Spreading Processes with Mutations over Multi-layer Networks Mansi Sooda Anirudh Sridharb Rashad Eletrebyc Chai Wah Wud Simon A. Levine H. Vincent Poorb and Osman Yagana

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Spreading Processes with Mutations over

Multi-layer Networks

Mansi Sooda, Anirudh Sridharb, Rashad Eletrebyc, Chai Wah Wud, Simon A. Levine, H. Vincent Poorb, and Osman Yagana

aDepartment of Electrical and Computer Engineering, Carnegie Mellon University, Pittsburgh, PA 15213 USA; bDepartment of Electrical Engineering, Princeton University,

Princeton, NJ 08544 USA; cRocket Travel, Inc, Chicago, IL 60661 USA; dThomas J. Watson Research Center, IBM, Yorktown Heights, NY 10598 USA; eDepartment of

Ecology and Evolutionary Biology, Princeton University, Princeton, NJ 08544 USA

A key scientiﬁc challenge during the outbreak of novel infectious

diseases is to predict how the course of the epidemic changes un-

der different countermeasures that limit interaction in the popula-

tion. Most epidemiological models do not consider the role of mu-

tations and heterogeneity in the type of contact events. However,

pathogens have the capacity to mutate in response to changing en-

vironments, especially caused by the increase in population immu-

nity to existing strains and the emergence of new pathogen strains

poses a continued threat to public health. Further, in light of differing

transmission risks in different congregate settings (e.g., schools and

ofﬁces), different mitigation strategies may need to be adopted to

control the spread of infection. We analyze a multi-layer multi-strain

model by simultaneously accounting for i) pathways for mutations in

the pathogen leading to the emergence of new pathogen strains, and

ii) differing transmission risks in different congregate settings, mod-

eled as network-layers. Assuming complete cross-immunity among

strains, namely, recovery from any infection prevents infection with

any other (an assumption that will need to be relaxed to deal with

COVID-19 or inﬂuenza), we derive the key epidemiological parame-

ters for the proposed multi-layer multi-strain framework. We demon-

strate that reductions to existing network-based models that dis-

count heterogeneity in either the strain or the network layers can

lead to incorrect predictions for the course of the outbreak. In ad-

dition, our results highlight that the impact of imposing/lifting mit-

igation measures concerning different contact network layers (e.g.,

school closures or work-from-home policies) should be evaluated in

connection with their effect on the likelihood of the emergence of

new pathogen strains.

Network Epidemics |Multi-layer Networks |Mutations |Agent-based

Models |Branching Process

Introduction

The recent outbreak of the COVID-19 pandemic, fuelled

by the novel coronavirus SARS-CoV-2 led to a devas-

tating loss of human life and upended livelihoods worldwide

(1). The highly transmissible, virulent, and rapidly mutat-

ing nature of the SARS-CoV-2 coronavirus (2) led to an un-

precedented burden on critical healthcare infrastructure. The

emergence of new strains of the pathogen as a result of mu-

tations poses a continued risk to public health (3,4). More-

over, when a new strain is introduced to a host population,

pharmaceutical interventions often take time to be developed,

tested, and made widely accessible (5,6). In the absence of

widespread access to treatment and vaccines, policymakers

are faced with the challenging problem of taming the out-

break with nonpharmaceutical interventions (NPIs) that en-

courage physical distancing in the host population to suppress

the growth rate of new infections (7–9). However, the ensuing

socio-economic burden (10,11) of NPIs, such as lockdowns,

makes it necessary to understand how imposing restrictions

in diﬀerent social settings (e.g., schools, oﬃces, etc.) alter the

course of the epidemic outbreak.

Epidemiological models that analyze the speed and scale

of the spread of infection can be broadly classiﬁed under two

approaches. The ﬁrst approach assumes homogeneous mixing,

i.e., the population is well-mixed, and an infected individual

is equally likely to infect any individual in the population

regardless of location and social interactions (12,13). The

second is a network-based approach that explicitly models the

contact patterns among individuals in the population and the

probability of transmission through any given contact (14–

16). Structural properties of the contact network such as het-

erogeneity in type of contacts (17), clustering (e.g., presence

of tightly connected communities) (18), centrality (e.g., pres-

ence of super-spreaders) (19,20) and degree-degree correla-

tions (21) are known to have profound implications for disease

spread and its control (22,23). To understand the impact of

NPIs that lead to reduction in physical contacts, network-

based epidemiological models have been employed widely in

the context of infectious diseases, including COVID-19 (24–

26).

