Download Lesson/Week 4

Lesson/Week 4
Disease Diffusion, Spatial Analysis, and Geographic
Visualization
Text: pages 352-398 and 454-468
Types of Diffusion
Modeling Disease Diffusion
Geographical Approaches to Studying the Pandemics of AIDS
Spatial Diffusion of Disease
“There are only two ways in which anything – whether it be an idea, a type of
mosquito, a food crop, a sophisticated technology for hospital care, or an infectious
agent of disease – can come to be found in a particular location. Either it developed
there independently or it somehow moved there from another place” Text p. 351
Important terms:
A person can be infected but never develop symptoms (subclinical),
or can develop symptoms (clinical)
The person can be even clinically sick, but not pass the infection to another person
(insufficient vieremia, a dead end)
How likely is the disease to develop in an infected person (pathogenecity of the
agent, resistance of the host)?
How virulent is the infection that occurs (case fatality rate)?
How likely is the infection to be passed on (second attack rate)?
The basic reproductive rate is the average number of secondary infections
produced when an infection occurs in a virgin population (i.e one in which
everyone is susceptible)
How long is the time period between infection and when a person becomes
infectious (incubation period, which is actually a little different from period and
time to symptoms; latency, because people with some infections start to “shed” the
disease agent in respiratory droplets, etc, before they actually develops symptoms
such as fever)?
People who can be infected are susceptible.
Those who have already had the disease, in reality or though vaccination , and can
no longer be infected are immune.
Those who are actively able to pass the infection are infectious.
For some diseases it is possible for individuals to have passed though the course of
the disease and no longer have symptoms themselves, yet still shed disease agents
and so be carriers. “Typhoid Mary” is the infamous example.
Spread of contagion requires contact in space. Distance and Density are important
parameters.
Likelihood of contact can be related to social networking as well as “herding”
(into schools, prisons, nursing homes, etc.).
Herd immunity is the proportion of immunity in a specific population needed to
prevent the infectious contact from finding a susceptible host, thereby ending the
infection.
Epidemic originally meant a disease’s spreading in a population, but in recent
years has been used (so as to refer also to nontransmissible diseases) to mean a
disease’s occurring in excess in a population.
Pandemic means that a disease is spreading among populations across national
borders.
The World Health Organization (WHO) today technically uses the term pandemic
to mean sustained spreading within communities in two or more countries (stage
5 in the WHO’s rating system) or in two or more of WHO regions of the world
(stage 6, globally pandemic).
• Diffuse is defined as ‘to disperse or be dispersed from a centre; to spread
widely, disseminate’. (Oxford English Dictionary)
• In geography, the term diffusion, has distinct usages:
– Expansion diffusion
– Relocation diffusion
– Combined expansion and relocation diffusion
Expansion diffusion
• Expansion diffusion: the process whereby a phenomenon of
interest (this may be information, a material artefact, a
disease), spreads from one place to another.
• In this expansion process, the item being diffused remains,
and often intensifies, in the originating region, but new
areas are also occupied by the item in subsequent time
periods.
• t1, t2, t3 denotes time 1, 2 and 3 respectively
Relocation Diffusion
• Relocation diffusion: is a spatial spread process, but the items being
diffused leave the areas where they originated as they move to new areas.
Combined Expansion and Relocation Diffusion
•
The diagram below shows how the two processes of expansion and
relocation may be combined.
An Example of a Combined Diffusion Wave
• The spread of the E1 Tor cholera epidemic, 1960-71
Types of Expansion Diffusion
• Expansion diffusion occurs in two ways.
– Contagious spread
– Hierarchical spread
• Cascade diffusion
Contagious spread
• Contagious spread depends on direct contact.
• This process is strongly influenced by distance because nearby individuals
or regions have a much higher probability of contact than remote individuals
or regions.
• Therefore, contagious spread tends to occur in a centrifugal manner from the
source region outward.
Hierarchical spread
• Hierarchical spread involves transmission through an ordered sequence of
classes or places, for example from large metropolitan centres to remote
villages.
• For example, within socially structured populations, innovations may be
adopted first on the upper level of the social hierarchy and then trickle down
to the lower levels.
• Cascade diffusion is a term reserved for processes that are always assumed
to be downwards from larger to smaller centres.
Stages of Diffusion Waves
• Torsten Hägerstand (1953) identified four distinct stages in the
passage of an innovation through an area:
– Primary stage
– Diffusion stage
– Condensing stage
– Saturation stage
Primary stage
• The primary stage marks the beginning of the diffusion process.
• A centre of adoption is established at the origin.
• There is a strong contrast in the level of adoption between this
centre and remote areas which is reflected in the steep decline of
the level of adoption curve beyond the origin.
Diffusion Stage
• The diffusion stage signals the start of the actual spread process
• There is a powerful centrifugal effect, resulting in the rapid growth
of acceptance in areas distant from the origin and by a reduction in
the strong regional contrasts typical of the primary stage.
