treatment causing at least
one mutation. The expected number of infections caused
by a single infectious individual from a disease under its
epidemic threshold hni is a well-known result of network
epidemiology [12] and can be used to calculate the probability P of the emergence of resistance. In hni infections,
resistance develops only if at least one leads to a failed
treatment (probability "c):
0
P ¼ 1 ! ð1 ! "cÞhni ¼ 1 ! ð1 ! "cÞThki=ð1!Thk iÞ ;
(14)
where hki and hk0 i are the mean degree and excess degree
of the network, respectively, and T is the effective transmissibility of the treated wild strain. A more complete
analysis 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 by
either the wild strain (c ¼ 0), the resistant strain (c > 0),
or a combination of both (if above their respective
thresholds). In standard epidemic and percolation models,
the transition from the disease-free equilibrium to an epidemic is observed by keeping all parameters constant and
progressively raising the transmissibility. Once the epidemic threshold is achieved, the disease is able to spread
to an increasingly larger macroscopic fraction of the network [12].
To highlight certain features, we consider the case
!r > !u —corresponding biologically to the development
of compensatory mutations in the pathogen in response to
the fitness cost typically associated with treatment resistance [18,19]—and set #u ¼ #t ¼ #r ¼ # for simplicity.
We note that while compensatory mutations are rare, the
selective pressures exerted by treatment can still give a
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PHYSICAL REVIEW LETTERS
large advantage to uncompensated resistant strains. In fact,
our results are qualitatively similar with or without these
mutations as long as the resistant-strain epidemic undergoes a phase transition before the treated wild-type strain.
The phase transition from the disease-free state to the
epidemic state, dominated mostly by the resistant strain, is
demonstrated in Fig. 2. The main feature is the explosive
transition, where the observed maximal epidemic size
jumps suddenly from zero to almost 10%. While the transition is technically second order (i.e., continuous), the
probability of resistance emergence falls to zero when !u
diminishes such that some epidemic sizes are practically
impossible to observe. In fact, in the case of extremely rare
mutations (i.e., c ! 0), even the infinite system will feature a discontinuous jump as the probability of a mutation
becomes a step function (see Fig. 3). The system thus
features a first-order phase transition in the limit of rare
mutations.
This is of great interest for research in percolation
processes as discontinuous phase transitions in percolation
models on networks have been claimed before [20,21], but
disproven [22]. We show here for the first time that these
transitions can actually occur on a general network structure, as opposed to fractal networks [23]. While the mechanisms potentially leading to such transitions in percolation
on networks are generally not well understood [21], the
discontinuity in our biologically inspired model can be
explained by a classic phase transition concept. In short,
a first-order (or explosive) phase transition is achieved
FIG. 2 (color online). Explosive phase transition. Emergence
of the giant component (i.e., of epidemics) as infection rates
increase. The results for over 106 simulations of the percolation
process on fat-tailed networks with 2:5 ( 105 nodes are plotted
(points). Point color represents the proportion of cases landing
on either branch (lighter ¼ less likely, darker ¼ more likely).
This coloring highlights the second transition, or invasive threshold, which marks the end of bistability. Analytical curves are
obtained by integrating our equations with (c > 0) and without
(c ¼ 0) mutation for upper and lower branches, respectively. The
color code of the upper branch corresponds to logP (log of the
probability of resistance emergence) and encodes the probability
of reaching that branch: likely in color, unlikely in white.
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because the resistant strain must wait for the wild strain
to spread and then for treatment to allow resistance to
spread throughout the system. While this is the most likely
outcome above the threshold of the wild strain, this scenario is almost impossible for smaller infection rates. If
more transmissible than the treated wild strain, the resistant
FIG. 3 (color online). From continuous to discontinuous
epidemics. (a) Equation (14) for the probability of resistance
emergence for various mutation probabilities c. As c goes
to zero, the transition at threshold converges toward a step
function leading to a discontinuity in possible epidemic size.
(b) Probability of reaching a resistant strain epidemic as a
function of infection rate !u for various mutation probabilities
c in simulations. Notice how Eq. (14) correctly predicts that
probabilities below the treated-disease threshold (dotted line) are
more affected by variations in c than for above the threshold.