In addition to the contact structure within the host popu-

lation, the course of an infectious disease is critically tied to

evolutionary adaptations or mutations in the pathogen. There

is growing evidence for the zoonotic origin of disease out-

breaks, including COVID-19, SARS, and H1N1 inﬂuenza, as

a result of cross-species transmission and subsequent evolu-

tionary adaptations (27–31). When pathogens enter a new

species, they are often poorly adapted to the physiological

environment in the new hosts and undergo evolutionary mu-

tations to adapt to the new hosts (27). The resulting vari-

ants or strains of the pathogen have varying risks of transmis-

sion, commonly measured through the reproduction number

or R0, which quantiﬁes the mean number of secondary infec-

tions triggered by an infected individual (32,33). Moreover,

even when a sizeable fraction of the population gains immu-

nity through vaccination or natural infection, the emergence

of new variants that can evade the acquired immunity poses

a continued threat to public health (3,4). A growing body

of work (27,34–44) has highlighted the need for developing

multi-strain epidemiological models that account for evolu-

tionary adaptations in the pathogen. For instance, there is a

vast literature on phylodynamics (38–41) which examines how

Author contributions: M.S., H.V.P., and O.Y. designed research; M.S., A.S., C.W.W., and O.Y. per-

formed research; M.S., A.S., C.W.W., S.A.L, H.V.P., and O.Y. contributed new reagents/analytic

tools; M.S., A,S., R.E., S.A.L., H.V.P. and O.Y. analyzed data; and M.S., A.S., R.E., C.W.W., S.A.L.,

H.V.P., and O.Y. wrote the paper.

2To whom correspondence should be addressed. E-mail: msood@andrew.cmu.edu

arXiv:2210.05051v2 [physics.soc-ph] 24 Jan 2023

epidemiological and evolutionary processes interact to impact

pathogen phylogenies. The past decade has also seen the de-

velopment of network-based models to identify risk factors

for the emergence of pathogens in light of diﬀerent contact

patterns (27,34,42,43). Further, a recent study (34) demon-

strated that models that do not consider evolutionary adapta-

tions may lead to incorrect predictions about the probability

of the emergence of an epidemic triggered by a mutating con-

tagion.

Most existing network-based approaches that analyze the

spread of mutating pathogens center on single-layered con-

tact networks, where the transmissibility, i.e., the probabil-

ity that an infective individual passes on the infection to a

contact, depends on the type of strain but not on the na-

ture of link/contact over which the infection is transmitted

(27,34,43). However, diﬀerent congregate settings such as

schools, hospitals, oﬃces, and private social gatherings pose

varied transmission risks (45). Recently, multi-layer networks

have been used to model human contact networks (24–26,33),

where each layer represents a diﬀerent social setting in which

an individual participates. While multi-layer contact net-

works (20,46–54) and multi-strain contagions (27,34–37,43)

have been extensively studied in separate contexts, there has

been a dearth of analysis on simultaneously accounting for the

multi-strain network structure and multi-strain spreading.

In this paper, we build upon the mathematical theory for

the multi-strain model proposed in (27,34) to account for

the multi-layer structure typical to human contact networks,

where diﬀerent network layers correspond to diﬀerent social

settings in which individuals congregate. Speciﬁcally, we as-

sume that the transmissibility depends not only on the type

of strain carried by an infective individual but also on the so-

cial setting (modeled through a network layer) in which they

meet their neighbors. The proposed framework allows study-

ing how NPIs, such as lockdowns in diﬀerent network layers,

impact the course of the spread of a contagion.

While the bulk of our discussion is on mutating conta-

gions in the context of infectious diseases, our results also

hold promise for applications in modeling social contagions,

e.g., news items circulating in social networks (34,55). Sim-

ilar to diﬀerent strains of a pathogen arising through muta-

tions, diﬀerent versions of the information are created as the

content is altered on social media platforms (56). The result-

ing variants of the information may have varying propensi-

ties to be circulated in the social network. Moreover, with

the burst of social media platforms, potential applications of

our multi-layer analysis of mutating contagions are in ana-

lyzing the multi-platform spread of misinformation where the

information gets altered across diﬀerent platforms.

