A FURTHER INVESTIGATION ON THE PROPAGATION OF

INFECTION IN A SQUARE LATTICE.

by

KRISTOFER NAEL B. MAYORDOMO

An Undergraduate Thesis submitted to the Physics Division

Institute of Mathematical Sciences and Physics

College of Arts and Sciences

University of the Philippines Los Banos

In Partial Fulfillment of the Requirements

for the Degree of

Bachelor of Science in Applied Physics

November 2013

CERTIFICATION

This is to certify that this undergraduate thesis entitled, A Further Investigationon the Propagation of Infection in a Square Lattice. and submitted by KristoferNael B. Mayordomo to fulfill part of the requirements for the degree of Bachelor ofScience in Applied Physics was successfully defended and approved on 20 November2013.

CHRYSLINE MARGUS N. PINOL, Ph.D.

Thesis Adviser

JUNIUS ANDRE F. BALISTA, M.S..

Thesis Committee Chair

The Institute of Mathematical Sciences and Physics (IMSP) endorses acceptanceof this undergraduate thesis as partial fulfillment of the requirements for the degreeof Bachelor of Science in Applied Physics.

LOU SERAFIN M. LOZADA, M.S

HeadPhysics Division, IMSP

VIRGILIO P. SISON, Ph.D.

DirectorIMSP

This undergraduate thesis is hereby officially accepted as partial fulfillment of therequirements for the degree of Bachelor of Science in Applied Physics.

ZITA VJ. ALBACEA, Ph.D.

DeanCollege of Arts and Sciences

i

ABSTRACT

MAYORDOMO, KRISTOFER NAEL B. University of the Philippines LosBanos, November 2013. A Further Investigation on the Propagation ofInfection in a Square Lattice..

Adviser: Chrysline Margus N. Pinol, Ph.D.

We aim to investigate further the dynamics of an Ising-based susceptible-infected

model proposed by Crisostomo and Pinol in 2012; specifically, the effect of varying

the location and number of infectives to the spread of infection in a closed popula-

tion having homogeneous interaction. Results show that the rate of propagation of

infection is independent of the location of the first infective. Increasing the initial

number of contagions generally hastens the spread of the disease. However, there

exists a critical number after which we observe a change in the trend. Instead of

speeding up, a further increase in the number of initial infectives slows down the

propagation of infection.

PACS: 75.10.HK[Classical spin models], 87.10.HK [Lattice models], b7.10. Rt [Montecarlo simulation], 0.5.a[Computational methods in statistical physics and nonlineardynamics]

ii

Table of Contents

Abstract ii

List of Figures iv

1 Introduction 11.1 Significance of the study . . . . . . . . . . . . . . . . . . . . . . . . . 31.2 Objectives of the study . . . . . . . . . . . . . . . . . . . . . . . . . . 51.3 Scope and limitation of the study . . . . . . . . . . . . . . . . . . . . 51.4 Time and place of study . . . . . . . . . . . . . . . . . . . . . . . . . 5

2 Review of literature 62.1 The deterministic model . . . . . . . . . . . . . . . . . . . . . . . . . 62.2 Stochastic models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.3 Some real-world applications . . . . . . . . . . . . . . . . . . . . . . . 8

2.3.1 Viral infections . . . . . . . . . . . . . . . . . . . . . . . . . . 82.3.2 Disease propagation and intervention . . . . . . . . . . . . . . 92.3.3 Ecological systems . . . . . . . . . . . . . . . . . . . . . . . . 112.3.4 Spread of information . . . . . . . . . . . . . . . . . . . . . . . 11

2.4 The Ising-based approach . . . . . . . . . . . . . . . . . . . . . . . . 12

3 Methodology 163.1 Initialization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163.2 Ising-model approach . . . . . . . . . . . . . . . . . . . . . . . . . . . 163.3 Simulation specifics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

4 Results and Discussion 204.1 Single Infective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204.2 Multiple infectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214.3 The effect of clustering . . . . . . . . . . . . . . . . . . . . . . . . . . 24

