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