Epidemiology modeling Epidemiology modeling with Stellawith Stella

CSCI 1210CSCI 1210

Stochastic vs. deterministicStochastic vs. deterministic

Suppose there are 1000 Suppose there are 1000 individuals and each one individuals and each one has a 30% chance of has a 30% chance of being infected:being infected:

StochasticStochastic method: run method: run the model on the right the model on the right 1000 times1000 times

DeterministicDeterministic method: method: 1000 * 30% = 300 get 1000 * 30% = 300 get infectedinfected

(Law of Mass Action)(Law of Mass Action)

Stella Stocks and FlowsStella Stocks and Flows

A A flowflow takes “stuff” out from a stock or takes “stuff” out from a stock or puts stuff into a stockputs stuff into a stock

Result of simple flow modelResult of simple flow model

Simple Epidemic Flow modelsSimple Epidemic Flow models

A short-term illness with recovery and A short-term illness with recovery and permanent immunitypermanent immunity

Simple Epidemic Flow ModelsSimple Epidemic Flow Models

Short-term lethal illness with no recovery Short-term lethal illness with no recovery or immunityor immunity

Examples: “Martian flu”, measles in IncasExamples: “Martian flu”, measles in Incas Note the flow into a Note the flow into a sinksink outside the model outside the model

Simple Epidemic Flow ModelsSimple Epidemic Flow Models

Short-term illness with recovery and Short-term illness with recovery and temporary immunitytemporary immunity

Example: malariaExample: malaria

Filling out the modelFilling out the model

These are dynamic modelsThese are dynamic models The value of each stock depends only on The value of each stock depends only on

the initial value and the flows over timethe initial value and the flows over time The flows depend on the assumptions and The flows depend on the assumptions and

state of the model – this is what state of the model – this is what determines how the model worksdetermines how the model works

The Infection processThe Infection process

Simplest model: small population in which Simplest model: small population in which everyone is in contacteveryone is in contact

Each sick person has a certain constant Each sick person has a certain constant probability of infecting each susceptible probability of infecting each susceptible person in one time unitperson in one time unit

Size of infection flow depends on the Size of infection flow depends on the number of sick people and the number of number of sick people and the number of susceptibles.susceptibles.

Modeling infection in StellaModeling infection in Stella

The thin arrows represent influences. Note that The thin arrows represent influences. Note that all the influences affect the rate of infection.all the influences affect the rate of infection.

We leave out incubation for simplicity: everyone We leave out incubation for simplicity: everyone is either susceptible or ill.is either susceptible or ill.

Qualitative analysis of infectionQualitative analysis of infection

When there are few sick people, there can When there are few sick people, there can be little infectionbe little infection

When nearly everyone is sick, there can When nearly everyone is sick, there can be little infectionbe little infection

Maximum infection will occur when the Maximum infection will occur when the population is between these casespopulation is between these cases

Eventually, everyone will get sick.Eventually, everyone will get sick.

Results of simple SI modelResults of simple SI model

Results of simple SI modelResults of simple SI model

A model with recovery and A model with recovery and immunityimmunity

After recovery, people are neither susceptible After recovery, people are neither susceptible nor illnor ill

A certain fraction of ill people will recover A certain fraction of ill people will recover each time period.each time period.

The rate of recoveries depends on the The rate of recoveries depends on the number of ill peoplenumber of ill people..

Results of the SIS modelResults of the SIS model

Infection and recovery ratesInfection and recovery rates

Effect of immunizationEffect of immunization

Reduces the initial number of susceptiblesReduces the initial number of susceptibles This reduces the infection rate, but does This reduces the infection rate, but does

not alter the recovery ratenot alter the recovery rate If the infection rate is small enough, the If the infection rate is small enough, the

disease will die out without becoming an disease will die out without becoming an epidemic (epidemic (herd immunityherd immunity).).

Infection and recovery, with Infection and recovery, with herd immunityherd immunity

Results of immunization Results of immunization campaigncampaign

Notes on Herd immunityNotes on Herd immunity

Not necessary to vaccinate the entire Not necessary to vaccinate the entire population.population.

Even individuals who were not vaccinated Even individuals who were not vaccinated share the benefits.share the benefits.

HIVHIV

Human Immunodiciency Virus (HIV)Human Immunodiciency Virus (HIV) A retrovirusA retrovirus Originated in Africa, probably in 20Originated in Africa, probably in 20thth

centurycentury Descended from simian virus (SIV) which Descended from simian virus (SIV) which

“jumped hosts”“jumped hosts” Long, contagious incubation periodLong, contagious incubation period

From HIV to AIDSFrom HIV to AIDS

Virus attacks human immune systemVirus attacks human immune system Death is from opportunistic secondary Death is from opportunistic secondary

infections, not HIV itselfinfections, not HIV itself Anti-retroviral drugs can slow the virus and Anti-retroviral drugs can slow the virus and

prolong life.prolong life.

AIDS and AfricaAIDS and Africa

42 million HIV/AIDS cases worldwide42 million HIV/AIDS cases worldwide 29 million cases in Africa29 million cases in Africa Origin of the virusOrigin of the virus Anarchy in central Africa (Uganda, Anarchy in central Africa (Uganda,

Rwanda, Congo) helps spread the diseaseRwanda, Congo) helps spread the disease

AIDS: the “Gay Plague”?AIDS: the “Gay Plague”?

Initially, US AIDS cases were almost all in Initially, US AIDS cases were almost all in gay mengay men

However, African AIDS cases are mostly However, African AIDS cases are mostly heterosexualheterosexual

More US heterosexual AIDS cases as time More US heterosexual AIDS cases as time has passedhas passed

What gives?What gives?

A two-tier modelA two-tier model

High-risk group initially contracts the High-risk group initially contracts the diseasedisease

Low-risk group does not have the diseaseLow-risk group does not have the disease Slight interaction between groupsSlight interaction between groups Two Two submodelssubmodels proceed separately but proceed separately but

have a weak have a weak couplingcoupling

Two-tier modelTwo-tier model

Results of the two-tier modelResults of the two-tier model

AIDS and the “Martian Flu”AIDS and the “Martian Flu”

HIV/AIDS is incurable, fatal, and has no HIV/AIDS is incurable, fatal, and has no known immunityknown immunity

However, US AIDS epidemic may have However, US AIDS epidemic may have peaked peaked

So, “Martian Flu” model needs elaborationSo, “Martian Flu” model needs elaboration

Elaborated AIDS modelElaborated AIDS model

Add birth and death flows for susceptibles Add birth and death flows for susceptibles who do not get infectedwho do not get infected

Either die naturally, retire from sex, or Either die naturally, retire from sex, or enter monogamous relationshipsenter monogamous relationships

Creates a situation similar to “herd Creates a situation similar to “herd immunity” modelimmunity” model

Elaborated single-pool modelElaborated single-pool model

AIDS model with high riskinessAIDS model with high riskiness

AIDS model with low riskinessAIDS model with low riskiness