5 Summary and conclusion 30

iii

List of Figures

1.1 The SIR model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2 The SI model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2.1 The compartmental representation of the SIR model. . . . . . . . . . 72.2 The scale-free network. . . . . . . . . . . . . . . . . . . . . . . . . . 102.3 SEIRZ framework. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.4 Procedure for the Ising-based malware epidemiological model. . . . . 132.5 Procedure for the Ising-based SI model. . . . . . . . . . . . . . . . . . 14

3.1 Von Neumann (or nearest neighbor) interaction . . . . . . . . . . . . 173.2 Flowchart of the procedure. . . . . . . . . . . . . . . . . . . . . . . . 18

4.1 Infection curves associated with Case 1. . . . . . . . . . . . . . . . . . 214.2 Infection curves associated with Case 2. . . . . . . . . . . . . . . . . . 214.3 Infection rate as a function of temperature. . . . . . . . . . . . . . 214.4 Infection curves using randomly distributed I0 at N = 100. . . . . . . 224.5 Infection curves using randomly distributed I0 at N = 150. . . . . . . 234.6 Infection curves using randomly distributed I0 at N = 200. . . . . . . 234.7 Infection rate associated with Scenario 1 at N = 100. . . . . . . . . . 244.8 Infection rate associated with Scenario 1 at N = 150. . . . . . . . . . 244.9 Infection rate associated with Scenario 1 at N = 200. . . . . . . . . . 254.10 Infection curves using clustered I0 at N = 100. . . . . . . . . . . . . . 254.11 Infection curves using clustered I0 at N = 150. . . . . . . . . . . . . . 264.12 Infection curves using clustered I0 at N = 200. . . . . . . . . . . . . . 264.13 Infection rate asociated with Scenario 2 at N = 100. . . . . . . . . . . 274.14 Infection rate asociated with Scenario 2 at N = 150. . . . . . . . . . . 274.15 Infection rate asociated with Scenario 2 at N = 200. . . . . . . . . . . 28

iv

Chapter 1

Introduction

Epidemiology pertains to the study of diseases in a population - its patterns,

causes and effects [1]. The field emerged as a result of peoples desire to find a

way to predict the occurrence and rate of spread of infection within a population,

and if possible, prevent it from reaching a critical level. The conventional approach

to studying a certain phenomenon usually involves experiments implemented in a

smaller scale. This, however, when applied to diseases is often times impractical and

unethical. Scientists, therefore, resort to mathematical models when studying the

dynamics of the spread of infection within a population.

The simplest epidemic model was first introduced by William Oglivy Kermack

and Anderson Gray McKendrick in 1927 [2]. More popularly known as SIR, the model

describes the health of a population using three basic classes: (1) susceptible, the

class vulnerable to the disease; (2) infected/infective, when the individual acquires

the infection, and; (3) recovered or removed class, when the individual is either cured

from the infection or dies. Transition between these states is presented in Figure 1.1.

The rate of change from one state to another is constant and given by and for

the infection and recovery, respectively. This may be represented using the following

differential equations:

dS

dt= SI, (1.1)

dI

dt= SI I, and (1.2)

dR

dt= I. (1.3)

1

Figure 1.1: The SIR model.

The population is assumed to be a closed community. There are no births, migrations

or deaths. The total population is fixed to a constant number N(t) = S(t) + I(t) +

R(t). The SIR model applies to many cases of infection. The most common example

is a viral infection popularly known as smallpox [1]. A person acquires the disease.

Symptoms manidfest over time, after which the person recovers.

From this very simple model has emerged many different variants. It works

with the assumption that a person, although already infected, remains vulnerable

to the disease. Common examples are rubella and measles. Rigorous medical treat-

ment does not make a child immune to these diseases [1]. The case of influenza, on

the other hand, may be simulated using the SIRS (Susceptible-Infective-Recovered-

Susceptible) model. When a person recovers from influenza, it does not assure that

reinfection will not occur. Thus, he may again acquire the infection after recovery

[4].

Due to its robustness1, the SIR has been extended to model related systems.

One good example would be the study conducted by Bettencourt, et, al. [5]. They

used epidemiological models, including the SIR, to describe the diffusion of ideas

1in computational physics, robustness of a model is referred to as the stability and versatility ofthe model to describe a specific phenomena.

