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Transient Analysis of a Resource-limited Recovery

Policy for Epidemics: a Retrial Queueing Approach

Aresh Dadlani∗, Muthukrishnan Senthil Kumar†, Kiseon Kim∗and Faryad Darabi Sahneh‡

∗School of EECS, Gwangju Institute of Science and Technology, Gwangju 61005, South Korea

†Department of Applied Mathematics and Computational Science, PSG College of Technology, India

‡School of Computer Science, Georgia Institute of Technology, Atlanta, GA 30332, USA

Email: dadlani@gist.ac.kr, msk@amc.psgtech.ac.in, kskim@gist.ac.kr, fsahneh3@gatech.edu

Abstract—Knowledge on the dynamics of standard epidemic

models and their variants over complex networks has been well-

established primarily in the stationary regime, with relatively

little light shed on their transient behavior. In this paper, we

analyze the transient characteristics of the classical susceptible-

infected (SI) process with a recovery policy modeled as a state-

dependent retrial queueing system in which arriving infected

nodes, upon ﬁnding all the limited number of recovery units

busy, join a virtual buffer and try persistently for service in

order to regain susceptibility. In particular, we formulate the

stochastic SI epidemic model with added retrial phenomenon as

a ﬁnite continuous-time Markov chain (CTMC) and derive the

Laplace transforms of the underlying transient state probability

distributions and corresponding moments for a closed population

of size Ndriven by homogeneous and heterogeneous contacts.

Our numerical results reveal the strong inﬂuence of infection

heterogeneity and retrial frequency on the transient behavior of

the model for various performance measures.

I. INTRODUCTION

Epidemiological models have assumed new relevance in

assessing spreading processes over a broad interdisciplinary

spectrum. Ranging from modeling malware propagation on

the digital landscape [1], [2] to ‘word-of-mouth’ inﬂuence

in social networking platforms [3], [4], analysis of classical

stochastic epidemic models and their deterministic approx-

imations at microscopic and macroscopic levels have been

the subject of serious scientiﬁc inquiry in recent years [5].

Much of the existing works however, are mostly concerned

with the long-term characteristics of the epidemics rather than

their transient behavior upto some speciﬁed time. In fact,

time-dependent analysis provides deeper insight on the system

behavior when the primary parameters are perturbed and thus,

can serve crucial in devising effective control measures.

In regard to applications of queueing theory in quantitative

analysis of epidemic progression, the authors of [6] showed the

number of infected nodes at the moment of ﬁrst detection to

be geometrically distributed by formulating the susceptible-

infected-removed (SIR) epidemic as an M/G/1queue with

processor sharing service discipline. Approximations for the

quasi-stationary distribution (QSD) of the number of sus-

ceptibles in the susceptible-infected-susceptible (SIS) and

susceptible-latent-infected-susceptible (SEIS) models were de-

rived in [7], wherein each node was either a busy (infected)

or an idle (susceptible) server in a homogeneously mixing

population. Similarly, analytical derivations of the QSD and

transient distribution for the maximum number of infectives in

a generalized SIS model, described as a birth-death process,

were detailed in [8]. Furthermore, to investigate the number

of infectives resulting from a computer virus (CodeRed-II)

attack during time interval (0, t], the block-structured state-

dependent event (BSDE) approach was advocated consider-

ing non-exponential and correlated infection and recovery

ﬂows in an SIS-type model [9]. In view of heterogeneous

infectiousness and susceptibility in the SIS model, Economou

et al. [10] studied the behavior of the corresponding 2N-

state Markov chain formulation in the quasi-stationary regime.

Moreover, Sahneh et al. [11] deduced the state occupancy

probabilities of the SIS model with multiple contact layers

by reducing the exact MN-state Markov chain representation

to a system of MN non-linear differential equations using

mean-ﬁeld approximation. The above works nonetheless, do

not consider a recovery policy, which inherently has limited

resources, within their models.