Model

Contact Network. We consider a population of size nwith

members in the set N={1,...,n}. Patterns of interac-

tion in the host population are encoded in the contact net-

work where an edge is drawn between two nodes if they can

come in contact and potentially transmit the infection. To ac-

count for variability in transmission risks associated with dif-

ferent social settings (e.g., neighborhood, school, workplace),

we consider a multi-layer contact network (46), where each

network layer corresponds to a unique social setting. For

simplicity, we focus on the case where each individual can

participate independently in two network layers denoted by

Fand Wrespectively. For instance, the network layer Fcan

be used to model the spread of infection between friends re-

siding in the same neighborhood, while the network layer

Wcan model the spread of infections amongst individuals

who congregate for work. In order to model participation in

Network model: Multi-layer conﬁguration model

Union of networks independently constructed using the conﬁguration model

Transmissibility depends on both the type of link and type of strain

ℍ=𝔽 ⨿𝕎

𝔽

𝕎

Fig. 1. Multi-layer network model: An illustration of a two-layer

contact network for modeling the spread of an infection over the friend-

ship/neighborhood network Fand work network W. The resultant contact

network H=FqW. Neighboring nodes in Hcan transmit infections to their

neighbors either through links in the Fnetwork (i.e., through type-flinks)

or Wnetwork (through type-wlinks).

each network layer, we ﬁrst independently label each node

as non-participating in network layer-awith probability αa

and participating in network layer-awith probability 1−αa,

where 0≤αa≤1, and where a∈ {f, w}. Next, for each

node that participates in network layer-a, the number of its

neighbors in layer-ais drawn from a degree distribution, de-

noted by {˜pa

k, k = 0,1,...,n}, where a∈ {f, w}. Under

this formulation, the degree of a node in layer-a, denoted by

{pa

k, k = 0,1, . . .}, with a∈ {f, w}, is given by

k= (1 −αa)˜pa

k+αa1{k= 0}, k = 0,1,..., [1]

where 1{} denotes the indicator random variable, admitting

the value one when k= 0 and zero when k≥1. We gener-

ate both layers independently according to the conﬁguration

model (57,58) with the degree distribution given through

Eq. (1). For notational simplicity, we say that edges in net-

work F(resp., H) are of type-f(resp., type-w). The multi-

layer contact network, denoted as H, is constructed by taking

the disjoint union (q)of network layers Wand F(Figure 1).

We assume that the network His static and focus on the emer-

gent spreading behavior in the limit of inﬁnite population size

(n→ ∞).

Spreading Process. We adopt a multi-strain spreading pro-

cess (27) to the multi-layer network setting as follows. For

each layer, the evolutionary adaptations in the pathogen are

modeled by corresponding mutation matrices. Let mdenote

the number of pathogen strains co-existing in a population.

For network layer F(resp., W), the mutation matrix, denoted

by µ

µf(resp., µ

µw) is a m×mmatrix. The entry µf

ij (resp.,

µw

ij ) denotes the probability that strain-imutates to strain-j

within a host who got infected through a type-f(resp., type-

w) link, with Pjµf

ij = 1 (resp., Pjµw

ij = 1). Given that

an individual carrying strain-imakes an infectious contact

through a type-f(resp., type-w) edge, the newly infected in-

dividual acquires strain jwith probability µf

ij (resp., µw

ij ).

We note that for the context of infectious diseases where the

epidemiological and evolutionary processes occur at a similar

time-scale and mutations of the pathogen occur within the

host, the mutation matrices do not depend on the network

structure (27) and µ

µw=µ

µf∗.

In the succeeding discussion, we focus on the setting where

two strains of the pathogen are dominant and assume m= 2.

We denote

µf=µf

11 µf

µf

21 µf

22, µ

µw=µw

11 µw

µw

21 µw

22.

We model the dependence of transmissibility on the type of

links using m×mdiagonal matrices T

Tf(resp., T

Tw), with [Tf

(resp., [Tw

i]) representing the transmissibility of strain-iover

a type-flink (resp., type-wlink), for i= 1,...,m. We have

Tf=Tf

0Tf

2, T

Tw=Tw

0Tw

2.