• This results in a flattening of the slope of the proportion of
adopters curve.
Condensing Stage
• In the condensing stage, the relative increase in the numbers
accepting an item is equal in all locations, regardless of their
distance from the original innovation centre
• The acceptance curve moves in a parallel fashion.
Saturation stage
• The final saturation stage is marked by a slowing and eventual cessation of
the diffusion process, which produces a further flattening of the acceptance
curve.
• In this stage, the item being diffused has been adopted throughout the
country, so that there is very little regional variation.
The Shape of a Changing Diffusion Profile
• The shape of the changing diffusion profile in time and space has
been formally modelled.
• The temporal build-up in the number of adopters of an innovation
follows an S-shaped curve when plotted against time, with a
logistic curve as the mathematical form most commonly adopted.
The Nature of Epidemics
• The Oxford English Dictionary defines an epidemic as: ‘a disease prevalent
among a people or community at a special time, and produced by some
special causes generally not present in the affected locality’.
• In the standard handbook of human communicable diseases, Benenson
defines an epidemic more fully as:
– The occurrence in a community or region of cases of an illness (or an
outbreak) clearly in excess of expectancy. The number of cases
indicating presence of an epidemic will vary according to the
infectious agent, size and type of population exposed, previous
experience or lack of exposure to the diseases, and time and place of
occurrence; epidemicity is thus relative to usual frequency of disease
in the same area, among the specified population, at the same season
of the year.
• Benenson's account goes on to stress that what constitutes an epidemic does
not necessarily depend on large numbers of cases or deaths.
• A single case of a communicable disease long absent from a population, or
the first invasion by a disease not previously recognized in that area, requires
immediate reporting and epidemiological investigation.
• Two cases of such a disease associated in time and place are taken to be
sufficient evidence of transmission for an epidemic to be declared.
Epidemics of communicable disease
• Epidemics of communicable disease are of two main types.
– Propagated epidemic
– Common-vehicle epidemic
Propagated epidemic
• A propagated epidemic is one that results from the chain transmission of
some infectious agent.
• This may be directly from person-to-person as in a measles outbreak, or
indirectly via some intermediate vector (malaria) or a microparasite.
• In some cases, indirect transmission may occur via humans (e.g., a
mosquito-man-mosquito chain with malaria).
• In others, the survival of the parasite is independent of man (thus,
Pasteurella pestis, the cause of bubonic plague is continually propagated
through rodents and the infection of man by an infected flea is in this respect
an accidental diversion).
Common-vehicle epidemic
• Common-vehicle epidemic results from the dissemination of a causative
agent.
• In this case, the epidemic may result from a group of people being infected
from a common medium (typically, water, milk, or food) which has been
contaminated by a disease causing organism.
• E.g., cholera and typhoid
Barriers
Diffusion occurs over a wide web, or NETWORK of places, people the
communication links through which information, people, and goods
flow.
In opposition to NETWORKS which channel and support diffusion,
BARRIERS slow and shape the process.
Barriers are often physical – oceans, deserts, mountains, rivers, etc. or
cultural. Rapid transport today has almost eliminated the effectiveness
of physical barriers while globalization is eroding the protective effects
of cultural barriers.
Absorbing barriers: Stop the diffusion process
Population vaccinated against smallpox were an example of an
absorbing barrier. Also when a small population size and/or density does
not facilitate contact transmission.
Reflecting barriers: Channel and intensify the local impact of a
diffusion process while blocking its spread to another locale.
Example: the presence of lake or unbridged river, hostile ethnic group or
gang.
Permeable barriers: allowing some diffusion but slowing the process.
Most international borders are permeable barriers.
Modeling Disease Diffusion
http://plus.maths.org/issue14/features/diseases/
One parameter (almost) does it all
• Diseases are a ubiquitous part of human life.
• Many, such as the common cold, have minor symptoms and are purely an
annoyance; but others, such as Ebola or AIDS, fill us with dread.
• It is the unseen and seemingly unpredictable nature of diseases, infecting
some individuals while others escape, that has gripped our imagination.
• Over the past one hundred years, mathematics has been used to understand
and predict the spread of diseases, relating important public-health questions
to basic infection parameters.
• The mathematics of diseases is, of course, a data-driven subject. Although
some purely theoretical work has been done, the key element in this field of
research is being able to link mathematical models and data.
• Case reports from doctors provides us with one of the most detailed sources
of biological data; we often know the number of weekly disease cases for a
variety of communities over many decades.
• This data also contains the signature of social effects, such as changes in
birth rate or the increased mixing rates during school terms. Therefore, a
comprehensive picture of disease dynamics requires a variety of
mathematical tools, from model creation to solving differential equations to
statistical analysis.
Basic Models
• Almost all mathematical models of diseases start from the same basic
premise: that the population can be subdivided into a set of distinct classes,
dependent upon their experience with respect to the disease.