In this paper, we address the impact of infected nodes

retrying for recovery controlled by limited resources on the

transient behavior of the SI model considering both, homoge-

neous and heterogeneous1contacts in a ﬁnite population. We

build our time-homogeneous CTMC model based upon the

retrial queueing notion in which infected nodes return back

to become susceptible only after being served by one of the

idle recovery units. On ﬁnding all units busy, the infected node

then joins a virtual buffer, known as orbit, from which it retries

persistently for access until granted service by an idle unit.

Our main interest is to numerically investigate the transient

state probability distributions of the number of infected nodes

undergoing recovery and those residing in the orbit as well as

their related moments under varying parametric settings which,

to the best of our knowledge, has not yet been reported.

An immediate practical application of our work is the time

and workload dependency analysis of online computer scan-

ning services. Under a practically relevant scenario comprising

of a small number of networked devices, an infected client

device attempts to access a server that provides malware detec-

tion/eradication services. In reality, the number of communi-

cation ports on the server dedicated for this purpose is always

1Heterogeneous contact in here refers to the non-uniform infection rate

associated with each node in any arbitrary network topology.

arXiv:1607.08443v1 [cs.SY] 28 Jul 2016

Closed Pop

u

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݅ߤ

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1

2

. . .

Recovery U

n

ܿ

n

its

݆ߠ

Orb

i

ܰെ

.

.

i

t

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.

.

Fig. 1: Schema of the state-dependent retrial SI model for

network of size |V|=Nwith ﬁnite recovery units (c<N).

limited and thus, cannot serve all clients simultaneously. On

arrival, if a client ﬁnds an idle port on the server, it connects

successfully to receive the required service. Otherwise, the

client retries randomly and independently, expecting to gain

access to an idle port on the server. Therefore, for a ﬁnite op-

eration time horizon, statistical information on the proportion

of infected clients being scanned and those awaiting access

serves substantial to network administrators.

The remainder of this paper is organized as follows. The

Markovian retrial SI model with limited recovery resources is

formally introduced in Section II. Derivations for the transient

state and marginal probabilities under different contact types

are provided in Section III, followed by numerical results in

Section IV. Finally, we conclude the paper in Section V with

directions for potential future work.

II. PRO PO SE D MOD EL DESCRIPTION

To make the subsequent derivations systematic, we intro-

duce some graph-theoretical nomenclature and the Markov

chain representation of the retrial SI model in this section.

A. Contact Network Topology

Consider a ﬁxed network of size Nwithin which a particular

infection spreads. We represent such a contact network as

an undirected graph G= (V, E ), where V={1,2, . . . , N}

denotes the set of constituent nodes and E⊆V×Vis the set

of interaction links. The associated adjacency matrix of Gis

given as A,[ai,j]V×V, where ai,j = 1 if iand jare adjacent

neighbors in contact, and ai,j = 0 if otherwise. Following the

deﬁnition of matrix A, the degree of any node i∈Vcan be

easily computed to be di=PN

j=1 ai,j .

B. Retrial SI Model Formulation

We extend the standard stochastic SI compartmental model

by reinforcing the intrinsic retrial behavior of infected nodes

contending for limited treatment resources as shown in Fig. 1.

Speciﬁcally, each node transitions from being in the suscepti-

ble (S) sub-population to the infected (I) sub-population, and

back again to susceptible upon receiving treatment. Since the

population is closed, i.e. |S|+|I|=N, the system state at

time t≥0can be fully described by I(t)and R(t)which

represent the number of recovery units (c)being occupied by

infected nodes and the number of retrying infected nodes in

the orbit of size N−c, respectively. The arrival of infected

nodes is assumed to follow a Poisson process with state-

dependent arrival rate of λi,j ∈R+, where i∈ {0,1, . . . , c}

and j∈ {0,1, . . . , N −c}. Also, an arriving infected node

undergoes recovery at one of the units for an exponentially

distributed time with mean µ−1∈R+, after which it once again

becomes susceptible to the contagion. Infected nodes awaiting

in the orbit attempt for service at exponentially distributed

random time intervals with mean θ−1∈R+. Following these

deﬁnitions, the ﬁnite CTMC representation of the retrial SI

epidemic can be described as the following bi-variate process:

X(t) = I(t), R(t);t≥0,(1)

taking values on state space Ω = (i, j)|i∈ {0,1, . . . , c};j∈

{0,1, . . . , N −c}. For any arbitrary x, y ∈Ωwith x= (i, j)

indicating the state of the system having irecovery units busy

treating infected nodes and jnumber of infected nodes in

the orbit at time t, the inﬁnitesimal transition probabilities are

speciﬁed as follows, where qx,y ∈R≥0denotes the transition

rate from state xto state y:

Px,y(t, t + ∆t),PrX(t+ ∆t) = y|X(t) = x

=(qx,y∆t+o(∆t),if x6=y

0,if otherwise.

(2)

For complete characterization of the time evolution of X(t),

we deﬁne P(t)=[pi,j(t)]1×|Ω|to be the transient state proba-

bility vector, with element pi,j(t)denoting the probability of

process X(t)being in state (i, j)at time t, i.e. ∀(i, j )∈Ω:

pi,j (t),Pr[X(t)=(i, j)] = Pr[I(t) = iand R(t) = j].(3)

III. TRANSIENT STATE ANALYS IS

The nature of node-level interactions has been shown to

profoundly impact the process of contagion and its control

mechanisms [11]–[13]. In this section, we derive the transient

solution of the state occupancy probabilities.

A. Retrial SI Model with Homogeneous Contacts

In this setting, we assume the spread of a typical infection

to be driven by homogeneous mixing in a population wherein

each node makes contact with another node at random time

intervals which are i.i.d. random variables [7]. Subsequently,

the state-dependent arrival rate of infected nodes at the recov-

ery units is expressed as λi,j =α(N−i−j)/N, where αis

the contact rate in the population. With the time-homogeneous

process X(t)deﬁned over Ωof size (c+1)(N−c+1), the

transition rates dictating the retrial SI model dynamics are:

qx,y =

λi,j ,if y=(i+1, j ); i≤c−1, j ≤N−c

iµ, if y=(i−1, j); 1 ≤i≤c, j ≤N−c

jθ, if y=(i+1, j −1); i≤c−1,1≤j≤N−c

λc,j ,if y=(c, j +1); j≤N−c−1

0,if otherwise.

(4)

Equation (4) simply expresses the four possible cases of state

transitions shown in Fig. 2. The corresponding Chapman-

Kolmogorov forward differential equations are:

Fig. 2: State transitions of the stochastic retrial SI model.

•Case I: When the set of recovery units (0≤i≤c−1) and

the orbit (0≤j≤N−c−1) have vacancies:

p0

i,j (t) = −(λi,j +iµ+jθ)pi,j (t)+λi−1,j pi−1,j (t)

+(j+1)θpi−1,j+1(t)+(i+1)µpi+1,j (t).(5)

•Case II: When all the recovery units are occupied (i=c)

and the orbit is not full (0≤j≤N−c−1):

p0

c,j (t) = −(λc,j +cµ)pc,j (t)+λc−1,jpc−1,j(t)

+(j+1)θpc−1,j+1(t)+λc,j−1pc,j−1(t).(6)

•Case III: When at least one recovery unit is idle (0≤i≤

c−1) and the orbit is full (j=N−c):

p0

i,N−c(t) = −(λi,N−c+iµ+(N−c)θ)pi,N−c(t)

+λi−1,N−cpi−1,N−c(t)+(i+1)µpi+1,N−c(t).(7)

•Case IV: When the ﬁnite set of recovery units (i=c) and

the orbit (j=N−c) are all full:

p0

c,N−c(t) = −cµpc,N−c(t)+λc−1,N−cpc−1,N−c(t)

+λc,N−c−1pc,N−c−1(t).(8)

Given P(0), equations (5)-(8) can be written in the matrix

form as P0(t) = P(t)·Q, where Q: Ω ×Ω→R≥0is the

inﬁnitesimal generator matrix with elements qx,y given as:

Q(x, y) =

qx,y,if x6=y

−P

y∈Ω

y6=x

qx,y,if x=y. (9)

We now employ the Laplace Transform (LT) operator L[·]on

(5)-(8) to obtain P∗(s) = L[P(t)], which is then used to yield

the probability vector P(t)through inverse LT. As a result, we

arrive at the following system of equations:

pi,j (0) =(s+λi,j +iµ+jθ)p∗

i,j (s)−λi−1,j p∗

i−1,j (s)

−(j+1)θp∗

i−1,j+1(s)−(i+1)µp∗

i+1,j (s),(10)

pc,j (0) =(s+λc,j +cµ)p∗

c,j (s)−λc−1,j p∗

c−1,j (s)

−(j+1)θp∗

c−1,j+1(s)−λc,j−1p∗

c,j−1(s),(11)

pi,N−c(0) =(s+λi,N−c+iµ+(N−c)θ)p∗

i,N−c(s)

−λi−1,N−cp∗

i−1,N−c(s)−(i+1)µp∗

i+1,N−c(s),(12)

pc,N−c(0) =(s+cµ)p∗

c,N−c(s)−λc−1,N−cp∗

c−1,N−c(s)

−λn,N−c−1p∗

c,N−c−1(s).(13)

Re-arranging (10)-(13) in the vector-matrix form results in

P∗(s) = P(0)M−1, where the invertible matrix Mexhibits

the following block tridiagonal structure:

M=

A0B00. . . 0 0

C1A1B1. . . 0 0

0C2A2. . . 0 0

.

.

..

.

..

.

.....

.

..

.

.

0 0 0 . . . CcAc

(N−c+1)(c+1)

(14)

For 0≤i≤c−1, the diagonal sub-matrix Ai= (s+λi,j +

iµ +jθ)IN−c+1, with Idenoting the identity matrix, whereas

Ac,[ˆau,v]is an upper bidiagonal matrix featured as:

ˆau,v =

s+λc,j +cµ, if u=v6=N−c

−λc,j ,if u=v−1

s+cµ, if u=v=N−c

0,if otherwise.

(15)

where uand vare unique integer values returned by the label-

ing function f: Ω →N≥0, such that f(i, j)=(N−c+1)i+j.

The diagonal sub-matrix Ci=−iµIN−c+1 and Bi,[ˆ

bu,v]is

a lower bidiagonal matrix with entries:

ˆ

bu,v =

s+λi,j ,if u=v

−jθ, if u=v+ 1

0,if otherwise.

(16)

After computing P∗(s)as above, we now ﬁnd the marginal

distributions of I(t)and R(t), denoted respectively as pi(t)

and qj(t), and their moments using inverse LT. Once identiﬁed,

the stationary probability vector, Π=[πi,j ]1×|Ω|, can also

be determined using Π = lims→0P∗(s) = limt→∞ P(t). The

probability of ﬁnding iinfected nodes under recovery at time

instant tis given as below:

pi(t) =

N−c

X

j=0

pi,j (t),(17)

with corresponding moments E[In(t)] = Pc

i=0 inpi(t). Like-

wise, the marginal distribution of R(t)is expressed as:

qj(t) =

c

X

i=0

pi,j (t),(18)

with E[Rn(t)] = PN−c

j=0 jnqj(t)as its nth raw moment.

B. Retrial SI Model with Heterogeneous Contacts

Unlike the homogeneous counterpart, where an infected

node can equally infect any susceptible node, heterogeneity

in the retrial SI model can be reﬂected in terms of non-

homogeneous (node-dependent) infection rates over the con-

tact network as in the works of [10] and [14]. To this end, we

assume that any susceptible node, say k, can be infected in

two ways: (i) infection stemming from some external source

(outside the population) according to a Poisson process with

rate δk

i,j ∈R+such that δk

i,j =αk(N−i−j)/N and

(ii) internal nodal infection following a Poisson process with

rate βk,l ∈R+on kfor all susceptible nodes lsuch that

ak,l = 1. Without loss of generality, all involved processes

governing the external/internal infections, recovery, and retrial

times are assumed to be mutually independent. This scenario is

analogous to virus propagation in computer networks, where

node knot only receives the virus from its neighbors, but

can also generate and spread its own virus. For convenience,

we denote the state-dependent arrival rate due to node kas

follows, where Isymbolizes the sub-population of infected

nodes in the network:

λk

i,j =δk

i,j +X

l∈I

βl,kal,k .(19)

By replacing λi,j and λc,j in (4) with λk

i,j and λk

c,j , respec-

tively, the state transition rates for the proposed model with

heterogeneities can be expressed in a similar manner. Repeti-

tion of the expressions for the Kolmogorov forward equations

and operations in the LT domain are not included here due to

limited space. Thus, undertaking the same approach as in the

preceding sub-section, with pk

i,j (t)deﬁned as the probability

of having irecovering and jorbital infected nodes due to

kat time t(and the other notations varied accordingly), the

respective marginal distributions and moments can be obtained

in a straightforward manner.

IV. NUMERICAL RES ULT S AN D DISCUSSIONS

The objective of resorting to numerical simulations in this

section is to analyze the transient state behavior of the retrial SI

model with respect to the derived performance measures under

varying parametric values so as to visualize the solutions in

practical scenarios. Since inverting the computed LT is quite

tedious, particularly for larger values of Nand c, we adopt the

Jagerman-Stehfest method which is built upon the Post-Widder

inversion formula to numerically compute the approximate

results for L−1[P∗(s)] and L−1[Pk∗(s)] [15]. Throughout this

section, we assume the initial condition of the system to be

p0,0(0)=1 and k= 2.

A. Marginal Distributions at Different Time Points

In Fig. 3, the marginal probabilities pi(t)and qj(t)for a

well-mixed population are plotted with parameters (N, c) =

(10,5),α=5,µ=0.4, and θ= 2. Under this parametric set-up,

we observe in Fig. 3a that within time interval t∈[0,2], unlike

the rapid exponential decay of p0(t), the probability of having

at least one infected node under recovery gradually rises before

p0+t/

p1+t/

p2+t/

p3+t/

p4+t/

p5+t/

0

2

4

6

8

10

0.0

0.2

0.4

0.6

0.8

1.0

time +t/

Probability Distribution of I+t/

(a) Marginal distribution of I(t).

q0t

q1t

q2t

q3t

q4t

q5t

0

2

4

6

8

10

0.0

0.2

0.4

0.6

0.8

1.0

time t

Probability Distribution of Rt

(b) Marginal distribution of R(t).

Fig. 3: Transient behavior under homogeneous contacts for

N=10,c=5,α= 5,µ= 0.4,θ= 2, and p0,0(0) = 1.

approaching the steady-state probability. This clearly indicates

that with the infected nodes initially arriving with maximum

rate λ0,0=α, the probability of the recovery units being

occupied increases and eventually, the probability of having

all units busy (p5(t)) reaches its highest in long-term with the

least probability for them being all idle. On the other hand,

Fig. 3b shows that q0(t)=1upto t= 0.5and then reduces to

its minimum value at a slower rate. Dependent on the initial

conditions, such behavior in interval t∈[0,9] is not far from

expectation. As the infected nodes begin to arrive, they are

immediately served by the idle units and enter the orbit only if

all units are busy. Thus, decrease in q0(t)reﬂects the recovery

units’ availability for service and as the number of arrivals

outnumber the recovery units, the orbit gradually begins to ﬁll

up thus, reducing the probability of it being vacant.