2

to a network of people. Investigations of these models characteristics have more

important implications that transecend epidemiology alone.

1.1 Significance of the study

In this study, we analyze the characteristics of the SI model introduced in [3].

Here, the health of the population is described using only two states: susceptible and

infected (Fig.1.2). Mathematically,

Figure 1.2: The SI model.

dS

dt= SI, and (1.4)

dI

dt= SI. (1.5)

where S(t) is the susceptible state, while I(t) is the infective state. Following the

model proposed by Kermack-McKendrick in 1927 [2], we assume a closed community.

That is, at any given time, N(t) = S(t) + I(t) = constant. The above differential

equations are easily solvable:

3

S(t) =N

1 + e(ttc), and (1.6)

I(t) =N

1 + e(ttc). (1.7)

where tc is the time when half of the population is infected and is the rate of

infection.

Despite its simplicity, the SI model is not very popular to epidemiologists.

Quick internet search gives us papers focusing on more complicated cases. This is

inevitable due to the fact that most diseases are best simulated by higher order

models such as the SIR and the SIS. However, there are also actual scenarios that

can be best described using the simple, one-way SI model. An instant example

would be diseases that do not yet have a cure, [7]. Every individual is susceptible

to the HIV virus (AIDS). Once infected, the person stays in this condition for the

remainder of his life, or until a cure is found in his lifetime. In fact, this became the

central topic of the dissertation by Cruz [6]. The SI model also fits in describing an

information network characterized by two states: received or unreceived. Nodes in

received states are transmitters capable of spreading information to the unreceived

nodes (information sinks) [7]. In a related study, the SI model is used to simulate

the rate of collapse of areas covered by trees in wind-disturbed forests [8].

As illustrated, susceptible-infected systems are more than just epidemiological

models. They can also be used to describe similar related phenomena. In the case of

diseases, the SI model can provide more information when an infection is dangerous,

or when it reaches an optimum level. In the case of communication networks, knowing

the best way to spread information can help device better transmission strategies.

Identifying the areas that are highly dense can help us predict where similar species

will grow next. There are more related cases that can be characterized by the SI

model, which gives us a leverage in understanding these systems better by simulation.

Evidently, our study holds its mandate to gain a better understanding not just about

the spread of epidemics, but also the world.

4

1.2 Objectives of the study

In this work, we investigate further the dynamics of disease spread in the Ising-

based Susceptible-Infected (SI) model proposed by Crisostomo and Pinol [3]. Specif-

ically, we look at the effect of varying the position and number of initial infective/s

to the spread of infection in a population.

1.3 Scope and limitation of the study

The Ising-based (SI) model assumes a fixed population in an m m squarelattice. There are no births, migration or deaths. Latent and incubation2 between

the states are also disregarded, therefore, infection is instantaneous. Total number

of individuals is constant and is equivalent to the size of the lattice, N = m2.

1.4 Time and place of study

This study was conducted from June 2012 to October 2013 at the Computational

Physics Laboratory, Institute of Mathematical Sciences and Physics, University of

the Philippines Los Banos under the supervision of Dr. Chrysline Margus N. Pinol,

thesis adviser.

2latent and incubation periods are defined in epidemiological modelling as stages in the exposurestate [11].

5

Chapter 2

Review of literature

There are several approaches in which we can study the spread of an epidemic

through modelling, depending on the factor that we need to consider: it can be

by chance of infectivity (deterministic and stochastic), time of the disease spread

(discrete or continuous), spatial distribution, or population structure (homogeneous

or heterogeneous) [9] [10]. We first take into account the basic approaches, the

stochastic and the deterministic approach.

2.1 The deterministic model

The deterministic model is the earliest model proposed by Kermack and McK-

endrick in 1927 [1]. Characteristic parameters of the population are fixed. It is

then divided into classes or compartments that designate their condition (Figure

2.1). Since the transfer rate is fixed for each class, this can be described through

differential equations:

dS

dt= SI, (2.1)

dI

dt= SI I, and (2.2)

dR

dt= I. (2.3)

It also follows that, for a fixed population

6

Figure 2.1: The compartmental representation of the SIR model.