(c) Effect of population size N on the probability of reaching a
resistant strain epidemic. Scenarios above threshold feature
macroscopic epidemics (fractions of N) of the wild strain and
are thus significantly more affected by population size than those
below thresholds where epidemics are microscopic (independent
of N), also as predicted by Eq. (14). (d) Combining these
behaviors for c ! 0 and N ! 1, we can expect that a very
effective treatment in a very large population will feature a
discontinuity in the observed or expected total epidemic size
(as shown by the solid red line). Other simulation parameters are
set to the values of Fig. 2.
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PHYSICAL REVIEW LETTERS
strain then contains an ‘‘infection potential,’’ conceptually
equivalent to latent heat in classical phase transition theory,
resulting in a discontinuity at the transition.
Bistable and competitive regimes.—For c > 0, there
exists a regime of bistability where a given disease can
either stay in the disease-free state or reach the epidemic
stable branch (hysteresis). Interestingly, when integrating
our mean-field analysis with a finite precision, there exists
a critical manifold (roughly P ) precision) marking a limit
above which the initial conditions escape the disease-free
state towards the epidemic state. The analytical observation of the bistability was thus achieved by using different
initial conditions (all < 10!5 ). Though our finite simulations start with a single infectious individual, they can
stochastically tunnel through this manifold and reach the
epidemic state. Figures 2 and 4 color the simulation results
to illustrate the likelihood of such events for each transmissibility (points). As transmission rates increase, the
system features a second phase transition corresponding
to the epidemic threshold of the case without mutation
(c ¼ 0), after which all epidemics reach the highest branch
(see Fig. 3).
This final regime also differs from regular percolation, as
both strains end up competing for the potential infections.
That is, if the wild strain spreads to high degree nodes early
on, the system is less easily invaded by the resistant strain.
To illustrate this competition for high degree nodes, consider the narrower spread of results on the uniform network
of Fig. 4 as opposed to the large competitive regime
observed on the heterogeneous network of Fig. 2. Within
FIG. 4 (color online). Importance of state correlations and
heterogeneity. Using the uniform network (i.e., pk ¼ $k;4 ), we
see a very similar phase transition. For comparison, the dotted
line corresponds to the prediction of classic epidemiological
models, neglecting state correlations, as used in the original
study of the present model [2]. Results of over 2 ( 106 simulations on networks with 2:5 ( 105 nodes are plotted, and both
points and lines use the same color scheme as Fig. 2. The lower
branch of our model, corresponding to epidemics of the treated
wild strain, is barely visited above its threshold (at !u * #u ) as
the upper branch is by far the most likely outcome at this point.
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this competitive regime, the dynamics become highly
sensitive to the initial conditions. Although the different
strains compete for the highest degree node even in the
limit of an infinite population (i.e., the analytical system),
this competition and sensitivity is always stronger in a
finite system. Inthe limit of rare mutations and large
infection rates, our model is akin to previous models of
competing epidemics [24,25].
Finally, note that these results are valid as long as the
resistant strain propagates faster than the treated wild
strain, i.e., !r > ð1 ! "Þ!u þ "ð1 ! cÞ!t which is likely
in practice according to realistic estimates [2]. Otherwise,
the dynamics still feature competition, but lacks both
bistability and the explosive phase transition as the disease
never accumulates infection potential.
Discussion.—In light of recent studies on first-order
transitions in percolation on networks, our simple biologically inspired model of coevolutive competition provides
deep insight into how discontinuous transitions can emerge
in such systems due to the build up of potential connectivity (latent heat) from coevolution. Similar results had
previously been observed on adaptive networks whose
structure changes through time [26] and in jamming transitions for a network with traffic awareness where routing
protocols depend on the network’s state [27]. This arguably
hints at a new universality class corresponding to coevolutive 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, treatment effectiveness depends highly on the initial conditions
[28]. This is especially important given the relative ease of
many pathogens to evolve resistance to treatment [29–33]
and the potential morbidity and mortality associated with
treatment failure (for example, the neuraminidase inhibitors
oseltamivir and zanamivir for the treatment of influenza)
[2,34–36]. From that point of view, future work will study
the implications of resistance development for the optimal
targeting, timing, and scale of treatment strategies. Finally
and most importantly, the first-order phase transition indicates that a microscopic change in transmission rate can
lead to a severe macroscopic jump in the expected epidemic
size. It is thus primordial that future efforts focus not only on
reducing mutation probability in treatment, but also on
detecting and controlling the emergence of resistance.