Marginal distributions for the same system under infection

heterogeneity are illustrated in Fig. 4, where αk=αdk/N

and the internal infection spreading rate from node kto its

susceptible neighbors is βk,l =dl/N. The relatively longer

transient phase (t∈[0,14]) in this ﬁgure is evidence of the

impact of non-uniform infection over the contact network. In

other words, since αkand βk,l are node degree-bounded rates,

the infection spread is less spontaneous thus, resulting in a

p0t

p1t

p2t

p3t

p4t

p5t

0

2

4

6

8

10

0.0

0.2

0.4

0.6

0.8

1.0

time t

Probability Distribution of It

(a) Marginal distribution of I(t).

q0t

q1t

q2t

q3t

q4t

q5t

0

2

4

6

8

10

0.0

0.2

0.4

0.6

0.8

1.0

time t

Probability Distribution of Rt

(b) Marginal distribution of R(t).

Fig. 4: Transient behavior under heterogeneous contacts for

N=10,c=5,α= 5,µ= 0.4,θ= 2,βk,l = 1, and p0,0(0)= 1.

longer time for healthy nodes to get infected in comparison to

uniform infectivity in Fig. 3. Hence, a priori information on

the connectivity pattern of the network is required in order to

control the spread under heterogeneous infectivities.

B. Expected Number of Infectives at Different Time Points

Tables I and II summarize the expected number of infected

nodes under recovery and awaiting treatment in the orbit at

discrete time epochs for the two contact types. A common

trend visible in both models is that the expected number of

infectives under recovery increases with the value of cfor any

arbitrary population size. Increasing cimproves the chances

of ﬁnding an idle recovery unit which in turn, reduces the

average number of infected nodes in the orbit. The transient

behavior exhibited by the retrial SI model with heterogeneous

infection rates however, exists for a comparatively longer time

interval before reaching stationarity as shown in Table II. For

instance, the average number of orbital nodes for (N, c) =

(20,15) stabilizes at t= 10 in Table I, while heterogeneity

in the infection rate prolongs this convergence to t > 20 as

shown in Table II. It should also be noted that the system never

approaches an infection-free equilibrium (where all nodes are

susceptible) due to the state dependency of the infection rate.

TABLE I: First moments under homogeneous contacts.

E[I(t)],E[R(t)]N=10 N=20 N=40

c=5

t=0.5(2.81,1) (2.79,1) (2.78,1.03)

t=2 (2.36,1.65) (1.79,2.52) (1.53,3.14)

t=5 (1.72,2.52) (0.93,5.64) (0.63,8.46)

t=10 (1.62,2.69) (0.69,7.62) (0.37,13.98)

t=20 (1.62,2.69) (0.63,8.31) (0.28,18.46)

c=10

t=0.5(3.01,0) (3.13,1) (3.19,1)

t=2 (5.6,0) (6.29,1.02) (6.37,1.09)

t=5 (6.47,0) (6.77,1.25) (5.79,2.08)

t=10 (6.52,0) (6.67,1.4) (4.99,3.16)

t=20 (6.52,0) (6.67,1.4) (4.66,3.68)

c=15

t=0.5- (3.13,1) (3.19,1)

t=2 - (6.59,1) (7.18,1)

t=5 - (8.39,1) (9.57,1.03)

t=10 - (8.68,1) (10,1.07)

t=20 - (8.68,1) (10.04,1.08)

c=20

t=0.5- (3.13,0) (3.19,1)

t=2 - (6.59,0) (7.19,1)

t=5 - (8.39,0) (9.83,1)

t=10 - (8.68,0) (10.47,1)

t=20 - (8.69,0) (10.52,1)

TABLE II: First moments under heterogeneous contacts.

E[I(t)],E[R(t)]N=10 N=20 N=40

c=5

t=0.5(2.27,1) (2.51,1) (2.39,1)

t=2 (2.89,1.25) (2.26,1.94) (2.38,1.81)

t=5 (2.2,1.98) (1.14,5.1) (1.19,5.15)

t=10 (1.89,2.3) (0.67,8.19) (0.62,10.2)

t=20 (1.83,2.37) (0.48,9.71) (0.33,17.74)

c=10

t=0.5(2.09,0) (2.66,1) (2.51,1)

t=2 (4.03,0) (5.59,1.01) (5.34,1.01)

t=5 (5.23,0) (6.52,1.25) (6.36,1.29)