We consider the following multi-strain spreading process on

a multi-layer network (Figure 2) that accounts for pathogen

transmission when epidemiological and evolutionary processes

occur on a similar timescale and each new infection oﬀers an

opportunity for mutation (27). The process starts when a ran-

domly chosen seed node is infected with strain-1. We refer to

such a seed node as the initial infective and the nodes that are

subsequently infected as later-generation infectives. The seed

node independently infects their susceptible neighbors con-

nected through type-f(resp., type-w) links with probability

1(resp., Tw

1). We assume that co-infection is not possible

and after infection, the pathogen mutates to strain-iwithin

the hosts with probabilities given by mutation matrices µ

µf

and µ

µw. Further, in line with (27,34), we assume that once

an individual becomes recovered after being infected with ei-

ther strain, then they can not be reinfected with any strain.

The infected nodes in turn infect their neighbors indepen-

dently with transmission probabilities governed by the strain

that they are carrying (i.e, strain-1 or strain-2), and the type

of edge used to infect their neighbors (i.e., type-for type-w).

The process terminates when no further infections are possi-

ble. Additional details regarding the Materials and Methods

are presented in SI Appendix 1.

We note that this paper is the ﬁrst eﬀort to develop a

framework for the multiscale process discussed. In it, we as-

sume complete cross-immunity between strains: recovery from

any infection prevents infection with any other. This is a good

assumption for example for myxomatosis, but is not a good

assumption for inﬂuenza or COVID, for which the emergence

of new strains is driven by escape from population immunity.

The case of incomplete cross-immunity, which is an essential

feature of the current pandemic, therefore will be the subject

of a follow-up paper.

Results

Summary of key contributions. We provide analytical results

for characterizing epidemic outbreaks caused by mutating

pathogens over multi-layer contact networks using tools from

multi-type branching processes. In particular, we derive three

key metrics to quantify the epidemic outbreak: i) the prob-

ability of emergence of an epidemic, ii) the expected frac-

tion of individuals infected with each strain, and iii) the crit-

ical threshold of phase transition beyond which an epidemic

∗In the context of information propagation, different strains (34) may correspond to different versions

of the information. Therefore, to provide a more general contagion model, we let the mutation

probabilities depend on the network layer.

(a) (b) (c) (d)

Fig. 2. Multi-strain transmission model: An illustration of the multi-

strain model with 2 strains on a contact network comprising 2 layers– (a)

An arbitrary chosen seed node acquires strain-1; (b) The seed node in-

dependently infects their susceptible neighbors connected through type-f

(resp., type-w) links with probability Tf

1(resp., Tw

1); (c) After infection,

the pathogen mutates to strain-2 within the hosts with probabilities given

by mutation matrices µ

µfand µ

µw; (d) The infected nodes in turn infect their

neighbors with transmission probabilities governed by the strain that they

are carrying (i.e, strain-1 or strain-2), and the type of edge used to infect

their neighbors (i.e., type-1 or type-2). The process terminates when no

further infections are possible.

outbreak occurs with a positive probability. Speciﬁcally, the

probability of emergence is deﬁned as the probability that a

randomly chosen infectious seed node leads to an epidemic,

i.e., a positive fraction of nodes get infected in the limit of

large network size. The epidemic threshold deﬁnes the crit-

ical point at which a phase transition occurs, leading to the

possibility of an epidemic outbreak. In other words, the epi-

demic threshold deﬁnes a region in the parameter space in

which the epidemic occurs with a positive probability while

outside that region, the outbreak dies out after a ﬁnite num-

ber of transmissions. Finally, we derive the conditional mean

of the fraction of individuals who get infected by each type of

strain given that an epidemic outbreak has occurred.

We supplement our theoretical ﬁndings with analytical

case studies and simulations for diﬀerent patterns of inter-

action in the host population and diﬀerent types of mutation

patterns in the pathogens. The multi-layer multi-strain mod-

eling framework allows for understanding trade-oﬀs, such as

the relative impact of countermeasures, including lock-downs

that alter the network layers on the emergence of highly con-

tagious strains. For cases where the spread of infection starts

with a moderately transmissible strain, we study how im-

posing/lifting mitigation measures across diﬀerent layers can

alter the course of the epidemic by increasing the risk of muta-

tion to a highly contagious strain. We derive the probability

of mutation to a highly transmissible strain which in turn pro-

vides a lower bound on the probability of emergence. Through

a case study for one-step irreversible mutation patterns, our

results highlight that reopening a new layer in the contact net-

work may be considered low-risk based on the transmissibility

of the current strain. Still, even a modest increase in infec-

tions caused by the additional layer can lead to an epidemic

outbreak to occur with a much higher probability. Therefore,

it is important to evaluate mitigation measures concerning

diﬀerent network layers in connection with their impact on

the likelihood of the emergence of new pathogen strains.

