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Introduction

Branching model

SIR model

Cellular Automaton model

Introduction

Branching model

SIR model

Cellular Automaton model

The study of epidemic disease has always been a topic where biological issues

mix with social ones.

When we talk about epidemic disease, we will be thinking of contagious diseases

caused by biological pathogens, things like influenza, measles, and today Ebola

or Marburg fever diseases, which spread from person to person.

Epidemics can pass explosively through a population. In extreme cases, a single

disease outbreak can have a significant effect on a whole civilization, as with the

epidemics started by the arrival of Europeans in the Americas, or the outbreak of

bubonic plague that killed 20% of the population of Europe over a seven-year

period in the 1300s.

The patterns by which epidemics spread through groups of people is determined

not just by the properties of the pathogen carrying it, including its

contagiousness, the length of its infectious period, and its severity, but also by

network structures within the population it is affecting.

But more generally, the opportunities for a disease to spread are given by a

contact network: there is a node for each person, and an edge if two people come

into contact with each other in a way that makes it possible for the disease to

spread from one to the other.

The transmission from one person to another is a sufficiently complex and

unobservable process at the person-to-person level that it is most useful to model

it as random.

That is, we will generally assume that when two people are directly linked in the

contact network, and one of them has the disease, there is a given probability that

he or she will pass it to the other.

This use of randomness allows us to abstract away questions about the

mechanics of how one person catches a disease from another for which we have

no useful simple models.

Introduction

Branching model

SIR model

Cellular Automaton model

The simplest model of contagion, which we refer to as a branching process

is working as follows.

(First wave.) Suppose that a person carrying a new disease enters a population, and

transmits it to each person he meets independently with a probability of p. Further,

suppose that he meets k people while he is contagious; let’s call these k people the first

wave of the epidemic.

Based on the random transmission of the disease from the initial person, some of the

people in the first wave may get infected with the disease, while others may not.

(Second wave.) Now, each person in the first wave goes out into the population and

meets k different people, resulting in a second wave of k · k = k2 people. Each infected

person in the first wave passes the disease independently to each of the k second-wave

people they meet, again independently with probability p.

(Subsequent waves.) Further waves are formed in the same way, by having each

person in the current wave meet k new people, passing the disease to each

independently with probability p.

Thus the contact network for this epidemic can be drawn as in figure below

(with k = 3 land only the first three waves shown). We refer to such a network as a

tree.

This tree is a representation of the Ebola spreading process where each person

contaminated will also contaminate from 1 to 4 other persons.

Basic Reproductive Number and a Dichotomy for Branching Processes

So there are only two possibilities for a disease in the branching process model: If the disease ever reaches a wave where it fails to infect anyone, then it has died out: since people in future waves can only catch the disease from others higher up in the tree, no one in any future wave will be infected either. Or it continues to infect people in every wave, proceeding infinitely through the contact network. It turns out that there is a simple condition to tell these two possibilities apart, based on a quantity called the basic reproductive number of the disease. The basic reproductive number, denoted R0 , is the expected number of new cases of the disease caused by a single individual. Since in our model everyone meets k new people and infects each with probability p, the basic reproductive number here is given by R0 = pk. The outcome of the disease in a branching process model is determined by whether the basic reproductive number is smaller or larger than 1.

If R0 < 1, then with probability 1, the disease dies out after a finite number of waves. If R0 > 1, then with probability greater than 0 the disease persists by infecting at least one person in each wave.

Here below a real Ebola propagation with k = 10 in Gabon in 2001.

Ebola: Human transmission of the disease

Introduction

Branching model

SIR model

Cellular Automaton model

The SIR epidemic model can be applied to any network model structure.

To do this, we preserve the basic ingredients of the branching process model at the level of

individual nodes, but make the contact structure much more general. An individual node in the

branching process model goes through three potential stages during the course of the epidemic:

Susceptible: Before the node has caught the disease, it is susceptible to infection from

its neighbors.