L. H.-D. is grateful to the Natural Sciences and
Engineering Research Council of Canada and to Calcul
Québec for computing facilities. O. P.-L. is supported by
the WAESOBD LSAMPBD NSF Cooperative Agreement
HRD-1025879. G. M. G. was supported by an INET grant
(Grant No. IN01100005). B. M. A. holds an NSF Graduate
Research Fellowship (Grant No. DGE-0707427). The
authors also wish to thank the Santa Fe Institute and their
Complex Systems Summer School at which this work was
performed.
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PHYSICAL REVIEW LETTERS
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108103-5
Pathogen mutation modeled by competition
between site and bond percolation
Supplemental Material
Laurent Hébert-Dufresne
Département de Physique, de Génie Physique, et d’Optique, Université Laval, Québec, Canada G1V 0A6
Oscar Patterson-Lomba
Mathematical, Computational, and Modeling Sciences Center, School of Human Evolution and Social Change,
Arizona State University, Tempe, AZ, 85287
Georg M. Goerg
Department of Statistics, Carnegie Mellon University, Pittsburgh, PA, 15213
Benjamin M. Althouse
Department 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 considers five possibles states for individuals: susceptible (S), infectious and untreated (Iu ),
infectious and treated (It ), infectious with a resistant strain (Ir ), or recovered (R) [1]. The
model obeys the following rules:
• a link from Ix to S leads to an infection at rate
x;
• a wild strain infection (through Iu or It ) is untreated with probability 1
or treated with probability ⇢;
• treatment is e↵ective with probability 1
bility c, S ! Ir ;
⇢, S ! Iu ,
c, S ! It , or leads to mutation with proba-
• an infection caused by the resistant strain (through Ir ) can only transmit this strain,
S ! Ir ;
• infectious individuals of type Ix recover at rate
x.
The rules are iterated until no infectious individuals remain, and the final epidemic size is
calculated.
Network model
One of the main advantages of network modeling resides in the possibility to account for
heterogeneity 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.
HÉBERT-DUFRESNE et al. (2012)
10
Pathogen Mutation: SOM
0
uniform
heavy-tail
10-1
%
10
-2
10
-3
10-4
10-5
1
4
10
k
25
50 75
Figure 1: Degree distributions used in the main text. The uniform distribution corresponds
to a delta distribution, i.e. pk = 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 excess
degree hk 0 i = 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
P ki randomly taken from the degree
distribution {pk } under the unique constraint that N
i=1 ki is even. Degree (or stubs) are
then randomly matched without restrictions, to create a random network of size N with the
correct degree distribution. One unique network is created for every single simulation of the
dynamics and the mean-field analysis is then expected to reproduce the average behavior of
this network ensemble.
Complete mean field analysis
To accurately follow the consequences of heterogeneity in the chosen contact structure, one
must distinguish nodes not only by their states, but also by their degree [3]. For instance, the
mean fraction of susceptible nodes of degree k at time t, Sk (t) can be written as (dropping
explicit time dependencies):
Ṡk =
k
u hIu i
+
t hIt i
+
r hIr i
Sk
(1)
where hIx i is the probability that a randomly chosen links of a susceptible node leads to an
infectious individual of type x. Similarly for other node states, we can deduce:
I˙u,k = k
u hIu i
+
t hIt i
2
(1
⇢)Sk
u Iu,k
(2)
HÉBERT-DUFRESNE et al. (2012)
I˙t,k = k
I˙r,k = k
Pathogen Mutation: SOM
u hIu i
u hIu i
Ṙ =
+
X
+
t hIt i
t hIt i
u Iu,k
⇢(1
c)Sk
t It,k
⇢cSk + k r hIr iSk
+
t It,k
+
r Ir,k
(3)
r Ir,k
.
(4)
(5)
k
We must be careful in evaluating the mean-field quantities (hIx i) as a susceptible may be
less 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 at
least one susceptible node (denoted as [SX]):
˙ =
[SS]
[SI˙ u ] =
2
⇥
+2
˙ t] = ⇥
[SI
+2
⇥
[SI˙ r ] =
+2
˙ =
[SR]
+
u hIu i
+
t hIt i
+
r hIr i
u hIu i+ t hIt i+ r hIr i
0
u hIu i+ t hIt i hks i(1
u hIu i+ t hIt i+ r hIr i
0
u hIu i+ t hIt i hks i⇢(1
hks0 i[SS]
hks0 i+
u+ u
⇢)[SS]
hks0 i+ t + t
c)[SS]
⇤
⇤
(6)
[SIu ]
(7)
[SIt ]
⇤
0
u hIu i+ t hIt i+ r hIr i hks i+ r + r [SIr ]
0
0
u hIu i+ t hIt i hks i⇢c[SS] + 2 r hIr ihks i[SS]
+ t hIt i + r hIr i hks0 i[SR]
u [SIu ] + t [SIt ] + r [SIr ]
u hIu i
(8)
(9)
(10)
where hks0 i is the average excess degree of susceptible nodes. Equations (1-10) represent the
minimal set of equations required to describe the system with desired dynamics, in the sense
that they are sufficient to calculate all the mean field quantities on which they depend. A
simple averaging procedure yields:
P
k(k 1)Sk
0
hks i = k P
(11)
k kSk
hIx i =
[SIx ]
.