Next, we propose transformations to simpler epidemiolog-

ical models and unravel conditions under which we can re-

duce the multi-layer multi-strain model to simpler models for

accurately characterizing the epidemic outbreak. We show

that while a reduction to a single-layer model can accurately

predict the epidemic characteristics when the network lay-

ers are purely Poisson, a departure from Poisson distribu-

tion can lead to incorrect predictions with single-layer models.

Moreover, we show that the success of approaches that coa-

lesce the multi-layer structure to an equivalent single-layer is

critically dependent on the dispersion indices of the network

layers being perfectly matched. However, in practice, diﬀer-

ent network layers (representing diﬀerent congregate settings)

are expected to have diﬀerent structural characteristics, fur-

ther highlighting the need for considering multi-layer network

models for predicting the course of an outbreak. Our results

further underscore the need for developing epidemiological

models that account for heterogeneity among the pathogen

variants as well as the contact network layers.

Probability of Emergence. The ﬁrst question that we inves-

tigate is whether a spreading process started by infecting a

randomly chosen seed node with strain-1 causes an outbreak

infecting a positive fraction of individuals, i.e., an outbreak

of size Ω(n). Our results are based on multi-type branch-

ing processes (27,34,59,60). For computing the probability

of emergence, we ﬁrst deﬁne probability generating functions

(PGFs) of the excess degree distribution: Let g(zf, zw)de-

note the PGF for joint degree distribution of a randomly se-

lected node (initial infective/ seed) in the two network layers.

This corresponds to the PGF for the probability distribution

pd=pf

df·pw

dwand therefore,

g(zf, zw) = X

pdzfdf

(zw)dw.[2]

For a∈ {f, w}, we deﬁne, Ga(zf, zw)as the PGF for ex-

cess joint degree distribution for the number of type-fand

type-wcontacts of a node reached by following a randomly

selected type-aedge (later-generation infective/ intermediate

host). While computing Ga(zf, zw), we discount the type-a

edge that was used to infect the given node. We have

Gf(zf, zw) = X

dfpd

hdfizfdf−1(zw)dw,[3]

Gw(zf, zw) = X

dwpd

hdwizfdf

(zw)dw−1.[4]

The factor dfpd/hdfi(resp., dwpd/hdwi) gives the normalized

probability that an edge of type-f(resp., type-w) is attached

(at the other end) to a vertex with colored degree d= (df, dw)

(14).

Suppose, an arbitrary node ucarries strain-1 and trans-

mits the infection to one of its susceptible neighbors, denoted

as node v. Since there are two types of links/edges in the con-

tact network and two types of strains circulating in the host

population, there are four types of events that lead to the

transmission of infection from node uto v, namely, whether

edge (u, v)is

(i) type-fand no mutation occurs in host v;

(ii) type-fand mutation to strain-2 occurs in host v;

(iii) type-wand no mutation occurs in host v;

(iv) type-wand mutation to strain-2 occurs in host v.

In cases (i) and (iii) (resp., cases (ii) and (iv)) above, node

vacquires strain-1 (respectively, strain-2). For applying a

branching process argument (14,61) and writing recursive

equations using PGFs, it is crucial to keep track of both the

types of edges used to transmit the infection and the types of

strain acquired after mutation. Therefore, we keep a record of

the number of newly infected individuals who acquire strain-1

or strain-2, and the type of edge through which they acquired

the infection. We deﬁne the joint PGFs for transmitted in-

fections over four random variables corresponding to the four

infection events (i) - (iv) as follows.

γ1(zf

1, zf

2, zw

1, zw

2) =

g 1−Tf

1+Tf

1 2

j=1

µf

1jzf

j!,1−Tw

1+Tw

1 2

j=1

µw

1jzw

j!!.

For a∈ {f, w}and i∈ {1,2}, denote

Γa

i(zf

1, zf

2, zw

1, zw

2) =

Ga 1−Tf

i+Tf

i 2

j=1

µf

ij zf

j!,1−Tw

i+Tw

i 2

j=1

µw

ij zw

j!!.

