Modelling Infectious Disease
description
Transcript of Modelling Infectious Disease
Modelling Infectious Disease
… And other uses for Compartment Models
Plumbing
Tracking the concentration of dissolved particles through pipes
A simple conceptual model
rate rate
Volume
• Amount of solutes at the start = x(t=0)=x(0)=18
• Concentration of solutes at any time = x/V• Water coming in removes an amount of x at a constant rate• Need a model to calculate x(t)
A simple mathematical model
Vxr
dtdx
Vtrt
exVxrtx /
0
0)(
r r
V
The Solution
• X(0) = 18• r = 10• V = 100
Varying the rate of flow
Compartments & Flow
2
2
1
12
Vxr
Vxr
dtdx
3
3
2
23
Vxr
Vxr
dtdx
1
11
Vxr
dtdx
r r r r
V1 V2 V3
Changes in Concentration
Evaluate the Model
• Choose some parameters• V1 = 80• V2 = 100• V3 = 120• r = 20
• Define the initial conditions• x1(0) = 10• x2(0) = 0• x3(0) = 0
• http://math.fullerton.edu/mathews/N310/projects2/p14.htm (read from “More Background” onwards)
Results
General Framework
Any pattern you like…
Land
Sea
Air
From plumbing to infectious diseases
Infectious Disease
• Susceptible pool of people
• Infected pool of people
• Recovered pool of people
S
I
R
S I RbSI vI
Infection Rate:Contact rate
Infection probability
Recovery Rate
If D is the duration of infection:v = 1/D
bSIdtdS
vIbSIdtdI
vIdtdR
A “typical” flu epidemic
• Each infected person infects a susceptible every 2 days so bN=1/2 (N = S+I+R)
• Infections last on average 3 days so v=1/3
• London has 7.5 million people
• 10 infected people introduced
• See accompanying notes on parameter meanings
R0 as a useful statistic
• R0 is the basic reproductive number of the disease
• Similar to the r and R that appear in population models
• R0 = N*b*Duration = N(b/v)• If R0 > 1 epidemic• If R0 < 1 disease dies out naturally
Changes to Infection Rate
b=0.5/Nv=1/3
b=2/Nv=1/3
0 10 20 30 40 500
1
2
3
4
5
6
7
8x 10
6
Days
Peo
ple
SIR
Modifications are (almost) endless
Susceptible
Exposed
Infected
Recovered
SEIR
Susceptible
CarrierInfected
Recovered
Carrier Type Diseases: TB, Typhoid
Typhoid Mary• 1869-1938• Healthy carrier of
typhoid• Infected 47 people in
the US• Quarantined twice
under the mental health act
• We still do this!!– e.g. TB
Smallpox (Variola)
• Enveloped DNA virusgenus Orthopox
• Eradicated 1979
• Remains a biological threat– Huge vaccine stocks are held by many
Governments
Legrand et al. 2004, Epidemiol Infect, vol 132, pp19-25
Uninfectedcontacts(located)
Vaccinated successfully
Exposed contacts(missed)
Susceptible
Infectious
Removed
Exposed contacts(located)
Quarantine
Time to Invervention is crucial
Endemic Infections
• These are persistent infections in the population that tick along at a relatively stable level, never going extinct.
• This happens when the number of Infectious people remains constant
0 ISIdtdI
ISI
10 SR
1S
Minimum Vaccination Number
• Also known as Herd Immunity• At equilibrium (stable state)
R0S = 1
• Vaccinate proportion q of populationR0(1-q)=11-q=1/R0
qc=1-(1/R0)
• This is the minimum % of the pop that have to be vaccinated in order to stop the spread of the disease
Immunisation Thresholds
Disease R0 Thresholdqc=1-(1/R0)
Measles 15 93%
Smallpox 7 86%
Mumps 5 80%
Conclusions
• Compartment models are versatile– Flow of liquids between tanks– Diffusion of nutrients across sediment boundaries– Spread of disease through populations
• Endless elaborations can be made– Spatial structure– Population structure
Further Reading• The bible and for a link from SIR to population models:
Anderson & May. 1979. Population biology of infectious diseases: Part 1. Nature 280, 361-367.May & Anderson. 1979. Population biology of infectious diseases: Part 2. Nature 280, 455-461.
• For an evolutionary spin:Brown et al. 2008. Evolution of virulence: triggering host inflammation allows invading pathogens to exclude competitors.
• Fitting models to real data:Keeling & Grenfell, 2001. Understanding the persistence of measles: reconciling theory, simulation and observation. Proc Roy Soc B 269, 335-343.Indeed, anything by Bryan Grenfell is worth reading: http://www.cidd.psu.edu/people/bio_grenfell.html
• Foot-and-mouth disease:Tildesley et al. 2006. Optimal reactive vaccination strategies for a foot-and-mouth outbreak in the UK. Nature 440, 83-86. (and refs therein, esp the first 2)
• The original article:Kermack & McKendrick 1927. http://links.jstor.org/sici?sici=0950-1207%2819270801%29115%3A772%3C700%3AACTTMT%3E2.0.CO%3B2-Z
Tasks for next tutorial
• Why do some infectious diseases such as measles epidemics cycle?– Intrinsic (properties of the infective process itself)– Extrinsic (environmental)
• See Bryan Grenfell’s research on measles as a starter http://www.princeton.edu/eeb/people/display_person.xml?netid=grenfell&display=All