Network Modeling of Epidemics

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Network Modeling of Epidemics. 8-13 July 2013. Prof. Martina Morris Prof. Steven Goodreau Samuel Jenness Sponsored by: NICHD and the University of Washington. The lesson plan for the week. Intuition building: Poker chip simulation. Blue chips = susceptible Red chips = infected - PowerPoint PPT Presentation

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Network Modeling of Infectious Disease

Network Modeling of Epidemics8-13 July 20138-13 July 20131UW - NME WorkshopProf. Martina MorrisProf. Steven GoodreauSamuel Jenness

Sponsored by: NICHD and the University of WashingtonThe lesson plan for the weekDayContent1Susceptible-Infected-Recovered/Susceptible modelsIntuition and basic propertiesExploring simple SIR/S models in R2Cross-sectional network analysisExponential Random Graph Models (ERGMs) for networks3Dynamic network analysisSeparable Temporal ERGMs (STERGMs) for dynamic nets4Simulating disease transmission on dynamic networksWhen network dynamics are independent of disease dynamics5Simulating disease transmission on dynamic networksWhen network dynamics are dependent on disease dynamics6Discussion of projects28-13 July 2013UW - NME Workshop3Intuition building: Poker chip simulationBlue chips = susceptible

Red chips = infected

White chips = recovered We will track the epidemic by hand using a prevalence chart grid.8-13 July 2013UW - NME Workshop4Note:We will simulate an individual level process

Poker chips represent personsDrawing poker chips from the bag represents the contact processReplacing blue chips with red represents transmissionReplacing red chips with white represents recovery

And record population-level outcomes

Prevalence = number of infectedsSome qualitative properties also8-13 July 2013UW - NME Workshop5

Prevalence Worksheet8-13 July 2013UW - NME Workshop6

Prevalence WorksheetWe begin with one infected person

At time = 0, the start of the process8-13 July 2013UW - NME Workshop7

Prevalence WorksheetAt each subsequent time point we record the current number of infected persons8-13 July 2013UW - NME WorkshopTo the lab88-13 July 2013UW - NME Workshop9Constant growth modelINSTRUCTIONS: Start with 1 red chip (red = I)

For each round:

Add 1 more red chip

Mark outcome on prevalence tracking worksheet

Repeat

8-13 July 2013UW - NME Workshop10What does the graph look like?What disease might this represent?What would change if 3 new people were infected every time step?

Insight 1: Always the same growth rate.

Insight 2: The slope of the line equals the number of new infections every day

Insight 3: The number of new infections does not depend on the number currently infectedConstant growth: Implications8-13 July 2013UW - NME WorkshopQUESTIONS: what disease may this represent?11UW - NME WorkshopThis example is not an infectious process more like chronic disease

The model assumes there is an infinite susceptible population (ie. Assume a hidden bag, with an infinite number of blue susceptibles becoming red infecteds)

Each step is some unit of time (i.e. minute, hour, day, etc.)Only one state (infected)Only one transition (the infection process)

InfectedTransition:constant rate8-13 July 2013

Constant growth: Implications12I: Infected model (proportional growth)INSTRUCTIONS: Start with 1 red chip

For each round:

Add 1 more red chip for each red chip already on the table

Mark outcome on prevalence tracking worksheet

Repeat

8-13 July 2013UW - NME Workshop13I model: ImplicationsEach red chip infects 1 new case at each time stepWhat does the graph look like?What disease might this represent?Is this realistic? What is missing?

Insight 1: The number of infecteds grows exponentially.

Insight 2: The population size is infinite, so the number of infected is unbounded8-13 July 2013UW - NME Workshop14UW - NME WorkshopI model: ImplicationsThe simplest true infection process

Still only one state (infected)An implicit state of blue susceptibles of infinite sizeStill only one transition, but now the rate depends on the number currently infectedTransmission:Proportional rate8-13 July 2013Infected

15SI: Susceptible-Infected modelINSTRUCTIONS: Now we will use the bag (it represents the population)Prepare a bag with 1 red chip and 9 blue chips (10 total)

For each round: S=blue, I=red

Pick two chips If the chips are the same color, no infection occurs.Return both chips to bag, go to step (2)If the chips are different colors, infection occursReplace blue chip with red chip and return to bag

Mark outcome on tracking sheet

Are there any more blue chips in the bag?YES: Return to (1)NO: Stop8-13 July 2013UW - NME Workshop16SI model: ImplicationsWhat does the graph look like?Why do each of your graphs look different?What is the same and different across all of your runs?Will everyone eventually become infected?How long until everyone is infected?8-13 July 2013UW - NME Workshop17UW - NME WorkshopSI modelSSSSSSSSSSSSSSSI