We show that the quantity γ1(zf

1, zf

2, zw

1, zw

2)represents the

PGF for the number of infection events of each type induced

among the neighbors of a seed node when the seed node is

infected with strain-1; see SI Appendix 1.A. Furthermore, for

a∈ {f, w}and i∈ {1,2}, we show that Γa

i(zf

1, zf

2, zw

1, zw

2)is

the PGF for number of infection events of each type caused by

alater-generation infective (i.e., a typical intermediate host in

the process) that received the infection through a type-aedge

and carries strain-i. Building upon the PGFs for the infection

events caused by the seed and later-generation infectives, our

ﬁrst main result characterizes the probability of emergence

when the outbreak starts at an arbitrary node infected with

strain-1.

Theorem 1 (Probability of Emergence): It holds that

P[Emergence] = 1 −γ1(qf

1, qf

2, qw

1, qw

2),[5]

where, qf

1, qf

2, qw

1, qw

2are the smallest non-negative roots of

the ﬁxed point equations:

1= Γf

1(qf

1, qf

2, qw

1, qw

2)[6]

2= Γf

2(qf

1, qf

2, qw

1, qw

2)[7]

1= Γw

1(qf

1, qf

2, qw

1, qw

2)[8]

2= Γw

2(qf

1, qf

2, qw

1, qw

2).[9]

Here, for a∈ {w, f }and i∈ {1,2}, the term qa

ican be inter-

preted as the probability of extinction starting from one later-

generation infective carrying strain-i(after mutation) which

was infected through a type-aedge; see SI Appendix 1.A for a

detailed proof. Therefore, the probability of emergence of an

epidemic is given by the probability that at least one of the

infected neighbors of the seed triggers an unbounded chain of

transmission events. We note that Theorem 1 provides a strict

generalization for the probability of emergence of multi-strain

spreading on a single layer (27) and we can recover the proba-

bility of emergence for the case of single layer by substituting

Tf=T

Twand µ

µf=µ

µwin Equations Eq. (6)- Eq. (9).

Epidemic Threshold. Next, we characterize the epidemic

threshold, which deﬁnes a boundary of the region in the pa-

rameter space inside which the outbreak always dies out after

infecting only a ﬁnite number of individuals; while, outside

which, there is a positive probability of a positive fraction of

infections. The epidemic threshold is commonly studied as a

metric to characterize and epidemic and ascertain risk factors

(62). Let λfand λwdenote the ﬁrst moments of the dis-

tributions {pf

df}and {pw

dw}, respectively. Let hd2

fiand hd2

denote the corresponding second moments for distributions

{pf

df}and {pw

dw}. Further, deﬁne βfand βwas the mean of

the excess degree distributions respectively in the two layers.

We have

βf:= hd2

fi − λf

λf

and βw:= hd2

wi − λw

λw

.[10]

Theorem 2 (Epidemic Threshold): For

J=





1µf

11βfTf

1µf

12βfTw

1µw

11λwTw

1µw

12λw

2µf

21βfTf

2µf

22βfTw

2µw

21λwTw

2µw

22λw

1µf

11λfTf

1µf

12λfTw

1µw

11βwTw

1µw

12βw

2µf

21λfTf

2µf

22λfTw

2µw

21βwTw

2µw

22βw





,[11]

let σ(J)denote the spectral radius of σ(J). The epidemic

threshold is given by σ(J) = 1.

The above theorem states that the epidemic threshold is

tied to the spectral radius of the Jacobian matrix J, i.e.,

if σ(J)>1then an epidemic occurs with a positive prob-

ability, whereas if σ(J)≤1then with high probability the

infection causes a self-limited outbreak, where the fraction of

infected nodes vanishes to 0 as n→ ∞.The matrix Jis ob-

tained while determining the stability of the ﬁxed point of

the recursive equations in Theorem 1 by linearization around

1=qf

2=qw

1=qw

2= 1 (SI Appendix 1.B).

We note that when the mutation matrix is indecomposable,

meaning that each type of strain eventually may have lead

to the emergence of any other type of strain with a positive

probability, the threshold theorem for multi-type branching

processes (27) guarantees if σ(J)≤1, then qa

i= 1; whereas

if σ(J)>1, then 0≤qa

i<1, where i∈ {1,2}and a∈

{f, w}. For decomposable processes, the threshold theorem

(27) guarantees extinction (qa

i= 1) if σ(J)≤1; however

the uniqueness of the ﬁxed-point solution does not necessarily

hold when σ(J)>1.