S(t) + I(t) +R(t) = N, and (2.4)dS

dt+dI

dt+dR

dt= 0 (2.5)

This is the basic framework for most epidemiological models. However, the esti-

mates obtained using the deterministic approach are only suitable for large populations[12].

2.2 Stochastic models

Stochastic or probabilistic models are useful when we want to take into account

the probability of change and variability of a parameter in the population [11]. There

are multiple approaches that can possibly be used in stochastic modelling, one of

which is the Markov probabilistic model. Here, the transfer of states are given by

S + I 2I, and (2.6)I R. (2.7)

It can be seen that the combination of a susceptible and an infective results to two

infectives, and the infective will become recovered in the future. Rates of change

7

between the states are given by and , respectively. To be able to derive the

dynamics of the population, we apply the Markovian process given by

tp(S, I; t) =

N(S + 1)(I 1)p(S + 1, I 1; t)

+(I + 1)p(S, I + 1; t)

( NSI + pI)p(S, I; t). (2.8)

In Trapmans dissertation [13], he cited another basic model used in the

stochastic approach, the Reed-Frost model. The method assumes that the popu-

lation is fixed. The probability to become infected depends on the number of case

of infection at a certain time t. In [14], this is denoted by Ct.

Ct+1 = St(1 qCt) (2.9)

where St is the number of susceptibles, and (1 qCt) is the probability that an indi-vidual will make contact with an infective at least once. Furthermore, this expression

is used to define the probability of Ct+1 cases occuring at time (t+ 1) given by

P (Ct+1) =St!

Ct+1!St+1!(1 qCt)Ct+1(qCt)St+1 . (2.10)

Since stochastic modelling doesnt assume constant values, it provides a bet-

ter way of simulating real-world phenomenon, as it takes into consideration that the

individual characteristic might change. However, since the method relies on prob-

ability, it becomes too complicated for analysis and arriving at an explicit solution

may become difficult.

2.3 Some real-world applications

2.3.1 Viral infections

Epidemiological modelling is commonly implemented to determine the behavior

of a particular disease, usually on commonly occuring diseases.

Influenza is a popular option when studying epidemics. The fluctuating sea-

sonality of its occurrence became the topic of study implemented by Dushoff, et, al.

8

[4]. They have reported that the fluctuations might have been caused by inconsis-

tency in its transmission rate. Moreover, the unpredictable behavior of contacts can

lead to differences in the rate of infection. To be able to simulate this phenomenon,

the authors implemented a deterministic and a stochastic SIRS model. The solutions

obtained from both methods were presented graphically to compare which approach

yielded a more efficient result.

Models in epidemiology are best utilized in studying incurable diseases, one

of which HIV is an example [6]. The interest of this study started from the fact

that even though control measures of these kinds of diseases are always accessible,

the pathogen seem to evolve into new strains that are capable of being invasive.

Ordinary and delay differential equations were utilized to define various cases of

becoming infected, which is done through a rigorous mathematical approach. The

scope of the study is divided into four major parts: structured epidemic model

with two-ages; structured two-aged model with intraspecific competition; epidemic

models with time-delay; and two-sex epidemic model incorporationg socio-economic

and cultural factors.

In 2003, we can recall that the SARS epidemic from China that caused a world-

wide alarm. The desire to prevent and control this disease led to numerous studies,

one of which is conducted by Wang, et, al. [15]. Their work revolved on studying the

spatial component of disease spread. This is accomplished using available data from

reported cases of SARS in Beijing implemented in an SEIR (Susceptible-Exposed-

Infected-Removed) model. Afterwards, the geographical pattern of infection was

identified using a nearest neighbor heirarchal clustering technique.

2.3.2 Disease propagation and intervention

The rate of infection varies according to the change in its characteristic param-

eters. One of the factors can be the type of network used to model the epidemic

[16]. In epidemiology, network with more realistic structure tend to yield relatable

real-world data. An example of a reliable network is the scale-free network.