2[SS] + [SIu ] + [SIt ] + [SIr ] + [SR]
3
(12)
HÉBERT-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˙Iu ] =
2 u [Iu Iu ]
⇥
0
+
u hIu i+ t hIt i hks i +
u
hks0 i +
t
[Iu˙It ] =
+
+
[Iu˙Ir ] =
+
+
[Iu˙R] =
⇥
⇥
⇥
u
+
t
[Iu It ]
u hIu i+ t hIt i
u hIu i+ t hIt i
u
+
r
u hIu i+ t hIt i
[It˙It ] =
u
hks0 i
+
+
[It˙R] =
+
[Ir˙Ir ] =
t
⇥
+
r
+
˙ =
[RR]
⇢)[SIt ]
⇢(1
c)[SIu ]
u hIu i+ t hIt i
hks0 i(1
t [Iu It ]
+
⇢)[SR]
t
⇤
⇢(1
2 r [Ir Ir ] +
(16)
c)[SIt ]
hks0 i⇢(1
u [Iu It ] + r [It Ir ]
0
hks i⇢(1 c)[SR]
u hIu i+ t hIt i
r [SIr ]
(17)
(18)
hks0 i⇢c[SIr ]
+
t [It R]
u [Iu Ir ] + t [It Ir ]
0
hks i⇢c[SR] + r hIr ihks0 i[SR]
(19)
(20)
+ 2 r [Ir Ir ] +
u hIu i+ t hIt i
u [Iu R]
(15)
+ 2 t [It It ] +
u hIu i+ t hIt i
r [Ir R]
(14)
r [Iu Ir ]
c)[SIr ]
⇤
0
u hIu i+ t hIt i hks i + u ⇢c[SIt ]
t [It R]
(13)
[It Ir ] + hIr ihks0 i r [SIt ]
+ hIt ihks0 i + 1
[Ir˙R] =
(1
⇢)[SIr ]
⇤
+ u ⇢c[SIu ]
2 t [It It ]
⇥
0
+
u hIu i+ t hIt i hks i +
[It˙Ir ] =
⇤
⇢)[SIu ]
hks0 i(1
+ 2 u [Iu Iu ] +
u hIu i+ t hIt i
+
⇤
(1
[Iu Ir ] + hIr ihks0 i r [SIu ]
u hIu i+ t hIt i
u [Iu R]
hks0 i +
⇤
+
r [Ir R]
.
(21)
(22)
It is easily shown that the complete system conserves the density of both node and link
states, i.e.
i
Xh
Ṙ +
Ṡk + I˙u,k + I˙t,k + I˙r,k = 0
(23)
k
and
X
˙ ]=0.
[XY
{X,Y }
4
(24)
HÉBERT-DUFRESNE et al. (2012)
Pathogen Mutation: SOM
Mapping to percolation
In this model, there are only four possible scenarios for each node: S ! Iu ! R, S ! It ! R,
S ! Ir ! R, or S for all t. None of these scenarios can be traveled in reverse. This implies
that the considered continuous time model can be mapped unto a percolation process, or
more precisely, a coevolutive competition between site and bond percolation. The bond
percolation represents the propagation of the disease, while the site percolation represents
both treatment and mutation.
The di↵erent percolation probabilities involved can be easily evaluated. In fact, treatment
and mutation are already modeled as site percolation in the original dynamics. One then
only needs to evaluate the total probability of infection Tx through an [SIx ] link during the
time ⌧x spent in a given infectious state Ix :
Tx (⌧x ) = 1
lim (1
x
t!0
t)⌧x / t = 1
x ⌧x
e
.