t=10 (5.48,0) (6.39,1.52) (6.12,1.77)

t=20 (5.49,0) (6.33,1.6) (5.94,2.07)

c=15

t=0.5- (2.66,1) (2.51,1)

t=2 - (5.75,1) (5.48,1)

t=5 - (7.88,1) (7.79,1)

t=10 - (8.42,1) (8.51,1.02)

t=20 - (8.47,1) (8.6,1.02)

c=20

t=0.5- (2.66,0) (2.51,1)

t=2 - (5.75,0) (5.48,1)

t=5 - (7.9,0) (7.84,1)

t=10 - (8.47,0) (8.61,1)

t=20 - (8.52,0) (8.72,1)

Therefore, λi,j reaches its maximum value when (i, j) = (0,0)

and becomes zero when all the nodes are infected, i.e. (i, j) =

(c, N −c).

C. Impact of Retrial Rates at Different Time Points

The evolution of expected values of I(t)and R(t)in terms

of the retrial rate θare shown in Fig. 5 for (N, c) = (20,8)

and µ= 1. Fig. 5a depicts that the difference in the average

number of recovering nodes is very small for given θvalues

in both contact types. However, the transient state for the

homogeneous case lasts for a much shorter time than its

heterogeneous counterpart. Additionally, for θ= 0,E[I(t)]

reaches a slightly higher steady-state value than that for θ > 0.

Intuitively, for θ=0, the rate at which nodes get infected in the

q = 0

q = 1

q = 2

Homogeneous

Heterogeneous

1

2

3

4

5

3.88

3.90

3.92

3.94

3.96

3.98

4.00

4.02

EIt

HomogeneousContact Rate

0

1

2

3

4

5

6

1.0

1.5

2.0

2.5

3.0

3.5

4.0

time t

Expected Value of It

(a) Expected number of infectives under recovery.

Homogeneous

Heterogeneous

q = 0

q = 1

q = 2

0

1

2

3

4

5

6

1.0

5.0

2.0

3.0

1.5

time t

Expected Value of Rt

(b) Expected number of infectives in orbit.

Fig. 5: Expected values under varying retrial rates for N=20,

c=8,α=5,µ= 1,βk,l = 1, and p0,0(0) = 1.

reduced SI model with limited recovery units solely depends

on the number of busy recovery units. The impact of retrial

on the number of orbital nodes is given in Fig. 5b. While

the model with heterogeneities follows an identical trend, it

reveals lower values for E[I(t)] and E[R(t)]. The rationale

behind such variation is the high pace of infectivity caused by

the subsumed heterogeneous infection rates.

V. CONCLUSIONS AND FUTURE WO RK S

We presented an exact Markov chain model for the SI

epidemic process incorporated with the retrial attempts of

infected nodes instigated by a recovery policy. Motivated by

the signiﬁcance of transient behavior statistics in practical

online scanning services, we numerically obtain the marginal

probability distributions of the number of recovering and

orbital infectives and their corresponding moments for the

proposed model under homogeneous contacts. Accounting for

the possibility of infections caused by external and internal

sources, the study was further extended to unravel the impact

of infection heterogeneity on the network characteristics.

We believe utilizing the retrial notion under resource con-

straints is very promising and novel in epidemic modeling

thus, breeding several open problems. A prospective follow-

up on this work is a reasonably accurate mean-ﬁeld approxi-

mation geared for asymptotic steady-state distribution analysis.

Moreover, numerical results for cases of interest, namely large

Nand cvalues as well as quasi-stationary distribution of the

number of infected nodes can be further investigated. From the

viewpoint of resource budgeting and optimization, statistics

obtained in the transient regime can be used to ﬁnd the optimal

number of recovery units required to prevent an endemic in

its early stages.

ACKNOWLEDGMENT

This research was a part of the project titled “Development

of an Automated Fish-counter System and Measurement of

Underwater Farming-ﬁsh”, funded by the Ministry of Oceans

and Fisheries, South Korea.

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