The reliability of this network became the motivation of the paper by [7],

where they used a scale-free network to construct an SI epidemic model with identical

infectivity. In this model, the probability of one individual - represented by nodes -

9

Figure 2.2: The scale-free network.

to infect one another is given by

x(t) = 1 1(1 )(x,t1), (2.11)

where is the spreading rate, x is the susceptible individual, and (x, t 1) isthe number of contacts between x and the infectives at time t 1. The nodes haveidentical infectivity A, and the first infective is selected randomly. Furthermore, they

also determined faster spreading strategies and the effect of targetized immunization.

Another factor that can affect the behavior of disease spread is the homogen-

ity or heterogenity of the disease spread. This is explicitly discussed in the work done

by [17], where they explained how the models that rely on differential equations ac-

count for homogenity in mixing, and therefore fails to become a realistic model in the

long run. This is observed through a lattice-gas cellular automata model (LGCA).

In this model, a lattice composed of hexagonal cells is used. The center of

the cells in the lattice are connected altogether, which becomes the connection of

each node. In each cell, there can be a number of individuals that can randomly be

susceptible (S), infected (I) or recovered (R), which interact freely. The propagation

is governed by a time evolution probability given by E given by

10

E = PoRoC (2.12)

where P is the propagation step, R is the randomization step and C is the contactstep. Moreover, they used mean-field approximation to determine the dynamics of

disease spread in the LGCA model.

2.3.3 Ecological systems

Epidemiological models can also extend to non-disease related topics, especially

to fields where population models are used. In ecology, criticality is described as

sudden change in the characteristic of a system undergoing change [8]. It has three

types: classical, or the normal type where sudden change occur at the system when

it is undergoing change; self-organized criticalities that are self-inflicted or without

varying initial parameters, and; robust criticality that contains both of the charac-

teristics of the first and the second. Here, lattice models were used to explore events

that causes criticality, such as predator-prey and disturbance-recovery relations. The

transition of these states are similar to the dynamics of a susceptible-infective sys-

tem. Furthermore, the interaction between the relations listed are modelled using a

stochastic spatial model.

2.3.4 Spread of information

Information in wired networks are dissipated through different protocols. These

protocols are most of the time deterministic, which implies lower reliability. To solve

this, [18] presented a stochastic model based on the structure used in epidemiology.

The protocol was described as gossip driven, as it works by the nodes of the network

communicating with the others and relaying information from its memory. The model

created assumes a fixed number of nodes in its network. Each node attempts to send

the packet of data stored in its memory to the nodes connected to it. Since it is

a stochastic model, the transfer rate is determined by probability, governed by the

route length of one node to another.

Bettencourt, et al. [5] infered that the population dynamics of diffusion of

ideas is relatable to the standard models in epidemiology. However, the knowledge

11

of this fact has not been utilized as much. This became the framework of his study,

where he used the data on the physicists that adapted the use Feynman diagrams

in USA, Japan and USSR. The basic framework is in Figure 2.3, where S is the

susceptible state, E is the exposed state, I is the infective state, R is the recovered

state, and Z is an additional skeptic state. From here, he derived SIR, SIZ and

SIRZ models.

Figure 2.3: SEIRZ framework.

2.4 The Ising-based approach

Antonio, et al. [19] used the Ising model in malware epidemiology. In their

paper, they explained that the transmission of computer virus can be dependent on

whether the recepient in the network is oine or online. This is relatable to the

spin-up and spin-down configuration in the Ising model. The network is modelled

using an NN lattice, with the online nodes having a value of +1 and 1 for oine.The energy equation is computed using

Ei,j = si,jJi,j

snearestneighbors. (2.13)

In which the oine nodes change into online. Also, these nodes are associated with

12

a probability of becoming infected, given by

Pinf =number of online infective nodes

number of online nodes(2.14)

to be able to organize the steps of the procedure, we look into the flowchart in Figure

2.4:

Figure 2.4: Procedure for the Ising-based malware epidemiological model.

The Ising-framework is used in determining the network status of a node

(online or oine). The change is determined by the enery equation described in

Equation 2.13, and the probability given by

p = eEkBT (2.15)

where E is the change in energy, kb is the Boltzmann constant, and T defined as

the network traffic.