(25)
To evaluate the probability of a given ⌧x , first consider its cumulative distribution
F (⌧x ) = 1
lim (1
t!0
x
t)⌧x / t = 1
x ⌧x
e
.
(26)
from which the distribution of infectious period f (⌧x ) is straightforwardly obtained,
dF (⌧x )
= xe
d⌧x
The total probability of transmission thus becomes
Z 1
Tx =
Tx (⌧x )f (⌧x )d⌧x =
f (⌧x ) =
0
x ⌧x
.
(27)
x
x
+
.
(28)
x
The disease spread dynamics thus map to a bond percolation process using this probability of occupation for links between infectious and susceptible individuals. However, map
might not be the most appropriate expression here, as one distinction exist between the
two processes: if an individual stays infectious for a short/long period of time, all of his
links will have a lower/higher e↵ective Tx . Bond percolation does not consider these correlations between links sharing the same infectious nodes. Hence, the SIR epidemic model
is not isomorphic to the bond percolation model. We can however convince ourselves that
these correlations do not have any significant impact on our results. Consider Fig. 2 which
compares results obtained with our mean-field formalism to simulations of both the bond
percolation 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] and
we here rely on previous results to estimate the probability of resistance emergence.
5
HÉBERT-DUFRESNE et al. (2012)
Pathogen Mutation: SOM
Figure 2: Comparison between bond percolation and continuous SIR dynamics. The
curve is obtained
P by integrating the mean-field model (with initial conditions as small as possible
(in this case k It,k (0) = 10 8 ) and its color is given by log P with blue for low probabilities and
green for high probabilities. Notice how the results of the continuous SIR dynamics are qualitatively
similar to those of the bond percolation process. Both use roughly 106 simulations.
6
HÉBERT-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 undergo
a phase transition for a given value of u [4]. In the thermodynamic limit, macroscopic
epidemics can occur only above this threshold. Assuming that the treated disease roughly
behaves as if it propagated with an e↵ective rate e↵ = (1 ⇢) u + ⇢(1 c) t (or total
transmissiblity Te↵ ⌘ T = e↵ / ( + e↵ )), the epidemic threshold (critical point) is given by
[4]:
1
⇤
Te↵
= 0
(29)
hk i
or
⇤
◆✓
◆
(30)
u = ✓
0
hk i 1 1 ⇢ + ⇢ t / u
where hk 0 i is the mean excess degree and where we once again assumed that both the ratio
are kept fixed at all times.
t / u and the recovery rates t = u =
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 this
regime, the size sn of the microsopic epidemics (i.e. epidemics of finite size which correspond
to 0% of the infinite population) follow a known distribution [5]. However, assuming that
a number I0 & 1 of individuals are infectious when treatment begins, we can neglect the
full distribution and simply expect a number I0 hsi I0 of new infections by considering the
average epidemic size [4]
T hki
.
(31)
hsi = 1 +
1 T hk 0 i
From this result, we can write the probability PT <Tc of getting at least one mutation in
the I0 (hsi 1) new infections as
PT <Tc = 1
I0 1
(1
⇢c)
Te↵ hki
Te↵ hk0 i
,
(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 the
infinite system (starting with S = 0 at threshold) which is easily calculated from the degree
distribution (see Ref. [2]). The probability PT Tc of getting at least one mutation can then
be written as
PT Tc = 1 (1 ⇢c)SN .
(33)
In the limit of very rare mutations (i.e. c ! 0), it is easily shown that the probability of
getting at least one mutation also undergoes a (continuous/second-order) phase transition
at the epidemic threshold since
PT <Tc
0
lim
⇠
=0
(34)
c!0 PT Tc
N
and
PT Tc
N
lim
⇠
=1,
(35)
c!0 PT <Tc
0
7
HÉBERT-DUFRESNE et al. (2012)
Pathogen Mutation: SOM
easily evaluated from l’Hospital’s rule as the order of PT <Tc is finite (no matter how close u
is to u⇤ ) and the order of PT Tc is infinite due to the phase transition of the classic epidemic
dynamics.
In short, for a given c
0, one can calculate the (non-zero) probability of getting at
least one mutation for any given u (32). However, in the limit of very rare mutations, the
non-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: one
corresponding 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 feature
a regime of bistability where the epidemics can reach either branch; and the probability of
observing 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 is
a possible final state of the system only above the epidemic threshold of the wild strain; the
probability of reaching this state being zero below the threshold. Hence, there is a discontinuity 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 control
of 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 arbitrary
degree 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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