• The most simple of these models classifies individuals as one of susceptible,
infectious or recovered. This is termed the SIR model.
• Individuals are born into the susceptible class.
• Susceptible individuals have never come into contact with the disease and
are able to catch the disease, after which they move into the infectious class.
• Infectious individuals spread the disease to susceptibles, and remain in the
infectious class for a given period of time (the infectious period) before
moving into the recovered class.
• Finally, individuals in the recovered class are assumed to be immune for life.
• We can make this description more mathematical by formulating a
differential equation for the proportion of individuals in each class.
• Although initially this model shows large epidemics occurring at regular
intervals, eventually the level of the disease reaches a constant value.
The SIR model
The Epidemiological Parameter R0
Many interesting and useful results have been proved for the simple SIR model.
One fundamental parameter governs the spread of diseases, and is also related to
the long term behavior and the level of vaccination necessary for eradication.
This parameter is called the basic reproductive ratio, .
is defined by
epidemiologists as "the average number of secondary cases caused by an
infectious individual in a totally susceptible population".
As such
tells us about the initial rate of increase of the disease over a
generation.
When is greater than 1, the disease can enter a totally susceptible population and
the number of cases will increase, whereas when is less than 1, the disease will
always fail to spread.
Therefore, in its simplest form
given disease.
tells us whether a population is at risk from a
The value of R0 for some well-known diseases
Disease
R0
AIDS
2 to 5
Smallpox
3 to 5
Measles
16 to 18
Malaria
> 100
The endemic steady state
• An infectious disease is said to be endemic when it can be sustained in a
population without the need for external inputs.
• This means that, on average, each infected person is infecting exactly one
other person (any more and the number of people infected will grow
exponentially and there will be an epidemic, any less and the disease will die
out). In mathematical terms, that is:
• The basic reproduction number (R0) of the disease, assuming everyone is
susceptible, multiplied by the proportion of the population that is actually
susceptible (S) must be one (since those who are not susceptible do not
feature in our calculations as they cannot contract the disease).
• Notice that this relation means that for a disease to be in the endemic steady
state, the higher the basic reproduction number, the lower the proportion of
the population susceptible must be, and vice versa; a mathematical basis for
a result that might have been intuitively obvious.
The first assumption (above) lets us say that everyone in the population lives to age
L and then dies. If the average age of infection is A, then on average individuals
younger than A are susceptible and those older than A are immune (or infectious).
Thus the proportion of the population that is susceptible is given by:
But the mathematical definition of the endemic steady state can be rearranged to give:
And therefore, since things equal to the same thing are equal to each other:
This provides us with a simple way to estimate the parameter R0 using easily available data.
Complications
The theory outlined here has only scratched the surface of the research
done into epidemic spread and persistence. The study of diseases has
been such a successful application of mathematical theory because most
diseases conform to the assumptions behind the simple models.
However, many complications have been introduced into the SIR-type
models which allow them to better capture the observed dynamics and
answer more applied questions. The following are a list of practical
issues that have been implemented into SIR-type models.
•
Many diseases, such as measles or chickenpox, are primarily
disease of children. By further subdividing the population into
•
•
•
differing age-classes researchers have been able to capture agestructured transmission in more detail.
For such childhood infections there is greater mixing (the contact
rate is larger) during school terms. Such seasonal forcing leads to
regular epidemics or more complex dynamics, as the disease
oscillates between the high-contact and low-contact solutions.
When modelling the spread of HIV it is vital to subdivide the
population by sexual orientation and drug use.
For some diseases other organisms are involved in the
transmission, e.g. the mosquito is essential for transmission of
malaria, and together rats and fleas are responsible for the majority
of bubonic plague cases. For such diseases we need to couple an
SIR model for humans with an SIR model for the other organisms.
Geographical Approaches to the Pandemics of
AIDS
Simulation
http://www.xjtek.com/anylogic/demo_models/?application_area=Spread++of++Disease
http://www.xjtek.com/anylogic/demo_models/22/
SIR simulation
http://www.xjtek.com/anylogic/demo_models/23/
GIS
Example: HIV/AIDS Africa
“The aim of this article is to offer an alternative method that utilizes sentinel
data from HIV/AIDS testing centers to derive interpolated rates over space
and time and show spatial-temporal trends using the powerful tool of
GIScience and to use the interpolated rates to predict future HIV
prevalence.”
“Spatial analysis with GIS was used to generate continuous surface data
from point data of HIV infection rates for the entire African continent.”
“Interpolation methods are based on Waldo Tobler’s first law of geography:
“everything is related to everything else, but near things are more related
than distant things” (Tobler 1970, 236).”
http://pdfserve.informaworld.com/904097__789658469.pdf
Two GIS interpolation methods - IDW (inverse distance weighted) and
Kriging – applied (see results below)