The results explained that a more congested network, or a network where

there are more online users, are more susceptible to infectibility rather than a lightly

congested network.

The Ising-model was also used in epidemiology. In [3], this model was used to

model the behavior of a susceptible-infective system. Similar to [19], the dual state

13

characteristic of the SI model is related to the spin-up( = 1) and spin-down( =

1). The model is limited to a square lattice with periodic boundary conditions. Tobe able to determine the change of a susceptible to infected, the Hamiltonian energy

of the system per time step is compared, given by H is calculated as

H = Ji,j

x,y

ijxy. (2.16)

where J is the interaction parameter. If the recent energy is less than or equal to

the previous, the change is accepted. If otherwise, the change will be dictated by the

probability given by

p = eH/T (2.17)

The adaptation of the Ising-model approach is presented graphically in Fig. 2.5

Figure 2.5: Procedure for the Ising-based SI model.

In the study, the authors graphed of the resulting infection-time relation and

observed that it follows the logistic behavior. This similar trend can be observed

14

in existing standard models for the SI. Furthermore, the effect of varying the initial

parameters were also observed. Firstly, it was seen that increasing the interaction

parameter, J , causes a decrease in the rate of infection. This is related to the

inverse contact rate, or the frequency of interaction of one individual to the rest of

the population. Next, it was also observed that varying the parameter, T , causes

a directly proportional change to the rate of infection. That is, an increase in this

parameter also causes an increase to the rate of infection. However, at extremely

high values, the effect diminishes and ceases to cause change. This, on the other

hand, is related to the scaled temperature of the population.

15

Chapter 3

Methodology

A population of N individuals is modelled using an mm lattice. Each latticesite corresponds to one invididual. The population is divided into susceptibles, S

and infected/infectives, I. A susceptible individual is represented by a value of

1; for an infected individual, = 1. Here, we assume a closed community. Thatis, there are no births, deaths or migrations. The population size is held constant,

N(t) = m2 = S(t) + I(t).

3.1 Initialization

We begin the simulation with a population of susceptibles,

xy = 1 (3.1)for all xs and ys, where x and y denote individual positions in the lattice (row and

column numbers, respectively). The first infective is then planted by changing the

of a randomly chosen site from 1 to +1.

3.2 Ising-model approach

To simulate the spread of infection or the transition from S I, we adopt theprocess outlined in [3]. During a particular iteration, a random susceptible is chosen.

This susceptible will be infected (S I, 1 +1)

if such change will result to a Hamiltonian H that is less than or equal to theHamiltonian H of the previous configuration (prior to the infection),

16

or, for the case when H > H, according to the Monte Carlo probability

p = eH/T . (3.2)

In the Ising model, T represents the scaled temperature and the Hamiltonian, H is

the energy of the system given by

H = Ji,j

x,y

ijxy, (3.3)

where J is the interaction parameter and

x,y ijxy considers only the interaction

within the Von Neumann neighborhood (x, y){(i, j+1), (i, j1), (i+1, j), (i1, j)}surrounding ij (see also Fig. 3.1).

Figure 3.1: Von Neumann (or nearest neighbor) interaction

To avoid duplicity, we consider only the neighbors at the top and to the left of each

site. Equation 3.3 reduces to

H = Ji,j

ij(i1,j + i,j1

). (3.4)

We also present a graphical summary of the procedures in Figure 3.2.

17

Figure 3.2: Flowchart of the procedure.

3.3 Simulation specifics

In this study, we use the following parameters sets:

Lattice size, 100 100, 150 150, 200 200;

Interaction parameter, J = 1.0, and;

Temperature range, 0.5 T 8.0.

Furthermore, we consider two cases. First, we begin our simulation with a single

infective. Second, we investigate the effect of introducing multiple infectives.

For the case of single infective, we have two scenarios: first, the initial infective

is planted at the middle; and second, the initial infective at the edge of the lattice.

We analyze the behavior of these configurations as a function of time, and compare

their infection rates. Circular boundary conditions will not be applied on the lattice.