SSSSSISSSSSIIISI

SISIIISISSIISIII

IISIIIIIIIIIII II

Every draw has three possible outcomes:

SS: concordant negative SI: discordant II: concordant positive

The probability of each outcome changes as the process evolves.8-13 July 2013

18SI model: ImplicationsWhat does the graph look like?Why do each of your graphs look different?What is the same and different across all of your runs?Will everyone eventually become infected?How long until everyone is infected?8-13 July 2013UW - NME WorkshopInsight 1: Everyone will eventually become infectedInsight 2: The rate of infection depends on the proportion of both infecteds and susceptibles. Therefore, the rate of infection starts low, reaches its max halfway through, then decreases againInsight 3: The characteristic time signature for prevalence in a SI model is a logistic curve.What does this model assume about the duration of infection?19UW - NME WorkshopSI model: ImplicationsTwo states: susceptible and infected

One transition: transmission process

Now, we have a finite population, with total size N = S + ISusceptibleInfectedTransmission:Rate depends on both S & I8-13 July 2013IntroductionsWho we are

Who are you?208-13 July 2013UW - NME Workshop21Objectives for the 1 week courseGain intuition about population dynamics of infectious disease transmission, focusing on HIVStrengths and limitations of the different modeling frameworksUnderstand the basic principles and methods of network analysis relevant to infectious disease epidemiologyClassical network analysisModern (statistical) network analysis with ERGMsEmpirical study designs for networksDevelop the knowledge and software skills to run your own simple network transmission models.Using R, statnet and the EpiModel package 8-13 July 2013UW - NME Workshopdeterministic vs. stochastic model continuous or discrete timeanalytically or computational solutions22Objectives for todayGet an intuitive sense of epidemic modeling, including:

Elements of the transmission systemSignature dynamics of classic systems: the SIR/S familyThe roles that chance can playKey qualitative properties of a transmission system

Learn to explore simple SIR/S models in R using the EpiModel package, including:

Deterministic, compartmental models (ODEs)Stochastic, individual-based models

8-13 July 2013UW - NME Workshop23Models have three basic componentsElements actors in the model

States attributes of system elements

Transitions rates of movement between statesAll models, simple and complex, are built on these same building blocks8-13 July 2013UW - NME WorkshopNo matter the level of complexity, models are all built upon the same building blocks24Model component: ElementsElements can be:Persons AnimalsPathogens (microparasites, macroparasites)Environmental reservoirs (water, soil) Vectors (e.g., mosquitoes)Example: Measles transmission requires peoplePerson 1Person 2Person 3We will not be tracking the measles virus elements explicitly just the infection status of the persons8-13 July 2013UW - NME Workshop25UW - NME WorkshopStates attributes of elements. For example:Person/animal statesInfection status (Susceptible, Infected, Recovered)Demographic characteristics (male, female, )Pathogen states life cycle stage (e.g., larvae, reproducing adults, )

Model component: StatesExample: A simple measles model will have three person states: Person 1: susceptiblePerson 2:infectedPerson 3:recovered8-13 July 2013susceptible (S)infected (I)recovered w/ immunity (R)becoming infected26UW - NME WorkshopModel components: TransitionsTransitions movement between states

Deterministic: fixed rate of transition between states. Uses a population mean rate to govern process of movement

Stochastic: random probability that an element transitions between states. Uses a full probability distribution of rates to govern the process of movement.Example: The simple measles model has two transitions: Person 1: susceptiblePerson 2:infectedPerson 3:recoveredtransmissionrecovery8-13 July 2013recovering from infectionand 27Transmission: the heart of systemExamples of transmission types:

STD/HIV: direct body fluid contact (sex, needles, MTC)Measles, Influenza: respiratory, air-borneDiarrheal diseases: fecal-oralMalaria: vector-borne (mosquitoes)Schistosomiasis: water and vector-borne, via snails and nematodesCholera: water and food-borne

8-13 July 2013UW - NME Workshop28UW - NME WorkshopA typical system requires:IInfected personSSusceptible personaAn act of potential transmission between themtTransmission given actSusceptible InfectedDirect Transmission Parameters 8-13 July 2013acts and transmission are sometimes combined into a single parameterb = a t Force of infection29UW - NME WorkshopWhy act and not contact? Why a and not c? Why t and not b?

Contact most often means the specific act that may cause transmissione.g. in HIV, a sexual actIn this case: contact rate c means the frequency of sexual actsthe probabili