Our next result provides a decoupling of the epidemic

threshold into causal factors pertaining pathogen and mu-

tation, and structural properties of diﬀerent layers in the

contact network.

Lemma 1: When Tw

1/T f

1=Tw

2/T f

2=c, where c > 0, and let

µ=µ

µf=µ

µw, we get,

σ(J

J) = σβfcλw

λfcβw×σ(Tf

Tfµ

µ).[12]

Lemma 1 follows from the observation that with Tw

1/T f

2/T f

2=c, we can express J

Jas a Kronecker product of two

matrices (denoted by ⊗), as below.

J=





1µ11βfTf

1µ12βfTw

1µ11λwTw

1µ12λw

2µ21βfTf

2µ22βfTw

2µ21λwTw

2µ22λw

1µ11λfTf

1µ12λfTw

1µ11βwTw

1µ12βw

2µ21λfTf

2µ22λfTw

2µ21βwTw

2µ22βw







=βfcλw

λfcβw⊗(Tf

Tfµ

µ).[13]

We note that the ﬁrst assumption Tw

1/T w

2=Tf

1/T f

2is consis-

tent with scenarios where the ratio of the transmissibility of

the two strains in each layer is expected to be a property of

the pathogen and not the contact networks. This assumption

is supported by the typical modeling assumption (26) that so-

cial distancing measures such as increasing distance between

individuals lead to a reduction in the transmissibility of the

disease by a speciﬁc coeﬃcient for the entire network layer.

And therefore, when each network layer has speciﬁc restric-

tions in place (and corresponding coeﬃcients for reduction in

transmissibility), the ratio of the transmissibility of the two

strains in each layer ends up being a property of the hetero-

geneity in the strains. The second assumption (µ

µf=µ

µw)

in Lemma 1 is motivated by the assumption that mutations

occur within individual hosts, which is typical to multi-strain

spreading models; see (27) and the references therein.

We note that the decoupling obtained through Eq. (12)

reveals the delicate interplay of the network structure and

the transmission parameters in determining the threshold for

emergence of an epidemic outbreak. Lastly, we observe that

Lemma 1 provides a uniﬁed analysis for the spectral radius in-

cluding the case with a single-strain or a single-layer. For the

multi-strain spreading on a single-layer network, the spectral

radius can be derived by substituting Tf

i=Tw

iin Eq. (12)

and setting mean degree of one of the layers as 0, for in-

stance, setting λw=βw= 0, yielding the epidemic threshold,

denoted as ρMS−SL,

ρMS−SL =βf×σ(Tf

Tfµ

µ),[14]

where βcorresponds to the mean of the excess degree distri-

bution for the single-layered contact network. For the case of

the spread of a single strain on a multi-layer contact net-

work, we substitute Tf

1=Tf

2in Eq. (12), which implies

ρ(Tf

Tfµ

µ) = Tfρ(µ

µ) = Tf, yielding the epidemic threshold, de-

noted as ρSS−ML,

ρSS−ML =σβfcλw

λfcβw×Tf.[15]

It is easy to verify that the spectral radius as obtained from

Eq. (14) and Eq. (15) is consistent with the results in (34)

and (27).

Mean Epidemic Size. Next, we compute the mean epidemic

size and the mean fraction of nodes infected by each type of

strain. The knowledge of the fraction of individuals infected

by each strain is vital for cases when diﬀerent pathogen strains

have diﬀerent transmissibility and virulence. In such cases,

predicting the expected fraction of the population hit by the

more severe strain can help scale healthcare resources in time.

For computing the mean epidemic size, we consider the

zero-temperature random-ﬁeld Ising model on Bethe lattices

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SpreadingProcesseswithMutationsoverMulti-layerNetworksMansiSooda,AnirudhSridharb,RashadEletrebyc,ChaiWahWud,SimonA.Levine,H.VincentPoorb,andOsmanYaganaaDepartmentofElectricalandComputerEngineering,CarnegieMellonUniversity,Pittsburgh,PA15213USA;bDepartmentofElectricalEngineering,PrincetonUniversity,P...

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Spreading Processes with Mutations over Multi-layer Networks Mansi Sooda Anirudh Sridharb Rashad Eletrebyc Chai Wah Wud Simon A. Levine H. Vincent Poorb and Osman Yagana

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