For the case of multiple infectives, we start the simulation by assigning an

arbitrary number of infectives, x. We let

18

I0 = I(t = 0) =x

m2. (3.5)

Furthermore, we also consider two scenarios: infectives are clustered at the center;

and infectives are randomly placed throughout the population. Lattices we have

used are periodically bounded. We analyze the rate of infection for different values

of I0.

To estimate the value for , we fit the values of I and t

I(t) =Imax

1 + e(ttc)(3.6)

where:

I(t) is the number of infectives at time t;

Imax is the maximum number of infectives1;

is the rate of infection, and;

tc is the critical time , or the time when 50% of the population is infected.

1In a closed community, the maximum number of infectives is Imax = N = mm.

19

Chapter 4

Results and Discussion

4.1 Single Infective

We investigate the effect of varying the position of the first contagion on the rate

of spread of infection. We also removed circular boundary conditions to invoke the

effect of a real-world closed population.

First, we begin the simulation by planting the first infective in the middle of

the lattice. Figure 4.1 show plots the fraction of infected individuals against time,

for different values of the parameter T . Infection curves are logistic. Furthermore,

an increase in the value of parameter T causes a relative rise in the rate of spread of

infection. The maximum value for the fraction of infectives (y-axis) is reached faster

at lower T s.

For the second case, infection starts from an infective that is placed at the

edge of the lattice. Associated plots are presented in Figure 4.2. We recover the

same S-curves as in the first case.

To estimate the infection rate , we use a logistic fit,

I(t) =1

1 + e(ttc). (4.1)

Again, tc is to the point of inflection. This corresponds to the time when 50%

of the total population are already infected by the disease. Calculated values are

presented in Figure 4.3, for both cases. Each represents the average of 10 independent

trials. Notice that the points overlap. The location of the first contagion, therefore,

has no significant effect on the rate of propagation of infection.

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Figure 4.1: Infection curves asso-ciated with Case 1.

Figure 4.2: Infection curves asso-ciated with Case 2.

Figure 4.3: Infection rate as a function of temperature.

4.2 Multiple infectives

This time, we begin our simulations with an arbitrary number of infectives, x.

We let

I0 = I(t = 0) =x

N=

x

m2. (4.2)

Fixing T at 2.0, we consider two scenarios.

Scenario 1 I0 is distributed randomly throughout the lattice.

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Scenario 2 I0 is clustered at the center of the lattice.

Furthermore, we simulated the effect in three lattice sizes: 100 100, 150 150,200 200.

We investigate the effect of spreading I0 randomly across the lattice (popu-

lation). Figure 4.4, 4.5 and 4.6 shows infection curves with shape more similar to

the Figures ?? and 4.2, as well as the ones obtained by Crisostomo and Pinol [3]. As

expected, Imax is reached faster when there are more infectives to begin with. As the

lattice size is increased, the S-shape of the curves are becoming less prevalent. This

might have been caused by the fact that larger lattices require more iterations to be

completely affected by the virus. Figure 4.7 , 4.8 and 4.9 contains corresponding

values. Calculated infection rates increase with I0. However, we observe a decline in

beginning at 0.5 I0 0.6. The fall may be attributed to an increased likelihoodclustering at larger values of I0.

Figure 4.4: Infection curves using randomly distributed I0 at N = 100.

Figure 4.10, 4.11 and 4.12 displays I versus t curves obtained using Scenario 2.

This is for different values of I0. Compared to previous results (for the case of a single

infective), the S here is more pronounced. However, full infection is not obtained at

bigger lattice sizes. Normally, we would expect the infection to spread faster when

there are more contagions at the beginning of the simulation. The contrary can be

22

Figure 4.5: Infection curves using randomly distributed I0 at N = 150.

Figure 4.6: Infection curves using randomly distributed I0 at N = 200.

observed in Figure 4.13, 4.14 and 4.15 . Calculated values decrease with increasing

I0.

23

Figure 4.7: Infection rate associated with Scenario 1 at N = 100.

Figure 4.8: Infection rate associated with Scenario 1 at N = 150.

4.3 The effect of clustering

The discussion on the implications of structural patterns are commonly tackled

in spatial models in epidemiology [20]. The implications of clustering, specifically,

are discussed in some studies.