Infectious: Once the node has caught the disease, it is infectious and has some probability of

infecting each of its susceptible neighbors.

Recovered: After a particular node has experienced the full infectious period, this node is

removed from consideration, since it no longer poses a threat of future infection.

This model was for the first time proposed by O. Kermack and Anderson Gray

McKendrick as a special case of what we now call Kermack-McKendrick theory, and

followed work McKendrick had done with the Ronald Ross.

The dynamics of the SIR model are given by the system of Ordinary Differential

Equations:

Where b is the rate at which an infected person infects a susceptible, and g is the

recovery rate of infected people.

Where: - S(t) : Number of persons susceptible to be infected by the pathogen agent - I(t) : Number of infectious persons - R(t) : Number of persons that recovered

𝒅𝑺

𝒅𝒕= − 𝜷𝑺𝑰

𝒅𝑰

𝒅𝒕= 𝜷𝑺𝑰 − 𝜸𝑰

𝒅𝑹

𝒅𝒕= 𝜸𝑰

This system is non-linear, and does not admit a generic analytic solution.

Nevertheless, significant results can be derived analytically.

Firstly note that from:

it follows that:

expressing in mathematical terms the constancy of population N. Note that the

above relationship implies that one need only study the equation for two of the three

variables.

Secondly, we note that the dynamics of the infectious class depends on the

following ratio:

𝒅𝑺

𝒅𝒕+

𝒅𝑰

𝒅𝒕+

𝒅𝑹

𝒅𝒕= 𝟎

𝑺 𝒕 + 𝑰 𝒕 + 𝑹(𝒕) = N = constant

𝑹𝟎 =𝜷

𝜸 N

The SIR model developed in Python is set with the following data for simulating Ebola outbreak in a total population of 1.000 people. Assuming that the population is quarantined and no additional population is added during the period.

For Ebola: 𝜸 = 1/10 (Recovery rate) N = S + I + R = 1.000 (Total population ) I(0) = 1 (Initial infected population ) 𝜷 = 0.3 (Infection rate) 𝑹𝟎 = 3 ( 1< 𝑹𝟎 <4 )

. Resolution of SIR Ordinary Differential Equation system with python:

. Resolution of SIR Ordinary Differential Equation system with python:

Introduction

Branching model

SIR model

Cellular Automaton model

Cellular Automaton is more appropriate when it comes to visualize the

propagation of a disease on a map.

Cellular automata (CA) consist of dicrete agents or particules, which occupy some or all sites

of a regular lattice.

These particules have a discrete or continuous internal state variables and a set of rules

describing the evolution of their state and position.

The change of state of particule depends on the current state of the particule and those of

neighboring particles.

Concerning my Ebola CA developped in Python, 𝑹𝟎 ∈ {1;4} meaning that each

contaminated cell is going to contaminate between 1 to 4 cells during its contagious state.

In my model I am using a probabilistic method when deciding which particules are going to

be contaminated thus using Normal law and a Monte-Carlo method.

More each contaminated cell will have a probability of 70% to die thus allowing only 30% of

recovery rate as per current Ebola disease in west Africa.

Cellular Automaton principles

The CA starts with a single infected cell (in red). The first step will

test all its eigth neighbors in order to check if they are susceptible

(in blue) to be infected.

Assuming that all the eight neighbors are susceptible we launch a

random process in order to contaminate between 1 to 4 of these

eight susceptible cells.

After that we are starting a new random process implementing a

normal law N(0,1) in order to change the internal state of the

Infected cell thus killing it with a 70% death rate probability.

In order to take into account that contaminated cells are not statics

nor quarantined the CA is simulating some movements and

contaminates randomly some cells elsewhere in the city.

Presentation of the Cellular Automaton

The CA is representing a city composed by 7.000 cells or houses. Some of them are free (in white), orange cells

are representing streets, green cells are for parks, dark blue cell is for susceptible to be infected, red is the

infected, and cyan is for recovered .

Visualization of the propagation

Visualization of the propagation