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Modelling Infectious Disease. … And other uses for Compartment Models. Plumbing. Tracking the concentration of dissolved particles through pipes. A simple conceptual model. Amount of solutes at the start = x(t=0)=x(0)=18 Concentration of solutes at any time = x / V - PowerPoint PPT Presentation

Modelling Infectious Disease

Modelling Infectious Disease And other uses for Compartment Models1PlumbingTracking the concentration of dissolved particles through pipesA simple conceptual modelraterateVolumeAmount of solutes at the start = x(t=0)=x(0)=18Concentration of solutes at any time = x/VWater coming in removes an amount of x at a constant rateNeed a model to calculate x(t)A simple mathematical model

rrVThe SolutionX(0) = 18r = 10V = 100

Varying the rate of flow

Compartments & Flow

rrrrV1V2V3Changes in Concentration7Evaluate the ModelChoose some parametersV1 = 80V2 = 100V3 = 120r = 20Define the initial conditionsx1(0) = 10x2(0) = 0x3(0) = 0http://math.fullerton.edu/mathews/N310/projects2/p14.htm (read from More Background onwards)

8Results

9General Framework10Any pattern you likeLandSeaAir11From plumbing to infectious diseasesInfectious DiseaseSusceptible pool of people

Infected pool of people

Recovered pool of peopleSIR13SIRbSIvIInfection Rate:Contact rateInfection probabilityRecovery Rate

If D is the duration of infection:v = 1/D

14A typical flu epidemicEach infected person infects a susceptible every 2 days so bN=1/2 (N = S+I+R)Infections last on average 3 days so v=1/3London has 7.5 million people10 infected people introduced See accompanying notes on parameter meanings

15R0 as a useful statisticR0 is the basic reproductive number of the diseaseSimilar to the r and R that appear in population modelsR0 = N*b*Duration = N(b/v)If R0 > 1 epidemicIf R0 < 1 disease dies out naturally16Changes to Infection Rate

b=0.5/Nv=1/3

b=2/Nv=1/3

17Modifications are (almost) endlessSusceptibleExposedInfectedRecoveredSEIRSusceptibleCarrierInfectedRecoveredCarrier Type Diseases: TB, Typhoid18Typhoid Mary1869-1938Healthy carrier of typhoidInfected 47 people in the USQuarantined twice under the mental health actWe still do this!!e.g. TB

19Smallpox (Variola)Enveloped DNA virusgenus Orthopox

Eradicated 1979

Remains a biological threatHuge vaccine stocks are held by many Governments

20Legrand et al. 2004, Epidemiol Infect, vol 132, pp19-25Uninfectedcontacts(located)Vaccinated successfullyExposed contacts(missed)SusceptibleInfectiousRemovedExposed contacts(located)Quarantine21Time to Invervention is crucial

22Endemic InfectionsThese 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

23Minimum Vaccination NumberAlso known as Herd ImmunityAt equilibrium (stable state)

R0S = 1

Vaccinate proportion q of population

R0(1-q)=1

1-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 disease24Immunisation ThresholdsDiseaseR0Thresholdqc=1-(1/R0)Measles1593%Smallpox786%Mumps580%25ConclusionsCompartment models are versatileFlow of liquids between tanksDiffusion of nutrients across sediment boundariesSpread of disease through populationsEndless elaborations can be madeSpatial structurePopulation structure26Further ReadingThe 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

27Tasks for next tutorialWhy do some infectious diseases such as measles epidemics cycle?Intrinsic (properties of the infective process itself)Extrinsic (environmental)See Bryan Grenfells research on measles as a starter http://www.princeton.edu/eeb/people/display_person.xml?netid=grenfell&display=All