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Figure 4.9: Infection rate associated with Scenario 1 at N = 200.

Figure 4.10: Infection curves using clustered I0 at N = 100.

In [21], they presented the effect of local neighborhood structure in epidemic

processes. The interest came from the limitation of the models that are derived

from the stochastic and deterministic approaches, as there is no spatially explicit

discussion on its effects. To explore this, the authors constructed a cellular automata

25

Figure 4.11: Infection curves using clustered I0 at N = 150.

Figure 4.12: Infection curves using clustered I0 at N = 200.

model and implemented two types of neighborhood: Moore1 and activity-space, or

a constantly interacting neighborhood. Also, they have considered two cases of initial

infection: a centralized group and a uniformly distributed one. In the results, they

1Let x, y be the selected cell. The Moore neighborhood is {(x, y + 1), (x, y 1), (x 1, y), (x +1, y), (x 1, y 1), (x + 1, y 1), (x 1, y + 1), (x + 1, y + 1)}.

26

Figure 4.13: Infection rate asociated with Scenario 2 at N = 100.

Figure 4.14: Infection rate asociated with Scenario 2 at N = 150.

have reported that for both cases - Moore and activity-space - the propagation of

disease in a centralized group tends to cause a lower rate of infection.

Clustering also have significant implications in creating vaccination strategies.

Earlier, we have presented the study by [17] regarding the lattice-based model using

LGCA. The results of that study included the outcome of two kinds of vaccination

27

Figure 4.15: Infection rate asociated with Scenario 2 at N = 200.

strategies: the barrier type, where the vaccination shield a centralized group; and the

uniformly distributed type, where the vaccination were distributed in the population.

It was reported that the uniform vaccination caused an increase in the number of

the recovered. However, the rate of infection also increased in the same mode of

vaccination.

There are also implications of clustering in real-world epidemic scenarios.

In Australia, the recurrence of foot-and-mouth disease (FMD) on livestock led to

the study by [22]. They created a spatio-temporal SIR cellular automata model to

simulate the dynamics of the epidemic and its behavior over time and space in two

areas in Queensland, Australia. They used available data on recorded livestock and

feral pigs on both the regions of interest. It was discussed that besides the time

of initial infection, the density of the infectious animals can affect the cause of an

outbreak.

Earlier, the study on the spatial dynamics of SARS in urban areas in China

was presented [15], where they modelled the outcome of the epidemic using data from

reported cases and clustering techniques. They have reported that even the disease

will continue to spread over time, it will remain in clusters. Moreover, if preventive

measures are continued, such as control of mobility and withholding the infectives in

28

a region, it will cause a significant decrease in the propagation of the disease.

The clustering phenomenon is relatable to control measures done to prevent

the spread of an epidemic. A more popular name for this method is quarantine.

Sattenspiel and Herring [23] discussed that quarantines are measures that have been

implemented during occurrences of an outbreak. In the paper, they simulated the

potential effectivity of quarantine using data from the 1918-1919 flu epidemic that

occurred in Northern Canada through a compartmental model. The results presented

show that effectivity of quarantine depends on the mobility of an individual. That

is, when there is low mobility in the population, quarantines work as first preventive

measure. However, if the mobility is high, quarantines are less likely recommended.

29

Chapter 5

Summary and conclusion

We have investigated two cases of single infectivity: first, in the middle of the

lattice, then, on any random point at the edge of the lattice. Observing the behavior

of the rate of infection as a function of temperature, we have seen that the graphs

were similar. Hence, varying the initial infectives location does not affect the spread

of infection, despite having the circular boundary conditions removed.

We have also explored the effect of varying the initial number in the popula-

tion in two cases: randomly and grouped in the center. For the randomly distrbuted

initial infection, we have observed that the spreading of disease is faster when there

infected individuals are randomly placed in the population. However, beginning at

50% - 60%, the rate of infection slows down. For the clustered case, we have seen that

the spread of infection becomes slower, thus, providing a possibility of controlling

the infection. In the actual world, this can be related to the localization of infectious

individuals as a control measure to avoid an epidemic outbreak.

30

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